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[
"THE POPULATION OF CLOSE DOUBLE WHITE DWARFS IN THE GALAXY",
"THE POPULATION OF CLOSE DOUBLE WHITE DWARFS IN THE GALAXY"
] |
[
"Lev R Yungelson ",
"Gijs Nelemans ",
"Simon F Portegies Zwart ",
"Frank Verbunt ",
"\nInstitute of Astronomy\nRussian Academy of Sciences\nMoscowRussia\n",
"\nAstronomical Institute \"Anton Pannekoek\"\nAmsterdamthe Netherlands\n",
"\nAstronomical Institute\nMassachusetts Institute of Technology\nCambridgeUSA\n",
"\nUtrecht University\nUtrechtthe Netherlands\n"
] |
[
"Institute of Astronomy\nRussian Academy of Sciences\nMoscowRussia",
"Astronomical Institute \"Anton Pannekoek\"\nAmsterdamthe Netherlands",
"Astronomical Institute\nMassachusetts Institute of Technology\nCambridgeUSA",
"Utrecht University\nUtrechtthe Netherlands"
] |
[] |
We present a new model for the Galactic population of close double white dwarfs. The model accounts for the suggestion of the avoidance of a substantial spiral-in during mass transfer between a giant and a main-sequence star of comparable mass and for detailed cooling models. It agrees well with the observations of the local sample of white dwarfs if the initial binary fraction is ∼ 50 % and an ad hoc assumption is made that white dwarfs with M < ∼ 0.3 M⊙ cool faster than the models suggest. About 1000 white dwarfs with V < ∼ 15 have to be surveyed for detection of a pair which has M1 + M2 > ∼ M Ch and will merge within 10 Gyr.
|
10.1007/978-94-015-9723-4_25
|
[
"https://arxiv.org/pdf/astro-ph/0011248v1.pdf"
] | 14,317,598 |
astro-ph/0011248
|
895d79a67f15fddb67650c03365396c5c8eaa383
|
THE POPULATION OF CLOSE DOUBLE WHITE DWARFS IN THE GALAXY
13 Nov 2000
Lev R Yungelson
Gijs Nelemans
Simon F Portegies Zwart
Frank Verbunt
Institute of Astronomy
Russian Academy of Sciences
MoscowRussia
Astronomical Institute "Anton Pannekoek"
Amsterdamthe Netherlands
Astronomical Institute
Massachusetts Institute of Technology
CambridgeUSA
Utrecht University
Utrechtthe Netherlands
THE POPULATION OF CLOSE DOUBLE WHITE DWARFS IN THE GALAXY
13 Nov 2000stars: white dwarfs -stars: statistics -binaries: close
We present a new model for the Galactic population of close double white dwarfs. The model accounts for the suggestion of the avoidance of a substantial spiral-in during mass transfer between a giant and a main-sequence star of comparable mass and for detailed cooling models. It agrees well with the observations of the local sample of white dwarfs if the initial binary fraction is ∼ 50 % and an ad hoc assumption is made that white dwarfs with M < ∼ 0.3 M⊙ cool faster than the models suggest. About 1000 white dwarfs with V < ∼ 15 have to be surveyed for detection of a pair which has M1 + M2 > ∼ M Ch and will merge within 10 Gyr.
INTRODUCTION
The interest in the close double white dwarfs (hereafter CDWD) stems from several reasons: (i) white dwarfs are the endpoints of stellar evolution; (ii) CDWD experienced at least two stages of mass exchange and thus provide an important tool for testing the evolution of binaries; (iii) merging double CO white dwarfs are a model for SNe Ia; (iv) CDWD may be the most important contributors to the gravitational wave noise at ν < ∼ 0.01 Hz, possibly burying signals of other sources. We review the data on the currently known CDWD and present results of the modelling of the population of white dwarfs, which involves several new aspects. The most important are the treatment of the first stage of mass loss without significant spiral-in (Nelemans et al., 2000a), the use of detailed models for the cooling of white dwarfs and the consideration of different star formation histories for the Galactic disc. Table 1 lists the currently known CDWD for which the orbital period and the mass of at least one component are measured Maxted et al., 2000a). The accuracy of the mass determinations is certainly not better than ±0.05 M ⊙ . With the exception of WD 0135−052 and 1204+450, the brighter components of the pairs are likely to be He white dwarfs, since their mass is lower than 0.46 M ⊙ , the limiting mass for the helium ignition in a degenerate stellar core (Sweigart et al., 1990). In the range of M ≃ (0.35 − 0.45) M ⊙ white dwarfs could have CO cores and thick He envelopes (Iben and Tutukov, 1985). However, the probability of the formation of these so called "hybrid" WD is 4 -5 times lower than for helium WD with the same mass (Tutukov and Yungelson, 1993;Nelemans et al., 2000b). The binary dwarfs WD 0957−666, 1101+364, 1704−481A, and 2331+290 are expected to merge because of angular momentum loss via gravitational wave radiation. If both components are He dwarfs, the merger may result in the formation of a nondegenerate star or a supernova-scale explosion (Nomoto and Sugimoto, 1977). In the case of CO companions, the formation of an R CrB star is expected (Webbink, 1984;Iben et al., 1996).
OBSERVED CLOSE DOUBLE WHITE DWARFS
We included in Tab. 1 also data on two sdB stars with white dwarf companions (Orosz and Wade, 1999;Maxted et al., 2000b). Subdwarfs are supposed to be helium-burning stars. In these particular systems central helium burning will be completed before components merge. This makes these binaries candidates for future merging CO+CO white dwarfs. Remarkably, for KPD 1930+2752 the total mass of the system exceeds M Ch , making it a possible SN Ia candidate! About 10 more WD and sdB stars with suspected close WD companions are known (Marsh, 2000;Maxted et al., 2001). However, for these systems the orbital periods or the masses of the components are not yet determined.
FORMATION OF HELIUM WHITE DWARFS
Three CDWD are of special interest -WD 0136+768, 0957−666, and 1101+364. In these systems the masses of both components suggest that they are helium dwarfs and, thus, descend from degenerate cores of low-mass (sub)giants. If we designate main-sequence stars as MS, red (sub)giants as RG, white dwarfs as WD, the stage of mass exchange, either stable or unstable, as RLOF, and refer by subscripts 1 and 2 to the initial primary and secondary star, the evolutionary sequence which results in the formation of a double helium WD may be described as follows: MS 1 + MS 2 → RG 1 +MS 2 → RLOF 1 → WD 1 +MS 2 →WD 1 +RG 2 → RLOF 2 → WD 1 +WD 2 . If the mass transfer is unstable, the change of component separation a is usually calculated by balancing the binding energy of the envelope of the mass-losing star with the change of the orbital energy (Webbink, 1984):
M 1 (M 1 − M c ) λ r 1 = α M c M 2 2 a f − M 1 M 2 2 a i .(1)
Here M c is the mass of the core of the mass-losing star, r 1 is its radius, subscripts i and f refer to the initial and final values of the orbital separation, α is the efficiency of the deposition of orbital energy into the common envelope and λ is a parameter which depends on the density distribution in the stellar envelope; the usual assumption is λ = 0.5 (de Kool et al., 1987). It has to be noticed, however, that for stars more massive than 3 M ⊙ the values of λ for red giants may be significantly lower (Dewi and Tauris, 2000). It would be worthwhile to investigate also lower mass stars, which form most of the observed CDWD. Giants with degenerate helium cores obey a unique core-mass -radius relation (Refsdal and Weigert, 1970). Neglecting a slight dependence on the total mass of the star, this dependence (for solar chemical composition objects) may expressed as (Iben and Tutukov, 1985):
R ≈ 10 3.5 M 4 c ,(2)
where radius of the giant R and mass of the core M c are in solar units. At the instant when the star fills its Roche lobe, the radius given by Eq.
(2) is equal to the radius of the Roche lobe. If the mass loss is unstable, the mass of the core of the mass-losing star doesn't grow during this stage and the mass of the new-born white dwarf is equal to M c . Applying Eq.
(2) together with restrictions on the masses of the progenitors of helium white dwarfs (0.8 -2.3 M ⊙ ), Nelemans et al. (2000a) reconstructed the evolution of WD 0136+768, 0957−666, and 1101+364. Their conclusions may be summarized as follows: (i) in the first episode of mass loss, when the companion of the Roche lobe filling star was a main-sequence star of a comparable mass, no substantial spiral-in occurred, see Fig. 1; (ii) in the second mass loss episode, which resulted in the formation of the currently brighter white dwarf, the separation of the components was strongly reduced. The deposition of the energy into common envelope in this episode was highly efficient: in Eq. (1) the product αλ < ∼ 3. A note has to be made in relation with the latter statement: since Eq. (1) provides nothing more than an order of magnitude estimate, a formal solution which gives α ≥ 1 indicates that energy deposited into common envelope has to be comparable to the orbital energy of the binary.
Since the first mass loss episode is neither stable mass transfer nor a spiral-in in a common envelope and the physical picture of the process is absent, Nelemans et al. (2000a) suggested to use in the population Periods after the first phase of mass loss as a function of the mass of the secondary component at this time. Solid, dashed, and dotted lines are for WD 0957-666, 1101+364, and 0136+768 respectively. The top three lines are periods needed to explain the mass of the last formed white dwarf. The middle three lines give the maximum period if the first white dwarf would be formed by conservative mass exchange. The lower three lines give the periods for the case when the formation of the dwarf was accompanied by a spiral-in (Eq. (1) with αλ = 2).
Figure 2.
Dependence of the relative variation of the separation of the components a f /ai on the fraction of the mass lost by the star for the cases of spiral-in in a common envelope and AML regulated variation of a. Pairs of solid, dashed, and dash-dotted curves correspond to the initial mass ratio of 1.0, 0.5, and 0.2, respectively. The upper curve of every pair is for the "AML formalism" [Eq. (3)].
synthesis studies a simple linear equation for the angular momentum balance:
J i − J f = γJ i ∆M M ,(3)
where J i and J f are the angular momenta of the pre-and post-masstransfer binary, respectively, ∆M is the amount of mass lost by the binary and M is the total initial mass of the system. The parameter γ is adjusted by fitting the orbital periods and masses of the three abovementioned helium CDWD. It follows that 1.4 < ∼ γ < ∼ 1.8. As Fig. 2 illustrates, when Eq. (3) is applied, the separation between the components changes much less drastically, compared to the case when the "standard" common envelope formalism [Eq. (1)] is applied. For typical values of the fractional mass of the core M f /M i both formalisms give similar a f /a i when q decreases to ≃ 0.2. In the actual calculations, for a given M f /M i , the larger of two values of a f /a i was used.
COOLING OF WHITE DWARFS AND OBSERVATIONAL SELECTION
The observed sample of CDWD is biased. Some objects were selected for study since their low mass already suggested binarity. White dwarfs must be sufficiently bright for the mass determination and the measurement of the radial velocities. This suggests to compare a magnitude limited model sample of dwarfs with the observations. Hitherto, following Iben and Tutukov (1985), it was assumed in all population synthesis studies that all WD can be observed for 10 8 yr, unless close pairs merge in a shorter time. The actual cooling curves were never applied. However, recent studies (Driebe et al., 1998;Hansen, 1999;Benvenuto and Althaus, 1999;Sarna et al., 2000) show, that helium WD cool more slowly than carbon-oxygen ones. This is due to the higher heat content of the helium WD (Hansen, 1999) and residual nuclear burning in the relatively thick hydrogen envelopes (first noticed by Webbink (1975)). In our basic model, we take cooling curves for CO dwarfs from Blöcker (1995) and for He ones from Driebe et al. (1998, hereafter DSBH), see Fig. 3. The initial models for these curves are obtained by mimicking the mass loss by stars and therefore may be considered as the most realistic. However, as we show below, cooling times given by them probably can not be taken at face value and need revision downward. Mass loss in stellar wind and thermal flashes, which extinguishes hydrogen burning (Iben and Tutukov, 1986;Sarna et al., 2000;Althaus et al., 2000), may provide the necessary mechanism. Cooling curves for 0.179, 0.300, 0.414 M⊙ white dwarfs from Driebe et al. (1998) and for 0.6 and 0.8 M⊙ ones from Blöcker (1995), from right to left. Straight lines are the fits to the curves used in the simulations.
In addition to the sufficient brightness of the components, CDWD must have such orbital periods that radial velocities would be large enough to be measured, but small enough not to be smeared out during the integration. Following estimates of Maxted and Marsh (1999), we model this selection effect assuming that CDWD with 0.15 hr ≤ P orb ≤ 8.5 day will be detected with 100% probability and that above 8.5 day the detection probability decreases linearly to 0 at P orb = 35 day.
Our other basic assumptions may be listed as follows: we use Miller and Scalo (1979) IMF, flat distributions over initial mass ratios of the components and the logarithm of the orbital separation, and a thermal distribution of orbital eccentricities.
AN OVERVIEW OF THE MODELS
For the modelling of the population of double white dwarfs we used the numerical code SeBa (Portegies Zwart and Verbunt, 1996) with modifications for low-and intermediate mass stars described by Nelemans et al. (2000b). Some of the important points are discussed below.
Since most progenitors of white dwarfs are low-mass stars, the Galactic star formation history influences the current birthrate of CDWD and the properties of their population. This factor was not studied before. For the present study we take a time-dependent star formation rate in the Galactic disk as
SFR(t) = 15 exp(−t/τ ) M ⊙ yr −1 ,(4)
where τ = 7 Gyr. Equation (4) gives current SFR of 3.6 M ⊙ yr −1 , which is compatible with the observational estimates (Rana, 1991;van den Hoek and de Jong, 1997). With this equation, the amount of matter that has been turned into stars over the lifetime of the Galactic disk (10 Gyr in our model) is ∼ 8×10 10 M ⊙ . It is higher than the current mass of the disk, since a part of this matter is returned to the ISM by supernovae and stellar winds. We also compute several models with constant SFR, to allow comparison with previous work (see Tab. 2). The distribution of stars in the Galactic disk is taken as
ρ(R, z) = ρ 0 e −R/H sech(z/h) 2 pc −3 ,(5)
where we use H = 2.5 kpc (Sackett, 1997) and h = 200 pc. The age and mass dependence of h is neglected. Table 2 gives an overview of our assumptions and model results and a comparison with some models of other authors. Model A is our basic model with an exponential star formation rate in the disk [Eq. (4)], initial fraction of binaries equal to 50% (i.e. with 2/3 of all stars in binaries) and cooling of white dwarfs according to DSBH, but modified as described below, in Sec. 6. The 50% fraction of binaries is suggested as a lower limit to their actual occurrence by the studies of normal mainsequence stars (Abt, 1983;Duquennoy and Mayor, 1991). This model, The columns list the identifiers of the models, type of star formation rate assumed for the model (exponential or constant), initial fraction of binaries, total Galactic number of WD, rates of formation and merger of CDWD per 100 yr, rate of merger of CDWD with M 1 +M 2 ≥ M Ch per 1000 yr (SNe Ia), rate of formation of Interacting double WD (AM CVn type stars) per 1000 yr, total number of Close double WD in the Galaxy, local density of all WD per pc 3 , local rate of formation of planetary nebulae per pc 3 per yr. HAN and ITY denote (Iben et al., 1997) and (Han, 1998) models, OBS -observational data.
as well as all our models presented in the Table, were calculated with αλ = 2 in Eq. (1) and γ = 1.75 in Eq. (3). Model A ′ is similar to the model A, but assumes that the first stage of mass loss is a common envelope described by Eq.
(1) instead of Eq. (3). Model B is similar to the model A, but has initially all stars in binaries, while model C has a constant star formation rate and 50% binaries. Model D has a constant star formation rate and 100% binaries. Models C and D were normalised to SFR of 4 M ⊙ yr −1 , to allow comparison with the models of Iben et al. (1997, ITY) and Han (1998). The former model was recalculated by LRY for the disk age of 10 Gyr, the age assumed in this study. Note, that ITY assume that unstable mass loss always results in the formation of a common envelope, and their formulation of the equation for the energy balance is different from Eq. (1). Briefly, the comparison of models shows the following. Models with an exponential SFR compared to the models with a constant SFR (mod. A vs. mod. C) have a higher number of old stars and a higher mass of the disk. This gives higher birthrates of CDWD and AM CVn stars. The rate of mergers giving SNe Ia is similar, since it is determined mainly by the SFR in the past ∼ 300 Myr (Tutukov and Yungelson, 1994).
Model A with Eq. (3) for the first mass loss episode gives less mergers than model A ′ with Eq. (1), since it produces wider pairs. For the same reason the occurrence of SNe Ia is lower in model A ′ . Formation rate of interacting systems (AM CVn's) is higher in model A ′ , because, if two common envelope phases occur, the second-formed WD is typically less massive than companion; if such a pair is brought into contact due to angular momentum loss via gravitational wave radiation, unequal masses of components favour stable mass transfer (Nelemans et al., 2001).
Model D has an IMF rather similar to Han's model, but treats the first mass loss episode differently , does not have companion reinforced stellar wind, and has a higher αλ value. This results in relatively less mergers in the first stage of mass loss and in higher birthrate and total number of CDWD and higher occurrence rate of SNe Ia.
The Iben et al. model differs from all other models by applying a different equation for the evolution in the common envelope, which is, in practice, equivalent to the usage of much higher value of αλ. This results in less frequent mergers in both stages of mass loss, hence, a higher total number of CDWD. The total number of the Galactic WD in this model is lower, compared to our models. This is due to the ITY assumption that close and wide binaries obey different distributions over the mass ratios of the components: flat for close systems and ∝ q −2.5 for wide systems (see Tutukov & Yungelson (1993) for details).
Our models share with the Han and ITY models the same assumptions on the initial distributions of close binaries over mass ratios of components and their separations and have rather similar initial IMF for the primary components. The variations of the birthrates and numbers of objects in different classes (within factor ∼ 3) arise mainly from the differences in the treatment of mass loss and transfer, in initial-final mass relations and other more minor details of population synthesis codes. Since the assumptions in the different studies are generally in agreement with results obtained from the modelling of stellar evolution, Tab. 2 illustrates the limits of the accuracy of predictions by the state of the art population synthesis studies for binary stars.
Observational data which may help to constrain the models are rather scarce and uncertain, e.g., the estimates of local space density of white dwarfs ρ wd,⊙ differ by a factor 5 (Knox et al., 1999;Festin, 1998;Oswalt et al., 1995;Ruiz and Takamiya, 1995). Our basic model A, as well as model C, complies with these observational limits. Model A gives the annual birthrate of planetary nebulae in somewhat better agreement with the observations (Pottasch, 1996) than model C.
Model A, as well as the rest of the models in Tab. 2, agrees reasonably with the occurrence of SNe Ia in galaxies similar to the Milky Way (E. Cappellaro, this volume). The difference in SNe Ia rates between the models has to be attributed mainly to the different treatment of mass loss and to the different initial-final mass relations for the components of binaries. The model population of CDWD as a function of the orbital period and mass of the brighter dwarf of the pair. Top left: distribution of the currently born CDWD in model A. Top right: "observed sample" (V lim = 15), with cooling according to DSBH and Blöcker (1995). Bottom right: the same sample for the modified DSBH cooling. Bottom left: the total Galactic population of CDWD younger than 100 Myr. Dots with error bars: observed CDWD.
6.
MODELS VS. OBSERVATIONS 1. Orbital periods and masses of close double white dwarfs. The parameters which are determined for all known CDWD are the orbital period and the mass of the brighter component of the pair. We plot in Fig. 4 the P orb − m distributions of the occurrence rate for the currently born CDWD and for the simulated magnitude limited samples (V lim = 15) for the models with different cooling prescriptions. Cumulative distribution over the periods. The solid line is for the model with reduced cooling time for low-mass WD. The dashed line is for DSBH cooling without modifications and the dash-dotted line is for a model with constant "observability" time of 100 Myr. Open squares give distribution of observed CDWD, filled circles give the distribution including the sdB+wd binaries.
In general, the white dwarf observed as the brighter member of the pair, is the one that was formed last, but occasionally, it may be the one that was formed first, due to the effect of differential cooling, see Fig. 3. The top right panel of Fig. 4 shows that if DSBH cooling curves are taken at face value, the observed sample contains predominantly low mass (M < ∼ 0.3 M ⊙ ) white dwarfs, in contradiction to observations. However, as we mentioned above, low-mass WD may experience thermal flashes, which may reduce the mass of their hydrogen envelopes and extinguish hydrogen burning. This may be especially true for WD in close pairs (a ∼ R ⊙ ), where one easily expects the formation of a common envelope during a flash. Since estimates for the amount of mass which may be lost in a flash is not yet available , we make an ad hoc extreme assumption that white dwarfs with masses below 0.3 M ⊙ cool like the most massive helium (0.46 M ⊙ ) white dwarfs (hereafter we call this modified DSBH cooling). As can be seen from the bottom right panel of Fig. 4, this assumption brings the model in a much better agreement with observations. All model distributions which follow, are given for the modified DSBH cooling.
For comparison, we plot in the left bottom panel of Fig. 4 the "observed" distribution assuming (like in the studies of other authors) that all WD are visible for 10 8 yr, unless a pair merges earlier due to GWR. Since in this case cooling curves are not applied, i.e. no magnitudes are computed for the white dwarfs, we can not construct a magnitude limited sample and this panel gives the total number of "potentially visible" CDWD in the Galaxy. The better agreement with observations compared to the modified DSBH case is only apparent, as can be easily seen from the cumulative distributions (Fig. 5). Figure 5 shows a deficiency of observed systems between ∼ 5 hr and ∼ 1 day. No selection effects are known that prevent detection of white dwarf binaries with such periods. This "gap" may be partially filled if we plot also subdwarf B stars with suspected white dwarf companions, thus assuming that the current sdB star is a white dwarf in the making. In addition to systems listed in Tab. 1, we include the binaries for which only P orb is determined. However, the number of detected systems is still too small to decide whether the "gap" is real and whether revisions of the stellar evolution models or CDWD formation scenarios are required.
The merger of white dwarfs is one of the models for SNe Ia (see, e.g., Livio 1999 for a review). We estimate that one merger candidate with M 1 + M 2 ≥ 1.4 M ⊙ may be found in a WD sample complete to V ≈ 15 which contains ∼ 200 CDWD among a total of ∼ 1000 WD. In view of the data in Tab. 2, this estimate is probably uncertain within a factor of at least 2 ÷ 3. 2. Period -mass ratio distribution. Our treatment of the first phase of mass transfer between a giant and a main-sequence star, which doesn't result in a significant spiral-in, leads to a concentration of the mass ratios of the model systems around q ∼ 1. In practice, q in the observed systems can only be determined if the ratio of the luminosities of the components is < ∼ 5 (Moran et al., 2000). The P orb − q distribution for the theoretical magnitude limited sample which obeys this criterion, is shown in Fig. 6. In model A ′ (with a common envelope in the first mass transfer) CDWD have predominantly q ∼ (1.75 − 2), which is not observed and there are hardly any systems with q ∼ 1.
3. Mass spectrum of white dwarfs. The left panel of Fig. 7 shows the spectrum of white dwarf masses in a sample limited by V = 15 and based on model A. It includes white dwarfs in close pairs which are brighter than their companions and genuine single objects, white dwarfs which are components of the initially wide pairs, merger products, white dwarfs which became single due to disruption of binaries by SNe explosions. In the same Figure we plot the cumulative distribution for DA white dwarfs masses, as estimated by Bergeron et al. (1992) and Bragaglia et al. (1995)
1 .
The expected fraction of CDWD among all "observed" WD in our preferred model A is ∼ 23%, slightly higher than the upper limit of the range suggested by Maxted and Marsh (1999) for DA white dwarfs: 1.7 to 19% with 95% confidence. Since lowering the initial fraction of bina- ries below 50% would contradict the observations, the high percentage of CDWD in the model sample may mean that, even after our modification, DSBH results overestimate cooling times of the lowest mass white dwarfs. The right panel of Fig. 7 shows results of a simple numerical experiment in which we assign to all helium white dwarfs the cooling curve of a 0.414 M ⊙ dwarf from DSBH and a cooling curve of a 0.605 M ⊙ dwarf (Blöcker, 1995) to all CO white dwarfs. This gives 17% for the fraction of CDWD. Even so, the contribution of the lowest mass WD to the total model sample still seems to be overestimated. However, there may exist yet unknown selection effects which prevent their detection.
Yet another problem is the deficiency of (0.45−0.50) M ⊙ white dwarfs in the model sample. In this part of the theoretical mass distribution only hybrid white dwarfs, descending via case B mass transfer from the stars in a narrow range of ∼ (2.5 − 5) M ⊙ do occur (Iben and Tutukov, 1985). Their formation rate is relatively low. On the other hand, cooling of these objects which have almost pure helium envelopes with mass ∼ (0.01 − 0.1) M ⊙ was not yet studied and it's possible that the simple interpolation between cooling curves for CO and He white dwarfs is not appropriate and gives misleading results.
CONCLUSIONS
1. The standard Algol-type evolution and spiral-in in common envelopes cannot explain the wide orbits of the progenitors of close double WD after the first stage of mass loss. In low-mass binaries with similar masses of components, the loss of the envelope of the initial primary probably doesn't cause a strong spiral-in. In the absence of a physical picture of this process, we suggest to describe the change of separation of the components by an angular momentum loss law [Eq. (3)]. An attempt to model the parameters of three observed CDWD with two helium components results in 1.4 < ∼ γ < ∼ 1.8. 2. Modelling of the observed sample of close binary WD shows that low-mass WD have to cool faster than is suggested by the recently published cooling curves for helium WD with thick hydrogen envelopes (Driebe et al., 1998;Sarna et al., 2000). We suggest that an additional mass loss by WDs in winds or thermal flashes may strongly diminish the masses of envelopes and reduce the cooling times.
3. A reasonable agreement of the population synthesis results with observations in the estimates of the local space density of WD, and the P orb −m, P orb −q, and N (P orb ) distributions for CDWD may be achieved if the parameter γ in Eq. (3) is ∼ 1.7 and it's assumed that WD with M ≤ 0.3 M ⊙ cool like the most massive helium WD. Further reduction of cooling times for the helium WD may even improve agreement. 4. The model with an initial binary fraction of 50% (2/3 of stars are in binaries) fits the observations better than the model in which initially 100% of stars are in binaries.
5. We estimate that the detection of at least one WD pair with M 1 + M 2 ≥ M Ch may require a survey of ∼ 1000 white dwarfs. But the uncertainties in the input data for the population synthesis studies and badly known selection effects, most probably make this estimate accurate only within a factor of ∼ 2 ÷ 3.
Figure 1 .
1Figure 1. Periods after the first phase of mass loss as a function of the mass of the secondary component at this time. Solid, dashed, and dotted lines are for WD 0957-666, 1101+364, and 0136+768 respectively. The top three lines are periods needed to explain the mass of the last formed white dwarf. The middle three lines give the maximum period if the first white dwarf would be formed by conservative mass exchange. The lower three lines give the periods for the case when the formation of the dwarf was accompanied by a spiral-in (Eq. (1) with αλ = 2).
Figure 3 .
3Figure 3. Cooling curves for 0.179, 0.300, 0.414 M⊙ white dwarfs from Driebe et al. (1998) and for 0.6 and 0.8 M⊙ ones from Blöcker (1995), from right to left. Straight lines are the fits to the curves used in the simulations.
Figure 4 .
4Figure 4. The model population of CDWD as a function of the orbital period and mass of the brighter dwarf of the pair. Top left: distribution of the currently born CDWD in model A. Top right: "observed sample" (V lim = 15), with cooling according to DSBH and Blöcker (1995). Bottom right: the same sample for the modified DSBH cooling. Bottom left: the total Galactic population of CDWD younger than 100 Myr. Dots with error bars: observed CDWD.
Figure 5 .
5Figure 5. Cumulative distribution over the periods. The solid line is for the model with reduced cooling time for low-mass WD. The dashed line is for DSBH cooling without modifications and the dash-dotted line is for a model with constant "observability" time of 100 Myr. Open squares give distribution of observed CDWD, filled circles give the distribution including the sdB+wd binaries.
Figure 6 .
6Model sample of CDWD as a function of the orbital period and mass ratio of components for V lim = 15 and the ratio of the luminosities of the components ≤ 5. Left panel is for model A with the first phase of mass transfer treated by Eq. (3), right panel is the same for the model A ′ with standard Eq. (1) with αλ = 2 in both mass transfer phases. Dots with error bars are observed systems.
Figure 7 .
7Left panel -mass spectrum of all white dwarfs in model A with modified DSBH cooling. Right panel -the same for a model in which all helium white dwarfs cool like a ∼ 0.414 M⊙ dwarf and all CO white dwarfs cool like a 0.605 M⊙ dwarf. In both panels the theoretical cumulative distribution is shown as a solid black line and the cumulative distribution of observed systems as a grey line.
Table 1 .
1Known close double white dwarfs and subdwarfs with WD companions Objects 1 -14 are CDWD, objects 15 and 16 are sdB stars with suspected white dwarf companions. M 1 is the mass of the brighter component and M 2 is the mass of the companion. See the text for references.N
Name
P (d)
M1/M⊙
M2/M⊙
1
WD 0135−052
1.556
0.47
0.52
2
WD 0136+768
1.407
0.44
0.34
3
WD 0957−666
0.061
0.37
0.32
4
WD 1101+364
0.145
0.31
0.36
5
WD 1204+450
1.603
0.51
0.51
6
WD 1704+481A
0.145
0.39
0.56
7
WD 1022+050
1.157
0.35
8
WD 1202+608
1.493
0.40
9
WD 1241−010
3.347
0.31
10
WD 1317+453
4.872
0.33
11
WD 1713+332
1.123
0.38
12
WD 1824+040
6.266
0.39
13
WD 2032+188
5.084
0.36
14
WD 2331+290
0.167
0.39
15
KPD 0422+5421
0.09
0.51
0.53
16
KPD 1930+2752
0.095
0.5
0.97
Table 2 .
2Summary of modelsMod. SFR
bin WD
ν
νm
SN Ia IWD CWD ρ wd,⊙
νPN,⊙
%
10 9
10 −2 10 −2
10 −3
10 −3
10 8
10 −3
10 −12
A
Exp
50
9.2
4.8
2.2
3.2
4.6
2.5
19
2.3
A ′
Exp
50
3.0
2.4
2.2
11.0
1.0
B
Exp 100
8.1
3.6
5.4
8.8
4.1
C
Cns
50
4.0
3.2
1.6
3.4
3.6
1.2
8.5
1.9
D
Cns 100
5.3
2.8
5.8
6.1
1.9
HAN
Cns 100
2.9
2.8
2.6
23.0
0.9
ITY
Cns 100
4.0
8.7
1.7
1.9
8.5
3.5
8.3
1.5
OBS
4 ± 2
4 ÷ 20
3
These masses may have to be increased by about 0.05 M ⊙ , if one uses models of white dwarfs with thick hydrogen envelopes for the estimates(Napiwotzki et al., 1999).
LRY acknowledges the support of the LOC and Astronomical Institute "Anton Pannekoek" which enabled him to attend the conference. This work was supported by NWO Spinoza grant 08-0 to E.P.J. van den Heuvel, RFBR grant 99-02-16037, "Astronomy and Space Research" Program (project 1.4.4.1), and NASA through Hubble Fellowship grant HF-01112.01-98A awarded (to SPZ) by the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., for NASA under contract NAS 5-26555.
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|
[] |
[
"U(2) Flavor Physics without U(2) Symmetry Typeset using REVT E X",
"U(2) Flavor Physics without U(2) Symmetry Typeset using REVT E X"
] |
[
"Alfredo Aranda \nDepartment of Physics\nNuclear and Particle Theory Group\nCollege of William and Mary\n23187-8795WilliamsburgVA\n",
"Christopher D Carone \nDepartment of Physics\nNuclear and Particle Theory Group\nCollege of William and Mary\n23187-8795WilliamsburgVA\n",
"Richard F Lebed \nJefferson Lab\n12000 Jefferson Avenue23606Newport NewsVA\n"
] |
[
"Department of Physics\nNuclear and Particle Theory Group\nCollege of William and Mary\n23187-8795WilliamsburgVA",
"Department of Physics\nNuclear and Particle Theory Group\nCollege of William and Mary\n23187-8795WilliamsburgVA",
"Jefferson Lab\n12000 Jefferson Avenue23606Newport NewsVA"
] |
[] |
We present a model of fermion masses based on a minimal, non-Abelian discrete symmetry that reproduces the Yukawa matrices usually associated with U(2) theories of flavor. Mass and mixing angle relations that follow from the simple form of the quark and charged lepton Yukawa textures are therefore common to both theories. We show that the differing representation structure of our horizontal symmetry allows for new solutions to the solar and atmospheric neutrino problems that do not involve modification of the original charged fermion Yukawa textures, or the introduction of sterile neutrinos.
|
10.1016/s0370-2693(99)01497-5
|
[
"https://arxiv.org/pdf/hep-ph/9910392v2.pdf"
] | 17,098,285 |
hep-ph/9910392
|
5b57ab72c96e7c5d4caef847dc61baf6006230ef
|
U(2) Flavor Physics without U(2) Symmetry Typeset using REVT E X
5 Jan 2000 (October, 1999)
Alfredo Aranda
Department of Physics
Nuclear and Particle Theory Group
College of William and Mary
23187-8795WilliamsburgVA
Christopher D Carone
Department of Physics
Nuclear and Particle Theory Group
College of William and Mary
23187-8795WilliamsburgVA
Richard F Lebed
Jefferson Lab
12000 Jefferson Avenue23606Newport NewsVA
U(2) Flavor Physics without U(2) Symmetry Typeset using REVT E X
5 Jan 2000 (October, 1999)
We present a model of fermion masses based on a minimal, non-Abelian discrete symmetry that reproduces the Yukawa matrices usually associated with U(2) theories of flavor. Mass and mixing angle relations that follow from the simple form of the quark and charged lepton Yukawa textures are therefore common to both theories. We show that the differing representation structure of our horizontal symmetry allows for new solutions to the solar and atmospheric neutrino problems that do not involve modification of the original charged fermion Yukawa textures, or the introduction of sterile neutrinos.
I. INTRODUCTION
One path toward understanding the observed hierarchy of fermion masses and mixing angles is to assert that at some high energy scale all Yukawa couplings, except that of the top quark, are forbidden by a new symmetry G f that acts horizontally across the three standard model generations. As this symmetry is spontaneously broken to smaller subgroups at successively lower energy scales, a hierarchy of Yukawa couplings can be generated. The light fermion Yukawa couplings originate from higher-dimension operators involving the standard model matter fields and a set of 'flavon' fields φ, which are responsible for spontaneously breaking G f . The higher-dimension operators are suppressed by a flavor scale M f , which is the ultraviolet cut-off of the effective theory; ratios of flavon vacuum expectation values (vevs) to the flavor scale, φ /M f , provide a set of small symmetry-breaking parameters that may be included systematically in the low-energy effective theory. Many models of this type have been proposed, with G f either gauged or global, continuous or discrete, Abelian or non-Abelian, or some appropriate combination thereof [1]. Non-Abelian symmetries are particularly interesting in the context of supersymmetric theories, where flavor-changing neutral current (FCNC) processes mediated by superparticle exchange can be phenomenologically unacceptable [2]. If the three generations of any given standard model matter field are placed in 2⊕1 representations of some non-Abelian horizontal symmetry group, it is possible to achieve an exact degeneracy between superparticles of the first two generations when G f is unbroken. In the low-energy theory, this degeneracy is lifted by the same small symmetry-breaking parameters that determine the light fermion Yukawa couplings, so that FCNC effects remain adequately suppressed, even with superparticle masses less than a TeV. A particularly elegant model of this type considered in the literature assumes the continuous, global symmetry G f = U(2) [3][4][5]. Quarks and leptons are assigned to 2⊕1 representations, so that in tensor notation, one may represent the three generations of any matter field by F a + F 3 , where a is a U(2) index, and F is Q, U, D, L, or E. A set of flavons is introduced consisting of φ a , S ab , and A ab , where φ is a U(2) doublet, and S (A) is a symmetric (antisymmetric) U(2) triplet (singlet). If one assumes the pattern of vevs
φ M f = 0 ǫ , S M f = 0 0 0 ǫ , and A M f = 0 ǫ ′ −ǫ ′ 0 , (1.1)
which follows from the sequential breaking
U(2) ǫ → U(1) ǫ ′ → nothing ,(1.Y D ∼ 0 d 1 ǫ ′ 0 −d 1 ǫ ′ d 2 ǫ d 3 ǫ 0 d 4 ǫ 1 ,(1.3)
where ǫ ≈ 0.02, ǫ ′ ≈ 0.004, and d 1 , . . . , d 4 are O(1) coefficients that can be determined from Ref. [5]. Differences between hierarchies in Y D and Y U can be obtained by embedding the model in a grand unified theory [5]. For example, in an SU(5) GUT, one obtains differing powers of ǫ and ǫ ′ in the up quark Yukawa matrix by assuming that S ab transforms as a 75; combined GUT and flavor symmetries prevent A ab and S ab from coupling to the up and charm quark fields, unless an additional flavor singlet field Σ is introduced that transforms as an SU(5) adjoint. With Σ /M f ∼ ǫ, it is possible to explain why m d :: m s :: m b = λ 4 :: λ 2 :: 1, while m u :: m c :: m t = λ 8 :: λ 4 :: 1, where λ = 0.22 is the Cabibbo angle. The ratio m t /m b is assumed to be unrelated to U(2) symmetry breaking, and is put into the low-energy theory by hand.
In this letter we show that the properties of the U(2) model leading to the successful Yukawa textures described above are also properties of smaller discrete symmetry groups. To reproduce all of the phenomenological successes of the U(2) model, we require a candidate discrete symmetry group to have the following properties:
• 1, 2, and 3 dimensional representations.
• The multiplication rule 2⊗2=3⊕1.
• A subgroup H f such that the breaking pattern G f → H f → nothing reproduces the canonical U(2) texture given in Eq. (1.3). This implies that an unbroken H f -symmetry forbids all Yukawa entries with O(ǫ ′ ) vevs, but not those with O(ǫ) vevs.
In the next section we show that the smallest group satisfying these conditions is a product of the double tetrahedral group T ′ and an additional Z 3 factor. Since U(2) is isomorphic to SU(2)×U(1), it is not surprising that our candidate symmetry involves the product of a discrete subgroup of SU(2), T ′ , and a discrete subroup of U(1), Z 3 . At this point, the reader who is unfamiliar with discrete group theory may feel somewhat uneasy. 1 We stress that the group T ′ is in fact a very simple discrete symmetry, a spinorial generalization of the symmetry of a regular tetrahedron (see Section II). It is worth noting that the charge assignments in the model we present render T ′ a nonanomalous discrete gauge symmetry, while the Z 3 factor is anomalous. Models based on non-Abelian discrete gauge symmetries have yielded viable theories of fermion masses, as have models based on discrete subgroups of anomalous U(1) gauge symmetries [1]. In the latter case it is generally assumed that the U(1) anomalies are cancelled by the Green-Schwarz mechanism in string theory [6]. It is interesting that our model turns out to be a hybrid of these two ideas.
One of the virtues of the model discussed in this letter is that it allows for elegant extensions that explain the solar and atmospheric neutrino deficits, while maintaining the original quark and charged lepton Yukawa textures. This distinguishes our model from the modified version of the U(2) model presented in Ref. [7]. Preserving the U(2) charged fermion textures is desirable since they lead to successful mass and mixing angle relations such as |V ub /V cb | = m u /m c , which are 'exact' in the sense that they contain no unknown O(1) multiplicative factors. Since we succeed in explaining solar and atmospheric neutrino oscillations without sacrificing the predictivity of the original model, we need not introduce sterile neutrinos, as in Ref. [8]. However, we do not try to explain simultaneously the more controversial LSND results [9] in this paper. We will consider versions of our model that include sterile neutrinos in a longer publication [10].
II. THE SYMMETRY
We seek a non-Abelian candidate group G f that provides the 2⊕1 representation (rep) structure for the matter fields described in the previous section. In order for the breaking of G f to reproduce the U(2) charged fermion Yukawa texture in Eq. (1.3), one must have flavons that perform the same roles as φ a , S ab , and A ab in the U(2) model. Since these are doublet, triplet, and nontrivial singlet reps, respectively, we require G f to have reps of the same dimensions. Nontrivial singlets appear in all discrete groups of order < 32 [12], so we seek groups G f with doublet and triplet representations.
The order 12 tetrahedral group T , the group of proper symmetries of a regular tetrahedron (which is also the alternating group A 4 , consisting of even permutations of four objects), is the smallest containing a triplet rep, but has no doublet reps. A number of groups with orders < 24 possess either doublet or triplet reps, but not both (See, for example, [12]).
It turns out that two groups of order 24 possess both doublet and triplet reps. One is the symmetric group S 4 of permutations on four objects, which is isomorphic to the group O of proper symmetries of a cube as well as the group T d of all proper and improper symmetries of a regular tetrahedron. S 4 possesses two triplets 3 ± , two singlets 1 ± , and one doublet 2. However, in this case one encounters another difficulty: The combination rule for doublets in S 4 is 2 ⊗ 2 = 2 ⊕ 1 − ⊕ 1 + , which implies that the triplet flavon cannot connect two doublet fields such as those of the first two generations of Q and U. Thus, S 4 is not suitable for our purposes.
The unique group of order < 32 with the combination rule 2 ⊗ 2 ⊃ 3 is the double tetrahedral group T ′ , which is order 24. The character table, from which one may readily generate explicit representation matrices, is presented in Table I. Geometrically, T ′ is the group of symmetries of a regular tetrahedron under proper rotations (Fig. 1). These symmetries consist of 1) rotations by 2π/3 about an axis connecting a vertex and the opposite face (C 3 ), 2) rotations by π about an axis connecting the midpoints of two non-intersecting edges (C 2 ), and 3) the rotation R by 2π about any axis, which produces a factor −1 in the even-dimensional reps, exactly as in SU (2). Indeed, this feature is a consequence of T ′ ⊂ SU(2), and the rotations C 3 and C 2 are actually of orders 6 and 4, respectively. Also, T ′ is isomorphic to the group SL 2 (F 3 ), which consists of 2 × 2 unimodular matrices whose elements are added and multiplied as integers modulo 3.
T ′ has three singlets 1 0 and 1 ± , three doublets, 2 0 and 2 ± , and one triplet, 3. The triality superscript describes in a concise way the rules for combining these reps: With the identification of ± as ±1, the trialities add under addition modulo 3. In addition, the following rules hold:
1 ⊗ R = R ⊗ 1 = R for any rep R, 2 ⊗ 2 = 3 ⊕ 1, 2 ⊗ 3 = 3 ⊗ 2 = 2 0 ⊕ 2 + ⊕ 2 − , 3 ⊗ 3 = 3 ⊕ 3 ⊕ 1 0 ⊕ 1 + ⊕ 1 − . (2.1)
Trialities flip sign under Hermitian conjugation. Thus, for example, 2 + ⊗ 2 − = 3 ⊕ 1 0 , and (2 + ) † ⊗ 2 − = 3 ⊕ 1 + .
One must now determine whether it is possible to place a sequence of vevs hierarchically in the desired elements of the Yukawa matrices. Notice if G f is broken to a subgroup H f that rotates the first generation matter fields by a common nontrivial phase, then H f symmetry forbids all entries with O(ǫ ′ ) vevs in Eq. (1.3). Therefore, we require that the elements of G f defining this subgroup have two-dimensional rep matrices of the form diag{ρ, 1}, with ρ = exp(2πin/N) for some N that divides the order of G f and some integer n relatively prime with respect to N. This form for ρ follows because reps of finite groups may be chosen unitary, and must give the identity when raised to the power of the order of G f . Such elements generate a subgroup H f = Z N of G f . Whether such elements exist in G f can be determined since the rep of any element can be brought to diagonal form by a basis transformation, while the eigenvalues ρ, 1 are invariant under such basis changes.
Even if a given element C ∈ G f has the diagonal form diag{ρ 1 , ρ 2 }, ρ i = exp(2πin i /N) (and thus generates a subgroup, Z C N , of G f ), a phase rotation of the form diag{ρ, 1} can be achieved if the original G f is extended by forming a direct product with an additional factor Z N . We then identify H f as a subgroup of Z C N × Z N . We choose one element of the additional Z N to compensate the phase of the 22 element of C, and similarly for the other elements of the Z C N . The element corresponding to C in G f × Z N then effectively acts upon the doublet as diag{exp[2πi(n 1 −n 2 )/N], 1}, and the remaining symmetry is Z N/gcf(N,|n 1 −n 2 |) . In the case that |n 1 − n 2 | and N are relatively prime, this reduction amounts to forming the diagonal subgroup Z D N of Z C N × Z N . Similar arguments apply to the singlet and triplet reps. In the particular case of G f = T ′ , one finds elements C that generate either Z 2 or Z 3 subgroups. By introducing an additional Z n (with n = 2 or 3) one can arrange for a Z n subgroup that affects only the first generation fields. In the case of Z 2 , the nontrivial element of the diagonal subgroup is of the form diag{−1, 1}, which leaves the 11 and 22 entries of the Yukawa matrices invariant. The incorrect relation m u = m c then follows. On the other hand, Z 3 prevents an invariant 11 entry, so we are led to adopt
G f = T ′ × Z 3 .
(2.
2)
The reps of G f are named by extending the notation for T ′ to include a superscript indicating the Z 3 rep. These are the trivial rep 0, which takes all elements to the identity, and two complex-conjugate reps + and −. Like the trialities, these indices combine via addition modulo 3. We adopt the convention that the T ′ × Z 3 reps 1 00 , 1 +− , 1 −+ , 2 0− , 2 ++ and 2 −0 are special, in that these singlet reps and the second component of the doublets remain invariant under Z D 3 . Thus any 2⊕1 combination of these reps is potentially interesting for model building.
III. A MINIMAL MODEL
The minimal model has the three generations of matter fields transforming as 2 0− ⊕ 1 00 under G f = T ′ × Z 3 . The Higgs fields H U,D are pure singlets of G f and transform as 1 00 . Given these assignments, it is easy to obtain the transformation properties of the Yukawa matrices,
Y U,D,L ∼ [3 − ⊕ 1 0− ] [2 0+ ] [2 0+ ] [1 00 ] . (3.1)
Eq. (3.1) indicates the reps of the flavon fields needed to construct the fermion mass matrices. They are 1 0− , 2 0+ , and 3 − , which we call A, φ, and S, respectively. Once these flavons acquire vevs, the flavor group is broken. We are interested in a two-step breaking controlled by two small parameters ǫ, and ǫ ′ , where
T ′ ⊗ Z 3 ǫ −→ Z D 3 ǫ ′ −→ nothing . (3.2)
Since we have chosen a doublet rep for the first two generations that transforms as diag{ρ, 1} under Z D 3 , only the 22, 23, and 32 entries of the Yukawa matrices may develop vevs of O(ǫ), which we assume originate from vevs in S and φ. The symmetry Z D 3 is then broken by a 1 0− vev of O(ǫ ′ ). The Clebsch-Gordan coefficients that couple a 1 0− to two 2 0− doublets is proportional to σ 2 , so the ǫ ′ appears in an antisymmetric matrix. We therefore produce the U(2) texture of Eq. (1.3). Since the 1 0− and 3 − flavon vevs appear as antisymmetric and symmetric matrices, respectively, all features of the grand unified extension of the U(2) model apply here, assuming the same GUT transformation properties are assigned to φ, S, and A. One can also show readily that the squark and slepton mass squared matrices are the same as in the U(2) model.
It is worth noting that we could construct completely equivalent theories had we chosen to place the matter fields in reps like 2 ++ ⊕1 00 or 2 −0 ⊕1 00 , which have the same transformation properties under Z D 3 as our original choice. The reps 2 0− ⊕ 1 00 are desirable in that they fill the complete SU(2) representations 2 ⊕ 1, if we were to embed T ′ in SU (2). Since anomaly diagrams linear in this SU(2) vanish (and hence the linear Ibáñez-Ross condition is satisfied [13]), we conclude that T ′ is a consistent discrete gauge symmetry [14]. The additional Z 3 may also be considered a discrete gauge symmetry, providing its anomalies are cancelled by the Green-Schwarz mechanism.
IV. NEUTRINOS
In this section, we show that the model presented in Section III can be extended to describe the observed deficit in solar and atmospheric neutrinos. We consider two cases:
Case I: Here we do not assume grand unification, so that all flavons are SU(5) singlets. This case is of interest, for example, if one is only concerned with explaining flavor physics of the lepton sector. We choose
ν R ∼ 2 0− ⊕ 1 −+ . (4.1)
Note that the only difference from the other matter fields is the representation choice for the third generation field. The neutrino Dirac and Majorana mass matrices then have different textures from the charged fermion mass matrices. Their transformation properties are given by
M LR ∼ [3 − ⊕ 1 0− ] [2 +0 ] [2 0+ ] [1 +− ] , M RR ∼ [3 − ] [2 +0 ] [2 +0 ] [1 −+ ] . (4.2)
Note that we obtain the same triplet and nontrivial singlet in the upper 2 × 2 block as in the charged fermion mass matrices, as well as one of the same flavon doublets, the 2 0+ ; the rep 1 0− is not present in M RR , since Majorana mass matrices are symmetric. In addition we obtain the reps 2 +0 , 1 +− , and 1 −+ , which did not appear in Eq. (3.1). New flavon fields can now be introduced with these transformation properties, and their effects on the neutrino physics can be explored. Let us consider introducing a single 2 new flavon φ ν transforming as a 2 +0 and with a vev
φ ν ∼ σ 2 ǫ ′ ǫ , (4.3)
where σ 2 is the Clebsch that couples the two doublets to 1 0− . The introduction of this new flavon is the only extension we make to the model in order to describe the neutrino phenomenology. After introducing φ ν , the neutrino Dirac and Majorana mass matrices read
M LR = 0 l 1 ǫ ′ l 3 r 2 ǫ ′ −l 1 ǫ ′ l 2 ǫ l 3 r 1 ǫ 0 l 4 ǫ 0 H U , M RR = r 4 r 2 2 ǫ ′ 2 r 4 r 1 r 2 ǫǫ ′ r 2 ǫ ′ r 4 r 1 r 2 ǫǫ ′ r 3 ǫ r 1 ǫ r 2 ǫ ′ r 1 ǫ 0 Λ R , (4.4)
where Λ R is the right-handed neutrino mass scale, and we have parameterized the O(1) coefficients. Furthermore, the charged lepton Yukawa matrix including O(1) coefficients reads
Y L ∼ 0 c 1 ǫ ′ 0 −c 1 ǫ ′ 3c 2 ǫ c 3 ǫ 0 c 4 ǫ 1 . (4.5)
The factor of 3 in the 22 entry is simply assumed at present, but originates from the Georgi-Jarlskog mechanism [15] in the grand unified case considered later. The left-handed Majorana mass matrix M LL follows from the seesaw mechanism
M LL ≈ M LR M −1 RR M T LR ,(4.6)
which yields
M LL ∼ (ǫ ′ /ǫ) 2 ǫ ′ /ǫ ǫ ′ /ǫ ǫ ′ /ǫ 1 1 ǫ ′ /ǫ 1 1 H U 2 ǫ Λ R ,(4.7)
where we have suppressed the O(1) coefficients. We naturally obtain large mixing between second-and third-generation neutrinos, while the 12 and 13 mixing angles are O(ǫ ′ /ǫ). However, taking into account the diagonalization of Y L , the relative 12 mixing angle can be made smaller, as we discuss below. Explanation of the observed atmospheric neutrino fluxes by ν µ -ν τ mixing suggests sin 2 2θ 23 > ∼ 0.8 and 10 −3 < ∼ ∆m 2 23 < ∼ 10 −2 , while the solar neutrino deficit may be accommodated assuming the small-angle MSW solution 2×10 −3 < ∼ sin 2 2θ 12 < ∼ 10 −2 for 4 × 10 −6 < ∼ ∆m 2 12 < ∼ 10 −5 , where all squared masses are given in eV 2 . We display below an explicit choice of the O(1) parameters that yields both solutions simultaneously; a more systematic global fit will be presented in Ref. [10].
If M LL and Y L are diagonalized by M LL = V M 0 LL V † , Y L = U L Y 0 L U † R ,
then the neutrino CKM matrix is given by
V ν = U † L V. (4.8)
We aim to reproduce the 12 and 23 mixing angles, as well as the ratio 10 2 < ∼ ∆m 2 23 /∆m 2 12 < ∼ 2.5 × 10 3 suggested by the data. Obtaining this ratio is sufficient since Λ R is not de- which fall in the desired ranges. While all our coefficients are of natural size, we have arranged for an O(15%) cancellation between 12 mixing angles in U L and V to reduce the size of sin 2 2θ 12 to the desired value.
Case II: Here we assume that the flavons transform nontrivially under an SU(5) GUT group, namely A ∼ 1, S ∼ 75, φ ∼ 1, and Σ ∼ 24. Note that since H ∼ 5, the products SH and AH transform as a 45 and 5, respectively, ultimately providing a factor of 3 enhancement in the 22 entry of Y L (the Georgi-Jarlskog mechanism). In addition, two 2 +0 doublets are introduced, φ ν1 and φ ν2 , since the texture obtained for the neutrino masses by adding only one extra doublet is not viable. Both doublets φ ν have vevs of the form displayed in Eq. (4.3). Crucially, the presence of these two new doublets does not alter the form of any charged fermion Yukawa texture.
The neutrino Dirac and Majorana mass matrices now take the form
M LR = 0 l 1 ǫ ′ l 5 r 2 ǫ ′ −l 1 ǫ ′ l 2 ǫ 2 l 3 r 1 ǫ 0 l 4 ǫ 0 H U , M RR = r 3 ǫ ′ 2 r 4 ǫǫ ′ r 2 ǫ ′ r 4 ǫǫ ′ r 5 ǫ 2 r 1 ǫ r 2 ǫ ′ r 1 ǫ 0 Λ R ,(4.10)
while the charged fermion mass matrix is the same as in Eq. (4.5). Using Eq. (4.6) one obtains the texture: Again these values fall in the desired ranges to explain the atmospheric and solar neutrino deficits, assuming an appropriate choice for Λ R .
M LL ∼ (ǫ ′ /ǫ) 2 ǫ ′ /ǫ ǫ ′ /ǫ ǫ ′ /ǫ 1 1 ǫ ′ /ǫ 1 1 H U 2 Λ R .
V. CONCLUSIONS
In this letter we have shown how to reproduce the quark and charged lepton Yukawa textures of the U(2) model in their entirety, using a minimal non-Abelian discrete symmetry group. We showed that the representation structure of T ′ × Z 3 , in particular the existence of three distinct 2-dimensional irreducible representations, allows for solutions to the solar and atmospheric neutrino problems that require neither a modification of the simple charged fermion Yukawa textures of the U(2) model nor the introduction of singlet neutrinos. The simplicity of the symmetry structure of our model suggests that a more comprehensive investigation of the space of possible models is justified. Work on alternative neutrino sectors as well as a more detailed phenomenological analysis of the models described here will be presented elsewhere [10].
termined by symmetry considerations and may be chosen freely. Assuming the previous values ǫ = 0.02 and ǫ ′ = 0.004 and the parameter set (l 1 , . . . , l 4 , r 1 , . . . , r 4 , c 1 , . . . , c
now choose (l 1 , . . . , l 5 , r 1 , . . . , r 5 , c 1 , . . . , c 4 ) = (sin 2 2θ 12 = 6 × 10 −3 , sin 2 2θ 23 = 0.995.(4.12)
FIG. 1 .
1Geometrical illustration of the group T ′ . The rotations C 2 and C 3 are defined in the text.
For a review of basic terms, see Ref.[11].
Assuming more than one φ ν leads to the same qualitative results.
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|
[] |
[
"FOURIER BASES AND FOURIER FRAMES ON SELF-AFFINE MEASURES",
"FOURIER BASES AND FOURIER FRAMES ON SELF-AFFINE MEASURES"
] |
[
"Ervin Dorin ",
"Chun-Kit Dutkay ",
"Yang Lai ",
"Wang "
] |
[] |
[] |
This paper gives a review of the recent progress in the study of Fourier bases and Fourier frames on self-affine measures. In particular, we emphasize the new matrix analysis approach for checking the completeness of a mutually orthogonal set. This method helps us settle down a long-standing conjecture that Hadamard triples generates self-affine spectral measures. It also gives us non-trivial examples of fractal measures with Fourier frames. Furthermore, a new avenue is open to investigate whether the Middle Third Cantor measure admits Fourier frames. 2010 Mathematics Subject Classification. Primary 42B05, 42A85, 28A25. 1 2 DORIN ERVIN DUTKAY, CHUN-KIT LAI, AND YANG WANG exponential functions. In 1974 Fuglede, in a study [Fug74] that is quite well known today, proposed the following infamous conjecture, widely known today as Fuglede's Conjecture: Fuglede's Conjecture: A measurable set K is a spectral set in R d if and only if K tiles R d by translation. Fuglede coined the term spectral set to denote sets Ω that admit admit an orthogonal basis of exponential functions. Fuglede's Conjecture has been studied by many investigators, including the authors of this paper, Jorgensen, Pedersen, Lagarias, Laba, Kolountzakis, Matolcsi, Iosevich, Tao, and others ([JP98, JP99, IKT01, IKT03, Kol00, KM06b, Lab01, LRW00, LW96a, LW97, Tao04]), but it had baffled experts for 30 years until Terence Tao [Tao04] gave the first counterexample of a spectral set which is not a tile in R d , d ≥ 5. The example and technique were refined later to disprove the conjecture in both directions on R d for d ≥ 3 [Mat05, KM06b, KM06a]. It has remained open in dimensions d = 1 and d = 2.Although the Fuglede's Conjecture in its original form has been disproved, there is a clear connection between spectral sets and tiling that has remained a mystery. Furthermore, spectral sets are apparently only one of the problems among an extremely broad class of problems involving complex exponential functions as either bases or more generally, frames.The rest of this section will focus on Fourier bases and frames for measures. Basic concepts and some of the important recent results will be introduced and reviewed. They will be the gateway to further discussions on the subject in later sections, which include open questions and more recent progresses.
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10.1007/978-3-319-57805-7_5
|
[
"https://arxiv.org/pdf/1602.04750v1.pdf"
] | 119,260,550 |
1602.04750
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259199d19d7ab8628a44726ae2b9804f2c138e0e
|
FOURIER BASES AND FOURIER FRAMES ON SELF-AFFINE MEASURES
15 Feb 2016
Ervin Dorin
Chun-Kit Dutkay
Yang Lai
Wang
FOURIER BASES AND FOURIER FRAMES ON SELF-AFFINE MEASURES
15 Feb 2016arXiv:1602.04750v1 [math.FA]
This paper gives a review of the recent progress in the study of Fourier bases and Fourier frames on self-affine measures. In particular, we emphasize the new matrix analysis approach for checking the completeness of a mutually orthogonal set. This method helps us settle down a long-standing conjecture that Hadamard triples generates self-affine spectral measures. It also gives us non-trivial examples of fractal measures with Fourier frames. Furthermore, a new avenue is open to investigate whether the Middle Third Cantor measure admits Fourier frames. 2010 Mathematics Subject Classification. Primary 42B05, 42A85, 28A25. 1 2 DORIN ERVIN DUTKAY, CHUN-KIT LAI, AND YANG WANG exponential functions. In 1974 Fuglede, in a study [Fug74] that is quite well known today, proposed the following infamous conjecture, widely known today as Fuglede's Conjecture: Fuglede's Conjecture: A measurable set K is a spectral set in R d if and only if K tiles R d by translation. Fuglede coined the term spectral set to denote sets Ω that admit admit an orthogonal basis of exponential functions. Fuglede's Conjecture has been studied by many investigators, including the authors of this paper, Jorgensen, Pedersen, Lagarias, Laba, Kolountzakis, Matolcsi, Iosevich, Tao, and others ([JP98, JP99, IKT01, IKT03, Kol00, KM06b, Lab01, LRW00, LW96a, LW97, Tao04]), but it had baffled experts for 30 years until Terence Tao [Tao04] gave the first counterexample of a spectral set which is not a tile in R d , d ≥ 5. The example and technique were refined later to disprove the conjecture in both directions on R d for d ≥ 3 [Mat05, KM06b, KM06a]. It has remained open in dimensions d = 1 and d = 2.Although the Fuglede's Conjecture in its original form has been disproved, there is a clear connection between spectral sets and tiling that has remained a mystery. Furthermore, spectral sets are apparently only one of the problems among an extremely broad class of problems involving complex exponential functions as either bases or more generally, frames.The rest of this section will focus on Fourier bases and frames for measures. Basic concepts and some of the important recent results will be introduced and reviewed. They will be the gateway to further discussions on the subject in later sections, which include open questions and more recent progresses.
Introduction
It is well known that the set of exponential functions {e 2πinx : n ∈ Z} is an orthonormal basis for L 2 (I) where I = [0, 1]. An interval is not the only set on which an orthogonal basis consisting of exponential functions exists for L 2 (Ω). For example, for the set Ω = [0, 1]∪[2, 3] the set of exponential functions {e 2πiλx : λ ∈ Z ∪ Z + 1 4 } is an orthogonal basis for L 2 (Ω). It is an interesting question to ask what sets Ω admit an orthogonal basis consisting of 1.1. Fourier bases and spectral measures. One extension of spectral sets concerns the more general spectral measures. A spectral measure is a bounded Borel measure µ such that there exists a set of complex exponentials E(Λ) := {e λ } λ∈Λ , where e λ := e 2πi λ,x , that forms an orthogonal basis of L 2 (µ). If such Λ exists, it is called a spectrum for µ. A spectral set can be viewed as a special case of spectral measure by considering the Lebesgue measure χ K dx on K. Of particular intrigue are measures that are singular, especially self-similar measures that are closely tied to the field of fractal geometry and self-affine tiles.
The study of Fourier bases on singular measures started with the Jorgensen-Pedersen paper [JP98] in which they proved the following surprising result: consider the construction of a Cantor set with scale 4: take the unit interval, divide it into 4 equal pieces, keep the first and the third piece. Then for the two remaining intervals, perform the same procedure and keep repeating the procedure for the remaining intervals ad infinitum. The remaining set is the Cantor-4 set in the Jorgensen-Pedersen example. On this set consider the appropriate Hausdorff measure of dimension 1 2 , which we denote by µ 4 .
Jorgensen and Pedersen proved that the Hilbert space L 2 (µ 4 ) has an orthonormal family of exponential functions, i.e., a Fourier basis, namely e λ : λ = n k=0 4 k l k , l k ∈ {0, 1}, n ∈ N .
Moreover, they also proved that the Middle-Third-Cantor set, with its corresponding Hausdorff measure cannot have more than two mutually orthogonal exponential functions.
Definition 1.1. Let µ be a compactly supported Borel probability measure on R d and ·, · denote the standard inner product. We say that µ is a spectral measure if there exists a countable set Λ ⊂ R d , which we call a spectrum, such that E(Λ) := {e 2πi λ,x : λ ∈ Λ} is an orthonormal basis for L 2 (µ). The Fourier transform of µ is defined to be µ(ξ) = e −2πi ξ,x dµ(x).
It is easy to verify that a measure is a spectral measure with spectrum Λ if and only if the following two conditions are satisfied:
(i) (Orthogonality) µ(λ − λ ′ ) = 0 for all distinct λ, λ ′ ∈ Λ and (ii) (Completeness) If for f ∈ L 2 (µ), f (x)e −2πi λ,x dµ(x) = 0 for all λ ∈ Λ, then f = 0.
The Jorgensen-Pedersen Cantor-4 set and the measure µ 4 can be seen as the attractor and invariant measure of the iterated function system τ 0 (x) = x 4 , τ 2 (x) = x + 2 4 .
Definition 1.2. For a given expansive d × d integer matrix R and a finite set of integer vectors B in Z d , with #B =: N, we define the affine iterated function system (IFS) τ b (x) = R −1 (x + b), x ∈ R d , b ∈ B. The self-affine measure, also called the invariant measure, (with equal weights) is the unique probability measure µ = µ(R, B) satisfying
(1.1) µ(E) = b∈B 1 N µ(τ −1 b (E)), for all Borel subsets E of R d .
This measure is supported on the attractor T (R, B) which is the unique compact set that satisfies
T (R, B) = b∈B τ b (T (R, B)).
The set T (R, B) is also called the self-affine set associated with the IFS. See [Hut81] for details.
The starting idea in Jorgensen-Pedersen's construction is to find first orthogonal exponential functions at a finite level and then iterate by rescaling and translations. If we start with the atomic measure δ1 4 B := δ1 4 {0,2} = 1 2 (δ 0 + δ 2 ), we find that {e ℓ : ℓ ∈ L := {0, 1}} is an orthonormal basis for L 2 (δ1 4 B ). In other words, (R = 4, B = {0, 2}, L = {0, 1}) form a Hadamard triple. Definition 1.3. Let R ∈ M d (Z) be an d × d expansive matrix (all eigenvalues have modulus strictly greater than 1) with integer entries. Let B, L ⊂ Z d be a finite set of integer vectors with N = #B = #L (# means the cardinality). We assume without loss of generality that 0 ∈ B and 0 ∈ L. We say that the system (R, B, L) forms a Hadamard triple (or (R −1 B, L) forms a compatible pair in [ LW02] ) if the matrix
(1.2) H = 1 √ N e 2πi R −1 b,ℓ ℓ∈L,b∈B
is unitary, i.e., H * H = I. Denote by
B n = B + RB + · · · + R n−1 B, Λ n := L + R T L + · · · + (R T ) n−1 L.
If (R, B, L) form a Hadamard triple, then a simple computation (see e.g. [JP98]) shows that (R n , B n , Λ n ) is also a Hadamard triple, for all n ∈ N. In other words, the measures
ν n = δ R −n (Bn) = δ R −1 B * δ R −2 B * ... * δ R −n B
have a spectrum Λ n These measures approximate the singular measure µ(R, B) in the following sense,
(1.3) µ(R, B) = δ R −1 B * δ R −2 B * δ R −3 B ... = µ n * µ >n
where, by self-similarity, the measure µ >n (·) = µ(R n ·). One should expect that, under the right conditions, the set Λ = ∞ n=1 Λ n forms a spectrum for the measure µ(R, B) and this is the case for the Jorgensen-Pedersen example. However, this is not true in general, even though it always yields an orthonormal set, but, in some cases, this set can be incomplete. Here is a simple counterexample: consider the scale R = 2, and the digits B = {0, 1}. It is easy to see that µ(R, B) is the Lebesgue measure on [0, 1]. One can pick L = {0, 1} to make a Hadamard triple. But when we construct the set Λ we notice that it is actually Λ = N ∪ {0} and we know that the classical Fourier series are indexed by all the integers Z, so the exponential functions with frequencies in Λ are orthonormal, but not complete.
The natural conjecture was raised to see if for any Hadamard triple (R, B, L) the measure µ(R, B) is spectral. This conjecture was settled on R 1 [ LW02,DJ06]. The situation becomes more complicated on high dimension. Dutkay and Jorgensen showed that the conjecture is true if (R, B, L) satisfies a technical condition called reducibility condition [DJ07]. There are some other additional assumptions proposed by Strichartz guaranteeing the conjecture is true [Str98,Str00]. Some low-dimensional special cases were also considered by Li [Li14,Li15]. We eventually proved this conjecture in [DL15a,DHL15].
Theorem 1.4. If (R, B, L) is a Hadamard triple, then the self-affine measure µ = µ(R, B) is a spectral measure.
The proof of this result highlights some of the useful techniques for the study of spectral self-affine measures. We will give a sketch of proof of this result as part of this review.
1.2. Fourier frames. A natural generalization of orthonormal basis is the notion of frames. It allows expansion of functions in a non-unique way, but is robust to perturbation of frequencies [DHSW11]. Recall that a frame is a family of vectors {e i : i ∈ I} in a Hilbert space H with the property that there exist constants A, B > 0 (called the frame bounds) such that
(1.4) A f 2 ≤ i∈I | f , e i | 2 ≤ B f 2 , (f ∈ H).
A Borel measure µ is called a frame-spectral measure if there exists a family of exponential functions {e λ : λ ∈ Λ} forming a frame for L 2 (µ).
Soon after Jorgensen and Pedersen showed in [JP98] that the Middle-Third-Cantor measure µ 3 is not a spectral measure, a natural question was proposed by Strichartz [Str00] who asked whether the measure µ 3 is frame-spectral. The question is still open.
We proved in [DL14] that if a measure has a Fourier frame, then it must have a certain homogeneity under local translations (so it must look the same locally at every point). This condition excludes the possibility of Fourier frames on self-affine measures with unequal weights, but not for the Middle-Third-Cantor measures. Some fairy large Bessel sequences of exponential functions (i.e., only the upper bound holds in (1.4)) were constructed in [DHW11] for the Middle-Third-Cantor set. Some weighted Fourier frames were built by Picioroaga and Weber for the Cantor-4 set in [PW15].
The following condition generalizing Hadamard triples was introduced in [DL15a] and [LW15]. We state the definition on R 1 , but it can be defined on any dimension. Definition 1.5. Let ǫ j be such that 0 ≤ ǫ j < 1 and ∞ j=1 ǫ j < ∞. We say that {(N j , B j )} is an almost-Parseval-frame tower associated to {ǫ j } if (i) N j are integers and N j ≥ 2 for all j; (ii) B j ⊂ {0, 1, ..., N j − 1} and 0 ∈ B j for all j;
(iii) Let M j := #B j . There exists L j ⊂ Z (with 0 ∈ L j ) such that for all j,
(1.5) (1 − ǫ j ) 2 b∈B j |w b | 2 ≤ λ∈L j 1 M j b∈B j w b e −2πibλ/N j 2 ≤ (1 + ǫ j ) 2 b∈B j |w b | 2 for all w = (w b ) b∈B j ∈ C M j . Letting the matrix F j = 1 √ M j
e 2πibλ/N j λ∈L j ,b∈B j and · the standard Euclidean norm, (1.5) is equivalent to
(1.6) (1 − ǫ j ) w ≤ F j w ≤ (1 + ǫ j ) w for all w ∈ C M j .
Whenever {L j } j∈Z exists, we call {L j } j∈Z a pre-spectrum for the almost-Parseval-frame tower.
We define the following measures associated to an almost-Parseval-frame tower.
ν j = 1 M j b∈B j δ b/N 1 N 2 ...N j and (1.7) µ = ν 1 * ν 2 * .... := µ n * µ >n
where µ n is the convolution of the first n discrete measures and µ >n is the remaining part.
When all ǫ j = 0, (N j , B j , L j ) forms a Hadamard triple. We note that this class of measures includes self-similar measures because if we are given an integer N ≥ 2 and a set B ⊂ {0, 1, ..., N − 1} such that
N j = N n j , B j = B + NB + ... + N n j −1 B,
then the associated measure µ is the self-affine measure µ(N, B). In particular if N = 3 and B = {0, 2}, µ is the standard Middle Third Cantor measure. In such situation, the almost-Parseval-frame tower is called self-similar. We prove that Theorem 1.6. (i) If the self-similar almost-Parseval-frame tower as in Definition 1.5 exists, then the associated self-similar measure is frame-spectral.
(ii) There exists almost-Parseval-frame tower with ǫ j > 0 and the associated fractal measure is frame-spectral but not spectral.
In the rest of the paper we go into more details on spectral self-affine measures as well as frame-spectral measures. We consider the explicit construction of self-affine frame-spectral measures that are not spectral. It is our hope that this survey summarizes not only recent results on the subject, but also some of the key techniques used to tackle problems. The open questions we discuss here should serve to quickly lead people into this area.
Spectral Self-Affine Measures
We begin with the following definitions.
Definition 2.1. We call a finite set B ⊂ Z d , a simple digit set for R, if distinct elements of B are not congruent (mod R(Z d )). We define Z[R, B] to be the smallest lattice in R d that contains the set B and is invariant under R, i.e., R(Z[R, B]) ⊂ Z[R, B].
We say that µ(R, B) satisfies the no-overlap condition if
µ(τ b (T (R, B)) ∩ τ b ′ (T (R, B))) = 0, ∀b = b ′ ∈ B.
We have the following preliminary reductions that we can do to solve the problem. The following proposition perhaps gives us the main idea on how to prove the completeness of an orthogonal set of exponentials and the proof are readily generalized to give our various results.
Proposition 2.2. Suppose that (R, B, L) is a Hadadmard triple and Λ = ∞ n=1 Λ n where Λ n = L + R T L + ... + (R T ) n−1 L. Assume that δ(Λ) := inf n≥1 inf λ∈Λn | µ((R T ) −n λ)| 2 > 0
Then Λ is a spectrum for L 2 (µ(R, B)).
Proof. The proof of mutual orthogonality follows directly from the fact that
H n = 1 √ N n e −2πi R −n b , λ λ∈Λn,b∈Bn .
is a unitary matrix. We now show the completeness by showing that the following frame bounds hold: for any f ∈ L 2 (µ),
(2.1) δ(Λ) f 2 ≤ λ∈Λ f (x)e −2πi λ,x dµ(x) 2 ≤ f 2 .
The positive lower bound implies the completeness. To prove (2.1), we just need to check it for a dense set of functions in L 2 (µ). Let 1 E be the indicator function of the set E and
S n = b∈Bn w b 1 τ b (R,B) : w b ∈ C .
Thus, S n denotes the collection of all n th level step functions on T (R, B). It forms an increasing union of sets. Let also S = ∞ n=1 S n . This S forms a dense set in L 2 (µ). Now for
any f = b∈Bn w b 1 τ b (R,B) ∈ S, a direct computation shows that (2.2) |f | 2 dµ = 1 N n b∈Bn |w b | 2 = 1 N n w 2 where w = (w b ) b∈Bn and (2.3) f (x)e −2πi λ , x dµ(x) = 1 N n µ((R T ) −n λ) b∈Bn w b e −2πi R −n b , λ which means that (2.4) λ∈Λn f (x)e −2πi λ , x dµ(x) 2 = 1 N n λ∈Λn | µ((R T ) −n λ)| 2 b∈Bn 1 √ N n w b e −2πi R −n b , λ 2 As δ(Λ) ≤ | µ((R T ) −n λ)| 2 ≤ 1, we obtain 1 N n δ(Λ) H n w 2 ≤ λ∈Λn f (x)e −2πi λ , x dµ(x) 2 ≤ 1 N n H n w 2 .
But H n is a Hadamard matrix, so we have H n w = w and hence
δ(Λ) |f | 2 dµ ≤ λ∈Λn f (x)e −2πi λ , x dµ(x) 2 ≤ |f | 2 dµ.
As S n ⊂ S m for any n < m, we have
δ(Λ) |f | 2 dµ ≤ λ∈Λm f (x)e −2πi λ , x dµ(x) 2 ≤ |f | 2 dµ.
By taking m to infinity, we have (2.1).
As we have discussed in the introduction, we cannot expect the standard orthogonal set to be always a spectrum. We have to look for some other alternatives. The main important observation is to distinguish two cases depending on whether the periodic zero set
(2.5) Z := {ξ ∈ R d : µ(ξ + k) = 0, for all k ∈ Z d } of µ(R, B) is empty or not.
For the case Z = ∅, we will see that Theorem 1.4 follows along the same lines as Proposition 2.2. The case Z = ∅ is more subtle but it can appear in higher dimensions d, but, in this case, the system (R, B, L) has a special quasi-product form structure.
2.1. The case Z = ∅. Given a sequence of integers n k and let m k = n 1 + ... + n k . The self-affine measure can be rewritten as
µ(R, B) = δ R −n 1 Bn 1 * δ R −m 2 Bn 2 * ... * δ R −m k Bn k * ...
Then note that if we have another set J n k of integer vectors, with J n k ≡ L n k (mod (R T ) n k (Z d )), then it is easy to see that (R n k , B n k , J n k ) still form Hadamard triples. In this sequel, we produce many other mutually orthogonal sets, by:
(2.6) Λ k = J n 1 + (R T ) m 1 J n 2 + (R T ) m 2 J n 3 + ... + (R T ) m k−1 J n k , (2.7) Λ = ∞ k=1 Λ k .
Repeating the argument in Proposition 2.2, the following holds true.
Proposition 2.3. Suppose that (R, B, L) is a Hadadmard triple and Λ if given in (2.7). Assume that
(2.8) δ(Λ) := inf k≥1 inf λ∈Λ k | µ((R T ) −m k λ)| 2 > 0
Then Λ is a spectrum for L 2 (µ(R, B)).
The following proposition guarantees some Λ will satisfy δ(Λ) > 0.
Proposition 2.4. Suppose that Z = ∅. Then there exists Λ built as in (2.6) and (2.7) such that δ(Λ) > 0.
We now give the proof of this proposition. We start with a lemma.
Lemma 2.5. Suppose that Z = ∅ and let X be any compact set on R d . Then there exist
ǫ 0 > 0, δ 0 > 0 such that for all x ∈ X, there exists k x ∈ Z d such that for all y ∈ R d with y < ǫ 0 , we have | µ(x + y + k x )| 2 ≥ δ 0 . In addition, we can choose k 0 = 0 if 0 ∈ X. Proof. As Z = ∅, for all x ∈ X there exists k x ∈ Γ such that µ(x + k x ) = 0. Since µ is continuous, there exists an open ball B(x, ǫ x ) and δ x > 0 such that | µ(y + k x )| 2 ≥ δ x for all y ∈ B(x, ǫ x ). Since X is compact, there exist x 1 , . . . , x m ∈ X such that X ⊂ m i=1 B(x i , ǫ x i 2 ).
Let δ := min i δ x i and ǫ := min i ǫx i 2 . Then, for any x ∈ X, there exists i such that x ∈ B(x i , ǫx i 2 ). If y < ǫ, then x + y ∈ B(x i , ǫ x i ), so | µ(x + y + k x i )| 2 ≥ δ, we can redefine k x to be k x i to obtain the conclusion. Clearly, we can choose k 0 = 0 if 0 ∈ X since µ(0) = 1.
Proof of Proposition 2.4. Suppose that (R, B, L) is a Hadamard triple (R, B, L). Then we take X = T (R T , L), the self-affine set generated by R T and digit set L. Define J n = L + R T L + ... + (R T ) n−1 L By the definition of self-affine sets,
(R T ) −(n+p) J n ⊂ X, (n ∈ N, p ≥ 0).
Fix the ǫ 0 and δ 0 in Lemma 2.5. We now construct the sets Λ k and Λ as in (2.6) and (2.7), by replacing the sets J n k by some sets J n k to guarantee that the number δ(Λ) in (2.8) is positive.
We first start with Λ 0 := {0} and m 0 = n 0 = 0. Assuming that Λ k has been constructed, we first choose our n k+1 > n k so that
(2.9) (R T ) −(n k+1 +p) λ < ǫ 0 , ∀ λ ∈ Λ k , p ≥ 0.
We then define m k+1 = m k + n k+1 and
Λ k+1 = Λ k + (R T ) m k J n k+1 where J n k+1 = {j + (R T ) n k+1 k(j) : j ∈ J n k+1 , k(j) ∈ Z d }
where k(j) is chosen to be k x from Lemma 2.5, with x = (R T ) −n k+1 j ∈ X. As 0 ∈ J n k and k 0 = 0 for all k, the sets Λ k are of the form (2.6) and form an increasing sequence. For these sets Λ k , we claim that the associated Λ in (2.7) satisfies δ(Λ) > 0.
To justify the claim, we note that if λ ∈ Λ k , then
λ = λ ′ + (R T ) m k−1 j + (R T ) m k k(j),
where λ ′ ∈ Λ k−1 , j ∈ J n k . This means that
(R T ) −m k λ = (R T ) −m k λ ′ + (R T ) −n k j + k(j). By (2.9), (R T ) −m k λ ′ < ǫ 0 . From Lemma 2.5, since (R T ) −n k j ∈ X, we must have | µ((R T ) −m k λ)| 2 ≥ δ 0 > 0.
As δ 0 is independent of k, the claim is justified and hence this completes the proof of the proposition. . Combining Proposition 2.3 and Proposition 2.4, we settle the case Z = ∅.
Theorem 2.6. Suppose that Z = ∅ and (R, B, L) is a Hadamard triple. Then the self-affine measure µ(R, B) is a spectral measure.
2.2. The case Z = ∅. When Z = ∅, it means that there is an exponential function e ξ such that it is orthogonal to every exponential function with integer frequencies. This implies that none of the subsets of integers can be complete. Therefore, any construction of orthogonal sets within integers must fail to be spectral. We illustrate the situation through a simple example.
Example 2.7. Let R = 4 0 1 2 ,
B = 0 0 , 0 3 , 1 0 , 1 3 and L = 0 0 , 2 0 , 0 1 , 2 1 .Z(M B ) = 1 2 + n y : n ∈ Z, y ∈ R ∪ x 1 6 + 1 3 n : x ∈ R, n ∈ Z . Let (R T ) j = 4 j a j 0 2 j , for some a j ∈ Z. As µ(ξ) = ∞ j=1 M B ((R T ) −j (ξ))
, the zero set of µ, denoted by Z( µ), is equal to
Z( µ) = ∞ j=1 (R T ) j Z(M B ) = ∞ j=1 4 j ( 1 2 + n) + a j y 2 j y : n ∈ Z, y ∈ R ∪ 4 j x + a j ( 1 6 + 1 3 n) 2 j 1 6 + 1 3 n : x ∈ R, n ∈ Z .
We claim that the points in 0 . We now rewrite the second term in the union in Z( µ) as R × { 2 j−1 (1+2n) 3 }. As any integer can be written as 2 j p, for some j ≥ 0 and odd number p, this means that m 1+3n 3 ∈ Z( µ), justifying the claim. As Z = ∅, this shows that there is no spectrum in Z 2 for this measure. In fact,
T (R, B) = x∈K 1 {x} × ([0, 3] + g(x)),
where K 1 is the Cantor set of 1/4 contraction ratio and digit {0, 1} and g : [0, 1] → R is a measurable function obtaining from the off-diagonal entries.
To overcome this obstruction, as we will see, we prove that Z has a dynamical structure and, from this structure, we obtain that such Hadamard triples have to have a special quasi-product form, and moreover, in one of the factors, the digits form complete set of representatives. is spectral with spectrum (M T ) −1 Λ. (iv) There exists y 0 ∈ R d−r such that (R T 2 ) m y 0 ≡ y 0 (mod(R T 2 )Z d ) for some integer m ≥ 1 such that the unionS
Z := x ∈ R d : µ(x + k) = 0 for all k ∈ Z d ,= m−1 k=0 (R r × {(R T 2 ) k y 0 } + Z d )
is contained in the set
Z := x ∈ R d : μ(x + k) = 0 for all k ∈ Z d , whereμ = µ(R,B).
The setS is invariant (with respect to the system (mB,R T ,L, ), see the definition below and Definition 4.1) whereL is a complete set of representatives (modR T Z d ). In addition, all possible transitions from a point in
R r × {(R T 2 ) k y 0 } + Z d , 1 ≤ k ≤ m lead to a point in R r × {(R T 2 ) k−1 y 0 } + Z d .
The key fact in the proof of this proposition is that the set Z is invariant in the sense defined by Conze et al. in [CCR96]. That means that if x ∈ Z, k ∈ Z d and m B ((R T ) −1 (x + k)) = 0
then (R T ) −1 (x + k) is in Z, where m B (x) = 1 N b∈B e 2πi b , x .
Then, the results from [CCR96] show that Z must have a very special form, and this implies the proposition.
Theorem 2.9. Suppose that
R = R 1 0 C R 2 ,
(R, B, L 0 ) is a Hadamard triple and µ = µ(R, B) is the associated self-affine measure and Z = ∅. Then the set B has the following quasi-product form:
(2.11) B = (u i , v i + Qc i,j ) T : 1 ≤ i ≤ N 1 , 1 ≤ j ≤ | det R 2 | , where (i) N 1 = N/| det R 2 |, (ii) Q is a (d − r) × (d − r) integer matrix with | det Q| ≥ 2 and R 2 Q = QR 2 for some (d − r) × (d − r) integer matrix R 2 , (iii) the set {Qc i,j : 1 ≤ j ≤ | det R 2 |} is a complete set of representatives (mod R 2 (Z d−r )),
for all 1 ≤ i ≤ N 1 .
Moreover, one can find some L ≡ L 0 (mod R T (Z d )) so that (R, B, L) is a Hadamard triple and (R 1 , π 1 (B), L 1 (ℓ 2 )) and (R 2 , B 2 (b 1 ), π 2 (L)) are Hadamard triples on R r and R d−r respectively for all b 1 ∈ π 1 (B) and l 2 ∈ π 2 (L).
Here π 1 , π 2 are the projections onto the first and second components in R d = R r ×R d−r , and for b 1 ∈ π 1 (B), B 2 (b 1 ) := {b 2 : (b 1 , b 2 ) ∈ B} and for l 2 ∈ π 2 (L), L 1 (l 2 ) := {l 1 : (l 1 , l 2 ) ∈ L}.
In Example 2.7, it is easy to see that the digit set is in a quasi-product form with Q = 3. Suppose now the pair (R, B) is in the quasi-product form
(2.12) R = R 1 0 C R 2 (2.13) B = (u i , d i,j ) T : 1 ≤ i ≤ N 1 , 1 ≤ j ≤ N 2 := | det R 2 | ,
and {d i,j : 1 ≤ j ≤ N 2 } (d i,j = v i +Qc i,j as in Theorem 2.9) is a complete set of representatives (mod R 2 Z d−r ). We will show that the measure µ = µ(R, B) has a quasi-product structure.
Note that we have
R −1 = R −1 1 0 −R −1 2 CR −1 1 R −1 2
and, by induction,
R −k = R −k 1 0 D k R −k 2 , where D k := − k−1 l=0 R −(l+1) 2 CR −(k−l) 1 .
For the invariant set T (R, B), we can express it as a set of infinite sums,
T (R, B) = ∞ k=1 R −k b k : b k ∈ B .
Therefore any element (x, y) T ∈ T (R, B) can be written in the following form
x = ∞ k=1 R −k 1 a i k , y = ∞ k=1 D k a i k + ∞ k=1 R −k 2 d i k ,j k .
Let X 1 be the attractor (in R r ) associated to the IFS defined by the pair (R 1 , π 1 (B) = {u i : π 1 (B))). Let µ 1 be the (equal-weight) invariant measure associated to this pair.
1 ≤ i ≤ N 1 }) (i.e. X 1 = T (R 1 ,For each sequence ω = (i 1 i 2 . . . ) ∈ {1, . . . , N 1 } N = {1, . . . , N 1 } × {1, . . . , N 1 } × ..., define (2.14) x(ω) = ∞ k=1 R −k 1 u i k .
As (R 1 , π 1 (B)) forms Hadamard triple with some L 1 (ℓ 2 ), the measure µ(R 1 , π 1 (B)) has the no-overlap property. It implies that for µ 1 -a.e. x ∈ X 1 , there is a unique ω such that x(ω) = x. We define this as ω(x). This establishes a bijective correspondence, up to measure zero, between the set Ω 1 := {1, . . . , N 1 } N and X 1 . The measure µ 1 on X 1 is the pull-back of the product measure which assigns equal probabilities 1 N 1 to each digit. For ω = (i 1 i 2 . . . ) in Ω 1 , define
Ω 2 (ω) := {(d i 1 ,j 1 d i 2 ,j 2 . . . d in,jn . . . ) : j k ∈ {1, . . . , N 2 } for all k ∈ N}.
For ω ∈ Ω 1 , define g(ω) := ∞ k=1 D k a i k and g(x) := g(ω(x)), for x ∈ X 1 . Also Ω 2 (x) := Ω 2 (ω(x)). For x ∈ X 1 , define
X 2 (x) := X 2 (ω(x)) := ∞ k=1 R −k 2 d i k ,j k : j k ∈ {1, . . . , N 2 } for all k ∈ N .
Note that the attractor T (R, B) has the following form
T (R, B) = {(x, g(x) + y) T : x ∈ X 1 , y ∈ X 2 (x)}.
For ω ∈ Ω 1 , consider the product probability measure µ ω , on Ω 2 (ω), which assigns equal probabilities 1 N 2 to each digit d i k ,j k at level k. Next, we define the measure µ 2 ω on X 2 (ω). Let r ω : Ω 2 (ω) → X 2 (ω),
r ω (d i 1 ,j 1 d i 2 ,j 2 . . . ) = ∞ k=1 R −k 2 d i k ,j k . Define µ 2 x := µ 2 ω(x) := µ ω(x) • r −1 ω(x) .
Note that the measure µ 2 x is the infinite convolution product
δ R −1 2 B 2 (i 1 ) * δ R −2 2 B 2 (i 2 ) * . . . , where ω(x) = (i 1 i 2 . . . ), B 2 (i k ) := {d i k ,j : 1 ≤ j ≤ N 2 } and δ A := 1 #A a∈A δ a , for a subset A of R d−r .
The following lemmas were proved in [DJ07].
f dµ = X 1 X 2 (x) f (x, y + g(x)) dµ 2 x (y) dµ 1 (x).
Lemma 2.11. [DJ07, Lemma 4.5] If Λ 1 is a spectrum for the measure µ 1 , then
F (y) := λ 1 ∈Λ 1 | µ(x + λ 1 , y)| 2 = X 1 | µ 2 s (y)| 2 dµ 1 (s), (x ∈ R r , y ∈ R d−r ).
The two lemmas lead to the following proposition.
Proposition 2.12. [DHL15] For the quasi-product form given in (2.12) and (2.13), there exists a lattice Γ 2 such that for µ 1 -almost every x ∈ X 1 , the set Γ 2 is a spectrum for the measure µ 2 x .
Finally, the proof of the Theorem 1.4 follows by induction on the dimension d: we know it is true for d = 1 from [DJ06]. Then assume it is true for dimensions up to d − 1. The case Z = ∅ was treated before; if Z = ∅, then the measure µ is in the quasi-product described above. With Proposition 2.12, and using induction, the measure µ 1 has a spectrum Γ 1 and then the measure µ has the spectrum Γ 1 × Γ 2 .
Non-spectral singular measures with Fourier frames
Suppose that instead of the Hadamard triple, we are given the almost-Parseval-frame tower in Definition 1.5. A similar approach in Proposition 2.2 (for details see [LW15]) allows us to prove the following:
Proposition 3.1. Suppose that {(N j , B j )} is an almost-Parseval-frame tower associated to {ǫ j } with {L j } as its pre-spectrum. Let Λ = ∞ n=1 Λ n where Λ n = L 1 + N 1 L 2 + ... + (N 1 ...N n−1 )L n and let µ be the measure defined in (1.7). Assume that
δ(Λ) := inf n≥1 inf λ∈Λn | µ >n (λ)| 2 > 0
Then Λ is a frame spectrum for L 2 (µ) and for any f ∈ L 2 (µ),
(3.1) δ(Λ) ∞ j=1 (1 − ǫ j ) 2 f 2 ≤ λ∈Λ f (x)e −2πi λ,x dµ(x) 2 ≤ ∞ j=1
(1 + ǫ j ) 2 f 2 .
If we have a self-similar almost-Parseval-frame tower (all N j = N), then the measure µ >n (·) can be written as µ(N n ·). In this case, we can produce new candidates of frame spectra as in (2.6) and (2.7) and the consideration for Z = ∅ works in a similar way as in Proposition 2.4. We have Theorem 3.2. Suppose that (N j , B j ) is a self-similar almost-Parseval-frame tower and the associated measure µ satisfies Z = ∅. Then µ is a frame-spectral measure.
For a self-similar measure on R 1 as defined in Definition 1.5, Z = ∅ can be obtained without additional assumption.
Proof of Theorem 1.6 (i). By Theorem 3.2, it suffices to show that Z = ∅ for self-similar measures µ(N, B) defined by the almost-Parseval-frame tower in Definition 1.5. In fact, as B ⊂ {0, 1, ..., N −1}, the self-similar set T (N, B) is a compact set inside [0, 1]. By the Stone-Weierstrass theorem, the linear span of exponentials e n with integer frequencies is complete in the space of continuous functions on T (N, B). This shows that Z = ∅, completing the proof.
In the end of this section, we demonstrate the existence of an almost-Parseval-frame tower with ǫ j > 0, which gives a proof of Theorem 1.6(ii). More precisely, we prove Theorem 3.3. Let N n and M n be positive integers satisfying
(3.2) N j = M j K j + α j for some integer K j and 0 ≤ α j < M j with (3.3) ∞ j=1 α j M j K j < ∞. Define (3.4) B j = {0, K j , ..., (M j − 1)K j }, L j = {0, 1, ..., M j − 1}.
Then (N j , B j ) forms an almost-Parseval-frame tower associated with
ǫ j = 2πα j M j K j and its pre-spectrum is {L j }.
Proof. Let
F j = 1 M j e 2πibλ/N j λ∈L j ,b∈B j , H j = 1 M j e 2πibλ/M j K j λ∈L j ,b∈B j .
Then H j is a unitary matrix (in fact Hadamard matrices). We first show that for any j > 0, the operator norm ( A := max x =1 Ax )
(3.5) F j − H j ≤ 2πα j M j K j .
To see this, We note that by Cauchy-Schwarz inequality,
(3.6) F j − H j 2 ≤ 1 M j b∈B j λ∈L j e 2πibλ/N j − e 2πibλ/M j K j 2 .
We now estimate the difference of the exponentials inside the summation using an elementary estimate
|e iθ 1 − e iθ 2 | = |e i(θ 1 −θ 2 ) − 1| ≤ |θ 1 − θ 2 |.
This implies that
e 2πibλ/N j − e 2πibλ/M j K j 2 ≤ 2πbλ N j − 2πbλ M j K j 2 =4π 2 b 2 λ 2 α 2 j M 2 j K 2 j N 2 j (by N j = M j K j + α j ) ≤4π 2 M 2 j α 2 j N 2 j (by b ≤ M j K j and λ ≤ M j )
Hence, from (3.6),
F j − H j 2 ≤ 1 M j b∈B j λ∈L j 4π 2 M 2 j α 2 j N 2 j = 4π 2 M 3 j α 2 j N 2 j = 4π 2 M j α 2 j (K j + α j /M j ) 2 (3.7)
As α j ≥ 0, F j − H j 2 ≤ 4π 2 α 2 j M j /K 2 j and thus (3.5) follows by taking square root.
We now show that {(N j , B j )} forms an almost-Parseval-frame tower with pre-spectrum L j . The first two conditions for the almost-Parseval-frame tower are clearly satisfied. To see the last condition, we recall that ǫ j = 2π M j α j /K j . From the triangle inequality and (3.5), we have
F j w ≤ H j w + F j − H j w ≤ 1 + 2πα j M j K j w = (1 + ǫ j ) w .
Similarly, for the lower bound,
F j w ≥ H j w − F j − H j w ≥ 1 − 2πα j M j K j w = (1 − ǫ j ) w .
Thus, from (1.6), the last condition follows and (N j , B j ) satisfies the almost-Parseval-frame condition associated with {ǫ j } and ∞ j=1 ǫ j < ∞ is guaranteed by (3.3) in the assumption.
Some of the fractal measures induced by the almost-Parseval-frame tower were found to be non-spectral, one can refer to [LW15] for detail.
Explicit construction of spectrum
In general, the canonical orthogonal set in Proposition 2.2 is not necessarily a spectrum. But in some cases, we can complete this set by adding some more points, and, in some cases, it is possible to gives an explicit formula for the spectrum of the measure µ(R, B). Such a description can be given in the following definition. This is always true in dimension one, as explained in [DJ06].
Definition 4.1. Let (R, B, L) be a Hadamard triple. We define the function
m B (x) = 1 #B b∈B e 2πi b , x , (ξ ∈ R d ).
The Hadamard triple condition implies that δ R −1 B is a spectral measure with a spectrum L and m R −1 B is the Fourier transform of this measure. The Hadamard condition implies that
(4.1) ℓ∈L |m B ((R T ) −1 (x + ℓ))| 2 = 1, or ℓ∈L |m B (τ ℓ (x))| 2 = 1,
where we define the maps
τ ℓ (x) = (R T ) −1 (x + ℓ), (x ∈ R d , ℓ ∈ L), and τ ℓ 1 ...ℓm = τ ℓ 1 • ... • τ ℓm .
A closed set K in R d is called invariant (with respect to the system (R, B, L)) if, for all x ∈ K and all ℓ ∈ L |m B (τ ℓ (x))| > 0 =⇒ τ ℓ (x) ∈ K.
We say that the transition, using ℓ, from x to τ ℓ (x) is possible, if ℓ ∈ L and m B (τ ℓ (x)) > 0. When this set Λ is a spectrum (see Theorem 4.3 below), we call it the dynamically simple spectrum.
More generally, the set generated by an invariant subset A of R d , is the smallest set which contains −A and satisfies (ii).
R = 2 1 0 2 , B = 0 0 , 3 0 , 0 1 , 3 1 , L = 0 0 , 1 0 , 0 1 , 1 1 .
Note that B is a complete set of representatives mod RZ 2 and L is a complete set of representatives mod R T Z 2 ; thus we have a Hadamard triple. As shown in [LW96b, Example 2.3], the measure µ(R, B) is the normalized Lebesgue measure on the attractor T (R, B) which tiles R 2 with 3Z × Z. Hence 1 3 Z × Z is a spectrum for µ(R, B) (by [Fug74]). (1 + e 2πi3x + e 2πiy + e 2πi(3x+y) ).
If we want |m B (x, y)| = 1, then we must have equality in the triangle inequality and we get that 3x, y, (3x + y) ∈ Z. So (x, y) ∈ 1 3 Z × Z. For an extreme cycle (x, y), we must also have that ( Note that these form a complete set of representatives mod R T Z 2 . The set generated by the extreme cycles is then just Z 2 which is a proper subset of the spectrum 1 3 Z × Z, so the Hadamard triple is not dynamically simple.
Open problems
The major open problem in the study of Fourier analysis on fractals is to see whether the non-spectral self-affine measures are still frame-spectral. The idea of almost-Parseval-frame towers turns this problem into a problem of matrix analysis. Given an integral expanding matrix R and a set of simple digits B with N = #B < | det R|, the condition of almost-Parseval-frame towers can be reformulated equivalently as for any ǫ > 0, there exists n ∈ N and a set of L n ⊂ Z d such that the matrix
F n (B n , L n ) = 1 √ N n e 2πi R −n b,ℓ ℓ∈Ln,b∈Bn satisfies (1 − ǫ) w 2 ≤ F n w 2 ≤ (1 + ǫ) w 2 for any vectors w ∈ C N n . (Recall that B n = B + RB + ... + R n−1 B)
We observe that if we let B n and L n be respectively the complete representative class (mod R n (Z d )) and (mod (R T ) n (Z d )). Then the matrix F n w = w , ∀w ∈ C | det R| n As B n ⊂ B n , we can take the vectors w such that they are zero on the coordinates which are not in B n . This implies that F n (B n , L n )w = | det R| n N n w .
In other words, This shows that the collection of vectors { 1 √ N n e −2πi R −n b,λ b∈Bn : λ ∈ L n } forms a tight frame for C N n with frame bound | det R| n N n . Our problem is to extract a subset L n from L n such that we have an almost tight frame with frame constant nearly 1. This reminds us about the Kadison-Singer problem that was open for over 50 years and solved recently in [MSS15].
Theorem 5.1. [MSS15, Corollary 1.5] Let r be a positive integer and let u 1 , ..., u m ∈ C d such that m i=1 | w, u i | 2 = w 2 ∀w ∈ C d and u i ≤ δ for all i. Then there exists a partition S 1 , ..., S r of {1, ..., m} such that
i∈S j | w, u i | 2 ≤ 1 √ r + √ δ 2 w 2 ∀w ∈ C d .
This statement says that we can partition a tight frame into r subsets such that the frame constant of each partition is almost 1/r. Iterating this process allowed Nitzan et al [NOU14] to establish the existence of Fourier frames on any unbounded sets of finite measure. One of their lemmas states:
Lemma 5.2. [NOU14, Lemma 3] Let A be an K × L matrix and J ⊂ {1, ..., K}, we denote by A(J) the sub-matrix of A whose rows belong to the index J. Then there exist universal constants c 0 , C 0 > 0 such that whenever A is a K × L matrix, which is a sub-matrix of some K × K orthonormal matrix, such that all of its rows have equal ℓ 2 -norm, one can find a subset J ⊂ {1, ..., K} such that
c 0 L K w 2 ≤ A(J)w 2 ≤ C 0 L K w 2 , ∀w ∈ C n .
This lemma leads naturally to the following:
Proposition 5.3. With (R, B) as in Definition 1.2, there exist universal constants 0 < c 0 < C 0 < ∞ such that for all n, there exists J n such that
c 0 b∈Bn |w b | 2 ≤ λ∈Jn 1 √ N n b∈Bn w b e −2πi R −n b,λ 2 ≤ C 0 b∈Bn |w b | 2 for all (w b ) b∈Bn ∈ C N n .
Proof. Let F n = 1 | det R| n/2 e 2πi R −n b,ℓ ℓ∈Ln,b∈Bn where B n is a complete coset representative (mod R(Z d )) containing B n and L n is a complete coset representative (mod R T (Z d )). It is well known that F n is an orthonormal matrix. Let K = | det R| n and A n = 1 | det R| n/2 e 2πi R −n b,ℓ ℓ∈Ln,b∈Bn . Then A n is a sub-matrix of F n whose columns are exactly the ones with index in B n so that the size L is L = N n . By Lemma 5.2, we can find universal constants c 0 , C 0 , independent of n, such that for some J n ⊂ L n , we have c 0 N n | det R| n w 2 ≤ A(J n )w 2 ≤ C 0 N n | det R| n w 2 , ∀w ∈ C N n .
As | det R| n/2 N n/2 A(J n ) = 1 | det R| n/2 e 2πi R −n b,ℓ ℓ∈Jn,b∈Bn := F n , this shows c 0 w 2 ≤ F n w 2 ≤ C 0 w 2 , ∀w ∈ C N n .
This is equivalent to the inequality we stated.
This proposition shows that there always exists some subsets J n in which the norm of F (B n , J n ) is uniformly bounded by universal constants c 0 , C 0 , this indicates that the existence of almost-Parseval-frame pairs B n , L n is possible.
The system (R, B, L) forms a Hadamard triple if and only if the Dirac measure δ R −1 B = 1 #B b∈B δ R −1 b is a spectral measure with spectrum L. Moreover, this property is a key property in producing examples of singular spectral measures, in particular spectral selfaffine measures.
(i) If (R, B, L) is a Hadamard triple, then B is a simple digit set for R and L is a simple digit set for R T .(ii) We can assume without loss of generality that Z[R, B] = Z d . (iii) If B is a simple digit for R, then µ(R, B) satisfies the no-overlap condition. Property (i) is a simple consequence of mutually orthogonality. If Z[R, B] = Z d , we can conjugate some matrix to produce another Hadamard triple which satisfies with the desired property [DL15a, Proposition 4.1]. The proof of the no-overlap condition of the self-affine measure can be referred to [DL15a, Section 2].
Then (R, B, L) forms a Hadamard triple and Z[R, B] = Z 2 . However, the set defined in (2.5) Z = ∅ for the measure µ = µ(R, B). Proof. It is a direct check to see (R, B, L) forms a Hadamard triple and Z[R, B] = Z 2 . As M B (ξ 1 , ξ 2 ) = 1 4 (1 + e 2πiξ 1 )(1 + e 2πi3ξ 2 ). It follows that the zero set of M B , denoted by Z(M B ), is equal to
+
Z 2 are in Z( µ) which shows Z = ∅. Indeed, for any m 1 3 + n , m, n ∈ Z, we can write it as m 1+3n 3
Proposition 2 . 8 .
28Suppose that (R, B, L) forms a Hadamard triple and Z[R, B] = Z d and let µ = µ(R, B) be the associated self-affine measure µ = µ(R, B). Suppose that the set
is non-empty. Then there exists an integer matrix M with det M = 1 such that the following assertions hold:(i) The matrixR := MRM −1 is of the form (2.10)R = R 1 0 C R 2 , with R 1 ∈ M r (Z), R 2 ∈ M d−r (Z) expansive integer matrices and C ∈ M (d−r)×r (Z). (ii) IfB = MB andL = (M T ) −1 L, then (R,B,L) is a Hadamard triple. (iii) The measure µ(R, B) is spectral with spectrum Λ if and only if the measure µ(R,B)
Lemma 2 .
210. [DJ07, Lemma 4.4] For any bounded Borel functions on R d , T (R,B)
A compact invariant set is called minimal if it does not contain any proper compact invariant subset.For ℓ 1 , . . . , ℓ m ∈ L, the cycle C(ℓ 1 , . . . , ℓ m ) is the setC(ℓ 1 , . . . , ℓ m ) = {x 0 , τ ℓm (x 0 ), τ ℓ m−1 ℓm (x 0 ), . . . , τ ℓ 2 ...ℓm (x 0 )},where x 0 := ℘(ℓ 1 , . . . , ℓ m ) is the fixed point of the map τ ℓ 1 ...ℓm . i.e. τ ℓ 1 ...ℓm (x 0 ) = x 0 . The cycle C(ℓ 1 , . . . , ℓ m ) is called an extreme cycle for (R, B, L) if |m B (x)| = 1 for all x ∈ C(ℓ 1 , . . . , ℓ m ).
Definition 4. 2 .
2We say that the Hadamard triple (R, B, L) is dynamically simple if the only minimal compact invariant set are extreme cycles. For a Hadamard triple (R, B, L), the orthonormal set Λ generated by extreme cycles is the smallest set such that (i) it contains −C for all extreme cycles C for (R, B, L) (ii) it satisfies R T Λ + L ⊂ Λ.
Theorem 4.3.[DL15b] Let (R, B, L) be a dynamically simple Hadamard triple. Then the orthonormal set Λ generated by extreme cycles is a spectrum for the self-affine measure µ R,B and Λ is explicitly given by Λ = {ℓ 0 +R T ℓ 1 +. . . (R T ) n−1 ℓ n−1 +(R T ) n (−c) : ℓ 0 , . . . , ℓ n−1 ∈ L, n ≥ 0, c are extreme cycle points}.Moreover, if (R, B, L) is a Hadamard triple on R 1 , it must be dynamically simple.
Example 4. 4 .
4There are Hadamard triples which are not dynamically-simple. For example let
Figure 1 .
1T (R, B) Figure 2. T (R T , L) We look for the extreme cycles: we have m B (x, y) = 1 4
x, y) is in the attractor of the IFS (R T , L), and this is contained in [0, 1] × [−1, 1] (on can check the invariance of this rectangle for the IFS). So, one can check which of the points in 1 3 Z × Z ∩ [0, 1] × [−1, 1] are in an extreme cycle. The only extreme cycles are
Acknowledgements. This work was partially supported by a grant from the Simons Foundation (#228539 to Dorin Dutkay).
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|
[] |
[
"TOP QUARK PRODUCTION",
"TOP QUARK PRODUCTION"
] |
[
"Andreas B Meyer [email protected] \nON BEHALF OF THE ATLAS\nCMS AND D0 COLLABORATIONS DESY\nCDF\nNotkestr.8522603HamburgGermany\n"
] |
[
"ON BEHALF OF THE ATLAS\nCMS AND D0 COLLABORATIONS DESY\nCDF\nNotkestr.8522603HamburgGermany"
] |
[] |
Recent measurements of top quark pair and single top production are presented. The results include inclusive cross sections as well as studies of differential distributions. Evidence for single top quark production in association with a W-boson in the final state is reported for the first time. Calculations in perturbative QCD up to approximate next-to-next-to-leading order show very good agreement with the data.
| null |
[
"https://arxiv.org/pdf/1212.3957v1.pdf"
] | 119,163,794 |
1212.3957
|
9382dbee8708bb84402b1203e5ebc5963f21fb7d
|
TOP QUARK PRODUCTION
Andreas B Meyer [email protected]
ON BEHALF OF THE ATLAS
CMS AND D0 COLLABORATIONS DESY
CDF
Notkestr.8522603HamburgGermany
TOP QUARK PRODUCTION
Recent measurements of top quark pair and single top production are presented. The results include inclusive cross sections as well as studies of differential distributions. Evidence for single top quark production in association with a W-boson in the final state is reported for the first time. Calculations in perturbative QCD up to approximate next-to-next-to-leading order show very good agreement with the data.
Introduction
The top quark is by far the heaviest known elementary particle. Due to its large mass the top quark decays within 5 · 10 −25 s, before hadronisation, and thus gives direct access to its properties such as spin and charge. With its large mass, the top quark plays a crucial role in electroweak loop corrections, providing indirect constraints on the mass of the Higgs boson. Not least, top quark measurements provide important input to QCD calculations. The measurements help discriminate between different perturbative approaches, and have the potential to constrain QCD parameters. Moreover, various scenarios of physics beyond the Standard Model expect the top quark to couple to new particles. In many super-symmetric models the super-symmetric partner of the top quark is expected to be relatively light, such that it could be produced at the LHC. Experimentally, Standard Model top quark processes are a dominant background to many searches for new physics.
The top quark was discovered in 1995 at the CDF and D0 experiments at the Tevatron [1,2]. Until its shut-down in September 2011, several tens of thousands of top quark events were recorded. The Tevatron collaborations CDF and D0 are now finalizing the measurements with the full statistics. At the LHC at CERN, top quark events are produced at significantly larger rates, and to-date several million top quark events have been recorded by the ATLAS and CMS experiments. The large amount of data gives rise to a wealth of new results and continuous updates. In this report an overview of recent measurements of top quark production cross sections is given. Measurements of top quark angular distributions, and properties are reported in [3,4]. A recent review article can be found in [5].
Theoretical Status
At LHC energies, top quark pair production proceeds dominantly through gluongluon fusion. This is in contrast to the Tevatron where the dominant production process is qq annihilation at larger values of Bjorken-x. Several groups have t ½a þ bðm t À m 0 Þ þ cðm t À m 0 Þ 2 þ dðm t À m 0 Þ 3 ;
where t t and m t are in pb and GeV, respectively, and m 0 ¼ 170 GeV, a ¼ 5:78874 Â 10 9 pb GeV 4 , b ¼ À4:50763 Â 10 7 pb GeV 3 , c ¼ 1:50344 Â 10 5 pb GeV 2 , and d ¼ À1:00182 Â 10 3 pb GeV.
In Fig. 6 we compare this parametrization to three approximations to t t at next-to-next-to-leading-order (NNLO) QCD that include all next-to-next-to-leading logarithms (NNLL) in NNLO QCD [1,2,4]. atic and b-tagging method ow how the mean value of d due to each source of l but the considered source . The AE give the impact the nuisance parameters re changed by AE1 SD of t (pb) þ (pb) À (pb) [1,2,4] values of t t as a function of m t . The point shows t t measured using the combined method, the black line shows the fit with Eq. (10), and the gray band with its dashed delimiting lines shows the corresponding total experimental uncertainty. Each curve is bracketed by dashed lines of the corresponding color that represent the theoretical uncertainties due to the choice of PDF and the renormalization and factorization scales (added linearly).
012008-16
Top quark pole mass from cross section CMS Preliminary, √s=7 TeV, L=1.14 fb [15]. Right: Summary of indirect measurements of m(pole) [17].
performed calculations of the top quark pair production cross section up to approximate next-to-next-to-leading order in perturbative QCD. Most recently, full next-to-next-to-leading order calculations were achieved for the qq and gq initial states [6]. The resulting cross section is consistent with next-to-leading order calculations, but the scale uncertainties are significantly reduced, and are now of order 2%. Calculations for gg are expected very soon. A recent overview of the various available QCD calculations is available here [7]. Up-to-date Monte Carlo event generators implement the production process at matrix element level up to next-to-leading order [8,9]. In tree-level event generators, matrix elements are implemented beyond leading order, containing up to 3 or more hard final state partons [10,11,12]. In these generators the hadronisation and modeling of the full event final state is generally achieved using parton showers as provided by PYTHIA [13] or HERWIG [14].
In fixed-order perturbative top quark pair cross section calculations, a theoretically well-defined top quark mass (pole mass or M S mass) is used. This is in contrast to calculations as implemented in Monte Carlo generators where the top quark mass does not correspond to a specific renormalization scheme, and soft interactions and parton showers are used to simulate the hadronic final state. The dependence of the cross section prediction on the top quark mass can be exploited to extract the theoretically well-defined pole mass. Such studies have been performed by the D0, CMS and ATLAS experiments, yielding results as depicted in Figure 1 [15,16,17,18].
Top Quark Production
Experimental Signatures and Systematics
In the Standard Model top quarks are expected to decay to almost 100% into a W boson and a b-quark. Top quark pair events are thus characterized by the presence of two b-jets, which can experimentally be tagged, i.e. identified, using decay lifetime reconstruction techniques. Top quark pair events are classified experimentally based on the decays of the two W bosons. In the fully hadronic decay channel both W bosons decay into quarks. This final state has the largest branching ratio, but is also most difficult to identify and reconstruct experimentally. The channels where one or both of the W bosons decay into leptons are simpler to measure, and are referred to as +jets or dilepton channel, respectively. Triggering and reconstruction details depend on the flavour of the charged leptons (electron, muon or τ ). Analyses with τ -leptons use the decays of the τ into low multiplicity jets. In leptonic W boson decays large momentum neutrinos are produced which escape detection, thus giving rise to large missing transverse momentum.
Inclusive Cross Section Measurements
Measurements of the inclusive top quark pair cross section have been performed using all decay channels (except the one with two τ -leptons in the final state). The combination of the most recent measurements at the Tevatron, using data of an integrated luminosity of up to 8.8 fb −1 , yields a cross section of 7.65 ± 0.20 ± 0.36 pb, with a relative total uncertainty of 5.5%. An overview of the various analyses is given in Figure 2(left). The two single most precise results from D0 and CDF are obtained in the +jets channel [15,19]. Both results achieve optimal separation of the top quark signal from the background by combining results from analyses based on b-tagging with multivariate techniques. In Figure 3 the output distributions from the multivariate algorithms are shown. The D0 measurement, σ tt = 7.78 +0.77 −0.64 pb, is obtained from a simultaneous fit of signal and background distributions. In the fit, a scale factor for the contribution from events with W-bosons and heavy quark jets is also determined. The dominant systematics originate from uncertainties on the integrated luminosity as well as jet-identification. The result from CDF is σ tt = 7.70 ± 0.54 pb. Here, the luminosity uncertainty is minimized by normalizing the measurement to the measured number of Drell-Yan events where Z 0 decays into two leptons. For the latter measurement, the same triggers and lepton identification cuts are applied, such that the uncertainties are further reduced. The results are in good agreement with each other and with theory calculations at NNLO+NNLL which yield σ tt = 7.24 +0. 15 −0.24 (scale) +0.18 −0.12 (P DF ) pb [6]. In this calculation, performed for a top quark mass of 172.5 GeV, the contribution qq → tt is calculated to full NNLO [6]. Both collaborations D0 and CDF are now in the process of finalizing the top quark analyses using their full datasets, corresponding to about 10 fb −1 .
A compilation of measurements at the LHC using the dataset recorded in the year 2011 at a pp centre-of-mass energy of 7 TeV are presented in Figure 2(right) [21]. The most precise results from the LHC were achieved in the +jets channel (ATLAS) [22] and the dilepton channel (CMS) [23]. The latter is not yet included in Figure 2. The ATLAS analysis is based on a template fit to a systematic uncertainties in the maximum ssigning a parameter to each independent on. These ''nuisance'' parameters are althe maximization of the likelihood functainties, therefore the measured t t cross section can be different from the value obtained if the parameters for the systematic uncertainties are not included in the fit. The effects of a source of systematic uncertainty that is fully correlated among several channels are controlled by a single parameter in these channels. Figure 3. Output distributions from multivariate analyses from D0 (left) [15] and CDF (right) [19].
likelihood discriminant using event kinematic information as input (Figure 4(left)). The measured cross section is σ tt = 179.0 ± 3.9(stat.) ± 9.0(syst.) ± 6.6(lumi.) pb corresponding to a total relative uncertainty of 6.5%. Dominant systematic uncertainties arise from the modeling of the signal, the jet energy scale and the lepton identification uncertainties. The CMS analysis uses a profile likelihood fit to the 2dimensional jet and b-tag multiplicity distribution (Figure 4(right)). The measured cross section is σ tt = 161.9 ± 2.5(stat.) ± 5.1(syst.) ± 3.6(lumi.) pb corresponding to a total relative uncertainty of 4.2%. This is the single most precise top quark cross section measurement so far. Dominant systematic uncertainties are related to the jet energy scale and the lepton identification uncertainty. Both CMS and ATLAS have presented first results at a pp centre-of-mass energy √ s = 8 TeV [24,25,26]. A summary of the inclusive top quark pair cross section σt t = 179.0 ± 3.9stat ± 9.0syst ± 6.6l umi pb 6.5% 9% σt t = 164.4 ± 2.8stat ± 11.9syst ± 7.4l umi pb : Same as Fig. 1, but for the dilepton invariant-mass distribution of (a) the sum of the d µ + µ channels, and (b) the e ± µ ⌥ channel. The gap in the former distribution reflects irement that removes dileptons from Z decay.
signal t t ) b-jets ,N jets (N (2,0) (2,1) (2,2) (3,0) (3,1) (3,2) (3,3) 4,0) ≥ ( 4,1) ≥ ( 4,2) ≥ ( Data/Sim.
4.2%
σt t = 176 ± 5stat +14-11syst ± 8lumi pb 8% combination of cross sections from samples with and w/o b-tag most precise measurement so far measurements at the Tevatron and the LHC is shown in Figure 5. Good agreement with QCD calculations up to approximate NNLO is observed.
Differential Distributions
Additional information about the physics of top quark production can be gained from measurements of differential distributions. The measurement of kinematic top quark distributions does not only probe QCD predictions and provide input to an improved choice of QCD model and scale parameters. Differential distributions also have the potential to constrain the parton distribution functions of gluons at large x. Moreover, the distributions are sensitive to possible new physics which are especially expected to occur at high tt invariant masses, for example decays of massive Z-like bosons into top quark pairs.
The kinematic properties of the top quark pair are determined from the fourmomenta of all final-state objects by means of kinematic reconstruction algorithms. In the +jets channels constrained kinematic fitting algorithms are applied to obtain the kinematics of both top quarks. In the dilepton channels, due to the presence of two neutrinos, the kinematic reconstruction is underconstrained, even after imposing the full set of possible kinematic constraints such as that of the W-boson invariant mass of 80.4 GeV, the equality of the top quark and antiquark masses and assuming that the missing energy originates solely from the neutrinos in the event. Ambiguities between several solutions are resolved by prioritization e.g. by use of the expected neutrino energy distribution.
First differential measurements of tt cross sections were performed by CDF and D0 [27,28,29]. In general, good agreement of theoretical calculations with the data was observed. In Figure 6 (left) the differential cross section as a function of transverse momentum of the top quark is shown as measured by D0 [29]. Calculations including higher order corrections (up to approximate NNLO) are found to give an improved description of the data, especially w.r.t. their normalization.
At the LHC, the large tt production rate leads to a substantial reduction of the statistical uncertainties in each bin, and in turn helps reduce systematics. In Figure 6(right) the measurement of the transverse momentum distribution of top quarks by CMS is presented [30]. The measurement makes use of the full statistics accumulated in the year 2011 at 7 TeV and is limited by systematic uncertainties. Several models are confronted with the data. They all give a good description of the data. A yet improved description is achieved by the prediction to approximate NNLO.
A large number of distributions of the top quark and the tt system, as well as their decay products, has been measured at the LHC [30,31]. ATLAS and CMS report normalized differential cross sections, i.e. shape measurements, in which normalization uncertainties are removed. In Figure 7 the distributions of the invariant mass of the tt system are displayed. Both the ATLAS and the CMS data are very well described by the various calculations up to an energy scale of more than 1 TeV.
Jet Multiplicity Distributions
At LHC energies, the fraction of top quark pair events with additional hard jets in the final state is large, about half of the total number of events [32]. For the correct description of events with additional jets, contributions from higher order QCD processes are required which take into account additional radiation in the initial or final state. The understanding of these processes is important not least because multijet processes constitute important backgrounds for many new physics searches. In recent measurements from ATLAS and CMS the distributions of jet multiplicities [29] and the LHC (right) [30]. Figure 9: Normalised differential tt production cross section in the`+jets channels as a func of the p t T (top left) and y t (top right) of the top quarks, and the p tt T (middle left), y tt (mid right), and m tt (bottom) of the top-quark pairs. The superscript 't' refers to both top qu and antiquarks. The inner (outer) error bars indicate the statistical (combined statistical systematic) uncertainty. The measurements are compared to predictions from MADGRA POWHEG, and MC@NLO, and to an approximate NNLO calculation [11,12], when availa The MADGRAPH prediction is shown both as a curve and as a binned histogram. [31] and from CMS(right) [30]. and additional jets due to QCD radiation are studied in detail [33,34,35,36,37]. In Figure 8(left) the multiplicity distribution of jets for top quark pair events in the +jets channel is shown. The data are generally well described by the Monte Carlo generators. Towards large multiplicity the MC@NLO generator interfaced with parton shower from HERWIG is seen to predict significantly less events than MADGRAPH or POWHEG which use PYTHIA to generate the parton showers. An alternative way of investigating additional activity in the event is to study the gap fraction distribution. Events are vetoed if they contain an additional jet with transverse momentum above threshold in a central rapidity interval. The fraction of events surviving the jet veto, the gap fraction, is presented as a function of this threshold. The gap fraction distribution for jets in the central rapidity range is displayed in Figure 8(right). A qualitatively similar trend is observed as in the multiplicity distribution in that the MC@NLO generator predicts a larger fraction of events that have no jet activity beyond the jets originating directly from the top quark decays. As the data are able to discriminate between the predictions from different models, these results can be used to optimize the choice of models.
Differential Cross Section in the Number of Additional Partons
where A is used to correct for detector efficiencies and acceptances and L is the integrated luminosity. Table 1 and Fig. 2 show the measured normalized differential cross section of tt in different jet bins for each individual channel. Due to the normalization, those systematic uncertainties that are correlated across all bins of the measurement, e.g. the one for the integrated luminosity as well as all other normalization uncertainties, cancel out. The two measurements are combined using the BLUE (Best Linear Unbiased Estimator) method [25] assuming that all systematic uncertainties, except the ones related to the lepton selection, are 100% correlated. The results are included in Fig. 2. Good agreement with various generators and scales (except small discrepancies for MC@NLO and MADGRAPH down variations at high jet multiplicities) is found.
Differential Cross Section in the Number of Additional Partons
In this section an alternative method to extract information about additional radiation in tt events is proposed. Using MC generator information, the tt cross-section is measured as a function of the number of hard partons radiated in addition to the tt decay products. This measurement is only performed with the µ + jets channel. The resulting cross-section provides checks and constraints of the factorization and renormalization (Q 2 ) and matrix-element/parton-shower matching scale parameters used in the MADGRAPH MC generator. This measurement may
Single Top
Single top quark production, in the Standard Model, is expected to proceed through charged-current electro-weak interactions. Depending on whether the W-boson is time-like, space-like or real, one distinguishes between the s-channel, the t-channel and the tW -channel. In the latter channel, single top quarks are produced in association with a W boson in the final state. Measurements of single top production constitute a unique test of the electroweak interactions and quark-flavour dynamics as described by the CKM matrix. The measured single top production cross sections can be used for direct constraints of the CKM matrix element |V tb | and of possible contributions from new physics, e.g. arising from feed-down from potential fourth-generation quarks. The ratio of top and anti-top production is sensitive to b-quark parton density distributions as well as the ratio of u and d valence quarks.
Single top production was first discovered at the Tevatron in 2009, much later than top quark pair production due to the significantly larger backgrounds as well as the smaller cross sections of electro-weak single top production in comparison to the strongly produced top quark pairs. The Tevatron analyses are performed for a combination of t-channel and s-channel, using multi-variate analysis techniques to separate the signals from the background [38,39,40].
Depending on the center-of-mass energy, and the corresponding initial state parton distributions and phase space, the expected cross sections for the three production channels are very different between the Tevatron and the LHC. At the Tevatron, only the s and t-channel are expected to have measurable rates of similar order of magnitude, while at the LHC, the t and tW channel have large cross sections, while the s-channel is significantly smaller in rate. Experimentally, for trigger and background reasons, single top measurements are performed using the leptonic decays of the W-boson from the top quark.
In the case of D0, a combination of three multi-variate analyses is used. Each MVA method is trained separately for the two single top quark production channels: for the t-channel discriminants, with t-channel considered signal and s-channel treated as a part of the background, and for the s-channel discriminants, with s-channel considered signal and t-channel contributions treated as a part of the background. For the combined measurement of the sum of s-and t-channel, the Standard Model prediction for the ratio between s-and t-channel is used as input. The results from the three different analyses are combined, yielding a sum of cross sections of σ s+t = 3.43 +0.73 −0.74 pb. The main systematic uncertainties arise from uncertainties related to the jet energy scale, the b-jet identification and the integrated luminosity. Figure 9(left) shows the result of the analysis in which the t-channel is treated as signal.
The CDF Experiment performs a multi-variate analysis based on a neural network. For the sum of s and t-channel, the Standard Model prediction for the ratio between s-and t-channel is used as constraint, and the cross section is measured to be σ s+t = 3.04 +0.57 −0.53 pb. The result for the case in which the s-channel is treated as signal is shown in Figure 9(right). Both CDF and D0 establish a clear observation of single top quark production, in good quantitative agreement with the Standard Model prediction.
At the LHC, the cross section for single top production is significantly larger than at the Tevatron. At a centre-of-mass energy of 7 TeV, the cross section for t-channel production alone is predicted to be σ t = 64 +3.3 −2.6 pb [41], more than one third of the top quark pair cross section. Due to the large LHC luminosity and excellent background suppression capabilities, CMS and ATLAS have been able to perform measurements of the t-channel and tW -channel cross sections, with already very good precision.
The measurements in the t-channel are performed using events with exactly one isolated lepton (electron or muon) and two or three jets. One of the jets is identified as b-jet. Additional cuts on kinematic observables are applied to further remove background. Results are available from ATLAS and CMS for center-of-mass energies √ s of both 7 TeV and 8 TeV. Figure 10 gives an overview of the results.
In the CMS analysis of the dataset with √ s = 7 TeV, in the electron channel,
VII. CONCLUSIONS
We presentd a measurement of single top quark production in lepton plus jets final state using 7.5 fb −1 of pp collision data collected by CDF II experiment. We select events in the W +jets topology consistent with the signature of a charged lepton (electron or muon), large missing transverse energy ( E T ) from the W boson decay and two or thre jets, at least one of them is required to be identified as originating from a bottom quark. We use the new POWHEG Monte Carlo generator for single top signal samples in s-channel, t-channel and Wt-channel, which are extended a NLO accuracy, with an assumed top quark mass of 172.5 GeV/c 2 . The Neural Network multivariate method is used to discriminate signal against comparatively large backgrounds. We measure a single top cross section of 3.04 +0.5 −0.5 (stat+syst) and set a lower limit |V tb | > 0.78 at the 95% confidence level, assuming m t = 172.5 GeV/c 2 . With a two-dimensional fit for σ s and σ t , we obtain σ s = 1.81 +0.63 −0.58 pb and σ t = 1.49 +0.47 −0.42 pb.
31
6 lyses are performed exclusive subsamples methods the output ith a binning chosen nts to limit the uncerhniques use the same 70% correlated with these methods using NComb) that takes discriminants of the nd produces a single ure 2 shows comparal, the background discriminant, which t of the cross section. ion cross section is ach as in [4,17,18]. nd construct a twoability density as a tqb and tb processes. signals, backgrounds, ned likelihood as a els and all bins. No tive rates of tb and isson distribution for uniform prior probathe two signal cross tematic uncertainties priors that preserve channels. The tqb m a one-dimensional ained from this 2D axis, thus not making f the s-channel cross ction is obtained by nsembles of datasets ss section values are oss section extraction procedure. Figure 3 shows the 2D posterior probability density for the combined discriminant together with predictions from the SM [9] and various beyond-the-SM scenarios: four-quark-generations with CKM matrix element |V ts | = 0.2 [10], top-flavor model with new heavy bosons at a scale m x = 1 TeV [11], and FCNC with an up-quark/topquark/gluon coupling κ u /Λ = 0.036 [12]. SM [2] Four generations [3] Top-flavor The measured cross sections of σ(pp → tqb + X) = 2.90 ± 0.59 pb and σ(pp → tb + X) = 0.98 ± 0.63 pb are in good agreement with the SM expectation for a top quark mass of 172.5 GeV [9]. The uncertainty includes both statistical and systematic sources. The cross section for t-channel single top quark production is the most precise measurement of an individual single top quark production channel to date with an uncertainty of 20%.
s-channel cross section [pb]
The significance of the t-channel cross section measurement is computed using a log-likelihood ratio approach [5,19] which tests the compatibility of the data with two hypotheses: a null hypothesis where there is only background and a background plus signal hypothesis, where the number of signal events corresponds to the theoretical cross section. New for this analysis is the computation of the distributions for these two hypotheses given by an asymptotic Gaussian approximation [35]. With this approximation we compute for the first time, the significance of the measured tqb cross section independently of any assumption on the production rate of tb. We estimate the probability of the background to fluctuate and produce a signal as large as the one observed to be 1.
VII. CONCLUSIONS
We presentd a measurement of single top quark production in lepton plus jets final state using 7.5 fb −1 of pp collision data collected by CDF II experiment. We select events in the W +jets topology consistent with the signature of a charged lepton (electron or muon), large missing transverse energy ( E T ) from the W boson decay and two or three jets, at least one of them is required to be identified as originating from a bottom quark. We use the new POWHEG Monte Carlo generator for single top signal samples in s-channel, t-channel and Wt-channel, which are extended at NLO accuracy, with an assumed top quark mass of 172.5 GeV/c 2 . The Neural Network multivariate method is used to discriminate signal against comparatively large backgrounds. We measure a single top cross section of 3.04 +0.57 −0.53 (stat+syst) and set a lower limit |V tb | > 0.78 at the 95% confidence level, assuming m t = 172.5 GeV/c 2 . With a two-dimensional fit for σ s and σ t , we obtain σ s = 1.81 +0.63 −0.58 pb and σ t = 1.49 +0.47 −0.42 pb.
31
6
, the analyses are performed mutually exclusive subsamples all three methods the output istograms with a binning chosen enough events to limit the unceristics. e MVA techniques use the same only ≈ 70% correlated with re combine these methods using orithm (BNNComb) that takes l output discriminants of the methods, and produces a single inant. Figure 2 shows comparhannel signal, the background e combined discriminant, which easurement of the cross section. rk production cross section is sian approach as in [4,17,18]. h of [19] and construct a twoerior probability density as a tions for the tqb and tb processes. ts for the signals, backgrounds, form a binned likelihood as a alysis channels and all bins. No on the relative rates of tb and sume a Poisson distribution for events and uniform prior probaalues for the two signal cross ver the systematic uncertainties Gaussian priors that preserve n bins and channels. The tqb tracted from a one-dimensional ensity obtained from this 2D over the tb axis, thus not making the value of the s-channel cross tb cross section is obtained by b axis. Ensembles of datasets ifferent cross section values are ity of the cross section extraction procedure. Figure 3 shows the 2D posterior probability density for the combined discriminant together with predictions from the SM [9] and various beyond-the-SM scenarios: four-quark-generations with CKM matrix element |V ts | = 0.2 [10], top-flavor model with new heavy bosons at a scale m x = 1 TeV [11], and FCNC with an up-quark/topquark/gluon coupling κ u /Λ = 0.036 [12]. SM [2] Four generations [3] Top-flavor The measured cross sections of σ(pp → tqb + X) = 2.90 ± 0.59 pb and σ(pp → tb + X) = 0.98 ± 0.63 pb are in good agreement with the SM expectation for a top quark mass of 172.5 GeV [9]. The uncertainty includes both statistical and systematic sources. The cross section for t-channel single top quark production is the most precise measurement of an individual single top quark production channel to date with an uncertainty of 20%.
The significance of the t-channel cross section measurement is computed using a log-likelihood ratio approach [5,19] which tests the compatibility of the data with two hypotheses: a null hypothesis where there is only background and a background plus signal hypothesis, where the number of signal events corresponds to the theoretical cross section. New for this analysis is the computation of the distributions for these two hypotheses given by an asymptotic Gaussian approximation [35]. With this approximation we compute for the first time, the significance of the measured tqb cross section independently of any assumption on the production rate of tb. We estimate the probability of the background to fluctuate and produce a signal as large as the one observed to be 1.6 × 10 −8 , corresponding to a significance of 5.5 standard deviations (SD). The expected significance is 4.6 SD.
The presence of the t-channel signal is visible in t and s-channel Figure 9. Results of the single top production cross section measurements in the t-channel and in the s-channel from the D0 Experiment (left) [39] and the CDF Experiment (right) [40]. the missing transverse energy is required to be larger than 35 GeV. For the muon channel, the transverse mass of the W-boson is required to be larger than 40 GeV. The cross section measurement is obtained from the combination of three different analyses, of which two make use of multi-variate techniques. In the third analysis, the signal is determined by a template fit to the rapidity distribution η j of the (untagged) recoil jet using the event category with one lepton and two jets of which one is b-tagged. Other event categories are used to control the backgrounds. frame. The observed charge asymmetry and the cos q ⇤ distribution are presented in Fig. 4 for muon+electron events in the SR, for |h j 0 | > 2.8.
Neural Network Analysis
In the NN analysis, several kinematic variables, which are characteristic of SM single-top-quark production, are combined into a single discriminant by applying an NN technique. The NEU-ROBAYES package [26,27] used for this NN analysis combines a three-layer feed-forward NN with a complex, but robust, preprocessing. To reduce the influence of long tails in distributions, to the choice of the single top-quark t-channel signal generator is estimated from the difference between Ac-erMC and MCFM predictions [44]. The modelling uncertainty for the t t background is evaluated by comparing the generators MC@NLO and POWHEG [45,46] (with HERWIG showering). For the W+jets background a shape uncertainty is assigned based on the variation of the choices of the matching scale and of the functional form of the factorisation scale in ALP-GEN. Systematic uncertainties related to the parton distribution functions are taken into account for the signal and for all background processes which are modelled by simulated events. In addition to the nominal PDF set the alternative MSTW2008nlo68cl [47] and CTEQ6.6 PDF sets are investigated. Events are reweighted according to each of the PDF uncertainty eigenvectors and the total uncertainty is evaluated following the procedure described in Ref. [36]. An additional uncertainty 11]. The uncertainty on the dibos 5% [38].
Background normalisation to data. T ground estimate has an uncertainty analysis places an uncertainty of 50 events with W+heavy flavour jets an of W+light jets events. These unce as constraints on the predictions wh determining the W+jets rates and the tion. The cut-based analysis does not certainty on the W+heavy flavour an rates, but considers separately the im nant sources of uncertainty on the da normalisation factors. This treatment tion between each component of unce malisation factors and the uncertaint rates to be taken into account. The Z normalisation has an uncertainty of 6 The results from the three analyses are combined to yield a measured cross section of σ t = 67.2±6.1 pb [42]. Dominant uncertainties come from statistical limitations, the modeling of signal and backgrounds, as well as the b-tagging.
In the ATLAS analysis, events with one lepton and two or three jets are selected, if the missing transverse energy is larger than 25 GeV. The t-channel cross section is measured by fitting the distribution of a multivariate discriminant, constructed with a neural network, yielding a result σ t = 83 ± 4(stat.) +20 −19 (syst.) pb [43]. Dominant uncertainties arise from model uncertainties, such as the ISR/FSR scale, as well as the b-jet identification efficiency. The result is cross-checked by an independent analysis using a cut-based selection. In Figure 11(right) the distribution of the invariant mass of the lepton and the b-tagged jet is shown for two-jet events with |η j | > 2.0. Here, the scalar sum of lepton, jets and missing transverse energy, H T , is required to be larger than 210 GeV. ATLAS also measures the ratio R t between the cross sections for top and anti-top quark production in the t-channel. The result is R t = 1.81 ± 0.10(stat.) +0.21 −0.20 (syst.) [44]. The ratio is sensitive to the ratio of up-quark and down-quark parton distribution functions of the proton. In the measurement of the ratio, the uncertainties common to both channels cancel.
Both ATLAS and CMS already presented t-channel cross section measurements using the data recorded in 2012 at a centre-of-mass energy of 8 TeV [45,46]. In these analyses, the selection criteria are somewhat tightened with respect to the 7 TeV analyses described above, in order to cope with more severe backgrounds due to the increase of both pile-up events and center-of-mass energy. The results of the measurements are included in Figure 10.
The single top cross section in the tW -channel, inaccessibly small at the Teva-tron, is sizable at the LHC. Theoretical calculations predict the cross section to be σ tW = 15.7 +1.3 −1.4 pb [47]. Single top events in the tW channel form an important background to Higgs searches in the decay to two W bosons. The ATLAS and CMS experiments have performed first measurements of the single top cross section and observe signals at significances of 3.3 and 4.0 standard deviations, respectively [48,49]. Events with two leptons (electron or muon) and at least one jet are selected. In addition, the missing transverse energy in the event is required to be larger than 50 (30) GeV in ATLAS (CMS). Events in the same-flavour channels (with two electrons or two muons) are rejected if the invariant mass is between 81 GeV and 101 GeV, thereby removing backgrounds from Z+jet events. In the CMS analysis, the jet is required to be b-tagged. The dominant background originates from top quark pair production where both W bosons from the top quarks decay leptonically. Both ATLAS and CMS extract the cross section using a template fit to a boosted decision tree (BDT) discriminant distribution. In Figure 12(left) the distribution of the BDT discriminant is shown for the event category with two leptons and one jet. The ATLAS measurement yields σ tW = 16.8 ± 2.9(stat) ± 4.9(syst)pb [48]. The cross section as measured by CMS is σ tW = 16 +5 −4 pb [49], in good agreement with the Standard Model expectation. In Figure 12(right) the distribution of jet and lepton multiplicities in a cut-based cross check analysis, as performed by CMS, is displayed. A clear signal is seen in the two lepton and one jet category. described below. The main experimental source of systematic uncertainties comes from the knowledge of the jet energy scale (JES), which carries an uncertainty of 2% to 7% parameterised as a function of jet p T and η [31]. The presence of a b-jet in the event is also taken into account and an extra uncertainty of 2% to 5% depending on jet p T is added in quadrature to the non-bjet uncertainty. Other experimental uncertainty sources which have been considered are the jet energy resolution, the jet reconstruction efficiency, the lepton identification efficiency, the lepton energy scale determination and resolution as well as the multiple proton-proton collision and underlying event modelling. The uncertainty in the luminosity determination is 3.7% [10,11].
Uncertainties in the simulation include the effects of the MC generator choice, the scheme used in the hadronisation and showering and models of the initial and final state radiation (ISR/FSR). Generator choice uncertainty is estimated by comparing AcerMC with MC@NLO generators for single top-quark Wt events, and comparing POWHEG with MC@NLO generators for top quark pair events. Hadronisation and showering effects are estimated using the differences seen in generated events interfaced with either PYTHIA [36] or HERWIG. Finally, ISR/FSR modelling effects are assessed on MC signal and background samples interfaced with PYTHIA. Specific tunes are used to separately vary ISR and FSR modelling via changes to 1/Λ ISR QCD , the maximum parton virtuality in a space-like parton shower, the Λ FSR QCD scale and the FSR infrared cutoff [37]. 6 References the model as nuisance parameters. The same methods for the cross section measurement and the significance calculation as in the BDT analysis have been used. Figure 5 shows the event yields selected by the count-based analysis for each region, in data and simulation, in which the simulation yields have been normalized to the outcome of the maximum likelihood fit. The observed significance of the tW signal obtained with the count-based analysis is 3.5 s, with an expected significance of 3.2 ± 0.9 s. The count-based analysis measures a cross section of 15 ± 5 pb. These results are consistent with those obtained with the BDT analysis. In summary, using 4.9 fb 1 of data collected with the CMS experiment at the LHC, evidence has been found for the associated production of a single top quark and W boson in pp collisions at p s = 7 TeV with a significance of 4.0 s and a measured cross section of 16 +5 4 pb. We congratulate our colleagues in the CERN accelerator departments for the excellent performance of the LHC machine. We thank the technical and administrative staff at CERN and other CMS institutes, and acknowledge support from: BMWF and FWF (Austria); FNRS and [48] and the distribution of jet and lepton multiplicities in the CMS analysis (right) [49].
No evidence has been reported so far for single top production in the s-channel. The Standard Model cross section prediction is σ s = 4.6 ± 0.3 pb [50]. The event signature in this channel consists of one lepton and jets, very similar to that for the top quark pair, QCD and W+jet events. ATLAS reports an analysis in which an upper limit on the s-channel production cross section of σ s < 26.5(20.5) pb Top Quark Production 13 observed (expected) is set [51].
Conclusions
Top quark production is a field of rapid progress. The measurements provide important information about the production process as described in QCD, as well as sensitivity to possible new physics. At the Tevatron, precise final results are becoming available. At the LHC, large statistics and center-of-mass energy give access to a new realm of top quark precision physics. Measurements of inclusive cross sections have been performed for both top quark pair and single top production processes, exploiting all accessible decay channels. For top quark pair production, precise differential cross section results as well as studies of jet multiplicity distributions are available. In the sector of single top production, evidence for the associate production of top quarks with a W boson is reported for the first time. All measurements are in good agrement with Standard Model expectations.
c
Institute of Experimental Physics SAS, Košice, Slovakia 1 arXiv:1212.3957v1 [hep-ex] 17 Dec 2012 ndency for the hod lead to a dependence of simulated samples of t t of m t using the ALPGEN IA for the simulation of The resulting measure-II and can be parame-
FIG. 6
6(color online). Experimental and theoretical
- 1 valueFigure 1 .
11± theo ⊗ exp ± α s (Left: Dependence of the top quark pair cross section measurement by D0, as well as of various theory calculations, on the mass of the top quark
Figure 2 .
2Overview and Combination of top quark pair cross section measurements at the Tevatron (left)[20] and at the LHC (right)[21].
. Output of the RF discriminant for (a) and (b) ' þ 2 jets, (c) and (d) ' þ 3 jets, and (e) and (f) ' þ >3 jets unds and a t t signal based on the cross section obtained with the kinematic method. The ratio of data over MC own. The left plots (a), (c), and (e) show the results with the nuisance parameters fixed at a value of zero. The right show the results when the nuisance parameters are determined simultaneously with the t t cross section in the fit. In ts the contribution from the t t signal is normalized to the results of the cross section measurement, t t ¼ 7:00 and y. Contributions of the e þ jets and þ jets channels are summed.al. PHYSICAL REVIEW D 84, 4.3 fb -1 ,m t = 172.5 GeV), pre-tagged: tt =7.82±0.38(stat)±0.37(syst)±0.13(theo)pb D0 (5.3 fb -1 ,m t = tt =8.13±0.25(s Combined kinematical + b-tagging: D0 (5.3 fbtt =7.78±0.25(s CDF (4.3 fb -1 ,m t = 172.5 GeV): tt =7.70 ±0.52(stat+syst) pb
Figure 6 :
6from η(lepton), pT(jet), aplanarity, HT, 3p no b-tag W+jets background shape from ALPGEN, normalization fitted with constraint from W-charge asymmetry In-situ fit of systematics Dominant Systematics signal modeling (MC@NLO vs POWHEG), jet energy scale, lepton-ID Luminosity (not yet using final number) fit to 2ndary vertex mass in bins of jet and b-tag multiplicity, in-situ fit of systematics The multiplicity of b-tagged jets in events passing full event selections for (a) the summed e + e and µ + µ channels, and (b) the e ± µ ⌥ channels.
Figure 7 :
7Number of events selected for the three combined dilepton channels, as a function of the number of jets and b-tagged jets (N t extjets,N b-jets ) in each event. The data are shown by the dots, while the predicted tt and the contributing backgrounds are shown by the histograms. The hatched area corresponds to statistical and systematic uncertainties on the tt and on the background predictions taken in quadrature. The ratios of data to the sum of the tt and background predictions are given at the bottom, with the error bars again giving the statistical and systematic uncertainties taken in quadrature.
Same asFig. 1, but for the second-largest p T electrons, muons and jets in each event.
Figure 4 .
4Distributions of top quark pair cross section measurements from ATLAS(left) [22]and CMS(right)[23].
Moch, Uwer, Phys. Rev. D80 (2009) 054009 MSTW 2008 NNLO PDF, 90% C.L. uncertainty
Figure 5 .
5Top quark pair cross section measurements as a function of the pp and pp centre-ofmass energy. The bands represent result from calculations in perturbative QCD up to approximate NNLO
Figure 9 :
9Normalised differential tt production cross section in the`+jets of the p t T (top left) and y t (top right) of the top quarks, and the p tt T (m right), and m tt (bottom) of the top-quark pairs. The superscript 't' refe and antiquarks. The inner (outer) error bars indicate the statistical (co systematic) uncertainty. The measurements are compared to predictio POWHEG, and MC@NLO, and to an approximate NNLO calculation[11 The MADGRAPH prediction is shown both as a curve and as a binned hi
Figure 6 .
6Distributions of the top quark transverse momentum in top quark pair events at the Tevatron (D0) (left)
Figure 7 .
7Distributions of the invariant mass of the tt system from ATLAS(left)
0.268 ± 8 ⇥ 10 5 0.281 ± 8 ⇥ 10 5 0.174 ± 6 ⇥ 10 5 0.070 ± 4 ⇥ 10 5 0.021 ± 2 ⇥ 10 5 0.007 ± 1 ⇥ 10 5
Figure 2 :
2Measured normalized differential cross section of tt as a function of the number of jets in the individual channels and their combination. A comparison with MC expectations from different generators (left) and with Q 2 , matching threshold up and down MC samples (right) is also shown. The errors on the data points indicate the statistical (inner bars) and the total uncertainty.
Figure 8 .
8Normalized differential cross section as a function of jet multiplicity (left)[36] and the gap fraction, i.e. the fraction of events surviving the jet veto as a function of the transverse momentum of the jet (right)[35].
FIG. 4 :
4and the SM predictions of σ s and σ t are shown inFig. 4. We compare these with the NNNLO predictions o σ t+wt = 2.32 ± 0.27 pb and σ s = 1.05 ± 0.The results of the two-dimensional fit for σ s and σ t . The black point shows the best fit value, and the 68.3%, 95.5%, and 99.7% credibility regions are shown as shaded areas. The SM predictions are also indicated with their theoretical uncertainties.
models iscriminant for (a) the signal region. The bins B. The single top quark easured cross sections. bove the hatched bands ground prediction.
FIG. 3 :
3Posterior probability density for tqb vs tb single top quark production in contours of equal probability density. The measured cross section and various theoretical predictions are also shown.
FIG. 4 :
46 × 10 −8 , corresponding to a significance of 5.5 standard deviations (SD). The expected significance is 4.6 SD.The presence of the t-channel signal is visible in and the SM predictions of σ s and σ t are shown inFig. 4. We compare these with the NNNLO predictions of σ t+wt = 2.32 ± 0.27 pb and σ s The results of the two-dimensional fit for σ s and σ t . The black point shows the best fit value, and the 68.3%, 95.5%, and 99.7% credibility regions are shown as shaded areas. The SM predictions are also indicated with their theoretical uncertainties.
and background models t-channel discriminant for (a) the and (b) the signal region. The bins expected S:B. The single top quark zed to the measured cross sections. n is visible above the hatched bands on the background prediction.
FIG. 3 :
3Posterior probability density for tqb vs tb single top quark production in contours of equal probability density. The measured cross section and various theoretical predictions are also shown.
Phys.Rev.D 83 (2011) 091503 t-channel single top quark production
Figure 10 .
10Overview of t-channel measurements at the Tevatron and the LHC. The measurements are displayed as a function of the center of mass energy.
Figure 3 :
3Distributions of m`n b requiring |h j 0 | > 2.8, for muons (left) and electrons (right), obtained by normalising each process yield to the value from the fit. Because of limited simulated data, the background distribution is smoothed by using a simple spline curve.
Figure 4 :
4Distinct single-top-quark t-channel features in the SR for |h j 0 | > 2.8, for the electron and muon final states combined. The charge of the lepton (left) and cos q ⇤ (right). All processes are normalised to the fit results. Because of limited simulated data, the background distribution is smoothed by using a simple spline curve (right).
Figure 4 :
4(a) Distribution of the lepton charge after the full cut-based selection for 2-jet and 3-jet events. (b,c) invariant m the charged lepton, and the neutrino, m( νb), for the b-tagged sample for 2-jet (b) and 3-jet (c) events after applying all cut-b for the cut on m( νb). In all three distributions the t-channel single top-quark contribution is normalised to the observed cros the cut-based analysis. The last histogram bin includes overflows.
Figure 11 .
11Single top quark production in the t-channel at 7 TeV at the LHC. The distributions of invariant mass of the b-jet and the isolated lepton are displayed for CMS (left)[42] and ATLAS (right)[43].
Figure 11 (
11left) shows the distribution of the invariant mass of the lepton and the b-tagged jet for events with |η j | > 2.8. A clear signal of top quark events is seen.
Figure 5 :
5Event yields in data and simulation in the signal region (1j1t) and the two tt-enriched control regions for the count-based analysis. Simulation yields are scaled to the outcome of the statistical fit.
FWO (Belgium); CNPq, CAPES, FAPERJ, and FAPESP (Brazil); MES (Bulgaria); CERN; CAS, MoST, and NSFC (China); COLCIENCIAS (Colombia); MSES (Croatia); RPF (Cyprus); MoER, SF0690030s09 and ERDF (Estonia); Academy of Finland, MEC, and HIP (Finland); CEA and CNRS/IN2P3 (France); BMBF, DFG, and HGF (Germany); GSRT (Greece); OTKA and NKTH (Hungary); DAE and DST (India); IPM (Iran); SFI (Ireland); INFN (Italy); NRF and WCU (Korea); LAS (Lithuania); CINVESTAV, CONACYT, SEP, and UASLP-FAI (Mexico); MSI (New Zealand); PAEC (Pakistan); MSHE and NSC (Poland); FCT (Portugal); JINR (Armenia, Belarus, Georgia, Ukraine, Uzbekistan); MON, RosAtom, RAS and RFBR (Russia); MSTD (Serbia); SEIDI and CPAN (Spain); Swiss Funding Agencies (Switzerland); NSC (Taipei); TUBITAK
Figure 12 .
12Single top quark production in the tW -channel at 7 TeV at the LHC. The output distribution from the boosted-decision tree of the ATLAS analysis (left)
Table 1 :
1Measured normalized differential cross section for each jet multiplicity bin, compared with the MC expectations. Uncertainties in data are the full uncertainties (stat+syst+lumi) and the MC uncertainties include MC statistics only.
t t /dm t t d t t 1/ -3 10 -2 10 -1
AcknowledgementsI would like to thank the organizers of the PIC2012 conference for the realization of a very useful and enjoyable conference.
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[] |
[
"MASA'S AND CERTAIN TYPE I CLOSED FACES OF C * −ALGEBRAS",
"MASA'S AND CERTAIN TYPE I CLOSED FACES OF C * −ALGEBRAS"
] |
[
"Lawrence G Brown \nGeorge W. Mackey\n"
] |
[
"George W. Mackey"
] |
[] |
Let A be a separable C * −algebra and A * * its enveloping W * −algebra. A result of Akemann, Anderson, and Pedersen states that if {p n } is a sequence of mutually orthogonal, minimal projections in A * * such that P ∞ k p n is closed, ∀k, then there is a MASA B in A such that each ϕ n |B is pure and has a unique state extension to A, where ϕ n is the pure state of A supported by p n . We generalize this result in two ways: We prove that B can be required to contain an approximate identity of A, and we show that the countable discrete space which underlies the result cited can be replaced by a general totally disconnected space. We consider two special kinds of type I closed faces, both related to the above, atomic closed faces and closed faces with nearly closed extreme boundary. One specific question is whether an atomic closed face always has an "isolated point". We give a counterexample for this and also show that the answer is yes if the atomic face has nearly closed extreme boundary. We prove a complement to Glimm's theorem on type I C * −algebras which arises from the theory of type I closed faces. One of our examples is a type I closed face which is isomorphic to a closed face of every non-type I separable C * −algebra and which is not isomorphic to a closed face of any type I C * −algebra.
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10.1090/conm/449/08707
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[
"https://arxiv.org/pdf/0708.2290v1.pdf"
] | 2,507,398 |
0708.2290
|
cca938572bedcb67db82bf51860536aa09f7e90a
|
MASA'S AND CERTAIN TYPE I CLOSED FACES OF C * −ALGEBRAS
16 Aug 2007
Lawrence G Brown
George W. Mackey
MASA'S AND CERTAIN TYPE I CLOSED FACES OF C * −ALGEBRAS
16 Aug 2007Typeset by A M S-T E X 1AMS subject classification: 46L05 Key words and phrases C * −algebraclosed projectionclosed faceMASAq-continuouspure stateatomictype I
Let A be a separable C * −algebra and A * * its enveloping W * −algebra. A result of Akemann, Anderson, and Pedersen states that if {p n } is a sequence of mutually orthogonal, minimal projections in A * * such that P ∞ k p n is closed, ∀k, then there is a MASA B in A such that each ϕ n |B is pure and has a unique state extension to A, where ϕ n is the pure state of A supported by p n . We generalize this result in two ways: We prove that B can be required to contain an approximate identity of A, and we show that the countable discrete space which underlies the result cited can be replaced by a general totally disconnected space. We consider two special kinds of type I closed faces, both related to the above, atomic closed faces and closed faces with nearly closed extreme boundary. One specific question is whether an atomic closed face always has an "isolated point". We give a counterexample for this and also show that the answer is yes if the atomic face has nearly closed extreme boundary. We prove a complement to Glimm's theorem on type I C * −algebras which arises from the theory of type I closed faces. One of our examples is a type I closed face which is isomorphic to a closed face of every non-type I separable C * −algebra and which is not isomorphic to a closed face of any type I C * −algebra.
Introduction.
This paper was inspired by the paper [5] of C. Akemann, J. Anderson, and G. Pedersen. Much of the terminology used in this section is explained in later sections. To explain the connection with [5], we begin with:
Proposition 0.1. Let A be a C * −algebra and (p n ) a sequence of mutually orthogonal, minimal (rank one) projections in A * * . Let p = ∞ 1 p n , and let ϕ n be the pure state supported by p n . Then either of the following hypotheses implies that p is closed:
(i) ( [5, 2.7(1)⇒(2)]). There is a strictly positive element e such that each ϕ n is definite on e and ϕ n (e) → 0. (ii) ([12, Lemma 3]). There is a strictly positive element e such that ∞ 1 ϕ n (e) < ∞.
In circumstances similar to 0.1, [5] proves the existence of a MASA B such that each ϕ n | B has the unique extension property. The hypotheses require that A be non-unital. It is known (see [6, §4]) that a non-unital C * −algebra A may have MASA's which do not hereditarily generate A, or equivalently which do not contain an approximate identity of A. If the MASA constructed in [5] does not hereditarily generate A, the situation is intuitively unsatisfactory. (See the first paragraph of [5, §2].)
To investigate strengthening the result of [5], consider A, the result of adjoining an identity to A, and the pure state ϕ ∞ defined by ϕ ∞ (λ1 e A +a) = λ. The existence of a MASA B of A such that each ϕ n | B , 1 ≤ n < ∞, has the unique extension property and such that B hereditarily generates A is equivalent to the existence of a MASA B 1 of A such that each ϕ n | B 1 , 1 ≤ n ≤ ∞, has the unqiue extension property (B = B 1 ∩ A, B 1 = B).
Now the hypotheses of [5] imply that n∈I p n is closed for every subset I of N. Thus {p n : 1 ≤ n < ∞} has properties analogous to those of the discrete topological space N. But when p ∞ , the support projection of ϕ ∞ , is added to the set, the new set resembles the non-discrete space N ∪ {∞}. Thus we seek a generalization of the MASA result of [5] based on a class of topological spaces which includes N ∪ {∞}. We accomplish this in Corollary 2.4: Let A be a separable C * −algebra and X a totally disconnected, second countable, locally compact Hausdorff space. Assume that for each x in X, p x is a minimal projection in A * * , with associated pure state ϕ x , such that the p x 's are mutually orthogonal and for each closed (compact) subset S of X, x∈S p x is the atomic part of a closed (compact) projection, p S , in A * * . Then there is a MASA B of A such that B hereditarily generates A and each ϕ x | B has the unique extension property. Moreover, each p S is in B * * .
If X is general, the hypotheses of the above result may seem rather stringent. Partly in order to justify the generality of the result, we attempt to investigate the circumstances in which the hypotheses will be satisfied. A first observation is that every element of C 0 (X) (respectively, C b (X)) gives rise to an element of p X A * * p X which is strongly q-continuous (respectively q-continuous) on p X . (The concept "q-continuous on p" was defined in [7]. In [11], "strongly q-continuous on p" was defined and "Tietze extension theorems" for both kinds of relative qcontinuity were given.) Thus in Section 3 we give some basic results and examples on the subject of how many relatively q-continuous elements are supported by a given closed projection.
We also focus on a more specific question suggested by the theory of scattered C * −algebras: Suppose that p is an atomic closed projection in A * * and that pA * p is norm separable. Is there a minimal projection p 0 such that p 0 ≤ p and p − p 0 is closed? Such a p 0 would give an "isolated point" of the closed face F (p) supported by p. This question is related to the special case of 2.4 where X is countable. Clearly, if we seek to prove that certain conditions imply the hypotheses of 2.4, then we must be able to prove that these conditions imply a positive answer to our isolated point question. Note also that when X in 2.4 is countable, then p X is atomic, and the words "the atomic part of" can be omitted.
We give a counterexample for the isolated point question in Section 3, but we also give a positive result which has the following hypothesis (nearly closed extreme boundary):
(NCEB) [P (A) ∩ F (p)] − ⊂ {0} ∪ [t, 1]P (A) for some t in (0, 1].
Here P (A) is the pure state space of A, and (NCEB) holds in particular if the set of extreme points of F (p) is (weak * ) closed. Lest (NCEB) seem unnatural or excessively strong, we point out a connection with [5, §4]. Circumstances not covered by 0.1 are actually considered in [5]. Suppose {ϕ n : 1 ≤ n < ∞} is a collection of mutually orthogonal pure states such that ϕ n w * −→ 0 and each equivalence class is finite. With the additional assumption that there is a uniform bound on the size of the equivalence classes, the authors of [5] show in §4 that the needed conditions ( [5, 2.7(2)]) are satisfied. Without the uniform boundedness hypothesis, we can show easily that [5, 2.7(2)] is equivalent to (NCEB).
Our positive result, which is in Section 4, is roughly that if p is a closed projection satisfying (NCEB), then equivalence of pure states gives a proper closed map from [P (A) ∩ F (p)] − \ {0} onto a locally compact Hausdorff space X. If pA * p is norm separable, then X is countable and hence scattered. In general, X need not even be totally disconnected, of course.
There are some technicalities involving direct integral theory required in order to prove that closed subsets of X give rise to closed projections. This is what leads us to the study in Section 5 of type I closed faces, where the face F (p) is called type I if pA * * p is a type I W * −algebra. Obviously every atomic face is type I, and also F (p) is type I when p (is closed and) satisfies (NCEB), at least if A is separable. Our results on type I closed faces are only rudimentary, and we think the concept is worthy of further study.
Partly because theorems are not always discovered in logical order, our efforts to expand on the results of [5] have led us in several directions. The different parts of this paper, though closely related, do not mesh perfectly. In Section 7 we attempt to exhibit the formal relationships among the previous sections. The earlier sections can in large part be read independently of one another, except that Section 6 is a continuation of Section 4 relying on Section 5. The promised complement to Glimm's theorem is Proposition 5.11.
A preliminary preprint of this paper was circulated several years ago. Some results overlapping with Section 3 have been independently found by E. Kirchberg (cf. [22,Lemma 2.3]).
Preliminaries.
A will always denote a C * −algebra and A * * its enveloping W * −algebra. For h in A * * sa and F a Borel set in R E F (h) denotes the spectral projection of h for F . For many of our proofs A must be separable, but we rarely require that A be unital. [25, 3.10.7]) that p is closed if and only if F (p) is closed, and if so we follow the usual abuse of notation and call F (p) a "closed face of A". Also, p is called compact ([4]) if F (p) ∩ S(A) is closed or equivalently if p is closed in A * * , where A is the result of adjoining an identity to A.
The reduced atomic representation, π, of A is ⊕ i π i , where {π i } contains one representative of each unitary equivalence class of irreducible representations. Denote by z at the central projection in A * * that supports π. Thus z at A * * ∼ = ⊕ i B(H π i ), and (1 − z at )A * * has no type I factor direct summands. The atomic part of an element x of A * * is z at x, x is atomic if x = z at x, F (p) is atomic if p is atomic, etc. Also pure states ϕ and ψ are called equivalent if the irreducible representations π ϕ , π ψ are unitarily equivalent.
We want to comment further on Proposition 0.1. In fact 2.7(1) of [5] actually states that ϕ n (e) < ∞ rather than ϕ n (e) → 0. However, the proof in [5] that (1) implies (2) uses only the weaker hypothesis, so that it is correct to attribute 0.1(i) to [5]. (Unfortunately, when he was writing [12], the author had not yet read the proofs in [5].) Here is a generalization of 0.1: Lemma 1.1. Let (p n ) be a sequence of mutually orthogonal minimal projections in A * * and p = ∞ 1 p n . If, ∀a ∈ A, π * * (p)π(a)π * * (p) is a compact operator on H π , where π is the reduced atomic representation of A, then p is closed.
The proof of 1.1 and the fact that it implies 0.1(ii) is identical to the proof of Lemma 3 in [12]. Lemma 1.1 implies 0.1(i) because in that case π * * (p)π(e)π * * (p) is a diagonal operator whose matrix elements approach zero. (If A is σ-unital, it is enough to verify the compactness for a strictly positive element of A, as shown in [12].) Lemma 1.1 also applies under the Standing Assumptions of [5, §4], since then π * * (p)π(a)π * * (p) is a block-diagonal operator with bounded block size -in particular it is a (2N + 1)-diagonal operator.
Despite this, we offer the following new proof of 0.1(i), which may be instructive:
The hypothesis that ϕ n is definite on e is equivalent to p n e = ep n . Thus if λ n = ϕ n (e), then p n ≤ E {λ n } (e). Let ǫ k = sup{λ n : n > k} and q k = k 1 p n ∨ E [0,ǫ k ] (e). Since [0, ǫ k ] is a closed set, E [0,ǫ k ] (e) is closed, and thus [1,Theorem II.7] implies that q k is closed. Since E {0} (e) = 0, p = ∧ ∞ 1 q k , and [1, Proposition II.5] implies p is closed. Theorem 1.2. Let (p n ) be a sequence of mutually orthogonal minimal projections in A * * and p = ∞ 1 p n . Then the following are equivalent:
(i) Every subprojection of p in A * * is closed. (ii) n∈I p n is closed for each subset I of N. (iii) ∞ k p n is closed, ∀k (cf. [5, 2.7(2)]). (iv) π * * (p)π(A)π * * (p) ⊂ K(H π ), where π is the reduced atomic representation of A. Proof. (i)⇒(ii)⇒(iii) is obvious. (iii)⇒(iv): Let q k = ∞ k p n . Then F (q k ) is a closed subset of Q(A) and ∞ 1 F (q k ) = {0}
. By this and [5, 2.3], (q k ) approaches infinity in the sense of [5]. By definition, aq k → 0, ∀a ∈ A. Therefore π(a)π * * (p − q k ) → π(a)π * * (p) in norm.
Since π * * (p − q k ) is a finite rank operator, this implies π(a)π * * (p), and a fortiori π * * (p)π(a)π * * (p), is compact. (iv)⇒(i): Assume p ′ ∈ A * * and p ′ ≤ p. Then π * * (p ′ )π(A)π * * (p ′ ) ⊂ K(H π ). Clearly there are mutually orthogonal minimal projections p ′ n such that p ′ = ∞ 1 p ′ n . Thus p ′ is closed by 1.1. Perhaps it should also be mentioned that if I in 1.2(ii) is finite, then n∈I p n is finite rank and hence compact ([1, Corollary II.8]).
If ϕ is in P (A) and B is a C * −subalgebra of A, we say that ϕ| B has the unique extension property (UEP) if ϕ| B ∈ P (B) and ϕ is the only element of S(A) which extends ϕ| B . The next proposition is probably not original (see [5, p. 267]).
Proposition 1.3. Assume p is minimal projection in A * * and ϕ is the associated pure state. If B is a C * −subalgebra of A, then ϕ| B has (UEP) if and only if p is in B * * .
Proof. Of course B * * is identified with the weak* closure of B in A * * . First assume (UEP) and let q be the support projection of ϕ| B , so that q is a minimal projection in B * * . If ψ is in S(A) and ψ(q) = 1, then ψ| B is in F (q) ∩ S(B), and hence ψ| B = ϕ| B . By (UEP), ψ = ϕ. Thus we have shown that F (q), computed in A * , is one dimensional, and this clearly implies q = p.
Conversely, assume p ∈ B * * . Since p is minimal in A * * , it is clearly minimal in B * * . Since ϕ| B (p) = 1 and ϕ| B ≤ ϕ = 1, ϕ| B is a state supported by p. Therefore ϕ| B ∈ P (B). If ψ ∈ S(A) and ψ| B = ϕ| B , then ψ and ϕ agree also on B * * . Thus ψ(p) = ϕ(p) = 1, ψ is supported by p, and hence ψ = ϕ.
Recall the condition (NCEB), which was defined in Section 0 for any projection p in A * * . It is also convenient to have a name for the special case of (NCEB) where t = 1.
(CEB) [P (A) ∩ F (p)] − ⊂ {0} ∪ P (A).
The phrase "closed extreme boundary", is accurate only when p is closed, but the main uses of (NCEB) and (CEB) are for projections known a priori to be closed.
Theorem 1.4. Let (p n ) be a sequence of mutually orthogonal minimal projections in A * * , (ϕ n ) the associated sequence of pure states, and p = ∞ 1 p n . If the equivalence classes of {ϕ n } are finite and ϕ n w * −→ 0, then the following are equivalent:
(i) ∞ k p n is closed, ∀k. (ii) p satisfies (NCEB). (iii) p satisfies (CEB). (iv) [F (p) ∩ P (A)] − ⊂ {0} ∪ [t, 1]S(A) for some t in (0, 1].
Proof. Let Γ 1 , Γ 2 , . . . be the equivalence classes of {ϕ n }, and let q i = ϕ n ∈Γ i p n . Thus each q i is a finite rank, and hence compact, projection in A * * . (iv)⇒ (i): For this it is clearly permissible to simplify the notation by assuming k = 1. Thus we need to show that p, which is i q i , is closed. According to Proposition 4.2 of [5], for this it is sufficient to show that (q i ) approaches infinity. Let U be a convex neighborhood of 0 in A * . We need to find i 0 such that
F (q i ) ⊂ U for i ≥ i 0 . By the Krein-Milman theorem, it is sufficient to show F (q i ) ∩ P (A) ⊂ U for i ≥ i 0 .
If this is false we can find nets (ψ j ) and (i j ) such that ψ j ∈ F (q i j )∩P (A), i j → ∞, ψ j → ψ, and ψ = 0.
Let π be the reduced atomic representation of A and H j the range of π * * (q i j ). Thus dim H j = |Γ j |. If dim H j = 1 for arbitrarily large j, then ψ j = ϕ n for ϕ n ∈ Γ i j ; and we already know ϕ n → 0. Thus we may assume dim H j ≥ 2, ∀j. Then we can find unit vectors
u j , v ′ j , v ′′ j in H j such that (v ′ j , v ′′ j ) = 0, the pure states (π(·)v ′ j , v ′ j ) and (π(·)v ′′ j , v ′′ j )
are in Γ i j , and ψ j = (π(·)u j , u j ). Choose a unit vector v j in span{v ′ j , v ′′ j } such that (v j , u j ) = 0, and let θ j = (π(·)v j , v j ). Then θ j → 0. This follows from Lemma 4.1 of [5], with the N of [5] being 2, or it can be proved directly using an argument similar to the one below. Let f j = (π(·)u j , v j ), which is an element of A * . By the Schwarz inequality, |f j (a)| ≤ π(a * )v j = θ j (aa * ) 1/2 . Therefore f j → 0. Then if w j = ru j +sv j , with |r| 2 +|s| 2 = 1, and ρ j = (π(·)w j , w j ), we see that ρ j ∈ F (p) ∩ P (A), and ρ j → |r| 2 ψ. We can choose r, s such that 0 < |r| 2 ψ < t, in contradiction to (iv).
(i)⇒(iii): Suppose ψ j ∈ F (p) ∩ P (A) and ψ j → ψ. Then for each j there is i j such that ψ j ∈ F (q i j ). If i j ∞, then by passing to a subnet we may assume i j = i, ∀j. Then it is easy to see that ψ ∈ F (q i ) ∩ P (A). (Each ψ j is a vector state coming from H j and the unit sphere of H j is norm compact.) If i j → ∞, then for each k, ψ j ∈ F ( ∞ k p n ) for sufficiently large j.
By (i), ψ ∈ F ( ∞ k p n ). Since ∞ k=1 ∞ k p n = 0, ψ = 0. (iii)⇒(ii)⇒(iv)= A ⊕ C if A is already unital). Let ϕ ∞ in P ( A) be defined by ϕ ∞ (λ1 e A + a) = λ. Assume B 1 is a unital C * −subalgebra of A such that ϕ ∞ | B 1 has (U EP )
, and let B = B 1 ∩A. Then B hereditarily generates A and B * * = B * * 1 ∩A * * . Proof. That B * * = B * * 1 ∩ A * * follows, for example, from general Banach space theory and the fact that B 1 /B is finite dimensional. Now let p ∞ be the support projection of ϕ ∞ . Then p ∞ ∈ B * * 1 by 1.3. Since 1 e A ∈ B * * 1 , 1 e A − p ∞ is also in B * * 1 , and of course 1 e A − p ∞ is the identity of A * * . Thus 1 e A − p ∞ ∈ B * * , and this implies that B hereditarily generates A.
Lemma 2.2. Let A be a separable unital C * −algebra, p a closed projection in A * * , and q an open projection in A * * such that q ≥ p. Let B = her(q), the hereditary C * −subalgebra of A supported by q, and let U be a neighborhood of F (p) ∩ S(A) in S(A). Then there is a closed projection p ′ in B * * such that p ′ p = 0 and ϕ(p ′ ) = 0 implies ϕ ∈ U for ϕ in S(B).
Proof. As usual, we identify B * * with qA * * q and S(B) with {ϕ ∈ S(A) : ϕ(q) = 1}. (The weak * topologies of A * and B * agree on S(B).) By Akemann's Urysohn lemma, [2, Theorem 1.1], there is a in A sa such that p ≤ a ≤ q. Then a ∈ B. Let C = her(q − p), and let e be a strictly positive element of C. Let b = a − aea. Then, by an argument of Akemann [3,
1.1], E {1} (b) = p. Let p ′ n = E (−∞,1−n −1 ] (b),
where the spectral projection is computed in B * * . We claim that for n sufficiently large the choice p ′ = p ′ n suffices. If not, for each n there is ϕ n in S(B) such that ϕ n (p ′ n ) = 0 and ϕ n / ∈ U . Then ϕ n is supported by
q − p ′ n = E (1−n −1 ,1] (b) ≤ E [1−n −1 ,1] (b)
. Let ϕ be a cluster point of (ϕ n ) in S(A). Then since each
E [1−n −1 ,1] (b) is closed in A * * , ϕ is supported by ∞ n=1 E [1−n −1 ,1] (b) = E {1} (b) = p. Therefore ϕ ∈ F (p) ∩ S(A), a contradiction since ϕ n / ∈ U .
Theorem 2.3. Let A be a separable C * −algebra and X a second countable, totally disconnected, locally compact Hausdorff space. Assume that for each x in X, p x is an atomic projection in A * * , the p x 's are mutually orthogonal, and for every closed (compact) subset S of X there is a closed (compact) projection p S such that z at p S = x∈S p x . Then there is a MASA B in A such that B hereditarily generates A and each p S is in B * * .
Proof. First we reduce to the case A unital, X compact. To do this, let A be the result of adjoining a new identity to A, and let X = X ∪ {∞} be the one point compactification. If we let p ∞ be as in 2.1, all hypotheses of the theorem are satisfied for X, A. (If S is a compact subset of X, then p S is compact in A * * and hence closed in A * * . Any other closed subset of X is S ∪ {∞} for some closed subset S of X. The fact that p S is closed in A * * implies that p S + p ∞ is closed in A * * . Since A is unital, "closed" and "compact" mean the same for projections in A * * .) If B 1 satisfies the conclusion of the theorem for A, X, then by 2.1, B 1 ∩ A satisfies the conclusions of the theorem for A, X. Thus from now on we assume A unital and X compact.
Let C be the usual middle-thirds Cantor set in [0, 1]. Then there is a one-to-one continuous function f : X → C. We will let α and β denote finite strings of +'s and −'s, and |α| denote the length of α. Let C + , C − be the right and left halves of C, C ++ , C +− the right and left halves of C + , etc. Let p α = p f −1 (C α ) , and let F α = S(A) ∩ F (p α ). Note that p α+ p α− = 0 and p α = p α+ + p α− . This follows, for example, from the theory of universally measurable elements of A * * , [25, 4.3] and the fact that the relations are satisfied by the atomic parts of the projections. Let e be a strictly positive element of her(1 − p X ). We are going to construct recursively b α in A + and an open projection q α in A * * such that:
1. b α+ b α− = 0 2. p α ≤ q α ≤ E {1} (b α ) 3. b α+ , b α− ∈ her(q α ). (Thus b α b α± = b α± .) 4. If ϕ in S(A) is supported by E {1} (b α± ), then ϕ(e) < |α| −1 2 −|α| .
Fix non-negative functions g + , g − in C([−1, 1]) such that g + = 1 on 2 3 , 1 , g + is supported on 1 3 , 1 , g − = 1 on −1, − 2 3 , and g − is supported on −1, − 1 3 . Step 1, |α| = 1. Then 3 and 4 are vacuous. Choose a in A sa such that −1 ≤ a ≤ 1,
p − ≤ E {−1} (a), and p + ≤ E {1} (a)
. This is easily accomplished by [2, Theorem 1.1] and the continuous functional calculus. Let b ± = g ± (a), q + = E ( 2 3 ,1] (a), and
q − = E [−1,− 2 3 ) (a).
Step k, |α| = k > 1. We construct b β± , q β± for each β with |β| = k − 1, assuming of course that b β , q β have already been constructed. Apply 2.2 to find a closed projection p ′ in her(q β ) * * such that p ′ p β = 0 and if ϕ in S(A) is supported by q β and ϕ(p ′ ) = 0, then ϕ(e) < |β| −1 2 −|β| . Next choose a in her(q β ) such that −1 ≤ a ≤ 1, p ′ ≤ E {0} (a), and p β± ≤ E {±1} (a). The existence of a could be deduced from [11, 3.43], but it is more elementary to apply Akemann's Urysohn lemma for her(q β ) twice to obtain a 1 and a 2 with p β+ ≤ a 1 ≤ 1 − (p ′ + p β− ) and p β− ≤ a 2 ≤ 1 − (p ′ + p β+ ). Then let a = a 1 − a 2 . Then let q β+ = E ( 2 3 ,1] (a),
q β− = E [−1,− 2 3 ) (a), and b β± = g ± (a). Now {b α } is commutative, since for α = α ′ either b α b α ′ = 0, b α b α ′ = b α ′ , or b α b α ′ = b α . Let B be any MASA containing all b α 's. If p ′ α = E {1} (b α ), then p ′ α ∈ B * * . Note that p ′ α 1 p ′ α 2 = 0 if |α 1 | = |α 2 | and α 1 = α 2 and that p ′ α ≥ p α . We show that p X ∈ B * * by proving p X = ∞ n=1 |α|=n p ′ α .
Clearly the latter is at
least p X . Suppose ϕ ∈ S(A) ∩ F ( |α|=n p ′ α ). Let ϕ α = p ′ α ϕp ′ α . Then |α|=n ϕ α = 1,
ϕ α (e) < (n − 1) −1 2 −(n−1) ϕ α , by 4, and ϕ ≤ 2 n |α|=n ϕ α . Therefore ϕ(e) < 2(n − 1) −1 . If the above is true for all n, then ϕ(e) = 0 and hence ϕ ∈ F (p X ). Finally, to show that every p S is in B * * , note that every closed subset of C is the intersection of a sequence of clopen sets and every clopen set is the union of finitely many C α 's. Thus it is sufficient to show that each p α is in B * * . We do this by showing that p α = p X ∧ p ′ α . This follows from p ′ α ≥ p α , p ′ α p β = 0 if |α| = |β| and α = β, and p X = |β|=|α| p β .
Corollary 2.4. Assume the hypotheses of 2.3 and also that each p x is a minimal projection in A * * . Let ϕ x be the pure state supported by p x . Then if B is the MASA of 2.3, ϕ x |B has the unique extension property, ∀x ∈ X.
Proof. Combine 2.3 and 1.3, and note that p x = p S for S = {x}.
Remark 2.5. Since the construction of the MASA in 2.3 requires only the p S 's, we could start with a more general, but also more abstract, setup, an assignment S → p S , for S closed, such that:
(i) p S 1 p S 2 = p S 2 p S 1 , (ii) p ∅ = 0, (iii) p S 1 ∪S 2 = p S 1 ∨ p S 2 , (iv) p ∩ ∞ 1 S n = ∞
1 p S n , and (v) p S is closed and S compact implies p S compact. Because of our assumption that X is totally disconnected, condition (i) is redundant. These conditions do not imply that z at p S = x∈S z at p {x} , and this last property is not needed to construct the MASA. It was used in the proof of 2.3 to prove conditions (iii) and (iv).
Another alternative formulation, using relative q−continuity, appears below in 7.1 (see also 7.5). The hypotheses actually used in 2.3 and 2.4 imply a stronger relationship between the structure of F (p X ) and the space X.
Relative q-continuity
Let p be a closed projection in A * * and h an element of pA * * sa p. Then h is called q-continuous on p ( [7]) if E F (h) is closed for every closed subset F of R, where the spectral projection is computed in pA * * p, and h is called strongly q-continuous on p ( [11]) if in addition, E F (h) is compact when F is closed and 0 / ∈ F . It was shown in [11, 3.43] that h is strongly q-continuous on p if and only if h = pa for some a in A sa such that pa = ap, and if A is σ-unital, then h is q-continuous on p if and only if h = px for some x in M (A) sa such that px = xp.
It was neglected in [11] to give any serious examples or discussion of how extensive is the set of relatively q-continuous elements. For general h in pA * * p let us say that h is q-continuous or strongly q-continuous on p if both Re h and Im h are. Let SQC(p) = {h ∈ pA * * p : h is strongly q-continuous on p}, and let QC(p) = {h ∈ pA * * p : h is q-continuous on p}. By [11, 3.45
], SQC(p) is a C * −algebra, and if A is σ-unital, QC(p)
is also a C * −algebra. We say that p satisfies (M SQC) (many strongly q-continuous elements) if SQC(p) is σ-weakly dense in pA * * p and p satisfies (M QC) if QC(p) is σ-weakly dense in pA * * p. The von Neumann and Kaplansky density theorems give many equivalent formulations of (M SQC), and also (M QC) if A is σ-unital. As for the other extreme, we always have Cp ⊂ QC(p) and 0 ∈ SQC(p). We will show that QC(p) and SQC(p) need not be any bigger. Of course, QC(p) = SQC(p) if and only if p is compact.
Theorem 3.1. If p is a closed projection in A * * , then the following are equivalent:
1. p satisfies (M SQC). 2. pAp = SQC(p). 3. pAp is an algebra. 4. pAp is a Jordan algebra. 5. F (p) is isomorphic to the quasi-state space of a C * −algebra.
Remarks. If F 1 and F 2 are closed faces of C * −algebras, we say they are isomorphic if there is a 0-preserving affine isomorphism which is also a (weak * ) homeomorphism. An intrinsic characterization of pAp was observed in [11] (a portion of the proof of 3.5 for which no originality was claimed): pAp is the set of continuous affine functionals vanishing at 0 on F (p). With help of [15] one can find intrinsic characterizations of QC(p) and SQC(p). One of the consequences of [7, 4.4, 4.5] is that pA * * p is the bidual of the Banach space pAp. In [14] we will give an intrinsic characterization of pM (A)p. Thus many questions concerning a closed face of a C * −algebra A can be treated intrinsically, without knowing what A is. The C * −algebra of 5 is determined only up to Jordan * −isomorphism. Then if a ′ = a−r−r * , pa ′ p = pap and a ′ p = pa ′ . Thus pap ∈ SQC(p). This shows 2, but since pAp is σ-weakly dense in pA * * p, it is obvious that 2 implies 1.
Proof. 1 ⇒ 2: Since SQC(p) ⊂ pAp and pA * * p is the bidual of pAp, SQC(p) is dense in pAp in the weak Banach space topology. Therefore SQC(p) is norm dense in pAp. But SQC(p) is norm closed (since it is a C * −algebra, for example). 2 ⇒ 3 ⇒ 4: Obvious. 4 ⇒ 1: Let a ∈ A sa . Then papap ∈ pAp. Let (e i ) i∈D be an approximate identity of her(1 − p). Then pa(1 − e i )ap → papap. By Dini's theorem for continuous functions on F (p), this convergence is uniform. Thus pa(1 − e i − p)ap → 0, (1 − e i − p) 1/2 ap → 0, and (1 − e i − p)ap → 0. It follows that (1 − p)ap ∈ Ap,
That 3 implies 5 is obvious from previous remarks and is also essentially included in the proof of [7, 4.5].
That 5 implies 2 is also obvious from previous remarks and the fact ([4, Theorem III.3]) that 2 is true when p = 1.
Theorem 3.2. Let A be a σ-unital C * −algebra, p a closed projection in A * * , and let B = SQC(p). If B is non-degenerately embedded in pA * * p, then M (B) is naturally isomorphic to QC(p).
Remarks. 1. When B = pAp (i.e., when the conditions of 3.1 hold), this result was partly proved in [7, 4.5].
2. It follows from 3.2 that if SQC(p) is non-degenerate in pA * * p and if p does not satisfy (MSQC), then p does not satisfy (MQC). This is so because M (B) ⊂ B ′′ .
Proof. Let A * * be represented on H via the universal representation of A. The non-degeneracy hypothesis means that B is non-degenerately represented on pH. Therefore M (B) is isomorphic to the idealizer of B in B(pH). It follows that if F is a closed subset of R and h is in M (B) sa then there is a hereditary C * −subalgebra B 0 of B such that any approximate identity of B 0 converges to p − E F (h), where the spectral projection is computed in B(pH). Let B = {a ∈ A : ap = pa}, and let B 0 be the inverse image of B 0 in B. If q is the limit in B(H) of an approximate identity of B 0 , then q is an open projection in A * * , qp = pq, and qp = p − E F (h).
Thus E F (h) is p ∧ (1 − q), a closed projection in A * * , and h is in QC(p). Conversely, if x ∈ QC(p) and b ∈ B, then x = px and b = pb where x ∈ M (A), xp = px, and b ∈ B. Then xb = pxb ∈ B and bx = pbx ∈ B. Thus x ∈ M (B).
Remark. The σ-unitality was used only in the second part of the proof.
Theorem 3.3. If A in 3.1 is σ-unital, then the following conditions are equivalent to 1-5 of 3.1:
6. pAp ⊂ QC(p). 7. pM (A)p = QC(p).
Proof. That 2 ⇒ 6 is obvious. 6 ⇒ 3: Let x be in pAp and let a be in
A. Write x = px where x ∈ M (A) and xp = px. Then xpap = pxpap = p 2 xap ∈ pAp.
That 7 implies 6 is obvious. 2 ⇒ 7: Clearly we have the non-degeneracy required for 3.2. Let x be in pAp and let y be in M (A). Write x = px where x is in A and px = xp. Then xpyp = p(xy)p ∈ pAp, and pypx = p(yx)p ∈ pAp. Thus, in the notation of 3.2, pyp ∈ M (B), and hence pyp ∈ QC(p).
Example 3.4. In this example p is closed, infinite rank, abelian, and atomic, and pA * p is norm separable. Also SQC(p) = {0} but p satisfies (M QC). In particular, p is a counterexample for the question raised in Section 0 about isolated points. In fact, if p 0 is a minimal projection, p 0 ≤ p, and p − p 0 is closed, then obviously p 0 ∈ SQC(p).
Let
A = C([0, 1]) ⊗K.
Here K is the algebra of compact operators on a separable infinite dimensional Hilbert space H, {e 1 , e 2 , . . . } is an orthonormal basis of H, and P n is the projection on span{e 1 , . . . , e n }. A criterion for weak semicontinuity from [11, §5.G] will be used to describe closed projections in A * * . A closed projection is given by a projection-valued function P : [0, 1] → B(H) such that if h is any weak cluster point of P (y) as y → x, then h ≤ P (x). More precisely, P describes the atomic part of a closed projection p, and P determines p since a closed projection is determined by its atomic part. (In our case p will equal its atomic part.) We will construct a countable subset S of [0, 1] and unit vectors v(x) for each x in S. For x in S, P (x) is the rank one projection on Cv(x), and for x not in S, P (x) = 0.
The following trivial lemma is stated for purposes of reference:
3.4.1. Let {x i } be a sequence of distinct points in [0, 1] and let D be a countable subset of [0, 1]. Then there are distinct points y ij in [0, 1] \ ({x i } ∪ D) such that |y ij − x i | ≤ 2 −(i+j) .
We will take S = ∞ 0 S n , a disjoint union, where S n and v| S n will be constructed recursively so that P n v(x) ≤ n − 1 2 for x in S n .
Step 0:
Take S 0 = { 1 2 }, v( 1 2 ) = e 1 .
Step 1: Take
S 1 = {x i } where the x i 's are distinct, x i = 1 2 , and x i → 1 2 as i → ∞. Let v(x i ) = 2 − 1 2 e 1 + 2 − 1 2 e i+1 for i = 1, 2, . . . . . . .
Step n (n > 1, step n − 1 already completed):
Write S n−1 = {x 1 , x 2 , . . . }. Choose y ij 's as in 3.4.1 with D = ∪ n−2 0 S k . Let S n = {y ij : i, j = 1, 2, . . . } and v(y ij ) = n − 1 2 v(x i ) + (1 − n −1 ) 1 2 w ij , where w ij is a unit vector such that (w ij , v(x i )) = 0 and P i+j+n w ij = 0.
The first step in the proof is to show that we get a closed projection. Thus we may assume given a sequence (t r ) in [0, 1] such that t r → t and P (t r ) w −→ h. We must show h ≤ P (t). We have no difficulty if P (t r ) = 0. Thus we may assume, after passing to a subsequence, that t r ∈ S n(r) . If n(r) → ∞, then since P n(r) P (t r )P n(r) ≤ n(r) −1 , we must have h = 0. Thus, after again passing to a subsequence, we may assume n(r) = n, ∀r. Now it is easy to see by induction that n 0 S k is closed. In fact, every cluster point of S n is in S n−1 = n−1 0 S k . The proof that h ≤ P (t) will be left to the reader in the cases n = 0, n = 1. If n > 1, write t r = y i(r)j(r) in the notation of step n. If i(r) + j(r) ∞, we may assume, after passing to a subsequence, that t r = t, ∀r, a trivial case. If i(r) + j(r) → ∞, then t ∈ S m for some m < n. We use induction on n − m. First suppose i(r) ∞. Then, passing to a subsequence, we assume i(r) = i, ∀r. Then t = x i , and the construction shows that
h = n −1 P (x i ). If i(r) → ∞, let t ′ r = x i(r) . Then t ′ r → t, and [v(t r ) − n − 1 2 v(t ′ r )] w −→ 0. This shows that h = n −1 h ′ , where P (t ′ r ) w −→ h ′ . Since h ′ ≤ P (t) by induction, we conclude that h ≤ P (t), as desired.
Now since A is separable, every state in F (p) is the resultant of a probability measure on F (p) ∩ P (A). Since F (p) ∩ P (A) is countable, the integral is a Bochner integral and thus the resultant is an atomic state. This shows that p is atomic, as claimed. Also, pA * p is norm separable, being isometrically isomorphic to ℓ 1 (S). That p is abelian, in other words pA * * p is abelian, is now obvious (cf [10]).
Now if h is in pA * * p, h is determined by a function λ in ℓ ∞ (S), where h(x) = λ(x)P (x), x ∈ S. If h is in SQC(p), then h = ph, where h ∈ A and ph = hp. In particular, h(·) is a norm continuous function from [0, 1] to K. If x ∈ S, there is a sequence (x n ) in S such that x n → x and P (x n ) w −→ tP (x), where 0 < t < 1. Since P (·)h(·)P (·) = λ(·)P (·), we conclude that λ(x n ) → tλ(x). Since also h 2 ∈ SQC(p), we also have λ(x n ) 2 → tλ(x) 2 . This implies λ(x) = 0. Since x is arbitrary, h = 0. (The only property of h actually used is that h, h 2 ∈ pAp.)
Finally we note that any continuous function on [0, 1] gives rise to an element h of the center of M (A). Thus ph ∈ QC(p). It is easy to see that such elements of QC(p) generate pA * * p as a W * −algebra, and hence p satisfies (M QC).
Example 3.5. By modifying the previous example, we can obtain either of the following:
(a) a compact projectionp such that QC(p) = Cp (b) a closed projection p 1 such that SQC(p 1 ) = {0} and QC(p 1 ) = Cp 1 . In both cases we will still have p infinite rank, abelian, and atomic and pA * p norm separable, and of course A will still be separable.
(a) Let A and p be as in 3.4, and consider A andp = p + p ∞ . A * * is identified with A * * ⊕ C and p ∞ has its usual meaning, so that p ∞ = 0 ⊕ 1 in A * * ⊕ C and p = p ⊕ 1. Thenp is closed, and hence compact, in A * * . Suppose x = λ1 e A + a, λ ∈ C, a ∈ A, and xp =px. Then ap = pa, and hence by 3.4, ap = 0. It follows easily thatpx = λp. Therefore QC(p) = Cp.
(b) We will use a C * −algebra A 1 which is a maximal hereditary C * −subalgebra of A. Let p 0 be the minimal projection in A * * (actually in A * * ) corresponding to the projection P ( 1 2 ) in the notation of 3.4 (p 0 corresponds to the pure state ϕ 0 where ϕ 0 (a) = (a ( 1 2 )e 1 , e 1 )).
Then p 0 ≤ p ≤p. Let A 1 = her(1 − p 0 ) and p 1 =p − p 0 in A * * 1 . Then A * * 1 is identified with (1 − p 0 ) A * * (1 − p 0 ). It is easy to see that p 1 is closed in A * * 1 : The complementary projection to p 1 in A * * 1 is 1 − p 0 − p 1 = 1 −p,
and this supports a hereditary C * −subalgebra of A which happens to be contained in A 1 also. If x is in A 1 and xp 1 = p 1 x, then x is also in A and xp =px. Thus by (a),px = λp and hence
p 1 x = λp 1 . But x ∈ A 1 implies xp 0 = p 0 x = 0. Sincẽ px = λp implies p 0 x = λp 0 , λ = 0. Therefore SQC(p 1 ) = {0}.
Now A 1 can be regarded as the set of all norm continuous functions f : (1 e K − P ( 1 2 ))g(t)(1 e K − P ( 1 2 )) exists in norm.
[0, 1] → K such that f ( 1 2 )P ( 1 2 ) = P ( 1 2 )f ( 1 2 ) = 0 and the image of f in K/K is constant. Since
(iii) lim
t→ 1 2 P ( 1 2 )g(t)(1 e K − P ( 1 2 )) = lim t→ 1 2 (1 e K − P ( 1 2 ))g(t)P ( 1 2 ) = 0. (iv) If we write g(t) = λ(t)1 e K + x(t), λ(t) ∈ C, x(t) ∈ K, then λ(·)
is a constant. To see these, the main thing to note is that the constant function 1 e
K − P ( 1 2 ) is in A 1 .
Now assume g, as above, commutes with p 1 . Then x(t) commutes with P (t) for all t in S \ { 1 2 }, in the notation of 3.4. Just as in 3.4, this implies
P (t)x(t) = 0 for t in S \ { 1 2 }; i.e., p 1 x = 0 and p 1 g = λp 1 . Thus QC(p 1 ) = Cp 1 .
Example 3.6. Here we show, by a simpler example, how badly Theorem 3.2 can fail when the non-degeneracy hypothesis is eliminated. By [7, Theorem 2.7], if B is a non-unital separable C * −algebra, then M (B) is non-separable. Thus if A is separable and SQC(p) is non-unital (in particular non-trivial), and if the conclusion of 3.2 is true, then QC(p) is much larger than SQC(p). In this example, SQC(p) is (infinite dimensional and) non-unital and QC(p) = SQC(p) + Cp. Let A = c ⊗ K. Thus A * * can be identified with the set of bounded collections {h n : 1 ≤ n ≤ ∞}, h n ∈ B(H). Let v n = 2 − 1 2 e 1 +2 − 1 2 e n+1 , n < ∞, v ∞ = e 1 , let p n be the projection with range Cv n , and let p = {p n } in A * * . Then p is closed since p n w −→ 1 2 p ∞ , and clearly p is abelian. Any element of pA * * p is given by h n = λ n p n , 1 ≤ n ≤ ∞, {λ n } bounded. An easy argument, which is part of 3.4, shows that if h ∈ SQC(p) then λ ∞ = 0 and λ n → 0 as n → ∞. Conversely, any such h is in SQC(p); in fact h ∈ A ∩ pA * * p. Thus SQC(p) ∼ = c 0 . Next we show that h ∈ QC(p) implies λ n → λ ∞ . If this is false for h = h * , then there is a closed subset F of R such that λ ∞ / ∈ F and λ n ∈ F for infinitely many n. If q = E F (h), then q ∞ = 0 and q n = p n for infinitely many n. Since p n → 1 2 p ∞ = 0, q is not closed and h is not q-continuous on p. Thus QC(p) ∼ = c and QC(p)/SQC(p) is one dimensional.
Closed faces with (NCEB).
If A is the spectrum of A and p is a projection in A * * , we will denote by X the set of all [π] in A such that π * * (p) = 0. For [π] in X let p [π] be the atomic projection in A * * corresponding to π * * (p). Thus z at p = x∈X p x . If p is closed, or even universally measurable, then p is determined by the p x 's. If ϕ and ψ are in (0, ∞)P (A), we will say that ϕ and ψ are equivalent, and write ϕ ∼ ψ, if the pure states ϕ ϕ and ψ ψ are equivalent.
The proof of the next theorem and some of the other geometric arguments in this paper were inspired by Glimm [16].
Theorem 4.1. If p is a projection in A * * and if p satisfies (NCEB), then p x is finite rank, ∀x ∈ X.
Proof. Let π be an irreducible representation belonging to x, and let H x be the range of π * * (p). If the conclusion is false, there is an infinite orthonormal sequence,
{e 1 , e 2 , . . . }, in H x . Choose t > 0 such that [P (A) ∩ F (p)] − ⊂ [t, 1]P (A) ∪ {0}
and choose s such that 0 < s < t. Let v n = s 1/2 e i +(1−s) 1/2 e n , n > 2, where i is 1 or 2. Define ϕ n , ψ n in P (A) ∩ F (p) by ϕ n (a) = (π(a)v n , v n ), ψ n (a) = (π(a)e n , e n ). Let θ be any cluster point of (ψ n ) in Q(A). Since e n w −→ 0, (π(a)e i , e n ) → 0, ∀a ∈ A. Therefore sψ i + (1 − s)θ is a cluster point of (ϕ n ). If θ = 0, we have a contradiction to (NCEB), since 0 < s < t. Therefore θ ∈ [t, 1]P (A). Since we must also have sψ i + (1 − s)θ ∈ [t, 1]P (A), it follows that θ = r i ψ i for some r i ≥ t > 0. We have shown that θ = r 1 ψ 1 and θ = r 2 ψ 2 , a contradiction.
For the rest of this section we assume that p is closed and satisfies (NCEB). Let
X = [P (A) ∩ F (p)] − \ {0}. Then X ⊂ F (p) ∩ [t, 1]P (A) and X is locally compact, since X ∪ {0} is closed. We identify X with the set of equivalence classes in X via f : X → X, where f (ϕ) = [π ϕ ]
. Give X the quotient topology arising from f . Proof. The main point is to show the following: If (ϕ i ) i∈D and (ψ i ) i∈D are nets in X such that ϕ i ∼ ψ i , ϕ i → ϕ, and ψ i → ψ, then either ϕ = ψ = 0 or ϕ, ψ ∈ X and ϕ ∼ ψ. Assume this is false and consider first the case ϕ = 0, ψ ∈ X. Let π be the reduced atomic representation of A, H = H π , and choose vectors u i , v i in π * * (p)H of norm at most 1 such that ϕ i = (π(·)u i , u i ), ψ i = (π(·)v i , v i ). If g i (a) = (π(a)u i , v i ), then |g i (a)| ≤ π(a)u i = ϕ i (a * a) 1/2 → 0. Therefore
g i → 0. Now choose r i in R such that w i = 1, where w i = r i u i + ( t 2 ) 1/2 v i . Since u i 2 ≥ t, {r i } is bounded. Let θ i = (π(·)w i , w i ).
It follows from the above that θ i ∈ F (p) ∩ P (A) and θ i → t 2 ψ. Since 0 < t 2 ψ < t, this contradicts (NCEB). Next assume ϕ, ψ ∈ X and ϕ ∼ ψ. Then there are invariant subspaces H 1 and H 2 of H, corresponding to inequivalent irreducible representations, and non-zero vectors u in H 1 , v in H 2 such that ϕ = (π(·)u, u) and ψ = (π(·)v, v). Let u i , v i , and g i be as above with the extra condition that Re(u i , v i ) ≥ 0. Passing to a subnet, we may assume g i → g, g ∈ A * . Since |g i (a)| ≤ ϕ i (a * a) 1/2 , ∀a ∈ A, then |g(a)| ≤ ϕ(a * a) 1/2 . From the Hahn-Banach and Riesz-Fisher theorems we see that g = (π(·)u, u ′ ) for some u ′ in H. Clearly, we may assume u ′ ∈ H 1 . Similarly, |g i (a)| ≤ ψ i (aa * ) 1/2 , and hence |g(a)| ≤ ψ(aa * ) 1/2 . Therefore g = (π(·)v ′ , v) for some v ′ in H 2 . It follows that g = 0 ( [20]). Now choose r i in R + such that
w i = 1, where w i = r i (u i + v i ). Since 2t ≤ u i + v i 2 ≤ 4
, {r i } is bounded and bounded away from 0. If θ i = (π(·)w i , w i ), then θ i ∈ F (p) ∩ P (A) and every cluster point of (θ i ) is of the form r 2 (ϕ + ψ) for some cluster point r of (r i ). Since this last functional is not a multiple of a pure state, this contradicts (NCEB).
To complete the proof of the lemma, we have to show that the saturation of a closed set is closed. Suppose Y is a closed subset of X,
ϕ i ∈ f −1 (f (Y )), and ϕ i → ϕ in X. Choose ψ i in Y such that ϕ i ∼ ψ i .
Passing to a subnet if necessary, we may assume ψ i → ψ. By what has already been proved ψ ∈ X and ψ ∼ ϕ. Since Y is closed, ψ ∈ Y and hence ϕ ∈ f −1 (f (Y )). Thus f −1 (f (Y )) is closed (relative to X).
Theorem 4.3. X is a locally compact Hausdorff space and f is a proper map from X to X.
Proof. The fibers of f , i.e., the sets f −1 ({x}), x ∈ X, are compact (even norm compact) by 4.1. This, 4.2, and the fact that X is locally compact Hausdorff imply that X is locally compact Hausdorff, by standard point set topology. Any closed map with compact fibers is proper; i.e., the inverse image of a compact set is compact.
Remarks. 1. It follows from 4.3, or it could be deduced directly from the proof of 4.2, that the saturation of a compact subset of X is compact.
2. The topology of X is stronger than, and in general unequal to, the relative topology that X inherits from the usual hull-kernel topology of A. In fact, using [5] and 1.4, we can easily construct a closed projection satisfying (NCEB) and even (CEB) such that X is a countably infinite discrete space and the image of X in prim A consists of one point. Thus the relative topology is trivial on X.
Lemma 4.4. Assume p is an atomic closed projection satisfying (NCEB) and that pA * p is norm separable. Then for every closed subset S of X, x∈S p x is a closed projection.
Proof. Since pA * p has a linear subspace isometric to ℓ 1 (X), X must be countable. countably many fibers of f , since each of these fibers is contained in F (p x ) for some x in S, and since each p x is finite rank and hence closed, it is easy to see that any such resultant is in F (p S ). Thus F (p S ) is closed and hence p S is closed.
Corollary 4.5. Under the same assumptions, if p = 0, there is a minimal projection p 0 such that p 0 ≤ p and p − p 0 is closed. Also for every non-zero closed subprojection p ′ of p, there is a minimal projection p 0 such that p 0 ≤ p ′ and p ′ − p 0 is closed.
Proof. Since X is countable and locally compact Hausdorff, the Baire category theorem implies that X has an isolated point x 0 . Let p 0 be any minimal subprojection of p x 0 . Then p − p 0 = (p x 0 − p 0 ) + p X\{x 0 } , the sum of two orthogonal closed projections. Therefore p − p 0 is closed ([1, Theorem II.7]).
Remarks. 1. In Section 6 we will generalize 4.4 and 4.5 by dropping the requirement that pA * p be norm separable, but we will add the assumption that A is separable. We are not sure what technical assumptions are really needed.
2. Corollary 4.5 and Examples 3.4 and 3.5(a) constitute our results on the "isolated point" question raised in Section 0. The second sentence of 4.5 is closely analogous to the definition of a scattered topological space and less closely analogous to the definition of scattered C * −algebras. Obviously we have not found a necessary and sufficient condition for this to hold. . Any closed face of a a scattered C * −algebra satisfies the conclusion of 4.5 but not necessarily the hypothesis. Example 5.12 below, whose primary purpose is something else, is a closed face satisfying the conclusion of 4.5 (the proof of this is in 7.9), but not (NCEB), and which is not isomorphic to a closed face of any scattered C * −algebra.
We now consider the geometry of F (p) in more detail.
Theorem 4.6. If p is a closed projection satisfying (NCEB) and if (x i ) i∈D is a net in X converging to x, then there is a subnet (x j ) j∈D such that one of the following holds:
1. We have rank p x j = k ≤ n = rank p x , ∀j; and there are orthonormal bases {e j 1 , . . . , e j k } of range π * * j (p x j ) and {e 1 , . . . , e n } of range π * * (p x ) and an n×k matrix T such that tI k ≤ T * T ≤ I k and ∀z ∈ C k , ϕ j (z) → ϕ(w), where π j and π are irreducible representations belonging to x j and x, v j = k 1 z m e j m , v = n 1 w m e m , w = T z, ϕ j (z) = (π j (·)v j , v j ), and ϕ(w) = (π(·)v, v).
2. There is ϕ in P (A) ∩ F (p x ) such that every cluster point of (ϕ j ) is a multiple of ϕ, ∀ϕ j ∈ P (A) ∩ F (p x j ).
Proof. If rank p x i ∞, we first pick a subnet such that rank p x j = k, ∀j. If rank p x i → ∞, we must show there is a subnet satisfying 2; and we do this by contradiction. Thus assume there are a subnet (x j ) and pure states θ j , ψ j in F (p x j ) such that (θ j ) and (ψ j ) converge to non-proportional elements of F (p x ). In the first case choose an arbitrary orthonormal basis {e j 1 , . . . , e j k } of range π * * (p x j ). In the second case let k = n+1 and choose an orthonormal set {e j 1 , . . . , e j k } in range π * (p x j ) such that θ j = (π j (·)v j , v j ) and ψ j = (π j (·)v ′ j , v ′ j ) with v j , v ′ j unit vectors in span {e j 1 , . . . , e j k }. In both cases define f j ℓm in A * by f j ℓm = (π j (·)e j m , e j ℓ ), 1 ≤ ℓ, m ≤ k. Passing to a subnet, we may assume f j ℓm → f ℓm , ∀ℓ, m. Since the matrix [f j ℓm ] represents a positive linear functional on A ⊗ M k , the same must be true of the matrix [f ℓm ]. The GNS representation of A ⊗ M k induced by [f ℓm ] must be of the formπ ⊗ id for some respresentationπ of A, and [f ℓm ] must be the vector state induced by a vector (u 1 , . . . , u k ) in Hπ ⊕ · · · ⊕ Hπ. In other words, f ℓm = (π(·)u m , u ℓ ). Since f ℓℓ ∈ [t, 1][P (A) ∩ F (p x )],π ∼ = π ⊕ · · · ⊕ π. Thus we may write u ℓ = (u ℓ1 , . . . , u ℓr ), r ≤ k, where u ℓp ∈ range π * * (p x ). Now f ℓℓ = r 1 (π(·)u ℓp , u ℓp ). Since f ℓℓ ∈ [t, 1]P (A), there must be a non-zero vector y ℓ in range π * * (p x ) such that u ℓp = λ ℓp y ℓ with (λ ℓ· ) a non-zero element of C r . If z ∈ C k and ϕ j (z) is as above, then ϕ j (z) = z ℓ f j ℓm z m and hence ϕ j (z) → z ℓ f ℓm z m = (π(·) z ℓ u ℓ , z ℓ u ℓ ). Since this functional is a multiple of a pure state, the vectors z ℓ u ℓp , 1 ≤ p ≤ r, must be proportional. Suppose, for example, that y 1 and y 2 are linearly independent. Then the choice z = (1, 1, 0, . . . , 0) shows that (λ 1· ) and (λ 2· ) are proportional. For ℓ > 2, y ℓ cannot be a multiple of both y 1 and y 2 . Therefore all (λ ℓ· ) are proportional. Changing notation, we may write u ℓp = λ p y ℓ . Then ϕ j (z) → ( r 1 |λ p | 2 )(π(·) z ℓ y ℓ , z ℓ y ℓ ). Now choose any orthonormal basis of range π * * (p x ) and let T be the matrix of z → ( r 1 |λ p | 2 ) 1/2 z ℓ y ℓ . Since t z 2 2 ≤ lim ϕ j (z) ≤ z 2 2 , we must have tI k ≤ T * T ≤ I k . This implies k ≤ n so that 1 is proved. The other alternative is that span{y ℓ } is one dimensional. Then let ϕ ′ = (π(·)y 1 , y 1 ) and ϕ = ϕ ′ ϕ ′ . Since each f ℓm is proportional to ϕ, (ϕ j (z)) converges to a multiple of ϕ, ∀z ∈ C k , and more generally every cluster point of (ϕ j (z j )) is a multiple of ϕ for any bounded net (z j ) in C k . If k = rank p x j , this proves 2. In the original second case, rank p x j → ∞, k = n + 1, this proves the contradiction that establishes 2.
We say that a C * −algebra A satisfies (CEB) or (NCEB) if the closed projection 1 in A * * satifies (CEB) or (NCEB). In [17, §5] Glimm proved a necessary and sufficient condition for A to satisfy a property weaker than (NCEB), P (A) ⊂ [0, 1]P (A). His condition is:
(i) A is CCR, (ii) A is Hausdorff, and (iii) [π] ∈ A and dim π > 1 implies [π] is regular. Given (i) and (ii), (iii) can be restated as follows: If I is the ideal of A such that I = {[π] ∈ A : dim π > 1}, then I is a continuous trace C * −algebra. (See [27] for the theory of continuous trace C * −algebras.) It is presumably an easy exercise to derive a characterization of C * −algebras satisfying (CEB) or (NCEB) (they are equivalent for C * −algebras) from Glimm's result. In Corollary 4.7 below we derive such a characterization instead from 4.1-4.6. The purpose is not to put this result on the record, so long after [17]. The purpose is as follows: The class of closed faces of C * −algebras admits more varied behavior than the class of C * −algebras. One illustration of this is the contrast between the facts on the isolated point question for atomic closed faces of C * −algebras and the facts on scattered C * −algebras ( [18], [19]). Another illustration is the contrast between 4.6 and 4.7. (We will show by example that all of the behavior contemplated by 4.6 really occurs.) The exercise of deriving 4.7 from 4.1 to 4.6 gives some insight into why the behavior of closed faces is more varied than that of C * −algebras.
If A is a CCR C * −algebra with Hausdorff spectrum, then A is isomorphic to the set of continuous sections vanishing at ∞ of a continuous field, A(x), x ∈ A, of elementary C * −algebras. If x 0 ∈ A and A(x 0 ) is one dimensional, then there is a continuous section e(·) such that e(x 0 ) = 1 A(x 0 ) and e(x) is a projection for x in some neighborhood of x 0 ([17]). We will say A is locally unital at x 0 if e(x) = 1 A(x) in some neighborhood of x 0 .
Corollary 4.7. The following are equivalent for a C * −algebra A:
A satisfies (CEB) 2. A satisfies (NCEB) 3. (i) Every irreducible representation of
A is finite dimensional, (ii) A is Hausdorff, (iii) ∀n > 1, {[π] : dim π = n} is
an open subset of A, and (iv) A is locally unital at each [π]
with dim π = 1.
Remark. Condition 3(iii) says that the ideal I discussed above is the c 0 direct sum of n-homogeneous C * −algebras for various values of n. Thus the comparison of 3 with Glimm's condition is clear.
Proof. 2 ⇒ 3: (i) follows from 4.1 with p = 1. Since p = 1, X = A. Since the map from P (A) to A is open for the hull-kernel topology ([17]), the hull-kernel topology is the quotient topology; i.e., our topology on X agrees with the usual one when p = 1. Thus (ii) follows from 4.3. Again since the map from P (A) to A is open, if dim π > 1 and [π i ] → [π], then after passing to a subnet, we can find ϕ i , ψ i in P (A) associated to π i such that the nets (ϕ i ), (ψ i ) converge to distinct pure states associated to π. Thus alternative 2 of 4.6 cannot hold, and lim sup(dim π i ) ≤ dim π. It is always true in a C * −algebra that lim inf(dim π i ) ≥ dim π (but for a closed face we can have lim inf(rank p x i ) < rank p x ). This shows (iii). If x 0 , e(·) are as above and A is not locally unital at x 0 , then we can find (x i ) such that x i → x 0 and e(x i ) = 1, ∀i. Then we can find ϕ i in P (A) associated to x i such that ϕ i (e) = t 2 . It follows that ϕ = t 2 for any cluster point ϕ of (ϕ i ), in contradiction to (NCEB). This proves (iv).
That 1 implies 2 is obvious, and the proof that 3 implies 1 is left to the reader.
Examples 4.8. (a) We can illustrate alternative 1 of 4.6 with A = c ⊗ K. Choose k and n with k ≤ n, t > 0, and an n × k matrix T such that tI k ≤ T * T ≤ I k . Let S = (1 − T * T ) 1/2 , a k × k matrix. Let p ∞ be the projection on span{e 1 , . . . , e n } and for j < ∞ let p j be the range projection of T S , where the matrix is regarded as a linear isometry from C k to span{e 1 , . . . , e n , e n+j , . . . , e n+j+k−1 }. If p = {p j : 1 ≤ j ≤ ∞}, then p is a closed projection in A * * , p satisfies (NCEB) ((CEB) if t = 1) and 4.6.1 holds with the given matrix T . (Here we think of x j as j and x as ∞, and {e j 1 , . . . , e j k } corresponds to the columns of T S .) If we want a more complicated example, say one where two different subsequences give two different matrices, we can easily modify the above. Choose k ′ ≤ n and an n × k ′ matrix T ′ such that tI k ′ ≤ T ′ * T ′ ≤ I k ′ . Letp 2j−1 be the above p j , and let p 2j be the above p j constructed from T ′ instead of T .
(b) As a first example for alternative 2 of 4.6, consider A 1 = {(a n ) ∞ 1 : a n ∈ K and (a n ) converges to a scalar in norm}. Then A * * 1 can be identified with the set of bounded collections {h n : 1 ≤ n ≤ ∞} such that h n ∈ B(H) ⊕ C for n < ∞ and h ∞ ∈ C. Choose any sequence (n j ) of positive integers and define a closed projection p in A * * 1 by: p = {p j }, p ∞ = 1 e K , and p j is a rank n j projection in B(H) for j < ∞. It is easy to see that p satisfies (CEB) and 4.6.2. This easy example shows that there is no restriction on rank p x j when 4.6.2 holds, but this is all that it shows.
(c) For more complicated examples, in particular examples where some subsequences satisfy 4.6.1 and others 4.6.2, we can use
A 2 = A 1 ⊗ K. Then A * * 2 ∼ = A * * 1 ⊗B(H).
The construction in (a) above can also be used for A 2 . Letp ∞ = 1 ⊗ p ∞ and p j = q 0 ⊗ p j for j < ∞, where the p j 's are as in (a) and q 0 is a rank 1 projection in the B(H)-component of K * * . It is easy to see thatp is closed in A * * 2 and that F (p) is isomorphic to the closed face F (p) of (c ⊗ K) * * .
We can also construct examples of 4.6.2 using A 2 . Let T be a positive k × k matrix such that tI k ≤ T 2 ≤ I k , and let u be a unit vector in span{e 1 , . . . , e n } where k and n are arbitrary. Let S = (1 − T 2 ) 1/2 and define a closed projectioñ p in A * * 2 by:p = {p j : 1 ≤ j ≤ ∞},p ∞ = 1 ⊗ p ∞ for p ∞ the projection on span{e 1 , . . . , e n }, andp j is the range projection of T S where now T S sends C k to span{e 1 ⊗ u, e 2 ⊗ u, . . . , e k ⊗ u, e 1 ⊗ e n+j , . . . , e 1 ⊗ e n+j+k−1 }.
Then 4.6.2 holds with ϕ given by ϕ(a) = (a ∞ u, u). Also the columns of T S give an orthonormal basis {e j 1 , . . . , e j k } of rangep j , and, using the notation of 4.6.1, ϕ j (z) → T z 2 ϕ. It is easy to see thatp satisfies (NCEB).
By using the idea of the second paragraph of (a), we can construct a closed projection such that different subsequences exhibit different behavior. Some subsequences can satisfy 4.6.1, with different choices of T and k, and some can satisfy 4.6.2 with ϕ j (z) → T z 2 ϕ for different choices of T , k, and ϕ.
Remark. In 4.6.2 we showed only that every cluster point of (ϕ j ) is a multiple of ϕ and did not describe which multiples arise. When rank p x j is bounded, the same methods can easily be used to construct a subnet and a positive k × k matrix T such that tI k ≤ T 2 ≤ I k and ϕ j (z) → T z 2 ϕ.
Type I Closed Faces and Atomic Closed Faces
If p is a projection in A * * , we say that p or F (p) is type I if pA * * p is a type I von Neumann algebra. Clearly p is type I if and only if c(p), the central cover of p, is type I. Now F (p) is the normal quasi-state space of pA * * p, and for ϕ in F (p) the kernel of π ϕ contains (1 − c(p))A * * . Therefore p is type I if and only if π ϕ is a type I representation for all ϕ in F (p). (It doesn't matter whether we look at π ϕ or π * * ϕ .) Because z at A * * is a type I W * −algebra, every atomic projection is type I. However, if we also require that p be closed, or just universally measurable (say), it seems that the property of being type I may be useful.
Lemma 5.1. Let A be a separable C * −algebra and p a type I closed projection in A * * . Let µ be a probability measure on F (p), and let π = ⊕ π ω dµ(ω), the direct integal. Then π is a type I representation.
Proof. Let ϕ = ωdµ(ω), the resultant of µ. Then ϕ ∈ F (p), since F (p) is closed. Therefore π ϕ is type I, and π ϕ is a subrepresentation of π. We claim π and π ϕ have the same central support in A * * (i.e. π is quasi-equivalent to π ϕ ). Therefore π is also type I. To see the claimed quasi-equivalence, let v ω be the cyclic vector in H ω produced by the GNS construction, and let v = ⊕ v ω dµ(ω), a vector in H π . Then (π(a)v, v) = ϕ(a). For every µ-measurable subset S of F (p) (µ is a Borel measure, and "µ-measurable" means measurable with respect to the completion of µ) there is a projection P S in π(A) ′ such that the corresponding subrepresentation of π is ⊕ S π ω dµ(ω). It is easy to see that H π is the smallest closed invariant subspace containing P S v for all such S. Moreover the cyclic subrepresentation of π generated by P S v is equivalent to a subrepresentation of π ϕ . These remarks complete the proof.
The main fact needed from direct integral theory is something that the author learned from G. W. Mackey and is expressed as a lemma. For the ideas in the proof see [23], pages 112-117, and [24], especially page 159. The basic point is that the direct integral decomposition into irreducibles of a type I representation is almost unique.
Lemma 5.2 (Mackey). Let A be a separable C * −algebra, let π ′ · and π ′′ · be measurable fields of irreducible representations of A defined over standard measure spaces S ′ and S ′′ , and let
π ′ = ⊕ S ′ π ′ s dµ ′ (s), π ′′ = ⊕ S ′′ π ′′ s dµ ′′ (s). Assume that π ′ s ′ is in- equivalent to π ′′
s ′′ , ∀s ′ ∈ S ′ , ∀s ′′ ∈ S ′′ and that π ′ and π ′′ are type I representations. Then π ′ and π ′′ are disjoint (i.e., their central supports in A * * are orthogonal). Lemma 5.3. Let A be a separable C * −algebra and p a type I closed projection in A * * . Assume µ and ν are positive finite measures on F (p) ∩ P (A) such that ωdµ(ω) = ωdν(ω). Let E be a saturated Borel subset (or, more generally, a saturated (µ + ν)-measurable subset) of F (p) ∩ P (A). Then E ωdµ(ω) = E ωdν(ω) and in particular µ(E) = ν(E).
Proof. Let ϕ = ωdµ(ω) = ωdν(ω), π ′ = ⊕ π ω dµ(ω), and π ′′ = ⊕ π ω dν(ω).
As in the proof of 5.1, there are vectors v ′ in H π ′ and v ′′ in H π ′′ which induce the functional ϕ. Thus there is a partial isometry U which intertwines π ′ and π ′′ such that v ′ is in the initial space of U and U v ′ = v ′′ . Let P ′ E and P ′′ E be the projections in π ′ (A) ′ and π ′′ (A) ′ defined from E. Thus
µ(E) = (P ′ E v ′ , v ′ ) and ν(E) = (P ′′ E v ′′ , v ′′ ). By 5.2 and 5.1, (1−P ′′ E )U P ′ E = P ′′ E U (1− P ′ E ) = 0. Therefore P ′ E v ′ is in the initial space of U and U P ′ E v ′ = P ′′ E v ′′ .
The conclusion follows. Lemma 5.4. Let A be a separable C * −algebra, p a type I closed projection in A * * , and E a saturated Borel subset of F (p) ∩ P (A). Then there is a projection p E in A * * such that p E ≤ p and F (p E ) is the set of resultants of probability measures on E ∪ {0}.
Proof. Let
tϕ 1 + (1 − t)ϕ 2 ∈ F 1 , 0 < t < 1. By 5.3, tµ 1 (E ′ ) + (1 − t)µ 2 (E ′ ) = 0. Therefore µ 1 (E ′ ) = µ 2 (E ′ ) = 0, and ϕ 1 , ϕ 2 ∈ F 1 . Thus F 1 is a face.
To see that F 1 is norm closed, assume ϕ = ωdµ(ω) where µ(E ′ ) = δ > 0. We claim that dist(ϕ, F 1 ) ≥ δ. Suppose ψ = ωdν(ω) where ν(E ′ ) = 0 and ϕ − ψ = r. Then ϕ − ψ = λ 1 − λ 2 where λ 1 , λ 2 ≥ 0 and λ 1 + λ 2 = r. If λ i = ωdµ i (ω), for positive measures µ 1 , µ 2 on F (p) ∩ P (A), then µ + µ 2 and ν + µ 1 have the same resultant. Therefore by 5.3,
µ(E ′ ) + µ 2 (E ′ ) = ν(E ′ ) + µ 1 (E ′ ). Therefore µ(E ′ ) ≤ µ 1 (E ′ ) ≤ r.
This proves the claim and completes the proof of the lemma.
Remarks. Although the conclusion of 5.4 has what we need, more is true. Also F (p E ) ∩ S(A) is a split face of F (p) ∩ S(A), the complement being F (p E ′ ) ∩ S(A). This means that p E and p E ′ are centrally disjoint projections and p E + p E ′ = p. Also p E satisfies the barycenter formula. (The barycenter formula is discussed below before 5.13). A related statement is that F (p E ) is closed under resultants. The hypotheses of 5.4 could be weakened. We could assume that p satisfies the barycenter formula instead of that p is closed, and we could assume E universally measurable instead of Borel.
Lemma 5.5. Let A be a separable C * −algebra and p a closed projection in A * * . If π * * (p) has finite rank for every irreducible representation π of A, then p is type I.
Proof. Let π = ⊕ π s dµ(s) be a standard direct integral, where each π s is irreducible. Since p is closed, π * * (p) = ⊕ π * * s (p)dµ(s), where π * * · (p) is a Borel operator field. Therefore rank (π * * · (p)) is a Borel function, and by hypothesis it is everywhere finite-valued.
From the above it follows that any representation of A in a separable Hilbert space can be written as a direct sum, π = ∞ 0 π n , such that π n = ⊕ S n π s dµ(s) and rank (π * * s (p)) = n, ∀s ∈ S n . It was shown by A. Amitsur and J. Levitzki in [9] that there is a non-commutative polynomial G n of 2n variables such that G n vanishes on M 2n n but not on M 2n n+1 (cf. [21, Lemma 2], where a weaker but adequate result is proved). Also if G n vanishes on M 2n for a W * −algebra M , then M is a direct sum of type I k algebras for k ≤ n. Clearly G n vanishes on [π * * n (p)π n (A)π * * n (p)] 2n , n > 0, and hence by strong continuity G n vanishes on [π * * n (pA * * p)] 2n . Therefore π * * n (pA * * p) is type I, ∀n. (For n = 0, π * * 0 (p) = 0). If z n is the central support of π n in A * * , and z(π) is the central support of π, then z(π) = sup n z n . Since we have shown that z n pA * * p is type I, ∀n, then z(π)pA * * p is type I. Since sup{z(π) : π as above} = 1, pA * * p is type I. Corollary 5.6. If A is a separable C * −algebra, p is a closed projection in A * * , and if p satisfies (NCEB), then p is type I.
Proof. Combine 4.1 and 5.5.
We have already defined the concept of an atomic projection in A * * . We say that p is strongly atomic if p is atomic and pA * p is norm separable. If A is separable the separability of pA * p can be rephrased: There are only countably many points [π] in A such that π * * (p) = 0.
Question 5.7. If A is separable, is every closed atomic projection in A * * strongly atomic?
If p is closed and atomic and if µ is a probability measure on F (p) ∩ P (A), then ωdµ(ω) is in F (p) and hence is an atomic state. If A is separable, it follows from 5.3, for example, that µ is supported by the union of countably many equivalence classes. If p is not strongly atomic, this means that there are uncountably many equivalence classes in F (p) ∩ P (A) but every finite measure is concentrated on the union of countably many. It follows that the relation of equivalence of pure states is not countably separated on F (p) ∩ P (A). (If it were countably separated, the quotient Borel space would be an uncountable analytic Borel space ([24, Theorem 5.1]) and hence would support a continuous measure. This measure could be lifted to F (p)∩P (A) by the von Neumann selection lemma.) In particular A is not type I. Also p does not satisfy (NCEB), since the space X of Section 4 is second countable and hence countably separated when A is separable. This reasoning suggests the following:
Question 5.8. If A is a separable C * −algebra and p is a type I closed projection in A * * , is equivalence of pure states countably separated on F (p) ∩ P (A)?
Obviously 5.8 is analogous to Mackey's conjecture ( [23, p. 85] or [24, p. 163]), which was proved by Glimm in [17]. Of course [17] proved much more than Mackey's conjecture. We do not know whether there is a structure theorem for type I closed faces of similar power to Glimm's theorem. Because the variety of closed faces of C * −algebras is so great, there is not enough evidence to support a conjecture on any of these questions.
If the answer to 5.8 is yes for a particular p, then a standard form for elements of F (p) ∩ S(A) can be established. Let X be the set of equivalence classes of F (p) ∩ P (A), an analytic Borel space which is in one-one corespondence with a subset {[π x ] : x ∈ X} of A. Then an element ϕ of F (p) ∩ S(A) is determined by a probability measure µ on X and a measurable function f : X → S(A) such that f (x) is supported by π * * x (p). In fact ϕ is the resultant of a probability measure µ on F (p) ∩ P (A). Even though µ is not unique, 5.3 implies its pushforward to X is unique. The function f is obtained by writing µ = X ν x dµ(x), where ν x is supported on the equivalence class x, and f (x) = ωdν x (ω). It can be shown that f is unique modulo null sets. Thus, under the hypotheses given, the Choquet decomposition of ϕ is almost unique in a sense roughly analagous to Mackey's result that the direct integral decomposition of a type I representation into irreducibles is almost unique.
There is a converse question to 5.7 which we can answer. The proof is valid even for A nonseparable.
Proposition 5.9. If A is a C * −algebra and p is a closed projection in A * * such that z at pA * p is norm separable, then p is atomic and hence strongly atomic.
Proof. There is an increasing sequence (p n ) of finite rank projections such that p n → z at p. By 4.5.12 or 4.5.15 of [25], z at p is universally measurable. Since (1−z at )p is a universally measurable operator whose atomic part is 0, (1 − z at )p = 0 ( [25, 4.3.15]).
The following lemma, or the ideas in its proof, might be useful in connection with questions 5.7, 5.8. It will also be used to prove a complement to Glimm's theorem.
Lemma 5.10. If p is an atomic projection in A * * such that pA * p is norm separable, then F (p) ∩ P (A) is an F σ set relative to P (A).
Proof. The lemma can be rephrased more concretely: Let π : A → B(H) be an irreducible representation, let H 0 be a separable closed subspace of H, and let P 0 = {(π(·)v, v) : v is a unit vector in H 0 }. Then P 0 is an F σ set relative to P (A).
The proof is similar to that of 4.1. Let H 1 , H 2 , . . . be an increasing sequence of finite dimensional subspaces such that H 0 = (∪ ∞ 1 H n ) − , and let p n be the projection on H n . Let V n = {v ∈ H 0 : v = 1 and p n v ≥ 1 2 } and P n = {(π(·)v, v) : v ∈ V n }. Then P 0 = ∪ ∞ 1 P n , and we will show P n closed relative to P (A). Suppose v i ∈ V n , ϕ i = (π(·)v i , v i ), and the net (ϕ i ) converges to a pure state ϕ. Passing to a subnet if necessary, we may assume v i
w −→ v for some v in H 0 . Clearly v ≤ 1 and p n v ≥ 1 2 . Then v i = u i + w i , where u i → v in norm, w i w −→ 0, and (u i , w i ) = 0. Therefore (π(a)u i , w i ) → 0, ∀a ∈ A.
Passing to a further subnet, we may assume (π(·)w i , w i ) converges to some ψ in Q(A). Then ϕ = (π(·)v, v) + ψ. Since ϕ is pure, ψ must be proportional to (π(·)v, v). Therefore
ϕ = (π(·)v 1 , v 1 ) where v 1 = v/ v . Since p n v 1 ≥ p n v , ϕ ∈ P n .
Proposition 5.11. If A is a separable C * −algebra and π : A → B(H) is an irreducible representation such that π(A) ⊃ K(H), then there are uncountably many inequivalent irreducible representations of A with the same kernel as π.
Remark. Glimm's theorem implies that there are uncountably many irreducibles with the same kernel, but so far as we know, it was not previously known that that kernel can be taken to be the same as the kernel of the given π.
Proof. By replacing A with its quotient by the kernel of π, we may reduce to the case π faithful. Assume that A has only countably many faithful irreducible representations. Since A is second countable, there is a countable set {I n } of non-zero (closed, two-sided) ideals such that every non-faithful representation of A vanishes on some I n . Then since [π] is a dense point in A, the hull of I n has empty interior in A. Let F n = {ϕ ∈ P (A) : ϕ|I n = 0}. Since the map from P (A) to A is open, we have that F n is a closed nowhere dense set relative to P (A). It now follows from the Baire category theorem, applied to P (A), and 5.10 that there is a faithful irreducible representation π ′ whose associated pure states have non-empty interior in P (A). From the openness of the map from P (A) to A, we conclude that A has an open point, whence A has an ideal K, necessarily essential, such that K has only one point. The proof is concluded by showing K ∼ = K(H), and this can be done in at least two ways. There is a simple way to prove that every separable C * −algebra whose spectrum is a single point must be elementary (i.e., the affirmative answer to Naimark's question in the separable case), or one can apply Glimm's theorem to K.
The following example demolishes one naive conjecture with regard to the structure of type I closed faces.
Example 5.12. If A is any non-type I separable C * −algebra, then A has a type I closed face F (p) (p is even compact) such that F (p) is not isomorphic to a closed face of any type I C * −algebra.
If A is not unital, we consider A * * as a subalgebra of A * * and construct p as a projection in A * * closed in A * * , so that p will be compact. Since A is not type I, there is an irreducible representation π such that π(A) ⊃ K(H π ). For the natural extension of π to A, also denoted π, we also have π( A) ⊃ K(H π ). Let v 0 be a unit vector in H π , ϕ 0 = (π(·)v 0 , v 0 ) and p 0 the support projection in A * * of ϕ 0 . By a result of Glimm [16,Theorem 2], there is a sequence {v n } of unit vectors in H π such that v n w −→ 0 and (π(·)v n , v n ) → ϕ 0 in A * . By using the Gram-Schmidt process, we can find a subsequence {v n i } and an orthonormal sequence {w n i } such that (w n i , v 0 ) = 0 and w n i − v n i → 0. Let ϕ i = (π(·)w n i , w n i ).
Since p 0 is a minimal projection in A * * , it is closed in A * * . Let B be the hereditary C * −subalgebra of A supported by 1 − p 0 , and let e be a strictly positive element of B. Since ϕ i → ϕ 0 in A * and ϕ 0 | B = 0, ϕ i (e) → 0. Passing to a subsequence, we may assume ϕ i (e) < ∞. Let p i be the support projection of ϕ i . p i is in B * * ∩ A * * , considered as a subalgebra of A * * . By 0.1(ii),
∞ 1 p i is closed in B * * . Thus if p = ∞ 0 p i , p is closed in A * * .
Since p ∈ A * * , p is a compact projection in A * * . Since pA * * p ∼ = B(H 0 ) where H 0 = span{v 0 , w n 1 , w n 2 . . . }, p is a type I projection.
Suppose F (p) were isomorphic to a closed face, F (p ′ ), of a type I C * −algebra A ′ . Since p ′ (A ′ ) * * p ′ can be identified with the space of bounded affine functionals vanishing at 0 on F (p ′ ), p ′ (A ′ ) * * p ′ is Jordan * −isomorphic to pA * * p. Therefore p ′ (A ′ ) * * p ′ is a type I factor, and p ′ is associated with a single irreducible representation, π ′ , of A ′ . Since A ′ is type I, π ′ (A ′ ) ⊃ K(H π ′ ). Let ϕ ′ i , i ≥ 0, be the element of F (p ′ ) corresponding to ϕ i . Then ϕ ′ i → ϕ ′ 0 in A ′ * . This contradicts the facts that {ϕ ′ i } arises from an orthonormal sequence of vectors in H π ′ and π ′ (A ′ ) ⊃ K(H π ′ ).
We think it is fairly obvious from the proof of 0.1(ii) ([12, Lemma 3]), that the faces F (p) constructed above are all isomorphic. In Section 7 we will determine the structure of pAp, and this will be our formal proof of this fact.
Finally, we want to generalize 5.5 for use in connection with a remark in Section 7. If h ∈ A * * , we say that h satisfies the barycenter formula if, when regarded as a function on Q(A), h is measurable with respect to (the completion of) any regular Borel measure and ϕ(h) = h(ω)dµ(ω) whenever µ is a regular Borel measure on Q(A) and ϕ is the resultant of µ. If A is separable, it is sufficient to verify the formula for measures supported on P (A). Also when A is separable, the barycenter formula is equivalent to: π * * s (h) is a measurable field of operators and π * * (h) = ⊕ π * * s (h)dµ(s), whenever π = ⊕ π s dµ(s), a standard direct integral; and again it is sufficient to verify this in the special case where each π s is irreducible. Thus for A separable the set of elements of A * * satisfying the barycenter formula is a C * −algebra closed under weak sequential convergence. This C * −algebra is at least as large as {h : Re h, Im h are universally measurable} and appears to be a good thing to use, though the monotone sequential closure of A (discussed in [25, §4.5]) would do for our purposes. For A non-separable, we know of nothing more general than the space of universally measurable operators ([26]).
Theorem 5.13. If A is a separable C * −algebra, p is a projection in A * * satisfying the barycenter formula, and if π * * (p)π(A)π * * (p) ⊂ K(H π ) for every irreducible representation π of A, then p is type I.
Proof. First note that the proof of 5.5 is equally valid if p satisfies the barycenter formula instead of being closed. Let e be a strictly positive element of A. Then for ǫ > 0, E [ǫ,∞) (pep) satisfies the barycenter formula and π * * (E [ǫ,∞) (pep)) has finite rank for π irreducible. Therefore each p n is type I where p n = E [n −1 ,∞) (pep). Since p n ր p, p is type I.
6. More on Closed Faces with (NCEB) for A Separable.
The notations X, p x , X, and f have the same meanings as in Section 4. if ϕ n → tϕ, for ϕ n , ϕ in F (p) ∩ P (A), then there is a compact set S such that f (ϕ n ) ∈ S for n sufficiently large. Since F (p S ) ∩ S(A) is closed, it follows that t = 1.
Remarks. 1. If p satisfies only (NCEB) and S is compact, then p S is nearly relatively compact in the sense of [13].
2. The hypothesis of 4.4 included the assumption that p be strongly atomic, though this term was not used. Theorem 6.1 shows that this assumption can be dropped if A is separable. Also the discussion after 5.7 shows that if A is separable, p is closed and atomic, and p satisfies (NCEB), then p is strongly atomic. Thus for A separable, the hypothesis of 4.5 can be weakened by replacing strongly atomic with atomic.
3. In view of the remarks after 5.4, it is not hard to calculate the facial topology on the extreme boundary of F (p), when p is closed and satisfies (CEB). Its T 0ification is the compact Hausdorff space X ∪{∞}. If p satisfies only (NCEB), we still see that the closed split faces of F (p) containing 0 are in one-to-one correspondence with the closed subsets of X.
Some Relationships among Prior Sections and Concluding Remarks.
Each of the three main parts of this paper (Sections 2, 3, and 4-6) studies a different generalization of the situation considered in [5] (1.4 and 7.2 below are used to justify this statement). Sections 3-6 were motivated by our desire to investigate the circumstances in which 2.4 applies, but the detailed discussion below makes it clear that we have not solved this problem -if it can be called a "problem". There is a broader "problem" to which all three parts of this paper are relevant: Study the structure of those closed faces of C * −algebras which are closely modeled on locally compact Hausdorff spaces. We now discuss the relationships among the prior sections.
First we consider the relationship between Sections 2 and 3. The next result and the remarks following show that if we were willing to use the theory of relative q-continuity in the construction of MASA's, it would have been sufficient to prove the special case of 2.4 in which the projection p X is central and abelian. However, so far as we know, this special case is no easier.
Theorem 7.1. Let A be a separable C * −algebra and p a closed projection in A * * . Suppose B is a commutative C * −subalgebra of SQC(p) which is non-degenerately embedded in pA * * p. If B is totally disconnected, then there is a commutative C * −subalgebra C of A such that C contains an approximate identity of A, p ∈ C * * , and pC = B. Suppose p is a closed projection in A * * such that SQC(p) is non-degenerate in pA * * p (cf. 3.2) and that B is a MASA in SQC(p) which hereditarily generates SQC(p). If A is separable and B is totally disconnected, then 7.1 gives a commutative algebra C (which could be assumed a MASA in A). For each x in B we have a pure state ϕ x of B (or C) which is supported by a minimal projection p x in B * * , and it follows from pC = B and p ∈ C * * that also p x ∈ C * * . If p x is minimal in A * * , then ϕ x satisfies (UEP) relative to the inclusion of C in A. If pAp is an algebra (cf 3.1 and 7.2 below), then we need only start with a MASA B in pAp which hereditarily generates pAp and such that each pure state of B satisfies (UEP) relative to pAp. It was pointed out in Section 0 that under the hypotheses of 2.4 every element of C 0 (X) gives an element of SQC(p X ). It can be shown that C 0 (X) is thus embedded as a MASA in SQC(p X ) and that C 0 (X) is nondegenerate in p X A * * p X . Thus the above discussion applies.
Proposition 7.2. Conditions (i)-(iv) of 1.2 imply that pAp is an algebra, and p satisfies:
(G) [P (A) ∩ F (p)] − ⊂ [0, 1]P (A).
Proof. The reduced atomic representation, π, of A is faithful on pA * * p. Moreover, π * * (pA * * p) ∩ K(H π ) is a C * −algebra, and by 1.2(iv), π * * (pAp) is contained in this algebra. We show equality. Let h be an element of pA * * sa p such that π * * (h) is compact. It is sufficient to show h ∈ SQC(p). If F is a closed subset of R, then 1.2(i) implies that E F (h), computed in pA * * p, is closed. (In fact we don't need F closed for this.) If 0 / ∈ F (and F is closed), then π * * (E F (h)) is a finite rank operator on H π , and by [1, Corollary II.8] this implies E F (h) is compact. Thus h ∈ SQC(p). Then (G) follows from [17, §5] for example.
The same reasoning shows pA * * p = QC(p). Next we consider the relationship between Sections 2 and 4. By 6.1 if A is separable and p is a closed projection in A * * satisfying (CEB) then we have the hypotheses of 2.3 (except for total disconnectedness of X). By 4.1 each p x is of finite rank. For 2.4, we would want each p x to be of rank 1. This happens for the p x 's of Section 4 if and only if p is abelian. If p is not abelian, it might be possible to write p x = p x,1 + · · · + p x,n x where the p x,i 's are minimal and {p x,i } satisfies the hypotheses of 2.4 with X replaced by some space X. Then X would map onto X by a closed continuous map with finite fibers. However, Example 7.6(a) below shows that this is not always possible.
Conversely, suppose the hypotheses of 2.4 are satisfied. By 1.4, if X is countable and discrete and each equivalence class of {ϕ x : x ∈ X} is finite, then p X satisfies (CEB). If p X is abelian, we can deduce (7.3 below) that p X satisfies (CEB) even for X not discrete; but it is fairly obvious (cf 7.6(b) below) that in general p X need not satisfy (NCEB) or even (G).
In retrospect it seems that (G) is worthy of more study in the present context despite the fact, as pointed out in the remark following 4.5, that it does not imply a positive answer to the isolated point question. One reason is mentioned in 7.2 above. However, the conclusion of 1.4 is definitely false if we drop the hypothesis of finite equivalence classes. (This follows from 4.1.) It may be that (G) is part of the hypothesis of a nice result. Also, even though, by Example 3.4, (G) does not imply that F (p) is associated with a locally compact Hausdorff space, we do not know whether (G) implies that F (p) is associated with a Hausdorff space. We will show below that (G) does imply that p is type I.
One could also consider weaker conditions than (G):
(1) [P (A) ∩ F (p)] − ⊂ {type I factorial quasi-states} (2) [P (A) ∩ F (p)] − ⊂ z at A * .
(2) is suggested by the theory of perfect C * −algebras ( [8]). With regard to the relationship between Sections 3 and 4, we note that a closed projection p satisfying (CEB) need not satisfy (MSQC) (cf. 7.6(c) below). However, it follows from 6.1 that p does satisfy the hypothesis of 3.2 (A separable). Also if p is closed and abelian and satisfies (CEB), then p does satisfy (MSQC). (It follows from 5.3 that F (p) is isomorphic to the set of probability measures on X ∪ {0}. Is there a less technical proof?) It may be that there are other useful concepts on the extensiveness of SQC(p).
At the end of this section we return to Example 5.12, partly to show that it does satisfy the conclusion of 4.5. A complete theory for closed faces of C * −algebras analogous to the theory of scattered C * −algebras might have to be quite complicated. Proposition 7.3. Assume the hypotheses of 2.4 and also that p X is abelian. Then p X satisfies (CEB).
Remark. The hypothesis that p X is abelian can be rephrased more concretely:
The ϕ x 's are mutually inequivalent. ( [10]).
Proof. Since p X is abelian, P (A) ∩ F (p X ) = {ϕ x : x ∈ X}. Suppose ϕ x i → ψ in Q(A).
Passing to a subnet, we may assume x i → x in X or x i → ∞. If x i → x, let {S j } be a set of compact neighborhoods of x such that j S j = {x}. Since p S j is compact and ϕ x i ∈ F (P S j ) for i large, we conclude that ψ = 1 and ψ ∈ j F (p S j ) = F ( j p S j ). Since a closed projection is determined by its atomic part, j p S j = p x , and hence ψ = ϕ x . If x i → ∞, let {U j } be a set of relatively compact open subsets of X such that j U j = X, and let S j = X \U j . Since x i is eventually in S j and since p S j is closed, we see that ψ ∈ j F (P S j ) = F ( j p S j ) = {0}.
It would be desirable if the hypotheses in 2.4 that certain projections are atomic parts of closed projections could be stated entirely in terms of pure states (or equivalently, minimal projections). This can be done in a situation of intermediate generality. Consider the following conditions for a projection p in A * * :
(3) ∃t ∈ (0, 1] such that [P (A) ∩ F (p)] − ⊂ {0} ∪ [t, 1][P (A) ∩ F (p)]. (4) {0} ∪ [P (A) ∩ F (p)] is closed. (5) [P (A) ∩ F (p)] is closed.
Conditions (3) and (4) Then every element of C is the resultant of a probability measure on X. Since X is norm separable, the resultant is actually a Bochner integral. Hence C ⊂ F (p). The reverse inclusion follows easily from the structure of atomic von Neumann algebras.
Remark. The same argument works if (3) is replaced by a similar modification of (2). Remarks. 1. Let ϕ x be the pure state supported by p x . If p = x∈X p x is abelian, i.e. if the ϕ x 's are mutually inequivalent, then the hypotheses on x∈S p x can be stated more concretely:
(6)
If x n → x, then ϕ x n → ϕ x , and if x n → ∞, then ϕ x n → 0.
We can actually replace the hypotheses on x∈S p x by (6) p satisfies (3).) The proof of this uses Akemann's result in [1] that the supremum of finitely many mutually commuting closed projections is closed. 2. Even when X is uncountable, the hypotheses on x∈S p x in 2.4 can be modified somewhat: If x∈X p x is the atomic part of a closed projection, and if x∈S p x satisfies (3) for S closed and (5) for S compact, then we have the hypotheses of 2.4. We will not provide a complete proof of this because it would be rather technical and it is not clear that the result is a big improvement on 2.4. The main lemma is the following: Let p be a closed projection satisfying (NCEB) and q a subprojection of z at p. If A is separable and q satisfies (3), then q is the atomic part of a closed projection. The proof of this uses 5.6, the other results of Section 5 (in particular the discussion following 5.8), and the von Neumann selection lemma.
Examples 7.6. (a) Let A = c ⊗ K and define a closed projection p in A * * by letting p ∞ be the projection on span{e 1 , e 2 } and p n the projection on Ce 1 , n = 3k + 1
Ce 2 , n = 3k + 2 C(2 − 1 2 e 1 + 2 − 1 2 e 2 ), n = 3k
It is easy to see that p satisfies (CEB). Let ϕ n be the pure state of A supported by p n , n < ∞, and suppose B is a MASA of A such that each ϕ n | B satisfies (UEP). Thus each p n is in B * * . If b ∈ B, then e 1 is an eigenvector of each b 3k+1 and hence e 1 is an eigenvector of b ∞ . Similarly e 2 and 2 − 1 2 e 1 + 2 − 1 2 e 2 are eigenvectors of b ∞ . Therefore all three eigenvalues are the same and b ∞ p ∞ = λp ∞ . It follows that p ∞ is a minimal projection of B * * . Thus no matter how we write p ∞ = p ′ + p ′′ , with p ′ and p ′′ rank one projections, we cannot achieve the conclusion of 2.4, let alone the hypotheses.
(b) First note that if p is the projection of 5.12, then we have the hypotheses of 2.4 with X = N ∪ {∞} and p = p X . Since all of the ϕ n 's, 1 ≤ n ≤ ∞, are equivalent, it is easy to see that the non-pure state 1 2 ϕ 1 + 1 2 ϕ ∞ is in [P (A) ∩ F (p)] − (cf [16]), so that p does not satisfy (G).
It is better to give an example where the equivalence classes of {ϕ x : x ∈ X} are finite, since by 4.1 there is no hope of (NCEB) without this finiteness. A standard example suffices for this. Let A = {a ∈ c ⊗ M 2 : a ∞ is diagonal}. Let B = {a ∈ A : a n is diagonal, ∀n}. Then B is a MASA in A, and we let X = B, the disjoint union of two copies of N ∪ {∞}. It is clear that for x in X the pure state ψ x of B satisfies (UEP); and if p x is the support projection of ψ x , we have the hypothesis of 2.4 with p X = 1 A . Since A is not Hausdorff, it follows from [17, Thm. 6] that p X does not satisfy (G). Of course, this is also easy to see explicitly.
It is possible to give a similar example in which A is Hausdorff, but a different condition of [17, Theorem 6] is violated. Let A = {a ∈ c⊗M 2 ⊗M 2 : a ∞ ∈ M 2 ⊗I 2 }. If B = {a ∈ A : a n ∈ D 2 ⊗ D 2 , n < ∞; a ∞ ∈ D 2 ⊗ I 2 }, where D 2 = {d ∈ M 2 : d is diagonal}, then B is a MASA in A and we can proceed similarly to the above. Again X is the disjoint union of two copies of N ∪ {∞} (arising more naturally as N ∪ N ∪ {∞}).
(c) Consider one of the examples of alternative 2 of 4.6 constructed in 4.8(c). Let k = n = 2 and T = 1. We then get a projection p satisfying (CEB) where the space X of Section 4 is N ∪ {∞}, rank p n = 2, 1 ≤ n ≤ ∞, and there is a single element ϕ of F (p ∞ ) ∩ P (A) such that every sequence (ψ n ) with ψ n in F (p n ) ∩ P (A) converges to ϕ. In this case we can write p n = p n,1 + p n,2 so that the hypotheses of 2.4 are satisfied. All we have to do is take p ∞,1 to be the support projection of ϕ. X will be homeomorphic to N ∪ {∞}, but it arises as the disjoint union of N ∪ N ∪ {∞} with an isolated point. This example does not satisfy (MSQC). One way to see this is to note that the saturation of an open subset of F (p) ∩ P (A) need not be open, and hence F (p) is not isomorphic to the quasi-state space of a C * −algebra. Explicitly, any element h of SQC(p) (or QC(p)) must have ϕ definite on h ∞ . Lemma 7.7. If A is a C * −algebra, p is a projection in A * * , and p satisfies (G), then π * * (pAp) ⊂ K(H π ) for every irreducible representation π of A.
Proof. Part of the proof of 4.1 applies: If (e n ) is an orthonormal sequence in the range of π * * (p) and ψ n = (π(·)e n , e n ), we can conclude that ψ n → 0. If E [ǫ,∞) (π * * (pap)) has infinite rank for some a in A + and ǫ > 0, then we can obtain a contradiction by taking the e n 's in the range of E [ǫ,∞) (π * * (pap)).
Corollary 7.8. If A is a separable C * −algebra, p is a projection in A * * satisfying the barycenter formula (in particular if p is closed), and p satisfies (G), then p is type I.
Proof. Apply 5.13. 7.9. Continuation of Example 5.12.
(a) A subprojection p ′ of p is closed if and only if p ′ has finite rank or p ′ ≥ p 0 .
Proof. If p ′ has finite rank, then p ′ is closed by [1]. If p ′ ≥ p 0 , then p ′ − p 0 is closed in B * * by 0.1(ii). Therefore p ′ is closed in A * * and a fortiori in A * * .
If p ′ has infinite rank, then the range of p ′ contains an infinite dimensional subspace H ′ of span{w n 1 , w n 2 , . . . } = range(p − p 0 ). (This last is a codimension 1 subspace of H 0 = range p.) Let (u n ) be a sequence of unit vectors in H ′ such that u n w −→ 0 and ψ n = (π(·)u n , u n ). Then ψ n | B → 0, since π * * (pBp) ⊂ K(H π ). Since p is compact, it follows that ψ n → ϕ 0 . If p ′ is closed, this implies ϕ 0 ∈ F (p ′ ) and hence p ′ ≥ p 0 .
(b) For any non-zero closed subprojection p ′ of p there is a minimal projection p 1 such that p 1 ≤ p ′ and p ′ − p 1 is closed.
Proof. If p ′ has infinite rank, then p ′ ≥ p 0 . We can find a minimal projection p 1 such that p 1 ≤ p ′ − p 0 . Then p 0 ≤ p ′ − p 1 so that p ′ − p 1 is closed. If p ′ has finite rank then p ′ − p 1 is closed for any choice of p 1 . Proof. Let H 1 = H 0 ⊖ Cv 0 . By construction and the proof of 7.2, applied to B and p − p 0 , pBp = K(H 1 ). Note that since p is compact, pAp = p Ap. To show that pAp is contained in the set indicated, it is enough to show pap is compact when a ∈ A and ϕ 0 (a) = 0. By [25, 3.13.6], a = l + r, where l ∈ AB and r ∈ B A. Since x is compact if and only if x * x is compact, Bp ⊂ K(H π ); and similarly pB ⊂ K(H π ). Therefore pap ∈ K(H 0 ).
For the reverse inclusion, since p ∈ pAp, K(H 1 ) ⊂ pAp, and pAp is self-adjoint, it is sufficient to show that pAp contains every rank 1 operator x of the form v → (v, v 0 )v 1 for v 1 ∈ H 1 . By the Kadison transitivity theorem ( [20]) there is a in A such that av 0 = v 1 and a * v 0 = 0. By the above, since (av 0 , v 0 ) = 0, pap is compact. Hence (pap − x) ∈ K(H 1 ) = pBp. This implies that x is in p Ap.
Since by [7, 4.4], the bidual of pAp is pA * * p, and since the predual of a W * −algebra is unique, it follows from (c) that the dual space of pAp is T (H 0 ), the set of trace class operators on H 0 . A concrete statement of this reads:
If v 0 is a unit vector in the (separable, infinite dimensional) Hilbert space H 0 and
A v 0 = {x ∈ B(H 0 ) : x − (xv 0 , v 0 )I H 0 is compact},
then the dual space of A v 0 is naturally isometrically isomorphic to T (H 0 ). In particular, for T ∈ T (H 0 ), T 1 = sup{|T r(T x)| : x ∈ A v 0 , x ≤ 1}. It is amusing to give a direct proof of this statement (which removes the parenthetical part of the hypothesis). The main step is to prove the second sentence.
Finally, we note that 5.12 gives another example of how the behavior of closed faces of C * −algebras differs from that of C * −algebras. If π is an irreducible representation of a C * −algebra A, then π(A) ∩ K(H π ) is either 0 or K(H π ). The analogous statement for Example 5.12 (replacing A by pAp) is false.
S(A) is the state space of A, P (A) the pure state space, and Q(A) the quasi-state space (the set of positive functionals of norm at most 1). If p is a projection in A * * , F (p) = {ϕ ∈ Q(A) : ϕ(1 − p) = 0}, the norm closed face of Q(A) supported by p. (Elements of A * are regarded as functionals on A * * without notice.) Topological terminolgy regarding A * refers to the weak * topology unless the contrary is explicitly indicated. A projection p in A * * is called open ([1]) if it is the support projection of a hereditary C * −subalgebra of A and closed if 1 − p is open. Effros proved in [15, Theorem 4.8] (cf.
since Ap is closed by an argument similar to [7, 4.4]. If (1 − p)ap = xp for x in A, then pxp = 0. Therefore x ∈ L + R, where L = {b ∈ A : bp = 0} and R = L * = {b ∈ A : pb = 0}, (proof of [7, 4.4]). Since Lp = 0, (1 − p)ap = rp for some r in R.
1 2
1is not an isolated point of [0, 1], M (A 1 ) can be regarded as a set of functions g : [0, 1]\{ 1 2 } → K (cf. [7, Theorem 3.3] and note that K is unital). The requirements on g are: (i) g is norm continuous and bounded.
Lemma 4.2. f is a closed map.
Let p S = x∈S p x . Then every element of F (p S ) is the resultant of a probability measure supported by [F (p S ) ∩ P (A)] ∪ {0}, and a fortiori supported by f −1 (S) ∪ {0}. Since f −1 (S) ∪ {0} is compact, every element of F (p S ) − is the resultant of a probability measure on f −1 (S) ∪ {0}. Since f −1 (S) is the disjoint union of
Example 3.4 shows that we cannot replace (NCEB) by the weaker condition [F (p) ∩ P (A)] − ⊂ [0, 1]P (A), and 3.5(a) shows we cannot weaken (NCEB) to [F (p) ∩ P (A)] − ⊂ {0} ∪ [t, 1]S(A)
F 1 be the set of resultants of probability measures on E ∪ {0}. We claim that F 1 is a norm closed sub-face of F (p). The result then follows from [15, Theorem 4.4 and p. 396] (cf. [25, 3.6.11]). To see the claim, note that by Choquet theory every element of F (p) is the resultant of a probability measure on [F (p)∩P (A)]∪{0}. Let E ′ = [F (p)∩P (a)]\E. Then F 1 = { ωdµ(ω) : µ(E ′ ) = 0}. Suppose ϕ i = ωdµ i (ω), i = 1, 2, and
Theorem 6. 1 .
1If A is a separable C * −algebra, p is a closed projection in A * * , and if p satisfies (NCEB), then x∈S p x is the atomic part of a closed projection p S for every closed subset S of X. Also p satisfies (CEB) if and only if p S is compact for S compact. Proof. By 5.6, p is type I. Let S = f −1 (S), a closed subset of X, let E = S ∩P (A) = S ∩ S(A), a saturated subset of F (p) ∩ P (A), and let p S be the projection called p E in 5.4. By 5.4, F (p S ) = { ωdµ(ω) : µ is a probability measure on E ∪ {0}} = { ωdµ(ω) : µ is a probability measure on S ∪ {0}}. Since S ∪ {0} is a compact subset of A * , F (p S ) is closed, and hence p S is closed by [15, Theorem 4.8]. By 5.4, F (p S ) ∩ P (A) = E, and this implies that the atomic part of p S is x∈S p x .If p satisfies (CEB) and S is compact, then E = S, and S is compact by 4.3. Thus F (p S )∩S(A) = { ωdµ(ω) : µ is a probability measure on S}, a closed subset of A * . Therefore p S is compact. Conversely, if S compact implies p S compact and
Proof. Let B = {a ∈ A : ap = pa and pa ∈ B}. Then her(1 − p) is an ideal of B and B/her(1 − p) ∼ = B. We can apply 2.4 (or 2.3) with B playing the role of A and with X = B. For x in X, p x is the support projection in B * * of the pure state of B given by x. Let C be the MASA of B given by 2.4. Since B is non-degenerate in pA * * p, B hereditarily generates A. Since C hereditarily generates B by 2.4, C also hereditarily generates A. One way to deduce from 2.4 that pC, which is the image of C in B/her(1 − p), is all of B is to quote the classical Stone-Weierstrass theorem.
are strengthenings of (NCEB) and (CEB) respectively, and are equivalent to (NCEB) and (CEB) if p is closed. But we are interested in the case where p is atomic. If p is the atomic part of a closed projection q, then q satisfies (NCEB) or (CEB) if and only if p satisfies (3) or (4). If p is atomic and satisfies (3) or (4), is p necessarily the atomic part of a closed projection? We can prove this if p is strongly atomic, in which case p itself is closed. (In general let C be the closed convex hull of {0} ∪ [P (A) ∩ F (p)]. If q exists, then F (q) = C. The tricky thing is to prove that C is a face of Q(A).) Lemma 7.4. If p is a strongly atomic projection satisfying (3), then p is closed. Proof. Let X = {0} ∪ [P (A) ∩ F (p)] − and let C be the closed convex hull of X.
(c) If pA * * p is identified with B(H 0 ), then pAp = {x ∈ B(H 0 ) : x − ϕ 0 (x)I H 0 ∈ K(H 0 )} = {x ∈ B(H 0 ) : x − (xv 0 , v 0 )I H 0 ∈ K(H 0 )}.
is obvious. 2. Existence of MASA's Lemma 2.1. Let A be a C * −algebra and A the result of adjoining a new identity to A (i.e., A ∼
Corollary 7.5. Let X be a locally compact Hausdorff space with only countably many points, and let {p x : x ∈ X} be a family of mutually orthogonal minimal projections in A * * . If x∈S p x satisfies (3) whenever S is a closed subset of X and (5) when S is compact, and if A is separable, then we have the hypotheses and conclusions of Corollary 2.4.
if we assume only that the equivalence classes have bounded finite cardinality. (Note that they have to be finite by 4.1 if
Approximate units and maximal abelian C * −subalgebras. C A Akemann, 543-550. 4J. Funct. Anal. 4Pac. J. Math.C. A. Akemann, The general Stone-Weierstrass problem, J. Funct. Anal. 4 (1969), 277-294. 2. , Left ideal structure of C * −algebras, J. Funct. Anal. 6 (1970), 305-317. 3. , Approximate units and maximal abelian C * −subalgebras, Pac. J. Math. 33 (1970), 543-550. 4. , A Gelfand representation theory for C * −algebras, Pac. J. Math. 39 (1971), 1-11.
Approaching infinity in C * −algebras. C A Akemann, J Anderson, G K Pedersen, J. Operator Theory. 21Canad. J. Math.C. A. Akemann, J. Anderson, and G. K. Pedersen, Approaching infinity in C * −algebras, J. Operator Theory 21 (1989), 255-271. 6. , Excising states of C * −algebras, Canad. J. Math. 38 (1986), 1239-1260.
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Determination of A from M(A) and related matters. L G Brown, C Semicontinuity, Nearly relatively compact projections in operator algebras. 40Canad. J. Math.. unpublished manuscript. 14. , Semicontinuity and closed faces of C * −algebras. in preparationL. G. Brown, Semicontinuity and multipliers of C * −algebras, Canad. J. Math. 40 (1988), 865-988. 12. , Determination of A from M(A) and related matters, C.R. Math. Rep. Acad. Sci. Canada 10 (1988), 273-278. 13. , Nearly relatively compact projections in operator algebras, unpublished manuscript. 14. , Semicontinuity and closed faces of C * −algebras, in preparation.
Order ideals in a C * −algebra and its dual. E G Effros, Duke Math. J. 30E. G. Effros, Order ideals in a C * −algebra and its dual, Duke Math. J. 30 (1963), 391-412.
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308-314. 19. , Scattered C * −algebras II. H E Jensen, Math. Scand. 41Math. Scand.H. E. Jensen, Scattered C * −algebras, Math. Scand. 41 (1977), 308-314. 19. , Scattered C * −algebras II, Math. Scand. 43 (1978), 308-310.
Irreducible operator algebras. R V Kadison, Proc. Nat. Acad. Sci. USA. 43R. V. Kadison, Irreducible operator algebras, Proc. Nat. Acad. Sci. USA 43 (1957), 273-276.
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G W Mackey, The Theory of Unitary Group Representations. University of Chicago Press24Chicago and LondonBorel structures in groups and their dualsG. W. Mackey, The Theory of Unitary Group Representations, University of Chicago Press, Chicago and London, 1976. 24. , Borel structures in groups and their duals, Trans. Amer. Math. Soc. 85 (1957), 134-165.
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Morita Equivalence and Continuous Trace C * −algebras. I Raeburn, D P Williams, Mathematical Surveys and Monographs. 60American Mathematical SocietyProvidence R.I.I. Raeburn and D. P. Williams, Morita Equivalence and Continuous Trace C * −algebras, Mathematical Surveys and Monographs, vol. 60, American Mathematical Society, Providence R.I., 1998.
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[] |
[
"Heterotic horizons, Monge-Ampère equation and del Pezzo surfaces",
"Heterotic horizons, Monge-Ampère equation and del Pezzo surfaces"
] |
[
"J Gutowski \nDepartment of Mathematics King's College London Strand\nWC2R 2LSLondonUK\n",
"G Papadopoulos \nDepartment of Mathematics King's College London Strand\nWC2R 2LSLondonUK\n"
] |
[
"Department of Mathematics King's College London Strand\nWC2R 2LSLondonUK",
"Department of Mathematics King's College London Strand\nWC2R 2LSLondonUK"
] |
[] |
Heterotic horizons preserving 4 supersymmetries have sections which are T 2 fibrations over 6-dimensional conformally balanced Hermitian manifolds. We give new examples of horizons with sections S 3 × S 3 × T 2 and SU (3). We then examine the heterotic horizons which are T 4 fibrations over a Kähler 4-dimensional manifold. We prove that the solutions depend on 6 functions which are determined by a nonlinear differential system of 6 equations that include the Monge-Ampére equation. We show that this system has an explicit solution for the Kähler manifold S 2 × S 2 . We also demonstrate that there is an associated cohomological system which has solutions on del Pezzo surfaces. We raise the question of whether for every solution of the cohomological problem there is a solution of the differential system, and so a new heterotic horizon. The horizon sections have topologies which include ((k − 1)S 2 × S 4 #k(S 3 × S 3 )) × T 2 indicating the existence of exotic black holes. We also find an example of a horizon section which gives rise to two different near horizon geometries.
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10.1007/jhep10(2010)084
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[
"https://arxiv.org/pdf/1003.2864v3.pdf"
] | 115,169,944 |
1003.2864
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45b3c75457b71bcc74d55305866294a20322dc4c
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Heterotic horizons, Monge-Ampère equation and del Pezzo surfaces
4 Nov 2010
J Gutowski
Department of Mathematics King's College London Strand
WC2R 2LSLondonUK
G Papadopoulos
Department of Mathematics King's College London Strand
WC2R 2LSLondonUK
Heterotic horizons, Monge-Ampère equation and del Pezzo surfaces
4 Nov 2010
Heterotic horizons preserving 4 supersymmetries have sections which are T 2 fibrations over 6-dimensional conformally balanced Hermitian manifolds. We give new examples of horizons with sections S 3 × S 3 × T 2 and SU (3). We then examine the heterotic horizons which are T 4 fibrations over a Kähler 4-dimensional manifold. We prove that the solutions depend on 6 functions which are determined by a nonlinear differential system of 6 equations that include the Monge-Ampére equation. We show that this system has an explicit solution for the Kähler manifold S 2 × S 2 . We also demonstrate that there is an associated cohomological system which has solutions on del Pezzo surfaces. We raise the question of whether for every solution of the cohomological problem there is a solution of the differential system, and so a new heterotic horizon. The horizon sections have topologies which include ((k − 1)S 2 × S 4 #k(S 3 × S 3 )) × T 2 indicating the existence of exotic black holes. We also find an example of a horizon section which gives rise to two different near horizon geometries.
Introduction
In the past few years much work has been done to understand the topology and geometry of higher dimensional black holes following earlier work in four dimensions [1]- [7]. It has been realized that the four dimensional uniqueness theorems fail to hold in higher dimensions, and that the horizon sections can have many different topologies, including S n × S m , n ≥ 0, m ≥ 1 [8]- [19].
More recently, the near horizon geometry of supersymmetric heterotic black holes has been investigated [20] utilizing the solution of the Killing spinor equations of heterotic supergravity in [21]. It was found that the heterotic horizons are either products AdS 3 ×X for a suitable manifold X or (AdS 3 × Y )/S 1 , where AdS 3 twists over a base space with a U(1) connection. The heterotic horizons preserving 8 supersymmetries are isometric to AdS 3 × S 3 × T 4 or AdS 3 × S 3 × K 3 with constant dilaton and have horizon section S 1 × S 3 × T 4 or S 1 × S 3 × K 3 , respectively.
In this paper, we shall give new heterotic horizons which preserve N = 4 supersymmetries. Moreover, we shall provide evidence that heterotic black holes can have increasingly involved horizon topologies which are distinct from those expected from lifting 4-and 5dimensional black holes to 10 dimensions. It is known that N = 4 horizon sections S are holomorphic T 2 fibrations over a complex conformally balanced 6-dimensional manifold B. First, we shall reorganize the differential conditions which arise from supersymmetry in a form which is amenable to a cohomological analysis. Then we shall demonstrate that the differential system has solutions provided that certain conditions on the cohomology of B are satisfied. We present explicit heterotic horizon solutions with B = SU(3)/T 2 equipped with the balanced Hermitian structure and B = (S 3 × S 3 )/S 1 × S 1 . The horizon sections are S = SU(3) and S = S 3 ×S 3 ×T 2 , and the spacetime is M = SL(2, R) ×SU(3) /U (1) and M = AdS 3 × S 3 × S 3 × S 1 , respectively. In the former case, AdS 3 twists over B with a U(1) connection. In the latter case, the radius of AdS 3 is twice that of S 3 . The half supersymmetric horizon AdS 3 × S 3 × K 3 is also of this type with B = P 1 × K 3 Kähler and so Hermitian and balanced.
To construct more examples of heterotic horizons, one can take B to be a holomorphic T 2 -fibration over a 4-dimensional Kähler manifold X. Such complex manifolds with skewsymmetric torsion and holonomy contained in SU (3) have been considered before in [22,23,24,25]. For such B, the horizon section S is a T 4 fibration over X. We show that the resulting field equations and supersymmetry conditions lead to a non-linear system of 6 differential equations for 6 functions 1 . The 6 functions include the dilaton, the deformation of the Kähler metric of X within its Kähler class as well as the deformations of connections of holomorphic U(1) bundles within their Chern classes. The deformation of the metric within its Kähler class leads to a complex Monge-Ampère equation similar to that which appears in Yau's proof of the Calabi conjecture. The system also includes a conformally rescaled Hermitian-Einstein equation. However, the six differential equations do not separate. As in the general case, the differential equations give rise to some cohomological conditions for classes in X. These are necessary for the differential system to have a solution. We raise the question of whether for every solution of the cohomological problem there is also a solution of the differential system.
The non-linear differential system of equations has solutions. One explicit solution is X = P 1 × P 1 , B = P 1 × P 1 × T 2 and S = S 3 × S 3 × T 2 . The heterotic horizon spacetime is M = SL(2, R) × S 3 × S 3 × T 2 /S 1 . Another example is the half supersymmetric horizon AdS 3 × S 3 × T 4 with X = P 1 × T 2 , and either B = P 1 × T 2 × T 2 or B = S 3 × T 3 , and S = S 3 × T 5 . One of the consequences of the explicit examples we have constructed is that the same horizon section S = S 3 × S 3 × T 2 gives rise to two different near horizon geometries M = AdS 3 × S 3 × S 3 × S 1 and M = SL(2, R) × S 3 × S 3 × T 2 /S 1 . As far as we know, this is the first time that such a phenomenon has been observed. As such it represents an additional difficulty in the classification of black holes in higher dimensions as well as in the investigation of their thermodynamical properties.
To give evidence that there are more heterotic horizons, we demonstrate that the associated cohomological conditions have solutions for X a del Pezzo surface. Considering the del Pezzo surfaces as P 2 blown up at k < 9 points, we find that if k is odd the cohomological conditions are met provided that the Kähler class is identified with the anti-canonical class. If k is even, then a different choice for a Kähler class has to be made. There are solutions for all del Pezzo surfaces. We also investigate the topology of the associated heterotic horizons. For this we compute the de Rham cohomology of S. We find that in some cases the horizon sections have the same de Rham cohomology as that of (k − 1)(S 2 × S 4 )#k(S 3 × S 3 ) × T 2 , and under an additional assumption they are diffeomorphic to them. We also point out that the cohomology of these black hole horizons, and in particular some intersection matrices, are related to exceptional groups. This is inherited from the relation of the intersection matrix of the second cohomology of del Pezzo surfaces to the Cartan matrix of the exceptional groups. Our cohomological conditions do not have a solution on P 2 blown up at more than 9 points. However this restriction can be removed if the Bianchi identity of the 3-form field strength is modified either by adding point sources or by taking into account the heterotic anomaly.
We also explore the possibility of extending our near horizon geometries to full black hole solutions. One may expect that there is a continuous interpolation between a horizon section and a section of the asymptotic geometry of a black hole, ie the two sections are cobordant. We argue that in the presence of fermions, and in particular supersymmetry, the two sections must represent the same class in the spin co-bordism ring Ω spin * . Similar tests have been proposed elsewhere [19] using oriented cobordism. We find that most of our horizons can be associated with asymptotically flat or AdS black holes.
Since for all our solutions the 3-form field strength is closed, they can also be interpreted as solutions IIA, IIB and 11-dimensional supergravity. Lifting our heterotic solutions to 11-dimensional supergravity and then reducing them in a different direction, we show that new solutions can be constructed in IIA supergravity which have an AdS 2 component and RR fluxes. These can also be further T-dualized to find new solutions in IIB supergravity. Interpreting these new solutions as near horizon geometries, we provide evidence that both IIA and IIB supergravity admit black hole solutions with non-trivial topology and with all form fluxes non-vanishing.
In the cases that have been investigated so far, the near horizon supersymmetric solutions also arise as near brane geometries. For the 1/2 supersymmetric near horizon heterotic geometries this has been demonstrated in [20]. This is also the case for the 1/4 supersymmetric AdS 3 × S 3 × S 3 × S 1 solution. It turns out that this is the near brane geometry of two 5-branes with a localized string superposition. It is likely that the rest of the solutions have a near brane interpretation. This paper is organized as follows. In section two, we use the results of [20] to reconstruct the geometry of the horizon M and that of the horizon section S from geometric data on B and those of the toric fibration over B. We also summarize and extend some results of [23] on the relation between Hermitian conformally balanced manifolds and toric fibrations. In section 3, we give the differential system needed to construct heterotic horizon sections as T 2 fibrations over a conformally balanced Hermitian 6-dimensional manifold. We present an explicit solution with horizon section SU(3). In section 4, we construct heterotic horizon sections as T 4 fibrations over Kähler 4-dimensional manifolds X. We show that this leads to a differential system of 6 non-linear equations for 6 functions and has a cohomological analogue in the cohomology ring of X. We give an explicit example that solves the differential system. In section 5, we find solutions of the cohomological problem on del Pezzo surfaces. In section 6, we find brane configurations with near brane geometries similar to those that arise as near horizon geometries. In section 7, we explore our solutions in type II and 11-dimensional supergravities. In section 8, we use spin cobordism to provide evidence that our near horizon geometries can be extended to black hole solutions and in section 9, we give our conclusions. In appendix A, we compute the cohomology of horizon sections, and in appendix B we generalized the differential system that arises in the context of heterotic horizons.
Geometric conditions
Heterotic horizons which preserve 4 supersymmetries, and their associated sections, are fibrations over a 6-dimensional manifold B 6 . In particular, the horizon spacetime M is a SL(2, R) × U(1) fibration while the section S is a T 2 fibration. The fibre directions twist over B 6 with a non-trivial connection. In what follows, we shall reconstruct both M and S from geometric data given on B 6 and on the twisting of the fibration.
Reconstruction of the horizon sections
The geometry of a horizon spacetime M is completely determined in terms of the geometric data on the horizon section S. Because of this, we first reconstruct S as a T 2 fibration over B 6 . Let ds 2 (6) be the metric and H (6) be the torsion of B 6 . The metric and torsion of S can be written as
ds 2 (8) = k −2 h ⊗ h + k −2 ℓ ⊗ ℓ + ds 2 (6) ,H (8) = k −2 h ∧ dh + k −2 ℓ ∧ dℓ + H (6)
, (2.1) respectively, where h and ℓ are 1-forms on S which are interpreted as principal bundle connections associated with the fibration, and k 2 is the constant length of h and ℓ. Both ds 2 (8) andH (8) are invariant under the rotations of the T 2 fibre generated by the vector fields ξ and η dual to h and ℓ, respectively, relative to the metric 2 ds 2 (8) . In particular, one has
h = dτ + α i e i , ℓ = dσ + β i e i (2.2)
where 0 ≤ τ, σ ≤ 2π are the coordinates of T 2 , α and β are connections on B 6 and (e i ) is a local frame, ds 2 (6) = δ ij e i e j . The components of the metric and torsion depend only on the coordinates of B 6 .
Supersymmetry restricts both the connection h and ℓ, and the geometry of the base space B 6 as follows. B 6 is a Hermitian manifold with Hermitian form ω (6) which is compatible with the metric connection,∇ (6) , with skew-symmetric torsion, H (6) , iê
∇ (6) ω (6) = 0 . (2.3)
This is equivalent to setting
H (6) = −i I (6) dω (6) , dH (6) = 0 ,(2.4)
where I (6) is the complex structure on B 6 . Moreover, it is required that B 6 is conformally balanced. This means that the Lee form of B 6 is
θ ω (6) = 2dΦ , (2.5)
where Φ is the dilaton that depends only on the coordinates of B 6 . This summarizes the conditions on B 6 . The restriction on the twisting of the T 2 fibration over B 6 is made by putting appropriate conditions on the connections h and ℓ. In particular, N = 4 supersymmetry requires that
dh 2,0 = dℓ 2,0 = 0 , dh ij ω ij (6) = 0 , dℓ ij ω ij (6) = −2k 2 . (2.6)
This means that the curvature of the torus fibration is (1,1) with respect to the complex structure of B 6 and in addition one of the components of the connection is traceless while the other has constant trace. The first two conditions on the curvature of the fibration can be solved by requiring that S is a holomorphic fibration. It is straightforward to see that the Hermitian form
ω (8) = 1 k 2 h ∧ ℓ + ω (6) ,(2.7)
gives rise to an integrable complex structure on S.
Collecting the above data, one finds that
∇ (8) ω (8) = 0 ,∇ (8) h =∇ (8) ℓ = 0 ,(2.8)
where∇ (8) is the metric connection on S with skew-symmetric torsion H (8) . Consistency therefore requires thatH
(8) = −i I (8) dω (8) , (2.9)
which is equivalent to (2.1) and (2.4). So far the data imply that the holonomy of∇ (8) is
contained in U(3), hol(∇ (8) ) ⊆ U(3).
Furthermore, supersymmetry requires that the holonomy of∇ (8) must be contained in SU(3). One way to enforce this is to require that the Ricci form of the∇ (8) connection vanishesρ
(8) = 1 4 (R (8) ) kℓ, i j I j (8) i e k ∧ e ℓ = 0 ,(2.10)
where k, ℓ are S frame indices andR (8) is the curvature of the connection∇ (8) . This requirement gives several conditions. The only one which is independent from those that have already been stated is
(ρ (6) ) ij − dℓ ij = 0 ,(2.11)
whereρ (6) is the Ricci form of the∇ (6) connection. If B 6 is simply connected, this and (2.8) are necessary and sufficient conditions for hol(∇ (8) ) ⊆ SU(3). In particular, this implies that there is a (3,0) form χ such that
∇ (8) χ = 0 , i ξ χ = i η χ = 0 ,(2.12)
as stated in [20]. The geometries of B 6 and S are summarized in table 1.
Geometry B 6 S Hermitian yes yes Conformally balanced yes no θ = 2dΦ hol(∇) ⊆ SU (3) no yes hol(∇) ⊆ U (3) yes no dH (n) = 0 no yes The only remaining condition that needs to be satisfied in order to find a solution for both the Killing spinor and field equations of the theory is dH (8) = 0. This leads to
dH (8) = k −2 dh ∧ dh + k −2 dℓ ∧ dℓ + dH (6) = k −2 dh ∧ dh + k −2 dℓ ∧ dℓ − di I (6) dω (6) = 0 .
(2.13)
One can easily modify this condition if the heterotic anomaly is taken into account. But we shall not investigate this case here. To summarize, if S is taken to be a holomorphic torus fibration over a 6-dimensional Hermitian manifold, the conditions that must be satisfied to find a solution are
dh ij ω ij (6) = 0 , dℓ ij ω ij (6) = −2k 2 , θ ω (6) = 2dΦ , (ρ (6) ) ij − dℓ ij = 0 , k −2 dh ∧ dh + k −2 dℓ ∧ dℓ − di I (6) dω (6) = 0 .
(2.14)
In what follows, we investigate these conditions and give some explicit solutions.
Reconstruction of spacetime
The spacetime M of heterotic horizons preserving 4 supersymmetries is a SL(2, R) × U(1) fibration over B 6 . The base space B 6 satisfies all the properties mentioned in the previous section for constructing S. It remains to give the connection λ of M. This is expressed in terms of the connections h and ℓ of S as follows:
λ − = e − , λ + = e + − 1 2 k 2 u 2 e − − uh , λ 1 = k −1 h + k 2 ue − , λ 6 = k −1 ℓ ,(2.15)
where e − = dr + rh , e + = du .
(2.16)
The spacetime metric and torsion are given as
ds 2 ≡ 2λ − λ + + (λ 1 ) + (λ 6 ) 2 + ds 2 (6) = 2du(dr + rh) + k −2 h ⊗ h + k −2 ℓ ⊗ ℓ + ds 2 (6) , H ≡ CS(λ) + H (6) = du ∧ dr ∧ h + rdu ∧ dh + k −2 h ∧ dh + k −2 ℓ ∧ dℓ − i I (6) dω (6) . (2.17)
It is clear that given the geometric data on S, the geometry of M is completely described.
Toric fibrations
It is clear from the conditions that arise from supersymmetry as well as the examples that we shall investigate later that toric fibrations are central in the examination of near horizon geometries. Because of this, we shall derive some useful formulae for the analysis which shall follow. Toric fibrations in the context of manifolds with skew symmetric torsion and SU holonomy have been investigated before [22,23,24,25]. Suppose that 2n-dimensional manifold Y is a T 2(n−m) fibration over 2m-dimensional manifold X. Write the metric ds 2 and torsion H on Y as
ds 2 (2n) = δ ab λ a λ b + ds 2 (2m) , H (2n) = δ ab λ a ∧ dλ b + H (2m) ,(2.18)
where ds 2 (2m) and H (2m) are the metric and torsion on X. Suppose now that in addition X is a Hermitian manifold with complex structure I (2m) compatible with ds 2 (2m) . As is well known, the conditioñ
∇ (2m) I (2m) = 0 ,(2.19)
implies that
H (2m) = −i I (2m) dω (2m) ,(2.20)
where ω (2m) is the Hermitian form of X.
In turn, Y admits an almost Hermitian form ω (2n) compatible with ds 2 (2n) given by
ω (2n) = − n−m k=1 λ k ∧ λ n−m+k + ω (2m) . (2.21)
The associated almost complex structure I (2n) is integrable provided that the curvatures
F a = dλ a (2.22)
are (1,1)-forms on X,
(F a ) 2,0 = 0 . (2.23)
Furthermore, the connection with torsion,∇ (2n) , on Y is compatible with the complex structure,
I (2n) , on Y , ie∇ (2n) I (2n) = 0 ,(2.24)
and so
H (2n) = −i I (2n) dω (2n) . (2.25) In addition 3∇ (2n) λ a = 0 . (2.26)
As a result, the holonomy of∇ (2n) is contained in U(m),
hol(∇ (2n) ) ⊆ U(m) . (2.27)
Now let us investigate the conditions under which the holonomy of∇ (2n) reduces further to a subgroup of SU(m). For this, we express the curvature of the∇ (2n) connection in terms of that of∇ (2m) to find
R kℓ, i j =R kℓ, i j − δ ab F a kℓ F bi j , R ab, i j = F a i k F b k j − F b i k F a k j , R ak, i j =∇ k F a i j .
(2.28)
A necessary condition 4 for the holonomy of∇ (2n) to reduce to SU(n) is that the Ricci formρ
(2n) = 1 4R kℓ, i j I j i e k ∧ e ℓ ,(2.29)
of Y vanishes. This in turn gives
(ρ (2m) ) kℓ − 1 2 δ ab F a kℓ F bi j (I (2m) ) j i = 0 , ∇ k F a i j (I (2m) ) j i = 0 .
(2.30)
The first condition above can be simplified somewhat provided that X is conformally balanced, ie θ ω (2m) = 2dΦ. In particular after a bit of computation, one finds that
ρ (2m) = 1 4R kℓ, i j (I (2m) ) j i e k ∧ e ℓ = 1 2 dd I log det g − 2dd I Φ = −i∂∂ log det g (2m) + 4i∂∂Φ . (2.31)
where det g (2m) is the determinant of the Hermitian metric of X, ie det g (2m) = det(g (2m)αβ ).
Observe that the expression for the Ricci form of Hermitian conformally balanced manifolds is very similar to that of Kähler manifolds.
It is important to notice that even if X is conformally balanced, Y may not be. In particular, if X is conformally balanced a sufficient condition for Y to be conformally balanced is
F a ij ω ij (2m) = 0 . (2.32)
We shall use this when we consider horizon sections which are T 4 fibrations over Kähler 4-dimensional manifolds.
3 Horizons with 4 supersymmetries
Forms and geometric conditions
To solve the conditions (2.14), we shall take B 6 to be a Hermitian conformally balanced manifold and the T 2 -torus fibration to be holomorphic. Write S = P ⊠ Q, where P and Q are principal circle bundles over B 6 associated with the curvatures dh and dℓ, respectively. It is convenient to rewrite the conditions stated in (2.14) in form notation. After some straightforward computation, one finds that
dh ∧ ω 2 (6) = 0 , dℓ ∧ ω 2 (6) = − k 2 3 ω 3 (6) , d e −2Φ ω 2 (6) = 0 , ρ (6) − dℓ = 0 , k −2 dh ∧ dh + k −2 dℓ ∧ dℓ − di I (6) dω (6) = 0 . (3.1)
The first two conditions can be easily recognized as the Hermitian-Einstein conditions without and with cosmological constant, respectively, appropriately generalized for Hermitian conformally balanced manifolds. The third condition is the conformal balanced condition. The fourth condition identifies Q with the canonical bundle K over B 6 . Moreover observe that for conformally balanced manifoldŝ
ρ (6) = 1 2 dd I log det g (6) − 2dd I Φ = −i∂∂ log det g (6) + 4i∂∂Φ ,(3.2)
where det g (6) = det(g (6)αβ ) is the determinant of the Hermitian metric of B 6 . The conditions stated in (3.1) can be easily adapted to the cases for which B 6 admits some additional structure. Several possibilities are available. For example, one can assume that B 6 with respect to ω (6) is Kähler, conformally Kähler or balanced. In particular, the latter condition requires that the dilaton is constant. We shall demonstrate that the differential system (3.1) admits solutions. Another class of examples can be generated by taking B 6 to be a T 2 fibration over a Hermitian 4-manifold. This case will be investigated separately. The general case leads to a non-linear system that contains equations of Monge-Ampére type.
Cohomological conditions
All conditions in (3.1) lead to restrictions on the cohomology ring of B 6 . To see this observe that the conformal balanced condition for B 6 implies that e −2Φ ω 2 (6) is closed. Denote the cohomology class of e −2Φ ω 2 (6) with [e −2Φ ω 2 (6) ]. Then (3.1) implies that the cohomology classes are restricted as
c 1 (P ) ∧ [e −2Φ ω 2 (6) ] = 0 , c 1 (Q) ∧ [e −2Φ ω 2 (6) ] = − k 2 6π [e −2Φ ω 3 (6) ] , c 1 (B 6 ) − c 1 (Q) = 0 , c 1 (P ) ∧ c 1 (P ) + c 1 (Q) ∧ c 1 (Q) = 0 , (3.3)
where c 1 denotes the first Chern class of the appropriate circle bundle and c 1 (B 6 ) = 1 2π [ρ (6) ] = K is the first Chern class of the canonical bundle of B 6 . The conditions on the cohomology stated above are necessary for the existence of solutions. This means one should first seek manifolds B 6 and T 2 -bundles over them that satisfy the cohomological conditions (3.3) and then try to solve the differential conditions (3.1). One can also show that some of the cohomological conditions are also sufficient to find solutions to some of the equations, eg solutions for the first equation in (3.1). However, it is not known in general that holomorphic T 2 fibrations over conformally balanced Hermitian manifolds that satisfy all the cohomological conditions (3.3) are also solutions of (3.1). We shall demonstrate though that (3.1) admits solutions by constructing explicit examples.
New solution with horizon section SU (3)
We
shall demonstrate that S = SU(3) is a solution of (3.1) with B 6 = SU(3)/T 2 , where T 2 is identified with a maximal torus of SU(3). The metric of S is written ds 2 (8) = δ ij e i e jde i = − 1 2 c ijk e j ∧ e k (3.5)
where c are the structure constants of the Lie algebra SU(3) with respect to a real basis.
In particular, we work with a normalization with respect to which
c 126 = c 135 = c 368 = −c 234 = −c 258 = −c 456 = − 1 √ 2 , c 148 = − √ 2, c 257 = c 367 = − 3 2 . (3.6)
The hermitian form on SU(3) is taken to be
ω (8) = e 1 ∧ e 4 + e 2 ∧ e 5 + e 3 ∧ e 6 + e 7 ∧ e 8 ,(3.7)
and it is straightforward to check that the Maurer-Cartan equations imply that ω (8) is integrable. This complex structure has been introduced on SU (3) in [30] written in a different basis. The 3-form flux (8) is given by
H (8) = −di I (8) dωH (8) = −c , (3.8)
which is covariantly constant with respect to the Levi-civita connection and hence closed, and∇ (8) is flat. In the conventions we have adopted, SU (3) can be considered as a T 2 fibration over B 6 , where the T 2 lies in the directions spanned by e 7 and e 8 , so
ds 2 (6) = δ ij e i e j , ω (6) = e 1 ∧ e 4 + e 2 ∧ e 5 + e 3 ∧ e 6 (3.9)
for i, j = 1, . . . 6, e i = e i . Note that the Hermitian form ω (6) is different from the Kirillov symplectic form put on regular co-adjoint orbits of semi-simple groups. In particular, ω (6) defined above is not closed, however SU(3)/T 2 is balanced since
dω 2 (6) = 0 ,(3.10)
and so the dilaton is constant. The connections h, ℓ are defined as
h = √ 6e 8 − √ 2e 7 , ℓ = − √ 6e 7 − √ 2e 8 (3.11)
and so for these solutions, k = 2 √ 2. One can use the Maurer-Cartan equations to verify that the associated curvatures dh and dℓ satisfy the conditions (3.1), with flux H (6) = −i I (6) dω (6) given by
H (6) = 1 √ 2 e 1 ∧ e 2 ∧ e 3 + e 1 ∧ e 5 ∧ e 6 − e 2 ∧ e 4 ∧ e 6 − e 3 ∧ e 4 ∧ e 5 .
Horizon section
S 1 × S 3 × K 3
The S 1 × S 3 × K 3 horizon section is a T 2 fibration over B = S 2 × K 3 . There are two ways of demonstrating this. One is to begin from S 1 × S 3 × K 3 and project down on
B = S 2 × K 3 or alternatively reconstruct S 1 × S 3 × K 3 from S 2 × K 3 .
It is convenient to do the former operation. Of course both procedures give to the same result. To begin introduce the left invariant 1-forms on the 3-spheres as σ i , and the 1-form τ along S 1 . The Maurer-Cartan equations give 13) and cyclically in the indices 1,2,3. Then write the metric on S as
dσ 3 = σ 1 ∧ σ 2 , dτ = 0 ,(3.ds 2 (8) = (σ 3 ) 2 + (σ 1 ) 2 + (σ 2 ) 2 + (τ ) 2 + ds 2 (K 3 ) ,(3.14)
where
ds 2 (K 3 ) is the hyper-Kähler metric on K 3 . The Hermitian form on S 1 × S 3 × K 3 is ω (8) = −σ 3 ∧ τ − σ 1 ∧ σ 2 + ω(K 3 ) ,(3.15)
where ω(K 3 ) is a (1,1)-Kähler form on K 3 . It is easy to see that the complex structure is integrable. Moreover, one finds that
H (8) = σ 1 ∧ σ 2 ∧ σ 3 . (3.16)
Since the connection with torsion on the group manifold S 1 × S 3 is flat, the only contribution in the holonomy of∇ (8) comes from the Levi-Civita connection of K 3 and so hol(∇ (8) ) = SU(2). Thereforeρ
(8) = 0 . (3.17) Observe though that S 3 × S 3 × T 2 is not (conformally) balanced.
To investigate the geometry of B = S 2 × K 3 consider the 2-form
ω = −σ 1 ∧ σ 2 + ω(K 3 ) , (3.18) on S 1 × S 3 × K 3 .
If the T 2 directions of the fibre are along σ 3 and τ , ω descends to a Hermitian form on B as
i σ 3 ω = i τ ω = 0 ,(3.19)
and
L σ 3 ω = i σ 3 dω = 0 ,(3.20)
and similarly
L τ ω = 0 . (3.21)
So we set ω (6) = ω. Since dω = 0, B = S 2 × K 3 is Kähler and so balanced as required.
The dilaton is constant. The curvatures of the principal bundle connections are
dℓ = σ 1 ∧ σ 2 , dh = 0 . (3.22)
In particular the connection which twists AdS 3 is flat and so the spacetime is a product AdS 3 × S 3 × K 3 . Moreover dℓ is (1,1) and its trace is constant. Note in addition that ρ (6) does not vanish. It receives a contribution from S 2 . However, this cancels the contribution of dℓ so that (3.17) is satisfied. The solution preserves 1/2 of the supersymmetry [20].
New solution with horizon section
S 3 × S 3 × T 2
The S 3 ×S 3 ×T 2 horizon section is a T 4 fibration over X = S 2 ×S 2 . As in the S 1 ×S 3 ×K 3 case investigated previously, we shall begin from S 3 × S 3 × T 2 and project down to B 6 . To begin introduce the left invariant 1-forms on the two 3-spheres as σ i and ρ i which satisfy the Maurer-Cartan equations
dσ 3 = σ 1 ∧ σ 2 , dρ 3 = ρ 1 ∧ ρ 2 ,(3.23)
and cyclically in the indices 1,2,3, respectively. Then write the metric on S as
ds 2 (8) = (σ 3 ) 2 + (σ 1 ) 2 + (σ 2 ) 2 + (ρ 3 ) 2 + (ρ 1 ) 2 + (ρ 2 ) 2 + (τ 1 ) 2 + (τ 2 ) 2 ,(3.24)
where
dτ 1 = dτ 2 = 0 , (3.25) are the standard 1-forms on T 2 . The Hermitian form on S 3 × S 3 × T 2 is ω (8) = 1 √ 2 τ 1 ∧ (σ 3 + ρ 3 ) − σ 1 ∧ σ 2 − ρ 1 ∧ ρ 2 − 1 √ 2 τ 2 ∧ (σ 3 − ρ 3 ) ,(3.26)
leading toH
(8) = σ 1 ∧ σ 2 ∧ σ 3 + ρ 1 ∧ ρ 2 ∧ ρ 3 , (3.27)
which is the expected 3-form field strength for group manifolds. It is easy to see using the Maurer-Cartan equations that the associated complex structure is integrable. Moreover ∇ (8) is flat and so its holonomy is contained in SU (3). Observe though that S 3 × S 3 × T 2 is not (conformally) balanced. The T 2 fibration of the horizon section is chosen along the vector fields dual to τ 1 and σ 3 + ρ 3 . Moreover set
ω = −σ 1 ∧ σ 2 − ρ 1 ∧ ρ 2 − 1 √ 2 τ 2 ∧ (σ 3 − ρ 3 ) . (3.28)
After a straightforward calculation, it is easy to see that
i σ 3 +ρ 3 ω = i τ 1 ω = 0 ,(3.29)
and
L σ 3 +ρ 3 ω = i σ 3 +ρ 3 dω = 0 ,(3.30)
and similarly
L τ 1 ω = 0 . (3.31)
Therefore ω descends to a Hermitian form on B 6 = (S 3 × S 3 )/S 1 × S 1 . So ω = ω (6) and the metric on B 6 is
ds 2 (6) = 1 2 (σ 3 − ρ 3 ) 2 + (σ 1 ) 2 + (σ 2 ) 2 + (ρ 1 ) 2 + (ρ 2 ) 2 + (τ 2 ) 2 . (3.32)
Observe that B is balanced
dω 2 (6) = 0 , (3.33) as expected. Choosing h = √ 2τ 1 , ℓ = σ 3 + ρ 3 , (3.34) so that k = √ 2, one finds that dh = 0 , dℓ = σ 1 ∧ σ 2 + ρ 1 ∧ ρ 2 . (3.35)
So h, ℓ satisfy the properties for the curvatures of the T 2 fibration as required by supersymmetry. Since dh = 0, the spacetime is AdS 3 × S 3 × S 3 × S 1 . As we shall demonstrate later, this is the near horizon geometry of two 5-branes intersecting on a string with the string localized on the transverse space.
Horizons as torus fibrations over a 4-manifold 4.1 Geometric conditions
A large class of heterotic horizons can be found by taking B 6 to be a T 2 fibration over a 4-dimensional balanced Hermitian manifold X. All 4-dimensional conformally balanced Hermitian manifolds are conformally Kähler. Since the conformal balanced condition is with respect to the dilaton Φ, one writes for the Hermitian form of X ω X = e 2Φ κ , dκ = 0 , (4.1)
where κ is the Kähler form. Therefore S is a T 4 fibration over X. As before we consider the two principal bundle connections ℓ and h, set h = h 1 , and introduce two connections h 2 and h 3 along the two additional torus directions. It follows from the properties of torus fibrations in section 2 that B 6 is conformally balanced provided that the curvatures dh 2 and dh 3 are (1,1) and traceless with respect to κ. Using this, we summarize the conditions required for a spacetime to preserve 4 supersymmetries as follows
dh 1 ∧ κ = 0 , dh 2 ∧ κ = 0 , dh 3 ∧ κ = 0 , dℓ ∧ κ = − k 2 2 e 2Φ κ 2 , dκ = 0 , −i∂∂ log det(iκ) − dℓ = 0 , k −2 dh 1 ∧ dh 1 + k −2 dh 2 ∧ dh 2 + k −2 dh 3 ∧ dh 3 + k −2 dℓ ∧ dℓ + 2i∂∂e 2Φ ∧ κ = 0 . (4.2)
As in the general case the above conditions lead to restrictions on the cohomology of X. Here, in addition, the above differential system can be rewritten as a system of six equations for six functions. Before we do this, we shall first describe the conditions on the cohomology on X.
Cohomological conditions
It is clear that S = P 1 ⊠ P 2 ⊠ P 3 ⊠ Q, where P 1 , P 2 , P 3 , Q are principal circle bundles over X and P 1 = P . The conditions (4.2) imply the cohomology classes are restricted as
c 1 (P 1 ) ∧ [κ] = 0 , c 1 (P 2 ) ∧ [κ] = 0 , c 1 (P 3 ) ∧ [κ] = 0 , c 1 (Q) ∧ [κ] = − k 2 4π [e 2Φ κ 2 ] , c 1 (X) − c 1 (Q) = 0 , c 1 (P 1 ) ∧ c 1 (P 1 ) + c 1 (P 2 ) ∧ c 1 (P 2 ) + c 1 (P 3 ) ∧ c 1 (P 3 ) + c 1 (Q) ∧ c 1 (Q) = 0 , (4.3)
where c 1 denotes the first Chern class of the appropriate circle bundle and c 1 (X) is the first Chern class of the canonical bundle of X. It is clear from the cohomology conditions above that c 1 (Q) must be identified with the canonical class of X. Moreover, the class c 1 (Q) ∧ [κ] must be a negative multiple of the volume class of X.
The above conditions can also be rewritten in terms of the intersection form of X
[α] · [β] = X α ∧ β . (4.4)
The only difference is that the ring product in H 2 (X) is replaced with the product as defined by the intersection form.
Differential system
To express the differential conditions as a system of six equations for six functions, we shall use the ∂∂-lemma and the cohomological conditions stated above. It is assumed that X is chosen such that the conditions (4.3) have a solution. The procedure resembles the background field method used for quantum calculations in field theory. Thus one splits the fields, which are represented by the connections, metric, hermitian form and dilaton, into a background part and a fluctuation. Though here, the fluctuations are not restricted to be small. In particular, introduce fixed background fieldsh 1 ,h 2 ,h 3 ,l,Φ and κ which satisfy the cohomological conditions (4.3). Then using the ∂∂-lemma, there are (1,1)-forms α 1 , α 2 , α 3 , χ and ψ, and a function f , which depend only on the background data, such that
dh 1 ∧κ = i∂∂α 1 , dh 2 ∧κ = i∂∂α 2 , dh 3 ∧κ = i∂∂α 3 , dl ∧κ = − k 2 2 e 2Φκ2 + i∂∂χ , − i∂∂ log det(iκ) + i∂∂f − dl = 0 , k −2 (dh 1 ∧ dh 1 + dh 2 ∧ dh 2 + dh 3 ∧ dh 3 + dl ∧ dl) + i∂∂ψ = 0 . (4.5)
Since we are seeking solutions that preserve the chosen cohomological classes, and using again the ∂∂-lemma, there are functions s 1 , s 2 , s 3 , v and w such that
dh 1 = dh 1 + i∂∂s 1 , dh 2 = dh 1 + i∂∂s 2 , dh 3 = dh 1 + i∂∂s 3 , dℓ = dl + i∂∂v , κ =κ − i∂∂w , Φ =Φ + ϕ ,(4.6)
where we have also expressed the dilaton 5 in terms of the background field and a fluctuation ϕ.
Substituting (4.6) into (4.2) and subtracting from the resulting expressions (4.5), one finds
i∂∂s 1 ∧κ − dh 1 ∧ i∂∂w + ∂∂s 1 ∧ ∂∂w = −i∂∂α 1 , i∂∂s 2 ∧κ − dh 2 ∧ i∂∂w + ∂∂s 2 ∧ ∂∂w = −i∂∂α 2 , i∂∂s 3 ∧κ − dh 3 ∧ i∂∂w + ∂∂s 3 ∧ ∂∂w = −i∂∂α 3 , i∂∂v ∧κ − dl ∧ i∂∂w + ∂∂v ∧ ∂∂w = − k 2 2 e 2(Φ+ϕ) (κ − i∂∂w) 2 + k 2 2 e 2Φκ2 − i∂∂χ , det iκ + ∂∂w iκ = e −f −v+c , k −2 [2idh 1 ∧ ∂∂s 1 − ∂∂s 1 ∧ ∂∂s 1 + 2idh 2 ∧ ∂∂s 2 − ∂∂s 2 ∧ ∂∂s 2 + 2idh 3 ∧ ∂∂s 3 −∂∂s 3 ∧ ∂∂s 3 + 2idl ∧ ∂∂v − ∂∂v ∧ ∂∂v] + 2i∂∂e 2(Φ+ϕ) ∧ (κ − i∂∂w) = i∂∂ψ , (4.7)
where c is a constant. This is a non-linear system of six equations for the six functions s 1 , s 2 , s 3 , v, w and ϕ. It contains a Monge-Ampére type of equation. It is easy to see that each equation can be solved for one unknown function treating the remaining functions as sources. For example if v is a source, the Monge-Ampére equation is identical to the one solved by Yau for the proof of the Calabi conjecture. However, it is less clear that the full non-linear system has solutions. Therefore the question rises as to whether for every solution of the cohomological conditions (4.3), there is a smooth solution of (4.7).
There is also the possibility that (4.7) that does not have any solutions at all. This is not the case. We shall give explicit examples below which solve all the conditions. Moreover, we shall explore the conditions on the cohomology. We shall demonstrate that these have solutions for X a del Pezzo surface.
New solution with horizon section S 3 × S 3 × T 2
The S 3 ×S 3 ×T 2 horizon section is a T 4 fibration over X = S 2 ×S 2 . As in the S 1 ×S 3 ×K 3 case investigated previously, we shall begin from S 3 × S 3 × T 2 and project down to B and to X. To begin introduce the left invariant 1-forms σ i and ρ i on the two 3-spheres which satisfy the Maurer-Cartan equations as in (3.23). Then write the metric on S as
ds 2 (8) = (σ 3 ) 2 + (σ 1 ) 2 + (σ 2 ) 2 + (ρ 3 ) 2 + (ρ 1 ) 2 + (ρ 2 ) 2 + (τ 1 ) 2 + (τ 2 ) 2 ,(4.8)
where
dτ 1 = dτ 2 = 0 ,(4.9)
are the standard 1-forms on T 2 . The Hermitian form on S 3 × S 3 × T 2 is
ω (8) = −σ 3 ∧ ρ 3 − σ 1 ∧ σ 2 − ρ 1 ∧ ρ 2 − τ 1 ∧ τ 2 ,(4.10)
leading toH
(8) = σ 1 ∧ σ 2 ∧ σ 3 + ρ 1 ∧ ρ 2 ∧ ρ 3 ,(4.11)
which is the expected 3-form field strength for group manifolds. It is easy to see using the Maurer-Cartan equations that the associated complex structure is integrable. Moreover ∇ (8) is flat and so its holonomy is contained in SU(3). Observe though that S 3 × S 3 × T 2 is not (conformally) balanced. First, we shall investigate the geometry of B = S 2 × S 2 × T 2 and then of X. For this, consider the 2-form
ω = −σ 1 ∧ σ 2 − ρ 1 ∧ ρ 2 − τ 1 ∧ τ 2 ,(4.12)
on S 3 × S 3 × T 2 . If the fibre directions are along σ 3 and ρ 3 , ω descends to a Hermitian form on B as
i σ 3 ω = i ρ 3 ω = 0 ,(4.13)
and
L σ 3 ω = i σ 3 dω = 0 ,(4.14)
and similarly
L ρ 3 ω = 0 . (4.15)
So we set ω (6) = ω. Since dω = 0, B = S 2 ×S 2 ×T 2 is Kähler and so balanced as required.
The dilaton is constant. The canonical bundle of S 2 × S 2 × T 2 is not trivial. However, it becomes trivial after pulling it back on S 3 × S 3 × T 2 . To see this consider the (3,0) form
χ = 1 2 √ 2 (σ 1 + iσ 2 ) ∧ (ρ 1 + iρ 2 ) ∧ (τ 1 + iτ 2 ) , (4.16) on S 3 × S 3 × T 2 . Clearly i σ 3 χ = i ρ 3 χ = 0 , (4.17) but L σ 3 +ρ 3 χ = 2iχ , L σ 3 −ρ 3 χ = 0 . (4.18)
Since χ is transformed up to a phase in the σ 3 + ρ 3 direction, the trivial bundle with section χ over S 3 × S 3 × T 2 is projected down to a non-trivial bundle over S 2 × S 2 × T 2 . Furthermore, one can set
h = −(σ 3 − ρ 3 ) , ℓ = (σ 3 + ρ 3 ) . (4.19) Then dh = −(σ 1 ∧ σ 2 − ρ 1 ∧ ρ 2 ) (4.20)
which is (1,1) and traceless and
dℓ = (σ 1 ∧ σ 2 + ρ 1 ∧ ρ 2 ) (4.21)
which is (1,1) but not traceless. The trace of dℓ is constant as required.
Since B = S 2 × S 2 × T 2 , the 4-dimensional Kähler manifold X is X = S 2 × S 2 . The projection from B to X is along the trivial T 2 fibration. The Kähler form on X is κ = −σ 1 ∧ σ 2 − ρ 1 ∧ ρ 2 . Clearly, S 3 × S 3 × T 2 is a T 4 fibration over S 2 × S 2 with principal bundle connection ℓ = ℓ, h 1 = h, h 2 = τ 1 and h 3 = τ 2 . The spacetime is isomorphic to (AdS 3 × S 3 × S 3 )/S 1 × T 2 .
Horizon section S 3 × T 5
To describe the geometry, we introduce the left invariant 1-forms σ on S 3 as in the previous example. We also consider the 1-forms τ 3 , τ 4 and τ 5 spanning the T 3 directions that replace the second S 3 . So now
dτ 3 = dτ 4 = dτ 5 = 0 . (4.22)
The analysis is identical to the one presented in the previous example. The only difference is that the 1-forms τ which replace the ρ's are closed instead of satisfying (3.23). Thus one finds that B = S 2 × T 4 , which is Kähler, and X = S 2 × T 2 .
The principal bundle connections are ℓ = σ 3 , h 1 = τ 1 , h 2 = τ 2 and h 3 = τ 3 . The curvatures of the principal bundle connections are
dℓ = σ 1 ∧ σ 2 , dh 1 = dh 2 = dh 3 = 0 . (4.23)
Clearly all the curvatures are (1,1) and dℓ has constant trace. The spacetime is AdS 3 × S 3 × T 4 and preserves 1/2 of supersymmetry [20]. Since dh = 0, h = h 1 , AdS 3 does not twist over B.
Solutions of cohomology conditions
We have demonstrated that the differential system (4.7) admits solutions. Now we shall provide evidence that there may be a large class of solutions to (4.7) by proving that there are many manifolds that satisfy the cohomological conditions (4.3). We shall take X to be P 2 blown up at k points 6 . This surface has been used before in the context of manifolds with skew-symmetric torsion [24]. If k < 9, then these manifolds are del Pezzo surfaces dP 9−k . These have found applications in the context of mysterious duality [27], which relates the U-duality [26] 1/2 BPS states of M-theory toroidal compactifications to rational curves in del Pezzo surfaces, and supergravity [28]. Appropriately choosing the points that P 2 is blown up, the cohomology H 2 (X, Z) is generated by the hypersurface class H and the exceptional divisors E i , i = 1, . . . , k [36]. The intersection form is
H · H = 1 , H · E i = 0 , E i · E j = −δ ij . (5.1)
The anti-canonical class, −K, of X is
−K = 3H − E 1 − E 2 − · · · − E k . (5.2)
There is another basis in the cohomology which consists of the anti-canonical class −K and
α i = E i − E i+1 , i = 1, . . . , k − 1 ,
6 Throughout this section k is the number of points that P 2 is blown up and it should not be confused with the normalization factor that appears in the definition of metric and fluxes in previous sections, see eg (2.1).
α k = H − E 1 − E 2 − E 3 .
(5.3)
The anti-canonical class is orthogonal to the rest of the generators. The intersection matrix in the (α i ), i = 1, . . . , k, basis is
K · α i = 0 , α i · α j = −A ij , i, j = 1, . . . , k ,(5.4)
where (A ij ) is the Cartan matrix of exceptional Lie algebras E k , see table 1. As a result, the intersection matrix can be represented with the associated Dynkin diagram. To solve the remaining conditions in (4.3), one has to choose appropriately the Chern classes of the bundles P 1 , P 2 and P 3 , and [κ]. We shall consider the cases for which X is a del Pezzo surface and for which k > 9 separately.
k E k 1 A 1 2 A 1 ⊕ A 1 3 A 2 ⊕ A 1 4 A 4 5 D 5 6 E 6 7 E 7 8 E 8 k > 9 E k
del Pezzo
5.1.1 k < 9 odd
There are two cases to consider here depending on the choice of Kähler class. Not all classes in H 2 (X, R) can be represented by a Kähler form. Those that can obey the Nakai-Moishezon criteria. These will be stated later. In this case, the most straightforward choice is The solution for k = 1 is unique, modulo overall sign changes in the c 1 (P s ) and permutations of the P s . The Chern class can be chosen, without loss of generality, as
[κ] = −K. Observe that c 1 (Q) · [κ] = −K 2 = k − 9 < 0, ie c 1 (Q) ∧ [κ] isc 1 (P 1 ) = H − 3E 1 , c 1 (P 2 ) = c 1 (P 3 ) = 0 . (5.9) k = 3
There are seven distinct types of solutions up to permutations in the P s and overall changes of sign in c 1 (P s ). In particular, one finds There are seven distinct types of solutions up to permutations in the P s and overall changes of sign in c 1 (P s ). In all cases, one can take, without loss of generality c 1 (P 3 ) = 0, and the remaining Chern classes c 1 (P 1 ), c 1 (P 2 ) are
c 1 (P 1 ) = E a 1 − E a 2 , c 1 (P 2 ) = E b 1 − E b 2 , c 1 (P 3 ) = E f 1 − E f 3 , c 1 (P 1 ) = H − E 1 − E 2 − E 3 , c 1 (P 2 ) = E a 1 − E a 2 , c 1 (P 3 ) = E b 1 − E b 2 , c 1 (P 1 ) = ±c 1 (P 2 ) = H − E 1 − E 2 − E 3 , c 1 (P 3 ) = E a 1 − E a 2 , c 1 (P 1 ) = ±c 1 (P 2 ) = ±c 1 (P 3 ) = H − E 1 − E 2 − E 3 , c 1 (P 1 ) = −2E a 1 + E a 2 + E a 3 , c 1 (P 2 ) = c 1 (P 3 ) = 0 , c 1 (P 1 ) = H − 2E a 1 − E a 2 , c 1 (P 2 ) = E b 1 − E b 2 , c 1 (P 3 ) = 0 , c 1 (P 1 ) = H − 2E a 1 − E a 2 , c 1 (P 2 ) = H − E 1 − E 2 − E 3 , c 1 (P 3 ) = 0 , (5.10) where {a i } are distinct, {b i } are distinct, {f i } are distinctc 1 (P 1 ) = −H − E a 1 + E a 2 + E a 3 + E a 4 + E a 5 , c 1 (P 2 ) = 0 , c 1 (P 1 ) = −E a 1 − E a 2 + E a 3 + E a 4 , c 1 (P 2 ) = 0 , c 1 (P 1 ) = −H + E a 1 + E a 2 + E a 3 , c 1 (P 2 ) = E b 1 − E b 2 , c 1 (P 1 ) = E a 1 − E a 2 , c 1 (P 2 ) = E b 1 − E b 2 , c 1 (P 1 ) = −H + E a 1 + E a 2 + E a 3 , c 1 (P 2 ) = −H + E b 1 + E b 2 + E b 3 , c 1 (P 1 ) = H − 2E a 1 − E a 2 , c 1 (P 2 ) = 0 , c 1 (P 1 ) = 2H − 2E a 1 − E a 2 − E a 3 − E a 4 − E a 5 , c 1 (P 2 ) = 0 . (5.11) k = 7
There are three distinct types of solutions again up to permutations in the P s and overall changes of sign in c 1 (P s ). One can take, without loss of generality, c 1 (P 2 ) = c 1 (P 3 ) = 0. The remaining Chern class c 1 (P 1 ) is
c 1 (P 1 ) = −2H + E a 1 + E a 2 + E a 3 + E a 4 + E a 5 + E a 6 , c 1 (P 1 ) = −H + E a 1 + E a 2 + E a 3 , c 1 (P 1 ) = E a 1 − E a 2 .
(5.12)
k < 9 even
As we have mentioned, if the Kähler class is identified with the anti-canonical class −K, one cannot solve the cohomological conditions when k is even. However, this problem can be circumvented by choosing another Kähler class. For this write
[κ] = pH − i q i E i . (5.13)
The requirement that [κ] is represented by a Kähler class restricts the components p, q i . In particular according to Nakai-Moishezon criteria, [κ] must satisfy
• [κ] · [κ] > 0.
• [κ] · D > 0 for any irreducible curve D with negative self intersection.
• [κ] · C > 0 for an ample divisor C.
The divisor class nH − E 1 − E 2 · · · − E k is ample for sufficiently large n. On the blow up of distinct points on a smooth cubic in P 2 , the irreducible curves with negative intersection are E i , H − E i − E j , i = j for k ≤ 9. The above conditions lead to the restrictions
p 2 > k i=1 q 2 i , p > q i + q j , q i > 0 , i = 1, . . . , k . (5.14)
Writing
c 1 (P s ) = n s H − i m si E i , (5.15) the conditions c 1 (P s ) · [κ] = 0 imply n s p − i m si q i = 0 . (5.16) In addition, −c 1 (Q) · [κ] > 0 gives 3p − k i=1 q i > 0 . (5.17)
Furthermore, the condition associated with the closure of H (8) gives
9 − k + s n 2 s − s i (m si ) 2 = 0 . (5.18)
Examples of solutions to the cohomological conditions are as follows
k = 2 [κ] = 4H − E 1 − E 2 , c 1 (P 1 ) = H − 2E 1 − 2E 2 , c 1 (P 2 ) = c 1 (P 3 ) = 0 . (5.19) k = 4 [κ] = 4H − E 1 − E 2 − E 3 − E 4 , c 1 (P 1 ) = H − E 1 − E 2 − E 3 − E 4 , c 1 (P 2 ) = E 1 − E 2 , c 1 (P 3 ) = 0 . (5.20) k = 6 [κ] = 8H − 3E 1 − 3E 2 − 3E 3 − 3E 4 − 3E 5 − 3E 6 , c 1 (P 2 ) = c 1 (P 3 ) = 0 , c 1 (P 1 ) = 3H − 2E 1 − 2E 2 − E 3 − E 4 − E 5 − E 6 . (5.21) k = 8 [κ] = 17H − 6 8 i=1 E i , c 1 (P 2 ) = c 1 (P 3 ) = 0 , c 1 (P 1 ) = 6H − 3E 1 − 2E 2 − 2E 3 − 2E 4 − 2E 5 − 2E 6 − 2E 7 − 2E 8 . (5.22)
A direct observation reveals that c 1 (P s ) 2 ≤ 0 as required for classes represented by anti-self-dual 2-forms. Moreover in all cases −c 1 (Q) · [κ] > 0.
It is clear that there are many solutions of the cohomological conditions on del Pezzo surfaces. So it is likely that the differential system (4.2) admits solutions as well. This will give a new geometry on del Pezzo surfaces as it will be different from the Einstein metrics of [31]. This is because if the Einstein metric is chosen as a solution of the differential system (4.2), then dℓ will be proportional to the Kähler form and so the dilaton has to be constant. As a result, it is not possible to solve the equation which arises from dH = 0 as the wedge product of harmonic forms may not be harmonic.
k > 9
The cohomological system does not have solutions for k > 9 for any choice of Kähler class. This is because now K 2 = 9 − k < 0 and c 2 1 (P s ) ≤ 0 since the latter are represented by anti-self-dual forms. Therefore the cohomological condition associated with dH (8) = 0 cannot be satisfied. This can be confirmed using Nakai-Moishezon criteria for choosing the Kähler class and the rest of the conditions of the cohomological system. It is worth noting that for k ≥ 10, apart from E i , H − E i − E j , i = j, −K is also an irreducible curve with negative self-intersection. Then (5.17) arises as part of the Nakai-Moishezon criteria.
To include solutions with k > 9, the cohomological condition associated with dH (8) = 0 must be modified. One option is to include contributions from the anomaly cancelation which schematically leads to Choosing these appropriately, one can cancel the negative contributions arising from the canonical class and the rest of the Chern classes of the circle bundles.
dH (8) = − α ′ 4 (trR 2 − trF 2 ) .
Topology of heterotic horizons
In all cases, the horizon spacetime is contractible to the horizon section S. In turn the horizon sections are T 2 fibrations over B 6 . We shall focus on the case for which B 6 is a T 2 -fibration over a Kähler 4-manifold X. Then we shall adapt the calculation for a del Pezzo surface X = dP 9−k . We shall compute the cohomology in three different scenarios depending on the number of non-trivial line bundles which appear in the construction of S from X. First suppose that S = T 2 × B, and B is a non-trivial T 2 fibration over X. To simplify the computations, let us focus on de Rham cohomology. To compute the de Rham cohomology of S, it suffices to find the cohomology of B. Since B is a nontrivial fibration over X, the Chern classes c 1 (Q) = b 1 and c 1 (P ) = b 2 can be chosen as the first two basis elements in H 2 (X, R), ie H 2 (X, R) = R b 1 , . . . , b m . Moreover let H * (T 2 , R) = R θ 1 , θ 2 , θ 1 ∧ θ 2 . Using the spectral sequence for a fibration E p,q 2 = H p (X, H q (T 2 , R)) , p = 0, . . . , 4 , q = 0, . . . , 2 .
(5.25)
To find the cohomology of B, it suffices to calculate the action of the d 2 differential. This has been done in appendix A. It turns out that
H 0 (B, R) = H 6 (B, R) = R , H 2 (B, R) = H 4 (B, R) = R m−2 , H 3 (B, R) = R 2m−2 . (5.26)
We can also give the intersection matrices of the cohomology of B. For this suppose that the intersection matrix A of X is
X b e ∧ b f = A ef , e, f = 1, . . . , m. (5.27)
A basis in the cohomology of B 6 is
H 2 (B, R) = R b a , a = 1, 2 , H 3 (B, R) = R θ r ∧ b a , θ 1 ∧ b 1 + θ 2 ∧ b 2 , θ 1 ∧ b 2 + θ 2 ∧ b 1 . (5.28)
Observe that one of the generators of H 3 (B, R) is the Chern-Simons form. It is straightforward to compute the integrals
B ω (6) ∧ b a ∧ b b = −A ab , B θ r ∧ b a ∧ θ s ∧ b b = −ǫ rs A ab , B θ r ∧ b a ∧ (θ 1 ∧ b 1 + θ 2 ∧ b 2 ) = −ǫ r1 A a1 − ǫ r2 A a2 , B θ r ∧ b a ∧ (θ 1 ∧ b 2 + θ 2 ∧ b 1 ) = −ǫ r1 A a2 − ǫ r2 A a1 , B (θ 1 ∧ b 1 + θ 2 ∧ b 2 ) ∧ (θ 1 ∧ b 2 + θ 2 ∧ b 1 ) = −A 11 + A 22 . (5.29)
Apart from the first, the rest give the intersection matrix of B. Note that even though dω (6) = 0, the first integral does not depend on the representatives of the classes. Now suppose that X = dP 9−k . For the solutions for which [κ] = −b 1 = −K, one can choose, up to an appropriate rescaling, α = −K and β = b 2 as K · b 2 = 0 and b 2 2 = 0 to satisfy the assumptions stated in appendix A to calculate the cohomology. In the case that [κ] = −K, one chooses α = [κ]. Moreover, there is always a class with the properties of β. Clearly, the intersection matrices of B 6 inherit the exceptional structure of the cohomology of del Pezzo surfaces.
Moreover, it turns out that if X = dP 9−k , B 6 has the same de Rham cohomology as (k − 1)(S 2 × S 4 )#k(S 3 × S 3 ). In fact under some additional conditions, it is diffeomorphic to it [24]. The black holes that can arise from such horizons have non-trivial topology.
Next suppose that S = S 1 × Y and Y is a non-trivial T 3 fibration over X. In such case, the calculation explained in appendix A reveals that
H 0 (Y, R) = H 7 (Y, R) = R , H 2 (Y, R) = H 5 (Y, R) = R m−3 , H 3 (Y, R) = H 4 (Y, R) = R 3m−4 . (5.30)
Furthermore, if S is a non-trivial fibration over X, then the calculation in appendix A gives
H 0 (S, R) = H 8 (S, R) = R , H 2 (S, R) = H 6 (S, R) = R m−4 , H 3 (S, R) = H 5 (S, R) = R 4m−7 , H 4 (S, R) = R 6m−8 . (5.31)
It is clear that in all cases the cohomology of the horizon section is non-trivial indicating the existence of exotic heterotic black holes.
Near brane geometries for localized brane intersections
In all known cases, black hole horizons also arise as near brane geometries. It is likely that our heterotic horizon solutions can also be interpreted as the near brane geometry of a brane configuration. Since the only fluxes that are switched on are the dilaton and the 3-form field strength, it is expected that they correspond to the near brane geometry of a configuration of 5-branes and fundamental strings. In particular, we shall show that the AdS 3 × S 3 × S 3 × S 1 solution is the near brane geometry of two 5-branes intersecting on a string with the latter localized in all transverse directions. The metric, 3-form field 5brane 0 1 2 3 4 5 5brane 0 1 6 7 8 9 string 0 1 strength and dilaton for this configuration can be written as
ds 2 = h −1 s (x, y)(−dt 2 + dσ 2 ) + h 5 (x)dx 2 + h ′ 5 (y)dy 2 , H = dt ∧ dσ ∧ dh −1 s + ⋆ x dh 5 + ⋆ y dh ′ 5 , e 2Φ = h −1 s h 5 h ′ 5 ,(6.1)
where dx 2 and dy 2 is the Euclidean metric on the transverse spaces R 4 of the corresponding 5-brane and all the Hodge duals have been taken with the flat metric. If
h 5 = 1 + q 5 |x| 2 , h ′ 5 = 1 + q ′ 5 |y| 2 ,(6.2)
then dH = 0. To determine h s , one has to solve the field equation for the 2-form gauge potential which gives
h ′ 5 δ ab ∂ a ∂ b h s + h 5 δ a ′ b ′ ∂ a ′ ∂ b ′ h s = 0 . (6.3) A solution for h s is h s = 1 + q 1 |x| 2 + q ′ 1 |y| 2 + s |y| 2 |x| 2 ,(6.4)
see also [32,33]. The geometry near the common intersection is recovered in the limit |y| 2 , |x| 2 → 0 with the ratio |y|/|x| fixed. In this limit, the last term in the above equation dominates. Evaluating the metric near this limit after a change of co-ordinates one finds
AdS 3 × S 3 × S 3 × S 1 with constant dilaton. Moreover H = 2dvol(AdS 3 ) + dvol(S 3 ) + dvol(S 3 ) . (6.5)
This is the near horizon example which we have found in section 3.5.
Near horizon geometries for type II black holes
The solutions we have found have closed 3-form field strength and so they are solutions of type IIA and IIB supergravities. As solutions to IIA supergravity, they can be lifted to 11 dimensions. In particular the general form of the solution in (2.17) lifted to 11 dimensions reads ds 2 (11) = e − 2Φ 3 ds 2 (10) + e 4Φ 3 dy 2 F = dy ∧ H .
(7.1)
Reducing the solution back to IIA along the non-trivial fibre direction h, we find that
ds 2 (A) = e −Φ [−k 2 r 2 du 2 + 2dudr + k −2 ℓ ⊗ ℓ + ds 2 (6) + e 4Φ 3 dy 2 ] F 2 = k −2 (dh + dr ∧ du) , H = −k −2 dy ∧ dh , F 4 = dy ∧ [k −2 ℓ ∧ dℓ + H (6) ] , e 2Φ (A) = e −Φ .(7.2)
One may notice that the metric e Φ ds 2 (10) has an AdS 2 spanned by the coordinates r, u. Therefore if the dilaton is constant, and this is the case for many of our solutions, the resulting spacetime can be interpreted as the near horizon geometry of IIA black holes. The solution preserves all 4 supersymmetries as the reduction is done along h and dh is (1,1) and traceless. This is because the spinorial Lie derivative of all the Killing spinors along the vector field associated to h vanishes [34].
The solution can be further be T-dualized to IIB along the direction of ℓ. In this case, the metric of the IIB background is Again for constant dilaton, the metric has an AdS 2 component. Therefore such spacetimes can be interpreted as the near horizon geometries of IIB black holes. However in this case, all 4 supersymmetries of the IIA background will be broken after T-duality to IIB as dℓ although (1,1) is not traceless. The spinorial Lie derivative along the vector field associated to ℓ on the Killing spinor will not vanish. Supersymmetry under T-duality can be preserved provided that B 6 satisfies additional properties and the T-duality operation is taken along another direction. In particular if B 6 is a T 2 -fibration over a Kähler manifold, then the two additional fibre directions are associated with curvatures dh r , r = 2, 3, which are (1,1) and traceless. In this case, after T-duality along any of these two directions one obtains a IIB background with an AdS 2 factor preserving 4 supersymmetries. This is because again the spinorial Lie derivative of the IIA Killing spinor vanishes when is taken along one of the two fibre directions. So even though IIB supergravity does not have a 1-form gauge potential, there are supersymmetric black hole near horizon geometries.
From horizons to black holes
Having found a near horizon geometry, it is natural to ask whether this can be extended to a full black hole geometry. It is not expected that all near horizon geometries will give rise to black hole solutions. In the absence of a solution to the field equations, some qualitative tests have been devised. One such test is based on the expectation that there is a continuous interpolation of the horizon section to the compact section of the asymptotic geometry via a Cauchy surface which lies outside the horizon. Such a test suggests that the horizon and asymptotic sections are cobordant. The cobordism mostly used for this is the oriented cobordism ring Ω * , see [19]. However, it seems to us that in the presence of spinors which must be defined at both the horizon and the asymptotic region of spacetime, and in particular supersymmetry, the most relevant equivalence is that of spin cobordism ring Ω spin * . Ω spin n , n ≤ 8, has been computed in [35], see also references within. Here we shall use that Moreover Ω spin 1 is generated by the circle with the periodic spin structure, and Ω spin 8 is generated by P 2 (H), ie the space of quaternionic lines in H 3 , and a manifold L 8 satisfying the relation 4L 8 = K 3 × K 3 . Clearly heterotic horizons lie in Ω spin 8 . These data indicate that all the solutions 7 we have found, including a large class which may arise from del Pezzo surfaces, can be the near horizon geometries of asymptotically flat or asymptotically AdS black holes. For such black holes the asymptotic section is a sphere S 8 and so it represents the trivial class in Ω spin 8 . This is also the case for most Kaluza-Klein black holes. On the other hand all our explicit horizon sections, apart from SU(3), are products S 3 × Z. Since S 3 represents the trivial class, because Ω spin 3 vanishes, one can consider D 4 ×Z which has boundary S 3 ×Z and so S 3 ×Z also represents the trivial class in Ω spin 8 . Therefore all such horizon sections can be associated with asymptotically flat or AdS black holes. The same argument applies for the examples based on del Pezzo surfaces since for most of them the horizon section is S = S 1 × M 7 . Since Ω spin 7 = 0, one can again argue that S 1 × M 7 represents the trivial class in Ω spin 8 . It may appear that the cobordism equivalence between the horizon and asymptotic sections do not impose much restriction. This is mostly the case but not always. One of the near horizon geometries that arises in heterotic theory has section K 3 × K 3 [20]. As we have mentioned this represents a non-trivial class in Ω spin 8 . Therefore it is not in the same cobordism class as S 8 and so it cannot be the near horizon geometry of an asymptotically flat or AdS black hole. Moreover, it cannot be the near horizon geometry of most Kaluza-Klein black holes which arise from lifting a 4-or 5-dimensional black hole to 10 dimensions. This is because the asymptotic section of such Kaluza-Klein black holes is expected to be either products of spheres or S × H, where again S is a product of spheres and H is a special holonomy manifold, ie H is either K 3 or a 6-dimensional Calabi-Yau. However such spaces represent the trivial class in Ω spin 8 and so cannot be cobordant to K 3 × K 3 .
Concluding Remarks
We have constructed explicit examples of near horizon geometries of heterotic supergravity which preserve 4 supersymmetries. Amongst the horizon sections we have found are SU(3) and S 3 ×S 3 ×T 2 . The near horizon geometry in the SU(3) case is (SL(2, R)×SU (3))/U(1) and SL(2, R) is twisted with respect to a U(1) connection. The horizon section S 3 × S 3 × T 2 gives rise to two different near horizon geometries. One near horizon geometry is AdS 3 × S 3 × S 3 × S 1 . But there is also the possibility that the near horizon geometry is (AdS 3 × S 3 × S 3 )/S 1 × T 2 . Therefore a horizon section does not determine the near horizon geometry uniquely. To our knowledge, it is the first time that such a possibility has been observed.
We have also demonstrated that a large class of solutions can arise provided that the horizon section is chosen to be a T 4 fibration over 4-dimensional a Kähler manifold X. The resulting differential system contains 6 equations for 6 unknown functions which include the Monge-Ampére equation and a conformally rescaled Hermitian-Einstein equation. We have shown that the non-linear system has solutions for X = P 1 ×P 1 . We have also given a set of conditions in the cohomology of X which are necessary for the existence of solutions. We have found many solutions of this cohomological system when X is a del Pezzo surface. We have also raised the question of whether for every solution of the cohomological system there is a solution of the differential equations and so a new heterotic horizon.
We have investigated the topology of the horizons we have found by computing their de Rham cohomology. Those horizons that are associated with del Pezzo surfaces exhibit cohomology with intersection matrices which are closely related to exceptional groups. Of course this is inherited from the intersection matrix of del Pezzo surfaces which is the Cartan matrix of exceptional Lie algebras. The cohomological properties of our horizons point to a relation to U-duality invariant brane configurations but we have not been able to make this more precise.
There is not an apriori reason to believe that a near horizon solution is associated with a black hole. However, there are some tests that one can do. One of them is to argue that the asymptotic section of a black hole is cobordant to the horizon section. In the presence of spinors, this means that both sections are spin cobordant. We have found that all our horizon sections which preserve 4 supersymmetries are in the trivial class and so they may be near horizons of asymptotically flat or AdS black holes.
The plethora of near horizon solutions we have found suggest that there are many exotic supersymmetric black holes in heterotic supergravity. Some of them will be Kaluza-Klein black holes but there is a possibility that some will have a purely 10-dimensional origin. The existence of such black holes will give new insights into string theory and M-theory.
To compute the cohomology of the total space of the torus fibration, we shall use the spectral sequences method explained in [38], see also [24]. Suppose that S = T 2 × B, where B is a non-trivial T 2 fibration over X. X is simply connected. In such a case, the E * , * 2 part of the spectral sequence defined in (5.25) is summarized in table 4. To compute the action of the d 2 differential first observe that d 2 : R θ r → R b a with d 2 (θ r ) = b r which is clearly 1-1. Next d 2 : R θ 12 → R θ r ∧b a with d 2 (θ 12 ) = θ 1 ∧b 2 −θ 2 ∧b 1 which is also 1-1. Moreover d 2 :
R θ 12 0 R θ 12 ∧ b a 0 R θ 12 ∧ v R θ r 0 R θ r ∧ b a 0 R θ r ∧ v R 0 R b a 0 R vR θ r ∧ b a → R v with d 2 (θ r ∧ b a ) = b r ∧ b a ,
where v is the volume class of X. Suppose now that there are α, β ∈ H 2 (X, R) such that
b 1 ∧ α = v , b 2 ∧ β = v , b 1 ∧ β = b 2 ∧ α = 0 . (A.1)
In such a case, d 2 : R θ r ∧ b a → R v is onto. Furthermore, d 2 : R θ 12 ∧ b a → R θ r ∧ v with d 2 (θ 12 ∧ b a ) = θ 1 ∧ b 2 ∧ b a − θ 2 ∧ b 1 ∧ b a is also onto. As a result, E * , * 3 is given in table 5 and converges in the cohomology of the bundle space.
Therefore the cohomology of B is given by Suppose that S = S 1 × Y , where Y is a non-trivial T 3 fibration over X. To find the cohomology of Y , the E * , * 2 part of the the spectral sequence is given in table 6. Moreover observe that d 2 : R θ r → R b a with d 2 (θ r ) = b r which is clearly 1-1. Next 0 0 R m−2 0 R 0 0 R 2m−2 0 0 R 0 R m−2 0 0 Table 6: The entries are the elements of E * , * 2 d 2 : R θ rs → R θ r ∧ b a with d 2 (θ rs ) = θ r ∧ b s − θ s ∧ b r which is also 1-1. Moreover d 2 : R θ r ∧b a → R v with d 2 (θ r ∧b a ) = b r ∧b a . Suppose now that there are α s ∈ H 2 (X, R) such that b r ∧ α s = δ rs v .
R θ 123 0 R θ 123 ∧ b a 0 R θ 123 ∧ v R θ rs 0 R θ rs ∧ b a 0 R θ rs ∧ v R θ r 0 R θ r ∧ b a 0 R θ r ∧ v R 0 R b a 0 R v(A.3)
In such a case, d 2 :
R θ r ∧ b a → R v is onto. Furthermore, d 2 : R θ rs ∧ b a → R θ r ∧ v with d 2 (θ rs ∧ b a ) = θ r ∧ b s ∧ b a − θ s ∧ b r ∧ b a
is also onto. Next d 2 : R θ 123 → R θ rs ∧ b a with d 2 (θ 123 ) = θ 12 ∧ b 3 + θ 31 ∧ b 2 + θ 23 ∧ b 1 , and d 2 : R θ rs ∧ b a → R θ r ∧ v with d 2 (θ rs ∧ b a ) = θ r ∧ b s ∧ b a − θ s ∧ b r ∧ b a . The former map is 1 − 1 and the second is onto. As a result, E * , * 3 is given in table 7 and converges in the cohomology of the bundle space. 0 0 R m−3 0 R 0 0 R 3m−4 0 0 0 0 R 3m−4 0 0 R 0 R m−3 0 0 In particular, ones finds that
H 0 (Y, R) = H 7 (Y, R) = R , H 2 (Y, R) = H 5 (Y, R) = R m−3 , H 3 (Y, R) = H 4 (Y, R) = R 3m−4 . (A.4)
Suppose that S is a non-trivial T 4 fibration over X. The E * , * 2 of part of the spectral sequence is given in table 8.
A similar analysis to the one we have presented for the two similar cases above gives that the elements of E * , * 3 are given in table 9. In particular, one finds that H 0 (S, R) = H 8 (S, R) = R , H 2 (S, R) = H 6 (S, R) = R m−4 , H 3 (S, R) = H 5 (S, R) = R 4m−7 , H 4 (S, R) = R 6m−8 .
(A.5) Appendix B A generalization of the differential system
R θ 1234 0 R θ 1234 ∧ b a 0 R θ 1234 ∧ v R θ rst 0 R θ rst ∧ b a 0 R θ rst ∧ v R θ rs 0 R θ rs ∧ b a 0 R θ rs ∧ v R θ r 0 R θ r ∧ b a 0 R θ r ∧ v R 0 R b a 0 R v0 0 R m−4 0 R 0 0 R 4m−7 0 0 0 0 R 6m−8 0 0 0 0 R 4m−7 0 0 R 0 R m−4 0 0
The differential (4.2) and cohomological (4.3) systems can be easily generalized to holomorphic T 2n -fibrations Y over 4-dimensional Kähler manifolds X. First consider a conformal scaling of the Kähler form κ of X as ω X = e 2Φ κ , dκ = 0 . (B.1)
Then introduce connections h r adapted to the T n fibration for which the curvature dh r is (1,1). The metric and 3-form field strength on Y are ds 2 = G rs h r h s + e 2Φ ds 2 (X)
H = G rs h r ∧ dh s − i I de 2Φ ∧ κ (B.2)
where G rs is a constant fibre metric and I is the complex structure on X. The Hermitian form on Y is
ω(Y ) = φ rs h r ∧ h s + e 2Φ κ (B.3)
where φ is a compatible constant Hermitian form which together with G gives rise to a complex structure on the fibre. Requiring that the connection∇ on Y has holonomy contained in SU(2) ⊂ SU(2 + n), one finds that dh r ∧ κ = v r 2 e 2Φ κ 2 −i∂∂ log det(iκ) + 2i∂∂Φ + G rs v r dh s = 0 , G rs dh r ∧ dh s + 2i∂∂e 2Φ ∧ κ = 0 , (B.4)
where v r are constants. One can rewrite the differential equations in terms of a non-linear system for 2n + 2 functions. The calculation is an adaptation of the one we have done already for heterotic horizons and we shall not repeat it here. As for heterotic horizons, there is an associated cohomological system given by
c 1 (P r ) ∧ [κ] = 1 2 [e 2Φ κ 2 ]
, c 1 (X) + v s c 1 (P s ) = 0 , G rs c 1 (P r ) ∧ c 1 (P s ) = 0 , (B.5)
where c 1 (P r ) denotes the first Chern class of the P r circle bundle, c 1 (X) is the first Chern class of the canonical bundle of X and v s = G rs v r . Since c 1 (P r ), c 1 (X) ∈ H 2 (X, Z), it is required that G rs v r c 1 (P s ) ∈ H 2 (X, Z) imposing restrictions on the fibre metric and v.
= 1, . . . , 8, and e i satisfy the Maurer-Cartan equations
H(6) is not closed. Thus SU(3) satisfies all the properties of a heterotic horizon. The associated spacetime is SL(2, R) × SU(3) /U(1).
2 :
2The intersection matrix of a del Pezzo surface, k < 9, is given by the Cartan matrix of exceptional Lie algebras. The intersection matrix of of P 2 blown up at k > 9 in general position are also given by the Cartan matrix of exceptional algebras E k .The (4.3) conditions require that c 1 (Q) = K .(5.5)
8
8a negative multiple of the volume class of X as required. Settingc 1 (P s ) = n s H − i m si E i , s = 1, 2, 3 ,(5.6)we find that the condition c 1 (P s ) · [cohomological condition which arises from the closure of H (clear that for the above choice of Kähler class, there are solutions only if k is odd. Moreover, it is straightforward to show that for k = 3, 5, 7, (5.7) and (5.8) imply that |m si | ≤ 2 for all s, i. So there are only finitely many solutions. Furthermore one also finds, for k = 3, 5, 7, that at most one of the m si can take the value ±2, the remaining m si must be ±1 or 0. One can then enumerate all possible solutions. k = 1
taking values in the set {1, 2, . . . , k}, k = 3. Though we do allow for a i = b j and so on.
one can add point anti-5-brane sources as dH (8) = nδ(NS5) . (5.24)
ds 2 (B) = e −Φ [−k 2 r 2 du 2 + 2dudr + ds 2(6) + e 4Φ 3 dy 2 ] + k 2 e Φ dx 2 . (7.3)
H 0 (
0B, R) = H 6 (B, R) = R , H 2 (B, R) = H 4 (B, R) = R m−2 , H 3 (B, R) = R 2m−2 . (A.2)
Table 1 :
1The geometry of B 6 and S is summarized. Observe that both geometries elegantlyavoid the conditions of the no-go theorem of [29] and so compact examples exist.
Table
Table 3 :
3The worldvolume directions of the branes are given.
Table 4 :
4The entries are the elements of of E * , *2
Table 5 :
5The entries are the elements of of E * , *3
Table 7 :
7The entries are the elements of of E * , *3
Table 8 :
8The entries are the elements of E * , *2
Table 9 :
9The entries are the elements of of E * , *3
The system we derive is different from that of[25] as in our case the torsion is closed on the horizon section S.
One could write a more general metric on S by introducing a non-trivial constant metric on the fibre directions. The choice we have made suffices for our purpose.
This has not been observed in[23]. As a result, the holonomy of the connection with skew-symmetric torsion of 6-dimensional manifolds which are T 2 fibrations is contained in SU (2) rather than SU (3).4 It is also sufficient if the base space X is simply connected.
It is not necessary to split the dilaton in a background and a fluctuation but this was done for uniformity.
All our horizon sections admit metrics with positive scalar curvature and so satisfy the positive Yamabe type condition of[37].
Acknowledgments: We would like to thank Geo Grantcharov for correspondence and Fabio Riccioni for helpful discussions. JG is supported by the EPSRC grant EP/F069774/1. GP is partially supported by the EPSRC grant EP/F069774/1 and the STFC rolling grant ST/G000/395/1.Appendix A Topology of horizon sections
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"Uniwersytetu Łódzkiego\nUl. Pomorska 149/153PL-90-236ŁódźPoland",
"Uniwersytetu Łódzkiego\nUl. Pomorska 149/153PL-90-236ŁódźPoland",
"Uniwersytetu Łódzkiego\nUl. Pomorska 149/153PL-90-236ŁódźPoland"
] |
[] |
A derivation of the ionization rate for the hydrogen-like ion in the strong linearly polarized laser field is presented. This derivation utilizes the famous Keldysh probability amplitude in the length gauge (in the dipole approximation) and without Coulomb effects in the final state of the ionized electron. No further approximations are being made, because the amplitude has been expanded in the double Fourier series in a time domain (with the help of the generalized Bessel functions). Thus, our theory has no other limitations characteristic of the original Keldysh theory. We compare our "exact" theory with the original Keldysh one, studying photoionization energy spectra and total ionization rates. We show breakdown of the original Keldysh theory for higher frequencies. In the barrier-suppresion regime the "exact" Keldysh theory gives results closer to well-known numerical or other analytical results. _______________________________ *[email protected]
| null |
[
"https://arxiv.org/pdf/1509.03994v1.pdf"
] | 118,362,490 |
1509.03994
|
a4604df35bd2f769139db92eee2745ab19a1d1e5
|
Keldysh theory re-examined: Application of the generalized Bessel functions
Jarosław H Bauer
Uniwersytetu Łódzkiego
Ul. Pomorska 149/153PL-90-236ŁódźPoland
Fizyki Katedra
Uniwersytetu Łódzkiego
Ul. Pomorska 149/153PL-90-236ŁódźPoland
Wydział Teoretycznej
Uniwersytetu Łódzkiego
Ul. Pomorska 149/153PL-90-236ŁódźPoland
Fizyki
Uniwersytetu Łódzkiego
Ul. Pomorska 149/153PL-90-236ŁódźPoland
Informatyki Stosowanej
Uniwersytetu Łódzkiego
Ul. Pomorska 149/153PL-90-236ŁódźPoland
Keldysh theory re-examined: Application of the generalized Bessel functions
A derivation of the ionization rate for the hydrogen-like ion in the strong linearly polarized laser field is presented. This derivation utilizes the famous Keldysh probability amplitude in the length gauge (in the dipole approximation) and without Coulomb effects in the final state of the ionized electron. No further approximations are being made, because the amplitude has been expanded in the double Fourier series in a time domain (with the help of the generalized Bessel functions). Thus, our theory has no other limitations characteristic of the original Keldysh theory. We compare our "exact" theory with the original Keldysh one, studying photoionization energy spectra and total ionization rates. We show breakdown of the original Keldysh theory for higher frequencies. In the barrier-suppresion regime the "exact" Keldysh theory gives results closer to well-known numerical or other analytical results. _______________________________ *[email protected]
I. INTRODUCTION
One of the most important and the most frequently used methods employed to study an ionization in a strong laser field are S -matrix theories [1][2][3]. They are usually called Keldysh, or Keldysh-Faisal-Reiss (KFR) theories [4][5][6][7]. Also the name "strongfield approximation" (SFA) is sometimes preferred [7]. One usually makes the assumption that an ionized electron, due to an action of the laser field, makes a transition from a given initial bound state to a final free state, which can be treated as dominated only by the laser field. In light of the presented considerations the simplest version of the theory neglects completely an effect of a binding potential (short-range or long-range) and uses the Gordon-Volkov wave function [8] as a final state of the ionized electron. In his pioneering work [4] Keldysh identified the so-called adiabaticity parameter (termed after him) and obtained relatively simple analytical expressions for ionization rates and arbitrary . The Keldysh parameter is often used to distinguish between multiphoton ionization (when 1 ) and tunneling ionization (when 1
). nF Z F E B 2 ,(1)
where is the laser frequency, F -the amplitude of the laser field, and
2 2 2 / n Z E B
-the binding energy of the atom (of the nuclear charge Z ) initially in the state described by the well-known m l n , , quantum numbers (without spin), in the nonrelativistic approach. In Ref. [4] and in the present paper we consider only the hydrogen-like ion (i.e. the initial state is the ground state: 0 , 0 , 1 n ) in the linearly polarized monochromatic laser field. Such a simple theory can be applied only when the laser field is strong enough. An extensive and a very good discussion of applicability conditions of the SFA can be found in Refs. [1,7] (see also Sect. III in Ref. [9]). Based on this, we restrict our interest here only to the domain of the field parameters [4] which describes ionization rate for 1 has only a character of an estimation and a more accurate theory should take into account Coulomb effects in the final state of the ionized electron. As it was noted by Keldysh and others [10,11], there are two conditions that are necessary to derive final expressions for the ionization rate. These conditions limit the applicability range of the theory: (i) the saddle point approximation to perform the contour integral, and (ii) the small final kinetic momentum approximation. According to Ref. [10], the adiabatic assumption B E is necessary, to fulfill the condition (i). There have been attempts to derive ionization rate formulas (starting from Eq.(2)) without building on (i) or (ii) [10][11][12]. However, we are aware of no expressions involving the generalized Bessel functions in the length gauge (for linear polarization). In this way our present work fills some gap in the literature. On the other hand, the saddle-point method or semi-analytical methods enabled much progress in the last few decades (see, for example, Refs. [2,3,[13][14][15][16]).
In contrast, the well-known expressions derived in the velocity gauge for H ion and the s H 1 atom long ago by Reiss [7] include Bessel functions (the ordinary ones for circular polarization and the generalized ones for linear polarization). These formulas do not rely on the assumptions (i) and (ii). The main aim of the present work is to derive expressions analogous to those in Ref. [7] in the length gauge and investigate their main properties. In the case of circularly polarized laser field such theory has been developed for arbitrary initial states m l n , , with 1 n and 2 n of the hydrogen-like ion [9,17,18]. S -matrix theories in the above-mentioned two gauges give qualitatively similar results (particularly this concerns the shape of photoelectron energy spectra) for initial states of even parity [19], like the ground states of (16). In Sec. IV we discuss the results of our numerical calculations (including photoelectron energy spectra and total ionization rates) and we make a comparison of these results with other results, mainly with the original Keldysh theory. In the Appendix we show in details how the Fourier coefficients and the generalized Bessel functions have been calculated in this paper. In the present work we consistently use atomic units (a.u.):
1 e m e
, substituting explicitly -1 for the electronic charge. We keep any nuclear charge Z in all the equations given below, but finally, in our numerical calculations, we put Z 1 for the hydrogen atom.
II. KELDYSH THEORY
In the Keldysh theory [4] one starts from the following matrix element, which is the approximate probability amplitude of strong-field ionization in the length gauge:
t r t F r t r r d dt i S i GV Keldysh fi , , 1 3 ,(2)
where the initial, ground state of a hydrogen-like ion is described by the well-known wave function
t Z i Zr Z t iE r t r B i i 2 exp exp exp , 2 3 (3)
in position space. In the length gauge the laser-atom interaction Hamiltonian is
t F r H I , with t A c t F 1
being the electric field vector, and the Gordon-Volkov state [8] in this gauge is
5 t GV d t i r t i t r 2 2 / 3 2 exp 2 1 , ,(4)
where p is kinetic (asymptotic) momentum of the ionized outgoing electron and its canonical momentum is given by
t A c p t 1 .
(5)
In the dipole approximation magnetic-field component of the laser is zero and the
electric-field one is 0 sin t F t F (where
is the polarization vector and 0 some arbitrary initial phase of the laser field). Starting from Eq. (1), Keldysh [Eq. (20) in Ref. [4]] obtained the following ionization rate formula (for 1
): 2 3 4 / 7 10 1 1 3 2 exp 2 3 F Z ZF W Keldysh .(6)
In the above expression n -photon contributions have been summed up owing to the procedure valid when 0 (going over from summation over n to integration). Without this procedure, the ionization rate (6) has the general form
0 , , n B n Keldysh F E W W ,(7)
where n W are partial ionization rates corresponding to absorption of exactly n photons above threshold. n W are given by Eqs. (16), (18), and (19) in Ref. [4]. To obtain Eq. (6) one also has to make a Taylor expansion around 0 , leaving only the lowestorder term in the pre-exponential factor and the two lowest-order terms in the exponent of Eq. (16) in Ref. [4]. Now then, one can also write down "an intermediate" expression, which does not involve "a multiphoton" summation, but contains all the complicated dependence on the Keldysh parameter :
results. We have also verified numerically that for const F and 0 Eq. (8) goes to finite value (like Eq. (6)), as should be. To obtain the most accurate expression (7), one has to include the missing factor / 2 as well. Let us denote these three ionization rates as "Keldysh 1" (Eq. (7)), the most accurate), "Keldysh 2" (Eq. (8)), and "Keldysh 3" (Eq. (6), the least accurate), respectively.
III. KELDYSH THEORY WITH THE GENERALIZED BESSEL FUNCTIONS
Using Eqs. (3)-(5) one can show [9] that the ionization probability amplitude (2) may be presented as follows
t B B i Keldysh fi t iE d i E t t dt i S 2 2 2 exp 2 1 1 ,(9)
where the momentum-space wave function of the initial ground state of the s H 1 atom is equal to
2 2 2 5 2 / 3 3 1 8 exp 2 p Z Z r r p i r d p i i .(10)
7
To calculate the ionization probability amplitude (9) we assume that the laser field propagates along the x axis and the polarization vector is parallel to the z axis. Following Reiss [1,7] (and our earlier works, for example [9]) we introduce the z parameter such that
2 4 / I z U P ,/ 10 51 . 3 . . 1 cm W u a
). The product of the first two factors in the integrand of Eq.
(9) is proportional to k t ik k B e p A E t 0 1 2 2 1 ,(11)
and may be expanded in the above Fourier series. It appears that the coefficients can be calculated analytically with the help of the residue theorem (see the Appendix). The exponential factor in Eq. (9) may be also expanded, in the standard way, using the Fourier-Bessel expansion: for linearly polarized field (we use the same convention with respect to b a J n , as in Ref. [7]).
n n t in b a J t ib t ia 0 0 0 exp , 2 sin sin exp ,(12)
When we apply expansions (11) and (12), the amplitude (9) turns out to be proportional to
k n B k n Keldysh fi t E k n z p i dt k n i p A b a J i S , 2 0 2 exp exp , 1 ,(13) k N B k k N Keldysh fi t E N z p i dt iN p A b a J i S , 2 0 2 exp exp , 1 ,(14)
The integral upon time leads to the well-known distribution (a particular model of Dirac function), which enables applying standard procedure. The differential ionization rate p w , which is the transition probability per unit time and unit volume in the canonical momentum ( p ) space, can be found from
t S p w fi t 2 1 lim .(15)
To obtain the total ionization probability per unit time W , one has to integrate the differential ionization rate over all the possible final momenta of the outgoing electron.
The final result is
2 0 5 3 0 0 , , 1 sin 8 N N k k N N k k N N N N b a J E A d E Z W p d p w W ,(16)
where
cos / 8 N zE a and 2 / z b
. The minimal number of photons absorbed is
1 / 0 B E z N
, and the kinetic energy of the ionized outgoing electron is
B N N E z N p E 2 / 2 .(17)
The symbol .. denotes integer part of the (positive) number inside. In deriving equation (16) we have used the following property of the generalized Bessel functions:
b a J b a J n n n , 1 ,
, and the relation:
p A p A k k
(see the Appendix for some other details).
Let us note that analogous expression in the velocity gauge is much simpler [7] and contains only one summation. In our notation it is given below ( a and b as in Eq. (16)):
0 0 2 2 5 3 , sin 8 N N N B N N SFA SFA b a J d E E E Z p d p w W .(18)
Equations (16) and (18) show total ionization rates as a sum over partial N -photon ionization rates, where the kinetic energy of outgoing electron is given by Eq. (17).
The same concerns Eq. (7), but there the index n denotes the number of photons above threshold. Thus
0 N N n
. This is obvious, if we look at the argument of the Dirac function from Eq. (14) in Ref. [4].
IV. NUMERICAL RESULTS AND CONCLUSION
The Fig. 1 to Fig. 3. Comparing upper two curves each time, one can see that agreement between "Keldysh 1" and "Exact Keldysh" is quite satisfactory, particularly for lower photoelectron energies. This is understandable, because the small final kinetic momentum approximation is utilized in "Keldysh 1". According to this approximation only electrons with sufficiently small final kinetic energy E should mostly contribute to the total ionization rate ( .
. 5 . 0 2 / 2 u a E p E B
). This is hardly satisfied only in Fig. 3 This line is shown in our log-log plots as a slanted dashed line. Figures 8 and 9 resemble Fig. 6 from Ref. [22]. (However, let us notice that numerical ionization rates
for . . 1 . 0 u a and . . 2 . 0 u a
in the latter figure indicate that the proper ionization rate should be a concave function in log-log plot, but not a ruled line.) It seems that a significant theoretical progress was attained about ten years later [14]. In Figs. 8 and 9 for comparison we show also the ionization rate of Popruzhenko et al. (see Eqs. (5)- (7) 11 from Ref. [14]), which is valid for any Keldysh parameter and takes into account Coulomb effects in the final state of the ionized electron. Curiously enough, our "Exact Keldysh" curve typically lies about one order of magnitude below the curve denoted as "Ref. [14]" and the distance between these two curves decreases with increasing F . In Figs. 8 and 9 we show also for comparison the above-mentioned three " Keldysh
ACKNOWLEDGMENTS
The present paper has been supported by the University of Łódź.
APPENDIX
T 2 / we obtain T B n E c t A p dt t in p A 0 2 0 2 / / exp 2 , (A1) p A n
is always a real number. Let us introduce new parameters:
0 2 / 2 B E z p A , cos 2 p z B , and 0 z C . Then Eq. (A1) takes the form T n t C t B A dt t in p A 0 0 0 0 2 cos cos exp 2 .(A2)2 1 ,(A3)
where C denotes a circle of a unit radius, with the center in the origin, in the complex s plane. The "plus" means that we circulate the circle in a counter-clockwise direction.
To utilize the residue theorem, we have to solve the equation:
the motion of the ionized electron remains nonrelativistic. Currently Eq. (21) from Ref.
.
, we have derived the ionization rate formula for the hydrogen-like ion in the strong linearly polarized laser field, using the Keldysh probability amplitude in the length gauge (in the dipole approximation) and without Coulomb effects in the final state of the ionized electron. This calculation is exact in the sense that no further analytical approximations are used. As one could expect, it appears that the original Keldysh theory[4] (where additional analytical approximations have been done) leads to satisfactory photoelectron energy spectra only in its low-energy part and only when The total ionization rate is affected moderately by these approximations, which cause an underestimation of this rate by a factor of 2-3 at best. The price to be paid for the lack of additional analytical approximations are numerical calculations connected with the generalized Bessel functions. These calculations become really time-consuming for low frequencies or large Keldysh parameters for the method applied by us (see the Appendix for more details). Another price which we pay is connected with analytical calculations of the Fourier coefficients from Eq. (11). However, since it is possible for the s H 1 atom, it is likely possible for other bound states of this atom. There have been many attempts to take into account Coulomb effects in the final state of the ionized electron (see, for example, Refs. [9,20] and references therein) by replacing the Gordon-Volkov wave function ( GV ) by various more complicated Coulomb-Volkov wave functions ( CV ). These latter are approximate solutions of the full time-dependent Schrödinger equation. It is obvious, that the general scheme of calculations presented here could be readily repeated, if CV is used instead of GV . This would only change the Fourier coefficients . Eq. (11)). Our investigations in this direction are under way.
FIGFIG.
(a.u.) FIG. 1. (Color online) Photoelectron energy spectra for "Keldysh 1" theory [Eq. (7) and Ref. [4]] (solid blue line), "Exact Keldysh" theory [Eq. (16)] (solid red line) and "SFA (Reiss)" theory [Eq. (18) and Ref. [7]] (solid black line). Laser field parameters and the Keldysh parameter are given in the text frame of the plot. (See the main text for more details.) . 3. (Color online) Same as Fig. 1, but for . (a.u.) FIG. 5. (Color online) Same as Fig. 1, but for . . 7. (Color online) Same as Fig. 1, but for . Dashed black (slanted) line: numerical fit obtained in Ref. [22]. Solid black line: the theory from Ref. [14]. Original Keldysh theories are denoted (from top to bottom in the plot) as "K3" (dotted green line), "K1" (solid blue line), and "K2" (dashed magenta line). Solid red line: "Exact Keldysh" theory. Two vertical dashed lines show values of the Keldysh parameter (for a given F ), which decreases from left to
H . However, several earlier calculations have convinced us that the length gauge gives ionization rates closer to experimental results, and theoretical static-field results and references therein). It seems that in the so-called barrier-suppression regime relevance of Coulomb corrections in the final state of the ionized electron should not be great and should decrease with increasing the laser field F . Thus it appears reasonable to study the case without Coulomb corrections in details. This could be a kind of some benchmark result and a starting point for future improvements (including Coulomb corrections).Our paper is organized as follows. In Sec. II we present the basis of the original . In Sec. III we present our analytical calculations. The main result is given in Eq.s
H 1 and
(when
B
E
) [9,20,21] (4
Keldysh theory [4] and its main results for
1
whereP
U denotes the ponderomotive energy
of the electron and
2
F
I
denotes the laser intensity in atomic units
(
2
16
). In Figs. 1-7 we present several photoionization energy spectra of outgoing electrons for laser frequencies obeying this condition and the condition 1 . In each of these plots we show three curves, corresponding, respectively, to the most accurate original "Keldysh 1" [Eq.(7)] spectrum, to present "Exact Keldysh" [Eq.(16)] and to its velocity gauge counterpart "SFA (Reiss)" [Eq.(18) and Ref.[7]]. InFigs. 1-3original Keldysh theory [4] has a high-frequency limitation. In our case (
1
Z
)
the theory is expected to be valid, if
.
.
5
.
0
2
/
2
u
a
Z
E
B
(preferably when
B
E
.
.
01
.
0
u
a
and the peak laser field
F decreases from
With increasing the laser frequency (and the peak laser field) the "Keldysh 1" curves become more and more different from respective "Exact Keldysh"curves. This is particularly visible for higher energies, where small momentum overestimates the number of high-energy electrons in a significant way. It is worth to recall that the SFA has no frequency limitation[1,7] (except the fact that should not be in resonance with excited bound states of the ionized atom), so its length gauge counterpart derived here ("Exact Keldysh") does not have this limitation as well.In Figs. 8 and 9 we show various total ionization rates as a function of the peak laser field in the range of two orders of magnitude:respectively. Thus, the barrier-suppression ionization range is included here. Several years ago Bauer and Mulser in Ref.[22] concluded that no analytical theory is able to properly describe the ionization rate in this range for. In
Figs. 1, 2 and 4-7 the range of significant kinetic energies is much wider. With
increasing
F more and more electrons have
.
.
5
.
0
u
a
E
(see Figs. 2 and 1). This
concerns all three curves, but velocity gauge partial ionization rates are at least one
order of magnitude (or more) smaller than length gauge ones. Let us note that
oscillations that appear in the "Exact Keldysh" and "SFA (Reiss)" curves are not of
constant period, which is much greater than photon energy. This shows some affinity
between these two results, which have been obtained with the help of the generalized
Bessel functions. In Figs. 4-7 we keep the Keldysh parameter constant:
1
33
.
0
,
which is quite typical in many experiments. In Figs. 4-7 we increase the laser
frequency from
.
.
02
.
0
u
a
, through
.
.
057
.
0
u
a ,
.
.
1
.
0
u
a
up to
.
.
25
.
0
u
a ,
respectively. approximation fails. For
.
.
1
.
0
u
a
and
.
.
25
.
0
u
a
"Keldysh 1" theory
.
.
2
.
.
02
.
0
u
a
F
u
a
(this
corresponds to four orders of magnitude in intensity) for
.
.
1
.
0
u
a
and
.
.
2
.
0
u
a
,
B
E
. They proposed a simple
numerical fit (based on ab initio calculations) for the
s
H 1 atom, namely
2
4
.
2 F
W
.
Let us consider the Fourier expansion of the expression
1
2 2
/
B
E
t
as a
function of time. Multiplying both sides of Eq. (11) by
0
exp
t
in
, and
integrating this equation from
0 to
is normalized to unity in the entire space.Hence, the following equation is satisfied for any instant of time(A7)Making the Fourier expansion of, like in Eq.(11), substituting it to Eq. (A7), and integrating both sides of the resultant equation over time from 0 to) then (see, for example,is simply equal to the left-hand side of Eq.(11)(A11)When either of the numbers: n , a , b is large, the integrand is highly oscillatory and passes zero many times in the interval , 0 . Especially in this case the so-called linear approximation model is very useful[27]. Let us consider the following integralwhere .(A14)The above expression behaves well when . The fraction in Eq. (A14) is then simply equal to . We were also using the efficient Gaussian procedure[28]to claculate numerically the integral from Eq. (A11) and to verify the results obtained from Eq. (A14) in an independent way.
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|
[] |
[
"A Stochastic Analysis of a Brownian Ratchet Model for Actin-Based Motility and Integrate-and-Firing Neurons",
"A Stochastic Analysis of a Brownian Ratchet Model for Actin-Based Motility and Integrate-and-Firing Neurons",
"A Stochastic Analysis of a Brownian Ratchet Model for Actin-Based Motility and Integrate-and-Firing Neurons",
"A Stochastic Analysis of a Brownian Ratchet Model for Actin-Based Motility and Integrate-and-Firing Neurons"
] |
[
"Hong Qian \nDepartment of Applied Mathematics\nUniversity of Washington\n98195SeattleWA\n",
"Hong Qian \nDepartment of Applied Mathematics\nUniversity of Washington\n98195SeattleWA\n"
] |
[
"Department of Applied Mathematics\nUniversity of Washington\n98195SeattleWA",
"Department of Applied Mathematics\nUniversity of Washington\n98195SeattleWA"
] |
[] |
In recent single-particle tracking (SPT) measurements on Listeria monocytogenes motility in vitro, the actin-based stochastic dynamics of the bacterium movement is analyzed statistically(Kuo and McGrath, 2000). The mean-square displacement (MSD) of the detrended trajectory exhibit a linear behavior; it has been suggested that a corresponding analysis for the Brownian ratchet model(Peskin, Odell, & Oster, 1993)leads to a non-monotonic MSD. A simplified version of the Brownian ratchet, when its motion is limited by the bacterium movement, is proposed and analyzed stochastically. Analytical results for the simple model are obtained and statistical data analysis is investigated. The MSD of the stochastic bacterium movement is a quadratic function while the MSD for the detrended trajectory is shown to be linear. The mean velocity and effective diffusion constant of the propelled bacterium in the long-time limit, and the short-time relaxation are obtained from the MSD analysis. The MSD of the gap between actin and the bacterium exhibits an oscillatory behavior when there is a large resistant force from the bacterium. The stochastic model for actin-based motility is also mathematically equivalent to a model for integrate-and-firing neurons. Hence our mathematical results have applications in other biological problems. For comparison, a continuous formalism of the BR model with great analytical simplicity is also studied.
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10.3970/mcb.2004.001.267
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"https://arxiv.org/pdf/cond-mat/0106409v1.pdf"
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cond-mat/0106409
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680bad71a1e799d4ddb6babedcf81ff7617fb45b
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A Stochastic Analysis of a Brownian Ratchet Model for Actin-Based Motility and Integrate-and-Firing Neurons
20 Jun 2001 February 1, 2008
Hong Qian
Department of Applied Mathematics
University of Washington
98195SeattleWA
A Stochastic Analysis of a Brownian Ratchet Model for Actin-Based Motility and Integrate-and-Firing Neurons
20 Jun 2001 February 1, 2008actin polymerizationexit problemmean first passage timenano-biochemistrysingle-particle trackingstochastic processes
In recent single-particle tracking (SPT) measurements on Listeria monocytogenes motility in vitro, the actin-based stochastic dynamics of the bacterium movement is analyzed statistically(Kuo and McGrath, 2000). The mean-square displacement (MSD) of the detrended trajectory exhibit a linear behavior; it has been suggested that a corresponding analysis for the Brownian ratchet model(Peskin, Odell, & Oster, 1993)leads to a non-monotonic MSD. A simplified version of the Brownian ratchet, when its motion is limited by the bacterium movement, is proposed and analyzed stochastically. Analytical results for the simple model are obtained and statistical data analysis is investigated. The MSD of the stochastic bacterium movement is a quadratic function while the MSD for the detrended trajectory is shown to be linear. The mean velocity and effective diffusion constant of the propelled bacterium in the long-time limit, and the short-time relaxation are obtained from the MSD analysis. The MSD of the gap between actin and the bacterium exhibits an oscillatory behavior when there is a large resistant force from the bacterium. The stochastic model for actin-based motility is also mathematically equivalent to a model for integrate-and-firing neurons. Hence our mathematical results have applications in other biological problems. For comparison, a continuous formalism of the BR model with great analytical simplicity is also studied.
Introduction
Actin polymerization plays an important role in nonmuscle cell mechanics, motility, and functions (Pollard et al, 2000;Pantaloni et al., 2000). In recent years, quantitative analyses of the molecular mechanism for actin-based motility are made possible by both laboratory experiments on Listeria monocytogenes (see van Oudenaarden and Theriot, 1999, and the references cited within) and a series of insightful mathematical models (Hill, 1981, 1987, Peskin et al., 1993, Mogilner and Oster 1996. The interaction between experimental observations and theoretical ideas has generated exciting research in biophysics and mathematical biology.
Following the seminal work of Peskin et al. (1993), a sizable literature now exists on mathematical models and analyses of the polymerization-based motility, known as Brownian ratchet (BR). Even though the original model on fluctuations is clearly a probabilistic one, it was cast mathematically in terms of the difference and differential equations with only a minimal stochastic interpretation. In the subsequent development, this stochastic nature of the model often has been obscured. In experimental laboratories, on the other hand, researchers often use Monte Carlo simulations to model the biological problem, partly because the data are inevitably stochastic.
This situation has prevented a truly quantitative understanding of the actin-based motility and a closer interaction between the experimental measurements and mathematical modeling. In a recent experiment, Kuo and McGrath (2000) used the highly sensitive single-particle tracking (SPT) methodology to measure the stochastic movement of L. monocytogenes propelled by actin polymerization. The seemingly random data are then analyzed statistically in terms of the meansquare displacement (MSD). The exquisite data with nanometre precision reveals the discrete steps in the bacteria movement, presumably due to the actin polymerization, one G-actin monomer at a time.
The stochastic nature of the BR, and the statistical treatment employed in experimental data analyses, necessitate a mathematical analysis of the BR model in fully stochastic terms. This is the main objective of the present work. Furthermore, Kuo and McGrath (2000) suggested that the BR movement, after detrending, exhibits a non-monotonic MSD. We shall investigate these practical issues as well. The significance of the stochastic interpretation is that one needs only to think about a single BR, and can derive theoretical MSD to compare with experiments.
In order to clearly present the stochastic approach to the BR, we study only a special, but relevant, case of the generic BR model proposed by Peskin et al (1993). This restriction makes the model easily analyzed analytically. Interestingly, the mathematical model is also identical to one for integrate-and-firing neuron proposed many years ago by Gerstein and Mandelbrot (1964).
In recent years, integrate-and-firing model has become one of the essential components in neural modeling (Hopfield and Herz, 1995). The stochastic model is quite basic; therefore we expect that our mathematical results also have applications in other branches of mathematical biology. The fractal nature of such model has also been discussed recently (Qian et al., 1999).
All the mathematical background on stochastic processes used in this work can be found in the excellent text by Taylor and Karlin (1998). To help the readers who are not familiar with some of the stochastic mathematics, italic font is used for the key words when they first appear in the paper.
Stochastic Formulation of a Brownian Ratchet Model
(i) We consider an F-actin polymerizes in a 1-dimensional fashion with the rate of monomer addition α and the rate of depolymerization β. α is a pseudo-first order rate constant which is proportional to the G-actin monomer concentration. Each G-actin monomer has a size of δ. Hence the actin polymerization is modeled as a continuous-time random walk (Hill, 1987). We shall take the growing direction as positive, and denote the position of the tip of the actin filament by X(t) which is a stochastic process taking discrete values kδ, where k is an integer.
(ii) We assume that a bacterium is, in the front of the growing actin filament, located at Y(t):
X(t) ≤ Y(t).
The bacterium has an intrinsic diffusion constant D b , and experiencing (or exerting) a resistant force F in the direction against the actin polymerization. In the absence of the actin filament, the bacterium movement is a Brownian motion with a constant drift rate −F/η b . Since a bacterium is a living organism, the D b and the η b are not necessarily related by the Einstein
relation η b D b = k B T for inert equilibrium objects.
(iii) The F-actin and the bacterium interact only when they encounter: X(t) = Y(t). The actin filament, however, can not penetrate the bacteria wall. Therefore, the motion of the bacterium and the actin polymerization are coupled via a reflecting boundary condition at X(t) = Y(t).
(i)-(iii) are the basic assumptions of the generic BR model first proposed by Peskin et al. (1993).
In the present work, we shall further assume that (iv) the α is sufficiently large and (v) β ≈ 0.
Therefore, whenever the gap ∆(t) Y(t) − X(t) = δ, the gap will be immediately filled by a G-actin monomer, and the polymer does not depolymerize. These two assumptions correspond to a rapid polymerization condition under which the bacteria movement is the rate-limiting process in the overall kinetics. Fig. 1 shows the basic, stochastic behavior of X(t), Y(t), and ∆(t). Kuo and McGrath (2000) also introduced a detrended Y(t). Let v be the mean velocity of the bacterium movement Y(t), then the detrendŶ(t) is defined asŶ(t) Y(t) − vt.
Let ξ k be the time for incorporating the kth G-actin monomer. Then at time ξ k , X(ξ k ) = Y(ξ k ) = kδ. When t > ξ k , Y(t) follows a Brownian motion with diffusion constant D, drift rate −F/η b , and reflecting boundary at kδ. Y(t) moves stochastically and when it reaches (k + 1)δ, denoted the time by ξ k+1 , the (k+1)th G-actin monomer is incorporated. Then the process repeats.
The waiting time for the next monomer to be incorporated is a random variable, we shall denote it by T: ξ k+1 = ξ k + T. This is our stochastic formalism for the BR model. Our analysis focuses on the stochastic properties of the random variable T.
M SD(τ ) = E (X(τ + t) − X(t)) 2(1)
which is a powerful analytical tool for analyzing stochastic processes with independent increments or stationarity. The E[. . .] in Eq. 1 denotes the expectation of random variables. For a stochastic process with independent increments, MSD(τ ) is further simplified into E (X(τ ) − X(0)) 2 . In the case of a stationary process, its MSD is directly related to the correlation function:
E[X(τ )X(0)] = E[X 2 ] − 1 2 M SD(τ ).(2)
The significance of MSD is that it can be obtained through a statistical analysis of stochastic experimental data (Qian et al., 1991). It is the essential link between the experimental measurements on fluctuations and stochastic mathematical models. For an experimental time series {x n |0 ≤ n ≤ N }, the MSD is defined as:
M SD(m) = 1 N − m + 1 N −m k=0 (x k+m − x k ) 2 .(3)
The statistical relation between the experimentally determined MSD in Eq. 3 and the theoretical MSD in Eq. 1 can be found in the paper by Qian et al. (1991).
Basic Properties of the Model: Analytical Results
Mean Waiting Time and Waiting Time Distribution. The time interval T between the repeated incorporation of successive actin monomer is the exit time of a diffusion process. By exit time T z , we mean the time a Brownian particle takes to reach δ the first time, starting at z
(0 ≤ z ≤ δ). Clearly T z is a random variable; its expectation T (z) = E[T z ]
, known as mean first passage time, is the solution to the differential equation (Taylor and Karlin, 1998)
D b T ′′ zz − (F/η b )T ′ z = −1(4)
with boundary conditions T ′ z (0) = 0 and T (δ) = 0. Hence
T (z) = η 2 b D b F 2 e F δ/η b D b − e F z/η b D b + η b (z − δ) F .(5)
Therefore,
E[T] = T (0) = δ 2 D b e ω − 1 − ω ω 2 ,(6)
where
ω = F δ/(η b D b ) is the nondimensionalized resistant force.
The probability density function f Tz (t) for the waiting time, T z can be obtained in terms of its Laplace transform, also known as the characteristic function of the random variable T z , Q T (z, ν)
= ∞ 0 f Tz (t)e −νt dt which satisfies the following differential equation (Weiss, 1966)
D b ∂ 2 Q T (z, ν) ∂z 2 − F η b ∂Q T (z, ν) ∂z = νQ T (z, ν)(7)
with boundary condition ∂Q T (0, ν)/∂z = 0 and Q T (δ, ν) = 1. Note Eq. 4 is a special case of Eq.
7 for T (z) = −∂Q T (z, 0)/∂ν.
Eq. 7 can be analytically solved:
Q T (0, ν) = λ − − λ + λ − e λ + δ − λ + e λ − δ(8)
where
λ ± = ω 2δ ± ω 2δ 2 + ν D b .
Therefore, the variance in the waiting time
V ar[T] = δ 4 D 2 b 3e 2ω − (10ω − 6)e ω + ω 2 − 2ω − 9 ω 4 .(9)
If there is no resistant force from the bacteria, ω = 0 and we have a simple expression
Q T (0, ν) = cosh δ 2 ν/D b −1 .(10)
Renewal Processes, The Statistical Properties of X(t) and Y(t). With T as the waiting time, the tip of the rapid growing actin filament, X(t), is a renewal process. There is a large literature on this subject. The most relevant result to our model is the elementary renewal theorem
for large t E[X(t)] ≈ δ E[T] t(11)
Therefore as a renewal process, a BR executes successive steps with size δ and average time E[T].
The mean velocity of the BR, thus, is
v = δ E[T] = D b δ ω 2 e ω − 1 − ω .(12)
This result is in agreement with that of Peskin et al. (1993).
Furthermore from the theory of renewal process (Taylor and Karlin, 1998)
V ar[X(t)] ≈ δ 2 V ar[T] E 3 [T] t σ 2 t,(13)
where, according to Eqs. 6 and 9,
σ 2 = 3e 2ω − (10ω − 6)e ω + ω 2 − 2ω − 9 (e ω − 1 − ω) 3 ω 2 D b .(14)
The MSD for X(t), therefore, is
E (X(t) − X(0)) 2 ≈ σ 2 t + (vt) 2(15)
which is a quadratic function of t. The expression for σ 2 is a new result of the present work, which is comparable with experimental data. Fig. 3 shows the dependence of v and σ 2 as functions of
ω = F δ/η b D b , the resistant force from the bacterium. If ω = 0, then v = 2D b /δ and σ 2 = 14D b /3. Since Y(t) − X(t) < δ while both increase linearly with t, for large t Y(t) ≈ X(t)
with an error less than δ, the size of a single actin monomer. Strictly speaking, the Y(t) is not a stochastic process with independent increments. However, the error involved, again, is only on the order of the size of a single G-actin. To understand the statistical correlation of Y(t) within each "step", see the section below on the gap.
Detrend Y(t) and Its Statistical Properties. In the recent experimental work (Kuo and McGrath, 2000), the detrended Y(t) has also been reported, which can be defined asŶ(t)
Y(t) − vt, where v is the mean velocity. For nδ ≤ Y(t) ≤ (n + 1)δ, Y(t) Y(t) − vt = Y(t) − Y(ξ n ) + nδ − vξ n − v(t − ξ n ) = Y(τ ) − vτ + nδ − vξ n(16)
in which random variable ξ n is the time for Y(t) to reach nδ the first time, τ = t − ξ n , and v is given in Eq. 12. Hence, 0 ≤ τ ≤ ξ n+1 − ξ n = ξ 1 = T, with its expectation, variance, and characteristic function given in Eqs. 6, 9, and 8, respectively.
The statistical properties ofŶ(t) are readily to be calculated:
E Ŷ (t) = E (Y(τ )] − vτ ≈ 0,(17)V ar Ŷ (t) = V ar [Y(τ )] + nvV ar [T] ≈ σ 2 t.(18)
Thus, we see that the detrendŶ(t) does not become stationary with increasing time. While its expectation is zero, its variance increases linear with the time t, the epitome of a symmetric random movement. The parameter σ 2 2 is the effective diffusion constant of the BR.
Statistical Properties of the Gap. The gap between the bacteria, Y(t), and the tip of the
actin filament X(t), ∆(t) Y(t) − X(t)
behaves completely different from the detrendŶ(t). It reaches asymptotically to stationarity.
We can provide a reasonable estimation for the relaxation time for the gap to reach its stationarity from the largest nonzero eigenvalues (µ) of the diffusion operator
D b d 2 dx 2 + F η b d dx u(x) = µu(x)(19)
under the boundary condition D b u ′ (0) + (F/η b )u(0) = u(δ) = 0. See Appendix for details. All the eigenvalues are real and ≤ 0,
µ(z) = − D b 4δ 2 (z 2 + ω 2 )
, where the z are the roots of the transcendental equation cos z = ω 2 −z 2 ω 2 +z 2 . Fig. 4 suggests that the largest eigenvalue corresponds to the exit time which increases with the resistant force. For resistant force F ≫ 2η b D b /δ, there is a separation between the time scale for the exit and the time scale for establishing a quasi-stationary distribution for ∆(t) (Appendix). Fig. 5 shows the MSD for ∆(t), which is directly related to the correlation function for the stationary process (Eq. 2). After normalized by 2V ar[∆], the MSD are approximately the same for 0 ≤ ω ≤ 6. The correlation time decreases with ω for ω = 2, 4, 6, and 12 (i.e., p = 0.55, 0.6, 0.65, and 0.8), corresponding to the second largest eigenvalue in Fig. 4, When there is a large resistant force F , the exit time T has a small relative variance (Eqs. 9, and 6) and the exit becomes an event with sufficient regularity. This is reflected in the oscillation of the MSD in Fig. 5.
An Analytical Analysis of a Continuous Stochastic Formalism of BR
We can replace the assumptions (iv) and (v) in the Section 2 with a continuous model for the discrete polymerization. In other word, we approximate the random walk by a diffusion with diffusion constant and drift rate (Feller, 1957;Hill, 1987):
D a = (α + β)δ 2 /2, V a = (α − β)δ.(20)
where β and α are first-order and pseudo-first-order rate constants for the depolymerization and polymerization, δ is the size of a G-actin monomer. The dynamic equation governing the probability density function (pdf) P X (x, t) for the stochastic processes X(t) is:
∂P X (x, t) ∂t = D a ∂ 2 P X (x, t) ∂x 2 − V a ∂P X (x, t) ∂x(21)
where P X (x, t) has the probabilistic meaning of P X (x)dx = P rob{x ≤ X < x + dx}. Similarly, the dynamical equation for Brownian motion of the bacterium Y with resisting force F is, as before,
∂P Y (y, t) ∂t = D b ∂ 2 P Y (y, t) ∂y 2 + F η b ∂P Y (y, t) ∂y .(22)
These two equations are coupled since X ≤ Y. We call Eqs 20, 21, and 22 the continuous formalism for the BR. It represents a two-dimensional diffusion in the triangle region of x ≤ y:
∂P XY (x, y, t) ∂t = D a ∂ 2 P XY (x, y, t) ∂x 2 + D b ∂ 2 P XY (x, y, t) ∂y 2 − V a ∂P XY (x, y, t) ∂x + F η b ∂P XY (x, y, t) ∂y .(23)
The advantage of this version of the BR is its analytical simplicity. A coordinate transformation can be introduced:
∆ = Y − X, Z = D a Y + D b X D a + D b(24)
where ∆ represents the gap between the tip of the actin filament and the bacterium, Z represents an averaged position of X and Y, we shall call it the center of mass of the BR. With this transformation, the two differential equations are decoupled:
∂P ∆ (∆, t) ∂t = (D a + D b ) ∂ 2 P ∆ (∆, t) ∂∆ 2 + V a + F η b ∂P ∆ (∆, t) ∂∆ , (∆ ≥ 0) (25) ∂P Z (z, t) ∂t = D a D b D a + D b ∂ 2 P Z (z, t) ∂z 2 − D b V a − D a F/η D a + D b ∂P Z (z, t) ∂z (−∞ < z < +∞).(26)
It can be immediately concluded from these two equations that the gap ∆(t) approaches to its stationary, exponential distribution (see Appendix)
P ∆ (∆) = V a + F/η b D a + D b e Va+F /η b Da+D b ∆ , ∆ ≥ 0.(27)
Z however increases steadily with an effective diffusion constant D z , 1 Dz = 1 Da + 1 D b , and a mean velocity
V z = D b Va−DaF/η b Da+D b
. This result can be understood in terms of Newtonian mechanics: the driving force from actin polymerization is F a = η a V a = k B T V a /D a , and the resistant force is F , and hence the net force on the BR is F z = F a − F with the frictional coefficient the BR (center of mass) being k B T /D z . Hence
V z = D z F z k B T = D a D b D a + D b V a D a − F k B T = D b V a − D a F/η b D a + D b .(28)
The parameter D a and V a are defined in terms of the α, β, and δ in Eq. 20. α is a pseudo-first order rate constant which is proportional to the G-actin concentration c 0 , as well as the probability of the gap ∆ being greater than δ. Therefore, α is a function of external force F ; it can be determined in a self-consistent manner by the transcendental equation:
α(F ) = α 0 c 0 ∞ δ P ∆ (s)ds = α 0 c 0 exp − (α − β)δ + F/η b (α + β)δ 2 /2 + D b δ(29)
where α 0 is the intrinsic, second-order rate constant for polymerization. We see that V z in Eq. 28
is a linear function of resistant force F explicitly; however, nonlinearity arises since D a and V a are implicit functions of the resistant force F , via α(F ). In other words, the resistant force F slows down the BR by two different mechanisms: a linear Newtonian resistance and also a reduction in the rate of polymerization via a diminished gap. Eq. 29 has the general form α(F ) ∝ e −rF δ/k B T with an entropic barrier.
V z in Eq. 28 is necessarily smaller than V a , indicating that the bacterium retards the polymeriza-
tion. When F = η b D b V a /D a = η b D b δ 2(α−β)
α+β , the bacterium completely stalls the polymerization. This yields the critical stalling force which agrees with the well known result of Hill (1987) for an inert object: (k B T /δ)ln(α/β). Furthermore, if we note that η b D b = k B T , then the rate of polymerization against a resisting force F is at its maximal
V z = V a − F D a /k B T when D b → ∞.
This is a result of Peskin et al. (1993) who first elucidated the crucial role played by the fluctuating "barrier" in the filamental growth. In a more general context, the dynamic characteristics of the "force transducer" by which the resisting force is applied to the growing tip of the filament is an integral part of the molecular process. 1 We now show that in the limit of δ → 0, our V z from the continuous model is equivalent to the ratchet velocity derived by Peskin, Odell, and Oster (POO, 1993). Note that in the previous work the definition for the ratchet velocity was rather convoluted; The V z in the present model is more 1 In the work of Hill (1981), this issue is not considered because of its quasi-thermodynamic approach. The applied force was assumed to have an instantaneous dynamic characteristics. From equilibrium thermodynamics one knows that polymerization under resisting force F has α(F )/β(F ) = e −F δ/k B T . So how does F contribute to α(F ) and β(F ) individually? A splitting parameter r is defined as α(F ) = α(0)e −rF δ/k B T and consequently β(F ) = β(0)e (1−r)F δ/k B T , where α(0) and β(0) are the α and β in the main text. The rate of polymerization under force F , therefore, is and β = 1 2 − Va δ + 2Da δ 2 . Substituting these two expressions into
[α(F ) − β(F )]δ = α(0)e −rF δ/k B T − β(0)e (1−r)F δ/k B T δ.(30)V poo = δ α ∞ δ P ∆ (∆)d∆ − β ∞ 0 P ∆ (∆)d∆ we have lim δ→0 V poo = V a − D a δ δ 0 P ∆ (∆)d∆ = V a − D a P ∆ (0) = D b V a − D a F/η b D a + D b = V z .
In the derivation we have used Eq. 27. Note that the basic molecular parameters for polymerization, α, β and δ are actually contained in the parameter D a and V a . In the mathematical limit of δ → 0, there is at the same time α and β → ∞ such that D a and V a are finite (Feller, 1957).
Discussion
Nanometre precision measurements on L. monocytogenes movement (Kuo and McGarth, 2000) have There could be several explanations for the small D b . a) Kuo and McGrath (2000) suggested an association between the bacterium and the actin structure, which leads to endorsing the twodimensional BR with bending (Mogilner and Oster, 1996). b) It should be noted, however, that association-dissociation can also be introduced into the one-dimensional BR in the form of an attractive force between the actin and the bacterium; thus a nonzero F as function of (y − x). This, we suspect, will also lead to a reduced apparent D b on a longer time scale. c) As we have pointed out, the D b of a living bacterium is not necessarily related to its physical size and frictional coefficient η b .
A bacterium could have an internal mechanism, by utilizing its biochemical free energy, to localize itself near the tip of the actin filament with diminished Brownian motion. Finally, all existing models on BR have only dealt with single filaments. The continuous formalism we proposed here is in fact our initial step to extend the BR to a filamentous bundle. All these topics are currently under investigation.
In a special Science issue on Movement: Molecular to Robotic, two articles reviewed recent progress on force and motion generated on the molecular level by two completely different bio- Qian, 2000b, and references cited within). Since the motion of a single motor protein is Brownian, it has to be characterized in terms of probability distribution. The simplest model is that of Huxley (1957). This model corresponds to one on polymerization with nucleotide hydrolysis proposed by Dogterom and Leibler (1993). Both models addressed the important issue of nucleotide hydrolysis, but neglected the stochastic nature in the movement of motor protein and actin polymerization, respectively. Without the diffusion term, such mathematical model is known as random evolution (Pinsky, 1991). (Qian, 1998(Qian, , 2000b(Qian, , 2001a.
The biological systems discussed in the two Science articles (Vale and Milligan, 2000;Mahadevan and Matsudaira, 2000) and the mechanistic, molecular models proposed are completely different.
Yet they share fundamentally the same physiochemical principle which units both models in a quantitative fashion. The mathematical model seems to capture the basic principle for molecular movements and forces in cell biology.
To mathematical biologists, BR is a class of models which is based on a similar physical model but can have many different mathematical representations and different degree of approximations.
We have shown two such analyses in the present work. The essential feature of all the models can, and should, be presented in terms of their MSD, which provides the BR, in steady-state, with an effective diffusion constant(D z ) and a mean velocity (V z ), both as functions of the resistant force.
More subtle differences between models can be found in the transient behavior. The comparison between our analyses is summarized in the Table, in which the effective diffusion constant for the continuous model
D z = (α + β)δ 2 D b /2 D b + (α + β)δ 2 /2 −→ D b
when α → ∞, and the BR velocity
V z = D b (α − β)δ − D b (α + β)δω/2 D b + (α + β)δ 2 /2 −→ (D b /δ)(2 − ω)
is a linear force-velocity relationship.
D z /D b V z δ/D b discrete model 3e 2ω −(10ω−6)e ω +ω 2 −2ω−9 (e ω −1−ω) 3 ω 2 ω 2 e ω −1−ω continuous model 1 2 − ω
It is seen that in the continuous model, the effective diffusion constant D z is always less than the D b , while in the discrete model, the D z = σ 2 /2 is a function of the resistant force. When the force is small, D z can in fact be greater than D b . This is a type of facilitated diffusion.
It is important to point out that the results from our discrete analysis is invalid when the resistant F is sufficiently large, when the polymerization is near its stalling force. This is due to the assumption of infinite large α. This explains why there is no critical force in Fig. 3 fundamental difference is the nucleotide hydrolysis which leads to an irreversible thermodynamic nonequilibrium steady-state, with heat dissipation (Qian, 2001c), in the latter rather than the usual equilibrium. Thus, the BR model is also a natural generalization of the standard polymer theory for passive materials (Doi and Edward, 1986) to active materials for which T.L. Hill (1987) has coined the term "steady-state polymer". There is a continuous intellectual thread in all these mathematical theories. , the movement of the bacterium, X(t), the movement of the tip of the actin filament, ∆(t) Y(t) − X(t), the gap, andŶ(t), the detrended Y(t). Among the four types of data, only the ∆(t) approaches stationarity.Ŷ(t) has zero expectation but with a linear MSD, a characteristic of Brownian motion without drift. Both X(t) and Y(t) show the typical diffusion with a drift (Qian et al., 1990). Fig. 1. Y(t), the movement of the bacterium, X(t), the movement of the tip of the actin filament, ∆(t), the gap, and the detrend Y(t). As expected, after a brief period of time, the Y(t) and X(t) are almost indistinguishable; the gap between them quickly reaches stationarity. The detrendŶ(t) shows a linear MSD, as observed in the experiments (Kuo and McGarth, 2000). In the SPT experiments, the velocity can be obtained as the quadratic term in the MSD of Y(t), the σ 2 can be obtained either as the linear term in the MSD of Y(t), or the MSD of the detrendŶ(t). The velocity v is given in Eq. 12 and the σ is given in Eq. 14. t gap(t) 1-gap(t) (1-gap(t))/10,omega<0 Figure 6: The function gap(t), defined in Eq. 34, is the time course for the the mean gap size to approach to its stationarity. For ω > 0, the mean gap size approaches to a finite size, while for ω < 0, the mean gap size grows without bound.
Acknowledgement
Fig. 2
2shows the mean-square displacement (MSD) of the stochastic data inFig. 1. The MSD for a stochastic processes X(t) is defined as
With very small δ, this gives [α(0) − β(0)]δ − F δ 2 [r α(0) + (1 − r)β(0)]/kBT which should be compared with Vz = (α − β)δ − F δ 2 (α + β)/2kB T from the main text. This indicates that the BR has a splitting factor of r = 1/2. However, Hill's analysis does not have the contribution from the dynamic characteristics of the barrier, i.e., D b . Eq. 30 is the starting point of the recent work ofKolomeisky and Fisher (2001).
shown that the bacteria move with steps. Considering there are many actin filaments in a bundle which propels a bacterium, this observation indicates a synchronized filamental growth in the bundle. The synchronization is not inconsistent with a bundle of actin filaments propelling a bacterium with sufficiently small Brownian movement (i.e., small D b ). The significantly reduced Brownian motion is indeed observed experimentally, both in the direction parallel and perpendicular to the actin growth.
logical systems: motor protein movement and cytoskeletal filamental polymerization(Vale and Milligan, 2000;Mahadevan and Matsudaira, 2000). Both systems can move against resistant force by utilizing chemical free energy. In the abstract of the second article, it was stated "Not all biological movements are caused by molecular motors sliding along filaments or tubules. Just as springs and ratchets can store or release energy and rectify motion in physical systems, their analogs can perform similar functions in biological systems." While there has been much work done on motor proteins and protein polymerization in connection to various cellular phenomena such as motility, less has been discussed about the fundamental physical principles of these two processes. It turns out that both molecular processes have a single, unified mathematical model which accounts for their chemomechanical energy transduction.Theoretical formalism for motor proteins are now well established (seeJülicher et al., 1997;
With the ATP cap and hydrolysis, the model for the stochastic dynamics of actin polymerization will be precisely in the same class of the models for single motor proteins (S.-D. Liang, G. Martinez, G.M. Odell, and H. Qian, work in progress). The experimental measurements on both systems also proceed with parallel paths, as demonstrated byDogterom and Yurke (1997), and more recentlyKuo and McGrath (2000). Similar to the measurements on load-velocity curves for motor proteins, Dogterom and Yurke measured the velocity as a function of resistant force for single microtubules growing in vitro. Analysis of their data suggests that under the stalled (critical) condition, polymerization is in a nonequilibrium steady-state rather than a thermodynamic equilibrium(Kolomeisky and Fisher, 2001;Hill, 1987).All these experimental evidences indicate that the class of BR model (or augmented Huxley model) is a fundamental mathematical model for chemomechanical energy transduction. Recent work on the nonequilibrium statistical mechanics and thermodynamics of BR, in the context for single macromolecules in aqueous solution, also provided the mathematical model with a solid foundation in statistical physics of Boltzmann, Gibbs, and Onsager
, at which the velocity v = 0. The more realistic model with finite α and β does lead to a finite, positive stalling force (Peskin et al., 1993; Kolomeisky and Fisher, 2001). The valid regime for our discrete model is a rapid growing actin filament with the bacteria viscous drag being the limiting factor in the overall BR movement. Finally, it is worth pointing out that mechanical studies of cellular properties and functions can be approximately classified as for passive and active materials. The former can be understood in terms of the theories of viscoelasticity and polymer dynamics, see Qian (2000a) for a general approach to the problem. Materials with chemomechanical energy transduction are active. The
I thank Scot Kuo, Gilbert Martinez, and Gary Odell for many helpful discussions, Elliot Elson and Charles Peskin for helpful comments on the manuscript.
Figure 1 :
1A set of examples, from Monte Carlo simulations, for the stochastic trajectories of Y(t)
Figure 2 :
2The MSD calculated for the four types of data in
Figure 3 :
3The velocity vδ/D b and MSD σ 2 /D b , nondimensionalized, as functions of the resistant force on bacterium, ω = F δ/η b D b .
Figure 4 :
4Numerical computation for the three smallest eigenvalues (in magnitude, all eigenvalues are negative) of Eq. 19, µδ 2 /D b , as function of the resistant force ω. These are the most relevant modes in the relaxation (and correlation function) of the gap, ∆(t), approaching to stationarity. The smallest eigenvalue corresponds to the exit time, shown in the figure (labeled 1/E[T]
Figure 5 :
5The normalized MSD for the ∆(t), from Monte Carlo simulation, for different resistant force F which is related to the probability (p) shown in the figure: F/(η b δ) = 0.1(2p − 1). A standard MSD for a stationary process is directly related to its time correlation function 2 E[∆ 2 (t)] − E[∆(t)∆(0)] , with its asymptote being the 2V ar[∆] when t → ∞. In the simulations, the diffusion constant is D b /δ 2 = 0.005. 0 ≤ ∆(t) ≤ 1; the stationary E[∆] = 0.33, 0.27, 0.21, 0.16, and 0.08 for p = 0.5 − 0.8 respectively; the corresponding relative variances V ar[∆]/E 2 [∆] = 0.53, 0.64, 0.79, 0.91, and 0.96. The relative variance for an exponential distribution is 1.
AppendixDiffusion with Drift in Semi-infinite Space with Noflux Boundary. To understand the dynamics of the gap in the continuous model, one needs to solve the time-dependent diffusion equation, Eq. 25. In nondimensionalized form:with boundary condition u x + ωu = 0 at x = 0 and x = ∞. Amazingly, classic texts on diffusion(Carslaw and Jaeger, 1959;Crank, 1975)did not give an explicit solution to the problem. Becasue of its central importance in the theory of BR, we give some explicit results below.The eigenfunction of the problem associated with the eigenvalue µ(z) = − 1and for µ = 0, √ ωe −ωx . Note there is a gap in µ between µ = 0 and the continuous spectrumTherefore, the solution to Eq. 31 with initial data δ(x) iswhich approaches to the exponential distribution ωe −ωt when t → ∞. From Eq. 33 we havewhere τ = ω 2 t/4. The curve in the square bracket is a universal curve, we shall denote it by 1 − gap(τ ). SeeFig. 6. It is interesting to point out that there is a sharp transition between an unlimited growth of x for ω < 0 and a stationary state for x when ω > 0. This mathematical result is similar to that ofDogterom and Leibler (1993). Finally,The Dynamics of Gap ∆(t) in the Discrete Model. Eq. 19 has a second boundary condition u = 0 at x = δ, which corresponds to nondimensionalized Eq. 31 with 0 ≤ x ≤ 1 and u(1) = 0. It has a set of discrete eigevalues. The eigenfunctions to the nondimensionalized Eq.19 with eigenvalue µ(z) = − 1 4 (ω 2 + z 2 ) ≤ 0, are still given in Eq. 32, but the z's are now the discrete roots of the transcendental equation cos z = ω 2 −z 2 ω 2 +z 2 . When ω < 2, the equation for z has only real roots; hence the largest eigenvalues is µ < −ω 2 /4. If, however, ω > 2, then there is a pair of imaginary ±iz * , |z * | < ω. Then the largest eigenvalue is µ = − ω 2 − (z * ) 2 /4.Fig. 4shows how the largest eigenvalue (smallest in magnitude) changes as functions of ω. It is seen that the largest eigenvalue can be well represented by 1/E[T]The solution to the time-dependent Eq. 19 with initial data δ(x) can be obtained in terms of the u(x, t; z)'s in Eq. 32, and approximated by the first term:Therefore,which tends to zero because the exit probability. However, the remaining probability for ∆(t)quickly approaches to a quasi-stationary distributionFor large ω, this distribution approaches to the exponential distribution ωe −ωx as expected (data not shown).
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|
[] |
[
"LFI 30 and 44 GHz receivers Back-end Modules",
"LFI 30 and 44 GHz receivers Back-end Modules"
] |
[
"E Artal [email protected] \nDpt. Ingeniería de Comunicaciones\nUniversidad de Cantabria\n39005Plaza de la Ciencia, SantanderSpain\n",
"B Aja \nDpt. Ingeniería de Comunicaciones\nUniversidad de Cantabria\n39005Plaza de la Ciencia, SantanderSpain\n",
"M L De La Fuente \nDpt. Ingeniería de Comunicaciones\nUniversidad de Cantabria\n39005Plaza de la Ciencia, SantanderSpain\n",
"J P Pascual \nDpt. Ingeniería de Comunicaciones\nUniversidad de Cantabria\n39005Plaza de la Ciencia, SantanderSpain\n",
"A Mediavilla \nDpt. Ingeniería de Comunicaciones\nUniversidad de Cantabria\n39005Plaza de la Ciencia, SantanderSpain\n",
"E Martinez-Gonzalez \nInstituto de Física de Cantabria\nCSIC-Universidad de Cantabria\nAvda. de los Castros s/n39005SantanderSpain\n",
"L Pradell \nDepartament de Teoría del Senyal i Comunicacions\nUniversitat Politécnica de Catalunya\nBarcelonaSpain\n",
"P De Paco \nDepartament de Teoría del Senyal i Comunicacions\nUniversitat Politécnica de Catalunya\nBarcelonaSpain\n",
"M Bara \nMier Comunicaciones S.A. La Garriga\nBarcelonaSpain\n",
"E Blanco \nMier Comunicaciones S.A. La Garriga\nBarcelonaSpain\n",
"E García \nMier Comunicaciones S.A. La Garriga\nBarcelonaSpain\n",
"R Davis \nJodrell Bank Centre for Astrophysics\nUniversity of Manchester\nUnited Kingdom\n",
"D Kettle \nJodrell Bank Centre for Astrophysics\nUniversity of Manchester\nUnited Kingdom\n",
"N Roddis \nJodrell Bank Centre for Astrophysics\nUniversity of Manchester\nUnited Kingdom\n",
"A Wilkinson \nJodrell Bank Centre for Astrophysics\nUniversity of Manchester\nUnited Kingdom\n",
"M Bersanelli \nDepartment of Physics Via Celoria 16\nUniversitá degli studi di Milano\nMilanoItaly\n",
"A Mennella \nDepartment of Physics Via Celoria 16\nUniversitá degli studi di Milano\nMilanoItaly\n",
"M Tomasi \nDepartment of Physics Via Celoria 16\nUniversitá degli studi di Milano\nMilanoItaly\n",
"R C Butler \nINAF/IASF -Bologna\nVia P. Gobetti 101I-40129BolognaItaly\n",
"F Cuttaia \nINAF/IASF -Bologna\nVia P. Gobetti 101I-40129BolognaItaly\n",
"N Mandolesi \nINAF/IASF -Bologna\nVia P. Gobetti 101I-40129BolognaItaly\n",
"L Stringhetti \nINAF/IASF -Bologna\nVia P. Gobetti 101I-40129BolognaItaly\n"
] |
[
"Dpt. Ingeniería de Comunicaciones\nUniversidad de Cantabria\n39005Plaza de la Ciencia, SantanderSpain",
"Dpt. Ingeniería de Comunicaciones\nUniversidad de Cantabria\n39005Plaza de la Ciencia, SantanderSpain",
"Dpt. Ingeniería de Comunicaciones\nUniversidad de Cantabria\n39005Plaza de la Ciencia, SantanderSpain",
"Dpt. Ingeniería de Comunicaciones\nUniversidad de Cantabria\n39005Plaza de la Ciencia, SantanderSpain",
"Dpt. Ingeniería de Comunicaciones\nUniversidad de Cantabria\n39005Plaza de la Ciencia, SantanderSpain",
"Instituto de Física de Cantabria\nCSIC-Universidad de Cantabria\nAvda. de los Castros s/n39005SantanderSpain",
"Departament de Teoría del Senyal i Comunicacions\nUniversitat Politécnica de Catalunya\nBarcelonaSpain",
"Departament de Teoría del Senyal i Comunicacions\nUniversitat Politécnica de Catalunya\nBarcelonaSpain",
"Mier Comunicaciones S.A. La Garriga\nBarcelonaSpain",
"Mier Comunicaciones S.A. La Garriga\nBarcelonaSpain",
"Mier Comunicaciones S.A. La Garriga\nBarcelonaSpain",
"Jodrell Bank Centre for Astrophysics\nUniversity of Manchester\nUnited Kingdom",
"Jodrell Bank Centre for Astrophysics\nUniversity of Manchester\nUnited Kingdom",
"Jodrell Bank Centre for Astrophysics\nUniversity of Manchester\nUnited Kingdom",
"Jodrell Bank Centre for Astrophysics\nUniversity of Manchester\nUnited Kingdom",
"Department of Physics Via Celoria 16\nUniversitá degli studi di Milano\nMilanoItaly",
"Department of Physics Via Celoria 16\nUniversitá degli studi di Milano\nMilanoItaly",
"Department of Physics Via Celoria 16\nUniversitá degli studi di Milano\nMilanoItaly",
"INAF/IASF -Bologna\nVia P. Gobetti 101I-40129BolognaItaly",
"INAF/IASF -Bologna\nVia P. Gobetti 101I-40129BolognaItaly",
"INAF/IASF -Bologna\nVia P. Gobetti 101I-40129BolognaItaly",
"INAF/IASF -Bologna\nVia P. Gobetti 101I-40129BolognaItaly"
] |
[] |
The 30 and 44 GHz Back End Modules (BEM) for the Planck Low Frequency Instrument are broadband receivers (20% relative bandwidth) working at room temperature. The signals coming from the Front End Module are amplified, band pass filtered and finally converted to DC by a detector diode. Each receiver has two identical branches following the differential scheme of the Planck radiometers. The BEM design is based on MMIC Low Noise Amplifiers using GaAs P-HEMT devices, microstrip filters and Schottky diode detectors. Their manufacturing development has included elegant breadboard prototypes and finally qualification and flight model units. Electrical, mechanical and environmental tests were carried out for the characterization and verification of the manufactured BEMs. A description of the 30 and 44 GHz Back End Modules of Planck-LFI radiometers is given, with details of the tests done to determine their electrical and environmental performances. The electrical performances of the 30 and 44 GHz Back End Modules: frequency response, effective bandwidth, equivalent noise temperature, 1/f noise and linearity are presented.
|
10.1088/1748-0221/4/12/t12003
|
[
"https://arxiv.org/pdf/1001.4771v2.pdf"
] | 56,126,939 |
1001.4771
|
0109ad10ae98c73da7c9a4252c6f942aec69cdde
|
LFI 30 and 44 GHz receivers Back-end Modules
27 Jan 2010
E Artal [email protected]
Dpt. Ingeniería de Comunicaciones
Universidad de Cantabria
39005Plaza de la Ciencia, SantanderSpain
B Aja
Dpt. Ingeniería de Comunicaciones
Universidad de Cantabria
39005Plaza de la Ciencia, SantanderSpain
M L De La Fuente
Dpt. Ingeniería de Comunicaciones
Universidad de Cantabria
39005Plaza de la Ciencia, SantanderSpain
J P Pascual
Dpt. Ingeniería de Comunicaciones
Universidad de Cantabria
39005Plaza de la Ciencia, SantanderSpain
A Mediavilla
Dpt. Ingeniería de Comunicaciones
Universidad de Cantabria
39005Plaza de la Ciencia, SantanderSpain
E Martinez-Gonzalez
Instituto de Física de Cantabria
CSIC-Universidad de Cantabria
Avda. de los Castros s/n39005SantanderSpain
L Pradell
Departament de Teoría del Senyal i Comunicacions
Universitat Politécnica de Catalunya
BarcelonaSpain
P De Paco
Departament de Teoría del Senyal i Comunicacions
Universitat Politécnica de Catalunya
BarcelonaSpain
M Bara
Mier Comunicaciones S.A. La Garriga
BarcelonaSpain
E Blanco
Mier Comunicaciones S.A. La Garriga
BarcelonaSpain
E García
Mier Comunicaciones S.A. La Garriga
BarcelonaSpain
R Davis
Jodrell Bank Centre for Astrophysics
University of Manchester
United Kingdom
D Kettle
Jodrell Bank Centre for Astrophysics
University of Manchester
United Kingdom
N Roddis
Jodrell Bank Centre for Astrophysics
University of Manchester
United Kingdom
A Wilkinson
Jodrell Bank Centre for Astrophysics
University of Manchester
United Kingdom
M Bersanelli
Department of Physics Via Celoria 16
Universitá degli studi di Milano
MilanoItaly
A Mennella
Department of Physics Via Celoria 16
Universitá degli studi di Milano
MilanoItaly
M Tomasi
Department of Physics Via Celoria 16
Universitá degli studi di Milano
MilanoItaly
R C Butler
INAF/IASF -Bologna
Via P. Gobetti 101I-40129BolognaItaly
F Cuttaia
INAF/IASF -Bologna
Via P. Gobetti 101I-40129BolognaItaly
N Mandolesi
INAF/IASF -Bologna
Via P. Gobetti 101I-40129BolognaItaly
L Stringhetti
INAF/IASF -Bologna
Via P. Gobetti 101I-40129BolognaItaly
LFI 30 and 44 GHz receivers Back-end Modules
27 Jan 2010Preprint typeset in JINST style -HYPER VERSIONRadiometer-Cosmic Microwave Background -Instrumentation: receivers -MMIC LNA -Schottky diode detector * Corresponding author
The 30 and 44 GHz Back End Modules (BEM) for the Planck Low Frequency Instrument are broadband receivers (20% relative bandwidth) working at room temperature. The signals coming from the Front End Module are amplified, band pass filtered and finally converted to DC by a detector diode. Each receiver has two identical branches following the differential scheme of the Planck radiometers. The BEM design is based on MMIC Low Noise Amplifiers using GaAs P-HEMT devices, microstrip filters and Schottky diode detectors. Their manufacturing development has included elegant breadboard prototypes and finally qualification and flight model units. Electrical, mechanical and environmental tests were carried out for the characterization and verification of the manufactured BEMs. A description of the 30 and 44 GHz Back End Modules of Planck-LFI radiometers is given, with details of the tests done to determine their electrical and environmental performances. The electrical performances of the 30 and 44 GHz Back End Modules: frequency response, effective bandwidth, equivalent noise temperature, 1/f noise and linearity are presented.
Introduction
The Low Frequency Instrument (LFI) is one of the two instruments of the ESA Planck satellite. The objective of the satellite is to achieve precision maps of the Cosmic Microwave Background with a very good sensitivity to observe sky temperature anisotropies [1]. LFI contains very high sensitive receivers based on cryogenic radiometers with their Front-End Modules (FEM) operating at 20 K [2]. LFI covers three adjacent frequency bands centred at 30, 44 and 70 GHz, with a fractional bandwidth of 20%, and uses cryogenic very low noise amplifiers with Indium Phosphide (InP) High Electron Mobility Transistors (HEMT) in the FEM. The Front-End Modules are physically and thermally separated from the Back-End Modules by over a meter of copper and stainless steel rectangular waveguide. The signals originating in the FEM are then further amplified and finally detected in the BEM that operate at 300 K.
Planck-LFI radiometers are based on a differential scheme [3], [4], [5], shown in Figure 1. Signals from the sky and from a 4 K reference load are combined using a (180 • ) hybrid. Both signals are amplified by Low Noise Amplifiers (LNA) in the Front-End Module. Two identical phase switches introduce a differential (180 • ) phase shift in one branch with relation to the other. The second (180 • ) hybrid delivers a signal proportional to sky temperature in one output, and a signal proportional to the reference load temperature in the other.
The balanced structure of the FEM permits to obtain two signals, sky and reference, contaminated by the same amount of noise because both have passed through the same LNA and phase switches. By differencing signals across the two outputs using post-processing, the noise can be cancelled. Phase switching at 4 kHz is done also to remove the 1/f noise of the BEM, because it does not have a balanced structure like the FEM. This BEM noise is noticeable at low frequencies, at some hundreds of Hz. Noise cancellation can only be perfect if the two FEM branches are identical and the balance is perfect as well. In practice a little unbalance is tolerated and it was tested as a leakage from one input to the theoretically isolated output.
Next
Design of the Back-End Modules
In this section, we will describe all the fundamentals of each RF component designed and their individual performance before integration.
Low Noise Amplifiers
Each Low Noise Amplifier (LNA) consists of two cascaded amplifiers in order to have enough gain and to provide a signal in the square law region of the diode detector. Monolithic Microwave Integrated Circuits (MMIC) on GaAs technology have been chosen at 30 and 44 GHz. The MMIC at Ka-band are commercial circuits, model HMC263 from Hittite. They have four stages of Pseudomorphic-High Electron Mobility Transistors (PHEMT), with an operating bandwidth from 24 GHz to 36 GHz, and offer 23 dB of gain and 3 dB of noise figure from a self-biasing supply of +3 Volt, 52 mA. A picture of the MMIC at Ka-band, its noise figure, gain and input and output return loss are shown in Figure 2. At Q-band two custom designed MMICs have been assembled [6]. They have been manufactured with the process ED02AH from OMMIC, which employs a 0.2 µm gate length PHEMT on GaAs. The first MMIC has four stages of depletion mode transistors (Normally ON: N-ON) with gate widths of 4x15 µm and 6x15 µm, shown in Figure 3 (a). The second MMIC, in Figure 3 (b), has four-stages of enhancement mode transistors (Normally-OFF: N-OFF) with gate widths of 6x15 µm. The N-ON LNA preceding the N-OFF, was found to be the best in terms of input and output matching and noise performance. The main requirements of the two LNAs are to provide low noise and low power consumption with enough gain. Using as first amplifier an LNA, based on N-ON PHEMT transistors, a good noise performance was obtained with a power consumption of only 90 mW . The second LNA, with N-OFF PHEMT transistors, was added to increase the gain with a very low impact in the noise figure and a very low power consumption of 32 mW . All the circuits were measured on wafer using a coplanar probe station. Noise figure, return loss and associated gain for the Q-band depletion and enhancement transistor LNAs are plotted in Figure 4.
Band Pass Filters
A band pass filter was used to define an effective bandwidth of 20% and to reject undesired signals out of the band of interest. Low bandpass losses, more than 10 dB out of band losses and small size were considered the main objectives to fulfil. The filter was based on microstrip coupled line structure that was chosen because it provides inherently good band pass characteristics. The design is a three-order Chebyshev resonator filter. Electrical models of coupled lines and an electromagnetic simulator was used in the design phase. The design method was based on the classic prototype filter tables provided by [7], but a design methodology has been developed to achieve a predictable frequency response in microstrip filters using commercial CAD software. After a careful evaluation of the validity of the CAD models, comparing simulated and accurately measured results, the design was restricted to microstrip elements than could be well characterized [8]. The selection of the substrate became critical due to the gaps and widths of the microstrip lines because it sets the line-etching precision required and the minimum losses achievable. Microstrip lines were made on Duroid 6002 substrate with 10 mils thickness and dielectric constant of 2.92. Several units of microstrip band pass filters were fabricated and tested. Typical test results for the 30 GHz filter are insertion losses lower than 0.84 dB in the band, and for the 44 GHz filter, insertion losses better than 1.5 dB and return losses better than 10 dB across the operating bandwidth. A photograph of a 30 GHz filter is shown in Figure 5. The response of the filters when they have been measured with coplanar to microstrip transitions using a coplanar probe station is depicted in Figure 6.
Diode Detectors
The diode for the detector was a GaAs planar doped barrier Schottky diode. The specific model chosen was a zero-bias beam-lead diode HSCH-9161 from Agilent Technologies. This diode has suitable characteristics to be used at microwave frequencies. Among the main specifications in the detector design are the input matching, the sensitivity and the tangential sensitivity. The diode equivalent circuit is not a good match to 50 Ohm, so it was necessary to synthesize a network that would transform it to something close with an input matching network. Thus the detector is composed of a hybrid reactive/passive matching network, and the Schottky diode. Both detectors, for 30 GHz and 44 GHz, were mounted with a coplanar-to-microstrip transition to make on-wafer tests as a previous step to the BEM integration. The practical implementation is performed with transmission lines printed on a standard dielectric substrate, Alumina with 10 mils thickness and dielectric constant of 9.9. A view of the detectors is shown in Figure 7. The output voltage sensitivity has been measured at 30 GHz for the Ka band detector and it is shown in Figure 8 (a). Figure 8 (b) depicts the rectification efficiency of the Q-band detector at the frequency of 44 GHz for −30 dBm of input power.
DC Amplifier
The detector diode output is connected to a low noise DC-amplifier, with an adequate voltage gain to have the required detected signal level for the data acquisition electronic module. A schematic of the DC-amplifier is shown in Figure 9. The first stage has an OP27 precision operational amplifier that combines low offset and drift characteristics and low noise, making it ideal for precision instrumentation applications and accurate amplification of a low-level signal. A second balanced stage, implemented with an OP200, provides a balanced and bipolar output. DC amplifier total power consumption with a high impedance load is 37 mW .
The amplifier gain is given by the ratio between the differential output voltage V out , and V in (input voltage from the detector), where V out is given by:
V out = V out+ −V out− = 2 · R 2 R 1 + R 2 · 1 + R 4 R 3 ·V in (2.1)
The detector resistive load is R 1 + R 2 . In order not to affect the detector RF response, the condition R 1 + R 2 ≥ 50 kOhm was fulfilled.
Since the phase switch frequency rate in the receiver is 4096 Hz, the output signal has to provide a video bandwidth of at least 50 kHz, which means that the output signal contains more than ten harmonics in order not to degrade the information. The DC amplifier measured gainbandwidth product was 5.7 MHz. This operational amplifier gain bandwidth product has been taken into account, and the maximum achievable balanced gain without loosing output bandwidth was 100. A voltage gain of 50 is due to the OP27 and a further of 2 due to unbalanced to balanced conversion of the OP200 with unit individual gain. Another constraint of this DC amplifier is to provide an output voltage in a window between 0.2 Volt and 0.8 Volt, where the data acquisition electronics (DAE) works properly. The designed DC amplifier was adjusted for each channel, taking into account small RF gain differences, in order to have the output DC voltage inside the window and to achieve the output bandwidth requirement. The resulting bandwidth was 163 kHz. Figure 10 shows the DC amplifier measured low-frequency noise referred to the amplifier input. The white noise voltage level is 8 nV / √ Hz (= −162 dBV / √ Hz), and the flicker noise kneefrequency, defined as the point where the noise voltage is √ 2 times the white noise voltage, is 2.8 Hz.
Manufacturing
Elegant breadboard prototype
The objective of the Elegant Breadboard (EBB) prototypes was to experimentally demonstrate the radiometer concept, by integrating a fully representative unit in the laboratory. An EBB demonstrator must have the same electrical functionality as the final flight unit. The internal architecture, electrical scheme, and the electronic components were the same as for flight. The electronics components do not need to be space qualified. The size and weight of the EBB demonstrator can be different from the flight unit. To make easier the integration with the EBB of the Front End Module, the EBB version of the 30 and 44 GHz BEM included only one branch. In fact the EBB version is a quarter of one full BEM. Figure 11 shows a photograph of one EBB at 30 GHz. This EBB branch contains a Low Noise Amplifier (LNA), a Band Pass Filter (BPF), a Schottky diode detector (DET) and a low frequency amplifier (DC Amp). The central frequency is 30 GHz and the nominal fractional bandwidth is 20%. Figure 12 shows two EBB BEM units connected to the Prototype Demonstrator (PD) FEM in the laboratory at Jodrell Bank Observatory.
The first functional element of each BEM is a waveguide-to-microstrip transition that was designed using a ridge waveguide [9]. The input waveguide flange is WR-28 for the 30 GHz BEM and WR-22 for the 44 GHz BEM. The transition is a four sections stepped ridge waveguide to microstrip line transition which provides a broadband performance. This transition has been chosen for its broad bandwidth, low insertion loss, and repeatable performance. Figure 13
Qualification and Flight Model units
The final BEM mechanical configuration for the Qualification Model and Flight Model has four RF branches, providing signal amplification and detection for two complete radiometers. Mechanical design had been carried out by Mier Comunicaciones S.A. within an allowed envelope of 70 x 60 x 39 mm 3 including all the RF and DC circuitry. Mass is 305 g for 30 GHz BEM and 278 g for the 44 GHz BEM. Figure 14 shows the external view of the BEM. Internally there are 5 different levels of PCB circuits, from top to bottom, as follows:
Electrical characterization tests
A set of basic tests were performed at three different temperatures in the range of possible operating temperature; T low (−25 • C), T nom (26 • C) and T high (48 • C), to verify proper operation of the FM Back-End Modules and fulfillment of the specified electrical requirements. They were assembled and tested in a clean room equipped with a vacuum chamber which enables us to achieve low pressure levels and to vary the base plate and shroud temperature within the acceptance ranges. The principal tests performed were frequency response, equivalent noise temperature, stability and linearity. Test equipment available included vector network analyzers, swept frequency sources, noise sources and low frequency spectrum analyzers.
Frequency response and effective bandwidth
The frequency response (RF to DC) measurements were done injecting a CW small signal into the waveguide input, for a constant power level. The level of the signal was set to yield a readily measurable response, but not so large as to cause non-linear effects. This test was needed in order to know the bandpass response. It was measured stepping the synthesizer source through 201 frequencies and recording the output voltage for each frequency, with the RF output of the synthesized enabled and disabled. Figure 18 shows the RF to DC response for the flight model BEMs at 30 GHz and 44 GHz.
The measured results obtained for the RF to DC response are used to calculate the effective bandwidth. The effective bandwidth is defined according to next expresion.
BW e f f = | G( f )d f | 2 |G( f )| 2 d f (4.1)
G( f ) is the RF power gain of the BEM, including the RF detector response. This G( f ) gain is obtained by RF to DC response test. This test is performed with a microwave sweep generator providing a constant input power versus frequency, so the effective bandwidth can be calculated as (4.2), using only the output voltage values taken at discrete frequencies.
BW e f f = ∆ f · N N + 1 ∑ N i=1 V out (i) −V outo f f 2 ∑ N i=1 (V out (i) −V outo f f ) 2 (4.2)
where N is the number of frequency points, ∆ f is the frequency step, V out (i) the DC output voltage at each frequency and V outo f f is the DC output voltage when the sweep generator is off. V outo f f is typically about 25 mV for the 30 GHz BEM and lower values for the 44 GHz BEM. Table 1 shows the values of the effective bandwidth for each flight model BEM at three different temperatures in the range of possible operating temperature.
Channel A Channel B Channel C Channel D BEM
T low T nom T high T low T nom T high T low T nom T high T low T nom T high 30 GHz FM1 9.11 9.16 9.07 9.17 9.14 9.02 9.34 9.36 9.27 9.54 9.45 9.28 30 GHz FM2 9.23 9.26 9.29 9.35 9.29 9.32 9.16 9.10 9.06 9.84 9.77 9.72 30 GHz FM3 9.34 9.26 9.13 9.04 9.09 9.02 9.03 9.00 8. 88
Equivalent noise temperature
A method has been developed to achieve an accurate and unique equivalent noise temperature of the whole receiver. This method takes into account commercial noise sources, which have not a flat Excess Noise Ratio (ENR) versus frequency in millimetre-wave range, and RF to DC receiver performance along the band. Because the hot temperature of the used noise source and the BEM RF gain show variations across the operating bandwidth, the next expression was used to obtain the global equivalent temperature (T rec ) [10]:
T rec = ∑ f 2 f 1 T h ( f )V det ( f ) −Y T c ∑ f 2 f 1 V det ( f ) (Y − 1) ∑ f 2 f 1 V det ( f ) (4.3)
Where T h and T c are the hot and cold temperature of the noise source, Y is the noise Y − f actor, and V det is the detected voltage at each frequency when a CW signal, with a constant power sufficiently above white noise level, is applied at the BEM input. The Y − f actor is given by:
Y = V det | h V det | c (4.4)
Where V det | h and V det | c are the receiver output voltages when two known source temperatures, hot and cold loads, are connected at the BEM input.
The Y − f actor was tested with a cold load and a hot load using a commercial noise source Q347B from Agilent. The equivalent noise temperature as a total power radiometer was slightly worse than on wafer measured noise figure of a naked MMIC due mainly to losses in the waveguide to microstrip transition and to the readjustment of the bias point to decrease the ripple in the operating band, trading off noise temperature and effective bandwidth. This noise temperature has minimum impact on the global radiometer performance due to the high gain of the FEM. Table 2 shows the values of the equivalent noise temperature for each flight model BEM at three different temperatures in the range of possible operating temperature. The large variability of the equivalent noise temperature of 44 GHz BEM units was due to their large dependence on the input matching network result, which was observed to be a very critical parameter, not easy to control during the assembly process of MMIC.
Stability: 1/f noise
The raw measurements of the output spectrum are used for the determination of the 1/f knee frequency. First, the output spectrum data are handled in a logarithmic scale in frequency. This way, it is possible to plot the spectral noise density composed by a white noise constant value, at high frequencies, plus the contribution of the 1/f noise at low frequencies. According to the definition, the 1/f noise knee frequency is the frequency at which the noise voltage spectrum density is at a level of √ 2 of the white noise voltage spectrum density level, for instance in V rms/sqrt(Hz). Using power density instead of voltage density then the level is 3 dB above the white noise level (for instance in a scale of dBm/Hz).
The BEM low frequency power spectrum was characterized with a Hewlett Packard Vector Signal Analyzer HP81490A when a wave-guide matched load is connected to the input. The test was done at three temperatures: nominal (299 K), low (273 K) and high (326 K). The 1/f knee frequency was below 400 Hz in all BEM units, much lower than the phase switching of the FEM (4096 Hz), so gain fluctuations of the back-end module did not impact on the global performance of the radiometer. The dominant 1/f noise source is attributed to the Schottky diode detector, since it refers directly to the diode current. The 1/f noise spectrum of each LNA alone was tested and the knee-frequency was about 13 Hz for the N-ON LNA and about 15 Hz for the N-OFF LNA. These results make evident that diode detector is mainly responsible for the knee-frequency of the BEM. The results for the four channels of a 30 GHz BEM FM unit are given in Table 3. Figure 19 shows a typical noise spectrum of a flight model BEM at 30 GHz.
Linearity
The detected voltage versus input power was measured in order to get the BEM dynamic range. A HP83650B generator was used as CW source. Results for the 44 GHz QM representative BEM are depicted in Figure 20 (solid line) for a 40.3 GHz CW signal input. In order to use a more realistic input signal, a wide band noise stimulus was used to measure the BEM sensitivity. This noise was obtained from a broadband white noise source, Q347B from Agilent, accordingly filtered and amplified by another 44 GHz BEM branch with only the radiofrequency chain (detector and DC amplifier not included). Results are also plotted in Figure 20 (circles). Taking into account the FEM gain and noise temperature, the BEM is always working under compression regime. The BEM non-linearity is a combination of the second MMIC LNA gain compression and the nonlinearity of the diode detector. Given the dynamic range and the compression response of the BEM, it is not possible to identify the 1 dB compression point. Due to this compression effect, measured signal must be converted using the calibration curves, according to the procedures described in detail in in [11].
Results of compression for channel A of a FM unit at 44 GHz are shown in Figure 21. The test was done at three temperatures: nominal, high and low.
Verification tests
Vibration and thermal vacuum
Comprehensive vibration tests were performed, at different planes and frequency profiles, from 5 Hz to 2000 Hz. The sequence applied in the QM BEM unit test was the following one for each axis: Low sine vibration, sine vibration, random vibration and low sine vibration surveys. The low level sinusoidal vibration surveys were conducted per each axis, prior and after performing sinusoidal and random vibration, at a sweep of 2 oct/min and an acceleration of 0.5 g, in the range 5-2000 Hz, only one sweep per run. The sinusoidal vibration test consisted of a single sweep per axis, at a rate of 2 oct/min. In all vibration tests the BEM units were not operating.
The units were tested in 3 mutually perpendicular axes, 2 of them parallel to the base plate and the third one perpendicular to it. Figure 22 shows the axes orientation with relation to the BEM unit.
As the specified vibration levels were defined with relation to the spacecraft axis system, a stiff fixture providing the right inclination for the BEM units was envisaged in order to match to the shaker axis. The BEM units were hard mounted on the fixture. This fixture guaranteed that the major modes of the BEM were not modified, in the sense that frequency shifts were below 5% for lower frequency modes. The units were tested at the environmental temperature expected during satellite launch. A view of one BEM unit mounted on the shaker is in Figure 23. For the FM units the acceptance sinusoidal vibration test consisted of a single sweep per axis, at a rate of 4 oct/min. The acceptance levels applied in this test are indicated in Table 4. The acceptance random vibration test was done according to the levels and durations presented in Table 5.
The proper BEM performance, after the vibration of each unit, was checked by post-dynamic performance verification tests. These checks were performed by measuring a small number of indicative requirements of the BEM, like power consumption, in order to be sure that the unit was still alive and performing well. The electrical behaviour, in terms of noise and RF to DC conversion, did not change after vibration and thermal vacuum tests. The tested deviations were within the range of the test equipment measurement accuracy.
Thermal vacuum tests for BEM flight units were done basically by six thermal cycles, with a total duration of about 25 hours, between −35 • C and +55 • C. The thermal profile is depicted in Figure 24. During thermal cycling, a set of Reduced Performance Tests (RPT) have been performed. As in the vibration tests, after thermal vacuum test the survival condition and functionality of each unit was checked.
Each unit in turn was screwed to the base plate of the vacuum chamber and low thermal resistance ensured by thermal silcone compound. Temperature cycling in the acceptance range was thus achieved by changing the base plate temperature. The DC feedthroughs of the chamber were connected to the BEM, in order to have available the DC power lines outside the chamber. Output channel signals and DC consumption were continuously monitored by the Agilent 34970A Data Logger with the 34901A switching unit.
As it is shown in Figure 25, the BEM waveguide inputs were left open. In order to avoid the detection of any noise signal, an aluminium wall, covered with a layer of microwave absorber, has been located at a distance of 2.5 cm from the BEM.
Electromagnetic compatibility
Very strict EMC tests were performed on BEM units, according to specifications. Both conducted and radiated emission and susceptibility tests were made. Interfering signals covering the range from 30 Hz to 18 GHz depending on the individual test were used. Radiated tests included electric and magnetic fields emission and susceptibility. In the case of conducted EMC tests the emission and the susceptibility were tested through the power supply lines ± 5 Volt. The most difficult test to fulfil was the conducted susceptibility, because the BEM units do not have DC to DC converters inside, and power supply fluctuations appeared at the BEM detected output. Special filtering on the power supply input lines was used to avoid susceptibility at low frequencies.
Conclusions
Back End Modules at 30 and 44 GHz for Planck Low Frequency Instrument have been designed, manufactured and tested. They have successfully fulfilled the electrical, mechanical and environmental requirements of Planck satellite mission. Qualification model units and Flight Model units have demonstrated the compliance with the required performances.
Figure 1 .
1Planck Mission radiometer scheme.
sections describe in detail the 30 and 44 GHz BEM units of Planck-LFI, from their design principles and individual subsystems performance, to the final Flight Model units which are integrated in the Planck satellite. Prototype and Elegant Breadboard units are presented and described showing their main characteristics. Electrical and environmental tests have been performed in all the Qualification and Flight Model units delivered to the Planck satellite Project System Team.
Figure 2 .
2(a) Ka-band LNA MMIC (b) MMIC performance.
Figure 3 .
3Q-band LNAs, (a) Normally ON; (b) Normally OFF.
Figure 4 .
4Q-band LNAs on wafer performance, (a) Normally ON; (b) Normally OFF.
Figure 5 .
5Microstrip bandpass filter.
Figure 6 .
6Band-pass filter performance, (a) 30 GHz; (b) 44 GHz.
Figure 7 .
7Diode detectors, (a) 30 GHz ; (b) 44 GHz.
Figure 8 .
8(a) Ka-band detector sensitivity at 30 GHz ; (b) Q-band detector rectification efficiency at 44 GHz for -30 dBm of input power.
Figure 9 .
9DC-amplifier schematic.
Figure 10 .
10DC amplifier low frequency noise, referred to the amplifier input.
Figure 11 .
11General view of the EBB 30 GHz BEM branch with LNA, band pass filter and detector.
shows the experimental results of back to back rectangular waveguide to microstrip transitions using ridge waveguide transformer. Microstrip line losses are included in the result. The length of microstrip 50 Ohm line, on Alumina substrate 10 mils thickness, was 10 mm in both units (Ka band and Q band). The estimated insertion loss of one single ridge waveguide transition is lower than 0.2 dB.
Figure 12 .
12The EBB 30 GHz BEM branches connected to the Prototype Demonstrator FEM in Jodrell Bank Observatory.
Figure 13 .
13Insertion loss and return loss of back to back rectangular waveguide to microstrip transitions through stepped ridge sections.(a) Ka band unit, (b) Q band unit.
Figure 14 .
14Back End module external view.
Figure 15 .
15(a) DC Amp PCB contains operational amplifiers; (b) DC PCB contains voltage regulators. 1. DC PCB: It contains voltage regulators for a half BEM (one receiver, two branches) 2. DC amplifiers PCB: It contains the DC operational amplifiers for a half BEM (two signal detected outputs) 3. RF part: It contains two RF branches (one receiver) 4. DC amplifiers PCB: It contains the DC operational amplifiers for the other half BEM (two signal detected outputs) 5. DC PCB: It contains voltage regulators for the other half BEM (one receiver, two branches) Pictures of the DC and RF circuits are in Figure 15 and Figure 16. Qualification Model (QM) units have been manufactured using identical electrical and mechanical components as for the Flight Model (FM) units. The QM and FM components have identical quality level and have been space qualified following the same procedures. The only difference between QM and FM units is the different environmental tests done on each case. In
Figure 16 .
16RF parts (two branches) in: (a) 30 GHz BEM, (b) 44 GHz BEM.
Figure 17 .
17(a) FM units of 30 GHz BEM; (b) QM unit of 44 GHz BEM. particular vibration levels and thermal cycling tests for QM units have been more demanding than for FM units. Summarising: QM units are identical to the FM units, but they are not intended to be installed in the satellite. In order to be ready to deal with limited unexpected component failure in the FM units, before launching the satellite, spare FM units of the 30 and 44 GHz BEM have been also manufactured and tested. Pictures of FM or QM units of 30 and 44 GHz BEM are showed inFigure 17
.
Figure 18 .
18RF to DC response at three temperatures for a CW power input of -60 dBm, (a) BEM 30 GHz Flight Model; (b) BEM 44 GHz Flight Model.
Figure 19 .
19Typical low frequency noise power spectrum at the BEM output. Tested in a 50 Ohm spectrum analyser.
Figure 20 .
2044 GHz QM BEM dynamic range.
Figure 21 .
21Compression at 44 GHz (FM unit).
Figure 22 .
22BEM planes.
Figure 23 .
2330 GHz QM BEM attached to the shaker.
Figure 24 .
24Thermal profile for a FM unit test (RPT instants highlighted with squares).
Figure 25 .
25BEM fixation to base plate of thermal vacuum chamber.
Table 1 .
1Effective bandwidth of the BEM Flight models in GHz. (One unit of each band is a Flight Spare). Estimated error = ± 1%.
Table 2 .
2Equivalent noise temperature of the BEM Flight models in Kelvin. (One unit of each band is a Flight Spare). Estimated error: ± 20 K.Channel A
Channel B
Channel C
Channel D
BEM
T low T nom T high T low T nom T high T low T nom T high T low T nom T high
30 GHz FM1 196 317 365
159 294 349
231 349 413
150 292 346
30 GHz FM2 176 288 324
179 299 332
202 316 357
167 307 350
30 GHz FM3 164 286 347
129 272 323
257 342 410
185 301 364
44 GHz FM1 923 856 1006
734 676 745
798 662 744
546 643 881
44 GHz FM2 346 494 397
513 674 587
349 426 386
299 405 509
44 GHz FM3 419 467 525
520 595 755
385 437 591
392 437 484
44 GHz FM4 342 561 939
464 459 609
271 380 609
342 510 598
Table 3 .
31/f knee frequency (Hz) of 30 GHz FM2 BEM unit.
Channel T low T nom T high
A
75
90
103
B
75
100
94
C
74
100 117
D
87
100 100
Table 4 .
4Acceptance sinusoidal vibration test levels.Axis
Frequency range (Hz)
Level
Out-of-plane (OOP)
5 -21.25
+/-8 mm
21.25 -100
20 g
In-plane (IP − NFP, IP − PFP)
5 -21.25
+/-8 mm
21.25 -100
16 g
Table 5 .
5Acceptance random vibration test levels. Frequency (Hz) Power Spectral Density (g 2 /Hz) G rms Time per axis (s)Axis
20
0.0482
80
0.192
Out − o f − plane
220
6.912
(OOP)
240
6.912
25.05
60
300
0.832
500
0.832
2000
0.00822
20
0.0241
80
0.096
In − plane
150
3.456
(IP − NFP, IP − PFP)
170
3.456
20.01
60
300
0.704
500
0.704
2000
0.00411
AcknowledgmentsThis work has been supported by the Spanish "Plan Nacional de I+D+i", Programa Nacional de Espacio, grants ESP2002-04141-C03-01/02/03 and ESP2004-07067-C03-02. The authors would like to thank Eva Cuerno and Alexandrina Pana for the assembly of the BEM prototypes. Planck is a project of the European Space Agency with instruments funded by ESA member states, and with special contributions from Denmark and NASA (USA). The Planck-LFI project is developed by an International Consortium lead by Italy and involving Canada, Finland, Germany, Norway, Spain, Switzerland, UK, USA.
The Planck-LFI programme. N Mandolesi, A&A. SubmittedN. Mandolesi et al., "The Planck-LFI programme", 2009, A&A, Submitted.
Design, development and verification of the 30 and 44 GHz front-end modules for the Planck Low Frequency Instrument. R Davis, JINST. This issueR. Davis et al., "Design, development and verification of the 30 and 44 GHz front-end modules for the Planck Low Frequency Instrument", 2009, JINST , This issue.
Sensibilité des Radiotélescopes et Récepteurs à Correĺation. E J Blum, Annales d'Astrophysique. 22E.J. Blum, "Sensibilité des Radiotélescopes et Récepteurs à Correĺation", Annales d'Astrophysique,Vol.22, pp. 140-163. Feb. 1959.
Multi-band Radiometer for Measuring the Cosmic Microwave Background. M Bersanelli, N Mandolesi, J Marti-Canales, Proc. 32nd European Microwave Conference. 32nd European Microwave ConferenceMilan, ItalyM. Bersanelli, N. Mandolesi, J. Marti-Canales, "Multi-band Radiometer for Measuring the Cosmic Microwave Background", Proc. 32nd European Microwave Conference, pp. 547-550, Milan, Italy, Sept. 2002.
Planck-LFI Instrument Description. M Bersanelli, A&A. SubmittedM. Bersanelli, et al., "Planck-LFI Instrument Description", 2009, A&A, Submitted
Q-Band Monolithic GaAs PHEMT Low Noise Amplifiers: Comparative Study of Depletion and Enhancement Mode Transistors. B Aja, Proc. GAAS 2002 Symposium. GAAS 2002 SymposiumMilan, ItalyB. Aja et al., "Q-Band Monolithic GaAs PHEMT Low Noise Amplifiers: Comparative Study of Depletion and Enhancement Mode Transistors", Proc. GAAS 2002 Symposium, pp. 53-56. Milan, Italy, Sept. 2002.
Microwave Filters Impedance-Matching Networks and Coupling Structures. G L Matthaei, L Young, E M Jones, Mc.Graw-HillG.L. Matthaei, L. Young, E.M. Jones, "Microwave Filters Impedance-Matching Networks and Coupling Structures". Mc.Graw-Hill, 1964.
Millimetre Wave Broadband Bandpass Microstrip Filters. Design and Test. M Dettrati, Proc. 32nd European Microwave Conference. 32nd European Microwave ConferenceMilan, ItalyM. Dettrati et al., "Millimetre Wave Broadband Bandpass Microstrip Filters. Design and Test". Proc. 32nd European Microwave Conference, pp. 573-575, Milan, Italy, Sept. 2002.
Closed-Form Expressions for the Parameters of Finned and Ridged Waveguides. W J R Hoefer, M Burton, IEEE MTT. 3012W.J.R. Hoefer, M. Burton, "Closed-Form Expressions for the Parameters of Finned and Ridged Waveguides". IEEE MTT, Vol. 30, no. 12, pp. 2190-2194, Dec. 1982.
A New Method to Obtain Total Power Receiver Equivalent Noise Temperature. B Aja, Proc. 33rd European Microwave Conference. 33rd European Microwave ConferenceMunich, GermanyB. Aja et al., "A New Method to Obtain Total Power Receiver Equivalent Noise Temperature", Proc. 33rd European Microwave Conference, pp. 355-358, Munich, Germany, Oct. 2003.
A Mennella, The linearity response of the Planck-LFI flight model receivers. This issueA. Mennella, et al., "The linearity response of the Planck-LFI flight model receivers", 2009, JINST , This issue.
|
[] |
[
"Compactifying Spec Z",
"Compactifying Spec Z"
] |
[
"Satoshi Takagi "
] |
[] |
[] |
In this paper, we introduce a new algebraic type of 'convexoid rings', and we give the definition of (weak) convexoid schemes, which share similar properties with ordinary schemes. As a result, we give a purely-algebraic construction of the compactification Spec Z = Spec Z∪ {∞}, which is realized as the Zariski-Riemann space of Spec Z in the category of weak convexoid schemes.
| null |
[
"https://arxiv.org/pdf/1203.4914v2.pdf"
] | 118,342,674 |
1203.4914
|
88f98440e360a551d2cbd4426e743906fa5c23b5
|
Compactifying Spec Z
23 Mar 2012
Satoshi Takagi
Compactifying Spec Z
23 Mar 2012arXiv:1203.4914v2 [math.AG]
In this paper, we introduce a new algebraic type of 'convexoid rings', and we give the definition of (weak) convexoid schemes, which share similar properties with ordinary schemes. As a result, we give a purely-algebraic construction of the compactification Spec Z = Spec Z∪ {∞}, which is realized as the Zariski-Riemann space of Spec Z in the category of weak convexoid schemes.
Introduction
In this paper, we give a purely-algebraic construction of the compactification Spec Z = Spec Z ∪ {∞}.
The philosophy of Arakelov tells that the correct compactification of Spec Z should be the space which consists of finite places together with the infinite place ∞. However, the conventional theories could not obtain this space canonically, since Spec Z is the final object in the category of schemes. Therefore, Arakelov geometers and number theorists had to give ad hoc definition for the desired spaces: in Arakelov geometry, we endow an hermitian metric on vector bundles, as a substitute for the information on the infinite place; in number theory, places (finite or infinite) are defined by valuations, and is not defined algebraically.
Here it might be valuable to ask why these concepts and definitions behave so nicely, not how. Also, we might ask why the infinite place cannot be realized within the category of schemes.
The key point is simple. Here, we give a new type of algebra which we call convexoid rings: these have two binary operators ⊞ and ×, and are (commutative) monoids with respect to ×: however, we do not assume the associativity of ⊞. The category of convexoid rings contains that of rings as a full subcategory (and also, multiplicative monoids with absorbing elements), and we can consider 'convexoid schemes' as a generalization of schemes. We can go further, and define 'weak convexoid schemes' so that we can treat Zariski-Riemann spaces properly. As a corollary, we obtain the main theorem:
Theorem 0.1. The compactification Spec Z = Spec Z∪{∞} of Spec Z can be realized as a weak convexoid scheme. It is defined by the universal property, namely the Zariski-Riemann space of Spec Z over Proj R 0 , where R 0 is the initial object in the category of convexoid rings. The stalk O ∞ of Spec Z at the infinity place is the valuation convexoid ring DQ, which is the unit disk in Q consisting of rationals the absolute value of which is not more than 1.
This theorem can also be extended to the ring of integers O K of any algebraic field K. We remark that the set Γ(Spec Z, O) of global sections is {0, ±1}, which some F 1 -geometers denote by F 1 2 . This does not have the ⊞-structure, but only the multiplicative monoid structure as expected. We dare not say that we have obtained the correct definition of F 1 (or F 1 2 ); many people are hoping for too many dreams on F 1 , and we just gave a partial answer for this.
We also remark that this compactification of Spec Z is almost identical to that of Haran's [H], or even that of Durov's [Du]; of course they are not mentioning the convexoid structure, but at least the stalks of the structure sheaves on the infinite places coincide as a multiplicative monoid. However, we emphasize the fact that the construction given in this paper canonically induces the archimedean norm structure from the algebraic structure, and therefore Spec Z is determined by the universal property; while the other two bring the archimedean norm structure outside the algebraic world and therefore their definitions are ad hoc.
This paper is organized as follows: In §1, we illustrate how we come up with convexoids, since the reader may wonder why he or she has to be involved with it instead of sticking to the classical world of rings.
In §2, we give the definition of multi-convexoids and multi-convexoid rings, and see that their behaviour is quite similar to those of rings.
In §3, we prove a variant of the classical Ostrowski's theorem. The crucial difference is that it is formulated in completely algebraic terms, and this theorem lies at the heart of the main result of this paper.
In §4, we give the definition of convexoid schemes. It is already assured that we can define them analogously as the theory of ordinary schemes [T1]. However to reach the goal, we must weaken the condition of what a 'patching' should be, by admitting certain kinds of twists, or in other words, weak homomorphisms. A twist does not affect the semiring of ideals, hence we can safely run the construction of the underlying space of the spectrum. We also give the construction of the 'fake closure' of Spec Z: this is a convexoid scheme with the underlying space homeomorphic to Spec Z, and is close to our answer. Still, it is mal-behaved on the infinite place, hence we will seek for further improvement in the latter sections.
In §5, we give the definition of graded convexoid rings, and the convexoid schemes Proj A for a graded convexoid ring A. These constructions are completely analogous to those of rings, and we claim that the above 'fake closure' can be expressed as Proj R 0 , which turns out to be a very natural object.
In §6, we define the notion of weak convexoid schemes, which is a variant of weak C -schemes introduced in [T1]. This enables us to treat Zariski-Riemann spaces, and as a result, we finally reach the correct definition of Spec Z.
The last section §7 is an appendix, which shows that the analogy of linear systems and projective morphisms in algebraic geometry is also valid for Spec R 0 , and that we have an immersion Proj R 0 → P, where P is a proprojective space over F 1 2 . The concepts and statements introduced in this section are by no means precise: these will be affirmed in the forthcoming papers.
Notation and conventions: Any ring is unital. We denote by (CMnd 0 ) (resp. (CRing)) the category of commutative monoids with absorbing elements (resp. commutative rings) and their homomorphisms. For any subring R of C, we denote by DR the unit disk {x ∈ R | |x| ≤ 1}. This has a structure of a convexoid ring (see Definition 2.1 and Theorem 3.1). When given a commutative (convexoid) ring R, we denote by Ω(R) the distributive lattice of finitely generated ideals of R modulo the congruence a 2 = a. Two ideals a and b is equal in Ω(R) if and only if √ a = √ b. We frequently use the terminologies of category theory, based on the textbook [CWM]. The theory of convexoid schemes shares most of the part with the one already exposited in [T1] and [T2]; we will not repeat the argument here, and many basic facts will be referred to the above mentioned articles.
Preliminary Observations
The development of the theory of schemes over F 1 arose recently, motivated by the Riemann hypothesis.
Weil's conjecture, which is an analogy of the Riemann hypothesis for the positive characteristic case, has been proved by Deligne [Del], by considering the multiple zeta function on the n-fold product X × Fq X × Fq · · · × Fq X of the given projective variety X over F q .
Many people are hoping to imitate this method to prove the original Riemann hypothesis up to now. The essential part is to find a correct 'base field' F 1 , which is called the field with one element, so that we can regard Spec Z as an open curve defined over F 1 , and consider the n-fold object
Z ⊗n = Z ⊗ F 1 Z ⊗ F 1 · · · ⊗ F 1 Z
with a multiple zeta function defined over it. Although Z ⊗n is not defined appropriately yet, the preferred multiple zeta function is constructed [K].
Also, we would like to obtain the compactification of Spec Z over F 1 so that we could formulate Lefschetz-type formula for the complete zeta function [Den]:ζ
(s) = 2 i=0 det ∞ 1 2π (s − Θ)|H i (X, R) (−1) i+1 .
Connes has shown the determinantal representation of the Riemann zeta function [C1], and furthermore gave a geometric representation, by considering a function space on a projective line over F 1 [C2]. These results are suggesting the importance of the theory of schemes over F 1 , apart from the philosophy of Arakelov. However, the definition of F 1 has not reached a full agreement yet. See [PL] for the survey in this topic.
Let us go back to try for the compactification of Spec Z.
Recall that, when we are given a (non-compact) smooth curve X over a base field k, we can construct its universal compactification as a Zariski-Riemann space ZR(X, k) , namely the spaces of valuation rings over k:
Spec k(X) / / X = ZR(X, k) v v ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ Spec k
Therefore, if we wish some analogy to hold, Spec Z should then be the Zariski-Riemann space of Spec Z over F 1 :
Spec Z / / Spec Z = ZR(Spec Z, F 1 ) u u ❦ ❦ ❦ ❦ ❦ ❦ ❦ ❦ ❦ ❦ ❦ ❦ ❦ ❦ ❦ Spec F 1 .
However, since Z is the initial object in the category of rings, we cannot have a 'base field' F 1 in the category of rings; we must widen our perspectives.
Some experts say that F 1 -algebras should be regarded as a monoid, and schemes over F 1 is a geometric object constructed from monoids. One way to look at is that F 1 is the initial element in the category of commutative monoids with absorbing elements: F 1 = {0, 1}. (We can further attach an idempotent additive structure so that F 1 becomes a Boolean algebra, but this is not essential.) However, it is doubtful that we can recover the infinite place, only by considering the multiplicative monoid structure: indeed, we can define and consider Zariski-Riemann spaces for a morphism of commutative monoids. When applying this to F 1 → Z, we obtain a 'proper space' X over Spec F 1 . However, this has infinitely many infinite places. This happens since we ignore the additive structure, and hence also the archimedean norm structure of Z.
This observation shows that we cannot totally abandon the additive structure.
Let us look more closely. On a finite place p of Z, we obtain a local ring (more precisely, a discrete valuation ring) Z (p) , and by completion we obtain the ring Z p of p-adic integers. This is the 'unit disk' {x ∈ Q p | |x| p ≤ 1} of of the p-adic field Q p . If we wish to have an analogy of this for the infinite place, then the objects corresponding to
Z (p) (resp. Z p , Q p ) should be DQ = {x ∈ Q | |x| ∞ ≤ 1}, (resp. DR = {x ∈ R | |x| ∞ ≤ 1}, R).
However, we run into a problem since the unit disks DQ and DR are not rings: they are multiplicative monoids, but are not closed under addition. This is the central motivation of introducing a new algebra in this paper: we want to have an algebraic type V with a multiplicative monoid structure such that,
(1) V -algebras share good properties with those of rings, and
(2) the category of V -algebras includes the unit disks DQ, DR shown above.
The V -algebras are what we call convexoid rings in this paper. Let us review DQ. This multiplicative monoid is not closed under addition; however, we can always think of taking the mean value (a + b)/2 of two elements a, b ∈ DQ. This binary operation (a, b) → (a + b)/2 will be denoted by ⊞. This operation does not satisfy associativity. However, the distribution law holds, and its behaviour resembles to that of rings very much. Moreover, we can think of convexoid ring spectra and schemes, just as in the case of rings. This is because the general scheme theory does not require associativity of the underlying operators [T1], such as ⊞.
Fortunately, the theorem of Ostrowski (Theorem 3.1) tells that we can obtain the preferred valuations of Q only by assuming the condition 1 ⊞ 1 ∈ R × , where R is the valuation convexoid ring corresponding to the valuation. This condition corresponds to the triangular inequality: |a + b| ≤ |a| + |b|. Hence, if we denote by R 0 the initial object of convexoid rings, then the compactification Spec Z can be obtained over R 0 [(1 ⊞ 1) −1 ], except for the finite place p = 2, which is the antipode of the infinite place.
Until now, we don't need multi-convexoid rings. However, some problem arise when considering projective convexoid schemes. Let us review the construction Proj A for a commutative ring A. Proj A is covered by the open sets of the form D + (f ), where f is a homogeneous element of A, and D + (f ) is isomorphic to
Spec A (f ) = {a/f n | deg a = n deg f }.
When we try to apply this theory to graded convexoid rings, we cannot give an appropriate convexoid structure on A (f ) when deg f > 1, but only a multi-convexoid structure. This is why we introduced the notion of multiconvexoids. However, multi-convexoids still behave fairly well, and does not bother when constructing schemes. This is the rough idea of the theory introduced in this paper.
Convexoids
An algebraic type V is commutative (in the sense of [T1]), if for any m-ary operator φ and any n-ary operator ψ, the following holds:
φ(ψ(x 11 , · · · , x 1n ), · · · , ψ(x m1 , · · · , x mn )) = ψ(φ(x 11 , · · · , x m1 ), · · · , φ(x 1n , · · · , x mn )).
Definition 2.1. Let d be a positive integer. A d-convexoid is a quadruple (G, ⊞ d , −, 0) where G is a set, ⊞ d (resp. −, 0) is a 2 d -ary (resp. unary, constant) operator on G such that (a) G is commutative, (b) ⊞ d is symmetric, namely ⊞ d (a σ(1) , · · · , a σ(2 d ) ) = ⊞ d (a 1 , · · · , a 2 d )
for any element σ of the symmetric group S 2 d .
(c) ⊞ d (a 1 , · · · , a 2 d−1 , −a 1 , · · · , −a 2 d−1 ) = 0 for any element a 1 , · · · , a 2 d−1 ∈ G.
We denote by (Cxd d ) the category of d-convexoids and its homomorphisms. When d = 1, we simply say 'convexoids' and drop the superscript. Also, we
write a ⊞ b instead of ⊞ 1 (a, b) = ⊞(a, b).
By the commutativity, the Hom set (Cxd d )(M, N) canonically becomes a convexoid for any convexoid M, N, and the composition becomes bilinear. Also, we can define the tensor product M ⊗ (−) : (Cxd d ) → (Cxd d ) for any convexoid M as the left adjoint of (Cxd d )(M, −). This gives a closed symmetric monoidal structure on (Cxd d ). When we do not specify d, we merely say multi-convexoids instead of d-convexoids.
A d-convexoid ring is a monoid object in (Cxd d ). We denote by (CxdRing d ) the category of d-convexoid rings and their homomorphisms.
Definition 2.2. Let A be a d-convexoid ring. (1) The constant γ A = ⊞ d (1, 0, · · · , 0) is the fundamental constant of A.
Note that this is in the center of A.
(
2) A is normalized, if γ A = 1.
This fundamental constant plays the key role in this algebra. Let (A, ⊞ d ) be a d-convexoid ring, and u ∈ A be any element. Then, we can define another d-convexoid structure⊞ d by setting
⊞ d (a 1 , · · · , a 2 d ) = u · ⊞ d (a 1 , · · · , a 2 d ).
This means that, we have many choices of a ⊞ d -structure on a convexoid ring A and it is crucial to preserve this flexibility when we consider convexoid schemes.
Definition 2.3. (1) A map f : A → B between two d-convexoid rings is a weak homomorphism if (a) f is a homomorphism of multiplicative monoids, (b) γ B f (⊞ d (a 1 , · · · , a 2 d )) = f (γ A ) ⊞ d (f (a 1 ), · · · , f (a 2 d )), and (c) γ B and f (γ A ) generates the same ideal in B. (2) Two d-convexoid structures ⊞ d 1 and ⊞ d 2 on a d-convexoid ring A are equivalent, if the identity map (A, ⊞ d 1 ) → (A, ⊞ d 2 ) is a weak isomor- phism.
It is easy to see that weak homomorphisms are closed under compositions.
Proposition 2.4. Let A be a d-convexoid ring.
(1) A is equivalent to a normalized d-convexoid ring if and only if γ A is invertible.
(2) In particular if d = 1, then A is equivalent to a ring if and only if γ A is invertible.
Proof.
(2) It suffices to show that a normalized convexoid ring is a ring. Associativity of ⊞ is given by
(a ⊞ b) ⊞ c = (a ⊞ b) ⊞ (1 ⊞ 0)c = (a ⊞ b) ⊞ (c ⊞ 0) = (a ⊞ 0) ⊞ (b ⊞ c) = a ⊞ (b ⊞ c).
Also, 0 is the unit with respect to ⊞:
a = (1 ⊞ 0)a = a ⊞ 0.
Corollary 2.5. The left adjoint of the underlying functor U : (Ring) → (CxdRing) is given by R → R/ ≡, where ≡ is the congruence generated by 1 ⊞ 0 = 1.
Corollary 2.6. Let A be a convexoid ring. Then, A[γ −1 A ] is equivalent to a ring.
This implies that, rings are in a sense, 'localizations' of convexoid rings.
Corollary 2.7. The initial object of (CxdRing) can be realized as the smallest subset R 0 of the polynomial ring Z[γ] satisfying (a) 0, 1 ∈ R 0 , and
(b) f, g ∈ R 0 ⇒ f g, γ(f + g), −f ∈ R 0 .
The natural functor F : (CxdRing) → (Ring) gives a surjective convexoid ring homomorphism R 0 → Z between the initial objects defined by γ → 1.
Lemma 2.8. (1) mγ n ∈ R 0 for any non-negative integers m, n such that |m| ≤ 2 n .
(2) In particular, the homomorphism
R 0 → DZ[1/2] (γ → 1/2) is surjec- tive.
This is an easy calculation, and the proof is left to the reader.
Proposition 2.9. Let (Mnd 0 ) be the category of (multiplicative) monoids with absorbing elements. For M ∈ (Mnd 0 ), we can define the trivial dconvexoid structure on M by ⊞ d (a 1 , · · · , a 2 d ) ≡ 0 for any a 1 , · · · , a 2 d ∈ M.
Hence, we have a fully faithful functor (Mnd 0 ) → (CxdRing d ).
This proposition shows that, any monoid can be regarded as a convexoid ring. However, the ⊞-structure has no information in this case.
Next, we will compare d-convexoid rings and e-convexoid rings, where e is a divisor of d. Let (A, ⊞ e ) be an e-convexoid ring. Then we can canonically define a d-convexoid structure by induction on d:
⊞ d (a 1 , · · · , a 2 d ) = ⊞ e (⊞ d−e (a 1 , · · · , a 2 d−e ), · · · , ⊞ d−e (a 2 d −2 d−e +1 , · · · , a 2 d )).
Proposition 2.10. Suppose d = er for some positive integers e, r. Let (A, ⊞ d ) be a d-convexoid ring, and suppose the fundamental constant γ A is a r-th power of an invertible element µ. Then, we can define el-convexoid structure on A for 1 ≤ l ≤ r by an descending induction on l:
⊞ el (a 1 , · · · , a 2 el ) = µ −1 ⊞ e(l+1) (a 1 , · · · , a 2 el , 0, · · · , 0). Moreover, the d-convexoid structure⊞ d induced from ⊞ e coincides with ⊞ d .
Note that
⊞ el (a 1 , · · · , a el ) = µ l−r ⊞ d (a 1 , · · · , a 2 el , 0, · · · , 0).
Proof. It is a straightforward calculation that ⊞ el gives a el-convexoid structure on A. We will see that⊞ d coincides with ⊞ d by induction:
⊞ d (a 1 , · · · , a 2 d ) = µ 1−r ⊞ d (µ −1 ⊞ d (a 1 , · · · , a 2 e(r−1) , 0, · · · , 0), · · · , µ −1 ⊞ d (a 2 d −2 e(r−1) +1 , · · · , a 2 d , 0, · · · , 0), 0, · · · , 0) = µ −r ⊞ d (⊞ d (a 1 , · · · , a 2 d ), 0, · · · , 0) = ⊞ d (a 1 , · · · , a 2 d ).
Definition 2.11. Suppose e is a divisor of a positive integer d. Let A, B be an e-convexoid ring and a d-convexoid ring, respectively. A map f :
A → B is a weak homomorphism if f decomposes into a sequence of maps Af → B ′ j → B such that (1) B ′ is a d-convexoid ring,
(2) j is a weak isomorphism in the sense of Definition 2.3,
(3) the d-convexoid structure of B ′ is induced by an e-convexoid structure ⊞ e B , and
(4)f is a weak homomorphism (in the sense of Definition 2.3) with respect to this e-convexoid structure.
A d-convexoid structure ⊞ d and an e-convexoid structure ⊞ e on A are equiv-
alent, if the identity maps (A, ⊞ e ) → (A, ⊞ m ) and (A, ⊞ d ) → (A, ⊞ m ) are a
weak isomorphisms for some m-convexoid structure ⊞ m , with m a common multiple of d and e.
We can verify that a composition of two weak homomorphism becomes again a weak homomorphism.
With the aid of Corollary 2.6, we obtain:
Corollary 2.12. Let (A, ⊞ d ) be a d-convexoid ring. Then, A[γ −1 A ] is equiv- alent to a ring.
Next, we will investigate the spectrum for commutative convexoid rings. Here, we will restrict our attention to the Zariski topology.
In the sequel, we assume that any convexoid ring is commutative, and its fundamental constant is a non-zero divisor.
Proposition 2.13. Let (R, ⊞ d ) be a commutative d-convexoid ring, and⊞ d another d-convexoid structure on R. (1) Suppose⊞ d is defined by⊞ d = L u • ⊞ d for some u ∈ R, where L u is the left multiplication of u.
Then, there is an immersion Spec R → SpecR.
(2) If ⊞ d and⊞ d are equivalent in the sense of Definition 2.3, then ideals of (R, ⊞ d ) are exactly those of (R,⊞ d ). In particular, Spec(R, ⊞ d ) and
Spec(R,⊞ d ) are canonically homeomorphic.
In particular for a commutative ring R we have immersions
Spec (Ring) R → Spec (Cxd) (R, ⊞) → Spec (CMnd 0 ) R, where a ⊞ b = u(a + b)
is the convexoid structure on R defined by a constant u ∈ R, and Spec (Cxd) (R, ⊞) (resp. Spec (CMnd 0 ) R) is the spectrum obtained by regarding R as a convexoid ring (resp. commutative monoid with an absorbing element).
Proof.
(1) We have a natural map Ω(R) → Ω(R), sending a to the ideal generated by a. This is a surjective lattice homomorphism, hence induces an immersion Spec R → SpecR of the corresponding morphism of coherent spaces.
(2) Let γ,γ be the fundamental constants of (R, ⊞ d ), (R,⊞ d ), respectively.
Since ⊞ and⊞ are equivalent, R[γ −1 ] and R[γ −1 ] are equal, and γ = uγ for some u ∈ R. Suppose a is an ideal of (R, ⊞ d ), and a 1 , · · · , a 2 d ∈ a. Then,
⊞ d (a 1 , · · · , a 2 d ) = γ i a i = uγ i a i = u⊞ d (a 1 , · · · , a 2 d ).
This shows that a is also an ideal of (R,⊞).
Proposition 2.14. Let e be a divisor of a positive integer d = er, (A, ⊞ e ) an e-convexoid ring, and and ⊞ d the d-convexoid structure induced by ⊞ e . Then, the natural map
Ω(A, ⊞ e ) → Ω(A, ⊞ d )
is an isomorphism. Its inverse is given by a → a , where a is the ideal generated by a.
This shows that the underlying topological space of the spectrum of a multi-convexoid ring is invariant under weak isomorphisms (in the sense of Definition 2.11).
Proof. Let a ∈ Ω(A, ⊞ d ) be an finitely generated ideal. It suffices to show that √ a = {a ∈ A | a n ∈ a (∃n)} is an ideal in (A, ⊞ e ), hence equal to a . For any a 1 , · · · , a 2 e ∈ √ a, set
b = ⊞ e (a 1 , · · · , a 2 e ) r = ⊞ d (a p ) p ∈ a,
where p runs through all maps {1, · · · , r} → 2 e and a p = 2 e i=1 a #p −1 (i) i . This shows that b N ∈ a for sufficiently large N.
Ostrowski's Theorem
A commutative convexoid ring is integral, if 0 is a prime ideal. As in the case of rings, we can define the fractional field Q(R) of an integral convexoid ring R. A commutative integral convexoid ring R is a valuation convexoid ring, if for any non-zero element x of the fractional field K = Q(R), either x or x −1 is in R. As in the case of rings, the set I(R) of R-submodules of K becomes totally ordered by the inclusion relations. We have a group homomorphism | · | : K × → I(R) \ 0 (a → aR), and this satisfies the following properties:
(1) ker | · | = R × , and
(2) |a ⊞ b| ≤ max{|a|, |b|}.
The following theorem is a variant of the classical Ostrowski's theorem:
Theorem 3.1. Let R be a non-trivial valuation convexoid ring with Q(R) weak-isomorphic to Q and 1 ⊞ 1 invertible. Then, R is either (1) the local ring Z (p) at the finite place p = 2 with ⊞ = +, or
(2) the unit disk DQ = {x ∈ Q | |x| ∞ ≤ 1}, with a ⊞ b = ±(a + b)/2.
Proof. Let | · | be the valuation corresponding to R. Since R is a subring of Q, the value group I(R) is automatically archimedean. Hence, we may regard I(R) as a multiplicative submonoid of positive real numbers. Also, note that | · | is determined by the value on N.
• Case |N| ≤ 1: We will show that R is equal to Z (p) for some prime p.
It suffices to show that there is a unique prime p such that |p| < 1.
First, we will show the uniqueness: suppose there exist two primes p, q such that |p|, |q| < 1. Then |p| e , |q| e < |1/2| for sufficiently large integer e. Also, there are two integers m, n such that γ −1 (mp e ⊞ nq e ) = mp e + nq e = 1
since p e and q e are coprime. Since 2γ = 1 ⊞ 1 is invertible, we have
|1| = |2||1/2γ||mp e ⊞ nq e | ≤ |2| max{|m||p| e , |n||q| e } ≤ |2| max{|p| e , |q| e } < |2||1/2| = |1|,
a contradiction. Hence such a prime p is unique.
Since R is non-trivial, there exists an integer n with |n| < 1. Hence, there is a prime divisor p of n such that |p| < 1.
• Case |N| ≤ 1: Let ⊞ m : R 2 m → R be the m-convexoid structure induced from ⊞. For three integers a, b, n ∈ N bigger than 1, b n can be expanded in the form
b n = 0≤i<2 m c i a i ,
where m is an integer satisfying |2| m−1 ≤ n log a b < |2| m , and c i = {0, 1, · · · , a − 1} and is zero for i ≥ j for some j ≤ n log a b + 1. Note that |2| > 1 from the assumption |N| ≤ 1. Also, there is an integer l such that a ≤ 2 l . Then b n = 2 m (2γ) −m ⊞ m (c 1 a, c 2 a 2 , · · · , c j−1 a j−1 , 0, · · · 0), and |c i | ≤ |2| l . These yield
|b n | ≤ |2| m max i |c i a i | ≤ |2| m+l max{|a| j , 1} ≤ (n log a b) · |2| l+1 max{|a| n log a b , 1}.
Taking the n-th root of both sides, we obtain |b| ≤ (n log a b · |2| l+1 ) 1/n max{|a| log a b , 1}.
Taking the limit n → ∞, we have |b| ≤ max{|a| log a b , 1}. Now, take b as |b| > 1. Then |a| must also be bigger than 1, hence log |b|/ log b ≤ log |a|/ log a. By symmetry, this inequality is in fact equal. Hence, the valuation | · | is equivalent to the absolute norm, and |2γ| = 1 shows that γ = ±1/2.
Theorem 3.2. The above Ostrowski's theorem is valid for any algebraic field: let K be an algebraic field, and R be a non-trivial valuation convexoid ring with 1 ⊞ 1 invertible. Then, R is either
(1) the local ring O K,p , where the characteristic of the residue field κ(p) is not 2, or
(2) D σ K = {x ∈ K | |σ(x)| ≤ 1}, where σ : K → C is an immersion of fields.
Proof. Here, we will only check the key points which are different from the proof of original Ostrowski's theorem. (cf. [BS], pp. 278-280) Let | · | be the valuation associated to R.
(1) The valuation ν associated to R is non-trivial on Q. Indeed, suppose the valuation is trivial on Q, and let b 1 , · · · , b n be a basis of K over Q.
Then it turns out that ν(x) ≤ 2 n max i ν(b i ) for any x ∈ K, which is a contradiction since a non-trivial valuation is never bounded.
(2) Let R be a valuation convexoid ring of C, which satisfies R ∩ R = DR. Then R = DC. Indeed, suppose ν(z) > 1 for some z ∈ C such that |z| = 1. Then for any n,
ν(z n ) ≤ 2 max{ℜ(z), |ℑ(z)| · ν( √ −1)} ≤ 2 max{1, ν( √ −1)}
which is a contradiction since ν(z n ) → ∞ as n → ∞. For general 0 = z ∈ C, we have ν(z) = ν(|z|)ν(z/|z|) = |z|.
The remaining part of the proof is completely identical to the original one, hence we will omit it.
Convexoid schemes
The main goal of this paper is to obtain the compactification Spec Z = Spec Z ∪ {∞} in the form of Zariski-Riemann space. We can consider (weak) 'convexoid ring schemes' in the sense of [T1]; the definitions are analogous to those of schemes and weak C -schemes.
However, this is not sufficient for our purpose, since we need to admit weak isomorphisms for restriction maps and transition maps.
Definition 4.1. A convexoid scheme is a pair (X, O X ) such that X is a coherent space and O X is a (CMnd 0 )-valued sheaves, such that (1) X is locally isomorphic (as a monoid-valued space) to the spectrum of a multi-convexoid ring: namely, there is a finite open covering X = ∪ i U i and isomorphisms φ i : Spec (Cxd) R i → U i of monoid-valued spaces, where R i is a commutative multi-convexoid ring.
(
2) If φ −1 i (V ) is affine for some open subset V of U i ∩ U j , then φ −1 j (V )
is also affine, and the transition map
σ ji : φ −1 i (V ) → φ −1 j (V )
is a weak isomorphism.
(3) σ kj σ ji = σ ki for any i, j, k.
A morphism f : X → Y of convexoid schemes is a morphism of monoidvalued spaces such that the induced homomorphism
O Y,f (x) → O X,x
is a weak homomorphism of multi-convexoid rings and is local for any x ∈ X, namely the inverse image of the maximal ideal of O X,x coincides with that of O Y,f (x) .
Here, we defined the appropriate notion of convexoid schemes for our purpose. Note that for the set of sections O X (U) for an open set U of a convexoid scheme X may not be a multi-convexoid ring, but only a multiplicative monoid.
Before we construct the compactification Spec Z, we need to define the base convexoid scheme S 0 . Let R 0 be the initial object of (CxdRing), and set
U 1 = Spec R 0 [γ −1 ], U 2 = Spec R 0 [(2γ) −1 ].
These are open subsets of Spec R 0 . Note that R 0 [γ −1 ] has a ring structure,
isomorphic to Z[γ ±1 ].
There is an automorphism on
U 1 ∩ U 2 = Spec (Cxd) R 0 [(2γ 2 ) −1 ] ≃ Spec (Cxd) Z[1/2][γ ±1 ]
induced by the automorphism
φ : Z[1/2][γ ±1 ] → Z[1/2][γ ±1 ] (γ → γ/2).
We have a diagram of open immersions
U 1 ← U 1 ∩ U 2 φ * → U 1 ∩ U 2 → U 2 ,
which gives a 'twist' patching S of U 1 and U 2 , namely S is defined by the pushout diagram
U 1 ∩ U 2 / / φ * U 1 U 2 / / S 0 .
We have a closed immersion f : Spec Z → U 1 defined by
R 0 [γ −1 ] ≃ Z[γ ±1 ] → Z (γ → 1).
If we stick to convexoid schemes, we obtain a 'fake closure' of Spec Z: we have a homomorphism
R 1 = R 0 [(2γ) −1 ] → DZ[1/2] (γ → 1/2)
which is surjective by Lemma 2.8. This induces a morphism g : Spec DZ[1/2] → U 2 , which is a closed immersion since the complement of the image is
Spec R 1 [(γ ⊞ (−1/2)) −1 ] = Spec R 1 [(γ − 1/2) −1 ].
Lemma 4.2. The underlying space of the spectrum Spec DZ[1/2] is settheoretically isomorphic to Spec Z[1/2] ∪ {∞}, where ∞ is the pullback of the maximal ideal {a ∈ Q | |a| < 1} of DQ. As a coherent space, ∞ is the unique closed point.
Proof. The inclusion Spec Z[1/2] ∪ {∞} ⊂ Spec DZ[1/2] is obvious. Let p be a prime ideal of Spec DZ[1/2]. If p does not contain 1/2, then p is a pullback of a prime ideal of Z[1/2]. Suppose p contain 1/2. Then p contains any a ∈ DZ[1/2] such that |a| ≤ 1/2. For any element a ∈ DZ[1/2] with its absolute value less than 1, we have |a n | < 1/2 for sufficiently large n. Since p is prime, a must be in p. It is obvious that this p is the unique maximal ideal, since its complement is the unit group {±1}.
Using the above lemma, we define Y 0 by the pushout:
Spec Z[1/2] / / Spec Z Spec DZ[1/2] / / Y 0 .
We have a commutative diagram
Spec Z[1/2] f / / U 1 ∩ U 2 φ * Spec Z[1/2] g / / U 1 ∩ U 2 ,
which shows that f and g patch up to give a closed immersion Y 0 → S 0 . This can be regarded as the closure of Spec Z in S 0 with its reduced induced subscheme structure.
(Y 0 , O Y 0 ) = F 1 2 = {0,
Graded convexoid rings and Proj
In the previous section, we gave an ad hoc definition of the 'fake closure' Y 0 of Spec Z. At first sight, this seems to be a very artificial object. But some reader may have noticed that it resembles to the construction of projective line P 1 in algebraic geometry. Indeed, Y 0 can be realized as Proj R 0 , under a suitable definition.
Definition 5.1. (1) A convexoid ring A is (N-)graded, if: (a) B = A[γ −1 A ] is a (Z-)graded ring: say B = ⊕ d∈Z B d , where B d is the degree d part. (b) A is in the positive part: A ⊂ ⊕ d≥0 B d .
(c) For any a ∈ A, a d is also in A for any d, where a = d a d is the homogeneous decomposition of a in B.
We will denote A ∩ B d by A d , and the set of homogeneous elements of
A by A h . Note that A d = A ∩ B d in general.
(2) An ideal a of a graded convexoid ring is homogeneous, if a ∈ a implies a d ∈ a for any d, where a = a d is the homogeneous decomposition of a.
(3) For any graded convexoid ring A, set A + = A ∩ (⊕ d>0 B d ).
We will assume the following convention for any graded convexoid ring A:
A + is finitely generated as a homogeneous ideal of A.
(5.1)
For a homogeneous element a, its degree will be denoted by |a|.
Definition 5.2. Let A be a commutative graded convexoid ring.
(1) Proj A is the set of all homogeneous prime ideals of A which does not contain A + . The open basis of Proj A is given by the form
D + (f ) = {p ∈ Proj A | f / ∈ p},
where f is a homogeneous element of A. Then Proj A becomes a coherent space, and has a finite open covering Proj A = ∪ f ∈A h D + (f ), by the assumption (5.1).
(2) For f ∈ A d , a d-convexoid ring A (f ) is defined by
A (f ) = a/f n ∈ A[f −1 ] | a is homogeneous of degree dn .
The d-convexoid structure on A (f ) is given by
⊞ d a 1 f n 1 , · · · , a 2 d f n 2 d = 1 f N +|γ A | ⊞ d A (a 1 f m 1 , · · · , a 2 d f m 2 d ), where N = 2 d i=1 n i and m i = N − n i . (3) We have a homeomorphism D + (f ) → Spec A (f ) given by p → {a/f n ∈ A (f ) | a ∈ p}.
Its inverse is given by q →q, whereq is the homogeneous ideal of A generated by a ∈ A h such that a |f | /f |a| ∈ q.
(4) We can define a monoid-valued sheaf O on Proj A, so that (D + (f ), O| D + (f ) ) is isomorphic to Spec A (f ) as a monoid-valued space. This is well defined, since we have a natural isomorphism ϕ :
A (f ) [f |g| /g |f | ] ≃ A (g) [g |f | /f |g| ]
of multiplicative monoids for any two homogeneous elements f, g ∈ A h . Also, (Proj A, O) becomes a convexoid scheme, since ϕ is a weak isomorphism of multi-convexoid rings.
Remark 5.3. Note that A (f ) does not have a convexoid structure in general when |f | > 1, since the ⊞-operation shifts the degree.
We will apply Proj to the initial object R 0 . Note that R 0 is graded, by setting deg γ = 1.
Lemma 5.4. The degree 1 part (R 0 ) 1 of R 0 consists of ±γ, ±2γ. Also, (R 0 ) + is generated by (R 0 ) 1 .
Again, this is an easy exercise, and the proof is left to the reader. By the above lemma, Proj R 0 is covered by two affines
D + (γ) ≃ Spec A (γ) ≃ Spec Z, D + (2γ) ≃ Spec A (2γ) ≃ Spec DZ[1/2].
It is obvious to see that the patching of D + (γ) and D + (2γ) coincides with that of Y 0 introduced in the previous section, which shows that Y 0 is isomorphic to Proj R 0 .
Remark 5.5. As we have mentioned in Remark 4.3, the global section Γ(Proj R 0 , O) of Proj R 0 is F 1 2 , which is only a monoid. We might want to formulate a morphism π : Proj A → Spec F 1 2 in some sense and say that π is proper, but there lies a technical difficulty: since ∞ is the unique closed point in Spec DZ[1/2], there are two homomorphisms Spec Z (p) → Spec DZ[1/2] for odd prime p, sending the closed point to either ∞ or (p) ∈ Z[1/2]. This shows that we cannot say that Proj A is proper.
Weak convexoid schemes
The appropriate compactification Spec Z is realized only as a more general object, the construction of which is given by an analogue of that of Aschemes [T2].
For a coherent space X, we denote by Ω(X) the distributive lattice of quasi-compact open subsets of X. There is a natural (DLat)-valued sheaf τ X on X defined by U → Ω(U) for each quasi-compact open U.
Definition 6.1. (1) A weak convexoid scheme is a quadruple (X, O X , U, β X )
where X is a coherent space, O X is a sheaf of commutative multiplicative monoids on X, and U = {(U i , ⊞ d i i )} i is a set of pairs of a quasicompact open subset U i of X and a d i -convexoid ring structure ⊞ d i i on O X (U i ), such that (a) U is a covering of X, namely ∪ U ∈U U = X, and (b) U is a lower set, namely if V ⊂ U and U ∈ U, then V ∈ U and Γ(U) → Γ(V ) is a weak homomorphism.
We can define a (DLat)-valued sheaf ΩO X as follows: recall that the correspondence R → Ω(R) (see the end of §0 for the definition) gives a functor Ω : (CxdRing) → (DLat), where (DLat) is the category of distributive lattices. Note that Ω(R) does not change when we replace the ⊞ by another equivalent multi-convexoid structure by Proposition 2.13 and Proposition 2.14.
For an open subset V of X, let U V be the subset of U consisting of all quasi-compact open subsets contained in V . Then ΩO X (V ) is defined as the equalizer of
U ∈U ΩΓ(U, O X ) ⇒ U 1 ,U 2 ∈U ΩΓ(U 1 ∩ U 2 ).
β X is a morphism ΩO X → τ X of (DLat)-valued sheaves on X, which satisfies the following: for any inclusion V ⊂ U of open subsets of X, the restriction map
O X (U) → O X (V ) factors through T −1 O X (U), where T is the multiplicative system of O X (U) defined by T = {f | β X (U)(f ) ≥ V }.
Here, f is identified with the principal ideal (f ) ∈ ΩO X (U) generated by f . We refer to β X as the support morphism of X.
(2) For a weak convexoid scheme X and a point x ∈ X, the stalk O X,x need not have a canonical choice of a ⊞-structure. However, we can define the notion of a finitely generated radical ideal of O X,x : it is independent of the choice of the ⊞-structure. Also, O X,x [γ −1 x ] has a natural structure of a commutative ring, where γ x is the fundamental constant of any O X (U), x ∈ U. (This constant depends on the choice of U, but the localization O X,x [γ −1 x ] is independent. Hence, we will call γ x the fundamental constant of O X,x .) Then O X,x becomes local, in the sense that the complement of the set of units forms the maximal ideal.
(3) A morphism f : X → Y of weak convexoid schemes is a morphism of monoid-valued spaces such that for any x ∈ X,
(a) f x : O Y,f (x) → O X,x induces a ring homomorphism O Y,f (x) [γ −1 f (x) ] → O X,x [γ −1 x ], and (b) f x is local : f −1 x (m x ) = m f (x) where m x (resp. m f (x) ) is the unique maximal ideal of O X,x (resp. O Y,f (x) ).
We will first recall what a Zariski-Riemann space should be. where R is a valuation convexoid ring and K its fraction field, there exist a unique morphism Spec R → X making the whole diagram commutative.
(2) Let X be a (weak) convexoid scheme over a base (weak) convexoid scheme S. The Zariski-Riemann space of X over S is a S-morphism X → ZR(X, S) where ZR(X, S) is a proper (weak) convexoid scheme over S and is universal: namely, any S-morphism f : X → Y with Y → S proper factors uniquely through ZR(X, S):
X f / / Y ZR(X, S)
: :
If the Zariski-Riemann space exists, then it is unique up to isomorphism. However, this may not be constructed within the category of (weak) convexoid schemes.
Let O K be the integer ring of an algebraic field K. Note that the Zariski-Riemann space ZR(Spec K, Spec Z) is isomorphic to Spec O K . Since Spec Z is a closed subscheme of U 1 , we see that ZR(Spec K, U 1 ) coincides with Spec O K .
We will prove the following:
Theorem 6.3. The Zariski-Riemann space
X = ZR(Spec K, S 0 ) = ZR(Spec O K , S 0 ) = ZR(Spec O K , Proj R 0 )
exists as a weak convexoid scheme. Its underlying space is set-theoretically isomorphic to Spec Z ∪ {∞ σ } σ , where ∞ σ is the absolute valuation corresponding to an immersion σ :
K → C of fields. The stalk O X,∞ is isomorphic to D σ K = {x ∈ K | |σ(x)| ∞ ≤ 1}.
This is what we wanted to construct.
Remark 6.4. Let A be the algebraic type of commutative rings. In the category of (profinite) A -schemes, the existence of the Zariski-Riemann space is assured [T2]. However, we do not have a general theory of Zariski-Riemann spaces for convexoid schemes. Therefore, we will content ourselves by constructing the Zariski-Riemann space X explicitly for this specific case.
Proof. First, we will construct X. We only have to construct its restriction X 2 to the fiber on U 2 , since Zariski-Riemann spaces are local with respect to the base. Let |X 2 | be the set of all valuation convexoid rings of K such that 2γ = 1 ⊞ 1 is invertible. We endow a topology on |X 2 | which is generated by the open basis of the form
U(S) = {R ∈ |X 2 | | S ⊂ R},
where S is any finite subset of K.
The theorem of Ostrowski 3.2 tells that any valuation convexoid ring in |X 2 | is either (a) the trivial one, (b) the (non-complete) discrete valuation ring O K,(p) such that the characteristic of the residue field κ(p) is not 2, (c) the disk D σ K associated to an immersion σ : K → C of fields.
The above topology makes |X 2 | into a coherent space: a non-empty open subset U of |X 2 | is a subset whose complement is a finite set not containing the trivial valuation ring. The structure sheaf O X | X 2 is defined by
U → {a ∈ K | a ∈ R (∀R ∈ U)}.
The support morphism β X : ΩO X | X 2 → τ X | X 2 is defined by
(f 1 , · · · , f n ) → {R ∈ U | f i ∈ m R (∀i)} c .
It is straightforward to see that β X is well defined and that X 2 = (|X 2 |, O X | X 2 , β X ) becomes a weak convexoid scheme. Let us denote by ∞ σ the point of X 2 corresponding to D σ K. Then we see that X 2 \ {∞ σ } σ is isomorphic to Spec O K [1/2]. Therefore, we obtain X by the pushout
Spec O K [1/2] / / X 2 Spec O K / / X.
We have a convexoid ring homomorphism
R 0 [(2γ) −1 ] → Γ(X 2 , O X ) = ∩ σ:K→C D σ O K [1/2]
by γ → 1/2. This induces a morphism X 2 → U 2 , and patches up with Spec O K → U 1 to give the morphism ν : X → S 0 . We see that ν is proper, since ν| Spec O K is a closed immersion, and ν| X 2 is obviously proper from the construction.
Finally, we will see that X has the universal property. It suffices to show that X 2 → U 2 satisfies the property. Let f :
Spec O K [1/2] → Y be a U 2 -morphism,(U) → O X 2 (f −1 U), wheñ f −1 U contains some infinite places ∞ σ . Since ∞ σ ∈f −1 U implies that the map O Y (U) → K factors through D σ K,
this weak homomorphism also factors through O X 2 (f −1 U). Therefore, we have constructed the morphism f : X 2 → Y of monoid-valued spaces, and it is straightforward to check that this is indeed a morphism of weak convexoid schemes. The uniqueness off is obvious from the construction.
7 Appendix: Embedding of Proj R 0
As we have seen, the initial object R 0 in the category of convexoid rings has a natural grading structure, and the convexoid scheme Proj R 0 is the 'fake closure' of Spec Z.
Once we have a projective scheme, algebraic geometers would ask what the projective embedding associated to a very ample line bundle might be. We will seek for an analogy of the projective embedding for Proj R 0 . This can be realized, and the result can be summarized as follows:
Theorem 7.1. Let R 0 be the initial object in the category of convexoid rings and d a positive integer.
(1) Each line bundle O(d) of Proj R 0 gives a morphism Proj R 0 → P 2 d −1 F 1 2 of monoid-valued spaces.
(2) In particular, we have an immersion Proj R 0 → P into a proprojective space P over F 1 2 .
Before we proceed, we will review what the projective space P n F 1 (or, P n F 1 2 ; the construction is essentially the same) is. As we have mentioned in the introduction, F 1 -algebras are regarded as a monoid: for example, a polynomial ring F 1 [x 1 , · · · , x n ] over F 1 is a free commutative monoid N n ∪{0} with an absorbing element 0, generated by x 1 , · · · , x n . In this sense, we know that schemes over F 1 (namely, 'monoid schemes') can be constructed, and there is an adjunction Spec : (CMnd 0 ) ⇄ (Sch/F 1 ) op : Γ.
(cf. [T1]). In the sequel, we only consider coherent schemes and quasicompact morphisms. We have a left adjoint of the underlying functor U : For example, a fan (in the sense of toric geometry [O]) ∆ together with an absorbing element gives a scheme Spec ∆ over F 1 , and X = Z × F 1 Spec ∆ is just the toric scheme over Z associated to the fan ∆. The F 1 -scheme Spec ∆ has its underlying space as a subset of X consisting of the generic points of the images of T-invariant sections Spec Z → X, where T is the maximal torus of X. In particular, for each fiber F of X → Spec Z, the points of Spec ∆ correspond to T-invariant points of F . The natural morphism π X : X → Spec ∆ sends each point x of X to the generic point of the closure of the T-orbit of {x}.
(0) Figure 1: Configuration of the points of P 2 F 1 .
(x 0 ) (x 2 ) (x 1 ) (x 0 , x 2 ) (x 0 , x 1 ) (x 1 , x 2 )
In particular, the projective space P n F 1 corresponds to the fan ∆ representing the projective space P n , and its points correspond to prime ideals generated by monomials over homogeneous coordinates; therefore, the configuration of points can be described as a n-simplex; each l-dimensional face of the n-simplex corresponds to a l-dimensional point of P n F 1 (Figure 1). If we replace F 1 by F 1 2 , then the underlying space does not change, but only the structure sheaf becomes the sheaf of F 1 2 -algebras, namely each section admits its minus. Now, we go back to Proj R 0 . We will imitate the construction of projective morphisms in algebraic geometry, with an exception that we forget the additive structures.
The set L d = (R 0 ) d of homogeneous elements of R 0 ⊂ Z[γ] of degree d consists of 0, ±γ d , ±2γ d , ±3γ d · · · , ±2 d γ d .
Unlike the case of rings, this set does not have an additive structure, but only the F 1 2 = (R 0 ) 0 -action; namely, L d is an F 1 2 -module. However, we can still regard it as a linear system, and consider the line bundle O(d) and even a rational map, associated to L d as follows.
The line bundle O(d) is a O Proj R 0 -submodule of the locally constant sheaf Q * , generated by mγ d /nγ d = m/n (1 ≤ m, n ≤ 2 d ).
Indeed, this canonically becomes a line bundle, and L d can be regarded as the set of global sections of O(d). In other words, O(d) is globally generated.
The linear system L d is free as an F 1 2 -module, hence we will fix a basis γ d , 2γ d , · · · , 2 d γ d to construct the morphism associated to L d in the sequel.
Let F 1 2 [x 1 , · · · , x 2 d ] be a polynomial ring (in fact, a monoid) with coefficients in F 1 2 , with the canonical grading. For each 1 ≤ n ≤ 2 d , we have a morphism of monoids F 1 2 [x 1 , · · · , x 2 d ] → (R 0 ) (nγ d ) (x l → lγ d /nγ d = l/n), which extends to F 1 2 [x 1 /x n , · · · , x 2 d /x n ] → (R 0 ) (nγ d ) . These patch up to give a morphism f d : Proj R 0 → P 2 d −1 F 1 2 of monoid-valued spaces. For a finite place p ∈ Spec Z ⊂ Proj R 0 , f d sends p to the point corresponding to the prime (x p , x 2p , · · · , x [2 d /p]p ), where {x i } i are homogeneous coordinates of P 2 d −1 . The infinity place ∞ goes to the point corresponding to (x 1 , · · · , x 2 d −1 ), which is one of the closed points of P 2 d −1 .
Note that f d never becomes an immersion, since a finite place p goes to the generic point of P 2 d −1 when p is larger than 2 d . This is just one translation of the fact that the multiplicative monoid Z \ {0} is not finitely generated.
However, we can consider the infinite product
P = d P 2 d −1 = P 2 1 −1 F 1 2 × F 1 2 P 2 2 −1 F 1 2 × F 1 2 · · ·
and a morphism f : Proj R 0 → P in the category of weak schemes over F 1 (cf. [T1]). Then, f becomes an immersion.
Remark 4. 3 .
3The monoid of sections O Y 0 (U) for an open set U of Y 0 becomes a convexoid ring, if and only if U does not contain both two points 2 and ∞: 2 and ∞ are antipodes to each other. In particular, Γ
Definition 6. 2 .
2(1) A morphism f : X → S of convexoid schemes is proper, if it satisfies the following condition:
(CRing) → (CMnd 0 ), which we denote by Z[·]/0: for M ∈ (CMnd 0 ), Z[M]/0 is the monoid ring Z[M] divided by the ideal generated by the absorbing element 0 M of M. The functor Z[·]/0 patches up to give a functor Z × F 1 (−) : (Sch/F 1 ) → (Sch). The unit morphism M → Z[M]/0 induces a morphism Spec Z[M]/0 → Spec M, and this extends to give a natural morphism π X : Z × F 1 X → X of monoid-valued spaces.
±1} is only a multiplicative monoid. This happens since the local fundamental constant 1/2 and 1 (they are defined on Spec DZ[1/2] and Spec Z, respectively) cannot be extended globally.However, since the infinity place ∞ of Spec DZ[1/2] is the maximal ideal, we cannot have DQ on the stalk O Y 0 ,∞ ; we have DZ[1/2] instead. This is not what we want.
where Y is a weak convexoid scheme, proper over U 2 . For each valuation convexoid ring R ∈ |X 2 |, we have the following commutative diagram Spec K and the properness of Y tells that there is a unique arrow Spec R → Y making the whole diagram commutative. This gives a unique set-theoretic mapf : |X 2 | → |Y |. This becomes continuous, since it is continuous onX 2 \ {∞ σ } σ → Y .It remains to construct the morphism between the structure sheaves. We only have to consider O Y/ /
Y
Spec R
/ /
; ;
U 2 ,
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A -schemes and Zariski-Riemann spaces, preprint arXiv: math.AG/1101.2796, to appear in Rendiconti del Seminario Matematico della. S Takagi, KyotoUniversità di Padova S. Takagi: Department of Mathematics, Faculty of Science, Kyoto UniversityJapan E-mail address: [email protected], S.: A -schemes and Zariski-Riemann spaces, preprint arXiv: math.AG/1101.2796, to appear in Rendiconti del Seminario Matematico della Università di Padova S. Takagi: Department of Mathematics, Faculty of Science, Ky- oto University, Kyoto, 606-8502, Japan E-mail address: [email protected]
|
[] |
[
"Machine-learning Skyrmions",
"Machine-learning Skyrmions",
"Machine-learning Skyrmions",
"Machine-learning Skyrmions"
] |
[
"Vinit Kumar Singh \nDepartment of Physics\nIndian Institute of Technology\n721302KharagpurIndia\n",
"Jung Hoon Han \nDepartment of Physics\nSungkyunkwan University\n16419SuwonKorea\n",
"Vinit Kumar Singh \nDepartment of Physics\nIndian Institute of Technology\n721302KharagpurIndia\n",
"Jung Hoon Han \nDepartment of Physics\nSungkyunkwan University\n16419SuwonKorea\n"
] |
[
"Department of Physics\nIndian Institute of Technology\n721302KharagpurIndia",
"Department of Physics\nSungkyunkwan University\n16419SuwonKorea",
"Department of Physics\nIndian Institute of Technology\n721302KharagpurIndia",
"Department of Physics\nSungkyunkwan University\n16419SuwonKorea"
] |
[] |
Principles of machine learning (ML) are applied to models that support skyrmion phases in two dimensions. Most successful predictions were found when a convolutional neural network (CNN) layer was inserted as well as several layers of neural networks. A new training scheme based on features of the input configuration such as magnetization and spin chirality is introduced to make reliable predictions on the mixed phases, consisting of either a mixture of spiral and skyrmions or of skyrmions and ferromagnets. It proved possible to further train external parameters such as the external magnetic field and temperature and make reliable predictions on them. The predictive capacity of the ML continued to apply to configurations that are not generated by the original Hamiltonian used in the training stage, but a different Hamiltonian adiabatically connected to the original one.The basic strategy behind teaching ML algorithm to recognize various phases of many-body systems 1-14 , whether classical or quantum, is to train it on many examples of many-body configurations together with answers to the phases to which they belong. After the successful implementation of supervised learning as such, the ML algorithm can predict the phase of a new configuration, not drawn from the previous training set. If the input state is drawn from near the phase transition, the prediction is either one or the other side of the transition with certain probability. The gradual change in the probability with temperature or tuning parameter can be used to locate the critical temperature (classical) or the interaction strength (quantum) of the continuous phase transition point. Most studies in recent years have focused on the transition between ordered and disordered phases separated with a second-order critical point. Following the natural progression in the level of sophistication, models studied with the ML method have evolved from Ising 1-11 to planar (XY)7,9,12,13, and most recently to Heisenberg 14 spins.Here we address various aspects of the Heisenberg-Dzyaloshinskii-Moriya-Zeeman (HDMZ) spin Hamiltonian by the ML method:This lattice model, usually solved in two-dimensional L × L square lattice, describes the magnetic interaction at the interface of a magnetic layer with a non-magnetic layer, or a magnetic layer exposed to vacuum. Its phase diagram, by now well-known, includes the skyrmion crystal over some intermediate field range, flanked by spiral phase at low field and ferromagnetic phase at high field 15-19 .Classifying mixed phases:As an initial application of ML ideas to the HDMZ model, we created a training set of configurations drawn from deep inside the spiral, skyrmionic, and ferromagnetic phases of the model (1). Our training set was generated by the Monte Carlo (MC) method with D/J = √ 6, corresponding to the spiral period λ = 6. (J will be set to unity from now on.) The inter-skyrmion distance in the skyrmion phase is also of the same order. The average spin chirality χ, defined in (2), over the (T, B) plane is presented inFig. 1(a). Judging from the lack of significant spin chirality, the B ∈ [0, 1.2] region over the temperature T ∈ [0.03, 0.25] (box 1 in the chirality map,Fig. 1)can be said to belong to the spiral phase. Likewise, one can say with confidence that B ∈ [3.4, 4.2] over T ∈ [0.03, 0.25] belongs to the ferromagnetic phase (box 5 inFig. 1). Here T = 0.03 is the lowest temperature reached in our MC calculation. The robust skyrmion phase can be found at B ∈ [1.8, 2.6] (box 3). For each B, the temperature interval T ∈ [0.03, 2.0] was divided into 40 steps using adaptive scheduling, i.e. exponentially decaying step size with a decay rate of 0.1. That gave us a total of 20 steps from the interval T ∈ [0.03, 0.25], and with each step we drew 100 MC configurations. Finally, 17 different magnetic field values were selected, for a total of 20 × 100 × 17 = 34, 000 training configurations.The ML architecture 21 used for the training involved an initial CNN layer of 6×6 filter size (since the skyrmion diameter and period of spirals are both 6) and a second CNN layer with 3 × 3 filter size both accompanied with Max Pool filters, followed by flatten and two dense neural network layers containing 512, 1024 neurons respectively, which then led to the output layer. Batch normalization and dropout regularization are applied to outputs from each layer. The schematic diagram of the architecture can be found in the Supplementary Material (SM). The final outcome was then compared to one of three values, 0, 1, 2, corresponding to spiral, skyrmion, and ferromagnetic phases, respectively. The training data was initially prepared in terms of spin angles (θ i , φ i ), which gave far poorer results than if the data was prepared in terms of the magnetization n i = (sin θ i cos φ i , sin θ i , sin φ i , cos θ i ). All the ML analysis presented in this paper is thus based arXiv:1806.03749v1 [cond-mat.dis-nn]
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10.1103/physrevb.99.174426
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[
"https://arxiv.org/pdf/1806.03749v2.pdf"
] | 91,180,209 |
1806.03749
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f3aecec0e2013cf9b38fc4380cadd00a7ea4c1fe
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Machine-learning Skyrmions
11 Jun 2018
Vinit Kumar Singh
Department of Physics
Indian Institute of Technology
721302KharagpurIndia
Jung Hoon Han
Department of Physics
Sungkyunkwan University
16419SuwonKorea
Machine-learning Skyrmions
11 Jun 2018(Dated: June 12, 2018)
Principles of machine learning (ML) are applied to models that support skyrmion phases in two dimensions. Most successful predictions were found when a convolutional neural network (CNN) layer was inserted as well as several layers of neural networks. A new training scheme based on features of the input configuration such as magnetization and spin chirality is introduced to make reliable predictions on the mixed phases, consisting of either a mixture of spiral and skyrmions or of skyrmions and ferromagnets. It proved possible to further train external parameters such as the external magnetic field and temperature and make reliable predictions on them. The predictive capacity of the ML continued to apply to configurations that are not generated by the original Hamiltonian used in the training stage, but a different Hamiltonian adiabatically connected to the original one.The basic strategy behind teaching ML algorithm to recognize various phases of many-body systems 1-14 , whether classical or quantum, is to train it on many examples of many-body configurations together with answers to the phases to which they belong. After the successful implementation of supervised learning as such, the ML algorithm can predict the phase of a new configuration, not drawn from the previous training set. If the input state is drawn from near the phase transition, the prediction is either one or the other side of the transition with certain probability. The gradual change in the probability with temperature or tuning parameter can be used to locate the critical temperature (classical) or the interaction strength (quantum) of the continuous phase transition point. Most studies in recent years have focused on the transition between ordered and disordered phases separated with a second-order critical point. Following the natural progression in the level of sophistication, models studied with the ML method have evolved from Ising 1-11 to planar (XY)7,9,12,13, and most recently to Heisenberg 14 spins.Here we address various aspects of the Heisenberg-Dzyaloshinskii-Moriya-Zeeman (HDMZ) spin Hamiltonian by the ML method:This lattice model, usually solved in two-dimensional L × L square lattice, describes the magnetic interaction at the interface of a magnetic layer with a non-magnetic layer, or a magnetic layer exposed to vacuum. Its phase diagram, by now well-known, includes the skyrmion crystal over some intermediate field range, flanked by spiral phase at low field and ferromagnetic phase at high field 15-19 .Classifying mixed phases:As an initial application of ML ideas to the HDMZ model, we created a training set of configurations drawn from deep inside the spiral, skyrmionic, and ferromagnetic phases of the model (1). Our training set was generated by the Monte Carlo (MC) method with D/J = √ 6, corresponding to the spiral period λ = 6. (J will be set to unity from now on.) The inter-skyrmion distance in the skyrmion phase is also of the same order. The average spin chirality χ, defined in (2), over the (T, B) plane is presented inFig. 1(a). Judging from the lack of significant spin chirality, the B ∈ [0, 1.2] region over the temperature T ∈ [0.03, 0.25] (box 1 in the chirality map,Fig. 1)can be said to belong to the spiral phase. Likewise, one can say with confidence that B ∈ [3.4, 4.2] over T ∈ [0.03, 0.25] belongs to the ferromagnetic phase (box 5 inFig. 1). Here T = 0.03 is the lowest temperature reached in our MC calculation. The robust skyrmion phase can be found at B ∈ [1.8, 2.6] (box 3). For each B, the temperature interval T ∈ [0.03, 2.0] was divided into 40 steps using adaptive scheduling, i.e. exponentially decaying step size with a decay rate of 0.1. That gave us a total of 20 steps from the interval T ∈ [0.03, 0.25], and with each step we drew 100 MC configurations. Finally, 17 different magnetic field values were selected, for a total of 20 × 100 × 17 = 34, 000 training configurations.The ML architecture 21 used for the training involved an initial CNN layer of 6×6 filter size (since the skyrmion diameter and period of spirals are both 6) and a second CNN layer with 3 × 3 filter size both accompanied with Max Pool filters, followed by flatten and two dense neural network layers containing 512, 1024 neurons respectively, which then led to the output layer. Batch normalization and dropout regularization are applied to outputs from each layer. The schematic diagram of the architecture can be found in the Supplementary Material (SM). The final outcome was then compared to one of three values, 0, 1, 2, corresponding to spiral, skyrmion, and ferromagnetic phases, respectively. The training data was initially prepared in terms of spin angles (θ i , φ i ), which gave far poorer results than if the data was prepared in terms of the magnetization n i = (sin θ i cos φ i , sin θ i , sin φ i , cos θ i ). All the ML analysis presented in this paper is thus based arXiv:1806.03749v1 [cond-mat.dis-nn]
Principles of machine learning (ML) are applied to models that support skyrmion phases in two dimensions. Most successful predictions were found when a convolutional neural network (CNN) layer was inserted as well as several layers of neural networks. A new training scheme based on features of the input configuration such as magnetization and spin chirality is introduced to make reliable predictions on the mixed phases, consisting of either a mixture of spiral and skyrmions or of skyrmions and ferromagnets. It proved possible to further train external parameters such as the external magnetic field and temperature and make reliable predictions on them. The predictive capacity of the ML continued to apply to configurations that are not generated by the original Hamiltonian used in the training stage, but a different Hamiltonian adiabatically connected to the original one.
The basic strategy behind teaching ML algorithm to recognize various phases of many-body systems 1-14 , whether classical or quantum, is to train it on many examples of many-body configurations together with answers to the phases to which they belong. After the successful implementation of supervised learning as such, the ML algorithm can predict the phase of a new configuration, not drawn from the previous training set. If the input state is drawn from near the phase transition, the prediction is either one or the other side of the transition with certain probability. The gradual change in the probability with temperature or tuning parameter can be used to locate the critical temperature (classical) or the interaction strength (quantum) of the continuous phase transition point. Most studies in recent years have focused on the transition between ordered and disordered phases separated with a second-order critical point. Following the natural progression in the level of sophistication, models studied with the ML method have evolved from Ising 1-11 to planar (XY) 7,9,12,13 , and most recently to Heisenberg 14 spins.
Here we address various aspects of the Heisenberg-Dzyaloshinskii-Moriya-Zeeman (HDMZ) spin Hamiltonian by the ML method:
H HDMZ = −J i∈L 2 n i · (n i+x + n i+ŷ ) +D i (ŷ · n i ×n i+x −x · n i × n i+ŷ ) − B · i n i . (1)
This lattice model, usually solved in two-dimensional L × L square lattice, describes the magnetic interaction at the interface of a magnetic layer with a non-magnetic layer, or a magnetic layer exposed to vacuum. Its phase diagram, by now well-known, includes the skyrmion crystal over some intermediate field range, flanked by spiral phase at low field and ferromagnetic phase at high field [15][16][17][18][19] .
Classifying mixed phases: As an initial application of ML ideas to the HDMZ model, we created a training set of configurations drawn from deep inside the spiral, skyrmionic, and ferromagnetic phases of the model (1). Our training set was generated by the Monte Carlo (MC) method with D/J = √ 6, corresponding to the spiral period λ = 6. (J will be set to unity from now on.) The inter-skyrmion distance in the skyrmion phase is also of the same order. The average spin chirality χ, defined in (2), over the (T, B) plane is presented in Fig. 1(a). Judging from the lack of significant spin chirality, the B ∈ [0, 1.2] region over the temperature T ∈ [0.03, 0.25] (box 1 in the chirality map, Fig. 1) can be said to belong to the spiral phase. Likewise, one can say with confidence that B ∈ [3.4, 4.2] over T ∈ [0.03, 0.25] belongs to the ferromagnetic phase (box 5 in Fig. 1). Here T = 0.03 is the lowest temperature reached in our MC calculation. The robust skyrmion phase can be found at B ∈ [1.8, 2.6] (box 3). For each B, the temperature interval T ∈ [0.03, 2.0] was divided into 40 steps using adaptive scheduling, i.e. exponentially decaying step size with a decay rate of 0.1. That gave us a total of 20 steps from the interval T ∈ [0.03, 0.25], and with each step we drew 100 MC configurations. Finally, 17 different magnetic field values were selected, for a total of 20 × 100 × 17 = 34, 000 training configurations.
The ML architecture 21 used for the training involved an initial CNN layer of 6×6 filter size (since the skyrmion diameter and period of spirals are both 6) and a second CNN layer with 3 × 3 filter size both accompanied with Max Pool filters, followed by flatten and two dense neural network layers containing 512, 1024 neurons respectively, which then led to the output layer. Batch normalization and dropout regularization are applied to outputs from each layer. The schematic diagram of the architecture can be found in the Supplementary Material (SM). The final outcome was then compared to one of three values, 0, 1, 2, corresponding to spiral, skyrmion, and ferromagnetic phases, respectively. The training data was initially prepared in terms of spin angles (θ i , φ i ), which gave far poorer results than if the data was prepared in terms of the magnetization n i = (sin θ i cos φ i , sin θ i , sin φ i , cos θ i ). All the ML analysis presented in this paper is thus based arXiv:1806.03749v1 [cond-mat.dis-nn] 11 Jun 2018 on the magnetization inputs. The architecture consisting purely of the deep neural network layer did not work as well as the one we used, involving the CNN filter layer as well. Further minute changes in the architecture had little impact on the overall quality of final results. Only the z-component was used for training in earlier work 14 . After training, the validation procedure gave nearly 100% correct values for the phase labels. This is not surprising given the fact that validation sets also came from deep inside one of the three phases.
It is well-known both experimentally and from simulations that a substantial mixed phase region exists in two-dimensional skyrmion matter 20 . They are mixed spiral and skyrmion (SpSk) regions at low fields, and mixed skyrmion and ferromagnetic (SkFm) regions at higher fields. Because of their presence, a sharp phase boundary separating one phase from another is difficult to define. The next logical step in our investigation is therefore to generate configurations that have such mixed characters, and ask the ML program to predict their phases. For testing purpose we generate a new batch of MC configurations at T ∈ [0.03, 0.25] and B ∈ [1.2, 1.8] (box 2 in Fig. 1), which admittedly belongs to the SpSk mixed phase, and another batch at T ∈ [0.03, 0.25] and B ∈ [2.6, 3.4] (box 4 in Fig. 1) belonging to the SkFm mixed phase.
At each (T, B), 100 configurations were generated and fed to the machine for prediction. The answers given by the machine are then averaged over and shown as probabilities for spiral, skymion, and ferromagnetic phase, in Fig. 1 (b) and (c). Since the temperature range was quite small and the variations in the configurations were minor, the answers were averaged over all the 20 temperature steps in the interval T ∈ [0.03, 0.25] as well. Despite the fact that each data point in Fig. 1(b) and (c) represents an average over 100 × 20 = 2, 000 configurations, and that extremely fine steps in magnetic field ∆B = 0.01 was used, the final results are far from being smooth. Our results are in stark contrast to earlier attempts on a vast array of models exhibiting second-order phase transition. There, the ML program trained exclusively on the configurations deep inside the ordered and disordered phases could successfully predict the phases near the critical point in a continuously varying manner 2,3,6-8,11-13 . The ML program trained on the three distinct phases of the skyrmion model, on the other hand, fails quite dramatically to make continuously varying predictions for its mixed phases. Rather than declaring it as a failure, we view it as the way the neural network can successfully perceive the first-order phase transition -with a substantial mixed-phase regiondifferently from the sharp second-order transitions. At the same time, it is a telling suggestion that one must seek other means of characterizing mixed phases. This problem is taken up next.
Feature predictions:
The main characteristics of the skyrmion and the ferromagnetic phases are the average spin chirality and the magnetization, respectively, defined as (N =number of lattice sites)
χ = (1/N ) i (n i · n i+x × n i+ŷ ), m = (1/N ) i n z i .(2)
The spiral phase is the one where none of these features takes on significant values. Instead of training the ML algorithm on the labels of configurations such as spiral, skyrmion, or ferromagnet, we train it on their features such as χ and m. Once the ML algorithm has been trained to predict those values correctly, the problem of labeling a given configuration is as good as solved. For instance an input configuration whose (χ) m is predicted to be near the the maximum allowed value can have no other label than skyrmion (ferromagnet). Intermediate values of both χ and m signify the SkFm mixed phase. Finally, a configuration with small but non-negligible χ and m are likely associated with the SpSk mixed phase. The labeling problem is delegated to the human decision, while the machine is left to do its work at predicting quantitative features of the input. We carried out supervised learning of the (χ, m) features on configurations drawn from a wide temperature range 0.1 ≤ T ≤ 2.0 (∆T = 0.1) and magnetic field 0 ≤ B ≤ 4.0 (∆B = 0.2) corresponding to the entire chirality map in Fig. 1. For each (T, B) we collect 200 MC-annealed configurations for training purpose. The temperature and magnetic field intervals were sufficiently fine, as can be seen from the high quality of the presmearing data of χ in Fig. 1. After training with a total of 200 × 20 × 21 = 82, 000 configuration, a fresh set of 40,000 configurations were generated to compare the machine-predicted (χ, m) against the actual values. As shown in Fig. 2, the agreement between predicted and actual values is very good across the entire phase diagram.
Encouraged by the success of feature prediction in terms of χ and m, we next ask if the machine can be trained to recognize the particular temperature and magnetic field from which the input configuration originated. Figure 2 shows predicted values of (T, B) to be very close to the actual values 22 . In contrast to (χ, m) which are the mechanical variables of the configuration, (T, B) are thermodynamic variables. It turns out that ML works well in predicting both kinds of features. Prediction for adiabatically connected phases: Various modifications can be added to the HDMZ Hamiltonian (1) to represent the realistic situation of the material. One that reflects the disorder in the material can be given, for instance, by [16][17][18]
H K = −K i∈random (S z i ) 2 .(3)
The magnetic anisotropy term of strength K is added at the random sites occupying a fraction p of the whole lattice. The model H(K, p) = H HDMZ + H K represents an adiabatically connected family of Hamiltonians as long as K is sufficiently small compared to other energy scales. It is interesting to ask whether the ML algorithm, trained solely on configurations drawn from H(0, 0) = H HDMZ , can have predictive power over those generated from arbitrary H(K, p). It is also a pragmatic question, when it comes to addressing the machine's predictive power over the experimental data, as real materials are never free of inhomogeneities and one does not have the a priori knowledge of the governing Hamiltonian. A large number of configurations at K = 1.0 and p = 0.5 was generated by MC and tested by the ML algorithm, previously trained solely on the pristine Hamiltonian H HDMZ . As shown in Fig. 3, very good fits of all features (χ, m, B, T ) were obtained. Similar plots for several (K, p) values can be found in SM. The error in the prediction can be quantified by measuring ∆X ≡ i=T,B |X pred. −X act. |/400, where 400 refers to the total number of (T, B) steps used in the generation of the test set, and X = χ, m, B, T . Table I shows the mean errors in (χ, m, B, T ) for several (K, p) values. Both ∆χ and ∆m remain less than 0.05 as K grows from 0 to 2 (recall J = 1 and D = √ 6). Note that χ and m have the maximum size of 1. On the other hand, there is a systematic growth in ∆B and ∆T as K becomes larger.
We plot predicted values B pred. against the actual B in Fig. 4(a) for several (K, p)'s. There is an approximate linear relationship in B pred. against B, at least until B pred. reaches saturation, with the slope that grows almost linearly with K, as shown in Fig. 4(b). The effect of the added anisotropy can be qualitatively understood within the mean-field picture by replacing K i∈random (n z i ) 2 with 2Kpm i∈L 2 n z i , where p is the impurity fraction. Assuming the magnetization m de-pending linearly on B, m = αB, the overall effect of the random anisotropy term is to replace the external field B by the effective one, B eff = (1 + 2Kpα)B. The machine, having been trained solely on pristine H HDMZ , knows nothing of the impurity effect a priori and "erroneously" predicts the renormalized B eff for the input, thereby incidentally divulging the discrepancy between the training and testing Hamiltonian.
Such expectations are consistent with our numerical analysis of Fig. 4(b), showing almost linear increase in the slope of B pred. with K. To further prove this picture we obtain α independently from linear fits to predicted m values such as shown in Fig. 3. The two ways of extracting the susceptibility α agree very well. The ∼ 2 times difference in the estimated slopes for p = 0.5 and p = 1 data are consistent with the mean-field picture of B eff , as shown in Fig. 4(b). The under-estimation of the temperature by the machine, as shown in Fig. 3 and SM figure 2, can be also understood, qualitatively, as a result of K having the tendency to stiffen the spins and align them. At the same bare temperature T , configurations generated at finite K tend to have more alignment of spins, which is "erroneously" seen by the machine to be the consequence of lesser effective temperature T eff < T .
Concluding remarks: Ideas of machine learning have been applied to the skyrmion model with quite accurate predictions for both mechanical and thermodynamic features of the input. Especially the mechanical quantities can be predicted reliably for images produced by different Hamiltonians, indicating that the machine has learned the formulas for calculating them. The architecture design we found optimal for training of the skyrmion matter is a deep layer with CNN (SM figure 1).
FIG. 1 .
1(top) Spin chirality [χ in Eq. (2)] in the (T, B) plane obtained by MC calculation on the HDMZ Hamiltonian (1). Color scale represents the normalized value of the chirality. Boxes 1, 3, and 5 (2 and 4) represent regions where training (testing) data were taken for label predictions. Two configurations on the left show a typical SpSk and SkFm mixed state, respectively. The z-component of the local magnetization is used for the plots. (bottom) Probability of phase predictions in the SpSk and SkFm phases. The numbers represent averages over the testing set in the temperature interval T ∈ [0.03, 0.25] at the same B value. The irregularities are not artifacts of the small data size.
FIG. 2 .
2Machine-predicted values of (χ, m, B, T ) in blue curves, compared to their actual values in red. The training and the testing were done on MC configurations generated by the Hamiltonian (1). Different curves are offset for clarity.
FIG. 3 .
3Machine prediction of (χ, m, B, T ) for MC configurations drawn from HHDMZ + HK with K = 1 and p = 0.5. The training itself was done on MC configurations generated by HHDMZ alone.
FIG. 4 .
4(a) Predicted magnetic field B pred at T = 0.1, for several values of K and p = 1. The reference line in black is the actual B. A linear fit (dashed line) to the transient part of the curve gives the slope s. (b)The slope s deduced from the B pred. fit is shown as blue (p = 0.5) and green (p = 1) dots. They follow linear relationship as shown by lines of the same color. Twice the magnetic susceptibility α deduced from linear fits to m pred , shown in yellow (p = 0.5) and magenta (p = 1), also gives the similar slope. Slopes for p = 1 is larger than the slopes for p = 0.5 in roughly obedience of the mean-field relationship s = 2Kpα.
TABLE I :
IAveraged variance between predicted and actual values of (χ, m, B, T ).
ACKNOWLEDGMENTSThis work was supported by Samsung Science and Technology Foundation under Project Number SSTF-BA1701-07.
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Shown in the figures 2 and 3 are the averages of the machine-predicted values, and the averages of the actual values. Averaging is done over the MC configurations. There is a greater degree of fluctuation if the predictions of an individual configuration is compared with the actual value of that configurationAveraging is done over the MC configurations generated at a fixed (T, B). Shown in the figures 2 and 3 are the averages of the machine-predicted values, and the averages of the actual values. There is a greater degree of fluctuation if the predictions of an individual configuration is compared with the actual value of that configuration.
Supplementary Material. Supplementary Material
|
[] |
[
"Band structure of n-and p-doped core-shell nanowires",
"Band structure of n-and p-doped core-shell nanowires"
] |
[
"Andrea Vezzosi \nDipartimento di Scienze Fisiche, Informatiche e Matematiche\nUniversità di Modena e Reggio Emilia\nVia Campi 213/a41125ModenaItaly\n\nCentro S3\nCNR-Istituto Nanoscienze\nVia Campi 213/a41125ModenaItaly\n",
"Andrea Bertoni \nCentro S3\nCNR-Istituto Nanoscienze\nVia Campi 213/a41125ModenaItaly\n",
"Guido Goldoni \nDipartimento di Scienze Fisiche, Informatiche e Matematiche\nUniversità di Modena e Reggio Emilia\nVia Campi 213/a41125ModenaItaly\n\nCentro S3\nCNR-Istituto Nanoscienze\nVia Campi 213/a41125ModenaItaly\n"
] |
[
"Dipartimento di Scienze Fisiche, Informatiche e Matematiche\nUniversità di Modena e Reggio Emilia\nVia Campi 213/a41125ModenaItaly",
"Centro S3\nCNR-Istituto Nanoscienze\nVia Campi 213/a41125ModenaItaly",
"Centro S3\nCNR-Istituto Nanoscienze\nVia Campi 213/a41125ModenaItaly",
"Dipartimento di Scienze Fisiche, Informatiche e Matematiche\nUniversità di Modena e Reggio Emilia\nVia Campi 213/a41125ModenaItaly",
"Centro S3\nCNR-Istituto Nanoscienze\nVia Campi 213/a41125ModenaItaly"
] |
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We investigate the electronic band structure of modulation-doped GaAs/AlGaAs core-shell nanowires for both n-and p-doping. We developed an 8-band Burt-Foreman k · p Hamiltonian approach to describe coupled conduction and valence bands in heterostructured nanowires of arbitrary composition, growth directions, and doping. Coulomb interactions with the electron/hole gas are taken into account within a mean-field self-consistent approach. We map the ensuing multiband envelope function and Poisson equations to optimized, non-uniform real-space grids by the finite element method. Self-consistent charge density, single-particle subbands, density of states and absorption spectra are obtained at different doping regimes. For n-doped samples, the large restructuring of the electron gas for increasing doping results in the formation of quasi-1D electron channels at the core-shell interface. Strong heavy-hole/light-hole coupling of hole states leads to non parabolic dispersions with mass inversion, similarly to planar structures, which persist at large dopings, giving rise to direct LH and indirect HH gaps. In p-doped samples the hole gas forms an almost isotropic, ring-like cloud for a large range of doping. Here, as a result of the increasing localization, HH and LH states uncouple, and mass inversion takes place at a threshold density. A similar evolution is obtained at fixed doping as a function of temperature. We show that signatures of the evolution of the band structure can be singled out in the anisotropy of linearly polarized optical absorption.
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10.1103/physrevb.105.245303
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"https://arxiv.org/pdf/2202.01572v3.pdf"
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2202.01572
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56e1aa4ba65e5e39cf167223f7a7a3ee6d7bdb13
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Band structure of n-and p-doped core-shell nanowires
Andrea Vezzosi
Dipartimento di Scienze Fisiche, Informatiche e Matematiche
Università di Modena e Reggio Emilia
Via Campi 213/a41125ModenaItaly
Centro S3
CNR-Istituto Nanoscienze
Via Campi 213/a41125ModenaItaly
Andrea Bertoni
Centro S3
CNR-Istituto Nanoscienze
Via Campi 213/a41125ModenaItaly
Guido Goldoni
Dipartimento di Scienze Fisiche, Informatiche e Matematiche
Università di Modena e Reggio Emilia
Via Campi 213/a41125ModenaItaly
Centro S3
CNR-Istituto Nanoscienze
Via Campi 213/a41125ModenaItaly
Band structure of n-and p-doped core-shell nanowires
(Dated: May 26, 2022)
We investigate the electronic band structure of modulation-doped GaAs/AlGaAs core-shell nanowires for both n-and p-doping. We developed an 8-band Burt-Foreman k · p Hamiltonian approach to describe coupled conduction and valence bands in heterostructured nanowires of arbitrary composition, growth directions, and doping. Coulomb interactions with the electron/hole gas are taken into account within a mean-field self-consistent approach. We map the ensuing multiband envelope function and Poisson equations to optimized, non-uniform real-space grids by the finite element method. Self-consistent charge density, single-particle subbands, density of states and absorption spectra are obtained at different doping regimes. For n-doped samples, the large restructuring of the electron gas for increasing doping results in the formation of quasi-1D electron channels at the core-shell interface. Strong heavy-hole/light-hole coupling of hole states leads to non parabolic dispersions with mass inversion, similarly to planar structures, which persist at large dopings, giving rise to direct LH and indirect HH gaps. In p-doped samples the hole gas forms an almost isotropic, ring-like cloud for a large range of doping. Here, as a result of the increasing localization, HH and LH states uncouple, and mass inversion takes place at a threshold density. A similar evolution is obtained at fixed doping as a function of temperature. We show that signatures of the evolution of the band structure can be singled out in the anisotropy of linearly polarized optical absorption.
I. INTRODUCTION
Among III-V compound semiconductor nanostructures, radially heterostructured nanowires represent an increasingly investigated, silicon-compatible perspective for applications in transistor-based electronic devices [1] and opto-electronic devices [2,3]. From the point of view of material quality, several issues have already been settled on the route to technological exploitation of nanowires or as a platform for coherent quantum phenomena. These include self-assisted growth [4,5], order and polytypism [6,7], high-quality interfaces [8], and multi-layer growth [9]. One critical issue bridging between material science and device nanofabrication is the control of doping, for example in modulation doped heterostructures [10,11] and radial p-n junctions [12]. This is still a concern in terms of reproducibility between nanowires and homogeneity within each nanowire [13,14].
As in the realm of planar heterostructures, GaAs-based nanomaterials play a special role also for nanowires. Ultra-high-mobility devices in planar GaAs/AlGaAs heterojunctions build on the modulation doping concept [15], whereby dopants are incorporated in a higher gap AlGaAs layer, physically separated from the lower-gap layers, where carriers are confined, suppressing carrierionized impurity scattering. A corresponding, modulation doped radial heterostructure is schematically shown * [email protected] † [email protected] ‡ [email protected] in Fig. 1 [10], which can be seen as a planar heterojunction with wrapped around layers. Carriers are confined in the GaAs core, while dopants are incorporated in an outer AlGaAs layer. Typically, a thin GaAs capping layer is included to prevent Al oxidation. While mobility is still improving in planar systems [16], where background impurities are the limiting factor, high mobility is more difficult to achieve in core-multi-shell nanowires, though [11,17], and experimental and theoretical characterization is needed.
Due to comparable kinetic and Coulomb energies, in doped core-shell nanowires (CSNWs) electronic states [18] and ensuing response functions [19,20] are determined by the self-consistent field of free carriers, which, in turn depends on the concentration and type of doping [18], together with the Fermi level pinning at surface states [21,22]. Hence, different doping regimes may result in distinct charge localization patterns [23]. The ability to predict the band structure in doped CSNWs is therefore a complex task.
Among the methods used, the envelope function approach stands out for its versatility and computational efficiency. Single-band descriptions have been widely used, including non-perturbative electric and magnetic fields [18,[24][25][26]. Multi-band k · p descriptions, which include spin-orbit coupling arising from valence states that are crucial to describe, e.g., optical properties [27,28], have been employed for several classes of materials, taking into account composition modulations, crystallographic details and mesoscopic symmetries [29][30][31][32][33][34][35]. Spin-orbit coupling in the conduction band has been evaluated, also in presence of strong magnetic fields [36][37][38]. However, a full description of the band structure of doped CSNWs in arXiv:2202.01572v3 [cond-mat.mes-hall] 25 May 2022 the different doping regimes including the self-consistent field arising from the free charge is still missing.
In this paper we investigate the electronic band structure of modulation-doped GaAs/AlGaAs CSNWs with nor p-type doping. We employ an 8-band Burt-Foreman k · p Hamiltonian approach, with Coulomb interactions with the electron/hole gas taken into account within a mean-field self-consistent approach. The numerical burden arising from the self-consistent solution of multi-band envelope function and Poisson equations is minimized by the use of the finite element method (FEM) with nonuniform real-space grids, optimized to different doping regimes. Self-consistent charge density, single-particle subbands, density of states and absorption spectra are then obtained. For strong n-doping, quasi-1D channel tend to form at the corners of the core-shell interface. Heavy-hole (HH)/light-hole (LH) couplings lead to nonparabolic dispersions with mass inversion in the valence band, similarly to planar structures, giving rise to direct LH and indirect HH gaps persisting at any doping density. In strongly p-doped samples, on the contrary, the hole gas forms an almost isotropic, ring-like cloud. As a result of the increasing localization, HH and LH states uncouple, and mass inversion takes place at a threshold density. Similar evolutions are obtained at fixed doping as a function of temperature. We suggest that signatures of the evolution of band structure can be traced in the anisotropy of linearly polarized optical absorption.
In Sec. II we outline our theoretical-computational methods, with detailed derivations reported in the Appendix. Emphasis is on the generality of the method, and mapping on optimized FEM grids. Band structures, density of states, projected charge densities and optical anisotropy are discussed in Sec. III, as a function of the doping density, separately for both n-and p-doped samples.
II. THEORETICAL AND COMPUTATIONAL METHODS
A. The k · p description
To obtain the band structure of a modulation-doped CSNW, we have developed an 8-band k · p envelope function approach. Assuming translational invariance along the nanowire growth axis z, and the position vector r = (r ⊥ , z), the n−th eigenstate at the in-wire wavevector k z are written as
Ψ n (r, k z ) = 8 ν=1 e ikzz ψ ν n (r ⊥ , k z )u ν (r) ,(1)
where u ν (r) = |J, J z is a Bloch basis function in the total angular momentum representation (see Eq. A18). We choose the quantization axis of J parallel to z. The coefficients ψ ν n (r ⊥ , k z ) are the ν-th component of the n-th solution of the multi-band envelope-function equation
8 ν=1 Ĥ µν BF (r ⊥ , k z ) − eV el (r ⊥ )δ µν ψ ν n (r ⊥ , k z ) = E n (k z )ψ µ n (r ⊥ , k z ) ,(2)
whereĤ BF is the 8 × 8 Burt-Foreman Hamiltonian operator, with material-dependent parameters and including the band offsets, and V el represents the electrostatic potential generated by free carriers and fully ionized dopants. The operatorĤ BF is obtained from the k · p bulk Hamiltonian by replacing k x and k y with the corresponding differential operators. Material modulations are included by keeping track of the correct non-symmetrized operator ordering, as described in Appendix A. This procedure yields a set of second order coupled partial differential equations which we numerically solve using FEM on an appropriate 2D grid [39][40][41] with Dirichlet boundary conditions. The strongly non-parabolic subbands E n (k z ) and the corresponding envelope functions ψ ν n (r ⊥ , k z ) are finally determined on a uniform grid of
wave vectors k z ∈ [−k M , k M ].
From the solutions of Eq. (2), we evaluate the total charge density
ρ(r ⊥ ) = e [n h (r ⊥ ) − n e (r ⊥ ) + n D (r ⊥ ) − n A (r ⊥ )] (3)
from the fully ionized donor or acceptor profiles, n D (r ⊥ ) or n A (r ⊥ ), respectively, and the free electron and hole charge densities given by, respectively, (5) where the first summation runs over the conduction (valence) subband indices for electrons (holes). Here, f (E, µ, T ) is the Fermi-Dirac distribution function, µ is the chemical potential, T is the temperature and k B is the Boltzmann constant.
n e (r ⊥ ) = n∈c.s. 8 ν=1 k M −k M dk 2π f (E n (k), µ, T ) × × |ψ ν n (r ⊥ , k)| 2 , (4) n h (r ⊥ ) = n∈v.s. 8 ν=1 k M −k M dk 2π (1 − f (E n (k), µ, T )) × × |ψ ν n (r ⊥ , k)| 2 ,
In practice, Eq. (2) needs to be solved only in [0, k M ], since eigenstates at negative wave vectors can be obtained (up to an arbitrary phase factor) applying the time reversal symmetry operator
T = e −iπJy K ,(6)
where J y is the y−component of the total angular momentum and K is the complex conjugate operator. The electrostatic potential V el (r ⊥ ) is the solution of the Poisson equation with the source term given by the total charge density of the system, possibly with a material and position-dependent relative dielectric constant,
∇ (r ⊥ )∇V el (r ⊥ ) = − ρ(r ⊥ ) 0 .(7)
Again, Eq. (7) is solved using FEM on a 2D grid with Dirichlet boundary conditions. The potential at the outer boundary of the CSNW is fixed to zero at the six edges of the outer layer of the structure. Note that the computational protocol allows to include arbitrary voltages at gates surrounding the nanowire [42], although we will not investigate this configuration here. The steps described above are iterated self-consistently until convergence in a seemingly Schrödinger-Poisson cycle, here generalized to a multi-band Hamiltonian. We stop iterations when the relative change of the charge density between two successive iterations falls below 10 −3 at any node of the grid.
To characterize bands states, a k z -dependent spinorial analysis is useful. The contribution of any of the component of the envelope function can be estimated as
C ν n (k z ) = |ψ ν n (r ⊥ , k z )| 2 dr ⊥ ,(8)
with the normalization condition
8 ν=1 C ν n (k z ) = 1 ,
at each subband index n and wave-vector k z . When analysing electronic states, we shall classify states in terms of EL, HH, LH characters (see Appendix A, Eq. (A18)), which are computed as
C EL (k z ) = C 1 n (k z ) + C 2 n (k z ) , C HH (k z ) = C 3 n (k z ) + C 4 n (k z ) , C LH (k z ) = C 5 n (k z ) + C 6 n (k z ) .(9)
We shall also plot the projected probability distributions at k z = 0, defined as
φ EL (r ⊥ ) = ν∈{1,2} C ν n (0) |ψ ν n (r ⊥ , 0)| 2 ξ ν n , φ HH (r ⊥ ) = ν∈{3,4} C ν n (0) |ψ ν n (r ⊥ , 0)| 2 ξ ν n , φ LH (r ⊥ ) = ν∈{5,6} C ν n (0) |ψ ν n (r ⊥ , 0)| 2 ξ ν n ,(10)
where ξ ν n = max r ⊥ |ψ ν n (r ⊥ , 0)| 2 . Additionally, we compute the projected density of states (PDOS) for any given component ν of the wave function,
g ν (E) = 1 N subbands n k C ν n (k)δ(E − E n (k)) ,(11)
where N is the total number of points in k-space considered in the summation. Furthermore, for n/p-doped samples we evaluate the self-consistent linear charge density of electrons/holes as
ρ lin = n e/h (r ⊥ )dr ⊥ .(12)
The calculation of the optical anisotropy proceeds as follows. In the dipole approximation, the interband absorption intensity of photons with energy ω and light polarization vector ε reads:
I ε ( ω) ∝ n∈v.s. m∈c.s. k |M ε n→m,k | 2 × × [f (E n (k)) − f (E m (k))] δ[E m (k) − E n (k) + ω] ,(13)
where M ε n→m,k is the interband optical matrix element, [45] M ε n→m,kz
8 µν=1 u µ |ε · p|u ν × × dr ⊥ ψ µ * m (r ⊥ , k z )ψ ν n (r ⊥ , k z ) .(14)
Note that doping, in addition to determine the envelope functions via the self-consistent field, enters Eq. (13) through Fermi-Dirac distributions, which account for band filling effects when electron/hole subband edges approach the Fermi energy due to doping. For undoped structures the Fermi energy is well within the gap, and this term is almost equal to unity. In heavily doped structures, however, it inhibits interband transitions to the lowest subbands which may be non-negligibly occupied. Finally, we compute the relative optical anisotropy β between linearly polarized light along the wire axis, I εz , and perpendicular to it along the x direction, I εx :
β = I εz − I εx I εz + I εx .(15)
B. Numerical implementation details
The above self-consistent 8-band k · p equations may result in a computationally intensive task, but a number of strategies can be implemented to keep the computational burden low and avoid the use of massively parallel architectures. Most of the strategies mentioned below take advantage of the flexibility of FEM which allows the use of non-uniform grids, which we generate by the Free FEM library [46].
The k · p Hamiltonian is represented on a 2D hexagonal domain, partitioned in a D 6 symmetry-compliant, unstructured mesh of triangular elements. Since at different doping levels the charge density forms substantially dissimilar localization patterns [18], different densityoptimized grids are used at different doping levels. A typical grid used for a high density regime is shown in Fig. 1(b), showing that the grid is denser where the charge density is localized.
We emphasize that the use of centro-symmetric grid is critical to correctly reproduce the expected degeneracies, without the need of extremely dense grids. Breaking the inversion symmetry of the grid would not only artificially split the orbital degeneracy expected in the conduction band (see Sec. III A), but also split the spin degeneracy, particularly in the strongly spin-orbit coupled valence band [47]. The need to maintain the inversion symmetry discourages the use of automatic adaptive grid methods. Hence, we use fixed, although optimized, non-uniform grids.
In CSNWs which are at stage here, the charge density is confined to the GaAs core (although in a non trivial manner) and rapidly goes to zero inside the shell material; therefore, we use larger elements inside the shell with respect to the core and we require the envelope function to vanish somewhere inside the shell, typically at the doping layer. A typical grid used in the calculations is shown in Fig. 1(b). Finally, we found it convenient to use coarser grids during the self-consistent cycle, with optimized, finer grid used only in the last iterations.
For the 8-band k · p model the bound states of interest around the gap correspond to interior eigenvalues of the Hamiltonian matrix. To compute the charge density via Eqs. (4), (5) the sum is restricted to few tens of subbands (usually n max =60 for the electrons and n max =100 for holes), and iterative methods are preferable. We use the Arnoldi method [48], implemented in the ARPACK library [49], together with the shift-and-invert approach, where the original problem is recast to target the largest eigenvalues. This approach provides faster convergence and enables the search for n max eigenvalues around an energy value E search . Thus, since for both n-and pdoping the occupation of the minority carrier is negligible, during the self-consistent cycle one needs to solve only for the conduction or the valence band structure, respectively, by properly choosing E search [50]. The full band structure is then calculated only in the final converged self-consistent potential.
The Poisson equation is solved on a single specific mesh extending over the entire 2D domain of the heterostructure. To go back-and-forth between the grids of the envelope function and Poisson solver, as well as between different grids used during the self-consistent cycle, we make use of 2D linear interpolation. To achieve the convergence of the self-consistent protocol we rely on the modified second Broyden's method [51][52][53][54] when updating the electrostatic potential at the current iteration. The inverse Jacobian is updated using the information from M = 8 previous iterations. We fixed the weight corresponding to the first iteration to w 0 = 0.01, while all the other weights w m , with m = 1, ..., M − 1, are computed as suggested in Ref. [52]. The simple mixing parameter α is fixed to 0.05.
Before the simulation starts, the mesh is processed through a bandwidth reduction procedure leveraging the reverse Cuthill-McKee algorithm [55] implemented within the SciPy library [56]. This is done in order to obtain tightly banded sparse matrices from the FEM discretization.
The above self-consistent numerical protocol and ancillary calculations have been implemented in a python library. A typical run uses a grid of about 7000 triangular elements and 3500 nodes for the k · p problem and 10-20 self-consistent iterations. A run on a single node architecture equipped with 16 2.60 GHz Intel Xeon E5-2670 processor cores takes about 6 hours CPU time.
III. RESULTS
We simulate a typical modulation-doped structure [11] consisting of a GaAs hexagonal core with an edge-toedge distance of 80 nm surrounded by a 50-nm-wide Al 0.3 Ga 0.7 As shell and a GaAs capping layer of thickness 10 nm [see Fig. 1(a)]. The 2D-coordinate system has the x-and y-axes directed along the [112] and [110] crystallographic directions respectively. Buried inside the shell, at a distance of 20 nm from the core-shell interface, a 10-nm-thick layer is doped at a constant density n D of donors or n A of acceptors. All calculations discussed below are performed at T = 20 K, except in Sec. III D. The chemical potential µ is fixed at the mid-gap value of GaAs [22].
A. Band structure of the undoped material As a reference for calculations of the band structure of doped CSNWs to be discussed in the next sections, we first consider an undoped sample and analyze the conduction and valence bands subbands, which are shown in Fig. 2(left), together with the corresponding PDOSs (right). These are best analyzed together with the projected probability distributions of the EL, HH, and LH spinor components [see Eqs. (9), (10)] at k z = 0, which are shown separately in Fig. 3 [57].
We first consider conduction states. Due to the large gap of GaAs, which disentangles conduction and valence bands in the k · p Hamiltonian, conduction subbands [ Fig. 2(a)] show an almost pure EL character with parabolic dispersion and ensuing 1/ √ energy PDOS [ Fig. 2(b)]. In a system with D 6h symmetry, assuming a perfectly isotropic band structure, we expect
FIG. 2.
Conduction (a) and valence (c) subbands of an undoped GaAs/AlGaAs CSNW (see text for parameters). In (c) the hue/color represents the spinorial character in terms of HH and LH, according to Eqs. (9). Conduction (b) and valence (d) PDOS for different spinor components. The zero of the energy in each panel is taken at the bulk band edge of GaAs for conduction and valence band, respectively.
the ground state to be non degenerate, while the second/third and fourth/fifth doublets are degenerate [58]. Here, anisotropic residual interactions with the valence band remove the degeneracies by ∼ 10 −3 meV, a quantity which cannot be distinguished in Fig. 2. Indeed, the single/double degeneracy of the levels is easily recognized in the height of the peaks of the PDOS.
As shown in Fig. 3(left column), the lowest conduction state is 1s-like in the center, while the nearly degenerate doublets are ring-like states with an increasing modulation in the corners of the hexagonal confining potential. The 6-th state is again a non-degenerate state with a 2s character. Higher levels (not shown here) have maxima on the corners of the hexagon and nodes along the facets or vice versa. [58,59] The valence subbands [ Fig. 2(c)] are of course denser in energy than conduction subbands, due to the larger mass of holes. The LH-HH mixing, which is small but finite also at Γ, leads to a strongly non-parabolic dispersion of the subbands with k z . As shown by the color code of the lines, the two highest subbands have a predominant LH character at Γ, which is also shown by the corresponding distribution functions in Fig. 3(center and right columns) [60]. In between several subbands in Fig. 2(c) pointing downward and with a strong LH character at Γ, we recognize a mixed character state (the 3rd subband) and an almost HH subband (the 6-th state) (see also Fig. 3). These two subbands strongly couple at finite wave vec- tors (note from the hue that these two subbands exchange their HH-LH character), causing a strong camel's back dispersion of the third subband and a corresponding peak in the PDOS at ∼ −0.54meV with 50% character of either HH and LH components. All in all, the LH character dominates the PDOS, which agrees with Ref. 33. Note that band crossings of the third subband can be traced to states belonging to different irreducible representations of the C 3v double symmetry group of [111] oriented nanowires with hexagonal cross-section [61][62][63].
The probability distributions of HH and LH states shown in Fig. 3 are either s-like or ring-like (arising from a quadrupolar symmetry of the real/imaginary parts of the envelope functions), similarly to corresponding conduction states, although of course the ordering is different, as HH-and LH-like states interlace. No orbital degeneracies are expected, since the strongly anisotropic bulk valence band structure does not share the hexagonal symmetry of the confinement.
We finally note that all electronic states are doubly spin-degenerate, due to the centro-symmetric symmetry of the system (which is carefully preserved by the FEM grid), which will hold true in all calculations throughout [64].
B. n-doping
We now consider n-doped samples with increasing doping density n D , up to high-doping regimes. As shown in Fig. 4(a), the self-consistent linear charge density [Eq. (12)] increases almost linearly for large doping, while an increasing number of conduction subbands fall below the Fermi energy [ Fig. 4(b)]. The evolution of the (unoccupied) valence band states at Γ is also shown in Fig. 4(c).
The evolution of the localization of the self-consistent charge density and the corresponding electrostatic potential, shown in Fig. 5(left), is not trivial. With increasing doping, the charge density evolves from a small, isotropic charge distribution in the core of the structure to a larger, ring-like charge density distribution, and finally to a charge density which is primarily located in the corners of the core, as can be inferred by comparing the edge-to-edge and corner-to-corner profiles in Fig. 5. This is in agreement with single-band self-consistent calculations [18,65], as expected from the nearly pure EL character of conduction subbands.
Conduction subbands retain a trivial parabolic dispersion regardless of the doping level (and type), which is therefore not shown here. However, it is still interesting to consider the evolution of the localization of the conduction envelope functions shown in terms of the projected density distribution φ EL (r ⊥ ) in Fig. 6 (left columns in each panel), with increasing doping (panels from left to right). For each of the seven lowest levels, the larger the doping, the more localized is φ EL (r ⊥ ) at the core-shell interface. For the largest doping shown here, all subbands feature a clear six-fold symmetry induced by the heterostructure confining potential. Note that the ordering of the levels in terms of symmetry depends on the level of doping, as seen from the "exchange" of the 6-th and 7-th levels with increasing doping.
Although for n-doping the charge density is determined by conduction band states, the valence band structure does have an evolution as well, due to the restructuring of the free charge density and ensuing change in the self-consistent confining electrostatic potential shown in Fig. 5. The valence band structure shown in Fig. 5 (second column) shows a downward shift of the subbands and an increase of the inter-subband gaps, due to the increased localization energy at the core-shell interface. The k z = 0 character (Fig. 5, right column) at low doping is ∼ 10 ÷ 30% LH for most states, except for the ground level which is almost completely LH, and two states which stand out with a strong HH character. Increasing doping increases the gaps, but does not change much the subband dispersions. At the largest doping shown here, the PDOS is dominated by i) a LH peak near the gap, and ii) two overlapping peaks, one arising from a LH band and one from a HH band. Note, however, that the latter HH peak arises from the camel's back subband with a maximum at a finite k z and, therefore, an indirect gap with the conduction band. Figure 6 shows that as doping increases holes tend to be more localized in the center, with a mostly isotropic distribution. This is at difference with conduction states which move towards the GaAs/AlGaAs interface at larger doping densities, and it is due to the opposite sign of the electrostatic energy. Note that, as already noted for EL states, also for HH and LH states the order in terms of symmetry is not preserved as doping is swept. For example the 7-th level changes both character and orbital symmetry as n D moves from 1.75 to 1.80 10 18 cm −3 .
C. p-doping
We next discuss the results for p-doped materials, focusing on the effects of an increasing acceptor density n A on the band structure and the hole charge density localization. Figure 7 shows a linear increase of the free charge density after a threshold density of dopants. Note that the range of densities is similar with respect to the n-doping case, despite the very different parameters and, as we shall see below, charge localization.
Indeed, the free charge n h , shown in Fig. 8(left) at selected values of the acceptor density n A , shows a dip in the center already at weak doping, which is consistent with the larger mass and lower confinement energy of holes with respect to conduction electrons. As the acceptor doping density n A increases, the charge progressively moves toward the interfaces to minimize Coulomb energy, in analogy with the n-doping case, but at difference with the latter case the hole gas remains remarkably isotropic, a seen by comparing the edge-toedge and corner-to-corner profiles which nearly coincide in Fig. 8(left). In other words, the hole charge forms a uniform gas with a cylindrical shape and little resemblance to the host hexagonal confining potential up to these doping densities.
As n A is swept, the conduction levels [ Fig. 7(b)] shift in energy with respect to the Fermi level and finally stabilize, while an increasing number of hole subbands approach the Fermi energy and contribute to the free charge. Note that at large dopings, hole levels separate in a low-energy and a high-energy branch, which correspond to increasingly LH-and HH-like levels, respectively.
At difference with the n-doping case, the hole band structure is strongly affected by p-doping, as exemplified in Fig. 8. This is due to the different localization energies of HHs and LHs in the increasingly localizing self-consistent potential. A prominent effect can be seen by comparing Figs. 8 and 9. The only strongly HH level (the 6-th level in Fig. 9, left panel) moves to lower energy due to the light mass. As a result, HH-LH mixing and related anticrossings are removed, the mass of the camel's back subband changes sign, and all bands point downward with a small mass at the large densities. Note that the PDOS at large doping is dominated by far by LH states. Furthermore, as a consequence of the reduced k · p coupling in the valence band at high doping densities, the hole energy levels at Γ tend to group in 6-fold clusters [see Fig. 7(c)] separated by gaps that increase with increasing n A [58]. Figure 9 shows that all highest valence subbands become strongly localized at the interfaces at high doping. Contrary to conduction electrons, however, which always tend to localize at the six corners, holes alternate subbands localized at the corners and at the facets, which is again in agreement with single-band calculations in Ref. [18]. Since the charge density is a convolution of these levels, the isotropy of the hole cloud noted above is justified.
We also note that, as doping is increased, there is no definite order of LH-and HH-like levels in term of symmetry/localization, due to the increasing hole confinement energy towards the core-shell interface which is different for HH and LH components.
Finally, we note that, similarly to n-doped samples, minority carriers localize in the opposite direction, due to the opposite sign of the self-consistent potential. However, conduction electron are much more rigid and stable due to the light mass, hence showing little evolution with doping density, and in particular no symmetry inversion takes place.
D. Temperature dependence
The electronic states discussed above are the result of the competition between comparable energy scales in the meV range. As temperatures of ∼ 10 K are in the same energy range, we expect that small changes in temperature at this scale may bring about strong restructuring of the electronic system. As we shall see below, in general the effect of a temperature variation on the freecarrier charge density and the valence band structure are qualitatively analogous to the effects of a varying doping density.
In Fig. 10 we consider an n-doped sample with donor density n D = 1.76×10 18 cm −3 at T = 10 K (top row) and T = 30 K (bottom row), which are above and below the temperature used in Sec. III B. Such temperature variations respectively increase or decrease the bulk-band gap values of 1 meV with respect to the values in Tab. I for both the core and the shell materials. As a result, the band-offset are unchanged, while the band structure parameters that are affected by a rescaling procedure are slightly modified. Starting at the lower temperature, the electronic charge density (left column) evolves from an isotropic charge density centered in the core to a ring-like density. This is similar to the effect of increasing doping, as in Fig. 5, as the occupation probability of the levels above the chemical potential increases with temperature, and more charge populates the nanowire. Consistently with Fig. 5, the valence band structure and PDOS are little affected by temperature in this range. However, the subbands are shifted in the opposite direction with respect to Fig. 5.
In Fig. 11 we consider a p-doped sample with acceptor density n A = 1.75 × 10 18 cm −3 at the same two temperature as above. Again, increasing the temperature results in a greater hole charge density and a more pronounced charge depletion in the center due to the Coulomb interaction. Clearly, valence band states are more sensitive to changes in the charge density for p-doping. Indeed, Fig. 11 shows that as temperature is increased, HH-like states move to lower energies, while HH-like subbands change their curvature downward. As a consequence, the PDOS undergoes a substantial restructuring, as all main features are LH-like. Note that in contrast to the case of a doping density variation, the valence band structure is shifted downward when the temperature increases.
E. Optical anisotropy
Optical absorption in quasi-one-dimensional systems is dominated by excitonic and polarization effects induced by Coulomb interactions, not included in Eqs. (13), (14) [66,67]. However, the optical anisotropy between linearly polarized light along and transverse the nanowire axis, should be less sensitive to Coulomb effects [27,28]. On the other hand, while x-polarized light couples to HH states [see Eq. (A18)], z-polarized light does not. Hence, β is a sensitive probe of the orbital composition of valence band states [27].
In Fig. 12 we show the calculated relative optical anisotropy β [Eq. (15)] at selected doping concentrations for n-(left) and p-doped (right) samples, respectively. Doping concentration increases from top to bottom in both panels. To emphasize the anisotropy of the more intense absorption peaks, the line darkness is modulated with the intensity of the absorption spectrum at the given photon energy. For reference, we also show in the inset single-particle absorption spectra in the two polarizations (for the undoped sample) with optical transitions from the n-th valence state to the m-th conduction state labelled mn .
As a reference, we shall first describe the spectral anisotropy of the undoped sample [top panels in Figs. 12(a),(b)]. The first positive structure, labelled a , arises from the fundamental optical transition 11 [see inset of Fig. 12(a)] which involves the almost purely LH state. This is also an intense transition due to the overlapping envelope functions components (see Fig. 3, first row). The positive anisotropy is β 3/5, which is expected from the ratio between the momentum matrix element in the z and x directions,
S, ± 1 2 p z 3 2 , ± 1 2 2 = 4 S, ± 1 2 p x 3 2 , ± 1 2 2 , hence β = 4−1 4+1
. The next two negative dips in the anisotropy structure a involve the m = 1 EL subband, and arise from the HH components of transitions 13 and 16 (see Fig. 3, third and sixth row). As HH components do not couple to EL states for light linearly polarized along z, we indeed expect the anisotropy to be negative for these optical transitions.
A second, positive anisotropy set of structures at higher photon energies, labelled b , involves transitions to the m = 2 conduction subband with predominantly LH initial states, namely 22 and 24 transitions (see also Fig. 3, second and fourth row).
As the optical anisotropy discriminates specific transitions, it is interesting to discuss how the anisotropy spectra evolves with doping concentration. As seen in Fig. 12(a,b) in both n-and p-doped samples the absorption edge experiences a red-shift with increasing doping, due to band gap renormalization. In Fig. 13 we compare the energy difference ∆E between the ground state energy of the conduction and the valence band, respectively, showing that the effective energy gap decreases almost linearly for both kind of samples in the range of a few meV as doping concentration rises.
In n-doped samples, Fig. 12(a), the absorption intensity of the lowest transitions gradually vanishes with doping, which is due to two concomitant effects, i) bandfilling due to electron subbands falling below the Fermi level, which inhibits inter-band absorption to these levels, and ii) optical matrix element reduction, which is due to Coulomb repulsion: the free charge distribution in the occupied band tends to localize towards the core-shell interfaces as doping concentration is increased, while confining states in the center in the other band, lessening the optical matrix element between initial and final states [Eq. (14)]. Both effects contribute to suppress low-energy absorption at high-doping, finally moving the absorption edge to the strongly anisotropic structure c , originated by transitions to the m = 7 EL subband, namely 71 , mainly LH with positive anisotropy, and 73 , mainly HH, hence with negative anisotropy.
For p-doped samples, see Fig. 12(b), the band-filling effect is less pronounced within the examined range of doping. In fact, even at the highest acceptor density shown in Fig. 12(b), the highest valence subband does not cross the Fermi level (see Fig. 7). Here, the suppression of the absorption intensity with positive anisotropy at a is mainly due to reduction of the initial and the final states' overlap, due to an increasing localization towards the core-shell interfaces of the hole ground state envelope function (see Fig. 9, first row). The first negative dip gradually disappears because the third valence subband loses its HH character with increasing doping (see Fig. 9, third row and Fig. 8, second column). The opposite occurs for the second negative anisotropy peak, which persists at high doping, due to the increasing HH character of the sixth hole subband with doping, as already pointed out in Sec. III C, which in turn increases the optical matrix element for x-polarized light.
IV. CONCLUSIONS
We have thoroughly investigated the band structure of doped GaAs-based CSNWs, with an emphasis on the evolution of spin-orbit coupled valence band states with doping, either of n-or p-type. This is an important piece of information for the characterization of such materials, where doping is still an issue.
Our calculations, performed with a state-of-the-art Burt-Foreman 8-band k · p description, treat many-body effects at the mean-field level, and extend previous investigations to realistic descriptions of doped materials. The use of a flexible FEM approach, which allows to use non-uniform grids, proved to be numerically efficient at different doping levels. This is clearly an advantage in view of multi-parameter optimization, e.g., by stochastic methods [35,68,69].
In particular, we have investigated a proto-typical CSNW with remote doping. As in corresponding planar heterojunctions, the conduction subbands feature a parabolic in-wire dispersion, while hole subbands have a complex dispersion, with inverted masses, which has been rationalized in terms of HH-LH mixing. In large core nanowires, with small confinement energies, increasing doping density moves the majority carriers to the core-shell interface in order to reduce the Coulomb energy. Correspondingly, the states of the minority carrier band are confined to the core by the self-consistent electrostatic field, and in general the overlap of conduction and valence states decreases. While this is qualitatively true for both types of dopings, our calculations allow to identify several differences between the two type of samples which may have an impact, in particular, on optical absorption. In particular, for p-doping the valence band structure is strongly reshaped by confinement of holes at the core-shell interface, and all low energy excitations have a strong LH character.
It may be expected that band structure affects optical absorption and, in particular, optical anisotropy for light polarized along or normal to the nanowire axis. Hence, we have evaluated the doping-density dependent optical anisotropy, which is able to distinguish the spin-orbital character of the transition. In addition to the expected band-filling effects, specific signatures can be identified in the anisotropy patterns which distinguish between nand p-doping.
H λ = H 0 + V + H so .(A2)
The k · p Hamiltonian H 0 , neglecting bulk inversion asymmetry terms, is given by [70]
H 0 = k T A c k 0 iP k T 0 T 0 k T A c k 0 T iP k T −ikP 0 H v 0 3×3 0 −ikP 0 3×3 H v ,(A3)
where k = (k x , k y , k z ) T , P is the optical matrix parameter, related the the Kane energy parameter by
P = 2m 0 E p (A4)
and A c is the renormalized conduction band effective mass parameter, For many relevant semiconductors, including the present case, the standard parameters lead to a negative value for A c . This fact induces spurious solutions [71] that bend within the band gap for large wave vectors. To avoid these unphysical results we set A c = 1 and rescale the E p parameter in order to still get the correct conduction band dispersion: [72,73]
A c = 2 2m e − 2 3 P 2 E g − 1 3 P 2 E g + ∆ so .(E rsc p = E g (E g + ∆ so ) E g + 2 3 ∆ so 1 m e − 2 .(A6)
Foreman rigorously showed that this approach is not an approximation and is equivalent to a change of the Bloch basis [74]. We checked that for the regimes investigated and with the relatively coarse grids permitted by the use of FEM, no highly oscillatory, discretization-related spurious solutions appears in our calculations [71].
In the above expression the matrix H v reads
H v = 2 k 2 2m 0 I 3×3 + k x Lk x + k y M k y + k z M k z k x N + k y + k y N − k x k x N + k z + k z N − k x k y N + k x + k x N − k y k x M k x + k y Lk y + k z M k z k y N + k z + k z N − k y k z N + k x + k x N − k z k z N + k y + k y N − k z k x M k x + k y M k y + k z Lk z . (A7)
where L, M , N + and N − are the Dresselhaus-Kip-Kittel parameters which read
L = 2 2m 0 (−γ 1 − 4γ 2 − 1) , M = 2 2m 0 (2γ 2 −γ 1 − 1) , N + = 2 2m 0 (−6γ 3 − (2γ 2 −γ 1 − 1)) , N − = 2 2m 0 (2γ 2 −γ 1 − 1) .(A8)
Here, the modified Luttinger parametersγ i arẽ
γ 1 = γ 1 − E rsc p 3E g , γ 2 = γ 2 − E rsc p 6E g , γ 3 = γ 3 − E rsc p 6E g ,(A9)
where E g is the bulk band gap and E rsc p the rescaled Kane energy.
In Eq. (A2) the last two terms represent respectively the in-plane potential profile due to different band edges of adjacent layer materials
V = diag[E c , E c ,Ē v ,Ē v ,Ē v ,Ē v ,Ē v ,Ē v ] ,(A10)
withĒ v = E v − ∆so 3 , and the spin-orbit interaction Hamiltonian:
H so = ∆ so 3 .(A11)
The Hamiltonian H λ can be rewritten in the form
H λ = α,β=x,y,z k α D αβ k β + α=x,y,z F α L k α + k α F α R + G ,(A12)
where D α,β , F α L(R) and G are 8 × 8 matrices that can be directly obtained from Eq. (A2) by properly collecting terms involving the same powers of the wave vector's components. In particular, D α,β = D β,α and F α L = F α R . It should be also noted that these matrices are not Hermitian. Nevertheless, the sums D α,β +D β,α and F α L +F α R are indeed Hermitian. To treat nanowires oriented along an arbitrary direction we define k in Eq. (A12) in the rotated coordinate system. We have r = R r and k = R k, where R is the orthogonal rotation matrix
R(θ, φ) = cos θ cos φ sin φ cos θ − sin θ − sin φ cos φ 0 cos φ sin θ sin φ sin θ cos θ . (A13)
The matrices in Eq. (A12), when expressed in terms of the rotated wave vector, k = R k, transform according to
D αβ (θ, φ) = α β R αα D α β R −1 β β , (A14) F α L(R) (θ, φ) = α F α L(R) R −1 α α ,(A15)
where D αβ and F α L(R) are the matrices defined in the original coordinate system with principal axis directed along the [001] direction. For convenience, from now on we will omit the (θ, φ) notation, implicitly assuming that each of the matrices D αβ and F α L(R) is defined in the rotated coordinate system.
The transformation that connects the cartesian basis rotates the spin. We now chose the following symmetry adapted basis that diagonalizes spin-orbit interaction: [75] {χ} = 1 2 , 1 2 EL = |S ↑ ,
1 2 , − 1 2 EL = i |S ↓ , 3 2 , 3 2 HH = 1/2 |(X + iY ) ↑ , 3 2 , − 3 2 HH = i 1/2 |(X − iY ) ↓ , 3 2 , 1 2 LH = i 1/6 |(X + iY ) ↓ − i 2/3 |Z ↑ , 3 2 , − 1 2 LH = 1/6 |(X − iY ) ↑ + 2/3 |Z ↓ , 1 2 , 1 2 SO = 1/3 (|(X + iY ) ↓ + |Z ↑ ) , 1 2 , − 1 2 SO = −i 1/3 (|(X − iY ) ↑ − |Z ↓ ) .(A18)
Note that here the total angular momentum is defined with respect to the principal axes in the rotated coordinate system. It follows that the transformation matrix to go from {γ} to {χ} is
Q = 1 0 0 0 0 0 0 0 0 i 0 0 0 0 0 0 0 0 i 1 √ 2 1 √ 2 0 0 0 0 0 0 0 0 0 1 √ 2 i 1 √ 2 0 0 0 0 0 −i 2 3 i 1 √ 6 − 1 √ 6 0 0 0 1 √ 6 −i 1 √ 6 0 0 0 2 3 0 0 0 0 1 √ 3 1 √ 3 i 1 √ 3 0 0 0 −i 1 √ 3 − 1 √ 3 0 0 0 i 1 √ 3 .
(A19) Defining P = QAU to be the transformation matrix from {λ} to {χ}, the matrices in H λ expressed in terms of the rotated wave vector k transform according to [33] D αβ → P * D αβ P T ,
F α L(R) → P * F α L(R) P T , G → P * GP T .(A20)
To obtain the operatorĤ BF appearing in the envelope function equations we now perform the replacements (k x → −i ∂ ∂x , k y → −i ∂ ∂y ) in Eq. (A12) paying attention to preserve the correct operator ordering. Since k z is now just a parameter, the Hamiltonian operator after the replacement has the following form
H BF = α,β=x,y ∂ αD αβ ∂ β + α=x,yF α L ∂ α + ∂ αF α R +Ḡ , (A21) whereD αβ = −D αβ , F α L = −i(F α L + k z D zα ) , F α R = −i(F α R + k z D αz ) , G = G + k 2 z D zz + k z (F z R + F z L ) .
(A22)
Appendix B: FEM implementation
Equations (2) and (7) are solved within the FEM framework [39]. Here, one writes the proper action integral A that generates the above set of coupled differential equations through a variational procedure. For the multi-band k · p equations we have [35,70]
A = µν dr ⊥ ψ * µ αβ=x,y − ∂ αD αβ µν ∂ β + α=x,y F α L,µν ∂ α − ∂ αF α R,µν +Ḡ µν − E δ µν ψ ν ,(B1)
where µ and ν indicate the components of the envelope function. Here, the correct operator ordering is retained if we take the differential operator to act on the left (right) whenk x,y multiplies F x,y R (F x,y L ). It is easy to check that Eq. (B1) is equivalent to the original eigenvalue problem Eq. (2) by performing a functional variation of A with respect to ψ * µ and invoking the principle of least action. Here, surface terms arising from the integration by parts can be eliminated using the continuity of the envelope function and of the probability current across the interfaces [76,77]. If the wave function is set to zero on the domain boundaries, the boundary surface term vanishes too.
The action integral A is discretized into n el triangular finite elements of the 2D domain,
A = n el i el A (i el ) .(B2)
Within the i el element, each component of the unknown envelope function is approximated using Lagrange linear interpolation polynomials [40] N j (r ⊥ ) so that
ψ µ (r ⊥ ) = 3 j=1 ψ µj N j (r ⊥ ) ,(B3)
where the expansion coefficient ψ µj represent the value of the µ-th component of the envelope function at the j-th triangle's vertex, also called nodal point. Using Eqs. (B1), (B3), we obtain
A (i el ) = µν 3 ij=1 ψ * µi dr ⊥ N i (r ⊥ )L µν N j (r ⊥ ) ψ νj = µν 3 ij=1 ψ * µi M (i el ) µνij ψ νj ,(B4)
where L µν represents the operators appearing in the integrand of Eq. (B1), namely the Lagrangian density.
The total action is given by the sum of each element's contribution. This can be written in a very natural manner in matrix form by imposing inter-element continuity through carefully overlaying the element matrices M (i el ) [39]. To understand how to construct a global matrix starting from element matrices it is convenient to make a simple example. Let's consider a single component of the envelope function ψ µ = ψ and two adjacent triangular elements (i el = 1, 2) having two nodes (and one edge) in common. The action integral for these two elements reads A = Since the two elements share two nodes and we require inter-element continuity, we set ψ
2 = ψ (2) 2 ,(1)
where it is implicitly assumed that the first and the second node of both elements respectively overlap. Using the above conditions, it is possible to rewrite Eq. (B5) in the global form
A = 4 IJ=1 ψ * I M IJ ψ J ,(B6)
where I and J now stand for global node indices and M is obtained from M (1) and M (2) by summing the contributions from the same nodes and collecting the envelope functions on common vertices, e.g., for I, J = 1, 2 we have M IJ = M
(1)
IJ + M(2)
IJ . From this example it is now easy to see that the action integral in its global form can be written as
A = µν n glob IJ ψ * µI M µνIJ ψ νJ ,(B7)
where n glob is the total number of nodes on the discretization domain. We now invoke the principle of stationary action and obtain the equation of motion in algebraic form. We vary the action integral with respect to ψ * µI to obtain simultaneous equation for the coefficients ψ νJ , Here, H µνIJ represents the discretized form of the Burt-Foreman operatorĤ µν BF in Eq. (2), while S µνIJ is an overlap matrix which is present due to the non-orthogonality of the basis functions N j . From Eq. (B9) it is clear that the dimension of the problem is given by the number of nodes n glob in the simulation domain, times the number of components of the envelope function.
For the Poisson equation the energy functional to be minimized is given by
A = dr ⊥ 1 2 (r ⊥ )∇V el (r ⊥ ) · ∇V el (r ⊥ ) − 1 0 V el (r ⊥ )ρ(r ⊥ ) .(B10)
A functional variation of A with respect to V el followed by an integration by parts gives the Poisson equation Eq. (7). Expressing the electrostatic potential again in terms of Lagrange linear interpolation polynomials inside each triangular element,
V el (r ⊥ ) = 3 j=1 V j el N j (r ⊥ ) ,(B11)
and following the same procedure described for the k · p problem, a linear system of n glob equations is obtained:
n glob J C IJ V J el = b I .(B12)
After the inclusion of proper boundary conditions, Eqs. (B9) and (B12) are finally solved with standard library routines.
FIG. 1 .
1(a) Sketch of the section of the simulated CSNWs.
FIG. 3 .
3Projected probability distributions [Eqs.(10)] of the six lowest conduction band (1st column) and six highest valence band states (2nd and 3rd column) at Γ for the undoped material ofFig. 2.
FIG. 4 .
4(a) Linear free charge density [Eq.(12)], (b) conduction subband energies at kz = 0, and (c) valence subband energies at kz = 0 as a function of the doping density nD. Energies are referred to the Fermi level.
FIG. 5 .
5Left column: free charge density distribution ne (blue) and self-consistent conduction-band profile CB (red) shown along the edge-to-edge (dashed line) and corner-to-corner (full line) directions of the CSNW section for T = 20 K at selected values of nD, as indicated. Doping increases from top to bottom. Middle left and middle right columns: valence subbands and PDOS, respectively, corresponding to the doping density and self-consistent potential of the left panels. The hue/color represents the spinorial character in terms of HH and LH, according to Eqs.(9). Right column: LH-character of each subband at Γ.
FIG. 6 .
6Projected probability distributions [Eqs. (10)] for the seven lowest conduction and seven highest valence subbands at the same selected doping densities of Fig. 5, as indicated. Each column corresponds to the EL, LH, HH component, as indicated.
FIG. 7 .
7(a) Linear free charge density [Eq. (12)], (b) conduction subband energies at kz = 0, and (c) valence subband energies at kz = 0 as a function of the doping density nA. Energies are referred to the Fermi level.
FIG. 8 .
8Left column: free charge density distribution n h (blue) and self-consistent valence-band profile VB (red) shown along the edge-to-edge (dashed line) and corner-to-corner (full line) directions of the CSNW section for T = 20 K at selected values of nA, as indicated. Doping increases from top to bottom. Middle left and middle right columns: valence subbands and PDOS, respectively, corresponding to the doping density and self-consistent potential of the left panels. The hue/color represents the spinorial character in terms of HH and LH, according to Eqs.(9). Right column: LH-character of each subband at Γ.
FIG. 9 .
9Projected probability distributions [Eqs. (10)] for the seven lowest conduction and seven highest valence subbands at the same selected doping densities of Fig. 8, as indicated. Each column corresponds to the EL, LH, HH component, as indicated. FIG. 10. Left column: free charge density distribution ne (blue) and self-consistent conduction-band profile CB (red) shown along the edge-to-edge (dashed line) and corner-to-corner (full line) directions of the CSNW section for T = 10 K (top row) and T = 30 K (bottom row) at the single doping density nD = 1.76 × 10 18 cm −3 . Middle left and middle right columns: valence subbands and PDOS, respectively, corresponding to the doping density, temperature and self-consistent potential of the left panels. The hue/color represents the spinorial character in terms of HH and LH, according to Eqs. (9). Right column: LH-character of each subband at Γ. FIG. 11. Left column: free charge density distribution n h (blue) and self-consistent valence-band profile VB (red) shown along the edge-to-edge (dashed line) and corner-to-corner (full line) directions of the CSNW section for T = 10 K (top row) and T = 30 K (bottom row) at the single doping density nA = 1.75 × 10 18 cm −3 . Middle left and middle right columns:valence subbands and PDOS, respectively, corresponding to the doping density, temperature and self-consistent potential of the left panels. The hue/color represents the spinorial character in terms of HH and LH, according to Eqs.(9). Right column: LH-character of each subband at Γ.
ACKNOWLEDGMENTS
Discussions with Pawe l Wójcik are gratefully acknowledged. AB acknowledges partial financial support from the EU project IQubits (Call No. H2020-FETOPEN-2018-2019-2020-01, Project ID No. 829005). The authors acknowledge CINECA for HPC computing resources and support under the ISCRA initiative (No. IsC87 ES-QUDO-HP10CXQWD5).
FIG. 12 .
12(a) Optical anisotropy β [Eq. (15)] for n-doped samples at different donor concentrations. From top to bottom: undoped, 1.75, 1.77, 1.79, 1.82, 1.87, 1.90 ×10 18 cm −3 . Horizontal dashed lines indicate the zero reference, each panel extends vertically from -1 to +1. The gray hue represents the intensity of the corresponding absorption spectrum [Eq. (13)] at the given photon energy. Eg=1.518 eV is the band gap of GaAs at T = 20 K. Inset: calculated absorption spectra of the undoped structure for linearly polarized light. Peaks are labelled with mn , where m is the index of the final conduction subband and n the index of the initial valence subband involved in the optical transition. (b) Same as panel (a) but for p-doped samples. From top to bottom: undoped, 1.73, 1.75, 1.77, 1.79, 1.80, 1.82 ×10 18 cm −3 . Appendix A: k · p Hamiltonian In this appendix we show how to obtain the operator H BF in Eq. (2). In the cartesian basis λ = {|S ↑ , |S ↓ , |X ↑ , |Y ↑ , |Z ↑ , |X ↓ , |Y ↓ , |Z ↓ } (A1) the Burt-Foreman Hamiltonian with the principal axis along the [001] direction can be written as
A5) in FIG. 13. Effective energy gap ∆E as a function of doping concentration for n-and p-doped samples.
{λ} to a new one with the principal axis directed along the (θ, φ) direction is given by {γ} = AU {λ}, {γ} = {|S ↑ , |S ↓ , |X ↑ , |Y ↑ , |Z ↑ , |X ↓ , |Y ↓ , |Z ↓ } , (A16) where U = diag[1, 1, R, R] is a standard rotation operator and A = diag[Ā,Ā ⊗ I 3×3 ], wherē A = e −iφ/2 cos θ/2 e iφ/2 sin θ/2 −e −iφ/2 sin θ/2 e iφ/2 cos θ/2 . (A17)
M
µνIJ ψ νJ = 0 . (B8)Given the particular form of the integrand in Eq. (B1), the above expression results in a generalized eigenvalue problem: µνIJ ψ νJ − E δ µν S µνIJ ] ψ νJ = 0 (B9)
in grey scale arbitrary units). The grid stops at the doping layer, where the charge density is assumed to vanish. The grid used for the Poisson equation is different and extends to the outer boundary of the structure.The principal axes of the 2D-coordinate system are directed
along the [112] and [110] crystallographic directions. (b) An
example of an optimized FEM grid used to solve the enve-
lope function equation (2) with superimposed self-consistent
charge density (
TABLE I .
IMaterial parameters used in the simulations at T = 20 K. Eg is the energy gap, ∆Ec, ∆Ev are the conduction and valence band offset values at the GaAs/Al0.3Ga0.7As interface, ∆so is the split-off energy, Ep is the bare Kane energy, E rsc p the rescaled Kane energy (Eq. A6), me is the effective conduction electron mass, γi are the bare Luttinger parameters,γi are the rescaled values (Eq. A9), r is the relative dielectric constant and a lc is the lattice constant. The band structure parameters are taken from Ref.[43] except for the band offset values. The latter have been determined assuming an offset ratio of ∆Ec : ∆Ev = 63 : 37, as recommended in Ref.[44].GaAs
Al0.3/Ga0.7/As
Eg [eV]
1.518
1.936
∆Ec [eV]
0.263
∆Ev [eV]
0.155
∆so [eV]
0.341
0.323
Ep / E rsc
p
[eV]
28.8 / 20.9
26.5 / 18.0
me
0.067
0.092
γ1 /γ1
6.98 / 2.39
6.01 / 2.91
γ2 /γ2
2.06 / -0.235
1.69 / 0.138
γ3 /γ3
2.93 / 0.635
2.48 / 0.928
r
13.18
12.24
a lc [nm]
0.56
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[
"An infinite-dimensional representation of the Ray-Knight theorems",
"An infinite-dimensional representation of the Ray-Knight theorems"
] |
[
"Elie Aïdékon ",
"Yueyun Hu ",
"Zhan Shi "
] |
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The classical Ray-Knight theorems for Brownian motion determine the law of its local time process either at the first hitting time of a given value a by the local time at the origin, or at the first hitting time of a given position b by Brownian motion. We extend these results by describing the local time process jointly for all a and all b, by means of stochastic integral with respect to an appropriate white noise. Our result applies to µ-processes, and has an immediate application: a µ-process is the height process of a Feller continuous-state branching process (CSBP) with immigration (Lambert [10]), whereas a Feller CSBP with immigration satisfies a stochastic differential equation driven by a white noise (Dawson and Li [7]); our result gives an explicit relation between these two descriptions and shows that the stochastic differential equation in question is a reformulation of Tanaka's formula.1The µ-process, also referred to as perturbed reflecting Brownian motion, has attracted much attention in the nineties: Lévy's arc sine law, Ray-Knight theorems as well as pathwise uniqueness of doubly perturbed Brownian motion, see for example[12,22,3,4,18,21,5,6,14,15].The local time of the µ-process at suitable stopping times, as a process of the space variable, turns out to be a squared Bessel process. This is referred to as a Ray-Knight theorem. More precisely, let us fix µ > 0 from now on, so the process (X t , t ≥ 0) is recurrent on R. Since X is a continous semimartingale, we may define L(t, r) := lim ε→0 1 ε t 0
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2012.01761
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075f0fcf0a35f836e8ec3d1296f5adc4b0074da4
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An infinite-dimensional representation of the Ray-Knight theorems
3 Dec 2020
Elie Aïdékon
Yueyun Hu
Zhan Shi
An infinite-dimensional representation of the Ray-Knight theorems
3 Dec 2020Ray-Knight theoremµ-processwhite noiseTanaka's for- mula 2010 Mathematics Subject Classification 60J6560J55
The classical Ray-Knight theorems for Brownian motion determine the law of its local time process either at the first hitting time of a given value a by the local time at the origin, or at the first hitting time of a given position b by Brownian motion. We extend these results by describing the local time process jointly for all a and all b, by means of stochastic integral with respect to an appropriate white noise. Our result applies to µ-processes, and has an immediate application: a µ-process is the height process of a Feller continuous-state branching process (CSBP) with immigration (Lambert [10]), whereas a Feller CSBP with immigration satisfies a stochastic differential equation driven by a white noise (Dawson and Li [7]); our result gives an explicit relation between these two descriptions and shows that the stochastic differential equation in question is a reformulation of Tanaka's formula.1The µ-process, also referred to as perturbed reflecting Brownian motion, has attracted much attention in the nineties: Lévy's arc sine law, Ray-Knight theorems as well as pathwise uniqueness of doubly perturbed Brownian motion, see for example[12,22,3,4,18,21,5,6,14,15].The local time of the µ-process at suitable stopping times, as a process of the space variable, turns out to be a squared Bessel process. This is referred to as a Ray-Knight theorem. More precisely, let us fix µ > 0 from now on, so the process (X t , t ≥ 0) is recurrent on R. Since X is a continous semimartingale, we may define L(t, r) := lim ε→0 1 ε t 0
Introduction
Let (B t ) t≥0 be standard one-dimensional Brownian motion associated with its completed natural filtration (B t ) t≥0 . Denote by (L t ) t≥0 a continuous version of local times of (B t ) at position 0. Let µ ∈ R\{0}. The µ-process X := (X t ) t≥0 is defined as follows:
X t := |B t | − µL t , t ≥ 0.
There are two important special cases of µ-processes: Brownian motion (µ = 1, this is seen using Lévy's identity), and the three-dimensional Bessel process (µ = −1, seen by means of Lévy's and Pitman's identities).
1 {r≤Xs≤r+ε} ds, t ≥ 0, r ∈ R,
as the local time of X at time t and position r. Moreover, we may and will take a bicontinuous version of local times L(·, ·), see [17], Theorem VI.1.7. Let [4] (see also Yor [22], Chapter 9) and by Le Gall and Yor [12] respectively. Theorem 1.1. Fix µ > 0.
(i) ( [4], [22]) Let a > 0. The process L(τ 0 a , −h), h ≥ 0 is a squared Bessel process of dimension (2 − 2 µ ), starting from a and absorbed at 0. (ii) ( [12]) Let b < 0. The process L(T b , b+h), 0 ≤ h ≤ |b| is a squared Bessel process of dimension 2 µ , starting from 0 and reflected at 0. In the special case µ = 1: the process X is Brownian motion by Lévy's identity, so Theorem 1.1 boils down to the classical Ray-Knight theorem for Brownian motion, originally proved by Ray [16] and Knight [9] independently. Werner [21] gave an alternative proof of Theorem 1.1 using a result of Lamperti [11] on semi-stable Markov processes. Perman [14] gave another proof of (i) by establishing a path-decomposition result of X.
The aim of this work is to describe the underlying Brownian motion, jointly for all a and b, in the local time processes in Theorem 1.1. We do this by means of Tanaka's formula and Walsh's stochastic integral with respect to a white noise W ; see Theorem 1.2 below. The idea of using Tanaka's formula to prove Ray-Knight theorems is not new, and can be found for example in Jeulin [8] (for diffusion processes) and in Norris, Rogers and Williams [13] (for Brownian motion with a local time drift); our main contribution is to show how the white noise W explicitly gives the Brownian part jointly for all a and b in Theorem 1.1.
The aforementioned white noise W is defined as follows. For any Borel function g :
R + × R → R such that R + dℓ R g 2 (ℓ, x)dx < ∞, let (1.3) W (g) := ∞ 0 g L(t, X t ), X t sgn(B t )dB t .
It is easily seen that W is a white noise on R + ×R; indeed, by the occupation time formula (Exercise VI.1.15 in [17]),
∞ 0 g 2 L(t, X t ), X t dt = R dx ∞ 0 g 2 L(t, x), x d t L(t, x) = R dx ∞ 0 g 2 (ℓ, x)dℓ.
The exponential martingale for Brownian motion implies that
E e W (g)− 1 2 ∞ 0 g 2 (L(t,Xt),Xt)dt = 1,
showing that W (g) is a centered Gaussian random variable with variance R + R g 2 (ℓ, x)dxdℓ.
The main result of this work is the following theorem:
Theorem 1.2. Fix µ > 0. Let W be the white noise defined via (1.3). (i) Almost surely for all a > 0, (1.4) L(τ 0 a , −h) = a − 2 0 −h W [0, L(τ 0 a , x)], dx + 2 − 2 µ h, h ∈ [0, |I τ 0 a |],
where I t := inf 0≤s≤t X s , t ≥ 0, denotes the infimum process of X.
(ii) Almost surely for all b < 0,
(1.5) L(T b , b + h) = 2 b+h b W [0, L(T b , x)], dx + 2 µ h, h ∈ [0, |b|].
The precise meaning of stochastic integrals with respect to W is given in Section 3. Indeed, we will show that almost surely (1.4) holds for any fixed a > 0, hence for all a belonging to a countable dense set of R + . By using the regularity of local times, we may and will choose a version of stochastic integral such that (1.4) holds simultaneously for all a > 0. The same remark applies to (1.5) as well as to Theorem 5.1 in Section 5.
It is not surprising, at least in the case (ii), that the local times of a µ-process can be represented as solution of an SDE driven by a white noise. As a matter of fact, by duality (see [21]), the process (X T b −t − b, t ∈ [0, T b ]) has the same law as the process |B t | + µL t , stopped when leaving |b| for the last time. On one hand, the process (|B t | + µL t , t ≥ 0) is the height process of a Feller CSBP with immigration (see [10], remark p. 57 in Section 4). On the other hand, Bertoin and Le Gall showed in [1] that general CSBPs are related to flows of subordinators, constructed in [2] critical CSBPs without Gaussian coefficient as solutions of SDEs driven by compensated Poisson random measures. Dawson and Li [7] generalized this SDE to include a Gaussian coefficient and possible immigration. Applied to our setting, it is shown that a Feller CSBP with immigration can be constructed as a solution of (1.5). Theorem 1.2 connects directly the local times of the µ-process to equation (1.5), without making use of the framework of CSBPs, and in Section 4 we are going to see Theorem 1.1 as a consequence of Tanaka's formula for X.
The rest of the paper is as follows. In Section 2, we follow Walsh [19] by introducing the excursion filtration, then make an enlargement of the filtrationà la Jeulin [8]. Section 3 is devoted to study of the martingale measure associated with the white noise W . In particular, stochastic integration with respect to W is defined. Theorems 1.2 and 1.1 are proved in Section 4. Sections 5 presents analogous results for the µ-process defined on R.
The excursion filtration
We first introduce some notation which will be used throughout the paper.
Notation 2.1. Let x ∈ R.
We define the process X −,x obtained by gluing the excursions of X below x as follows. Let, for t ≥ 0,
A −,x t := t 0 1 {Xs≤x} ds, α −,x t := inf{u > 0, A −,x u > t},
with the usual convention inf ∅ := ∞. Define
X −,x t := X α −,x t , t < A −,x ∞ := ∞ 0 1 {Xs≤x} ds.
Similarly, we define A +,x t , α +,x t and X +,x by replacing X s ≤ x by X s > x. When the process is denoted by X with some superscript, the analogous quantities keep the same superscript. For example, L +,x (t, y) denotes the local time of X +,x at position y and time t, and I +,x t = inf 0≤s≤t X +,x s .
Remark 2.2. Let x ∈ R. One can reconstruct X from X −,x and X +,x by gluing the excursions of X −,x and of X +,x , indexed by their local time.
The following proposition is adapted from Section 8.5 of [22].
Proposition 2.3. Let x ≤ 0. (i) Define the filtration (F +,x u ) u≥0 by F +,x u := σ(X +,x s , s ∈ [0, u]
) and the process
β +,x u := α +,x u 0 1 {Xs>x} sgn(B s )dB s , u ≥ 0.
Then β +,x is (F +,x u )-Brownian motion and X +,x is an (F +,x u )-semimartingale with decomposition
(2.1) X +,x u = β +,x u − 1 − µ µ I +,x u + 1 2 L +,x (u, x), u ≥ 0. (ii) Define the filtration (F −,x u ) u≥0 by F −,x u := σ(X −,x s , s ∈ [0, u]) and the process β −,x u := α −,x u 0 1 {Xs≤x} sgn(B s )dB s , u ≥ 0. Then β −,x is (F −,x u )-Brownian motion and X −,x is an (F −,x u )-semimartingale with de- composition (2.2) X −,x u = x + β −,x u − 1 − µ µ (I −,x u − x) − 1 2 L −,x (u, x), u ≥ 0.
(iii) The Brownian motions β +,x and β −,x are independent.
Proof. By Tanaka's formula,
(X t − x) + = (X 0 − x) + + t 0 1 {Xs>x} dX s + 1 2 L(t, x). Take t = α +,x u . We get (2.3) X +,x u = α +,x u 0 1 {Xs>x} dX s + 1 2 L(α +,x u , x).
Moreover, dX s = d|B s | − µdL s = sgn(B s )dB s + (1 − µ)dL s by another application of Tanaka's formula. Also observe that I t = −µL t , hence
(2.4) dX s = sgn(B s )dB s − 1 − µ µ dI s , s ≥ 0.
Therefore,
X +,x u = α +,x u 0 1 {Xs>x} sgn(B s )dB s − 1 − µ µ α +,x u 0 1 {Xs>x} dI s + 1 2 L(α +,x u , x).
We notice that L +,x (u, r) = L(α +,x u , r) for any r ∈ [x, ∞) and u ≥ 0. On the other hand,
α +,x u 0 1 {Xs>x} dI s = I α +,x u ∧Tx which is also the infimum of X +,x on the time interval [0, u]
. This yields (2.1). This equation also implies that β +,x is adapted to F +,x . Moreover, from the definition of β +,x u and Proposition V.1.5 of [17]
, β +,x u is a (B α +,x u )-continuous martingale with β +,x , β +,x u = u, hence (B α +,x u )-Brownian motion. Since F +,x u ⊂ B α +,x u , we deduce that β +,x u is also (F +,x u )-Brownian motion. This proves (i). The proof of (ii) is similar. Tanaka's formula applied to (X t+Tx −x) − with t = α −,x u −T x implies that X −,x u = x + α −,x u Tx 1 {Xs≤x} sgn(B s )dB s − 1 − µ µ α −,x u Tx 1 {Xs≤x} dI s − 1 2 L(α −,x u , x). We observe that L(α −,x u , x) = L −,x (u, x) and α −,x u Tx 1 {Xs≤x} dI s = I α −,x u −x while I α −,x u = I −,x u which gives (2.
2). We conclude as for (i). The statement (iii) is a consequence of Knight's theorem on orthogonal martingales.
The following result is well-known. It has been proved in Section 8.5 of [22] when µ ∈ (0, 2), in [21] and in [15]. Here, following [22], we choose to see it as a consequence of Proposition 2.3.
Corollary 2.4. Let x ≤ 0. The processes X +,x and X −,x are independent.
Proof. By Proposition 2.3 (iii), the martingale parts of X +,x and X −,x , namely β +,x and β −,x , are independent. It remains to see that X +,x is measurable with respect to β +,x and X −,x with respect to β −,x , which was established by Chaumont and Doney [5] and Davis [6].
The excursion filtration, introduced by Walsh [19], is defined as
E + x := F +,x ∞ = σ(X +,x s , s ≥ 0), x ∈ R. Similarly we define E − x := F −,x ∞ = σ(X −,x s , s ≥ 0) for x ∈ R. It is routine to check, using the time-changes α −,x and α +,x , that E − x is increasing in x whereas E + x is decreasing. 4 Define, for u ≥ 0, G +,x u := σ(F +,x u , E − x ), G −,x u := σ(F −,x u , E + x ).
The idea of such an enlargement of filtrations goes back at least to Jeulin [8].
Corollary 2.5. Consider a random function g(ℓ, y) = g(ℓ, y, ω) such that the process
t → g(L(t, X t ), X t ) is (B t )-progressively measurable and E R + dℓ R g(ℓ, y) 2 dy < ∞. Fix x ≤ 0. (i) The process u → g L +,x (u, X +,x u ), X +,x u is (G +,
x u )-progressive and almost surely,
(2.5) ∞ 0 g L(t, X t ), X t 1 {Xt>x} sgn(B t )dB t = ∞ 0 g L +,x (u, X +,x u ), X +,x u dβ +,x u . (ii) The process u → g L −,x (u, X −,x u ), X −,x u is (G −,
x u )-progressive and almost surely, 4 We will be implicitly working with a right-continuous (and complete) version of the filtrations (E − x ) x∈R and (E + x ) x∈R -if necessary, by means of the procedure of usual augmention, as described in Section I.4 of Revuz and Yor [17]. In our work, we only study continuous martingales, which are also martingales with respect to augmented filtrations. The same remark applies to the µ-process defined on R in Section 5.
(2.6) ∞ 0 g L(t, X t ), X t 1 {Xt≤x} sgn(B t )dB t = ∞ 0 g L −,x (u, X −,x u ), X −,x u dβ −,x u .
Proof. We prove (i). The process u → g(L +,x (u, X +,x u ), X +,x u ) is (B α +,x u )-progressive (Proposition V.1.4, [17]). Therefore it is also progressive with respect to (G +,x u ) because the latter filtration is larger. We prove now (2.5). By a time-change (Proposition V.1.4, [17]),
α +,x t 0 g L(s, X s ), X s 1 {Xs>x} sgn(B s )dB s = t 0 g L +,x (u, X +,x u ), X +,x u dβ +,x u .
Letting t → ∞ yields (2.5). Statement (ii) is proved similarly. Proposition 3.1. In the setting of Walsh [20], (M r , r ≥ 0) is a continuous martingale measure with respect to the filtration (E + −r , r ≥ 0). Proof. Since W is a white noise, it suffices to show that M r is measurable with respect to E + −r and that M s − M r is independent of E + −r for any 0 ≤ r < s. The first statement comes from (2.5) applied to x = −r and g(ℓ, y) = 1 A×[−r,0] (ℓ, y) for a Borel set A with finite Lebesgue measure. The second statement comes from (2.6) applied to x = −r and g(ℓ, y) = 1 A×[−s,−r) (ℓ, y). Since the processes X −,−r and β −,−r are independent of E + −r , the proposition follows.
We are going to extend (1.3), seen as an equality for deterministic functions g, to random functions. To this end, we first recall the construction by Walsh in [20] of stochastic integral with respect to the martingale measure M. A (random) function f is said to be elementary if it is of the form f (ℓ, x) := Z 1 [a,b) (x) 1 A (ℓ), where a < b ≤ 0, A ⊂ R + is a Borel set of finite Lebesgue measure, and Z is a bounded E + b -measurable real-valued random variable. Denote by f · M the stochastic integral with respect to M:
f · M := Z (M |a| (A) − M |b| (A)) = Z W (1 A×[a,b) ).
A simple function is a (finite) linear combination of elementary functions. We extend by linearity the definition of f · M to simple functions f and furthermore by isometry to any f ∈ L 2 , where L 2 denotes the space of (E + −r , r ≥ 0)-predictable and squareintegrable functions, defined as the closure of the space of simple functions under the norm:
f := E R + dℓ R − f 2 (ℓ, x)dx 1/2
. For any f ∈ L 2 , f · M is a centered random variable with E[(f · M) 2 ] = f 2 . We write f · M ≡ R + ×R − f (ℓ, x)W (dℓ, dx) and for any r ≥ 0,
(f 1 R + ×[−r,0] ) · M ≡ R + ×[−r,0] f (ℓ, x)W (dℓ, dx). The latter, if furthermore f is of form f (ℓ, x) = 1 {0≤ℓ≤σx} η x ,
will be re-written as 0 −r η x W [0, σ x ], dx . By the construction of stochastic integral and Proposition 3.1, R + ×[−r,0] f (ℓ, x)W (dℓ, dx) is a continuous martingale with respect to the filtration (E + −r , r ≥ 0), of quadratic variation process R + ×[−r,0] f 2 (ℓ, x)dℓdx.
Proposition 3.2. Take g ∈ L 2 such that s → g L(s, X s ), X s admits a version which is progressive with respect to the Brownian filtration (B s ). 5 Then
g · M = ∞ 0 g L(s, X s ), X s sgn(B s )dB s a.s.
Proof. By definition of L 2 , there exists a sequence of simple functions g n such that g − g n → 0 as n → ∞. By isometry, g · M − g n · M → 0 in L 2 . Since g n is a simple function, g n is of the form
g n (ℓ, x) = ∞ k,j=1 Z n k,j 1 [a n k ,a n k−1 ) (x)1 A n j (ℓ),
where for each n, 0 = a n 0 > ... > a n k > a n k+1 > ... is a decreasing sequence such that a n k → −∞ as k → ∞, (A n j ) j≥1 is a collection of (nonrandom) pairwise disjoint Borel subsets of R + with finite Lebesgue measures, and for any k, j ≥ 1, Z n k,j is a bounded E + a n k−1 -measurable random variable. Moreover for all large k, j, Z n k,j = 0, which means the above double sum runs in fact over a finite index set of k and j.
Note that for a.e. z ≤ 0, g(·, z) is measurable with respect to E + z (as g n is). We may (and will) take a version of g such that g(·, z) is measurable with respect to E + z for all z ≤ 0.
By applying (1.3) and (3.1), we deduce from the linearity of the integral that
g n · M = ∞ k,j=1 Z n k,j ∞ 0
1 [a n k ,a n k−1 ) (X s )1 A n j (L(s, X s ))sgn(B s )dB s .
Note that we can (and we will) take (a n k , a n k−1 ) instead of [a n k , a n k−1 ) without changing the value of g n · M. By (2.5) and (2.6) respectively, ∞ 0 1 (a n k ,a n
k−1 ) (X s )1 A n j (L(s, X s ))sgn(B s )dB s = ∞ 0 1 (a n k ,a n k−1 ) (X −,x u )1 A n j (L −,x (u, X −,x u ))dβ −,x u (3.2) = ∞ 0
1 (a n k ,a n k−1 ) (X +,y u )1 A n j (L +,y (u, X +,y u ))dβ +,y with x = a n k−1 and y = a n k . Similarly, ∞ 0 g L(s, X s ), X s 1 (a n k ,a n
k−1 ) (X s )sgn(B s )dB s = ∞ 0 g L −,x (u, X −,x u ), X −,x u 1 (a n k ,a n k−1 ) (X −,x u )dβ −,x u , (3.4) = ∞ 0
g L +,y (u, X +,y u ), X +,y u 1 (a n k ,a n k−1 ) (X +,y u )dβ +,y u (3.5) with x = a n k−1 and y = a n k as in (3.2) 1 (a n k ,a n k−1 ) (X +,y u )1 A n j (L +,y (u, X +,y u ))dβ +,y u − ∞ 0 g L +,y (u, X +,y u ), X +,y u 1 (a n k ,a n k−1 ) (X +,y u )dβ +,y u ,
where as before y = a n k . Since Z n k,j is E + a n k−1 -measurable, hence E + a n k -measurable, we see that the sum over j in the definition of ∆ n (k) is E + a n k -measurable. For the last integral in ∆ n (k), we use the fact that for z ≤ 0, g(·, z) is measurable with respect to E + z . Note that X +,y u and L +,y (·, ·) are measurable with respect to E + y . Since X +,y u ≥ y and E + z decreases on z, we deduce that g(·, X +,y u ) is E + y -mesurable, and so is g L +,y (u, X +,y u ), X +,y u . It follows that ∆ n (k) is measurable with respect to E + a n k . Now we prove that (∆ n (k)) k≥1 is a martingale difference sequence with respect to the filtration (E + a n k ) k≥1 . Indeed, using (3.2) and (3.4) instead of (3.3) and (3.5), one can also write
∆ n (k) = ∞ j=1 Z n k,j ∞ 0 1 (a n k ,a n k−1 ) (X −,x u )1 A n j (L −,x (u, X −,x u ))dβ −,x u − ∞ 0 g L −,x (u, X −,x u ), X −,x u 1 (a n k ,a n k−1 ) (X −,x u )dβ −,x u ,
with x = a n k−1 . Recall from Proposition 2.3 that β −,x is (F −,x u )-Brownian motion, which is independent of E + x by Corollary 2.4. Then β −,x can be seen as (G −,x u )-Brownian motion.
By Corollary 2.5 (ii), g L −,x (u, X −,x u ), X −,x u is progressive with respect to the filtration (G −,x u ), while 1 (a n k ,a n
k−1 ) (X −,x u ) g n L −,x (u, X −,x u ), X −,x u = ∞ j=1
Z n k,j 1 (a n k ,a n
k−1 ) (X −,x u )1 A n j (L −,x (u, X −,x u ))
is (G −,x u )-progressive as well. Therefore, one can write
(3.6) ∆ n (k) = ∞ 0 1 (a n k ,a n k−1 ) (X −,x u ) (g n − g) L −,x (u, X −,x u ), X −,x u dβ −,x u , with x = a n k−1 . It follows that (since G −,x 0 = E + x ) E[∆ n (k) | E + a n k−1 ] = 0.
In other words, the process j → j k=1 ∆ n (k) is a martingale and we have
E[(g n · M − I(g)) 2 ] = ∞ k=1 E[∆ n (k) 2 ].
From (3.6), we get
E[∆ n (k) 2 ] = E ∞ 0 1 (a n k ,a n k−1 ) (X −,x u ) (g n − g)(L −,x (u, X −,x u ), X −,x u ) 2 du = E a n k−1 a n k (g n (ℓ, z) − g(ℓ, z)) 2 dℓdz ,
where in the second equality we have used the occupation time formula. It follows that E[(g n · M − I(g)) 2 ] = g n − g 2 which goes to 0 as n → ∞. Thus we get that g · M = I(g). . Moreover, we may define in a similar way the stochastic integral f · M for any f ∈ L 2 , where L 2 denotes the space of (E − b+r ) r≥0predictable functions f such that E R + dℓ [b,∞) f 2 (ℓ, x)dx < ∞. Then for any g ∈ L 2 such that s → g(L(s, X s ), X s ) admits a version which is progressive with respect to the Brownian filtration (B s ), we have
(3.7) g · M = ∞ 0 g L(s, X s ), X s dX s − 1 − µ µ 0 −∞ g(0, x)dx, a.s.
With a slight abuse of notation, we shall write g · M ≡ R + ×[b,∞) g(ℓ, x)W (dℓ, dx) and
(g1 R + ×[b,t) )· M ≡ R + ×[b,t) g(ℓ, x)W (dℓ, dx) for any t ≥ b. Then R + ×[b,b+r) g(ℓ, x)W (dℓ, dx),
is an (E − b+r )-continuous martingale with quadratic variation process R + ×[b,b+r) g 2 (ℓ, x)dℓdx for r ≥ 0.
L(τ 0 a , x) = L +,x (A +,x τ 0 a , x). Since {A +,x τ 0 a ≥ t} = {L(α +,x t , 0) ≤ a} = {L +,x (t, 0) ≤ a}, we obtain that L(τ 0 a , x) is measurable with respect to E + x . Let g(ℓ, x) := 1 {0≤ℓ≤L(τ 0 a ,x)}∩{−h<x≤0} .
Using the continuity of local times L(τ 0 a , x) on x and the fact that E[ R + ×R g 2 (ℓ, x)dℓdx] < ∞, we get that g ∈ L 2 .
Observe that a.s., g(L(s, X s ), X s ) = 1 {0≤s≤τ 0 a , −h<Xs≤0} , ds-a.e. This follows from the fact that In view of (4.2), this yields Theorem 1.2 (i) for each fixed a > 0 and h ∈ [0, |I τ 0 a |]. Since the processes are continuous in h and càdlàg in a, they coincide except on a null set.
L(T b , b + h) = 2h + 2 T b 0 1 {Xs≤b+h} dX s = 2 T b 0 1 {Xs≤b+h} sgn(B s )dB s + 2 µ h, (4.3)
where the second equality follows from (2.4) again and the fact that
T b 0 1 {Xs≤b+h} dI s = T b 0 1 {Is≤b+h} dI s = −h.
The main difference with Part (i) is the measurability. As a matter of fact, for any
x ≥ b, L(T b , x) is E − x -measurable: observe that L(T b , x) = L −,x (A −,x T b , x)
and for any t ≥ 0,
{A −,x T b > t} = {T b > α −,x t } = {inf 0≤s≤α −,x t X s > b} = {inf 0≤s≤t X −,x s > b} is E − x -mesurable. Let g(ℓ, x) := 1 {0<ℓ≤L(T b ,x)}×{b≤x≤b+h} .
We can check as in (i) that we may apply (3.7) to get that
Extension to the two-sided µ-process
In this Section, we shall explore the strong Markov property at the hitting times of a µ-process defined on R and present an analogue of Theorem 1.2. This result, apart from its own interest, will be useful in a forthcoming work on the duality of Jacobi stochastic flows.
Let (B t ) t∈R be a two-sided Brownian motion, which means that for t ≤ 0, B t = B ′ −t , where B ′ is a standard Brownian motion independent of (B t ) t≥0 . Denote by (L ′ t ) t≥0 the local time process at position zero of B ′ .
Recall that X t = |B t | − µL t , for t ≥ 0. For t ≤ 0, we let X t := |B ′ −t | + µL ′ −t . We call (X t ) t∈R a two-sided µ-process. Fix µ > 0. Notice that X t → ∞ as t → −∞, and T r < 0 when r > 0. We naturally extend the notation T r := inf{t ∈ R : X t = r} for r ∈ R,
L(t, x) := lim ε→0 1 ε t −∞ 1 {x≤Xs≤x+ε} ds, x ∈ R,
= inf{s ≥ 0 : L(s, r) > t}, be the inverse local time of X. Denote by (1.2) T r := inf{t ≥ 0 : X t = r}, the hitting time of r. The following Ray-Knight theorems were established by Carmona, Petit and Yor
3
The martingale measure associated with W Recall the definition of the white noise W in (1.3). For any Borel set A of R + with finite Lebesgue measure and r ≥ 0, we define (3.1) M r (A) := W (1 A×[−r,0] ).
5
By version we mean a (B s )-progressive process (h s ) such that ∞ 0 1 {g(L(s,Xs),Xs) =hs} ds = 0 a.s.
s, X s ), X s sgn(B s )dB s .Then g n · M − I(g)
Remark 3. 3 .
3Fix b < 0. Similarly to (3.1) we may define a martingale measure M by M r (A) := W (1 A×[b,b+r] ), for any Borel set A ⊂ R + of finite Lebesgue measure and r ≥ 0. The analogs of Propositions 3.1 and 3.2 hold for M . Specifically, ( M r , r ≥ 0) is a martingale measure with respect to the filtration (E − b+r , r ≥ 0)
's formula, for any r ≥ 0 and x ∈ R,
Part (i) of Theorems 1.2 and 1.1: Applying (4.1) to r = τ 0 a gives that L(τ {Xs≤x} dX s for all x ∈ R. Let h ≥ 0. Taking x = −h and x = 0, and using the fact that L(τ 0 a , 0) = a, we obtain thatL(τ 0 a , −h) {Is>−h} dI s = I min(T −h ,I τ 0 a ) = − min(h, |I τ 0 a |). To deal with the stochastic integral with respect to (B s ) in (4.2), we shall use Proposition 3.2. First we remark that for x ≤ 0, L(τ 0 a , x) is measurable with respect to E + x . In fact, let u := A +as τ 0 a is an increasing time for A +,x . Therefore
1
{L(s,x)≤L(τ 0 a ,x)} d s L(s, x) = 0, by the occupation time formula. Then s → g(L(s, X s ), X s ) admits a version which is (B s )progressive and we are entitled to apply Proposition 3.2 to see thatτ 0 a 0 1 {−h<Xs≤0} sgn(B s )dB s = R + ×(−h,0] 1 {0≤ℓ≤L(τ 0 a ,x)} W (dℓ, dx) = 0 −h W [0, L(τ 0 a , x)], dx .
W [0, L(τ 0 a , x)], dx is an (E + −h ) h≥0 -continuous martingale with quadratic variation process 0 −h L(τ 0 a , x)dx, it follows from the Dambis-Dubins-Schwarz theorem that there exists (E + −h )-Brownian motion γ such that , s)dγ s . Going back to (4.2), we see that for all 0 ≤ h ≤ inf{s ≥ 0 : L(τ 0 a , −s) = 0} = |I τ Part (ii) of Theorems 1.2 and 1.1: Let b < 0 and T b := inf{t ≥ 0 : X t = b}. For h ∈ [0, |b|], we get from (4.1) that
sgn(B s )dB s = R + ×[b,b+h] 1 {0≤ℓ≤L(T b ,b+h)} W (dℓ, dx) = b+h b W [0, L(T b , x)], dx ,proving, in view of (4.3), Theorem 1.2 (ii). Furthermore by Remark 3.3, the process h → b+h b W [0, L(T b , x)], dx is an (E − b+h )-continuous martingale with quadratic variation process b+h b L(T b , x)dx. This easily yields Theorem 1.1 (ii).
the local time accumulated by (X t , t ∈ R) at position x up to time t, andτ x a := inf{t ∈ R : L(t, x) > a}, a ≥ 0, Taking t = τ r a , and x = r then x = r + h, ′ 1 {r<Xs≤r+h} dI s = Tr T r ′ 1 {Is≤r+h} dI s = −h. By Proposition 3.2 applied to (X (r ′ ) , B (r ′ ) ), −r ′ <X (r ′ ) s ≤r+h−r ′ } sgn(B (r ′ ) s )dB (r ′ ) to r = r ′ .This proves (5.3) and completes the proof of the theorem.
A×[x,y] (L (r) (t, X (r) t ), X (r) t )sgn(B (r) t )dB (r) twhere we recall that B (r) t = B t+Tr , t ≥ 0. We deduce that if x ≤ y ≤ 0 and A is a bounded Borel set of R + ,
the inverse local time at position x. We define now for bounded Borel functions g with compact support,The stochastic integral has to be understood as an integral with respect to the Brownian motion B (r) := (B t+Tr , t ≥ 0) where r is any positive real such that g(ℓ, x) = 0 for all x ≥ r (that B (r) is a standard Brownian motion comes from the fact that (B ′ Tr+t , t ∈ [0, |T r |]) is distributed as (B t , t ∈ [0, T −r ])). We will see in the following theorem that W defines a white noise.Similarly to Notation 2.1, for x ∈ R, we can consider the process (X −,x u , u ≥ 0) obtained by gluing the excursions of X below x: that is we set for t ∈ R, A −,xwhere I t := inf −∞<s≤t X s , t ∈ R, denotes the infimum process of X. Proof. Notice that for any r ∈ R, X (r) := (X Tr+t − r, t ≥ 0) is distributed as (X t , t ≥ 0). Therefore we can apply Theorem 1.2 to X (r) . As in (1.3), we define W (r) the white noise associated to X (r) : for any Borel function g :where L (r) (·, ·) denote the local times of X (r) . Let τx,(r) a be the associated inverse local times. By Theorem 1.2 (i) applied to X (r) , for a > 0, (5.4)We deduce that (5.1) defines a white noise on R + × (−∞, r), hence on R + × R since r can be made arbitrary large. Equation (5.4) becomesIt is (5.2). We prove now (5.3). Let r ′ > r + h arbitrary. Using Tanaka's formula applied to (X T r ′ +t , t ≥ 0), we have for any x ∈ R and t ≥ T r ′ ,x) − L(T r ′ , x)).
The Bolthausen-Sznitman coalescent and the genealogy of continuous-state branching processes. J Bertoin, Le Gall, J.-F , 117Probab. Theory Relat. FieldsBertoin, J. and Le Gall, J.-F. (2000). The Bolthausen-Sznitman coalescent and the genealogy of continuous-state branching processes. Probab. Theory Relat. Fields, 117, 249-266.
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[] |
[
"Theoretical description of optical and x-ray absorption spectra of MgO including many-body effects",
"Theoretical description of optical and x-ray absorption spectra of MgO including many-body effects"
] |
[
"Vijaya Begum \nDepartment of Physics and Center for Nanointegration Duisburg-Essen (CENIDE)\nUniversity of Duisburg-Essen\nDuisburgGermany\n",
"Markus E Gruner \nDepartment of Physics and Center for Nanointegration Duisburg-Essen (CENIDE)\nUniversity of Duisburg-Essen\nDuisburgGermany\n",
"Christian Vorwerk \nInstitute für Physik and IRIS Adlershof\nHumboldt-Universität zu Berlin\nBerlinGermany\n",
"Claudia Draxl \nInstitute für Physik and IRIS Adlershof\nHumboldt-Universität zu Berlin\nBerlinGermany\n",
"Rossitza Pentcheva \nDepartment of Physics and Center for Nanointegration Duisburg-Essen (CENIDE)\nUniversity of Duisburg-Essen\nDuisburgGermany\n"
] |
[
"Department of Physics and Center for Nanointegration Duisburg-Essen (CENIDE)\nUniversity of Duisburg-Essen\nDuisburgGermany",
"Department of Physics and Center for Nanointegration Duisburg-Essen (CENIDE)\nUniversity of Duisburg-Essen\nDuisburgGermany",
"Institute für Physik and IRIS Adlershof\nHumboldt-Universität zu Berlin\nBerlinGermany",
"Institute für Physik and IRIS Adlershof\nHumboldt-Universität zu Berlin\nBerlinGermany",
"Department of Physics and Center for Nanointegration Duisburg-Essen (CENIDE)\nUniversity of Duisburg-Essen\nDuisburgGermany"
] |
[] |
Here we report the optical and x-ray absorption (XAS) spectra of the wide-band-gap oxide MgO using density functional theory (DFT) and many-body perturbation theory (MBPT). Our comprehensive study of the electronic structure shows that while the band gap is underestimated with the exchange-correlation functional PBEsol (4.58 eV) and the hybrid functional HSE06 (6.58 eV) compared to the experimental value (7.7 eV), it is significantly improved (7.52 eV) and even overcompensated (8.53 eV) when quasiparticle corrections are considered. Inclusion of excitonic effects by solving the Bethe-Salpeter equation (BSE) yields the optical spectrum in excellent agreement with experiment. Excellent agreement is observed also for the O and Mg K-edge absorption spectra, demonstrating the importance of the electron-hole interaction within MBPT. Projection of the electron-hole coupling coefficients from the BSE eigenvectors on the band structure allows us to determine the origin of prominent peaks and identify the orbital character of the relevant contributions. The real space projection of the lowest energy exciton wavefunction of the optical spectrum indicates a Wannier-Mott type, whereas the first exciton in the O K-edge is more localized. arXiv:2012.08960v2 [cond-mat.other] 4 Apr 2021
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10.1103/physrevb.103.195128
|
[
"https://arxiv.org/pdf/2012.08960v2.pdf"
] | 229,188,338 |
2012.08960
|
10fd2febda2f41a296ff91167d44844339ef1723
|
Theoretical description of optical and x-ray absorption spectra of MgO including many-body effects
Vijaya Begum
Department of Physics and Center for Nanointegration Duisburg-Essen (CENIDE)
University of Duisburg-Essen
DuisburgGermany
Markus E Gruner
Department of Physics and Center for Nanointegration Duisburg-Essen (CENIDE)
University of Duisburg-Essen
DuisburgGermany
Christian Vorwerk
Institute für Physik and IRIS Adlershof
Humboldt-Universität zu Berlin
BerlinGermany
Claudia Draxl
Institute für Physik and IRIS Adlershof
Humboldt-Universität zu Berlin
BerlinGermany
Rossitza Pentcheva
Department of Physics and Center for Nanointegration Duisburg-Essen (CENIDE)
University of Duisburg-Essen
DuisburgGermany
Theoretical description of optical and x-ray absorption spectra of MgO including many-body effects
and European Theoretical Spectroscopy Facility (Dated: April 6, 2021)
Here we report the optical and x-ray absorption (XAS) spectra of the wide-band-gap oxide MgO using density functional theory (DFT) and many-body perturbation theory (MBPT). Our comprehensive study of the electronic structure shows that while the band gap is underestimated with the exchange-correlation functional PBEsol (4.58 eV) and the hybrid functional HSE06 (6.58 eV) compared to the experimental value (7.7 eV), it is significantly improved (7.52 eV) and even overcompensated (8.53 eV) when quasiparticle corrections are considered. Inclusion of excitonic effects by solving the Bethe-Salpeter equation (BSE) yields the optical spectrum in excellent agreement with experiment. Excellent agreement is observed also for the O and Mg K-edge absorption spectra, demonstrating the importance of the electron-hole interaction within MBPT. Projection of the electron-hole coupling coefficients from the BSE eigenvectors on the band structure allows us to determine the origin of prominent peaks and identify the orbital character of the relevant contributions. The real space projection of the lowest energy exciton wavefunction of the optical spectrum indicates a Wannier-Mott type, whereas the first exciton in the O K-edge is more localized. arXiv:2012.08960v2 [cond-mat.other] 4 Apr 2021
I. INTRODUCTION
MgO is one of the most extensively studied oxides which is used as a substrate material and in various heterostructures with applications related to tunneling magnetoresistance [1][2][3]. As a wide band gap insulator with a measured optical band gap of 7.7 [4] from absolutereflectance measurements with UV radiation and 7.83 eV [5] from thermoreflectance spectroscopy, this material is employed e.g. in transient x-ray spectroscopy and time-dependent density-functional theory (DFT) calculations aiming to unravel the propagation of excitations across the interface in metal-insulator heterostructures [6][7][8][9]. Understanding spectroscopic features from firstprinciples requires accurate modeling beyond the ground state properties including excitations of different origin and energy scale.
The structural and electronic properties of MgO have been widely studied with first-principles calculations [10][11][12]. DFT calculations with semilocal functionals yield a fundamental band gap of 4.88, 4.50 and 4.76 eV [10,12,13], respectively. Many-body perturbation theory (MBPT) calculations employing Hedin's GW approximation [14] render an increased fundamental gap of 6.8 and 7.25 eV [11,12], respectively, which is still lower than the experimental one. The optical spectrum, calculated by Wang et al. [13] using the local density approximation (LDA) as the exchange-correlation functional for the DFT calculation and subsequently including GW and excitonic corrections agrees with experiment [15] w.r.t. peak positions up to 12 eV whereas the amplitude of the peaks beyond the first one is overestimated due to the limited number of unoccupied bands employed in the BSE corrections. Schleife et al. [10] studied the frequency-dependent di-electric function for different MgO polymorphs -wurzite, zinc blende, and rocksalt -in the independent particle (IP) approximation using the generalized gradient approximation in the PW91 parametrization [16]. Good agreement with experiment concerning the peak positions was obtained by including excitonic corrections with BSE, based on the Kohn-Sham (KS) eigenenergies and a scissors operator to describe the QP eigenenergies [17,18].
While optical spectroscopy probes excitations from valence bands, x-ray absorption spectroscopy (XAS) probes those from the strongly localized core states. A common approach to model XAS is the final state rule (FSR) [19] based on Fermi's Golden rule, where the effects of screening of the core-hole (the so-called final-state effects) are calculated in a supercell. Alternatively, XAS can be described by considering quasiparticle and excitonic effects within MBPT by using GW and solving the BSE. Rehr et al. [20] showed that while both approaches led to similar overall features in the O and Mg K-edge spectra of MgO, BSE calculations result in better agreement with experiment at high transition energy due to the non-local treatment of the exchange interaction. Recent implementations of BSE in all-electron codes [21,22] with explicit treatment of core states have demonstrated very good agreement with experiment for the XAS spectra of TiO 2 (rutile and anatase), PbI 2 , and CaO [22]. The latter approach is adopted in this work.
Here we describe both the optical and x-ray absorption spectra of bulk MgO including many-body effects. As a first step, we perform the G 0 W 0 corrections starting from Kohn-Sham (KS) wavefunctions. We show that careful consideration of the electron-hole interaction with BSE is essential to achieve agreement with experiment for both valence and core excitation spectra. In particular, the op-tical spectrum calculated with two different DFT functionals (PBEsol and HSE06) including the G 0 W 0 and BSE corrections are consistent with experiment [4,23] and previous theoretical work [17,18] and yield an improved agreement regarding the intensity of the peaks at higher energies, highlighting the importance of quasiparticle and excitonic effects.
Previous studies have shown that a dense k-mesh is required for the sampling of the Brillouin zone to describe sufficiently the localization of the excitonic wave function and the fine structure in the vicinity of the absorption edge [11]. Here, we use a model for the static screening with parameters fitted to the G 0 W 0 calculation, to solve the BSE (so-called model BSE [11,24]) starting directly from DFT wavefunctions on a denser k-mesh, which improves in particular the low energy range (7−11 eV). The results for the model BSE are presented in Appendix B. Beyond previous work we provide a thorough analysis of interband transitions contributing to the peaks in the optical spectrum. Further insight into the nature of the first bound exciton is given by the real-space visualization of its wave function.
Employing the exciting code, the O and Mg K-edge XAS spectra calculated with BSE show very good agreement with the experimental spectra [25] and with previous theoretical results using the FSR [20]. Knowledge of the origin of peaks is essential for the interpretation of x-ray spectra. The main incentive of this study is to identify the nature of transitions which contribute to the peaks and analyze the character of the first exciton in the O K-edge both in real and reciprocal space.
The paper is structured as follows: the details of the calculations are presented in Section II, followed by the discussion of the results in Section III. We start with the electronic properties of MgO in III A and then compare the optical spectra calculated with two different starting exchange-correlation functionals in III B. Subsequently, we analyze the transitions in reciprocal space to derive the origin of contributions to the peaks in the spectrum.
In subsection III C, we present the XAS spectra of the O and Mg K-edge and identify the underlying transitions in reciprocal space for the prominent peaks. Finally, subsection III D is dedicated to the real-space visualization of the first exciton of the optical and the O K-edge xray absorption spectrum. The results are summarized in Section IV, followed by two appendices showing a comparison of the optical spectra obtained with VASP and exciting and the optical spectrum with the model BSE.
II. COMPUTATIONAL DETAILS
The DFT calculations are performed with the VASP code (version 5.4.4) [26,27], using pseudopotentials in combination with the projector augmented wave (PAW) method [28], and the exciting code [29] (version Nitrogen) employing the all-electron full-potential (linearized) augmented planewave + local orbital [(L)APW+lo] method. For the exchange-correlation functional we chose the generalized gradient approximation (GGA) in the implementation of Perdew, Burke, and Ernzerhof (PBE96) [30], PBEsol [31,32], and the hybrid functional, HSE06 [33,34]. The equilibrium lattice constant determined with the different functionals amounts to 4.24Å (PBE96), 4.21Å (PBEsol), and 4.20Å (HSE06), the experimental one being 4.212Å [35].
For the calculation of the optical spectrum with VASP, we have performed single-shot G 0 W 0 on top of the KS wavefunctions obtained with two DFT functionals, PBEsol and HSE06 and subsequently included excitonic corrections by solving the BSE. For all the BSE calculations the Tamm-Dancoff-approximation (TDA) [36] is adopted. The calculations are performed for a two-atom unit cell with a Γ-centered 15×15×15 k-mesh (unless otherwise specified) with a plane-wave cut-off energy of 650 eV. GW PAW pseudopotentials for excited properties were employed in all the calculations with two valence electrons for Mg: 3s 2 and six for O: 2s 2 , 2p 4 . 192 unoccupied bands are used for both the DFT and single-shot G 0 W 0 calculations with 100 frequency-grid points. For the optical spectrum a Lorentzian broadening of 0.3 eV is used.
Employing PBEsol [31,32] as the starting exchangecorrelation functional for the ground-state calculation, single-shot G 0 W 0 calculations are also performed with the exciting code [29] together with BSE [37] (within TDA) for the optical and x-ray absorption spectra [22]. A Γ-centered 11×11×11 mesh shifted by (0.09, 0.02,0.04) is employed for the calculations. Muffin-tin radii of 1.058 and 0.767Å for Mg and O, respectively, are used with a basis set cut-off R M T |G + k| max = 7, and the lattice constant is set to the PBEsol value of 4.21Å. The energy threshold to include the local field effects in the excited properties, |G + q| max , is set to 4.5 a.u. −1 for the optical and O K-edge, and 1.5 a.u. −1 for the Mg K-edge absorption spectra. The exchange-correlation functional PBEsol is employed for the Kohn-Sham (KS) states and a total of 192 unoccupied bands are considered in the ground state and G 0 W 0 calculation for the optical and O and Mg K-edge x-ray absorption spectra. For the optical spectrum in the BSE calculation, four occupied and five unoccupied bands are considered, while eight unoccupied bands were taken into account for the XAS spectra. A Lorentzian broadening with a width of 0.55 eV is applied to the spectra to mimic the excitation lifetime. The atomic structures and isosurfaces are visualized with the VESTA software [38] and the band structure is calculated with the Wannier90 [39] package in VASP.
III. RESULTS
A. Electronic properties
We start our analysis by comparing the electronic properties obtained from DFT calculations with three Table I presents the band gap calculated with VASP. With PBE96 (4.49 eV) and PBEsol (4.58 eV), the band gaps are considerably underestimated, consistent with previous calculations [10][11][12]. On the other hand, HSE06 renders a band gap of 6.58 eV closest but still below the experimental value of 7.7 and 7.83 eV [4,5]. The G 0 W 0 band gap obtained with PBEsol (7.52 eV) is closest to experiment, whereas a somewhat lower value (7.26 eV) is obtained with PBE96 which is in agreement with the value of 7.25 eV from Ref. [12]. The latter study [12] has also addressed the effect of selfconsistent quasiparticle correction cycle on the optical properties: self-consistency in G while keeping W 0 constant (GW 0 ) increased the band gap to 7.72 eV, while fully self-consistent GW led to an overestimated band gap of 8.47 eV and was attributed to the missing vertex corrections in the self-consistency cycle. While the size of the band gap may be reproduced by considering (partial) self-consistency in GW , our results show that the inclusion of excitonic effects (see Sections III B and III C) is essential in order to describe the relevant features and the shape of the spectrum. We note that the G 0 W 0 band gap with the hybrid HSE06 functional is also overcorrected (8.53 eV). This is consistent with previous findings that the effects of the starting exchange-correlation functional are large at the independent-particle level, but the differences are reduced when considering quasiparticle [40] and eventually excitonic effects [41].
Since PBEsol and HSE06 provide better electronic properties as compared to PBE96 we continue the analysis with those. In Fig. 1a In the range of 4.5 -11 eV beyond the CB minimum, 3s, 3p, and Mg 3s states prevail, whereas above 11 eV O 3p states become predominant, followed by Mg 3p and 3d states above 15 eV (cf. Fig. 1d and Figs. 2e, f). We will further analyze the ion-and orbital projections in the band structure in Section III B 3 and Section III C to correlate the contributions with the optical and XAS spectra.
B. Optical properties
Starting from the electronic structure presented in the last section, we determine the optical spectrum including also many-body effects. We discuss the effect of approximations to the exchange-correlation functional, namely PBEsol and HSE06, on the spectra and the role of inclusion of G 0 W 0 and excitonic corrections with BSE. In ad- dition, the interband transitions responsible for the spectral features are analyzed in reciprocal space.
Optical spectrum within IP approximation and inclusion of G0W0 corrections
The calculated optical spectra are plotted in Fig. 3 together with the experimental ones [4,23]. The imaginary part of the experimental dielectric function shows four prominent peaks (marked in Fig. 3b): the first two at ∼7.7 eV and 10.7 eV are of nearly equal intensity, the third and fourth peak are at 13.32 and 16.9 eV, respectively, with a smaller intensity of the latter. We start our analysis by considering the results from the independent particle (IP) approximation using the KS eigenvalues calculated with the functionals PBEsol and HSE06. The imaginary part of the dielectric function has its onset at 4.58 and 6.58 eV for PBEsol and HSE06, respectively, below the experimental spectrum, due to the underestimation of the band gap (cf . Table I). Moreover, prominent peaks in the imaginary part of the spectrum are observed at ∼8.5, 11, and 15.5 eV for PBEsol and at around 11, 13, and 17.5 eV for HSE06, corresponding to pronounced band transitions that coincide with points of inflection in the real part of the spectrum.
Inclusion of many-body effects within the G 0 W 0 approximation results in a blue shift by ∼3 eV (PBEsol) and 2 eV (HSE06) compared to the IP spectra, due to the increased band gaps within G 0 W 0 . This strong effect is attributed to the weak dielectric screening in MgO [13]. In Figs. 3b, d sharper features emerge in 2 with peaks at ∼11.8, 14, and 19 eV (PBEsol), that are shifted to higher energies at ∼ 12.5, 15, and 20.5 eV (HSE06). The real part of the spectrum in Figs. 3a, c exhibits weaker/smoothened features compared to experiment [4] (Fig. 3b, d). The macroscopic static electronic dielectric constant, ∞ =Re (ω = 0) obtained with PBEsol and HSE06 is presented in Table II. Within the IP approximation, ∞ is overestimated for PBEsol (3.29) compared to the experimental value 2.94 [4], similar to previous results with GGA-PW91 (3.16) [17]. We also included the local field effects in the IP calculation and find that the dielectric constant decreases from 3.29 to 3.04 (PBEsol) and 2.76 to 2.57 (HSE06). A similar trend was observed in the work of Gajdoš et al. [42] for semiconductors as Si, SiC, AlP, GaAs, and for insulator as diamond (C). On the other hand, with the hybrid functional, ∞ is underestimated (2.76). Upon including quasiparticle effects (G 0 W 0 ), the values are substantially reduced to 2.78 (PBEsol) and 2.53 (HSE06).
Optical spectrum with excitonic corrections
Additional to the quasiparticle corrections, we consider the effects arising from electron-hole interaction by solving the Bethe-Salpeter equation. The calculations are performed with four occupied and five unoccupied bands which are sufficient to evaluate the optical spectrum up to 30 eV.
The inclusion of excitonic effects leads to a redistribution of the spectral weight to lower energies w.r.t. the G 0 W 0 spectrum and the emergence of a sharp peak at the absorption onset. With PBEsol as the starting point in Fig. 3b, the agreement with experiment w.r.t. spectral shape is improved, but the onset of the imaginary part of the dielectric function is ∼0.7 eV lower than experiment [4,23]. The prominent peaks are at ∼7.0 eV, 10 eV, 12.4 eV, and 16 eV. In the real part of the spectrum, the sign reversal at 12.8 eV indicates a plasmonic resonance.
On the other hand, using HSE06 as the starting func-tional, the real and imaginary part of the spectrum are in excellent agreement with experiment. The peak positions of the distinct features and the plasmonic resonance at 13.4 eV coincide with the experimental ones [23]. The four peaks of 2 at ∼8, 10.5, 13, and 17 eV are largely aligned with experiment, as shown in Figs. 3c and d. Further analysis of the origin of the peaks in reciprocal space and the real-space projection of the first exciton are provided in Section III B 3 and Section III D, respectively. The improved description w.r.t. the energetic positions and, to a lesser extent, intensity of the peaks can be attributed to the description of the ground state with a hybrid functional HSE06, the larger number of unoccupied bands considered in the BSE calculation and to performing BSE on top of G 0 W 0 . Furthermore, the agreement to experiment concerning the macroscopic static electronic dielectric constant ∞ is improved after BSE to 3.08 (PBEsol) and 2.81 (HSE06), respectively (cf . Table II), also consistent with a previous value of 3.12 [17], where excitonic corrections were included using the KS eigenenergies and a scissor shift approach. We note that increasing the number of unoccupied bands to 9, leads to slightly higher values for ∞ 3.16 (PBEsol) and 2.90 (HSE06), the latter being in excellent agreement with experiment. In particular, more empty states are necessary for the calculation of the real part of the dielectric function from the Kramers-Kronig relation. Due to the high computational cost and an enhanced memory demand with more unoccupied bands, we proceed with five unoccupied bands for the further analysis. Furthermore, for the binding energy of the first exciton we obtain 442 meV with PBEsol and 596 meV with HSE06. A similar value of 429 meV was obtained by Fuchs et al. [11] employing the KS eigenenergies (GGA) with a scissor shift of 2.98 eV and subsequently including excitonic corrections. The overestimation of the binding energy w.r.t experiment (80 meV [4]) may be attributed to the fact that the ionic contributions to the static screening is not considered [43,44].
Analysis of spectral features in reciprocal space
In order to identify the origin of the most prominent peaks, we have performed calculations with the allelectron exciting code. The real and imaginary part of the dielectric function for the G 0 W 0 +BSE with PBEsol as the DFT functional and similar parameters (four occupied and five unoccupied bands) are plotted in Figs. 4a, b, and show good agreement with experiment as well as the VASP result w.r.t. the energetic positions of the peaks (a comparison of the spectra obtained with the two codes is provided in Appendix A and Fig. 8). The PBEsol band gap is 4.60 eV at the KS level and increases to 7.25 eV after quasiparticle corrections with G 0 W 0 are included. The most prominent peaks are marked in Fig. 4b and the corresponding e-h contributions are studied in Figs. 4c-f. We recall that the Bethe-Salpeter equation represents an eigenvalue problem for an effective two-particle Hamiltonian [37,45]:
v c k H vck,v c k A λ v c k = E λ A λ vck .(1)
where E λ are the transition energies and A λ vck are the corresponding states in terms of vk → ck transitions. The e-h coupling coefficients for a particular transition displayed as circles in Figs. 4c-f are calculated from the BSE eigenvector A λ as:
w λ ck = c | A λ vck | 2 , w λ vk = v | A λ vck | 2 .(2)
The first exciton at 6.82 eV has a binding energy of 435 meV, close to the value obtained with VASP, as discussed in the previous section. This bound exciton contributes to the shoulder at the onset of the spectrum. The interband transitions responsible for the exciton and its real-space distribution are discussed in detail in Section III D. In Fig. 4 a, the green line marks the fundamental band gap, below which the bound excitons lie.
The first peak at 7.3 eV (cf. Fig. 4c) arises due to transitions from the top of the valence band (VB) to the bottom of the conduction band (CB) around the Γpoint in reciprocal space. A comparison with the site and orbital-projected DOS (Fig. 1) and band structure (Fig. 2) reveals a mixed O 3s, 3p, and Mg 3s character. The second peak at 9.4 eV involves interband transitions from the topmost VB to the lowest CB along L − Γ − X and Γ − K. The CB is more dispersive along L − Γ and has mixed O 3s and 3p character with Mg 3p contributions near L. The next peak at 10.4 eV stems from transitions to the CB from deeper-lying valence bands along L − Γ − X and Γ − K. The final peak at 12.2 eV, plotted in Fig. 4f, results from transitions from the top of the VB to the higher-lying CB around X as well as along K − Γ. In this energy range, the CB consists of O 3p and Mg 3s and 3d xy ,d xz states along Γ − X and K − Γ.
The influence of lattice screening on the optical spectrum is a topic of current research and so far there is no established framework to treat the exciton-phonon coupling that renormalizes the absorption spectra. Such effects should be assessed in future work.
C. X-ray absorption spectra
We now turn to the x-ray absorption spectra of the O and Mg K-edge of bulk MgO calculated with the exciting code. The ground state calculations were performed with the PBEsol exchange-correlation functional and the quasiparticle corrections were included with the single-step G 0 W 0 . Finally, the excitonic corrections were accounted for by solving BSE.
O K-edge
The theoretical XAS spectrum of the O K-edge is plotted in Fig. 5d together with the experimental spectrum from Luches et al. [25], who performed x-ray absorption measurements on MgO films of varying thickness grown epitaxially on Ag(001) as well as on polycrystalline bulk samples. The G 0 W 0 +BSE spectrum is characterized by six prominent peaks with high oscillator strength (cf. Fig. 5a). Their origin in terms of transitions to the conduction bands is visualized in Fig. 5b-g. We find that for the BSE calculation, a total of eight unoccupied bands are sufficient to obtain agreement with experiment and converge the oscillator strengths in the energy interval up to 30 eV.
While the spectrum within the independent quasiparticle aprroximation (IQPA) obtained after the G 0 W 0 correction captures the four peak-feature, a very good agreement to experiment concerning the spectral shape and the relative positions of the three prominent peaks at ∼ 537, 546, and 557 eV is obtained only after the G 0 W 0 +BSE corrections. The spectrum is also consistent with earlier work of Rehr et al. [20] using FSR and BSE. The reduced intensity of the third peak in the G 0 W 0 +BSE spectrum can be attributed to the limited number of unoccupied bands considered in the calculation. Typically, the GW approximation is not widely explored for core-level states, only recently there are first promising reports for its application to molecular 1s lev- els [46]. In our calculations we do not correct the core energies obtained from DFT. Thus we shift the BSE spectrum to align with the experiment, as done in earlier works [22,47]. For the O K-edge a shift of 34.4 eV is applied to the G 0 W 0 +BSE spectrum to align the first peak with experiment. The same shift is also applied to the IQPA spectrum.
The green line in Fig. 5 a marks the direct band gap and the states below it correspond to bound excitons.The first bound exciton at the O K-edge is found at 534.5 eV with a binding energy of 690 meV, its real-space distribution is discussed in Section III D. This value is comparable to previous results for other oxides, e.g. 0.5 eV was reported for β-Ga 2 O 3 [48], and 285 meV, 345 meV, and 323 meV for the α-, β-, and ε phase of Ga 2 O 3 [49], respectively. Six prominent features in the XAS spectrum with high oscillator strength are marked and their origin is analyzed further in Figs. 5b-g. Transitions at the onset of the spectrum at 535.3 eV are localized around Γ at the bottom of the CB (Fig. 5b) and comprise predominantly O 3p character hybridized with O 3s (cf. Figs. 2a-c), and Mg 3s character (cf. Fig. 2d). The second peak at 536.9 eV also arises from transitions to the lowest CB, but is more dispersive along L − Γ − X and K − Γ. The subsequent peak at 537.7 eV stems from transitions to the lowest CB, but is localized midway along the L − Γ with some contribution along K − Γ and has hybridized O 3s, 3p, and Mg 3s character. Furthermore, the peak at 539.5 eV results from transitions to the second lowest unoccupied band localized at X and dispersive along K − Γ with Mg 3d xz (cf. Fig. 2f) as well as O 3p character. Transitions to higher unoccupied bands around W and Γ with mixed O 3p and 3d and Mg 3p, t 2g character result in a peak at 546.5 eV. The final peak at 557.2 eV arises from transitions to CB at energies above 25 eV with O 3p and Mg 3p, e g contributions along X − W − K − Γ. Fig. 6a displays the Mg K-edge from the G 0 W 0 +BSE calculation and the experimental spectrum from Luches et al. [25]. The experimental spectrum has four prominent peaks at 1308.3, 1314.4, and 1316.2 eV, followed by a broader peak at 1326.6 eV, with a noticeable difference in peak intensities for normal and grazing incidence of the MgO film on Ag(001) and for the polycrystalline sample. While the IQPA spectrum for the O K-edge shows overall agreement with the G 0 W 0 +BSE result, for the Mg K-edge the IQPA spectrum fails to describe the general features of the experimental spectrum. On the other hand, including the core-hole -electron interaction leads to a large redistribution of the spectral weight, accompanied by the emergence of a high intensity peak at the onset of the BSE spectrum. Overall, the G 0 W 0 +BSE Mg K-edge is in very good agreement with the experimental spectrum of [25] and with previous BSE and FSR calculations [20] concerning peak positions and relative intensity. Similar to the O K-edge, we applied a shift of 58.8 eV to the G 0 W 0 +BSE spectrum to align it to the first peak in the experimental spectrum and the same shift is applied to the IQPA spectrum.
Mg K-edge
The green line in Fig. 6a denotes the presence of bound excitons below this energy, the first bound exciton being at 1305.81 eV with a binding energy of 760 meV. Ten prominent peaks with high oscillator strength are marked in the spectrum which are analyzed further in Figs. 6bk. The first peak at 1307.4 eV arises from transitions to the bottom of CB with Mg 3p character, hybridized with Mg 3s states (cf. Figs. 2a,b). The second peak at 1308.2 eV comprises transitions along the L − Γ − X with Mg 3p character and along K − Γ with mixed Mg 3s and 3p character. The third peak at 1310.1 eV includes transitions to the lowest CB concentrated halfway between Γ − X with hybridized Mg 3s and O 3s and 3p character (cf. Figs. 2a,b). Moreover, the peaks at 1313.8, 1315.6, and 1316.7 eV arise from transitions to higher energy CB (>10 eV) and are dispersive along the whole k-path with hybridized O 3p and Mg 3s and 3p as well as Mg 3d xz character (cf. Figs. 2d-f). The peaks at 1323.6, 1325.7, 1327.6, and 1329.5 eV stem from the transitions to the unoccupied bands with E >20 eV predominantly along X − W − K with prevailing hybridized Mg 3p and e g character.
D. Real-space projection of the first exciton
The real-space wavefunction of the excited electron for a given exciton can be obtained from the BSE eigenvectors A λ as:
Ψ λ (r h , r e ) = vck A λ vck ψ * vk (r h )ψ ck (r e ).(3)
For more details see Ref. [37,47] and references therein.
For the analysis we fixed the hole slightly off the oxygen The first bound exciton of the optical spectrum (Fig. 7a) consists of transition from the valence band maximum (VBM) to the conduction band minimum (CBM) that are strongly localized around Γ. Since the excited electron is distributed solely over the lowest, highly dispersive conduction band, the bound exciton was previously described in the Wannier-Mott two-band model by Fuchs et al. [11]. In Fig. 7c we display a cut along the (561) plane through the center of the spread of the wave function near the fixed position of the hole, that shows that the exciton is delocalized over several unit cells which supports the Wannier-Mott character. Moreover the intensity of the spread has a maximum at the oxygen sites and is weaker at the Mg sites. The reciprocal space projection in conjunction with the orbitally projected band structure (cf. Fig. 2) shows a main contribution of hybridized O 3s and Mg 3s states at the CBM.
For comparison, we have also analyzed the real space projection of the first exciton in the O K-edge XAS spectrum. As shown in Fig. 7b this exciton involves transitions to the CBM, but is more dispersive in reciprocal space along L − Γ − X and K − Γ. This goes hand in hand with a stronger localization in real space, visible from the real space projection in Fig. 7d along the (516) plane that exhibits a significant decrease in the spread of the wavefunction. Compared to the exciton of the optical spectrum, here the spread is confined to two to three unit cells only. The 2D cut through the center of the wavefunction spread also illustrates the orbital contributions with s and p character near the O sites, whereas the contributions around the Mg sites have s-like character. This can be attributed to the strong hybridization of the O 3s, 3p, and Mg 3s states around the CBM, discussed above.
IV. SUMMARY
We have provided a comprehensive study of the optical and x-ray absorption spectra of bulk MgO with the VASP and exciting codes. The results indicate that the quasiparticle, and in particular excitonic effects are crucial to describe the spectra, concerning peak positions and to a lesser extent intensity.
For the optical spectrum, the effect of two different functionals (GGA-PBEsol and the hybrid HSE06) are studied: an excellent agreement with the experiment is obtained with HSE06 w.r.t. the energetic positions of the peaks. Analysis of the electron-hole coupling coefficients in reciprocal space allows us to identify the valence to conduction band transitions contributing to the peaks in the spectrum. In particular, the peak at 7.3 eV arises due to transitions localized around Γ from the top of the VB to the bottom of the CB with mixed O 3s, 3p, and Mg 3s character, followed by a peak at 9.4 eV stemming from similar interband transitions but along L − Γ − X and Γ − K with mixed O 3s, 3p, and Mg 3p character near L. The third peak at 10.4 eV is from transitions to the bottom of CB from deeper lying valence bands and the final peak at 12.2 eV results from a transition to higher lying conduction bands with hybridized O 3p and Mg 3s and 3d xy ,d xz character.
The inclusion of core-hole electron interaction by solving the BSE is found to be essential also for the XAS Mg and O K-edge. By visualizing the transitions to the unoccupied bands in reciprocal space, we determine the origin of the relevant peaks in the spectra. In the O Kedge spectrum, the peak at ∼ 537 eV originates from the transitions to unoccupied bands with hybridized O 3s, 3p, and Mg 3s character, the peak at 546 eV stems from O 3p, 3d hybridized with Mg 3p and t 2g states, and the peak at 557 eV emerges from transitions to the CB with hybridized O 3p and Mg 3p and 3d character. The real space projection of the electronic part of the wavefunction of the first exciton in the optical spectrum shows it has a delocalized Wannier-Mott character, consistent with previous studies in reciprocal space [11]. On the other hand, the wavefunction of the first exciton in the O K-edge spectrum is stronger localized and spreads up to only three unit cells. We believe that our detailed analysis of the optical and x-ray excitations in this paradigmatic oxide material regarding their orbital character and extension in real and reciprocal space based on state-of-the-art many-body approaches serves as an important benchmark and provides useful background information for the interpretation of experimental data both from static but also time-dependent investigations. in the energy range of 7 − 11 eV we even observe a better agreement with experiment for mBSE when compared with full BSE associated with the increase in k-point density. For energies higher than 11 eV, the positions of the peaks are also well reproduced with the mBSE approach with slight increase in the intensity when compared with the full BSE spectrum. The good agreement between the mBSE and full BSE spectra is attributed to the fact that the electronic structure does not change significantly from DFT to G 0 W 0 (cf. Fig. 1), the prevailing effect being a rigid shift of the conduction band. The binding energy of the first exciton from mBSE is 610.8 meV.
FIG. 1 .
1(a) Kohn-Sham and G0W 0 band structure and (b-d) total and projected density of states (PDOS) of MgO calculated with PBEsol within VASP. different functionals, namely PBE96, PBEsol and HSE06.
the Kohn-Sham and G 0 W 0 band structure with the PBEsol functional is plotted along high-symmetry points, showing a direct (Γ − Γ) band gap. The inclusion of quasiparticle effects in the G 0 W 0 calculation leads to a nearly rigid shift of the unoccupied Kohn-Sham bands to higher energies. The top of the valence band (VB) consists mainly of O 2p states (cf. the projected density of states in Figs. 1b-d) with low dispersion along the L−Γ−K direction, whereas the lower bands are more dispersive. Further insight into the orbital-resolved contributions of O and Mg on the band structure is provided in Fig. 2. The bottom of the conduction band (CB) comprises hybridized O 3s, 3p, and Mg 3s states that are highly dispersive along the L−Γ−X and K − Γ directions (cf. Figs. 1c, d and Figs. 2a, b, and d).
FIG. 2 .
2Oxygen (a-c) and Mg (d-f) orbital-resolved contributions projected on the ground state band structure within VASP.
FIG. 3 .
3Optical spectrum of bulk MgO obtained with VASP: (a), (c) real part and (b), (d) imaginary part of the dielectric function for PBEsol and HSE06 as the starting functional, respectively. A Lorentzian broadening of 0.3 eV is employed for all the calculated spectra. The IP, IP+G0W0, and G0W0+BSE results are shown by brown solid, green dash-dotted, and red solid lines, respectively. Additionally, the experimental data from Exp. 1 [4] (black solid line), and Exp. 2 [23] (black dashed line) are displayed.
FIG. 4 .
4Optical spectrum with PBEsol including many-body corrections calculated with the exciting code: (a) real and (b) imaginary part of the dielectric function. A Lorentzian broadening 0.3 eV is employed for the calculated spectrum for the G0W0+BSE corrections (red line) and the red vertical bars represent the oscillator strength (arb. units). The direct band gap at 7.26 eV is marked by a vertical green line. Experimental spectra from Roelssler et al. [4] (black solid line) and Bortz et al. [23] (black dashed line) are shown for comparison. (c−f) electron-hole coupling coefficients represented as circles in reciprocal space for the peaks at different energies marked in (b), where the size of the circle is proportional to the magnitude of the e-h contribution.
FIG. 5 .
5XAS spectrum of the O K-edge using G0W0+BSE calculated with the exciting code: (a) calculated absorption spectra with G0W0+BSE (red line) and within the independent quasiparticle approximation (IQPA, brown shaded area) are compared with experimental spectra from Luches et al. [25] (black line). A shift of 34.4 eV was applied to the calculated spectra to align to the first peak of the experiment and a Lorentzian broadening of 0.55 eV is adopted to mimic the excitation lifetime. The green line at 535.2 eV marks the direct band gap. The red vertical bars represent the oscillator strength (arb. units). (b-g) excitonic contributions to the final states in the CB of the peaks marked in (a).
FIG. 6 .
6XAS spectrum of Mg K-edge including G0W0+BSE corrections calculated with the exciting code: (a) calculated absorption spectra with G0W0+BSE (red line)) and within IQPA (brown shaded area) are compared with experimental spectra from Luches et al. [25] on a MgO film on Ag(001), grazing/normal incidence of photon beam (black dashed-dotted/solid line); polycrystalline MgO (black dashed line). A shift of 58.8 eV was applied to the calculated spectra to achieve coincidence with the first peak of the experiment and a Lorentzian broadening of 0.55 eV is adopted for the theoretical curve to mimic the excitation lifetime. The vertical green line at 1306.6 eV marks the direct band gap. The red vertical bars represent the oscillator strength (arb. units). (b-k) excitonic contributions to the final states in the CB of the peaks marked in (a).position (0.52,0.52,0.52) and plotted the electronic part of the wavefunction in real-space for the first exciton of the optical and the O K-edge XAS spectrum inFig. 7.
FIG. 7 .
7Analysis of the first exciton in the optical spectrum (a) and O K-edge XAS (b) in reciprocal space. The lower panels show the density associated with the electronic part of the excitonic wavefunctions for a selected cross section in real space: In (c) along (561) for the optical excitation shown in (a) and (d) along (516) for the O K-edge XAS in (b). The color code visualizes the spacial extension of the wave functions: Blue colors refer to vanishing or low densities, orange to red colors to elevated densities. The hole is fixed near the oxygen (fractional coordinate: 0.52, 0.52, 0.52) and is marked by a white cross.
FIG. 9 .
9Comparison of the spectrum obtained with mBSE on top of DFT (k-mesh of 21×21×21) and with G0W0+BSE corrections (k-mesh of 15×15×15). The spectra were calculated with HSE06 as the starting functional and compared with the experiments of Roessler et al. [4] (black solid line) and Bortz et al. [23] (black dashed line). A Lorentzian broadening of 0.3 eV is employed for the theory spectra.
TABLE I .
IComparison of the fundamental band gap from the DFT and the G0W0 calculation with different starting functionals.Exc
DFT
G0W0
Experiment
Eg (eV)
(Γ − Γ)
PBE96
4.49
7.26
7.7 a ,
7.83 b
PBEsol
4.58
7.52
HSE06
6.58
8.53
a Reference 4
b Reference 5
TABLE II .
IIComparison of the macroscopic static electronic dielectric constant ∞ in the IP approximation and after G0W0 and BSE with different DFT functionals.Exc
IP
G0W0
BSE
Experiment
PBEsol
3.29
2.78
3.08
2.94[4]
HSE06
2.76
2.53
2.81
ACKNOWLEDGMENTSWe thank Caterina CocchiAppendix A: Comparison of the optical spectrum calculated with exciting and VASPThe optical spectra obtained with exciting and VASP with the same exchange-correlation functional (PBEsol) inFig. 8show very good agreement in the overall shape and peak positions with some smaller differences in peak heights. Fuchs et al.[11]have pointed out the importance of using a dense k-mesh for BSE in order to obtain convergence of the optical spectrum in the lower energy range. However, this goes hand in hand with a high computation cost in the G 0 W 0 calculation. One way to circumvent this is to perform a model BSE calculation directly using the DFT wave functions and considering only the required number of bands which cover the energy range of interest[11]. The omission of the G 0 W 0 step allows us to use a higher k-mesh and thus improve the convergence. Here, we discuss the result obtained by using a model for the static screening with parameters fitted to the screened Coulomb kernel diagonal values obtained from G 0 W 0 calculation, a detailed description can be found in Ref.[11,24,50,51]and was previously used in Ref.[24,50]. Good agreement between mBSE and the full BSE (G 0 W 0 +BSE) was recently obtained for bulk SrTiO 3[41].In our study, mBSE is performed on a 21×21×21 kmesh with a range separation parameter λ = 1.44Å −1 , ion-clamped static dielectric function −1 ∞ = 0.38974, and a scissor shift of 1.95 eV, corresponding to the difference between the KS energies obtained with HSE06 and the quasiparticle band gap. InFig. 9the spectrum from mBSE and full BSE obtained with HSE06 as the starting functional on a k-mesh of 15×15×15 are displayed. Overall, very good agreement is obtained. In particular,
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. J P Perdew, A Ruzsinszky, G I Csonka, O A Vydrov, G E Scuseria, L A Constantin, X Zhou, K Burke, Phys. Rev. Lett. 100136406J. P. Perdew, A. Ruzsinszky, G. I. Csonka, O. A. Vydrov, G. E. Scuseria, L. A. Constantin, X. Zhou, and K. Burke, Phys. Rev. Lett. 100, 136406 (2008).
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C W Vorwerk, Theoretical Spectroscopy of Ga2O3. Humboldt-Universität zu Berlin, Mathematisch-Naturwissenschaftliche FakultätPh.D. thesisC. W. Vorwerk, Theoretical Spectroscopy of Ga2O3, Ph.D. thesis, Humboldt-Universität zu Berlin, Mathematisch-Naturwissenschaftliche Fakultät (2021).
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|
[] |
[
"Diffractive vector mesons at large momentum transfer from the BFKL equation *",
"Diffractive vector mesons at large momentum transfer from the BFKL equation *"
] |
[
"R Enberg \nHigh Energy Physics\nUppsala University\nSweden\n",
"L Motyka \nHigh Energy Physics\nUppsala University\nSweden\n\nInstitute of Physics\nJagellonian University\nKrakówPoland\n",
"G Poludniowski \nDepartment of Physics and Astronomy\nUniversity of Manchester\nUK\n"
] |
[
"High Energy Physics\nUppsala University\nSweden",
"High Energy Physics\nUppsala University\nSweden",
"Institute of Physics\nJagellonian University\nKrakówPoland",
"Department of Physics and Astronomy\nUniversity of Manchester\nUK"
] |
[
"RE at X International Workshop on Deep Inelastic Scattering (DIS2002) Cracow"
] |
Diffractive vector meson photoproduction accompanied by proton dissociation is studied for large momentum transfer. The process is described by the non-forward BFKL equation, for which an analytical solution is found for all conformal spins, giving the scattering amplitude. Results are compared to HERA data on ρ production.
| null |
[
"https://export.arxiv.org/pdf/hep-ph/0207034v1.pdf"
] | 5,425,802 |
hep-ph/0207034
|
2b883ea7b2d0890e38a21220cb7179d1cd273180
|
Diffractive vector mesons at large momentum transfer from the BFKL equation *
May 2002
R Enberg
High Energy Physics
Uppsala University
Sweden
L Motyka
High Energy Physics
Uppsala University
Sweden
Institute of Physics
Jagellonian University
KrakówPoland
G Poludniowski
Department of Physics and Astronomy
University of Manchester
UK
Diffractive vector mesons at large momentum transfer from the BFKL equation *
RE at X International Workshop on Deep Inelastic Scattering (DIS2002) Cracow
1May 2002
Diffractive vector meson photoproduction accompanied by proton dissociation is studied for large momentum transfer. The process is described by the non-forward BFKL equation, for which an analytical solution is found for all conformal spins, giving the scattering amplitude. Results are compared to HERA data on ρ production.
Diffractive production of vector mesons in γp collisions at large momentum transfer, γp → V X, is an experimentally clean process. The signal consists of an isolated vector meson with large transverse momentum, separated from the remnant of the incoming proton by a large rapidity gap. There are recent measurements of cross-sections and helicity amplitudes for this process [1,2].
The large momentum transfer involved makes it possible to describe the colour singlet exchange in terms of perturbative QCD. This is in contrast to vector meson production in diffractive processes with small momentum transfer, where the sensitivity to the infrared region is larger. The perturbative QCD description of hard colour singlet exchange across a large rapidity interval relies on the BFKL equation [3,4], which resums leading powers of the rapidity y to all orders in the perturbative expansion of the amplitude. The colour singlet system, or pomeron, is here a composite system of two reggeized gluons.
Let us mention that this process has been studied before; for heavy mesons in BFKL [5] using the Mueller-Tang [6] approximation, and for light mesons at the Born level [7]. The data on the cross-sections can be fitted with a BFKL calculation, but not by the fixed order formulae. The understanding of the helicity structure remains a challenge.
We have recently studied the production of heavy mesons in a BFKL framework [8], so here we concentrate on the case of light mesons.
At large momentum transfer the pomeron couples predominantly to a single parton, (see fig. 1), which means that the cross-section may be factorized into a convolution of the partonic cross-section with the parton distribution functions. We therefore calculate the amplitude for γq → V q (since γg → V g differs only by a colour factor).
In the BFKL framework the scattering amplitude is calculated as the convolution of three factors; schematically A = Φ γ→V ⊗ K BFKL ⊗ Φ q→q , where Φ are the impact factors describing the coupling of the pomeron to the indicated vertices, and K BFKL is the BFKL kernel describing the evolution of the gluon ladder.
The non-forward BFKL equation has a solution due to Lipatov [4]:
A = 1 (2π) 6 n dν ν 2 + n 2 4 [ν 2 + ( n−1 2 ) 2 ][ν 2 + ( n+1 2 ) 2 ] e ωn(ν)y I 1 n,ν (k, q) I 2 ⋆ n,ν (k ′ , q).
(1) This represents an expansion of the amplitude in the complete basis of eigenfunctions E n,v of the BFKL kernel. The functions I 1,2 n,ν are projections of the impact factors Φ 1,2 (k, q) on these eigenfunctions, see [4,8] for details.
The integer n in (1) is known as the conformal spin. The terms in the sum with non-zero n are exponentially suppressed by the factor e ωn(ν)y , with ω n (ν) < 0 for n = 0, and so the amplitude is usually approximated by the leading n = 0 term (the Mueller-Tang approximation [6]). This approximation, however, is only good for very large rapidities y. For moderate y the higher n terms can still be important, as was found in [9,10]. We therefore calculate the amplitudes including all n.
The quark impact factor is given in ref. [9] for all conformal spins, and we have to compute the vector meson impact factors. This is done separately for heavy [8] and light [11] mesons, using different approximations for the vector meson wave functions. In the heavy meson case, we used the nonrelativistic approximation, where the constituent quarks are assumed to each carry half of the meson momentum. Our results can be found in [8].
Ivanov et al. [7] give the helicity amplitudes for light meson production, M ++ , M +0 and M +− , where the first index corresponds to the polarization of the incoming photon and the second to the vector meson. These are referred to as the no-flip, single-flip and double-flip amplitudes, respectively. Their calculation assumes Born-level two-gluon exchange, and they use a relativistic approximation for the vector meson wave functions, with massless quarks. Note that the longitudinal and transverse degrees of freedom are factorized. For instance, taking r to be the transverse separation of the quarks in the vector meson, and u to be the lightcone momentum fraction of the quark, they give the single-flip amplitude as
M +0 = d 2 k k 2 (k − q) 2 C α 2 s d 2 r du 4π f dipole r · e + r 2 f ρ 2 (1 − 2u)φ (u) ,(2)
where φ (u) is the twist-2 vector meson wave function. Expressed in the impact factor picture, the factor in brackets is just the product of the impact factors Φ q→q and Φ γ→V +0 . Thus, we calculate I γ→V n,ν by projecting these expressions onto the BFKL eigenfunctions, and insert the result into (1).
The projection of the impact factor I γ→V n,ν is then proportional to [11] I = sin π (1/2 + β + µ) B + (α, q * , ξ * ) B + (β, q, ξ)
− sin π (1/2 + β − µ) B − (α, q * , ξ * ) B − (β, q, ξ)(3)
for even n and zero for odd n. Here µ = n/2 − iν, µ = −n/2 − iν, and we introduce the conformal blocks B ±
B ± (α, q, ξ) = 2i k 3 2 +α iq 4k ±µ Γ(3/2 + α ± µ)
Γ(1 ± µ) × 2 F 1 (3/2 + α ± µ , 1/2 ± µ ; 1 ± 2µ ; 2/(1 + ξ)) , (4) and B ± obtained by µ → µ. The constants α and β take different values for different helicity amplitudes.
We will now go on to evaluate the obtained expressions, concentrating on the case of light mesons and referring to [8] for our results on heavy mesons. ZEUS have measured the differential cross-section dσ/dt for ρ and φ mesons for momentum transfers up to 8 GeV 2 [1]. In fig. 1 we show a comparison of the data together with our calculation, including all conformal spins. Here we have chosen a fixed value of α s = 0.38 in the prefactor (see (2)) and α s = 0.24 in the eigenfunctions ω n (ν), defining the pomeron intercept. These choices of different α s reflect the impact of non-leading corrections to the BFKL intercept. The rapidity is defined as y = ln(ŝ/m 2 ρ ). Note that the turnover at |t| ∼ 2 GeV 2 is due to the too restrictive infrared cut-off u min = −m 2 ρ /t (taken from [7]) in the integration over u. Such a cut-off in u was required to regulate an unphysical divergence.
In addition, ZEUS have also measured the spin density matrix elements r 04 ij for the process. These parametrize the decay angular distributions of the mesons, and can be related to the helicity amplitudes M ij (see e.g. [1]). Note, however, that both the no-flip and double-flip amplitudes processes lead to the same polarization states of the vector meson, and therefore they cannot be distinguished in unpolarized experiment. The r 04 ij contain interferences, though, so it can be inferred from the data that all three helicity amplitudes are non-zero for ρ and φ, while for J/Ψ, only the no-flip amplitude seems to be non-zero.
In fig. 2, we show our BFKL predictions for r 04 ij compared to the ZEUS data. The qualitative features are well reproduced. The shapes of the curves depend somewhat on the choices of pomeron intercept and definition of the rapidity, but we find that (i ) the (+−) component of the amplitude dominates and (ii ) the (++) component is negative. The BFKL evolution thus changes the features from the fixed-order Born level results of [7].
These results should be interpreted with some care, however, because of some uncertainties. We have not included the chiral-odd components of the photon wave function of [7], but believe that these may be small [11]. Also, the endpoints in the u-integration have to be treated carefully and may change the results. Furthermore, the approximations used, with massless quarks and a factorized meson wave function, have to be understood.
In conclusion, we have studied the process γp → V X in a BFKL framework, obtaining exact solutions of the BFKL equation. Comparing the calculations to ZEUS data shows good agreement with both the differential cross-section and the spin density matrix elements for diffractive ρ production.
We thank Jeff Forshaw for useful discussions. This study was supported in part by the Swedish Natural Science Research Council and by the Polish Committee for Scientific Research (KBN) grant no. 5P03B 14420.
Fig. 1 .
1Feynman graph and differential cross-section for the process γp → ρX.
Fig. 2 .
2Spin density matrix elements for ρ meson production.
. K Klimek, these proceedingsK. Klimek, these proceedings;
. S Chekanov, ZEUS Collaborationhep- ex/0205081Eur. Phys. J. C. in pressS. Chekanov et al. [ZEUS Collaboration], hep- ex/0205081, Eur. Phys. J. C, in press.
. D P Brown, these proceedingsD. P. Brown, these proceedings.
. L N Lipatov, Sov. J. Nucl. Phys. 23338L. N. Lipatov, Sov. J. Nucl. Phys. 23, 338 (1976);
. E A Kuraev, L N Lipatov, V S Fadin, ibid. 45Sov. Phys. JETP. 44199E. A. Kuraev, L. N. Lipatov and V. S. Fadin, Sov. Phys. JETP 44, 443 (1976); ibid. 45, 199 (1977);
. I I Balitsky, L N Lipatov, Sov. J. Nucl. Phys. 28822I. I. Balitsky and L. N. Lipatov, Sov. J. Nucl. Phys. 28, 822 (1978).
. L N Lipatov, Sov. Phys. JETP. 63904L. N. Lipatov, Sov. Phys. JETP 63, 904 (1986);
. Phys. Rep. 286131Phys. Rep. 286, 131 (1997).
. J R Forshaw, M G Ryskin, Z. Phys. 68137J. R. Forshaw and M. G. Ryskin, Z. Phys. C68, 137 (1995);
. J Bartels, J R Forshaw, H Lotter, M Wüsthoff, Phys. Lett. 375301J. Bartels, J. R. Forshaw, H. Lotter and M. Wüsthoff, Phys. Lett. B375, 301 (1996);
. J R Forshaw, G Poludniowski, hep-ph/0107068Eur. Phys. J. C in press. J. R. Forshaw and G. Poludniowski, Eur. Phys. J. C in press, hep-ph/0107068.
. A H Mueller, W.-K Tang, Phys. Lett. 284123A. H. Mueller and W.-K. Tang, Phys. Lett. B284, 123 (1992).
. D Y Ivanov, Phys. Lett. B. 478295ibid. BD. Y. Ivanov et al. Phys. Lett. B 478, 101 (2000); ibid. B 498, 295 (2001).
. R Enberg, L Motyka, G Poludniowski, hep-ph/0207027R. Enberg, L. Motyka and G. Poludniowski, hep-ph/0207027.
. L Motyka, A D Martin, M G Ryskin, Phys. Lett. B. 524107L. Motyka, A. D. Martin and M. G. Ryskin, Phys. Lett. B 524, 107 (2002).
. R Enberg, G Ingelman, L Motyka, Phys. Lett. B. 524273R. Enberg, G. Ingelman and L. Motyka, Phys. Lett. B 524, 273 (2002).
. R Enberg, J Forshaw, L Motyka, G Poludniowski, in preparationR. Enberg, J. Forshaw, L. Motyka and G. Poludniowski, in preparation.
|
[] |
[
"Astrochemistry at work in the L1157-B1 shock: acetaldehyde formation",
"Astrochemistry at work in the L1157-B1 shock: acetaldehyde formation"
] |
[
"C Codella \nINAF-Osservatorio Astrofisico di Arcetri\nL.go E. Fermi 550125FirenzeItaly\n",
"F Fontani \nINAF-Osservatorio Astrofisico di Arcetri\nL.go E. Fermi 550125FirenzeItaly\n",
"C Ceccarelli \nUniv. Grenoble Alpes\nIPAG\nF-38000GrenobleFrance\n\nCNRS\nIPAG\nF-38000GrenobleFrance\n",
"L Podio \nINAF-Osservatorio Astrofisico di Arcetri\nL.go E. Fermi 550125FirenzeItaly\n",
"S Viti \nDepartment of Physics and Astronomy\nUniversity College London\nGower Str eetWC1E 6BTLondonUK\n",
"R Bachiller \nIGN\nObservatorio Astronómico Nacional\nCalle Alfonso XIII\n28004MadridSpain\n",
"M Benedettini \nIstituto di Astrofisica e Planetologia Spaziali\nINAF\nvia Fosso del Cavaliere 10000133RomaItaly\n",
"B Lefloch \nUniv. Grenoble Alpes\nIPAG\nF-38000GrenobleFrance\n\nCNRS\nIPAG\nF-38000GrenobleFrance\n"
] |
[
"INAF-Osservatorio Astrofisico di Arcetri\nL.go E. Fermi 550125FirenzeItaly",
"INAF-Osservatorio Astrofisico di Arcetri\nL.go E. Fermi 550125FirenzeItaly",
"Univ. Grenoble Alpes\nIPAG\nF-38000GrenobleFrance",
"CNRS\nIPAG\nF-38000GrenobleFrance",
"INAF-Osservatorio Astrofisico di Arcetri\nL.go E. Fermi 550125FirenzeItaly",
"Department of Physics and Astronomy\nUniversity College London\nGower Str eetWC1E 6BTLondonUK",
"IGN\nObservatorio Astronómico Nacional\nCalle Alfonso XIII\n28004MadridSpain",
"Istituto di Astrofisica e Planetologia Spaziali\nINAF\nvia Fosso del Cavaliere 10000133RomaItaly",
"Univ. Grenoble Alpes\nIPAG\nF-38000GrenobleFrance",
"CNRS\nIPAG\nF-38000GrenobleFrance"
] |
[
"Mon. Not. R. Astron. Soc"
] |
The formation of complex organic molecules (COMs) in protostellar environments is a hotly debated topic. In particular, the relative importance of the gas phase processes as compared to a direct formation of COMs on the dust grain surfaces is so far unknown. We report here the first high-resolution images of acetaldehyde (CH 3 CHO) emission towards the chemically rich protostellar shock L1157-B1, obtained at 2 mm with the IRAM Plateau de Bure interferometer. Six blueshifted CH 3 CHO lines with E u = 26-35 K have been detected. The acetaldehyde spatial distribution follows the young (∼ 2000 yr) outflow cavity produced by the impact of the jet with the ambient medium, indicating that this COM is closely associated with the region enriched by iced species evaporated from dust mantles and released into the gas phase. A high CH 3 CHO relative abundance, 2-3 × 10 −8 , is inferred, similarly to what found in hotcorinos. Astrochemical modelling indicates that gas phase reactions can produce the observed quantity of acetaldehyde only if a large fraction of carbon, of the order of 0.1%, is locked into iced hydrocarbons.
|
10.1093/mnrasl/slu204
|
[
"https://arxiv.org/pdf/1412.8318v1.pdf"
] | 118,693,234 |
1412.8318
|
a97690181b1d9db8e1e73f7b60f11d88b39b61b2
|
Astrochemistry at work in the L1157-B1 shock: acetaldehyde formation
2011. 30 December 2014
C Codella
INAF-Osservatorio Astrofisico di Arcetri
L.go E. Fermi 550125FirenzeItaly
F Fontani
INAF-Osservatorio Astrofisico di Arcetri
L.go E. Fermi 550125FirenzeItaly
C Ceccarelli
Univ. Grenoble Alpes
IPAG
F-38000GrenobleFrance
CNRS
IPAG
F-38000GrenobleFrance
L Podio
INAF-Osservatorio Astrofisico di Arcetri
L.go E. Fermi 550125FirenzeItaly
S Viti
Department of Physics and Astronomy
University College London
Gower Str eetWC1E 6BTLondonUK
R Bachiller
IGN
Observatorio Astronómico Nacional
Calle Alfonso XIII
28004MadridSpain
M Benedettini
Istituto di Astrofisica e Planetologia Spaziali
INAF
via Fosso del Cavaliere 10000133RomaItaly
B Lefloch
Univ. Grenoble Alpes
IPAG
F-38000GrenobleFrance
CNRS
IPAG
F-38000GrenobleFrance
Astrochemistry at work in the L1157-B1 shock: acetaldehyde formation
Mon. Not. R. Astron. Soc
0002011. 30 December 2014Accepted date. Received date; in original form date(MN L A T E X style file v2.2)Molecular data -Stars: formation -radio lines: ISM -submillimetre: ISM -ISM: molecules
The formation of complex organic molecules (COMs) in protostellar environments is a hotly debated topic. In particular, the relative importance of the gas phase processes as compared to a direct formation of COMs on the dust grain surfaces is so far unknown. We report here the first high-resolution images of acetaldehyde (CH 3 CHO) emission towards the chemically rich protostellar shock L1157-B1, obtained at 2 mm with the IRAM Plateau de Bure interferometer. Six blueshifted CH 3 CHO lines with E u = 26-35 K have been detected. The acetaldehyde spatial distribution follows the young (∼ 2000 yr) outflow cavity produced by the impact of the jet with the ambient medium, indicating that this COM is closely associated with the region enriched by iced species evaporated from dust mantles and released into the gas phase. A high CH 3 CHO relative abundance, 2-3 × 10 −8 , is inferred, similarly to what found in hotcorinos. Astrochemical modelling indicates that gas phase reactions can produce the observed quantity of acetaldehyde only if a large fraction of carbon, of the order of 0.1%, is locked into iced hydrocarbons.
INTRODUCTION
Complex organic molecules (COMs) have a key role among the many molecules so far detected in space: since they follow the same chemical rules of carbon-based chemistry, which terrestrial life is based on, they may give us an insight into the universality of life. Of course, large biotic molecules are not detectable in space, certainly not via (sub)millimeter observations. However, to determine whether pre-biotic molecules may form in space, we first need to understand the basic mechanisms that form smaller COMs. There is an extensive literature on the subject and still much debate on how COMs may form in space (e.g. Herbst & van Dishoeck 2009;Caselli & Ceccarelli 2012;Bergin 2013). Two basic processes are, in principle, possible: COMs may form on the grain surfaces or in gas phase. It is * E-mail: [email protected] possible and even probable that the two processes are both important in different conditions for different molecules.
Acetaldehyde (CH3CHO) has been detected in a large range of interstellar conditions and with different abundances, namely in hot cores (Blake et al. 1986), hot corinos (Cazaux et al. 2003), cold envelopes (Jaber et al. 2014), Galactic Center clouds (Requena-Torres et al. 2006) and prestellar cores (Öberg et al. 2010). Grain surface models predict that CH3CHO is one of the simplest COMs and can be formed either by the combination of two radicals on the grain surface, CH3 and HCO, which become mobile when the grain temperature reaches ∼30 K (Garrods & Herbst 2006), or by irradiation of iced CH4, CO2 and other iced species (Bennett et al. 2005). For the former route, the two radicals are predicted to be formed either because of the photolysis of more complex molecules on the grain mantles or, more simply, because of the partial hydrogenation of simple biatomic molecules on the grain mantles (Taquet et al. 2012). Conversely, gas phase models claim that acetaldehyde is easily formed by the oxidation of hydrocarbons, which are produced by the hydrogenation of carbon chain on the grain mantles (Charnley et al. 1992(Charnley et al. , 2004. Finally, a further possible mechanism involving formation in the very high density gas-phase immediately after ice mantles are sublimated has been proposed by Rawlings et al. (2013).
In general, it is very difficult to distinguish which of these three mechanisms are at work and, consequently, their relative importance. The chemically rich shocked region L1157-B1 offers a unique possibility to test these theories, as it is a place where the dust is not heated by the protostar, but some of the grain mantles are sputtered/injected in the gas phase because of the passage of a shock (see e.g. Fontani et al. 2014). The L1157-mm protostar (d = 250 pc) drives a chemically rich outflow (Bachiller et al. 2001), associated with molecular clumpy cavities (Gueth et al. 1996), created by episodic events in a precessing jet. Located at the apex of the more recent cavity, the bright bow shock called B1 has a kinematical age of 2000 years. This shock spot has been the target of several studies (e.g. the Large Programs Herschel/CHESS 1 (Chemical Herschel Surveys of Star forming regions; Ceccarelli et al. 2010; and IRAM-30m/ASAI 2 (Astrochemical Survey At IRAM). In this Letter we report high spatial resolution observations of acetaldehyde, with the aim to constrain and quantify the contribution of gas phase chemistry to the CH3CHO formation.
OBSERVATIONS
L1157-B1 was observed with the IRAM Plateau de Bure (PdB) 6-element array in April-May 2013 using both the C and D configurations, with 21-176 m baselines, filtering out structures 20 ′′ , and providing an angular resolution of 2. ′′ 5 × 2. ′′ 3 (PA = 90 • ). The primary HPBW is ∼ 37 ′′ . The observed CH3CHO lines (see Table 1) at ∼ 134-136 GHz were detected using the WideX backend which covers a 4 GHz spectral window at a 2 MHz (∼ 4.4 km s −1 ) spectral resolution. The system temperature was 100-200 K in all tracks, and the amount of precipitable water vapor was generally ∼ 5 mm. Calibration was carried out following standard procedures, using GILDAS-CLIC 3 . Calibration was performed on 3C279 (bandpass), 1926+611, and 1928+738 (phase and amplitude). The absolute flux scale was set by observing MWC349 (∼ 1.5 Jy at 134 GHz). The typical rms noise in the 2 MHz channels was ∼ 0.7 mJy beam −1 .
RESULTS: IMAGES AND SPECTRA
Acetaldehyde emission has been clearly (S/N 10) detected towards L1157-B1. Fig. 1 shows the map of the CH3CHO(70,7-60,6) E and A lines integrated emission. In order to verify whether the present CH3CHO image is altered by filtering of large-scale emission, we produced the CH3CHO(70,7-60,6) E+A spectrum summing the emission measured at PdBI in a circle of diameter equal to the half-power beam width (HPBW) of the IRAM-30m telescope (17 ′′ ). We evaluated the missing flux by comparing such emission with the spectrum directly measured with the single-dish (from the ASAI spectral survey, Lefloch et al., in preparation). As already found for HDCO by Fontani et al. (2014), with the PdBI we recover more than 80% of the flux, indicating that both tracers do not have significant extended structures. The spatial distribution reported in Fig. 1 shows that CH3CHO is mainly associated with two regions: (i) the eastern B0-B1 cavity opened by the precessing jet (called 'E-wall', see Fig. 1 in Fontani et al. 2014), and (ii) the archlike structure composed by the B1a-e-f-b clumps identified by CH3CN (called 'arch'). The red, turquoise, and magenta polygons shown in Fig. 1 sketch out these two regions, intersecting at the position of the B1a clump. Note that B1a is in turn located where the precessing jet is expected to impact the cavity producing a dissociative J-shock (traced by high velocity SiO, H2O, [FeII], [OI], and high-J CO emission: e.g. Gueth et al. 1998, Benedettini et al. 2012. Figure 2 shows the CH3CHO line spectrum observed with the 3.6 GHz WideX bandwidth towards the brightest clump, B1a. Up to six lines (Eu = 26-35 K, see Table 1) are detected with a S/N > 3. Using the GILDAS-Weeds package (Maret et al. 2011) and assuming optically thin emission and LTE conditions, we produced the synthetic spectrum (red line in Fig. 2) that best fits the observed one. Note that the CH3CHO lines are blue-shifted, by 2 km s −1 , with respect to the cloud systemic velocity (+2.6 km s −1 : Bachiller & Peréz Gutiérrez 1997), and have linewidths of 8 km s −1 . Similarly, we extracted the CH3CHO line spectrum towards the three B1 zones, 'E-wall', 'arch', and 'head', shown in Fig. 1. Table 2 reports the measured peak velocities, intensities (in TB scale), FWHM linewidths, and integrated intensities, for each of the three zones. Figure 1 compares the CH3CHO distribution with that of HDCO (Fontani et al. 2014), showing an excellent agreement, with weak or no emission at the head of the bow B1 structure (called 'head'). The acetaldehyde emission is concentrated towards the 'E-wall' and 'arch' zones, namely the part of B1 associated with the most recent shocked material, as probed by the HDCO emission. This is further supported by the fact that the brightest acetaldehyde emission comes where also CH3OH, another dust mantle product, and CH3CN, a 6-atoms COM, emission peak (Codella Table 2. Observed parameters (in T B scale) of the CH 3 CHO(7 0,7 -6 0,6 )E and A emission, and acetaldehyde column densities N CH 3 CHO derived in the 3 regions identified in Fig. 1 (E-wall, arch, and head) following Fontani et al. (2014), see Sect. 3. The (range of) excitation temperatures (Tex) used to derive N CH 3 CHO have been assumed equal to the rotation temperatures derived in Codella et al. (2012), Lefloch et al. (2012), and Fontani et al. (2014). The last columns report the X(CH 3 CHO)/X(CH 3 OH) and X(CH 3 CHO)/X(HDCO) abundance ratios using the CH 3 OH and HDCO data (and similar beams) by Benedettini et al. (2013) and Fontani et al. (2014).
CH3CHO ABUNDANCE
Transition T peak a V peak a F W HM a F int a N CH 3 CHO b CH 3 CHO/CH 3 OH CH 3 CHO/HDCO (mK)
(km s −1 ) (km s −1 ) (mK km s −1 ) (10 12 cm −2 ) (10 −2 ) 10 K -70 K 10 K -70 K 10 K -70 K E-wall 7 0,7 -6 0,6 E 30 ( . Finally, the CH3CHO observed emission is also confined in the low-velocity range (F W HM ∼ 8 km s −1 ) of the L1157-B1 outflow, which is dominated by the extended B1 bow-cavity, according to Lefloch et al. (2012) and Busquet et al. (2014). In summary, similarly to HDCO, CH3CHO traces the extended interface between the shock and the ambient gas, which is chemically enriched by the sputtering of the dust mantles.
To derive the column density, we used the LTE populated and optically thin assumption and best fitted the six detected lines of Tables 1-2. Towards the B1a peak, we find NCH 3 CHO = 9 × 10 13 cm −2 , and a rotational temperature of Trot = 15 K, in agreement with the value derived for the molecular cavity from single-dish CO and HDCO measurements (10-70 K; Lefloch et al. 2012, Codella et al. 2012. Assuming rotational temperatures between 10 and 70 K (Table 2) we derived a column density of 5-30 × 10 12 cm −2 in the 'E-wall' and 'arch' regions, and ∼ 2-3 10 12 cm −2 in the 'head'. The size of the regions (at 3σ level) is 9 ′′ ('E-wall'), 7 ′′ ('arch'), and 8 ′′ ('head'). An estimate of the CH3CHO abundance can be derived using the the CO column density ≃ 10 17 cm −2 derived by Lefloch et al. (2012) on a 20 ′′ scale. We derived NCH 3 CHO using the CH3CHO spectrum extracted on the same scale and assuming 10-70 K. We find NCH 3 CHO ∼ 0.9-1.6 × 10 13 cm −2 , which implies a high abundance, X(CH3CHO) ≃ 2-3 × 10 −8 , similar to what has been measured in hot-corinos (≃ 2-6 × 10 −8 , Cazaux et al. 2003), and larger than that measured in prestellar cores (∼ 10 −11 , Vastel et al. 2014) and towards high-mass star forming regions (∼ 10 −11 -10 −9 , Cazaux et al. 2003;Charnley 2004).
GAS PHASE FORMATION OF CH3CHO
The ratio between NCH 3 CHO and the column density of HDCO, i.e. a molecule which in L1157-B1 is predominantly released by grain mantles (Fontani et al. 2014), is higher (even if we consider the uncertainties, see Table 2) in the 'arch' with respect to the 'E-wall' by a factor ∼ 2-8. Assuming the same grain mantle composition and release mecha-nism, this difference suggests that, in the 'arch', a significative fraction of the observed CH3CHO is formed in the gas phase. In the gas phase, the injection from grain mantles of ethane (C2H6) is expected to drive first C2H5 and successively acetaldehyde (e.g. Charnley 2004;Vasyunin & Herbst 2013): the overlap between the HDCO (Fontani et al. 2014) and CH3CHO emitting regions supports this scenario. We can, therefore, use the measured CH3CHO abundance to constrain the quantity of C2H5 that has to be present in the gas phase in order to produce the observed quantity of CH3CHO. To this end, we use the chemical code AS-TROCHEM 4 , a pseudo time dependent model that follows the evolution of a gas cloud with a fixed temperature and density considering a network of chemical reactions in the gas phase. We followed the same 2-steps procedure adopted in Podio et al. (2014) and Mendoza et al. (2014), to first compute the steady-state abundances in the cloud (i.e. T kin =10 K, nH 2 =10 4 cm −3 , ζ=3 10 16 s −1 ); and then we follow the gas evolution over 2000 yr at the shocked conditions (i.e. T kin =70 K and nH 2 =10 5 cm −3 ). To estimate the influence of a possibly larger gas T kin during the passage of the shock, we also run cases with temperatures up to 1000 K. We adopt the OSU 5 chemical network and assume visual extinction of AV = 10 mag and grain size of 0.1 µm. We assume that the abundances of OCS and CO2 are also enhanced by the passage of the shock, namely their abundance in step 2 is X(CO2) = 6 10 −5 and X(OCS) = 6 10 −6 . Similarly, we assume that the abundance of methanol in step 2 is 2 10 −6 , in agreement with the most recent determination in L1157-B1 by Mendoza et al. (2014). Finally, we varied the C2H5 abundance from 2 × 10 −7 to 2 × 10 −5 . As expected, the predicted steady-state abundance of acetaldehyde in the cloud is very low (1.5 10 −15 ). However, once C2H5 is in the gas phase, it rapidly reacts with oxygen forming abundant acetaldehyde on timescale shorter than 100 years (Fig. 3). The CH3CHO abundance reaches the observed value, ≃ 2-3 × 10 −8 , at the shock age (2000 years), for C2H5 ∼ 2-6 × 10 −7 . Note that Figure 1. Chemical differentation in L1157-B1: the maps are centred at: α(J2000) = 20 h 39 m 09. s 5, δ(J2000) = +68 • 01 ′ 10. ′′ 0, i.e. at ∆α = +21. s 7 and ∆δ = -64. ′′ 0 from the driving protostar. Upper panel: CH 3 CHO(7 0,7 -6 0,6 )E+A integrated emission (green colour, black contours) on top of the HDCO(2 1,1 − 1 0,1 ) line (white contours; Fontani et al. 2014). First contour and steps of the CH 3 CHO image correspondes to 3σ (1 mJy beam −1 ). The ellipses show the synthesised HPBW (2. ′′ 5 × 2. ′′ 3, PA = 90 • ). The red, turquoise, and magenta polygons called 'E-wall', 'arch', and 'head' indicate the 3 portions of L1157-B1 selected by Fontani et al. (2014) to investigate H 2 CO deuteration. Bottom panel: CH 3 CN(8 K -7 K ) emission (green colour, black contours; Codella et al. 2009) on top of the CH 3 OH(3 2 -2 K ) emission (white; Benedettini et al. 2013). The HPBWs are: 3. ′′ 4 × 2. ′′ 1 (PA = 10 • ) for CH 3 CN and 3. ′′ 5 × 2. ′′ 3 (PA = 12 • ) for CH 3 OH. The labels indicate the L1157-B1 clumps identified using the CH 3 CN image (Codella et al. 2009). we obtain the same result if the gas temperature is 500 K, and a 30% higher value at 1000 K. Figure 3 shows also that the CH3CHO/CH3OH abundance ratio is expected to drop between 10 3 yr and 10 4 yr. A different age could, therefore, justify the slightly smaller CH3CHO/CH3OH ratio observed towards the 'head' region.
DISCUSSION AND CONCLUSIONS
We have shown that acetaldehyde is abundant, X(CH3CHO) ≃ 2-3 × 10 −8 , in the gas associated with the passage of a shock and enriched by iced species sputtered from grain mantles and injected into the gas phase. The measured acetaldehyde abundance could be consistent with the scenario of oxydation of gaseous hydrocarbons formed in a previous phase and released by the grain mantles. However, the abundance of the C2H5 required to reproduce the measured CH3CHO is very high, ∼ 2-6 × 10 −7 , namely less than 0.6% the elemental gaseous carbon. There are no observations of C2H5, hence it is impossible to compare with direct estimates of the abundance of this molecule. However, it has been argued that large quantities of frozen methane, of a few % of iced mantle water, is found around the L1527-mm protostar, where the detection of CH3D (Sakai et al. 2012) indicates X(CH4) ≃ 0.4-1.5 × 10 −5 . This large abundance has been attributed to a low density of the pre-collapse core from which L1527-mm originated (Aikawa et al. 2008). Interestingly, the analysis of the deuteration of water, methanol and formaldehyde in L1157-B1 led Codella et al. (2012) to conclude that also the mantles of L1157-B1 were formed in relatively low density (∼ 10 3 cm −3 ) conditions.
To conclude, in the specific case of L1157-B1, gas phase reactions can produce the observed quantity of acetaldehyde only if a large fraction of carbon, of the order of 0.1%, is locked into iced hydrocarbons. Further observations of the hydrocarbons abundance in L1557-B1 are needed to confirm or dismiss our hypothesis. Table 1). The red line shows the synthetic spectra which better reproduce the observations: it has been obtained with the GILDAS-Weeds package (Maret et al. 2011) assuming optically thin emission and LTE conditions with N CH 3 CHO = 9 × 10 13 cm −2 , Tex = 15 K, v LSR = +0.6 km s −1 , and FWHM linewidth = 8.0 km s −1 . . Evolution of acetaldehyde (CH 3 CHO, black), (C 2 H 5 , blue), and methanol (CH 3 OH, red) abundances in the shock as a function of time. Observed abundances (colour circles) are overplotted at the shock age (t shock ∼ 2000 years, vertical dotted line). The evolution is computed from steady-state values in the cloud (n H 2 = 10 4 cm −3 , T kin = 10 K, ζ = 3 × 10 −16 s −1 ) by enhancing the gas temperature and density (n H 2 = 10 5 cm −3 , T kin = 70 K), and the abundance of molecules which are thought to be sputtered off dust grain mantles. We set X CO 2 = 6 × 10 −5 and X OCS = 6 × 10 −6 as in Podio et al. 2014, X CH 3 OH = 2 × 10 −6 (Mendoza et al. 2014), and vary the abundance of C 2 H 5 between 2 × 10 −7 and 2 × 10 −5 .
a
The errors are the gaussian fit uncertainties. The spectral resolution is 4.4 km s −1 . b Derived using the (7 0,7 -6 0,6 ) E and A emissions.et al. , Benedettini et al. 2013
Figure 2 .
2CH 3 CHO emission (in T B scale) extracted at the B1a position (α(J2000) = 20 h 39 m 10. s 2, δ(J2000) = +68 • 01 ′ 12. ′′ 0). the three panels show the frequency intervals of the 4 GHz wide WideX where the six CH 3 CHO lines with S/N 3σ (33 mK) are located (see
Figure 3
3Figure 3. Evolution of acetaldehyde (CH 3 CHO, black), (C 2 H 5 , blue), and methanol (CH 3 OH, red) abundances in the shock as a function of time. Observed abundances (colour circles) are overplotted at the shock age (t shock ∼ 2000 years, vertical dotted line). The evolution is computed from steady-state values in the cloud (n H 2 = 10 4 cm −3 , T kin = 10 K, ζ = 3 × 10 −16 s −1 ) by enhancing the gas temperature and density (n H 2 = 10 5 cm −3 , T kin = 70 K), and the abundance of molecules which are thought to be sputtered off dust grain mantles. We set X CO 2 = 6 × 10 −5 and X OCS = 6 × 10 −6 as in Podio et al. 2014, X CH 3 OH = 2 × 10 −6 (Mendoza et al. 2014), and vary the abundance of C 2 H 5 between 2 × 10 −7 and 2 × 10 −5 .
Table 1 .
1List of CH 3 CHO transitions detected towards L1157-B1Transition
ν a
Eu a
Sµ 2a
log(A/s −1 ) a
(GHz)
(K)
(D 2 )
(7 0,7 -6 0,6 )E
133.830
26
88.5
-4.04
(7 0,7 -6 0,6 )A
133.854
26
88.4
-4.08
(7 2,6 -6 2,5 )A
134.694
35
81.3
-4.11
(7 2,6 -6 2,5 )E
134.895
35
79.7
-4.12
(7 2,5 -6 2,4 )E
135.477
35
79.7
-4.11
(7 2,5 -6 2,4 )A
135.685
35
81.3
-4.10
a From the Jet Propulsion Laboratory database (Pickett et al.
1998).
http://www-laog.obs.ujf-grenoble.fr/heberges/chess/ 2 http://www.oan.es/asai 3 http://www.iram.fr/IRAMFR/GILDAS
c 2011 RAS, MNRAS 000, 1-5
http://smaret.github.com/astrochem/ 5 http://faculty.virginia.edu/ericherb c 2011 RAS, MNRAS 000, 1-5
ACKNOWLEDGMENTSThe authors are grateful to P. Caselli for instructive comments and suggestions, as well as to the IRAM staff for its help in the calibration of the PdBI data. This research has received funding from the European Commission Seventh Framework Programme (FP/2007-2013, the PRIN INAF 2012 -JEDI, and the Italian Ministero dell'Istruzione, Università e Ricerca through the grant Progetti Premiali 2012 -iALMA. LP has received funding from the European Union Seventh Framework Programme (FP7/2007(FP7/ -2013. CC and BL acknowledge the financial support from the French Space Agency CNES, and RB from Spanish MINECO (FIS2012-32096).This paper has been typeset from a T E X/ L A T E X file prepared by the author.
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"EQUIVALENCE BETWEEN THE OSSERMAN CONDITION AND THE RAKIĆ DUALITY PRINCIPLE IN DIMENSION FOUR"
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[
"M Brozos-Vázquez ",
"E Merino "
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We show that 4-dimensional Riemannian manifolds which satisfy the Rakić duality principle are Osserman (i.e. the eigenvalues of the Jacobi operator are constant), thus both conditions are equivalent.The Jacobi operator of a two-point-homogeneous Riemannian manifold has constant eigenvalues. In[9]Osserman wondered if the converse is true. That problem, known in the literature as the Osserman problem, was solved by the contributions of several authors, see[2,5,7,8]and[3,4]for a broad exposition on the topic.More formally the Osserman condition can be phrased as: Pointwise Osserman condition. A Riemannian manifold (M, g) is pointwise Osserman if the eigenvalues of the Jacobi operator J (x) = R(·, x)x do not depend on the unit vector x ∈ T p M , for every point p ∈ M (the eigenvalues may vary from point to point).On the process of studying Osserman manifolds, it was shown that pointwise Osserman manifolds satisfy the following duality principle (see[10], [6] and [4]), which has also been investigated recently in higher signature in[1]. Rakić duality principle. A Riemannian manifold (M, g) satisfies the Rakić duality principle if for every point p ∈ M and for any unit vectors x, y ∈ T p M :where λ is a real number. Note that both definitions are pointwise. The pointwise Osserman condition is not equivalent to the global one (i.e. the eigenvalues do not depend on the point p) in dimension four[5]. This fact, together with some other features that we will recall in Section 1, makes dimension four a special dimension for the Osserman problem. It is an open problem in this context whether the Rakić duality principle implies the Osserman condition. The following is the main theorem of the paper and provides an affirmative answer to that question. Theorem 1. Let (M, g) be a Riemannian manifold of dimension 4. The following assertions are equivalent: (i) (M, g) is pointwise Osserman. (ii) (M, g) satisfies the Rakić duality principle.Outline of the paper. In Section 1 we recall some results and introduce the notation we will use in the proof of Theorem 1. In Section 2 we show that in Key words and phrases. Jacobi operator, Osserman manifold, rank-one symmetric space. 2010 Mathematics Subject Classification: 53C20. M. B.-V. is supported by projects MTM2009-07756 and INCITE09 207 151 PR (Spain). E. M. is supported by projects MTM2008-05861 and INCITE09 207 151 PR (Spain).Proof. That (i) implies (ii) was proved in[10]. We assume that (V, ·, · , A) satisfies the duality in (1) to prove the converse.Let x be a unit vector, J (x) is self-adjoint and hence diagonalizable. Denote by λ and µ the eigenvalues of the Jacobi operator J (x) restricted to x ⊥ . Let y and z be unit eigenvectors associated to λ and µ, respectively, so that J (x)y = λy and J (x)z = µz. By the duality principle we also have J (y)x = λx and J (z)x = µx .Since J (y) preserves the subspace span{x, y}, it also preserves span{x, y} ⊥ , so J (y)z = γz and, by duality, J (z)y = γy, for a certain eigenvalue γ. Compute J (cos θx + sin θy)(− sin θx + cos θy) = λ(− sin θx + cos θy)to see that J (cos θx + sin θy) leaves span{x, y} invariant. So z is an eigenvector for J (cos θx + sin θy) and there exists α such that J (cos θx + sin θy)z = αz. Hence, by (1), we have:α(cos θx + sin θy) = J (z)(cos θx + sin θy) = cos θµx + sin θγy .Lemma 4. Let (V, ·, · , A) be a 4-dimensional algebraic model which satisfies the Rakić duality principle. Then every unit vector x ∈ V can be completed to an orthonormal basis {x, y, z, w} such thatwhere λ 1 , λ 2 , λ 3 , λ 4 , λ 5 and λ 6 are real numbers.Proof. Let x be a unit vector and let y, z, w be a basis of unit eigenvectors associated to J (x), i.e.J (x)y = λ 1 y, J (x)z = λ 2 z, J (x)w = λ 3 w .Since J (y)x = λ 1 x and J (y)y = 0, we have that J (y)span{z, w} ⊂ span{z, w}. Hence there exist a, b ∈ R such that a 2 + b 2 = 1 and J (y)(az + bw) = α(az + bw) , J (y)(bz − aw) = β(bz − aw) .Proof. Adopt the notation of Lemmas 4 and 5 as concerns the basis {x, y, z, w} and the corresponding eigenvalues for the Jacobi operator. Moreover, assume that λ 1 = λ 6 , λ 2 = λ 5 and λ 3 = λ 4 . In summary, we consider a basis {x, y, z, w} such that J (x)y = λ 1 y, J (x)z = λ 2 z, J (x)w = λ 3 w, J (y)z = λ 3 z, J (y)w = λ 2 w, J (z)w = λ 1 w.
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10.1016/j.geomphys.2012.08.002
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"https://arxiv.org/pdf/1109.0386v1.pdf"
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1109.0386
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d458fe46569265519bb23bfdb0cab27b6a944420
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EQUIVALENCE BETWEEN THE OSSERMAN CONDITION AND THE RAKIĆ DUALITY PRINCIPLE IN DIMENSION FOUR
2 Sep 2011
M Brozos-Vázquez
E Merino
EQUIVALENCE BETWEEN THE OSSERMAN CONDITION AND THE RAKIĆ DUALITY PRINCIPLE IN DIMENSION FOUR
2 Sep 2011
We show that 4-dimensional Riemannian manifolds which satisfy the Rakić duality principle are Osserman (i.e. the eigenvalues of the Jacobi operator are constant), thus both conditions are equivalent.The Jacobi operator of a two-point-homogeneous Riemannian manifold has constant eigenvalues. In[9]Osserman wondered if the converse is true. That problem, known in the literature as the Osserman problem, was solved by the contributions of several authors, see[2,5,7,8]and[3,4]for a broad exposition on the topic.More formally the Osserman condition can be phrased as: Pointwise Osserman condition. A Riemannian manifold (M, g) is pointwise Osserman if the eigenvalues of the Jacobi operator J (x) = R(·, x)x do not depend on the unit vector x ∈ T p M , for every point p ∈ M (the eigenvalues may vary from point to point).On the process of studying Osserman manifolds, it was shown that pointwise Osserman manifolds satisfy the following duality principle (see[10], [6] and [4]), which has also been investigated recently in higher signature in[1]. Rakić duality principle. A Riemannian manifold (M, g) satisfies the Rakić duality principle if for every point p ∈ M and for any unit vectors x, y ∈ T p M :where λ is a real number. Note that both definitions are pointwise. The pointwise Osserman condition is not equivalent to the global one (i.e. the eigenvalues do not depend on the point p) in dimension four[5]. This fact, together with some other features that we will recall in Section 1, makes dimension four a special dimension for the Osserman problem. It is an open problem in this context whether the Rakić duality principle implies the Osserman condition. The following is the main theorem of the paper and provides an affirmative answer to that question. Theorem 1. Let (M, g) be a Riemannian manifold of dimension 4. The following assertions are equivalent: (i) (M, g) is pointwise Osserman. (ii) (M, g) satisfies the Rakić duality principle.Outline of the paper. In Section 1 we recall some results and introduce the notation we will use in the proof of Theorem 1. In Section 2 we show that in Key words and phrases. Jacobi operator, Osserman manifold, rank-one symmetric space. 2010 Mathematics Subject Classification: 53C20. M. B.-V. is supported by projects MTM2009-07756 and INCITE09 207 151 PR (Spain). E. M. is supported by projects MTM2008-05861 and INCITE09 207 151 PR (Spain).Proof. That (i) implies (ii) was proved in[10]. We assume that (V, ·, · , A) satisfies the duality in (1) to prove the converse.Let x be a unit vector, J (x) is self-adjoint and hence diagonalizable. Denote by λ and µ the eigenvalues of the Jacobi operator J (x) restricted to x ⊥ . Let y and z be unit eigenvectors associated to λ and µ, respectively, so that J (x)y = λy and J (x)z = µz. By the duality principle we also have J (y)x = λx and J (z)x = µx .Since J (y) preserves the subspace span{x, y}, it also preserves span{x, y} ⊥ , so J (y)z = γz and, by duality, J (z)y = γy, for a certain eigenvalue γ. Compute J (cos θx + sin θy)(− sin θx + cos θy) = λ(− sin θx + cos θy)to see that J (cos θx + sin θy) leaves span{x, y} invariant. So z is an eigenvector for J (cos θx + sin θy) and there exists α such that J (cos θx + sin θy)z = αz. Hence, by (1), we have:α(cos θx + sin θy) = J (z)(cos θx + sin θy) = cos θµx + sin θγy .Lemma 4. Let (V, ·, · , A) be a 4-dimensional algebraic model which satisfies the Rakić duality principle. Then every unit vector x ∈ V can be completed to an orthonormal basis {x, y, z, w} such thatwhere λ 1 , λ 2 , λ 3 , λ 4 , λ 5 and λ 6 are real numbers.Proof. Let x be a unit vector and let y, z, w be a basis of unit eigenvectors associated to J (x), i.e.J (x)y = λ 1 y, J (x)z = λ 2 z, J (x)w = λ 3 w .Since J (y)x = λ 1 x and J (y)y = 0, we have that J (y)span{z, w} ⊂ span{z, w}. Hence there exist a, b ∈ R such that a 2 + b 2 = 1 and J (y)(az + bw) = α(az + bw) , J (y)(bz − aw) = β(bz − aw) .Proof. Adopt the notation of Lemmas 4 and 5 as concerns the basis {x, y, z, w} and the corresponding eigenvalues for the Jacobi operator. Moreover, assume that λ 1 = λ 6 , λ 2 = λ 5 and λ 3 = λ 4 . In summary, we consider a basis {x, y, z, w} such that J (x)y = λ 1 y, J (x)z = λ 2 z, J (x)w = λ 3 w, J (y)z = λ 3 z, J (y)w = λ 2 w, J (z)w = λ 1 w.
dimension three the Rakić duality principle implies the Osserman condition, thus showing that an analogous of Theorem 1 is also true in a lower dimension. Finally, in Section 3, we prove Theorem 1.
Preliminaries
We work at a purely algebraic level. Let V be a vector space of dimension n, ·, · a positive definite inner product and A an algebraic curvature tensor, i.e. a (0, 4)-tensor which satisfies the following relations: (2) A(x, y, z, w) = −A(y, x, z, w) = A(z, w, x, y), A(x, y, z, w) + A(y, z, x, w) + A(z, x, y, w) = 0 .
We refer to the triple (V, ·, · , A) as an algebraic model. We use the inner product to upper indices and define the curvature operator by A(x, y)z, w := A(x, y, z, w) for any vectors x, y, z, w ∈ V . Thus the Jacobi operator is defined as J (x)y := A(y, x)x. Note that J (x)x = 0, so we restrict J (x) to x ⊥ henceforth.
Using analogy with the pointwise geometric setting, we say that the model (V, ·, · , A) is Osserman if the eigenvalues of J (x) do not depend on the unit vector x ∈ V . Similarly, we say that (V, ·, · , A) satisfies the Rakić duality principle if for any eigenvalue λ we have that J (x)y = λy if and only if J (y)x = λx.
A model is said to be Einstein if ρ(·, ·) = c ·, · , where ρ(x, y) := T r{z → A(z, x)y} is the Ricci tensor and c is a real number. Contract this identity to see that c = τ n , where τ denotes the scalar curvature. Every Osserman model is Einstein (see [3,4]).
A particular feature of 4-dimensional models is that the Hodge star operator ⋆ acts on the space of bi-vectors Λ = {x ∧ y : x, y ∈ V } satisfying ⋆ 2 = Id, where Id stands for the identity map. This induces a splitting Λ = Λ + ⊕ Λ − into the eigenspaces associated to the +1 and −1 eigenvalues. For an orthonormal basis {e 1 , e 2 , e 3 , e 4 } of V , an orthonormal basis of Λ ± is given by
(3) Λ ± = span E ± 1 = e 1 ∧e 2 ±e 3 ∧e 4 √ 2 , E ± 2 = e 1 ∧e 3 ∓e 2 ∧e 4 √ 2 , E ± 3 = e 1 ∧e 4 ±e 2 ∧e 3 √ 2 .
The Weyl tensor in dimension four is given by
W (x, y, z, w) = R(x, y, z, w) + τ 6 { x, w y, z − x, z y, w } − 1 2 {ρ(x, w) y, z + ρ(y, z) x, w − ρ(x, z) y, w − ρ(y, w) x, z }. Denote by W ± the restriction of the Weyl tensor acting on bi-vectors to Λ ± . A model (V, ·, · , A) is said to be self-dual if W − = 0 and anti-self-dual if W + = 0.
In Section 3 we will use the following well-known characterization of Osserman models in dimension four [5].
Theorem 2. A model (V, ·, · , A) of dimension 4
is Osserman if and only if it is
Einstein and self-dual (or anti-self-dual).
The Rakić duality principle in dimension 3
We begin by showing that the Rakić duality principle implies the Osserman condition for an algebraic model (V, ·, · , A) of dimension 3; thus we have the following: Theorem 3. Let (V, ·, · , A) be a 3-dimensional algebraic model. The following two conditions are equivalent:
(i) (V, ·, · , A) is Osserman. (ii) (V, ·, · , A) satisfies the Rakić duality principle.
Since x and y are linearly independent we get that α = µ = γ. We repeat the same argument for cos θy + sin θz to see that, indeed, λ = µ = γ. Since x was chosen arbitrarily we have just shown that J (x) = λ(x) Id for every unit vector x ∈ V , where λ(x) is a function of x. In order to finish the proof we must show that λ is constant. Take x and y unit vectors. There exists z ⊥ x, y so that J (x)z = λ(x)z and J (y)z = λ(y)z. By the Rakić duality principle we have that J (z)x = λ(z)x = λ(x)x and that J (z)y = λ(z)y = λ(y)y. Hence λ(x) = λ(z) = λ(y) and (V, ·, · , A) is Osserman.
Proof of Theorem 1
Theorem 1 will be a consequence of the following sequence of lemmas. We begin by choosing an appropriate basis for our subsequent analysis.
By the Rakić duality principle αy = J (az + bw)(y) = a 2 J (z)y + b 2 J (w)y + ab{A(y, z)w + A(y, w)z}, βy = J (bz − aw)(y) = b 2 J (z)y + a 2 J (w)y − ab{A(y, z)w + A(y, w)z}.
Sum both equation to get
(α + β)y = a 2 J (z)y + b 2 J (w)y + b 2 J (z)y + a 2 J (w)y = J (z)y + J (w)y .
We already had that J (z)y, x = J (w)y, x = 0 because x is an eigenvector for J (z) and J (w), and these are self-adjoint. Take the inner product of the previous expression with respect to z and w to see that J (z)y, w = J (w)y, z = 0. Therefore, J (z)y = λ 4 y and J (w)y = λ 5 y for certain λ 4 , λ 5 ∈ R. Now the Rakić duality principle implies that J (y)z = λ 4 z and J (y)w = λ 5 w. Finally, since x and y are eigenvectors for J (z), so is w which generates the orthogonal complement of span{x, y} in z ⊥ . Hence J (z)w = λ 6 w for a certain λ 6 ∈ R.
Lemma 5. Let (V, ·, · , A) be a 4-dimensional model. If (V, ·, · , A) satisfies the Rakić duality principle then it is Einstein. Proof. We adopt notation in Lemma 4. For any θ ∈ [0, 2π), set r θ = cos θx + sin θy. Note that J (r θ )(sin θx − cos θy) = λ 1 (sin θx − cos θy). Hence J (r θ ) span{z, w} ⊂ span{z, w} and there exist a, b ∈ R with a 2 + b 2 = 1 such that
J (r θ )(az + bw) = α(az + bw), J (r θ )(bz − aw) = β(bz − aw),
for certain α and β real numbers. Expand (5) J (r θ ) = cos 2 θJ (x) + sin 2 θJ (y) + cos θ sin θ{A(·, x)y + A(·, y)x} .
Now, we particularize θ = π 4 to set r = r π 4 = x+y √ 2 . Using expressions in (4) and equation (5) we compute J (r)(az + bw) = a 2 {λ 2 z + λ 4 z + A(z, x)y + A(z, y)x} + b 2 {λ 3 w + λ 5 w + A(w, x)y + A(w, y)x} and J (r)(bz − aw) = b 2 {λ 2 z + λ 4 z + A(z, x)y + A(z, y)x} − a 2 {λ 3 w + λ 5 w + A(w, x)y + A(w, y)x}. Taking the inner product with z and w we obtain the following equations:
(6) αa = a 2 λ 2 + a 2 λ 4 + b 2 {A(w, x, y, z) + A(w, y, x, z)}, βb = b 2 λ 2 + b 2 λ 4 − a 2 {A(w, x, y, z) + A(w, y, x, z)}, αb = b 2 λ 3 + b 2 λ 5 + a 2 {A(z, x, y, w) + A(z, y, x, w)}, βa = a 2 λ 3 + a 2 λ 5 − b 2 {A(z, x, y, w) + A(z, y, x, w)}.
On the other hand, apply the Rakić duality principle to see that J (az+bw)(x+y) = α(x + y) and J (bz − aw)(x + y) = β(x + y). Expanding we get α(x + y) = a 2 λ 2 x + b 2 λ 3 x + ab{A(x, z)w + A(x, w)z} +a 2 λ 4 y + b 2 λ 5 y + ab{A(y, z)w + A(y, w)z} and β(x + y) = b 2 λ 2 x + a 2 λ 3 x − ab{A(x, z)w + A(x, w)z} +b 2 λ 4 y + a 2 λ 5 y − ab{A(y, z)w + A(y, w)z}. Sum both expressions to see that λ 2 x + λ 3 x + λ 4 y + λ 5 y = (α + β)(x + y).
Since x and y are linearly independent we obtain α + β = λ 2 + λ 3 = λ 4 + λ 5 , so
(7) λ 2 + λ 3 − λ 4 − λ 5 = 0.
Take the inner product with x and y in the expressions above to obtain:
α = a 2 λ 2 + b 2 λ 3 + ab{A(y, z, w, x) + A(y, w, z, x)},(8)
α = a 2 λ 4 + b 2 λ 5 + ab{A(x, z, w, y) + A(x, w, z, y)},
(9) β = b 2 λ 2 + a 2 λ 3 − ab{A(y, z, w, x) + A(y, w, z, x)},(10)β = b 2 λ 4 + a 2 λ 5 − ab{A(x, z, w, y) + A(x, w, z, y)}.(11)
Compute (8)-(9)-(10)+(11) to obtain the following equation:
(12) (a 2 − b 2 )(λ 2 − λ 4 − λ 3 + λ 5 ) = 0.
Now, from (7) and (12) we get two possibilities:
• a 2 = b 2 , which implies λ 2 = λ 5 and λ 3 = λ 4 by (6), or • λ 2 = λ 4 and λ 3 = λ 5 .
Repeat the previous argument interchanging y by z to see that:
• λ 1 = λ 4 and λ 3 = λ 6 , or • λ 1 = λ 6 and λ 3 = λ 4 , and interchanging y by w to see that
• λ 1 = λ 5 and λ 2 = λ 6 , or • λ 1 = λ 6 and λ 2 = λ 5 .
In summary, combine the possibilities above to see that the possible eigenvalue structures are: a) λ 1 = λ 2 = λ 3 = λ 4 = λ 5 = λ 6 , b) λ 1 = λ 6 and λ 2 = λ 3 = λ 4 = λ 5 , c) λ 2 = λ 5 and λ 1 = λ 3 = λ 4 = λ 6 , d) λ 3 = λ 4 and λ 1 = λ 2 = λ 5 = λ 6 , e) λ 1 = λ 6 , λ 2 = λ 5 and λ 3 = λ 4 . Now note that ρ(u, v) = J (x)u, v + J (y)u, v + J (z)u, v + J (w)u, v = 0 for u, v ∈ {x, y, z, w} with u = v. Also, since ρ(v, v) = T r{J (v)}, it is straightforward to see that the diagonal components of the Ricci tensor ρ are given by
ρ(x, x) = λ 1 + λ 2 + λ 3 , ρ(y, y) = λ 1 + λ 4 + λ 5 = λ 1 + λ 2 + λ 3 , ρ(z, z) = λ 2 + λ 4 + λ 6 = λ 1 + λ 2 + λ 3 , ρ(w, w) = λ 3 + λ 5 + λ 6 = λ 1 + λ 2 + λ 3 .
Hence ρ(·, ·) = (λ 1 + λ 2 + λ 3 ) ·, · and (V, ·, · , A) is Einstein.
We continue our study of an Einstein model taking advantage of the previous results. Our current task is to find out the mixed terms of the curvature, this is, those which involve vectors x, y, z and w. Observe that Case e) in the proof of Lemma 5 is more general than Cases a), b), c), d). Thus, we assume henceforth that λ 1 = λ 6 , λ 2 = λ 5 and λ 3 = λ 4 so all the possible cases are considered at once. Lemma 6. Let (V, ·, · , A) be a 4-dimensional Einstein model which satisfies the Rakić duality principle. Then (V, ·, · , A) is self-dual or anti-self-dual.
Proof of Theorem 1. That (i) implies (ii) was proved in [10] (see [4] for an alternative proof). That (ii) implies (i) is a direct consequence of Theorem 2 and Lemmas 5 and 6.
Recall notation r θ = cos θx + sin θy from Lemma 5 and consider θ = π 6 so s = r π 6 = √ 3 2 x + 1 2 y. We repeat a previous argument to see that there exist a and b, with a 2 + b 2 = 1 and a, b > 0 (a and b cannot be 0 as a consequence of the Rakić duality principle; if a < 0 or b < 0 change z by −z or w by −w to rearrange signs) such that J (s)(az + bw) = α(az + bw)for α a real number. Expand the previous expression to get4 b{A(w, x)y + A(w, y)x}. Take the inner product with z and w to obtain the following equations:4 a{A(z, x, y, w) + A(z, y, x, w)}. Apply the Rakić duality principle to see that J (az + bw)s = αs and expand α(2 ab{A(x, z)w + A(x, w)z} + 1 2 ab{A(y, z)w + A(y, w)z}. Now take the inner product with x and y to obtain:2 ab{A(x, z, w, y) + A(x, w, z, y)}. On the one hand, from (13) we obtain the following relation on the eigenvalues:that we can write asOn the other hand, from (14) we getNow, use that b 2 = 1 − a 2 and substitute α in (15):Since we are assuming a, b > 0, the possible solutions are:Now, substitute in (13) or (14) to see that if α = λ 2 = λ 3 then A(x, z, w, y) + A(x, w, z, y) = 0, if α = λ 2 = λ 3 then A(x, z, w, y) + A(x, w, z, y) = λ 2 − λ 3 , and if α = λ 3 = λ 2 then A(x, z, w, y) + A(x, w, z, y) = λ 3 − λ 2 . We repeat this argument for √ 3 2 x + 1 2 z and √ 3 2 x + 1 2 w to see that: A(x, z, w, y) + A(x, w, z, y) = ±(λ 2 − λ 3 ), A(x, w, y, z) + A(x, y, w, z) = ±(λ 3 − λ 1 ), A(x, y, z, w) + A(x, z, y, w) = ±(λ 1 − λ 2 ).Changing the sign of a vector in {x, y, z, w} if necessary, we can assume without loss of generality that:Now compute. Use the previous relations and equations(2)to compute the other components of the curvature which involve all the elements of the basis {x, y, z, w}. Hence the curvature tensor is given by:A(x, y, y, x) = A(z, w, w, z) = λ 1 , A(x, z, z, x) = A(y, w, w, y) = λ 2 ,A(x, w, w, x) = A(y, z, z, y) = λ 3 , A(x, z, w, y) = −λ1+2λ2−λ3 Recall that ρ(·, ·) = (λ 1 + λ 2 + λ 3 ) ·, · and τ = 4(λ 1 + λ 2 + λ 3 ). The Weyl tensor is given by:Now we see that (V, ·, · , A) is Osserman by checking that, with the chosen orientation, it is anti-self-dual:
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Spain E-mail address: [email protected], [email protected]. Department of Mathematics, University of A CoruñaDepartment of Mathematics, University of A Coruña, Spain E-mail address: [email protected], [email protected]
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[] |
[
"A Minimalist Turbulent Boundary Layer Model",
"A Minimalist Turbulent Boundary Layer Model"
] |
[
"L Moriconi \nInstituto de Física\nUniversidade Federal do Rio de Janeiro\nC.P. 6852821945-970Rio de JaneiroRJBrazil\n"
] |
[
"Instituto de Física\nUniversidade Federal do Rio de Janeiro\nC.P. 6852821945-970Rio de JaneiroRJBrazil"
] |
[
"PACS numbers: 47.27.-i, 47.27.nb, 47"
] |
We introduce an elementary model of a turbulent boundary layer over a flat surface, given as a vertical random distribution of spanwise Lamb-Oseen vortex configurations placed over a non-slip boundary condition line. We are able to reproduce several important features of realistic flows, such as the viscous and logarithmic boundary sublayers, and the general behavior of the first statistical moments (turbulent intensity, skewness and flatness) of the streamwise velocity fluctuations. As an application, we advance some heuristic considerations on the boundary layer underlying kinematics that could be associated with the phenomenon of drag reduction by polymers, finding a suggestive support from its experimental signatures.Turbulent boundary layers have been a central topic of interest in fluid dynamic research for long years[1,2]. Nevertheless their obvious technological importance, a satisfactory description of the physical mechanisms which underlie the boundary velocity fluctuations remains elusive to date. As the result of intensive computational and experimental efforts carried out mainly along the last two decades, it is by now clear that the turbulent boundary layer is the stage for the production and the complex interaction of coherent structures [3], a fact that was formerly antecipated by Theodorsen [4] and Townsend[5].Standard phenomenological formulations of the turbulent boundary layer problem aim at solving selfconsistent equations for the expectation values of velocity and the Reynolds stress tensor components, relevant quantities in engineering applications[2,6]. At the very conception of these models no fundamental role is given to the whole boundary layer zoo of coherent structures, like streamwise and hairpin vortices, low speed streaks, etc. It is an open question, for instance, if a structural derivation of the Prandtl von-Karman logarithmic law of the wall is viable. In this sense, turbulent boundary layer modelling is a difficult problem of statistical physics, analogous to the derivation of thermodynamic equations of state from molecular kinematics/dynamics. The literature on the subject is still relatively small, although an initial discussion may be traced back to 1982 with Perry and Chong[7]. This work differs from previous attempts[7,8,9,10,11,12]essentially in its stronger simplifying assumptions, specific coherent structure modelling (the vortex-dipole model to be introduced below), and the analysis of higher order statistics for the streamwise velocity fluctuations. We do not seek, at the present level of mathematical treatment, quantitative agreement with experiments; instead, we compute a set of general profiles, which turn out to be clearly supported by observations.Our focus is on the streamwise fluctuations of the velocity field. Let us assume that close enough to the wall these fluctuations are mostly due to the flow generated by hairpin vortices[13,14,15], like the one depicted inFig.1, momentarily located in the surroundings of the measurement position. The main contribution to streamwise fluctuations would come from the spanwise sector of hairpin vortices (also called "hairpin's head" ), while subdominant contributions would be related to their necks and legs. B z y x A FIG. 1: A hairpin vortex which propagates along the positive x direction. Streamwise fluctuations of the velocity field, associated to the passage of the hairpin vortex, are due essentially to the flow generated by its spanwise sector, which lies between points A and B. The curved arrow indicates the vorticity orientation.Since streamwise fluctuations of velocity are the only ones we have in mind, it is natural to replace the hairpin vortex by a simpler and more mathematically tractable structure, which we take to be a spanwise Lamb-Oseen vortex, an exact and non-stationary solution of the Navier-Stokes equations, described in cilindrical coordinates aswhere Γ is the total circulation around the vortex and r 2 c = 2νt is the squared vortex core radius at time t, defined in terms of the kinematical viscosity ν.
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10.1103/physreve.79.046306
|
[
"https://arxiv.org/pdf/0809.0878v1.pdf"
] | 8,231,711 |
0809.0878
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0762048fef8ec6fb981a7d6355d1b69b076a8f0a
|
A Minimalist Turbulent Boundary Layer Model
DeCopyright De4 Sep 2008
L Moriconi
Instituto de Física
Universidade Federal do Rio de Janeiro
C.P. 6852821945-970Rio de JaneiroRJBrazil
A Minimalist Turbulent Boundary Layer Model
PACS numbers: 47.27.-i, 47.27.nb, 47
De274 Sep 2008arXiv:0809.0878v1 [physics.flu-dyn]
We introduce an elementary model of a turbulent boundary layer over a flat surface, given as a vertical random distribution of spanwise Lamb-Oseen vortex configurations placed over a non-slip boundary condition line. We are able to reproduce several important features of realistic flows, such as the viscous and logarithmic boundary sublayers, and the general behavior of the first statistical moments (turbulent intensity, skewness and flatness) of the streamwise velocity fluctuations. As an application, we advance some heuristic considerations on the boundary layer underlying kinematics that could be associated with the phenomenon of drag reduction by polymers, finding a suggestive support from its experimental signatures.Turbulent boundary layers have been a central topic of interest in fluid dynamic research for long years[1,2]. Nevertheless their obvious technological importance, a satisfactory description of the physical mechanisms which underlie the boundary velocity fluctuations remains elusive to date. As the result of intensive computational and experimental efforts carried out mainly along the last two decades, it is by now clear that the turbulent boundary layer is the stage for the production and the complex interaction of coherent structures [3], a fact that was formerly antecipated by Theodorsen [4] and Townsend[5].Standard phenomenological formulations of the turbulent boundary layer problem aim at solving selfconsistent equations for the expectation values of velocity and the Reynolds stress tensor components, relevant quantities in engineering applications[2,6]. At the very conception of these models no fundamental role is given to the whole boundary layer zoo of coherent structures, like streamwise and hairpin vortices, low speed streaks, etc. It is an open question, for instance, if a structural derivation of the Prandtl von-Karman logarithmic law of the wall is viable. In this sense, turbulent boundary layer modelling is a difficult problem of statistical physics, analogous to the derivation of thermodynamic equations of state from molecular kinematics/dynamics. The literature on the subject is still relatively small, although an initial discussion may be traced back to 1982 with Perry and Chong[7]. This work differs from previous attempts[7,8,9,10,11,12]essentially in its stronger simplifying assumptions, specific coherent structure modelling (the vortex-dipole model to be introduced below), and the analysis of higher order statistics for the streamwise velocity fluctuations. We do not seek, at the present level of mathematical treatment, quantitative agreement with experiments; instead, we compute a set of general profiles, which turn out to be clearly supported by observations.Our focus is on the streamwise fluctuations of the velocity field. Let us assume that close enough to the wall these fluctuations are mostly due to the flow generated by hairpin vortices[13,14,15], like the one depicted inFig.1, momentarily located in the surroundings of the measurement position. The main contribution to streamwise fluctuations would come from the spanwise sector of hairpin vortices (also called "hairpin's head" ), while subdominant contributions would be related to their necks and legs. B z y x A FIG. 1: A hairpin vortex which propagates along the positive x direction. Streamwise fluctuations of the velocity field, associated to the passage of the hairpin vortex, are due essentially to the flow generated by its spanwise sector, which lies between points A and B. The curved arrow indicates the vorticity orientation.Since streamwise fluctuations of velocity are the only ones we have in mind, it is natural to replace the hairpin vortex by a simpler and more mathematically tractable structure, which we take to be a spanwise Lamb-Oseen vortex, an exact and non-stationary solution of the Navier-Stokes equations, described in cilindrical coordinates aswhere Γ is the total circulation around the vortex and r 2 c = 2νt is the squared vortex core radius at time t, defined in terms of the kinematical viscosity ν.
We introduce an elementary model of a turbulent boundary layer over a flat surface, given as a vertical random distribution of spanwise Lamb-Oseen vortex configurations placed over a non-slip boundary condition line. We are able to reproduce several important features of realistic flows, such as the viscous and logarithmic boundary sublayers, and the general behavior of the first statistical moments (turbulent intensity, skewness and flatness) of the streamwise velocity fluctuations. As an application, we advance some heuristic considerations on the boundary layer underlying kinematics that could be associated with the phenomenon of drag reduction by polymers, finding a suggestive support from its experimental signatures. [1,2]. Nevertheless their obvious technological importance, a satisfactory description of the physical mechanisms which underlie the boundary velocity fluctuations remains elusive to date. As the result of intensive computational and experimental efforts carried out mainly along the last two decades, it is by now clear that the turbulent boundary layer is the stage for the production and the complex interaction of coherent structures [3], a fact that was formerly antecipated by Theodorsen [4] and Townsend [5].
Standard phenomenological formulations of the turbulent boundary layer problem aim at solving selfconsistent equations for the expectation values of velocity and the Reynolds stress tensor components, relevant quantities in engineering applications [2,6]. At the very conception of these models no fundamental role is given to the whole boundary layer zoo of coherent structures, like streamwise and hairpin vortices, low speed streaks, etc. It is an open question, for instance, if a structural derivation of the Prandtl von-Karman logarithmic law of the wall is viable. In this sense, turbulent boundary layer modelling is a difficult problem of statistical physics, analogous to the derivation of thermodynamic equations of state from molecular kinematics/dynamics. The literature on the subject is still relatively small, although an initial discussion may be traced back to 1982 with Perry and Chong [7]. This work differs from previous attempts [7,8,9,10,11,12] essentially in its stronger simplifying assumptions, specific coherent structure modelling (the vortex-dipole model to be introduced below), and the analysis of higher order statistics for the streamwise velocity fluctuations. We do not seek, at the present level of mathematical treatment, quantitative agreement with experiments; instead, we compute a set of general profiles, which turn out to be clearly supported by observations.
Our focus is on the streamwise fluctuations of the velocity field. Let us assume that close enough to the wall these fluctuations are mostly due to the flow generated by hairpin vortices [13,14,15], like the one depicted in Since streamwise fluctuations of velocity are the only ones we have in mind, it is natural to replace the hairpin vortex by a simpler and more mathematically tractable structure, which we take to be a spanwise Lamb-Oseen vortex, an exact and non-stationary solution of the Navier-Stokes equations, described in cilindrical coordinates as
u θ (r) = Γ 2πr [1 − exp(−r 2 /2r 2 c )] ,(1)
where Γ is the total circulation around the vortex and r 2 c = 2νt is the squared vortex core radius at time t, defined in terms of the kinematical viscosity ν.
Of course, (1) solves the fluid equations of motion in the absence of boundaries, so (1) is just a rough approximation to a real vortex parallel to the wall. In our modelling definitions, we postulate that the symmetry axis of the vortex lies in the plane (henceforth designed the "measurement plane") that contains the measurement point and is normal to the wall. We assume, then, that at equally spaced time intervals, a given vortex is replaced by another one, at a random distance y from the wall, with some prescribed probability distribution function (pdf) ρ(y).
Surface P V FIG. 2:
The vortex dipole construction. The dashed line is contained in the measurement plane. The upper plane vortex is the "real" one, while the other is its mirror image. A uniform backgroung flow with velocity V is superimposed to the velocity field produced by the vortices, so that P be a stagnant point.
As the model under consideration is effectively twodimensional, a mirror vortex is introduced "below the wall", so that streamlines do not cross the material surface. Furthermore, in order not to completely neglect the non-slip boundary condition, some improvement is attained if we make the velocity field to vanish at the intersection of the measurement plane with the wall. For this purpose, an external homogeneous velocity field is superimposed to the field generated by the vortex dipole. These definitions are shown in Fig. 2.
An interpretation of the time dependence in Eq. (1) is in order. The time variable t is taken to be the total time elapsed since the hairpin vortex was created at the wall. This assumption, however, is not of great help, if there is no way to relate the vortex vertical position y to the time t. To solve this problem, at least in a phenomenological fashion, we find inspiration in the scaling structure of the laminar boundary layer over a flat surface. It is well-established, and analytically predicted by the Blasius solution, that the laminar boundary layer thickness grows with the distance from the leading edge as δ ∼ √
x. This result can be understood in elementary terms as the fact that any small perturbation which is transported along the main direction of the flow (say, the horizontal one) follows a diffusive drift along the vertical direction. An analogy to the context of turbulent boundary layers can be drawn, replacing words like "perturbations" by "coherent structures" and "molecular viscosity" by "eddy viscosity". Actually, hairpin vortices are observed to grow in size as they get farther from the surface [15]. Therefore, we suppose that at time t, a diffusion-like relation t ∼ y 2 holds, and the vortex core radius can be written as r c = ay in (1), where a plays the role of a phenomenological parameter in the turbulent boundary layer modelling [16].
The flow depicted in Fig. 2 is assumed to describe a boundary layer with no pressure gradient. This is so because the velocity field is symmetric under reflections on the measurement plane. Pressure gradients are probably related to the flow induced by the whole gas of hairpin's vortices, appearing, therefore, as a "many-body" effect.
We are now ready to work out a few relevant equations. Suppose that the Lamb-Oseen vortices are centered at y and −y. The streamwise velocity field vanishes at point P in Fig. 2. Once the resulting streamwise velocity is given as the sum of three contributions (the external, the real and the mirror vortex velocity fields), we get, using Eq. (1),
0 = V − Γ πy [1 − exp(−1/2a 2 )] .(2)
The above equation holds for any y only if the total circulation Γ is a y−dependent quantity. We find
Γ(y) = πV y 1 − exp(−1/2a 2 ) .(3)
It is a straightforward task to evaluate expectation values of general y−dependent functionals F = F [u(y)] of the streamwise velocity field. It follows that
F [u(y)] = ∞ 0 dy ′ ρ(y ′ )F [u(y, y ′ )] ,(4)
where the "two-point velocity",
u(y, y ′ ) = V + Γ(y ′ ) 2π(y − y ′ ) [1 − exp(−(y − y ′ ) 2 /2a 2 y ′2 )] − Γ(y ′ ) 2π(y + y ′ ) [1 − exp(−(y + y ′ ) 2 /2a 2 y ′2 )] ,(5)
is nothing but the velocity field at y when the Lamb-Oseen vortices are placed at y ′ and −y ′ . We have applied (4) for a set of velocity functionals, comparing the y−dependent profiles so obtained with experimental results, as discussed below. As input parameters, we take a = 1.0, V = 1.0 (i.e., the streamwise velocity is computed in units of the external velocity V ). We use the pdf
ρ(y) = 2b π(y 2 + b 2 ) ,(6)
with b = 1.0, to model fluctuations of the vortex's height above the surface. It is important to note that the choice of the lorentzian distribution (6), although arbitrary, is by no means restrictive. We have checked out that smoothly decaying distributions lead to similar conclusions, if one is indeed interested in a qualitative understanding of turbulent boundary layer fluctuations.
• The viscous and logarithmic layers
The mean streamwise velocity is obtained from the expectation value of F [u(y)] = u(y). The overall profile is shown in Fig. 3, which interpolates between zero velocity at the wall and unit velocity at infinity. Even though u(y) seems to give a reasonable profile for the interval 0 ≤ y < ∞, the model does not apply to the outer layer, because of the stronger interactions between coherent structures and also for the high intermittency produced by the random entrainment of external laminar flow that take place in that region. In Figs. 4 and 5, viscous and log-layers are clearly noticed for certain ranges of vertical distances. The excellent fit of the data to the straight line in Fig. 5 is given by u(y) = 0.31 ln(y) + 0.55. The numerical coefficients have precision of 0.1%. Observe that a purely dimensional argument yields u(y) = V f (y/b) in our particular model. Therefore, the numerical verification of a log-layer forces us to identify the effective external velocity V to the friction velocity, up to numerical factors. We may conjecture, thus, that the friction velocity is "what is left" when a few dominant vortical structures near the wall are removed. In other words, the friction velocity can be interpreted here as a "mean field", while fluctuations are modeled by isolated vortex dipoles. • Higher order statistics
Let F n [u(y)] ≡ [u(y) − u(y) ] n . We introduce u rms = F 2 [u(y)](7)
and the hyperflatness functions S n (y), given by
S n (y) = F n [u(y)] F 2 [u(y)] n .(8)
Fine measurements of turbulent boundary layer fluctuations for S 3 (skewness) and S 4 (flatness or kurtosis), which can resolve the region very close to the surface, are reported in Ref. [17]. As we can see from Figs. 6, 7 and 8, there is a clear qualitative agreement with observations. In particular, the abrupt sign-changing transition of skewness is remarkably reproduced by the vortex dipole model. • Drag reduction by polymers
The phenomenon of drag reduction by polymers [18,19] is a long-standing puzzle of non-newtonian fluid mechanics. The broad picture is that dissipation is attenuated due to the interaction of polymers with the coherent structures created near walls.
Recent PIV (particle image velocimetry) experiments in flows with dilute polymers have shown that vorticity fluctuations -and probably coherent structures -are supressed at some point above the surface in turbulent boundary layers [20]. Also, by about the same time, interesting signatures of drag reduction have been found in the profiles of u rms (y) and S 3 (y) in connection with polymer dilution [21]. An additional peak is observed for u rms (y), while the skewness S 3 (y) shows further signchanging transitions. The vortex dipole model allows us to relate these apparently distinct features of drag reduction by polymers. Coherent structure supression can be naturally accounted for by changing the form of the vortex pdf ρ(y). We take, for instance, a distribution which is uniform for 0 ≤ y ≤ c, but vanishes for y > c, that is,
ρ(y) = c −1 [θ(y) − θ(y − c)] ,(9)
where θ(y) is the Heaviside function. Therefore, we suppose no vortex is found for y > c, as the result of polymer interactions. Setting c = 1.0 and keeping the previous definitions of a, b and V , we get the results shown in Figs. 9 and 10, which are in striking correspondence with real profiles [21].
We conclude with some general remarks. The present model does not take into account further aspects of the turbulent boundary layer phenomenology, which become important if the interest is shifted toward quantitative comparisons with experiments. An essential improvement, along the above lines, would be to introduce a pair of streamwise vortex configurations with opposite vorticity as a way to mimick hairpin legs. Only in this way it would be possible to compute the shear stress and, thus, define the physical scales of length and velocity which are necessary for the description of the inner boundary layer.
To summarize, we have introduced an elementary model of a turbulent boundary layer, focusing our attention on the streamwise fluctuations of velocity induced by hairpin vortices. The model's main scope is to provide qualitative insights on the velocity and hyperflatness profiles. Up to the knowledge of the author, this is the first time the profiles of skewness and flatness of usual turbulent boundary layers, as well as certain statistical signatures of the phenomenon of drag reduction by polymers have been theoretically reproduced by means of vortex methods.
PACS numbers: 47.27.-i, 47.27.nb, 47.27.De Turbulent boundary layers have been a central topic of interest in fluid dynamic research for long years
Fig. 1 ,
1momentarily located in the surroundings of the measurement position. The main contribution to streamwise fluctuations would come from the spanwise sector of hairpin vortices (also called "hairpin's head" ), while subdominant contributions would be related to their necks and legs.
vortex which propagates along the positive x direction. Streamwise fluctuations of the velocity field, associated to the passage of the hairpin vortex, are due essentially to the flow generated by its spanwise sector, which lies between points A and B. The curved arrow indicates the vorticity orientation.
FIG. 3 :
3The mean streamwise velocity.
FIG. 4 :FIG. 5 :
45The viscous layer, verified in the range 0.01 ≤ y ≤ 0.25. The straight line has slope 0.95. The logarithmic layer, verified for 0.6 ≤ y ≤ 1.85. The inset shows the same data plotted in linear scales.
FIG. 6 :
6The urms velocity. The inset shows the experimental measurements of urms by Lorkowski[17].
FIG. 7 :
7The skewness for 0.3 ≤ y ≤ 5.0. The inset shows the experimental measurements of S3(y) 1/3 /urms by Lorkowski[17].
FIG. 8 :
8The flatness for 0.1 ≤ y ≤ 5.0. The inset shows the experimental measurements of S4(y) 1/4 /urms by Lorkowski[17].
FIG. 9 :
9The urms velocity, as affected by coherent structure suppression, for 0.01 ≤ y ≤ 5.0. An additional peak is observed at y ≃ 2.5.
FIG. 10 :
10The skewness, as affected by coherent structure suppression, for 0.2 ≤ y ≤ 2.1. Note the additional sign-changing transitions that take place within 1.5 < y < 2.0.
AcknowledgmentsThis work has been partially supported by CNPq and FAPERJ. I thank Atila Freire for several interesting discussions and for calling my attention to Refs.[7,8,9,10,11]. I also thank Katepalli Sreenivasan for kindly providing me a copy of Ref.[9].
One Hundred Years of Boundary Layer Research. G.E.A. Meier and K.R. SreenivasanSpringer-VerlagIU-TAM SymposiumOne Hundred Years of Boundary Layer Research, IU- TAM Symposium, edited by G.E.A. Meier and K.R. Sreenivasan, Springer-Verlag (2004).
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A A Townsend, The Structure of Turbulent Shear Flow. Cambridge University PressA.A. Townsend, The Structure of Turbulent Shear Flow, Cambridge University Press (1976).
S B Pope, Turbulent Flows. Cambridge University PressS.B. Pope, Turbulent Flows, Cambridge University Press (2000).
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A Unified View of the Origin and Morphology of the Turbulent Boundary Layer Structure in Turbulence Management and Relaminarization. K R Sreenivasan, H.W. Liepmann and R. NarasimhaSpringer-VerlagK.R. Sreenivasan, A Unified View of the Origin and Mor- phology of the Turbulent Boundary Layer Structure in Turbulence Management and Relaminarization, edited by H.W. Liepmann and R. Narasimha, Springer-Verlag (1987).
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. I Marusic, A E Perry, J. Fluid Mech. 298398I. Marusic and A.E. Perry and , J. Fluid Mech. 298, 398 (1995).
. I Marusic, Phys. Fluids. 13735I. Marusic, Phys. Fluids 13, 735 (2001).
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. R J Adrian, Phys. Fluids. 1941301R.J. Adrian, Phys. Fluids 19, 041301 (2007).
There is no strong argument to rule out a general anomalous diffusive ansatz, rc = ay α . We would not find, however. important qualitative departures for the results derived hereThere is no strong argument to rule out a general anoma- lous diffusive ansatz, rc = ay α . We would not find, how- ever, important qualitative departures for the results de- rived here.
Small-Scale Forcing of a Turbulent Boundary Layer, FDRL TR 97-2, Fluid Dynamics Research Laboratory. T Lorkowski, Massachusetts Institute of Technology. T. Lorkowski, Small-Scale Forcing of a Turbulent Bound- ary Layer, FDRL TR 97-2, Fluid Dynamics Re- search Laboratory, Massachusetts Institute of Technol- ogy (1997).
B A Toms, Proceedings of the International Congress on Rheology. the International Congress on RheologyNorth-Holland, Amsterdam135B.A. Toms, in Proceedings of the International Congress on Rheology, p. 135, North-Holland, Amsterdam (1949).
. I Procaccia, V S Vov, R Benzi, Rev. Mod. Phys. 80225I. Procaccia, V.S. L'Vov and R. Benzi, Rev. Mod. Phys. 80, 225 (2008).
. C M White, V S R Somandepalli, M G , Exp. Fluids. 3662C.M. White, V.S.R. Somandepalli and M.G. Mungal, Exp. Fluids 36, 62 (2004).
. M Itoh, S Tamano, K Yokota, M Ninagawa, Phys. Fluids. 1775107M. Itoh, S. Tamano, K. Yokota and M. Ninagawa, Phys. Fluids 17, 075107 (2005).
|
[] |
[
"ANALYSIS OF DIFFERENT PRIVACY PRESERVING CLOUD STORAGE FRAMEWORKS",
"ANALYSIS OF DIFFERENT PRIVACY PRESERVING CLOUD STORAGE FRAMEWORKS"
] |
[
"Rajeev Bedi [email protected] \nDepartment of Computer Science and Engineering\nBeant College of Engineering and Technology\nGurdaspur\n",
"Mohit Marwaha \nDepartment of Information Technology\nBeant College of Engineering and Technology, Gurdaspur\n\n",
"Tajinder Singh [email protected] \nDepartment of Information Technology\nGlobal Institute of Engineering and Technology\nAmritsar\n",
"Harwinder Singh \nDepartment of Computer Science and Engineering, SSCET\nBadhani\n",
"Amritpal Singh [email protected] \nDepartment of Information Technology\nBeant College of Engineering and Technology, Gurdaspur\n\n"
] |
[
"Department of Computer Science and Engineering\nBeant College of Engineering and Technology\nGurdaspur",
"Department of Information Technology\nBeant College of Engineering and Technology, Gurdaspur\n",
"Department of Information Technology\nGlobal Institute of Engineering and Technology\nAmritsar",
"Department of Computer Science and Engineering, SSCET\nBadhani",
"Department of Information Technology\nBeant College of Engineering and Technology, Gurdaspur\n"
] |
[] |
Privacy Security of data in Cloud Storage is one of the main issues. Many Frameworks and Technologies are used to preserve data security in cloud storage.[1] Proposes a framework which includes the design of data organization structure, the generation and management of keys, the treatment of change of user's access right and dynamic operations of data, and the interaction between participants. It also design an interactive protocol and an extirpation-based key derivation algorithm, which are combined with lazy revocation, it uses multi-tree structure and symmetric encryption to form a privacy-preserving, efficient framework for cloud storage.[2] Proposes a framework which design a privacy-preserving cloud storage framework in which he designed an interaction protocol among participants, use key derivation algorithm to generate and manage keys, use both symmetric and asymmetric encryption to hide the sensitive data of users, and apply Bloom filter for cipher text retrieval. A system based on this framework is realized. This paper analyzes both the frameworks in terms of the feasibility of the frameworks, running overhead of the system and the privacy security of the frameworks.
|
10.5121/ijcsit.2011.3610
|
[
"https://arxiv.org/pdf/1205.2738v1.pdf"
] | 6,767,747 |
1205.2738
|
70e25ad2a84573fcf272d58d883ede064b387af2
|
ANALYSIS OF DIFFERENT PRIVACY PRESERVING CLOUD STORAGE FRAMEWORKS
Rajeev Bedi [email protected]
Department of Computer Science and Engineering
Beant College of Engineering and Technology
Gurdaspur
Mohit Marwaha
Department of Information Technology
Beant College of Engineering and Technology, Gurdaspur
Tajinder Singh [email protected]
Department of Information Technology
Global Institute of Engineering and Technology
Amritsar
Harwinder Singh
Department of Computer Science and Engineering, SSCET
Badhani
Amritpal Singh [email protected]
Department of Information Technology
Beant College of Engineering and Technology, Gurdaspur
ANALYSIS OF DIFFERENT PRIVACY PRESERVING CLOUD STORAGE FRAMEWORKS
Cloud storagekey derivationBloom filtercipher text retrievalsymmetric encryptionAsymmetric encryption
Privacy Security of data in Cloud Storage is one of the main issues. Many Frameworks and Technologies are used to preserve data security in cloud storage.[1] Proposes a framework which includes the design of data organization structure, the generation and management of keys, the treatment of change of user's access right and dynamic operations of data, and the interaction between participants. It also design an interactive protocol and an extirpation-based key derivation algorithm, which are combined with lazy revocation, it uses multi-tree structure and symmetric encryption to form a privacy-preserving, efficient framework for cloud storage.[2] Proposes a framework which design a privacy-preserving cloud storage framework in which he designed an interaction protocol among participants, use key derivation algorithm to generate and manage keys, use both symmetric and asymmetric encryption to hide the sensitive data of users, and apply Bloom filter for cipher text retrieval. A system based on this framework is realized. This paper analyzes both the frameworks in terms of the feasibility of the frameworks, running overhead of the system and the privacy security of the frameworks.
INTRODUCTION
Cloud storage provides scalable and Quality of service guaranteed resources for storage, users can store and compute their data from any location at anytime by a device which can be connected with Internet to visit that cloud. Besides these powerful advantages of cloud Storage, however, many people and companies is still feel hesitant to store their data in cloud. The reason behind this hesitancy is the fear of people and companies regarding loss of control on their data because there are some incidents of data loss and data leakage which make people to think about it. E.g. a cloud storage-provider named Linkup lost his business last year after losing 45% of stored client data due to an error of a system administrator [3]. In 2007, criminal's targeted Salesforce.com cloud service provider, and steal customer emails and addresses using a phishing attack [4]. Even Google's Docs was visited by unauthorized attacker, which caused data leakage [5]. Therefore, cloud storage providers must consider the privacy issue in priority. A lot of people doing work on outsourced storage. [6] Developed a Privacy-Preserving electronic health record system. On the basis of Symmetric and Asymmetric encryption, it developed two key derivation schemes and compared the advantages and disadvantages of these key derivations. But main drawback of this is it did not consider the effects of change of user access right and the run time operations of data time which greatly influence the effectiveness of key derivation according to the analysis of the following sections. [7] Developed a model named PDAS for preserving privacy and integrity of aggregate query results. It supports privacy protection by dividing the owner's database into M sections and sending a section to a service provider. Any N of them can cooperate to recover the entire database, but any smaller group cannot. Main drawback with PDAS is that it didn't encrypt the data, so in this case the service provider can get the full database by getting partial information. Another drawback is that it demands many service providers to cooperate, which is not realistic. [8]- [10] describes how to do some special calculations on encrypted databases, e.g. KNN(k-nearest neighbor), Boolean queries and keyword based queries etc. Main drawback with these is that they didn't provide a framework for data storage and access, and even didn't consider key management, dynamics of access right and data. [11] Focused on cloud data storage security by distributing the cloud data file F duplicable across a set of a=b+c distributed servers. A (b+c,c) Reed-Solomon erasurecorrecting code was used to create c redundancy parity vectors from b data vectors in such a way that the original b data vectors can be reconstructed from any m out of the b+c data and parity vectors. By placing each of the b+c vectors on a different server, the original data file can survive the failure of any c of the b+c servers without any data loss. It will be the future works of my research. [12] Proposed a scheme for efficient and secure access of outsourced data. It makes data index by binary tree, generates and managed keys by key derivation, it also deals with the dynamics of access right of user and data by over-encryption and/or lazy revocation. Its main drawback is that binary tree structure couldn't reflect the logical relation fully regarding organization of owners data; but this will increase the communication overhead as changing the user's access right would make other user whose access right doesn't change to update certificate, and it also occupy more storage space to store a control block on service provider which is uneconomical. Even it didn't consider how the dynamics of access right and data influences the effectiveness of key derivation. It didn't cover how to avoid collusive attack in which the revoked users cooperate with a cloud storage service provider.
The above analysis shows a privacy-preserving, efficient cloud storage framework is needed urgently. In this paper I am presenting the complete analysis of two Privacy Preserving cloud storage frameworks developed by [1] and [2]. In framework I, service providers and data owners manage data and build data index by multi-tree; for generating and managing keys, extirpation-based key derivation algorithm is designed to solve the ineffectiveness of key derivation; it deals with the dynamics of access right and data by lazy revocation; it ensures data confidentiality by symmetric encryption.
In framework II cipher text based retrieval means service provider can be an agent of data owner to retrieve the owner's data according to the user's query, and it doesn't know the content of the data and the query because they are encrypted, which protects the privacy of owner and user well is used. In this framework the owner is relieved from the overhead of data management, which reflects the main advantage of cloud storage. [13] Proposed symmetric encryption-based cipher text retrieval technique which support the owner to retrieve his own data and not allowed others to retrieve his data. [14] proposed an Asymmetric encryption-based cipher text retrieval scheme. In this the owner encrypted some keywords about his data in this the service provider support the owner to retrieve his own data by the keywords and it didn't not allow others to retrieve the data. [15] Proposed an Asymmetric encryption-based cipher text retrieval scheme which is used to help user M to put user N's data in service provider and only N can retrieve it. [16] Proposed a scheme based on bloom filter to retrieve data which matched with a Boolean query. But it was not fit for cloud storage because it needs the data owner to deal with the query of users and it always has a possibility of a false positive which is fatal to that situation. All, the above references are special cases of cloud storage and can't satisfy the demands of data sharing of cloud storage. Through the above analysis, we can see that a privacy preserving cloud storage framework supporting cipher text retrieval is needed urgently. In next sections of this paper, First of all I presented Privacy Preserving cloud storage framework I and framework II then several key issues in framework I and framework II and then performance evaluation of framework I and framework II. Figure 1 shows different functional modules of data owner, user and cloud storage service provider and interaction between them. The dashed lines described the functional correspondence of connected parts. The interaction protocol is as following:
Privacy-preserving Cloud Storage Framework I
1) Owner (A) sends data block b and which is encrypted by key k b to Cloud Service Provider(C) for storage. And EA indicates the encryption algorithm, k ac is the key between A and C, t mod reflects the time of last update of the data block, MAC (Message Authentication Code) is used to verify the integrity of message: 3) A verifies U's identity firstly, and searches on index and finds the data blocks which have the keyword and are satisfied the access control matrix after the verification is passed. Then, A sends the minimum key group key min of those data blocks and the certificate (certificate). Certificate includes the minimum number group datanumber min of those data blocks, kac is the key between A and C, t certificate indicates when the certificate is generated, and AR records the update times of the user's access right: MSG cu ={C, U, req_index, Eki (data ki ) ||…||E kt (data kt ), MAC} 6) U gets the data blocks, computes the keys of the data blocks from key min by key derivation algorithm, and then decrypts the data blocks.
MSG
There are two points about the framework needed to explain: first, the granularity of data block can be changed according to data owner's requirement, for example, a file or a 128K size data block. For the simplicity, the granularity of data block is file in the paper. Second, the files will not be encrypted if they needn't be kept secret. Figure 1 Privacy-preserving Cloud Storage framework I
SERVICE PROVIDERS
Privacy Preserving Cloud Storage Framework II
Privacy-preserving cloud storage framework Supporting cipher text retrieval is showed in figure 2, which reflects the functional modules of data owners, users and cloud storage service providers and the interaction among them. The dashed line describes the functional correspondences of connected parts. The interaction protocol is as following:
1) Owner (A) chooses a root key KEY root for file encryption by symmetric encryption, a pair of keys (k pub , k pri ) for keywords encryption of file by asymmetric encryption. Before file i is sent to Cloud Service Provider(C), owner generates the key k i of file i by key derivation algorithm and encrypts file i . Then he encrypts keywords {kw 1 , kw2,…,kw n } by k pub and produce Bloom filter BF i . At last, he sends encrypted files to service provider as following:
MSG ac ={A,C,E kos (A,C,E k1 (file 1 )||BF 1 And E indicates the symmetric encryption algorithm, k ac is the symmetric key between owner and service provider, t mod reflects the time of last update of the file, MAC (Message Authentication Code) is used to verify the integrity of message.
2) User (U) requests access authorization from owner. And kua indicates the symmetric key between user and owner, request Id is the serial number of request:
MSG Ua = {U, A, E kuo (U, A, requestId, MAC)}
3) Owner verifies user's identity firstly, and searches on access control list to determine the files which can be accessed by user, then sends the minimum key group key min of those files and the certificate (cert) to user. Certificate includes the minimum number group number min of those files, k os is the symmetric key between owner and service provider, t cert indicates when the certificate is generated, and AR records the update times of the user's access right: Asymmetric encryption algorithm which is used to encrypt keywords by owner and kpub is the public key of owner: MSG UC = {U, C, A, request Id , AEk pub (keyword), cert} 5) Service provider tests the certificate. If it is legitimate, service provider returns those requested files. E ki (file i) is the file which is encrypted by owner, and service provider never encrypts or decrypts owner's files.
Cert= {E kos (U,
MSG cu ={C, U, request Id , E ki (file i) ||…||E kt (file t), MAC} User gets the files, computes the keys of the files from key min by key derivation algorithm, and then decrypts the files. Of course, the files will not be encrypted if they needn't be kept secret.
Several Key Issues in Framework I
Data Organization Structure
This framework uses multi tree based data organization structure for organizing files of Owner. This multi tree structure is automatically generated by client software when these files are going to be stored in the servers of cloud service provider. Figure 3 shows an example. In this files are basically the leaf nodes and non-leaf nodes represent folders or different categories of files. The contents and name of the file is encrypted by owner as file_number$Ek file_number (file_name), for example 1_2_1$Ek 1_2_1 (Diary), before he sends the file Owner's file (1) Offices file (1_1)
Personal file (1_2)
File 1 (1_1_1) File 1 (1_1_2) Dairy (1_2_1) Study material (1_2_1) DAA (1_2_1_1) C++ (1_2_1_n)
to cloud service provider. So it prevents cloud service provider knowing the content and name of the file, which provides the privacy of owner. When the cloud service provider receives the file, it will construct an index for every owner according the file's number, which basically speeds up search on data.
Key Generation and key Management
For better access to data, every file must have different key. So framework I uses symmetric encryption to reduce the burden of encryption and decryption. Key derivation [17] can be used for key management. In this Owner chooses a 128-bit key as root key by a random function, then produces sub key by the following formula: knum=hash(kpar||number||kpar), and hash( ) is a public hash function. Owner only needs to store the root key, which is not only convenient to key management, but also saves the owner's storage space. When a user asks for some files which are satisfied with the keyword, the owner will return the minimum key group from which all requested files' keys can be derived and other unauthorized files' keys can't. Key derivation can reduce the communication overhead of participants efficiently. But the effectiveness of key derivation will be harmed in some case: when the access right of a user is changed, owner must use a new key to encrypt the files if owner don't want the user to access the files which could be accessed by the user before. The new key can't compute by Knum=hash(kpar||number||kpar), and the framework generates the new key by choosing a 128-bit number randomly. When there are a lot of files using new key or every penultimate level directory has a file using new key, the effect of key derivation is the same as the situation where key derivation is not used, namely owner must return N keys if there are N requested files.
To solve this problem, we design extirpation-based key derivation algorithm: owner labels the node with "update" which will use a new key because of the change of user access right in the index tree, and creates a new node in update tree. The new node has the same number with the original node and has a new key. The course is shown in figure 3. When the node needs to update the key again, it can change the key of the node in update tree. When user requests some files, owner will compute the minimum key group by extirpation-based key derivation algorithm. The algorithm is as following:
K(1_1) K(1_2) … K(1_n) K(1_1_1) K(1_2_1) K(1_2_2) ... K(1_2_m) K(1_2_1_1) Index Tree K(1) K(1_1) K(1_2) … K(1_n) K(1_1_1) K(1_2_2)
Update Tree Figure 4 the Correspondence between Index Tree and Update Tree
There is an example. When owner updates the key of file 1_2_2, he will get the file from service provider and decrypt it by the old key firstly. He asks service provider to delete the file and mark the node 1_2_2 with "updated". Then he encrypts the file's content and name with new key of the node in update tree, and sends the encrypted file to the service provider. When an authorized user requests the files which are in the folder 1_2, the owner searches the index tree and returns the minimum key group which includes key1_2 and key1_2_2。From the effect of the algorithm, node 1_2_2 seems to be extirpated from the index tree. The algorithm can reduce the number of returned key effectively.
Access Right Change
First, service provider builds an access right updating linked list updateAR[Owner_id] for every owner, and the node in linked list has two properties: node.id is the number of user, and node.times indicates how many times the access right of the user was updated. After Owneri updates the access right of Userj ， he sends the update massage to ServiceProviderk with the number of Userj. ServiceProviderk receives the massage and searchs the linked list updateAR[i]. If there is a node with node.id=j, then node.times++; otherwise ServiceProviderk inserts a new node into updateAR[i] and set node.id=j and node.times=1. When Userj requests files from ServiceProviderk, ServiceProviderk checks whether there is a node with node.id=j in updateAR[i]. If there is not such a node, ServiceProviderk returns the files; if there is such a node, ServiceProviderk will check whether node.time is equal to cert.AR. If node.time is equal to cert.AR, it will return the files; otherwise it will refuse to return the files and remind the Userj that his certification has expired. The above operations prevent revoked user getting files from service provider. Of course, a revoked user can steal files when they are transmitted. There are two methods to solve the problem: one is over-encryption [18] and the other is lazy revocation [19]. Over-encryption asks the service provider to encrypt the files before they are transmitted, which can prevent revoked user getting the files, but not all service providers are willing to provide such a service and encrypting a batch of files increases the economic burden of owner. Lazy revocation doesn't need owner and service provider to do anything before the file is updated because the stolen file is the same as the file which the revoked user had authorization to access. The framework adopts lazy revocation.
Dynamic Operations of Data
Owner has three dynamic operations on data: addition, deletion and update. When owner wants to add a new file, he will find a new number from the index tree according to the logical relation and compute the key by knumber=hash(kparent||number||kparent), and then encrypt the file and store it in service provider. When owner wants to delete a file, he will send a delete message to service provider to delete the file, then mark the node of the file in the index tree with "deleted". When there is a new file which wants to use the number of the deleted file, it will be treated as an updated file. When the file is updated, the key is valid if there is not a revoked user who could access the file before. Otherwise we need to do the following operations: owner marks the node of the file in index tree with "updated", and inserts a new node with same number and new key into update tree. Then he encrypts the content and name of the file with new key, and sends the encrypted file to service provider. Suppose tmodified indicates when the file was modified lasted and tcert indicates when the user's certificate was created. When an user requests the file, service provider compares tmodified and tcert, cert.AR and node.times of node whose node.id is equal to the user's number in updateAR[owner_id]. If tmodified>tcert and cert.AR==node.time, the user is an authorized user whose key is old, so service provider will return the file and remind him to get a new key; if tmodified≦tcert and cert.AR==node.time, the user is an authorized user who's key is new, so service provider will return the file; if cert.AR<node.time, the user is an revoked user, so service provider will refuse to return the file to him.
Several Key Issues in Framework II
Data Organization Structure, key derivation and management
Owner organizes his files in accordance with some logical relations. For reflecting the logical relations, the framework constructs the file index by multi-tree. Before those files are stored in service provider, the client software of owner will generate multi-tree index automatically according to their logical relation. Figure 2 shows an example. In such an index, only leaf nodes correspond to files, and non-leaf nodes represent folders or categories of files. Owner encrypts the content and name of a file and changes its' name as file_number$ Ekfile_number (file_name), for example 1_2_1$Ek1_2_1(Diary), before he sends the file to service provider. The pretreatment prevents service provider from knowing the content and name of the file, which protects the owner's privacy. The service provider will construct an index for every owner according the files' numbers, which can accelerate search on data. To have a flexible and fine-grained access control, every file has a unique key. The framework uses symmetric encryption to reduce the burden of encryption and decryption. But how to manage numerous keys? Key derivation can be used to solve the problem. Owner chooses a random 256-bit key as root key, then produces sub key by the following formula: keynumber=hash (keyparent||number||keyparent) and hash () is a public hash function. When a user asks for the access authorization, the owner will return the minimum key group from which all authorized files' keys can be derived and other unauthorized files' keys can't. For example, if user is authorized to access the files under the folder 1_2, owner just returns the key1_2. User can compute the keys of the files from key1_2 by key derivation algorithm. Key derivation not only facilitates key management, but also saves the owner's storage space and reduces the communication overhead of participants efficiently.
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Cipher text Retrieval Based on Bloom Filter
In cloud environment, owner stores his files in storage servers of service provider. Anybody who gets the authorization from owner can retrieve files by the help of service provider. Service provider will retrieve the owner's files according to the owner's authority scope and the user's query. The advantage of the design is that service provider undertakes the job of file retrieval, which reduces the computing pressure of owner and is convenient for file sharing. When owner's files are stored in plaintext and the query is expressed in plaintext too, file retrieval is easy. But when file and query are encrypted, it is not easy to retrieve files. Our scheme can solve the problem well. There are three key steps in our scheme: keyword extraction, Bloom filter generation and keyword retrieval.
(1) Keyword Extraction Owner finds some keywords to describe a file. When there are a lot of files, it is usually a miscellaneous and toilsome job. So we design a client software to extract keywords from filename of a file according the language character. For example, there is a file named "The Storage of Cloud Computing", and the keywords will be "storage", "cloud", "computing".
(2) Bloom Filter Generation: Owner chooses a pair of (3) Keyword Retrieval: User hopes to search the file whose keyword is str. He requests access authorization from owner firstly. Owner checks the identity of user and determines his access scope. For example, owner authorizes user to access the files under the file folder 1_2 in figure 2, so he puts numbermin=1_2 into certificate and sends it back to user. User encrypts str by owner's public key, namely w=AEkpub(str), and then sends w to service provider with the certificate. After receiving the query, service provider gets the numbermin from user's certificate, and then searches Bloom filter of every file in the scope of numbermin: array bits at positions h1(file_number||w),…, hk(file_number||w) are checked. If any selected bit is 0, str is definitely not a keyword of the file. On the other hand, if all the checked bits are 1, then w is considered as a keyword of the file. Through the above steps, service provider can find the files whose keyword is str even it doesn't know the content of the file and query, thereby ciphertext retrieval is realized. Adopting Bloom filter, owner needn't store real keywords in cloud, and he just store a bit array which carries the keywords' information, so it is efficient, safe and economic, which will be verified in the next section.
Performance Evaluation of framework I
Effectiveness of Extirpation-based Key Derivation Algorithm
To reflect the real cloud storage environment, experiment simulates the interactions among multiple users, multiple owners and multiple service providers. User requests files randomly, and owner changes user's access right and updates files randomly, too. Owner stores thirty files in different organization structures in service provider's server. Suppose the size of minimum key group of extirpation-based key derivation is size1, and the size of minimum key group of common key derivation is size2. By computing size1/size2, the effectiveness of extirpationbased key derivation can be verified, which is showed as Figure 6. , we can draw conclusions as following: (i) extirpation-based key derivation algorithm is very effective because size1/size2<1; (ii) the organization structure of files has a direct influence on the effectiveness of the algorithm; (iii) the position of the updated file has a direct influence on the effectiveness of the algorithm; (iv) when a file organization structure is fixed, the effectiveness of the algorithm fluctuates surrounding a value. The reason is that the effectiveness of the algorithm is 2/n if there are n files in an folder which has a updated file. So the Effectiveness will fluctuate around 2/n.
Run-time Overhead of the system
Run-time overhead is measured from three aspects: communication, computation and storage overhead. The system in [12] is as a reference system. Suppose the amount of requested files is nj by Userj; Owneri has mi users, the length of user_id is p, the size of file t before and after it is updated is lit and lit', the high of index tree is h, owner has f files; we adopt 128-bit key, hash() indicates the overhead of hash computing, and E(t) and D(t) is the computing overhead of encrypting and decrypting file t. The analysis is as following:As shown in Table 1, our system can reduce the communication, computing and storage overhead immensely.
Privacy Security
From Figure 1, we can find there are several hidden dangers which could leak user's privacy: (i)during the course of files transmitting, outside attacker can steal the files by eavesdropping;
(ii) inside attacker is easy to steal the files because the files are stored in service provider's servers; (iii)in collusive attacks, several revoked users or a revoked user and a service provider cooperate to steal the owner's files. Aimed at the first attack, attacker can't decrypt the file if he hasn't key. If he is a revoked user who has the key, he can decrypt the stolen file, but the file is the same as the file which he could has authorization to access before, which couldn't leak privacy. If the file was updated, the key of the file was changed too. So the key of the revoked user is invalid. To the second attack, owner encrypts the content and name of the files, and the keys are transmitted from owner to user. So the insider attacker has not the keys and couldn't decrypt the files. Against the third attack, there are two conditions: firstly, several revoked users cooperate to derive the key which couldn't be derived by the keys of one revoked user.
Following the proof in [20], we can show that the attacks have to have a non-negligible advantage in breaking the hash function to accomplish this task. Therefore, the proposed approach is robust against such a collusive attack if the hash function is considered safe. Secondly, a revoked user and a service provider cooperate to steal the files. It only works when the revoked user has a key of a folder and owner adds a new file into the folder with the key computed by knumber=hash(kparent||number||kparent). At this time, service provider transmits the file to revoked user, user can derive the file's key by his key and decrypt the file. To solve the problem, owner adds a new file by the way of updating a file. Through the above analysis, the framework has a excellent privacy security.
Performance Evaluation of framework II
Our research group is designing and developing a campus-level cloud computing platform named "Qing Cloud". The project is composed of cloud computing and cloud storage. Based on the above framework, we developed a system of cloud storage by Java. Now we will judge the feasibility of the framework by analyzing the performance of Bloom Filter, the run-time overhead of the system and privacy security.
Performance of Bloom Filter
False Positive Rate of Bloom Filter
Since the Bloom Filter is a probabilistic data structure, it always has a possibility of a false positive. The false positive rates are shown to be tunable by careful selection of parameters.
There are three key parameters which can affect the false positive rate: the number of hash functions k, the size of a bit array m and the number of keywords n. We will use the following formula to compute the false positive rate c:
(1)
Formula (1) is minimized for k=(m/n)*ln2, in which case it becomes:
(2)
Suppose the false positive rate is less than 0.01% , then r is set to more than 14 and m should be more than 2*n.
Overhead of Bloom Filter
We will analyze the performance of Bloom filter from computation, communication and storage overheads. In the framework, elliptic curve encryption algorithm(ECC) is used as the asymmetric encryption method which adopts 160-bit key. It encrypts keywords of files, and then transforms the cipher texts of a file's keywords into a Bloom filter. Suppose the false positive rate is less than 0.01%, there are five groups of keywords, which have different number of keywords and the keywords is generated randomly. The experiment is done in a computer with 1.86GH dual-core CPU and 2GB memory, and the result is
2 Run-time Overhead of the system
Run-time overhead is measured from three aspects: communication, computation and storage overhead, as showed in Table 2. Suppose the amount of files which is authorized to access by Userj is nj, the amount of files which satisfies Userj's query is sj; Owneri has mi users, the size of encrypted filek is fk and the length of its Bloom filter is bfk, r is the amount of hash function in Bloom Filter, the high of index tree is h and the nodes in index tree occupies q bits, Owneri has p files, and the average amount of keywords of every file is g; we adopt 128-bit key, hash( ) indicates the computation overhead of hash function, E( ) and D( ) is the computation overhead of encrypting and decrypting file by symmetric encryption, AE( ) is the computation overhead of encrypting keyword by asymmetric encryption; len is the key amount in minimum key group generated by key derivation. The analysis is as following:
In the computation overhead and storage overhead of Table 1, (O) indicates that owner undertakes the overhead, the rest may be deduced by analogy. Our system reduces run-time overhead immensely by the following measures: 1) Storage overhead of owner, communication overhead of number group and key group is reduced greatly by key derivation; 2) According to the framework, file retrieval is done by service provider instead of owner, which relieves the computation overhead of owner; 3) Bloom filter can store multiple keywords' information in a bit, which saves the storage space and reduces communication overhead; 4) To use multi-tree structure, the length of file's serial number is shorter than or equal to the height of the tree, which reduces the communication overhead of number group.
Privacy Security
From figure 1, we can find there are several potential threats to users' privacy: (i)during the course of files transmitting, outside attacker can steal the files by eavesdropping; (ii) inside attacker is easy to steal the files because the files are stored in service provider's servers; (iii) several malicious users or a malicious user and a service provider cooperate to steal the owner's files, which is called as collusive attack; (iv)when the user queries, service provider may take a peep at the content of query which is the privacy of user. Aimed at the first attack, attacker can't decrypt the file if he hasn't key. If he is a revoked user who has the key, he can decrypt the stolen file, but the file is the same as the file which he had authorization to access before, which couldn't leak privacy. If the file is updated, the key of the file will be changed too. So the key of the revoked user is invalid. To the second attack, owner encrypts the content and name of the files by symmetric encryption, encrypts keywords of files by asymmetric encryption, and transforms encrypted keywords into a Bloom filter by hash functions. So the encrypted files and the Bloom filters are stored in service provider. The symmetric keys are transmitted from owner to user, the private key of asymmetric key is only known by owner, and the cipher texts of the keywords are not stored in servers of service provider. So the insider attacker couldn't decrypt the files and keywords. Against the third attack, there are two conditions: firstly, several malicious users cooperate to derive the key which couldn't be derived by the keys of one of them. Because hash function is a one-way function, the proposed approach is robust against such a collusive attack. Secondly, owner's certificate limits the scope of files which can be accessed by a user. And certificate is encrypted by the symmetric key Kos, which just can be decrypted by owner and service provider, so it can prevent the user from retrieving other files which is out of the scope. Because of the false positive rate of Bloom filter, service provider may return the files which don't meet the query. But the file is in the scope of authorization, so it won't leak owner' privacy. When service provider is in collusion with malicious users and retrieves files which is out of the authorized scopes, service provider can find the files meeting the query, but he can't decrypt those files because he haven't keys. If service provider wants to know the content of user's query, it can only do that by exhaust algorithm. Support there are eighty-five letters of which a filename can be made in alphabet, when there is a five-letter keyword, it spends 30ms to encrypt a string and retrieve Bloom filter one time by a computer with 1.86GH dual-core CPU and 2GB memory. So, 2.11 years will be spent to find out the five letter keyword, which is considered as difficult calculation. So the privacy of users can be protected. From the above analysis, the framework does well in privacy security.
Conclusion
In the conclusion, analysis of two frameworks named Privacy Preserving cloud storage framework I and Privacy preserving cloud storage framework II supporting cipher text retrieval. These frameworks constructs data index by multi-tree, generate and manage keys by key derivation, realize cipher text retrieval by Bloom filter. These frameworks support the interactions among multiple users, multiple owners and multiple cloud service providers, but only supports owner-write-user-read. This paper analyzes the feasibility of the frameworks from the performance of Bloom filter, runtime overhead of the system and privacy security. And the result verifies that the framework II is good at managing keys, protecting owner privacy and reducing communication, storage and computation overhead.
AC = {A, C, EA ac (A, C, EA kb (data b ), t mod , MAC)} 2) User (U) requests data blocks from A. And k ua indicates the key between U and A, req_index is the index of request, and keyword reflects what the user is interested in: MSG UA = {U, A, Ek ua (U, A, req_index, keyword, MAC)}
Certificate= {E kos (U, req_index, datanumbermin, t certificate , AR, MAC)} MSG au = {A, U, Ekuo (A, U, req_index, keymin, certificate, MAC)} 4) U sends the certificate to C and asks for returning of those data blocks: MSG UC = {U, C, O, req_index, certificate} 5) C tests the certificate. If it is legitimate, C returns those requested data blocks:
number min , t cert , AR, MAC)} MSG au = {A, U, E kuo (A, U, requestId, number min , key min , cert, MAC)} 4) User sends the certificate to service provider and asks for some files which contain the keyword.
Figure 3
3Multi-tree based Data Organization Structure of Owner's Files
{ //when the key of a node is updated first time, and the node is represent as nodes[0] nodes [0].setUpdated( ); Node uNode=new Node(nodes[0].number,keyRandom()); //uNode is updated node updateTree.addNode(uNode
Figure 5 The
5Index Structure of Owner's Files
Kpub,Kpri), and the parameters of the Bloom filter, such as the number of hash functions k and the size of a bit array m. Then he encrypts keywordij of filei by kpub, namely KWij=AEkpub(keywordij), and AE() is asymmetric encryption. So Filei has an encrypted keyword set:{KWi1, KWi2,…, KWin}. Every element of keyword set is calculated as following: y1=hash1(i||KWij),y2=hash2(i||KWij),… yk=hashk(i||KWij), and then array bits of BFi at position y1, y2,…, yk are set to 1. Concatenation with the file number is necessary to make the bit pattern of Bloom filter BFi and BFm completely different even if the keywords of them are the same. At last, owner stores encrypted filei and BFi in service provider.
Figure 6 The
6Effectiveness of Extirpation-based Key Derivation Algorithm The Effectiveness of Extirpation-based Key Derivation Algorithm Figure (A) shows the effectiveness when updating the same file in three different file organization structures; figure (B) shows the effectiveness when updating another file in the above three structures. From figure (A) and (B)
||…||E ki (file i )||BF i ,tmodified, MAC)}Encryption Mechanism
Dynamic data operation
Index Management
Keys Management
Authorization Mechanism
Access Application
Encryption Mechanism
Key Derivation
Dynamic data operation
Data storage structure
Authorization Mechanism
Index Management
Storage servers
User
s
Owner
s
//when the key of a node is updated non-first time, the node is represent as nodes[0] String key=Nodes[0].getUpdatedNode().createNewKey();
encrypt (file, key);
}
else{ // compute the minimun key group
String key_min=" ";
Node pNode=null;
for(int i=0;i<nodes.length;i++)
{
if(nodes[i].updated==1)
key_min=key_min+nodes[i].getUpdateNode().getKey();
else
{
parentNode=findParentNode(i,nodes);
if(parentNode!=null)
{
key_min=key_min+pNode.getKey();
Node[] newNodes=nextNodes(nodes,pNode);
String s=extirpated_keyderivation(newNodes, 3);
key_min=key_min+s;
}
Else
key_min=key_min+nodes[i].getKey(); i++;
}
}
}
return key_min;
}
K(1)
Table 1
1Performance comparison of two systemsOur system
Reference system
Our system Reference
system
Communicat
ion
Overhead
minimum key
group
size1*128
size2*128
minimum
number
group
size1*(h/2)
size2*(h/2)
changing access
right
p+1
lit+lit'+(1/2) *mi *128
updating data
lit+lit'+(1/2)
*mi*128
lit+lit'+(1/2)*mi *128
Computing
overhead
key derivation
(Owner)
nj *(1/2)*h*hash() nj *(1/2)*h*hash()
key derivation
(Userj)
nj*(1/2)*h*hash() nj (1/2)*h*hash()
changing access
right
----
E(t')+D(t)
Storage
overhead
key
128
F*128
Control block
---
128+(h/2)+8
Update AR[i]
1/2) *mi*8
---
Table 2
2Computation and Storage Overhead of Bloom Filter Showed in table 1. From table 1, we can find that Bloom filter can reduce the communication and storage overheads. And at the same time, the computation overhead of Bloom filter is so small that it doesn't affect the performance of the framework, which is mainly used to calculate hash functions.
Table 3
3Run-time Overhead of the SystemType
Overhead
Communication
Overhead
minimum key group
len*128
minimum number group
len*(h/2)
file and bloom filter
p
∑ 1(fk + bfk)
k
Computation
Overhead
ciphertext retrieval(S)
(1/2)*nj*r*hash( )
key derivation (O)
nj*(1/2)*(1+h)*hash( )
key derivation (U)
sj*(1/2)*(1+h)*hash( )
file and keywords
encryption(O)
p*(E( )+g*(AE( )+
r*hash( )))
Storage
Overhead
key(O)
128
IndexTree(O)
p*h/2*q
IndexTree(S)
p*h/2*q*mi
file and bloom filter
p
∑ 1(fk + bfk)
k
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Engineering and Technology, Gurdaspur, Punjab since 2004. I did my B.Tech. and M.Tech. from Punjab Technical University, Jalandhar. I am doing PhD. M J Atallah, M Blanton, N Fazio, K B Frikken, ACM Trans. Inf. Syst. Secur. 123Dynamic and efficient key management for access hierarchies. in Cloud ComputingM.J.Atallah, M.Blanton, N.Fazio, and K.b.Frikken. Dynamic and efficient key management for access hierarchies. ACM Trans. Inf. Syst. Secur., 12(3), pages 1-43, 2009. of Engineering and Technology, Gurdaspur, Punjab since 2004. I did my B.Tech. and M.Tech. from Punjab Technical University, Jalandhar. I am doing PhD. in Cloud Computing.
|
[] |
[
"Proposed parameter-free model for interpreting the measured positron annihilation spectra of materials using a generalized gradient approximation",
"Proposed parameter-free model for interpreting the measured positron annihilation spectra of materials using a generalized gradient approximation"
] |
[
"Bernardo Barbiellini ",
"Jan Kuriplach ",
"\nDepartment of Physics\nDepartment of Low Temperature Physics\nFaculty of Mathematics and Physics\nNortheastern University\n02115BostonMassachusettsUSA\n",
"\nCharles University\nV Holešovičkách 2, CZ-180 00PragueCzech Republic\n"
] |
[
"Department of Physics\nDepartment of Low Temperature Physics\nFaculty of Mathematics and Physics\nNortheastern University\n02115BostonMassachusettsUSA",
"Charles University\nV Holešovičkách 2, CZ-180 00PragueCzech Republic"
] |
[] |
Positron annihilation spectroscopy is often used to analyze the local electronic structure of materials of technological interest. Reliable theoretical tools are crucial to interpret the measured spectra. Here, we propose a parameter-free gradient correction scheme for a local-density approximation obtained from high quality quantum Monte Carlo data. The results of our calculations compare favorably with positron affinity and lifetime measurements opening new avenues for highly precise and advanced positron characterization of materials.
|
10.1103/physrevlett.114.147401
|
[
"https://arxiv.org/pdf/1504.03359v1.pdf"
] | 9,425,785 |
1504.03359
|
c4e9833dc3f6f78de674ad4e685c2e93000afa72
|
Proposed parameter-free model for interpreting the measured positron annihilation spectra of materials using a generalized gradient approximation
13 Apr 2015
Bernardo Barbiellini
Jan Kuriplach
Department of Physics
Department of Low Temperature Physics
Faculty of Mathematics and Physics
Northeastern University
02115BostonMassachusettsUSA
Charles University
V Holešovičkách 2, CZ-180 00PragueCzech Republic
Proposed parameter-free model for interpreting the measured positron annihilation spectra of materials using a generalized gradient approximation
13 Apr 2015
Positron annihilation spectroscopy is often used to analyze the local electronic structure of materials of technological interest. Reliable theoretical tools are crucial to interpret the measured spectra. Here, we propose a parameter-free gradient correction scheme for a local-density approximation obtained from high quality quantum Monte Carlo data. The results of our calculations compare favorably with positron affinity and lifetime measurements opening new avenues for highly precise and advanced positron characterization of materials.
PACS numbers: 78.70.Bj, 71.60.+z, 71. 15.Mb The positron upon annihilation with its anti-particle, the electron, yields unique information about the electronic structure of bulk materials [1,2] and nanostructures [3]. The electron-positron density functional theory (DFT) [4] is used in order to obtain precise knowledge of the positron wave function and its overlap with the electron orbitals. The powerful combination of positron annihilation spectroscopy (PAS) and DFT calculations provides a highly accurate method for advanced characterization of materials [5].
Within the DFT framework, the generalized gradient approximation (GGA) method to describe electronpositron correlation effects in solids has shown a systematic improvement over the local density approximation (LDA) for positron affinities and annihilation characteristics [6][7][8][9]. Until now, a dimensional analysis has been used to determine the form of the lowest-order gradient correction with a semi-empirical coefficient α. So far, the pragmatic approach has been to fit α to large databases of positron lifetimes. Recently, both the LDA and the GGA [10,11] have been improved on the basis of new quantum Monte Carlo (QMC) data for the electron-positron correlation problem in a homogenous electron gas (HEG) [12].
However, one could claim that such good fits may be in some cases accidental [13]. Moreover, the present gradient corrections may also lead to some unphysical effects in the electron-positron correlation potential near the nuclei; namely, its too large oscillations due to the shell structure of core electrons. Therefore, here we propose to improve the GGA by extracting and deducing the α parameter from more fundamental physical principles. This more reliable derivation of α also reveals a gentle dependence of the local density reducing the gradient correction near the nuclei. Thus, α becomes a function of the local density as well.
In the case of the positron immersed in an electron gas the Coulomb attraction produces a cusp in the electron density at the positron site. The correlation potential describing the positron perturbation represents the electronic polarization due to the positron screening and can be obtained via the Hellmann-Feynman theorem using coupling-constant integration as follows [14] V
c (r) = − 1 0 dZ d 3 R ρ(R) [g(r, R, Z)−1] |r − R| ,(1)
where ρ(R) [g(r, R, Z)−1] is the screening cloud density around a positive particle with charge Z (g(r, R, Z) is the particle-electron pair distribution function). The effect of the density gradient on the correlation energy can be deduced from the distortion of the polarization cloud due to this gradient. For this purpose, one can use the dynamical structure factor S(q, ω) [15,16] of the HEG to show that in the high density limit the lowest order gradient correction is proportional to the parameter ǫ = (|∇ ln ρ|/q TF ) 2 (which depends on the ratio of the Thomas-Fermi length λ TF = 1/q TF and the inhomogeneity length 1/|∇ ln ρ|). This correction is given by the expression
∆V c (r) = β ǫ(r) 16 ,(2)
where the constant β = 0.066725 Hartree is linked to the coefficient of the term q 2 in the density response function wave vector expansion. The coefficient β has been calculated by Ma and Brueckner [17] and has been used by various authors [18][19][20]. Eq. (2) is in fact similar to that used to compute the correlation energy [20] for an electron gas with slowly varying density [21]. In order to interpolate to the case of rapid density variations (i.e. large ǫ), we use the formula
V c = V LDA c exp(−αǫ/3) ,(3)
from Ref. [6] [see Eq. (7) there]. This formula is based on the scaling relation for the correlation potential, as derived by Nieminen and Hodges [22]. But α is now a function of the local density (and thereby position). When we identify the first order expansion in ǫ with the Ma and Brueckner's result shown above, we find that
α(r) = − 3 16 β V LDA c (r) .(4)
The quantity α remains a gentle function of the density in the valence electron region and at low density it becomes very close to 0.05 [23] -a value found earlier within the empirical GGA [10,11]. Interestingly, α happens also to be of the same order as the fraction Z c of an electron displaced in electron-electron correlation effects which is typically of the order of 1/20 of the electron charge [24,25]. Like the potential V c , the positron annihilation rate depends on electron-positron correlation effects and must be enhanced over the independent particle model. The electron-positron enhancement theory [26] has some features in common with the interaction between a core hole and the conduction electrons treated both in X-ray emission [27] and in resonant inelastic X-ray scattering [28]. We can relate the correlation energy to the annihilation rate by using the scaling relation [22]. Therefore, one obtains an electron-positron enhancement annihilation factor γ given by
γ − 1 = (γ LDA − 1) exp(−αǫ) ,(5)
The enhancement term γ is used to calculate the total positron annihilation rate or the inverse lifetime 1/τ , which is expressed through the simple relation [5]
1 τ = πr 2 0 c d 3 r γ(r) ρ(r) |ψ + (r)| 2 ,(6)
where r 0 is the classical radius of the electron, c is the speed of light and ψ + (r) is the ground state positron wave function.
In this work, we have used the same accurate computational method described in Refs. [10,11]. Electronic structure calculations for selected materials were carried out using the self-consistent WIEN2k code [29], which imposes no shape restrictions for the electron density and the potential, while the positron wave function and energy were obtained using a Schrödinger equation solver based on a finite difference method. The exchangecorrelation potential for the electrons contains gradient corrections within the scheme proposed by Perdew, Burke and Ernzerhof [20] except in the case of the 4d and 5d elemental metals since some of their calculated properties (e.g. the lattice constant) become inappropriate when gradient corrections are used [30]. GGA corrections introduce cusps in the electron potential, negligible in the LDA, which reflect the atomic shell structure [30]. Numerical parameters of the WIEN2k code as well as of the positron solver were tested and optimized in order to obtain calculated positron lifetimes within a precision of 0.1 ps and positron affinities within 0.01 eV. Here we consider only systems in which the positron density is approaching zero in the limit of an infinite crystal. [31] (α = 0.22), DB = Drummond et al. [12], DG = gradient correction with DB (α = 0.05), PF = parameter-free gradient correction with DB (varying α). The last column gives experimental values taken from Refs. [32][33][34]. The exceptions are C and Si (see Ref.
[10] and references therein) and MgO (Ref. [35]). In the case of MgO an upper limit is given (see the text). DFT provides an excellent description of the Si electronic structure both in the solid and liquid phases [38]. It is therefore natural to start our tests of the parameterfree GGA positron potential in Si. A meaningful observable to check is the positron affinity A defined as the sum of the electron and positron chemical potentials. In the case of a semiconductor, the electron chemical potential is taken from the position of the top of the valence band. Recently, Cassidy et al. [39] have shown that the temperature invariant time of flight (TOF) component for Ps emitted from the surface of p-doped Si(100) has a kinetic energy equal to 0.6 eV. This TOF feature is explained by a bulk positron picking up a valence band electron just beneath the surface to form Ps with a kinetic energy of K = E Ps +A = 0.6 eV. Therefore, the experimental affinity for Si can be deduced to be A = −6.2 eV. When we use the GGA for both the electron and positron potentials, we find a theoretical value A = −6.35 eV, which is in excellent agreement with the value measured by Cassidy et al. while the corresponding LDA value shows a clear tendency to overestimate the magnitude of A. This LDA problem can be traced back to the screening effects. In the GGA, the value of A agrees with the experiment by reducing the screening charge. Calculated positron affinities within LDA and GGA against the corresponding experimental values for different materials are shown in Table I. The trends follow those of Si, nevertheless the experimental values of A are often of earlier date and not always reliable. The corresponding positron lifetimes are presented in Table II. Clearly, the trends of the parameter-free GGA are very similar to the empirical GGA [10,11]. In particular, one of the best result is given by Al which was problematic in the original GGA scheme [6]. Positron lifetime measurements in Li and Na were performed before the advent of reliable spectrometers and fitting procedures, as discussed in detail in Ref. [10], and may be affected by significant errors.
However, in the present scheme the positron has a slightly larger overlap with the core electrons as illustrated in Figs. 1 and 2 for Si and Cu, respectively. Some noticeable jumps of ǫ shown in Fig. 1 (d) and Fig. 2 (d) result in unphysically large local changes in the empirical GGA correlation potential depicted in Fig. 1 (c) and Fig. 2 (c). These problems are now cured by the variation of the function α in space illustrated by Fig. 1 (b) and Fig. 2 (b). Interestingly, α given by Eq. (4) seems to vary almost like the Thomas-Fermi length λ TF and becomes very small close to the nuclei. Therefore, the cusps in the parameter free GGA correlation potential become more damped because of the reduction of the screening length in the core region. This effect is further documented by exp(−αǫ/3) factor plots ( Fig. 1 (d) and Fig. 2 (d)) which define the reduction of the correlation potential in the core region. The exp(−αǫ/3) factor anticorrelates with the ǫ parameter; i.e. a large inhomogeneity corresponds to a small exponential factor. The variation of α in the core region should also improve the description of high-momentum annihilation spectra observed in coincidence Doppler broadening spectroscopy [40,41] and in angular correlation measurements [42].
The positron annihilation lifetime (PAL) provides a way to detect very small amounts of vacancy-defects in crystalline materials. Since thermalized positrons are trapped by vacancies before annihilating with electrons, their lifetime increases with respect the bulk values given the low electron density at the vacancy. For this reason, PAL has been widely used to characterize doped semiconducting samples of silicon and other technological relevant materials [2]. As shown by Table II the positron bulk lifetime of Si is very well described by the present theory. Therefore, deviations from the theoretical lifetime indicate the presence of imperfections in the sample. In a post-silicon electronics era, engineered doping of oxide electronics, which is similar to conventional doping in semiconductor technology, offers much greater functionality including electronic control of redox chemistry with applications to batteries, photovoltaics and catalysis. In particular, a well characterized material is MgO, which is a simple binary oxide with rock-salt structure. In MgO, a magnetic moment can arise from the unpaired 2p electrons at an oxygen site surrounding a cation vacancy with each nearest neighbor oxygen carrying a magnetic moment [43]. This magnetic property can be fine tuned to optimize spintronics devices. Concerning PAL studies, Tanaka et al. [44] have shown that MgO lifetime is significantly affected by Ga doping, which results in the creation of Mg vacancies. However, when the number of Mg vacancies decreases the lifetime converges to the bulk value 130 ps [45], which is in reasonable agreement with the present theory. A reliable experimental TOF study of MgO [35] reports a Ps emission peak energy of 2.6 eV. Since Ps is already formed in the bulk of MgO, the kinetic energy is given in this case by
K = E Ps + A − E B + E G ,
where E B is the Ps binding energy inside the MgO matrix and E G = 7.8 eV is the energy gap of MgO. Using our calculated affinity, we deduce that E B = 5.75 eV, which is consistent with typical values of Ps binding energy in the bulk [46]. In fact, this value must be smaller than E Ps because of screening effects in the bulk.
Ceria [47] is another oxide which has attracted considerable interest because of its applications in solid oxide fuel cells. It can be noted that by removing all the oxygen atoms, one recovers the fcc structure of Ce. Experimentally, positron seems only to detect the γ phase of Ce because of its stronger affinity with respect to the α phase. Interestingly, the experimental ceria lifetime 189 ps [48] appears to be much closer to theoretical value of γ-Ce rather than ceria. A possible reason for this discrepancy is that real samples can always contain patches of γ-Ce which strongly attract the positron because of their higher positron affinity. In this context, we should keep in mind that oxygen is very mobile in ceria.
As an example of advanced characterization, we now show that positron annihilation spectroscopy can be useful to understand the role of oxygen-related defects in high temperature superconductivity [49]. In practice, by comparing the experimental lifetimes [36] to an accurate theory it is possible to deduce that positrons are trapped at oxygen vacancies in the superconducting compound YBa 2 Cu 3 O 7−δ while this trapping becomes negligible in the non-superconducting compound where Y has been replaced by Pr. When positrons become completely delocalized for temperatures higher than 400 K, the lifetime becomes almost identical in the YBa 2 Cu 3 O 7 and PrBa 2 Cu 3 O 7 compounds in agreement with our calculations reported in Table II. Moreover, the calculated lifetime in the tetragonal YBa 2 Cu 3 O 6 lattice is 36 ps longer than in the orthorhombic YBa 2 Cu 3 O 7 . Such difference is consistent with experiments [37]. Curiously, the calculated positron affinity seems to indicate that Ps is emitted with about 0.15 eV higher kinetic energy from YBa 2 Cu 3 O 6 and PrBa 2 Cu 3 O 7 than from YBa 2 Cu 3 O 7 . Nevertheless, since the present DFT calculations fail in describing the insulating phase of YBa 2 Cu 3 O 6 and PrBa 2 Cu 3 O 7 we should take the positron affinity calculated values for these two compounds with caution.
In conclusion, we have demonstrated that the parameter-free GGA truly provides a simple, yet accurate step beyond LDA. It is also reassuring that the most reliable electron-positron LDA parametrization (based on the QMC simulations) combined with the parameter free gradient correction gives the best results compared with any of the older LDA potentials. Further studies combining the present approach with well-converged momentum densities calculations [50] are needed to check if first principle methods can soon improve the agreement over empirical approaches [42].
FIG. 1 .
1(Color online) One-dimensional profiles of (a) the electron density (including atomic orbitals), (b) the α parameter (and the Thomas-Fermi length λ TF ), (c) the positron correlation potential, and (d) the exponential factor exp(−αǫ/3) (and ǫ parameter) along the [100] direction in Si for LDA (DB), the empirical (DG) and the parameter-free (PF) GGA approaches. Si atoms are located at 0 and 10.26 au along [100]. λ TF and ǫ are shown for the purpose of observing correlations with corresponding quantities (the scales of λ TF and ǫ are different than those for α and exp(−αǫ/3), respectively).
FIG. 2 .
2(Color online) One dimensional profiles for Cu as explained in the caption of Fig. 1. Cu atoms are located at 0 and 6.83 au along the [100] direction.
We acknowledge fruitful discussions with A.P. Mills and Y.Nagashima. B.B. is supported by the U.S. Department of Energy (USDOE) Contract No. DE-FG0207ER46352 and has benefited for computer time from Northeastern University's Advanced Scientific Computation Center (ASCC) and USDOEs NERSC supercomputing center. J.K. acknowledges the support by the IT4Innovations Centre of Excellence project (CZ.1.05/1.1.00/02.0070), funded by the European Regional Development Fund and the national budget of the Czech Republic via the Research and Development for Innovations Operational Programme, as well as Czech Ministry of Education, Youth and Sports via the project Large Research, Development and Innovations Infrastructures (LM2011033).
TABLE I .
IPositron affinities (in eV) calculated according to various approaches: GC = original gradient correction with the Arponen and Pajanne potential
YBa2Cu3O7 orthorhombic −6.02 −6.78 −6.52 −6.58 PrBa2Cu3O7 orthorhombic −5.81 −6.57 −6.30 −6.36System
Structure
GC
DB DG
PF Exp.
Elements
Li
bcc
−7.31 −7.02 −6.95 −6.96
C
diamond
−1.33 −2.40 −1.87 −1.93 −2.0
Na
bcc
−7.18 −6.89 −6.80 −6.81
Al
fcc
−4.21 −4.04 −4.00 −4.01 −4.1
Si
diamond
−6.29 −6.47 −6.33 −6.35 −6.2
Fe
bcc
−3.40 −3.76 −3.62 −3.67 −3.3
Cu
fcc
−3.76 −4.23 −4.05 −4.11 −4.3
Nb
bcc
−3.61 −3.75 −3.65 −3.68 −3.8
Ce
fcc, α-Ce
−4.11 −4.16 −4.07 −4.09
Ce
fcc, γ-Ce
−5.34 −5.31 −5.23 −5.25
W
bcc
−1.72 −1.91 −1.82 −1.85 −1.9
Pt
fcc
−3.31 −3.77 −3.61 −3.67 −3.8
Compounds
MgO
rock salt
−5.56 −6.46 −6.17 −6.25 −5.2
Cu2O
cuprite
−5.88 −6.42 −6.21 −6.26
CeO2
fluorite
−6.55 −7.40 −7.12 −7.18
YBa2Cu3O6 tetragonal
−6.11 −6.65 −6.42 −6.46
TABLE II .
IIPositron lifetimes (in ps) calculated according to various approaches explained in the caption ofTable I. The last column gives experimental values discussed in Refs.[10,11]. The last experimental values for cuprates are extracted from Refs.[36,37].∼190 YBa2Cu3O7 orthorhombic 179.2 142.4 154.0 150.5 ∼165 PrBa2Cu3O7 orthorhombic 180.4 143.4 155.0 151.6 ∼165System
Structure
GC DB DG PF Exp.
Elements
Li
bcc
283.2 303.8 316.2 313.5 291
C
diamond
102.8 94.6 98.9 97.7 98+
Na
bcc
337.7 343.0 364.4 360.5 338
Al
fcc
154.2 161.0 164.1 163.0 160+
Si
diamond
222.7 208.1 217.3 215.9 216+
Fe
bcc
109.6 102.1 106.5 104.7 105+
Cu
fcc
120.0 107.4 113.3 110.9 110+
Nb
bcc
123.4 120.9 124.3 123.1 120+
Ce
fcc, α-Ce
169.5 165.0 170.5 169.0 233
Ce
fcc, γ-Ce
196.8 194.1 200.6 198.9 235
W
bcc
102.7 100.6 103.4 102.3 105
Pt
fcc
105.2 97.4 101.3 99.8 99+
Compounds
MgO
rock salt
146.2 119.0 128.5 125.4 130
Cu2O
cuprite
177.4 147.3 158.4 154.8 ∼174
CeO2
fluorite
173.7 138.2 149.1 146.0 <187
YBa2Cu3O6 tetragonal
224.5 175.4 190.8 186.5
. * B Amidei@neu, [email protected]. * [email protected] † [email protected]
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|
[] |
[
"Gaugino Condensation and Nonperturbative Superpotentials in Flux Compactifications",
"Gaugino Condensation and Nonperturbative Superpotentials in Flux Compactifications",
"Gaugino Condensation and Nonperturbative Superpotentials in Flux Compactifications",
"Gaugino Condensation and Nonperturbative Superpotentials in Flux Compactifications"
] |
[
"Lars Görlich \nTata Institute for Fundamental Research Homi Bhabha Road\n400 005MumbaiINDIA\n",
"Shamit Kachru \nDepartment of Physics\nSLAC Stanford University Stanford\n94305/94309CAUSA\n",
"Prasanta K Tripathy \nTata Institute for Fundamental Research Homi Bhabha Road\n400 005MumbaiINDIA\n",
"Sandip P Trivedi \nTata Institute for Fundamental Research Homi Bhabha Road\n400 005MumbaiINDIA\n",
"Lars Görlich \nTata Institute for Fundamental Research Homi Bhabha Road\n400 005MumbaiINDIA\n",
"Shamit Kachru \nDepartment of Physics\nSLAC Stanford University Stanford\n94305/94309CAUSA\n",
"Prasanta K Tripathy \nTata Institute for Fundamental Research Homi Bhabha Road\n400 005MumbaiINDIA\n",
"Sandip P Trivedi \nTata Institute for Fundamental Research Homi Bhabha Road\n400 005MumbaiINDIA\n"
] |
[
"Tata Institute for Fundamental Research Homi Bhabha Road\n400 005MumbaiINDIA",
"Department of Physics\nSLAC Stanford University Stanford\n94305/94309CAUSA",
"Tata Institute for Fundamental Research Homi Bhabha Road\n400 005MumbaiINDIA",
"Tata Institute for Fundamental Research Homi Bhabha Road\n400 005MumbaiINDIA",
"Tata Institute for Fundamental Research Homi Bhabha Road\n400 005MumbaiINDIA",
"Department of Physics\nSLAC Stanford University Stanford\n94305/94309CAUSA",
"Tata Institute for Fundamental Research Homi Bhabha Road\n400 005MumbaiINDIA",
"Tata Institute for Fundamental Research Homi Bhabha Road\n400 005MumbaiINDIA"
] |
[] |
There are two known sources of nonperturbative superpotentials for Kähler moduli in type IIB orientifolds, or F-theory compactifications on Calabi-Yau fourfolds, with flux:Euclidean brane instantons and low-energy dynamics in D7 brane gauge theories. The first class of effects, Euclidean D3 branes which lift in M-theory to M5 branes wrapping divisors of arithmetic genus 1 in the fourfold, is relatively well understood. The second class has been less explored. In this paper, we consider the explicit example of F-theory on K3 ×K3 with flux. The fluxes lift the D7 brane matter fields, and stabilize stacks of D7 branes at loci of enhanced gauge symmetry. The resulting theories exhibit gaugino condensation, and generate a nonperturbative superpotential for Kähler moduli. We describe how the relevant geometries in general contain cycles of arithmetic genus χ ≥ 1 (and how χ > 1 divisors can contribute to the superpotential, in the presence of flux). This second class of effects is likely to be important in finding even larger classes of models where the KKLT mechanism of moduli stabilization can be realized. We also address various claims about the situation for IIB models with a single Kähler modulus.
|
10.1088/1126-6708/2004/12/074
|
[
"https://arxiv.org/pdf/hep-th/0407130v2.pdf"
] | 7,124,151 |
hep-th/0407130
|
3686ec0ae47ad3a70fa0add0fb825409ca95d2d7
|
Gaugino Condensation and Nonperturbative Superpotentials in Flux Compactifications
20 Aug 2004 July 2004
Lars Görlich
Tata Institute for Fundamental Research Homi Bhabha Road
400 005MumbaiINDIA
Shamit Kachru
Department of Physics
SLAC Stanford University Stanford
94305/94309CAUSA
Prasanta K Tripathy
Tata Institute for Fundamental Research Homi Bhabha Road
400 005MumbaiINDIA
Sandip P Trivedi
Tata Institute for Fundamental Research Homi Bhabha Road
400 005MumbaiINDIA
Gaugino Condensation and Nonperturbative Superpotentials in Flux Compactifications
20 Aug 2004 July 20041 emails: goerlich, prasanta, sandip @theory.tifr.res.in, [email protected]
There are two known sources of nonperturbative superpotentials for Kähler moduli in type IIB orientifolds, or F-theory compactifications on Calabi-Yau fourfolds, with flux:Euclidean brane instantons and low-energy dynamics in D7 brane gauge theories. The first class of effects, Euclidean D3 branes which lift in M-theory to M5 branes wrapping divisors of arithmetic genus 1 in the fourfold, is relatively well understood. The second class has been less explored. In this paper, we consider the explicit example of F-theory on K3 ×K3 with flux. The fluxes lift the D7 brane matter fields, and stabilize stacks of D7 branes at loci of enhanced gauge symmetry. The resulting theories exhibit gaugino condensation, and generate a nonperturbative superpotential for Kähler moduli. We describe how the relevant geometries in general contain cycles of arithmetic genus χ ≥ 1 (and how χ > 1 divisors can contribute to the superpotential, in the presence of flux). This second class of effects is likely to be important in finding even larger classes of models where the KKLT mechanism of moduli stabilization can be realized. We also address various claims about the situation for IIB models with a single Kähler modulus.
Introduction and Review
For many years, it has been clear that string theory offers a plethora of choices for compactification to 4d with N ≥ 1 supersymmetry. With N = 1 supersymmetry, quantum effects can play an important role in breaking supersymmetry and/or changing the vacuum structure. It has long been hoped that such effects, perhaps in conjunction with early universe dynamics, would yield one or a few models as the preferred string compactifications -i.e., that there would be a simple vacuum selection principle. There is little evidence for such a picture (though of course our understanding of early universe cosmology in string theory is very limited). Instead, recent attempts to understand moduli stabilization have yielded mounting evidence that after including perturbative and nonperturbative contributions to the moduli potential, string theory manifests a tremendous landscape of vacua, including 4d (metastable) de Sitter and anti-de Sitter geometries with a wide range of different cosmological constants [1][2][3][4][5]. For nice reviews of this subject, see [6]. While a similar picture should emerge in each of the corners of the M-theory parameter space, we will concern ourselves here with the type IIB theory, where the story is best developed.
Because our goal in this paper is to clarify and extend some recent developments in this area, we will begin with a short review and status report on the subject.
If one wishes to obtain theories with 4d N ≤ 1 supersymmetry in the IIB setting, one large class of constructions was developed in [7] and references therein. These models are Calabi-Yau orientifolds with D3 and/or D7 branes, and also admit a description as Ftheory compactified on a Calabi-Yau fourfold. In a given such construction corresponding to compactification on the threefold M (with related fourfold X 4 ), one finds a tadpole condition which requires that the total D3 brane charge on M add up to zero [8]
N D3 + 1 (2π) 4 (α ′ ) 2 M H ∧ F = χ(X 4 ) 24 .
(1.1)
Here H and F are the NS and RR three-form field strengths of the IIB theory, and N D3 denotes the number of D3 branes one has chosen to insert transverse to M . So we see that in generic backgrounds, one will turn on RR and NS fluxes as part of tadpole cancellation.
The resulting class of 4d N = 1 supergravities was described in [7]. The fluxes generate a superpotential for the complex structure moduli and the dilaton τ . Defining G 3 = F − τ H, this superpotential is of the form [9]
W f lux = M G 3 ∧ Ω (1.2)
where Ω is the holomorphic three-form on M . For typical choices of the flux, the dilaton and complex structure moduli have isolated minima.
On the other hand, the Kähler moduli ρ i of M do not appear in W , and participate in a no-scale cancellation at leading orders in the α ′ and g s expansion [7]. One generally expects that quantum corrections will generate a potential for the ρ fields. If this potential arises only from corrections to the Kähler potential, any nontrivial vacua will typically occur for string scale compactification manifolds, and will be difficult to study.
As emphasized in [3], however, there are at least two classes of effects that lead one to expect ρ-dependent corrections to the superpotential in many models:
1) If M contains a 4-cycle Σ with the right topological properties, Witten argued that Euclidean D3 branes will generate a nonperturbative superpotential for the Kähler modes controlling the size of Σ [10]. These cycles lift, in the M-theory fourfold geometry, to "vertical" divisors of arithmetic genus 1 (where "vertical" roughly denotes that they wrap the fiber directions which shrink in the F-theory limit).
2) These models typically contain D7 branes. While as discussed in [11] the D7s often have numerous matter fields in the most naive Kaluza-Klein analysis of their 4d N = 1 supersymmetric worldvolume gauge theory, the three-form flux can give masses to many or all of these matter fields. If the fluxes do this while stabilizing the D7s at a coincident locus, gaugino condensation will ensue. For N D7s wrapping a 4-cycle Σ, the gauge coupling will satisfy 1 g 2 ∼ Vol(Σ), and therefore the gauge theoretic superpotential will generate a nonperturbative potential for Kähler moduli.
In the approximation that one keeps the leading contribution to this superpotential, one gets a schematic formula of the form
W = W f lux + e −aρ .
(1.
3)
If W f lux evaluated in the vacuum of the complex structure and dilaton moduli is small , i.e. W f lux = W 0 << 1, then one finds a resulting vacuum for ρ at moderately large volume. For instance in [3], a toy example was described that, with W 0 of 10 −4 and a of 1 10 , achieved a radius of just above 3 in string units (which translates to ρ ∼ 100). These moderately large radii can justify the neglect of more highly damped exponentials in the formula (1.3), yielding self-consistent solutions to the equations of the effective field theory.
The existence of solutions with W 0 << 1 was justified in [3] by explaining that given the number of flux vacua and naive estimates for how W 0 might vary in different solutions, very small values should arise in a small fraction of the solutions.
The vacua just described are, in the simplest cases, supersymmetric AdS vacua. It was further argued in [3] that by e.g. using warped solutions of the sort described in [7] (which incorporate a Klebanov-Strassler throat [12]), and including anti-D3 branes (whose dynamics in such throats was studied in [13]), one should be able to obtain de Sitter solutions to string theory. Instead of including anti-D3 branes, one could also imagine using anti-self dual field strengths in D7 branes [14]. Alternatively, one can simply start in a vacuum of the no-scale potential which is at positive V , and play off the tadpole for Kähler moduli against the nonperturbative corrections [15]. Another possibility was described in [16]. The end result, as argued in [3] and later references, is that the IIB superstring theory seems to admit a rich landscape of vacua with many de Sitter and antide Sitter critical points, exhibiting different values of the cosmological constant. While only a small fraction of the vacua will arise from small W 0 in the language above, and will therefore arise at moderately large radius, this small fraction was argued to yield a large absolute number of vacua.
Two recent lines of development have added considerable support to this picture. In the first, Ashok and Douglas and later authors [17][18][19] have studied the statistics of the flux vacua which arise in the no-scale approximation. The most basic result concerns the number of vacua, and finds (as one would expect from simple generalizations of [1]) that
N vac ∼ 1 b 3 (M )! χ(X 4 ) 24 b 3 (M )
.
(1.4)
Typical numbers yield χ 24 of order 1000 and b 3 (M ) of order 100, and in fact N vac can be in excess of 10 300 in simple examples. While these estimates neglect many possible further effects that could remove vacua, the basic picture seems robust against effects that have been neglected to date. Estimates of the attainable values of W 0 (or really e K |W | 2 ) at the level of the flux superpotential, fully support the assertion made in [3] that extremely small values will be attainable. In fact, values much smaller than the 10 −4 quoted in KKLT should arise -the fraction of vacua with e K |W | 2 ≤ ǫ seems to fall off only as the first power of ǫ [18,20]. The information available from the statistical studies is much more detailed than we have described here (predicting for instance clustering of vacua around the conifold [18,19] and other interesting dynamics), and it is heartening that the extremely complicated flux potentials admit such simple characterizations of the statistical properties of the resulting vacua.
However, one could still worry that although each of the ingredients in the KKLT construction and its relatives is quite reasonable, it might not be possible to assemble all of the ingredients simultaneously to make a working model. While this potential problem seems very unlikely from the viewpoint of low-energy effective field theory, the issue has now been settled directly from the string theory perspective. In a beautiful recent paper, Denef, Douglas and Florea (henceforth DDF) explicitly provided many examples of Calabi-Yau orientifolds where just Euclidean D3 instantons (effects of type 1) above) are present in sufficient numbers to stabilize all Kähler moduli [11]. Their constructions build on earlier important work of Grassi [21]. These models admit sufficiently many flux vacua that the full KKLT construction can be carried through, as long as the existing statistical studies are not grossly misleading. The fraction of models where just effects of type 1) suffice to carry out the KKLT program is not particularly small -in the simplest class DDF studied (fourfolds with Fano threefold base), 29 of 92 Calabi-Yau spaces could be stabilized this way. Each such manifold leads to a rich landscape of vacua. A subsequent work [22] also pointed to the existence of manifolds which cannot be stabilized by such effects alone, although this work did not address the large class, described in [11], which can be stabilized. Therefore, current evidence strongly supports the existence of a IIB landscape as envisioned in [3]. Among many issues which remain to be clarified is the question of the extent to which effects of type 2) above, low-energy field theory effects on D7 branes, can aid in the stabilization of Kähler moduli. In this work, we provide some explicit compact examples where flux potentials lift D7 translation modes, leaving a pure nonabelian N = 1 gauge theory on stacks of D7s. These will manifest gaugino condensation, although they do not fall into the class of models studied in geometric engineering in [23]-the latter work did not (and did not need to) account for D7 interactions with three-form fluxes.
Before proceeding with the technical analysis, we give a simple physical argument which explains why one should expect the phenomena we find to be rather generic. The gauge theories which arise on D7 brane worldvolumes are, in most simple examples, nonchiral gauge theories. Even if a naive analysis at the compactification scale indicates the presence of matter in the theory (the relevant KK scale analysis is described in [23] and other references), any further interactions at a lower scale can therefore give the matter a mass, leaving behind a pure gauge theory. In these systems, there is a clear source of such further perturbations -the presence of G 3 flux. Physically, it is then not surprising that one will often find pure gauge theories after including the effects of flux. We provide explicit examples on the fourfold K3 × K3 (the orientifold K3 × T 2 /Z 2 in IIB language) in the following; this example was chosen because it is one of the simplest models that includes D7 branes, and was analyzed in great detail in [24] (see also [25]). For the physical reason we explained above and the mathematical reasons we explain below, we also suspect this phenomenon occurs frequently in more complicated models, and will greatly enrich the class of models described so far [11] where the KKLT construction can be realized.
The organization of this note is as follows. In §2, we review the conditions for supersymmetry in M-theory and F-theory compactification on a Calabi-Yau fourfold. In §3, we describe what these conditions imply for K3 ×K3 models in more detail. In §4, we provide some examples of solutions to these conditions which yield 4d models with the D7 branes locked on loci of enhanced non-abelian gauge symmetry. In §5, we explore an orientifold example in detail and calculate the resulting non-perturbative superpotential. In §6, we discuss the kinds of divisors which arise in these singularities. In particular, there are no divisors of arithmetic genus 1 in the K3 × K3 examples, although the relevant divisors D i do satisfy χ(D i , O(D i )) > 1 (and in more general examples, would have χ ≥ 1). We also explain how these observations, when correctly generalized to other Calabi-Yau fourfolds, could relax some of the conditions stated in [22,11]. In particular, we briefly discuss the special case where the IIB theory has a single Kähler modulus, and argue that χ ≥ 1 divisors of the relevant type can arise there. We close in §7 with a discussion of future directions. Some relevant details about the geometry of elliptic K3s and the E 8 lattice are relegated to appendices, as is a discussion of how one can microscopically understand the relaxation of the arithmetic genus 1 condition [10] in the presence of flux.
The D7/D3 moduli in the K3×T 2 /Z 2 orientifold limit were studied earlier in [26] using the techniques of gauged supergravity. In particular it was shown that D7-brane moduli acquire a mass, in agreement with our discussion here 2 . Many of the solutions found below were also known to Greg Moore, from considerations similar to those he described in [25].
A complementary approach to deriving the flux-induced potentials for D7-brane moduli is described in the papers [28,29], and we thank those authors for informing us of their results prior to publication .
General Conditions For Supersymmetry
We are interested in F-theory compactifications on K3 × K3 with flux. In particular we are interested in asking whether the flux can stabilise all the complex structure moduli (including the 7-brane moduli) at points of enhanced gauge symmetry. In the discussion below we take the second K3 to be elliptically fibered and refer to it as K3 2 , the other is denoted as K3 1 and need not admit an elliptic fibration.
We will explore this issue by starting first in M theory on K3 1 × K3 2 in the presence of G 4 flux. This is dual to F-theory on K3 × K3 × S 1 . We will be mainly interested in Lorentz invariant 3 + 1 dimensional solutions in F-theory. These are obtained in the standard fashion by taking the size of the fiber torus in K3 2 to zero.
For N = 1 supersymmetry (four supercharges) G 4 must be of type (2,2). In addition G 4 must be primitive. For Lorentz invariant 3 + 1 dimensional solutions G 4 must have one leg along the base P 1 and another leg along the fiber of K3 2 . Primitivity then reduces to the condition
G 4 ∧ J 1 = 0, (2.1)
where J 1 is the Kähler form of K3 1 .
An elliptically fibered K3 can be described by an equation of Weierstraß form:
y 2 = x 3 + f 8 (z)x + g 12 (z), (2.2)
where f 8 (z) and g 12 (z) are polynomials of degree 8 and 12 respectively. This gives rise to 18 (complex) moduli which describe the complex structure of an elliptically fibered K3
surface. It is well known that the singularities which can occur in this equation are of A-D-E type. At a singularity the symmetry is enhanced to the corresponding A-D-E gauge group.
We will show in specific examples that for appropriate fluxes all the conditions of N = 1 supersymmetry are met and all the complex structure moduli are stabilised such that the elliptically fibered K3 is at a singularity resulting in an enhanced gauge symmetry.
For this purpose it is worth discussing the above conditions in some more detail. Some useful reference for the discussion below are [30], [7], [31], and [24].
The Conditions in More Detail
H 2 (K3, R) is a 22 dimensional vector space. An inner product can be defined on this space, given by (v 1 , v 2 ) ≡ K3 v 1 ∧ v 2 .
This has signature (3,19). H 2 (K3, Z) can be thought of as a lattice, Γ 3,19 , embedded in this vector space. In a suitable basis the inner product for the basis elements of this lattice takes the form of the matrix
U ⊕ U ⊕ U ⊕ (−E 8 ) ⊕ (−E 8 ),
where U is the 2 × 2 matrix
U = 0 1 1 0 (3.1)
and E 8 is the Cartan matrix for the E 8 lattice.
The holomorphic two-form Ω on K3 is given by a spacelike oriented two-plane in
H 2 (K3, R).
The moduli space of complex structures then corresponds to the space of all such distinct two-planes. Up to discrete identifications this is
G = O(3, 19)/O(2) × O(1, 19),
which is 40 dimensional. In the discussion below we will refer to both the holomorphic two-form and the associated space-like two plane as Ω. Clearly, all two-forms in H (1,1) must be orthogonal to Ω.
For the K3 surface to be elliptically fibered, a sublattice U ⊂ Γ (3,19) must be orthogonal to the two-plane 3 Ω. The moduli space of complex structures for elliptically fibered
(1, 1) type. That is P ic(K3) = H 2 (K3, Z) ∩ H (1,1) (K3).
So we see that for the K3 to be elliptically fibered U ⊂ P ic(K3).
How this requirement for an elliptic fibration comes about will be discussed further in the Appendix. For now we simply note that the two-forms dual to the base P 1 and the fibre torus both lie in U . We saw above that G 4 must have one leg along the base and 3 More accurately the sublattice has two basis elements whose inner product takes the form U , (3.1). In an abuse of notation we will refer to the sublattice itself as U below.
one along the fibre of the elliptically fibered K3. This means that G 4 cannot contain any element in U and so must be orthogonal to U . That is for any u ∈ U, G 4 · u = 0.
We can now restate the conditions for supersymmetry as follows. G 4 must be chosen to have two legs along K3 1 and two legs along K3 2 . A complex structure on K3 1 and K3 2 must exist such that G 4 is of type (2,2). The resulting Picard lattice for elliptically fibered K3 2 must contain the sublattice U . G 4 must be orthogonal to U . And finally, the Kähler form of K3 1 must satisfy the primitivity condition, (2.1), which can be restated in terms of the inner product defined above as J 1 · G 4 = 0.
Let us now discuss the singularities in the Weierstraß form, (2.2), in some more detail.
A root of Γ 3,19 is defined to be a vector α ∈ Γ 3,19 with α · α = −2. A singularity in the Weierstraß form arises if there is a root of Γ 3,19 which lies in P ic(K3) and which is orthogonal to U . That is, if there is a root orthogonal to both U and Ω. 4 The orthogonal roots form the root lattice of an A-D-E algebra. The singularity is of the corresponding
A-D-E type.
Before proceeding, let us make the following two comments. First, it is worth briefly recapitulating why one expects all complex structure moduli to be generically stabilised in an N = 1 susy solution. The requirement that G 4 is of type (2, 2) means that the Then G 4 can be expressed in terms of the three-form flux in IIB, G 3 = F 3 − φH 3 , as follows:
G 4 = − 1 φ −φ G 3 ∧ dz + 1 φ −φḠ 3 ∧ dz. 4
The requirement that the Einstein metric on K3 is at an orbifold singularity is somewhat different. The metric corresponds to a choice of space-like three-plane and an orbifold singularity occurs if a root is orthogonal to this three-plane. This ensures that orbifold has not been resolved either by Kähler deformations or complex structure deformations. For F-theory we are only interested in the complex structure deformations which preserve the elliptic form and we therefore require orthogonality with respect to Ω and U . G 4 can also give rise to two-form flux, F 2 , in the world-volume theory of the D7 branes.
For example, if two D7-branes come together giving rise to an A 1 singularity and if α is the corresponding root of Γ 3,19 that is orthogonal to Ω, then a non-trivial F 2 is turned on in the relative U (1) between the two D7-branes if G 4 has a component of the form,
G 4 = β ∧ α,
where β is an integral two-form in K3 1 . Note however that if P ic(K3 1 ) is trivial -as will be the case generically -such a component is not allowed by supersymmetry. This follows from noting that since α is orthogonal to Ω it must be of type (1,1). Supersymmetry requires that G 4 is of type (2,2), this means β must be an (1, 1) form in K3 1 and must therefore belong to P ic(K3 1 ). In any event, we shall avoid turning on such F 2 fluxes in our constructions.
Examples
Simplest examples
We will now construct an explicit example where all the complex structure moduli are stabilised at a point of enhanced symmetry.
Consider the six-dimensional subspace H 3,3 = U ⊕ U ⊕ U of H 2 (K3, Z)
. In a suitable basis, which we call (e 1 , · · · e 6 ) the inner product in this subspace takes the form, where e 1 , e 2 andẽ 1 ,ẽ 2 refer to vectors in the integral lattice of K3 1 and K3 2 respectively.
This flux satisfies the requirement of having two legs along the two K3s respectively.
It is easy to see that G 4 can be written as
G 4 2π = 1 2 [(e 1 + ie 2 ) ∧ (ẽ 1 − iẽ 2 ) + (e 1 − ie 2 ) ∧ (ẽ 1 + iẽ 2 )] (4.2)
So by choosing the complex structure Ω 1 , Ω 2 of K3 1 , K3 2 as follows,
Ω 1 = (e 1 + ie 2 ) (4.3) Ω 2 = (ẽ 1 + iẽ 2 ) ,(4.4)
we see that 5) and is therefore of type (2,2). We note that the identification (4.3), (4.4) is consistent with requiring that Ω 1,2 are space-like two-planes in H 2 (K3 1,2 , R). Also, since, e 1 · e 1 = e 2 · e 2 , e 1 · e 2 = 0 and similarly forẽ 1,2 , (4.3), (4.4) are consistent with the requirements that Ω 1 · Ω 1 = Ω 2 · Ω 2 = 0. In addition note thatẽ 3 ,ẽ 6 span a subspace U of H 2 (K3, Z)
G 4 2π = 1 2 Ω 1 ∧Ω 2 +Ω 1 ∧ Ω 2 ,(4.
and are orthogonal to Ω 2 . This ensures that K3 2 is elliptically fibered.
G 4 is orthogonal toẽ 3 ,ẽ 6 and therefore to U , this ensures that one leg of G 4 is along the base and the other along the fiber of K3 2 .
Finally a Kähler form for K3 1 can be chosen meeting the condition (2.1). The Kähler form corresponds to a space-like direction in H 2 (K3, R) orthogonal to Ω 1 . In the example above we could take this direction to be along e 3 , then we see that e 3 · G 4 = 0 so that the condition of primitivity is met.
Thus we see that all the requirements for an N = 1 solution are met in this example.
We will argue next that the complex structure moduli are all frozen about this point.
Consider a small deformation in the complex structure of
K3 1 . Under it Ω 1 → Ω 1 + χ 1 where χ 1 is a (1, 1) form in K3 1 .
Similarly Ω 2 → Ω 2 + χ 2 . Under this transformation,
G 4 2π = 1 2 Ω 1 ∧Ω 2 +Ω 1 ∧ Ω 2 + χ 1 ∧χ 2 +χ 1 ∧ χ 2 + K 4
with the four form K 4 defined as
K 4 = χ 1 ∧Ω 2 +χ 1 ∧ Ω 2 + χ 2 ∧Ω 1 +χ 2 ∧ Ω 1 .) × SU (2) × E 8 × E 8 .
Finally we note that the membrane tadpole condition is met in M-theory. Since
1 2 G 2π ∧ G 2π = 4 < χ 24 = 24,
one will need to add 20 M2 branes (D3 branes) in M-theory (F-theory).
An Orientifold example
By starting with the example above and changing the flux one can alter the complex structure Ω 2 so that the gauge symmetry is reduced. In particular the symmetry can be broken to SO(8) 4 . The resulting model then corresponds to taking the elliptically fibered K3 2 at the orientifold point, with 4 D7-branes at each orientifold plane. It is worth examining this orientifold limit of the example above in more detail. This will allow us to explicitly calculate the masses of the D7-branes. It will also allow us to make contact with the discussion in [24]. We will find below that the orientifold examples corresponds to solutions of the type (2+,0-) in the classification of [24] (section 3.3).
It is quite straightforward to find flux that will stabilise the complex structure at an orientifold singularity. The complex structure moduli space of the elliptically fibered K3 2 is 18 dimensional. At an orientifold point 16 of these moduli correspond to the location of D7-branes along the T 2 base of K3 2 and the remaining two moduli are the dilaton-axion and the complex structure of the T 2 . Requiring that the complex structure is at a (D 4 ) 4 singularity fixes the locations of the D7-branes while allowing the other two moduli to vary. We will proceed in two steps in the discussion below, first finding a particular point in the complex structure moduli space where the singularity is of (D 4 ) 4 type, and then determining a flux which fixes the complex structure at this point.
The model under consideration is dual to the heterotic string on K3 × T 2 . It is well known that by turning on Wilson lines on the heterotic side one can break the gauge symmetry down to SO(8) 4 . Using the duality map, one can then map this to a location in the complex structure moduli space of K3 2 . Before proceeding let us note that, upto a sign convention, our discussion of Wilson lines in the heterotic string will be based on [32].
For simplicity we take the heterotic theory with a square T 2 at the self-dual point, with no B field and with appropriate Wilson lines turned on. The resulting complex structure of K3 2 can then be described as follows. The complex structure of K3 2 corresponds to a twoplane Ω 2 . In §4.1 we described the basis vectors e 1 , · · · e 6 of the subspace H 3,3 ⊂ Γ 3,19 , with e 1 , e 4 spanning the first U subspace etc. It is easy to see that n 1 = e 1 +e 4 2 is a null vector, n 1 · n 1 = 0, meeting the condition n 1 · e 1 = 1. Furthermore it is a basis element for the U sublattice of Γ 3,19 . Similarly, we define the null vector, n 2 = e 2 +e 5 2 , which is a lattice basis element for the second U sublattice of H 3,3 . We will also need to introduce a basis in the (−E 8 ) ⊕ (−E 8 ) sublattice of Γ 3,19 . This is done by choosing vectors, E I , I = 1 · · · 16, E I · E J = −δ IJ . The roots of (−E 8 ) ⊕ (−E 8 ) are then given by q I E I for suitably chosen q I as discussed in Appendix B. The required spacelike two plane corresponds to a choice of two linearly independent space-like vectors. These are given bŷ e 1 ,ê 2 respectively, whereê
1 =ẽ 1 + W I E I +ñ 1 2 W I W I (4.6) e 2 =ẽ 2 +W I E I +ñ 2 2W IWI (4.7)
Here W I ,W I denote the two Wilson lines. Also as in the previous section, we are following conventions where the tilde superscript as inẽ 1 etc, refers to elements of
H 2 (K3 2 ), Γ 3,19 (K3 2 ). With W I = diag(1, 0 7 , 1, 0 7 ) (4.8) W I = diag(0 4 , 1 2 4 , 0 4 , 1 2 4 ),(4.9)
one can show that the roots of Γ 3,19 orthogonal toê 1 ,ê 2 correspond to the gauge group SO(8) 4 . It is easy to see thatê 1 ,ê 2 are linearly independent and therefore define a spacelike two-plane. Identifying this two-plane with Ω 2 gives the location of a (D 4 ) 4 singularity in the complex structure moduli space of K3 2 .
We now turn to determining the required flux which will stabilise the complex structure at this point in moduli space. As discussed in appendix B, 2ê 1 , 2ê 2 , are elements of the integral lattice, Γ 3,19 . So we can consider turning on the four-form flux,
G 4 = 2e 1 ∧ê 1 + 2e 2 ∧ê 2 .
(4.10)
Sinceê 1 ,ê 2 satisfy the relationsê 1 ·ê 1 = 2,ê 2 ·ê 2 = 2,ê 1 ·ê 2 = 0 , we see that the discussion in the previous section goes through unchanged showing that a complex structure for K3 1 × K3 2 exists given by (4.5), with G 4 being of type (2,2). Ω 1 is unchanged from (4.3),
and Ω 2 is given by
Ω 2 =ê 1 + iê 2 .
The orientifold model in more detail
It is worth exploring the orientifold model above in some more detail. We have given a general argument above that all the complex structure moduli are stabilised. Here we would like to explicitly verify this for the D7 moduli by calculating their mass. This will allow us to calculate in the next section the leading contribution (at large volume) to the non-perturbative superpotential due to gaugino condensation.
D7-moduli mass
The complex structure moduli space of the elliptically fibered
W I = diag(0, β, 0 2 , 1 2 4 , 0 4 , 1 2 4 ), (5.2)
where α, β are the locations of the D7-brane along the T 2 . The resulting complex structure of K3 2 is given by (4.11), whereê 1 ,ê 2 are now given bŷ We now turn to determining the mass for the D7-brane moduli. The superpotential is given by
e 1 =ẽ 1 + W I E I +ñ 1 2 W I W I +ñ 2 2 W IWI ,(5.W = G 4 ∧ Ω 4 . (5.5)
G 4 is fixed and given by (4.10), withê 1 ,ê 2 being given by (4.6), (4.7), with the Wilson lines, (4.8), (4.9), respectively. Ω 4 = Ω 1 ∧ Ω 2 . As the 7-brane moves away from the O7 plane the complex structure of K3 2 , Ω 2 , changes, as described in the previous paragraph. Ω 1 is fixed and given by (4.3). By substituting these expressions in W above it is straightforward to see that it takes the form,
W = 4(α + iβ) 2 . (5.6)
We saw above that the complex scalar α + iβ is the location of the D7-brane along the T 2 base of K3 2 . We will denote it by Φ = α + iβ in the discussion below. The quadratic term in the superpotential shows that this modulus acquires a mass. From (5.5), it is clear that the mass is linear in the flux.
One final comment. Our result above for the 7-brane moduli mass agrees with an earlier calculation using the methods of gauged supergravity, [26].
The Non-perturbative Superpotential
The low-energy dynamics of the orientifold model discussed above is that of an N = 1 SO(8) 4 gauge theory. The gauge group arises form the gauge fields on the D7-branes. The D7-branes wrap K3 1 and are transverse to the T 2 base of K3 2 . The D7-brane locations on the T 2 are moduli which are adjoint chiral superfields in the gauge theory. There is one adjoint field for each SO(8) group. We saw above that these moduli acquire a mass due to the flux. This mass scales with the radius of compactification, R, as 1/R 3 . At energy scales below this mass scale the low-energy dynamics is that of a pure N = 1 SYM theory with gauge group SO(8) 4 . It is well known that a non-perturbative superpotential is generated in pure SYM theory due to gaugino condensation. The form of this superpotential can be determined by standard field theory techniques. Since the gauge fields arise from the D7-brane world volume, the gauge coupling is given by
S = 8π 2 g 2 Y M + iθ = e 4u−φ + ib,
where e 4u , is the volume of K3 1 , φ is the dilaton and b is an axion that arises from the RR four-form. The gauge couplings of the four SO(8) groups are the same.
Standard field theory techniques then show that the non-perturbative superpotential is given by :
W NP = Ae −S/c 2 m (5.7)
where m is the mass of the adjoint chiral superfield. A above is a coefficient that depends on the expectation values of the frozen complex structure moduli, and c 2 is the dual Coxeter number in the adjoint representation of the gauge group. For SO(2n), c 2 = (2n−2), giving in particular c 2 = 6 for SO (8).
Let us briefly sketch how this result is obtained. We denote the tree-level superpotential, (5.6), as W tree in the discussion below. This tree-level superpotential has a U (1) R-symmetry under which Φ → Φe iθ . The R-symmetry corresponds to rotations in the plane of the T 2 in K3 × T 2 /Z 2 . We see that W tree has charge 2 under this R-symmetry.
This symmetry is anomalous in the quantum theory and it is easy to see that S → S −2ic 2 θ under it. Now W NP must transform in the same way as W tree under this symmetry, this fixes the dependence on S in (5.7). Similarly W tree has an R-symmetry under which m transforms with charge 2 and Φ is invariant. The R-symmetry can be shown to be non-anomalous so S is invariant under it. This determines the m dependence in (5.7) 5 .
The non-perturbative superpotential (5.7) results in a potential for the volume modulus of K3 1 . In general one would expect that (5.7) is not exact in string theory and there are corrections to it which are subleading at large volume. Requiring that W NP transforms correctly under full duality group, together with the large volume dependence determined above, might help fix the form of these subleading corrections.
Divisors contributing to the superpotential
Let us begin by reminding the reader of Witten's argument [10] that divisors D of arithmetic genus, χ = 1, are the relevant ones for superpotential generation in M-theory compactification on a Calabi-Yau fourfold X. For a Euclidean brane instanton to contribute to the superpotential, it should break half of the space-time supersymmetry, leaving precisely two fermion zero modes. For a Euclidean M5 brane instanton the zero modes 5 These two R symmetries act on the GVW superpotential as follows. Under the first symmetry, Ω → Ωe 2iθ , and the flux G 4 is invariant. Under the second symmetry,
G 4 → G 4 e 2iθ ,
and Ω is invariant. Towards the end of section 6 we will refer to a U (1) discussed in [10]. This symmetry is a combination of the two R-symmetries discussed above. Under it, with our choice of normalisation, Φ and m have charges 2, −2, respectively, while Ω, G 4 , have charges 4, −2.
are determined by the h k,0 (D) cohomology groups of the divisor wrapped by the Euclidean 5-brane. Witten considered the U (1) symmetry of the normal bundle N to D and argued that modes arising from the cohomology groups h 2p,0 (D) and h 2p+1,0 (D) have opposite U (1) charges and can pair up. The index obtained after grading the zero modes by the sign of their charge under this U (1) symmetry is 2χ. Thus a necessary (although not always sufficient) condition for the non-perturbative superpotential to arise is that χ = 1. We will see that after accounting for the effects of G-flux, this conclusion is modified, and in general χ > 1 divisors can also contribute to the space-time superpotential; the loophole was anticipated on p.10 of [10].
In the K3 × K3 examples
Here, we describe in slightly more detail the geometry of the singular K3 × K3 compactifications that must be yielding our gaugino condensates. In each of these cases, the singular elliptic K3 2 has an A-D-E singularity. For simplicity we will focus on the case of A N−1 here, but very similar remarks apply to the other two cases. Our discussion of the relevant geometries follows [23]. In general examples that would arise in elliptic Calabi-Yau fourfolds, this story would generalize as follows. One would look for singularities of the elliptic fibration of Kodaira I N type over some surface S in the base B (the singularities may worsen to I N+1 at codimension one in S, etc.). The D i (and D ′ ) are then nontrivial P 1 bundles over the surface S wrapped by the D7 branes π : D i → S. Our particular example is however quite simple: S is K3 1 and the fiber is constant along S. Therefore, the D i simply take the form
We have an
K3 1 × P 1 .
It is easy to see that the arithmetic genus of these cycles satisfies
χ(D i , O(D i )) = h 0,0 (K3) + h 2,0 (K3) = 2 . (6.1) 6
The additional P 1 which gives rise to the divisor D ′ can be understood as follows. Since is not an instanton effect in the 4d field theory picture, and it is not very surprising that there is no instanton with two fermion zero modes.
More general cases
More generally, our construction suggests the following. Consider any case where in a fourfold, there is a Kodaira type I N degeneration over S as described above (in physics language, this is the situation when there are N D7 branes wrapping S). In general there will be I N+1 curves in S also (physically, these are curves where another D7 intersects the stack of N D7s wrapping S). One can then consider an M5 brane wrapping the cycles D i → S and D ′ → S, which are fibrations over S with P 1 fibers. Using the Leray spectral sequence as on p.6 of [23], we see the following. Since the P 1 fiber has h 1,0 (P 1 ) = 0, and the only holomorphic function on P 1 is a constant, H i,0 (D i ) ≃ H i,0 (S). Therefore, these cycles will have arithmetic genus
χ(D i ) = h 0,0 (S) − h 1,0 (S) + h 2,0 (S) . (6.2)
Let us assume for simplicity that S is simply connected, so h 1,0 (S) = 0. This is not a serious restriction, as in many cases S inherits its first cohomology from the cohomology A recent paper [22] argued that vertical χ = 1 divisors cannot appear in elliptic fourfolds X → B with h 1,1 (B) = 1, and that hence such models cannot have nonperturbative superpotentials for the Kähler modulus. Since our comments suggest otherwise, let us address the contradiction. The arguments presented in [22] do not prove that vertical χ = 1 (or χ > 1) divisors cannot appear in models with h 1,1 (B) = 1. Equation (18) in §2.5 of [22], for the total Chern class of X, is in general incorrect for models where h 1,1 (X) > 2 but h 1,1 (B) = 1. This equation plays an important role in constraining the possible arithmetic genera of divisors. The argument of §2.6 in [22], which shows that no base B with with quark flavors arising from D7s are described in [34]. In many cases, more exotic theories which are not yet well understood can also arise.
In a forthcoming paper [36], the question of a non-perturbative superpotential in cases where the IIB compactification has a single Kähler modulus will be discussed in much greater depth. Several explicit examples of elliptic fourfolds X → B with h 1,1 (B) = 1 (and h 1,1 (X) > 2) and vertical divisors of arithmetic genus one have been found and will be presented in [36]. The simplest examples arise by working with the elliptic fourfold in W P 5 1,1,1,1,8,12 (which is elliptically fibered over P 3 ), constructing singularities of various Kodaira types over a P 2 in the P 3 , and blowing them up.
Finally, let us close this section by noting that KKLT discussed the single Kähler modulus case solely for simplicity of exposition. The arguments presented there are more general and do not depend in any important way on this condition. It would also be interesting to understand the story when our models are reduced to 3d directly in the language of M5 instantons. A 4d N = 1 pure gauge theory, reduced to 3d on a circle, exhibits superpotential generation due to abelian instantons on the Coulomb branch, and a microscopic version of this involving M5 instantons appeared in [23]. Our situation involves a 4d theory which has chiral multiplet matter fields, which are then given a mass. After the reduction to 3d, an instanton computation in the 3d field theory must still yield a nonzero result: the mass terms for the matter fields can be pulled down to absorb any extra zero modes which naively appear in the instanton calculation. This macroscopic phenomenon must have a microscopic analogue in M5 instanton calculations, and it would be interesting to understand this in detail. One comment in this regard is worth making. As discussed in [10] the arithmetic genus of the divisor wrapped by the M 5 brane corresponds to an index graded by a U (1) symmetry. This U (1) symmetry is the structure group of the normal bundle of the divisor. In the K3 1 × K3 2 case, as is discussed further in appendix C, one finds that the symmetry is broken by the flux. This suggests that in the M5 brane worldvolume theory, modes coming from h 2,0 can pair up amongst themselves and became massive in the presence of flux, allowing for a superpotential even when χ > 1.
Discussion
Finally, we note that in our K3 × K3 examples, G 4 fluxes which reduce purely to three-form fluxes in the IIB language (no field strengths F 2 turned on in the D7 branes) suffice to stabilize all D7 brane moduli. It would be interesting to see if this phenomenon persists in more generic examples. 8 We can obtain the intersection matrix for these two curves as follows. In general a genus g curve in K3 has self-intersection number equal to (2g −2) and hence for the section (which is a P 1 ) it is −2, whereas for the fiber it is zero. Since both the curves intersect each other transversely, their intersection number is one. Thus the intersection matrix is −2 1 1 0 which, after a change of basis, is identical to U . This means that the necessary condition for a K3 of Weierstraß type is that the Picard lattice must contain U as a sublattice. This also turns out to be a sufficient condition for the K3 to admit an elliptic fibration.
It is clear from the discussion above that the two elements of H 2 (K3, R) dual to the base and the fiber span U . Thus if G 4 must have one leg along the base and the other along the fibre of K3 2 it cannot contain any element of the sublattice U . Since the matrix U has eigenvalues (1, −1), this means G 4 must be orthogonal to all vectors in U . Here we discuss the (−E 8 )⊕(−E 8 ) lattice in more detail. The discussion will be based on section 11.6 in Polchinski's book [32] with some changes in convention, to account for the time-like nature of the lattice in our case etc. As mentioned in the discussion above we take 16 vectors, E I , I = 1, · · · 16 that are linearly independent and satisfy the the conditions E I · E J = −δ IJ . These vectors span the (−E 8 ) ⊕ (−E 8 ) vector space. The first (−E 8 ) lattice is given by vectors of the form 8 I=1 q I E I , where the (q 1 , · · · q 8 ) are either all integer or half integer and I q I ∈ 2Z. The second (−E 8 ) lattice is similarly given by appropriate linear combinations of E I , I = 9, · · · 16.
It is now easy to see why 2ê 1 , 2ê 2 , (4.6),(4.7), belong to the lattice Γ 3,19 .ẽ 1 , n 1 belong to the first U sublattice of Γ 3,19 , as we discussed in section 3 and 4, and similarlyẽ 2 , n 2 belong to the second U sublattice. For the choice (4.8), (4.9), we have W I W I =W IWI = 2.
So to show that 2ê 1 , 2ê 2 , belong to Γ 3,19 it is enough to show that 2W I E I and 2W I E I belong to it. But we see that both of these vectors are of the form q I E I with q I meeting the conditions mentioned above, so this is true.
Appendix C. Flux Breaks the M 5 Brane Worldvolume U (1) Symmetry
As discussed in §6 the arithmetic genus of the divisor wrapped by the Euclidean M 5 brane corresponds to an index graded by the U (1) symmetry which is the structure group of the normal bundle of the divisor. Here we show that in the K3 1 × K3 2 case this U (1) symmetry is broken by the G 4 flux.
Roughly speaking the argument is as follows. The divisor is K3 1 × P 1 , with the P 1 being an exceptional divisor of K3 2 associated with blowing up an ADE singularity. As we saw above the flux, G 4 , must have one leg along the fiber and one along the base of K3 2 . Now the singularity is located at a particular point in the base of K3 2 so the tangent along the base is normal to the divisor. Thus G 4 breaks the U (1) symmetry.
More precisely, let us illustrate this by taking the case of an A 1 singularity. In the vicinity of the singularity, K3 2 is described by the equation,
y 2 = x 2 − z 2 ,
in C 3 , with the coordinate along the base of K3 2 being say, z. Now the resolved K3 2 is partially covered by the coordinates z, s 2 , s 1 , where y = s 2 z, x = s 1 z, and s 2 , s 1 , satisfy the relation Away from z = 0, the tangent vector along the the base P 1 is given by ∂ z , and the tangent vector along the fiber T 2 by ∂ s 1 + ∂s 2 ∂s 1 ∂ s 1 . So it is clear that as z → 0, and we approach the divisor, the tangent vector along the fiber becames the tangent vector to the divisor and the tangent vector along the base becames the normal to the divisor. Since, as we mentioned above, G 4 has one leg along the base and one along the fiber of K3 2 , we see that the flux breaks the U (1) symmetry.
The argument above clearly generalises to other ADE singularities in K3 2 as well. 9 These coordinates miss two points in the resolved K3 2 . The divisor is actually the surface s 2 = s 2 1 − s 2 3 in P 2 , and the two points not included are s 3 = 0, s 2 = ±s 1 .
K3s is then given (again up to discrete identifications) by O(2, 18)/O(2) × O(18). This is 18 (complex) dimensional, the counting agrees with the moduli in the Weierstraß form,(2.2). The Picard lattice of K3 is defined as the lattice of integral two-forms which are of
components in G 4 must vanish. This imposes one more condition than the number of complex structure moduli. For a choice of flux where a susy solution does exist this implies that all the complex structure moduli should generically be lifted.Second, the four-form flux, G 4 , gives rise to three form flux in the IIB description. Let the holomorphic and anti-holomorphic differentials along the elliptically fibered torus of K3 2 be dz and dz, and φ be the modular parameter (the axion-dilaton in the IIB theory).
The basis elements are chosen so that e 1 , e 4 span the first U sublattice and so on. Also we note that e 1 , e 2 , e 3 are space-like and the rest are time-like. Now consider the flux G 4 2π = e 1 ∧ẽ 1 + e 2 ∧ẽ 2 (4.1)
end this section with a few comments. First, since G 4 is of form (4.5), our discussion in the previous section still goes through leading to the conclusion that the complex structure moduli are all stabilised. Second, we note that G 4 above does not have any component along the root lattice of SO(8) 4 , thus no gauge field flux F 2 is turned on in this case along the 7-brane world volumes. Third, the total contribution to the membrane tadpole condition is 16, this means 8 D3 branes would have to be added in the F-theory description. Fourth, the fact that this example maps to the (2+, 0−) case in[24] as mentioned above, follows simply by noting that there are only two linearly independent two-forms, e 1 , e 2 of K3 1 in G 4 . Finally, for simplicity here we have focused on one choice of flux. More generally other choices of flux can also stabilise the complex structure at a (D 4 ) 4 singularity for other values of the dilaton-axion and τ .
K3 2 is 18 complex dimensional. At the orientifold point, 16 of these 18 moduli correspond to Wilson lines that give the location of the D7-branes along the base T 2 of K3 2 . The remaining two moduli correspond to the dilaton-axion, and the modular parameter of the base T 2 . At the orientifold point, as was mentioned above, 4 D7 branes are located at each O7-plane. By symmetry one can see that all the D7-branes must have the same mass. Displacing one D7-brane from the O7-plane breaks the symmetry to SO(6) × SO(8) 3 . The corresponding Wilson lines are given by W I = diag(1, α, 0 6 , 1, 0 7 ), (5.1)
A N−1 singularity over K3 1 . Denote the exceptional divisor by D. D is the union of N −1 irreducible components D = ∪ i D i . After blowing up, the fiber over K3 1 will consist of N P 1 s, intersecting in such a way as to form the affine Dynkin diagram for SU (N ) (the D i have been supplemented by an additional divisor D ′ , which is the closure of the complement of the exceptional set D inside the resolved elliptic fiber) 6 .
K3 2
2is elliptically fibered there is a null vector, n, dual to the T 2 fiber. Like the roots of A N −1 , e i , n is also orthogonal to the the two-plane, Ω 2 . The additional P 1 is dual to n −e i . And e i together with n − i e i give rise to the Dynkin diagram of affine A N −1 . Hence, while it is completely clear from the 4d and 10d perspectives that our examples have pure Yang-Mills sectors which will undergo gaugino condensation, there need not be divisors of arithmetic genus 1. This is in keeping with the remarks in [10] about how infrared gauge theory effects may not give the correct naive zero mode count required for superpotential generation. It is suggestive that an M5-brane wrapping D (i.e., all N P 1 s) would be wrapping a cycle of χ(D) = 2N . This coincides with the number of fermion zero modes which are present in a naive instanton calculation in pure SU (N ) N = 1 Yang-Mills theory, where the gaugino condensate λλ scales like the N th root of a gaugino 2N-point correlator (see e.g. [33], pages 184-186). This underscores once again the fact that gaugino condensation
of the base of the F-theory elliptic fourfold (by the Lefschetz hyperplane theorem), and for simple examples this vanishes. We then see that χ(D i ) > 0, and all zero modes but the one arising from h 0,0 (S) are in correspondence to adjoint matter fields on the D7 branes wrapping S. (Recall a D7 brane wrapping S receives h 1,0 adjoint fields from Wilson lines on S, which vanishes for us, and h 2,0 adjoint fields from deformations of S inside the compactification manifold).7 Hence in all cases where the matter fields are lifted by three-form fluxes leaving an I N degeneration over S, there are also cycles present of the appropriate arithmetic genus (χ ≥ 1) to possibly contribute to the superpotential.We therefore expect that in many examples of IIB compactifications with flux, one will obtain contributions to the nonperturbative superpotential from stabilized coincidentD7 branes. This can happen sometimes even in the absence of cycles of arithmetic genus 1 in the related Calabi-Yau fourfold. However in all such cases, we expect (as in the K3×K3 examples) that divisors of χ ≥ 1 exist. We see no obstruction to such examples arising even in cases where the IIB compactification manifold has a single Kähler modulus. For instance, it is easy to write down examples in the elliptic fibration over P 3 which have singularities of various Kodaira types over a surface S (which has h 1,0 (S) = 0) in P 3 . The question is then whether appropriate fluxes can stabilize D7s on such a locus. If so, the resulting theory would exhibit gaugino condensation and a nonperturbative superpotential for the Kähler modulus. Our explicit examples of this in K3 × K3 give us confidence that the phenomenon will happen in more general examples.
h 1, 1
1(B) = 1 can be globally fibered over a surface S, is true. However, that fact is not relevant to the phenomena under discussion here. A global fibration of B over some S is not needed to obtain nonabelian gauge symmetries from coincident D7 branes in B, and it is straightforward to write down examples of I n singularities with n > 1 in the fourfolds of [35] with h 1,1 (B) = 1.7 In general there can also be fundamental matter fields arising from the intersections with other D7s. These also yield a nonchiral spectrum in simple examples, and analogues of the phenomena we are exploring here should be expected to occur. Some examples of SU (N c ) theories
There are several obvious directions for further work. Needless to say, it would be interesting to find explicit examples of the phenomena we have seen in K3 × K3 in other Calabi-Yau fourfolds. While explicit examples where all Kähler moduli can be stabilized already exist [11], we believe the kinds of effects described here will greatly broaden the class of examples.
Appendix B. The (−E 8 ) ⊕ (−E 8 ) Lattice
exceptional divisor in K3 2 is partially covered by z = 0 with s 2 , s 1 , satisfying (C.1) 9 .
and U . Thus the enhanced gauge symmetry in this example is SU (2Now since, Ω 1,2 ,Ω 1,2 are linearly independent, K 4 cannot vanish, so G 4 cannot be of type
(2, 2) after the small deformation. Thus for small deformations, G 4 no longer remains
(2, 2) and the complex structure moduli are all lifted.
Let us now turn to describing the enhanced symmetry. We note that the lattice vectors
of E 8 ⊕E 8 ⊂ Γ 3,19 are orthogonal to Ω 2 . In addition they are orthogonal to the U sublattice
spanned by (ẽ 3 ,ẽ 6 ). Similarly the vectorsẽ 4 ,ẽ 5 are roots which are orthogonal to both Ω 2
Masses for D7/D3 moduli have been extracted from string scattering amplitudes for type IIb orientifolds on T 6 /(Z 2 × Z 2 ) in[27].
It is clear that generic G 4 fluxes which reduce to both three-form fluxes and F 2 fluxes in IIB, do suffice to stabilize all D7 moduli in generic examples. This follows from simple counting of equations and fields given the GVW superpotential for a CY fourfold.
AcknowledgementsWe would like to thank P. AspinwallAppendix A. More on Elliptically Fibered K3Here we will describe how the requirement that U ⊂ P ic(K3) arises for an elliptically fibered K3.In general, as we know, the second cohomology group for K3 is given by the latticewhere U is the hyperbolic plane, (3.1), and E 8 is the Cartan matrix for the E 8 lattice.Being an elliptic fibration, the K3 obviously has, at least, two algebraic curves embedded in it, one being a nontrivial section of the bundle (which is guaranteed from the Weierstraß form) and the other being the elliptic fiber. Obviously, they are elements of the second homology group H 2 (K3, Z) (which is same as H 2 (K3, Z)), and since they are algebraic they also belong to H 1,1 (K3). Thus they belong to the Picard lattice of K3.
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[] |
[
"ON THE SEMIPRIMITIVITY AND THE SEMIPRIMALITY PROBLEMS FOR PARTIAL SMASH PRODUCTS",
"ON THE SEMIPRIMITIVITY AND THE SEMIPRIMALITY PROBLEMS FOR PARTIAL SMASH PRODUCTS"
] |
[
"Rafael Cavalheiro ",
"Alveri Sant'ana "
] |
[] |
[] |
In this paper we discuss about the semiprimitivity and the semiprimality of partial smash products. Let H be a semisimple Hopf algebra over a field and let A be a left partial H-module algebra. We study the H-prime and the H-Jacobson radicals of A and its relations with the prime and the Jacobson radicals of A#H, respectively. In particular, we prove that if A is H-semiprimitive, then A#H is semiprimitive provided that all irreducible representations of A are finite-dimensional, or A is an affine PI-algebra over and is a perfect field, or A is locally finite. Moreover, we prove that A#H is semiprime provided that A is an H-semiprime PI-algebra, generalizing for the setting of partial actions, the main results of [20] and [19].Theorem 4.33. Let H be a finite-dimensional Hopf algebra and let A be a partial H-module algebra. ThenIn particular, A is H-semiprimitive if and only if A#H is H › -semiprimitive.Proof. We will denote by R :" A#H. Let Φ and Ψ as in the Corollary 3.7. Since J H › pRq is H › -stable and J H › pRq Ď JpRq, by Proposition 4.25, we haveOn the other hand, also by Proposition 4.25,Since J H pAq#H is an H › -stable ideal, by Proposition 4.31, it follows that J H pAq#H Ď pJpRq : H › q " J H › pRq.Thus the second equality holds. The first one follows from the properties of Φ and Ψ because
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10.1142/s1005386718000020
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[
"https://arxiv.org/pdf/1510.05013v1.pdf"
] | 119,316,829 |
1510.05013
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00d8f126b7c4f05d81a061d0216f040fd65459e7
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ON THE SEMIPRIMITIVITY AND THE SEMIPRIMALITY PROBLEMS FOR PARTIAL SMASH PRODUCTS
16 Oct 2015
Rafael Cavalheiro
Alveri Sant'ana
ON THE SEMIPRIMITIVITY AND THE SEMIPRIMALITY PROBLEMS FOR PARTIAL SMASH PRODUCTS
16 Oct 2015
In this paper we discuss about the semiprimitivity and the semiprimality of partial smash products. Let H be a semisimple Hopf algebra over a field and let A be a left partial H-module algebra. We study the H-prime and the H-Jacobson radicals of A and its relations with the prime and the Jacobson radicals of A#H, respectively. In particular, we prove that if A is H-semiprimitive, then A#H is semiprimitive provided that all irreducible representations of A are finite-dimensional, or A is an affine PI-algebra over and is a perfect field, or A is locally finite. Moreover, we prove that A#H is semiprime provided that A is an H-semiprime PI-algebra, generalizing for the setting of partial actions, the main results of [20] and [19].Theorem 4.33. Let H be a finite-dimensional Hopf algebra and let A be a partial H-module algebra. ThenIn particular, A is H-semiprimitive if and only if A#H is H › -semiprimitive.Proof. We will denote by R :" A#H. Let Φ and Ψ as in the Corollary 3.7. Since J H › pRq is H › -stable and J H › pRq Ď JpRq, by Proposition 4.25, we haveOn the other hand, also by Proposition 4.25,Since J H pAq#H is an H › -stable ideal, by Proposition 4.31, it follows that J H pAq#H Ď pJpRq : H › q " J H › pRq.Thus the second equality holds. The first one follows from the properties of Φ and Ψ because
Introduction
Let H be a finite-dimensional Hopf algebra over a field and let A be a left H-module algebra. An important question in the theory of Hopf algebra actions is to know when the smash product A#H is semiprime. It is well known that if A is semiprime, then A#H is semiprime in the following cases: if H " G and |G|´1 P [13]; or if H " p Gq › [7]. In both cases H is semisimple Artinian. This suggests the following question (raised by Cohen and Fishman in [6]): Question 1. Let H be a semisimple Hopf algebra and let A be a semiprime H-module algebra. Is the smash product A#H also semiprime?
Many special cases of the Question 1 have been answered by adding hypotheses on H or on A (e.g. [24], [21], [20] and [19]). In [20] the authors tackled this question by studying the stability of the Jacobson radical. From this approach, naturally new related questions arisen about the semiprimitivity and the semiprimality of the smash product: Question 2. Let H be a semisimple Hopf algebra over a field and A an H-semiprimitive (resp. H-semiprime) H-module algebra. Is the smash product A#H semiprimitive (resp. semiprime)?
In [20] the authors showed that if has characteristic 0, then the Question 2 has an affirmative answer for the semiprimitivity case provided that A is a PI-algebra which is either affine or algebraic over , or all irreducible representations of A are finite-dimensional, or A is locally finite; if has positive characteristic then additional hypotheses were assumed. In [19] the authors showed that the answer for the case of semiprimality in Question 2 is 'yes' provided that A is a PI-algebra.
Partial group actions were first defined by R. Exel in the context of operator algebras in the study of C › -algebras generated by partial isometries on a Hilbert space [11]. In [9] partial group actions were defined axiomatically and in [5] the authors extended these concepts for the context of partial Hopf actions. Since then many papers were published in this subject and the partial actions became an independent area of research in ring theory. Significant advances were made in this area as, for example, Galois theory ( [10], [5]), Morita theory ( [2], [1]) and partial representations ( [9], [3]).
In this work we generalize the main results of [20] and [19] for the context of partial Hopf actions. More precisely, we consider a semisimple Hopf algebra H over a field and a left partial H-module algebra A. We prove that if A is H-semiprimitive then A#H is semiprimitive provided that all irreducible representations of A are finite-dimensional (Theorem 5.1); or A is an affine PI-algebra over and is a perfect field (Theorem 5.3); or A is locally finite (Theorem 5.5). Differently of [20] we not suppose additional hypotheses when has positive characteristic (except in the second case before). We also prove that A#H is semiprime when A is H-semiprime and satisfies a polynomial identity (Theorem 5.8).
For our purposes we study the concepts of H-prime and H-Jacobson radicals of a partial H-module algebra, which arise naturally from the concepts of H-stable ideal and partial pA, Hqmodules as a generalization of the concepts of prime and Jacobson radicals of an any algebra. This concepts of H-stable ideal and partial pA, Hq-modules are adaptations, for the case of partial actions, of the correspondent concepts of H-stable ideal and pA, Hq-modules studied, for example, in [12], [24] and [26] for global actions.
Partial (co)actions and partial smash products
Throughout this paper H will denote a Hopf algebra over a field (and, unless mentioned otherwise, all algebras will be over the same field ). Also we use the Sweedler's notation to denote the comultiplication of an element h P H, that is, ∆phq " ř h 1 b h 2 . We start recalling some well-known concepts which are fundamentals for this paper. For more details we refer [5].
Definition 2.1. A (left) partial action of a Hopf algebra H on an algebra A is a -linear map ξ : H b A Ñ A, h b a Þ Ñ h¨a, such that, for any a, b P A and h, g P H, we have (PA1) 1 H¨a " a (PA2) h¨papg¨bqq " ř ph 1¨a qpph 2 gq¨bq. In this case, A is called a (left) partial H-module algebra.
Of course, we can define right partial action of H on A, but in this paper we will consider only left actions of H on A so that the expression partial H-module algebra will means left partial H-module algebra. The same occur later for coactions, but on the right.
In this paper we will consider only unital algebras. In such a case, the condition (PA2) of the Definition 2.1 can be replaced for the following two conditions: for any a, b P A and h, g P H, (PA3) h¨pabq " ř ph 1¨a qph 2¨b q (PA4) h¨pg¨bq " ř ph 1¨1A qpph 2 gq¨bq. It is easy to verify that if A is a partial H-module algebra, then h¨1 A " εphq1 A , @ h P H ðñ h¨pg¨aq " phgq¨a , @ a P A, @ h, g P H.
When this is the case, A is called an H-module algebra (see [8,Definition 6.1.1]) and ξ is called global action. On the other hand, is immediate to see that any H-module algebra is a partial H-module algebra.
Example 2.2. [3, Proposition 1] Let B be an H-module algebra (global action) and let A be a right ideal of B with identity element 1 A . Then A becomes a partial H-module algebra by
h¨a :" 1 A ph Ż aq, @ a P A, @ h P H where Ż indicates the action of H on B.
As a particular case of this example, the authors in [2, p. 5] have considered a finite group G and the Hopf algebra H " p Gq › . Then B " G is an H-module algebra by
p g Ż h " δ g,h h , @ g, h P G.
Now consider a normal subgroup N G, N ‰ t1 G u, such that char ffl |N |. Thus, the element
e N " 1 |N | ÿ nPN
n is a central idempotent in G as it is easy to see. Hence, the ideal A " e N B is a unital -algebra with 1 A " e N and the action induced of H on A as in Example 2.2 is such that, for every g P N , p g¨eN " e N pp g Ż e N q " p1{|N |qe N g " p1{|N |qe N ‰ δ 1,g e N " εpp g qe N .
In particular, the partial action of H on A is not a global action.
Example 2.3. [10, Example 6.1] Let B be an algebra and let
A " BˆBˆB " Be 1 ' Be 2 ' Be 3 ,
where e 1 " p1 B , 0, 0q, e 2 " p0, 1 B , 0q and e 3 " p0, 0, 1 B q. If G " t1 G , g, g 2 , g 3 u is the cyclic group of order 4, then H " G acts partially on A by g¨e 1 " 0 , g¨e 2 " e 1 , g¨e 3 " e 2 , g 2¨e 1 " e 3 , g 2¨e 2 " 0 , g 2¨e 3 " e 1 , g 3¨e
1 " e 2 , g 3¨e 2 " e 3 , g 3¨e 3 " 0.
In the study of partial H-module algebras an important subalgebra is the so called invariant subalgebra.
Definition 2.4. Let A be a partial H-module algebra. The subalgebra of the invariant elements of A is defined as the set
A H " ta P A : h¨a " aph¨1 A q, @ h P Hu .
It is easy to see that A H is actually a subalgebra of A. When the action of H on A is global then we denote the invariant subalgebra by A H . If it is the case, we have A H " ta P A : h¨a " εphqa, @ h P Hu .
Proposition 2.5. Let A be a partial H-module algebra. If a P A H is invertible in A, then a´1 P A H . In particular, we have JpAq X A H Ď JpA H q.
Proof. For any h P H, we have h¨a´1 " a´1 ÿ aph 1¨1A qph 2¨a´1 q " a´1 ÿ ph 1¨a qph 2¨a´1 q " a´1ph¨paa´1qq " a´1ph¨1 A q and so a´1 P A H . As a consequence, I :" JpAq X A H is an ideal of A H such that 1´I is contained in the set of units of
A H . Hence JpAq X A H Ď JpA H q.
Dualizing the concept of (left) partial H-module algebra we obtain the definition of (right) partial H-comodule algebra as follows.
Definition 2.6. A (right) partial coaction of a Hopf algebra H on an algebra R is a -linear map ρ :
R Ñ R b H, x Þ Ñ ř x 0 b x 1 such that, for any x, y P R, (PC1) pid R b εqpρpxqq " x (PC2) ρpxyq " ρpxqρpyq (PC3) pρ b id H qpρpxqq " rρp1 R q b 1 H srpid R b ∆qpρpxqqs.
In this case, R is called (right) partial H-comodule algebra.
In the Sweedler's notation, the above conditions can be written as follows
(PC1) ř x 0 εpx 1 q " x (PC2) ř pxyq 0 b pxyq 1 " ř x 0 y 0 b x 1 y 1 (PC3) ř x 00 b x 01 b x 1 " ř 1 0 x 0 b 1 1 x 11 b x 12 .
for any x, y P R.
Analogous to the case of partial actions, if R is a partial H-comodule algebra and
ρp1 R q " 1 R b 1 H ,
then R is an H-comodule algebra (see [8,Definition 6.2.1]). In this case ρ is called global coaction. Also, it is immediate that any H-comodule algebra is a partial H-comodule algebra.
Definition 2.7. Let R be a partial H-comodule algebra. The subalgebra of the coinvariant elements of R is defined as the set
R coH " ! x P R : ρpxq " px b 1 H qρp1 R q " ÿ x1 0 b 1 1
) .
The fact that R coH is a subalgebra of R is an immediate consequence of (PC2).When the coaction is global we denote the coinvariant subalgebra by R coH , that is
R coH " tx P R : ρpxq " px b 1 H qu .
In the same sense of Proposition 2.5, we can prove the following result.
Proposition 2.8. Let R be a partial H-comodule algebra. If x P R coH is invertible in R, then x´1 P R coH . In particular, we have JpRq X R coH Ď JpR coH q.
Proof. In fact,
ρpx´1q " p1 R b 1 H qρp1 R x´1q " px´1 b 1 H qpx b 1 H qρp1 R qρpx´1q " px´1 b 1 H qρpxqρpx´1q " px´1 b 1 H qρpxx´1q " px´1 b 1 H qρp1 R q.
The rest is similar to the proof of Proposition 2.5.
We finish this section recalling the definition of the partial smash product. Let A be a partial H-module algebra. On the vector space A b H we consider the following multiplication:
pa b hqpb b gq " ÿ aph 1¨b q b h 2 g, @ a, b P A, @ h, g P H.
With this multiplication (and the usual addition) A b H becomes an associative algebra. This algebra is denoted by A#H and their (generators) elements by a#h instead of a b h. In general, A#H has not identity element unless the action of H on A is global (see [8,Proposition 6.1.7]). Therefore we consider the following subspace of A#H:
A#H :" pA#Hqp1 A #1 H q,
generated by elements of the form
a#h :" pa#hqp1 A #1 H q " ÿ aph 1¨1A q#h 2 .
Thus, A#H is an associative algebra with identity element
1 A #1 H " 1 A #1 H .
Definition 2.9. Let A be a partial H-module algebra. The algebra A#H is called the partial smash product of A by H.
Clearly, if A is an H-module algebra (global action) then A#H " A#H. Moreover, if A is a partial H-module algebra then there is a natural monomorphism of algebras given by
f : A Ñ A#H Ď A#H a Þ Ñ a#1 H " a#1 H ,
and we will identify A with the subalgebra f pAq " A#1 H " A#1 H of A#H. We further note that the partial smash product A#H has a structure of (right) H-comodule algebra (global coaction) given by
ρ : A#H Ñ A#H b H a#h Þ Ñ a#h 1 b h 2 ,
and that`A#H˘c oH " A.
The next remark will be useful later.
Remark 2.10. Suppose that A is a partial H-module algebra. Then the above observation and Proposition 2.8 give us the following interesting result
J`A#H˘X A Ď JpAq.
It is well known that if H is finite-dimensional, then its dual H › is a Hopf algebra. In this case, A#H has a structure of left H › -module algebra induced by its right H-comodule algebra structure, where the (global) action is given by
ϕ Ż`a#h˘:" ÿ a#h 1 ϕph 2 q, a P A, h P H, ϕ P H › .
Moreover,`A#H˘H › "`A#H˘c oH " A (see [8,Proposition 6.2.4]).
3.
On H-stable ideals and partial pA, Hq-modules
Hereafter, unless mentioned otherwise, A will denote a partial H-module algebra. In [12] and [26] the authors investigated the concepts of H-stable ideal of A and pA, Hq-module in the study of H-radicals when the action of H on A was global. These concepts can be adapted for the case of partial actions as we will see later. In the next section, we will explore these concepts in the study of the H-prime and H-Jacobson radicals of A in the setting of partial actions. In particular, we will establish a link between the H-prime radical of A and the H › -prime radical of A#H and also a link between the H-Jacobson radical of A and the H › -Jacobson radical of A#H, when H is finite-dimensional.
We begin with the concept of H-stable ideal. The aim here is to establish a link between the set of all H-stable ideals of A and the set of all ideals of A#H which are H-subcomodules (Theorem 3.6). When H is finite-dimensional this last set is just the set of all H › -stable ideals of A#H (Corollary 3.7). Let I be an H-stable ideal of A. Then the partial action of H on A induces a partial action of H on the quotient A{I given by h¨pa`Iq :" ph¨aq`I , a P A, h P H.
Moreover, we have the following algebra isomorphism:
A#H˘{`I#H˘-pA{Iq#H ,
where I#H :" ř x i #h i : x i P I, h i P H ( . As a consequence we have the following result.
Proposition 3.3. Suppose that the partial H-module algebra A is a subdirect product of partial H-module algebras A α -A{I α , where tI α u is a family of H-stable ideals of A such that Ş I α " 0. Then R :" A#H is a subdirect product of the algebras R α :" A α #H.
Proof. As above, for every α, there is an algebra isomorphism R{`I α #H˘-R α , thus it is enough to show that Ş`I α #H˘" 0. On the other hand, for every α we have that I α #H Ď I α #H. Thus, we will have achieved our goal if we show that Ş pI α #Hq " 0. Suppose that u P Ş pI α #Hq and write u " ř x i #h i , where tx i u Ď A and th i u Ď H, with th i u linearly independent. Then x i P I α for every i and every α. Thus, x i P Ş I α " 0, for every i, and so u " 0. Hence Ş pI α #Hq " 0, as desired.
The above proposition will be useful in the study of the semiprimitivity of the partial smash product because a subdirect product of semiprimitive algebras is also a semiprimitive algebra (see [14,Proposition 2.3.4]).
Given a subspace X Ď A, we will denote by pX : Hq :" tx P X : h¨x P X, @ h P Hu . Proof. For every x P pI : Hq, a P A and h P H, we have h¨pxaq " ÿ ph 1¨x qph 2¨a q P IA Ď I, so xa P pI : Hq. Analogously ApI : Hq Ď pI : Hq and therefore, pI : Hq is an ideal of A. Moreover, if x P pI : Hq and g P H then
h¨pg¨xq " ÿ ph 1¨1A qpph 2 gq¨xq P AI Ď I, @ h P H, so g¨x P pI : Hq. Thus pI : Hq is H-stable. Now we observe that pI : Hq contains every H-stable ideal of A which is contained in I and so the proof is complete.
Proposition 3.5. If I is an ideal of A#H, then I X A is an H-stable ideal of A.
Proof. Clearly I X A is an ideal of A. Moreover, for any x P I X A and h P H,
h¨x#1 H " ÿ h 1¨x #εph 2 q1 H " ÿ ph 1¨x qph 2¨1A q#h 3 Sph 4 q " ÿ ph 1¨x #h 2 qp1 A #Sph 3 qq " ÿ p1 A #h 1 qpx#1 H qp1 A #Sph 2 qq P I, so h¨x P I X A. Hence I X A is an H-stable ideal of A.
The next result generalizes, for the setting of partial actions, a known result about global actions (which is a consequence of [24,Lemma 1.3], because if the action of H on A is global, then A Ď A#H is a faithfully flat H-Galois extension).
Theorem 3.6. Let H be a Hopf algebra and let A be a partial H-module algebra. Then there exists an one-to-one correspondence between the sets
tH-stable ideals of Au Φ / / tIdeals of A#H which are H-subcomodulesu Ψ o o
given by ΦpIq " I#H, where I is an H-stable ideal of A, and ΨpIq " I X A, where I is an ideal of A#H which is H-subcomodule. Moreover, Ψ " Φ´1 and these maps preserve inclusions, sums, (finite) products and intersections.
Proof. We will denote by R :" A#H. Let I be an H-stable ideal of A and I an ideal of R which is H-subcomodule. For any a, b P A, x P I and h, g, k P H, we have pa#hqpx#gqpb#kq " ÿ aph 1¨x qpph 2 g 1 q¨bq#h 3 g 2 k P pApH¨IqAq#H Ď I#H.
Thus R`I#H˘R Ď I#H and so I#H is an ideal of R. Moreover, by Proposition 3.5 and using that I#H is an H-subcomodule of R we can deduce that Φ and Ψ are well defined. Now consider a basis th i u of H containing 1 H . If u P I#H XA Ď pI#HqXA, write u " ř x i #h i with tx i u Ď A. Since th i u is a basis of H, it follows that tx i u Ď I. On the other hand, 1 H P th i u and u P A, so x i " 0 if h i ‰ 1 H and therefore u P I. This shows that I#H X A Ď I. The other inclusion is clear and so it follows that ΨpΦpIqq " I.
To prove that ΦpΨpIqq " I, fix ř x i #h i P I. Then we have
ÿ x i #h i " ÿ`x i ph i1¨1A q#1 H˘`1A #h i2" ÿ`x i #h i11˘`1A #Sph i12 q˘`1 A #h i2˘. Since I is an H-subcomodule of R, ř x i #h i11 b h i12 b h i2 P I b H b H, so ř x i #h i " ř`x i ph i1¨1A q#1 H˘`1A #h i2˘P pI XAq`A#H˘" pI X Aq#H.
This shows that I Ď pI X Aq#H and the other inclusion is clear. Hence I " pI X Aq#H " ΦpΨpIqq and so Φ and Ψ are bijections with Ψ " Φ´1.
Now we need to prove that Φ and Ψ preserve inclusions, sums, (finite) products and intersections. For inclusions, this statement is clear from definitions of Φ and Ψ. With regard to the others, firstly note that sums, finite products and intersections of H-stable ideals of A (resp. H-subcomodules of A#H) are H-stable ideals of A (resp. H-subcomodules of A#H). Now fix a family tI α u αPΛ of H-stable ideals of A and a family tI β u βPΓ of ideals of A#H which are H-subcomodules. Clearly
Φ´ÿ I α¯" p ř I α q#H " ÿ I α #H " ÿ ΦpI α q
and consequently,
Ψ´ÿ I β¯" Ψ´ÿ ΦpΨpI β qq¯" Ψ´Φ´ÿ ΨpI β q¯¯" ÿ ΨpI β q.
Therefore Φ and Ψ preserve sums. Now observe that if I is a H-stable ideal of A, then I#H " IR and so IR is an ideal of R. In fact, the inclusion IR Ď I#H is clear. On the other hand, for any x P I and h P H, x#h " px#1 H qp1 A #hq P IR and also I#H Ď IR. Thus, for any α 1 , α 2 P Λ, ΦpI α1 I α2 q " pI α1 I α2 q#H " pI α1 I α2 qR " I α1 pI α2 Rq " I α1 pRI α2 Rq " pI α1 RqpI α2 Rq " pI α1 #HqpI α2 #Hq " ΦpI α1 qΦpI α2 q Consequently, for any β 1 , β 2 P Γ, ΨpI β1 I β2 q " Ψ´Φ`ΨpI β1 q˘Φ`ΨpI β2 q˘¯" Ψ´Φ´ΨpI β1 q ΨpI β2 q¯¯" ΨpI β1 qΨpI β2 q.
Hence, for finite products of H-stable ideals of A or ideal of R which are H-subcomodules, the result follows by induction.
Finally, we show that Φ and Ψ preserve intersections. Clearly
Ψ´č I β¯"´č I β¯X A " č pI β X Aq " č ΨpI β q,
and it follows that
Φ´č I α¯" Φ´č ΨpΦpI α qq¯" Φ´Ψ´č ΦpI α q¯¯" č ΦpI α q.
The proof is complete now.
If H is finite-dimensional, since the action of H › on A#H is global, then a subspace of A#H is an H-subcomodule if and only if it is an H › -submodule (see [23,Lemma 1.6.4]). Thus, our next result follows by Remark 3.2 .
Corollary 3.7. Let H be a finite-dimensional Hopf algebra and let A be a partial H-module algebra. There exists an one-to-one correspondence between the sets tH-stable ideals of Au
Φ / / tH › -stable ideals of A#Hu Ψ o o
given by ΦpIq " I#H, where I is an H-stable ideal of A, and ΨpIq " I X A, where I is an H ›stable ideal of A#H. Moreover Ψ " Φ´1 and these bijections preserve inclusions, sums, (finite) products and intersections. Now we present the definition of partial pA, Hq-modules. This concept is, in a certain sense, a generalization of the concept of A-module and it allows us to better understand the relation between A and A#H.
Definition 3.8. Let A be a (left) partial H-module algebra. A right (resp. left) partial pA, Hq- module is a right (resp. left) A-module M with a -linear map M b H Ñ M , m b h Þ Ñ m đ h (resp. H b M Ñ M , h b m Þ Ñ h § m) such that, for any m P M , a P A and h, g P H, we have (PM1) m đ 1 H " m presp. 1 H § m " mq (PM2) ppm đ hqaq đ g " ř pmph 1¨a qq đ ph 2 gq presp. h § papg § mqq " ř ph 1¨a qpph 2 gq § mqq.
Clearly, the conditions (PM1) and (PM2) are equivalent to (PM1) and the following two conditions: for any m P M , a P A and h, g P H,
(PM3) pm đ hqa " ř pmph 1¨a qq đ h 2 presp. h § pamqq " ř ph 1¨a qph 2 § mqq (PM4) pm đ hq đ g " ř pmph 1¨1A qq đ ph 2 gq presp. h § pg § mq " ř ph 1¨1A qpph 2 gq § mqq.
Remark 3.9. It is worth noting here that the apparent lack of symmetry between the definitions of right and left partial pA, Hq-module occurs because H acts on A by the left. For this reason, in some times, the argumentations in the proofs of our results about left and right partial pA, Hqmodules are quite different. For example, if V is an A-submodule of a right partial pA, Hq-module M , then the partial pA,
Hq-submodule of M generated by V is pV Aq đ H " V đ H, but if M is a left partial pA, Hq-module then the partial pA, Hq-submodule of M generated by V is ApH § V q.
The Proposition 3.11 will justify our interest in these objects and provides examples of right partial pA, Hq-modules. If A is a (left) partial H-module algebra, then A itself is a left partial pA, Hq-module. More generally, if I is a H-stable left ideal of A then I and A{I are left partial pA, Hq-modules. In the next section we will see how it is possible to build a partial pA, Hq-module from a given A-module.
As usual, we denote by ann A pM q (or simply annpM q) the annihilator of M . We recall that M is called faithful if annpM q " 0. The next result says about the annihilator of a partial pA, Hq-module. Proposition 3.10. If M is a partial pA, Hq-module then annpM q is an H-stable ideal of A.
Proof. It is clear that annpM q is an ideal of A. Moreover, for any m P M , x P annpM q and h P H, we have
mph¨xq " ÿ pmph 1¨x qq đ pεph 2 q1 H q " ÿ pmph 1¨x qq đ ph 2 Sph 3 qq " ÿ ppm đ h 1 qxq đ Sph 2 q P pM xq đ H " 0
if M is a right partial pA, Hq-module and also
ph¨xqm " ÿ ph 1¨x qppεph 2 q1 H q § mq " ÿ ph 1¨x qpph 2 Sph 3 qq § mq " ÿ h 1 § pxpSph 2 q § mqq P H § pxM q " 0
if M is a left partial pA, Hq-module. Thus, in any case, h¨x P annpM q. Hence annpM q is H-stable.
It is clear that if M is a right (resp. left) partial pA, Hq-module and I is an H-stable ideal of A such that I Ď annpM q, then M has a natural structure of right (resp. left) partial pA{I, Hqmodule.
Proposition 3.11. If M is a right (resp. left) partial pA, Hq-module then M is a right (resp. left) A#H-module, with action given by mpa#hq :" pmaq đ h presp. pa#hqm :" aph § mqq, m P M, a P A, h P H.
Conversely, if M is a right (resp. left) A#H-module then M is a right (resp. left) partial pA, Hqmodule, with actions given by
ma :" mpa#1 H q presp. am :" pa#1 H qmq, m P M, a P A m đ h :" mp1 A #hq presp. h § m :" p1 A #hqmq, m P M, h P H.
Moreover, in both cases we have that ann A pM q " ann A#H pM q X A.
Proof. Suppose that M is a right partial pA, Hq-module. The map
ζ : M b A#H Ñ M m b a#h Þ Ñ pmaq đ h
is well defined and -linear. In fact, if ς and ξ are the actions of A and H on M , respectively, then the -linear map
κ : M b A#H -M b A b H ς b idH ÝÝÝÝÑ M b H ξ ÝÑ M is such that, for any m P M , a P A and h P H, κ`m b a#h˘" κ`m b ř aph 1¨1A q#h 2˘" ÿ pmaph 1¨1A qq đ h 2 " ppmaq đ hq1 A " pmaq đ h.
Thus ζ is the restriction of κ to the subspace M b A#H, and therefore ζ is a well defined -linear map. Now, for any m P M , a, b P A and h, g P H, it is clear that m`1 A #1 H˘" m and also we have`m`a #h˘˘`b#g˘" pppmaq đ hqbq đ g " ÿ pmaph 1¨b qq đ ph 2 gq " m´ÿ aph 1¨b q#h 2 g" m``a#h˘`b#g˘˘, so ζ defines a right A#H-module structure on M . Conversely, suppose that M is a right A#H-module. Again we need verify that the maps
M b A Ñ M m b a Þ Ñ mpa#1 H q and M b H Ñ M m b h Þ Ñ mp1 A #hq
are well defined and -linear. The first one follows by the algebra inclusion A Ď A#H. The second one is exactly the composition of the -linear maps
M b H ÝÑ M b A b H idM b π Ý ÝÝÝÝ Ñ M b A#H ζ ÝÑ M m b h Þ Ñ m b 1 A b h where π : A b H Ñ A#H, a b h Þ Ñ pa#hqp1 A #1 H q " a#h,m đ hqa˘đ g "´`m`1 A #h˘˘`a#1 H˘¯`1A #g" m``1 A #h˘`a#1 H˘`1A #g˘" m´ÿ h 1¨a #h 2 g" ÿ`m ph 1¨a q˘đ ph 2 gq
Hence M is a right partial pA, Hq-module in this case.
To prove that M is a left pA, Hq-module if and only if M is a left A#H-module we proceed in the same way. Finally, we observe that the equality ann A pM q " ann A#H pM q X A is clear.
Note that the Proposition 3.10 is also a consequence of the Proposition 3.5 and of the equality ann A pM q " ann A#H pM q X A.
The H-prime and the H-Jacobson radicals
In this section we study the concepts of H-prime and H-Jacobson radicals of a partial H-module algebra A. These structures are analogous to the prime and Jacobson radicals of any algebra, and they are useful to study the semiprimality and the semiprimitivity problems for the partial smash product.
4.1.
H-(semi)prime ideals and the H-prime radical. We start by considering the (partial) H-prime radical. In [24] the authors studied the H-prime and the H-semiprime ideals of an H-module algebra A in the context of the global actions. Such concepts can also be considered when the action of H on A is a partial action. In the sequel we introduce this new concepts with the purpose to define the H-prime radical of A, when A is a partial H-module. After that, we establish a link between the H-prime radical of A and the H › -prime radical of A#H, when H is finite-dimensional. Our approach for H-prime and H-semiprime ideals, in a natural way, closely follows the classical one which can be found, for example, in [17, Section 10] about prime and semiprime ideals. We will present here the sketches of our proofs for the convenience of the reader. An example of H-prime ideal is given by a maximal H-stable ideal of A, whose existence is a consequence of Zorn's Lemma. The next result gives other useful characterizations of H-prime ideals. (1) p is H-prime;
(2) For any a, b P A, ApH¨aqApH¨bqA Ď p implies that a P p or b P p;
(3) For any a, b P A, ApH¨aqApH¨bq Ď p implies that a P p or b P p;
(4) For any a, b P A, pH¨aqApH¨bq Ď p implies that a P p or b P p; Following the terminology used in [17], we present our next definition.
Definition 4.3. We say that a nonempty subset M Ď A is an Hm-system if for any x, y P M , we have ApH¨xqApH¨yq X M ‰ H.
From the above definition we can give some more precise characterizations of H-prime ideal. The first one is an immediate consequence of Proposition 4.2 (3). Proof. Let I, J be H-stable ideals of A such that I, J Ę p. Then p Ł p`I and p Ł p`J, and it follows that there exist x, y P A such that
x P M X pp`Iq and y P M X pp`Jq.
Since M is an Hm-system, ApH¨xqApH¨yq X M ‰ H. On the other hand, we also have I :" tx P A : every Hm-system containing x meets Iu .
ApH¨xqApH¨yq Ď A`H¨pp`Iq˘A`H¨pp`Jq˘Ď pp`Iqpp`Jq Ď p`IJ, which implies that pp`IJq X M ‰ H. From p X M " H, it follows that IJ Ę p. Hence p is H-prime.
In the particular case when I " 0, we will call P H pAq :" H ? 0 the H-prime radical of A.
Clearly I Ď H ? I for any H-stable ideal I of A. We say that an ideal I of A is an H-radical ideal if I " H ? I. It is not clear from the Definition 4.6 that H ?
I is an H-stable ideal of A. This will be shown in the next result. Clearly every H-prime ideal is H-semiprime. Moreover, every intersection of H-semiprime ideals is also H-semiprime.
The proof of the next proposition is similar to that for Proposition 4.2.
Proposition 4.9. For any H-stable ideal s Ł A, the following statements are equivalent:
(1) s is H-semiprime;
(2) For any a P A, ApH¨aqApH¨aqA Ď s implies that a P s;
(3) For any a P A, ApH¨aqApH¨aq Ď s implies that a P s; (4) For any a P A, pH¨aqApH¨aq Ď s implies that a P s; (5) For any H-stable left ideal I Ď A, I 2 Ď s implies that I Ď s; (5') For any H-stable right ideal I Ď A, I 2 Ď s implies that I Ď s.
Definition 4.10. We say that a nonempty subset N Ď A is an Hn-system, if for any x P N , we have ApH¨xqApH¨xq X N ‰ H.
As an immediate consequence of the Proposition 4.9 (3) we also can characterize H-semiprime ideals by Hn-system. Proof. Fix a P N and define M " ta 1 , a 2 , ...u inductively as follows: a 1 " a, a 2 P ApH¨a 1 qApH¨a 1 q X N , a 3 P ApH¨a 2 qApH¨a 2 q X N , . . . To see that M is an Hm-system we first observe that, for any t ě 1, H¨a t`1 Ď H¨`ApH¨a t qApH¨a t q˘Ď ApH¨a t q, which implies H¨a t 1 Ď ApH¨a t q whenever t ď t 1 . Therefore, for any i, j ě 1, we have
" a j`1 P ApH¨a j qApH¨a j q Ď ApH¨a i qApH¨a j q if i ď j, a i`1 P ApH¨a i qApH¨a i q Ď ApH¨a i qApH¨a j q if i ě j,
in particular, ApH¨a i qApH¨a j q X M ‰ H. Hence M is an Hm-system such that a P M Ď N . Proposition 4.14. For any H-stable ideal s Ł A, the following statements are equivalent:
(1) s is H-semiprime;
(2) s is an intersection of H-prime ideals;
(3) s " H ? s.
Proof. In particular, the H-semiprime ideals of a partial H-module algebra are precisely the H-radical ideals. The following result is immediate from Propositions 4.7 and 4.14. (1) A is H-semiprime;
(2) P H pAq " 0;
(3) A has no nonzero nilpotent H-stable ideal;
(4) A has no nonzero nilpotent H-stable left ideal; (4') A has no nonzero nilpotent H-stable right ideal;
Proof.
(1) ô (2) follows from Corollary 4.15. For (1) ñ (4), let I be a nilpotent H-stable left ideal and let n ě 1 be the smallest positive integer such that I n " 0. If n ě 2, then pI n´1 q 2 Ď I n " 0 which implies that I n´1 " 0, a contradiction. Thus n " 1 and I " 0. The implication (1) ñ (4') is similar and the implications (3) ñ (1), (4) ñ (3) and (4') ñ (3) are easy to see.
The next proposition and its corollary establish a relation between the prime (resp. semiprime) ideals and the H-prime (resp. H-semiprime) ideals of a partial H-module algebra. Like in [17], we use the notation ?
I to indicate the (classical) radical of the ideal I of A, which is the intersection of all the prime ideals of A containing I. In the case which I " 0, we denote by P pAq :"
? 0 the prime radical of A.
Proposition 4.18. If P is a prime (resp. semiprime) ideal of A, then pP : Hq is H-prime (resp. H-semiprime). If H is finite-dimensional, then the converse is true, that is, every H-prime (resp. H-semiprime) ideal of A is of the form pP : Hq for some prime (resp. semiprime) ideal P of A.
Proof. Let P be a prime ideal of A. If I and J are H-stable ideals of A such that IJ Ď pP : Hq Ď P, then we have I Ď P or J Ď P. The H-stability of I and J implies that I Ď pP : Hq or J Ď pP : Hq. Hence pP : Hq is H-prime. The "semiprime" case is analogous. Now suppose that H is finite-dimensional and let p be an H-prime ideal of A. Consider the following two family of ideals of A:
G " tJ A : J is H-stable and J Ę pu F " tK A : p Ď K and J Ę K, @ J P Gu .
We will use the Zorn's Lemma to prove that F has a maximal element. Note that F ‰ H since p P F . Now, let tK λ u Ď F be a chain. Then K " Ť K λ is an ideal of A and p Ď K. We claim that J Ę K for every J P G. In fact, if J P G, then there exists a P J such that a R p, so ApH¨aqA Ę p and ApH¨aqA P G. This implies that ApH¨aqA Ę K λ for every λ. Because ApH¨aqA is a finitely generated ideal (since H is finite-dimensional) and tK λ u is a chain, we have that ApH¨aqA Ę K. In particular J Ę K (note that ApH¨aqA Ď J since a P J and J is an H-stable ideal). It follows that K P F is an upper bound for tK λ u. By Zorn's Lemma, there exists a maximal element P P F .
We claim that P is a prime ideal. In fact, suppose that I and J are ideals of A not contained in P. By maximality of P, there exist H-stable ideals K, L P G such that K Ď P`I and L Ď P`J, and so KL Ď P`IJ. Since KL P G (because p is H-prime), it follows that KL Ę P, and so IJ Ę P. Hence P is a prime ideal.
Since P P F and p is H-stable, we have p Ď pP : Hq. On the other hand, if J is an H-stable ideal such that p Ĺ J then J P G and so J Ę P. Hence p " pP : Hq and every H-prime ideal of A is of the form pP : Hq for some prime ideal P of A, when H is finite-dimensional.
Finally, assume that s is an H-semiprime ideal of A (and H is finite-dimensional). By Proposition 4.14, s " Ş p α , where p α runs over all the H-prime ideals of A containing s. From the above part, for every α, p α " pP α : Hq for some prime ideal P α of A. Thus s " Ş p α " Ş pP α : Hq " Ş P α : H˘with Ş P α a semiprime ideal of A. This completes the proof. Now we establish a link between the H-prime (resp. H-semiprime) ideals of A and the H ›prime (resp. H › -semiprime) ideals of A#H, when H is finite-dimensional. This allow us give a more precise description of the maps Φ and Ψ defined in Corollary 3.7. In particular, we establish a relation between the H-prime radical of A and the H › -prime radical of A#H.
Theorem 4.20. Let H be a finite-dimensional Hopf algebra and let A be a partial H-module algebra. Then the restrictions of the maps Φ and Ψ defined in Corollary 3.7 are bijections between the set of the H-prime (resp. H-semiprime) ideals of A and the set of the H › -prime (resp. H ›semiprime) ideals of A#H. In particular,
P H pAq " P H ›`A#H˘X A and P H ›`A#H˘" P H pAq#H , so A is an H-semiprime partial H-module algebra if and only if A#H is an H › -semiprime H › - module algebra.
Proof. We will denote by R " A#H. From Corollary 3.7, it is sufficient to prove that, for any H-prime (resp. H-semiprime) ideal p of A, its image Φppq is an H › -prime (resp. H › -semiprime) ideal of R and analogous for Ψ. In fact, let p be an H-prime ideal of A and let I, J be H › -stable ideals of R such that IJ Ď Φppq. Because Φ and Ψ preserve inclusions and products, ΨpIq and ΨpJ q are H-stable ideals of A such that ΨpIqΨpJ q " ΨpIJ q Ď ΨpΦppqq " p. Since p is H-prime, ΨpIq Ď p or ΨpJ q Ď p, which implies that I " ΦpΨpIqq Ď Φppq or J " ΦpΨpJ qq Ď Φppq. Hence Φppq is H › -prime. The proof that the H-semiprimality of an H-stable ideal p of A implies the H › -semiprimality of Φppq is analogous and the argumentation for Ψ is similar. Now the equalities P H pAq " P H ›pRq X A and P H ›pRq " P H pAq#H follow from P H pAq being the intersection of the all H-prime ideals of A, P H › pRq being the intersection of the all H › -prime ideals of R and also because Φ, Ψ preserve intersections.
č I α " J`A#H˘X A " č K β .
Proof. By Proposition 4.24, for each α, I α " P α X A for some right primitive ideal P α of A#H and, moreover, tP α u is the family of the right primitive ideals of A#H. Thus
č I α " č pP α X Aq "´č P α¯X A " J`A#H˘X A.
The proof of the second equality is similar.
Thus, the following definitions are completely natural. The next result in which we are interested says that if P is a right primitive ideal of A then pP : Hq is a right H-primitive ideal (Corollary 4.29). For this, we need to construct a right partial pA, Hq-module W from a given right A-module V such that V can be seen as an A-submodule of W which generates W as a partial pA, Hq-module. We will describe this construction below.
Let V be a right A-module. Consider the subspace W of V b H which is generated by the elements of the form ÿ vpk 1¨x q b k 2 , v P V, x P A, k P H. Then we define the right action of A on W as follows: for v P V , x, a P A and k P H,
ÿ vpk 1¨x q b k 2¯a :" ÿ vpk 1¨x qpk 2¨a q b k 3 " ÿ vpk 1¨p xaqq b k 2 .
It is clear that this action define an A-module structure on W . Moreover, we have V -V b 1 H Ď W as A-module because, for any v P V and a P A, we have
pv b 1 H qa " pvp1 H¨1A q b 1 H qa " vp1 H¨a q b 1 H " va b 1 H .
Now we define the right action of H on W in the following way: for v P V , x P A and k, h P H,
ÿ vpk 1¨x q b k 2¯đ h :" ÿ vpk 1¨x qppk 2 h 1 q¨1 A q b k 3 h 2 " ÿ vpk 1¨p xph 1¨1A qq b k 2 h 2 .
It is clear that w đ 1 H " w for any w P W . Moreover, if w " ř vpk 1¨x q b k 2 , with v P V , x P A and k P H, then, for any a P A and h, g P H, we have ppw đ hqaq đ g "´´ÿ vpk 1¨x q`pk 2 h 1 q¨1 A˘b k 3 h 2¯a¯đ g "´ÿ vpk 1¨x q`pk 2 h 1 q¨a˘b
k 3 h 2¯đ g " ÿ vpk 1¨x q`pk 2 h 1 q¨a˘`pk 3 h 2 g 1 q¨1 A˘b k 4 h 3 g 2 " ÿ v´k 1¨`x ph 1¨a q˘¯`pk 2 h 2 g 1 q¨1 A˘b k 3 h 3 g 2 "´ÿ v´k 1¨`x ph 1¨a q˘¯b k 2¯đ ph 2 gq "´´ÿ vpk 1¨x q b k 2¯p h 1¨a q¯đ ph 2 gq " pwph 1¨a qq đ ph 2 gq.
Hence W is a right partial pA, Hq-module which contain V as A-submodule. Since W is generated by elements of the form ř vpk 1¨x q b k 2 , we can see that
ÿ vpk 1¨x q b k 2 " ÿ vpk 1¨x qpk 2¨1A q b k 3 " ÿ pvpk 1¨x q b 1 H q đ k 2 " ÿ`p v b 1 H qpk 1¨x q˘đ k 2 P pV Aq đ H, v P V ,
x P A and k P H, and it follows that V generates W as partial pA, Hq-module.
We are particularly interested in the case when V is an irreducible right A-module. Proof. Let W be a right partial pA, Hq-module such that V is an A-submodule of W and V generates W as partial pA, Hq-module. Then W " pV Aq đ H " V đ H. Since V is an irreducible A-module, for any 0 ‰ u P V we have V " uA, and so W " puAq đ H. Thus every nonzero element of V generates W as partial pA, Hq-module. Now, by Zorn's Lemma, there exists a partial pA, Hq-submodule U of W maximal with respect to the property that U X V " 0. Then the quotient M " W {U has a natural right partial pA, Hq-module structure. Moreover, from U X V " 0, we have a natural monomorphism of right A-modules:
V ãÑ M " W {U v Þ Ñv " v`U .
We claim that M is irreducible (as partial pA, Hq-module). In fact, if N is a nonzero partial pA, Hq-submodule of M , then N " T {U for some partial pA, Hq-submodule T of W such that U Ł T . By maximality of U , we have T X V ‰ 0. Fix a nonzero element u P T X V . As observed above, W " puAq đ H, so M " pūAq đ H Ď N and N " M . Hence M is an irreducible right partial pA, Hq-module and V is isomorphic to an A-submodule of M .
Clearly M " pV Aq đ H " V đ H and the last statement follows.
Corollary 4.29. If P is a right primitive ideal of A, then pP : Hq is right H-primitive.
Proof. Let V be an irreducible right A-module such that P " annpV q and let M be an irreducible right partial pA, Hq-module such that V is an A-submodule of M , as in the Proposition 4.28. Denote I :" pP : Hq. For any v P V , x P I and h P H, we have pv đ hqx " ÿ pvph 1¨x qq đ h 2 P pV pH¨Iqq đ H Ď pV P q đ H " 0, therefore M I " pV đ HqI " 0 and pP : Hq " I Ď annpM q. On the other hand, annpM q is an Hstable ideal of A (Proposition 3.10) which is contained in P (since V Ď M ), so annpM q Ď pP : Hq.
Hence pP : Hq " annpM q is a right H-primitive ideal of A.
From now on we restrict ourselves to the case when H is finite-dimensional. In this situation, we can obtain more details about the H-Jacobson radical of A. First, however, as made before for right partial pA, Hq-module, we will present a construction of a left partial pA, Hq-module W from a given left A-module V , such that V can be seen as an A-submodule of W which generates W as a partial pA, Hq-module. This construction is an adaptation for partial actions of one which appeared in [19] and we will need a nonzero integral element in H › so that H need to be finite-dimensional.
Suppose that H is a finite-dimensional Hopf algebra and let A be a partial H-module algebra. In this case, analogously to the global case, the (left) partial H-module algebra structure of A induces a structure of (right) partial H › -comodule algebra ρ :
A Ñ A b H˚such that ρpaq " ÿ a 0 b a 1 ðñ h¨a " ÿ a 0 a 1 phq , @ h P H.(1)
(see [5] or [4]). Remember also that H › is an H-module algebra (global action) via h á ϕ " ř ϕ 1 ϕ 2 phq, for h P H and ϕ P H › . Then, for a P A and h P H, we have ρph¨aq " ÿ ρpa 0 qa 1 phq (by (1)) " ÿ a 00 b a 01 a 1 phq " ÿ 1 0 a 0 b 1 1 a 11 a 12 phq (by (PC3), Definition 2.6)
" ÿ 1 0 a 0 b 1 1 ph á a 1 q " ρp1 A qpid A b ph áqqρpaq
Now let V be a left A-module and consider the subspace W " ρpAqpV bH › q of V bH › generated by elements of the form
ρpxqpv b ϕq " ÿ x 0 v b x 1 ϕ , x P A, v P V, ϕ P H › .
We define the action of A and H on W as follows: for w P W , a P A and h P H,
a ‚ w " ρpaqw and h § w " ρp1 A qpid V b ph áqqpwq, that is, if w " ř x 0 v b x 1 ϕ with x P A, v P V and ϕ P H › , then a ‚ w :" ÿ a 0 x 0 v b a 1 x 1 ϕ " ρpaxqpv b ϕq (by (PC2), Definition 2.6)
and
h § w " ρp1 A q´ÿ x 0 v b h á px 1 ϕq¯" ÿ 1 0 x 0 v b 1 1 ph á px 1 ϕqq.
Evidently ‚ defines a left A-module structure on M . Also it is clear that (PM1) of the Definition 3.8 is satisfied.
For (PM2), let w " ř x 0 v b x 1 ϕ P W , with x P A, v P V , ϕ P H › ,h § pa ‚ pg § wqq " h §´a ‚´ÿ 1 0 x 0 v b 1 1 pg á px 1 ϕqq¯" h §´ÿ a 0 x 0 v b a 1 pg á px 1 ϕqq" ÿ 1 0 a 0 x 0 v b 1 1 h á pa 1 pg á px 1 ϕqqq " ÿ 1 0 a 0 x 0 v b 1 1 ph 1 á a 1 qpph 2 gq á px 1 ϕqq " ÿ ρph 1¨a qrx 0 v b ph 2 gq á px 1 ϕqs " ÿ ph 1¨a q ‚ pph 2 gq § wq.
Hence W is a left partial pA, Hq-module algebra. Now, let λ be a nonzero left integral in H › . Then we have an isomorphism of left A-modules V -V b λ. In fact, by (PC1) of Definition 2.6, for every v P V and a P A,
v b λ " 1v b λ " 1 0 v b ε H › p1 1 qλ " ÿ 1 0 v b 1 1 λ " ρp1 A qpv b λq P W
and, by the same reason, a ‚ pv b λq " av b λ. Moreover, by a result of Larson and Sweedler [18], the map H Ñ H › , given by h Þ Ñ ph á λq, is an isomorphism. Consequently, for every x P A, v P V and ϕ P H › , there exists h P H such that h á λ " ϕ, and then ÿ
x 0 v b x 1 ϕ " ρpxqpv b ph á λqq " x ‚ ph § pv b λqq P ApH § V q, via V -V b λ, so V generates W as left partial pA, Hq-module.
Remark 4.30. Observe that if V and W are as above and V is irreducible, then W " ApH §pvbλqq for any nonzero v P V and we can obtain the analogous of the Proposition 4.28 for irreducible left A-modules. The proof is essentially the same (with the appropriate adaptations), except for the inequality dim pM q ď dim pHq dim pV q. This latter follows since H is finite-dimensional and M " ApH § pv b λqq, so there exists an epimorphism of A-modules V n Ñ M . With regard to the Corollary 4.29 we can only deduce that annpM q Ď pP : Hq. However, this inclusion is enough for our purposes (see Theorem 5.1).
In the sequel we give an important characterization for the H-Jacobson radical when H is finite-dimensional. Proof. By Proposition 4.25 and Remark 2.10, we have that J H pAq " JpA#Hq X A Ď JpAq. Since J H pAq is H-stable then the inclusion J H pAq Ď pJpAq : Hq follows. Also, we observe that it is possible to deduce this same inclusion from Corollary 4.29.
To prove the reverse inclusion, we will show that J :" pJpAq : Hq Ď annpM q, for any irreducible left partial pA, Hq-module M . In fact, if 0 ‰ m P M , then ApH § mq is a nonzero partial pA, Hqsubmodule of M , so ApH § mq " M . Since H is finite-dimensional, this implies that M is finitely generated as A-module. where P is a right (resp. left) primitive ideal of A, to obtain J H pAq as it will be shown in the next result.
Corollary 4.32. Let tP α u be the family of all right (resp. left) primitive ideals of A. If H is finite-dimensional, then J H pAq " Ş pP α : Hq.
Proof. By Corollary 4.29 (resp. Remark 4.30), it is clear that J H pAq Ď Ş pP α : Hq. On the other hand, Ş pP α : Hq Ď Ş P α " JpAq. Since Ş pP α : Hq is H-stable, by Proposition 4.31, Ş pP α : Hq Ď pJpAq : Hq " J H pAq.
We finish this section by establishing a relation between the H-Jacobson radical of A and the H › -Jacobson radical of A#H, when H is finite-dimensional.
J H pAq " ΨpΦpJ H pAqqq " J H pAq#H X A " J H › pRq X A.
5.
On the semiprimitivity and the semiprimality problems for the partial smash products
In this section we apply the results of the Sections 3 and 4 to investigate the semiprimitivity and the semiprimality problems mentioned in the introduction, for partial smash products. Most of the results here generalize to the case of partial actions, the corresponding results on the semiprimitivity of the (global) smash product which appear in [20,Section 4], improving them when has positive characteristic. We also prove a result about the semiprimality of the partial smash product (Theorem 5.8), which generalizes [19,Theorem 3.4] for the case of partial actions. Proof. We will denote by R :" A#H. Let tV α u be the family of all irreducible (right) A-modules. By Proposition 4.28, for each α, there exists an irreducible (right) partial pA, Hq-module M α such that V α is an A-submodule of M α . For each α, we will denote by P α :" annpV α q, I α :" annpM α q Ď pP α : Hq (see the proof of the Corollary 4.29 and Remark 4.30) and A α :" A{I α . By Corollary 4.32, we have č I α Ď č pP α : Hq " J H pAq " 0.
Thus A is a subdirect product of the partial H-module algebras A α . By Proposition 3.3, R is a subdirect product of the algebras R α :" A α #H. Therefore, it is enough to show that each R α is semiprimitive (see [14,Proposition 2.3.4]).
Since V α is finite-dimensional, so is M α (Proposition 4.28 or Remark 4.30). Moreover, from I α " annpM α q it follows that A α " A{I α Ď End pM α q as algebras. Therefore dim pA α q ď pdim pM α qq 2 ă 8 and so R α is finite-dimensional.
Since A α is H-primitive it follows by Theorem 4.33 that R α is H › -semiprimitive. Moreover, because H › is cosemisimple Hopf algebra acting on R α globally and R α is finite-dimensional, we deduce that JpR α q is an H › -stable ideal, by [19,Corollary 3.2]. Hence JpR α q " pJpR α q : Hq " J H › pR α q " 0 and R α is semiprimitive as desired. Now we observe that if H is a Hopf algebra over a field , A is a partial H-module algebra and Ď is any field extension, then H :" H b becomes a Hopf algebra over with coalgebra structure given by
∆ H ph b αq " ÿ ph 1 b αq b ph 2 b 1 q " ÿ ph 1 b 1 q b ph 2 b αq
and ε H ph b αq " ε H phqα, for h P H and α P , where ε H is the counit of H. The antipode of H is given by
S H ph b αq " S H phq b α, h P H, α P ,
where S H is the antipode of H. Moreover, A :" A b becomes a partial H -module algebra by the H -action defined as ph b αq¨pa b βq :" ph¨aq b pαβq, α, β P , a P A, h P H.
With the above notations we can prove the following result.
Lemma 5.2. Let H be a finite-dimensional Hopf algebra over a field and let A be a partial H-module algebra. If Ď is a separable algebraic field extension, then
J H pA q " J H pAq b .
In particular, A is H-semiprimitive if and only if A is H -semiprimitive.
Proof. Since Ď is a separable algebraic extension we have JpA q " JpAq b (see [17,Theorem 5.17]). Thus
H ¨pJ H pAq b q " pH¨J H pAqq b Ď JpAq b " JpA q.
By Proposition 4.31, J H pAq b Ď pJpA q : H q " J H pA q. On the other hand, if u P J H pA q Ď JpA q " JpAq b , then there are x 1 , . . . , x n P JpAq and β 1 , . . . , β n P , with tβ i u linearly independent over , such that u " ř
x i b β i . So, for any h P H, ÿ ph¨x i q b β i " ph b 1 q¨u P J H pA q Ď JpAq b (since J H pA q is H -stable)
. From the linear independence of tβ i u, we can deduce that h¨x i P JpAq, for every 1 ď i ď n and every h P H. Therefore x i P pJpAq : Hq " J H pAq for every i, and u P J H pAq b . Hence, also the inclusion J H pA q Ď J H pAq b holds.
Now we are able to present our second result about the semiprimitivity of the partial smash product. [20,Theorem 4.2]) Let H be a semisimple Hopf algebra over a field and let A be an H-semiprime partial H-module algebra satisfying a polynomial identity. If is perfect and A is affine over , then R " A#H is semiprimitive.
Theorem 5.3. (See
Proof. Since A is a PI-algebra which is affine over , we have that JpAq " P pAq (see [25,Corollary 4.4.6]). Thus, A is actually H-semiprimitive. Denote by¯ the algebraic closure of . As is a perfect field it follows that the extension Ď¯ is separable. Taking "¯ in the Lemma 5.2, we have thatĀ isH-semiprimitive, whereH " H b ¯ andĀ " A b ¯ .
Moreover,Ā is a PI-algebra (see [25, Theorem 6.1.1]) which is affine over¯ , and so every irreducible rightĀ-module is finite-dimensional (see [20,Lemma 3.7] For the next result we will need one more lemma. Before, observe that if I is a right ideal of A, then so is H¨I. This follows because, for any h P H, x P I and a P A, ph¨xqa " ÿ ph 1¨x qpph 2 Sph 3 qq¨aq " ÿ h 1¨p xpSph 2 q¨aqq P H¨pIAq Ď H¨I, so pH¨IqA Ď H¨I. Proof. Fix an element y P H¨JpAq, and let x 1 , . . . , x n P JpAq and h 1 , . . . , h n P H such that y " ř h i¨xi . Since ř H¨x i is a finite-dimensional subspace of A, it generates a finite-dimensional subalgebra B of A. Clearly H¨B Ď B, so B is an H-module subalgebra of A.
Since A is locally finite, JpAq is a nil ideal of A, therefore each Bx i B Ď JpAq is a nil ideal of B and so it is contained in JpBq. In particular, each x i P JpBq. Since B is finite-dimensional, it follows from [19,Corollary 3.2] that H¨JpBq Ď JpBq. Thus, we have that y " ř h i¨xi P JpBq is a nilpotent element.
From this and the previous observation, it follows that H¨JpAq is a nil right ideal of A, so it is contained in JpAq. Hence JpAq is H-stable.
Theorem 5.5. (See [20,Corollary 4.4]) Let H be a semisimple Hopf algebra and let A be an H-semiprimitive partial H-module algebra. If A is locally finite, then A#H is semiprimitive.
Proof. By Theorem 4.33, the smash product A#H is an H › -semiprimitive H › -module algebra (global action). Also, H › is a finite-dimensional cosemisimple Hopf algebra, because H is semisimple. Moreover, A#H is locally finite, because if x 1 , . . . , x n P A#H and x i " ř j a ij #h ij , 1 ď i ď n, then there is a finite-dimensional H-stable subalgebra B of A generated by ř i,j pH¨1 A qpH¨a ij q, so that x 1 , . . . , x n are elements of the finite-dimensional subalgebra B#H of A#H. By Lemma 5.4, JpA#Hq is H › -stable. Hence, by Proposition 4.31, J`A#H˘"`J`A#H˘: H ›˘" J H ›`A#H˘" 0 and A#H is semiprimitive.
Corollary 5.6. (See [20,Theorem 4.2]) Let H be a semisimple Hopf algebra over a field and let A be an H-semiprime partial H-module algebra satisfying a polynomial identity. If A is algebraic over , then A#H is semiprimitive.
Proof. Under these conditions, we have JpAq " P pAq (see [17,Corollary 4.19] and [16, Theorem 3, p. 36]), thus A is actually H-semiprimitive. Moreover, A is locally finite (see [15,Theorem 6.4.3]), therefore the result follows from Theorem 5.5.
Corollary 5.7. Let H be a semisimple Hopf algebra over a field and let A be a partial H-module algebra. If
(1) every irreducible right A-module is finite-dimensional, or (2) is perfect and A is a PI-algebra which is affine over , or (3) A is locally finite (in particular, if A is a PI-algebra which is algebraic over ), then J`A#H˘" J H pAq#H.
Proof. By Theorem 4.33, J H pAq#H " J H ›`A#H˘Ď J`A#H˘. On the other hand, A{J H pAq is H-semiprimitive (so, in particular, H-semiprime). By Theorems 5.1, 5.3 or 5.5 (according the hypotheses), it follows that the factor algebra`A#H˘{`J H pAq#H˘-pA{J H pAqq#H is semiprimitive, so that J`A#H˘Ď J H pAq#H.
The next theorem generalizes [19,Theorem 3.4] for the case of partial actions. The proof follows almost the same steps as in [19], but we include here our proof for the sake of completeness.
Theorem 5.8. Let H be a semisimple Hopf algebra and let A be an H-semiprime partial H-module algebra satisfying a polynomial identity. Then A#H is semiprime.
Proof. Since A is a PI-algebra, if I is a nil ideal of A then I Ď P pAq, so pI : Hq Ď pP pAq : Hq " P H pAq " 0.
The polynomial ring Arts has a structure of partial H-module algebra if we extend the action of H by h¨t :" ph¨1 A qt, for h P H. Moreover, for any ideal I of A, pIrts : Hq " pI : Hqrts, since pptq " ÿ a i t i P pIrts : Hq ðñ H¨pptq Ď Irts ðñ H¨a i Ď I, @ i ðñ a i P pI : Hq, @ i ðñ pptq P pI : Hqrts.
By a theorem of Amitsur (see [17,Theorem 5.10]), JpArtsq " N rts for some nil ideal N of A, so J H pArtsq " pJpArtsq : Hq " pN rts : Hq " pN : Hqrts " 0 as observed above. Therefore Arts -A b rts is an H-semiprimitive partial H-module algebra which satisfies a polynomial identity (see [25, Theorem 6.1.1]). It follows from Theorem 4.33 that R :" Arts#H is H › -semiprimitive. Moreover, R satisfies a polynomial identity, because R is finitely generated as Arts-module (see [22,Corollary 13.4.9]). Since H › is cosemisimple, it follows from [19,Corollary 3.2] that if I is a nilpotent ideal of R, then H ›¨I Ď JpRq, so I Ď pJpRq : H › q " J H › pRq " 0. Thus R " Arts#H -A#Hrts is semiprime and therefore A#H is also semiprime (see [17,Proposition 10.18]). Corollary 5.9. Let H be a semisimple Hopf algebra and let A be a partial H-module algebra satisfying a polynomial identity. Then P`A#H˘" P H pAq#H.
Proof. By Theorem 4.20, P H pAq#H " P H ›`A#H˘Ď P`A#H˘. On the other hand, A{P H pAq is H-semiprime (and satisfies a polynomial identity). It follows from Theorem 5.8 that the factor algebra`A#H˘{`P H pAq#H˘-pA{P H pAqq#H is semiprime, so also P`A#H˘Ď P H pAq#H.
The hypothesis "H-semiprimitive" or "H-semiprime" in the above theorems are essentials (by Theorems 4.20 and 4.33, Corollary 4.19 and Proposition 4.31). Also, it is clear that the semisimplicity of H is a necessary condition for the Question 2 in the introduction, because if H acts trivially on A, then A#H " A b H is not semiprime if H is not semisimple.
Example 5.10. Let B, A and H be as in Example 2.3. If B is finite-dimensional and semiprimitive, then so is A. In particular, every irreducible A-module is finite-dimensional. By Theorem 5.1, the partial smash product A#H is semiprimitive. This also follows from Theorem 5.8 since, for finite-dimensional algebra, the Jacobson radical coincides with the prime radical.
Definition 3 . 1 .
31Let A be a partial H-module algebra. An ideal (resp. left, right ideal) I of A is said to be H-stable if H¨I Ď I.
Remark 3 . 2 .
32If the action of H on A is a global one, then the H-stable ideals of A are exactly the ideals of A which are H-submodules.
Proposition 3. 4 .
4Let I be an ideal of A. Then pI : Hq is the largest H-stable ideal of A contained in I. In particular, I is H-stable if and only if pI : Hq " I.
is the canonical projection and ζ is the action of A#H on M . The conditions (PM1) and (PM2) of the Definition 3.8 follow from the A#H-module structure of M : for any m P M , a P A and h, g P H, it is clear that m đ 1 H " m and also`p
Definition 4. 1 .
1An H-stable ideal p of A is called H-prime if p ‰ A and, for any H-stable ideals I, J Ď A, IJ Ď p ñ I Ď p or J Ď p.
Proposition 4. 2 .
2For any H-stable ideal p Ł A, the following statements are equivalent:
( 5 )
5For any H-stable left ideals I, J Ď A, IJ Ď p implies that I Ď p or J Ď p; (5') For any H-stable right ideals I, J Ď A, IJ Ď p implies that I Ď p or J Ď p.Proof. (1) ñ (2) follows because ApH¨aqA and ApH¨bqA are H-stable ideals and the implications (2) ñ (3) ñ (4) are easy. For (4) ñ (5) we assume that I and J are H-stable left ideals of A such that IJ Ď p, but I Ę p. Fix an element a P Izp. For any b P J, pH¨aqApH¨bq Ď IpAJq Ď IJ Ď p, so by (4), b P p. It follows that J Ď p. The implication (4) ñ (5') is analogous and the implications (5) ñ (1) and (5') ñ (1) are trivial.
Proposition 4. 4 .
4An H-stable ideal p of A is H-prime if and only if Azp is an Hm-system.
Proposition 4. 5 .
5Let M Ď A be an Hm-system. If p is an H-stable ideal of A maximal with respect to the property that p X M " H, then p is H-prime.
Definition 4. 6 .
6Let I be an H-stable ideal of A. We define the H-radical of I as the set H ?
Proposition 4. 7 .
7For any H-stable ideal I of A, H ? I is the intersection of all H-prime ideals of A containing I. In particular, H ?I is an H-stable ideal of A.Proof. Let p be an H-prime ideal of A containing I and let x P H ? I. Since Azp is an Hm-system (Proposition 4.4) which does not meet I it follows that x P p. Thus, H ?I is contained in the intersection of all H-prime ideals of A containing I.Conversely, if x R H ? I then there exists an Hm-system M Ď A such that x P M and M XI " H. By Zorn's Lemma, there exists an H-stable ideal p of A containing I which is maximal with respect to being p X M " H. By Proposition 4.5, p is an H-prime ideal (which contains I and x R p). The proof is complete now.
Definition 4. 8 .
8An H-stable ideal s of A is called H-semiprime if s ‰ A and, for any H-stable ideal I Ď A, I 2 Ď s ñ I Ď s.
Proposition 4 . 11 .
411An H-stable ideal s of A is H-semiprime if and only if Azs is an Hn-system. The next result can be proved in a similar way as Proposition 4.5.
Proposition 4 . 12 .
412Let N Ď A be an Hn-system. If s is an H-stable ideal of A maximal with respect to the property that s X N " H, then s is H-semiprime.The next result establishes a link between Hm-systems and Hn-systems which allow us a better understanding of the relations between H-prime and H-semiprime ideals.
Proposition 4 . 13 .
413Let N Ď A be an Hn-system. For every a P N , there exists an Hm-system M Ď N such that a P M . Equivalently, N is equal to the union of the Hm-systems contained in N .
(3) ñ (2) follows by Proposition 4.7 and (2) ñ (1) is clear. For (1) ñ (3), it is enough to prove that H ? s Ď s. Thus, consider a R s. Then Azs is an Hn-system containing a. By Proposition 4.13, there exists an Hm-system M Ď Azs such that a P M and so a R H ? s.
Corollary 4 . 15 .
415For any H-stable ideal I of A, H ? I is the smallest H-semiprime ideal of A containing I. In particular,Definition 4.16. A partial H-module algebra is called H-prime (resp. H-semiprime) if 0 is an H-prime (resp. H-semiprime) ideal.It is clear that an H-stable ideal p of A is H-prime if and only if A{p is an H-prime partial H-module algebra.
Proposition 4 . 17 .
417For a partial H-module algebra A, the following statements are equivalent:
4. 2 .
2H-primitive ideals and the H-Jacobson radical. Now we introduce the concept of Hprimitive ideal in the context of partial actions. The definition of H-Jacobson radical of a partial H-module algebra becomes natural from this concept (see Definition 4.26 below). The aim here is to establish relations (if any) between the H-Jacobson radical of A and the H › -Jacobson radical of A#H, when H is finite-dimensional. We start with the following definition.
Definition 4 . 21 .
421Let M be a partial pA, Hq-module. (i) M is called irreducible if M ‰ 0 and M has no other partial pA, Hq-submodules than 0 or M . (ii) M is called semisimple (or completely reducible) if M is a direct sum of irreducible partial pA, Hq-submodules.The next result is immediate from Proposition 3.11.Proposition 4.22. M is an irreducible (resp. semisimple) partial pA, Hq-module if and only if M is an irreducible (resp. semisimple) A#H-module.The concept of irreducible partial pA, Hq-module naturally lead us to the concept of H-primitive partial H-module algebra and H-primitive ideal.
Definition 4 . 23 .
423Let A be a partial H-module algebra.(i) A is called right (resp. left) H-primitive if there exists a faithful irreducible right (resp. left) partial pA, Hq-module.(ii) An H-stable ideal I of A is called right (resp. left) H-primitive ideal if A{I is a right (resp. left) H-primitive partial H-module algebra.Evidently an H-stable ideal I of A is right (resp. left) H-primitive if and only if I is the annihilator of an irreducible right (resp. left) partial pA, Hq-module. From Propositions 3.11 and 4.22 we have the following.
Proposition 4 . 24 .
424An H-stable ideal I of A is right (resp. left) H-primitive if and only if I " P X A for some right (resp. left) primitive ideal P of A#H.
Proposition 4 . 25 .
425Let tI α u be the family of the right H-primitive ideals of A and let tK β u be the family of the left H-primitive ideals of A. Then
Definition 4 . 26 .
426The H-Jacobson radical J H pAq of a partial H-module algebra A is defined as the intersection of all the right (or left) H-primitive ideals of A.
Definition 4. 27 .
27A partial H-module algebra A is called H-semiprimitive if J H pAq " 0.
Proposition 4 . 28 .
428Let V be an irreducible right A-module. Then there exists an irreducible right partial pA, Hq-module M , such that V is (isomorphic to) an A-submodule of M . Moreover, if H and V are finite-dimensional then M is also finite-dimensional and dim pM q ď dim pHq dim pV q.
Proposition 4. 31 .
31Suppose that H is finite-dimensional. Then J H pAq " pJpAq : Hq. In particular, A is H-semiprimitive if and only if JpAq does not contain nonzero H-stable ideals.
From J Ď JpAq, by Nakayama's Lemma, J M Ł M . Moreover, J is an H-stable ideal, so J M is a partial pA, Hq-submodule of M . Hence J M " 0 and pJpAq : Hq " J Ď annpM q. We observe that if H is finite-dimensional and the action of H on A is global, then we can recover [20, Corollary 2.6 (2)] by using the Proposition 4.31 combined with the Definitions 4.26 and 4.27. From Corollary 4.29 (resp. Remark 4.30) and Proposition 4.31, when H is finite-dimensional, it is sufficient to take the intersection of all H-stable ideals of A which are of the form pP : Hq,
Corollary 4.19. Suppose that H is finite-dimensional. For any H-stable ideal I of A we have Hq. In particular, P H pAq " pP pAq : Hq and A is H-semiprime if and only if P pAq does not contain nonzero H-stable ideals.H
?
I " p
?
I : Proof. By Proposition 4.18, p
?
I : Hq is an H-semiprime ideal. Moreover, I is an H-stable ideal
contained in
?
I, so I Ď p
?
I : Hq. By Corollary 4.15, H ?
I Ď p
?
I : Hq. On the other hand, again
by Proposition 4.18, H ?
I " pQ : Hq for some semiprime ideal Q of A. In particular I Ď Q, so
?
I Ď Q and therefore p
?
I : Hq Ď pQ : Hq " H ?
I.
Theorem 5.1. (See [20, Theorem 4.1]) Let H be a semisimple Hopf algebra and let A be an H-semiprimitive partial H-module algebra. If every irreducible right (or left) A-module is finitedimensional, then A#H is semiprimitive.
). SinceH is semisimple (see [23, Corollary 2.2.2]), it follows from Theorem 5.1 that R b ¯ -Ā#H is semiprimitive. Thus, by [17, Theorem 5.17], R is semiprimitive.
Lemma 5.4. (See[20, Remark 3.9]) Let H be a finite-dimensional cosemisimple Hopf algebra and let A be an H-module algebra (global action). If A is locally finite, then JpAq is H-stable.
Acknowledgments. The authors would like to thank Antonio Paques who gave us the definition of left partial pA, Hq-module.
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[
"Solar Neutrino Observables Sensitive to Matter Effects",
"Solar Neutrino Observables Sensitive to Matter Effects"
] |
[
"H Minakata \nDepartment of Physics\nTokyo Metropolitan University\n192-0397HachiojiTokyoJapan\n",
"C Peña-Garay \nInstitut de Física Corpuscular\nCSIC-Universitat de València\nApartado de Correos 22085E-46071ValènciaSpain\n"
] |
[
"Department of Physics\nTokyo Metropolitan University\n192-0397HachiojiTokyoJapan",
"Institut de Física Corpuscular\nCSIC-Universitat de València\nApartado de Correos 22085E-46071ValènciaSpain"
] |
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We discuss constraints on the coefficient A M SW which is introduced to simulate the effect of weaker or stronger matter potential for electron neutrinos with the current and future solar neutrino data. The currently available solar neutrino data leads to a bound A M SW = 1.47 −0.42 +0.54 ( −0.82 +1.88 ) at 1σ (3σ) CL, which is consistent with the Standard Model prediction A M SW = 1. For weaker matter potential (A M SW < 1), the constraint which comes from the flat 8 B neutrino spectrum is already very tight, indicating the evidence for matter effects. Whereas for stronger matter potential (A M SW > 1), the bound is milder and is dominated by the day-night asymmetry of 8 B neutrino flux recently observed by Super-Kamiokande. Among the list of observable of ongoing and future solar neutrino experiments, we find that (1) an improved precision of the day-night asymmetry of 8 B neutrinos, (2) precision measurements of the low energy quasi-monoenergetic neutrinos, and (3) the detection of the upturn of the 8 B neutrino spectrum at low energies, are the best choices to improve the bound on A M SW .
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10.1155/2012/349686
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"https://arxiv.org/pdf/1009.4869v3.pdf"
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1009.4869
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f941b575271b8f96d2e3237729874fa38ef54e86
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Solar Neutrino Observables Sensitive to Matter Effects
1 Aug 2012
H Minakata
Department of Physics
Tokyo Metropolitan University
192-0397HachiojiTokyoJapan
C Peña-Garay
Institut de Física Corpuscular
CSIC-Universitat de València
Apartado de Correos 22085E-46071ValènciaSpain
Solar Neutrino Observables Sensitive to Matter Effects
1 Aug 2012
We discuss constraints on the coefficient A M SW which is introduced to simulate the effect of weaker or stronger matter potential for electron neutrinos with the current and future solar neutrino data. The currently available solar neutrino data leads to a bound A M SW = 1.47 −0.42 +0.54 ( −0.82 +1.88 ) at 1σ (3σ) CL, which is consistent with the Standard Model prediction A M SW = 1. For weaker matter potential (A M SW < 1), the constraint which comes from the flat 8 B neutrino spectrum is already very tight, indicating the evidence for matter effects. Whereas for stronger matter potential (A M SW > 1), the bound is milder and is dominated by the day-night asymmetry of 8 B neutrino flux recently observed by Super-Kamiokande. Among the list of observable of ongoing and future solar neutrino experiments, we find that (1) an improved precision of the day-night asymmetry of 8 B neutrinos, (2) precision measurements of the low energy quasi-monoenergetic neutrinos, and (3) the detection of the upturn of the 8 B neutrino spectrum at low energies, are the best choices to improve the bound on A M SW .
Introduction
Neutrino propagation in matter is described by the Mikheyev-Smirnov-Wolfenstein (MSW) theory [1]. It was successfully applied to solve the solar neutrino problem [2], the discrepancy between the data [3,4,5,6,7] and the theoretical prediction of solar neutrino flux [8], which blossomed into the solution of the puzzle, the large-mixing-angle (LMA) MSW solution. The solution is in perfect agreement with the result obtained by KamLAND [9] which measured antineutrinos from nuclear reactors traversed essentially in vacuum.
The MSW theory relies on neutrino interaction with matter dictated by the standard electroweak theory and the standard treatment of refraction which is well founded in the theory of refraction of light. Therefore, it is believed to be on a firm basis. On the observational side it predicts a severer reduction of the solar neutrino flux at high energies due to the adiabatic transition in matter than at low energies where the vacuum oscillation effect dominates. Globally, the behavior is indeed seen in the experiments observing 8 B solar neutrinos at high energies [5,6] and in radiochemical experiments detecting low energy pp and 7 Be neutrinos [3,4], and more recently by the direct measurement of 7 Be neutrinos by Borexino [7]. For a summary plot of the current status of high and low energy solar neutrinos, see the review of solar neutrinos in this series. Therefore, one can say that the MSW theory is successfully confronted with the available experimental data.
Nevertheless, we believe that further test of the MSW theory is worth pursuing. First of all, it is testing the charged current (CC) contribution to the index of refraction of neutrinos of the Standard Model, which could not be tested anywhere else. Furthermore, in solar neutrinos, the transition from low to high energy behaviors mentioned above has not been clearly seen in a single experiment in a solar model independent manner. The Borexino and KamLAND experiments tried to fill the gap by observing 8 B neutrinos at relatively low energies [10,11]. SNO made an attempt to lower the threshold energy to 3.5 MeV [12,13] and the similar challenge is being undertaken by the Super-Kamiokande (SK) group [14]. Obviously, it is of interest to know to what extent these measurements make the test of the MSW theory stringent. Furthermore, the MSW theory is usually assumed to disentangle the genuine effect of CP phase δ from the matter effect in future experiments to determine δ and the mass hierarchy. Therefore, to prove the MSW theory to the accuracy required by such measurement is highly desirable to make these measurement robust. This reasoning has been spelled out in [15].
In this paper we discuss the question of to what extent tests of the MSW theory can be made stringent by various solar neutrino observables. We do this in the light of the past, ongoing and future experiments of solar neutrinos, anticipating the soon coming precision measurements. We adopt the framework for testing the MSW theory by using the measure A M SW [16], defined as the ratio of the effective coupling of weak interactions measured with coherent neutrino matter interactions in the forward direction to the Fermi coupling constant G F . 1 With various new experimental inputs, it is now quite timely to update the constraint on A M SW which was obtained in [16]. The new data include 7 Be and pep measurement by Borexino [7,18], the SNO low threshold data [12,13], Borexino and KamLAND measurement of 8 B flux [10,11], and most importantly, recent measurement of the day-night asymmetry by SK [14]. Moreover, we give qualitative understandings of the origin of the bound on A M SW from above and below unity. We then discuss how far we can go with a wealth of further solar neutrino observables.
Simple analytic treatment of matter effect dependences
In this section, we give a simple analytic description of how various solar neutrino observables depend upon the matter effect. It should serve for intuitive understanding of the characteristic features which we will see in the later sections. The reader will find a physics discussion in the flavor conversion review of this series. In the following, we denote the matter densities inside the Sun and in the Earth as ρ S and ρ E , respectively. Solar neutrino survival or appearance probabilities depend on three oscillation parameters: the solar oscillation parameters (θ 12 , ∆m 2 21 ≡ m 2 2 − m 2 1 ), and θ 13 . Smallness of the 1 Though the test involves the solar or the Earth electron number densities calculated by the standard solar model (SSM), or by the Preliminary Reference Earth Model (PREM) [17], respectively, we take the attitude that it primarily tests the standard electroweak theory charged current (CC) coupling constant in the forward direction. It is because the solar matter density calculated by SSM is cross checked by helioseismology to an accuracy much better than the one presented here. It is also due to that the Earth matter dependent observable, the day-night variation of solar neutrino flux, is insensitive to the precise profile of the Earth matter density.
recently measured value of θ 13 [19,20,21,22,23] and its small error greatly restricts the uncertainty introduced by this parameter on the determination of matter effects.
To quantify possible deviation from the MSW theory, following [16], we introduce the parameter A M SW by replacing the Fermi coupling constant G F by A M SW G F . The underlying assumption behind such simplified framework is that the deviation from the Fermi coupling constant is universal over fermions, in particular up and down quarks.
The survival probability in the absence of the Earth matter effect, i.e., during the day, is well described by [24,25,26] P D ee = cos 4 θ 13
1 2 + 1 2 · cos 2θ S · cos 2θ 12 + sin 4 θ 13(1)
Here θ S is the mixing angle at the production point inside the Sun:
cos 2θ S ≡ cos 2θ m (ρ S )(2)
where θ m (ρ) is the mixing angle in matter of density ρ S ,
cos 2θ S = cos 2θ 12 − ξ S (1 − 2ξ S cos 2θ 12 + ξ 2 S ) 1/2 .(3)
In (4), ξ S is defined as the ratio of the neutrino oscillation length in vacuum, l ν , to the refraction length in matter, l 0 :
ξ S ≡ l ν l 0 = 2 √ 2A M SW G F ρ S Y e cos 2 θ 13 m N E ∆m 2 = 0.203 × cos 2 θ 13 E 1 MeV ρ S Y e 100 g cm −3 ,(4)
where
l ν ≡ 4πE ∆m 2 , l 0 ≡ 2πm N √ 2A M SW G F ρ S Y e cos 2 θ 13 .(5)
In (4) and (5), ρ S is the matter density, Y e is the number of electrons per nucleon, and m N is the nucleon mass. In the last term we have used the best fit of the global analysis ∆m 2 21 = 7.5 × 10 −5 eV 2 . The average electron number densities ρ S Y e at the production point of various solar neutrino fluxes are tabulated in Table 1.
Energy spectrum
Solar neutrino observables taken in a single experiment have not shown an energy dependence yet. The neutrino oscillation parameters are such that we can not expect strong energy dependences. At low neutrino energies, small ξ S , Eq. (1) can be approximated by P D ee = cos 4 θ 13 1 − 1 2 sin 2 2θ 12 (1 + cos 2θ 12 ξ S ) + sin 4 θ 13
Notice that the correction to the asymptotic behavior is linear in A M SW at low energies while it is quadratic in A −1 M SW at high energies. It may mean that the energy spectrum at low energies could be more advantageous in tightening up the constraint on A M SW provided that these formulas with leading order corrections are valid.
It is well known that in the LMA MSW mechanism, 8 B neutrino spectrum must show an upturn from the asymptotic high energy (E ≫ 10MeV) to lower energies. The behavior is described by the correction term in (7) but only at a qualitative level. It indicates that the upturn component in the spectrum is a decreasing function of A M SW . On the other hand, at low energies populated by pp, 7 Be, and pep neutrinos, the solar neutrino energy spectrum display vacuum averaged oscillations or decoherence, (6). The deviation from this asymptotic low energy limit can be described by the correction term in (6) again at the (better) qualitative level. The term depend upon A M SW linearly so that the correction term is an increasing function of A M SW . Because of the negative sign in the correction term in (6), larger values of A M SW lead to smaller absolute values of P ee in both low and high energy regions.
To see how accurate is the behavior predicted by the above approximate analytic expressions, we have computed numerically (using the PREM profile) the average 1 − r µ/e P ee + r µ/e (E e,i ) as a function of electron energy. Here, O means taking average of P ee over neutrino energies with neutrino fluxes times the differential cross sections integrated over the true electron energy with response function. In the above expression, r µ/e ≡ σµ σe with σ e and σ µ being the cross sections of ν e e and ν µ e scattering, respectively. The computed results confirm qualitatively the behavior discussed above based on our analytic approximations. Thus, the energy spectrum of solar neutrinos at low and high energies can constrain A M SW in this way, as will be shown quantitatively in Sec. 0.3.
Day-night variation
The ν e survival probability at night during which solar neutrinos pass through the earth can be written, assuming adiabaticity, as [27] P N ee = P D ee − cos 2θ S cos 2 θ 13 f reg zenith (8) where P D ee is the one given in (1). f reg denotes the regeneration effect in the earth, and is given as f reg = P 2e − sin 2 θ 12 cos 2 θ 13 , where P 2e is the transition probability of second mass eigenstate to ν e . Under the constant density approximation in the earth, f reg is given by [27]
f reg = ξ E cos 2 θ 13 sin 2 2θ E sin 2 A M SW a E cos 2 θ 13 (1 − 2ξ −1 E cos 2 θ 12 + ξ −2 E ) 1 2 L 2 (9) for passage of distance L, where we have introduced a E ≡ √ 2G F N earth e = √ 2G F ρ E Y eE m N .
In (9), θ E and ξ E stand for the mixing angle and the ξ parameter [see (4)] with matter density ρ E in the earth. Within the range of neutrino parameters allowed by the solar neutrino data, the oscillatory term averages to 1 2 in good approximation when integrated over zenith angle. Then, the equation simplifies to
f reg zenith = 1 2 cos 2 θ 13 ξ E sin 2 2θ E .(10)
At E = 7 MeV, which is a typical energy for 8 B neutrinos, ξ E = 3.98×10 −2 and sin 2θ E = 0.940 for the average densityρ E = 5.6g/cm 3 and the electron fraction Y eE = 0.5 in the Earth. Then, f reg zenith is given as f reg zenith = 1.72 × 10 −2 for A M SW = 1 and sin 2 2θ 13 = 0.089. This result is in reasonable agreement with more detailed estimate using the PREM profile [17] for the Earth matter density. We now give a simple estimate of the day-night asymmetry A DN assuming constant matter density approximation in the earth, and its A M SW dependence. Under the approximation of small regeneration effect f reg ≪ 1, the day-night asymmetry A DN for the CC number of counts N CC measurement is approximately given by
A CC DN ≡ N N CC − N D CC 1 2 [N N CC + N D CC ] ≈ − 2 cos 2θ S 1 + cos 2θ 12 cos 2θ S f reg zenith(11)
where in the right-hand-side we have approximated A CC DN by the asymmetry of survival probabilities in day and in night at an appropriate neutrino energy, and ignored the terms of order f reg 2 zenith . Notice that the effects of the solar and the earth matter densities are contained only in cos 2θ S and f reg zenith , respectively.
At E = 7 MeV, ξ S = 1.31, cos 2θ 12 = 0.377, cos 2θ S = −0.710, and hence A CC DN = 3.41 × 10 −2 A M SW cos 4 θ 13 , about 3% day-night asymmetry for A M SW = 1. Note that cos 4 θ 13 = 0.95 for sin 2 2θ 13 = 0.1, so that the impact of θ 13 on A CC DN give only a minor modification. Though based on crude approximations, the value of A CC DN at A M SW = 1 obtained above is in excellent agreement with the one evaluated numerically for SNO CC measurement.
SNO and SK observes the day-night asymmetry by measurement of CC reactions and elastic scattering (CC+NC), respectively. We have computed A DN as a function of A M SW numerically (with PREM profile) without using analytic approximation. The result of A DN scales linearly with A M SW in a good approximation, A CC DN ≈ 0.044A M SW . Similarly, the day-night asymmetry for elastic scattering measurement can be easily computed. Its relationship to the A CC DN can be estimated in the similar manner as in (11),
A ES DN ≡ N N ES − N D ES 1 2 [N N ES + N D ES ] ≈ A CC DN × 1 + 2r µ/e 1 − r µ/e [P N ee + P D ee ] −1 ,(12)
taking into account the modification due to NC scattering. Using approximate values, r µ/e =
Constraints on A M SW by Solar Neutrino Observables
In this section we investigate quantitatively to what extent A M SW can be constrained by the current and the future solar neutrino data. The results of our calculations are presented in Fig. 1, supplemented with the relevant numbers in Table 2. We will discuss the results and its implications to some details in a step-by-step manner. We first discuss the constraints by the data currently available (Sec
Current constraint on A M SW
We include in our global analyses the KamLAND and all the available solar neutrino data [3,4,5,6,7,9,10,11,12,13,14]. To obtain all the results quoted in this paper we marginalize over the mixing angles θ 12 and θ 13 , the small mass squared difference ∆m 2 21 , and the solar neutrino fluxes f i [8,28] imposing the luminosity contraint [29]. We include in the analysis the θ 13 dependence derived from the analysis of the atmospheric, accelerator, and reactor data included in Ref. [30] as well as the recent measurement of θ 13 by [19,20,21,22,23]. The χ 2 used is defined by
χ 2 global (A M SW ) = M arg[χ 2 solar (∆m 2 21 , θ 12 , θ 13 , A M SW , f B , f Be , f pp , f CNO ) + χ 2 KamLAND (∆m 2 21 , θ 12 , θ 13 ) + χ 2 REACTOR+ATM+ACC (θ 13 )] ,(13)
where M arg implies to marginalize over the parameters shown but not over A M SW . Further details of the analysis methods can be found in Ref. [28]. The currently available neutrino data (blue solid line), which include SNO lower energy threshold data [12,13] and SK IV [14], do not allow a very precise determination of the A M SW parameter. A distinctive feature of the ∆χ 2 parabola shown in Fig. 1 is the asymmetry between the small and large A M SW regions. At A M SW < 1 the parabola is already fairly steep, and the "wall" is so stiff that can barely be changed by including the future data. While at A M SW > 1 the slope is relatively gentle. Figure 1: ∆χ 2 as a function of A M SW for the currently available solar neutrino data (shown in blue solid line) and the various solar neutrino observables expected in the near future (by color lines specified below). The current data include the one from SNO lower energy threshold analysis and SK I-IV. In addition to the current constraints on A M SW , we show the improved constraints when future solar neutrino data are added one by one: 3σ detection of the SK spectral upturn (magenta dashed line), low energy solar neutrino flux measurements of 7 Be at 5% and pep at 3% (red dash-dotted line), 3σ detection of the SK day-night asymmetry (black dotted line). The red dashed line shows the improved constraints by adding future spectral information at high and low energies. Finally, the global analysis by adding all the spectral information data and the day-night data produces the solid green line. Table 2: The ∆χ 2 minimum of A M SW , the allowed regions of A M SW at 1σ, and 3σ CL are shown in the first, second, and third columns, respectively, for the analyses with the currently available data (first row), the one with spectrum upturn of 8 B neutrinos at 3σ added to the current data (second row), the one with 7 Be and pep neutrinos with 5% and 3% accuracies, respectively, added to the current data (third row), the one with the new spectral information in the second and the third row added to the current data (fourth row), and the one with day-night asymmetry of 8 B neutrinos at 3σ added to the current data (fifth row). The last row presents results of global analysis with all the above data. The numbers in parentheses imply the ones obtained with improved knowledge of θ 12 , see text for details. LMA region preferred by the KamLAND data. The larger best fit value could also partly be due to an artifact of the weakness of the constraint in A M SW > 1 region. Notice that the Standard Model MSW theory value A M SW = 1 is off from the 1σ region but only by a tiny amount, as seen in Table 2. Let us understand these characteristics. The lower bound on A M SW mostly comes from the SK and the SNO data which shows that 8 B neutrino spectrum at high energies is essentially flat with P ee = sin 2 θ, the prediction of the adiabatic LMA MSW solution (see (7)). It is inconsistent with the vacuum oscillation, and hence the point A M SW = 0 is highly disfavored, showing the evidence for the matter effect.
One would think that the upper bound on A M SW should come from either the low energy solar neutrino data, or the deviation from the flat spectra at high energies. But, we still lack precise informations on low energy solar neutrinos, and the spectral upturn of 8 B neutrinos has not been observed beyond the level in [10,11]. Then, what is the origin of the upper bound A M SW < 2 at about 1σ CL? We argue that it mainly comes from the day-night asymmetry of 8 B neutrino flux which is contained in the binned data of SK and SNO. Recently, the SK collaboration reported a positive indication of the day-night asymmetry though the data is still consistent with no asymmetry at 2.3σ CL [14].
To show the point, we construct a very simple model for ∆χ 2 for the day-night asymmetry A ES DN . To what extent an improved knowledge of θ 12 affects A M SW ? It was suggested that a dedicated reactor neutrino experiment can measure sin 2 θ 12 to ≃2% accuracy [31,32]. It is also expected that precision measurement of pp spectrum could improve the accuracy of θ 12 determination to a similar extent [28]. Therefore, it is interesting to examine to what extent an improved knowledge of θ 12 affects the constraint on A M SW . Therefore, we re-compute the ∆χ 2 curves presented in Fig. 1 by adding the artificial term (sin 2 θ 12 − BEST ) 2 /0.02 in the ∆χ 2 assuming 2% accuracy in sin 2 θ 12 determination. The result of this computation is given in Table 2 in parentheses. As we see, size of the effect of improved θ 12 knowledge is not very significant.
Spectrum of solar neutrinos at high energies
We discuss next the impact of observing the spectral upturn of 8 B neutrinos at its relatively low energy part, E > ∼ 2 − 3 MeV. The evidence for the upturn must contribute to constrain the larger values of A M SW because A M SW could be very large without upturn, if day-night asymmetry is ignored. Unfortunately, the upturn has never been observed in the SK, SNO and Borexino measurement; The solar neutrino spectrum is consistent with a flat distribution. We assume that the upturn of 8 B neutrino energy spectrum can soon be established.
We discuss the impact on A M SW of seeing the upturn in recoil electron energy spectrum with 3 σ significance, which we assume to be in the region E e ≥ 3.5 MeV. To calculate ∆χ 2 we assume the errors estimated by the SK collaboration [14]. Adding the simulated data to the currently available data set produces the magenta dashed line in Fig. 1. We find a 25% reduction of the 3σ allowed range, A M SW = 1.34 −0.32 +0.45 ( −0.69 +1.66 ) at 1σ (3σ) CL. We can see that it does improve the upper bound on A M SW , for which the current constraint (blue solid line) is rather weak, but the improvement in the precision of A M SW is still moderate.
Some remarks are in order about the minimum point of ∆χ 2 . The best fit point with the present data is at A M SW > 1 as we saw above. For the analysis with simulated data, the ∆χ 2 minimum must be always at A M SW = 1 in all the cases discussed below. Therefore, the analysis with the present plus simulated data tends to pull the ∆χ 2 minimum toward smaller values of A M SW , and at the same time make the ∆χ 2 parabola narrower around the minimum. By conspiracy between these two features the current constraint (blue solid line) is almost degenerate to the other lines at A M SW < 1, the ones with spectral upturn (magenta dashed line) and low energy neutrinos (red dash-dotted line). These features can be observed in Fig. 1 and in Table 2.
Spectrum of solar neutrinos at low energies
Now, let us turn to the low energy solar neutrinos. The Borexino collaboration have already measured the 7 Be neutrino-electron scattering rate to an accuracy of ≃ ±5% [7], which we assume throughout this section. However, there is an important limitation on what can be learned from the very precise 7 Be flux. We have to use the SSM flux to determine the neutrino survival probability experimentally, and therefore, the uncertainties in the theoretical estimate [8] limit the precision with which the A M SW parameter can be determined from the 7 Be flux measurement.
The measurement of the pep flux has two important advantages, when compared to the 7 Be flux, in determining A M SW : a) the neutrino energy is higher, 1.44 MeV, so the importance of the solar matter effects is larger, b) the uncertainty in the theoretical estimate is much smaller. Firstly, the ratio of the pep to the pp neutrino flux is robustly determined by the SSM calculations, so it can be determined more accurately than the individual fluxes because the ratio depends only weakly on the solar astrophysical inputs. Secondly, a very precise measurement of the 7 Be flux, with all the other solar data and assuming energy conservation (luminosity constraint), leads to a very precise determination of the pp and pep flux, at the level of ∼ 1% accuracy [28]. See [18] for the first observation of pep neutrinos, and its current status of the uncertainties.
A measurement of the pep neutrino-electron scattering rate must be very accurate, ≃ ±3%, in order to significantly constrain our knowledge of the matter effects in the Sun. If this accuracy is achieved, then the uncertainties of the A M SW parameter in region A M SW > 1 will be significantly reduced. The red dash-dotted line in Fig. 1 shows the result of the combined analysis of future low energy data, an improved 7 Be measurement with 5% precision and a future pep measurement with 3% precision, added to the current data. The obtained constraint on A M SW is: A M SW = 1.25±0.28( −0.60 +1.09 ) at 1σ (3σ) CL. The resultant constraint on A M SW from above is much more powerful than the one obtained with spectrum upturn of high energy 8 B neutrinos at 3σ.
By having solar neutrino spectrum informations both at high and low energies it is tempting to ask how tight the constraint become if we combine them. The result of this exercise is plotted by the red dashed line in Fig. 1 and is also given in Table 2. The resultant constraint on A M SW is:
A M SW = 1.22 −0.25 +0.27 ( −0.57 +1.01 ) at 1σ (3σ) CL.
Day-night asymmetry
To have a feeling on to what extent constraint on A M SW can be tightened by possible future measurement, we extend the simple-minded model discussed in Sec. 0.3.1, but with further simplification of assuming A M SW = 1 as the best fit. Let us assume that the day-night asymmetry A ES DN can be determined with (2/N )% accuracy, an evidence for the day-night asymmetry at N σ CL. Then, the appropriate model ∆χ 2 is given under the same approximations as in Sec. 0.3.1 as ∆χ 2 = N 2 (A M SW − 1) 2 . We boldly assume that the day-night asymmetry at 3σ CL would be a practical goal in SK. It predicts ∆χ 2 = 9 (A M SW − 1) 2 , which means that A M SW can be constrained to the accuracy of 33% uncertainty at 1σ CL. Now, we give the result based on the real simulation of data. The black dotted line in Fig. 1 show the constraint on A M SW obtained by future 3σ CL measurement of the day-night asymmetry, which is added to the present solar neutrino data. As we see, the day-night asymmetry is very sensitive to the matter potential despite our modest assumption of 3σ CL measurement of A ES DN . The obtained constraint on A M SW is:
A M SW = 1.17 −0.21
+0.26 ( −0.51 +0.81 ) at 1σ (3σ) CL. The obtained upper bound on A M SW is actually stronger than the one expected by our simple-minded model ∆χ 2 . Apart from the shift of the bast fit to a larger value of A M SW , the behavior of ∆χ 2 is more like ∆χ 2 ≈ 14 (A M SW − 1) 2 in the region A M SW > 1. It can also been seen in Fig. 1 that the upper bound on A M SW due to the day-night asymmetry at 3σ CL (black dotted line) is stronger than the one from combined analysis of all the expected measurements of the shape of the spectrum (red dashed line) discussed at the end of Sec. 0.3.3.
Given the powerfulness of the day-night asymmetry for constraining A M SW , it is highly desirable to measure it at higher CL in the future. Of course, it would be a challenging task, and probably requires a megaton class water Cherenkov or large volume liquid scintillator detectors with solar neutrino detection capability. They include, for example, Hyper-Kamiokande [33], UNO [34], or the ones described in [35].
Global analysis
We now discuss to what extent the constraint on A M SW can become stringent when all the data of various observable are combined. The solid green line in Fig. 1 shows the constraint on A M SW obtained by the global analysis combining all the data sets considered in our analysis. The obtained sensitivity reads A M SW = 1.12 −0.17 +0.21 ( −0.45 +0.66 ) at 1σ (3σ) CL. Therefore, the present and the future solar neutrino data, under the assumptions of the accuracies of measurement stated before, can constrain A M SW to ≃15% (40%) at 1σ (3σ) CL from below, and to ≃20% (60%) at 1σ (3σ) CL from above. If we compare this to the current constraint A M SW = 1.47 −0.42 +0.54 ( −0.82 +1.88 ) the improvement of the errors for A M SW over the current precision is, very roughly speaking, a factor of ≃ 1.5 − 2 in region A M SW < 1, and it is a factor of ≃ 2 at A M SW > 1. Noticing that the efficiency of adding more data to have tighter constraint at A M SW < 1 is weakened by shift of the minimum of ∆χ 2 , improvement of the constraint on A M SW is more significant at A M SW > 1.
Summary
In this paper, we have discussed the question of to what extent tests of the MSW theory can be made stringent by various solar neutrino observables. First, we have updated the constraint on A M SW , the ratio of the effective coupling constant of neutrinos to G F , the Fermi coupling constant with the new data including SNO 8 B spectrum and SK day-night asymmetry. Then, we have discussed in detail how and to what extent the solar neutrino observable in the future tighten the constraint on A M SW .
The features of the obtained constraints can be summarized as follows:
• Interpretation of solar neutrino data at high energies by the vacuum oscillation is severely excluded by the SNO and SK experiments, which leads to a strong and robust lower bound of A M SW . On the other hand, the day-night asymmetry at ≃ 2σ level observed by SK dominates the bound at high A M SW side. We find that present data lead to A M SW = 1.47 −0.42 +0.54 ( −0.82 +1.88 ) at 1σ (3σ) CL. The Standard Model prediction A M SW = 1 is outside the 1σ CL range but only by tiny amount.
• We have explored the improvements that could be achieved by solar neutrinos experiments, ongoing and in construction. We discussed three observables that are sensitive enough to significantly improve the limits on A M SW , particularly in the region A M SW > 1: a) upturn of the 8 B solar neutrino spectra at low energies at 3σ CL, b) high precision measurement of mono-energetic low energy solar neutrinos, 7 Be (5% precision) and pep (3% precision) neutrinos, and c) day-night asymmetry of the 8 [36] and KamLAND [37] may detect spectrum modulation of B neutrinos at low energies at CL higher than 3σ.
Finally, by combining all the data set we have considered we obtain A M SW = 1.11 −0.18 +0.26 ( −0.47 +0.87 ) at 1σ (3σ) CL.
• In a wider context the constraints on effective neutrino matter coupling has been discussed in the framework of possible nonstandard interactions (NSI denoted as ε αβ ) of neutrinos [38], in which our A M SW may be interpreted as A M SW = 1 + ε ee (assuming vanishing of all the other ε αβ elements). With solar neutrinos see [39] for discussion of NSI. However, if NSI exist it is likely that the other elements will be detected first in long-baseline experiments; ε eµ and ε eτ are of second order in ǫ, while ε ee comes in only at third order in ǫ in a perturbative framework [40]. The analyses shows that the sensitivity to ε ee is indeed lower at least by an order of magnitude compared to the ones to ε eµ or ε eτ . See [41] and the references cited therein. Hence, the solar neutrinos are an alternative good probe for ε ee .
In conclusion, testing the theory of neutrino propagation in matter deserves further endeavor. The lack of an accurate measurement of the matter potential felt by solar neutrinos reflects the fact that solar neutrino data only do not precisely determine the mass square splitting. The good match of the independently determined mass square splitting by solar neutrino data and by reactor antineutrino data will confirm the Standard Model prediction of the relative index of refraction of electron neutrinos to the other flavor neutrinos. The lack of match of both measurements would point to new physics like the one tested here.
. 0.3.1). Then, we address the question of how the constraint on A M SW can be tightened with the future solar neutrino data, the spectral upturn of 8 B neutrinos (Sec. 0.3.2), the low energy 7 Be and pep neutrinos (Sec. 0.3.3), and finally the day-night asymmetry of the solar neutrino flux (Sec. 0.3.4). We pay special attention to the question of how the constraints on A M SW depend upon the significance of these measurements.
More quantitatively, A M SW = 1.47 −0.42 +0.54 ( −0.82 +1.88 ) at 1σ (3σ) CL. The best fit point with the present data is significantly larger than unity, A M SW = 1.47. It was 1.32 before and have driven to the larger value mostly by the new SK data which indicates a stronger matter effect than those expected by the MSW
It is made possible by the approximate linearity of A ES DN to A M SW . Let us start from the data of day-night asymmetry at SK I-IV: A ES DN = 4.0 ± 1.3 ± 0.8% [14], giving the total error 1.5% if added in quadrature. The expectation of A ES DN by the LMA solution is A ES DN = A M SW × 2.1% for ∆m 2 21 = 7.6 × 10 −5 eV 2 . Then, one can create an approximate model ∆χ 2 as ∆χ 2 = [(A DN − 2.8%) /1.5%] 2 = 2 (A M SW − 1.3) 2 . It explains very well, considering the extremely crude nature of the model, the behavior of ∆χ 2 with the current data (blue solid line) in Fig. 1 in the region A M SW > 1. Therefore, we find that about 2σ evidence of A ES DN in the SK data is the main cause of the sensitivity to A M SW in the region A M SW > 1.
Table 1 :
1Average electron density at the neutrino production region and energy of the relevant pp
solar neutrinos fluxes. Last column shows the ratio of the electron neutrino elastic scattering with
electrons cross section to the µ (or τ ) neutrino one. For this calculation, we have assumed a measured
electron kinetic energy range of [0.05,0.4], [1,1.4], [0,0.8], and [5,16] MeV for the pp, pep, 7 Be and 8 B
respectively.
Source ρ S Y e (g cm −3 ) Energy (MeV)
σµ
σe
pp
67.9
≤0.42
0.284
pep
73.8
1.44
0.203
7 Be
86.5
0.86
0.221
8 B
92.5
≤16
0.155
Whereas at high energies, small 1
ξ S , the oscillation probability (1) can be approximated, keeping only
the first energy dependent term as
P D
ee = cos 4 θ 13 sin 2 θ 12 +
1
4
sin 2 2θ 12 cos 2θ 12
1
ξ S
2
+ sin 4 θ 13
the factor in the square bracket can be estimated to be5 8 , giving a reasonable approximation for the ratio of A ES DN to A CC DN . A better approximation to the computed results of the A M SW dependence of the asymmetry is given by A ES DN = 0.02A M SW .1
6 and
1
2 [P N
ee + P D
ee ] = 1
3
B solar neutrino flux at 3σ CL. They lead to the improvement of the bound as follows: a) A M SW = 1.34 −0.32 +0.45 ( −0.69 +1.66 ) at 1σ (3σ) CL. b) A M SW = 1.25 ± 0.28( −0.60 +1.09 ) at 1σ (3σ) CL. c) A M SW = 1.17 −0.21 +0.26 ( −0.51 +0.81 ) at 1σ (3σ) CL. It could be expected that future measurement by SNO +
We thank the Galileo Galilei Institute for Theoretical Physics and the organizers of the workshop "What is ν?" for warm hospitality.
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[
"Fermi and Swift observations of the bright short GRB 090510",
"Fermi and Swift observations of the bright short GRB 090510"
] |
[
"V Pelassa \nLPTA\nCNRS\nIN2P3\nUniversité Montpellier 2\n\n"
] |
[
"LPTA\nCNRS\nIN2P3\nUniversité Montpellier 2\n"
] |
[
"Fermi Symposium"
] |
M. OhnoISAS/JAXA on behalf of the Fermi LAT and GBM collaborationsThe bright short-hard GRB 090510 was observed by both Swift and Fermi telescopes. The study of the prompt emission by Fermi revealed an additional high-energy spectral component, the largest lower limit ever on the bulk Lorentz factor in a short GRB jet, and brought the most stringent constraint ever on linear Lorentz invariance violation models. The fast repoint and follow-up by both telescopes allowed the first multiwavelength study of a GRB afterglow from optical range to several GeV. This long-lived emission has been studied in the framework of the internal shock and external shock models.
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"https://arxiv.org/pdf/1002.2863v1.pdf"
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1002.2863
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4499b183013997e6cd092602453bc8a7ed0a6e96
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Fermi and Swift observations of the bright short GRB 090510
2009. Nov. 2-5 1
V Pelassa
LPTA
CNRS
IN2P3
Université Montpellier 2
Fermi and Swift observations of the bright short GRB 090510
Fermi Symposium
Washington, D.C.2009. Nov. 2-5 1
M. OhnoISAS/JAXA on behalf of the Fermi LAT and GBM collaborationsThe bright short-hard GRB 090510 was observed by both Swift and Fermi telescopes. The study of the prompt emission by Fermi revealed an additional high-energy spectral component, the largest lower limit ever on the bulk Lorentz factor in a short GRB jet, and brought the most stringent constraint ever on linear Lorentz invariance violation models. The fast repoint and follow-up by both telescopes allowed the first multiwavelength study of a GRB afterglow from optical range to several GeV. This long-lived emission has been studied in the framework of the internal shock and external shock models.
The bright short-hard GRB 090510 was observed by both Swift and Fermi telescopes. The study of the prompt emission by Fermi revealed an additional high-energy spectral component, the largest lower limit ever on the bulk Lorentz factor in a short GRB jet, and brought the most stringent constraint ever on linear Lorentz invariance violation models. The fast repoint and follow-up by both telescopes allowed the first multiwavelength study of a GRB afterglow from optical range to several GeV. This long-lived emission has been studied in the framework of the internal shock and external shock models.
I. OBSERVATIONS
On May 10th, 2009 at 00:23:00.48 UT, the Burst Alert Telescope (BAT) onboard the Swift observatory detected a bright Gamma-Ray Burst (GRB) [1], and autonomously repointed towards it within 100 s, allowing for over 2 days of follow-up observation by the Ultra-Violet and Optical Telescope (UVOT) and the X-Ray Telescope (XRT). An afterglow was detected in ultra-violet and X-ray, which analysis provided a location RA = 333.55227, Dec = -26.5827, with an error of 1.4 arcsec (90% statistical) [2]. VLT afterglow observations provided a spectroscopic redshift z = 0.903 ± 0.003 [3].
On the same day at 00:22:59.97 UT, the Gammaray Burst Monitor (GBM) onboard Fermi detected a bright GRB [4]. A small and short spike at trigger time was followed 0.5 s later by 0.5 s-long emission made of 7 pulses. This GRB also triggered the Fermi Large Area Telescope (LAT) onboard algorithm [5] which located it at RA = 333.4 • , Dec = -26.767 • , with an error of 7 arcmin (statistical), consistent with the GBM and Swift detection. The spacecraft autonomously repointed towards the source for 5 hours of follow-up observation. The prompt emission was coincident with the GBM main emission. More than 150 photons above 100 MeV were observed, and more than 20 events above 1 GeV in the first minute after trigger (see fig. 1) [6]. The follow-up observation allowed the detection of a significant GeV emission up to 200 s after trigger.
Also AGILE [7], Integral-SPI and Suzaku-WAM [8] detected GRB 090510 prompt emission. According to all observations GRB 090510 was a very bright short GRB (table I).
In the following we focus on the prompt emission seen by Fermi (section II) and the multiwavelength afterglow study based on Swift and Fermi data (section III). Convention : spectral and temporal indices are defined according to to the fluence evolution The spectral analysis of the prompt emission was performed with RMfit, combining GBM and LAT data ( fig. 2).
The time-integrated spectrum shows a significant additional high-energy power-law component (N σ = 5.6, β P L = 0.62 ± 0.03), which carries 37% of the total γ-ray fluence.
The time-resolved spectroscopy shows the late arrival of the LAT emission and of the additional component. In bin 'a' (0.5s -0.6s after Fermi trigger), a Band function with steep β Band > 4 is fitted. In bin 'b' (0.6s -0.8s), a significant additional power-law component is found (β P L = 0.66 ± 0.04). In bin 'c' (0.8s -0.9s) this high-energy component can be fitted but not significantly. In bin 'd' (0.9s -1.0s) the LAT data only are fitted by a power-law (β P L = 0.9 ± 0.2). The observation of high-energy events allows to constrain the jet's initial bulk Lorentz factor Γ 0 . Lowand high-energy photons emitted in the same physical region of the jet can interact to produce e ± pairs. The opacity to pair-production is strongly reduced if the emission region moves towards the observer with a high bulk Lorentz factor. The optical depth to e ± -pair production at a given photon energy τ γγ (E) depends on the γ-ray emission spectral shape and variability time scale t v , the GRB redshift and the jet's bulk Lorentz factor. The condition τ γγ (E < E max ) < 1, where E max is the maximal observed photon energy, yields a lower limit on Γ 0 . For different epochs of the LAT emission, estimates of t v and spectral models lead to several possible lower limits, all of order Γ 0,min ∼ 10 3 (see table II). This is the highest lower limit ever set on a GRB, and the most rapid ejection ever observed from a short GRB.
C. Interpretation of the high-energy spectrum
The physical interpretation of the high-energy prompt spectrum of GRB 090510 is described in detail in [10]. The lower-energy part of the prompt emission spectrum (< 10 MeV) can be explained by nonthermal synchrotron processes, as well as photospheric thermal emission.
In a non-thermal leptonic model framework, Synchrotron Self-Compton (SSC) emission is expected at GeV-TeV energies and can result in the observed additional power-law component. Detailed simulations of SSC emission have been performed, using the observed Band component as the seed population, taking into account internal γγ opacity, self-absorption, radiative escapes, and considering either Thomson or Klein-Nishina scattering regimes. A low magnetic field (B << 10 3 G) favors Compton scattering and can explain the observed bright and hard high-energy component. It would also slow down the cascade formation which could explain the late onset of the high-energy emission. The interpretation of the highenergy emission as forward shock emission from an early afterglow was also considered to explain the delayed onset of the high-energy emission. However, this scenario assumes a very rapid ejection (Γ ≃ 2000 -4000), and implies a high-energy component spectral index of β P L ≃ 1. This spectrum is consistent with the long-lived spectrum, but not with the additional component found in bin 'b' (β P L = 0.66). In both cases, the hard spectral index of the soft photons (α Band = −0.48) is not explained by the standard synchrotron mechanism.
In hadronic models, secondary e ± pairs come from the decay of pions formed through photohadronic processes and by the γ−γ attenuation of synchrotron pho- tons radiated by protons and ions. Synchrotron radiation or inverse Compton scattering from these pairs can produce the observed high-energy γ-ray emission. In the case of photohadronic models, a stronger magnetic field makes secondary pion production more efficient through more rapid hadronic acceleration, but results in a high-energy component spectral index β P L ≃ 1, significantly softer than observed. On the other hand, inverse Compton scattering of secondary pairs can produce a high-energy component as hard as the one observed, but this scenario requires a very high isotropic-equivalent power P iso ∼ 10 55 erg.s −1 . Proton synchrotron models also require a high isotropic equivalent energy, and a high magnetic field. However, the apparent E iso problem can be solved if the jet is strongly collimated or if the true Lorentz factor Γ is a factor ≃2 less than the minimum values Γ 0,min estimated through the γ −γ opacity constraint. If Γ > Γ 0,min , then a proton synchrotron model is strongly disfavored, though note that GRBs with a very high isotropic energy have been observed.
D. Limits on Lorentz Invariance Violation (LIV)
Some quantum gravity models allow for a dependence of the photons speed v ph on their energy, i.e. their arrival time can be written as a development in their energy. In this framework, a low-energy (E ℓ ) and a high-energy (E h ) photons emitted together would
∆t = s n (1 + n) 2H 0 (E n h − E n ℓ ) (M QG,n c 2 ) n × z 0 (1 + z ′ ) n Ω m (1 + z ′ ) 3 + Ω Λ dz ′
where z is the GRB redshift, and n = 1 or 2. A standard cosmology (h, Ω m , Ω Λ ) = (0.71, 0.27, 0.73) has been assumed for the computations. The difference in arrival times can be of any sign s n , depending on the models. The observation of high-energy photons from distant sources allows to put constraints on the energy scales M QG,n that appear in the development.
GRB 090510 has a redshift of z = 0.903 ± 0.03, and its emission in the LAT energy range includes a photon of energy E h = 30.5 +5.8 −2.6 GeV detected 0.829 s after the GBM trigger. This event's topology makes it a very good photon candidate. The expected background rate and this event's direction proximity to GRB 090510 location allowed us to associate this event to the burst with high confidence (N σ =4.4 to 5.6 depending on the selections used).
The observed lightcurve allows different measurements of the temporal delay for this event, depending on which part of the lower energy emission it is associated to. As described in detail in [9], several lower limits on the linear term's energy scale M QG,1 could be derived (see table III). All require M QG,1 > M P lanck , which is a strong constraint on linear LIV models.
Lower limits on M QG,2 were also derived, but they are much less constraining. For more details on LIV limits derivation, see [12].
III. AFTERGLOW MULTI-WAVELENGTH STUDY
A. eV to GeV observations
The fast repoint by Swift and the ARR performed by Fermi provided a long-lasting observation of GRB 090510 afterglow, from the optical range to several GeV, interrupted by Earth occultation of the GRB to . XRT lightcurve is obtained as in [14,15]. All other data are shown with 68% error bars or 95% confidence level upper limits.
both observatories. The fluxes observed by all instruments are reported in fig. 3. UVOT and XRT observations of the afterglow started ∼ 100 s after the prompt emission trigger, and lasted over 2 days. The optical and UV lightcurve is best fit by a smoothly broken power-law with an initial rise : α Opt,1 = −0.50 +0. 11 −0.13 . The smooth break at t Opt = 1.58 +0.46 −0.37 ks is followed by a shallow decay α Opt,2 = 1.13 +0.11 −0.10 . The X-ray lightcurve is best fit by a broken power-law, with an initial shallow decay α X,1 = 0.74 ± 0.03, a break at t X = 1.43 +0.09 −0.15 ks and a steep late decay α X,2 = 2.18 ± 0.10.
The beginning of the temporally-extended emission in the LAT has been chosen as the end of the prompt emission seen by the GBM (0.9 s after GBM trigger, i.e. 0.38 s after BAT trigger). A significant emission was detected up to 200s after trigger and up to 4 GeV. This observation was divided in time bins, in all of them an unbinned likelihood spectral analysis was performed using diffuse class events [13]. The flux decay is best fit by a simple power-law of index Top : SEDs at different epochs, with the best fit shown (β2 and β3 constrained, see text). The butterfly at 100s indicates the 68% confidence level region for the LAT flux, obtained from an unbinned likelihood analysis (95% error bar at 100 MeV is shown). Successive SEDs in time order are rescaled by 1:1, 1:10, 1:100, 1:1000, 1:10000. Bottom : SED at 100s, with the best fit shown. The β2 -β3 constraint is relaxed. 68% error bars are shown for UVOT and XRT data. A 68% confidence level region is drawn in the LAT energy range. α γ = 1.38 ± 0.07, with no significant feature or break.
The power-law spectrum shows no significant evolution over time and has an average index β γ = 1.1±0.1.
B. Discussion on the afterglow origin
The interpretation of the long-lasting multiwaveband emission fron GRB 090510 is discussed in detail in [11]. Two scenarios are considered in the fireball model frame.
In the first scenario, X-ray and γ-ray emissions are due to internal shocks, and the optical and UV emis-sion to the forward shock. The fluxes measured at 100 s after trigger in the X-ray and γ-ray ranges are consistent with an internal shock origin, with some fine tuning required. For instance this scenario assumes that the initial rise of the optical emission is due to the forward shock onset, and therefore is steeper than what was observed. But the observation may have caught the end of the steep rise phase. This interpretation also requires a very low ambient density n ∼ 10 −6 cm −3 , which is low, even for short GRBs. The assumed initial bulk Lorentz factor Γ 0 ∼ 10 3 is consistent with the prompt emission analysis (see section II B).
Another possibility is that the full long-lasting emission comes from the forward shock region. This model predicts a broad spectrum with a doublybroken power-law shape and constraints on the indices : β 1 = −1/3, β 3 = β 2 + 1/2. Five successive Spectral Energy Distributions (SED) have been built for 5 different observation dates (100 s, 150 s, 700 s, 1000 s, 1400 s after trigger), including X-ray and UV data, as well as LAT data for the SED at 150 s (see fig. 4). A fit of these data yields a good agreement with the aforementioned model, with β 2 = 0.78 ± 0.04. The low-energy break decreases with time, from 0.43 keV to <0.01 keV. The high-energy break is poorly constrained (∈ [10−130] MeV) but is confirmed by a fit of the 100 s SED alone (N σ > 4.8) where the constraint on β 2 and β 3 is relaxed but still holds within error bars. As a result, the afterglow emission spectrum is well described by the forward shock model over 9 energy decades. The lightcurves observed before 1 ks after trigger are also consistent with this model. However, the early onset of the forward shock emission requires a very high bulk Lorentz factor Γ 0 > 5800, and some temporal emission properties are not well explained, e.g. α X,1 is too shallow and α Opt,2 = α X,2 . Theoretical extensions of the model can alleviate these problems. E.g., a phase of energy injection or an evolution of the microphysical parameters of the blast wave may cause an early shallow decay of the X-ray flux ; also the difference between X-ray and optical late decay slopes could be explained by dynamical effects.
IV. CONCLUSION
GRB 090510 observation was remarkable for several reasons. First, it was the first short GRB of known redshift (z = 0.903) observed at GeV energies. It yielded the highest energy photon ever observed from a short GRB (30.5 GeV). The observation of high-energy photons from this GRB allowed to put strong and robust constraints on linear LIV models (M QG,1 > M P lanck required) and on the jet's velocity (Γ 0,min ∼ 10 3 ).
This GRB also yielded the first clear evidence of an additional high-energy spectral component (N σ = 5.6 on the time-integrated prompt spectrum), which was to be observed in several other bursts [16,17]. This high-energy spectral component is a hint for SSC radiation or a possible UHECRs production in GRBs.
Finally, this energetic short GRB showed a bright optical and X-ray afterglow, as well as a long-lived GeV emission, which could be observed thanks to Swift and Fermi fast repoint abilities. The temporal and spectral properties of this long-lasting emission could be studied over 9 energy decades and can be explained in at least two ways : a combination of internal shock and forward shock emission reproduces well the fluxes observed although some fine tuning is required, and a forward-shock only origin can reproduce the observed spectra although it requires some theoretical extensions. Such joint Swift-Fermi observations are very promising for understanding GRB afterglow physics.
of the Fermi LAT and GBM collaborations
FIG. 1 :
1GRB 090510 observation by Fermi [9]. Panel (a) : LAT events passing the onboard (blue) or onground (red) photon selections. The lines show the arrival time dependence on photon energy for the events associated to the 31 GeV photon, according to linear (solid lines) or quadratic (dashed lines) energy dependence in the LIV model (see section II D). Different parts of the lower-energy emission are considered : full GBM emission (black), bulk GBM emission (red), bulk LAT emission (green), coicident GBM peak (blue). Panels (b -f ) : multi-instrument lightcurve. Curves (b -d) are background-subtracted. Curve (d) shows all LAT events passing the onboard photon selection. Curves (e -f) show on-ground selected photons inside the region of interest, used for the spectral analysis. Panel (f) also shows event energies. B. Limits on the jet's initial bulk Lorentz factor Γ0
FIG. 2 :
2GRB 090510 prompt spectrum[10].Panel (b) : time-integrated Spectral Energy Distribution (SED). The Band function and power-law components are shown. Their sum is drawn in thick line with 68% confidence level contours. Panel (c) : time-resolved SED with 68% confidence level contours. In bin 'a' no signal is detected in the LAT. Bins 'b' and 'c' show an additional power-law component. No signal is detected in the GBM after 0.9 s (bin 'd').
FIG. 3 :
3GRB 090510 Swift and Fermi observations[11].Top panel : LAT flux above 100 MeV and best fit to the flux decay (line). Bottom panel : energy flux densities averaged in the observed energy bands: BAT (15 keV -350 keV, stars); XRT (0.2 keV -10 keV, crosses); UVOT renormalised to white (diamonds); LAT (100 MeV -4 GeV, filled squares; the average spectral index was used to convert from photon to energy flux) with upper limits for β = 1.1 (triangles). The prompt emission is shown for comparison: GBM (8 keV -1 MeV, circles), LAT (100 MeV 4 GeV, empty squares)
FIG. 4 :
4GRB 090510 UVOT-XRT-LAT SEDs[11].
TABLE I :
IDuration (T90 and T50) of GRB 090510 prompt emission observed by different instruments.[10] Instrument
T90 (s) T50 (s)
GBM/NaI 3,6,7 2.1
0.2
Swift/BAT
4.0
0.7
Integral-SPI
2.5
0.1
Suzaku-WAM
5.8
0.5
description : F ∝ t −α ν −β
II. THE γ-RAY PROMPT EMISSION
A. Spectroscopy
TABLE II :
IIΓ0,min values from GRB 090510 observations.Time bin Spectral model tv (ms) Emax (GeV) Γ0,min
b
Band + PL 14 ± 2
3.43
951 ± 38
d
Band
12 ± 2
30.5
1324 ± 50
d
Band + PL 12 ± 2
30.5
1218 ± 61
TABLE III :
IIILimits on linear LIV derived from GRB 090510 observations[9].|∆t/∆E| or |∆t| MQG,1/M P lanck snMethod
< 30 ms/GeV
> 1.22
± 1 LAT lag analysis
< 859 ms
> 1.19
1
not emitted
before GBM onset
< 10 ms
> 102
± 1 GBM pulse width
arrive at different times, with the dominant LIV de-
lay term :
. E A Hoversten, GCN Circular. 9331E.A. Hoversten, et al., GCN Circular 9331
. M R Goad, GCN Circular. 9339M.R. Goad, et al., GCN Circular 9339
. A Rau, GCN Circular. 9353A. Rau, et al., GCN Circular 9353
. S Guiriec, GCN Circular. 9336S. Guiriec, et al., GCN Circular 9336
. M Ohno, GCN Circular. 9334M. Ohno, et al., GCN Circular 9334
. N Omodei, GCN Circular. 9350N. Omodei, et al., GCN Circular 9350
. F Longo, GCN Cicular. 9343F. Longo, et al., GCN Cicular 9343
. N Ohmori, GCN Circular. 9355N. Ohmori, et al., GCN Circular 9355
A limit on the variation of the speed of light arising from quantum gravity effects. A A Abdo, Nature. 462331A.A. Abdo, et al., "A limit on the variation of the speed of light arising from quantum gravity effects", Nature, 462, 331 (2009)
Fermi Observations of GRB 090510: a short hard Gamma-Ray Burst with powerlaw emission from 10 keV to GeV energies. A A Abdo, ApJ. submittedA.A. Abdo, et al. "Fermi Observations of GRB 090510: a short hard Gamma-Ray Burst with power- law emission from 10 keV to GeV energies", ApJ, submitted (2010)
. M. De Pasquale, P Schady, N P M Kuin, M J Page, P A Curran, S Zane, S R Oates, S T Holland, A A Breeveld, E A Hoversten, G Chincarini, D Grupe, M. De Pasquale, P. Schady, N.P.M. Kuin, M.J. Page, P.A. Curran, S. Zane, S.R. Oates, S.T. Holland, A. A. Breeveld, E.A. Hoversten, G. Chincarini, D.Grupe ;
Swift and Fermi observations of the early afterglow of the short Gamma-Ray Burst 090510. A A Abdo, ApJL. 709146A.A. Abdo et al. "Swift and Fermi observations of the early afterglow of the short Gamma-Ray Burst 090510", ApJL, 709, 146 (2009)
Constraining Lorentz Invariance Violation with Fermi. V Vasileiou, these proceedingsV. Vasileiou, "Constraining Lorentz Invariance Viola- tion with Fermi", these proceedings
Fermi observations of high-energy gamma-ray emission from GRB 080825C. A A Abdo, ApJ. 707580A.A. Abdo, et al., "Fermi observations of high-energy gamma-ray emission from GRB 080825C", ApJ, 707, 580 (2009)
An online repository of Swift/XRT light curves of γ-ray bursts. P A Evans, A P Beardmore, K L Page, A&A. 469Evans, P.A., Beardmore, A.P., Page, K.L. et al., "An online repository of Swift/XRT light curves of γ-ray bursts" A&A, 469, 379-385 (2007)
Methods and results of an automatic analysis of a complete sample of Swift-XRT observations of GRBs. P A Evans, A P Beardmore, K L Page, arXiv:08123662MNRAS submitted. Evans P.A., Beardmore, A.P., Page K.L., et al., "Methods and results of an automatic analysis of a complete sample of Swift-XRT observations of GRBs", MNRAS submitted, (2009) (arXiv:08123662)
Fermi GBM observations of three intense short GRBs. S Guiriec, these proceedingsS. Guiriec, "Fermi GBM observations of three intense short GRBs", these proceedings
Fermi observations of GRB 090902B: a distinct spectral component in the prompt and delayed emission. A A Abdo, ApJL. 706138A.A. Abdo, et al., "Fermi observations of GRB 090902B: a distinct spectral component in the prompt and delayed emission", ApJL, 706, 138 (2009)
|
[] |
[
"Anisotropy of the Superconducting State in Sr 2 RuO 4",
"Anisotropy of the Superconducting State in Sr 2 RuO 4"
] |
[
"C Rastovski \nDepartment of Physics\nUniversity of Notre Dame\nNotre Dame\n46556IndianaUSA\n",
"C D Dewhurst \nInstitut Laue-Langevin\n6 Rue Jules HorowitzF-38042GrenobleFrance\n",
"W J Gannon \nDepartment of Physics and Astronomy\nNorthwestern University\n60208EvanstonIllinoisUSA\n",
"D C Peets \nDepartment of Physics\nGraduate School of Science\nKyoto University\n606-8502KyotoJapan\n\nMax Planck Institute for Solid State Research\nD-70569StuttgartGermany\n",
"H Takatsu \nDepartment of Physics\nGraduate School of Science\nKyoto University\n606-8502KyotoJapan\n\nDepartment of Physics\nTokyo Metropolitan University\n192-0397TokyoJapan\n",
"Y Maeno \nDepartment of Physics\nGraduate School of Science\nKyoto University\n606-8502KyotoJapan\n",
"M Ichioka \nDepartment of Physics\nOkayama University\n700-8530OkayamaJapan\n",
"K Machida \nDepartment of Physics\nOkayama University\n700-8530OkayamaJapan\n",
"M R Eskildsen \nDepartment of Physics\nUniversity of Notre Dame\nNotre Dame\n46556IndianaUSA\n"
] |
[
"Department of Physics\nUniversity of Notre Dame\nNotre Dame\n46556IndianaUSA",
"Institut Laue-Langevin\n6 Rue Jules HorowitzF-38042GrenobleFrance",
"Department of Physics and Astronomy\nNorthwestern University\n60208EvanstonIllinoisUSA",
"Department of Physics\nGraduate School of Science\nKyoto University\n606-8502KyotoJapan",
"Max Planck Institute for Solid State Research\nD-70569StuttgartGermany",
"Department of Physics\nGraduate School of Science\nKyoto University\n606-8502KyotoJapan",
"Department of Physics\nTokyo Metropolitan University\n192-0397TokyoJapan",
"Department of Physics\nGraduate School of Science\nKyoto University\n606-8502KyotoJapan",
"Department of Physics\nOkayama University\n700-8530OkayamaJapan",
"Department of Physics\nOkayama University\n700-8530OkayamaJapan",
"Department of Physics\nUniversity of Notre Dame\nNotre Dame\n46556IndianaUSA"
] |
[] |
Despite intense studies the exact nature of the order parameter in superconducting Sr2RuO4 remains unresolved. We have used small-angle neutron scattering to study the vortex lattice in Sr2RuO4 with the field applied close to the basal plane, taking advantage of the transverse magnetization. We measured the intrinsic superconducting anisotropy between the c axis and the Ru-O basal plane (∼ 60), which greatly exceeds the upper critical field anisotropy (∼ 20). Our result imposes significant constraints on possible models of triplet pairing in Sr2RuO4 and raises questions concerning the direction of the zero spin projection axis.
|
10.1103/physrevlett.111.087003
|
[
"https://arxiv.org/pdf/1302.4810v3.pdf"
] | 28,358,336 |
1302.4810
|
8986ebc34889be4265cdbe69c5b24df2f9e6e409
|
Anisotropy of the Superconducting State in Sr 2 RuO 4
26 Aug 2013
C Rastovski
Department of Physics
University of Notre Dame
Notre Dame
46556IndianaUSA
C D Dewhurst
Institut Laue-Langevin
6 Rue Jules HorowitzF-38042GrenobleFrance
W J Gannon
Department of Physics and Astronomy
Northwestern University
60208EvanstonIllinoisUSA
D C Peets
Department of Physics
Graduate School of Science
Kyoto University
606-8502KyotoJapan
Max Planck Institute for Solid State Research
D-70569StuttgartGermany
H Takatsu
Department of Physics
Graduate School of Science
Kyoto University
606-8502KyotoJapan
Department of Physics
Tokyo Metropolitan University
192-0397TokyoJapan
Y Maeno
Department of Physics
Graduate School of Science
Kyoto University
606-8502KyotoJapan
M Ichioka
Department of Physics
Okayama University
700-8530OkayamaJapan
K Machida
Department of Physics
Okayama University
700-8530OkayamaJapan
M R Eskildsen
Department of Physics
University of Notre Dame
Notre Dame
46556IndianaUSA
Anisotropy of the Superconducting State in Sr 2 RuO 4
26 Aug 2013(Dated: August 27, 2013)
Despite intense studies the exact nature of the order parameter in superconducting Sr2RuO4 remains unresolved. We have used small-angle neutron scattering to study the vortex lattice in Sr2RuO4 with the field applied close to the basal plane, taking advantage of the transverse magnetization. We measured the intrinsic superconducting anisotropy between the c axis and the Ru-O basal plane (∼ 60), which greatly exceeds the upper critical field anisotropy (∼ 20). Our result imposes significant constraints on possible models of triplet pairing in Sr2RuO4 and raises questions concerning the direction of the zero spin projection axis.
The superconducting state emerges due to the formation and condensation of Cooper pairs, although the exact microscopic mechanism responsible for the pairing in different materials varies and in many cases remains elusive. In the prominent case of strontium ruthenate multiple experimental and theoretical studies provide compelling support for triplet pairing of carriers (electrons and/or holes) and an odd-parity, p-wave order parameter symmetry in superconducting Sr 2 RuO 4 [1,2]. At the same time, seemingly contradictory experimental results have left important open questions concerning the detailed structure and coupling of the orbital and spin parts of the order parameter. One example of this predicament is conflicting evidence as to whether the p-wave order parameter is chiral [3,4].
The motivation for the present work is the unresolved question regarding the anisotropy of the superconducting state of Sr 2 RuO 4 . The Fermi surface in this material consists of three largely two-dimensional sheets with Fermi velocity anisotropies ranging from 57 to 174 [1,5], and one would expect an upper critical field (H c2 ) anisotropy within this range [6,7]. Experiments, however, find a much smaller Γ Hc2 = H ⊥c c2 /H c c2 ≃ 20 at low temperature and a near constant upper critical field when the applied field is within ±2 • of the basal plane [8]. Within the same angular range the superconducting transition at H c2 becomes first order, leading to suggestions of a subtle coupling between the magnetic field and the triplet order parameter [9], or Pauli limiting, which is inconsistent with triplet pairing with the Cooper pair zero spin projection along the c axis [10].
In this Letter we report on measurements of the intrinsic anisotropy of the superconducting state (Γ ac ) in Sr 2 RuO 4 , which is found to be ∼ 3 times greater than Γ Hc2 . A successful model for the superconducting state in strontium ruthenate must be able to account for the large difference between these two anisotropies.
The anisotropy Γ ac was determined by small-angle neutron scattering (SANS) studies of the vortex lattice (VL). The experiment was performed using a single crystal of Sr 2 RuO 4 grown by the floating zone method and carefully annealed, yielding a critical temperature T c = 1.45 K and no indication of a 3 K phase [1]. Measurements were performed at T = 40 − 60 mK using a dilution refrigerator inserted into a horizontal-field cryomagnet. Magnetic fields of µ 0 H = 0.5 and 0.7 T were applied close to the sample a axis. A motorized Ω stage could rotate the dilution refrigerator within the magnet, allowing in situ sample alignment and measurements as the crystalline basal plane was rotated with respect to H. A schematic of the experimental configuration is shown in Fig. 1(a). The VL was prepared by changing H and Ω at the base temperature, followed by a damped small-amplitude field modulation. This method produces a well-ordered VL and eliminates the need for a field-cooling procedure before each measurement. The SANS experiment was carried out on the D11 and D22 instruments at Institut Laue-Langevin, using a neutron wavelength λ n = 1.7 nm and a wavelength spread ∆λ n /λ n = 10%. Part of the measurements were performed using polarized incident neutrons and a 3 He analysis cell to allow discrimination between spin-flip and non-spin-flip scattering.
In order to determine Γ ac it is necessary to study the VL with the magnetic field oriented parallel or very close to the crystalline basal plane. Such measurements are challenging and require a novel approach to VL SANS studies in order to be feasible. Briefly, the VL scattered intensity is determined by the amplitude of the field modulation and is proportional to |h| 2 , where h(q) is the Fourier transform of the magnetic field B(r) [11]. Using state-of-the-art SANS instruments at a high-flux neutron source such as Institut Laue-Langevin, it is possible to measure the diffraction from a well-ordered VL with a longitudinal Fourier coefficient |h z | as low as 0.1 − 1 mT, depending on the amount of background scattering [12].
Here |h z | ∝ λ −2 ⊥ , where λ ⊥ is the average penetration depth in the screening current plane perpendicular to the applied field. Previous SANS studies with H c found a VL form factor for Sr 2 RuO 4 no greater than a few mT [13]. This indicates that measurements with H ⊥ c should not be possible as |h ⊥c
z | 2 /|h c z | 2 ∝ (λ ab /λ c ) 2 = Γ −2
ac , and with Γ ac ≥ 20 we estimate |h ⊥c z | ≤ 3 µT, at least 2 orders of magnitude below what is required for a VL SANS experiment. However, in highly anisotropic superconductors such as Sr 2 RuO 4 , there is a strong preference for the vortex screening currents to run within the basal ab plane. A small "misalignment" angle Ω between the applied field and the basal plane will thus lead to a significant transverse Fourier coefficient (h x ). Estimates based on an extended London model which includes an effective mass anisotropy yields |h x /h z | 2 ∝ Γ 2 ac [14], and thus predict h ⊥c x to be comparable in magnitude to h c z . As a result, scattering due to the transverse field modulation should be observable. This is confirmed by the VL diffraction pattern shown in Fig. 1(b) which shows Bragg peaks aligned with the crystalline b direction (y axis).
Scattering from the transverse field modulation leads to a flipping of the neutron spin (σ ⊥ h x ) and a Zeeman splitting of the VL rocking curves shown in Fig. 2 [15]. Two maxima are observed for both the top [positive Q y in Fig. 1(b)] and bottom (negative Q y ) VL reflection, as the angle (ϕ) between the scattering vector Q and the direction of the incident neutron beam is varied to satisfy the Bragg condition. As expected, no scattering from the otherwise more commonly observed longitudinal VL field modulation (h z ) could be measured in Sr 2 RuO 4 . A more detailed discussion of the spin-flip scattering can be found in Ref. [16], where a similar but much less extreme effect was observed in yttrium barium copper oxide (YBCO).
To verify that the observed diffraction is due to spinflip scattering, measurements with a polarized neutron beam were performed (shown in the Supplemental Material [15]). In this case only one maximum is observed for each Bragg reflection, selected according to the direction of the neutron spin. Furthermore, the scattered intensity normalized to the incident neutron flux is doubled relative to the unpolarized beam as expected. Moreover, using polarization analysis it is possible to measure only the spin-flip scattering as shown in Fig. 1(b).
Dividing the integrated intensity by the incident neutron flux yields the integrated VL reflectivity
R = 2πγ 2 λ 2 n t 16Φ 0 Q |h x | 2 ,(1)
where γ = 1.913 is the neutron magnetic moment in nuclear magnetons, t is the sample thickness and Φ 0 = h/2e = 2069 T nm 2 is the flux quantum [12]. As shown in Fig. 2 each peak is fitted to the sum of three Gaussians due to the asymmetry of the rocking curves [17]. Moreover, the integrated intensity for the two maxima (top, bottom) for a given reflection are added, as each corresponds to half the incident flux (one direction of the neutron spin). The form factor obtained in this fashion is shown in Fig. 3, for all measured fields and Ω's. Figure 3 illustrates how the VL SANS measurements are possible within a narrow angular range, with H close to, but not perfectly aligned with, the basal plane. The width of the measurement "window" decreases with increasing field due to the rapidly decreasing H c2 (Ω) [8]. In addition, the overall form factor decreases with increasing field. While the anisotropic London model provides a qualitative description of the enhanced field modulation [14], it does not provide a good quantitative fit to the data. As shown in Fig. 3, an extended London model that includes a so-called core correction by multiplying the calculated |h x | by exp(−c Q 2 (Ω) ξ 2 ab ) still does not yield a good fit to the data. Here the constant c is of the order unity, Q(Ω) is the magnitude of the VL scattering vector (see below), and ξ ab = (Φ 0 /2πH c c2 ) 1/2 is the in-plane coherence length [12]. A quantitatively accurate model for the VL form factor is highly desirable as it would allow a determination of both λ and ξ.
We now turn to the main result of this Letter, which is the measurement of the VL anisotropy. In an anisotropic superconductor the VL Bragg peaks are expected to lie on an ellipse with a major-to-minor ratio given by [6] Γ VL = Γ ac cos 2 Ω + (Γ ac sin Ω) 2
as shown in Fig. 4(a). This Ω dependence was derived for anisotropic (but still three-dimensional) superconductors, and was verified in early VL SANS measurements on 2H-NbSe 2 with Γ ac = 3.2 [18]. Although Sr 2 RuO 4 is a layered material, the coherence length along the c axis [1], and we expect Eq. (2) to be applicable [19].
Because of the large anisotropy in Sr 2 RuO 4 , VL Bragg peaks which are not on the vertical axis have scattering vectors essentially parallel to h x , making them unmeasurable as only components of the magnetization perpendicular to Q will give rise to scattering [20]. Instead, we determine the VL anisotropy based on flux quantization. Assuming that each vortex carries one flux quantum Φ 0 , the area of the reciprocal space ellipse in Fig. 4(a) is determined uniquely by the applied magnetic field. This yields Γ VL = (Q 0 /Q) 2 , where Q is the magnitude of the measured VL scattering vector and Q 0 = 2π(2µ 0 H/ √ 3Φ 0 ) 1/2 corresponds to an undistorted hexagonal VL (Γ ac = 1). The magnitude of Q can be determined either from the position of the VL Bragg peaks on the detector as shown in Fig. 1(b) or from the peak positions ϕ 1 , . . . , ϕ 4 in Fig. 2 [15]. The two methods yield nearly identical results, and using the average Q we obtain Γ VL (Ω) shown in Fig. 4(b). Within the scatter in the data the results for both fields collapse onto a single curve, increasing upon approaching the a axis and reaching a value slightly higher than 50 before the intensity vanishes. Theoretical predictions of a fielddependent Γ VL and possibly a rotation of the VL are thus not observed [21,22]. If one assumes a quantization of Φ 0 /2, as recently reported for mesoscopic rings of Sr 2 RuO 4 [23], the deduced values for Γ VL would double. However, we consider this an unrealistic scenario in the present case, with a macroscopic, homogenous sample.
Fitting the data in Fig. 4(b) to Eq. (2) yields Γ ac = 58.5 ± 2.3. Only for angles within ±1.3 • does the measured anisotropy deviate from that expected for an infinite ac anisotropy. Also shown for comparison is Γ VL expected from the low temperature Γ Hc2 = 20 [8] and which provides a very poor fit to the data. We note that Γ Hc2 increases with temperature and reaches a value of ∼ 60 ≈ Γ ac at T c [24]. In addition, the fitted value of Γ ac coincides with the anisotropy of the β Fermi surface sheet (57) [1,5].
The large difference between Γ Hc2 and the intrinsic anisotropy of the superconducting state deep within the mixed phase measured by Γ ac indicates a strong suppression of the upper critical field in Sr 2 RuO 4 for H ⊥ c. One possible explanation for this difference is Pauli limiting due to the Zeeman splitting of spin-up and spindown carrier states by the applied magnetic field and the resulting reduction of the superconducting condensation energy [25]. In spin-triplet superconductors the order parameter is most conveniently described in terms of the d vector, directed along the zero spin projection axis where the configuration of the Cooper pairs is given by 1 √ 2 (|↑↓ + |↓↑ ) [1,2,4]. Consequently, Pauli limiting in the triplet case can only occur when H d. If one assumes Pauli limiting our results are thus inconsistent with the chiral superconducting state with d c proposed for Sr 2 RuO 4 [2,4]. It should be noted, however, that Pauli limiting itself appears to be in disagreement with Nuclear Magnetic Resonance and Nuclear Quadrupole Resonance Knight-shift measurements (summarized in Ref. [2]), which suggest that the d vector rotates in the presence of a magnetic field such that d ⊥ H.
Also remaining are a number of other models for the superconducting state in strontium ruthenate which are (or may be) consistent with our results. Among these are several possible ways to achieve a subtle coupling between the magnetic field and the triplet order parameter as discussed in some detail in Ref. 9. Other alternatives include (chiral) triplet pairing with d ⊥ c [26] that could possibly be locked along certain in-plane directions, recent multiband p-wave models [27], a field-dependent mixing of singlet and triplet states [28], or singlet superconductivity [10,29]. It should be noted, however, that s-wave superconductivity does not provide a satisfactory explanation for the extreme sensitivity of T c to impurities or to the chiral properties of Sr 2 RuO 4 [1,2,29]. Further experimental and theoretical work will be necessary to provide a definitive determination of the order parameter in this material.
In conclusion, we have used SANS to measure the anisotropy of the superconducting state in Sr 2 RuO 4 , taking advantage of the transverse VL field modulation which allows measurements in a narrow range of field angles close to, but not perfectly aligned with, the Ru-O basal plane. The superconducting anisotropy greatly exceeds that of the upper critical field and imposes significant constraints on the possible pairing of carriers in this material. Any model aimed at describing the superconducting phase must provide a satisfactory explanation for this observation.
We The two different directions of the neutron spin with respect to the applied field correspond to different nuclear Zeeman energies and lead to opposite shifts of the neutron momentum vector
k ↑(↓) = k 0 1 ± ∆ε/ε 0 ,(1)
where the subscript in parentheses henceforth corresponds to the lower (in this case minus) sign in the ±∆ε term. Here the nominal neutron wavevector k 0 = 2π/λ n , ε 0 =h 2 k 2 0 /2m n , ∆ε = γµ N B and γ = 1.913 is the neutron magnetic moment in nuclear magnetons µ N = eh/2m n = 31.5 neV/T. With λ n = 1.7 nm and µ 0 H = 0.5 T one finds k 2
↑ − k 2 ↓ = 2 × 10 −4 k 2 0 .
Due to the short Q in the range 0.003 − 0.01 k 0 , the small shift in the neutron wavevector nonetheless leads to a significant difference in the angle (ϕ 1(2) ) required to satisfy the Bragg condition as shown schematically in Fig. 1. In this case Bragg's law is replaced by
k 2 ↑ − k 2 ↓ ± Q 2 = 2k ↑(↓) Q sin ϕ 1(2) .(2)
The magnitude of Q can be determined from the peak positions ϕ 1 , . . . , ϕ 4 by Q 1(2) = ∓k ↑(↓) sin ϕ 1(2) ∓ k 2 ↓(↑) − k 2 ↑(↓) cos 2 ϕ 1(2)
Q 3(4) = ∓k ↓(↑) sin ϕ 3(4) ± k 2 ↑(↓) − k 2 ↓(↑) cos 2 ϕ 3(4) . Rocking curves obtained using a polarized neutron beam are shown in Fig. 2. These demonstrate how a single peak can be selected for each Bragg reflection according to the direction of the neutron spin. for neutrons polarized with their magnetic moment parallel (a) and antiparallel (b) to the applied field. Except where shown, error bars are equal to or smaller than the symbols. The intensity was normalized to the incident neutron flux. The curves are fits to the data as described in the main text.
FIG. 1 .
1Experimental geometry. (a) The coordinate system is defined with z along H and y in the Ru-O basal plane (along b). The applied magnetic field H is rotated away from the Ru-O (spanned by a and b) by an angle Ω. Neutron spins (σ) are parallel or antiparallel to the magnetic field. The incident neutron beam is in the yz plane, at an angle ϕ relative to the field direction. The observed VL scattering vector is denoted Q and the longitudinal and transverse component of the field modulation by hz and hx, respectively. (b) Diffraction pattern showing spin-flip scattering from the VL due to the transverse field modulation (hx). The two Bragg peaks correspond to ±Q in panel (a). No background subtraction was performed, but a small remnant of the undiffracted beam close to Q = 0 due to the finite flipping ratio (∼ 8) is masked off
FIG. 2 .
2Vortex lattice rocking curves showing the scattered intensity as a function of the angle ϕ, for an unpolarized neutron beam. Error bars are equal to or smaller than the symbols. Two maxima are observed for both the bottom (ϕ 1/3 ) and top (ϕ 2/4 ) VL Bragg reflections (rocking curves obtained with a polarized neutron beam can be found in the Supplemental Material[15]). The intensity was normalized to the incident neutron flux. The curves are fits to the data as described in the text.
FIG. 3 .
3Vortex lattice form factor at 40 mK as a function of applied field and angle Ω with the a axis. The statistical error is roughly the size of the symbols. The solid lines are guides to the eye. The dashed line shows an extended London model fit to the 0.5 T data as discussed in the text, with λ ab = 167 nm, ξ ab = 66 nm, c = 1/4, and Γac = 58.5.
FIG. 4 .
4Vortex lattice anisotropy. (a) Schematic of VL Bragg reflections lying on an ellipse with major-to-minor axis ratio, ΓVL = 6. Only the filled (red) peaks are observed. The reciprocal space area of the ellipse is πQ 2 0 = 8π 3 µ0H/ √ 3Φ0. (b) Measured VL anisotropy at 40 mK as a function of applied field and angle with the a axis (Ω). Except where shown explicitly the statistical error is the size of the symbols. The lines show the VL anisotropy calculated using Eq. (2) and Γac = 20 (dotted line), 58.5 (dashed line), and ∞ (full line).
ξ c = 3.3 nm is still several times greater than the Ru-O interlayer spacing (0.64 nm)
acknowledge discussions with W. P. Halperin, V. G. Kogan, I. Mazin, J. A. Sauls and S. Yonezawa, and assistance with sample alignment by G. Sigmon. Research support was provided by the U.S. Department of Energy, Office of Basic Energy Sciences, under Award No. DE-FG02-10ER46783 (neutron scattering) and by the MEXT of Japan KAKENHI No. 22103002 (crystal growth and characterization).
FIG. 1 .
1Schematics showing the scattering geometries corresponding to the reflection at ϕ = ϕ 1 (a) and ϕ 2 (b). The colors of the incident neutron wavevector (blue) and the scattering vector (red) correspond to those used inFig. 1of the main text.
FIG. 2 .
2Vortex lattice rocking curves showing the scattered intensity as a function of the angle ϕ,
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See Supplemental Material for more details concerning spin-flip scattering and Zeeman splitting of the rocking curves. See Supplemental Material for more details concerning spin-flip scattering and Zeeman splitting of the rocking curves.
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The reason for the rocking curves asymmetry is a field inhomegeneity, which is a well-known problem with the particular cryomagnet used for the experiment. The asymmetry does not affect the analysis or conclusions of this Letter. The reason for the rocking curves asymmetry is a field inhomegeneity, which is a well-known problem with the particular cryomagnet used for the experiment. The asymmetry does not affect the analysis or conclusions of this Letter.
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The value for ξc is obtained from the upper critical field assuming orbital limiting, H ⊥c c2 = Φ0/2πξ ab ξc. HoweverThe value for ξc is obtained from the upper critical field assuming orbital limiting, H ⊥c c2 = Φ0/2πξ ab ξc. However,
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"Orbifold Euler ",
"Zhiyuan Wang ",
"Jian Zhou "
] |
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We solve the problem of the computation of the orbifold Euler characteristics of Mg,n. We take the works of Harer-Zagier[30]and Bini-Harer [9] as our starting point, and apply the formalisms developed in[47]and [56] to this problem. These formalisms are typical examples of mathematical methods inspired by quantum field theories. We also present many closed formulas and some numerical data. In genus zero the results are related to Ramanujan polynomials, and in higher genera we get recursion relations almost identical to the recursion relations for Ramanujan polynomials but with different initial values. We also show that the generating series given by the orbifold Euler characteristics of Mg,n is the logarithm of the KP tau-function of the topological 1D gravity evaluated at the times given by the orbifold Euler characteristics of Mg,n. Conversely, the logarithm of this tau-function evaluated at the times given by certain generating series of the orbifold Euler characteristics of Mg,n is a generating series of the orbifold Euler characteristics of Mg,n. This is a new example of open-closed duality.
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24 Aug 2021
Orbifold Euler
Zhiyuan Wang
Jian Zhou 24 Aug 2021
We solve the problem of the computation of the orbifold Euler characteristics of Mg,n. We take the works of Harer-Zagier[30]and Bini-Harer [9] as our starting point, and apply the formalisms developed in[47]and [56] to this problem. These formalisms are typical examples of mathematical methods inspired by quantum field theories. We also present many closed formulas and some numerical data. In genus zero the results are related to Ramanujan polynomials, and in higher genera we get recursion relations almost identical to the recursion relations for Ramanujan polynomials but with different initial values. We also show that the generating series given by the orbifold Euler characteristics of Mg,n is the logarithm of the KP tau-function of the topological 1D gravity evaluated at the times given by the orbifold Euler characteristics of Mg,n. Conversely, the logarithm of this tau-function evaluated at the times given by certain generating series of the orbifold Euler characteristics of Mg,n is a generating series of the orbifold Euler characteristics of Mg,n. This is a new example of open-closed duality.
. The problem of the computation of the orbifold Euler characteristics χ(M g,n ) of the Deligne-Mumford moduli spaces M g,n of stable curves is a long standing problem in algebraic geometry. In this work we will solve this problem using some methods we have developed in an earlier work [47] inspired by the quantum field theory techniques developed by string theorists in the study of holomorphic anomaly equations. Such equations have played a crucial role in the theory of mirror symmetry in higher genera. By extracting the salient features of the theory of holomorphic anomaly equations, we formulate a notion of abstract quantum field theory in [47] and develop some recursion relations in this formalism. It turns out that this general formalism includes various problems and related recursion relations as special cases. We will show that this formalism is also applicable to the problem of computing χ(M g,n ) and it provides an effective algorithm for their computations. We will present various closed formulas which are not accessible by other methods in the literature. The idea of using quantum field theory techniques to solve problems in algebraic geometry is certainly not new. There are many well-known examples in the literature, and some of them are famous work on the problem we deal with in this work. Let us briefly review some of them.
The moduli space M g,n of smooth Riemann surfaces of genus g with n marked points (2g − 2 + n > 0) is an orbifold of dimension 3g − 3 + n, and the orbifold Euler characteristic of M g,n is given by the famous Harer-Zagier formula:
χ(M g,n ) = (−1) n · (2g − 1)B 2g (2g)! (2g + n − 3)!,
which is first proved by Harer and Zagier [30], and by Penner [41]. These authors have introduced the methods of Hermitian matrix models into the study of moduli spaces of algebraic curves. A different proof is given by Kontsevich [34,Appendix D], in an appendix of his famous work on the proof of Witten's Conjecture [50].
That work can be considered as a development of [30] and [41] in the sense that quantum field theory techniques such as summations over Feynman graphs also play an important role.
In [13], Deligne and Mumford construct a natural compactification M g,n of M g,n by allowing double points on curves. The compactification M g,n is the moduli space of stable curves of genus g with n marked points, and it carries a natural stratification which can be described by the dual graphs of stable curves:
M g,n = Γ M Γ ,
where the disjoint union is indexed by the set of all connected stable graphs of genus g with n external edges, and M Γ is the moduli space of stable curves whose dual graph is Γ. A natural question is the computations of the orbifold Euler characteristic and ordinary Euler characteristic of M g,n . Some results about the ordinary Euler characteristics for small g have already been given by different authors. For g = 0, see [24,28,31,37]. For g = 1, see [25]. For g = 2, see [8,22,26]. For g = 3, see [27]. There are also some excellent results on the orbifold Euler characteristics χ(M g,n ) in literatures. In Manin's computation of χ(M 0,n ), quantum field theory techniques have been applied to get the following result: let
χ(t) := t + ∞ n=2 χ(M 0,n+1 )
t n n! be the generating series at genus zero, then χ(t) is the unique solution of either one of the following two equations:
(1 + χ) log(1 + χ) = 2χ − t,
(1 + t − χ)χ t = 1 + χ.
See [28] for some combinatorial results related to the solution of the first equation. The common features of the works of Harer-Zagier, Penner and Manin are the following. They all formulated the geometric problem as finding a summation over Feynman graphs. In the case of Harer-Zagier and Penner, the relevant graphs are the fat graphs in the sense of 't Hooft [46] and these graphs lead them naturally to generalized Gaussian integrals over the space of Hermitian matrices; and in the case of Manin the relevant graphs are marked trees and this leads to a Gaussian integral on a one-dimensional space. The next step of all these works is to apply the Laplace method of finding asymptotic expansions of these formal integrals.
The orbifold Euler characteristic of M g,n /S n has been studied by this strategy by Bini-Harer [9]. First, they notice that M g,n /S n can also be written in terms a sum over stable graphs
χ(M g,n /S n ) = Γ∈G c g,n 1 | Aut(Γ)| v∈V (Γ) χ(M gv,valv ).
It follows that the generating series of χ(M g,n ) has a formal integral representation for which one can apply the Laplace method to obtain the asymptotic behaviors (see Theorem 3.2,Theorem 3.3]).
Unfortunately very few numerical data and no closed formula for χ(M g,n ) have been obtained by the above methods. The two ways to compute χ(M g,n ) in that work, i.e., summing over all stable graphs or expanding formal Gaussian integrals, are both too complicated to carry out specific computations. In fact, the complexity of finding all the relevant Feynman graphs and their automorphism groups makes it impossible to compute χ(M g,n /S n ) using the above graph sum formula directly unless g and n are extremely small (see eg. [47,Appendix] for all graphs with (g, n) = (2, 2) and (3, 0)); and expanding the logarithm of a formal Gaussian integral is no less complicated than writing down all graphs since it involves summing over all partitions of some integers. It is not surprising that rewriting the Feynman sum as a logarithm of a formal integral may not easily lead to the solution of the problem of finding closed expressions.
In quantum field theory, deriving some recursion relations is a general strategy to solve the problem of evaluating the Feynman sums. This will also be our strategy to solve the problem of computing χ(M g,n ). We will take the works of Harer-Zagier [30] and Bini-Harer [9] as our starting point and develop a more computable method to calculate χ(M g,n ). Because this problem can be formulated both as a sum over graphs and as a formal Gaussian integrals, we will attack this problem from both points of view. A general theory for deriving recursion relations for sums over graphs have been developed by us in an earlier work [47]. A general theory of evaluating formal Gaussian integrals has been developed by the second author in [56]. In this article, we will solve the problem of computing χ(M g,n ) by applying the results in these works.
1.2. Abstract quantum field theory and its realizations. Let us first review the formalism of the abstract quantum field theory and its realizations, which yields various types of recursion relations that provide effective algorithm for both the concrete computations and for deriving closed formulas involving summation over stable graphs. This formalism is developed by the authors in a previous work [47]. In that work, we have construct an abstract QFT using the diagrammatics of stable graphs. Our original intention in that work is to understand the mathematical structures of the BCOV holomorphic anomaly equation [6,7], see also [1,19,20,29,32,51,52]. Inspired by these physics literatures, we define the abstract free energy F g (g ≥ 2) to be formal summations of stable graphs of genus g without external edges, and the abstract n-point functions F g,n (2g − 2 + n > 0) to be formal summations of stable graphs of genus g with n external edges (see (12) and (13)). We also derive some recursion relations for these functions in terms of the edge-cutting operator K and edge-adding operators D = ∂ + γ (see Lemma 2.1 and Theorem 2.2). It is worth mentioning that the edge-cutting and edge-adding operators are constructed inspired by the stratification of M g,n .
Another part of such a general formalism is the notion of a 'realization' of the abstract QFT. As input we need a collection of functions F g,n (t) with 2g −2+n > 0, and a suitable formal variable κ we take as propagator, and from which we use the Feynman rules to construct a collection of functions F g,n (t, κ). Under suitable conditions, the recursion relations for the abstract n-point functions F g,n will induce the recursion relations for F g,n (t, κ). In this way one can unify various recursion relations in the literature as special examples.
We will review some preliminaries on this formalism in §2.
1.3.
Operator formalism for evaluations of formal Gaussian integrals. One important feature of our formalism of realization of abstract QFT is that we can express the resulting partition function as a formal Gaussian integral in one dimension (similar to the integral of Bini-Harer). In [56] various methods have been developed to calculate the partition functions. One of these methods is to expand the exponential in the integrand. This converts the summation over Feynman graphs into a summation over partitions, or pictorially, over Young diagrams. For the application of this idea to our problem, see (141). Using a recursion developed in the abstract QFT formalism, we derive an operator formalism to compute the summation over Young diagram in §4. 6. This involves operations on Young diagrams similar to that in Littlewood-Richardson rule. It turns out that this operator formalism appears in [56] in the setting of deriving W-constraints for the partition function of topological 1D gravity, i.e., formal Gaussian integrals in one dimension. There are some easily derived constraints of this partition function, called the flow equations and the polymer equation, from which one can derive the Virasoro constraints (see e.g. [56, §5- §6]). They can also be used to derive operator formulas for computing the partition functions. For their applications to χ(M g,0 ), see §4.10.
1.4.
Descriptions of our results. Let us briefly describe some of the results we obtain in this paper using the methods reviewed in last two subsections. [30,41], we will use them to construct F g,n (t) in our formalism of realization of the abstract QFT. Of course if one has a way to compute other orbifold characteristics of M g,n , then our method is applicable to the corresponding orbifold characteristics of M g,n . We consider the following realization of the abstract QFT. The abstract n-point function F g,n is realized by:
χ g,n (t, κ) := Γ∈G c g,n κ |E(Γ)| | Aut(Γ)| · v∈V (Γ) χ(M gv ,nv ) · t χ(C gv ,valv ) ,
where g v and val v are the genus and valence of the vertex v respectively, and χ(C gv ,valv ) = 2 − 2g − n is the Euler characteristic of a Riemann surface C gv ,valv of genus g with n punctures. By Bini-Harer's graph sum formula we have:
(1) χ g,n (1, 1) = χ(M g,n /S n ) = 1 n! χ(M g,n ), and so we can reduce the computations of χ(M g,n ) to that of χ g,n (t, κ). The functions χ g,n (t, κ) will be called the 'refined orbifold Euler characteristics' in this paper. By our general formalism we derive the following recursion relations: For 2g − 2 + n > 0, we have Dχ g,n = (n + 1)χ g,n+1 ,
(2) ∂ ∂κ χ g,n = 1 2 (DDχ g−1,n + g1+g2=g, n1+n2=n
Dχ g1,n1 Dχ g2,n2 ).
where D is a differential operator defined by:
(4) Dχ g,n = ∂ ∂t + κ 2 t −1 · ∂ ∂κ + n · κt −1 χ g,n .
Using these equations, one can either reduce the number n or reduce the genus g, hence the problem of computations of χ(M g,n ) is solved. It is very easy to automate the concrete computations by a computer algebra system. We will present various examples in the text (see eg. §(3.2)).
1.4.2.
Connection to configuration spaces and Ramanujan polynomials. Using our recursion relations (2) and (3), we will derive the following functional equation in genus zero:
(5) κ(1 + χ) log(1 + χ) = (κ + 1)χ − y
for the following generating series of refined orbifold Euler characteristics of M 0,n :
(6) χ(y, κ) := y + ∞ n=3
nκy n−1 · χ 0,n (1, κ).
When κ = 1, this recovers Manin's result [37] mentioned above. It is very striking that when κ is taken to be a positive integer m, the above functional equation is related to the formula for Euler characteristics of Fulton-MacPherson compactifications of configuration spaces of points on complex manifold X, also due to Manin [37], where m is the complex dimension of X. We do not have an explanation for this mysterious coincidence. Furthermore, we derive similar functional equations by the same method for higher genera. The results are formulated in Theorem 3.9 and Theorem 3.10.
We also make a connection of the linear recursion relation 2 to Ramanujan polynomials [4,5] and Shor [44] polynomials, hence to their related combinatorial interpretations in terms of counting trees that dates back to Cayley. See §3.5 for details.
1.4.3.
Results for χ(M g,0 ). By (3) one may get:
χ g,0 = B 2g 2g(2g − 2) + 1 2 κ 0 (3 − 2g) + κ 2 ∂ ∂κ + 2κ − 1 6 (4 − 2g) + κ 2 ∂ ∂κ χ g−1,0 + g−2 r=2
(2 − 2r) + κ 2 ∂ ∂κ χ r,0 · (2 − 2g + 2r) + κ 2 ∂ ∂κ χ g−r,0 dκ for g ≥ 3, where χ g,n (κ) := χ g,n (1, κ) is a polynomial in κ of degree 3g − 3. This provides a very effective algorithm for specific computations.
The refined orbifold Euler characteristics χ g,n (t, κ) has the following form:
(7) χ g,n (t, κ) = 3g−3+n k=0 a k g,n κ k · ( 1 t ) n−2+2g .
Consider the generating series G k (z) := g≥2 a k g,0 · z 2−2g of the coefficients in the case of n = 0. We express their generating series as a formal Gaussian integral (see Corollary 4.2):
(8) k≥0 λ 2k G k (z) = log 1 √ 2πλ 2 G(x + z + 1) · exp( S(x, z))dx ,
where G(z) is the Barnes G-function, and
S(x, z) = − λ −2 2 x 2 − ζ ′ (−1) + z 2 log(2π) + z 2 2 − 1 12 log z − 3z 2 4 + x − 1 2 log(2π) − z log z + z − x 2 2 log z.
Moreover, we prove that G k (z) can be expressed in terms of some generating series {V n (z)} of χ(M g,n ) in a simple way for every k ≥ 1, where V n (z) is defined by:
(9) V n (z) = ∞ g=0 χ(M g,n )z 2−2g−n
for n ≥ 3 (for the modifications when n < 3, see Definition 4.1). Using the Harer-Zagier formula, we are able to show that V n can be expressed in terms of the Barnes G-function G(z):
V 0 (z) ∼ log G(z + 1) − ζ ′ (−1) + z 2 log(2π) + z 2 2 − 1 12 log z − 3z 2 4 , V 1 (z) ∼ d dz log G(z + 1) − 1 2 log(2π) − z log z + z, V 2 (z) ∼ d 2 dz 2 log G(z + 1) − log z, V n (z) ∼ d n dz n log G(z + 1), ∀n ≥ 3.
Here ∼ indicates that even though as power series, the radii of convergence of V n (z) are all zero, nevertheless as z → ∞, they are the asymptotic series of some analytic functions. Let v n be a family of formal variables with deg(v n ) := n, then there exists a family of polynomials H k (v 1 , · · · , v 2k ) such that deg(H k ) = 2k and:
G k (z) = H k (v 1 , v 2 , · · · ) vn=Vn(z) .
See §4 for various methods to compute these polynomials based on the above formal Gaussian integral representation of G k (z) and the theory of topological 1D gravity.
1.4.4. Orbifold Euler characteristics of M g,0 and KP hierarchy. Another remarkable consequence of (8) is that since it is the specialization of the partition function of the topological 1D gravity, it is the evaluation of a τ -function of the KP hierarchy evaluated at the following time variables:
(10) T n = 1 n! V n (z).
See §6 for details and the generalization to χ(M g,n ) for general n. It is interesting to find a geometric model that describes all the KP flows away from this special choice of values of the time variables.
1.4.5. Explicit formulas for χ(M g,n ). In §3.7 we have derived an operator formalism that computes χ(M g,n+1 ) from χ(M g,n ) by reformulating a linear recursion relation from abstract QFT. We are able to use it to derive some closed formulas for χ(M g,n ). For example, in genus zero, we have:
χ(M 0,n ) = n! · n−3 k=0 A k (x) n ,
where [·] n means the coefficient of x n , and A k (x) is sequence of power series in x independent of n, defined as follows:
A 0 = 1 2 (1 + x) 2 log(1 + x) − 1 2 x − 3 4 x 2 , A 1 = 1 2 + (1 + x)(log(1 + x) − 1) + 1 2 (1 + x) 2 (log(1 + x) − 1) 2 ,
and for k ≥ 2,
A k = 2 m=2−k (x + 1) m (m + 2)! k−m−1 l=0 log(x + 1) − 1 l l! e l+m (1 − m, 2 − m, · · · , k − m − 1),
where e l are the elementary symmetric polynomials. It turns out that for every g ≥ 1, χ(M g,n ) all has an expression of this form. In fact, χ(M g,n ) is given by the following formula:
χ(M g,n ) = n! · 3g−3+n k=0 a k g,n = n! · 3g−3+n k=0 3g−3 p=0 a p g,0 A p g,k (x) n for g ≥ 2,
where the closed formulas for A p g,k (x) can be written down explicitly. See §5 for details.
1.4.6. Open-closed duality of orbifold Euler characteristics. We describe a new version of open-closed duality that represents the orbifold Euler characteristics of M g,n and M g,n in terms of each other via inversion formulas. More precisely, we have the following formula:
(11) χ(M g,n ) = n! · Γ∈G c g,n (−1) |E(Γ)| | Aut(Γ)| v∈V (Γ) χ(M gv ,valv ).
In a subsequent work [49], we give an interpretation of this open-closed duality as a generalization of the Möbius inversion formula (see Rota [42]).
1.5. Plan of the paper. The rest of this paper is arranged as follows. In §2 we review our formalism of abstract QFT and its realizations. We apply this formalism to derive some recursion relations for the refined Euler characteristics of M g,n in §3, and show how these recursions provide us a large number of numerical data. In §4 we present our results on χ(M g,0 ); in particular, we show that the theory for χ g,0 is equivalent to the topological 1D gravity. The results for χ(M g,n ) with n > 0 obtained by solving the linear recursion directly are presented in §5. In §6, we related the orbifold Euler characteristics of M g,n to the KP hierarchy. Finally in §7, we describe the open-closed duality.
Preliminaries on the Abstract QFT and Its Realizations
In this section, let us introduce some preliminaries on the formalism of the abstract quantum field theory for stable graphs and its realizations developed in [47]. The free energies of this 'abstract theory' satisfy a family of quadratic recursion relations, and we will recall the realization of such recursions under some particular Feynman rules.
2.1. Edge-cutting and edge-adding operators on stable graphs. In this subsection we introduce some geometric backgrounds of stable graphs, and recall the constructions of the edge-cutting and edge-adding operators on stable graphs inspired by these geometric backgrounds (see [47, §2]).
Let M g,n be the Deligne-Mumford moduli space of stable curves [13,33]. If we do not distinguish the marked points on a stable curve, then the equivalent classes of such stable curves of genus g with n marked points will be parametrized by M g,n /S n . The stratification of M g,n /S n can be described in terms of dual graphs of stable curves.
A stable graph Γ is a graph consisting of some vertices and edges satisfying the stability condition. Each vertex v is marked by a nonnegative integer g v ≥ 0 called the 'genus' of this vertex. There are two types of edges in stable graphs: an internal edge connects two vertices (the two vertices may be a same one and such an edge is called a loop on this vertex); and an external edge is attached to a vertex. A 'half-edge' is either an internal edge together with a choice of one endpoint of it, or an external edge. The 'valence' of a vertex is defined to be the number of half-edges attached to it. The stability condition is that, the valence of each vertex of genus 0 is at least three, and the valence of each vertex of genus 1 is at least one. Given a graph Γ, We will denote by V (Γ), E(Γ), E ext (Γ) the sets of vertices, internal edges, and external edges of Γ throughout this paper.
The genus of a connected stable graph Γ is defined to be
g(Γ) := h 1 (Γ) + v∈V (Γ) g v ,
where h 1 (Γ) is the number of independent loops in Γ. Let G c g,n be the set of all connected stable graphs of genus g with n external edges, then the stratification of M g,n /S n is given by:
M g,n /S n = Γ∈G c g,n M Γ ,
where M Γ is the moduli space of stable curves whose dual graph is Γ.
There are two families of natural maps on these moduli spaces of curves-the gluing maps and forgetful maps. By considering 'inverses' of these maps, the authors have constructed two families of operators on stable graphs in [47], called the edgecutting K and edge-adding operator D = ∂ + γ.
Denote by G g,n the set of all stable graphs of genus g with n external edges (not necessarily connected), and denote
V := Γ∈Gg,n 2g−2+n>0 QΓ.
The edge-cutting operator K is defined to be
K : V → V, Γ ∈ G g,n → K(Γ) = e∈E(Γ) Γ e ,
and Γ e is obtained from Γ by cutting the internal edge e (and so obtaining two new external edges). If the stable graph Γ has no internal edge, then we assign K(Γ) = 0. Such a procedure inverses the 'gluing' of stable maps. The construction of D : V → V is a little subtle but still natural. The operator ∂ consists two parts, the first one is to attach an external edge to a vertex and sum over all vertices:
g . . . → g . . .
, and the other one is to break up an internal edge and insert a trivalent vertex of genus 0 and then sum over all internal edges:
g 1 g 2 . . . . . . → g 1 g 2 . . . . . . 0 , g . . . → g . . . 0 .
And the operator γ acts on Γ by attaching a trivalent vertex of genus 0 to an external edge of Γ and summing over all external edges. If the stable graph Γ has no external edge, then we assign γ(Γ) = 0. See [47, §2.2] for more details about the definitions of these operators. The procedures in ∂ and γ inverse the procedure of forgetting a marked point on a stable graph, and here we have taken all unstablecontractions into consideration. It is not hard to see that D := ∂ + γ preserves the subspace of connected graphs: Definition 2.1. The free energy of genus g for the abstract QFT of stable graphs is defined to be:
V c := Γ∈G c g,n 2g−2+n>0 QΓ ⊂ V.(12) F g = Γ∈G c g,0 1 | Aut(Γ)| Γ, g ≥ 2.
The abstract n-point function of genus g is defined to be:
(13) F g,n = Γ∈G c g,n 1 | Aut(Γ)| Γ, 2g − 2 + n > 0.
Here Aut(Γ) is the group of automorphisms of Γ.
Remark 2.1. The stability condition ensures that the right-hand-sides of the above formulas are all finite summations.
Example 2.2. We have: Using the edge-cutting and edge-adding operators, we are able to formulate some recursion relations for F g,n . The followings are [47, Theorem 2.1; Lemma 2.1]: Theorem 2.1. For 2g − 2 + n > 0, we have (14) K F g,n = n + 2 2 F g−1,n+2 + 1 2 g1+g2=g, n1+n2=n+2, n1≥1,n2≥1
F 0,3 = 1 6 0 , F 1,1 = 1 + 1 2 0 ,(n 1 F g1,n1 ) · (n 2 F g2,n2 ),
where the sum on the right hand side is taken over all stable cases.
Remark 2.2. Given two graphs Γ 1 and Γ 2 , the 'product' Γ 1 · Γ 2 will be understood as the (disconnected) graph obtained by taking the disjoint union of Γ 1 and Γ 2 throughout this paper. D F g1,n1 · D F g2,n2 ).
In particular, by taking n = 0 we obtain a quadratic recursion for F g (g ≥ 2):
(17) K F g = 1 2 (D∂ F g−1 + g−1 r=1 ∂ F r · ∂ F g−r )
.
Here we use the conventions:
(18) ∂ F 1 = D F 1 := F 1,1 , D F 0,2 := 3 F 0,3 , DD F 0,1 := 6 F 0,3 ,
by formally applying Lemma 2.1.
Remark 2.3. Notice that the graphs appearing in the expression of F r do not have external edges, thus γ( F r ) = 0 and then D F r becomes ∂ F r when we derive (17) from (16).
Example 2.
3. Using the expression of F 2 in the previous example, we have:
K F 2 = 1 2 1 + 1 2 1 1 + 1 4 0 + 1 2 1 0 + 1 2 1 0 + 1 4 0 0 + 1 8 0 0 + 1 4 0 0 .
Then using the expression
F 1,2 = 1 2 1 + 1 4 0 + 1 2 0 1 + 1 4 0 0 + 1 4 0 0 ,
one can easily check:
K F 2 = F 1,2 + 1 2 F 1,1 · F 1,1 = 1 2 DD F 1 + D F 1 · D F 1 .
2.3.
Realization of the abstract QFT by Feynman rules. In this subsection we discuss the realizations of the abstract QFT. Here we present a generalization of the realization given in [47, §4].
Our input data are a sequence of formal functions {F g,n (t)} 2g−2+n>0 in a formal variable t, together with a formal variable κ called the propagator. Using these data, we will construct a sequence of functions { F g,n (t, κ)} 2g−2+n>0 by assigning the following Feynman rules to stable graphs. Let Γ be a stable graph, then we assign a weight w Γ to it:
(19) Γ → w Γ = v∈V (Γ) w v · e∈E(Γ) w e ,
where the weight of a vertex v ∈ V (Γ) of genus g v and valence val v is given by (20) w v := F gv ,valv (t), and the weight of an internal edge e ∈ E(Γ) is given by (21) w e = κ.
Then the function F g,n (t, κ) is defined to be the weight of F g,n (2g − 2 + n > 0). Such a procedure will be called a 'realization' of the abstract QFT. More specifically, the abstract free energy F g will be realized by:
(22) F g (t, κ) := Γ∈G c g,0 w Γ | Aut(Γ)| , g ≥ 2,
and the abstract n-point functions F g,n are realized by:
(23) F g,n (t, κ) := Γ∈G c g,n w Γ | Aut(Γ)| , 2g − 2 + n > 0.
Example 2.4. We give some examples of F g,n (t, κ) for small g and n:
F 0,3 = 1 6 F 0,3 , F 0,4 = 1 24 F 0,4 + 1 8 κF 2 0,3 , F 1,1 = F 1,1 + 1 2 κF 0,3 , F 2 = F 2,0 + κ 1 2 F 1,2 + 1 2 F 2 1,1 + κ 2 1 8 F 0,4 + 1 2 F 1,1 F 0,3 + 5 24 κ 3 F 2 0,3 .(24)
The above construction can be represented as a formal Gaussian integral. The following result is well-known in literatures (see eg. [47,Theorem 4.1]):
Theorem 2.3. Define a partition function Z(t, κ) := exp ∞ g=2 λ 2g−2 f g (t, κ) by (25) Z(t, κ) = 1 (2πκλ 2 ) 1 2 exp 2g−2+n>0 λ 2g−2 n! F g,n (t) · η n − λ −2 2κ η 2 dη, then f g (t, κ) = F g (t, κ) for every g ≥ 2.
2.4. Realization of the operators and recursion relations. We recall the realizations of recursion relations in Lemma 2.1 and Theorem 2.2 in this subsection. In order to achieve this, we need to construct the realizations of the operators K, ∂, and D = ∂ + γ first. Clearly the edge-cutting operator K will be realized by the partial derivative ∂ κ = ∂ ∂κ ; that means, (26) w
K(Γ) = ∂ κ (w Γ )
for every stable graph Γ. Moreover, the operator γ will be realized by multiplying by |E ext (Γ)| · κF 0,3 (t), since
w γ(Γ) = |E ext (Γ)| · κF 0,3 (t)w Γ .
The realization of the operator ∂ can be given as follows.
Definition 2.
2. An operator∂ is called a formal differential operator compatible with Feynman rules (19)- (21), if it satisfies the following conditions:
1) Given a stable graph Γ, the operator∂ acts on w Γ = κ |E(Γ)| · v∈V (Γ)
w v via 'Leibniz rules':
∂(w Γ ) = κ |E(Γ)| v∈V (Γ) ∂ (w v ) · v ′ =v w v ′ + |E(Γ)|κ |E(Γ)|−1∂ (κ) · v∈V (Γ) w v .
2)∂ acts on w v for a vertex v of genus g v and valence val v bŷ ∂(F gv ,valv ) = F gv ,valv +1 .
3)∂ acts on the propagator κ bŷ
∂(κ) = κ 2 · F 0,3 .
It is clear that for the given Feynman rules (19)- (21), the operator ∂ can be realized by a compatible formal differential operator∂, i.e., w ∂(Γ) =∂(w Γ ).
Example 2.5. Let {F g (t)} g≥0 be a sequence of holomorphic functions, and F g,0 := F g ;
F g,n := F (n) g (t) = ( ∂ ∂t ) n F g (t), n > 0,
and we choose the propagator κ to be the following function in t:
κ = 1 C − F ′′ 0 (t) ,
where C is either a constant or an anti-holomorphic function in t. Then clearly the operator ∂ t = ∂ ∂t is compatible with this Feynman rule, since
∂κ ∂t = κ 2 F 0,3 .
This is the original realization introduced in [47, §4].
Example 2.6. Let {F g (t)} be a sequence of holomorphic functions for g ≥ 0, and F g,0 := F g ;
F g,n := F (n) g (t) = ( ∂ ∂t ) n F g (t), n > 0.
And now we choose the propagator κ to be a formal variable independent of t. Then the operator ∂ ∂t + κ 2 F 0,3 · ∂ ∂κ is compatible with this Feynman rule, thus it realizes the operator ∂. Now let us recall the realization of the recursion relations. Fix a family of input data {F g,n } 2g−2+n>0 and κ, and suppose that we have found a suitable realization ∂ of the operator ∂. Denote byD the realization of the operator D, then:
(27)D(w Γ ) = (∂ + |E ext (Γ)|κF 0,3 )w Γ .
Now applying such a Feynman rule to Lemma 2.1 and Theorem (2.2), we obtain:
Lemma 2.2. For 2g − 2 + n > 0, we have (28)D F g,n = (n + 1) F g,n+1 .
Or more explicitly,
F g,n = 1 n! ∂ + (n − 1)κF 0,3 · · · ∂ + 2κF 0,3 ∂ + κF 0,3 ∂ F g (t, κ)
for g ≥ 2; and for g = 0, 1, F 1,n = 1 n! ∂ + (n − 1)κF 0,3 · · · ∂ + 2κF 0,3 ∂ + κF 0,3 F 1,1 (t, κ), n ≥ 1;
F 0,n = 3! n! ∂ + (n − 1)κF 0,3 · · · ∂ + 4κF 0,3 ∂ + 3κF 0,3 F 0,3 (t, κ), n ≥ 3.
Theorem 2.4. For 2g − 2 + n > 0, we have
(29) ∂ κ F g,n = 1 2 DD F g−1,n + g1+g2=g, n1+n2=nD F g1,n1 ·D F g2,n2 .
In particular, by taking n = 0 we obtain:
(30) ∂ κ F g = 1 2 D∂ F g−1 + g−1 r=1∂ F r ·∂ F g−r , g ≥ 2.
Here we use the convention
(31)∂ F 1 =D F 1 := F 1,1 ,D F 0,2 := 3 F 0,3 ,DD F 0,1 := 6 F 0,3 ,
by formally applying Lemma 2.2.
Refined Orbifold Euler Characteristic of M g,n
In this section, we introduce a refined orbifold Euler characteristic χ g,n (t, κ) of the moduli space M g,n . The orbifold Euler characteristic of M g,n will be recovered by χ(M g,n ) = χ g,n (1, 1). This construction is a particular realization of the abstract QFT for stable graphs, thus we are able to use the formalism recalled in the previous section to derive various recursion relations to compute χ g,n (t, κ). The last two sections of this paper will be devoted to explicit computations using these recursion relations.
3.1.
A realization of the abstract QFT. In this subsection, we first recall some well-known results about the orbifold Euler characteristic of M g,n (see [9,30,41]), and then construct a particular realization of the abstract QFT by assigning Feynman rules inspired by these works to the stable graphs. The refined orbifold Euler characteristic χ g,n (t, κ) will be defined to be the realization of F g,n .
Let M g,n be the moduli space of smooth stable curves of genus g with n marked points. The orbifold Euler characteristic of M g,n has been first computed by Harer and Zagier [30], and then also by Penner [41] and Kontsevich [34]. This problem can be reformulated as a enumeration of fat graphs, and thus can be studied using methods in Hermitian matrix models. See also [36] for an introduction for these results. The conclusion is as follows:
Theorem 3.1 ( [30]
). The orbifold Euler characteristic of M g,n is given by:
(32) χ(M g,n ) = (−1) n · (2g − 1)B 2g (2g)! (2g + n − 3)!, 2g − 2 + n > 0, where B 2g is the (2g)-th Bernoulli number.
Using the stratification of M g,n (and thus M g,n /S n ), Bini and Harer [9] have derived the following formula for their orbifold Euler characteristics:
Theorem 3.2 ( [9]
). Assume 2g − 2 + n > 0, then the orbifold Euler characteristic of M g,n /S n is given by the following graph sum formula:
(33) χ(M g,n /S n ) = Γ∈G c g,n 1 | Aut(Γ)| v∈V (Γ) χ(M gv,valv ),
where g v is the genus of a vertex v, and val v is the valence of v.
In what follows, we will use the two formulas (32) and (33) to formulate a realization of the abstract QFT for stable graphs.
In the formula (33) of Bini-Harer, the weight of a vertex of genus g and valence n is χ(M g,n ), and the propagator is just 1. However, as mentioned in the Introduction, it is very difficult to carry out concrete calculations directly using this Feynman rule.
In order to apply our formalism of abstract QFT and its realizations, we will modify the above Feynman rule by introducing two formal variables t and κ. We take κ to be the propagator, and take
(34) F orb g,n (t) := χ(M g,n )t −2g+2−n = (−1) n (2g − 1)B 2g (2g)! (2g + n − 3)! · t −2g+2−n
to be the weight of vertices. More precisely, for a stable graph Γ, our Feynman rule defines the weight w Γ of a stable graph Γ as follows:
(35) w Γ = v∈V (Γ) w v · e∈E(Γ) w e ,
where the weight of an internal edge is (36) w e := κ, and the weight of a vertex v of genus g v and valence val v is:
ω v = F orb gv ,valv (t) =χ(M gv ,valv ) · t χ(C gv ,valv ) =(−1) val v (2g v − 1)B 2gv (2g v )! (2g v + val v −3)! · t −2gv +2−valv ,(37)
here χ(C g,n ) is the Euler characteristic of a curve C g,n of genus g with n punctures.
It is clear that if we take:
F orb g (t) := (2g − 1)B 2g (2g)! (2g − 3)! · t −2g+2 , g ≥ 2; F orb 1,1 (t) := − B 2 2 · t −1 ; F orb 0,3 (t) := t −1 ,(38)
then we have:
F orb g,n (t) = ( d dt ) n F orb g (t), g ≥ 2; F orb 1,n (t) = ( d dt ) n−1 F orb 1,1 (t), n ≥ 1; F orb 0,n (t) = ( d dt ) n−3 F orb 0,3 (t), n ≥ 3.(39)
Definition 3.1. Assume 2g − 2 + n > 0. We define the refined orbifold Euler characteristic χ g,n (t, κ) of M g,n /S n to be the realization of F g,n under the Feynman rule (35), i.e.,
χ g,n (t, κ) := Γ∈G c g,n κ |E(Γ)| | Aut(Γ)| · v∈V (Γ) F orb gv ,valv (t),(40)
where g v and val v are the genus and valence of the vertex v respectively.
Clearly χ g,n (t, κ) is a polynomial in κ and t −1 , and the degree of κ encodes the codimension of the strata M Γ in M g,n .
For every (g, n) with 2g − 2 + n > 0, the orbifold Euler characteristic of M g,n /S n is obtained from χ g,n (t, κ) by simply taking t = 1 and κ = 1:
(41) χ(M g,n /S n ) = χ g,n (1, 1), thus we have (42) χ(M g,n ) = n! · χ g,n (1, 1).
The above construction can be represented in terms of a formal Gaussian integral by Theorem 2.3 as follows.
Theorem 3.3. Define a partition function Z orb (t, κ) by the following formal Gaussian integral:
(43) Z orb (t, κ) = 1 (2πλ 2 κ) 1 2 exp 2g−2+n>0 λ 2g−2 n! F orb g,n (t)η n − λ −2 2κ η 2 dη,
then its logarithm (i.e., the free energy) equals to
∞ g=2 χ g,0 (t, κ)λ 2g−2 . In particular,
we can take t = 1, then this integral becomes:
(44) Z orb (1, κ) = 1 (2πλ 2 κ) 1 2 exp 2g−2+n>0 λ 2g−2 n! χ(M g,n ) · η n − λ −2 2κ η 2 dη,
and its free energy is g≥2
χ g,0 (1, κ)λ 2g−2 .
Remark 3.1. The sum 2g−2+n>0 λ 2g−2 n! χ(M g,n ) · η n in the formal Gaussian integral (44) can be rewritten in the following form:
2g−2+n>0 λ 2g−2 n! χ(M g,n ) · η n = 2g−2+n>0 (−1) n (2g − 1)B 2g (2g)! (2g + n − 3)! · λ 2g−2 · η n n! = ∞ n=3 λ −2 (−1) n+1 η n n(n − 1)(n − 2) + ∞ n=1 B 2 2! · (−1) n η n n + ∞ g=2 (2g − 1)B 2g (2g)! · λ 2g−2 ∞ n=0 (2g + n − 3)! · (−1) n η n n! = 1 2 (1 + η) 2 log(1 + η) − η 2 − 3η 2 4 λ −2 − 1 12 log(1 + η) + ∞ g=2 B 2g 2g(2g − 2) · λ 2g−2 (1 + η) 2−2g .
Similarly, we have
2g−2+n>0 λ 2g−2 n! F orb g,n (t) · η n =λ −2 − 3 4 η 2 − 1 2 ηt + 1 2 (η + t) 2 log(1 + η t ) − 1 12 log(1 + η t ) + ∞ g=2 λ 2g−2 · B 2g 2g(2g − 2) (η + t) 2−2g .
See [15] for a similar computation. Here we are taking summation over n first. Later in §?? we will see that it is also important to take summation over g first.
By evaluating the formal Gaussian integrals in (43) and (44) above directly from their definitions, we get the following two formulas for generating series for {χ g,0 (t, κ)} g≥2 and {χ g,0 (1, κ)} g≥2 respectively:
g≥2 λ 2g−2 χ g,0 (t, κ) = log k≥0 2gi−2+li>0 l 1 k! · λ 2(g1+···+g k )−2k+2l l 1 ! · · · l k ! × (2l − 1)!! · k i=1 F orb gi,li (t) · κ l · δ l1+···+l k ,2l ,(45)g≥2 λ 2g−2 χ g,0 (1, κ) = log k≥0 2gi−2+li>0 l 1 k! · λ 2(g1+···+g k )−2k+2l l 1 ! · · · l k ! × (2l − 1)!! · k i=1 χ(M gi,li ) · κ l · δ l1+···+l k ,2l .(46)
The complexity of the right-hand side of these formulas make it unpractical to use them for computations or deriving simpler expressions.
In the remaining of this section, we will apply our formalism to derive a quadratic recursion for χ g,n (t, κ) (and also for χ g,n (1, κ)); and furthermore, we will present a linear recursion for χ g,n (t, κ) with fixed genus g.
Quadratic recursion relation.
In this subsection we derive the quadratic recursion relations for χ g,n (t, κ).
Consider the realizations of Theorem 2.2. As pointed out in §2.4, the edgecutting operator in this case is just the partial derivative ∂ κ = ∂ ∂κ . The operator ∂ is realized by a formal differential operator which takes F orb g,n to F orb g,n+1 , and takes κ to κ 2 · t −1 (see Definition 2.2). Here since we are regarding t and κ as two independent formal variables, we can simply take the realization of ∂ to be: (47) d := ∂ ∂t + κ 2 t −1 · ∂ ∂κ as a realization of the operator ∂. Then the operator D = ∂ + γ is realized by
(48) D = ∂ ∂t + κ 2 t −1 · ∂ ∂κ + |E ext (Γ)| · κt −1 ,
where |E ext (Γ)| is the number of external edges of Γ. Now the quadratic recursion in Theorem 2.2 is realized as follows:
Theorem 3.4. Assume 2g − 2 + n > 0, then we have:
(49) ∂ ∂κ χ g,n = 1 2 (DDχ g−1,n + g1+g2=g, n1+n2=n
Dχ g1,n1 · Dχ g2,n2 ).
In particular, by taking n = 0 we get:
(50) ∂ ∂κ χ g,0 = 1 2 (Ddχ g−1,0 + g−1 r=1 dχ r,0 · dχ g−r,0 ), g ≥ 2.
Here we use the convention:
(51) dχ 1,0 = Dχ 1,0 := χ 1,1 , Dχ 0,2 := 3χ 0,3 , DDχ 0,1 := 6χ 0,3 .
Integrating the recursion formula (49) with respect to κ, we can determine χ g,n up to a term independent of κ. From the definition (40) we may easily see that this term is just 1 n! F orb g,n (t) where F orb g,n (t) is given by (37). Thus we get an algorithm that finds {χ g,n (t, κ)} recursively:
Theorem 3.5. For 2g − 2 + n > 0, we have (52) χ g,n = 1 n! F orb g,n (t) + 1 2 κ 0 DDχ g−1,n + g1+g2=g, n1+n2=n
Dχ g1,n1 · Dχ g2,n2 dκ.
For our original problem of the computations of orbifold Euler characteristic χ(M g,n /S n ) = χ g,n (1, 1), the above recursion can be simplified. In fact, we can apply Theorem 2.1 directly, and in this way we do not need the variable t or the operator D in our quadratic recursion relation. In other words, we may define
(53) χ g,n (κ) := χ g,n (1, κ) = Γ∈G c g,n κ |E(Γ)| | Aut(Γ)| · v∈V (Γ) χ(M gv ,valv ),
then the following result is obtained by directly applying the Feynman rule
v → χ(M gv ,valv ), v ∈ V (Γ), e → κ, e ∈ E(Γ),
to the relation (14):
Theorem 3.6. For 2g − 2 + n > 0, we have χ g,n (κ) = 1 2 κ 0 (n + 2)(n + 1) χ g−1,n+2 + g1+g2=g, n1+n2=n+2 n1≥1,n2≥1 n 1 n 2 χ g1,n1 χ g2,n2 dκ + 1 n! χ(M g,n ),(54)
where the sum is taken over all stable cases.
The refined orbifold Euler characteristic χ g,n (t, κ) can be recovered from χ g,n (κ) = χ g,n (1, κ) using the following relation (thus sometimes we will also call χ g,n (κ) the refined orbifold Euler characteristic):
Proposition 3.1. Assume 2g − 2 + n > 0. We have: (55) χ g,n (t, κ) = t 2−2g−n · χ g,n (κ).
Proof. Let Γ be a connected graph of genus g with n external edges. Its weight w Γ (under the Feynman rule (35)) is a monomial in t −1 and κ, and it suffices to show that the degree of t −1 is 2g − 2 + n. In fact, the Euler's formula tells us:
1 − h 1 = |V (Γ)| − |E(Γ)|,
where h 1 is the number of independent loops in Γ. Thus the degree of t −1 is:
v∈V (Γ) (2g v − 2 + val v ) =2 v∈V (Γ) g v + v∈V (Γ) val v −2|V (Γ)| =2 v∈V (Γ) g v + 2|E(Γ)| + n − 2|V (Γ)| =2 v∈V (Γ) g v + n + 2h 1 − 2 =2g − 2 + n.
This proves the conclusion.
Remark 3.2. In [16], Do and Norbury give the following quadratic recursion ( [16, Prop 6.1]) for χ(M g,n ):
χ(M g,n+1 ) = (2 − 2g − n)χ(M g,n ) + 1 2 χ(M g−1,n+2 ) + 1 2 g h=0 n k=0 n k χ(M h,k+1 )χ(M g−h,n−k+1 ).
To apply such recursion relations to compute χ(M g,n ), one needs the formulas for χ(M g,0 ) for g ≥ 2 as initial values. Such formulas do not seem to be accessible by their method. For g = 0 and g = 1, the fact that χ(M 0,3 ) = 1 and χ(M 1,1 ) = 5 12 can be used to recursively compute χ(M 0,n )(n ≥ 4) and χ(M 1,n )(n ≥ 2), but the problems of finding their explicit formulas are not addressed by these authors.
Example 3.1. Consider the special case g = 0. This has already been studied by Keel [31] and Manin [38]. Now let us study this case using our recursion. Taking g = 0 in (54), we get a recursion formula for { χ 0,n } n≥3 :
(56) χ 0,n (κ) = 1 2 κ 0 n−1 i=3 i(n + 2 − i) χ 0,i χ 0,n+2−i dκ + 1 n! χ(M 0,n )
for every n ≥ 3. The Harer-Zagier formula (32) provides the initial data:
χ(M 0,n ) = (−1) n+1 · (n − 3)!, n ≥ 3; χ 0,3 = 1 6 χ(M 0,3 ) = 1 6 .
Then we easily obtain the following data:
χ 0,3 = 1 6 ; χ 0,4 = − 1 24 + 1 8 κ; χ 0,5 = 1 60 − 1 12 κ + 1 8 κ 2 ; χ 0,6 = − 1 120 + 1 18 κ − 7 48 κ 2 + 7 48 κ 3 ; χ 0,7 = 1 210 − 7 180 κ + 5 36 κ 2 − 1 4 κ 3 + 3 16 κ 4 ; χ 0,8 = − 1 336 + 41 1440 κ − 181 1440 κ 2 + 5 16 κ 3 − 55 128 κ 4 + 33 128 κ 5 ; χ 0,9 = 1 504 − 109 5040 κ + 97 864 κ 2 − 451 1296 κ 3 + 385 576 κ 4 − 143 192 κ 5 + 143 384 κ 6 ; χ 0,10 = − 1 720 + 853 50400 κ − 6061 60480 κ 2 + 1903 5184 κ 3 − 15301 17280 κ 4 + 1001 720 κ 5 − 1001 768 κ 6 + 143 256 κ 7 ;
· · · · · · By taking κ = 1 and multiplying by n!, we get: These numbers coincide with the results obtained in [9,16].
χ
Example 3.2. Now let us move on to the case g = 1. Taking g = 1 in (54), we obtain:
χ 1,n (κ) = 1 2 κ 0 (n + 2)(n + 1) χ 0,n+2 + 2 n+1 i=3 i(n + 2 − i) χ 0,i χ 1,n+2−i dκ + 1 n! χ(M 1,n )(57)
for every n ≥ 1. Here by (32) the initial values are:
χ(M 1,n ) = (−1) n · (n − 1)! 12 .
Then explicit computations give us:
χ 1,1 = − 1 12 + 1 2 κ; χ 1,2 = 1 24 − 7 24 κ + 1 2 κ 2 ; χ 1,3 = − 1 36 + 2 9 κ − 5 8 κ 2 + 2 3 κ 3 ; χ 1,4 = 1 48 − 3 16 κ + 199 288 κ 2 − 41 32 κ 4 + κ 4 ; χ 1,5 = − 1 60 + 1 6 κ − 533 720 κ 2 + 89 48 κ 3 − 83 32 κ 4 + 8 5 κ 5 ; χ 1,6 = 1 72 − 11 72 κ + 677 864 κ 2 − 5203 2160 κ 3 + 2669 576 κ 4 − 1003 192 κ 5 + 8 3 κ 6 ; χ 1,7 = − 1 84 + 1 7 κ − 277 336 κ 2 + 3197 1080 κ 3 − 1131 160 κ 4 + 799 72 κ 5 − 2015 192 κ 6 + 32 7 κ 7 ; χ 1,8 = 1 96 − 13 96 κ + 2323 2688 κ 2 − 425491 120960 κ 3 + 341639 34560 κ 4 − 223829 11520 κ 5 + 39673 1536 κ 6 − 32339 1536 κ 7 + κ 8 ;
· · · · · · By taking κ = 1 and multiplying by n!, we get:
χ(M 1,1 ) = 5 12 , χ(M 1,2 ) = 1 2 , χ(M 1,3 ) = 17 12 , χ(M 1,4 ) = 35 6 , χ(M 1,5 ) = 389 12 , χ(M 1,6 ) = 1349 6 , χ(M 1,7 ) = 22489 12 , χ(M 1,8 ) = 36459 2 ,
· · · · · · Example 3.3. Next consider the case g = 2. The recursion is as follows:
χ 2,n (κ) = 1 2 κ 0 (n + 2)(n + 1) χ 1,n+2 + 2 n+2 i=3 i(n + 2 − i) χ 0,i χ 2,n+2−i + n+1 j=1 χ 1,j χ 1,n+2−j dκ + 1 n! χ(M 2,n )(58)
for every n ≥ 0. Here by (32) we have:
χ(M 2,n ) = (−1) n+1 · (n + 1)! 240 .
Then explicit computations give us · · · · · · By taking κ = 1 and multiplying by n!, we get:
χ 2,0 = − 1 240 + 13 288 κ − 1 6 κ 2 + 5 24 κ 3 ; χ 2,1 = 1 120 − 13 144 κ + 109 288 κ 2 − 3 4 κ 3 + 5 8 κ 4 ; χ 2,χ(M 2,0 ) = 119 1440 , χ(M 2,1 ) = 247 1440 , χ(M 2,2 ) = 413 720 , χ(M 2,3 ) = 89 32 , χ(M 2,4 ) = 12431 720 , χ(M 2,5 ) = 189443 144 , χ(M 2,6 ) = 853541 720 , · · · · · ·
In this subsection we have found an algorithm that effectively computes χ(M g,n ) and provided some examples and numerical data. We will show that our method can also be used to derived some closed formula in §4 and §5.
3.3. Linear recursion relation. In this subsection we show that there is a simple linear recursion for χ g,n (κ) at each fixed genus g.
It is clear that when 2g − 2 + n > 0, χ g,n (κ) is a polynomial in κ. We have:
Lemma 3.1. χ g,n (κ) is a polynomial in κ of degree 3g − 3 + n.
Proof. It is not hard to see that the maximum of |E(Γ)| for a connected Γ of genus g with n external edges is obtained when the vertices of Γ are all trivalent and of genus 0. Assume Γ is such a graph, then
1 − g = |V (Γ)| − |E(Γ)| = 1 3 n + 2|E(Γ)| − |E(Γ)|,
and this completes the proof.
Throughout this paper, we will denote by {a i g,n } the coefficients of this polynomial:
(59) χ g,n (κ) = 3g−3+n i=0 a i g,n κ i .
Then it is clear that:
a i g,n = Γ 1 | Aut(Γ)| v∈V (Γ) χ(M gv ,valv ),
where the summation is over all connected stable graphs of genus g, with i internal edges and n external edges.
For a fixed g ≥ 0, we now derive a linear recursion relation that computes χ g,n (t, κ) as well as the coefficients {a i g,n } from the initial data
{a i g,0 } (or {a i 0,3 }, {a i 1,1 } for g = 0, 1 respectively).
Theorem 3.7. For every 2g − 2 + n > 0, we have
(60) Dχ g,n = (n + 1)χ g,n+1 ,
where D is given by (48). Or equivalently,
(61) (n + 1)a i g,n+1 = (2 − 2g − n)a i g,n + (n + i − 1)a i−1 g,n .
Here all possible unstable terms appearing in the right hand side is set to be zero.
Proof. The recursion (60) is simply the realization of Lemma 2.1. Recall that Proposition (3.1) tells us χ g,n (t, κ) is of the form:
χ g,n (t, κ) = 3g−3+n i=0 a i g,n κ i · ( 1 t ) n−2+2g .
Now compare the coefficients of κ k in two sides of (60), where the left-hand-side is:
Dχ g,n = ∂ ∂t + κ 2 t −1 · ∂ ∂κ + n · κt −1 χ g,n ,
then we may obtain:
(62) (2 − 2g − n)a i g,n + (i − 1) · a i−1 g,n + n · a i−1 g,n = (n + 1)a i g,n+1 .
Theorem 3.7 tells us that for every k ≥ 1, the coefficient a i g,n+1 in χ g,n+1 is uniquely determined by two coefficients a i g,n and a i−1 g,n in χ g,n . Therefore this suggests us to write down the coefficients in a triangle (just like the YangHui's Triangle or Pascal's Triangle) for a fixed genus g ≥ 0, such that an element in this triangle is a linear combination of the two elements above it.
The genus zero linear recursion and Manin's functional equation.
Let us study the linear recursion at genus 0 in this subsection. We will recover a functional equation of Manin for χ(M 0,n ) using this linear recursion.
For g = 0, we put the coefficients {a k 0,n } for n ≥ 3 in a triangle as follows. · · · · · ·
The first row of this triangle is given by χ 0,3 = 1 6 , and the first element of each row is given by 1 n! χ(M 0,n ) for n ≥ 3. Then by Theorem 3.7 all the other elements are determined uniquely by the two elements above it via the linear recursion relation
(63) (n + 1)a k 0,n+1 = (2 − n)a k 0,n + (n + k − 1)a k−1 0,n . Theorem 3.8. Define a generating series χ(y, κ) of χ 0,n (κ) by (64) χ(y, κ) := y + ∞ n=3 nκy n−1 · χ 0,n (κ),
then χ satisfies the equation:
(65) κ(1 + χ) log(1 + χ) = (κ + 1)χ − y.
Proof. Recall that χ 0,n (κ) = n−3 i=0 a i 0,n κ i . The linear recursion (63) gives us:
(66) (n + 1) χ 0,n+1 = (2 − n) χ 0,n + nκ χ 0,n + κ 2 d dκ χ 0,n ,
where d dκ χ 0,n can be rewritten as d dκ
χ 0,n = 1 2 n−1 i=3 i(n + 2 − i) χ 0,i χ 0,n+2−i
by the quadratic recursion (56). Now it is easy to check that (66) is equivalent to the following equation for χ:
χ = 2 y 0 χdy − yχ + 1 2 κχ 2 + y.
Applying d dy on both sides of this equation, we get:
χ ′ = κχχ ′ + χ − yχ ′ + 1,
where χ ′ = ∂χ ∂y . Therefore we have:
(67) χ ′ = 1 + χ 1 + y − κχ ,
and the conclusion follows from this equation by integrating with respect to y.
By Taking κ = 1 in the above theorem, we recover the following result of Manin (see [37, (0.9)]):
Corollary 3.1. The generating series
χ(y, 1) = y + ∞ n=3 y n−1 (n − 1)! χ(M 0,n )
satisfies the functional equation:
(68) 1 + χ(y, 1) log 1 + χ(y, 1) = 2χ(y, 1) − y.
Remark 3.3. In our formalism, the equation (65) is a generalization of Manin's equation (68) to the case of refined orbifold Euler characteristics. Notice that the equation (65) also appear in [37] and [28], as a generalization of equation (68) [38, p. 197]), he proved that for a compact smooth algebraic manifold X of dimension m,
1 + n≥1 χ(X[n]) t n n! = (1 + η) χ(X) ,
where X[n] is the Fulton-MacPherson compactification [23] of the configuration space associated to X, and η is the unique solution in t + t 2 Q[[t]] to the equation:
(69) m(1 + η) log(1 + η) = (m + 1)η − t.
This is of exactly the same form as our equation (65) for the refined orbifold Euler characteristic. In Manin's result, the integer m is the dimension of X, while in our formalism κ is a formal variable encoding the codimensions of the strata in M g,n .
In the work [28] of Goulden, Litsyn and Shevelev, they study the generalized equation (69) and give some results about the coefficients of the solution (see [28, §2.1]):
a n−3 0,n = (2n − 5)!! n! ; a n−4 0,n = − n − 3 3 · (2n − 5)!! n! ; a n−5 0,n = (n − 2)(n − 3)(n − 4) 3 2 · (2n − 7)!! n! ; a n−6 0,n = − (n − 4)(n − 5)(5(n − 2) 2 + 1) 3 4 · 5 · (2n − 7)!! n! ; a n−7 0,n = (n − 4)(n − 5)(n − 6)(5(n − 2) 3 + 4n − 5) 2 · 3 5 · 5 · (2n − 9)!! n! ; · · · · · ·(70)
Our a n−k 0,n is related to their notation µ j (n) in the following way:
a n−k 0,n = 1 n! µ k−2 (n − 1).
Solving the equation (65) directly does not provide us a simple way to obtain an explicit expression for the solution. We will see in §5.1 that one is able to obtain an explicit formula for the generating series of coefficients {a k 0,n } with the help of the linear recursion (63).
3.5.
Relationship to Ramanujan polynomials. Note that the genus zero coefficients {(−1) k+n+1 · n! · a k 0,n } are all integers. They are the sequence A075856 on Sloane's on-line Encyclopedia of Integer Sequences [45]. The references listed at this website leads us to note the relationship of χ(y, κ) to many interesting works in combinatorics, in particular, to Ramanujan psi polynomials [4,5].
This relationship also holds in the higher genus case. Indeed, if we define b k g,n := (−1) k+n+1 · n! · a k g,n , then our recursion (61) for {a k g,n } with fixed g becomes:
(71) b k g,n = (n + 2g − 3)b k g,n−1 + (n + k − 2)b k−1 g,n−1 .
This is a special case of the following recursion relation for x = 2g − 1:
(72) Q n,k (x) = (x + n − 1)Q n−1,k (x) + (n + k − 2)Q n−1,k−1 (x)
discovered by Shor [44, §2] in his proof of Cayley's formula for counting labelled trees. In fact, the recursion for the special values P k g,n := Q n−1,k+1 (2g − 1) is:
(73) P k g,n = (n + 2g − 3)P k g,n−1 + (n + k − 2)P k−1 g,n−1 . Shor shows that (74) n−1 k=0 Q n,k (x) = (x + n) n−1 .
Zeng [54,Proposition 7] establishes the following remarkable connection:
(75) Q n,k (x) = ψ k+1 (n − 1, x + n),
where ψ k (r, x) are the Ramanujan polynomials (1 ≤ k ≤ r + 1) defined by:
(76) ∞ j=0 (x + j) r+j e −u(x+j) u j j! = r+1 k=1 ψ k (r, x) (1 − u) r+k .
Ramanujan gives the following recursion relation of ψ k (r, x):
(77) ψ k (r + 1, x) = (x − 1)ψ k (r, x − 1) + ψ k−1 (r + 1, x) − ψ k−1 (r + 1, x − 1).
Berndt et al. [4,5] obtain the following recursion relation:
(78) ψ k (r, x) = (x − r − k + 1)ψ k (r − 1, x) + (r + k − 2)ψ k−1 (r − 1, x).
This is used by Zeng [54] to connect Ramanujan polynomials to Shor polynomials. He also gives the combinatorial interpretations of such a connection. For some other related works, see [10,17,18]. In particular, Chen and Yang give a contextfree grammar for the Ramanujan-Shor polynomials in [11]. However, we do not have b k g,n = P k g,n = Q n−1,k+1 (2g − 1) for g > 0 because the initial values for Q n,k (x) are Q 1,0 (x) = 1, Q n,−1 (x) = 0, n ≥ 1,
Q 1,k (x) = 0, k ≥ 1,
and these correspond to
b −1 g,2 = 1, b −2 g,n = 0, n ≥ 1, b k−1
g,2 = 0, k ≥ 1, but these are not satisfied by b k g,n . Nevertheless, when g = 0, since a k 0,n makes sense only for n ≥ 3 and 0 ≤ k ≤ n − 3, these conditions are automatically satisfied, so we get:
(79) (−1) k+n+1 · n! · a k 0,n = Q n−1,k+1 (−1) = ψ k+2 (n − 2, n − 2)
. This relates our refined orbifold Euler characteristics of M 0,n to special values of the Shor polynomials and Ramanujan polynomials, hence to their combinatorial interpretations related to counting trees. In particular, one can use [54, (35), (21)] to compute a k 0,n . When g > 0, such connection is partially lost, but it is still desirable to find combinatorial interpretations of the refined orbifold Euler characteristics of M g,n that explains the recursion relation (61).
3.6. Generalization of Manin's functional equation to higher genera. In this subsection, we derive functional equations for generating series of { χ g,n (κ)} for g ≥ 1.
First let us consider the case of genus one. For g = 1, we put the coefficients {a k 1,n } for n ≥ 1 in a triangle (without the first row) as follows.
· · · · · ·
The first row is given by χ 1,1 = − 1 12 + 1 2 κ, and the first element of each row is given by 1 n! χ(M 1,n ) for n ≥ 1. Then all the other elements are determined by the two elements above it via the linear recursion relation (80) (n + 1)a k 1,n+1 = −n · a k 1,n + (n + k − 1)a k−1 1,n . Similarly to the case of genus zero, we define:
(81) ψ(y, κ) := ∞ n=1
nκy n−1 χ 1,n (κ), then ψ can be determined by the following:
Theorem 3.9. We have
(82) (1 + y − κχ)ψ = 1 2 κ 2 χ ′ − 1 12 κ,
where χ(y, κ) is defined by (64) and χ ′ = ∂χ ∂y .
Proof. The linear recursion (80) tells us:
(83) (n + 1) χ 1,n+1 = −n χ 1,n + nκ χ 1,n + κ 2 d dκ
χ 1,n ,
where d dκ χ 1,n can be rewritten as
d dκ χ 1,n = 1 2 (n + 2)(n + 1) χ 0,n+2 + n+1 i=3 i(n + 2 − i) χ 0,i χ 1,n+2−i
by the quadratic recursion (57). Therefore the recursion (83) is equivalent to the following equation for χ(y, κ) and ψ(y, κ):
ψ = −yψ + 1 2 κ 2 χ ′ + κχψ − 1 12 κ.
Notice that we have (67), thus the above result can be rewritten as:
ψ = 1 2 κ 2 (1 + χ) (1 + y − κχ) 2 − 1 12 κ (1 + y − κχ)
.
In particular, take κ = 1 in the above results, we ontain:
Corollary 3.2.
We have:
ψ(y, 1) = (1 + χ(y, 1)) 2(1 + y − χ(y, 1)) 2 − In general, for a fixed genus g ≥ 2, we can put the coefficients {a k g,n } into a triangle (without fisrt 3g − 3 rows for g ≥ 2). The first row is the coefficients of χ g,0 (κ), which can be determined either by the quadratic recursion relation (54) using lower genus data, or by expanding (46) directly (actually the first method is more efficient than the second one). Then every other element in this triangle can be determined by the two elements above it via the linear recursion (61). In §5 we will present explicit solutions to such linear recursion relations in all genera g ≥ 0.
Now let us define ϕ 0 (y, κ) := χ(y, κ), and
(84) ϕ g (y, κ) := ∞ n=1
nκy n−1 χ g,n (κ), g ≥ 1.
In particular, it is clear that ϕ 1 = ψ. Then we have:
Theorem 3.10. For every g ≥ 1, we have
(85) (y + 1)ϕ ′ g + (2g − 1)ϕ g = 1 2 κ 2 ϕ ′′ g−1 + κ · g1+g2=g ϕ ′ g1 ϕ g2 .
where ϕ ′ g := ∂ ∂y ϕ g .
Proof. Similar to the case of genus zero and genus one, we combine the linear recursion (61) and the quadratic recursion (54) to get:
(n + 1) χ g,n+1 =(2 − 2g − n) χ g,n + nκ χ g,n + 1 2 κ 2 (n + 2)(n + 1) χ g−1,n+2
+ 1 2 κ g1+g2=g n1+n2=n+2 n1≥1,n2≥1 n 1 n 2 χ g1,n1 χ g2,n2 .
This equation is equivalent to
ϕ g = (2 − 2g) y 0 ϕ g dy − yϕ g + 1 2 κ 2 ϕ ′ g−1 + 1 2 κ g1+g2=g ϕ g1 ϕ g2 + ϕ g (κ),
where ϕ g (κ) is a term independent of y. Then the conclusion holds by applying ∂ ∂y to the above equation.
In particular, we can take κ = 1, then then the above theorem gives us:
Corollary 3.3.
We have:
(y + 1)ϕ ′ g (y, 1) + (2g − 1)ϕ(y, 1) = 1 2 ϕ ′′ g−1 (y, 1) + g1+g2=g ϕ ′ g1 (y, 1)ϕ g2 (y, 1), where ϕ 0 (y, 1) = y + ∞ n=3 y n−1 (n − 1)! χ(M 0,n ), ϕ g (y, 1) = ∞ n=1 y n−1 (n − 1)! χ(M g,n ), g ≥ 1.
3.7. Operator formalism for the linear recursion. In this subsection we present a reformulation of the results in last subsection using an operator formalism. First recall that the definition (48) of the operator D depends on the number of external edges |E ext (Γ)| (which is just n when acting on χ g,n (t, κ)). Here in order to derive an operator formalism, we are supposed to modify D first to get an operator which does not depend on n.
Notice that the refined orbifold Euler characteristic χ g,n (t, κ) is of the form:
χ g,n (t, κ) = t 2−2g−n · χ g,n (κ), it follows that ∂ ∂t χ g,n (t, κ) = (2 − 2g − n)t 1−2g−n χ g,n (κ) = (2 − 2g)t −1 · χ g,n (κ) − nt −1 · χ g,n (κ).
Therefore the operator D acts on χ g,n (t, κ) by:
Dχ g,n (t, κ) = ∂ ∂t + κ 2 t −1 · ∂ ∂κ + n · κt −1 χ g,n (t, κ) = (1 − κ) ∂ ∂t + κ 2 t ∂ ∂κ + (2 − 2g)κ t χ g,n (t, κ).
Define D to be the operator:
(86) D := (1 − κ) ∂ ∂t + κ 2 t ∂ ∂κ + (2 − 2g)κ t ,
then D is a suitable modification of D, and the recursion (60) is equivalent to:
(87) χ g,n+1 (t, κ) = 1 n + 1 Dχ g,n (t, κ),
or,
χ 0,n = 3! n! · D n−3 χ 0,3 , n ≥ 3; χ 1,n = 1 n! · D n−1 χ 1,1 , n ≥ 1; χ g,n = 1 n! · D n χ g,0 , g ≥ 2.(88)
Now let us define:
χ 0 := ∞ n=3 n! · χ 0,n (t, κ) = t 2 · ∞ n=3
n! t n χ 0,n (κ);
χ 1 := ∞ n=1 n! · χ 1,n (t, κ) = ∞ n=1
n! t n χ 1,n (κ);
χ g := ∞ n=0 n! · χ g,n (t, κ) = t 2−2g · ∞ n=0
n! t n χ g,n (κ), g ≥ 2, then clearly they are generating series of { χ g,n (κ)}. Similarly, we can define another type of generating series:
χ 0 := ∞ n=3 χ 0,n (t, κ) = t 2 · ∞ n=3
1 t n χ 0,n (κ);
χ 1 := ∞ n=1 χ 1,n (t, κ) = ∞ n=1
1 t n χ 1,n (κ);
χ g := ∞ n=0 χ g,n (t, κ) = t 2−2g · ∞ n=0 1 t n χ g,n (κ), g ≥ 2.
Then the linear recursion (87) gives us the following:
Theorem 3.11. We have
χ g (t, κ) = 1 1 − D χ g,0 (t, κ); χ g (t, κ) = e D χ g,0 (t, κ),
where D is given by (86). Here we use the convention: 3.8. Motivic realization of the abstract quantum field theory. The Euler characteristic is an example of motivic characteristic classes. In this subsection we speculate on the possibility of a realization of our abstract quantum field theory by using the orbifold motivic classes of the Deligne-Mumford moduli space M g,n of stable curves. The theory of motivic measures and motivic integrals was first introduce by Kontsevich [35], and generalized to singular spaces by Denef and Loeser [14]. Let VAR be the category of complex algebraic varieties of finite type, and R be a commutative ring with unity. A motivic class is a ring homomorphism
[·] : K 0 (VAR) → R,
where K 0 (VAR) is the Grothendieck ring of complex varieties. In other words, a motivic class is a map [·] satisfying:
(1) [X] = [X ′ ], for X ∼ = X ′ ; (2) [X] = [X \ Y ] + [Y ], for a closed subvariety Y ⊂ X; (3) [X × Y ] = [X] · [Y ]; (4) [pt] = 1.
For our purpose we need to consider the orbifold motivic class of the Deligne-Mumford moduli space M g,n of stable curves. We need to have a relation Such consideration would give us a natural realization of our abstract quantum field theory. We assign the Feynman rule to a stable graph Γ ∈ G c g,n as follows. The contribution of a vertex v ∈ V (Γ) is defined to be the orbifold motivic class of M g(v),val (v) :
ω v = [M g(v),val(v) ],
and the contribution of an internal edge e ∈ E(Γ) is set to be ω e = 1. Thus the Feynman rule is
Γ → ω Γ = v∈V (Γ) [M g(v),val(v) ].
Therefore the abstract n-point function F g,n is realized by
F g,n = Γ∈G c g,n 1 | Aut(Γ)| v∈V (Γ) [M g(v),val(v) ] ,
i.e., F g,n = [M g,n /S n ] is the orbifold motivic class of M g,n . Once this has been done, we can apply our formalism introduced in [47] to this case to derive some quadratic recursion relations for the orbifold motivic class [M g,n /S n ].
Structures of χ g,0 (t, κ)
In §3.3 and §3.7 we have reduced the computations of χ g,n (t, κ) to the problem of computing χ g,0 (t, κ). In this section we will present various methods to solve χ g,0 (t, κ). In particular, we show that the generating series G k (z) of the coefficients {a k g,0 } can be represented as some polynomials in certain generating series V n (z) of χ(M g,n ), and the explicit formulas for V n (z) can be given in terms of the Barnes G-function. We show that finding the expression of G k (z) in terms of V n (z) is equivalent to the topological 1D gravity (with a genus-shift).
4.1.
Computations of χ g,0 (t, κ) by quadratic recursions. In this subsection we specialize the quadratic recursion in §3.2 to the case n = 0.
Taking n = 0 in Theorem 3.5, we obtain a recursion:
χ g,0 (t, κ) =F orb g,0 (t) + 1 2 κ 0 DDχ g−1,0 + g−2 r=2
Dχ r,0 Dχ g−r,0 + 2Dχ g−1,0 · χ 1,1 dκ
=F orb g,0 (t) + 1 2 κ 0 ∂ ∂t + κ 2 t ∂ ∂κ + κ t ∂ ∂t + κ 2 t ∂ ∂κ χ g−1,0 + g−2 r=2 ∂ ∂t + κ 2 t ∂ ∂κ χ r,0 ∂ ∂t + κ 2 t ∂ ∂κ χ g−r,0 + 2 ∂ ∂t + κ 2 t ∂ ∂κ χ g−1,0 · (− 1 12 + 1 2 κ)t −1 dκ(91)
for g ≥ 3, where F orb g,0 (t) is given by (34). Recall that the refined orbifold Euler characteristic χ g,n (t, κ) is of the following form:
χ g,n (t, κ) = χ g,n (κ) · t 2−2g−n = 3g−3+n k=0 a k g,n κ k · ( 1 t ) n−2+2g , then we have (92) ∂ ∂t χ g,n t=1 = (2 − 2g − n) χ g,0 .
Thus by taking t = 1 in the above quadratic recursion, we get the following:
Theorem 4.1. For g ≥ 3, we have:
χ g,0 = B 2g 2g(2g − 2) + 1 2 κ 0 (3 − 2g) + κ 2 ∂ ∂κ + 2κ − 1 6 (4 − 2g) + κ 2 ∂ ∂κ χ g−1,0 + g−2 r=2
(2 − 2r) + κ 2 ∂ ∂κ χ r,0 · (2 − 2g + 2r) + κ 2 ∂ ∂κ χ g−r,0 dκ.
It follows that the recursion for the coefficients {a k g,0 } is: . In particular, the recursion for the sequence {a 3g−3 g,0 } g≥2 is:
Corollary 4.1. We have: a k g,0 = 1 2k · ( 17 6 − 2g)(4 − 2g)a k−1 g−1,0 + k(4 − 2g) + (k − 2)( 17 6 − 2g) a k−2 g−1,0 + (k 2 − 3k)a k−3 g−1,0 + g−2 r=2 l+m=k−1 (2 − 2r)(2 − 2g +a 3g−3 g,0 = 1 2 (3g − 6)a 3g−6 g−1,0 + 1 6g − 6 g−2 r=2 (3r − 3)(3g − 3r − 3)a 3r−3 r,0 a 3g−3r−3 g−r,0 for g ≥ 3.
Example 4.1. Using the quadratic recursion relation in the above theorem and the initial value
χ 2,0 (κ) = − 1 240 + 13 288 κ − 1 6 κ 2 + 5 24 κ 3 ,
one can recursively compute χ g,0 (κ). · · · · · · 4.2. The generating series of a k g,0 for fixed k. Each χ g,0 (κ) is the generating function of a k g,0 for fixed g. In this subsection let us consider the generating function of a k g,0 for fixed k. Define G k (z) to be such generating functions:
(93) G k (z) := g≥2 a k g,0 · z 2−2g .
The recursion relations for a k g,0 in last subsection can be translated into the following recursion relations for G k (z) (k ≥ 1): Proposition 4.1. We have:
G k (z) = a k 2,0 z −2 + z −2 2k (θ − 7 6 )θG k−1 (z) + (2k − 2)θ − 7 6 k + 7 3 G k−2 (z) + (k 2 − 3k)G k−3 (z) + k−1 l=0 θG l (z) · θG k−1−l (z) + 2 k−2 l=0 θG l (z) · (k − 2 − l)G k−2−l (z) + k−3 l=0 l(k − 3 − l)G l (z)G k−3−l (z) ,(94)
where θ := z d dz , and the initial values are a k 2,0 = 0 except for:
(95) G 0 (z) = ∞ g=2 B 2g 2g(2g − 2) z 2−2g .
At the Digital Library of Mathematical Functions (NIST) ( §24.11) one can find the formula
(96) B 2n ∼ (−1) n−1 4 √ πn n πe 2n
, so the series on the right-hand of (95) has zero radius of convergence, and it should only be understood as an asymptotic series for now. The quadratic recursion (94) gives us the following expressions in terms of Bernoulli numbers:
G 1 (z) = g≥3 1 2 − 5 24(g − 1) B 2g−2 z 2−2g + g≥4 g1+g2=g g1,g2≥2 B 2g1 B 2g2 8g 1 g 2 z 2−2g + 13 288 z −2 , G 2 (z) = g≥4 ( g 2 2 − 23 12 g + 493 288 + 13 288(2g − 4) )B 2g−4 z 2−2g − g≥3 B 2g−2 4(g − 1) z 2−2g + g≥5 g1+g2=g−1 g1≥2,g2≥2 g 2 2g 1 + 1 4 − 11 24g 1 + 5 96g 1 g 2 B 2g1 B 2g2 z 2−2g + g≥6 g1+g2+g3=g g1,g2,g3≥2 1 8g 1 g 2 − 1 16g 1 g 2 g 3 B 2g1 B 2g2 B 2g3 z 2−2g + 247 3456 z −4 − 1 6 z −2 .
4.3.
Generating series of χ(M g,n ) in terms of Barnes G-function. By the method of last subsection it is clear that one can express each G k (z) in terms of Bernoulli numbers. A priori, they are just series whose radii of convergence are zero. We will relate them to the Barnes G-function G(z) in this subsection. First let us recall a result due to Distler-Vafa [15] which represents the generating series of χ(M g,0 ) (i.e., the series G 0 (z)) in terms of the Euler Gamma-function:
(97) Γ(z) := ∞ 0 t z−1 e −t dt.
Recall that Gamma-function Γ(z) has the following Weierstrass product:
(98) 1 Γ(z) = ze γz ∞ n=1 1 + z n e − z n ,
where γ is the Euler-Mascheroni's constant defined by:
(99) γ := lim
n→∞ 1 + 1 2 + · · · + 1 n − log n .
The Stirling series is the following asymptotic expansion of log Γ(z):
(100) log Γ(z) ∼ (z − 1 2 ) log z − z + 1 2 log(2π) + ∞ k=1 B 2k 2k(2k − 1)z 2k−1 .
By (95) we have:
d dz G 0 (z) = − ∞ g=2 B 2g 2g z 1−2g
is the asymptotic series of
(−z) − d dz log Γ(z + 1) + log z + 1 2 z −1 − 1 12 z −2 = z d dz log Γ(z + 1) − z log z − 1 2 + 1 12 z −1 .
Recall that Γ(z) is singular at z = 0, in order to carry out the integration from 0 to z, in the above we have used the fact that (101) Γ(z + 1) = zΓ(z).
Thus we obtain the following: Then up to some constant C, we have for z ≫ 0:
(103) z 0 z d dz log Γ(z + 1) dz − 1 2 z 2 log z + 1 4 z 2 − 1 2 z + 1 12 log z − C ∼ G 0 (z).
Remark 4.1. This Lemma is essentially due to Distler and Vafa [15]. Our modification is based on the following observation. In [15, (10)], the following formula is used:
(104) F (µ) = µ x d dx log Γ(x) dx.
Note Γ(x) is not define at x = 0, so this expression needs some suitable regularization at x = 0. Our formula (103) avoids this problem.
Remark 4.2. We will later fix the constant C to be ζ ′ (−1). On the left-hand side of (103), 1 2 z 2 log(z) − 1 4 z 2 + 1 2 z should be understood as 'the contribution of M 0,0 ', and − 1 12 log(z) should be understood as 'the contribution of M 1,0 '. In other words, we define
F 0,0 (z) = 1 2 z 2 log(z) − 1 4 z 2 + 1 2 z + ζ ′ (−1),(105)F 1,0 (z) = − 1 12 log(z).(106)
With these understood, (103) can be rewritten as:
(107) F 0,0 (z) + F 1,1 (z) + ∞ g=2 χ(M g,0 )z 2−2g = z 0 z d dz log Γ(z + 1) dz. Definition 4.1. Define V 0 (z) := ∞ g=2 χ(M g,0 )z 2−2g ; V n (z) := ∞ g=1 χ(M g,n )z 2−2g−n , n = 1, 2; V n (z) := ∞ g=0 χ(M g,n )z 2−2g−n , n ≥ 3,(108)
to be the total contributions of all stable vertices of valence n.
By (32), one can see
(109) χ(M g,n+1 ) = (2 − 2g − n) · χ(M g,n ).
It follows that:
V ′ 0 (z) = V 1 (z) + 1 12 z −1 , V ′ 1 (z) = V 2 (z), V ′ 2 (z) = V 3 (z) − z −1 , V ′ n (z) = V n+1 (z), n ≥ 3.(110)
The series V n (z) can be related to the Gamma function as follows:
V 0 (z) =G 0 (z) ∼ z 0 z d dz log Γ(z + 1) dz − 1 2 z 2 log z + 1 4 z 2 − 1 2 z − C + 1 12 log z, V 1 (z) = d dz G 0 (z) − 1 12 z −1 ∼ z d dz log Γ(z) − z log z + 1 2 , V 2 (z) = d 2 dz 2 G 0 (z) + 1 12 z −2 ∼ z d 2 dz 2 log Γ(z) + d dz log Γ(z) − log z − 1,
and for n ≥ 3, It satisfies the relations
V n (z) = d n dz n G 0 (z) + (−1) n+1 (n − 3)! · z 2−n + (−1) n 12 (n − 1)! · z −nG(z + 1) = Γ(z)G(z), G(1) = 1.(112)
One can show that:
d dz log G(z + 1) = 1 2 log(2π) − z − 1 2 − γz + ∞ n=1 1 1 + z n − 1 + z n = 1 2 log(2π) − z − 1 2 − γz + ∞ n=1 ∞ m=2 (−1) m z m n m ,
and similarly, by (98) and (101) we have:
d dz log Γ(z + 1) = −γ − ∞ n=1 1 n 1 + z n − 1 n = −γ − ∞ n=1 ∞ j=1 (−1) j z j n j+1 ,
and so we also have:
(113) d dz log G(z + 1) = 1 2 log(2π) − 1 2 − z + z d dz log Γ(z + 1).
Therefore,
log G(z + 1) = z 2 log(2π) − z 2 − 1 2 z 2 + z 0 z d dz log Γ(z + 1) dz = z 2 log(2π) − z 2 − 1 2 z 2 + z log Γ(z + 1) − z 0 log Γ(z + 1)dz.(114)
Barnes [3] obtained the following asymptotic expansion of log G(z + 1):
log G(z + 1) ∼ ζ ′ (−1) + z 2 log(2π) + z 2 2 − 1 12 log z − 3z 2 4 + ∞ k=1 B 2k+2 2k(2k + 2)z 2k ,
where ζ is the Riemann zeta function. (For a modern derivation, see [21].) In particular, we have
z 0 z d dz log Γ(z + 1) dz = log G(z + 1) − z 2 log(2π) + z 2 + z 2 2 ∼ζ ′ (−1) + z 2 2 − 1 12 log z − z 2 4 + z 2 + ∞ k=1 B 2k+2 z −2k 2k(2k + 2) ,(115)
so by (95) or (103) we get:
(116) G 0 (z) = log G(z + 1) − ζ ′ (−1) + z 2 log(2π) + z 2 2 − 1 12 log z − 3z 2 4 ,
and the constant C in (103) is ζ ′ (−1).
Similarly, one can rewrite V n (z) in terms of the Barnes G-function, and the result is as follows:
Lemma 4.2. We have:
V 0 (z) ∼ log G(z + 1) − ζ ′ (−1) + z 2 log(2π) + z 2 2 − 1 12 log z − 3z 2 4 , V 1 (z) ∼ d dz log G(z + 1) − 1 2 log(2π) − z log z + z, V 2 (z) ∼ d 2 dz 2 log G(z + 1) − log z,(117)
and
(118) V n (z) = d n dz n log G(z + 1), ∀n ≥ 3.
Proof. The conclusion follows form (116), (110) and direct computations. (105) and (106), we can rewrite the above results in the following form:
(119) log G(z + 1) − z 2 log(2π) + z 2 + z 2 2 ∼ F 0,0 (z) + F 1,0 (z) + g≥2 χ(M g,0 )z 2−2g , (120) d dz log G(z + 1) − 1 2 log(2π) + 1 2 + z ∼ F ′ 0,0 (z) + g≥1 χ(M g,1 )z 1−2g , (121) d 2 dz 2 log G(z + 1) + 1 ∼ F ′′ 0,0 (z) + g≥1 χ(M g,2 )z −2g , and (122) d n dz n log G(z + 1) ∼ g≥0 χ(M g,n )z 2−2g−n , ∀n ≥ 3.
Remark 4.4. There is a probability interpretation of the left-hand side of (119) by [39,Theorem 1.2]. Recall that a gamma variable γ a with parameter a > 0 is a random variable with distribution density function
(123) t a−1 exp(−t) Γ(a) , t > 0.
If {γ n } n≥1 are independent gamma random variables with respective parameters n, then for z ∈ C such that ℜ(z) > −1,
lim N →∞ 1 N z 2 /2 E exp −z N n=1 γ n n − N = exp z 2 + z 2 − z 2 log(2π) + log G(z + 1) −1 .(124)
As a consequence of (119), we then have the following asymptotic expansion:
lim N →∞ 1 N z 2 /2 E exp −z N n=1 γ n n − N ∼F 0,0 (z) + F 1,0 (z) + g≥2 χ(M g,0 )z 2−2g .(125)
This is a very surprising connection that relates a probability problem to algebraic geometry and mathematical physics. Now if we multiply (119)-(122) by x 0 , x 1 , x 2 /2! and x n /n! respectively and sum them up we get:
log G(z + x + 1) − z 2 log(2π) + z 2 + z 2 2 + (− 1 2 log(2π) + 1 2 + z)x + x 2 2 ∼F 0,0 (z) + F 1,0 (z) + F ′ 0,0 (z)x + x 2 2 F ′′ 0,0 (z) + 2g−2+n>0 χ(M g,n )z 2−2g−n x n .(126)
4.4. Expressions of G k (z) in terms of Barnes G-function. Now let us find a way to express G k (z) in terms of the generating series V n (z). Then using the above lemma, this will give us a way to represent G k (z) in terms of the Barnes G-function. First we need:
Theorem 4.2.
Define a partition function
(127) Z = 1 √ 2πλ 2 exp λ −2 − 1 2 x 2 + n≥0 λ 2 V n (z) · x n n! dx,
then:
(128) log( Z) = k≥0 λ 2k G k (z).
the coefficient of λ 2k in log( Z) equals G k (z) for every k ≥ 0.
Proof. Recall in Theorem 3.3 we have shown that if
Z orb (t, κ) = 1 (2πλ 2 κ) 1 2 exp 2g−2+n>0 λ 2g−2 n! χ(M g,n )t 2−2g−n η n − λ −2 2κ η 2 dη,
then its logarithm equals to:
∞ g=2 χ g,0 (t, κ)λ 2g−2 = ∞ g=2 i≥0 a i g,0 t 2−2g κ i λ 2g−2 = i≥0 G i ( t λ ) · κ i .
Now let us rewrite Z orb (t, κ) in terms of V n as follows:
Z orb (t, κ) = 1 (2πλ 2 κ) 1 2 exp n≥0 1 n! V n ( t λ ) · η λ n − 1 2κ η λ 2 dη.
We make the following change of variables: z = t λ , x = η λ , and after that change κ toλ 2 . Then we see that the logarithm of 1 (2πλ 2 )
1 2 exp n≥0 1 n! V n (z) · x n − 1 2λ 2 x 2 dx is k≥0 G k (z) ·λ 2k = k≥0 G k (z) · κ k .
Now plugging Lemma 4.2 into this theorem, we obtain:
exp k≥0 λ 2k G k (z) = 1 √ 2πλ 2 dx · exp − λ −2 2 x 2 + log G(x + z + 1) − ζ ′ (−1) + z 2 log(2π) + z 2 2 − 1 12 log z − 3z 2 4 + x − 1 2 log(2π) − z log z + z − x 2 2 log z .(129)
Thus:
Corollary 4.2.
We have:
(130) k≥0 λ 2k G k (z) = log 1 √ 2πλ 2 G(x + z + 1) · exp( S(x, z))dx , where S(x, z) = − λ −2 2 x 2 − ζ ′ (−1) + z 2 log(2π) + z 2 2 − 1 12 log z − 3z 2 4 + x − 1 2 log(2π) − z log z + z − x 2 2 log z.(131)
Now Theorem 4.2 gives us a way to express G k (z) in terms of the generating series {V n (z)}. In the next two subsections, we will see that G k (z) can be represented as a polynomial in V 1 (z), V 2 (z), · · · , V 2k (z), and we can use the formalism recalled in §2 to present a simple recursion to compute these polynomials. Furthermore, the formal Gaussian integral (127) relates this problem to the theory of topological 1D gravity, and this will be explained in §4.7- §4.10 and §6.
4.5.
Representing G k (z) in terms of V n (z). Now in this subsection we show that G k (z) can be represented as a polynomial in the generating series {V n (z)}. The following result is a consequence of Theorem 4.2:
Theorem 4.3. For k ≥ 1, we have (132) G k (z) = Γ∈ G k 1 | Aut( Γ)| v∈V ( Γ) V val v (z),
where G k is the set of all connected graphs with unmarked vertices (not necessarily stable), which has k internal edges and no external edge; and V ( Γ) is the set of vertices of Γ. The weight of vertices {V n (z)} n≥1 are given by Lemma 4.2.
Proof. By Theorem 4.2, G k (z) can be expressed by a Feynman sum over graphs with vertex contributions given by V n (z) for a vertex of valence n, and with 1 as propagator. As in It follows that G k is the sum over graphs with k internal edges.
Corollary 4.3. If we assign V n (z) to be of degree n for every n ≥ 0, then G k (z) is of degree 2k.
Now let us see a few examples.
Example 4.3. We can directly check the expressions for G 1 (z) and G 2 (z) can be reformulated in the following form:
G 1 (z) = 1 2 V 2 (z) + 1 2 V 1 (z) 2 , G 2 (z) = 1 8 V 4 + 1 4 V 2 2 + 1 2 V 1 V 3 + 1 2 V 2 1 V 2 .
The graphs for G 1 (z) are:
A first sight, these two graphs have different genus: o ne has genus one and the other has genus zero. It seems odd that both contribute to G 1 (z). To explain this discrepancy, we need to let each vertex of graphs have genus 1, then both of the above graphs have genus 2. Similarly, the graphs for G 2 (z) are:
When the vertices are assigned to have genus 1, then all these graphs have genus 3.
4.6.
Recursive computations for G k (z) and Young diagram representation. In the previous subsection we have seen that G k (z) can be expressed in terms of {V 1 (z), · · · , V 2k (z)} by the formula (132), and the explicit formulas for V n (z) are given in terms of the Barnes G-function. In this subsection we present a simple quadratic recursion to compute the expressions G k (z) in terms of V n (z).
Here we need first to clarify some possible confusions of our notations. Although G k (z) and V k (z) are some Laurent series in z which have precise expressions, in the following subsections in order to find simple polynomial expressions for G k = G k (V 1 , · · · , V 2k ) we will sometimes treat V k as some formal variables and use operations such like taking partial derivatives with respect to V k .
To be more precise, one is supposed to introduce a family of new formal variables {v n } and a family of polynomials H k = H k (v 1 , · · · , v 2k ) such that H k are represented using {v n } in such a way that is similar to the formal Gaussian integral (127) or the Feynman rule (132). In other works, we study a 'theory' with coulpling constants {v n }. And after finding the recursion formulas for H k , we take the special values v n = V n (z) and then
H k (v 1 , v 2 , · · · ) vn=Vn(z) = G k (z).
But for simplicity, in what follows simply regard V n as some formal variables by abuse of notations. Now let us present a simple quadratic recursion for the polynomial expressions of G k = G k (V 1 , V 2 , · · · ) using the formalism of abstract QFT and its realization. Although the graph sum (132) seems not to be over stable graphs, here we can again use the trick of 'shifting the genus of vertices' (see Example 4.3). By assigning genus g v = 1 for every vertex v ∈ V ( Γ), we understand the problem of counting connected graphs Γ ∈ G k for k ≥ 1 as counting connected stable graphs Γ ∈ G c k+1,0 of genus k + 1 with no external edges, whose vertices are all of genus one. Here the Feynman rule we need is:
v → V val v , for v ∈ V (Γ) and g v = 1; v → 0, for v ∈ V (Γ) and g v = 1; e → 1,
for e ∈ E(Γ).
Now consider the realization of the edge-cutting operator K and the edge-adding operators D = ∂ + γ. Since all graphs Γ ∈ G k has k internal edges, the operator K is realized by simply multiplying by k. Moreover, since in this case the weight of all genus zero vertices are zero, every procedure involving 'adding a trivalent vertex of genus zero' in the definition of the operator D = ∂ + γ will be realized to zero. Therefore, the only procedure we need to consider is adding an external edge on one of the vertices of genus one. Now it is clear that D is realized by an operator d V with:
1) d V (V l ) = V l+1 , for every l ≥ 1; 2) d V acts on polynomials in {V l } l≥1 via Leibniz rule.
Therefore, we can simply take:
(135) d V := ∞ l=0 V l+1 ∂ ∂V l .
Now the realization of Theorem 2.2 gives us:
Theorem 4.4. For every k ≥ 1, we have:
(136) k · G k = 1 2 d 2 V G k−1 + k r=1 d V G r−1 · d V G k−r .
Inductively, one might see that 2 k · k! · G k is a polynomial in V 1 , V 2 , · · · , V 2k with integer coefficients. Example 4.4. Using this recursion, one can easily compute:
G 3 = 1 2 V 2 1 V 2 2 + 1 6 V 3 2 + 1 6 V 3 1 V 3 + V 1 V 2 V 3 + 5 24 V 2 3 + 1 4 V 2 1 V 4 + 1 4 V 2 V 4 + 1 8 V 1 V 5 + 1 48 V 6 ; G 4 = 1 2 V 2 1 V 3 2 + 1 8 V 4 2 + 1 2 V 3 1 V 2 V 3 + 3 2 V 1 V 2 2 V 3 + 1 2 V 2 1 V 2 3 + 5 8 V 2 V 2 3 + 1 24 V 4 1 V 4 + 3 4 V 2 1 V 2 V 4 + 3 8 V 2 2 V 4 + 2 3 V 1 V 3 V 4 + 1 12 V 2 4 + 1 12 V 3 1 V 5 + 3 8 V 1 V 2 V 5 + 7 48 V 3 V 5 + 1 16 V 2 1 V 6 + 1 16 V 2 V 6 + 1 48 V 1 V 7 + 1 384 V 8 ; · · · · · ·
The operator d V also appears in an earlier work of the second author [56, §7.6]. In that work, d V is interpreted using some rules in terms of Young diagrams, which is similar to the Littlewood-Richardson rule. Let us recall these rules in our case. First by Corollary 4.3, we can rewrite the Feynman rule as:
(137) G k = |µ|=2k Γ∈ Gµ 1 | Aut(Γ)| V µ ,
where the sum is over all partitions µ = (µ 1 , · · · , µ l ) of 2k, and G µ ⊂ G k is the set of all graphs with vertices which have valences µ 1 , · · · , µ l respectively, and
V µ := V µ1 V µ2 · · · V µ l .
The partition µ can be represented in terms of a Young diagram which has l rows. Then the action of d V on V µ can be interpreted as adding a box to this Young diagram on the right of one of the l rows. If the new diagram is not a Young diagram any more, then we switch the rows of this diagram to make it a Young diagram. Then we have
d V (V µ ) = ν∈Yµ V ν ,
where Y µ is the set of Young diagrams obtained from µ using the above rule. For example:
→ + + = + 2 .
Therefore in the recursion (136), the term d 2 V G k−1 corresponds to adding two boxes to all Young diagrams corresponding to partitions of 2(k − 1) via the above rule; and the term d V G r−1 · d V G k−r is to add a box to a Young diagram corresponding to a partition of 2(r − 1), and add another box to a Young diagram corresponding to a partition of 2(k − r), and then put one of the resulting diagrams underneath the other and switch the rows if necessary to get a new Young diagram.
4.7.
Relationship to topological 1D gravity. The integral representation (127) of the partition function Z establishes a connection to some earlier work on topological 1D gravity by the second author [56].
By Corollary 4.3 we know that G k is of the form:
G k = |µ|=2k a µ V µ1 V µ2 · · · V µ l ,
where the summations is over all partitions µ = (µ 1 , µ 2 , · · · , µ l ) of 2k. Now let us use another way to represent this partition:
µ = (1 m1 2 m2 · · · (2k) m 2k ),
where m i is the number of i appearing in the sequence (µ 1 , µ 2 , · · · , µ l ). Then G k can be expanded in the following way:
G k = |µ|=2k G µ m 1 ! · · · m 2k ! V m1 1 · · · V m 2k 2k ,
where by (137) the coefficients G µ can be expressed as:
(138) G µ = Γ∈ Gµ 1 | Aut(Γ)| · 2k i=1 m i !.
Given a graph Γ ∈ G µ where µ = (µ 1 , · · · , µ l ) is a partition of 2k, by Euler's formula we know that the genus g of Γ is determined by:
l − 1 2 (µ 1 + · · · + µ l ) = 1 − g, therefore (139) g = k + 1 − l.
Now comparing the graph sum (138) with [56,Proposition 4.4], one easily obtain the following:
Theorem 4.5. Given a partition µ = (µ 1 , µ 2 , · · · , µ l ) = (1 m1 · · · (2k) m 2k ) of 2k, we have:
G µ = τ µ1−1 τ µ2−1 · · · τ µ l −1 1D k+1−l = τ m1 0 τ m2 1 · · · τ mn n−1 1D k+1−l ,(140)
where the right-hand-sides of the above equation are correlators of the topological 1D gravity.
The coefficients G µ will understood as the correlators of the theory defined by the partition function (127), and the equation (139) is equivalent to the selection rule for the topological 1D gravity [56, (119)].
Now we have seen that the study of {G k (V 1 , V 2 , · · · )} are equivalent to the topological 1D gravity (up to a shift of the genus). Then it follows that the results developed in the works [39,40,56] can also be applied to this problem.
For example, similar to the formula [56, (116)], by expanding the formal Gaussian integral in Theorem 4.2 one knows that:
(141) G k = n≥1 (−1) n+1 n n j=1 kj =k n j=1 mj >0 m j i=1 l (j) i =2kj (2k j − 1)!! · V l (j) 1 · · · V l (j) m j m j ! · l (j) 1 ! · · · l (j) mj !
for every k ≥ 1. However, this formula is not convenient for practical computations since the right-hand-side are too complicated. In the following subsections, we will follow [56, §5- §6] and derive the Virasoro constraints (or equivalently, the flow equations and the polymer equation) for the partition function Z.
4.8.
Operator formalism for computing Z. The recursion relations (136) can be solved in an operator formalism for Z:
Theorem 4.6. The partition function Z = exp k≥0 λ 2k G k is given by the following formula:
(142) Z = e 1 2 λ 2d2 V e G0 ,
whered V is an operator defined by:
(143)d V = V 1 ∂ ∂G 0 + ∞ l=1 V l+1 ∂ ∂V l .
Proof. Denote := λ 2 , then the quadratic recursion (136) is equivalent to:
∂ ∂ Z = 1 2d 2 V Z.
Thus we have:
Z = ∞ n=0 n n! ∂ n Z ∂ n =0 = ∞ n=0 n n! · 1 2d 2 V n Z =0 .
Notice that for every k ≥ 0 we always have:
d k V Z =0 =d k V (e G0 )
since Z is of the form Z = exp k≥0 k G k . Therefore
(144) Z = ∞ n=0 n n! · 1 2d 2 V n e G0 = exp 1 2 d 2 V exp G 0 .
One can also derive other solutions in operator formalism for Z using the methods in the study of the topological 1D gravity. Now let us introduce the flow equations and the polymer equation for Z (see [56, §6.1]).
Theorem 4.7. We have the flow equations:
(145) ∂ Z ∂V k = 1 k! ∂ k ∂V k 1 Z, k ≥ 1,
and the polymer equation:
(146) k≥0 V k+1 − λ −2 δ k,1 k! · ∂ k ∂V k 1 Z = 0.
These equations can be proved by directly taking partial derivatives to the partition function (127) and here we omit the details. A straightforward consequence of the flow equations is the following: Corollary 4.4. We have the following expression for Z in operator formalism:
(147) Z = exp ∞ k=2 1 k! V k d k dV k 1 exp V 0 + 1 2 λ 2 V 2 1 .
Proof. Take V 2 = V 3 = · · · = 0 in the partition function (127), we have
Z(V 0 , V 1 ) = 1 √ 2πλ 2 exp − 1 2 λ −2 x 2 + V 0 + V 1 x dx = exp V 0 + 1 2 λ 2 V 2 1 .
Then the conclusion follows from (145).
We now expand the exponentials on the right-hand side of (147) to get:
Z e V0 = exp ∞ k=2 1 k! V k d k dV k 1 exp 1 2 λ 2 V 2 1 = n≥0 k≥2 km k =n k≥2 V m k k m k !(k!) m k · d n dV n 1 m1≥0 λ 2m1 2 m1 m 1 ! V 2m1 1 = n≥0 k≥2 km k =n k≥2 V m k k m k !(k!) m k m1≥0 (2m 1 ) · · · (2m 1 − n + 1)λ 2m1 2 m1 m 1 ! V 2m1−n 1 .
Using selection rule for correlators one can check that this matches with (141).
4.9.
Operator formalism for computing G k (z). In last subsection we have given an operator formulation for computing the partition function Z. In this subsection we present an operator formalism for G k (z). The basic tools we need are the Virasoro constraints for the partition function Z. We will see that the Virasoro constraints in this case enable us to obtain G l from G l−1 directly by applying an operator A.
Combining the flow equations (145) and polymer equation (146), we can get the puncture equation:
V 1 + k≥1 (V k+1 − λ −2 δ k,1 ) ∂ ∂V k Z = 0.
Applying ∂ ∂V1 m+1 and using the flow equations again, we have:
(148) (m + 1) ∂ m ∂V m 1 + k≥0 1 k! (V k+1 − λ −2 δ k,1 ) ∂ m+k+1 ∂V m+k+1 1 Z = 0, or equivalently,(149)(m + 1)! ∂ ∂V m + k≥0 (m + k + 1)! k! (V k+1 − λ −2 δ k,1 ) ∂ ∂V m+k+1 Z = 0.
This gives us the Virasoro constraints (see [56, Theorem 6.1]):
Theorem 4.8. For every m ≥ −1, we have L m Z = 0, where L −1 = V 1 + k≥1 (V k+1 − λ −2 δ k,1 ) ∂ ∂V k ; L 0 = 1 + k≥0 (k + 1)(V k+1 − λ −2 δ k,1 ) ∂ ∂V k+1
;
L m = (m + 1)! ∂ ∂V m + k≥0 (m + k + 1)! k! (V k+1 − λ −2 δ k,1 ) ∂ ∂V m+k+1
, m ≥ 1.
These operators satisfy the Virasoro commutation relation:
[L m , L n ] = (m − n)L m+n , ∀m, n ≥ −1.
This theorem can be checked by a direct computation. We can also write the Virasoro constraints in terms of the free energy log Z = k λ 2k G k . By doing this, we obtain:
Proposition 4.2.
For every l ≥ 2 and m ≥ −1, we have:
(150) (m + 2) ∂ ∂V m+2 G l = k≥0 m + k + 1 k V k+1 ∂ ∂V m+k+1 G l−1 + ∂ ∂V m G l−1 .
In particular, for m = −1, we have
(151) ∂ ∂V 1 G l = d V G l−1 .
Here we use that convention ∂ ∂V−1 := 0. Therefore, for every l ≥ 1 and m ≥ 1, we have
(152) m ∂ ∂V m G l = k≥0 m + k − 1 k V k+1 ∂ ∂V m+k−1 + ∂ ∂V m−2 G l−1 .
Theorem 4.9. Define an operator
(153) A := m≥1 V m k≥0 m + k − 1 k V k+1 ∂ ∂V m+k−1 + ∂ ∂V m−2 ,
where we use that convention ∂ ∂V−1 := 0. Then we have:
(154) G k = 1 2 k · k! A k G 0 , k ≥ 1,
and the free energy of Z is
log Z = k≥0 λ 2k G k = e 1 2 λ 2 A (G 0 ).
Proof. By the homogeneity condition given in Corollary 4.3 and the relation (152), we have:
2k · G k = m≥1 mV m ∂ ∂V m G k = AG k−1 .
Then the conclusion is clear.
Notice that although the operator A in the above theorem is a summation of an infinite numbers of terms, one only needs to evaluate a finite number of terms while computing G l by applying A to G l−1 since G l−1 is a polynomial in {V k } 1≤k≤2l−2 . (136) and (154) we easily see that G k are polynomials in V l , while from (94) we can not see this polynomiality. Now let us summarize the formulas for the orbifold Euler characteristics χ (M g,0 ).
From the definition of G k (z) and {a k g,0 }, one can see:
(155) χ(M g,0 ) = 3g−3 k=0 G k (z) z 2−2g = ∞ k=0 G k (z) z 2−2g ,
where [·] z 2−2g means the coefficient of z 2−2g . Now by taking λ = 1 in Theorem 4.6 and 4.9, we obtain the following formulas for χ(M g,0 ): 2 V e G0(z)
z 2−2g = e 1 2 A G 0 (z) z 2−2g
, whered V and A are given by (143) and (153) respectively.
5.
Results for χ g,n (t, κ) and χ(M g,n )
In §3.3 we have showed that given initial values {a k 0,3 }, {a k 1,1 } and {a k g,0 } (g ≥ 2), all the coefficients {a k g,n } are computed by the linear recursion (61); and the structures of the initial values {a k 0,3 }, {a k 1,1 } and {a k g,0 } (g ≥ 2) have been discussed in §4. Now in the present section, let us solve the linear recursion (61) directly and give explicit formulas for the solutions. This completes our calculations of {a k g,n } and thus of χ g,n (t, κ) and χ(M g,n ). We will first solve the genus zero case, and it turns out that the results in higher genera exhibit similar patterns. 5.1. Explicit expressions for generating series of a k 0,n and χ(M 0,n ). In this subsection, we present an explicit expression for the generating series of χ(M 0,n ) (n ≥ 3) by solving the linear recursion (61) of the coefficients {a k 0,n } at genus zero. Take g = 0. Recall that the refined orbifold Euler characteristic χ 0,n (t, κ) is of the form (Proposition 3.1):
χ 0,n (t, κ) = t 2−n · χ 0,n (κ) = t 2−n · n−3 i=0 a i 0,n κ i , n ≥ 3,
and the orbifold Euler characteristic of M 0,n is given by:
χ(M 0,n ) = n! · χ 0,n (1, 1) = n! · n−3 i=0 a i 0,n .
Now let us consider the following generating series of the coefficients {a i 0,n }:
(158) A i (x) := ∞ n=3 a i 0,n · x n , i ≥ 0,
where we formally denote a i 0,n := 0 for i > n − 3. Then it is not hard to see that the generating series of the orbifold Euler characteristics of M 0,n is given by:
(159) ∞ n=3 χ(M 0,n ) · x n n! = ∞ k=0 A k (x),
and we have
(160) n−3 i=0 a i 0,n = n−3 i=0 A i (x) n ,
where [·] n means taking the coefficient of x n . Our main result in this subsection is the following:
Theorem 5.1. For n ≥ 3, we have:
(161) χ(M 0,n ) = n! · n−3 k=0 A k (x) n ,
where [·] n means the coefficient of x n . The functions A k (x) are given by:
A 0 = 1 2 (1 + x) 2 log(1 + x) − 1 2 x − 3 4 x 2 , A 1 = 1 2 + (1 + x)(log(1 + x) − 1) + 1 2 (1 + x) 2 (log(1 + x) − 1) 2 ,
and for k ≥ 2,
A k = 2 m=2−k (x + 1) m (m + 2)! k−m−1 l=0 log(x + 1) − 1 l l! e l+m (1 − m, 2 − m, · · · , k − m − 1),
where e l are the elementary symmetric polynomials (162) e l (x 1 , x 2 , · · · , x n ) = 1≤j1<···<j l ≤n
x j1 · · · x j l for l ≥ 0, and we use the convention e l := 0 for l < 0.
Before giving a proof to the above theorem, let us first do some explicit calculations of A k for small k. From the Harer-Zagier formula (32) one knows that A 0 (x) is given by:
A 0 (x) = ∞ n=3 1 n! χ(M 0,n ) · x n = ∞ n=3 (−1) n+1 B 0 · (n − 3)! · x n n! = 1 2 (1 + x) 2 log(1 + x) − 1 2 x − 3 4 x 2 .(163)
Now let us derive a recursion relation for {A k (x)}. The linear recursion (63) can be converted into the following:
(164) d dx A k = 2A k − x d dx A k + (x d dx + k − 1)A k−1 .
This is an ordinary differential equation of first order, thus a unique solution A k (x) is determined by this equation together with the initial value A k (0) = 0. Solving this ODE simply gives us:
(165) A k (x) = (1 + x) 2 · x 0 x d dx A k−1 + (k − 1)A k−1 dx (1 + x) 3 .
Then all A k (x) are recursively computed using this recursion and the initial value A 0 (x) given by (163).
In what follows, we will make a tricky change of variable:
(166) x = e s+1 − 1, and the recursion (164) in this new variable s is:
(167) d ds A k = 2A k + (1 − e −s−1 ) d ds + (k − 1) A k−1 ,
or after solving the ODE,
(168) A k (s) = e 2(s+1) · s −1 1 − e −(s+1) d ds A k−1 + (k − 1)A k−1 e −2(s+1) ds.
Here are examples of first a few A k computed using (163) and (168): Now apply the change of variables (166) to the statement in Theorem 5.1. In order to prove that theorem, we only need to prove the following:
A 0 = − 1 4 + e s+1 + e 2(s+1) (− 1 4 + 1 2 s), A 1 = 1 2 + e s+1 s + 1 2 e 2(s+1) s 2 , A 2 = 1 2 (s + 1) + e s+1 (s 2 + s) + e 2(s+1) ( 1 2 s 2 + 1 2 s 3 ), A 3 = 1 6 e −(Proposition 5.1. A k is of the form: (169) A k = e 2(s+1) a k,2 + e s+1 a k,1 + · · · + e −(k−2)(s+1) a k,−(k−2) , k ≥ 1,
where a k,j are polynomials in s:
a k,−m = 1 (m + 2)! k−m−1 l=0 s l l! e l+m (−m + 1, −m + 2, · · · , k − m − 1), k ≥ 2.
Proof. By the uniqueness of the solution, we only need to check that such an A k (s) satisfies the equation (167), and the initial value is A k (s = −1) = 0. First let us check the equation (167). It is equivalent to the following sequence of differential equations in a k,j :
(170) a ′ k,2 = a ′ k−1,2 + (k + 1)a k−1,2 ; a ′ k,1 − a k,1 = a ′ k−1,1 + ka k−1,1 − 2a k−1,2 − a ′ k−1,2 ; · · · · · · a ′ k,−k+3 − (k − 1)a k,−k+3 = a ′ k−1,−k+3 + 2a k−1,j +(k − 2)a k−1,−k+2 − a ′ k−1,−k+2 ; a ′ k,−k+2 − ka k,−k+2 = (k − 3)a k−1,−k+3 − a ′ k−1,−k+3 .
Here the notation a ′ k,j means d ds a k,j . These equations can be checked directly case by case. For example, the first one holds since: a ′ k−1,2 + (k + 1)a k−1,2 = k l=2 s l−1 (l − 1)! e l−2 (3, · · · , k) + (k + 1) k l=2 s l l! e l−2 (3, · · · , k) = k−1 l=2 s l l! e l−1 (3, · · · , k) + (k + 1)e l−2 (3, · · · , k) + s k k! e k−2 (3, · · · , k) + s
= k−1 l=2 s l l! · e l−1 (3, · · · , k + 1) + s k k! e k−2 (3, · · · , k) + s = k+1 l=2 s l−1 (l − 1)! e l−2 (3, · · · , k + 1) = a ′ k,2 .
Here we have used a combinatorial identity: e m+1 (u 1 , · · · , u n ) + u · e m (u 1 , · · · , u n ) = e m+1 (u, u 1 , · · · , u n ).
This identity follows directly from the definition of the elementary symmetric polynomials (162), and we will use this identity quite often throughout the proofs in this section.
Now let us check the second equation in (170). This equation holds because:
a ′ k−1,1 + ka k−1,1 − 2a k−1,2 − a ′ k−1,2 = k−1 l=1 s l−1 (l − 1)! e l−1 (2, 3, · · · , k − 1) + k k−1 l=1 s l l! e l−1 (2, 3, · · · , k − 1) − 2 k l=2 s l l! e l−2 (3, 4, · · · , k) − k l=2 s l−1 (l − 1)! e l−2 (3, 4, · · · , k) = k−2 l=1 s l l! e l (2, · · · , k − 1) + ke l−1 (2, · · · , k − 1) + ks k−1 (k − 1)! e k−2 (2, · · · , k − 1) + 1 − 2s k k! e k−2 (3, · · · , k) − s − k−1 l=2 s l l! 2e l−2 (3, 4, · · · , k) + e l−1 (3, 4, · · · , k) = k−2 l=1 s l l! · e l (2, · · · , k) + ks k−1 (k − 1)! e k−2 (2, · · · , k − 1) + 1 − k−1 l=2 s l l! · e l−1 (2, 3, · · · , k) − 2s k k! e k−2 (3, · · · , k) − s = k−1 l=0 s l l! · e l (2, · · · , k) − k l=1 s l l! · e l−1 (2, 3, · · · , k) = a ′ k,1 − a k,1 .
The rest of the equations in (170) all hold for the same reason. Therefore A k (s) defined by (169)
+ 1 2 · k−1 l=0
(−1) l l! e l (1, 2, · · · , k − 1) + k l=1 (−1) l l! e l−1 (2, 3, · · · , k) + k+1 l=2 (−1) l l! e l−2 (3, 4, · · · , k + 1) = 0.
This is equivalent to
k−2 l=1 l m=1 1 (m + 2)! (−1) l−m (l − m)! e l (−m + 1, −m + 2, · · · , k − m − 1) + 1 2 · k−1 l=0 (−1) l l! e l (1, 2, · · · , k − 1) − k−1 l=0 (−1) l (l + 1)! e l (2, 3, · · · , k) + k−1 l=0
(−1) l (l + 2)! e l (3, 4, · · · , k + 1) = 0.
Since we have
1 2 · (−1) 0 0! − (−1) 0 1! + (−1) 0 2! = 0
for l = 0, and
(−1) k−1 (k + 1)! e k−1 (3, 4, · · · , k + 1) − (−1) k−1 k! e k−1 (2, 3, · · · , k) + 1 2 · (−1) k−1 (k − 1)! e k−1 (1, 2, · · · , k − 1) = (−1) k−1 (k + 1)! (k + 1)! 2! − (−1) k−1 k! · k! + 1 2 · (−1) k−1 (k − 1)! · (k − 1)! = 0
for l = k − 1, it now suffices to show that (−1) l (l + 2)! e l (3, 4, · · · , k + 1) − (−1) l (l + 1)! e l (2, 3, · · · , k) + 1 2 · (−1) l l! e l (1, 2, · · · , k − 1)
0 = l+2 j=0 (−1) j l + 2 j l m=0 k − 1 − m l − m k k − m (−j + 2) l−m = l m=0 l+2 j=0 (−1) j l + 2 j (−j + 2) l−m · k − 1 − m l − m k k − m .(172)
Now it suffices to prove (172). Applying the operator (−x d dx ) l−m to the identity The refined orbifold Euler characteristic χ 1,n (t, κ) is given by:
x −2 (1 − x) l+2 =l+2χ 1,n (t, κ) = t −n · χ 1,n (κ) = t −n · n i=0
a i 1,n κ i , and the orbifold Euler characteristic χ(M 1,n ) is given by:
χ(M 1,n ) = n! · χ 1,n (1, 1) = n! · n k=0
a k 1,n .
Now for g = 1, define the following generating series of {a k 1,n }:
B k (x) := ∞ n=1 a k 1,n x n ,
then the linear recursion (80) gives us the following recursion for B k (x):
d dx (B k − δ k,1 · x 2 ) + x d dx B k = x d dx B k−1 + (k − 1)B k−1 , B k (0) = 0,
and by (32) B 0 (x) is given by:
B 0 (x) = ∞ n=1 1 n! χ(M 1,n ) · x n = − 1 12 log(1 + x).
Recall that all the coefficients {a k 1,n } are determined by
χ 1,1 (κ) = − 1 12 + 1 2 κ,
i.e., by two numbers a 0 1,1 = − 1 12 and a 1 1,1 = 1 2 using (80) Now since the recursion is linear, here we can separate this initial data χ 1,1 into two parts in the following way. First, we replace by the initial data χ 1,1 by 1 = 1 + 0 · κ and run the recursion, and the generating series B k (x) will be replaced by new generating series C k (x); and then we replace by the initial data χ 1,1 by κ = 0 + 1 · κ, and obtain new generating series D k (x) similary. The it is clear that
B k = − 1 12 C k + 1 2 D k , where C k (x) and D k (x) are determined by: d dx C k + x d dx C k = x d dx C k−1 + (k − 1)C k−1 , C 0 (x) = log(1 + x); C k (0) = 0; d dx D k + x d dx D k = x d dx D k−1 + (k − 1)D k−1 , D 0 (x) = 0, D 1 (x) = log(1 + x); D k (0) = 0.(173)
Now it is not hard to see that the generating series of the orbifold Euler characteristics of M 1,n is given by:
(174) ∞ n=1 χ(M 1,n ) · x n n! = ∞ k=0 − 1 12 C k (x) + 1 2 D k (x) .
Our main result in this subsection is the following:
Theorem 5.2. For every n ≥ 1, we have:
χ(M 1,n ) = n! · n k=0 − 1 12 C k (x) + 1 2 D k (x) n ,
where [·] n means the coefficient of x n . Let s := log(x + 1) − 1, then the explicit formulas for C k are given by:
C 0 = s + 1; C k = c k,−k e −k(s+1) + c k,−k+1 e (−k+1)(s+1) + · · · + c k,−1 e −(s+1) + c k,0 , k ≥ 1, where c k,0 = k l=1 s l l! e l−1 (1, 2, · · · , k − 1), k ≥ 1; c k,−m = 1 m! k−m−1 l=0 s l l! e l (−m + 1, −m + 2, · · · , k − m − 1), m > 0, k ≥ m + 1.
And the explicit formulas for D k are given by: D 0 = 0,
D 1 = s + 1; D k = d k,−k+1 e (−k+1)(s+1) + d k,−k+2 e (−k+2)(s+1) + · · · + d k,−1 e −(s+1) + d k,0 , k ≥ 2, where d k,0 = 1 k + ks + k l=2 s l l! k−l+1 j=1
j 2 · e l−2 (j + 1, j + 2, · · · , k − 1) ,
d k,−1 = (k − 1) + k−2 l=1 s l l! k−l j=0 j 2 · e l−1 (j + 1, j + 2, · · · , k − 2) , d k,−m = (−1) m+1 1 m · (k − m) + (−1) m+1 m k−m−1 l=0 s l l! m−1 h=0 (−1) h e h (1, 1 2 , 1 3 , · · · , 1 m − 1 ) × k−l−m−h j=1 j 2 e l−1+h (j + 1, · · · , k − m − 1) , m ≥ 2.
These functions d k,−m can be rewritten in the following way:
d k,−m = (−1) m+1 1 m (k − m) + (−1) m+1 md n,−m (s), m ≥ 2,
where the functionsd n,−m (s) are given by:
d k,−2 = k−3 l=0 s l l! k−l−2 j=1 j 2 e l−1 (j + 1, · · · , k − 3) − k−l−3 j=1 j 2 e l (j + 1, · · · , k − 3) , d k,−m =d k−1,−m+1 − 1 m − 1 d dsd k−1,−m+1 , m ≥ 3.
For example,
d k,−3 = k−4 l=0 s l l! k−l−3 j=1 j 2 e l−1 (j + 1, · · · , k − 4) − k−l−4 j=1
j 2 e l (j + 1, · · · , k − 4)
− 1 2 k−l−4 j=1 j 2 e l (j + 1, · · · , k − 4) − k−l−5 j=1 j 2 e l+1 (j + 1, · · · , k − 4) . d k,−4 = k−5 l=0 s l l! k−l−4 j=1 j 2 e l−1 (j + 1, · · · , k − 5) − k−l−5 j=1 j 2 e l (j + 1, · · · , k − 5) − 1 2 k−l−5 j=1 j 2 e l (j + 1, · · · , k − 5) − k−l−6 j=1 j 2 e l+1 (j + 1, · · · , k − 5) − 1 3 k−l−5 j=1 j 2 e l (j + 1, · · · , k − 5) − k−l−6 j=1 j 2 e l+1 (j + 1, · · · , k − 5) − 1 2 k−l−6 j=1
j 2 e l+1 (j + 1, · · · , k − 5) − k−l−7 j=1 j 2 e l+2 (j + 1, · · · , k − 5) .
We will omit the proof of this theorem for genus one, since it can be proved using the same method as the case of genus zero given in the previous subsection. One only needs to check that C k and D k given in this theorem satisfy the recursions and initial conditions in (173). See also the next subsection where we will present a similar proof in a general case for g ≥ 2.
5.3. Solutions of the linear recursion in general case g ≥ 2. Similar to the cases of genus zero and genus one, in the general case g ≥ 2 one can also solve the linear recursion (61) explicitly to compute the generating series of χ(M g,n ). Let us do this in the present subsection.
For g ≥ 2, recall that χ g,n (t, κ) = t 2−2g−n · χ g,n (κ) and χ g,0 (κ) = a 0 g,0 + a 1 g,0 κ · · · + a 3g−3 g,0 κ 3g−3 . The coefficients a i g,0 (0 ≤ i ≤ 3g − 3) are the initial data for the linear recursion, and the structures of these initial data have been discussed in §4. Similar to the case g = 1, we decompose A g,k (x) := ∞ n=0 a k g,n x n into the following summation:
A g,k (x) = a 0 g,0 A 0 g,k (x) + a 1 g,0 A 1 g,k (x) · · · + a 3g−3 g,0 A 3g−3 g,k (x), where the sequence {A p k (x)} k≥0 is the solution of the linear recursion if we replace the initial data χ g,0 (κ) by κ p (0 ≤ p ≤ 3g − 3). Then from (61) we know that these sequences are determined by:
(175) A p g,0 (x) = · · · = A p g,p−1 (x) = 0; A p g,p (x) = ∞ n=0 (−1) n · (2g−3+n)! n!·(2g−3)! · x n = (1 + x) 2−2g ; A p g,k (0) = 0, k > p; d dx A p g,k + x d dx A p g,k + (2g − 2)A p g,k = x d dx A p g,k−1 + (k − 1)A p g,k−1 .
Recall that the orbifold Euler characteristic of M g,n is given by
χ(M g,n ) = n! · χ g,n (1, 1) = n! · 3g−3+n p=0 a p g,n ,
therefore the generating series of χ(M g,n ) is given by:
(176) ∞ n=0 χ(M g,n ) · x n n! = 3g−3 p=0 a p g,0 ∞ k=0 A p g,k (x).
Our main result in this subsection is the following:
Theorem 5.3. The orbifold Euler characteristic χ(M g,n ) is given by:
χ(M g,n ) = n! · 3g−3+n k=0 a k g,n = n! · 3g−3+n k=0 3g−3 p=0 a p g,0 A p g,k (x) n ,
where [·] n means the coefficient of x n . Let x = e s+1 − 1, then A p g,k (g ≥ 2) are given by:
A p g,0 = · · · = A p g,p−1 = 0; A p g,p = e (2−2g)(s+1) ; A p g,k = a p g,k,−k−2g+2+p e (−k−2g+2+p)(s+1) + · · · + a p g,k,−2g+2 e (−2g+2)(s+1) , k > p,
where for p = 0 we have:
a 0 g,k,−2g+2 = (2 − 2g) k l=1 s l l! e l−1 (−2g + 3, −2g + 4, · · · , −2g + k + 1), a 0 g,k,−2g+2−m = 2 − 2g m! k−m l=0 s l l! e l+m−1 (−2g + 3 − m, · · · , −2g + k + 1 − m), m ≥ 1.
For p = 1, we have:
a 1 g,k,−2g+2−m = 1 m! k−m−1 l=0
s l l! e l+m (−2g + 3 − m, · · · , −2g + k + 1 − m).
For p = 2, we have:
a 2 g,k,−2g+2 = k − 1 + k−2 l=1 s l l! k−l−2g+2 j=−2g+4 j(j + 2g − 3)e l−1 (j + 1, · · · , k − 2g + 1) , a 2 g,k,−2g+2−m = (−1) m 2g − 3 + m m · (k − 1 − m) + (−1) m m! · k−2−m l=0 s l l! m h=0 (−1) h e m−h (2g − 2, · · · , 2g − 3 + m)× k−l−2g+2−m−h j=−2g+4 j(j + 2g − 3)e l−1+h (j + 1, · · · , k − 2g − m + 1) , m ≥ 1.
And for p ≥ 3, we have
a p g,k,−2g+2−m = (−1) m m! k−p−m l=0 s l l! m h=0 (−1) h · e m−h (2g − 2, · · · , 2g − 3 + m) × k−l−2g+1−m−h j=−2g+p+1 j + 2g − 3 p − 2 e l+h (j + 1, · · · , k − 2g − m + 1) ,(177)
or equivalently,
a p g,k,−2g+2−m = 1 m! k−p−m l=0 s l l! −2g+k+1−m−l j=−2g+p+1 j + 2g − 3 p − 2
× e m+l (−2g + 3 − m, · · · , −2g + 2; j + 1, · · · , −2g + k + 1 − m) .
(178)
Proof. We will only prove the case p ≥ 3. The proofs for other cases are all similar. The equivalence of (177) and (178)
(180) d ds A p g,k + (2g − 2)A p g,k = 1 − e −(s+1) d ds A p g,k−1 + (k − 1)A p g,k−1 , or equivalently,(181)
d ds a p g,k,−2g+2 = d ds a p g,k−1,−2g+2 + (k − 2g + 1)a p g,k−1,−2g+2 , d ds a p g,k,−2g+1 − a p g,k,−2g+1 = (2g − 2)a p g,k−1,−2g+2 − d ds a p g,k−1,−2g+2 + d ds a p g,k−1,−2g+1 + (k − 2g)a p g,k−1,−2g+1 , · · · · · · Similar to the equations (170) in the case g = 0, these equations for g ≥ 2 can also be checked case by case using the expression (177), For example, the first equation 6.1. Topological 1D gravity and KP hierarchy. First let us recall a basic result concerning the topological 1D gravity.
The partition function of the topological 1D gravity (with λ = 1) is defined to be ( [56, (93)]):
(187) Z 1D := 1 √ 2π dx exp − 1 2 x 2 + n≥1 t n−1 x n n! ,
where t n are the coupling constants. The relation between the correlators (i.e., coefficients of each term t n1 i1 · · · t n k i k in log Z 1D ) and the coefficients {a k g,n } of the refined orbifold Euler characteristics has been discussed in §4. 7.
It has been shown by Nishigaki and Yoneya that: 40]). Z 1D is a tau-function of the KP hierarchy with respect to the time variables (T 1 , T 2 , · · · ) where:
Theorem 6.1 ( [
(188) T n = t n−1 n! , n ≥ 1.
6.2. χ(M g,0 ) and KP hierarchy. Now the connection of the partition function Z defined by (127) with the topological 1D gravity leads to a connection of χ(M g,0 ) to KP hierarchy.
Recall that although in §4 we have 'pretended' that V k are some formal variables in order to derive recursion relations for the expressions of G k = G k (V 1 , V 2 , · · · ), they are actually generating series of χ(M g,n ) and have definite expressions (see (108)). Now if we take λ = 1 in (127), then we get:
1 √ 2π exp − 1 2 x 2 + n≥0 V n (z) · x n n! dx = exp k≥0 G k (z) = exp g≥2 χ(M g,0 )z 2−2g ,(189)
in other words,
exp g≥2 χ(M g,0 ) − χ(M g,0 ) z 2−2g = 1 √ 2π exp − 1 2 x 2 + n≥1 V n (z) · x n n! dx.(190)
So by Theorem 6.1 we have obtained our main result in this section:
Theorem 6.2. The generating series g≥2 χ(M g,0 ) − χ(M g,0 ) z 2−2g is the logarithm of the tau-function Z 1D of the KP hierarchy, evaluated at time:
(191) T n = 1 n! V n (z), n ≥ 1,
where V n (z) are the generating series of the orbifold Euler characteristics of M g,n :
V n (z) := ∞ g=1 χ(M g,n )z 2−2g−n , n = 1, 2; V n (z) := ∞ g=0 χ(M g,n )z 2−2g−n , n ≥ 3,(192)
whose explicit formulas are given in Lemma 4.2.
6.3. Generalization to χ(M g,n ). The above result can be generalized to a generating series of all χ(M g,n ) with 2g − 2 + n > 0. First let us recall the following formula observed by Bini and Harer [9, (11)]:
Lemma 6.1 ( [9]
). Let y, z be two formal variables, then:
exp 2g−2+n>0 1 n! χ(M g,n )y n z 2−2g = 1 √ 2π exp − 1 2 (x − yz) 2 + n≥0 V n (z) · x n n! dx.(193)
Proof. Let us give a brief proof of this equation by analyzing the Feynman graph expansion. The formal integral in the right-hand side can be expanded as:
1 √ 2π exp − 1 2 (x − yz) 2 + n≥0 V n (z) · x n n! dx = e − 1 2 y 2 z 2 √ 2π exp − 1 2 x 2 + yz · x 1! + 2g−2+n>0 χ(M g,n )z 2−2g−n · x n n! dx = e − 1 2 y 2 z 2 · exp Γw Γ | Aut(Γ)| .
Here the graphs Γ are connected graphs with no external edges, and each vertex of Γ can be one of the following:
1) A stable vertex, i.e., vertex of genus g and valence n with 2g − 2 + n > 0; 2) A vertex of genus 0 and valence 1.
Let the weight w v of a stable vertex of genus g and valence n be χ(M g,n )z 2−2g−n , and the weight w v of a vertex of genus 0 and valence 1 be simply yz, then the weight of a graphw Γ in the above formula is:
(194)w Γ = v: vertex w v .
Now notice that a vertex v of genus 0 and valence 1 can be regarded as an external edge in such graphs, thus we can rewrite the above graph sum as follows: z v (2−2gv −val(v)) · (yz) n · 1 n! χ(M g,n ) = 1 n! · y n z 2−2g χ(M g,n )
for every 2g − 2 + n > 0, where the second equality holds by Euler's formula:
1−(g − v g v ) = |V (Γ)|−|E(Γ)| = |V (Γ)|− 1 2 ( v val(v)−n) = n 2 + v (1− val(v) 2 ).
Thus the conclusion is proved. Now again by Theorem 6.1, we have: Theorem 6.3. The generating series 2g−2+n>0 y n z 2−2g n! · χ(M g,n ) − V 0 (y, z)
of the orbifold Euler characteristics of M g,n is the logarithm of the tau-function Z 1D of the KP hierarchy, evaluated at time:
(195) T n = 1 n! V n (y, z), n ≥ 1.
where V n (y, z) are the following generating series of χ(M g,n ): In this section we discuss a duality between the orbifold characteristic of the moduli spaces M g,n and M g,n . The main results are proved in [49].
In the previous work [48], the authors have introduced the notion of Fourier-like transforms for stable graphs, which are a family of linear transformations on the infinite-dimensional vector space spanned by all stable graphs. Given a number ǫ (in R or C, etc.) and a stable graph Γ, one can construct a 'stable graph Γ ǫ of type ǫ' whose underlying graph is Γ. Roughly speaking, this Γ ǫ stands for a linear combination of stable graph in the usual sense, obtained by suitably gluing some 'vertices of type ǫ' together, where a vertex of type ǫ of genus g and valence n is defined to be the linear combination n! · F g,n (see (13)). The linear map Φ ǫ on the space spanned by all stable graphs which takes Γ to Γ ǫ for every Γ is called a Fourier-like transform. See [48, §6] for this construction, and we will not describe the details here. Now fix a propagator κ. When a Feynman rule of the form (19) is assigned to the stable graphs, the Fourier-like transforms will be realized by a transformation which takes the input data {F g,n } 2g−2+n>0 to the output data { F g,n } 2g−2+n>0 of the realization of abstract QFT where the propagator is taken to be ǫκ: F gv ,valv , 2g − 2 + n > 0.
We have interpreted this procedure as a transformation on the space of 'field theories', see [48, §6] for details. One of the main results of that work is the following duality theorem:
Φ ǫ • Φ −ǫ = id .
As a corollary, taking κ = 1 and ǫ = 1, we know that the inverse of the transformation {F g,n } → { F g,n } given by the graph sum formula Applying this duality theorem to Theorem 3.2, we obtain the following graph sum formula which inverses (33):
Theorem 7.2 ( [49]). Assume 2g − 2 + n > 0, then the orbifold Euler characteristic of M g,n /S n is given by the following graph sum formula: Now comparing this theorem with Theorem 3.2, we see that the graph sum formulas that represents χ(M g,n ) in terms of χ(M g,n ) and represents χ(M g,n ) in terms of χ(M g,n ) is almost the same, and the only difference is an additional factor (−1) in the propagator. This duality between the two types of orbifold Euler characteristics is a new example of the open-closed duality.
Similar to the integral formula (193), we can derive the following formula from the above graph sum formula:
Notice that the quadratic term in x in the exponential on the right-hand side is 1 2 x 2 . This tells us that we need to understand this integral formally as the above summation over graphs where the weight of an internal edge is (−1). Now denotẽ x = −ix, then the above formal integral formula becomes:
exp 2g−2+n>0 1 n! χ(M g,n )y n z 2−2g = i √ 2π exp − 1 2 (x + iyz) 2 + 2g−2+n>0 1 n! χ(M g,n )(ix) n z 2−2g−n dx,(199)
thus by Theorem 6.1 one has the following dual version of Theorem 6.3: is the tau-function Z 1D of the KP hierarchy evaluated at time:
T n = 1 n! δ n,0 2 · y 2 z 2 − δ n,1 · iyz + g≥0 g>1− n 2 χ(M g,n ) · i n z 2−2g−n , n ≥ 1.
Remark 7.1. In [49], we have interpreted the open-closed duality (197) as an analogue of the Möbius inversion formula (see Rota [42] for an introduction). Recall that given a locally finite partially-ordered set P , the zeta function ζ(x, y) on P is defined by:
ζ(x, y) := 1, if x ≤ y; 0, otherwise, and the Möbius function µ is defined to be the inverse of ζ in the incidence algebra.
Here µ is an integer-valued function. Then given a real-valued function f on P , let g be defined by: In the work [49], we presented a similar construction to interpret (197) in the following way. First we introduce a partial ordering on the set of stable graphs using the edge-contraction procedures. Next, we modify the zeta function such that the information about the orders of automorphism groups of stable graphs are encoded in it. We denote byζ this generalized zeta function, and byμ its inverse in the incidence algebra. Thenζ andμ are rational-valued functions on the set of graphs. Moreover, we have an inversion formula for a pair of functions (f, g) which are related to each other via (ζ,μ). In [49], we prove that this generalized Möbius inverse formula becomes the open-closed duality formula (197) if one takes f (Γ) := w Γ where w Γ = v∈V (Γ) χ(M gv ,valv ).
Concluding Remarks
In this work, we have studied the problem of computing the orbifold Euler characteristics using two (mathematical) formalisms inspired by quantum physics. We have obtained the following results: 1) We have introduced the refined orbifold Euler characteristics of M g,n , and derive a quadratic recursion and a linear recursion using the formalism of abstract QFT for stable graphs developed in [47], which enables us to obtain the numerical data completely. 2) We have studied the structures of the refined orbifold Euler characteristics of M g,0 , and showed that this problem is equivalent to the topological 1D gravity. This enables us to solve χ(M g,0 ) using techniques developed in [56] (such as the Virasoro constraints). Moreover, this method leads to a relation between χ(M g,n ) and the KP hierarchy. 3) We have solved the linear recursion and given the explicit formulas for the solutions. This gives a way to represent χ(M g,n ) using the coefficients of the refined orbifold Euler characteristics of M g,0 for fixed g. 4) We have described a version of open-closed duality that represents the orbifold Euler characteristics of M g,n and M g,n in terms of each other via inversion formulas.
There are some more unexpected applications of physics ideas on this geometric problem. In fact, in [57,58] the second author has developed another formalism called the emergent geometry of KP hierarchy, which allows one to construct some geometric structures emerging from a tau-function τ and derive some algorithms to compute the n-point functions associated to τ . Now due to the results in §6, this formalism will enable us to study such a problem in algebraic geometry using techniques from integrable systems. This is what the Witten Conjecture/Kontsevich Theorem [34,50] has inspired us to do. We will report the applications of emergent geometry of KP hierarchy to the computations of χ(M g,n ) in a subsequent work.
Acknowledgements. The authors thank Professor Di Yang for helpful discussions. The second author is partly supported by NSFC grant 11661131005 and 11890662.
A. Tables of Notations
There are a lot of notations appearing in this work. Here we make tables of some notations that have appeared in several different subsections, and list out the locations of their definitions to avoid confusion. Notation Meaning Location F orb g,n (t) weight of vertices (using Harer-Zagier formula) (34) κ weight of an internal edge (a formal variable) (36) χ g,n (t, κ) refined orbifold Euler characteristic (40) χ g,n (κ) χ g,n (κ) := χ g,n (1, κ) (53) Z orb (t, κ) partition function for χ g,0 (t, κ)
on the Abstract QFT and Its Realizations 9 2.1. Edge-cutting and edge-adding operators on stable graphs 9 2.2. Abstract free energies and their recursion relations 11 2.3. Realization of the abstract QFT by Feynman rules 12 2.4. Realization of the operators and recursion relations 13 3. Refined Orbifold Euler Characteristic of M grealization of the abstract quantum field theory 32 4. Structures of χ g,0 (t, κ) 33 4.1. Computations of χ g,0 (t, κ) by quadratic recursions 33 4.2. The generating series of a k g,0 for fixed k 34 4.3. Generating series of χ(M g,n ) in terms of Barnes G-function 35 1. Introduction 1.1. The computations of χ(M g,n )
Example 2. 1 .
1Here we give some examples of these operators: Abstract free energies and their recursion relations. In this subsection we recall the construction of the abstract QFT, including of the definition of the abstract free energies and their recursion relations (see[47, §2.3-2.4]).The following definition is [47, Definition 2.1]:
(M 0,3 ) = 1, χ(M 0,4 ) = 2, χ(M 0,5 ) = 7, χ(M 0,6 ) = 34, χ(M 0,7 ) = 213, χ(M 0,8 ) = 1630, χ(M 0,9 ) = 14747, χ(M 0,10 ) = 153946, · · · · · ·
D 3 χ 0
0,0 := 6χ 0,3 ; Dχ 1,0 := 2χ 1,1 ; χ 1,0 := 0; D j χ 0,0 := 0, j = 0, 1, 2.
[
finite group G acting on X. We also need to make sense of the orbifold motivic class of M g,n . (The work[53] might be useful for this purpose.) Then we understand the orbifold motivic class of M g,n /S M g(v),val(v) ] ,where [M g(v),val(v) ] denotes the orbifold motivic class of M g(v),val(v) (whatever that means).
Example 4. 2 .
2By the Harer-Zagier formula (32), we have:
Lemma 4. 1 .
1Let G 0 (z) be the generating series of χ(M g,0 ) for g ≥
n−1 log Γ(z). The main result of this subsection is that V n (z) can be represented in terms of the Barnes G-functions. The Barnes G-function, also called the double Gamma function, is defined by the equation:
Remark 4. 3 .
3With the notations in
Example 4.3, the genera of the relevant graphs are defined by assigning every vertex to have genus 1, then g( Γ) := h 1 ( Γ) + |V ( Γ)|, where h 1 ( Γ) is the number of independent loops in Γ. By Euler's formula, (133) h 1 ( Γ) = 1 + |E( Γ)| − |V ( Γ)|, and so we have (134) g( Γ) = |E( Γ)| + 1.
Remark 4 . 5 .
45This operator formalism is inspired by the work[2] of Alexandrov on a similar result for the Witten-Kontsevich tau function. See also[55] for the result in the case of r-spin curves.4.10. The orbifold Euler characteristics of M g,0 . Now we have derived three types of recursions (94), (136) and (154) for the generating series G k (z) of the coefficients {a k g,0 }. From
is indeed a solution of the recursive equation(167). Now we only need to check A k (s = −1) = 0, i.e., l+m (−m + 1, −m + 2, · · · , k − m − 1)
us prove (171). Notice that the integers e l (1, 2, · · · , k−1) are the Stirling numbers of the first kind k k − l . We have:
(172), thus A(s = −1) = 0 indeed holds, and we have finished the proof. 5.2. Explicit expressions for generating series of {a k 1,n } and χ(M 1,n ). In this subsection we present explicit expressions for the generating series of {a k 1,n } and χ(1, n) by solving the linear recursion (61) at genus one. The result is similar to the case of genus zero.
Open-Closed Duality for the Orbifold Euler Characteristics of M g,n and M g,n
F
F
gv ,valv .
gv ,valv ), where g v is the genus of a vertex v, val v is the valence of v, and |E(Γ)| is the number of internal edges of Γ.
g,n )x n z 2−2g−n dx.
1.4.1. Recursion relations. Since the orbifold Euler characteristics of M h,m have been explicitly computed in the literatures
The following are the first few examples:χ 3,0 (κ) =
1
1008
−
19
1440
κ +
1307
17280
κ 2 −
2539
10368
κ 3 +
35
72
κ 4 −
55
96
κ 5 +
5
16
κ 6 ,
χ 4,0 (κ) = −
1
1440
+
6221
604800
κ −
17063
241920
κ 2 +
187051
622080
κ 3 −
2235257
2488320
κ 4
+
182341
92160
κ 5 −
66773
20736
κ 6 +
8549
2304
κ 7 −
1045
384
κ 8 +
1105
1152
κ 9 ,
χ 5,0 (κ) =
1
1056
−
181
12096
κ +
32821
290304
κ 2 −
667199
1209600
κ 3 +
114641981
58060800
κ 4
−
578872613
104509440
κ 5 +
374564131
29859840
κ 6 −
229328099
9953280
κ 7 +
2805265
82944
κ 8
−
3182161
82944
κ 9 +
145883
4608
κ 10 −
26015
1536
κ 11 +
565
128
κ 12 ,
is clear. By the uniqueness of solutions to first order ODE's, we only need to check that such A p g,k satisfy and the initial condition A p g,k (s = −1) = 0 for k > p. First let us check the equation (179). It is equivalent to(179)
d
dx
A p
g,k + x
d
dx
A p
g,k + (2g − 2)A p
g,k = x
d
dx
A p
g,k−1 + (k − 1)A p
g,k−1
where Γ 0 is the graph consisting of two vertices of valence one, with an internal edge connecting them. Moreover, using (33) one has:Γw
Γ
| Aut(Γ)|
=w
Γ0
2
+
Γ: connected
stable graphw
Γ
| Aut(Γ)|
=
y 2 z 2
2
+
Γ: connected
stable graphw
Γ
| Aut(Γ)|
,
Γ∈G c
g,nw
Γ =
Γ∈G c
g,n
Theorem 7.1 ( [48]). We have:
Table 1 .
1Notations in the abstract QFT and its realizations the set of connected stable graphs of genus g §2.1 with n external edges G g,n the set of stable graphs of genus g §2.1 with n external edgesV (Γ) the set of vertices of the graph Γ §2.1 E(Γ)the set of internal edges of the graph Γ §2.1 E ext (Γ) the set of external edges of the graph Γ F g,n (t) weight of vertices in a realization (20) κ weight of an internal edge in a realization(21) F g (t, κ) realization of F g (22) F g,n (t, κ) realization of F g,n (23) Z(t, κ)formal Gaussian integral representation (25) ∂ κ realization of K (partial derivative w.r.t κ)Notation Meaning
Location
G c
g,n
§2.1
Aut(Γ)
group of automorphisms of the graph Γ
§2.2
g v
genus of the vertex v
§2
val v
valence of the vertex v
§2
F g
abstract free energy of genus g
(12)
F g,n
abstract n-point function of genus g
(13)
K
edge-cutting operator
§2.2
D, ∂, γ
edge-adding operators
§2.2
(26)
∂
realization of ∂
§2.4
D
realization of D
(27)
Table 2 .
2Notations in computations of χ(M g,n )
holds because: d ds a p g,k−1,−2g+2 + (k − 2g + 1)a p g,k−1,−2g+2p − 2 e l (j + 1, · · · , k − 2g)j + 2g − 3 p − 2 e l+1 (j + 1, · · · , k − 2g)Now it suffices to prove the initial value conditions A p g,k (s = −1) = 0 for k > p, i.e., to prove:This is equivalent to say:To prove the equation (183), we first claim the following combinatorial identity:for k > p ≥ 3 and 0 ≤ r ≤ k − p. This combinatorial identity can be proved by induction on r. First notice that the case r = 0 is trivial for every p ≥ 3 and k > p. Now let us assume that (184) holds for r, and consider the case r + 1. Using the property r+1 m = r m−1 + r m , we have:Notice here2g + 3 − m, · · · , −2g + 2; j + 1, · · · , −2g + k + 1 − (m + 1)) = (1 − k)e r (−2g + 3 − m, · · · , −2g + 2; j + 1, · · · , −2g + k − m), thus (185) equals to:×e r (−2g + 3 − m, · · · , −2g + 2; j + 1, · · · , −2g + k − m) .By the induction hypothesis for (k − 1, p, r), this becomes:which proves the claim (184). Now applying (184) to the left-hand-side of (183), we see that in order to prove (183) we only need to show:for p ≥ 3 and k > p. Using the identity k−1 r = k−2 r + k−2 r−1 , the left-hand-side of this equation becomes:Again using k−2 r = k−3 r + k−3 r−1 , we have:Inductively, we get:thus (183) indeed holds. This completes the proof.6. Generating Series of Orbifold Euler Characteristics of M g,n and M g,n as Tau-Functions of the KP HierarchyIn the §4 we have seen that χ(M g,0 ) are related to the topological 1D gravity. Now in this section let us relate them to the KP hierarchy via the topological 1D gravity, and generalize this result to χ(M g,n ) for 2g − 2 + n > 0. See[12,43]for an introduction to the KP hierarchy.
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|
[] |
[
"DETECTION OF TOOL BASED EDITED IMAGES FROM ERROR LEVEL ANALYSIS AND CONVOLUTIONAL NEURAL NETWORK",
"DETECTION OF TOOL BASED EDITED IMAGES FROM ERROR LEVEL ANALYSIS AND CONVOLUTIONAL NEURAL NETWORK"
] |
[
"EngineerAbhishek Gupta [email protected] \nUniversity of Mumbai Mumbai\nUniversity of Mumbai Mumbai\n\n",
"Raunak Joshi [email protected] \nUniversity of Mumbai Mumbai\nUniversity of Mumbai Mumbai\n\n",
"Assistant ProfessorRonald Laban [email protected] \nUniversity of Mumbai Mumbai\nUniversity of Mumbai Mumbai\n\n",
"Sjcem Palghar \nUniversity of Mumbai Mumbai\nUniversity of Mumbai Mumbai\n\n"
] |
[
"University of Mumbai Mumbai\nUniversity of Mumbai Mumbai\n",
"University of Mumbai Mumbai\nUniversity of Mumbai Mumbai\n",
"University of Mumbai Mumbai\nUniversity of Mumbai Mumbai\n",
"University of Mumbai Mumbai\nUniversity of Mumbai Mumbai\n"
] |
[] |
Image Forgery is a problem of image forensics and its detection can be leveraged using Deep Learning. In this paper we present an approach for identification of authentic and tampered images done using image editing tools with Error Level Analysis and Convolutional Neural Network. The process is performed on CASIA ITDE v2 dataset and trained for 50 and 100 epochs respectively. The respective accuracies of the training and validation sets are represented using graphs.
|
10.48550/arxiv.2204.09075
|
[
"https://arxiv.org/pdf/2204.09075v1.pdf"
] | 248,266,684 |
2204.09075
|
1eb8da749254dca5f05ae7bcbbd69b142ba909d9
|
DETECTION OF TOOL BASED EDITED IMAGES FROM ERROR LEVEL ANALYSIS AND CONVOLUTIONAL NEURAL NETWORK
EngineerAbhishek Gupta [email protected]
University of Mumbai Mumbai
University of Mumbai Mumbai
Raunak Joshi [email protected]
University of Mumbai Mumbai
University of Mumbai Mumbai
Assistant ProfessorRonald Laban [email protected]
University of Mumbai Mumbai
University of Mumbai Mumbai
Sjcem Palghar
University of Mumbai Mumbai
University of Mumbai Mumbai
DETECTION OF TOOL BASED EDITED IMAGES FROM ERROR LEVEL ANALYSIS AND CONVOLUTIONAL NEURAL NETWORK
Convolutional Neural Network · Error Level Analysis · Image Forensics · Image Tampering
Image Forgery is a problem of image forensics and its detection can be leveraged using Deep Learning. In this paper we present an approach for identification of authentic and tampered images done using image editing tools with Error Level Analysis and Convolutional Neural Network. The process is performed on CASIA ITDE v2 dataset and trained for 50 and 100 epochs respectively. The respective accuracies of the training and validation sets are represented using graphs.
Introduction
The field of Deep Learning [1] has subsequently made groundbreaking research in various areas of vision, linguistics and audio but the basis of it can be traced back from the days of Machine Learning [2]. Basically the idea revolves around the prognostication of patterns from the data with differences. Machine Learning was certainly derived from the inferential statistic and later grew to become an independent area of research. The machine learning is derived in primary divisions of classification and regression, where we focus on classification being the problem of this paper. Binary Classification [3] is the precise type of classification that we want to focus on. The binary classification starts with various algorithms in field of machine learning, starting from logistic regression [4] to discriminant analysis [5], support vectors [6], nearest neighbors [7], bagging [8], boosting ensemble [9] and stacking generalization [10] methods, but the problem we focus on is much more in depth. The need of deep learning is evident as many state of the art machine learning algorithms we want to use are going to fall short on performance for our problem. The area of deep learning works with more samples of data and in more detailed fashion, almost replicating the human tendencies. The deep learning works with the concept of artificial neural network that replicates the biological neuron available in human brain. It is capable of deriving patterns with great depth and create quite a difference when compared with the tradition machine learning algorithms. The learning representations in done with the help of forward propagation [11] where the features are learnt by the network with the help of weights and arbitrarily declared biases. These later are given to activation functions for deeper learning and minimizing the load on the network. Later the loss is computed after completion of one forward propagation and it is optimized in the backward propagation [12] process where a loss optimizer is used that computes the gradients of the loss and retraces to weights for better learning. This entire process is considered as one epoch of training. This process is repeated several times till the deeper representation of the patterns from the data is done. The problem we focus on uses computer vision with deep learning, where a specialized neural network specially designed for vision problems is used, known as Convolutional Neural Network [13]. This recognizes patterns from the images considering the number of channels. The process starts by assigning various filters in the learning process that help the network to get greater details of the image. These are termed as the features for the network which are also compressed in dimensions and increased in depth with a fashion to consider only the important features. These features are later flattened and treated as an ordinary neural network to work with the output layer. The problem that we focus on is classification of an image after using editing tools. The editing tools in the area of deep learning [14] can be used but this is about detection of the edited image using deep learning. The similar concepts will be elaborated in further sections of the paper to work with problem of the forged images with editing tools.
arXiv:2204.09075v1 [cs.CV] 19 Apr 2022 2 Methodology
This section of paper focuses on the workflow design and implementation of the problem. The problem revolves around the detection of tampered image with error level analysis given to deep learning model. The dataset is the first and most important portion of the implementation.
Dataset
The dataset we used for this problem is CASIA dataset. The CASIA ground-truth [15] dataset is prominent version that contains 8 categories of images. CASIA ground-truth dataset has an extension that is being used as the tampered images dataset. Known as CASIA ITDE v.2 [16], basically the dataset contains 2 prime classes, authentic images and tampered images. These tampered images are done using photoshop image editing tool.
Error Level Analysis
Image forensics [17] is the branch where Error Level Analysis [18] also abbreviated as ELA is used for identification of the image portions with different compression levels is done. The representation of this can be done for detection of the tampered images using the editing tool. This can practically turn out to be a very good preprocessing step before proceeding with model. The authentic image can be observed below The application of the ELA can be done. It also considers a threshold value that determines the quality of image that highlights the edges. 90% is the quality value used which can be seen in the figure 2.
Network
The network used is a Convolutional Neural Network that has many elements involved. The input layer, hidden layers and output layer are the primary subdivisions of the entire structure. The input images are given to a 2-Dimensional Convolutional Layer which is responsible for feature extraction from the image by using a set of filters. The layer uses 32, 5x5 dimensional set of filters. ReLU [19] is the activation function used which is basically rectified linear unit. Similarly one more convolutional 2-dimensional layer with 32, 5x5 filters are used along with ReLU activation. The parameters learnt by first layer are 2432. The parameters learnt by second layer are 25632. Followed by the 2 convolutional layers, 2-dimensional max pooling [20] layer is used. This specifically removes the unnecessary feature information and reduces the dimension of the activation layer. Parameters learnt from this layer are zero as main motive of the layer is to reduce the dimension. Although the dimensions after reduction keep the adequate amount of information, very subsequent amount of information can be discarded because it affects the bias of the network. Much more information when held in the process of forward propagation can be responsible for poor learning representations. Even in the later stages the network during the backward propagation can run into vanishing gradient [21] problem where calculating the gradients with loss optimizers becomes a daunting task. In such a scenario Dropout [22] layers can be used. They discard a lot of useless information making the networks learn better. Certain amount of information to be dropped from a neuron needs to be specified. This amount is specified in percentages. The threshold we used is 25%, and parameters learnt are zero. Even the dimension of the layers stay the same in this layer. Now we flatten the entire conv layers for further process. Now the process works like an ordinary neural network. The flattened layer is passed to ReLU layer with 256 hidden neurons. This layer learns 29491456 parameters. Again one dropout layer is applied with 50% threshold. In the last layer we use a softmax [23] activation output layer that gives the probabilities of both classes. Since this is a binary classification problem, we could've used sigmoid [24] activation, but we used softmax as we wanted to distinguish between the confidence attained in both the classes after prediction. The backward propagation of the network uses binary cross-entropy [25] as the loss function and Adam [26] as loss optimizer function. The model has been trained for different set of epochs to observe the effect in different ways. One time for 50 epochs and other time for 100 epochs. These yield different observations which we will see in the Results section of this paper.
Results
The result section gives how well the implementation has taken the lead. The model is defined by the 2 metrics, viz. Accuracy and Loss. Accuracy focuses on how well the model has learnt from the data with training set and validation set. Loss focuses on its reduction rate, closer to zero, better the loss. This is also calculated for training and validation. The exact values for the loss over 100 epochs with training and validation sets are 10.27% and 20.50% respectively.
Conclusion
The tampering of images is done using image editing tools and this is a form of image forensics problem which in this paper is solved by leveraging deep learning. The Error Level Analysis technique is applied to every single image in the training data of CASIA that creates a good distinction between authentic and tampered images using editing tools. Later a simple Convolutional Neural Network is used to classify the 2 separate classes. Result section of the paper gives good insights on it and this is a very simple point we tried working with and later can be used to yield more effective applications for which we would be more than glad if this paper is used as a reference material.
Figure 1 :
1Authentic Image
Figure 2 :
2Authentic Image under ELAThe tampered images in similar fashion can be observed, the image and its ELA version.
Figure 3 :
3Tampered ImageThe tampered image under ELA can be seen infigure 4.
Figure 4 :
4Tampered Image under ELAThese images certainly point out the details from the dataset and their respective divisions. The images are used as the data for training the network.
Figure 5 :
5Network used for Implementation
Figure 6 :
6Accuracy for 50 EpochsThe accuracy for 50 epochs of training and validation can be seen infigure 6. The exact numbers for training set is 94.33% and validation set is 92.56%. Similarly the loss values for training and validation set can be seen infigure 7.
Figure 7 :
7Loss for 50 EpochsThe exact values for the loss of training and validation for 50 epochs is 16.31% and 19.63%. Similar way of analysis for 100 epochs can be seen for training and validation accuracy in figure 8.
Figure 8 :
8Accuracy for 100 EpochsThe exact values for the accuracy over 100 epochs with training and validation sets are 96.13% and 92.20% respectively. This same can be done for loss values over 100 epochs and can be seen infigure 9.
Figure 9 :
9Loss for 100 Epochs
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|
[] |
[
"Determining the interspecies interaction strength of a two-species Bose-Einstein condensate from the density profile of one species",
"Determining the interspecies interaction strength of a two-species Bose-Einstein condensate from the density profile of one species"
] |
[
"P Kuopanportti \nDepartment of Physics\nUniversity of Helsinki\nP.O. Box 43FI-00014HelsinkiFinland\n",
"Y M Liu \nDepartment of Physics\nShaoguan University\n512005ShaoguanChina\n",
"Y Z He \nSchool of Physics\nSun Yat-Sen University\n510275GuangzhouChina\n",
"C G Bao \nSchool of Physics\nSun Yat-Sen University\n510275GuangzhouChina\n"
] |
[
"Department of Physics\nUniversity of Helsinki\nP.O. Box 43FI-00014HelsinkiFinland",
"Department of Physics\nShaoguan University\n512005ShaoguanChina",
"School of Physics\nSun Yat-Sen University\n510275GuangzhouChina",
"School of Physics\nSun Yat-Sen University\n510275GuangzhouChina"
] |
[] |
We study harmonically trapped two-species Bose-Einstein condensates within the Gross-Pitaevskii formalism. By invoking the Thomas-Fermi approximation, we derive an analytical solution for the miscible ground state in a particular region of the system's parameter space. This solution furnishes a simple formula for determining the relative strength of the interspecies interaction from a measurement of the density distribution of only one of the two species. Accompanying numerical simulations confirm its accuracy for sufficiently large numbers of condensed particles. The introduced formula provides a condensate-based scheme that complements the typical experimental methods of evaluating interspecies scattering lengths from collisional measurements on thermal samples.
| null |
[
"https://arxiv.org/pdf/1804.11156v1.pdf"
] | 119,536,107 |
1804.11156
|
01443c4cf81fe5431b24c666a6d3405b59893d90
|
Determining the interspecies interaction strength of a two-species Bose-Einstein condensate from the density profile of one species
P Kuopanportti
Department of Physics
University of Helsinki
P.O. Box 43FI-00014HelsinkiFinland
Y M Liu
Department of Physics
Shaoguan University
512005ShaoguanChina
Y Z He
School of Physics
Sun Yat-Sen University
510275GuangzhouChina
C G Bao
School of Physics
Sun Yat-Sen University
510275GuangzhouChina
Determining the interspecies interaction strength of a two-species Bose-Einstein condensate from the density profile of one species
(Dated: May 1, 2018)numbers: 0375Mn6785Fg0375Hh Keywords: Bose-Einstein condensationMulticomponent condensateThomas-Fermi approximation
We study harmonically trapped two-species Bose-Einstein condensates within the Gross-Pitaevskii formalism. By invoking the Thomas-Fermi approximation, we derive an analytical solution for the miscible ground state in a particular region of the system's parameter space. This solution furnishes a simple formula for determining the relative strength of the interspecies interaction from a measurement of the density distribution of only one of the two species. Accompanying numerical simulations confirm its accuracy for sufficiently large numbers of condensed particles. The introduced formula provides a condensate-based scheme that complements the typical experimental methods of evaluating interspecies scattering lengths from collisional measurements on thermal samples.
Introduction-Binary mixtures of Bose-Einstein condensates (BECs) have been extensively studied in recent years, both experimentally and theoretically. In experiments to date, these so-called two-species BECs have been produced by using either two different elements [1][2][3][4][5][6][7][8][9][10], two distinct isotopes of the same element [11][12][13], or a single isotope in two different internal states [14][15][16][17][18][19]. Theoretical studies, in turn, have addressed diverse phenomena such as segregation [20][21][22][23][24] and the associated symmetry breaking [25][26][27][28][29], wetting phase transitions [30], and exotic vortex structures [31][32][33][34][35][36][37][38][39][40], to name but a few.
A key ingredient that gives rise to these phenomena and sets the two-species system apart from the single-species BEC is, quite obviously, the interspecies interaction, which is taken here to be of the zero-range density-density type. It can have a drastic effect on the ground-state density distributions, leading, for example, to segregation of the two condensates when it is strongly repulsive [6,10,11]. In this paper, we demonstrate how the ground-state shapes of the coupled condensates encode crucial information about the interspecies interaction, even when no phase separation occurs, and how the information can be conveniently extracted. Specifically, based on the analytical Thomas-Fermi (TF) formalism, we derive below a simple formula [Eq. (17)] that can be used to determine the relative strength of the interspecies interaction from a measurement of the density distribution of just one of the two miscible condensate species. Gross-Pitaevskii model-As the starting point of our theoretical treatment, let N A bosonic atoms of species A and mass m A and N B bosonic atoms of species B and mass m B be confined and Bose-Einstein condensed in three-dimensional, concentric harmonic traps. Atoms within each species are assumed to interact through repulsive contact interaction of strength c S = 4π 2 a S S /m S > 0, where S ∈ {A, B} and a S S is the s-wave scattering length between atoms of species S . The interspecies contact interaction strength c AB = 2π 2 a AB m −1 A + m −1 B , where a AB is the positive or negative interspecies swave scattering length, is taken to be weak enough for the two species to remain miscible [41]. The concentric harmonic traps are written as V S trap r) = m S (ω 2 S x x 2 + ω 2 S y y 2 + ω 2 S z z 2 /2, where the trap frequencies ω S l , l ∈ {x, y, z}, may all be different. It should be noted, however, that we have assumed the two traps to be co-aligned such that they can both be assigned the same symmetry axes (which we have selected as our Cartesian coordinate axes). For the sake of convenience and notational symmetry, we introduce a mass m and a frequency ω and hereafter use ω and a osc ≡ √ /(mω) as units of energy and length, respectively. Assuming that the temperature is close enough to zero, the ground state of the two-species BEC can be described accurately by the timeindependent coupled Gross-Pitaevskii (GP) equations [42][43][44] for the condensate wave functions φ S , S ∈ {A, B}:
− m 2m S ∇ 2 + 1 2 γ S x x 2 + γ S y y 2 + γ S z z 2 + α S |φ S (r)| 2 + β S |φ / S (r)| 2 − µ S φ S (r) = 0,(1)
where γ S l = m S ω 2 S l /(mω 2 ) is a dimensionless trap frequency, / S is defined such that / A = B and / B = A, the dimensionless coupling constants are
α S = 4πN S ma S S m S a osc ,(2)β S = 2πN S m (m A + m B ) a AB m A m B a osc ,(3)
and µ S are the chemical potentials that enter as Lagrange multipliers enforcing the unit normalizations R 3 |φ 2 S (r)|d 3 r = 1. Since we will only consider flowless ground states, we can assume φ S ∈ R.
Thomas-Fermi solution-We introduce the TF approximation (TFA) [44][45][46], which applies to sufficiently large numbers of condensed atoms and amounts to neglecting the kinetic energy terms. When both φ A and φ B are nonzero, the resulting TF versions of Eqs. (1) can be written as
α S φ 2 S + β S φ 2 / S = µ S − 1 2 γ S x x 2 + γ S y y 2 + γ S z z 2 .(If the determinant D ≡ α A α B − β A β B 0, we obtain φ 2 S = X S − Y S x x 2 − Y S y y 2 − Y S z z 2 ,(5)
where
X S ≡ (α / S µ S − β S µ / S )/D, (6) Y S l ≡ (α / S γ S l − β S γ / S l )/(2D).(7)
We will refer to the formal solution given by Eqs. (5) as Form II. Equations (6) can be solved for the chemical potentials:
µ S = α S X S + β S X / S .(8)
The parameters Y S l defined in Eqs. (7) are known once the input parameters are given, while X S remain unknown because they depend on µ S . If exactly one of the two wave functions, say φ A , is zero in a certain region of R 3 , the formal solution is
φ A = 0, (9a) φ 2 B = 1 α B µ B − 1 2 γ Bx x 2 + γ By y 2 + γ Bz z 2 .(9b)
A solution of this type is referred to as Form I B , where the subscript B indicates the nonvanishing species. Analogously, Form I A can be defined. Together with the vacuum φ A = φ B = 0, Forms I A , I B , and II exhaust all possible types of local TF solutions of Eqs. (1). If one of the wave functions in Form II, say φ A (r), reaches zero as we vary r, we arrive at a boundary surface of Form II (for instance, φ A will reach zero upon increasing x sufficiently if Y Ax > 0). Crossing the boundary will lead to a transformation from Form II to Form I B . It is emphasized that both wave functions are always continuous at the form boundaries; this is because the equations governing the two neighbouring forms become exactly the same for the boundary points. In this way the formal solutions, each with its own specific domain of definition, will be naturally and continuously linked up to form the complete piecewise-defined TF solution over entire R 3 . The complete TF wave functions, however, will not in general be differentiable at the form boundaries. The two unknowns µ S appearing in the entire solution can be obtained from the two additional equations
R 3 φ 2 S d 3 r = 1 for normal- ization.
The parameter space of the two-species model is fairly high-dimensional: even after all the redundancies are removed, one must specify the values of at least nine independent parameters in order to fix all the coefficients in Eqs. (1). Partly for this reason, we will not develop the general TF solution any further in what follows. Instead, for our purposes, it is sufficient to consider a specific type of TF solution satisfying the following assumptions: (i) The isosurfaces of φ 2 A are ellipsoids, and φ 2 A has its maximum at the origin.
(ii) φ 2 B > 0 if φ 2 A > 0. (iii)
The boundary surfaces of the two condensates do not have any points in common. A TF solution satisfying assumptions (i)-(iii) will approximate the ground state of the system in a particular region of the whole parameter space.
Note that these assumptions are different for the two species and hence should be used as the criteria for assigning the two labels A and B.
Due to assumptions (i) and (ii), the solution has Form II at the origin, and it follows from Eqs. (5)
that X A = φ 2 A (r = 0) > 0 and X B = φ 2 B (r = 0) > 0.
Assumption (i) also implies that all the three Y Al > 0. Let us refer to the region in which φ A remains nonzero as the inner region, Ω in ≡ (x, y, z) ∈ R 3 | l Y Al l 2 < X A . We know from assumption (ii) that the solution has Form II in Ω in and from assumption (iii) that φ 2 B remains positive on the boundary ellipsoid ∂Ω in . As we cross ∂Ω in to the outside, the solution acquires Form I B [Eqs. (9)]. Since all the three γ Bl are positive by definition, the isosurfaces of φ 2 B in Eq. (9b) are also ellipsoids, and φ 2 B will reach zero on the ellipsoid (x, y, z) ∈ R 3 | l γ Bl l 2 = 2µ B ≡ S. Because both wave functions vanish outside S, it is the boundary surface of the whole two-species BEC. The region between ∂Ω in and S is referred to as the outer region and denoted by Ω out .
The
normalization 1 = R 3 φ 2 A d 3 r = Ω in φ 2 A d 3 r yields X A = 15 Y Ax Y Ay Y Az 8π 2/5 .(10)
From the normalization 1 = (8) and (10), we obtain
R 3 φ 2 B d 3 r = Ω in ∪Ω out φ 2 B d 3 r and Eqs.µ 5/2 B 15 = l γ 1/2 Bl 16 √ 2π α B + 5 2 β B + l 2α B Y Bl − γ Bl 4Y Al .(11)
We can further use Eqs. (6) and (8) to write down closed analytical expressions for the remaining unknowns
X B = (µ B − β B X A ) /α B and µ A = (DX A + β A µ B ) /α B .
Thus, all the quantities involved in φ A and φ B have now been determined in terms of the model input parameters, and thereby the desired TF solution has been obtained.
In order for the solution to be self-consistent, it must satisfy the assumptions made in its design. Since we must necessarily have
Y Ax > 0, Y Ay > 0, Y Az > 0,(12)
the requirement X A > 0 is immediately satisfied by Eq. (10). By utilizing standard techniques of analytical minimization, we can cast the constraint φ 2 B (r) > 0 ∀r ∈Ω in , where φ 2 B is given by Eq. (5), as the inequality
X B X A > max 0, max l Y Bl Y Al .(13)
Furthermore, assumption (iii) implies that φ 2 B as given by Eq. (9b) must be positive on ∂Ω in , which in turn requires that
µ B > X A 2 max l γ Bl Y Al .(14)
As long as the ten input parameters γ S l , α S , and β S are chosen such that the inequalities (12)- (14) are satisfied, the TF solution derived above is self-consistent. Table I. RMS values of the coordinates x and z for ground-state density distribution of a harmonically trapped, three-dimensional two-species BEC with N B /N A = 2.5, c AB /c A = 1.032, ω Bx /ω Ax = 1.5, ω Az /ω Ax = 2, ω Bz /ω Bx = 1.5, and ω Ay /ω Ax = ω By /ω Bx = 1. The RMS values are given for both the numerically obtained GP solution and the Thomas-Fermi approximation (TFA) derived in the text. The second-to-last column shows the estimate (c AB /c B ) est for the relative interspecies interaction strength, which is obtained by evaluating the right-hand side of Eq. (17) for the numerically obtained φ A . The last column gives the relative error of (c AB /c B ) est . For the TFA, Eq. (17) is exact and yields the true value c AB /c B = 0.8. We have used the values m B /m A = 0.471 and c B /c A = 1.29, which correspond to species A being 87 Rb and species B being 41 K. The first column shows the value of the intraspecies interaction strength for species A, α A = N A c A /( ωa 3 osc ) = 4πN A a AA /a osc , where a osc = √ / (mω), m = m A , and ω = ω Ax . All lengths are given in units of a osc . (If we use ω = 2π × 100 Hz and a AA = 99 a B , we obtain a osc ≈ 1.1 µm and α A ≈ 0.061 × N A .) All the listed states satisfy inequalities (12)-(14), rendering our TF solution self-consistent. Table I, with α A = 10 5 . Panels (a) and (c) show n A = |φ A | 2 , while panels (b) and (d) are for n B = |φ B | 2 . Here a Ax = √ / (m A ω Ax ). Each atomic density is rotationally symmetric about the z axis and is presented here in the plane y = 0. Evaluating the right-hand side of Eq. (17) for the GP solution shown in panels (c) and (d) yields the estimate (c AB /c B ) est = 0.7907, which is 1.13% smaller than the true value 0.8.
x 2 1/2 A x 2 1/2 B z 2 1/2 A z 2 1/2 B (c AB /c B ) est α A Numer
Interaction strength formula-It turns out that information on V AB can be extracted simply by observing the inner cloud, i.e., the density distribution of species A. The mean square value of the atomic coordinate l ∈ {x, y, z} in the species-A cloud is
l 2 A ≡ |φ 2 A |l 2 d 3 r = 15 8π 2/5 Y 1/5 Ax Y 1/5 Ay Y 1/5 Az 7 Y Al .(15)
Furthermore, we define a 3-by-3 matrix δ A with elements
δ A ll ≡ l 2 A / l 2 A = Y Al /Y Al ,(16)
where l, l ∈ {x, y, z}. For the case γ Al /γ Al γ Bl /γ Bl , we can, by using Eqs. (7), rewrite Eq. (16) as
c AB c B = m A m B ω 2 Al − ω 2 Al δ A ll ω 2 Bl − ω 2 Bl δ A ll ω Al ω Al ω Bl ω Bl .(17)
If the atomic masses and trap frequencies are known, Eq. (17) can be used to determine the relative strength of the interspecies interaction by measuring the mean-square values of any two coordinates in the density distribution of species A only. As such, Eq. (17) provides a means to determine c AB that complements the standard methods involving collisional measurements on thermal samples [1,47]. We note that the masses and trap frequencies are typically known to a high accuracy (for example, by measuring the dipole oscillations of the center of mass of the atomic cloud, the relative uncertainty in determining the trap frequency can be as small as 0.001). Consequently, the uncertainties and possible errors in determining c AB /c B via Eq. (17) are likely to arise mainly from the uncertainties in the measurement of δ A ll . In the experiments, it is typically straighforward to vary any of the involved trap frequencies and, in this way, to obtain a large number of individual estimates for c AB /c B at different values of the frequencies. Such data will allow one to assess the consistency between Eq. (17) and the experimental data and to obtain an accurate estimate for c AB /c B with a well-defined uncertainty by averaging over the individual measurements.
Numerical results-The above formulae are generalizations to triaxial configurations of previously obtained expressions for spherically symmetric harmonic traps [48,49]. They are all based on the TFA and will therefore inherit its errors. However, the TFA is known to become more accurate with increasing number of atoms. It is therefore natural to ask how large condensates one would need in order for the approximation error to be negligible. To this end, we perform numerical calculations beyond the TFA to obtain the exact ground-state solutions of the GP equations (1). We further define (c AB /c B ) est as the estimate obtained from Eq. (17) by replacing the TF value of δ A ll with that of the numerical solution. When the relative error of this estimate is negligible, Eq. (17) is applicable for the determination of c AB /c B .
In our numerical calculations, we discretize Eqs. (1) by applying the standard three-point finite-difference stencil and solve the resulting equations iteratively with the successive overrelaxation algorithm. We use coordinate grids with step lengths ≤ 0.05 × √ /m A (ω Ax ) in each direction. To enable simple visualization, we set ω Ax = ω Ay and ω Bx = ω By and limit the simulations to cases where both φ 2 A and φ 2 B are cylindrically symmetric about the z axis. We stress, however, that our analytical treatment also applies to two-species BECs with no cylindrical symmetry.
Table I collects our numerical results for a two-species BEC where the two condensates are coupled through a repulsive interspecies interaction of relative strength c AB /c B = 0.8 and confined in cylindrically symmetric oblate harmonic traps (i.e., ω S z > ω S x = ω S y ). The table entries correspond to different values of α A ∝ N A , while the other system parameters are kept constant as described in the caption of Table I. The analytical TF and the numerical GP solutions for the entry with α A = 10 5 are shown in Fig. 1. For the smallest four values of α A in Table I, the root-mean-square (RMS) values
x 2 A and z 2 A show a noticeable discrepancy between the numerical GP solution and the TFA; consequently, for these states (c AB /c B ) est differs significantly from the true value 0.8. However, when α A increases above 10 4 , the accuracy of the TFA improves, the values of x 2 A and z 2 A computed for the numerical solution approach their TF limits, and (c AB /c B ) est becomes very close to 0.8. For α A = 10 6 , for instance, (c AB /c B ) est has a relative error of −0.239% only. Table II lists the corresponding results for a case where both harmonic traps are prolate (ω S z < ω S x ) and there is a fairly strong interspecies repulsion of c AB /c B = 1.06 [50]. The analytical and numerical solutions are presented in Fig. 2 for α A = 10 4 . Despite the prolate trap with ω Az /ω Ax = 0.8, the density φ 2 A shown in Fig. 2(c) is observed to have a highly oblate profile due to its coupling to species B; this suggests that the shape of the cloud carries a strong signal of the interspecies interaction, and consequently we may expect (c AB /c B ) est to be particularly accurate in this configuration. Indeed, its relative error (c AB /c B ) est c B /c AB − 1 is only −1.13% already at α A = 10 4 and becomes < 10 −4 at α A = 10 7 .
Conclusions-In summary, we have presented an analytical TF solution for a miscible two-species BEC confined in a three-dimensional harmonic trap; we derived a formula, given by Eq. (17), that enables one to determine the relative interspecies strength c AB /c B from the knowledge of the RMS values of two coordinates in the density distribution of species-A atoms. Since Eq. (17) holds only within the TFA, we tested its applicability to the numerical solution of the original GP equations. Although the resulting estimate (c AB /c B ) est was found to be highly inaccurate for small numbers of condensed atoms, Table II, with α A = 10 4 . Applying Eq. (17) to the numerically obtained GP solution in panels (c) and (d) yields the estimate (c AB /c B ) est = 1.0516, which is 0.79% smaller than the exact value 1.06. its relative error became smaller or comparable to typical experimental uncertainties at atom numbers achievable at stateof-the-art experiments. Hence, Eq. (17) may provide a useful way of determinining the interspecies interaction strength, complementary to the usual methods that do not necessarily involve BECs. In particular, it could also be used to crosscheck that the interspecies interactions between condensed atoms are similar to those between noncondensed (but still cold) atoms.
The basic approach of forming piecewise-defined TF solutions of multispecies BECs is obviously quite general [44] and can be applied to many more situations besides the one considered here. In future work, it would be beneficial to derive the detailed TF solutions for the entire parameter space and investigate whether similar convenient relations could be found outside the validity range of the present solution. Such a general solution would also facilitate a detailed study of the ground-state phase diagram of the system. Another possible Table II. RMS values of the coordinates x and z in the ground-state density distribution and the corresponding estimates (c AB /c B ) est for a harmonically trapped, three-dimensional two-species BEC with N B /N A = 20, c AB /c A = 0.8217, ω Bx /ω Ax = 0.65, ω Az /ω Ax = 0.8, ω Bz /ω Bx = 0.6, and ω Ay /ω Ax = ω By /ω Bx = 1. We have used the values m A /m B = 0.471 and c A /c B = 1.29, which correspond to species A being 41 K and species B being 87 Rb. The first column shows the value of the intraspecies interaction strength for species A, α A = N A c A /( ωa 3 osc ) = 4πN A a AA /a osc , where a osc = √ / (mω), m = m A , and ω = ω Ax . (If we use ω = 2π × 100 Hz and a AA = 60 a B , we obtain a osc ≈ 1.57 µm and α A ≈ 0.0254 × N A .) All lengths are given in units of a osc , and the true value for c AB /c B is exactly 1.06. future extension could also be to apply the present approach to mixtures of two spinor condensates [51][52][53][54] instead of scalar ones, or to consider a more general scenario where the two harmonic traps are not required to have pairwise parallel symmetry axes.
Figure 1 .
1(a)-(b) Analytical TF and (c)-(d) numerical GP solutions for the axisymmetric two-species BEC corresponding to the sixth entry in
Figure 2 .
2As Fig. 1, but for the fourth entry in
B (c AB /c B ) estx 2 1/2
A
x 2 1/2
B
z 2 1/2
A
z 2 1/2
α A
Numer.
TFA
Numer.
TFA
Numer.
TFA
Numer.
TFA
Value Error (%)
10 1
1.1053
0.9013
1.0411
1.0061
0.9695
0.3662
1.6973
1.7100
0.7827 −26.163
10 2
1.5755
1.4284
1.6021
1.5946
1.0151
0.5804
2.7023
2.7102
0.9624
−9.203
10 3
2.3155
2.2639
2.5292
2.5272
1.1663
0.9199
4.2918
4.2954
1.0276
−3.053
10 4
3.5987
3.5880
4.0062
4.0054
1.5629
1.4579
6.8064
6.8078
1.0516
−0.789
10 5
5.6867
5.6867
6.3485
6.3481
2.3478
2.3107
10.7891
10.7896
1.0581
−0.178
10 6
9.0117
9.0128
10.0611
10.0610
3.6742
3.6622
17.1002
17.1004
1.0596
−0.037
10 7
14.2838
14.2843
15.9456
15.9456
5.8078
5.8041
27.1022
27.1022
1.0599
−0.007
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"Some Heuristics about Elliptic Curves Some Heuristics about Elliptic Curves SOME HEURISTICS ABOUT ELLIPTIC CURVES",
"Some Heuristics about Elliptic Curves Some Heuristics about Elliptic Curves SOME HEURISTICS ABOUT ELLIPTIC CURVES"
] |
[
"Mark Watkins \nSURFACE SURFACE Mathematics -Faculty Scholarship Mathematics\nSyracuse University Syracuse University\nSyracuse University\n\n",
"Mark Watkins \nSURFACE SURFACE Mathematics -Faculty Scholarship Mathematics\nSyracuse University Syracuse University\nSyracuse University\n\n"
] |
[
"SURFACE SURFACE Mathematics -Faculty Scholarship Mathematics\nSyracuse University Syracuse University\nSyracuse University\n",
"SURFACE SURFACE Mathematics -Faculty Scholarship Mathematics\nSyracuse University Syracuse University\nSyracuse University\n"
] |
[] |
We give some heuristics for counting elliptic curves with certain properties. In particular, we re-derive the Brumer-McGuinness heuristic for the number of curves with positive/negative discriminant up to X, which is an application of lattice-point counting. We then introduce heuristics (with refinements from random matrix theory) that allow us to predict how often we expect an elliptic curve E with even parity to have L(E, 1) = 0. We find that we expect there to be about c 1 X 19/24 (log X) 3/8 curves with |∆| < X with even parity and positive (analytic) rank; since Brumer and McGuinness predict cX 5/6 total curves, this implies that asymptotically almost all even parity curves have rank 0. We then derive similar estimates for ordering by conductor, and conclude by giving various data regarding our heuristics and related questions.
|
10.1080/10586458.2008.10129019
|
[
"https://surface.syr.edu/cgi/viewcontent.cgi?article=1104&context=mat&httpsredir=1&referer="
] | 9,182,723 |
math/0608766
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9c2c43dd7b59823e5c1898b09e2dc2b40e13d452
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Some Heuristics about Elliptic Curves Some Heuristics about Elliptic Curves SOME HEURISTICS ABOUT ELLIPTIC CURVES
8-30-2006 30 Aug 2006
Mark Watkins
SURFACE SURFACE Mathematics -Faculty Scholarship Mathematics
Syracuse University Syracuse University
Syracuse University
Mark Watkins
SURFACE SURFACE Mathematics -Faculty Scholarship Mathematics
Syracuse University Syracuse University
Syracuse University
Some Heuristics about Elliptic Curves Some Heuristics about Elliptic Curves SOME HEURISTICS ABOUT ELLIPTIC CURVES
8-30-2006 30 Aug 2006Follow this and additional works at: https://surface.syr.edu/mat Part of the Mathematics Commons Recommended Citation Recommended Citation Watkins, Mark, "Some Heuristics about Elliptic Curves" (2006). Mathematics -Faculty Scholarship. 110. https://surface.syr.edu/mat/110 This Article is brought to you for free and open access by the Mathematics at SURFACE. It has been accepted for inclusion in Mathematics -Faculty Scholarship by an authorized administrator of SURFACE. For more information, please contact [email protected]. arXiv:math/0608766v2 [math.NT]
We give some heuristics for counting elliptic curves with certain properties. In particular, we re-derive the Brumer-McGuinness heuristic for the number of curves with positive/negative discriminant up to X, which is an application of lattice-point counting. We then introduce heuristics (with refinements from random matrix theory) that allow us to predict how often we expect an elliptic curve E with even parity to have L(E, 1) = 0. We find that we expect there to be about c 1 X 19/24 (log X) 3/8 curves with |∆| < X with even parity and positive (analytic) rank; since Brumer and McGuinness predict cX 5/6 total curves, this implies that asymptotically almost all even parity curves have rank 0. We then derive similar estimates for ordering by conductor, and conclude by giving various data regarding our heuristics and related questions.
Introduction
We give some heuristics for counting elliptic curves with certain properties. In particular, we re-derive the Brumer-McGuinness heuristic for the number of curves with positive/negative discriminant up to X, which is an application of lattice-point counting. We then introduce heuristics (with refinements from random matrix theory) that allow us to predict how often we expect an elliptic curve E with even parity to have L(E, 1) = 0. It turns out that we roughly expect that a curve with even parity has L(E, 1) = 0 with probability proportional to the square root of its real period, and, since the real period is very roughly 1/∆ 1/12 , this leads us to the prediction that almost all curves with even parity should have L(E, 1) = 0. By the conjecture of Birch and Swinnerton-Dyer, this says that almost all such curves have rank 0.
We then make similar heuristics when enumerating by conductor. The first task here is simply to count curves with conductor up to X, and for this we use heuristics involving how often large powers of primes divide the discriminant. Upon making this estimate, we are then able to imitate the argument we made previously, and thus derive an asymptotic for the number of curves with even parity and L(E, 1) = 0 under the ordering by conductor. We again get the heuristic that almost all curves with even parity should have L(E, 1) = 0.
We then make a few remarks regarding how often curves should have nontrivial isogenies and/or torsion under different orderings, and then present some data regarding average ranks. We conclude by giving data for Mordell-Weil lattice distribution for rank 2 curves, and speculating about symmetric power L-functions.
The Brumer-McGuinness Heuristic
First we re-derive the Brumer-McGuinness heuristic [3] for the number of elliptic curves whose absolute discriminant is less than a given bound X; the technique here is essentially lattice-point counting, and we derive our estimates via the assumption that these counts are well-approximated by the areas of the given regions.
Conjecture 2.1. [Brumer-McGuinness] The number A ± (X) of rational elliptic curves with a global minimal model (including at ∞) and positive or negative discriminant whose absolute value is less than X is asymptotically A ± (X) ∼ α± ζ(10) X 5/6 , where α ± = √ 3 10 ∞ ±1
dx √
x 3 ∓1 .
As indicated by Brumer and McGuinness, the identity α − = √ 3α + was already known to Legendre, and is related to complex multiplication. These constants can be expressed in terms of Beta integrals B(u, v) (u+v) as α + = 1 3 B(1/2, 1/6) and α − = B(1/2, 1/3). Recall that every rational elliptic curve has a unique integral minimal model y 2 + a 1 xy + a 3 y = x 3 + a 2 x 2 + a 4 x + a 6 with a 1 , a 3 ∈ {0, 1} and |a 2 | ≤ 1. Fix one of the 12 choices of (a 1 , a 2 , a 3 ). Since these are all bounded by 1 the discriminant is thus approximately −64a 3 4 − 432a 2 6 . So we wish to count the number of (a 4 , a 6 )lattice-points with |64a 3 4 + 432a 2 6 | ≤ X, noting that Brumer and McGuinness divide the curves according to the sign of the discriminant. The lattice-point count for a 1 = a 2 = a 3 = 0 is given by 0<−64a 3 4 −432a 2 6 <X 1.
= 1 0 x u−1 (1 − x) v−1 dx = Γ(u)Γ(v) Γ
We estimate this lattice-point count by the integral U du 4 du 6 for the region U given by |64u 3 4 + 432u 2 6 | < X. After splitting into two parts based upon the sign of the discriminant and performing the u 4 -integration, we get where the factor of 2 comes from the sign of u 6 . Changing variables u 6 = w X/432 and multiplying by 12 for the number of (a 1 , a 2 , a 3 )-choices we get 24 (64) 1/3 X 5/6 √ 432 ∞ 0 (w 2 + 1) 1/3 − (w 2 ) 1/3 dw+ + 24 (64) 1/3
X 5/6 √ 432 ∞ 0 (w 2 ) 1/3 − (w 2 − 1) 1/3 dw.
These integrals are probably known, but I am unable to find a reference. The first integral simplifies 1 to 3
5 ∞ 1
dx √ x 3 −1 = 1 5 B(1/2, 1/6), while the second becomes
3 5 ∞ −1 dx √ x 3 +1 = 3 5 B(1/2, 1/3).
This counts all models of curves; if we eliminate nonminimal models, for which we have p 4 |c 4 and p 12 |∆ for some prime p, we expect to accrue an extra factor 2 of ζ (10). From this, we get the conjecture of Brumer and McGuinness as stated above. 1 As N. D. Elkies indicated to us, we can write I(a) = ∞ 0 (t 2 + a) 1/3 − (t 2 ) 1/3 dt, differentiate under the integral sign, then substitute t 2 + a = ax 3 , and finally re-integrate to obtain I(1). 2 Note that some choices of (a 1 , a 2 , a 3 ) necessarily have odd discriminant, but the other choices compensate to give the proper Euler factors at 2 (and 3).
3.
Counting curves of even parity whose central L-value vanishes.
See [28, §15-16] for definitions of the conductor N and L-function L(E, s) of an elliptic curve E. Since rational elliptic curves are modular, we have that the completed L-function Λ(E, s) = Γ(s)( √ N/2π) s L(E, s) satisfies Λ(E, s) = ±Λ(E, 2−s). When the plus sign occurs, we say that E has even parity.
We now try to count elliptic curves E with even parity for which L(E, 1) = 0. Throughout this section, E shall be a curve with even parity, and we shall order curves by discriminant. Via the conjectural Parity Principle, we expect that, under any reasonable ordering, half of the elliptic curves should have even parity; in particular, we predict that there are asymptotically A ± (X)/2 curves with even parity and positive/negative discriminant up to X.
Our main tool shall be random matrix theory, which gives a heuristic for predicting how often L(E, 1) is small. We could alternatively derive a cruder heuristic by assuming the the order of the Shafarevich-Tate group is a random square integer in a given interval, but random matrix theory has the advantage of being able to predict a more explicit asymptotic. Our principal heuristic is the following:
Heuristic 3.1. The number R(X) of rational elliptic curves E with even parity and L(E, 1) = 0 and absolute discriminant less than X is given asymptotically by R(X) ∼ cX 19/24 (log X) 3/8 for some computable constant c > 0.
In particular, note that we get the prediction that almost all curves with even parity have L(E, 1) = 0 under this ordering.
3.1. Random matrix theory. Originally developed in mathematical statistics by Wishart [34] in the 1920s and then in mathematical physics (especially the spectra of highly excited nuclei) by Wigner [33], Dyson, Mehta, and others (particularly [21]), random matrix theory [23] has now found some applications in number theory, the earliest being the oft-told story of Dyson's remark to Montgomery regarding the pair-correlation of zeros of the Riemann ζ-function. Based on substantial numerical evidence, random matrix theory appears to give reasonable models for the distribution of L-values in families, though the issue of what constitutes a proper family is a delicate one (see particularly [6, §3], where the notion of family comes from the ability to calculate moments of L-functions rather than from algebraic geometry).
The family of quadratic twists of a given elliptic curve E : y 2 = x 3 + Ax + B is given by E d : y 2 = x 3 + Ad 2 x+ Bd 3 for squarefree d. The work (most significantly a monodromy computation) of Katz and Sarnak [17] regarding families of curves over function fields implies that when we restrict to quadratic twists with even parity, we should expect that the L-functions are modelled by random matrices with even orthogonal symmetry. Though we have no function field analogue in our case, we brazenly assume (largely from looking at the sign in the functional equation) that the symmetry type is again orthogonal with even parity. What this means is that we want to model properties of the L-function via random matrices taken from SO(2M ) with respect to Haar measure. Here we wish the mean density of zeros of the L-functions to match the mean density of eigenvalues of our matrices, and so, as in [18], we should take 2M ≈ 2 log N . We suspect that the L-value distribution is approximately given by the distribution of the evaluations at 1 of the characteristic polynomials of our random matrices. In the large, this distribution is determined entirely by the symmetry type, while finer considerations are distinguished via arithmetic considerations.
With this assumption, via the moment conjectures of [18] and then using Mellin inversion, as t → 0 we have (see (21) of [7]) that
(1) Prob[L(E, 1) ≤ t] ∼ α(E)t 1/2 M 3/8 .
This heuristic is stated for fixed M ≈ log N , but we shall also allow M → ∞. It is not easy to understand this probability, as both the constant α(E) and the matrix-size M depend on E. We can take curves with e M ≤ N ≤ e M+1 to mollify the impact of the conductor, but in order to average over a set of curves, we need to understand how α(E) varies. One idea is that α(E) separates into two parts, one of which depends on local structure (Frobenius traces) of the curve, and the other of which depends only upon the size of the conductor N . Letting G be the Barnes G-function (such that G(z + 1) = Γ(z)G(z) with G(1) = 1) and M = ⌊log N ⌋ we have that
α(E) = α R (M ) · α A (E) with α R (M ) →α R = 2 1/8 G(1/2)π −1/4 as M → ∞ and (2) α A (E) = p F (p) = p 1 − 1 p 3/8 p p + 1 1 p + L p (1/p) −1/2 2 + L p (−1/p) −1/2 2
where L p (X) = (1−a p X +pX 2 ) −1 when p ∤ ∆ and L p (X) = (1−a p X) −1 otherwise; see (10) of [7] evaluated at k = −1/2, though that equation is wrong at primes that divide the discriminant -see (20) of [8], where Q should be taken to be 1. Note that the Sato-Tate conjecture [31] implies that a 2 p is p on average, and this implies that the above Euler product converges.
3.2.
Discretisation of the L-value distribution. For precise definitions of the Tamagawa numbers, torsion group, periods, and Shafarevich-Tate group, see [28], though below we give a brief description of some of these. We let τ p (E) be the Tamagawa number of E at the (possibly infinite) prime p, and write τ (E) = p τ p (E) for the Tamagawa product and T (E) for the size of the torsion group. We also write Ω re (E) for the real period and X an (E) for the size of the Shafarevich-Tate group when L(E, 1) = 0, with X an (E) = 0 when L(E, 1) = 0.
We wish to assert that sufficiently small values of L(E, 1) actually correspond to L(E, 1) = 0. We do this via the conjectural formula of Birch and Swinnerton-Dyer [1], which asserts that
L(E, 1) = Ω re (E) · τ (E) T (E) 2 · X an (E).
Our discretisation 3 will be that
L(E, 1) < Ω re (E) · τ (E) T (E) 2 implies L(E, 1) = 0.
Note that we are only using that X an takes on integral values, and do not use the (conjectural) fact that it is square.
Using (1), we estimate the number of curves with positive (for simplicity) discriminant less than X and even parity and L(E, 1) = 0 via the lattice-point sum
W (X) = c 4 , c 6 minimal 0<c 3 4 −c 2 6 <1728X α R (M )α A (E) · Ω re (E)τ (E) T (E) 2 · M 3/8 .
We need to introduce congruence conditions on c 4 and c 6 to make sure that they correspond to a minimal model of an elliptic curve. The paper [30] uses the work of Connell [5] in a different context to get that there are 288 classes of (c 4 mod 576, c 6 mod 1728) that can give minimal models, and so we get a factor of 288/(576 · 1728), assuming that each congruence class has the same impact on all the entities in the sum. Indeed, this independence (on average) of various quantities with respect to c 4 and c 6 is critical in our estimation of W (X). There is also the question of non-minimal models, 4 from which we get a factor of 1/ζ(10).
Guess 3.2. The lattice-point sum W (X) can be approximated as X → ∞ bŷ
W (X) = 288 (576 · 1728) 1 ζ(10) ·α RᾱA β τ · 1≤ u 3 4 −u 2 6 1728 <X Ω re (E) 1/2 · (log ∆) 3/8 du 4 du 6 .
Hereα R is the limit 2 1/8 G(1/2)π −1/4 of α R (M ) as M → ∞, whileᾱ A is a suitable average of the arithmetic factors α A (E), and β τ is the average of the square root of the Tamagawa product. We have also approximated log N ≈ log ∆ and assumed the torsion is trivial; below we will give these heuristic justification (on average). Note that everything left in the integral is a smooth function of u 4 and u 6 .
We shall first evaluate the integral inŴ (X) given these suppositions, and then try to justify the various assumptions that are inherent in this guess. 5 3.3. Evaluation of the integral. Write E as y 2 = 4x 3 − (c 4 /12)x − c 6 /216, and put e 1 > e 2 > e 3 for the roots of the cubic polynomial on the right side. We have
1/Ω re = agm √ e 1 − e 2 , √ e 1 − e 3 /π.
We also have that (e 1 − e 2 )(e 1 − e 3 )(e 2 − e 3 ) = ∆/16 from the formula for the discriminant. We next write e 1 − e 2 = ∆ 1/6 λ and e 2 − e 3 = ∆ 1/6 µ so that we have µλ(λ+µ) = 1/4, while e 1 = ∆ 1/6 3 (µ+2λ), e 2 = ∆ 1/6 3 (µ−λ), and e 3 = − ∆ 1/6 3 (2µ+λ). Thus we get
−c 6 /864 = −e 1 e 2 e 3 = ∆ 1/2 27 (µ + 2λ)(µ − λ)(2µ + λ)
and −c 4 /48 = e 1 e 2 + e 1 e 3 + e 2 e 3 = −
∆ 1/3 3 (µ 2 + λµ + λ 2 ).
Changing variables in theŴ -integral gives a Jacobian of 432/∆ 1/6 µ 4 + µ so that
W (X) =c X 1 ∞ 0 (log ∆) 3/8 ∆ 1/12 agm( √ λ, √ λ + µ) dµ d∆ ∆ 1/6 µ 4 + µ ,
where λ = ( µ 4 + µ − µ 2 )/2µ. Thus the variables are nicely separated, and since the µ-integral converges, we do indeed getŴ (X) ∼ cX 19/24 (log X) 3/8 . A similar argument can be given for curves with negative discriminant. This concludes our derivation of Heuristic 3.1, and now we turn to giving some reasons for our expectation that the arithmetic factors can be mollified by taking their averages.
3.4.
Expectations for arithmetic factors on average. In the next section we shall explain (among other things) why we expect that log N ≈ log ∆ for almost all curves, and in section 5, we shall recall the classical parametrisations of X 1 (N ) due to Fricke to indicate why we expect the torsion size is 1 on average. Here we show how to compute the various averages (with respect to ordering by discriminant) of the square root of the Tamagawa product and the arithmetic factors α A (E).
For both heuristics, we shall make the assumption that curves satisfying the discriminant bound |∆| ≤ X behave essentially the same as those that satisfy |c 4 | ≤ X 1/3 and |c 6 | ≤ X 1/2 . That is, we approximate our region by a big box. We write D for the absolute value of ∆. First we consider the Tamagawa product.
We wish to know how often a prime divides the discriminant to a high power. Fix a prime p ≥ 5 with p a lot smaller than X 1/3 . We can estimate the probability that p k |∆ by considering all p 2k choices of c 4 and c 6 modulo p k , that is, by counting the number of solutions
S(p k ) to c 3 4 − c 2 6 = 1728∆ ≡ 0 (mod p k ). This auxiliary curve c 3 4 = c 2 6 is singular at (0, 0) over F p , and has (p − 1) non-singular F p -solutions which lift to p k−1 (p − 1) points modulo p k .
For p k sufficiently small, our (c 4 , c 6 )-region is so large that we can show that the probability that p k |∆ is S(p k )/p 2k . We assume that big primes act (on average) in the same manner, while a similar heuristic can be given for p = 2, 3. Curves with p 4 |c 4 and p 6 |c 6 will not be given by their minimal model; indeed, we want to exclude these curves, and thus will multiply our probabilities by κ p = (1 − 1/p 10 ) −1 to make them conditional on this criterion. For instance, the above counting of points says that there is a probability of (p 2 − p)/p 2 that p ∤ D, and so upon conditioning upon minimal models we get κ p (1 − 1/p) for this probability.
What is the probability P m (p, k) that a curve given by a minimal model has multiplicative reduction at p ≥ 5 and p k D for some k > 0? In terms of Kodaira symbols, this is the case of I k . For multiplicative reduction we need that p ∤ c 4 , c 6 . These events are independent and each has a probability (1 − 1/p) of occurring. Upon assuming these conditions and working modulo p k , there are (p k − p k−1 ) such choices for each, and of the resulting (c 4 , c 6 ) pairs we noted above that p k−1 (p − 1) of them have p k |D. So, given a curve with p ∤ c 4 , c 6 , we have a probability of 1/p k−1 (p−1) that p k |D, which gives 1/p k for the probability that p k D. In symbols, we have that (for p ≥ 5 and k ≥ 1)
Prob p k (c 3 4 − c 2 6 ) p ∤ c 4 , c 6 = 1/p k .
Including the conditional probability for minimal models, we get
(3) P m (p, k) = (1 − 1/p 10 ) −1 (1 − 1/p) 2 /p k (for p ≥ 5 and k ≥ 1).
Note that summing this over k ≥ 1 gives κ p (1 − 1/p)/p for the probability for an elliptic curve to have multiplicative reduction at p. What is the probability P a (p, k) that a curve given by a minimal model has additive reduction at p ≥ 5 and p k D for some k > 0? We shall temporarily ignore the factor of κ p = (1 − 1/p 10 ) −1 from non-minimal models and include it at the end. We must have that p|c 4 , c 6 , and thus get that k ≥ 2. For k = 2, 3, 4, which correspond to Kodaira symbols II, III, and IV respectively, the computation is not too bad: we get that p 2 D exactly when p|c 4 and p c 6 , so that the probability is (1/p) · (1 − 1/p)/p = (1 − 1/p)/p 2 ; for p 3 D we need p c 4 and p 2 |c 6 and thus get (1 − 1/p)/p · (1/p 2 ) = (1 − 1/p)/p 3 for the probability; and for p 4 D we need p 2 |c 4 and p 2 c 6 and so get (1/p 2 ) · (1 − 1/p)/p 2 = (1 − 1/p)/p 4 for the probability. Note that the case k = 5 cannot occur. Thus we have (for p ≥ 5) the formula P a (p, k)
= (1 − 1/p 10 ) −1 (1 − 1/p)/p k for k = 2, 3, 4.
More complications occur for k ≥ 6, where now we split into two cases depending upon whether additive reduction persists upon taking the quadratic twist by p. This occurs when p 3 |c 4 and p 4 |c 6 , and we denote by P n a (p, k) the probability that p k D in this subcase. Just as above, we get that P n a (p, k) 9,10. These are respectively the cases of Kodaira symbols IV ⋆ , III ⋆ , and II ⋆ . For k = 11 we have P n a (p, k) = 0, while for k ≥ 12 our condition of minimality implies that we should take P n a (p, k) = 0. We denote by P t a (p, k) the probability that p 6 |D with either p 2 c 4 or p 3 c 6 . First we consider curves for which p 7 |D, and these have multiplicative reduction at p upon twisting. In particular, these curves have p 2 c 4 and p 3 c 6 , and the probability of this is (
= (1 − 1/p 10 ) −1 (1 − 1/p)/p k−1 for k = 8,1 − 1/p)/p 2 · (1 − 1/p)/p 3 . Consider k ≥ 7.
We then take c 4 /p 2 and c 6 /p 3 both modulo p k−6 , and get that p k−6 (D/p 6 ) with probability 1/p k−6 in analogy with the above. So we get that P t a (p, k)
= (1 − 1/p 10 ) −1 (1 − 1/p) 2 /p k−1 for k ≥ 7.
This corresponds to the case of I ⋆ k−6 . Finally, for p 6 D (which is the case I ⋆ 0 ) we get a probability of (1/p 2 ) · (1/p 3 ) for the chance that p 2 |c 4 and p 3 |c 6 , and (since there are p points mod p on the auxiliary curve (c 4 /p 2 ) 3 ≡ (c 6 /p 3 ) 2 (mod p)) a conditional probability of (p 2 −p)/p 2 that p 6 D. So we get that P t a (p, 6) = (1 − 1/p 10 ) −1 (1 − 1/p)/p 5 . We now impose our current notation on the previous paragraphs, and naturally let P t a (p, k) = 0 and P n a (p, k) = P a (p, k) for k ≤ 5. Our final result is that
(4) P n a (p, k) = (1 − 1/p 10 ) −1 (1 − 1/p)/p k k = 2, 3, 4 (1 − 1/p 10 ) −1 (1 − 1/p)/p k−1 k = 8, 9, 10 (5) P t a (p, k) = (1 − 1/p 10 ) −1 (1 − 1/p)/p 5 k = 6 (1 − 1/p 10 ) −1 (1 − 1/p) 2 /p k−1 k ≥ 7
with P n a (p, k) and P t a (p, k) equal to zero for other k. We conclude by defining P 0 (p, k) to be zero for k > 0 and to be the
probability (1 − 1/p 10 ) −1 (1 − 1/p) that p ∤ D for k = 0.
We can easily check that we really do have the required probability relation ∞ k=0 P m (p, k) + P n a (p, k) + P t a (p, k) + P 0 (p, k) = 1, as: the cases of multiplicative reduction give κ p (1 − 1/p)/p; the cases of Kodaira symbols II, III, and IV give κ p (1/p 2 − 1/p 5 ); the cases of Kodaira symbols IV ⋆ , III ⋆ , and II ⋆ give κ p (1/p 7 − 1/p 10 ); the cases of I ⋆ k summed for k ≥ 1 give κ p (1 − 1/p)/p 6 ; the case of I ⋆ 0 gives κ p (1 − 1/p)/p 5 ; and the sum of these with P 0 (p, 0) = κ p (1 − 1/p) does indeed give us 1. We could do a similar (more tedious) analysis for p = 2, 3, but this would obscure our argument.
Given a curve of discriminant D, we can now compute the expectation for its Tamagawa number. We consider primes p|D with p ≥ 5, and compute the local Tamagawa number t(p). When E has multiplicative reduction at p and p k D, then t(p) = k if −c 6 is square mod p, and else t(p) = 1, 2 depending upon whether k is odd or even. So the average of t(p) for this case is ǫ m
(k) = 1 2 (1+ √ k), 1 2 ( √ 2+ √ k) for k odd/even respectively.
When E has potentially multiplicative reduction at p with p k D, for k odd we have t(p) = 4, 2 depending on whether (c 6 /p 3 ) · (∆/p k ) is square mod p, and for k even we have t(p) = 4, 2 depending on whether ∆/p k is square mod p. In both cases the average of t(p) is 1 2 ( √ 2 + √ 4). In the case of I ⋆ 0 reduction where we have p 6 D, we have that t(p) = 1, 2, 4 corresponding to whether the cubic
x 3 − (27c 4 /p 2 )x − (54c 6 /p 3 ) has 0, 1, 3 roots modulo p. So the average of t(p) is √ 1 (p − 1)(p + 1)/3 + √ 2 p(p − 1)/2 + √ 4 (p − 1)(p − 2)/6 (p − 1)(p + 1)/3 + p(p − 1)/2 + (p − 1)(p − 2)/6 = 2 3 + √ 2 2 − 1 3p .
in this case. For the remaining cases, when p 2 D or p 10 D we have t(p) = 1, while when p 3 D or p 9 D we have t(p) = 2. Finally, when p 4 D we have t(p) = 3, 1 depending on whether −6c 6 /p 2 is square mod p, and similarly when p 8 D we have t(p) = 3, 1 depending on whether −6c 6 /p 4 is square mod p, so that the average of t(p) in both cases is 1 2
(1 + √ 3). We get that ǫ n a (k) = 1, √ 2, 1 2 (1 + √ 3), 1 2 (1 + √ 3), √ 2, 1 for k = 2, 3, 4, 8, 9, 10, while (6) ǫ m (k) = 1 2 (1 + √ k), k odd 1 2 ( √ 2 + √ k), k even and ǫ t a (p, k) = 2 3 + √ 2 2 − 1 3p , k = 6 1 2 ( √ 2 + √ 4), k ≥ 7
with ǫ n a (k) and ǫ t a (p, k) equal to zero for other k. We define the expected square root of the Tamagawa number K(p) at p by
(7) K(p) = ∞ k=0 ǫ m (k)P m (p, k) + ǫ n a (k)P n a (p, k) + ǫ t a (p, k)P t a (p, k) + P 0 (p, k)
and the expected global 6 Tamagawa number to be β τ = p K(p). The convergence of this product follows from an analysis of the dominant k = 0, 1, 2 terms of (7), which gives a behaviour of 1 + O(1/p 2 ). So we get that the Tamagawa product is a constant on average, which we do not bother to compute explicitly (we would need to consider p = 2, 3 more carefully to get a precise value).
To compute the average value of α A (E) = p F (p) in (2) we similarly assume 7 that each prime acts independently; we then compute the average value for each prime by calculating the distribution of F (p) when considering all the curves modulo p (including those with singular reduction, and again making the slight adjustment for non-minimal models). This gives some constant for the averageᾱ A of α A (E), which we do not compute explicitly. Note that p F (p) converges if we assume the Sato-Tate conjecture [31] since in this case we have that a 2 p is p on average.
Relation between conductor and discriminant
We now give heuristics for how often we expect the ratio between the absolute discriminant and the conductor to be large. The main heuristic we derive is: 6 Note that the Tamagawa number at infinity is 1 when E has negative discriminant and else is 2, the former occurring approximately √ 3/(1 + √ 3) ≈ 63.4% of the time. 7 This argumentative technique can also be used to bolster our assumption that using Connell's conditions should be independent of other considerations.
Heuristic 4.1. The number B(X) of rational elliptic curves whose conductor is less than X satisfies B(X) ∼ cX 5/6 for an explicit constant c > 0.
To derive this heuristic, we estimate the proportion of curves with a given ratio of (absolute) discriminant to conductor. Since the conductor is often the squarefree kernel of the discriminant, by way of explanation we first consider the behaviour of f (n) = n/sqfree(n). The probability that f (n) = 1 is given by the probability that n is squarefree, which is classically known to be 1/ζ(2) = 6/π 2 . Given a prime power p m , to have f (n) = p m says that n = p m+1 u where u is squarefree and coprime to p. The probability that p m+1 n is (1 − 1/p)/p m+1 , and given this, the conditional probability that n/p m+1 is squarefree is (6/π 2 ) · (1 − 1/p 2 ) −1 . Extending this multiplicatively beyond prime powers, we get that Prob n/sqfree(n) = q = 6 π 2
p m q 1/p (m+1) (1 + 1/p) = 6 π 2 1 q p|q 1 p + 1 .
In particular, the average of f (n) γ exists for γ < 1; in our elliptic curve analogue, we will require such an average for γ = 5/6. We note that it appears to be an interesting open question to prove an asymptotic for n≤X n/sqfree(n).
4.1.
Derivation of the heuristic. We keep the notation D = |∆| and wish to compute the probability that D/N = q for a fixed positive integer q. For a prime power p v with p ≥ 5, the probability that p v (D/N ) is given by: the probability that E has multiplicative reduction at p and p v+1 D, that is P m (p, v + 1); plus the probability that E has additive reduction at p and p v+2 D, that is P a (p, v + 2); and the contribution from P 0 (p, v), which is zero for v > 0 and for v = 0 is the probability that p does not divide D. So, writing v = v p (q), we get that (with a similar modified formula for p = 2, 3)
(8) Prob D/N = q = p E p (v p (q)) = p P m (p, 1 + v) + P a (p, 2 + v) + P 0 (p, v) .
It should be emphasised that this probability is with respect to (as in the previous section) the ordering of the curves by discriminant. We have
(9) E:NE ≤X 1 ≈ ∞ q=1 E:D≤qX Prob D/N = q ∼ ∞ q=1 α(qX) 5/6 · Prob D/N = q ,
where α = α + + α − from the Brumer-McGuinness heuristic 2.1. If this last sum converges, then we get Heuristic 4.1.
To show the last sum in (9) does indeed converge, we upper-bound the probability in (8). We have that P m (p, v+1) ≤ 1/p v+1 and P a (p, v+2) ≤ 2/p v+1 , which implieŝ
f (q) = Prob D/N = q = p E p (v p (q)) ≤ 1 q p|q 3 p .
We then estimate
∞ q=1 q 5/6f (q) ≤ ∞ q=1 1 q 1/6 p|q 3 p = p 1 + ∞ l=1 3/p (p l ) 1/6 ≤ p 1 + 3/p p 1/6 − 1 ,
and the last product is convergent upon comparison to ζ(7/6) 3 . Thus we shown that the last sum in (9) converges, so that Heuristic 4.1 follows.
We can note that Fouvry, Nair, and Tenenbaum [13] have shown that the number of minimal models with D ≤ X is at least cX 5/6 , and that the number of curves with D ≤ X with Szpiro ratio log D log N ≥ κ is no more than c ǫ X 1/κ+ǫ for every ǫ > 0.
4.2.
Dependence of D/N and the Tamagawa product. We expect that D/N should be independent of the real period, but the Tamagawa product and D/N should be somewhat related. 8 We compute the expected square root of the Tamagawa product when D/N = q. As with (8) and using the ǫ defined in (6), we find that this is given by
η(q) = p ǫ m (v 1 )P m (p, v 1 ) + ǫ n a (v 2 )P n a (p, v 2 ) + ǫ t a (p, v 2 )P t a (p, v 2 ) + P 0 (p, v) P m (p, v 1 ) + P a (p, v 2 ) + P 0 (p, v) ,
where v 1 = v + 1 and v 2 = v + 2 and v = v p (q).
4.3.
The comparison of log ∆ with log N . We now want to compare log ∆ with log N , and explicate the replacement therein in Guess 3.2. In order to bound the effect of curves with large D/N , we note that
Prob D/N ≥ Y = q≥Yf (q) ≤ q≥Y 1 q p|q 3 p ,
and use Rankin's trick, so that for any 0 < α < 1 we have (using p α − 1 ≥ α log p)
Prob D/N ≥ Y ≤ ∞ q=1 q Y 1−α · 1 q p|q 3 p = Y α Y p 1 + 3 p 1+α + 3 p 1+2α + · · · = Y α Y p 1 + 3/p p α − 1 ≪ Y α Y exp pĉ /p α log p ≪ e c √ log Y Y
for some constantsĉ, c, by taking α = 1/ √ log Y (this result is stronger than needed). However, a more pedantic derivation of Guess 3.2 does not simply allow replacing log N by log ∆, but requires analysis (assuming Ω re (E) to be independent of q) of The above estimate on the tail of the probability and a simple bound on η(q) in terms of the divisor function shows that we can truncate the q-sum at Y with an error of O(1/Y 8/9 ), and choosing (say) Y = e √ log X gives us that log(∆/q) ∼ log ∆ (note that we restricted to ∆ > √ X). So the bracketed term becomes the desired q<Y (log ∆) 3/8 η(q) · Prob D/N = q ∼ β τ (log ∆) 3/8 , upon noting that the q-part of the sum converges to β τ as Y → ∞. 8 The size of the torsion subgroup should also be related to D/N , but in the next section we argue that curves with nontrivial torsion are sufficiently sparse so as to be ignored.
4.4.
Counting curves with vanishing L-value. We now estimate the number of elliptic curves E with even parity and L(E, 1) = 0 when ordered by conductor.
Heuristic 4.2. LetR(X) be the number of elliptic curves E with even parity and conductor less than X and L(E, 1) = 0. ThenR(X) ∼ cX 19/24 (log X) 3/8 for some explicit constant c > 0.
From Guess 3.2 we get that the number of even parity curves with 0 < ∆ < qX and D/N = q and L(E, 1) = 0 is given bŷ W (qX) · η(q)/β τ · Prob D/N = q , and we sum this over all q. As we argued above, the tail of the sum does not affect the asymptotic (and so we can take log ∆ ∼ log N inŴ ), and again we get that the q-sum converges. This then gives the desired asymptotic for the number of even parity curves with conductor less than X and vanishing central L-value (upon arguing similarly for curves with negative discriminant).
Torsion and isogenies
We can also count curves that have a given torsion group or isogeny structure. For instance, an elliptic curve with a 2-torsion point can be written as an integral model in the form y 2 = x 3 +ax 2 +bx where ∆ = 16b 2 (a 2 −4b); thus, by lattice-point counting, we estimate about √ X curves with absolute discriminant less than X. The effect on the conductor can perhaps more easily be seen by using the Fricke parametrisation c 4 = (t + 16)(t + 64)T 2 and c 6 = (t − 8)(t + 64) 2 T 3 of curves with a rational 2-isogeny, and then substituting t = p/q and V = T /q to get c 4 = (p+16q)(p+64q)V 2 and c 6 = (p−8q)(p+64q) 2 V 3 so that ∆ = p(p+64q) 3 q 2 V 6 . The summation over the twisting parameter V just multiplies our estimate by a constant, while ABC-estimates imply that there should be no more than X 2/3+ǫ coprime pairs (p, q) with the squarefree kernel of pq(p + 64q) smaller than X in absolute value. So we get the heuristic that almost all curves have no 2-torsion, even under ordering by conductor. Indeed, the exceptional set is so sparse that we can ignore it in our calculations. A similar argument applies for other isogenies, and more generally for splitting of division polynomials. Also, the results of Duke [12] for exceptional primes are applicable here, albeit with a different ordering.
Experiments
We wish to provide some experimental data for the above heuristics. In particular, the two large datasets of Brumer-McGuinness [3] and Stein-Watkins [30] for curves of prime conductor up to 10 8 and 10 10 show little drop in the observed average (analytic) rank. Brumer and McGuinness considered about 310700 curves with prime conductor less than 10 8 and found an average rank of about 0.978, while Stein and Watkins extended this to over 11 million curves with prime conductor up to 10 10 and found an average rank of about 0.964. Both datasets are expected to be nearly exhaustive 9 amongst curves with prime conductor up to the given limit. These results led some to speculate that the average rank might be constant. To test this, we chose a selection of curves with prime conductor of size 10 14 . It is non-trivial to get a good dataset, since we must account for congruence conditions on the elliptic curve coefficients and the variation of the size of the real period. 6.1. Average analytic rank for curves with prime conductor near 10 14 . As in [30], we divided the (c 4 , c 6 ) pairs into 288 congruence classes with (c 4 ,c 6 ) = c 4 mod 576, c 6 mod 1728 . Many of these classes force the prime 2 to divide the discriminant, and thus do not produce any curves of prime conductor. For each class (c 4 ,c 6 ), we took the 10000 parameter selections (c 4 , c 6 ) = 576(1000 + i) +c 4 , 1728(100000 + j) +c 6 for (i, j) ∈ [1..10] × [1..1000], and then of these 2880000 curves, took the 89913 models that had prime discriminant (note that all the discriminants are positive). This gives us good distribution across congruence classes, and while the real period does not vary as much as possible, below we will attempt to understand how this affects the average rank.
It then took a few months to compute the (suspected) analytic ranks for these curves. We got about 0.937 for the average rank. We then did a similar experiment for curves with negative discriminant given by (c 4 , c 6 ) = 576(−883 + i) +c 4 , 1728(100000 + j) +c 6 for (i, j) ∈ [1..10] × [1.
.1000], took the subset of 89749 curves with prime conductor, and found the average rank to be about 0.869. This discrepancy between positive and negative discriminant is also in the Brumer-McGuinness and Stein-Watkins datasets, and indeed was noted in [3]. 10 We do not average the results from positive and negative discriminant; the Brumer-McGuinness Conjecture 2.1 implies that the split is not 50-50.
In any case, our results show a substantial drop in the average rank, which, at the very least, indicates that the average rank is not constant. The alternative statistic of frequency of positive rank for curves with even parity also showed a significant drop. For prime positive discriminant curves it was 44.1% for Brumer-McGuinness and 41.7% for Stein-Watkins, but only 36.0% for our dataset -for negative discriminant curves, these numbers are 37.7%, 36.4%, and 31.3%. 6.2. Variation of real period. Our random sampling of curves with prime conductor of size 10 14 must account for various properties of the curves if our results are to possess legitimacy. Above we speculated that the real period plays the most significant rôle, and so we wish to understand how our choice has affected it.
To judge the effect that variation of the real period might have, we did some comparisons with the Stein-Watkins database. First consider curves of positive prime discriminant, and write E as y 2 = 4x 3 + b 2 x 2 + 2b 4 x + b 6 and e 1 > e 2 > e 3 for the real roots of the cubic. We looked at curves with even parity and considered the frequency of positive rank as a function of the root quotient t = e1−e2 e1−e3 , noting that 11 Ω re ∆ 1/12 = 2 1/3 π(t−t 2 ) 1/6 agm(1, √ t) . The curves we considered all had 0.617 < t < 0.629. However, similar to when we considered curves ordered by conductor, before counting curves with extra rank, we should first simply count curves. Figure 1 indicates the distribution of root quotient t for the curves of prime (positive) discriminant and even parity from the Stein-Watkins database (more than 2 million curves meet the criteria). The x-axis is divided up into bins of size 1/1000; there are more than 100 times as many curves with t < 0.001 as there with 0.500 < t < 0.501, with the most extremal dots not even appearing on the graph. 10 "An interesting phenomenon was the systematic influence of the discriminant sign on all aspects of the arithmetic of the curve." 11 The calculation follows as in the previous sections; via calculus, we can compute that this function is maximised at t ≈ .0388505246188 with a maximum just below 4.414499094. and Ω re ∆ 1/12 as a function of t.
Next we plot the frequency of L(E, 1) = 0 as a function of the root quotient in Figure 2. Since there are only about 1000 curves in some of our bins, we do not get such a nice graph. Note that the left-most and especially the rightmost dots are much below their nearest neighbors, the graph slopes down in general, and drops more at the end. We see no evidence that our results should be overly biased. In particular, the frequency of L(E, 1) = 0 is 41.7% amongst all even parity curves of prime discriminant in the Stein-Watkins database, and is 42.8% for the 12324 such curves with 0.617 < t < 0.629. The function plotted (labelled on the right axis) in Figure 2 is of Ω re ∆ 1/12 = 2 1/3 π(t−t 2 ) 1/6
agm(1, √ t)
as a function of t, and note that this goes to 0 as t → 0, 1; there is nothing canonical about the choice of our t-parameter, and we chose it more for convenience than anything else.
Similar computations can be made in the case of negative discriminant, which we briefly discuss for completeness (again restricting to curves with even parity where appropriate). Let r be the real root of the cubic polynomial 4x 3 + b 2 x 2 + 2b 4 x + b 6 , and Z > 0 the imaginary part of the conjugate pair of nonreal roots. Letting r = r + b 2 /12 and c =r/Z we then have 12
Ω re |∆| 1/12 = π √ 2 (1 + 9c 2 /4) 1/12 agm 1, 1 2 + 3c 4 √ 1+9c 2 /4 .
We renormalise via taking C = 1/2 + arctan(c)/π, and graph the distribution of curves versus C in Figure 3. The symmetry 13 of the graph might indicate that the coordinate transform is reasonable. All our curves have 0.555 < C < 0.557. This is maximised at c ≈ −33.58515148525, with the maximum a bit less than 8.82921518. 13 The blotches around 0.22-0.23 and 0.77-0.78 appear to come from the fact that curves with a 4 small (in particular ±1) tend to have C in these ranges (for our discriminant range), and this causes instability in the counting function.
Next we plot the frequency of L(E, 1) = 0 as a function of the root quotient in Figure 4. Again we also graph the function Ω re |∆| 1/12 on the right axis. Here the drop-off is more pronounced than with the curves of positive discriminant. Note the floating dot around C = 1/2. Indeed the 100 closest curves with C < 1/2 all have positive rank; this breaks down when crossing the 1/2-barrier. This is not particularly a mystery; these curves have a 6 = 0 and/or b 6 = 1, and thus have an obvious rational point. Recall that C = 1/2 corresponds to c = 0 =r. and Ω re |∆| 1/12 as a function of C.
We again see no evidence that our results should be biased. In particular, the frequency of L(E, 1) = 0 is 36.4% amongst all even parity curves of negative prime discriminant in the Stein-Watkins database, and is 37.0% for the 4695 such curves with 0.555 < C < 0.557. 6.3. Other considerations. The idea that the "probability" that an even parity curve possesses positive rank should be proportional to √ Ω re is perhaps overly simplistic -in particular, it is not borne out too precisely by the Stein-Watkins dataset. We consider positive prime discriminant curves with even parity; for those with 0.64 < Ω re < 0.65 we have 78784 curves of which 45.9% have positive rank, while of the 9872 with 0.32 < Ω re < 0.325 we have 36.0% with positive rank, for a ratio of 1.28, which is not too close to √ 2. One consideration here is that we have placed a discriminant limit on our curves, and there are curves with larger discriminant and 0.32 < Ω re < 0.325 that we have not considered. This, however, is extra-particular to the idea that only the real period should be of import.
One possibility is that curves with small discriminant and/or large real period have smaller probability of L(E, 1) = 0 that our estimate of c √ Ω re would suggest -indeed, it might be argued (maybe due to arithmetic considerations, or perhaps explicit formulae for the zeros of L-functions) that curves with such small discriminant cannot realise their nominal expected frequency of positive rank. Unfortunately, we cannot do much to quantify these musings, as the effect would likely be in a secondary term, making it difficult to detect experimentally. Note also that a relative depression of rank for small discriminant curves would give a reason for the near-constant average rank observed by Brumer-McGuinness and Stein-Watkins.
6.4. Mordell-Weil lattice distribution for rank 2 curves. We have other evidence that curves of small discriminant might not behave quite as expected. We undertook to compute generators for the Mordell-Weil group for all 2143079 curves of (analytic) rank 2 of prime conductor less than 10 10 in the Stein-Watkins database. 14 J. E. Cremona ran his mwrank programme [9] on all these curves, and it was successful in provably finding the Mordell-Weil group for 2114188 of these. For about 2500 curves, the search region was too big to find the 2-covering quartics via invariant methods, while around 8500 curves had a generator of large height that could not be found, and over 18000 had 2-Selmer rank greater than 2. We then used the FourDescent machinery of MAGMA [2] which reduced the number of problematic curves to 54. Of these, 19 have analytic X of 16.0 and we expect that either 3-descent or 8-descent [29] will complete (assuming GRH to compute the class group) the Mordell-Weil group verification; for the 35 other curves, there is likely a generator of height more than 225 which we did not attempt to find. 15 We then looked at the distribution of the Mordell-Weil lattices obtained from the induced inner product from the height pairing; since all of our curves have rank 2, we get 2-dimensional lattices. We are not so interested in the size of the obtained lattices, but more so in their shape. Via the use of lattice reduction (which reduces to continued fractions in this case), given any two generators we can find the point P of smallest positive height on the curve. By normalising P to be the unit vector, we then get a vector in the upper-half-plane corresponding to another generator Q. Via the standard reduction algorithm, we can translate Q so that it corresponds to a point in the fundamental domain for the action of SL 2 (Z). Finally, by replacing Q by −Q if necessary, we can ensure that this point is in the right half of the fundamental domain (in other words, we must choose an embedding for our Mordell-Weil lattice). In this manner, for each rank 2 curve we associate a unique point z = x + iy in the upper-half-plane with x 2 + y 2 ≥ 1 and 0 ≤ x ≤ 1/2.
With no other guidance, we might expect that the obtained distribution for the z is given by 16 the Haar measure (dx dy)/y 2 . We find, however, that this is not borne out too well by experiment. In particular, we should expect that 1/2 π/6 ≈ 95.5% of the curves should have y ≥ 1, while the experimental result is about 93.5%. Furthermore, we should expect that the proportion of curves with y ≥ Y should die off like 1/Y as y → ∞; however, we get that 35.4% of the curves have y ≥ 2, 14 We also computed the Mordell-Weil group for curves with higher ranks but do not describe the obtained data here. 15 A bit more searching might resolve a few of the outstanding cases, but the extremal case of [0, 0, 1, −237882589, −1412186639384] appears to have a generator of height more than 600, and thus other methods will likely be needed to try to find it. T. A. Fisher has recently used 6-descent to find some of the missing points. 16 Siegel [27] similarly uses Haar measure to put a natural measure on n-dimensional lattices of determinant 1.
only 9.4% have y ≥ 4, while 1.7% have y ≥ 8 and 0.2% have y ≥ 16. The validity of the vertical distribution data might be arguable based upon concerns regarding the discriminant cutoff of our dataset, but the horizontal distribution is also skewed. If we consider only curves with y ≥ 1, then we should get uniform distribution in the x-aspect; however, Table 1 shows that we do not have such uniformity. We cannot say whether these unexpected results from the experimental data are artifacts of choosing curves with small discriminant; it is just as probable that our Haar-measure hypothesis concerning the lattice distribution is simply incorrect. 6.5. Symmetric power L-functions. Similar to questions about the vanishing of L(E, s), we can ask about the vanishing of the symmetric power L-functions L(Sym 2k−1 E, s). We refer the reader to [22] for more details about this, but mention that, due to conjectures of Deligne and more generally Bloch and Beȋlinson [24], we expect that we should have a formula similar to that of Birch and Swinnerton-Dyer,
stating that L(Sym 2k−1 E, k)(2πN ) ( k 2 ) /Ω ( k+1 2 ) + Ω ( k 2 )
− should be rational with small denominator. Here, for k odd, Ω + is the real period and Ω − the imaginary period, with this reversed for k even. Ignoring the contribution from the conductor, and crudely estimating that Ω + ≈ Ω − ≈ 1/∆ 1/12 , an application of discretisation as before gives that the probability that L(Sym 2k−1 E, s) has even parity and L(Sym 2k−1 E, k) = 0 is bounded above (cf. the ignoring of N ) by c(log ∆) 3/8 · 1/∆ k 2 /12 . Again following the analogy of above, we can then upper-bound the number of curves with conductor less than X with even-signed symmetric (2k − 1)st power and L(Sym 2k−1 E, k) = 0 by c k (ǫ)X 5/6−k 2 /24+ǫ for every ǫ > 0. It could be argued that we should order curves according to the conductor of the symmetric power L-function rather than that of the curve, but we do not think such concerns are that relevant to our imprecise discussion. In particular, the above estimate predicts that there are finitely many curves with extra vanishing when k ≥ 5. It should be said that this heuristic will likely mislead us about curves with complex multiplication, for which the symmetric power L-function factors (it is imprimitive in the sense of the Selberg class), with each factor having a 50% chance of having odd parity. However, even ignoring CM curves, the data of [22] find a handful of curves for which the 9th, 11th and even the 13th symmetric powers appear (to 12 digits of precision) to have a central zero of order 2. We find this surprising, and casts some doubt about the validity of our methodology of modelling of vanishings.
6.6. Quadratic twists of higher symmetric powers. The techniques we used earlier in this paper have also been used to model vanishings in quadratic twist families, and we can extend the analyses to symmetric powers. 6.6.1. Non-CM curves. We fix a non-CM curve E and let E d be its dth quadratic twist, taking d to be a fundamental discriminant. From an analogue of the Birch-Swinnerton-Dyer conjecture we expect to get a small-denominator rational from the quotient 17
L(Sym 3 E d , 2)(2πN E )/Ω im (E d ) 3 Ω re (E d ). We have that Ω im (E d ) 3 Ω re (E d ) ≈ Ω im (E)/d 3/2
· Ω re (E)/d 1/2 and so we expect the number of fundamental discriminants |d| < D such that L(Sym 3 E d , s) has even parity with L(Sym 3 E d , 2) = 0 to be given crudely (up to log-factors) by d<D c/ √ d 2 . So we expect about (log D) b quadratic twists with double zeros for the 3rd symmetric power; generalising predicts finitely many extra vanishings for higher (odd) powers. We took the curves 11a:[0, −1, 1, 0, 0] and 14a:[1, 0, 1, −1, 0], and computed either L(Sym 3 E d , 2) or L ′ (Sym 3 E d , 2) for all fundamental discriminant d with |d| < 5000. We did the same for 15a:[1, 1, 1, 0, 0] for |d| < 4000. We then looked at the number of vanishings (to 9 digits of precision). For 11a we found 58 double zeros and one triple zero (indicated by a star in Table 2) while for 14a we found 88 double zeros and three triple zeros, and 15a yielded 83 double zeros and two triple zeros. 6.6.2. CM curves. Next we consider CM curves, for which we can compute significantly more data, but the modelling of vanishings is slightly different. Let E be a rational elliptic curve with CM, and ψ its Hecke character. We shall take ψ to be "twist-minimal" -this is not the same as the "canonical" character of Rohrlich [26], but rather we just take E to be a minimal (quadratic) twist. Indeed, we shall only 17 The contribution from the conductor actually comes from non-integral Tamagawa numbers from the Bloch-Kato exponential map, and in the case of quadratic twists, the twisting parameter d should not appear in the final expression. consider 11 different choices of E, given (up to isogeny class) by 27a, 32a, 36a, 49a, 121a, 256a, 256b, 361a, 1849a, 4489a, and 26569a, noting that 27a/36a and 32a/256b are respectively cubic and quartic twist-pairs. In our tables, these can appear in a briefer format, such as 67 2 for 4489a.
We normalise the Hecke L-function L(ψ, s) to have s = 1 be the center of the critical strip. For d a fundamental discriminant, we let ψ d be the Hecke Grössencharacter ψ twisted by the quadratic Dirichlet character of conductor d. Finally, note that the symmetric powers L(Sym k ψ, s) are just L(ψ k , s), where we must take ψ k to be the primitive underlying Grössencharacter.
We then expect L(ψ 3 , 2)(2π)/Ω im (E) 3 to be rational with small denominator. We can then use discretisation as before to count the expected number of fundamental discriminants |d| < D for which the L-function L(ψ 3 d , s) has even parity but vanishes at the central point -since we have Ω im (E d ) 3 ≈ Ω im (E)/d 3/2 , we expect the number of discriminants d that yield even parity and L(ψ 3 d , 2) = 0 is crudely given by d<D 1/ √ d 3/2 , so we should get about D 1/4 such discriminants up to D. For higher symmetric powers, we expect that L(ψ 2k−1 , k)(2π) k−1 /Ω + (E) 2k−1 is rational with small denominator, and thus get that there should be finitely many quadratic twists of even parity with vanishing central value.
We took the above eleven CM curves and took their (fundamental) quadratic twists up to 10 5 . We must be careful to exclude twists that are isogenous to other twists. In particular, we need to define a primitive discriminant for a curve with CM by an order of the field K -this is a fundamental discriminant d such that disc(K) does not divide d, expect for K = Q(i) when d > 0 is additionally primitive when 8 d. Note also that 27a and 36a have the same symmetric cube L-function. Table 3. Counts of double order zeros for primitive twists 27a 32a 36a 49a 121a 256a 256b 361a 1849a 4489a 26569a 3rd 59 32 -67 78 32 21 45 28 31 1 5th 3 1 5 2 1 2 2 0 0 0 0 7th 0 0 2 0 1 0 0 0 0 0 0 Table 3 lists the following results for counts of central double zeros (to 32 digits) for the L-functions of the 3rd, 5th, and 7th symmetric powers. 18 Tables 4 and 5 list the primitive discriminants that yield the double zeros. The notable signedness can be explained via the sign of the functional equation. 19 We are unable to explain the paucity of double zeros for twists of 26569a; Liu and Xu have the latest results [20] on the vanishing of such L-functions, but their bounds are far from the observed data. Similarly, the last-listed double zero for 4489a at 67260 seems quite small.
There appear to be implications vis-a-vis higher vanishings in some cases; for instance, except for 27a, in the thirteen cases that L(ψ 5 d , s) has a double zero at s = 3 then L(ψ d , s) also has a double zero at s = 1. Similarly, the 7th symmetric power for the 27365th twist of 121a has a double zero, as does the 3rd symmetric power, while the L-function of the twist itself has a triple zero. Also, the 22909th twist of 36a has double zeros for its first, third, and fifth powers (note that 36a does not appear in Table 4 as the data are identical to that for 27a). 18 We found no even twists with L(ψ 9 d , 5) = 0, and no triple zeros appeared in the data. 19 The local signs at p = 2, 3 involve wild ramification are more complicated (see [32,19,11] for a theoretical description), and thus there is no complete correlation in some cases. 20 For the 3rd symmetric power, the crude prediction is that we should have (asymptotically) many more extra vanishings for twists in the CM case than in the non-CM case, but this is not borne out by the data. Additionally, we have no triple zeros in the CM case (where the dataset is almost 100 times as large), while we already have six for the non-CM curves. This is directly antithetical to our suspicion that there should be more extra vanishings in the CM case. As before, this might cast some doubt on our methodology of modelling of vanishings.
Ω
re (E) · q<∆ η(q)(log ∆/q) 3/8 · Prob D/N = q du 4 du 6 .
Figure 1 .Figure 2 .
12∆ ∆ > 0: Positive rank frequency as a function of the root quotient t,
Figure 3 .
3∆ < 0: Distribution of curves as a function of C12
Figure 4 .
4∆ < 0: Positive rank frequency as a function of C,
Table 1 .
1Horiztonaldistribution of rank 2 lattices with y ≥ 1
0.00 ≤ x < 0.05 9.0% 0.25 ≤ x < 0.30 10.0%
0.05 ≤ x < 0.10 9.6% 0.30 ≤ x < 0.35 10.2%
0.10 ≤ x < 0.15 9.8% 0.35 ≤ x < 0.40 10.5%
0.15 ≤ x < 0.20 9.9% 0.40 ≤ x < 0.45 10.6%
0.20 ≤ x < 0.25 10.0% 0.45 ≤ x ≤ 0.50 10.5%
Table 2 .
2Fundamental d with ord
Table 4 .Table 5 .
45Primitive d with ord Primitive d with ord s=k L(ψ 2k−1 d , s) = 2 for some k ≥ 3 27a 5th: −13091 4040 18044 49a 5th: 437 19317 32a 5th: 1704 121a 5th: −183 7th: 27365 36a 5th: −856 −2104 −31592 −88580 22909 256a 5th: −79 −21252 36a 7th: −95 2488 256b 5th: −511 89320 6.6.3. Comparison between the CM and non-CM cases. For the twist computations for the symmetric powers, we can go much further (about 20 times as far) in the CM case because the conductors do not grow as rapidly.s=2
The precision of this discretisation might be the most-debatable methodology we use. Indeed, we are essentially taking a "sharp cutoff", while it might be better to have a more smooth transition function. For this reason, we do not specify the leading constant in our final heuristic.
At p = 2, 3, non-minimality occurs when c 4 /p 4 and c 6 /p 6 satisfy the congruences.5 Note that our methods do not readily generalise to higher rank, as there is no apparent way to model the heights of points (and thus the regulator).
This is one reason to take prime conductor curves; we also have |∆| = N with few exceptions.
AcknowledgementsThe author was partially supported by Engineering and Physical Sciences Research Council (EPSRC) grant GR/T00658/01 (United Kingdom). He thanks N. D. Elkies, H. A. Helfgott, and A. Venkatesh for useful comments, and N. P. Jones for the reference[12].
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|
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[
"Hydrodynamically synchronized states in active colloidal arrays",
"Hydrodynamically synchronized states in active colloidal arrays"
] |
[
"Loïc Damet \nCavendish Laboratory and Nanoscience Centre\nUniversity of Cambridge\nCB3 0HECambridgeU. K\n",
"Giovanni M Cicuta \nDip. Fisica\nUniversità di Parma\nINFN, Sez. Milano-Bicocca\ngruppo di ParmaItaly\n",
"Jurij Kotar \nCavendish Laboratory and Nanoscience Centre\nUniversity of Cambridge\nCB3 0HECambridgeU. K\n",
"Marco Cosentino Lagomarsino \nGenomic Physics Group\nCNRS \"Microorganism Genomics\" and University Pierre et Marie Curie\n15, rue de l'École de MédecineFRE 3214ParisFrance\n",
"Pietro Cicuta \nCavendish Laboratory and Nanoscience Centre\nUniversity of Cambridge\nCB3 0HECambridgeU. K\n"
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[
"Cavendish Laboratory and Nanoscience Centre\nUniversity of Cambridge\nCB3 0HECambridgeU. K",
"Dip. Fisica\nUniversità di Parma\nINFN, Sez. Milano-Bicocca\ngruppo di ParmaItaly",
"Cavendish Laboratory and Nanoscience Centre\nUniversity of Cambridge\nCB3 0HECambridgeU. K",
"Genomic Physics Group\nCNRS \"Microorganism Genomics\" and University Pierre et Marie Curie\n15, rue de l'École de MédecineFRE 3214ParisFrance",
"Cavendish Laboratory and Nanoscience Centre\nUniversity of Cambridge\nCB3 0HECambridgeU. K"
] |
[] |
Colloidal particles moving in a fluid interact via the induced velocity field. The collective dynamic state for a class of actively forced colloids, driven by harmonic potentials via a rule that couples forces to configurations, to perform small oscillations around an average position, is shown by experiment, simulation and theoretical arguments to be determined by the eigenmode structure of the coupling matrix. It is remarkable that the dynamical state can therefore be predicted from the mean spatial configuration of the active colloids, or from an analysis of the fluctuations near equilibrium. This has the surprising consequence that while 2 particles, or polygonal arrays of 4 or more colloids, synchronize with the nearest neighbors in anti-phase, a system of 3 equally spaced colloids synchronizes in-phase. In the absence of thermal fluctuations, the stable dynamical state is predominantly formed by the eigenmode with longest relaxation time.
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10.1039/c2sm25778e
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[
"https://arxiv.org/pdf/1110.1526v2.pdf"
] | 118,579,305 |
1110.1526
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ff1a8fbd486caf32753b771725d1b5c755c726d7
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Hydrodynamically synchronized states in active colloidal arrays
10 Oct 2011
Loïc Damet
Cavendish Laboratory and Nanoscience Centre
University of Cambridge
CB3 0HECambridgeU. K
Giovanni M Cicuta
Dip. Fisica
Università di Parma
INFN, Sez. Milano-Bicocca
gruppo di ParmaItaly
Jurij Kotar
Cavendish Laboratory and Nanoscience Centre
University of Cambridge
CB3 0HECambridgeU. K
Marco Cosentino Lagomarsino
Genomic Physics Group
CNRS "Microorganism Genomics" and University Pierre et Marie Curie
15, rue de l'École de MédecineFRE 3214ParisFrance
Pietro Cicuta
Cavendish Laboratory and Nanoscience Centre
University of Cambridge
CB3 0HECambridgeU. K
Hydrodynamically synchronized states in active colloidal arrays
10 Oct 2011
Colloidal particles moving in a fluid interact via the induced velocity field. The collective dynamic state for a class of actively forced colloids, driven by harmonic potentials via a rule that couples forces to configurations, to perform small oscillations around an average position, is shown by experiment, simulation and theoretical arguments to be determined by the eigenmode structure of the coupling matrix. It is remarkable that the dynamical state can therefore be predicted from the mean spatial configuration of the active colloids, or from an analysis of the fluctuations near equilibrium. This has the surprising consequence that while 2 particles, or polygonal arrays of 4 or more colloids, synchronize with the nearest neighbors in anti-phase, a system of 3 equally spaced colloids synchronizes in-phase. In the absence of thermal fluctuations, the stable dynamical state is predominantly formed by the eigenmode with longest relaxation time.
Hydrodynamic coupling at low Reynolds' number is an important feature in biological flows, with key consequences in apparently diverse phenomena such as the motility of microorganisms [1], circulation in the brain [2] and functioning of the ear [3]. In various biological tissues, a macroscopic number of units, for example cilia, display synchronized dynamics, leading to metachronal waves [4]. Nearby cilia may beat in-phase or out of phase, and indeed may be in a condition where is it possible to readily switch between the two dynamical states [1]. One outstanding question is what determines the character of the dynamical steady state. Recent progress in "hydrodynamic synchronization" is reviewed in [5], and an overview of low Reynolds number (Re) flows [6]. In an attempt to model (both experimentally and theoretically) the physics of hydrodynamic synchronization, two main ideas have emerged. The first is to consider the coupling of two or more objects driven by a constant force over general (pre-defined) orbits [7]. Within this idea of "rotor" cilia models, the phase of each rotor is free and there can be synchronization of different rotors under certain conditions. This has recently been studied very generally in [8], extending the case of circular driving force orbits [7]. A different model consists of a "geometric switch", and was proposed in [9]; here the force is discontinuous, and this is not described within the formalism of [8]. It is not yet established which of these ideas is most appropriate to describe a biological scenario. The discontinuity of force should not be discounted a-priori, considering the fact that molecular motors undergo discrete attachment/detachment events which couple to the force generation. In previous work the geometric switch model was investigated for the simple case of two active elements, showing the robustness of hydrodynamic coupling in the presence of noise [10]. A linear chain of oscillators, following the geometric switch rule with general actuation forces, was studied recently by numerical methods, in the absence of noise [11].
In this Letter we show how the non-equilibrium dynamical behavior of the "geometric switch" model can be understood from the eigenmode structure of the Oseen tensor, which depends on the geometrical arrangement oscillators. Systems of a small number of elements are considered, in the presence of thermal noise: between three and five colloidal particles are arranged on equally spaced average positions on a circumference, and each one is maintained in a driven oscillatory tangential orbit, with fixed amplitude but free phase and period. The main result is that the fundamental (longest lived) hydrodynamic mode (an easily derivable quantity, in contrast to finding full solutions of the system) dominates the collective motion in the driven steady state. As a consequence, the steady state can be strikingly different depending on the number of particles and their arrangement. The "dynamical motifs" observed here may guide the analysis of cilia coordination in complex biological systems.
This system is realized experimentally with optical traps, with fast video feedback to impose the fixed amplitude driven oscillation. The only interaction between the elements is through the hydrodynamic flow arising from the colloid movements. Brownian Dynamics (BD) simulations, in which the hydrodynamic interaction is calculated through Oseen's tensor [12], are compared to the experimental data and can also be performed readily with a larger number of particles in the system. The assumptions for this treatment [13] are a low Reynolds number, particles far relative to their diameter, and a steady flow, both of which are satisfied in the physical context. As a further consequence of hydrodynamic interaction, there are correlations in the Brownian fluctuations of different particles. The dynamics of systems where spheres are held by fixed harmonic potentials on the vertices of arbitrary planar regular polygons has been solved exactly within Oseen's description of hydrodynamics [14], giving a basis from which to understand the coupling in the active scenario.
Optical traps are used to confine colloidal beads within harmonic potentials, the system hardware is described in greater detail in [10,15]. In this work, a varying As the number of particles N is varied from N = 3 to N = 5 (and beyond, numerically), the circle radius R is increased to maintain constant the distance between bead centers d = 2R sin(π/N ) = 8 µm.
(b) Images showing one snapshot of the system, where the particles are highlighted in red. Particle positions after 20 frames (green) and 40 frames (blue) are overlayed. Videos are available as SM, and show clearly the N = 3 system performing in-phase oscillations, the N = 4 with neighbors in anti-phase, and N = 5 with phase locking between neighbors. The optical tweezers system performs image analysis on every frame to determine particle positions, and thus implement the "geometric switch" condition described in the text, at rates between 200 and 300 frames/s. Each colloidal particle is effectively a phase oscillator, undergoing a motion bound in amplitude but free in period and phase.
number of silica beads of radius a = 1.5 µm (Bangslabs) are trapped from below by a time-shared laser beam, focussed by a water immersion objective (Zeiss, Achroplan IR 63x/0.90 W). A pair of acousto-optical deflectors (AOD) allows the positioning of the laser beam in the (x, y) plane with sub-nanometer precision; time-sharing is at a rate ∼ 10 5 Hz, corresponding to negligible diffusion of the beads in each laser cycle. The solvent in which the beads are suspended is a glycerol (Fisher, Analysis Grade) in water (Ultrapure grade, ELGA) solution 50% w/w, giving a nominal viscosity of η = 6 mPa s at 20 o C [16]. Experiments are performed in a temperature controlled laboratory, T = 21 o C. The trapping plane is positioned (20 ± 1) µm above the flat bottom of the sample, in a sample volume that is around 100 µm thick.
To realize the "geometric switch" condition, an active driving of each colloid is implemented here, similarly to [10], but for 3,4 or 5 particles, and driving the colloids on segments tangential to the ring on which they are positioned on average, see Figure 1(a). A boundary is set at a pre-defined particle position, a distance ξ from the minimum of the currently active optical trap. The trapped particle moves (on average) towards the trap minimum and, when it crosses the boundary, the current trap is switched off and the other trap, with its minimum a distance λ away, is activated. Therefore the amplitude of oscillations is λ−2ξ. This process is implemented via image analysis and feedback to the AOD for laser deflection in the experiments (and as a condition in the numerical simulations). Colloidal particles are always being driven, and out of equilibrium in the sense that they do not reach the minimum of the active trap.
The optical trap potential is harmonic to a very good approximation, with stiffness κ in the range 1.0 to 2.6 pN/µm, depending on the number of beads trapped (each time, calibrated with precision ±0.2 pN/µm). The relaxation time τ 0 = γ/κ is of the order of 0.1s. The experiments have been performed with λ = 2 µm, ξ = 0.31 µm and d = 8 µm. The period of an isolated oscillator is T 0 = 2τ 0 log[(λ − ξ)/ξ] [10], which under the experimental conditions is about 0.3s. With image acquisition through an AVT Marlin F-131B CMOS camera, operating at shutter aperture time of 1.5 ms, and frame rate between 200 and 300 fps (depending on the ring size, hence captured region of interest) there are multiple frames captured within the relevant timescales τ 0 , T 0 . Since the configuration is analysed experimentally only at each frame, the corresponding time interval should be considered as a feedback time. Video is acquired for over 4 minutes, i.e. over 48000 frames. There is typically a transient lasting around a few periods before the systems reach the steady state discussed below. In addition to the driven motion, colloids are affected by stochastic thermal fluctuations fluctuations and by the net flow induced by all other moving particles. Experiments are performed increasing the number of beads N , maintaining constant the arc-distance between neighboring beads as shown in Fig 1(b). Figure 2(a) shows that for N = 3 the three beads move in phase with each other, whilst for N = 4 (Fig. 2(c)) the nearest neighbors are in anti-phase. The behavior of N = 5 (Fig. 2(e)) is an apparently more complex phase locking. To understand the character of the steady state solution we explored the properties of the Oseen coupling tensor, which depends on the geometry of the mean particle arrangement. Taking a step back, we recall that in passive systems the theoretical description of the fluctuations of many-bead ring systems held in fixed position optical traps was calculated recently [14], extending original ideas by Polin, Grier and Quake [17,18] and [19]. A system of N beads held in static traps in two dimensions has 2N normal modes, and these can be calculated ana- lytically for configurations with high symmetry [14,18]. The active driving of each bead along its tangent direction can be thought as introducing a constraint for each bead to move only along its fixed tangential direction. This reduces the number of modes by half, so that there are N normal modes for a N bead system. They can be obtained by projecting the two dimensional system of equations of motion [14] onto the tangent vectors. Similarly to previous analysis of passive geometries [14,18,19] the motion of the j th -particle originates a force on the
i−particle f i,j α = (H −1 ) α,β i,j v (j)
β that depends on the whole configuration, where H is the Oseen tensor [13]. This leads to a system of equations: in which the coupling force scales as a/d, and the force F i acting on the i th -particle is harmonic and tangent to the ring, with the "geometric switch" rule,
F i − n j=1 H −1 i,j d rj (t) dt + f i (t) = 0, i = 1, 2 · · · , n r i (t) · t( r i ) = 0, t(θ) = − sin θ cos θ ,(1)F i = −κ x i (t) ∓ λ 2 t 2π(i − 1) n ,(2)
where ± λ 2 is the coordinate of the bottom of the harmonic well. The stochastic force f i (t) in Eq. 1 represents the thermal noise on the i th -particle, and it can be assumed that [18]. t( r i ) is a versor tangent to the ring, at the position r i , with anti-clockwise direction. The system of eq. 1 can be linearized for small displacements. For symmetric arrangements the coupling matrix is particularly simple, and the eigenvectors can be obtained readily [20]. The eigenvectors are the normal modes of the coupled system.
< f i (t) >= 0, < f i (t 1 )f j (t 2 ) >= 2k B T (H −1 ) ij δ(t 1 − t 2 )
Knowing analytically the normal modes of the system constrained to tangential motion, it is possible to decompose the dynamical steady state into projections onto each of the modes, and look at the mode evolution in between switch events. Experimentally it is clear that for N = 3 and N = 4, there is a single mode which has very high amplitude (Fig 2(b,d)). In contrast, for N = 5 two modes have high amplitude, and they alternate periodically (Fig 2(f)). This behavior is confirmed by BD simulations performed as in [10], see figure in Supplementary Materials.
Let us recall that for every even value of N , the eigenmode with longest relaxation time is described by the pairs of adjacent spheres moving tangentially in an antiphase motion, as shown in the companion paper [20]. For odd N with N > 3, the longest relaxation time corresponds to two equivalent modes, in which the motion of neighbors is almost in anti-phase, but on cycling around the ring each particle is time-shifted by +∆T (or −∆T in the equivalent mode) relative to its neighbor. The case N = 3 is unique, in that the analytical theory shows that the longest relaxation mode is one with all the beads moving in phase. In Fig 2 the mode with the highest amplitude is consistently the one with longest relaxation time. Why is the steady state dynamics largely captured by a normal-mode analysis, and how does the longest lived mode determine the steady state dynamics? The hydrodynamic modes are a key step in constructing the solutions of the dynamical system, which are given for each stretch between consecutive switches by linear superposition of the eigenmodes, and can be obtained analytically for the deterministic (absence of noise) system [20]. The faster the mode, the more its amplitude has decayed before each geometric switch. In other words, it is the amplitude of the longest lived mode that "dominates" at the geometric switch, and thus enforces the overall character of the steady state solution.
In the presence of increasing thermal noise, the bead trajectories and mode amplitudes can deviate from the solutions of the deterministic system, as shown in Fig. 3, but the solutions remain stable. For N = 7, like N = 5 and all other odd N > 3, there is a degenerate fundamental mode [20], and the dynamic state shows an alternating amplitude of the projections onto these two modes. The trajectories display a fixed phase relation between beads, and follow each other in the order 1, 3, 5, 7, 2, 4, 6 in Fig. 3(a,c), and in the equivalent sequence under time reversal in the simulation of Fig. 3(e). The nearest neighbors are almost in anti-phase, but delayed by the small interval T N /N . Next-nearest neighbors are almost inphase, with a delay 2T N /N , and so on, describing a propagating wave. Depending on the initial conditions, at low noise the system will fall into a state where intervals are either positive or negative going around the ring. This dynamical state has a propagating phase. At higher noise (i.e. higher temperature, or weaker coupling), whilst the system remains overall synchronized, the propagating character is lost over long times because the system is able to flip between the two equivalent states.
The simple actively driven oscillator experiments presented here highlight the importance of geometry in determining the leading properties of the collective steady state. One may envision that in biological systems, which present complex disordered arrangements, and a vast number of oscillators, the behavior highlighted by these simple arrays could represent the local dynamics in tightly coupled sub-systems. In this perspective, the small-system patterns of behavior can be thought of "dynamical motifs", linked simply to the geometrical arrangement of beating elements, and that can be analyzed to infer the properties of the individual oscillators.
online) Regular arrays of actively driven colloidal particles synchronize into steady collective dynamical states. (a) sketch of experiment.
FIG
. 2. (Color online) The active tangential motion of beads is strongly correlated in experiments. Shown here are the displacements x(t) relative to the mean position, for systems of N = 3, 4, 5 in (a), (c), (e) respectively. The color code red, green, blue, cyan, magenta identifies different beads, anticlockwise. In (b), (c), (d) the instantaneous projections pm(t) m = 1, 2, ...N on the eigenvectors of the coupling matrix show the important characteristic that the mode with longest relaxation time (red) dominates the displacement configuration.For N = 5, as for larger odd-numbered systems, there are two degenerate modes (red, green), which are seen to alternate in amplitude. The color code indicates decreasing relaxation time: red, green, blue, cyan, magenta.
FIG
. 3. (Color online) The dynamic state converges to the analytical prediction in the limit of no Brownian noise. In numerical simulation it is straightforward to tune the level of noise. Here, the trajectories and mode projections of a system of 7 particles are shown for temperatures of T =1K (a,b), 273K (c,d) and 1000K (e,f), with other physical parameters close to the experiments and κ=2 pN/µm, ξ = 0.3 µm. Colors match those in Figures 2, plus yellow and black used for beads (and modes) 6 and 7 respectively.
Supplementary MaterialFigure 2in the Letter. These displacements obtained computationally show that the description of hydrodynamic coupling via Oseen's tensor is valid.
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|
[] |
[
"High energy properties of X-ray sources observed with BeppoSAX",
"High energy properties of X-ray sources observed with BeppoSAX"
] |
[
"F Frontera \nDipartimento di Fisica\nUniversità di Ferrara\nVia Paradiso 1244100FerraraItaly\n\nIstituto Tecnologie e Studio Radiazioni Extraterrestri (TeSRE)\nvia Gobetti 10140129BolognaC.N.RItaly\n",
"D Dal Fiume \nIstituto Tecnologie e Studio Radiazioni Extraterrestri (TeSRE)\nvia Gobetti 10140129BolognaC.N.RItaly\n",
"G Malaguti \nIstituto Tecnologie e Studio Radiazioni Extraterrestri (TeSRE)\nvia Gobetti 10140129BolognaC.N.RItaly\n",
"L Nicastro \nIstituto Tecnologie e Studio Radiazioni Extraterrestri (TeSRE)\nvia Gobetti 10140129BolognaC.N.RItaly\n",
"M Orlandini \nIstituto Tecnologie e Studio Radiazioni Extraterrestri (TeSRE)\nvia Gobetti 10140129BolognaC.N.RItaly\n",
"E Palazzi \nIstituto Tecnologie e Studio Radiazioni Extraterrestri (TeSRE)\nvia Gobetti 10140129BolognaC.N.RItaly\n",
"E Pian \nIstituto Tecnologie e Studio Radiazioni Extraterrestri (TeSRE)\nvia Gobetti 10140129BolognaC.N.RItaly\n",
"F Favata \nAstrophysics Division\nSpace Science Department of ESA\nESTEC\nKeplerlaan 12200 AGNoordwijkThe Netherlands\n",
"A Santangelo \nIstituto Fisica Cosmica e Applicazioni all'Informatica (IFCAI)\nC.N.R., via La Malfa 15390146PalermoItaly\n"
] |
[
"Dipartimento di Fisica\nUniversità di Ferrara\nVia Paradiso 1244100FerraraItaly",
"Istituto Tecnologie e Studio Radiazioni Extraterrestri (TeSRE)\nvia Gobetti 10140129BolognaC.N.RItaly",
"Istituto Tecnologie e Studio Radiazioni Extraterrestri (TeSRE)\nvia Gobetti 10140129BolognaC.N.RItaly",
"Istituto Tecnologie e Studio Radiazioni Extraterrestri (TeSRE)\nvia Gobetti 10140129BolognaC.N.RItaly",
"Istituto Tecnologie e Studio Radiazioni Extraterrestri (TeSRE)\nvia Gobetti 10140129BolognaC.N.RItaly",
"Istituto Tecnologie e Studio Radiazioni Extraterrestri (TeSRE)\nvia Gobetti 10140129BolognaC.N.RItaly",
"Istituto Tecnologie e Studio Radiazioni Extraterrestri (TeSRE)\nvia Gobetti 10140129BolognaC.N.RItaly",
"Istituto Tecnologie e Studio Radiazioni Extraterrestri (TeSRE)\nvia Gobetti 10140129BolognaC.N.RItaly",
"Astrophysics Division\nSpace Science Department of ESA\nESTEC\nKeplerlaan 12200 AGNoordwijkThe Netherlands",
"Istituto Fisica Cosmica e Applicazioni all'Informatica (IFCAI)\nC.N.R., via La Malfa 15390146PalermoItaly"
] |
[] |
We report on highlight results on celestial sources observed in the high energy band (> 20 keV) with BeppoSAX . In particular we review the spectral properties of sources that belong to different classes of objects, i.e., stellar coronae (Algol), supernova remnants (Cas A), low mass X-ray binaries (Cygnus X-2 and the X-ray burster GS1826-238), black hole candidates (Cygnus X-1) and Active Galactic Nuclei (Mkn 3). We detect, for the first time, the broad-band spectrum of a stellar corona up to 100 keV; for Cas A we report upper limits to the 44 Ti line intensities that are lower than those available to date; for Cyg X-2 we report the evidence of a high energy component; we report a clear detection of a broad Fe K line feature from Cyg X-1 in soft state and during its transition to hard state; Mkn 3 is one of several Seyfert 2 galaxies detected with BeppoSAX at high energies, for which Compton scattering process is important.
|
10.1016/s0920-5632(98)00227-8
|
[
"https://arxiv.org/pdf/astro-ph/9802078v1.pdf"
] | 16,719,409 |
astro-ph/9802078
|
b92b9d00e1ee02be2dcfd77fb567fd883862a344
|
High energy properties of X-ray sources observed with BeppoSAX
6 Feb 1998
F Frontera
Dipartimento di Fisica
Università di Ferrara
Via Paradiso 1244100FerraraItaly
Istituto Tecnologie e Studio Radiazioni Extraterrestri (TeSRE)
via Gobetti 10140129BolognaC.N.RItaly
D Dal Fiume
Istituto Tecnologie e Studio Radiazioni Extraterrestri (TeSRE)
via Gobetti 10140129BolognaC.N.RItaly
G Malaguti
Istituto Tecnologie e Studio Radiazioni Extraterrestri (TeSRE)
via Gobetti 10140129BolognaC.N.RItaly
L Nicastro
Istituto Tecnologie e Studio Radiazioni Extraterrestri (TeSRE)
via Gobetti 10140129BolognaC.N.RItaly
M Orlandini
Istituto Tecnologie e Studio Radiazioni Extraterrestri (TeSRE)
via Gobetti 10140129BolognaC.N.RItaly
E Palazzi
Istituto Tecnologie e Studio Radiazioni Extraterrestri (TeSRE)
via Gobetti 10140129BolognaC.N.RItaly
E Pian
Istituto Tecnologie e Studio Radiazioni Extraterrestri (TeSRE)
via Gobetti 10140129BolognaC.N.RItaly
F Favata
Astrophysics Division
Space Science Department of ESA
ESTEC
Keplerlaan 12200 AGNoordwijkThe Netherlands
A Santangelo
Istituto Fisica Cosmica e Applicazioni all'Informatica (IFCAI)
C.N.R., via La Malfa 15390146PalermoItaly
High energy properties of X-ray sources observed with BeppoSAX
6 Feb 19981
We report on highlight results on celestial sources observed in the high energy band (> 20 keV) with BeppoSAX . In particular we review the spectral properties of sources that belong to different classes of objects, i.e., stellar coronae (Algol), supernova remnants (Cas A), low mass X-ray binaries (Cygnus X-2 and the X-ray burster GS1826-238), black hole candidates (Cygnus X-1) and Active Galactic Nuclei (Mkn 3). We detect, for the first time, the broad-band spectrum of a stellar corona up to 100 keV; for Cas A we report upper limits to the 44 Ti line intensities that are lower than those available to date; for Cyg X-2 we report the evidence of a high energy component; we report a clear detection of a broad Fe K line feature from Cyg X-1 in soft state and during its transition to hard state; Mkn 3 is one of several Seyfert 2 galaxies detected with BeppoSAX at high energies, for which Compton scattering process is important.
INTRODUCTION
High energy properties of celestial X-ray sources give important information to understand their radiation mechanisms and the energetic processes occurring in them and/or in their environments.
Hard X-ray emission (> 20 keV) is currently observed from several classes of X-ray sources. Galactic X-ray sources that are known emitters of hard X-rays include black-hole candidates, X-ray pulsars, weak-magnetic-field neutron stars in Low Mass X-ray Binaries (LMXRBs), mainly X-ray bursters (XRBs), Cataclismic Variables (CV), in particular Polars, Crab-like supernova remnants.
High energy spectra of black-hole candidates (BHC) have permitted to infer the presence of Comptonization processes of soft photons occurring close to the black-hole (e.g., in a disk corona).
XRBs, that are weakly magnetized neutron stars (with surface field intensity B ≤ 10 10 -10 11 G), turned out to be hard X-ray emitters, once the sensitivity of the high energy instruments was increased at the 10 mCrab level (see [1] for a recent review). From their spectral properties, similarities with and diversities from BHCs have been inferred, like the presence of an accretion disk that can extend, as in the case of BHC, close to the surface of the compact object, and the presence of an additional component of soft photons that, unlike in BHCs, originates from the neutron star surface and can be a major source of thermal emission and electron cooling through Comptonization.
X-ray pulsars are well known emitters of hard X-rays. Observations in the hard X-ray band are relevant in order to get a measurement of the magnetic field intensity at the neutron star surface. Even if current models of the X-ray spectrum of these objects are still unsatisfactory at high energies, the measurement of cyclotron resonance features gives a direct estimate of the inten- sity of the neutron star magnetic field strength [2].
Emission from young shell-like supernova remnants mainly extends to low energies (< 20 keV). Detection of hard X-rays with determination of their spectral properties can provide important information on the emission mechanism (thermal vs. non thermal, like synchrotron radiation).
From stellar coronae, apart the Sun, hard Xrays have never been observed. As we will see, this gap has been filled with BeppoSAX .
Among the extragalactic X-ray sources, hard X-ray emission is observed from Active Galactic Nuclei (AGNs), that include Seyfert galaxies of both types (1 and 2), radio quiet QSOs and radio loud QSOs (which include blazars). A great effort is currently under way to interpret the different classes of AGNs in a unified scheme, which is sketched in Fig. 1. The basic energy production mechanism is accretion of matter onto a massive black hole (≈ 10 8 M ⊙ ) via an accretion disk. A massive toroid of larger radius (in the range from several parsecs to few tens of parsecs [3]) is assumed to surround the accretion disk. Depending on the configuration of the disk with respect to the toroid, on their relative sizes and distances, and on the viewing angle, an AGN should show different observational features and thus fall in one of the different classes above mentioned.
The above scheme is being tested also for stellar-mass black holes (e.g., Cygnus X-1). Thus the unified scheme can be a general picture to interpret galactic and extragalactic black holes, accreting matter via an accretion disk. Given many similarities in the X-ray emission from stellar mass BHs with low-magnetic-field neutron stars in LMXRBs, the unified scheme now applied to AGNs could be extended to several classes of Xray sources. Hard X-ray spectral properties of these sources can provide unique information to diagnose the presence of a black hole versus a weak-magnetic-field neutron star, to test the unified model for AGNs and its validity for stellar mass BHCs.
Thank to a broad energy band of operation (0.1-300 keV) and a uniform flux sensitivity in this range, BeppoSAX [5] has the unmatched capability of simultaneously sampling the spectrum of X-ray sources over more than three decades of energy.
The SAX/PDS instrument [6], with a sensitivity of about 1 mCrab at 100 keV, allows an accurate determination of the spectrum of many Xray sources at the highest energies (13-200 keV).
For the brightest sources (> 10 mCrab), the HPGSPC instrument (6-60 keV) [7] provides the spectral coverage necessary to match the information provided by the low energy instruments LECS (0.1-10 keV) [8] and MECS (2-10 keV) [9] telescopes and the PDS.
Here we review some relevant results obtained with BeppoSAX during its Performance Verification Phase and first year Core Program, with particular focus on the PDS instrument. The spectral deconvolution was performed with the XSPEC software package, by using the instrument response function distributed from the Bep-poSAX Scientific Data Center. BeppoSAX . Superposed is the best fit thermal model [12].
HIGHLIGHT RESULTS
The reviewed sources span a large range of intensities and include a stellar corona (Algol), a supernova remnant (Cas A), two LMXRBs, one of which (Cygnus X-2) is a Z source and the other (GS1826-634) is an X-ray burster, a BHC (Cygnus X-1) observed in two different spectral states and the Seyfert 2 galaxy Mkn 3.
Algol
The general nature of stellar coronae as a class of thermal soft X-ray emitters was already established with the Einstein observatory [10]. The extensive observations with soft X-ray telescopes (Einstein, ROSAT) has shown that the typical peak plasma temperatures in the more active sources are of the order of a few keV, although during intense flares their X-ray luminosity and coronal temperatures increases strongly, leading to the expectation that hard X-rays may be detected. This phenomenon has been confirmed during the recent observation with BeppoSAX of Algol [11]. Algol is a binary system composed by a B8V and a K2IV star, with a binary period of 2.9 days. BeppoSAX observed the complete evolution of a large flare, lasting about 1 day. During the flare the soft (0.1-10 keV) X-ray luminosity increased by a factor more than 20. The source spectrum measured with LECS and PDS instruments during the peak of the flare is shown in Fig. 2. The flare spectrum is reasonably described with a two component emission model from a hot diffuse gas (MEKAL model in XSPEC) [12]. The preliminary analysis shows no evidence for the presence of a non-thermal component, up to highest observed energies. The characteristic temperatures of the flare spectrum are of the order of ∼ 2 × 10 7 and ∼ 1 × 10 8 • K. Further details of this observation are described in [11].
Cas A
The supernova remnant Cas A was observed with BeppoSAX on August 6, 1996. A nonthermal high-energy component in the X-ray emission from the source has been clearly detected [13]. The broad-band (0.5-100 keV) X-ray spectrum of the source (see Fig. 3) is modeled using the sum of three components: one Non-Equilibrium of Ionization (NEI) plasma component representing the emission from the ejecta; one NEI component representing the emission from the shocked material surrounding the circumstellar medium; a power law (PL) component to model hard X-ray emission. Best fit parameter values of the PL are a photon index of 2.95 +0.10 −0.05 and a normalization parameter of 0.69 ph cm −2 s −1 keV −1 at 1 keV. The power law high energy component is very likely of synchrotron origin [14]. It gives a sizeable contribution at lower energies, being comparable in intensity to the thermal continuum at the position of the Fe K complex.
The search for radioactive emission lines due to 44 Ti formed during the supernova explosion gives a 2σ upper limit to the intensity of the line at 68 keV of 1.3 × 10 −5 ph cm −2 s −1 and an upper limit a factor about 10 times lower (4.4 × 10 −6 ph cm −2 s −1 ) at 78 keV. A 99% upper limit of 8.6 ×10 −5 ph cm −2 s −1 for both lines was given by [15] with OSSE and a similar value was obtained with RXTE [16].
CYGNUS X-2
The LMXRB Cygnus X-2 is a member of a binary system (binary period of 9.84 days), consisting of a low-magnetic-field neutron star and a late type low-mass (∼ 0.7 M ⊙ ) star V1341 Cyg [17,18,19]. It is classified as Z source, on the basis of its X-ray colour-colour spectral behaviour [20,21]. Its low-energy continuum spectrum, measured with the EXOSAT satellite, can be fitted with a Comptonization model (kT = 3.7 keV, optical depth τ = 9.4) plus a black body component (kT = 1.21 keV) [22]. An emission Fe K line was first detected with the Tenma [23] and EXOSAT [24] satellites and, later on, it was resolved with the Broad Band X-ray Telescope Past high energy observations of Cygnus X-2 [26] showed that only a small fraction (about 1%) of the total source luminosity is emitted in hard X-rays.
BeppoSAX observed Cygnus X-2 on July 23, 1996, during the Science Verification Phase. Results obtained with the LECS instrument have been already published [27]. Here we report preliminary results of the same observation obtained with MECS and PDS. The on-source exposure times were about 40 ks for MECS and about 20 ks for PDS. Figure 4a shows the source spectrum in the 2-200 keV energy band. The broad-band spectrum is not well fit by the low-energy model described above (see Fig. 4b), yielding a reduced χ 2 ν of 1.62 for 179 degrees of freedom (dof). By adding a power law component to fit the high energy excess χ 2 ν decreases to 1.51. The fit is still not satisfactory and thus a more suitable model has to be worked out, but this preliminary result shows that a high energy component is very likely present in the data. The preliminary model components and fit parameters are the following: soft blackbody with kT bb ≃ 1.5 keV, Sunyaev and Titarchuk [28] Comptonization model with kT ST = 3.3 keV and optical depth τ ST ≈ 9 keV, high energy power law model with photon index α = 1.9 ± 0.7.
We estimated a possible contribution from the galactic ridge to the observed hard tail in the Cygnus X-2 spectrum. Using the results from Yamasaki et al. [29], we conclude that this contribution is a very small fraction of our 10-100 keV flux.
GS1826-238
GS1826-238 was discovered in September 1988 with Ginga [30] at a flux level of 26 mCrab in the 1-40 keV energy band with a hard power law spectrum (photon index Γ = 1.7). Later on, the source was detected at a 7σ significance level with OSSE above 50 keV with a steep power law spectrum (Γ = 3.1 ± 0.5) [31]. The source was optically identified with a V19.3 star [32]. Given that its erratic flux time variability is reminiscent of that exhibited by Cygnus X-1, the source was classified by Tanaka and Lewin [33] as a black hole candidate. On March 31, 1997, the Wide Field Cameras (WFC) [34] aboard BeppoSAX detected three X-ray bursts from the source [35], suggesting that we are in presence of a weak magnetic field neutron star in a LMXRB.
BeppoSAX again observed the source on October 6-7, 17-18, and 27, 1997 as a target of opportunity (TOO). The first observation was triggered by a hard X-ray outburst with a peak flux of about 100 mCrab observed with the BATSE experiment aboard CGRO. We report here preliminary results obtained during the first observation. The exposure time was 7.7 ks for LECS, 21.7 ks for MECS and 10 ks for PDS. The 2-10 keV flux level from the source was 5.8 × 10 −10 erg cm −2 s −1 , while at higher energies (20-100 keV) was 7.9 × 10 −10 erg cm −2 s −1 . Figure 5 shows the broad-band count rate spectrum of the source. A best fit to the data is obtained with an absorbed blackbody (bb) plus a PL with an exponential cut-off. The best fit parameters are the following: N H = 5.1×10 21 cm −2 , kT bb ≈ 1.21 keV, PL photon index Γ ≈ 1.5, high energy cut-off parameter approximately equal to 40 keV. X-ray bursts were observed during both the first and the second TOOs.
CYGNUS X-1
Cygnus X-1 is the most convincing example of a binary system that hides a stellar mass BH. As discussed above, AGNs are the best candidate objects to contain massive BH. The Compton reflection model used for AGNs is suggested to hold for Cygnus X-1 as well (see, e.g., [36,37]). A test for the validity of this model for Cygnus X-1 is the presence of a broad Fe K emission line in the X-ray spectrum of the source as a result of fluorescence from the disk. This line was actually detected with EXOSAT/GSPC [38] during a hard X-ray state of the source, but not confirmed in high resolution observations of Cygnus X-1 with the BBXRT [39] and ASCA [40] missions.
BeppoSAX observed the source three times during 1996, on June 22 and 25, during which the source was in soft state (SS) and on September 12, when the source was going back to its normal hard X-ray state (HS). Final results of these observations will be reported elsewhere [41].
Here we report preliminary results on the Fe K line feature. A broad line is observed in all the above observations. Figure 6 shows the count rate spectrum in the 2-10 keV band measured with the MECS telescopes during the 25 June observation. Only during the first SS observation a Fe K edge is clearly detected (E edge = 7.6 keV, τ max = 0.24). A reflection component in the continuum spectrum is detected with the HPGSPC detector (8.5-60 keV) in both SS and HS. Using a reflection model for the MECS continuum spectrum around the Fe line, with the reflection parameters estimated from HPGSPC, one obtains for the spectral feature during the second SS observation a best fit with a disk-line model with rest energy at 6.4 keV (χ 2 ν = 1.1). In the other observations, a Gaussian line profile gives better fits, independently of the continuum used (power law with or without a reflection component). The equivalent width is highest in the first observation (about 1 keV) and lower (∼ 300 eV) in the other observations, independently of the source state. These results, if interpreted in terms of a Fe K fluorescence from an accretion disk around a BH, require a different dimension of the emission region.
Mkn 3
Mkn 3 (z = 0.0135) is a type 2 Seyfert first detected in X-rays by Ginga [42]. It was only marginally detected above 50 keV with the OSSE experiment aboard the Compton Gamma Ray Observatory [43].
It was observed with BeppoSAX on November 26,1996. More details of this observation and results can be found elsewhere [44]. The flux level measured in the low (2-10 keV) energy band is 6 × 10 −12 erg cm −2 s −1 , corresponding to 0.28 mCrab, while that in the high (20-100 keV) band is 1.3 × 10 −10 erg cm −2 s −1 , corresponding to 7.6 mCrab, with a ratio (High/Low) = 27. The high energy spectrum has been determined up to 200 keV. Figure 7 shows the count rate spectrum of the source in the 0. AGN [44].
CONCLUSIONS
Observations of celestial sources with Bep-poSAX show new key results on their high energy (> 20 keV) spectral properties.
Strong high energy emission has been detected from Algol during 1 day long flare. The emission appears to be the tail of the thermal X-ray radiation.
LMXRBs are known sources of low energy Xray emission, but the high energy emission is well known only for a small part of them (see a recent review by [1]). We detected for the first time a non thermal high energy component from the Z source Cygnus X-2 and derived, for the first time with the same satellite, the broadband (0.1-200 keV) photon spectrum of an X-ray burster during a transient hard X-ray outburst.
We expect to be able, using the three observations available, to study the relative behaviour of the high energy components with respect to the low-energy one as a function of the high energy flux of GS1826-238.
We have reported the clear detection of a broad Fe K line feature from Cygnus X-1 during a soft state of the source and we have discussed its behaviour for different spectral states. The Fe fluorescence emission does not appear to be consistent with a constant disk emission region.
As far as the AGNs are concerned, the detection of power laws with strong absorptions at lower energies and reflection components, is now observed for several Seyfert 2 galaxies. We have shown the outstanding example of Mkn 3.
Figure 1 .
1Unified model for AGNs. Adapted from[4].
Figure 2 .
2Count rate spectrum of Algol during the maximum of a long flare as observed by
Figure 3 .
3Deconvolved spectrum of Cas A. The different components are shown. From[13].
Figure 4 .
4Top: 2-200 keV count rate spectrum of Cygnus X-2 with superposed the best fit model (see text). Bottom: residuals with respect to the model. (BBXRT) which detected a broad (FWHM = 0.97 keV) line with centroid energy 6.71 keV, and EW = 60 ± 27 eV [25].
Figure 5 .
5Top: Broad band count rate spectrum of GS1826-238 (1st TOO) with superposed the best fit model (see text). Bottom: residuals with respect to the model.
Figure 6 .
6Top: Count rate spectrum of Cygnus X-1 measured by MECS during the June 25, 1996 observations. The other BeppoSAX instruments were switched off. Superposed is the best fit power law model to the continuum (excluded from the fit the 4.5-8.5 keV band). Bottom: excesses from the the model. The broad Fe K line is apparent with no clear evidence of a K edge.
6-200 keV energy band. Preliminary results indicate the presence of a soft excess, a prominent Iron Kα complex and a 15-50 keV hump. The heavily (N H ≈ 10 24 cm −2 ) absorbed primary ionization provides only a few percent of the 2-10 keV flux, while it dominates above 10 keV, in the PDS band. This result can be interpreted in terms of the unified model of
Figure 7 .
7BeppoSAX spectrum of Mkn 3 fitted with an absorbed power law. The residuals indicate a strong Fe K line plus a 15-50 keV hump.
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[] |
[
"X-RAY PROPERTIES OF INTERMEDIATE-MASS BLACK HOLES IN ACTIVE GALAXIES. II. X-RAY-BRIGHT ACCRETION AND POSSIBLE EVIDENCE FOR SLIM DISKS",
"X-RAY PROPERTIES OF INTERMEDIATE-MASS BLACK HOLES IN ACTIVE GALAXIES. II. X-RAY-BRIGHT ACCRETION AND POSSIBLE EVIDENCE FOR SLIM DISKS"
] |
[
"Louis-Benoit Desroches ",
"Jenny E Greene ",
"Luis C Ho "
] |
[] |
[] |
We present X-ray properties of optically-selected intermediate-mass (∼ 10 5 -10 6 M ⊙ ) black holes (BHs) in active galaxies (AGNs), using data from the Chandra X-Ray Observatory. Our observations are a continuation of a pilot study byGreene & Ho (2007c). Of the 8 objects observed, 5 are detected with X-ray luminosities in the range L 0.5−2 keV = 10 41 -10 43 erg s −1 , consistent with the previously observed sample. Objects with enough counts to extract a spectrum are well fit by an absorbed power law. We continue to find a range of soft photon indices 1 < Γ s < 2.7, where N (E) ∝ E −Γs , consistent with previous AGN studies, but generally flatter than other narrow-line Seyfert 1 active nuclei (NLS1s). The soft photon index correlates strongly with Xray luminosity and Eddington ratio, but does not depend on BH mass. There is no justification for the inclusion of any additional components, such as a soft excess, although this may be a function of the relative inefficiency of detecting counts above 2 keV in these relatively shallow observations. As a whole, the X-ray-to-optical spectral slope α ox is flatter than in more massive systems, even other NLS1s. Only X-ray-selected NLS1s with very high Eddington ratios share a similar α ox . This is suggestive of a physical change in the accretion structure at low masses and at very high accretion rates, possibly due to the onset of slim disks. Although the detailed physical explanation for the X-ray loudness of these intermediate-mass BHs is not certain, it is very striking that targets selected on the basis of optical properties should be so distinctly offset in their broader spectral energy distributions.
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10.1088/0004-637x/698/2/1515
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[
"https://arxiv.org/pdf/0903.2257v1.pdf"
] | 14,940,781 |
0903.2257
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10b3af4ac5cc737d0181dd9cba2530c961777aa8
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X-RAY PROPERTIES OF INTERMEDIATE-MASS BLACK HOLES IN ACTIVE GALAXIES. II. X-RAY-BRIGHT ACCRETION AND POSSIBLE EVIDENCE FOR SLIM DISKS
12 Mar 2009 REVISED MARCH 12, 2009;
Louis-Benoit Desroches
Jenny E Greene
Luis C Ho
X-RAY PROPERTIES OF INTERMEDIATE-MASS BLACK HOLES IN ACTIVE GALAXIES. II. X-RAY-BRIGHT ACCRETION AND POSSIBLE EVIDENCE FOR SLIM DISKS
12 Mar 2009 REVISED MARCH 12, 2009;revised March 12, 2009; submitted to the Astrophysical JournalSUBMITTED TO THE Astrophysical Journal Preprint typeset using L A T E X style emulateapj v. 10/09/06Subject headings: galaxies: active -galaxies: nuclei -galaxies: Seyfert -galaxies: statistics -X-rays: galaxies
We present X-ray properties of optically-selected intermediate-mass (∼ 10 5 -10 6 M ⊙ ) black holes (BHs) in active galaxies (AGNs), using data from the Chandra X-Ray Observatory. Our observations are a continuation of a pilot study byGreene & Ho (2007c). Of the 8 objects observed, 5 are detected with X-ray luminosities in the range L 0.5−2 keV = 10 41 -10 43 erg s −1 , consistent with the previously observed sample. Objects with enough counts to extract a spectrum are well fit by an absorbed power law. We continue to find a range of soft photon indices 1 < Γ s < 2.7, where N (E) ∝ E −Γs , consistent with previous AGN studies, but generally flatter than other narrow-line Seyfert 1 active nuclei (NLS1s). The soft photon index correlates strongly with Xray luminosity and Eddington ratio, but does not depend on BH mass. There is no justification for the inclusion of any additional components, such as a soft excess, although this may be a function of the relative inefficiency of detecting counts above 2 keV in these relatively shallow observations. As a whole, the X-ray-to-optical spectral slope α ox is flatter than in more massive systems, even other NLS1s. Only X-ray-selected NLS1s with very high Eddington ratios share a similar α ox . This is suggestive of a physical change in the accretion structure at low masses and at very high accretion rates, possibly due to the onset of slim disks. Although the detailed physical explanation for the X-ray loudness of these intermediate-mass BHs is not certain, it is very striking that targets selected on the basis of optical properties should be so distinctly offset in their broader spectral energy distributions.
INTRODUCTION
Supermassive black holes (BHs), with masses of ∼ 10 6 -10 9 M ⊙ , exist at the center of nearly all elliptical galaxies and galaxy bulges, as determined from stellar and gas dynamics, and from the presence of actively accreting galactic centers (active galactic nuclei; AGNs). An important problem yet to be resolved in cosmological galaxy evolution is understanding the creation and growth of "seed" BHs. Stellar-mass BHs, the end product of massive stars, have masses of only ≈ 10M ⊙ , leaving a gap of 5 orders of magnitude in BH mass. BHs in this unknown region are often dubbed intermediate-mass BHs (or low-mass galactic BHs). The recent discoveries of previously unknown ≈10 5 M ⊙ galactic BHs are beginning to constrain formation and evolution models of such seed BHs. NGC 4395, a bulgeless late-type spiral galaxy, and POX 52, a spheroidal galaxy, both contain Seyfert 1 AGNs with BH masses estimated to be ≈ 10 5 M ⊙ (Filippenko & Ho 2003;Peterson et al. 2005; Barth et al. 2004).
BH mass (M BH ) correlates strongly with various properties of spheroidal systems, such as luminosity (Kormendy & Richstone 1995) and stellar velocity dispersion (Gebhardt et al. 2000;Ferrarese & Merritt 2000;Tremaine et al. 2002). The evolution of bulges and the growth of BHs are thus likely coupled, although the possible existence of nuclear BHs in bulgeless or nearly bulgeless spiral galaxies (Desroches & Ho 2009) suggests that bulges are not a necessary condition for BH growth. Nuclear intermediate-mass BHs are unfortunately very difficult to detect; the gravitational sphere of influence of a 10 5 M ⊙ BH is unresolvable beyond the Local Group, even with the Hubble Space Telescope. We can rely on AGN signatures, however, to signify the presence of a BH, and use the observed broad emission lines to estimate BH mass ). Greene & Ho (2004) systemically searched the First Data Release of the Sloan Digital Sky Survey (SDSS) and found 19 galaxies with BH estimates of 10 6 M ⊙ , which forms the parent sample of this study.
The homogenous selection of the Greene & Ho sample allows for important broadband, multiwavelength investigations, to determine how spectral properties change with BH mass in this intermediate-mass regime. These objects were found to be radio-faint using the Very Large Array . The X-ray luminosity, from a pilot study of this sample (Greene & Ho 2007c; hereafter referred to as Paper I), ranges from L 0.5−2 keV ≈ 10 41 to 10 43 erg s −1 .
Here we present the rest of the X-ray results for the remaining objects in the sample.
The observations and data analysis are discussed in § 2. We present our results and discuss physical connections to other AGNs in § 3. Finally, we summarize our findings in § 4. We assume a cosmology such that H 0 = 71 km s −1 Mpc −1 , Ω m = 0.27, and Ω Λ = 0.75 (Spergel et al. 2003).
2. X-RAY SAMPLE AND DATA ANALYSIS We observed 8 intermediate-mass BHs from Greene & Ho (2004), which were not observed in Paper I, with the Advanced CCD Imaging Spectrometer (ACIS; Garmire et al. 2003) on board Chandra (Weisskopf et al. 1996). The observations were obtained during Guest Observer Cycle 8 between 2007 March and 2007 August. As in Paper I, images were obtained at the aim point of the S3 CCD in faint mode. We once again read out only 1/8 of the chip, with a minimum read-out time of 0.4 s, to reduce the effects of pile-up. Effective exposure times ranged from 4.98 ks to 5.49 ks.
We use standard type 2 event files, processed by the Chandra X-Ray Center, for further analysis. We use the CIAO (Chandra Interactive Analysis of Observations) task celldetect, with default parameters, to automatically detect and extract centroid positions for each source. Of the 8 objects observed, 5 are detected. One source (GH03) required setting the minimum signal-to-noise (S/N) threshold down to 2 from the default of 3 in order to detect it. The onaxis point-spread function (PSF) of Chandra contains 95% of the encircled energy within 1 ′′ , and so we adopt a 2 ′′ radius aperture and extract background-subtracted counts using the CIAO task dmextract. The background regions are annuli of inner radius 7 ′′ and outer radius 15 ′′ . Counts are extracted in two bands: the soft band (0.5-2 keV; C s ) and the hard band (2-8 keV; C h ). As in Paper I, 1 ′′ extractions are consistent with encircling 95% of the energy, justifying our choice of a 2 ′′ aperture. For those objects without a detection, we measured the background counts in the 2 ′′ aperture, and determined 5 the counts necessary for a theoretical source to be detected by celldetect with our required minimum S/N.
We calculate a "hardness ratio" for our detected objects, defined as H ≡ (C h − C s )/(C h + C s ), as a rough estimate of the spectral shape. We can then use this ratio to infer a soft photon index Γ s , where N (E) ∝ E −Γs , as described by Gallagher et al. (2005) and used in Paper I. To accomplish this, we build artificial spectra with known photon indices and Galactic absorption, and then "observe" them with the same instrumental response as the observations. The response is characterized by the redistribution matrix file (RMF), which modifies the input energy spectrum into the observed distribution of pulse heights due to finite energy resolution, and the auxiliary response file (ARF), which modifies the input spectrum due to the effective area and quantum efficiency of the detectors. The task fakeit within the spectral-fitting package XSPEC (Arnaud 1996) is used to generate artificial spectra with photon indices ranging from Γ = 1 to 3 and with the same Galactic absorption as the true observation, determined from Dickey & Lockman (1990) using WebPIMMS 6 . The hardness ratio is then measured from these artificial spectra and compared to the true ratio. By matching the hardness ratios, we can thus infer a photon index Γ HR . This photon index and known neutral column density N H are then used with WebPIMMS to calculate fluxes from the observed count rates. Results are presented in Table 1. We note that because we do not include an intrinsic neutral column in our calculations, our photon indices represent lower limits to the true value. Given our acceptable spectral fits, however, we see no compelling reason to suspect a significant contribution from such a component (see below and Paper I).
Spectral Fitting
Two of our observed targets (GH12 and GH17) have enough counts (>200) for a reliable spectral fit. We use the same aperture and background region as above to extract a spectrum using the CIAO task psextract, which also gen-erates the appropriate RMF and ARF files. We select a minimum of 20 counts for each energy bin to permit the use of χ 2 statistics. We limit our analysis to 0.3-5 keV and 0.5-5 keV for GH12 and GH17, respectively, to avoid large uncertainties due to detector response and low counts.
Using XSPEC, we fit a simple absorbed power-law model to each spectrum. The value for the absorption is fixed to the Galactic value (Dickey & Lockman 1990). These fits are shown in Figure 1 and Table 2, with 90% confidence errors quoted. In both cases, the reduced χ 2 ν is consistent with 1, indicating a reliable fit with no additional components necessary. The photon indices Γ HR and Γ s are similar in both cases. We also performed fits with the absorption parameter allowed to vary. For GH12, the resulting fit differs by more than the 90% confidence errors. The new fit parameters are N H = (8 ± 2) × 10 20 cm −2 , Γ s = 2.77 ± 0.15, normalization = (3.8 ± 0.7) × 10 −4 photons s −1 keV −1 at 1 keV, and χ 2 ν = 0.88. The differing neutral column density between the two fits may be tentative evidence for an intrinsic absorber, although with χ 2 ν formally less than 1, we may be simply overfitting the data. In the case of GH17, allowing N H to vary results in a similar fit (within uncertainties). In neither case do the data have sufficient depth nor spectral coverage to model any additional components, if present, such as a soft thermal excess. Our model is, as a result, oversimplified but acceptable given the spectral information available.
In Paper I, only GH04 displayed marginal evidence for a soft excess. The absence of evidence for soft excesses in our data may be a function of the relatively soft energies detectable by Chandra in short exposures, in addition to our limited S/N. Miniutti et al. (2009) observed 4 objects in our sample, detected by both Chandra and ROSAT, with XMM-Newton. They found an apparent break in the X-ray spectra of all 4 objects at ≈ 2 keV. An absorbed power-law fit, using only the hard 2-10 keV counts, produces a very noticeable soft excess when extrapolated to softer energies. The hard photon index Γ h (2-10 keV) is also much flatter than our observed soft photon index, with Γ h = 1.76. Thus, with long exposures capable of detecting significant hard X-ray counts, these intermediate-mass BHs appear to behave very similarly to other radio-quiet, type 1 Seyferts and quasars which exhibit very pronounced soft excesses below 1 keV (Boller et al. 1996). The physical explanation for such an excess is still unclear (Miniutti et al. 2009).
3. RESULTS AND DISCUSSION 3.1. Narrow-Line Seyfert 1 Galaxies The Greene & Ho (2004) sample can be classified as narrow-line Seyfert 1 (NLS1) galaxies, a subclass of AGNs, based on the width of the broad permitted lines, in particular FWHM Hβ < 2000 km s −1 . NLS1s are thought to be intermediate-mass BHs radiating at high Eddington ratios (Pounds et al. 1995), the "narrow" broad lines a result of the small virial velocities associated with the intermediate-mass BH. Indeed the M BH and Eddington ratio (L bol /L Edd ) estimates from Greene & Ho (2004) support the picture that this sample is at the low-mass end of the classical NLS1 subclass. Collin et al. (2006) postulate, however, that NLS1s arise instead from high-mass BH systems observed at high inclination, reproducing the "narrow" lines. As noted in Paper I, we cannot rule this out, but the agreement of the Greene & Ho objects with the extrapolation of the M BH -σ ⋆ relation suggests that these are true intermediate-mass BHs (Barth et al. 2005;. The host galaxies are similarly low-mass and low-luminosity . Although some optical spectroscopic properties, such as the strength of Fe II and [O III] lines, differ between classical NLS1s and the Greene & Ho sample, both groups tend to be exceptionally radio-quiet .
Characteristic NLS1 properties, such as FWHM Hβ and the soft X-ray photon index, have been suggested to depend on the BH mass. Boller et al. (1996) and Porquet et al. (2004) have argued in particular that low BH masses are necessary for steep photon indices. In Paper I, however, the observed sample covered a range of 600 ≤ FWHM Hβ ≤ 1800 km s −1 and 1 ≤ Γ s ≤ 3, with a mean Γ s = 2.1 ± 0.2, flatter than the XMM-Newton PG quasar sample of Porquet et al. (2004) despite having lower BH masses (10 5 < M BH /M ⊙ < 10 6.5 ). Including our current sample, the mean becomes Γ s = 2.2 ± 0.1 compared to Γ s = 2.6 ± 0.1 from Porquet et al. (2004) and Γ = 2.58 ± 0.05 measured for general Röntgensatellit (ROSAT) samples (Yuan et al. 1998). As was determined in Paper I, this clearly signifies that low BH mass is not sufficient for a steep soft X-ray power law. Unlike in classic NLS1 samples, the Greene & Ho (2004) sample is not soft-X-ray selected, but rather selected to have low BH mass, thus leading to a wider distribution of Eddington ratios. The high Γ s , low-BH-mass objects from Porquet et al. (2004), however, are also the objects with the highest Eddington ratios.
With X-ray properties of the full Greene & Ho (2004) sample, we can revisit the possible correlations of Γ s with other parameters, such as L 0.5−2 keV , and L bol /L Edd . This is shown in Figure 2. As we have already noted, M BH by itself is a poor indicator of soft photon index. The X-ray luminosity L 0.5−2 keV and Eddington ratio continue to be significantly correlated with Γ s . We compare our observations in Figure 2 with X-ray-weak NLS1s (Williams et al. 2004) and PG quasars (Porquet et al. 2004); the 0.3-2 keV XMM-Newton observations of Porquet et al. are converted to 0.5-2 keV using their derived spectral slopes. Eddington ratios are derived using L Hα to estimate L 5100Å , 2007b, and assuming L bol = 9λL 5100Å , which has a typical scatter of ≈ 0.4 dex (Ho 2008). We use optical luminosities to avoid potentially spurious correlations between Γ s and L X which may arise because we are more sensitive to soft sources with Chandra.
These results are consistent with other published results. For instance, Shemmer et al. (2006) (and references therein) argue for the hard X-ray spectral index Γ h (defined for energies greater than 2 keV) depending primarily on the accretion rate. They find that Γ h increases with increasing L bol /L Edd (ranging from 0.05 to 1.0), qualitatively similar to our results. This has also been discussed by, for instance, Brandt & Boller (1998) and Lu & Yu (1999). A typical explanation for this correlation invokes a high accretion rate driving up the disk temperature, producing more soft disk photons which could Compton cool the corona, reducing the hard X-ray emission and thus steepening the X-ray spectral index (Haardt & Maraschi 1993;Pounds et al. 1995).
NLS1s also exhibit pronounced X-ray variability on short timescales (Boller et al. 1996;Leighly 1999). Our data do not have long enough exposures to make any meaningful variability measurements, although we can use archival data to investigate long-term variability. In Paper I four objects had ROSAT All-Sky Survey (RASS) detections, but only GH01 showed significant variability (factor of ≈ 5) over this ≈ 10 yr timescale in soft X-rays. Two objects in our sample have archival RASS data; GH12 is currently ≈ 2 times brighter whereas GH17 is ≈ 4 times brighter. Thus half of the Greene & Ho sample detected by ROSAT exhibit factor of few variability over decadal timescales, with the other half limited to small-amplitude (<50%) variability.
X-ray-to-Optical Flux Ratio
The ratio of the optical-to-X-ray flux is an important broadband diagnostic of the broader spectral energy distribution. To characterize this ratio, we use α ox , the slope of a hypothetical power law extending from 2500Å to 2 keV (Tananbaum et al. 1979). We adopt the following definition: (Strateva et al. 2005). To obtain a flux density at 2500Å, we use Hα measurements from Greene & Ho (2007b) to determine the AGN flux density at 5100Å . We then assume a powerlaw optical continuum such that f λ ∝ λ −β , with an average β = 1.56 ± 0.1 (Vanden Berk et al. 2001;Greene & Ho 2007a), to calculate f 2500Å . This differs from Paper I in that we do not use the measured L 5100Å , which is potentially affected by galaxy starlight. In Figure 3 we plot α ox vs. the monochromatic luminosity at 2500Å, a well-known correlation (Avni & Tananbaum 1982;Bechtold et al. 2003). We include the X-ray-weak NLS1 sample of Williams et al. (2004), X-ray-selected NLS1s (Grupe et al. 2004), and the PG quasars that are classified as NLS1s according to FWHM Hβ (Boroson & Green 1992), with α ox given by Brandt et al. (2000). We also include upper limits for our non-detections, with the exception of GH19, which is not considered an intermediate-mass BH candidate according to the revised detection algorithm of Greene & Ho (2007a). Our sample continues to agree reasonably well with extrapolations to lower luminosity (and mass). We note that the Williams et al. (2004) sample is likely X-ray-weak as a result of intrinsic absorption (Brandt et al. 2000).
α ox ≡ −0.3838 log(f 2500Å /f 2 keV )
In a larger sample of 174 SDSS-selected intermediatemass BHs in active galaxies, 55 are detected by ROSAT (Greene & Ho 2007a). If we do not restrict ourselves to intermediate-mass BHs but consider all SDSS-selected AGNs (Greene & Ho 2007b) with cross-identifications in both the SDSS and the RASS, then our sample grows to 2235 objects (of which 658 are NLS1s with FWHM Hβ < 2000 km s −1 ). We include all these samples in Figure 3. For the SDSS-RASS sample, we estimate α ox by converting the ROSAT 0.1-2.4 keV counts to 0.5-2 keV using WebPIMMS, assuming an absorbed power law with index Γ s = 2, Galactic extinction of log(N H ) = 20.27 (the median of the Greene & Ho sample), and redshift z = 0.19 (median of the whole sample). The monochromatic luminosity L 2500Å is once again estimated from the Hα emission line.
Although intermediate-mass BHs follow extrapolations of α ox to lower mass, in general this sample appears to be Xray-bright. This can be seen by plotting the direct opticalto-X-ray flux ratio (where f opt = λf λ at 5100Å, and f X is the 0.5-2 keV flux) as a function of both the optical luminosity and the X-ray luminosity (Figure 4). The intermediatemass BH sample is at the lower end of the optical luminosity range exhibited by AGNs in SDSS, but has a comparable X-ray luminosity as other BHs many orders of magnitude higher in mass. This is an important result since these two samples are drawn from the same SDSS-RASS catalog and observed with the same instrument. Furthermore, although the bolometric corrections and BH mass estimators used suffer from large scatter and systematic uncertainty (for instance, with the geometry of the broad-line region), all RASS objects were treated uniformly. The systematically higher α ox for the intermediate-mass BHs is thus a real effect (supported by our Chandra observations). The optical-to-X-ray flux ratio is similar to that of Grupe et al. (2004), who selected their objects based on X-ray flux (and therefore represent the high-X-ray end of the distribution of NLS1s). Our sample has no such selection, is drawn from a uniform SDSS parent sample based solely on M BH , and yet displays significantly lower opticalto-X-ray flux ratios than NLS1s drawn from the same parent sample.
What could be driving the relative X-ray loudness of the low-mass sample? The three obvious physical parameters are L bol , L bol /L Edd , and M BH . As is the case for the spectral index, it has been suggested that α ox depends on the accretion rate of the BH (Kelly et al. 2008). Given the observed correlation between Γ s and L bol /L Edd (Figure 2), it is reasonable to expect a similar correlation with respect to α ox . In Figure 5 we plot α ox against L bol /L Edd and M BH . We clearly see, however, that accretion rate, as defined by the optical luminosity, is not a driver of α ox for the bulk of the AGNs, at least over this range in Eddington ratios, which span roughly a factor of 100 from L bol /L Edd ≈ 10 −2 to 1. If we assume that the intrinsic scatter is more dominant than the error on individual measurements, that the scatter is symmetric, and we force χ 2 per degree of freedom to equal 1, then using the fit routine fitexy results in α ox ∝ (0.002 ± 0.007) log(L bol /L Edd ) for the large SDSS AGN sample, consistent with no dependence. We also find only a weak correlation α ox ∝ −(0.052 ± 0.005) log(M BH ). Because we estimate L 2500Å and ultimately L bol based on the Hα emission line, Figure 5 also suggests that α ox is independent of L UV /L Edd . The higher α ox for the intermediate-mass BHs is therefore more of a discontinuous jump from the main AGN population, rather than a smooth correlation.
These results are somewhat at odds with Kelly et al. (2008), who find L UV /L Edd and M BH to be correlated with α ox over a wide range in mass and accretion rate. It should be noted that their range of L bol /L Edd is larger than ours and thus perhaps they see a correlation that we do not probe. Their sample also spans a much wider range in redshift. Furthermore, we employ different correlations to convert from emission-line fluxes to continuum fluxes. In our sample, however, we treat all the SDSS-RASS objects in the same manner; therefore the higher α ox for intermediate-mass BHs is not an artifact.
Our results are consistent with Paper I, in which α ox was independent of L bol /L Edd for the Chandra sample; this now appears to be a more general property of AGNs, with two distinct subgroups. The bolometric fraction of the X-ray emission increases with decreasing UV/optical continuum strength, independent of the Eddington ratio. This property can be explained via disk-corona models, where soft disk photons cool the X-ray-emitting corona via Compton cooling, as well as thermally reprocessing some fraction of hard X-rays (Haardt & Maraschi 1993). Of course, the lack of dependence on L bol /L Edd extends only so far; at low enough Eddington ratio, the accretion flow is thought to transition from an optically thick, geometrically thin disk (Shakura & Sunyaev 1973) to a radiatively inefficient, optically thin, and geometrically thick disk (Ho 1999(Ho , 2008Quataert 2001;Narayan 2005). At this transition, there will surely be a sharp change in α ox . Ho (2008) sees evidence for such a transition at around log(L bol /L Edd ) ≈ −3. In the case of low-ionization nuclear emission-line regions (LINERs), thought to be a result of radiatively inefficient accretion (Ho 2008), α ox falls below a naive extrapolation of the Steffen et al. best fit to lower luminosity (Maoz 2007). Perhaps an analogous transition is occurring for intermediate-mass BHs.
Possible Evidence for Slim Disks
Given that our low-BH-mass sample exhibits a systematically flatter α ox at a given optical luminosity and Eddington ratio, it may indicate an important physical distinction in the accretion flow associated with the low BH mass. One possibility is that intermediate-mass BHs form slim disks (Abramowicz et al. 1988), an accretion disk model that has historically been invoked for nearly-Eddington or super-Eddington accretion flows (the possibility of super-Eddington flows is discussed by Mineshige 2007 andOhsuga et al. 2005). In such disks, the very high temperature and luminosity of the inner accretion disk causes the geometrically thin disk to become inflated at small radii, creating a pronounced atmospheric structure. The transition radius between the inner slim disk and the outer thin disk increases with increasing L bol /L Edd (Bonning et al. 2007). At small radii, the accretion becomes radiatively inefficient because of photon-trapping effects and radial advection of material sets in. Locally radiated flux within this transition radius (and the associated effective temperature) may then be depressed when observing this disk. Slim-disk models have luminosities and effective temperatures (at fixed BH mass) that are nearly independent of accretion rate, since any extra energy emitted as a result of higher L bol /L Edd falls directly into the BH (Wang et al. 1999;Mineshige et al. 2000). The expected α ox is close to −1, flatter than typical AGNs (Mineshige et al. 2000). The relatively stronger X-ray emission may be a result of this inflated inner-disk atmosphere, with plenty of X-rayemitting, hot and diffuse gas, coupled with depressed optical emission due to the radial advection of energy at small radii.
Slim disks are thought to become important above L bol /L Edd ≈ 0.3 (Bonning et al. 2007); we indeed see a flare up in α ox above such an accretion rate in Figure 5. The low-redshift, ROSAT-detected AGNs, including classical NLS1s, exhibit a remarkably constant α ox , whereas ROSATdetected, intermediate-mass BHs have systematically higher values. The Grupe et al. NLS1 sample also lies above the main AGN population. Given that the Grupe et al. objects exhibit the highest accretion rates of objects considered here (a fraction of which are super-Eddington), it seems plausible that these are genuine slim disks. The similarity in α ox between intermediate-mass BHs and the Grupe et al. NLS1s suggests that slim disks might be important at intermediate mass as well. As Mineshige et al. (2000) discuss, the prediction that α ox ≈ −1 for slim disks is a potential problem, since most NLS1s have α ox ≈ −1.5. As is clear from Figure 5, however, we measure values close to −1 for intermediate-mass BHs and super-Eddington NLS1s; the mean α ox is −1.12 ± 0.02 for the ROSAT intermediate-mass BHs, and −1.18 ± 0.01 for the Grupe et al. sample, whereas the mean α ox is −1.39±0.01 for the low-redshift SDSS-RASS AGNs, and −1.36±0.01 for the low-redshift SDSS-RASS NLS1s.
As intriguing as these results might be, we must remember the inherent uncertainty associated with these BH-mass and Eddington-ratio estimates. This makes comparisons to other accreting BH systems, such as stellar BH binaries with more accurately measured mass functions and observed state changes, difficult at best. The important result, however, remains: intermediate-mass BHs exhibit a distinct spectral energy distribution compared to higher-mass NLS1s. Slim disks provide an interesting framework for the interpretation of this result. The change in energy distribution is clearly driven by some combination of BH mass and Eddington ratio, which unfortunately we cannot fully disentangle with our current data. For example, in Figure 6 we plot α ox versus (L bol /L Edd ) 1/4 M −1/4 BH , which is proportional to the disk effective temperature (Frank et al. 1992). The higher α ox objects are those with high disk temperatures, but not all hightemperature objects exhibit a high α ox (i.e. most NLS1s). Further study is clearly warranted, ideally with a comprehensive, homogenous, and detailed survey. Complete spectral energy distributions are needed to quantify the varying X-ray bolometric corrections between the different AGN classes.
SUMMARY
We present X-ray observations of the remaining intermediate-mass BHs found by Greene & Ho (2004) and not observed by Greene & Ho (2007c). We detect 5 out of 8 objects in 5 ks observations with Chandra. The mean observed properties, such as hardness ratio and soft photon index Γ s , are similar to the initial sample; we continue to find a range of indices 1 < Γ s < 3, consistent with previous AGN studies. Only 2 objects have sufficient counts to extract reliable spectra, and both are well fit with simple absorbed power-law models. The resulting χ 2 values do not justify any additional components, such as a soft excess, although this may be a function of relative inefficiency of detecting counts above 2 keV in short exposures. The soft photon index continues to be correlated strongly with X-ray luminosity and Eddington ratio, while the BH mass remains a poor indicator of the X-ray spectral slope.
Although the Greene & Ho sample shares many character-istics with classical NLS1s, there are important differences between the two. In particular, the X-ray-to-optical flux index α ox of these intermediate-mass AGNs is flatter, similar to NLS1s radiating near or above the Eddington limit. This may be evidence for a change in the accretion structure of such systems, perhaps due to the formation of a slim disk instead of a classical thin disk. There appears to be a sharp transition; within the two groups (intermediate-mass AGNs and super-Eddington NLS1s vs. normal AGNs and classical NLS1s) α ox is independent of the Eddington ratio. Additionally, we do not see evidence for very steep soft photon indices, as suggested by Boller et al. (1996) and Porquet et al. (2004), despite the very low BH masses. As was shown in the pilot study, the feasibility of detecting intermediate-mass BHs with short Chandra exposures is clearly established. Such observations are a vital component to understanding the broad spectral energy distribution and behavior of BHs in this previously unobserved intermediatemass regime, and to properly measure and calibrate their bolometric luminosities. Our single-epoch, short exposures do not allow us to study X-ray variability on short timescales, known to be very pronounced in NLS1s, although we note that variability of factors of a few can be seen when compared with ≈ 10 yr old ROSAT archival data. Further X-ray observations of such objects would help to clarify these issues.
We thank an anonymous referee for thoughtful comments. L.-B.D. would like to thank Eliot Quataert and J.E.G. Peek for helpful and insightful discussions. Support for this work was provided by NASA through grant SAO 08700135 issued by the Chandra X-ray Observatory Center, which is operated by the Smithsonian Astrophysical Observatory on behalf of NASA under contract NAS8-03060, and through grant HST-GO-11130.01-A from the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., for NASA, under contract NAS5-26555. Greene & Ho (2004). Col. (2) Table 1 (Dickey & Lockman 1990). For GH12 we also plot an absorbed power-law model with N H as a free parameter (dotted red line), which results in a marginally better fit. Fit parameters are given in Table 2. The bottom panel in each plot shows the residuals for each model normalized by the 1σ uncertainty in the measurement. Greene & Ho (2004) objects detected by Chandra (black circles) populate the intermediate-mass regime, but show that Γs is not strongly correlated with BH mass. We also include optically selected, X-ray-weak NLS1s (Williams et al. 2004; red triangles), PG quasars observed with XMM-Newton (Porquet et al. 2004; blue open squares), NGC 4395 (Moran et al. 2005;asterisk), and POX 52 (Thornton et al. 2008;cross). (b) Γs vs. L 0.5−2 keV . This relation exhibits the strongest correlation. (c) Γs vs. L bol /L Edd . L bol is estimated from optical observations only, to avoid any potential secondary correlations between X-ray luminosity and slope. We use L Hα to estimate L 5100Å , 2007b) for our Chandra sample, and then assume L bol = 9L 5100Å (Ho 2008). X-ray Properties of IMBHs 9 FIG. 3.-X-ray-optical spectral index αox vs. monochromatic 2500Å luminosity, in units of erg s −1 Hz −1 . Black circles are the Greene & Ho (2004) objects detected by Chandra, where L 2500Å is determined from Hα measurements , 2007b. We also plot the sample of 55 ROSAT-detected, intermediate-mass BHs (Greene & Ho 2007a; purple open circles), the full sample of ROSAT-detected, low-redshift (z < 0.35), type 1 AGNs from SDSS (Greene & Ho 2007b; gray points), NLS1s from that same sample (green points), X-ray-weak NLS1s (Williams et al. 2004; red triangles), PG NLS1s (blue open diamonds), X-ray-selected NLS1s (Grupe et al. 2004; orange pluses), NGC 4395 (Moran et al. 2005;asterisk), and POX 52 (Thornton et al. 2008;cross). The best-fit L 2500Å -αox relation from Steffen et al. (2006) is also included (solid line; 1σ width of line given by dashed lines).
FIG. 4.-Optical-to-X-ray flux ratio vs. (a) X-ray luminosity, and (b) optical luminosity. X-ray flux is defined over 0.5-2 keV. Optical flux is defined as λf λ at λ = 5100Å, determined from Hα emission-line measurements . Symbols and conventions as in Figure 3. NGC 4395 is not shown as it has very low X-ray and optical luminosity. (Ho 2008), with L 5100Å estimated from Hα emission-line measurements . The X-ray flux is defined over 0.5-2 keV. The BH masses in our sample are estimated via virial techniques . Symbols and conventions as in Figure 3. Masses for PG NLS1s, NGC 4395 and POX 52 are given by , Peterson et al. (2005), and Barth et al. (2004), respectively.
FIG. 6.-X-ray-optical spectral index αox vs. effective temperature of the disk (Frank et al. 1992). The bolometric luminosity is assumed to follow L bol = 9L 5100Å (Ho 2008), with L 5100Å estimated from Hα emission-line measurements . The BH masses are estimated via virial techniques . Symbols and conventions as in Figure 3. Masses for PG NLS1s, NGC 4395 and POX 52 are given by , Peterson et al. (2005), and Barth et al. (2004), respectively.
( 3 )
3: Logarithm of neutral column density, in cm −2 , from Dickey & Lockman 1990 using WebPIMMS. Col. (4): 0.5-2 keV count rate, in counts s −1 . Col. (5): 2-8 keV count rate, in counts s −1 . Col. (6): Hardness ratio, where H ≡ (C h − Cs)/(C h + Cs). Col. (7): Photon index, where N (E) ∝ E −Γ HR , determined from the hardness ratio (see text). Col. (8): 0.5-2 keV flux, in erg s −1 cm −2 . Col. (9): 2-8 keV flux, in erg s −1 cm −2 . Col. (10): 0.5-2 keV luminosity, in erg s −1 . Col. (11): 2-8 keV luminosity, in erg s −1 . Col. (12): Hα luminosity (Greene & Ho 2007b), in erg s −1 . GH19 is not included in the new sample (Greene & Ho 2007a), and thus has no updated Hα measurement. Col. (13): Ratio of optical to X-ray flux, where αox = −0.3838 log(f 2500Å /f 2 keV ). See text for details. Col. (14): BH mass (Greene & Ho 2007b). GH19 is not included in the new sample (Greene & Ho 2007a), and thus has no updated BH mass measurement. Col. (15): Reference for original observation: 1) Greene & Ho 2007c; 2) this work.
: Power-law index (where N (E) ∝ E −Γs ). Col. (3): Normalization at 1 keV in 10 −4 photons s −1 keV −1 . Col (4): Reduced χ 2 . Col. (5): Degrees of freedom. FIG. 1.-Extracted X-ray spectra for objects with > 200 counts. Energy bins are chosen to have a minimum of 20 counts. Also plotted are absorbed power-law models (solid black line) with neutral column density N H fixed to the value in
FIG. 2 .
2-(a) Γs (0.5-2 keV) vs. M BH . The
FIG. 5 .
5-X-ray-optical spectral index αox vs. (a) Eddington ratio; and (b) M BH . The bolometric luminosity is assumed to follow L bol = 9L 5100Å
TABLE 1 X
10013 ± 0.0008 −0.73 ± 0.13 2.4 ± 0.6 −13.49 +0.10 NOTE. -Col. (1): Identification number from Greene & Ho 2004. Col. (2): Luminosity distance, in Mpc, calculated using the observed SDSS redshift and our adopted cosmology. Col.-RAY PROPERTIES
TABLE 2 SPECTRAL
2FITSID
Γs
Norm.
χ 2
ν
dof
(1)
(2)
(3)
(4)
(5)
GH12 2.40 ± 0.06 3.0 ± 0.2 1.10 41
GH17 2.52 ± 0.13 1.0 ± 0.1 0.94 12
NOTE. -Col. (1): Identification number from
See the online celldetect documentation for a detailed description of how the S/N ratio is calculated. 6 http://heasarc.gsfc.nasa.gov/Tools/w3pimms.html
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[] |
[
"Lepton flavor violating Higgs decays induced by massive unparticle",
"Lepton flavor violating Higgs decays induced by massive unparticle"
] |
[
"E O Iltan .*e-mailaddress:[email protected] \nMiddle East Technical University\nNorthern Cyprus CampusGuzelyurt, Mersin 10TURKEY\n"
] |
[
"Middle East Technical University\nNorthern Cyprus CampusGuzelyurt, Mersin 10TURKEY"
] |
[] |
We predict the branching ratios of the lepton flavor violating Higgs decays H 0 → e ± µ ± , H 0 → e ± τ ± and H 0 → µ ± τ ± with the assumption that lepton flavor violation is due to the unparticle mediation. Here, we consider an effective interaction which breaks the conformal invariance after the electroweak symmetry breaking and causes that unparticle becomes massive. The new interaction results in a modification of the mediating unparticle propagator and brings additional contribution to the branching ratios of the lepton flavor violating decays with the new vertex including Higgs field and two unparticle fields. We observe that the branching ratios of the decays under consideration lie in the range of 10 −6 − 10 −4
| null |
[
"https://arxiv.org/pdf/1006.2095v2.pdf"
] | 118,433,357 |
1006.2095
|
e9d995c0387c4e53d5d6cf1a271abbd0d4e8a7da
|
Lepton flavor violating Higgs decays induced by massive unparticle
12 Oct 2010
E O Iltan .*e-mailaddress:[email protected]
Middle East Technical University
Northern Cyprus CampusGuzelyurt, Mersin 10TURKEY
Lepton flavor violating Higgs decays induced by massive unparticle
12 Oct 2010
We predict the branching ratios of the lepton flavor violating Higgs decays H 0 → e ± µ ± , H 0 → e ± τ ± and H 0 → µ ± τ ± with the assumption that lepton flavor violation is due to the unparticle mediation. Here, we consider an effective interaction which breaks the conformal invariance after the electroweak symmetry breaking and causes that unparticle becomes massive. The new interaction results in a modification of the mediating unparticle propagator and brings additional contribution to the branching ratios of the lepton flavor violating decays with the new vertex including Higgs field and two unparticle fields. We observe that the branching ratios of the decays under consideration lie in the range of 10 −6 − 10 −4
The standard model (SM) electroweak symmetry breaking mechanism which can explain the production of the masses of fundamental particles will be tested at the Large Hadron Collider (LHC) and, hopefully, the Higgs boson H 0 , which is responsible for this mechanism will be hunt soon. The possible decays of the Higgs boson to the SM particles are worthwhile to study and, among them, the lepton flavor violating (LFV) decays reach great interest [1,2,4,3,5] since the LF violation mechanism is sensitive to the physics beyond the SM. The addition of the new Higgs doublet to the SM particle spectrum is one of the possibility to switch on the LFV interactions, arising from the tree level LFV couplings. In [1,2,3], H 0 → τ µ decay has been analyzed and the branching ratio (BR) at the order of magnitude of 0.001 − 0.1 has been estimated. In [4], the observable BRs of LF changing H 0 decays have been obtained in the SM with right handed neutrinos. Another possibility to switch on the LF violation is to introduce the intermediate scalar unparticle (U) with the effective U-lepton-lepton vertex in the loop level.
In [5], the BRs of the LFV Higgs decays H 0 → e ± µ ± , H 0 → e ± τ ± and H 0 → µ ± τ ± have been estimated, by respecting the unparticle idea. Unparticles, introduced by Georgi [6,7], come out as new degrees of freedom due to the SM-ultraviolet sector interaction; they are massless and have non integral scaling dimension d u , around, Λ U ∼ 1 T eV .
In the present work we study the LFV SM Higgs decays by considering that the LF violation exists in the one loop level and it is carried by the effective U-lepton-lepton vertex. The effective interaction lagrangian, which is responsible for the LFV decays, is
L F V = 1 Λ du−1 U λ S ijl i l j + λ P ijl i iγ 5 l j U ,(1)
with the lepton field l and scalar (pseudoscalar) coupling λ S ij (λ P ij ). Here we consider the operators with the lowest possible dimension since their contributions are dominant in the low energy effective theory (see [8]). Furthermore, we consider that there exists an additional interaction which ensures a non-zero mass to unparticle after the electroweak symmetry breaking [9] as
L U = − λ Λ 2 du−2 U U 2 H † H ,(2)
and we get
L U = − 1 2 λ Λ 2 du−2 U U 2 H 0 2 + 2 v H 0 + v 2 ,(3)
when the Higgs doublet develops the vacuum expectation value. The interaction in eq.(3) leads to the lagrangian
L ′ U = − m 4−2 d U U v U 2 H 0 ,(4)
with the unparticle mass
m U = √ λ v Λ du−1 U 1 2−d U ,(5)
and this term results in an additional diagram driving the LFV decays with the help of Ulepton-lepton vertices (see Fig.1-(d)). Here, the non-zero unparticle mass m U is the sign of the broken conformal invariance and one expects that the unparticle propagator is modified. The propagator is model dependent (see [10]) and we consider the one in the simple model [11,12]
d 4 x e ipx < 0|T U(x) U(0) 0 >= i A du 2 π ∞ 0 ds s du−2 p 2 − µ 2 − s + iǫ ,(6)
with
A du = 16 π 5/2 (2 π) 2 du Γ(d u + 1 2 ) Γ(d u − 1) Γ(2 d u ) ,(7)
and the scale µ where unparticle sector becomes a particle sector. This choice has clues about the unparticle nature of the hidden sector, it carries the information on the effects of the broken scale invariance and ensures a possibility to estimate the scale invariance breaking effects 1 . In our calculations we choose µ = m U and d u ∼ 1.0 which is the case that unparticle behaves as if it is almost gauge singlet scalar 2 . Now, we are ready to present the BR for H 0 → l − 1 l + 2 decay,
BR(H 0 → l − 1 l + 2 ) = 1 16 π m H 0 |M| 2 Γ H 0 ,(8)
where M is the matrix element of the LFV H 0 → l − 1 l + 2 decay (see Fig.1) and Γ H 0 is the Higgs total decay width. The square of the matrix element |M| 2 reads
|M| 2 = 2 m 2 H 0 − (m l − 1 + m l + 2 ) 2 |A| 2 + 2 m 2 H 0 − (m l − 1 − m l + 2 ) 2 |A ′ | 2 ,(9)
with the amplitudes
A = 1 0 dx f S self + 1 0 dx 1−x 0 dy f S vert , A ′ = 1 0 dx f ′ S self + 1 0 dx 1−x 0 dy f ′ S vert .(10)
The
functions 3 f S self , f ′ S self , f S vert , f ′ S vert are f S self = −i c 1 (1 − x) 1−du 16 π 2 m l + 2 − m l − 1 (1 − d u ) 3 i=1 (λ S il 1 λ S il 2 + λ P il 1 λ P il 2 ) m l − 1 m l + 2 (1 − x) × L du−1 self − L ′du−1 self − (λ P il 1 λ P il 2 − λ S il 1 λ S il 2 ) m i m l + 2 L du−1 self − m l − 1 L ′du−1 self , f ′ S self = i c 1 (1 − x) 1−du 16 π 2 m l + 2 + m l − 1 (1 − d u ) 3 i=1 (λ P il 1 λ S il 2 + λ S il 1 λ P il 2 ) m l − 1 m l + 2 (1 − x) × L du−1 self − L ′du−1 self − (λ P il 1 λ S il 2 − λ S il 1 λ P il 2 ) m i m l + 2 L du−1 self + m l − 1 L ′du−1 self , f S vert = i c 1 m i (1 − x − y) 1−du 16 π 2 3 i=1 1 L 2−du vert (λ P il 1 λ P il 2 − λ S il 1 λ S il 2 ) (1 − x − y) × m 2 l − 1 x + m 2 l + 2 y − m l + 2 m l − 1 + x y m 2 H 0 − 2 L vert 1 − d u − m 2 i − (λ P il 1 λ P il 2 + λ S il 1 λ S il 2 ) m i m l − 1 (2 x − 1) + m l + 2 (2 y − 1) − i c 2 Γ[3 − 2 d u ] (x y) 1−du 16 π 2 Γ[2 − d u ] 2 3 i=1 1 L 3−2 du 2vert m i (λ P il 1 λ P il 2 − λ S il 1 λ S il 2 ) − (λ P il 1 λ P il 2 + λ S il 1 λ S il 2 ) m l − 1 x + m l + 2 y , f ′ S vert = i c 1 m i (1 − x − y) 1−du 16 π 2 3 i=1 1 L 2−du vert (λ S il 1 λ P il 2 − λ P il 1 λ S il 2 ) (1 − x − y) × m 2 l − 1 x + m 2 l + 2 y + m l + 2 m l − 1 + x y m 2 H 0 − 2 L vert 1 − d u − m 2 i + (λ S il 1 λ P il 2 + λ P il 1 λ S il 2 ) m i m l − 1 (2 x − 1) + m l + 2 (1 − 2 y) − i c 2 Γ[3 − 2 d u ] (x y) 1−du 16 π 2 Γ[2 − d u ] 2 3 i=1 1 L 3−2 du 2vert m i (λ S il 1 λ P il 2 − λ P il 1 λ S il 2 ) + (λ S il 1 λ P il 2 + λ P il 1 λ S il 2 ) m l − 1 x − m l + 2 y ,(11)
where L self , L ′ self , L vert , and L 2vert are
L self = x m 2 l − 1 (1 − x) − m 2 i + m 2 U (x − 1) , L ′ self = x m 2 l + 2 (1 − x) − m 2 i + m 2 U (x − 1) , L vert = (m 2 l − 1 x + m 2 l + 2 y) (1 − x − y) − m 2 i (x + y) + m 2 H 0 x y − m 2 U (1 − x − y) , 3 f S self , f ′ Sself
are the same as the functions presented in [5] except that the propagators L self and L ′ self contain the unparticle mass term m U . On the other hand f S vert , f ′ S vert include additional part proportional to the parameter c 2 which comes from the new interaction (see eq.(4)) leading to the vertex given in Fig.1
-d L 2vert = (m 2 l − 1 x + m 2 l + 2 y) (1 − x − y) − m 2 i (1 − x − y) + m 2 H 0 x y − m 2 U (x + y) ,(12)
with
c 1 = g A du e −i π du 4 m W sin (d u π) Λ 2 (du−1) u , c 2 = A 2 du m 4−2 du U e −2 i π du 4 v sin 2 (d u π) Λ 2 (du−1) u .(13)
Here λ S,P il 1(2) are the scalar and pseudoscalar couplings related to the U − i − l − 1 (l + 2 ) interaction where i, (i = e, µ, τ ) is the internal lepton and l − 1 (l + 2 ) the outgoing lepton (anti lepton). Notice that, in the numerical calculations, we consider the BR due to the production of sum of charged states, namely,
BR(H 0 → l ± 1 l ± 2 ) = Γ(H 0 → (l 1 l 2 +l 2 l 1 )) Γ H 0 .(14)
Discussion
This section is devoted to the analysis of the BRs of the LFV H 0 → l − 1 l + 2 decays in the case that the LF violation is carried by the U-lepton-lepton vertex. The LFV decays exist at least in the loop level with the help of the internal unparticle mediation. The interaction Lagrangian given in eq.(2) results in a nonzero mass for unparticle after the electroweak symmetry breaking and the propagator of unparticle existing in the loop should be modified. In the present work we take the propagator as (see eq. (6))
P (p 2 ) = i A du 2 sin π d u e −i du π (p 2 − m 2 U ) 2−du ,(15)
which becomes a massive scalar propagator for d u = 1.
The LF violation is carried by single unparticle mediation and two unparticles mediation in the loop (see Fig.1 couplings, the scalar λ S ij and pseudo scalar λ P ij , are among the free parameters which we choose λ S ij = λ P ij = λ ij . Furthermore, we first consider that the diagonal λ ii = λ 0 and off diagonal λ ij = κλ 0 , i = j couplings are family blind with κ < 1. Second we assume that the the diagonal couplings λ ii carry the lepton family hierarchy, namely, λ τ τ > λ µµ > λ ee , on the other hand, the off-diagonal couplings, λ ij are family blind, universal and λ ij = κλ ee . In our numerical calculations, we choose κ = 0.5 and we take the magnitude of the FV coupling(s) at most 1.0 in order to ensure that the calculations are the perturbative for d u = 1.0.
In order to estimate the BR of the LFV decays under consideration one needs the Higgs mass and its total decay width. The theoretical upper and lower bounds of Higgs mass read 1.0 T eV and 0.1 T eV [13], respectively. This is due to the fact that one does not meet the unitarity problem and the instability of the Higgs potential both. Furthermore, the electroweak measurements predict the range of the Higgs mass as m H 0 = 129 +74 −49 [14] which is not in contradiction with the theoretical results. The total Higgs decay width is another parameter which should be restricted and it is estimated by using the possible decays for the chosen Higgs mass 4 . Notice that throughout our calculations we choose m H 0 = 120 (GeV ) and we use the input values given in Table (1).
Parameter
Value Table 1: The values of the input parameters used in the numerical calculations.
m e 0.0005 (GeV) m µ 0.106 (GeV) m τ 1.780 (GeV) Γ(H 0 )| m H 0 =120 GeV 0.0029 (GeV) G F 1.1663710 −5 (GeV −2 )
In Fig.2 In Fig.6 (7) we present the BR(H 0 → τ ± e ± (τ ± µ ± )) with respect to λ for the scale parameter d u = 1. Here, the solid (long dashed-short dashed) line represents the BR for λ ee = λ µµ = λ τ τ = 1.0 (λ ee = 0.1, λ µµ = 0.5, λ τ τ = 1.0-λ ee = 0.01, λ µµ = 0.1, λ τ τ = 1.0). It is observed that the BR is suppressed more than one order in the range 0.0 < λ < 1.0 and this suppression is strong for λ < 0.3.
Conclusion
As a summary, the mass of unparticle which arises with unparticle Higgs scalar interaction results in that the BRs of the LFV H 0 → l ± 1 l ± 2 decays are suppressed. The BRs are of the order of 10 −6 for λ ∼ 1.0 and d u ∼ 1.0. If the unparticle-Higgs scalar interaction is switched off unparticle remains massless and the BRs of the decays studied reach to the values of the order of 10 −4 for FB U-lepton-lepton couplings. With the possible production of the Higgs boson H 0 at the LHC the theoretical results of the BRs of the LFV Higgs decays will be tested and the new physics which drives the flavor violation, including the unparticle sector will be searched.
). The possible two unparticles mediation brings an additional contribution to the LFV decays with the strength which is a function of unparticle mass m U , reaching 246 GeV when d u ∼ 1.0 for the coupling λ ∼ 1.0. In our numerical calculations we take the scaling parameter d u not far from 1.0, namely 1.0 ≤ d u ≤ 1.2. On the other hand we take the coupling λ as λ ≤ 1.0 in order to guarantee that the calculations are perturbative in case of d u ∼ 1.0 and we choose the energy scale Λ u as Λ u ∼ 1.0 (T eV ). The FV U-lepton-lepton
, we present the BR(H 0 → µ ± e ± ) with respect to the scale parameter d u for the flavor blind (FB) couplings λ ee = λ µµ = λ τ τ = 1.0. Here, the solid (long dashed-short dasheddotted) line represents the BR for λ = 0.0 (0.2−0.5−1.0). The possible interaction of unparticle with the Higgs scalar leads to a nonzero mass for unparticle after the spontaneous symmetry breaking and the mass term leads to a suppression in the BR. The additional term coming from the U − U − H 0 vertex does not result is an enhancement in the BR. The BR reaches to the values of the order of 10 −4 for λ = 0 and d u ∼ 1.0. For λ ∼ 1.0 and near d u ∼ 1.0 5 the BR is of the order of 10 −6 . Fig.3 represents the BR(H 0 → µ ± e ± ) with respect to λ for the scale parameter d u = 1. Here, the solid (long dashed-short dashed) line represents the BR for λ ee = λ µµ = λ τ τ = 1.0 (λ ee = 0.1, λ µµ = 0.5, λ τ τ = 1.0-λ ee = 0.01, λ µµ = 0.1, λ τ τ = 1.0). This figure shows the strong sensitivity of the BR to the U − U − H 0 interaction strength λ, the BR(H 0 → τ ± e ± (τ ± µ ± )) with respect to the scale parameter d u , for the FB couplings λ ee = λ µµ = λ τ τ = 1.0. Here, the solid-long dashed-short dashed-dotted lines represent the BR for λ = 0.0 − 0.2 − 0.5 − 1.0. In the case of d u ∼ 1.0, the BR is almost 5.0 × 10 −6 (6.0 × 10 −6 ) for λ ∼ 1.0 and enhances up to 4.0 × 10 −4 for λ = 0 and d u ∼ 1.0.Similar to the previous decay the mass term leads to a suppression in the BR and the additional term coming from the U − U − H 0 vertex is not enough to enhance the BR over the numerical values which is obtained for the massless unparticle case.
Figure 1 :
1One loop diagrams contribute to H 0 → l − 1 l + 2 decay with scalar unparticle mediator. Solid line represents the lepton field: i represents the internal lepton, l − 1 (l + 2 ) outgoing lepton (anti lepton), dashed line the Higgs field, double dashed line unparticle field.
Figure 2 :
2d u dependence of the BR (H 0 → µ ± e ± ) for λ ee = λ µµ = λ τ τ = 1.0. Here, the solid (long dashed-short dashed-dotted) line represents the BR for λ = 0.0 (0.2 − 0.5 − 1.0).
Figure 3 :
3λ dependence of the BR (H 0 → µ ± e ± ) for d u = 1. Here, the solid (long dashed-short dashed) line represents the BR for λ ee = λ µµ = λ τ τ = 1.0 (λ ee = 0.1, λ µµ = 0.5, λ τ τ = 1.0λ ee = 0.01, λ µµ = 0.1, λ τ τ = 1.0).
Figure 4 :
4The same asFig.2but for H 0 → τ ± e ± decay.
Figure 5 :
5The same asFig.2but for H 0 → τ ± µ ± decay.
Figure 6 :
6The same asFig.3but for H 0 → τ ± e ± decay.
Figure 7 :
7The same asFig.3but for H 0 → τ ± µ ± decay.
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|
[] |
[
"Constraints on decaying Dark Matter from XMM-Newton observations of M31",
"Constraints on decaying Dark Matter from XMM-Newton observations of M31"
] |
[
"Alexey Boyarsky \nPH-TH\nCERN\nCH-1211Geneve 23Switzerland\n\nBogolyubov Institute for Theoretical Physics\ncÉ cole Polytechnique Fédérale de Lausanne\nInstitute of Theoretical Physics\nFSB/ITP/LPPC\nBSP 72003780, CH-1015Kiev, LausanneUkraine, Switzerland\n",
"Dmytro Iakubovskyi \nBogolyubov Institute for Theoretical Physics\ncÉ cole Polytechnique Fédérale de Lausanne\nInstitute of Theoretical Physics\nFSB/ITP/LPPC\nBSP 72003780, CH-1015Kiev, LausanneUkraine, Switzerland\n",
"Oleg Ruchayskiy ",
"Vladimir Savchenko \nBogolyubov Institute for Theoretical Physics\ncÉ cole Polytechnique Fédérale de Lausanne\nInstitute of Theoretical Physics\nFSB/ITP/LPPC\nBSP 72003780, CH-1015Kiev, LausanneUkraine, Switzerland\n\nPhysics Department\nKiev National Taras Shevchenko University\n03022KievUkraine\n"
] |
[
"PH-TH\nCERN\nCH-1211Geneve 23Switzerland",
"Bogolyubov Institute for Theoretical Physics\ncÉ cole Polytechnique Fédérale de Lausanne\nInstitute of Theoretical Physics\nFSB/ITP/LPPC\nBSP 72003780, CH-1015Kiev, LausanneUkraine, Switzerland",
"Bogolyubov Institute for Theoretical Physics\ncÉ cole Polytechnique Fédérale de Lausanne\nInstitute of Theoretical Physics\nFSB/ITP/LPPC\nBSP 72003780, CH-1015Kiev, LausanneUkraine, Switzerland",
"Bogolyubov Institute for Theoretical Physics\ncÉ cole Polytechnique Fédérale de Lausanne\nInstitute of Theoretical Physics\nFSB/ITP/LPPC\nBSP 72003780, CH-1015Kiev, LausanneUkraine, Switzerland",
"Physics Department\nKiev National Taras Shevchenko University\n03022KievUkraine"
] |
[
"Mon. Not. R. Astron. Soc"
] |
We derive constraints on the parameters of the radiatively decaying Dark Matter (DM) particle, using the XMM-Newton EPIC spectra of the Andromeda galaxy (M31). Using the observations of the outer (5'-13') parts of M31, we improve the existing constraints. For the case of sterile neutrino DM, combining our constraints with the latest computation of abundances of sterile neutrinos in the Dodelson-Widrow (DW) scenario, we obtain the lower mass limit m s < 4 keV, which is stronger than the previous one m s < 6 keV, obtained recently by Asaka et al. (2007). Comparing this limit with the most recent results on Lyman-α forest analysis of Viel et al.(2007)(m s > 5.6 keV), we argue that the scenario in which all the DM is produced via the DW mechanism is ruled out. We discuss however other production mechanisms and note that the sterile neutrino remains a viable candidate for Dark Matter, either warm or cold.
|
10.1111/j.1365-2966.2008.13266.x
|
[
"https://arxiv.org/pdf/0709.2301v2.pdf"
] | 16,879,612 |
0709.2301
|
818531056d2f3255de77f67b39beee0189a7448e
|
Constraints on decaying Dark Matter from XMM-Newton observations of M31
10 Sep 2008. 2007
Alexey Boyarsky
PH-TH
CERN
CH-1211Geneve 23Switzerland
Bogolyubov Institute for Theoretical Physics
cÉ cole Polytechnique Fédérale de Lausanne
Institute of Theoretical Physics
FSB/ITP/LPPC
BSP 72003780, CH-1015Kiev, LausanneUkraine, Switzerland
Dmytro Iakubovskyi
Bogolyubov Institute for Theoretical Physics
cÉ cole Polytechnique Fédérale de Lausanne
Institute of Theoretical Physics
FSB/ITP/LPPC
BSP 72003780, CH-1015Kiev, LausanneUkraine, Switzerland
Oleg Ruchayskiy
Vladimir Savchenko
Bogolyubov Institute for Theoretical Physics
cÉ cole Polytechnique Fédérale de Lausanne
Institute of Theoretical Physics
FSB/ITP/LPPC
BSP 72003780, CH-1015Kiev, LausanneUkraine, Switzerland
Physics Department
Kiev National Taras Shevchenko University
03022KievUkraine
Constraints on decaying Dark Matter from XMM-Newton observations of M31
Mon. Not. R. Astron. Soc
00010 Sep 2008. 2007Received <date> ; in original form <date>Printed 10 (MN L A T E X style file v2.2)
We derive constraints on the parameters of the radiatively decaying Dark Matter (DM) particle, using the XMM-Newton EPIC spectra of the Andromeda galaxy (M31). Using the observations of the outer (5'-13') parts of M31, we improve the existing constraints. For the case of sterile neutrino DM, combining our constraints with the latest computation of abundances of sterile neutrinos in the Dodelson-Widrow (DW) scenario, we obtain the lower mass limit m s < 4 keV, which is stronger than the previous one m s < 6 keV, obtained recently by Asaka et al. (2007). Comparing this limit with the most recent results on Lyman-α forest analysis of Viel et al.(2007)(m s > 5.6 keV), we argue that the scenario in which all the DM is produced via the DW mechanism is ruled out. We discuss however other production mechanisms and note that the sterile neutrino remains a viable candidate for Dark Matter, either warm or cold.
INTRODUCTION
A vast body of evidence points to the existence of Dark Matter (DM) in addition to the ordinary visible matter in the Universe. The evidence includes: velocity curves of galaxies in clusters and stars in galaxies; observations of galaxy clusters in X-rays; gravitational lensing data; cosmic microwave background anisotropies. While the DM contributes some 22% to the total energy density in the Universe, its properties remain largely unknown. The Standard Model of particle physics (SM) does not provide a DM candidate. The DM cannot be made out of baryons, as such an amount of baryonic matter cannot be generated in the framework of an otherwise successful scenario of Big Bang nucleosynthesis (Dar 1995). In addition, current microlensing experiments exclude the possibility that MACHOs (massive compact halo objects) constitute the dominant amount of the total mass density in the local halo (Gates et al. 1995;Lasserre et al. 2000;Alcock et al. 2000). The only possible non-baryonic DM candidate in the SM could be the neutrino, however this possibility is ruled out by the present data on the large scale structure (LSS) of the Universe.
What properties of the DM particles can be deduced from existing observations? Some information comes from studies of structure formation. Namely, the velocity distribution of the DM particles at the time of structure formation affects greatly the power spectrum of density perturbations, as measured by a variety of experiments (see e.g. Tegmark et al. 2004). One of the parameters, characterizing the influence of the DM velocity dispersion on the power spectrum, is the free-streaming length λ FS -the distance traveled by the DM particle from the time when it became nonrelativistic until today. Roughly speaking, the free-streaming length determines the minimal scale at which the Jeans instability can develop, and therefore non-trivial free-streaming implies modifi-cation of the spectrum of density perturbations at wave numbers k λ −1 FS . If the DM particles have negligible velocity dispersion, they constitute the so-called cold DM (CDM), which forms structure in a "bottom-up" fashion (i.e. smaller scale objects formed first and then merged into the larger ones, see e.g. Peebles 1980). The neutrino DM represents the opposite case -hot DM (HDM). In HDM scenarios, structure forms in a top-down fashion (Zel'dovich 1970), and the first structures to collapse have size comparable to the Hubble scale (Bisnovatyi-Kogan 1980;Bond et al. 1980;Doroshkevich et al. 1981;Bond & Szalay 1983). In this scenario the galaxies do not have enough time to form, contradicting to the existing observations (see e.g. White et al. 1983;Peebles 1984).
Warm DM (WDM) represents an intermediate case, cutting structure formation at some scale, with the details being dependent on a particular WDM model.
Both WDM and CDM fit the LSS data equally well. The differences appear when one starts to analyze the details of structure formation for galaxy-size objects (modifications of the power spectrum at momenta k 0.5hMpc −1 ). It is usually said that WDM predicts "less power at smaller scales", meaning in particular that one expects smaller number of dwarf satellite galaxies and shallower density profiles than those predicted by CDM models (Navarro et al. 1997;Klypin et al. 1999;Ghigna et al. 2000). Thus WDM models can provide the way to solve the "missing satellite" problem and the problem of central density peaks in galaxy-sized DM halos (Klypin et al. 1999;Moore et al. 1999;Bode et al. 2001;Avila-Reese et al. 2001).
There exist a number of direct astrophysical observations which seem to contradict the N-body simulations of galaxy formations, performed in the framework of the CDM models (e.g. Diemand et al. 2007; Strigari et al. 2007). Namely, direct mea-surements of the DM density profiles in dwarf spheroidal (dSph) satellites of the Milky Way favour cored profiles (Gilmore et al. 2006Wu 2007;Gilmore 2007). 1 The number of dwarf satellite galaxies, as currently observed, is still more than an order of magnitude below the CDM predictions, in spite of the drastically improved sensitivity towards the search (see Gilmore et al. 2007;Koposov et al. 2007) and resolution of numerical simulations (Strigari et al. 2007). There seems to exist a smallest scale (∼120 pc) at which the DM is observed Gilmore 2007). However, as of now there is no definitive statement about the "CDM substructure crisis" (see Simon & Geha (2007) in regard to the smallest observed DM scale and Penarrubia et al. (2007) for an alternative solution of the "missing satellite problem").
The power-spectrum of density perturbations at scales of interest for the WDM vs. CDM issue can also be studied, analyzing the Lyman-α forest data (absorption feature by the neutral hydrogen at λ = 1216Å at different red-shifts in the distant quasar spectra, Hui et al. 1997). This involves comparison of the observed spectra of Ly-α absorption lines with those obtained as a result of numerical simulations in various DM models. In this way one arrives at an upper limit on the free-streaming length of the DM particles.
Various particle physics models provide WDM candidates. Possible examples include gravitinos and axinos in various supersymmetric models (see e.g. Baltz & Murayama 2003;Cembranos et al. 2006;Seto & Yamaguchi 2007). Another WDM candidate is the sterile neutrino with a mass in the keV range (Dodelson & Widrow 1994). Recently, this candidate received a lot of attention. Namely, an extension of the minimal SM (MSM) with the three right-handed neutrinos was suggested . This extension (called νMSM) explains several observed phenomena beyond the MSM under the minimal number of assumptions. Namely, apart from the absence of the DM candidate, the MSM fails to explain observed neutrino oscillations -the transition between neutrinos of different flavors (for a review see e.g. Fogli et al. 2006;Strumia & Vissani 2006;Giunti 2007). The explanation of this phenomenon is the existence of neutrino mass. The most natural way to provide this mass is to add right-handed neutrinos. Indeed, in the MSM, neutrinos are left-handed (all other fermions have both left-handed and right-handed counterparts) and strictly massless. The structure of the MSM dictates that right-handed neutrinos, if added to the theory, would not be charged with respect to any Standard Model interactions and interact with other matter only via mixing with the usual (left-handed) neutrinos (that is why right-handed neutrinos are often called sterile neutrinos to distinguish them from the left-handed active ones). Moreover, as demonstrated by , the parameters of added right-handed neutrinos can be chosen in such a way that such a model resolves another problem of the MSM -it explains the excess of baryons over antibaryons in the Universe (the baryon asymmetry), while at the same time it does not spoil the predictions of Big Bang nucleosynthesis. For this to be true, the masses of two of these sterile neutrinos should be chosen in the range 300 MeV M 2,3 20 GeV, while the mass of the third (lighter) sterile neutrino is arbitrary (as long as it is below M 2,3 ). In particular, its mass can be in the keV range, providing the WDM candidate. Such a sterile neutrino can be produced in the Early Universe in the correct amount via various mechanisms: via non-resonant oscillations with active neutrinos (Dodelson & Widrow 1994;Dolgov & Hansen 2002;Abazajian et al. 2001;Asaka et al. 2006Asaka et al. , 2007, via interaction with the inflaton (Shaposhnikov & Tkachev 2006), via resonant oscillations in the presence of lepton asymmetries (Shi & Fuller 1999), and have cosmologically long life-time.
Finally, the sterile neutrino with mass in the keVrange would have other interesting astrophysical applications (see e.g. Sommer-Larsen & Dolgov (2001); Kusenko (2006); Biermann & Kusenko (2006); Hidaka & Fuller (2006); Hidaka & Fuller (2007); Stasielak et al. (2007) and references therein).
Existing bounds on sterile neutrino DM
The mass of the sterile neutrino DM should satisfy the universal Tremaine-Gunn lower bound (Tremaine & Gunn 1979;Dalcanton & Hogan 2001): m s 300 − 500 eV. A stronger (although model dependent) lower bound comes from the Lyman-α forest analysis. Assuming a particular velocity distribution of the sterile neutrino 2 one can obtain a relation between the DM mass and λ FS and therefore convert an upper bound on the free-streaming length to a lower bound on the mass of the sterile neutrino. In the recent works of Seljak et al. (2006); Viel et al. (2006) this bound was found to be 14 keV (correspondingly 10 keV) at 95% CL in the Dodelson-Widrow (DW) production model (Dodelson & Widrow 1994). New results from QSO lensing give similar restrictions for the DW model: m s 10 keV (Miranda & Macciò 2007). For different models of production, the relation between the DM mass and the free-streaming length is different and the Lyman-α mass bound for sterile neutrinos can be as low as M s > 2.5 keV (see e.g. Ruchayskiy 2007). 3 The sterile neutrino DM is not completely stable. In particular, it has a radiative decay channel into an active neutrino and a photon, emitting a monoenergetic photon with energy E γ = m s /2 (where m s is the mass of the sterile neutrino). As a result, the (indirect) search for the DM decay line in the X-ray spectra of objects with large DM overdensity becomes an important way to restrict the parameters (mass and decay width) of sterile neutrino DM. During the last two years a number of papers appeared devoted to this task: Boyarsky et al. 2006b,c,d,e,a;Riemer-Sørensen et al. 2006;Watson et al. 2006;Boyarsky et al. 2007;Abazajian et al. 2007. The current status of these observations is summarized, e.g., in Ruchayskiy (2007). The results of the computation of sterile neutrino production in the early Universe (Asaka et al. 2007), combined with these X-ray bounds, puts an upper bound on the sterile neutrino mass of m s < 6 keV (Asaka et al. 2007). This is below the lower bound on the sterile neutrino DM mass from the Lyman-α forest analysis of Seljak et al. (2006); Viel et al. (2006). Thus it would seem that the scenario, in which all the sterile neutrino DM is produced via the DW mechanism, is ruled out (the recent work by Palazzo et al. (2007) also explored the possibility that the sterile neutrino, produced through DW scenario, constitutes but a fraction of DM and found this fraction to be below 70%). However, the results of Seljak et al. (2006); Viel et al. (2006) are based on the low-resolution SDSS Lyman-α dataset of McDonald et al. (2006). It was shown recently by Viel et al. (2007) that using high-resolution HIRES spectra ) one arrives at the lower limit m s > 5.6 keV. Thus, the small window of masses 5.6 keV < m s < 6 keV remains open in the DW model. Therefore further improvement of X-ray bounds is crucial for exploring (and possibly closing) this region of parameters.
It was shown in Boyarsky et al. (2006d) that the objects in the Local Halo (e.g. dwarf spheroidal galaxies) are the best objects in terms of the signal to noise ratio. The Andromeda galaxy (M31) is one of the nearest galaxies, excluding dwarves, that enables one to resolve most of its bright point sources and extract the spectrum of its diffuse emission. It also has a massive and well-studied dark matter halo (e.g. Klypin et al. 2002;Widrow & Dubinski 2005;Geehan et al. 2006;Tempel et al. 2007). The first step in such an analysis was done by Watson et al. (2006) (hereafter denoted by W06), who analyzed the diffuse emission from the 5 central arcmin, using the data processed by Shirey et al. (2001). We repeat the analysis of the central part of the M31, processing more observations, and extend the analysis to the off-centre region (5 ′ − 13 ′ ). We also analyze the uncertainties in the DM distribution in the central part of M31. The outer region of M31 has much fainter diffuse emission than its central part (c.f. e.g. Takahashi et al. 2004, Fig. 8), and uncertainties in the determining of the distribution of DM in this region are lower. All this allows us to strengthen the restrictions on the parameters of sterile neutrino DM, while using more conservative estimates of the DM signal.
The paper is organized as follows. We briefly summarize the properties of decaying DM in Section 2. The description of DM in M31 and expected DM decay flux is computed in Section 3. In Section 4 we describe the methodology of EPIC MOS and PN data reduction which we perform by using two different methods: Extended Sources Analysis Software (ESAS) and single background subtraction method (SBS). In Section 5 we fit the spectra and obtain the restrictions on sterile neutrino parameters. Finally, we discuss our results in Section 6.
DECAYING DARK MATTER MODEL
The flux of the DM decay from a given direction (in photons s −1 cm −2 ) is given by
F DM = ΓE γ m s f ov cone ρ DM (r) 4π|D L + r| 2 dr.
(1)
Here D L is the luminosity distance between an observer and the centre of an observed object, ρ DM (r) is the DM density, and the integration is performed over the DM distribution inside the (truncated) cone -solid angle, spanned by the field of view (FoV) of the X-ray satellite. In case of distant objects 4 , Eq.
(1) can be simplified:
F DM = M f ov DM Γ 4πD 2 L E γ m s ,(2)
where M f ov DM is the mass of DM within a telescope field of view, m s -mass of the sterile neutrino DM. In the case of small FoV, Eq. (2) simplifies to
F DM = ΓS DM ΩE γ 4πm s ,(3)
where
S DM = l.o.s. ρ DM (r)dr(4)
is the DM column density (the integral goes along the line of sight), Ω ≪ 1 -FoV solid angle. The decay rate of the sterile neutrino DM is equal to (Pal & Wolfenstein 1982;Barger et al. 1995)
5 Γ = 9αG 2 F 1024π 4 sin 2 (2θ)m 5 s ≈ 1.38 · 10 −30 s −1 sin 2 (2θ) 10 −8 m s 1 keV 5 . (5)
Here m s is the sterile neutrino mass, θ -mixing angle between sterile and active neutrinos. From a compact cloud of sterile neutrino DM we therefore obtain the flux:
F DM ≈ 6.38 · 10 6 keV cm 2 · s M f ov dm 10 10 M ⊙ kpc D L 2 sin 2 (2θ) m s 1 keV 5 .(6)
ANDROMEDA GALAXY (M31)
M31, or Andromeda galaxy, is one of the nearest galaxies, excluding dwarves; it is located at the distance D L = 784 ± 13 ± 17 kpc (Stanek & Garnavich 1998). Its proximity allows us to resolve most of its point sources and extract the spectrum of diffuse emission of its central part.
Available XMM-Newton (Jansen et al. 2001) observations cover the region of central 15 ′ of M31 with exposure time greater than 100 ksec (see Table 1). W06 used the XMM-Newton data on central 5 ′ of M31 (observation 0112570401 processed by Shirey et al. (2001), exposure time about 30 ksec) to produce restrictions on the parameters of sterile neutrino DM. The sufficient increase of photon statistics enables us to analyze the outer (5 ′ -13 ′ ) faint part of M31, which, however, has a significant mass of DM (see Section 3.1 below).
In this work we will analyze two different spatial regions of Andromeda galaxy: region circle5, which corresponds to 5 ′ circle around the centre of M31, and region ring5-13, which corresponds to the ring with inner and outer radii of 5 ′ and 13 ′ , respectively.
Calculation of DM mass
To obtain the restriction on parameters of the decaying DM, we should calculate the total DM mass M f ov dm , which corresponds to both spatial regions: circle5 and ring5-13, both with and without resolved point sources. To estimate the systematic uncertainties of the evaluation of the DM decay signal and to find the most conservative estimate for it, we analyze various available DM profiles (Kerins et al. 2001;Klypin et al. 2002;Widrow & Dubinski 2005;Geehan et al. 2006;Carignan et al. 2006;Tempel et al. 2007 Klypin et al. (2002) (before and after adiabatic contraction), Geehan et al. (2006) and Kerins et al. (2001) are marked as "K1", "K2", "GFBG" and "KER", respectively. The DM distributions from Tempel et al. (2007) are marked as "KING", "MOORE", "N04", "NFW" and "BURK" (see text). The DM distributions from Widrow & Dubinski (2005) are marked as "M31A", "M31B" and "M31C".
• (K1) Before 6 adiabatic contraction stage, Klypin et al. (2002) assume that DM distribution is purely Navarro-Frenk-White (NFW) (Navarro et al. 1997):
ρ DM (r) = 1 4π log(1 + C) − C/(1 + C) M vir r(r + r s ) 2 .(7)
The parameters of this NFW distribution (in terms of the favored C1 model of Klypin et al. 2002) are: M vir = 1.60 × 10 12 M ⊙ ; r s = 25.0 kpc; C = 12.
• (K2) This non-analytical model is the result of adiabatic contraction of the K1 profile, described above. To obtain it, we extract the data from the Fig. 4 of Klypin et al. (2002). In the top part of this figure the dot-dashed curve is the contribution of the DM halo to the total M31 mass distribution (C1 model of Klypin et al. 2002).
As the precise form of this mass distribution is not analytic, we scanned this curve and produced the file with numerical values of enclosed mass M DM (r) within the sphere of radius r. After that, we interpolated the M DM (r), and evaluated the radial density distribution
ρ DM (r) = 1 4πr 2 dM DM (r) dr .(8)ρ KER (r) = ρ h (0) a 2 a 2 +r 2 r R max , 0 r > R max .(9)
where ρ h (0) = 0.23M ⊙ pc −3 , a = 2 kpc, R max = 200 kpc.
• (M31A-C) Profiles of Widrow & Dubinski (2005). In this paper the authors propose several models, which differ by the relative disk/halo contribution. These non-analytical models (M31a-d) incorporate an exponential disk, a Hernquist model bulge, an NFW halo (before contraction) and a central supermassive black hole. The stability against the formation of bars was numerically studied. 7 We also use density distributions from the recent paper of Tempel et al. (2007). The main aim of this paper is to derive the DM density distribution in the central part of M31 (0.02-35 kpc from the centre).
• (KING) Modified isothermal profile (King 1962;Einasto et al. 1974):
ρ IS O (r) = ρ 0 1 + r 2 r 2 c −1 − 1 + r 2 0 r 2 c −1 r r 0 , 0 r > r 0 .(10)
where ρ 0 = 0.413M ⊙ pc −3 , r c = 1.47 kpc, r 0 = 117 kpc.
• (MOORE) Moore profile (Moore et al. 1999):
ρ MOORE (r) = ρ c r rc 1.5 1 + r rc 1.5 ,(11)
where ρ c = 4.43 · 10 −3 M ⊙ pc −3 , r c = 17.9 kpc.
• (N04) Density distribution of Navarro et al. (2004):
ρ N04 (r) = ρ c exp − 2 α r α r α c − 1 ,(12)
where parameter α, according to simulations, equals to 0.172 ± 0.032 (Navarro et al. 2004). For N04 we take α = 0.17, ρ c = 6.42 · 10 −3 M ⊙ pc −3 , r c = 11.6 kpc.
• (NFW) Navarro-Frenk-White profile: Table 3. DM mass (in 10 9 M ⊙ ) without point sources: results of our Monte Carlo integration. The fraction of DM, removed together with the point sources, is also shown. All notations are the same as in previous table. where ρ c = 5.20 · 10 −2 M ⊙ pc −3 , r c = 8.31 kpc.
ρ NFW (r) = ρ c r rc 1 + r rc 2 ,(13)
• (BURK) Burkert profile (Burkert 1995):
ρ BURK (r) = ρ 0 1 + r rc 1 + r 2 r 2 c ,(14)
where
ρ 0 = 0.335M ⊙ pc −3 , r c = 3.43 kpc.
The computed DM masses within the FoV for all these profiles are shown in Table 2. We see that for the model used by W06 (model K2 in our notations), our estimate of the DM mass within the central 5 ′ coincides with the value used in W06: M 5 = (1.3±0.2)·10 10 M ⊙ . Notice, however, that to obtain the diffuse spectrum, we extracted all point sources, resolved with the significance 4σ. Each source was removed with the circle of the radius of 36 ′′ (see Sec. 4.1 for details). This led to the reduction of the area of the FoV by about 70% in case of circle5 region (c.f. Fig. 1). As the density of the DM changes with the off-centre distance and this change can be significant (c.f. Fig. 2), we performed the integration of the DM density distribution over the FoV with excluded point sources. To calculate the DM mass in such "swiss cheese" regions ( Fig. 1), we used Monte Carlo integration. The results are summarized in the Table 3.
To check possible systematic effects of our Monte Carlo integration method, we also obtained the values of enclosed mass inside the 13 arcmin sphere, and compared them with analytical calculations (wherever possible). Such an error does not exceed the purely statistical error of numerical integration (see Table 2). As one can see from Tables 2-3, the most conservative DM model, describing regions circle5 and ring5-13, is the model M31B of Widrow & Dubinski (2005). Therefore, to obtain restrictions on the DM parameters in what follows, we will use the DM mass estimates based on this model. For the DM distributions listed above, we also build the DM column density S dm (given by Eq. (4)) versus off-centre angle. The result is shown on Fig. 2. It is clearly seen that, in the off-centre regions, there is still a lot of DM, and, together with the fact that the surface brightness of X-ray diffuse emission falls rapidly outside the central 5 ′ (c.f. Takahashi et al. 2004), improving the restrictions of W06 by analyzing the off-centre 5 ′ −13 ′ ring. Moreover, as one can see from Table 3 and Fig. 2, the uncertainty of DM in this region is less than in the circle5 region.
To estimate the additional contribution from the Milky Way DM halo in the direction of M31, we use an isothermal DM distribution (as e.g. in Boyarsky et al. 2006dBoyarsky et al. , 2007. The DM column density is equal to
S MW,DM = v 2 h 8πr c G N K(φ),(15)
where v h = 170 km s −1 , r c = 4 kpc -parameters of isothermal model, r ⊙ = 8 kpc -distance from Earth to the Galactic Centre, and
K(φ) = r c R(φ) π 2 + arctan r⊙ cos φ R(φ) , cos φ 0 arctan R(φ) r⊙ cos φ , cos φ < 0.(16)
Here φ is defined via cos φ = cos l cos b for an object with galactic coordinates (b, l), R(φ) = r 2 c + r 2 ⊙ sin 2 φ 1/2 . For Andromeda galaxy (l = 121.17 • , b = −21.57 • , i.e. φ = 118.77 • ) one obtains S MW,DM ≈ 6.2 · 10 −3 g · cm −2 = 3.5 × 10 27 keV · cm −2
According to Fig. 2, the MW contributes < 5% to the total DM column density along the central part of Andromeda galaxy, and therefore will be neglected in what follows.
DATA REDUCTION AND BACKGROUND SUBTRACTION
To obtain restrictions on the parameters of the sterile neutrino, we need to analyze diffuse emission from faint extended regions of M31. There exist several well-developed background subtraction procedures for the diffuse sources (see, for instance, XMM-Newton SAS User Guide 8 , Nevalainen et al. 2005, Read & Ponman 2003. In this paper we use two methods of background subtraction:
Extended Sources Analysis Software (ESAS)
This method, recently developed by ESAC/GSFC team 9 , allows one to subtract instrumental and cosmic backgrounds separately. It seems to be better than the subtraction of the scaled blank-sky background, averaged through the entire XMM-Newton Field of View (see next subsection for details), as instrumental and cosmic backgrounds (due to their different origin) have different vignetting correction factors. ESAS models instrumental background from "first principles", using filter-wheel closed data and data from the unexposed corners of archived observations. Using this software, we are assured that no DM line can be in our background, in contrast with the "black sky" background subtraction method and, especially, local background subtraction (used e.g. in Shirey et al. (2001) to produce the diffuse spectrum of central 5 ′ of M31). The price to pay is the necessity of modelling cosmic background.
To prepare the EPIC MOS (Turner et al. 2001) event lists, we used the ESAS script mos-filter. After running mos-filter, we produced cleaned MOS images in sky coordinates, which were used to obtain the mosaic image (with the help of SAS v.7.0.0 tool emosaic). We used these event lists and images to find the point sources using SAS task edetect chain. Source detections were accepted with likelihood values above 10 (about 4σ). We found 243 point sources in this way. After that, we excluded each of them within the circular region of the radius 36 ′′ , which corresponds to the removal of ∼ 70 − 85% of total encircled energy, depending on the on-axis angle (see XMM users handbook 10 for details). The constructed mosaic image with detected point sources and selected regions is shown in Fig. 1. We obtained the MOS1 and MOS2 spectra and constructed the corresponding background with the help of ESAS scripts mos-spectra 11 and xmm-back, respectively.
Finally, we grouped the spectra with corresponding response and background files with the help of FTOOL grppha, a part of HEASOFT v6.1. To ensure Gaussian statistics, the minimum number of counts per bin was set to be 50.
The ESAS method of background subtraction, however, has several difficulties. The number of fitting parameters substantially increases, hence it is harder to find true minimum of χ 2 . The quantitative analysis of the 1.3−1.8 keV energy range is also not possible, because of the presence of two strong unmodelled instrumental lines (see Figs. 3,4). EPIC-PN (Strüder et al. 2001) data reduction is not yet implemented in ESAS. Therefore, to cross-check the results obtained with the help of ESAS software, we also processed EPIC data with the help of the blank-sky data subtraction (SBS) method (Read & Ponman 2003).
Blank-sky background subtraction (SBS)
We processed the same M31 observations (Table 1) as in the previous Section, using both MOS and PN data. To subtract the blanksky background we firstly cast it at the position of M31 with the help of the script skycast 12 , written by the XMM-Newton group in Birmingham. The scaling coefficient was derived by comparing count rates for E 10 keV from source regions and background sample. To produce spectra, ARF, RMF and to group them correctly (we needed to extract them from non-circular regions), we modified the Birmingham script createspectra. 13 The spatial regions were chosen similarly to those in Sec. 4.1, so it would be possible to compare the results of the two different methods (see Sec.5.3). When analyzing PN data, we found that the role of out-oftime (OOT) events was significant. This is due to the fact that the rate of OOT events is proportional to the total rate inside the full PN FoV rather than the rate of diffuse emission (outside excluded point sources). Therefore, it was necessary to remove the OOT events from the PN event lists. Most of the OOT events (from the bright point sources) form strips in the images and can be easily removed with the help of spatial filtering. This additional filtering also slightly reduced the possible DM signal, which was (in this outer region) nearly proportional to BACKSCALE keyword. This was accounted for when producing SBS PN restrictions.
FITTING THE SPECTRA IN XSPEC AND PRODUCING RESTRICTIONS
After we have prepared the data (with ESAS and SBS background subtraction methods) we fitted obtained spectra with realistic model (using Xspec spectral fitting package version 11.3.2, Arnaud 1996). The results of our fits are shown in Tables 4, 5, 6.
Notice that the fit results obtained by two background subtraction methods (ESAS and SBS) coincide within the 90% confidence interval (Table 4). 14 Also shown in Table 4 are the results of Takahashi et al. (2004), who analyzed diffuse emission in the central 6 ′ of M31. 15 Below we discuss separately the fitting of ESAS and SBS spectra.
ESAS spectra
We build 0.5 − 10.0 keV MOS spectra of circle5 and ring5-13 regions for 3 observations from Table 1. 16 Thus for each spatial region we have 6 spectra to fit -from observations with MOS1 and MOS2 cameras. We fix the model parameters to be equal for all six spectra from the same spatial region (except for normalization of the remaining soft proton background, as the spectra from different observations are slightly different).
Since ESAS software subtracts only the instrumental background component, the remaining cosmic background should be modelled. The cosmic background component is modelled with the help of Xspec model apec+(apec+pow)*wabs, according to the ESAS manual. A cool (∼ 0.1 keV), unabsorbed apec (Smith et al. 2001) component represents the thermal emission from the Local Hot Bubble. The hot (∼ 0.25 keV), absorbed apec component represents emission from the hotter halo and/or intergalactic medium. The last, absorbed powerlaw component with powerlaw index Γ = 1.41 represents the unresolved background from cosmological sources. We kept its normalization fixed for each region; it corresponds to 8.88 · 10 −7 Xspec units per square arcmin, or to 10.5 photons keV −1 s −1 cm −2 sr −1 . The corresponding hydrogen column density in wabs was left to vary below its Galactic value n H = 6.7 · 10 20 cm −2 (Morrison & McCammon 1983). To model the soft proton contamination, we used bknpow/b model (we fix its break energy at 3.3 keV), where index /b means that this component is not folded through the instrumental effective area (in Xspec versions 11 and earlier).
The diskbb+bbody (the same as the LMXB model in Taka hashi et al. 2004) component describes the point sources, which were not excluded. We fitted the diffuse M31 component in outer regions with the help of the sum of three vmekal (Mewe et al. 1986;Liedahl et al. 1995) models with fixed temperatures and abundances. The wabs column density was fixed at its Galactic value.
SBS spectra
We fitted the data from MOS and PN cameras, processed using SBS method (both separately and combined). As both cosmic and instrumental background is subtracted in SBS method, we fitted MOS and PN spectra on wabs*(diskbb+bbody+vmekal+vmekal+vmekal) Xspec model at the energy range 0.6-10.0 keV (0.6-12.0 keV in case of PN camera). The reduced χ 2 obtained by fitting our spectra are shown in Table 5; fit parameters are shown in Table 4. 14 The value of norm bb also coincides within 90% confidence interval if one propagates the uncertainty of blank-sky background normalization. 15 The appreciable difference between our errors and those of Takahashi et al. (2004) is due to the fact that we did not fix the metal abundances equal to each other. This was essential for our purposes, because of the
Region
Reduced χ
Producing restrictions on sterile neutrino parameters
In this subsection we describe two different techniques of searching for the narrow (compared to the spectral resolution of XMM-Newton) decay line in the spectra, processed by ESAS and SBS methods.
As shown on Fig. 5, above 2.0 keV there are few emission lines in the model of the spectrum of M31, and continuum emission dominates. In this case, it is possible to apply the "statistical" method, discussed e.g. in Boyarsky et al. (2006d). Namely, after fitting the spectra with the selected models (Secs. 5.1-5.2 above), we add an extra Gaussian line with the help of Xspec command addcomp. We then freeze its energy E γ , leave the line width σ to vary within 0-10 eV, and repeat the fit. For each line energy, we refit the model and derive an upper limit on the flux in the Gaussian line, allowing all other model parameters to vary. In particular we allow the abundances of heavy elements, that produce the thermal emission lines to vary. This produces the most conservative restrictions as the added line could account for some of the flux from the thermal components. After that we calculate the 3σ error with the help of Xspec command error line norm 9.0. To obtain conservative upper limits, we allow as much freedom as possible for the parameters of the thermal model. The 3σ upper limit on the DM line flux is shown in Fig. 8. These flux restrictions can be turned into constraints on parameters of the sterile neutrino (m s and sin 2 (2θ)), using Eq. (6) and the value of the M f ov dm from the Table 3 for the model M31B.
Below 2.0 keV, there are a lot of strong emission lines, which dominate over the continuum, creating a "line forest". As the intrinsic widths of these lines are much more narrow than the spectral resolution of EPIC cameras of XMM-Newton, and the abundances of various elements are known with large uncertainties, it is very hard to reliably distinguish these emission lines from a possible DM decay line. Therefore, to produce robust constraints, we apply the "full flux" method below 2 keV. In this method, we equate the DM line flux to the full flux plus 3 flux uncertainties over the energy interval ∆E equal to the spectral resolution of the instrument. 17 We also produce model-dependent "statistical" constraints below 2.0 keV. To reduce model uncertainty, we fix most metal abundances at their values known from optical observations of M31 (Jacoby & Ciardullo 1999; Jacoby & Ford 1986; Dennefeld & Kunth clear presence of the "line forest" at energies below 2.0 keV (see Sec. 5.3 and Fig. 5). 16 We exclude the region 1.3-1.8 keV due to the presence of two strong unmodelled instrumental lines, see Sec. 4.1. 17 To find the proper value of ∆E, we fold thin Gaussian line with appropriate RMF, and then evaluate FWHM of obtained broadened line. The FWHM ∆E, calculated in such a manner, slowly increases with line energy and changes from 0.18 keV to 0.21 keV in the 0.5 − 2.0 keV energy region. Table 6. Abundances from optical observations (in solar units). Our allowed range of abundances, used for construction the model-dependent restriction (see Sec. 5.3), is also shown.
1981; Blair et al. 1982). The confidence ranges of these abundances are shown in Table 6.
To compare our results with previous work on M31 (Watson et al. 2006, hereafter W06) we performed full flux analysis in the whole region of energies of the MOS camera of XMM-Newton. The results are shown in Fig. 9. One can see that our full flux results from circle5 region are somewhat weaker that the corresponding results of W06 (by a factor 2-3 in the region m s ∼ 4 keV; more than an order of magnitude at m s 2 keV and m s 12 keV). There are several reasons for this. As discussed in Sec. 3.1 we use an ∼ 8 times lower estimate for the DM mass within the FoV, because we use the more recent and more conservative DM profile of Widrow & Dubinski (2005) and compute the amount of DM by explicit integration over the FoV with removed point sources. At the same time, comparing our diffuse spectrum (Figs. 6-7) with Fig. 1 in W06, we see that the intensity of our diffuse spectrum is ∼ 2 − 3 times lower (due to the ∼ 4 times larger number of point sources removed). Therefore, one would expect a factor 2-3 difference between our results (as indeed is seen at m s ∼ 4 keV).
An additional discrepancy at low energies is due to the different choice of the energy bin intervals. In W06 the energy bin interval was chosen according to the empirical formula ∆E = E γ /30 = m s /60, while we have determined it using the XMM-Newton response matrices (as described in footnote 17 above). The difference is most prominent at low energies: e.g. at E ∼ 1 keV we obtain ∆E ≈ 0.2 keV, which is ∼ 6 times bigger than the value, used by W06. Therefore, at small energies we would expect constraints about an order of magnitude lower than those of W06, as Fig. 9 indeed demonstrates.
The other important effect, seen in Fig. 9, is the high-energy behaviour. Our restrictions remain nearly constant for m s 12 keV (E γ 6 keV), in contrast to the steeply decreasing results of W06. This is due to the fact that W06 used an energy-averaged countrate-to-flux conversion factor (i.e., the telescope effective area): see Sec. IV of W06. However, the effective area of the XMM-Newton MOS cameras declines sharply with energy, essentially going to zero at 9-10 keV. 18 Therefore, after a proper conversion, a constant count rate at high energies, assumed by W06 would correspond to a sharply rising physical flux in photons/(s · cm 2 ), which is of course incorrect. We performed a full data analysis, taking into account the dependence of the effective area on the energy and our constraints weaken sharply at high energies. This effect is well-known and is present in many papers that perform spectral analysis of XMM-Newton or Chandra data.
Our final constraints are shown in Fig. 10. At masses m s 4 keV (energies E γ 2 keV) we use the results of statistical constraints from the ring5-13 region. To produce the final restriction, we choose, for each value of m s , the minimal value of sin 2 (2θ). For m s < 4 keV (E γ < 2 keV) we plot both the model-independent (full flux) and the model-dependent constraints. The restrictions of Boyarsky et al. (2007) and Watson et al. (2006) are shown for comparison.
The high-energy behaviour of our final statistical constraints differs from that of in Fig. 9. There are several reasons for this. Firstly, in Fig. 9 we showed the full flux restrictions from the MOS camera (to compare our results with those of W06), while in Fig. 10 we used the combined constraints from both MOS and PN cameras. The PN camera has a wider energy range: its effective area decreases only above E ≈ 10 keV 19 , which explains the weak- Figure 10. Restrictions on (m s , sin 2 (2θ)) plane. The strongest previous limits of Boyarsky et al. (2007) as well as results of W06 are shown for comparison. The region above the curve is excluded. ening of constraints on Fig.10 for m s 20 keV. The "peak" at m s ≈ 16 − 18 keV, is due to the presence of strong Cu instrumental lines in the PN background spectrum (Strüder et al. 2001, see also Fig. 8). This region has, thus, higher errors, which weaken the constraints. Finally, we used several jointly fitted spectra (up to 9 in MOSPN-OOT dataset) in our "statistical" method, as opposed to the restrictions in Fig.9 where we used only one spectrum. The combination of several spectra improves the bounds, as statistical errors decrease.
RESULTS AND CONCLUSIONS
Using available XMM-Newton data on the central region of the Andromeda galaxy (M31), we obtained new restrictions on sterile neutrino Dark Matter parameters. We analyzed various DM distributions for the central part of M31, and obtained a conservative estimate of the DM mass inside the central 13 ′ , using the model M31B of Widrow & Dubinski (2005). This DM distribution turned out to be the most conservative among those which studied the DM distribution in the inner part of M31. 20 We found that exclusion of numerous point sources from the central part significantly improves our limits, therefore we have also calculated the DM mass in such "cheesed" regions with the help of Monte Carlo integration.
As the surface brightness is low in the selected regions, the choice of the background subtraction method is important. We processed XMM-Newton data from these regions with the help of two different background subtraction techniques -the Extended Sources Analysis Software (ESAS), and the blank-sky background subtraction (SBS), using the blank-sky background dataset of Read & Ponman (2003). We have shown that these totally different background subtraction methods give similar results.
To compare our results with the previous work on M31 (Watson et al. 2006, W06), we obtained the full flux restriction from the central 5 ′ of M31. Our full flux results (shown in Fig. 9) mostly reproduce the results of W06, up to differences arising from our more conservative estimate of expected DM signal and proper data analysis (see Sec. 5.3 for detailed discussion).
Our final upper limits (both model-dependent and modelindependent) are shown in Fig. 10. We improved the previous bounds of W06 on sin 2 (2θ) by as much as an order of magnitude for masses 4 keV m s 8 keV. Due to the significant low-energy thermal component in M31 diffuse emission, to produce the modelindependent constraints, we have used the "full flux" method for m s < 4.0 keV (i.e. E γ < 2.0 keV). In this region, the strongest constraints remain those of Boyarsky et al. (2007). We have also produced model-dependent constraints for E γ < 2.0 keV, using the "statistical" method; in this case we found the best-fit model by fixing the metallic abundances at the level of optical observations. The comparison of our upper limit with the lower bound on sterile neutrino pulsar kick mechanism (Fuller et al. 2003) improves the previous bounds and can exclude part of the parameter region (for 4 keV< m s < 20 keV).
Finally, it should be noticed that although throughout this paper we were writing about the sterile neutrino DM, the results of this work are equally applicable to any decaying DM candidate (e.g. gravitino), emitting photon of energy E γ and having decay width Γ. Our final results in this case are presented in Fig. 11. For other works discussing cosmological and astrophysical effects of decaying DM see de Rujula & Glashow (1980); Berezhiani et al. (1987); Doroshkevich et al. (1989); ; . An extensive review of the results can also be found in the book by Khlopov (1997).
Sterile neutrino in Dodelson-Widrow model
The results of this work have important consequences to one of the production models for the sterile neutrino, the so-called "Dodelson-Widrow" (DW) scenario -production through (non-resonant) oscillations with an active neutrino (Dodelson & Widrow 1994). The computation of the abundance is complicated in this case by the fact that the production mainly happens around the QCD transition and therefore QCD contributions are hard to compute (see Asaka et al. 2006, and refs. therein). A first-principles computation, taking into account all QCD contributions in a proper way, was performed in Asaka et al. (2007).
We compare the results of this computation with X-ray bounds obtained in this work and previous works in Fig. 12. The upper and lower dashed lines, bounding the grey area, correspond to the DW production scenario when all hadronic uncertainties are pushed in one or another direction; the thick central line corresponds to the most probable relation between m s and sin 2 (2θ). Upon comparison with X-ray bounds, we find that the upper bound on the DM mass in the DW scenario is reliably below m s < 4 keV (even if we push our X-ray bounds up by a factor of 2, to account for some yet unknown systematics and push all the uncertainties in hadronic contributions to the DW production in one direction).
This improves by 50% the previous bound m s < 6 keV of Asaka et al. (2007). Notice that other bounds on m s , that appeared in the literature (e.g. m s < 3.5 keV of Watson et al. (2006) and m s < 3 keV of Boyarsky et al. (2006d)) were based on the computations of Abazajian (2006), which did not take into account all QCD contributions.
Our present results may be combined with the Lyman-α analysis of Seljak et al. (2006); Viel et al. (2006); Viel et al. (2007). As follows from the most recent analysis of Viel et al. (2007), if one uses only the high-resolution high-redshift Lyman-α spectra of Becker et al. (2007) then one finds the lower bound on the sterile neutrino DM mass in the DW scenario to be m s > 5.6 keV, which is in contradiction with our current upper bound m s < 4 keV (but would have left a narrow allowed window for m s if one had used the previous bound m s < 6 keV of Asaka et al. 2007). If one takes into account the low-resolution SDSS Lyman-α dataset (Mc-Donald et al. 2006), used in Seljak et al. (2006; Viel et al. (2006), this contradiction becomes much stronger. Although the Lyman-α method relies on a very complicated analysis with (???) some unknown systematic uncertainties, it seems that the model in which all of the DM is produces through the DW scenario is ruled out.
However, there is another way to produce the sterile neutrino through oscillations with active neutrinos (resonant production in the presence of lepton asymmetries, Shi & Fuller 1999 (SF)). In this case, one qualitatively expects that the results of the Lyman-α analysis can be lowered by a significant amount, as for the same mass, the mean velocity (free-streaming length) in the SF model can be much lower than in the DW model. However, as sterile neutrinos are produced in the non-equilibrium way and their spectrum differs significantly from the thermal one, the actual Lyman-α bounds may depend not only on the free-streaming but also on the detailed shape of the spectrum. The detailed analysis of the SF production and corresponding re-analysis of the Lyman-α data is needed. Currently, the SF mechanism is not ruled out.
Finally, there is also the possibility of production of the sterile neutrino DM through the decay of the light inflaton (Shaposhnikov & Tkachev 2006), which cannot be ruled out by X-ray observations.
Therefore, the sterile neutrino remains a viable and interesting DM candidate, which can be either warm or cold. One of the most interesting ranges of parameters is that of low masses, which is also in the potential reach of laboratory experiments (Bezrukov & Shaposhnikov 2007) and will be probed with future X-ray spectrometers (Boyarsky et al. 2006a;den Herder et al. 2007). 21 However, the search for the sterile neutrino DM signal in all energy ranges above Tremaine-Gunn limit should also be conducted. creating wonderful atmosphere for young Ukrainian scientists, and to Ukrainian Virtual Roentgen and Gamma-Ray Observatory VIRGO.UA 23 and computing cluster of Bogolyubov Institute for Theoretical Physics 24 , for using their computing resources. This work was supported by the Swiss National Science Foundation and the Swiss Agency for Development and Cooperation in the framework of the programme SCOPES -Scientific co-operation between Eastern Europe and Switzerland. D.I. also acknowledges support from the INTAS project No. 05-1000008-7865. The work of A.B. was (partially) supported by the EU 6th Framework Marie Curie Research and Training network "UniverseNet" (MRTN-CT-2006-035863). O.R. would like to acknowledge support of the Swiss Science Foundation.
•
(GFBG) Preferred Navarro-Frenk-White distribution from Geehan et al. (2006): M vir = 6.80 × 10 11 M ⊙ ; r s = 8.18 kpc; C = 22. • (KER) Isothermal profile used in Kerins et al. (2001):
Figure 1 .
1Selected regions in the central part of M31 (shown in linear scale). Small circles correspond to excluded point source regions, large circles have radius of 5 and 13 arcmin.
Figure 2 .
2M31 DM column density versus off-centre angle as result of our Monte Carlo integration, based on DM profiles of Sec. 3.1. (Point sources are not excluded).
Figure 3 .
3Observed spectrum (top) and modelled instrumental background (bottom) MOS1 from ObsID 0112570101, region ring5-13. It can be seen that the spectrum and modelled background almost coincide for E > 7 keV.
Figure 4 .Figure 5 .
45Folded spectra from ring5-13 region (by ESAS method), with excluded point sources. The presence of two unsubtracted instrumental lines at 1.49 keVand 1.75 keVis clearly shown. Unfolded spectra and best-fit model from ring5-13 region (by ESAS method), with excluded point sources. The "line forest" at energies lower 2.0 keV is clearly visible.
Figure 6 .
6Folded MOS1 spectra from circle5 region, ObsID 0112570401, with (top) and without (bottom) point sources.
Figure 7 .
7Folded spectra and best-fit model from circle5 region, with excluded point sources.
Figure 8 .
83σ upper limit on the DM line flux (the region of parameter space above the curves is excluded). Left panel: upper limits from the different spatial regions for the spectra, processed by ESAS method. Right panel: upper limits for the ring5-13 region for both ESAS and SBS methods.
Figure 9 .
9Our limits on m s , sin 2 (2θ) parameters, obtained by using the full flux method from different spatial regions of M31 (a region of parameter space above a curve is excluded). The restriction from W06 is shown for comparison. this work, model-independent restriction) M31 (this work, model-dependent restriction) Boyarsky et al. 2007 M31(Watson et al. 2006)
Figure 11 .Figure 12 .
1112Constraints on the decay width Γ of any radiatively decaying DM from this work (marked "M31") andBoyarsky et al. (2007) (marked "MW"). The shaded region of parameters is excluded. Current X-ray constraints, combined with the DW production model. Colored regions are excluded. The grey region shows the range of parameters which give correct abundance in the DW model(Asaka et al. 2007). The color shaded regions mark the restrictions from "LMC"(Boyarsky et al. 2006d), "MW"(Boyarsky et al. 2007) and "M31" (this work). Model-dependent restrictions from M31 for m s < 2 keV are shown in (green) dashed line.
Table 1. Observations of the central part of M31, used in our analysis.Table 2. DM mass (in 10 9 M ⊙ ) inside regions, used in our analysis: results of our Monte Carlo integration. The point sources are not excluded here. The 95% statistical errors are also shown. The DM distributions of):
Obs. ID
Starting time, UTC
Filter
Cleaned MOS1/MOS2/PN exposure, ks
0112570401
2000-06-25 08:12:41
Medium
30.8/31.0/27.6
0109270101
2001-06-29 06:15:17
Medium
40.1/41.9/47.4
0112570101
2002-01-06 18:00:56
Thin
63.0/63.0/55.3
Model
circle5
ring5-13
13 arcmin sphere, MC result
13 arcmin sphere, analytical result
K1, with sources
3.27 ± 0.01
12.49 ± 0.03
5.84 ± 0.02
5.84
K2, with sources
11.88 ± 0.03
23.75 ± 0.09
20.76 ± 0.09
-
GFBG, with sources
6.59 ± 0.02
20.46 ± 0.06
13.40 ± 0.03
13.39
KING, with sources
6.68 ± 0.01
24.61 ± 0.05
14.80 ± 0.02
14.80
MOORE, with sources
7.34 ± 0.02
19.48 ± 0.02
13.79 ± 0.02
13.78
N04, with sources
7.68 ± 0.03
22.89 ± 0.07
15.16 ± 0.06
15.18
NFW, with sources
11.08 ± 0.04
40.5 ± 0.1
22.3 ± 0.1
22.25
BURK, with sources
6.71 ± 0.02
27.97 ± 0.03
15.90 ± 0.05
15.90
KER, with sources
5.35 ± 0.02
22.45 ± 0.04
11.56 ± 0.03
11.56
M31A, with sources
5.95 ± 0.01
16.45 ± 0.02
11.03 ± 0.02
-
M31B, with sources
4.99 ± 0.01
14.24 ± 0.01
9.40 ± 0.02
-
M31C, with sources
5.60 ± 0.01
16.12 ± 0.01
10.29 ± 0.02
-
7 We do not use the fourth model (M31d), because in Widrow & Dubinski (2005) it was found that this model develops a bar, which rules it out experimentally.Model
circle5
Removed from circle5, %
ring5-13
Removed from ring5-13, %
K1, without sources
0.767 ± 0.004
76.6
9.71 ± 0.02
22.3
K2, without sources
2.31 ± 0.02
80.4
18.09 ± 0.08
23.9
GFBG, without sources
1.48 ± 0.01
77.4
15.77 ± 0.06
23.0
KING, without sources
1.64 ± 0.01
75.5
18.99 ± 0.06
22.9
MOORE, without sources
1.52 ± 0.01
79.2
14.98 ± 0.03
23.1
N04, without sources
1.70 ± 0.02
77.7
17.62 ± 0.05
23.0
NFW, without sources
2.59 ± 0.01
76.7
31.34 ± 0.07
22.5
BURK, without sources
1.67 ± 0.02
75.1
21.68 ± 0.02
22.5
KER, without sources
1.33 ± 0.01
75.0
17.42 ± 0.04
22.5
M31A, without sources
1.24 ± 0.01
79.3
12.66 ± 0.02
22.9
M31B, without sources
1.04 ± 0.01
79.1
10.98 ± 0.01
23.0
M31C, without sources
1.21 ± 0.01
78.4
12.43 ± 0.01
22.9
For certain dSph cusped profiles are still admissible, but disfavored. Additional considerations rule out the possibility of existence of cusped profiles for the Ursa Minor and Fornax (Kleyna et al. 2003a,b; Goerdt et al. 2006; Sánchez-Salcedo et al. 2006).
Sterile neutrinos are not in thermal equilibrium in the early Universe and therefore their velocity distribution is non-universal and depends on the model of production. 3 Strictly speaking, in case of other models of production the power spectrum of density fluctuations is not only characterized by the free-streaming length. Therefore, the rescaling of the results ofSeljak et al. (2006);Viel et al. (2006) can be used only as the estimates and the reanalysis of the Lyman-α data for the case of each model is required.
Namely, if luminosity distance D L is much greater than the characteristic scale of the DM distribution.
Our decay rate is 2 times smaller than the one used in W06. This is due to the Majorana nature of the sterile neutrino, which we consider (c.f.Barger et al. 1995). The final constraints for a Dirac particle would thus be 2 times stronger. 4 A.Boyarsky et al.
In contrast to the other models, this model does not describe the current DM distribution, but helps our understanding the time evolution of DM mass inside constant FoV.
http://xmm.esac.esa.int/external/xmm user support/documentation/sas usg/USG 9 We use ESAS version 1.0. 10 http://xmm.esac.esa.int/external/xmm user support/documentation/uhb
To produce correct RMF file, we changed in the script mos-spectra option rmfgen detmaptype=psf to rmfgen detmaptype=dataset. 12 http://www.sr.bham.ac.uk/xmm3/skycast 13 http://www.sr.bham.ac.uk/xmm3/createspectra
For PN camera this happens at ∼ 12 keV (c.f.Fig. 8). 19 XMM-Newton Users Handbook, Sec. 3.2.2.1, http://xmm.esac.esa.int/external/xmm user support/documentation/uhb 2.5
We would like to notice, however, that in the workKerins (2004), a number of "extreme" (i.e. maximizing contributions of disk, spheroid or halo) models are considered. Some of these models would reduce an estimated DM signal from the inner 13 ′ (and correspondingly our limits) by a factor ∼ 2. We chose to use the family of models, shown onFig. 2, as they qualitatively agree with each other and do not contain any "extreme" assumptions. However, below, in deriving a model-dependent upper limit of the mass of the DM particle, we will introduce an additional penalty factor, to account for this and other possible systematic uncertainties.
See also EDGE Project: http://projects.iasf-roma.inaf.it/edge 22 http://sec.bitp.kiev.ua
ACKNOWLEDGEMENTSWe would like to thank B. Gripaios, A. Neronov, J. Nevalainen, M. Markevich, M. Shaposhnikov, C. Watson for useful comments. D.I. is grateful to ESAC team and especially to M. Kirsch, for granting his stay at ESAC and for useful discussions. D.I. and V.S. are also grateful to M. Ehle, R. Saxton and S. Snowden for useful discussions about ESAS software, to Scientific and Educational Centre 22 of the Bogolyubov Institute for Theoretical Physics in Kiev, Ukraine, and especially to V. Shadura, for
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"ON THE GEOMETRY OF NORMAL HOROSPHERICAL G-VARIETIES OF COMPLEXITY ONE",
"ON THE GEOMETRY OF NORMAL HOROSPHERICAL G-VARIETIES OF COMPLEXITY ONE"
] |
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"Kevin Langlois ",
"Ronan Terpereau "
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Let G be a connected simply-connected reductive algebraic group. In this article, we consider the normal algebraic varieties equipped with a horospherical G-action such that the quotient of a G-stable open subset is a curve. Let X be such a G-variety. Using the combinatorial description of Timashev, we describe the class group of X by generators and relations and we give a representative of the canonical class. Moreover, we obtain a smoothness criterion for X and a criterion to determine whether the singularities of X are rational or log-terminal respectively.
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"https://arxiv.org/pdf/1411.2480v2.pdf"
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1411.2480
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bee2408d9b129b5670b70d58fe1ffe5d798d0f5a
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ON THE GEOMETRY OF NORMAL HOROSPHERICAL G-VARIETIES OF COMPLEXITY ONE
10 Nov 2014
Kevin Langlois
Ronan Terpereau
ON THE GEOMETRY OF NORMAL HOROSPHERICAL G-VARIETIES OF COMPLEXITY ONE
10 Nov 2014
Let G be a connected simply-connected reductive algebraic group. In this article, we consider the normal algebraic varieties equipped with a horospherical G-action such that the quotient of a G-stable open subset is a curve. Let X be such a G-variety. Using the combinatorial description of Timashev, we describe the class group of X by generators and relations and we give a representative of the canonical class. Moreover, we obtain a smoothness criterion for X and a criterion to determine whether the singularities of X are rational or log-terminal respectively.
Introduction
The varieties and the algebraic groups that we consider are defined over an algebraically closed field k of characteristic zero. Let G be a connected simplyconnected reductive algebraic group and let B ⊂ G be a Borel subgroup. The aim of this article is to describe certain geometric properties of a family of G-varieties: the normal horospherical G-varieties of complexity one. Among other results, we obtain explicit criteria to characterize the singularities of these varieties, we describe their class group by generators and relations, and we give an explicit representative of their canonical class.
Recall that the complexity of a G-variety X is the transcendence degree of the field extension k(X) B over k, where k(X) B denotes the field of B-invariant rational functions on X; see [LV83,Vin86]. Since all the Borel subgroups of G are conjugated, this notion does not depend on the choice of B. Geometrically, the complexity is the codimension of a general B-orbit. For instance, the G-varieties of complexity zero contain an open B-orbit. Those which are normal are called spherical varieties; see [Kno91,Pez10,Hur11,Tim11,Per14] for more information.
The study of the G-varieties of complexity one is the next step (after the spherical case) towards the classification of normal G-varieties. Many important examples of G-varieties of complexity one motivate this study:
• The normal T -varieties of complexity one, where T is a torus; see [KKMS73,Tim08] for a combinatorial description, [Lan14] for a generalization over an arbitrary field, [FZ03] for the case of surfaces, and [AH06, AHS08, AIPSV12] for higher complexity. • The homogeneous spaces G/H of complexity one, where H is a connected reductive closed subgroup, are described in [Pan92,AC04]. • The embeddings of SL 2 /K into normal SL 2 -varieties, where K is a finite subgroup, are studied in [Pop73], [LV83,§9], [Mos90], and [Mos92]. More generally, see [Tim11,§16.5] for a classification of the embeddings of G/K where G is a semisimple group of rank 1 and K ⊂ G is a finite subgroup. • An example from classical geometry: Let T ⊂ SL 3 be the subgroup of diagonal matrices. The homogeneous space SL 3 /T can be identified with the set of ordered triangles on P 2 . The embeddings of SL 3 /T are studied in [War82] and [Tim11,16.5].
A combinatorial description of normal G-varieties of complexity one is obtained in [Tim97]. This description is inspired by the Luna-Vust theory of embeddings of homogeneous spaces G/H into normal varieties; see [LV83].
A G-action is called horospherical if the isotropy group of any point contains a maximal unipotent subgroup of G. Therefore a homogeneous space G/H is horospherical if and only if H contains a maximal unipotent subgroup; such an H is called a horospherical subgroup of G. It follows from the Bruhat decomposition of G that every horospherical G-homogeneous space is spherical, and thus a general G-orbit of a horospherical G-variety of complexity one has codimension one. We will recall in Section 1.1 how the set of horospherical subgroups of G containing the unipotent radical of B can be described from the set of simple roots of G.
The G-equivariant birational class of a horospherical G-variety X is determined by the invariant field k(X) G and by the isotropy subgroup H of a general point. Indeed, if r denotes the complexity of the G-variety X, then by [Kno90, Satz 2.2] there exist an r-dimensional variety C and a G-equivariant birational map φ : X Z := C × G/H, where G acts on Z = C × G/H by translation on G/H. The map φ induces field isomorphisms k(X) ≃ Frac (k(C) ⊗ k k(G/H)) and k(C) ≃ k(X) G . If r = 1, then we can assume that C is a smooth projective curve.
This article is structured as follows:
In the first part, we set up our framework by explaining the combinatorial description of normal horospherical G-varieties of complexity one (following Timashev). Let us be more precise. In Section 1.1 we describe the horospherical Ghomogeneous spaces. In Section 1.2 we recall the definition of the scheme of geometric localities Sch G (Z) whose normal separated G-stable open subsets of finite type, also called G-models of Z, are the normal G-varieties G-birational to Z. We also introduce the notion of a chart, which is a B-stable affine open subset of Sch G (Z), and of a germ, which is a (proper) G-stable closed subvariety of Sch G (Z). Then we consider the B-stable divisors on G/H, also called colors (of G/H), and we introduce the set of G-valuations of k(Z). Finally, we define the colored σpolyhedral divisors in Section 1.3 and explain how to obtain any G-model of Z from a finite collection of such polyhedral divisors.
The results of this paper are stated and proved in the second part. In Section 2.1 we explain how to obtain any simple G-model of Z as the parabolic induction of an affine L-variety, where L ⊂ G is a Levy subgroup. In particular, we obtain an effective construction of any simple G-model of Z. From this, we deduce several criteria to characterize the singularities of simple G-models of Z; see Theorem 2.2 for a criterion for rationality of singularities and Theorems 2.3 and 2.4 for smoothness criteria. As mentioned at the end of Section 2.1, our smoothness criteria are explicit thanks to the works of Pasquier and Batyrev-Moreau. In Section 2.2 we prove the existence of the decoloration morphism for normal horospherical G-varieties of complexity one and give an explicit description of this morphism in terms of germs; see Proposition 2.7 for a precise statement. The decoloration morphism was introduced by Brion for spherical varieties in [Bri91, §3.3] and plays a key-role in our study of normal horospherical G-varieties of complexity one. Until the end of this introduction, we let X be such a variety. In Section 2.3, the heart of this paper, we parametrize the G-stable prime Weil divisors of X and deduce from this a description of the class group of X by generators and relations; see Theorem 2.8 and Corollary 2.9. Then we obtain a criterion of factoriality for X; see Corollary 2.10. Also, we relate the description of stable Cartier divisors obtained by Timashev in [Tim00] to our description of stable Weil divisors; see Corollary 2.13. In Section 2.4 we give an explicit representative of the canonical class of X; see Theorem 2.14. From this, we deduce criteria for X to be Q-Gorenstein or log-terminal respectively; see Corollary 2.15 and Theorem 2.18. Finally, Proposition 2.17 provides an explicit resolution of singularities of X which factors through the decoloration morphism defined in Section 2.2.
Notation. The base field k is algebraically closed of characteristic zero. An integral separated scheme of finite type over k is called a variety. If X is a variety, then k[X] denotes the coordinate ring of X and k(X) denotes the field of rational functions of X. A point of X is always assumed to be closed. We denote by G a connected simply-connected reductive algebraic group (i.e., a direct product of a torus and a connected simply-connected semisimple group), by B ⊂ G a Borel subgroup, by U = R u (B) the unipotent radical of B, and by T ⊂ B a maximal (algebraic) torus. We denote by G m the multiplicative group over k. A subgroup of G is always a closed subgroup. If H is such a subgroup, then
N G (H) = {g ∈ G | gHg −1 ⊂ H} is the normalizer of H in G.
For an algebraic group K, we denote by
χ(K) = {algebraic group homomorphisms φ : K → G m }
the character group of K. A variety on which K acts (algebraically) is called a Kvariety. An algebra (over k) on which K acts by algebra automorphisms is called a K-algebra. If X is a K-variety, then k[X] and k(X) are K-algebras; in particular, k[X] and k(X) are linear representations of K.
Preliminaries
In this first part, we explain the combinatorial description of normal G-varieties of complexity one as given in [Tim11,§16] and specialized in the horospherical case. Let C be a smooth projective curve, let G/H be a horospherical G-homogeneous space, and let Z = C × G/H; the group G acts on Z by translation on the second factor. The approach of Timashev consists in giving a classification of all the normal G-varieties which are G-birational to Z.
1.1. Horospherical homogeneous spaces. In this section we enunciate a combinatorial description of the horospherical homogeneous spaces; see [Pas08,§2] for details.
Let S be the set of simple roots of G with respect to (T, B). There exists a well-known one-to-one correspondence I → P I between the powerset of S and the set of parabolic subgroups of G containing B; see [Spr98,Theorem 8.4.3]. Let us assume that the closed subgroup H ⊂ G contains the unipotent radical U of B. Then P = N G (H) is a parabolic subgroup containing B. Therefore, there exists a unique subset I ⊂ S such that P = P I . The quotient algebraic group K := P/H is a torus and M = χ(K) identifies naturally with a sublattice of χ(T ).
The next statement ([Pas08, Proposition 2.4]) explains how the pair (M, I) completely describes the horospherical homogeneous space G/H. • the set of closed subgroups of G containing U ; and • the set of pairs (M, I), where M is a sublattice of χ(T ) and I is a subset of S such that for every α ∈ I and every m ∈ M , we have m,α = 0 (hereα denotes the coroot of α). All varieties which are birational to Z may be glued together into a scheme over k that we denote by Sch(Z). More precisely, the schematic points of Sch(Z) are local rings corresponding to prime ideals of finitely generated subalgebras with quotient field k(Z) and the spectra of those subalgebras define a base of the Zariski topology on Sch(Z) (by identifying prime ideals with associated local rings). Furthermore, the abstract group G acts on the set Sch(Z) via its linear action on k(Z). We We now introduce the notions of chart and germ of a G-model X of Z. A chart (or affine chart or B-chart ) of X is an affine dense open subset of X which is B-stable. A germ (or a G-germ) of X is a non-empty G-stable irreducible closed subvariety Y X. By [Sum74, Theorem 1], for every germ Y X there exists a chart X 0 ⊂ X such that X 0 ∩ Y = ∅. The G-model X of Z is called simple if it has a chart intersecting all the germs. By [Tim00, §5, Lemma 2], every simple G-model of Z is quasi-projective. Moreover, X is a finite union of simple G-models of Z.
1.2.2. We now introduce the notion of color. Let K ′ be an algebraic group acting on a variety X ′ , then a K ′ -divisor on X ′ is an irreducible closed subvariety of X ′ which is K ′ -stable and has codimension one. A color of G/H is a B-divisor on G/H. Let us consider the natural map π : G/H → G/P.
Then each color (of G/H) is of the form D α = π −1 (E α ), where E α is the Schubert variety of codimension one corresponding to the root α ∈ S \ I. We represent colors as vectors of the lattice N = Hom Z (M, Z) as follows: For the natural action of B on k(G/H), the lattice M = χ(P/H) identifies with the lattice of B-weights of the B-algebra k(G/H). For every (non-zero) B-eigenvector
f ∈ k(G/H) of weight m ∈ M , we put m, ̺(D α ) = v Dα (f ),
where v Dα is the valuation associated with D α . The value ̺(D α ) does not depend on the choice of f and coincides with the restriction of the corootα to the lattice M ; see [Pas08,§2]. Denoting by F 0 the set of colors, we obtain a map ̺ : F 0 → N . Let us note that ̺ is not injective in general; for instance, if H = P is a parabolic subgroup, then N = {0} and thus ̺ is constant.
Let X be a G-model of Z. A B-divisor of X which is not G-stable is called a color of X. There is a one-to-one correspondence between the set of colors of G/H and the set of colors of X given as follows: X possesses a G-stable open subset of the form C ′ × G/H, where C ′ ⊆ C is a dense open subset. If D is a color of G/H, then the closure of C ′ × D in X is a color of X and vice-versa. In the following, we will always denote the colors of X and G/H in the same way.
1.2.3. A G-valuation of k(Z) is a function v : k(Z) ⋆ → Q such that: • v(a + b) ≥ min{v(a), v(b)}, for all a, b ∈ k(Z) ⋆ satisfying a + b ∈ k(Z) ⋆ ;
• v is a group homomorphism from (k(Z) ⋆ , ×) to (Q, +); • the subgroup k ⋆ is contained in the kernel of v; and • v(g · a) = v(a) for every g ∈ G and every a ∈ k(Z) ⋆ .
By [Tim11,Proposition 19.8], every G-valuation of k(Z) is proportional to a valuation v D for a G-divisor D on a G-model of Z. Let us denote by N Q = Q ⊗ Z N the Q-vector space associated with N . We follow [Tim11, Definition 16.1] and define the set E as the disjoint union of sets {z} × E z , where E z = N Q × Q ≥0 and z ∈ C, modulo the equivalence relation ∼ defined by
(1) (z, v, l) ∼ (z ′ , v ′ , l ′ ) if and only if z = z ′ , v = v ′ , l = l ′ or v = v ′ , l = l ′ = 0.
Therefore, E is the disjoint union indexed by C of copies of the upper half-space
N Q × Q ≥0 ⊂ N Q × Q with boundaries N Q × {0} identified as a common part.
There is a bijection between the set of G-valuations of k(Z) and the set E , which we now explain. Let A M denote the algebra generated by the B-eigenvectors of k(Z). Since M is a free abelian group, the exact sequence of abelian groups Let u = [(z, v, l)] ∈ E . We define a valuation w = w u of A M as follows:
w i∈I f i χ mi = min i∈I {v(m i ) + l · ord z (f i )} ,
where I is a finite set, the m i are pairwise distinct elements of M , and each f i belongs to k(C) ⋆ . By [Tim11,Corollary 19.13, Theorems 20.3, 21.10], for every u ∈ E , there exists a unique G-valuation of k(Z) such that the restriction to A M is w u . From now on, we will always identify E with the set of G-valuations of k(Z) and N Q as a part of E via the (well-defined) map v → [(·, v, 0)].
1.3. Colored polyhedral divisors. Let M Q = Q ⊗ Z M , then the Q-vector spaces M Q and N Q are dual to each other; we denote the duality by
M Q × N Q → Q, (m, v) → m, v .
We recall that a strongly convex polyhedral cone in N Q is a cone generated by a finite number of vectors and which contains no line.
1.3.1. We now introduce the notion of colored polyhedral divisor, which is equivalent to the one of colored hypercone introduced in [Tim11, Definition 16.18].
Definition 1.2. (i) Let σ ⊂ N Q be a strongly convex polyhedral cone. A σpolyhedron is a subset of N Q obtained as a Minkowski sum Q + σ, where Q ⊂ N Q is the convex hull of a non-empty finite subset. Let C 0 be a dense open subset of the curve C, let
D = z∈C0 ∆ z · [z]
be a formal sum over the points of C 0 , where each ∆ z is a σ-polyhedron of N Q and ∆ z = σ for all but a finite number of z ∈ C 0 , and let F ⊂ F 0 be a set of colors of G/H such that
• 0 does not belong to ̺(F ); and • ̺(F ) ⊂ σ.
We call such a pair (D, F ) a colored σ-polyhedral divisor on C 0 . If σ and F are clear from the context, then we write D instead of (D, F ) and call D a colored polyhedral divisor on C 0 . We say that D is trivial
(on C 0 ) if ∆ z = σ for every z ∈ C 0 . Let σ ∨ = {m ∈ M Q | ∀v ∈ σ, m, v ≥ 0}
denote the dual polyhedral cone of σ. To each m ∈ σ ∨ , we associate a Q-divisor on C 0 :
D(m) = z∈C0 min v∈∆z(0) m, v · [z],
where for every z ∈ C 0 we denote by ∆ z (0) the set of vertices of ∆ z .
(ii) To a given D on C 0 we associate the following M -graded normal k-algebra (see [AH06,§3] for details):
A[C 0 , D] := m∈σ ∨ ∩M A m χ m , where A m = H 0 (C 0 , O C0 (D(m))) := H 0 (C 0 , O C0 (⌊D(m)⌋))
and ⌊D(m)⌋ is the Weil divisor (with integer coefficients) on C 0 obtained by taking the integer part of each coefficient of D(m). The multiplication on A[C 0 , D] is constructed from the maps
τ m,m ′ : A m × A m ′ → A m+m ′ , (f 1 , f 2 ) → f 1 · f 2 .
Let us note that each map τ m,m ′ is well-defined since for all m, m ′ ∈ σ ∨ :
D(m) + D(m ′ ) ≤ D(m + m ′ ). (iii) Let w ∈ σ ∨ ∩ M such that A w = {0} and f ∈ A w \ {0}
, and let F w be the set of colors defined by the relation
F w = {D ∈ F | ̺(D) ∈ w ⊥ }. The localization of the colored σ-polyhedral divisor (D, F ) with respect to f χ w is the colored (σ ∩ w ⊥ )-polyhedral divisor (D w f , F w ) defined by D w f = z∈(C0) w f Face(∆ z , w) · [z],
where
(C 0 ) w f = C 0 \Z(f ) with Z(f ) = Supp(divf + D(w)), and Face(∆ z , w) := v ∈ ∆ z w, v ≤ min v ′ ∈∆z w, v ′ .
(iv) To ensure that the algebra A[C 0 , D] is finitely generated over k and has Frac A M as field of fractions (we recall that A M denotes the algebra generated by the B-eigenvectors of k(Z)), we now introduce the notion of properness for colored polyhedral divisors following [AH06,§2]. The colored σ-polyhedral divisor (D, F ) is called proper if either C 0 is affine or C 0 = C is projective and satisfies the following conditions:
• deg D := z∈C ∆ z σ. • If min v∈deg D m, v = 0, then rD(m) is a principal divisor for some r ∈ Z >0 .
In the following, σ ⊂ N Q always denotes a strongly convex polyhedral cone and F a set of colors satisfying the conditions of Definition 1.2; in particular, (σ, F ) is a colored cone of N Q (with respect to G/H); see [Tim11,Definition 15.3] for the definition of colored cones. Let us note that, if D is a proper colored polyhedral divisor on C 0 , then by [AHS08, Proposition 3.3] we have the relation
A[C 0 , D] f χ w = A[(C 0 ) w f , D w f ].
The next remark makes the link between the description of [Tim11] and the notions introduced in Definition 1.2.
A = (k(C) ⊗ k k[Bx 0 ]) ∩ D∈F O vD ∩ v∈C (D)(1) O v , where for a discrete G-valuation v of k(Z), O v = {f ∈ k(Z) ⋆ | v(f ) ≥ 0} ∪ {0}
is the corresponding local ring. By [Tim11, Theorem 13.8, §16.4], A is a normal algebra of finite type over k. Moreover, by [Tim11, Corollary 13.9, §16.4], the Bvariety X 0 = Spec A is an open subset of Sch G (Z). Also, by [Tim11, Theorem 12.6], the subscheme X(D) := G · X 0 is a G-model of Z and X 0 is a chart of X(D). Conversely, for every simple G-model X of Z there exists a colored polyhedral divisor D such that X = X(D); see [Tim11,Theorem 16.19]. By [Tim97, Theorem 3.1] and with the notation as in Definition 1.2, we have the relation:
X(D w f ) = G · (X 0 ) f χ w , where (X 0 ) f χ w is the distinguished Zariski open subset of f χ w ∈ k[X 0 ]. Lemma 1.4. Let (D, F ) be a proper colored σ-polyhedral divisor on a dense open subset C 0 ⊆ C, and let X 0 ⊂ X(D) be the corresponding chart. Then the invariant algebra k[X 0 ] U identifies with the algebra A[C 0 , D] introduced in Definition 1.2(ii).
Proof. The subalgebra
k[X 0 ] U ⊂ (k(C) ⊗ k k[Bx 0 ]) U = A M is generated by elements of the form f χ m , where f ∈ k(C) ⋆ and m ∈ M , satisfying v D (f χ m ) ≥ 0
and v(f χ m ) ≥ 0 for all D ∈ F and v ∈ C(D)(1). One may check that these conditions are equivalent to div(f ) + D(m) ≥ 0, which proves the lemma.
1.3.3. Let X be the simple G-model of Z associated with the proper colored σpolyhedral divisor (D, F ) as explained in Subsection 1.3.2. By [Tim11,Theorem 16.19 (2)], there is a combinatorial description of the set of germs of X (defined in Subsection 1.2.1). Each germ Y is described by a pair (C 1 , F 1 ), called the colored datum of Y , where C 1 is an hyperface of C (D) (see [Tim11,Definition 16.18] for the definition of an hyperface of an hypercone) and
F 1 := {D α ∈ F | ̺(D α ) ∈ C 1 }.
Geometrically, F 1 is the subset of colors of F that contain the germ Y .
1.3.4. In this subsection we consider the general description of the non-necessarily simple G-models of Z.
A finite collection Σ = {(D i , F i )} i∈J of proper colored polyhedral divisors de- fined on dense open subsets of C is a colored divisorial fan if for all i, j ∈ J there exists l ∈ J such that C(D l ) = C(D i )∩C(D j ) and (C(D l ), F l ) is a common hyperface of (C(D i ), F i ) and (C(D j ), F j ); see [Tim11, §16
.4] for the definition of an hyperface of a colored hypercone.
Let us denote
|Σ| = i∈J C (D i ) ⊂ E
the support of the colored divisorial fan Σ.
The next theorem gives a description of the G-models of Z; see [Tim11,Theorem 16.19 (3)] and [Tim11, Corollary 12.14].
Theorem 1.5. If Σ = {(D i , F i )} i∈J is a colored divisorial fan on C, then the union X(Σ) := i∈J X(D i ) in the scheme Sch G (Z) (defined in Subsection 1.2.1) is a G-model of Z, and every G-model of Z is obtained in this way. Moreover, the G-model X(Σ) of Z is a complete variety if and only if |Σ| = E , where E is the set of G-valuations (defined in Subsection 1.2.3).
Main results
2.1. Parabolic induction and smoothness criteria. In this section we prove Lemma 2.1 which allows us to construct explicitly any simple G-model of Z as the parabolic induction of an affine L-variety, where L ⊂ G is a Levy subgroup. From this we deduce a criterion (Theorem 2.2) to determine whether the singularities of a simple G-model of Z are rational. Moreover, we obtain smoothness criteria (Theorems 2.3 and 2.4) for simple G-models of Z. Throughout this section we fix a proper colored σ-polyhedral divisor (D, F ) on a dense open subset C 0 ⊂ C. [Tim11,§28]. As before, T denotes a maximal torus of G contained in B. We consider the following set of simple roots
This subsection is inspired by
I ′ := {α ∈ S \ I | D α ∈ F } ∪ I
and we denote by P F the parabolic subgroup P I ′ containing B. We choose a Levy subgroup L ⊂ P F containing T and a Borel subgroup B L of L containing T such that I ′ is the set of simple roots of L (with respect to (T, B L )). We denote by Z L the horospherical L-variety C × L/H L , where H L = L ∩ H. By [Tim11, Corollary 15.6], the homogeneous space L/H L is quasi-affine. Moreover, denoting P L = N L (H L ), the torus P L /H L identifies with K = P/H, and M = χ(P/H) = χ(P L /H L ) through this identification. Let F L be the set of colors of L/H L , then one may check that the image of F L by the map ̺ in the lattice N is the same as the one of F . In particular, F L satisfies the conditions of Definition 1.2. We denote by (D L , F L ) the colored σ-polyhedral divisor on C 0 (relatively to Z L ) defined by the formal sum
D L := D = z∈C0 ∆ z · [z], and by X(D L ) ⊂ Sch L (Z L ) the associated L-variety; see Subsection 1.3.2.
The quotient morphism P F → P F /R u (P F ) ∼ = L makes X(D L ) a P F -variety. As the quotient morphism G → G/P F is locally trivial (for the Zariski topology), we can form the twisted product G × PF X(D L ). The latter is defined as the quotient (G × X(D L ))/P F , where P F acts as follows:
p.(g, x) := (gp −1 , p.x), where p ∈ P F , g ∈ G, and x ∈ X(D L ).
See [Jan87, §I.5] for more details on twisted products. Lemma 2.1. With the notation of Subsection 2.1.1, the L-variety X(D L ) is affine and
X(D) is G-isomorphic to the twisted product G × PF X(D L ). Moreover, the L-algebra k[X(D L )] identifies with the L-subalgebra A[C 0 , D L ] := m∈σ ∨ ∩M H 0 (C 0 , O C0 (D(m))) ⊗ k V (m) ⊂ k(C) ⊗ k k[L/H L ], where D(m) is the Q-divisor on C 0 as in Definition 1.2 and V (m) is the simple L-submodule of k[L/H L ] of highest weight m with respect to (T, B L ).
Proof. The G-isomorphism between G × PF X(D L ) and X(D) is a straightforward consequence of [Tim11, Propositions 14.4, 20.13]. As F L is the set of colors of L/H L , the B L -variety X(D L ) is the chart corresponding to D L (see the remark before [Tim11, Corollary 13.10]), and thus X(D L ) is affine. As X(D L ) is horospherical, we have the following L-algebra decomposition (see [Tim11,Proposition 7.6]): We recall that a normal variety X has rational singularities if there exists a resolution of singularities φ : Y → X such that the higher direct images of φ * applied to O Y vanish. This notion does not depend on the choice of the resolution of singularities. We recall that there is a correspondence between the colored cones of a horospherical homogeneous space and its simple equivariant embeddings; see [Kno91, Theorem 2.3. With the same notation as before, and assuming that C 0 is affine, the following statements are equivalent: Proof. Let us fix z ∈ C 0 . As C 0 is smooth, by [EGA IV, IV,17.11.4], there exist open subsets U z ⊂ C 0 and V z ⊂ A 1 containing z and 0 respectively, and there exists an étale morphism τ z :
k[X(D L )] ≃ m∈σ ∨ ∩M k[X(D L )] (BL) m ⊗ k V (m),U z → V z such that τ z (z) = 0. Let D ′ L := x∈A 1 ∆ ′ x ·[x] with ∆ ′ 0 = ∆ z and ∆ ′ x = σ for all x = 0 be a polyhedral divisor on A 1 . Let D L|U z and D ′
L|V z denote the polyhedral divisor obtained by restricting D L and D ′ L on U z and V z respectively. One may check that the morphism τ z induces an étale morphism of algebras φ z :
A[V z , D ′ L|V z ] → A[U z , D L|U z ]
. By Lemma 2.1, the morphism φ z in turn induces an étale morphism δ z : X(D L|U z ) → X(D ′ L|V z ). Let us denote by Γ L the colored cone (C (D L ) z , F L ) and by X ΓL the corresponding embedding of G m × L/H L ; then X ΓL is L-isomorphic to X(D ′ L ). Let γ z be the morphism which makes the diagram
X(D L|U z ) δz γz / / X ΓL ∼ = X(D ′ L|V z ) / / X(D ′ L )
commute, where the bottom arrow is an open embedding. In particular, γ z is an étale morphism. Let us prove (ii) ⇒ (i). Denote by Γ the colored cone (C (D) z , F ), and let X Γ be the corresponding embedding of G m × G/H. Then we have a G ′ -isomorphism
X Γ ∼ = G ′ × P ′ F X ΓL (see [Tim11, Theorem 28.2])
, where G ′ = G m × G and P ′ F ⊂ G ′ is the parabolic subgroup constructed as in Subsection 2.1.1. By assumption, X Γ is smooth and thus so is X ΓL . This implies that X(D L|U z ) is also smooth. Since Theorem 2.4. With the same notation as before, and assuming that C 0 is projective (i.e., C 0 = C), the following statements are equivalent:
(i) The G-variety X(D) is smooth. (ii) The curve C is P 1 , the polyhedral divisor D is equivalent to a proper colored polyhedral divisor z∈P 1 ∆ z ·[z] with ∆ z = σ except when z = 0 or ∞, and the simple embedding of the G m × G-homogeneous space G m × G/H associated with the colored cones (C, F ) is smooth, where C is the cone generated by
(σ × {0}) ∪ (∆ 0 × {1}) ∪ (∆ ∞ × {−1}).
Proof. Let us prove (i) ⇒ (ii). Suppose that X = X(D) is smooth. Then by Lemma 2.1 we can assume that X is affine and identify k
[X] with A[C 0 , D] = m∈σ ∨ ∩M H 0 (C 0 , O C0 (D(m))) ⊗ k V (m)
. Moreover, we can assume that σ ∨ is strongly convex; otherwise, there is a non-trivial torus D and a G-variety X ′ such that X ∼ = D × X ′ and then we replace X by X ′ . By Luna's slice theorem (see [Lun73, §III, Corollaire 2]), there is a G-isomorphism X ∼ = G × F V , where F ⊂ G is a reductive closed subgroup and V is a F -module. It follows from the proof of Luna's slice theorem that F is in fact the stabilizer of a point of X and thus F is a horospherical subgroup. Since F is reductive and contains a maximal unipotent subgroup of G, it contains the semisimple part of G, and thus G/F is a torus. Now the surjective map G
× F V → G/F induces an inclusion k[G/F ] ⊂ k[G × F V ] = k[X].
Since σ ∨ is strongly convex and D is proper, we have k[G/F ] = k and so G = F . Therefore we obtain that X is G-isomorphic to the G-module V . Let us identify X with V and let us denote by γ : G → GL(X) the corresponding homomorphism. If T ⊂ B is a maximal torus, then there exists a maximal torus T ⊂ GL(X) containing γ(T ) and normalizing γ(U ), where U is the unipotent radical of B. Therefore X//U is a toric variety for the action of T. It follows from [AH06, §11] that C 0 = P 1 and D is equivalent to a polyhedral divisor supported by 0 and ∞. Hence, X = Spec A[C 0 , D] identifies with the horospherical embedding associated with the colored cone (C, F ) (to do this, we need to change the splitting m → χ m so that the isomorphism at the level of U -invariant algebras extends to the corresponding G-algebras). The converse implication (ii) ⇒ (i) is easy and is left to the reader. where I is the subset of simple roots of G defined in Subsection 1.1, I F = {α ∈ S \ I | D α ∈ F }, W I is the subgroup of the Weyl group W = N G (T )/T generated by the simple reflexions s α (α ∈ I), a α = β∈Φ + \ΦI β,α , Φ + is the set of positive roots with respect to (T, B), and Φ I is the set of roots that are sums of elements of I.
Decoloration morphism.
In this section, we prove the existence of the decoloration morphism for the normal horospherical G-varieties of complexity one; see [Bri91, §3.3] for the spherical case. This will be used several times in a crucial way to prove our statements in the following, and also to construct an explicit resolution of singularities of X(Σ) in Proposition 2.17.
Definition 2.6. Let Σ = {(D i , F i )} i∈J be a colored divisorial fan, then the decoloration of Σ is the colored divisorial fan
Σ dec := {(D i , ∅)} i∈J .
A G-model of Z such that there exists a colored fan Σ ′ satisfying X = X(Σ ′ ) and Σ ′ dec = Σ ′ is called quasi-toroidal. As a consequence of the description of the germs of a G-model of Z, this notion does not depend on the choice of Σ ′ . Likewise, if (C 0 , F 0 ) is the colored datum of some germ Y ′ ⊂ X(Σ), then we say that the germ Y ′ ⊂ X(Σ dec ) corresponding to the colored datum
(C 0 , ∅) is the decoloration of Y ′ . Let Σ = {(D i , F i )} i∈J be a colored divisorial fan, then the collection of k- schemes Y (D i ) = Spec A[C i 0 , D i ]
, equipped with their K-action, glue together to give a normal K-variety of complexity one that we denote by Y (Σ); see [AHS08, Theorem 5.3, Remark 7.4 (ii)].
Before stating the next result, let us introduce some notation. Let Y ⊂ X be a germ of a G-model of Z; we say that Y is a geometric realization of O X,Y in the variety X. The support of Y , denoted by Supp(Y ), is the set of G-valuations v of k(Z) such that O v dominates the local ring O X,Y .
Proposition 2.7. Let Σ = {(D i , F i )} i∈J be a colored divisorial fan, and let X i 0 and X i dec be the charts corresponding to
(D i , F i ) and (D i , ∅) respectively. The inclusions k[X i 0 ] ⊆ k[X i dec ] induce a projective birational G-morphism π dec : X(Σ dec ) → X(Σ).
Moreover, for every germ Y ⊂ X, the subset π −1 dec (Y ) is the germ obtained by decoloring the colored datum of Y . Also, there exists a G-isomorphism between X(Σ dec ) and the twisted product G/H× K Y (Σ), where Y (Σ) is the K-variety defined above and K = P/H acts on G/H as follows: for every g ∈ G, for every pH ∈ K, we have pH · gH = gp −1 H.
Proof. Let us fix an index
i ∈ J. The inclusion k[X i 0 ] ⊆ k[X i dec ]
induces a birational B-morphism ι : X i dec → X i 0 . By [Tim11, Proposition 12.12], for ι to extend to a G-morphism
π i dec : X(D i , ∅) → X(D i , F i ),
it suffices to show that if Y ⊂ X(D i , ∅) is a germ, then there exists a (necessar-
ily unique) germ Y ′ ⊂ X(D i , F i ) such that the local ring O X(D i ,∅),Y dominates O X(D i ,F i ),Y ′ . Let us consider the colored datum (C i 0 , ∅) of the germ Y ⊂ X(D i , ∅), and denote F i 0 = {D ∈ F i | ̺(D) ∈ C i 0 }. Then (C i 0 , F i 0 ) is the colored datum of a germ Y ′ ⊂ X(D i , F i ). By [Tim11,, +∞[·[0], where N Q = Q, σ = Q ≥0 . The k-algebra A[A 1 , D] = m≥0 k[t]t −⌊ 1 2 m⌋ χ m
is generated by the homogeneous elements t, χ 1 , and 1 t χ 2 . Therefore, Y (D) = Spec A[A 1 , D] can be identified with the affine surface V (xz − y 2 ) ⊂ A 3 equipped with the G m -action defined by λ · (x, y, z) = (x, λ −1 y, λ −2 z) with λ ∈ G m . Denoting by (x 1 , x 2 , x 3 ) a system of coordinates of A 3 , the twisted action of G m on the product G/H × Y (D) is given by λ · (x 1 , x 2 , x 3 , x, y, z) = (λ −1 x 1 , λ −1 x 2 , λ −1 x 3 , x, λ −1 y, λ −2 z).
By Proposition 2.7, the G-variety X := X(D, ∅) identifies with the quotient G/H× K Y (D). Hence, X is the hypersurface xy − z 2 = 0 in the complement of
X = G/H × K Y (D) → G/P . The open orbit Bx 0 ⊂ P 2 is precisely P 2 \ {x 3 = 0} ≃ A 2 . Thus the variety q −1 (Bx 0 ) is the hypersurface xy − z 2 = 0 in X ′ \ {x 3 = 0} which is isomorphic to A 2 × V (xy − z 2 ).
2.3. Parametrization of the stable prime divisors. In this section, we start by describing in Theorem 2.8 the germs of codimension one of a normal horospherical G-variety of complexity one X. From this, we deduce a description of the class group of X by generators and relations; see Corollary 2.9. Next, we obtain a factoriality criterion for X; see Corollary 2.10. Finally, in Subsection 2.3.4, we relate the description of stable Cartier divisors obtained by Timashev in [Tim00] to our description of stable Weil divisors.
(z, v) where z ∈ C 0 and v ∈ ∆ z (0) is a vertex of ∆ z . If Σ = {(D i , F i )} i∈J , then we put Vert(Σ) := i∈J Vert(D i ) ⊆ C × N Q .
The set of extremal rays of D, denoted by Ray(D) or Ray(D, F ), consists in extremal rays ρ ⊆ σ such that ρ ∩ ̺(F ) = ∅, and satisfying ρ ∩ deg D = ∅ when C 0 = C. To simplify the notation, we denote by the same letter an extremal ray of a polyhedral cone of N Q and its primitive vector with respect to the lattice N . We also denote
Ray(Σ) := i∈J Ray(D i ) ⊆ N Q ,
where we recall that N Q naturally identifies with a subset of E ; see Subsection 1.2.3. Finally, we denote by C Σ the union of open subsets i∈J C i 0 ⊆ C, where C i 0 is the curve on which D i ∈ Σ is defined. ≥0 (v, 1))], ∅) where ρ ⊆ σ is an extremal ray, F 1 = {D α ∈ F | ̺(D α ) ∈ ρ}, z ∈ C 0 , and v ∈ ∆ z (0). We are going to examine in each case the maximal germs corresponding to G-divisors. We proceed in three steps.
Step 1: We consider the case where X(D) is quasi-toroidal, i.e., F = ∅. Then, by Proposition 2.7, X(D) identifies with G/H × K Y (D) as a G-variety. By [Jan87, I.5.21(1)], there is a natural bijection between the set of K-divisors on Y (D) and the set of G-divisors on G/H × K Y (D). By [PS11, Proposition 3.13], the set of K-divisors on Y (D) is parametrized by the set Vert(D) Ray(D):
(z, v) → Y (z,v) , ρ → Y ρ . The G-valuation v Dρ with D ρ = G/H × K Y ρ resp. v D (z,v) with D (z,v) = G/H × K Y (z,v) , is represented by [(·, ρ, 0)] ∈ E resp. by [(z, µ(v)(v, 1))] ∈ E , where µ(v) := inf{d ∈ Z >0 | dv ∈ N}.
This parametrization of the G-divisors on X(D) by the set Vert(D) Ray(D) is the one requested.
Step 2: We now consider an arbitrary simple G-model X(D, F ) of Z, and let ·, ρ, 0)], F 1 ). In this second step, we want to prove that D ′ ρ is contracted by π dec if and only if F 1 = ∅. Let ρ ⋆ = ρ ⊥ ∩ σ ∨ ⊆ σ ∨ denote the dual face of ρ. By properness of D (see [AH06,§2]) there exists a homogeneous element f χ m ∈ A[C 0 , D] of degree m belonging to the relative interior of ρ ⋆ such that
{z ∈ C 0 | ∆ z = σ} ⊂ Z(f ) = Supp(divf + D(m)) ⊂ C 0 .
The localization D m f of D with respect to f χ m is the colored ρ-polyhedral divisor trivial on the curve (C 0 ) m f = C 0 \Z(f ) with set of colors F 1 . We denote by X ′ the simple embedding of G/H associated with the colored cone (ρ, F 1 ). By computing in an appropriate chart, one checks that the product (C 0 ) m f × X ′ identifies with X(D m f , F 1 ) as a G-variety. We denote by Gx the (unique) closed orbit of X ′ . Then D ′′ ρ := (C 0 ) m f × Gx is a maximal germ (for the inclusion) of X(D m f , F 1 ) and corresponds thus to the colored datum (ρ, Step 3: It remains to study the germs of the form D (z,v) X(D, F ). Let us fix a vertex (z, v) ∈ Vert(D), and let us write σ ∨ as a union
σ ∨ = w∈∆z(0) σ ∨ w , where σ ∨ w = m ∈ σ ∨ m, w = min v ′ ∈∆z(0) m, v ′ .
The cones σ ∨ w are pairwise distinct, generate a quasi-fan of σ ∨ (see [AH06,§1]) and are all of full dimension in M Q . Let m ∈ σ ∨ v ∩ M be a lattice vector in the relative interior of σ ∨ v . By properness of D, and up to a change of m by a strictly positive integer multiple, we can choose a homogeneous element f χ m ∈ A[C 0 , D] of degree m such that
(C 0 ) m f = C 0 \Z(f ) contains z. Then D m f is of the form z ′ ∈(C0) m f Q z ′ · [z ′ ],
where Q z ′ ⊂ N Q is a polytope for every z ′ ∈ (C 0 ) m Corollary 2.9. The class group Cl(X(Σ)) is isomorphic to the abelian group
Cl(C Σ ) ⊕ (z,v)∈Vert(Σ) ZD (z,v) ⊕ ρ∈Ray(Σ) Z D ρ ⊕ α∈S\I Z D α ,
where D α ⊂ X(Σ) is the color associated with α ∈ S \ I, modulo the relations:
[z] = (z,v)∈Vert(Σ) µ(v) D (z,v) and (z,v)∈Vert(Σ) µ(v) m, v D (z,v) + ρ∈Ray(Σ) m, ρ D ρ + α∈S\I m, ̺(D α ) D α = 0, where m ∈ M , z ∈ C Σ , and µ(v) = inf{d ∈ Z >0 | dv ∈ N}.
Proof. By [Tim11, Proposition 17.1], every divisor X(Σ) is linearly equivalent to a B-stable divisor. Hence, by Theorem 2.8, we have a surjective homomorphism π Σ from the free abelian group
(3) Γ Σ := (z,v)∈Vert(Σ) ZD (z,v) ⊕ ρ∈Ray(Σ) Z D ρ ⊕ α∈S\I Z D α
onto Cl(X(Σ)). The kernel of π Σ is formed by principal divisors associated with the B-eigenvectors of k(X(Σ)), i.e., by elements of the form div(f χ m ) :
(z,v)∈Vert(Σ) v D (z,v) (f χ m ) · D (z,v) + ρ∈Ray(Σ) v Dρ (f χ m ) · D ρ + α∈S\I v Dα (f χ m ) · D α = (z,v)∈Vert(Σ) µ(v)( m, v + ord z f )D (z,v) + ρ∈Ray(Σ) m, ρ D ρ + α∈S\I m, ̺(D α ) D α ,
where f ∈ k(C) ⋆ and m ∈ M . Let us now consider the surjective homomorphism from
Cl(C Σ ) ⊕ (z,v)∈Vert(Σ) Z D (z,v) ⊕ ρ∈Ray(Σ) Z D ρ ⊕ α∈S\I Z D α onto Γ Σ / Ker π Σ defined by z∈CΣ a z · [z] + i a i D i → z∈CΣ a z (z,v)∈Vert(Σ) µ(v)[D (z,v) ] + i a i [D i ],
where the D i represent the B-divisors on X(Σ). Then one may check that the kernel of this homomorphism is exactly given by the relations stated above.
Example. Returning to the example of the SL 3 -variety X(D) = {xz − y 2 = 0} ∩ (P(−1, −1, −1, 0, −1, −2) \ P(0, −1, −2)) considered in Section 2.2, we can apply Corollary 2.9 to determine the class group of X(D). We obtain that Cl(X(D)) is the abelian group ZD (0, 1 2 ) ⊕ ZD ρ ⊕ ZD α , where ρ = Q ≥0 and D α is the unique color of X(D), modulo the following relations:
• 2D (0, 1 2 ) = 0; and • mD ρ + 2mD α = 0 for every m ∈ Z.
It follows that Cl(X(D)) ∼ = Z ⊕ Z/2Z.
2.3.3.
Let Y (Σ) be the K-variety defined at the beginning of Subsection 2.2. We denote by ρ → Y ρ , (z, v) → Y (z,v) the parametrization of the K-divisors on Y (Σ) given by Theorem 2.8. We recall that a normal variety is called factorial if its class group is trivial. The next corollary gives a criterion of factoriality for the G-variety X(D).
m α , ̺(D β ) = µ(v)( m α , v + ord z (f α )) = m α , ρ = 0.
Proof. The decoloration morphism induces an isomorphism of varieties
X(D, F )\Y 0 ≃ X(D, ∅)\π −1 dec (Y 0 ), where Y 0 is the G-stable closed subset x ∈ X(D, F ) Gx ⊂ α∈S\I D α .
As the codimension of Y 0 in X(D, F ) is at least two, if Y ′ ⊂ X(D, ∅) is the union of irreducible components of codimension one of π −1 dec (Y 0 ), then we have a group isomorphism
Cl(X(D, F )) ≃ Cl (X(D, ∅)\Y ′ ) .
By Proposition 2.7, we can identify X(D, ∅) with G/H × K Y (D) and Y ′ with G/H × K Y . Let us consider the chart X 0 = q −1 (Bx 0 ), where Bx 0 is the open B-orbit of G/P and
q : G/H × K Y (D) = G × P Y (D) → G/P
is the projection. The complement of the union of colors α∈S\I D α in X(D, ∅)\Y ′ is exactly
X 0 \(X 0 ∩ Y ′ ) ≃ Bx 0 × (Y (D)\Y ).
As Bx 0 is an affine space, by reiterating several times [Har77, Proposition II.6.6], we obtain
Cl (X 0 \Y ′ ) ≃ Cl (Y (D)\Y ) .
To sum up, we have an exact sequence:
α∈S\I ZD α → Cl (X(D, ∅)\Y ′ ) → Cl (Y (D)\Y ) → 0,
where the first arrow is induced by the surjective homomorphism from the group Γ Σ defined by (3) (i) For every z ∈ C 0 , there exists m z ∈ M and γ z ∈ Z such that θ(z, u, l) = u(m z ) + lγ z , for every (u, l) ∈ C (D) z .
If C 0 = C, then θ must satisfy the extra condition:
(ii) We have m := m z = m z ′ , for every z, z ′ ∈ C, and there exists f ∈ k(C) ⋆ such that
divf = z∈C γ z · [z].
Let us denote by F Σ the union of all the sets F i , where the (D i , F i ) run over Σ.
A colored integral piecewise linear function on Σ is a pair (θ, (r α )), where θ is a function
θ : |Σ| = i∈J C (D i ) → Q
such that the restriction θ |C (D i )∩C (D j ) is integral linear for every i, j ∈ J, and where (r α ) is a sequence of integers with α running over the set of roots α ∈ S \ I such that D α ∈ F Σ . The pair (θ, (r α )) is called principal if θ satisfies (ii) and r α = m, ̺(D α ) . We denote respectively by PL(Σ) and Prin(Σ) the abelian groups (for the natural additive law) of colored integral piecewise linear functions of Σ and of principal colored integral piecewise linear functions of Σ. If Σ has a single element D, then we will denote PL(D) and Prin(D) instead of PL(Σ) and Prin(Σ) respectively.
As a direct consequence of [Kno94], [ Corollary 2.13. With the notation above, if (θ, (r α )) ∈ PL(Σ), then
D θ := (z,v)∈Vert(Σ) θ(z, µ(v)(v, 1)) · D (z,v) + ρ∈Ray(Σ) θ(·, ρ, 0) · D ρ + Dα∈FΣ θ(·, ̺(D α ), 0) · D α + Dα ∈FΣ r α · D α
is a B-stable Cartier divisor on X(Σ). More precisely, the map θ → D θ is an isomorphism between the group PL(Σ) and the group of B-stable Cartier divisors on X(Σ), and there is a short exact sequence:
0 → Prin(Σ) → PL(Σ) → Pic(X(Σ)) → 0,
where Pic(X(Σ)) is the Picard group of X(Σ).
2.4.
Canonical class and log-terminal singularities. In this section, we give an explicit representative of the canonical class for X a normal horospherical Gvariety of complexity one; see Theorem 2.14. From this, we deduce a criterion for X to be Q-Gorenstein; see Corollary 2.15. Then we construct an explicit resolution of singularities of X; see Proposition 2.17. Finally, we obtain a criterion to determine whether the singularities of X are log-terminal; see Theorem 2.18. Theorem 2.14. Let Σ be a colored divisorial fan on C. Then with the notation of Subsection 2.3.1 every canonical divisor on X = X(Σ) is linearly equivalent to
K X = − ρ∈Ray(Σ) D ρ + (z,v)∈Vert(Σ) (µ(v)b z + µ(v) − 1)D (z,v) − α∈S\I a α D α , where K C = z∈C b z · [z]
is a canonical divisor on C, a α = β∈Φ + \ΦI β,α ≥ 2, Φ + is the set of positive roots with respect to (T, B), and Φ I is the set of roots that are sums of elements of I.
Proof. Let us consider the union
Y ′ = Y 0 ∪ Sing(X) ⊆ X,
where Y 0 is the biggest G-stable closed subset contained in the union of all the colors of X and Sing(X) is the singular locus. Then X(Σ ′ ) = X\Y ′ is smooth, quasitoroidal and the set Vert(Σ) Ray(Σ) identifies with the set Vert(Σ ′ ) Ray(Σ ′ ). Therefore, by Proposition 2.7, we can assume without loss of generality that X = G × P Y (Σ) is smooth and quasi-toroidal. Adapting the argument of the proof of [ST99, Proposition 4.2 c)] to our setting, we obtain an isomorphism of O X -modules
O X (K X ) ≃ O X (D) ⊗ q * O G/P (K G/P ),
where K G/P is a canonical divisor on G/P given by
K G/P = − α∈S\I a α D α and the divisor D is defined by D = (z,v)∈Vert(Σ) c (z,v) D (z,v) + ρ∈Ray(Σ) c ρ D ρ such that K Y (Σ) = (z,v)∈Vert(Σ) c (z,v) Y (z,v) + ρ∈Ray(Σ) c ρ Y ρ .
is a canonical divisor on Y (Σ) (see the beginning of Subsection 2.3.3 for the notation (i) For every ρ ∈ Ray(Σ), we have θ(·, ρ, 0) = −d.
(ii) There exists a canonical divisor
K C = z∈C b z · [z] on C such that, for every (z, v) ∈ Vert(Σ), we have θ(z, µ(v)v, 1) = d(µ(v)b z + µ(v) − 1). (iii) For every D α ∈ F Σ , we have θ(·, ̺(D α ), 0) = −da α .
Proof. It is a straightforward consequence of Corollary 2.13 and Theorem 2.14.
2.4.3. The remainder of this paper is dedicated to the study of log-terminal singularities of a simple G-model of Z.
Let X be a Q-Gorenstein variety, let φ : X ′ → X be a resolution of singularities, and let d ∈ Z >0 such that dK X is Cartier. Then the pull-back φ * (dK X ) is welldefined. The discrepancy of φ is the Q-divisor
K X ′ − φ * (K X ) := K X ′ − 1 d φ * (dK X ).
The discrepancy of φ does not depend either on the choice of the canonical divisors K X , K X ′ nor on the integer d ∈ Z >0 . We say that X has (purely) log-terminal singularities if each coefficient of K X ′ − φ * (K X ) is strictly bigger than −1. The property of having log-terminal singularities does not depend on the choice of the resolution of singularities φ. More generally, if D is a Q-divisor on X such that K X + D is Q-Cartier, then we say that the pair (X, D) is (purely) log-terminal if each coefficient of K X ′ − φ * (K X + D) is strictly bigger than −1. The next lemma will be useful to prove Theorem 2.18; see [Kol97,§3] for more details about log-terminal singularities and for a proof of the statement.
Lemma 2.16. Let φ : X ′ → X be a proper birational morphism between normal varieties. Let D be a Q-divisor on X such that K X + D is Q-Cartier, and let D ′ be the Q-divisor on X ′ defined by K X ′ + D ′ = φ * (K X + D).
Then (X, D) is log-terminal if and only if (X ′ , D ′ ) is log-terminal and the coefficients of the prime divisors of −D ′ (corresponding to exceptional divisors of φ) are strictly bigger than −1.
2.4.4. In this subsection, we give an explicit method to construct a resolution of singularities of the G-model X(D) of Z.
Let (D, F ) be a proper colored σ-polyhedral divisor on a dense open subset C 0 ⊆ C. With the notation of Definition 1.2, we denotẽ [LS13,§2]. The G-variety G × PỸ (D) identifies with X(Σ tor ), where Σ tor is the colored divisorial fan Σ tor = {(D |Ci , ∅)} i∈J ; the sequence (C i ) i∈J forming a covering of C 0 by affine open subsets. Moreover, if we consider a divisorial fan Σ that refines Σ tor and such that for all colored polyhedral divisor D ∈ Σ and z ∈ C with C(D) z = ∅, the polyhedral cone C(D) z is regular (i.e., a cone generated by a subset of a basis of the lattice N × Z), then X(Σ) → X(Σ tor ) is a resolution of singularities. The next result is a straightforward consequence of this discussion. (i) The curve C 0 is affine. (ii) The curve C 0 is the projective line P 1 and z∈C0 1 − 1 µz < 2, where for every z ∈ C 0 we denote µ z := max{µ(v) | v ∈ ∆ z (0)} and µ(v) := inf{d ∈ Z >0 | dv ∈ N}.
Proof. If C 0 is affine, then by the proof of Theorem 2.3, we obtain that X(D) is covered by étale open subsets of horospherical embeddings. Hence, by [Bri93,Theorem 4.1], X(D) has log terminal singularities. Therefore, we can assume that C 0 = C is projective. Let us consider a canonical divisor K X as in Theorem 2.14, let d ∈ Z >0 be such that dK X is a Cartier divisor, and let θ ∈ PL(D) such that dK X = D θ . By Corollary 2.13, we know that the restriction of dK X on the open subset
X 1 := X(D)\ Dα ∈FΣ D α
is a principal divisor. Moreover, since every color D α satisfying D α / ∈ F Σ does not contain a G-orbit of X(D), the open subset ψ −1 (X 1 ) intersects each exceptional divisor of the partial desingularization ψ : X(Σ tor ) → X(D) given by Proposition 2.17. Therefore we can replace X by X 1 and suppose that dK X is principal. It follows that ψ * (dK X ) is the principal divisor of a homogeneous element f χ m ∈ A M of degree m considered as a rational function of X(Σ tor ). Hence, we have the equality −D ′ := K X(Σtor) − ψ * K X(D) = ρ ∈Ray(D) (−1 − m, ρ )D ρ .
By Lemma 2.16, X(D) has log-terminal singularities if and only if −D ′ has its coefficients strictly bigger than −1 and (X(Σ tor ), D ′ ) has log-terminal singularities. Thus, by the same argument as in [LS13,§4] (see the sketch of proof before [LS13, Theorem 4.7]), we know that X(D) has log-terminal singularities if and only if m, ρ < 0, for every ρ ∈ Ray(D).
Let ρ ⊆ σ be an extremal ray such that ρ ∈ Ray(D), then ρ ∩ ̺(F ) = ∅ or ρ ∩ deg D = ∅. Let us suppose that ρ ∩ ̺(F ) = ∅. Then there exists D α ∈ F and λ ∈ Q >0 such that ρ = λ̺(D α ). Hence m, ρ = λ m, ̺(D α ) = −λda α < 0.
Let us now suppose that ρ ∩ deg D = ∅. Then ρ = λv for a vertex v ∈ deg D and for some λ ∈ Q >0 . Let (v z ) z∈C be a sequence of elements of ∆ z (0) such that v = z∈C v z . As the coefficients of K X and 1 d div f χ m at the prime divisor D (z,vz) corresponding to any (z, v z ) ∈ Vert(D) are the same, we have equalities
µ(v z )b z + µ(v z ) − 1 = 1 d µ(v z )( m, v z + ord z f ),
where K C = z∈C b z · [z] is a canonical divisor on C. Since C is projective, we have deg (div f ) = 0. Hence, summing over C on both sides gives the equality
deg K C + z∈C 1 − 1 µ(v z ) = 1 d m, v .
As deg D ⊂ σ, we conclude that X(D) has log-terminal singularities if and only if the condition (ii) is satisfied.
Proposition 1 . 1 .
11The map G/H → (M, I) is a bijection between
denote by Sch G (Z) the maximal normal open subscheme on which the action of G on Sch(Z) is regular; see [Tim11, Proposition 12.2]. A G-model of Z is a G-stable dense open subset of Sch G (Z) which is separated and Noetherian (or equivalently separated and of finite type over k).
0
→ k(Z) B ⋆ → A M → M → 0 splits. Let us fix once and for all a (non-canonical) splitting M → A M , m → χ m . Then A M admits an M -grading given by (2) A M = m∈M k(C)χ m .
Remark 1. 3 .
3Let (D, F ) be a colored σ-polyhedral divisor on a dense open subset C 0 ⊆ C. Let C (D) be the subset of E defined as the disjoint union ⊔ z∈C0 {z} × C (D) z modulo the equivalence relation ∼ defined by (1), where C (D) z is the cone generated by σ × {0} and ∆ z × {1}. Then the pair (C (D), F ) is called the the colored hypercone associated with (D, F ). This gives a one-to-one correspondence between the set of colored polyhedral divisors defined on a dense open subset of C and the set of colored hypercones of E . Moreover, through this correspondence, the properness of colored polyhedral divisors corresponds to the admissibility of colored hypercones; see [Tim11, Definition 16.12]. 1.3.2. In this subsection, we explain how to construct a simple G-model of Z (introduced in Subsection 1.2.1) starting from a proper colored polyhedral divisor; see [Tim11, §13] for details. Let (D, F ) be a proper colored σ-polyhedral divisor on a dense open subset C 0 ⊆ C. Let us denote by C (D)(1) the set of elements [(z, v, l)] ∈ C (D) such that (v, l) is the primitive vector of an extremal ray of C (D) z , and let Bx 0 be the open B-orbit of G/H. Let us consider the subalgebra A ⊂ k(Z) defined by:
vector space generated by the B L -eigenvectors of weight m on which L acts trivially. The vector space k[X(D L )] (BL) m identifies with the space of global sections of ⌊D(m)⌋. This proves the lemma. Example. Let us assume that C 0 = C is projective. Let G = SL 2 and let H be the unipotent radical of the subgroup of upper triangular matrices. Then M = N = Z and the set of colors F 0 of G/H is a singleton. Let σ = Q ≥0 and let D be a proper σ-polyhedral divisor on C such that the Weil Q-divisor D(1) on C is integral and very ample. Then by Lemma 2.1, the G-variety X(D, F 0 ) is affine and k[X(D, F 0 )] identifies with d≥0 H 0 (C, O C (d.D(1))) ⊗ k V (d) as a G-algebra, where V (d) is the irreducible representation of G of dimension d + 1 obtained by linearizing the line bundle O P 1 (d). Then the line bundle L := O C (D(1)) ⊠ O P 1 (1) on C × P 1 is very ample, and the G-variety X(D, F 0 ) can be realized as the affine cone over the G-equivariant embedding of C × P 1 in the projectivization of the space of global sections of L . 2.1.3. The next results are criteria to characterize the singularities of a simple G-model X(D) of Z; see [LS13, Proposition 5.5] for the case of normal T -varieties and [Tim00, §6, Theorem 7] for a criterion for rationality of singularities in the general setting of normal G-varieties of complexity one.
Theorem 2. 2 .
2Let (D, F ) be a proper colored σ-polyhedral divisor on C 0 . The simple G-model X(D) of Z has rational singularities if and only if one of the following assertions holds. (i) The curve C 0 is affine. (ii) The curve C 0 is the projective line P 1 and deg ⌊D(m)⌋ ≥ −1 for every m ∈ σ ∨ ∩ M , where D(m) is the Q-divisor on C 0 as in Definition 1.2. Proof. In this proof, we identify X(D) with the parabolic induction of X(D L ) using Lemma 2.1. Let us denote by Bx 0 the open B-orbit of G/P F and let q : X(D) = G × PF X(D L ) → G/P F be the (G-equivariant) projection. Then q −1 (Bx 0 ) ≃ Bx 0 × X(D L ) is a chart of X(D) intersecting all the germs of X(D). As Bx 0 is an affine space, X(D) has rational singularities if and only if X(D L )//U L ≃ Spec A[C 0 , D] has rational singularities; see [Tim11, Theorem D5 (3)]. We conclude by [LS13, Proposition 5.5].
Theorem 3.1] for details. The next theorem gives a smoothness criterion for X(D) when (D, F ) is a proper colored polyhedral divisor on an affine curve C 0 ; see [Bri91, §4.2] for the spherical case, [LS13, §5] for the case of normal T -varieties of complexity one, and [Mos90, §3] for the case of embeddings of SL 2 and PSL 2 . Our proof is inspired by a description of toroidal embeddings given in [KKMS73, §II].
(i) The G-variety X(D) is smooth. (ii) For every z ∈ C 0 , the simple embeddings of the G m × G-homogeneous space G m × G/H associated with the colored cones (C (D) z , F ) are smooth; see Remark 1.3 for the definition of C (D) z .
C 0 is affine, (X(D L|U z )) z∈C0 is an open covering of X(D L ). Therefore X(D L ) is smooth and thus, by parabolic induction, so is X(D). Let us prove (i) ⇒ (ii). If X(D) is smooth, then so is X(D L ) by parabolic induction. Hence, by the diagram above, there exists an open subsetV ⊂ A 1 containing 0 such that X(D ′ L|V ) is smooth. Denoting L ′ = G m × L and identifying X(D ′ L|V )with an open subset of X ΓL , we have L ′ · X(D ′ L|V ) = X ΓL . This implies that X ΓL is smooth, and thus so is X Γ .We say that two proper colored polyhedral divisors D and D ′ on C 0 are equivalent if A[C 0 , D] and A[C 0 , D ′ ] (introduced in Definition 1.2 (ii)) are isomorphic as M -graded algebras; see [AH06, Section 8] and [Lan14, Proposition 4.5] for a combinatorial description of the equivalence between two such polyhedral divisors.The next theorem gives a smoothness criterion for X(D) when (D, F ) is a proper colored polyhedral divisor on a projective curve C 0 = C.
The smoothness criteria of Theorems 2.3 and 2.4 can be made explicit by applying the smoothness criterion in the horospherical embedding case given by the following theorem; see[Pas06, §II] and[BM13, §5].Theorem 2.5. Let X be a G-equivariant embedding of the horospherical homogeneous space G/H associated with a colored cone (C, F ). Then X is smooth if and only if the following conditions are satisfied. (i) The elements of F have pairwise distinct images through the map ̺ defined in Subsection 1.2.2. (ii) The cone C is generated by a subset of a basis of N containing ̺(F ). (iii) We have the equality |W I | · α∈IF a α = |W I∪IF |,
Proposition 14.1 (2)] or[Kno93, §3.8], the support of a germ associated to a color datum (C ′ , F ′ ) depends only on C ′ , and thus Supp(Y ) = Supp(Y ′ ). This implies the equal-ity O X(D i ,∅),Y = O X(D i ,F i ),Y ′ ; see the proof of [Tim11, Proposition 14.1(1)]. In particular, O X(D i ,∅),Y dominates O X(D i ,F i ),Y ′ and thus ι extends to a G-morphism. Let now Y ′ ⊂ X(D i , F i ) be an arbitrary germ. Since the induced map (π i dec ) * is the identity on E (see the remark before [Tim11, Theorem 12.13]), the subset (π i dec ) −1 (Y ′ ) is irreducible and coincides with the decoloration of Y ′ . The properness of π i dec follows from [Tim11, Theorem 12.13]. Since the varieties X(D i , ∅) and X(D i , F i ) are quasi-projective, the morphism π i dec is projective. The existence and the properties of the morphism π dec : X(Σ dec ) → X(Σ) are obtained by gluing. For the last claim, it suffices to show that X(D, ∅) is G-isomorphic to G/H× K Y (D). Let us note that the latter identifies with G × P Y (D). We conclude by[Tim11, Propositions 14.4, 20.13].Example. We consider the natural action of G = SL 3 on A 3⋆ = A 3 \ {(0, 0, 0)}. LetH be the isotropy subgroup of the point (1, 0, 0) for this action. Then H is a horospherical subgroup of G and A 3 ⋆ ∼ = G/H. Also, the torus K = N G (H)/H ∼ = G m acts diagonally on A 3 ⋆ and the fibration G/H = A 3 ⋆ → G/P = P 2 is simply the quotient morphism for the G m -action. Let us consider the colored σ-polyhedral divisor on A 1 = Spec k[t] defined by F = ∅ and D = [ 1 2
{[ 0
0: 0 : 0 : x : y : z] | [x : y : z] ∈ P(0, −1, −2)} in the weighted projective space X ′ := P(−1, −1, −1, 0, −1, −2). Let B ⊂ G be the Borel subgroup of upper triangular matrices. To determine a chart of X, it suffices to determine the inverse image of the open B-orbit in P 2 = G/P through the projection q :
state our results we need first to introduce the set of vertices and the set of extremal rays of a colored polyhedral divisor. Let (D, F ) be an element of a colored divisorial fan Σ with D = z∈C0 ∆ z · [z]. The set of vertices of D, denoted by Vert(D), consists in pairs
2. 3 . 2 .
32In the next theorem we parametrize the set of G-divisors of a G-model X(Σ) of Z by the set Vert(Σ) Ray(Σ). This description is a natural generalization of the case of normal T -varieties specialized to the case of T -actions of complexity one; see [FZ03, Theorem 4.22] and [PS11, Proposition 3.13]. Theorem 2.8. Let Div(Σ) denote the set of G-divisors of X(Σ). With the notation above, the map Vert(Σ) Ray(Σ) → Div(Σ), (z, v) → D (z,v) , ρ → D ρ which to the vertex (z, v) associates the germ D (z,v) of X(Σ) defined by the colored datum ([(z, Q ≥0 (v, 1))], ∅) resp. to the ray ρ associates the germ D ρ of X(Σ) defined by the colored datum (ρ, ∅) = ([(·, ρ, 0)], ∅), is a bijection. Proof. Without loss of generality, we can suppose that X(Σ) = X(D) is given by a proper colored σ-polyhedral divisor (D, F ) on a dense open subset C 0 ⊆ C, where D = z∈C0 ∆ z · [z]. By [Tim11, Theorem 16.19 (2)], the maximal germs of X(D) have color data of the form (ρ, F 1 ) and ([(z, Q
π
dec : X(D, ∅) → X(D, F ) be the decoloration morphism. By Proposition 2.7, any G-divisor on X(D, F ) is the image of a (unique) G-divisor on X(D, ∅). Let us consider the G-divisor D ′ ρ on X(D, ∅) corresponding to the color datum ([(·, ρ, 0)], ∅), where ρ belongs to Ray(D, ∅). Then D ρ = π dec (D ′ ρ ) is the germ of X(D, F ) corresponding to the color datum ([(
F 1 ); see [Tim11, Theorem 16.19]. The germ D ′′ ρ is a geometric realization of O X(D,F ),Dρ in the open subset X(D m f , F 1 ) ⊂ X(D, F ), that is, D ρ is the closure of D ′′ ρ in X(D, F ). Therefore, D ρ is a divisor on X(D, F ) if and only if Gx is a divisor on X ′ if and only if F 1 = ∅; see [Kno91, Lemma 2.4] for the last equivalence.
f and Q z = {v}. The set of colors of D m f is empty. By Step 1 applied to the open subset X(D m f , ∅) ⊂ X(D, F ), the germ D ′ (z,v) X(D m f , ∅) corresponding to the colored datum ([(z, Q ≥0 (v, 1))], ∅) is of codimension one. Thus D (z,v) is of codimension one as the closure of D ′ (z,v) in X(D, F ). This proves the existence of the parametrization of the G-divisors.
Corollary 2 . 10 .
210Let (D, F ) be a proper colored σ-polyhedral divisor on a dense open subset C 0 ⊆ C. Then X(D) is factorial if and only if the two following conditions are satisfied. (i) The equality Cl(Y (D)) = Yρ⊂Y Z[Y ρ ] holds, where Y denotes the union of the K-divisors Y ρ with ρ satisfying ̺(F ) ∩ ρ = ∅. (ii) For every α ∈ S \ I, there exists m α ∈ M and f α ∈ k(C) ⋆ such that m α , ̺(D α ) = 1, and for all β ∈ S \ (I ∪ {α}), (z, v) ∈ Vert(D), and ρ ∈ Ray(D, F ):
onto Cl(X(D)). Therefore Cl(X(D)) = 0 if and only if Cl (Y (D)\Y ) = 0 and the first arrow above is zero. By Corollary 2.9, this corresponds exactly to the conditions (i) and (ii). This proves the Corollary.Remark 2.11. In general, the factoriality of Y (D) does not imply the one of G × P Y (D). For instance, we can consider G = SL 2 , H the unipotent subgroup of upper triangular matrices, C = P 1 , and (D, ∅) the colored Q ≥0 -polyhedral divisor trivial on A 1 ⊂ P 1 . The SL 2 -variety X(D) identifies with A 1 × Bl 0 (A 2 ), where Bl 0 (A 2 ) is the bowing-up of A 2 at the origin. We have Cl(X(D)) ≃ Z whereas Cl(Y (D)) = Cl(A 2 ) = 0.2.3.4. As a by-product of Theorem 2.8, we can refine the description of B-stable Cartier divisors of [Tim00] to our setting. Let us start with a definition.Definition 2.12. Let Σ = {(D i , F i )} i∈J be a colored divisorial fan on C, and let D = D i ∈ Σ be a colored polyhedral divisor on C 0 with set of colors F = F i . Recall that we denote by C (D) the hypercone associated with D. An integral linear function on D is a map θ : C (D) → Q satisfying the following properties:
2.4.1. The next result gives an explicit canonical divisor for any normal horospherical G-variety of complexity one; see [Mos90, §5] for the case of the embeddings of SL 2 , [Bri93] for the case of the spherical varieties, and [FZ03, Corollary 4.25], [PS11, Theorem 3.21] for the case of normal T -varieties.
Y
(D) = Spec OC m∈σ ∨ ∩M O C (D(m))χ m .Then the natural morphismỸ (D) → Y (D) induces a partial desingularization G × PỸ (D) → G × P Y (D) = X(D, ∅); see [AH06, Theorem 3.1 (ii)] and
Proposition 2 . 17 .
217The morphism φ : X(Σ) → X(Σ tor ) → X(D, ∅) → X(D, F ) obtained by composing the decoloration morphism of Section 2.2 with the morphisms defined above is a resolution of singularities of X(D, F ). Moreover, with the notation of Subsection 2.3.1, the exceptional divisors of φ correspond to the subsets Ray(Σ)\Ray(D) and Vert(Σ)\Vert(D). 2.4.5. The next statement gives a characterization of the normal horospherical Gvarieties of complexity one having log-terminal singularities. See [Bri93, Theorem 4.1] for the spherical case and [LS13, Theorem 4.7] for the case of normal T -varieties. Theorem 2.18. Let (D, F ) be a proper colored σ-polyhedral divisor on a dense open subset C 0 ⊆ C. We suppose that X(D) is Q-Gorenstein. Then X(D) has log-terminal singularities if and only if one of the following assertions holds.
1.2. Models, colors, and valuations. From now on, H is a closed subgroup of G associated with a pair (M, I) as in Proposition 1.1 and P = P I is the parabolic subgroup N G (H). Also, we recall that Z = C × G/H.1.2.1. We first introduce the scheme of geometric localities as in [Tim11, §12.2].
2.1.2. The next result is an adaptation of [PV72, §3] and[Pau81, §5] to the case of complexity one.
Tim00, §4], [Tim11, §17], and Theorem 2.8, we obtain the next result. See [Bri89, §3.1] for the spherical case and [PS11, Corollary 3.19] for the case of normal T -varieties.
). We conclude by [PS11, Theorem 3.21]. 2.4.2. A normal G-variety X is called Q-Gorenstein if one (and thus any) canonical divisor K X is Q-Cartier. The next corollary gives a combinatorial criterion for a normal horospherical G-variety of complexity one to be Q-Gorenstein; see [Bri93, Proposition 4.1] for the spherical case and [LS13, Proposition 4.4] for the case of normal T -varieties. Corollary 2.15. With the notation of Subsections 2.3.1 and 2.3.4, the variety X(Σ) is Q-Gorenstein if there exists d ∈ Z >0 and θ ∈ PL(Σ) such that the following conditions are all satisfied.
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Campus Cantoblanco. B Wargane, Détermination des valuations invariantes de SL(3)/T. Thèse, Grenoble, 1982. Instituto de Ciencias Matematicas (ICMAT). Madrid, Spain E-mail; Mainz, Germany E-mailJohannes Gutenberg -Universität Mainzaddress: [email protected] Fachbereich Physik, Mathematik und InformatikB. Wargane, Détermination des valuations invariantes de SL(3)/T. Thèse, Grenoble, 1982. Instituto de Ciencias Matematicas (ICMAT), Campus Cantoblanco, Madrid, Spain E-mail address: [email protected] Fachbereich Physik, Mathematik und Informatik, Johannes Gutenberg -Univer- sität Mainz, Mainz, Germany E-mail address: [email protected]
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[] |
[
"THE SIGNATURE PACKAGE ON WITT SPACES",
"THE SIGNATURE PACKAGE ON WITT SPACES"
] |
[
"Pierre Albin ",
"Eric Leichtnam ",
"ANDRafe Mazzeo ",
"Paolo Piazza "
] |
[] |
[] |
In this paper we prove a variety of results about the signature operator on Witt spaces. First, we give a parametrix construction for the signature operator on any compact, oriented, stratified pseudomanifold X which satisfies the Witt condition. This construction, which is inductive over the 'depth' of the singularity, is then used to show that the signature operator is essentially self-adjoint and has discrete spectrum of finite multiplicity, so that its index -the analytic signature of X -is well-defined. This provides an alternate approach to some well-known results due to Cheeger. We then prove some new results. By coupling this parametrix construction to a C * r Γ Mishchenko bundle associated to any Galois covering of X with covering group Γ, we prove analogues of the same analytic results, from which it follows that one may define an analytic signature index class as an element of the K-theory of C * r Γ. We go on to establish in this setting and for this class the full range of conclusions which sometimes goes by the name of the signature package. In particular, we prove a new and purely topological theorem, asserting the stratified homotopy invariance of the higher signatures of X, defined through the homology L-class of X, whenever the rational assembly map K * (BΓ) ⊗ Q → K * (C * r Γ) ⊗ Q is injective.
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10.24033/asens.2165
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[
"https://arxiv.org/pdf/1112.0989v1.pdf"
] | 119,296,702 |
1112.0989
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1e705d680dda17623bcff32c5da6c2708aef280d
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THE SIGNATURE PACKAGE ON WITT SPACES
5 Dec 2011
Pierre Albin
Eric Leichtnam
ANDRafe Mazzeo
Paolo Piazza
THE SIGNATURE PACKAGE ON WITT SPACES
5 Dec 2011
In this paper we prove a variety of results about the signature operator on Witt spaces. First, we give a parametrix construction for the signature operator on any compact, oriented, stratified pseudomanifold X which satisfies the Witt condition. This construction, which is inductive over the 'depth' of the singularity, is then used to show that the signature operator is essentially self-adjoint and has discrete spectrum of finite multiplicity, so that its index -the analytic signature of X -is well-defined. This provides an alternate approach to some well-known results due to Cheeger. We then prove some new results. By coupling this parametrix construction to a C * r Γ Mishchenko bundle associated to any Galois covering of X with covering group Γ, we prove analogues of the same analytic results, from which it follows that one may define an analytic signature index class as an element of the K-theory of C * r Γ. We go on to establish in this setting and for this class the full range of conclusions which sometimes goes by the name of the signature package. In particular, we prove a new and purely topological theorem, asserting the stratified homotopy invariance of the higher signatures of X, defined through the homology L-class of X, whenever the rational assembly map K * (BΓ) ⊗ Q → K * (C * r Γ) ⊗ Q is injective.
Introduction
Let X be an orientable closed compact Riemannian manifold with fundamental group Γ. Let X ′ be a Galois Γ-covering and r : X → BΓ a classifying map for X ′ . The signature package for the pair (X, r : X → BΓ) refers to the following collection of results:
(1) the signature operator with values in the Mishchenko bundle r * EΓ × Γ C * r Γ defines a signature index class Ind( ð sign ) ∈ K * (C * r Γ), * ≡ dim X (mod 2); (2) the signature index class is a bordism invariant; more precisely it defines a group homomorphism Ω SO * (BΓ) → K * (C * r Γ); (3) the signature index class is a homotopy invariant; (4) there is a K-homology signature class [ð sign ] ∈ K * (X) whose Chern character is, rationally, the Poincaré dual of the L-Class; (5) the assembly map β : K * (BΓ) → K * (C * r Γ) sends the class r * [ð sign ] into Ind( ð sign ); (6) if the assembly map is rationally injective, one can deduce from (1) - (5) that the Novikov higher signatures { L(X) ∪ r * α, [X] , α ∈ H * (BΓ, Q)} are homotopy invariant.
We call this list of results, together with the following item, the full signature package: (7) there is a (C * -algebraic) symmetric signature σ C * r Γ (X, r) ∈ K * (C * r Γ), which is topologically defined, a bordism invariant σ C * r Γ : Ω SO * (BΓ) → K * (C * r Γ) and, in addition, is equal to the signature index class.
For history and background see [16] [50] and for a survey we refer to [29].
The main goal of this paper is to formulate and establish the signature package for a class of stratified pseudomanifolds known as Witt spaces. In particular, we prove by analytic methods a new and purely topological result concerning the stratified homotopy invariance of suitably defined higher signatures under an injectivity assumption on the assembly map for the group Γ.
The origins of the signature package on a closed oriented manifold X can be traced back to the Atiyah-Singer proof of the signature formula of Hirzebruch, σ top (X) = L(X) := L(X), [X] . In this proof the central object is the Fredholm index of the signature operator which is proved to be simultaneously equal to the topological signature of the manifold σ top (X) and to its L-genus L(X):
σ top (X) = ind(ð sign ) = L(X) .
The idea of using index theory to investigate topological properties of X received new impetus through the seminal work of Lusztig, who used the family index theorem of Atiyah-Singer in order to establish the Novikov conjecture on the homotopy invariance of the higher signatures of X when π 1 (X) = Z k . Most of the signature package as formulated here can be seen as a noncommutative version of the results of Lusztig. Crucial in the formulation and proof of the signature package are the following issues:
• the Poincaré duality property for the (co)homology of X and more generally, the Algebraic Poincaré Complex structure of its (co)chain complex; • the possibility of defining bordism groups Ω SO (T ), T a topological space, with cycles given by closed oriented manifolds endowed with a reference map to T ; • an elliptic theory which allows one to establish the analytic properties of ð sign and then connect them to the topological properties of X; • the possibility of extending this elliptic theory to signature operators twisted by a bundle of finitely generated projective A-modules, where A is a C *algebra. The prototype is the signature operator ð sign twisted by the Mishchenko bundle r * EΓ × Γ C * r Γ.
Once one moves from closed oriented manifold to stratified pseudomanifolds, many of these issues need careful reformulation and substantially more care. First, it is well-known that Poincaré duality fails on a general stratified pseudomanifold X. Next, the bordism group Ω pseudo (T ), the cycles of which are arbitrary stratified pseudomanifolds endowed with a reference map to T , is not the right one; indeed, as explained in [4], the coefficients of such a theory, Ω pseudo (point), are trivial. Finally, the analytic properties of the signature operator on the regular part of a stratified pseudomanifold endowed with an 'incomplete iterated edge metric' (which is a particularly simple and natural type of metric that can be constructed on such a space) are much more delicate than in the closed case. In particular, this operator may not even be essentially self-adjoint, and the possibility of numerous distinct self-adjoint extensions complicates the possible connections to topology.
The first problem has been tackled by Goresky and MacPherson in the topological setting [19] [20] and by Cheeger in the analytic setting [11] [12] (at least for the particular subclass of stratified pseudomanifolds we consider below). The search for a cohomology theory on such spaces with some vestiges of Poincaré duality led Goresky and MacPherson to their discovery of intersection (co)homology groups, IH * p ( X, Q), where p is a 'perversity function', and to the existence of a perfect pairing IH * p ( X, Q) × IH * q ( X, Q) → Q where p and q are complementary perversities. Notice that we still do not obtain a signature unless the perversities can be chosen the same, i.e. unless there is a perfect pairing IH * m ( X, Q) × IH * m ( X, Q) → Q for some perversity function m. Witt spaces constitute a subclass of stratified pseudomanifolds for which all of these difficulties can be overcome.
A stratified pseudomanifold X is a Witt space if any even-dimensional link L satisfies IH dim L/2 m (L, Q) = 0, where m is the upper-middle perversity function. Examples of Witt spaces include any singular projective variety over C. We list some particularly interesting properties of Witt spaces:
• the upper-middle and lower-middle perversity functions define the same intersection cohomology groups, which are then denoted IH * m ( X); • there is a perfect pairing IH * m ( X, Q) × IH * m ( X, Q) → Q ; in particular, there is a well defined intersection cohomology signature;
• there are well-defined and nontrivial Witt bordism groups Ω Witt (T ) (for example, these are rationally isomorphic to the connected version of KOhomology, ko(T ) ⊗ Z Q); • there is a class of Riemannian metrics on the regular part of X for which -the signature operator is essentially self-adjoint -its unique self-adjoint extension has discrete spectrum of finite multiplicity -there is a de Rham-Hodge theorem, connecting the Hodge cohomology, the L 2 -cohomology and the intersection cohomology IH * m ( X, C). The topological results here are due to Goresky-MacPherson and Siegel. The analytic results are due initially to Cheeger, though there is much further work in this area, see, for example, [9], [38], [26], [53]. Cheeger's results on the signature operator are based on a careful analysis of the heat kernel of the associated Laplacian.
We have a number of goals in this article:
• we give a new treatment of Cheeger's result on the signature operator based on the methods of geometric microlocal analysis; • this approach is then adapted to the signature operator ð sign with value in the Mishchenko bundle r * EΓ × Γ C * r Γ; • we carefully analyze the resulting index class, with particular emphasis on its stability property; • we collect this analytic information and establish the whole range of results encompassed by the signature package on Witt spaces. In particular, we prove a Novikov conjecture on Witt spaces whenever the assembly map for the fundamental group is rationally injective. We note again that this is a new and purely topological result.
This article is divided into three parts. In the first, we give a detailed account of the resolution, through a series of blowups, of an arbitrary stratified pseudomanifolds (not necessarily satisfying the Witt condition) to a manifold with corners. This has been studied in the past, most notably by Verona [58]; the novelty in our treatment is the introduction of iterated fibration structures, a notion due to Melrose, as an extra structure on the boundary faces of the resolved manifold with corners. We also show that a manifold with corners with an iterated fibration structure can be blown down to a stratified pseudomanifold. In other words, the classes of stratified pseudomanifolds and of manifold with corners with iterated fibration structure are equivalent. Much of this material is based on unpublished work by Richard Melrose, and we are grateful to him for letting us use and develop these ideas here. We then describe the (incomplete) iterated edge metrics, which are the simplest type of incomplete metrics adapted to this class of singular space. We show in particular that the space of such metrics is nonempty and path-connected. We also consider, for any such metric, certain conformally related complete, and 'partially complete' metrics used in the ensuing analysis.
The second part of this article focuses on the analysis of natural elliptic operators, specifically, the de Rham and signature operators, associated to incomplete iterated edge metrics. Our methods are drawn from geometric microlocal analysis. Indeed, in the case of simply stratified spaces, with only one singular stratum, there is a very detailed pseudodifferential theory [40] which can be used for problems of this type, and in the even simpler case of manifolds with isolated conic singularities, one may use the somewhat simpler b-calculus of Melrose, see [42]. In either of these cases, a crucial step is to consider the de Rham or signature operator associated to an incomplete edge or conic metric as a singular factor multiplying an elliptic operator in the edge or b-calculus, and then to study this latter, auxiliary, operator using methods adapted to the geometry of an associated complete metric g on the interior of the resolved space X.
This idea was employed by Gil and Mendoza [18] in the conic case, where X is a manifold with boundary and g is a b-(or asymptotically cylindrical) metric, and also by Hunsicker and Mazzeo [26], for Witt spaces with simple edge singularities. We sketch this transformation briefly in these two cases.
First suppose that ( X, g) is a space with isolated conic singularity. Then we can write ð sign = r −1 D, where D is an elliptic differential b-operator of order 1; in local coordinates r ≥ 0 and z on F (so F = ∂ X),
(1.1) D = A(r, z) (r∂ r + ð sign,F ) .
The second term on the right is the signature operator on the link F . Thus D defines a b-operator on X. Mapping properties of the signature operator and regularity properties for solutions of ð sign u = 0 are consequences of the corresponding properties for D, which can be studied using the calculus of pseudodifferential boperators. Next, suppose that X has a simple edge singularity; then X is a manifold with fibered boundary and g = r −2 g is a complete edge metric, where r is the distance to the singular stratum in ( X, g). Here too, ð sign = r −1 D where D is an elliptic edge operator. Locally, using coordinates (r, y, z), where r is as above (hence is the radial variable in the cone fibres), and z ∈ F and y are coordinates on the edge, we have (1.2) D = A(r, y, z) r∂ r + B i (r, y, z)r∂ yi + ð sign,F .
Thus D is an elliptic differential edge operator on X in the sense of [40], and the pseudodifferential edge calculus from that paper can be used to obtain all necessary properties of ð sign . One of the main elements in the b-and edge calculi is the use of model operators associated to an operator such as D. In the b-calculus, D is modelled near the cone point by its indicial operator; in the edge calculus, D has two models: its indicial operator and its normal operator. The latter captures the tangential behaviour of D along the edge, as well as its asymptotic behaviour in the r and z directions. Their mapping properties, as determined by the construction of inverses for them, are key in understanding the analytic properties of D and hence of ð sign .
For iterated edge spaces, we proceed in a fairly similar way, using an inductive procedure. Let ( X, g) be an iterated edge space and Y a stratum of maximal depth, so that Y is a compact smooth manifold without boundary and some neighbourhood of Y in X is a cone bundle over Y with each fibre a cone over a compact space F . If this maximal depth is greater than one, then F is an iterated edge space with depth one less than that of X. If r is the radial coordinate in this cone bundle, then ð sign = r −1 D where D = A(r, y, z) (r∂ r + B i (r, y, z)r∂ yi + ð sign,F ). Here ð sign,F is the signature operator on F , and is an iterated edge operator. The gain is that since F is one step 'simpler' than X, by induction we can assume that the analytic properties of ð sign,F are already known, and from these we deduce the corresponding properties for ð sign on X. Notice that we are conformally rescaling in only the 'final' radial variable and appealing to the geometry of the partially complete metric r −2 g on the complement of Y in X.
Ideally, at this stage we could appeal to a complete pseudodifferential calculus adapted to this iterated edge geometry. Such a calculus does not yet exist, but we can take a shorter route for the problems at hand. Rather than developing all aspects of this pseudodifferential theory at each step of this induction, we develop only certain parts of the Fredholm and regularity theory for the signature operator, and phrase these in terms of a priori estimates rather than the sharp structure of the Schwartz kernel of a parametrix for it. By establishing the correct set of estimates at each stage of the induction, we can prove the corresponding estimates for spaces of one greater depth. This involves analyzing the normal and indicial operators of the partial completion of ð sign , and uses the Witt hypothesis in a crucial way.
As noted earlier, an important feature of this approach is that it carries over directly when ð sign is coupled to a C * bundle. Hence the main theorem in the higher setting can be deduced with little extra effort from the techniques used for the ordinary case. This is a key motivations for developing a geometric microlocal approach to replace the earlier successful methods of Cheeger. The fact that such techniques are well suited to this higher setting has already played a role, for example, on manifolds with boundary, cf. the work of Leichtnam, Lott and Piazza [34] on the Novikov conjecture on manifolds with boundary and the survey [36].
This leads eventually to our main analytic and topological Theorems: Theorem 1.1. Let X be any smoothly stratified pseudomanifold satisfying the Witt hypothesis. Let g be any adapted Riemannian metric on the regular part of X. Denote by ð = d + δ either the Hodge-de Rham operator ð dR or the signature operator ð sign associated to g. Then: 1) As an unbounded operator on C ∞ c (X, iie Λ * (X)) ⊂ L 2 iie (X; iie Λ * (X)), ð has a unique closed extension, hence is essentially self-adjoint.
2) For any ε > 0, the domain of this unique closed extension, still denoted ð, is contained in
ρ 1−ε L 2 iie (X; iie Λ * (X)) ∩ H 1 loc (X; iie Λ * (X)) which is compactly included in L 2
iie (X; iie Λ * (X)). 3) As an operator on its maximal domain endowed with the graph norm, ð is Fredholm. 4) ð has discrete spectrum of finite multiplicity.
Items 1), 3) and 4) have been proved by Cheeger [13] (using the heat-kernel) for metrics quasi-isometric to a piecewise flat ones.
Theorem 1.2.
There is a well defined signature class [ð sign ] ∈ K * ( X), * = dim X (mod 2), which is independent of the choice of the adapted metric on the regular part of X. When dim X is even, the index of the signature operator is well-defined.
If X ′ → X is a Galois covering with group Γ and r : X → BΓ is the classifying map, then the signature operator ð sign with coefficients in the Mishchenko bundle, together with the C * r Γ-Hilbert module L 2 iie,Γ (X; iie Λ * Γ X) define an unbounded Kasparov (C, C * r Γ)-bimodule and hence a class in KK * (C, C * r Γ) =K * (C * r Γ), which we call the index class associated to ð sign and denote by Ind( ð sign ) ∈ K * (C * r Γ). If
[[ð sign ]] ∈ KK * (C( X)⊗C * r Γ, C * r Γ)
is the class obtained from [ð sign ] ∈ KK * (C( X), C) by tensoring with C * r Γ, then Ind( ð sign ) is equal to the Kasparov product of the class defined by the Mishchenko bundle
[ C * r Γ] ∈ KK 0 (C, C( X) ⊗ C * r Γ) with [[ð sign ]]: (1.3) Ind( ð sign ) = [ C * r Γ)] ⊗ [[ð sign ]]
. In particular, the index class Ind( ð sign ) does not depend on the choice of the adapted metric on the regular part of X. Finally, if β : K * (BΓ) → K * (C * r Γ) denotes the assembly map in K-theory, then
(1.4) β(r * [ð sign ]) = Ind( ð sign ) in K * (C * r Γ)
These theorems establish property 1), the first part of property 4) and property 5) of the signature package on Witt spaces. The rest of the signature package is proved in the third part of this paper.
The Witt bordism invariance of the signature index class Ind( ð sign ) in K * (C * r Γ) is proved using KK-techniques, just as in the closed case.
The proof that Ind( ð sign ) ∈ K * (C * r Γ) is a stratified homotopy invariant is more difficult. In Section 9 we follow the strategy of Hilsum and Skandalis, but encounter extra complications caused by the singular structure of X. To deal with these we use the interplay between the compact singular space X with its incomplete metric and its resolution X with the conformally related complete metric.
The equality of the Chern character of the signature K-homology class [ð sign ] ∈ K * ( X) with the homology L-class L * ( X) had already been proved by Moscovici and Wu using Cheeger's methods, and we simply quote their result. The stratified homotopy invariance of the higher signatures, defined as the collection of numbers
{ α, r * (L * ( X)) , α ∈ H * (BΓ, Q)},
is proved in Section 10 under the hypothesis that the assembly map β is rationally injective. Finally, in Section 11 we prove the (rational) equality of our index class Ind( ð sign ) in K * (C * r Γ) with the C * r Γ-symmetric signature σ Witt C * r Γ ( X) obtained from the one recently defined by Banagl in L * (QΓ). The Witt-bordism invariance of Ind( ð sign ) and σ Witt C * r Γ ( X) plays a fundamental role in the proof of this last item in the signature package.
In the brief final section, we explain where the proof of each item in the signature package may be found in this paper.
Stratified spaces and resolution of singularities
This section describes the class of smoothly stratified pseudomanifolds. We first recall the notion of a stratified space with 'control data'; this is a topological space which decomposes into a union of smooth strata, each with a specified tubular neighbourhood with fixed product decomposition, all satisfying several basic axioms. This material is taken from the paper of Brasselet-Hector-Saralegi [7], but see Verona [58] and Pflaum [48] for more detailed expositions. We also refer the reader to [39], [25], [4] and [30]. Definitions are not entirely consistent across those sources, so one purpose of reviewing this material is to specify the precise definitions used here. A second goal here is to prove the equivalence of this class of smoothly stratified pseudomanifolds and of the class of manifolds with corners with iterated fibration structures, as introduced by Melrose. The correspondence between elements in these two classes is by blowup (resolution) and blowdown, respectively We introduce the latter class in §2.2 and show that any manifold with corners with iterated fibration structure can be blown down to a smoothly stratified pseudomanifold. The converse, that any smoothly stratified pseudomanifold can be blown up, or resolved, to obtain a manifold with corners with iterated fibration structure, is proved in §2.3; this resolution was already defined by Brasselet et al. [7], cf. also Verona [58], though those authors did not phrase it in terms of the fibration structures on the boundaries of the resolution. The proper definition of isomorphism between these spaces is subtle; we discuss this and propose a suitable definition, phrased in terms of this resolution, in §2.4. This alternate description of smoothly stratified pseudomanifolds also helps to elucidate certain notions such as the natural classes of structure vector fields, metrics, etc.
Smoothly stratified spaces.
Definition 1. A stratified space X is a metrizable, locally compact, second countable space which admits a locally finite decomposition into a union of locally closed strata S = {Y α }, where each Y α is a smooth (usually open) manifold, with dimension depending on the index α. We assume the following:
i) If Y α , Y β ∈ S and Y α ∩ Y β = ∅, then Y α ⊂ Y β . ii) Each stratum Y is endowed with a set of 'control data' T Y , π Y and ρ Y ; here T Y is a neighbourhood of Y in X which retracts onto Y , π Y : T Y −→ Y is a fixed continuous retraction and ρ Y : T Y → [0, 2) is a proper 'radial function' in this tubular neighbourhood such that ρ −1 Y (0) = Y . Furthermore, we require that if Z ∈ S and Z ∩ T Y = ∅, then (π Y , ρ Y ) : T Y ∩ Z −→ Y × [0, 2) is a proper differentiable submersion. iii) If W, Y, Z ∈ S, and if p ∈ T Y ∩ T Z ∩ W and π Z (p) ∈ T Y ∩ Z, then π Y (π Z (p)) = π Y (p) and ρ Y (π Z (p)) = ρ Y (p). iv) If Y, Z ∈ S, then Y ∩ Z = ∅ ⇔ T Y ∩ Z = ∅, T Y ∩ T Z = ∅ ⇔ Y ⊂ Z, Y = Z or Z ⊂ Y . v) For each Y ∈ S, the restriction π Y : T Y → Y is a locally trivial fibra- tion with fibre the cone C(L Y ) over some other stratified space L Y (called the link over Y ), with atlas U Y = {(φ, U)} where each φ is a trivialization π −1 Y (U) → U × C(L Y )
, and the transition functions are stratified isomorphisms (in the sense of Definition 4 below) of C(L Y ) which preserve the rays of each conic fibre as well as the radial variable ρ Y itself, hence are suspensions of isomorphisms of each link L Y which vary smoothly with the variable y ∈ U. If in addition we let X j be the union of all strata of dimensions less than or equal to j, and require that vi) X = X n ⊇ X n−1 = X n−2 ⊇ X n−3 ⊇ . . . ⊇ X 0 and X \ X n−2 is dense in X then we say that X is a stratified pseudomanifold.
Some of these conditions require elaboration:
• The depth of a stratum Y is the largest integer k such that there is a chain of strata Y = Y k , . . . , Y 0 with Y j ⊂ Y j−1 for 1 ≤ j ≤ k. A stratum of maximal depth is always a closed manifold. The maximal depth of any stratum in X is called the depth of X as a stratified space. (Note that this is the opposite convention of depth from that in [7]. ) We refer to the dense open stratum of a stratified pseudomanifold X as its regular set, and the union of all other strata as the singular set,
reg( X) := X \ sing( X), where sing( X) = Y ∈S depth Y >0 Y.
• If X and X ′ are two stratified spaces, a stratified isomorphism between them is a homeomorphism F : X → X ′ which carries the open strata of X to the open strata of X ′ diffeomorphically, and such that
π ′ F (Y )) • F = F • π Y , ρ ′ Y = ρ F (Y ) • F for all Y ∈ S(X).
(We shall discuss this in more detail below.)
• If Z is any stratified space, then the cone over Z, denoted C(Z), is the space Z × R + with Z × {0} collapsed to a point. This is a new stratified space, with depth one greater than Z itself. The vertex 0 := Z × {0}/ ∼ is the only maximal depth stratum; π 0 is the natural retraction onto the vertex and ρ 0 is the radial function of the cone.
• There is a small generalization of the coning construction. For any Y ∈ S,
let S Y = ρ −1 Y (1). This is the total space of a fibration π Y : S Y → Y with fibre L Y . Define the mapping cylinder over S Y by Cyl (S Y , π Y ) = S Y × [0, 2) / ∼ where (c, 0) ∼ (c ′ , 0) if π Y (c) = π Y (c ′ ). The equivalence class of a point (c, t) is sometimes denoted [c, t], though we often just write (c, t) for simplicity. Then there is a stratified isomorphism F Y : Cyl (S Y , π Y ) −→ T Y ;
this is defined in the canonical way on each local trivialization U × C(L Y ) and since the transition maps in axiom v) respect this definition, F Y is well-defined.
• Finally, suppose that Z is any other stratum of X with T Y ∩ Z = ∅, so by axiom iv), Y ⊂ Z. Then S Y ∩ Z is a stratum of S Y .
We have been brief here since these axioms are described more carefully in the references cited above. Axiom v) is sometimes considered to be a consequence of the other axioms. In the topological category (where the local trivializations of the tubular neighbourhoods are only required to be homeomorphisms) this is true, but the situation is less clear for smoothly stratified spaces, so we prefer to leave this axiom explicit. Let us direct the reader to [39] and [25] for more on this.
We elaborate further on the definition of stratified isomorphism. This definition is strictly determined by the control data on the domain and range, i.e. by the condition that F preserve the product decomposition of each tubular neighbourhood. It is nontrivial to prove that the same space X endowed with two different sets of control data are isomorphic in this sense. There are other even more rigid definitions of isomorphism in the literature. The one in [48] requires that the spaces X and X ′ are differentiably embedded into some ambient Euclidean space, and that the map F locally extends to a diffeomorphism of these ambient spaces. For example, let X be a union of three copies of the half-plane R × R + , as follows. The first and second are embedded as {(x, y, z) : z = 0, y ≥ 0} and {(x, y, z) : y = 0, z ≥ 0}, while the third is given by {(x, y, z) : y = r cos α(x), z = r sin α(x), r ≥ 0} where α : R → (0, π/2) is smooth. In other words, this last sheet is the union of a smoothly varying family of rays orthogonal to the x-axis, with slope α(x) at each slice. Requiring a stratified isomorphism to extend to a diffeomorphism of the ambient R 3 would make these spaces for different functions α(x) inequivalent. We propose a different definition below which has various advantages over either of the ones above.
Iterated fibration structures.
The definition of an iterated fibration structure was proposed by Melrose in the late '90's as the boundary fibration structure in the sense of [43] associated to the resolution of an iterated edge space (what we are calling a smoothly stratified space). It has not appeared in the literature previously (though we can now refer to [1], which was finished after the present paper), and we are grateful to him for allowing us to present it here. The passage to this resolution is necessary in order to apply the methods of geometric microlocal analysis. A calculus of pseudodifferential iterated edge operators, when it is eventually written down fully, will yield direct proofs of most of the analytic facts in later sections of this paper.
Let X be a manifold of dimension n with corners up to codimension k. This means that any point p ∈ X has a neighbourhood U ∋ p which is diffeomorphic to a neighbourhood of the origin V in the orthant (R + ) ℓ × R n−ℓ for some ℓ ≤ k, with p mapped to the origin. There are induced local coordinates (x 1 , . . . , x ℓ , y 1 , . . . , y n−ℓ ), where each x i ≥ 0 and y j ∈ (−ǫ, ǫ). There is an obvious decomposition of X into its interior and the union of its boundary faces of various codimensions. We make the additional global assumption that each face is itself an embedded manifold with corners in X, or in other words, that no boundary face intersects itself.
We shall frequently encounter fibrations f : X → X ′ between manifolds with corners. By definition, a map f is a fibration in this setting if it satisfies the following three properties: f is a 'b-map', which means that if ρ ′ is any boundary defining function in X ′ , then f * (ρ ′ ) is a product of boundary defining functions of X multiplied by a smooth nonvanishing function; next, each q ∈ X ′ has a neighbourhood U such that f −1 (U) is diffeomorphic to U × F where the fibre F is again a manifold with corners; finally, we require that each fibre F be a 'psubmanifold' in X, which means that in terms of an appropriate adapted corner coordinate system (x, y) ∈ (R + ) ℓ × R n−ℓ , as above, each F is defined by setting some subset of these coordinates equal to 0.
The collection of boundary faces of codimension one play a special role, and is denoted H = {H α } α∈A for some index set A. Each boundary face G is the intersection of some collection of boundary hypersurfaces,
G = H α1 ∩ . . . ∩ H α ℓ , which we often write as H A ′ where A ′ = {α 1 , . . . , α ℓ } ⊂ A.
Definition 2 (Melrose). An iterated fibration structure on the manifold with corners X consists of the following data: a) Each H α is the total space of a fibration f α : H α → B α , where both the fibre F α and base B α are themselves manifolds with corners. b) If two boundary hypersurfaces meet, i.e.
H αβ := H α ∩ H β = ∅, then dim F α = dim F β . c) If H αβ = ∅ as in b)
, and dim F α < dim F β , then the fibration of H α restricts naturally to H αβ (i.e. the leaves of the fibration of H α which intersect the corner lie entirely within the corner) to give a fibration of H αβ with fibres F α , whereas the larger fibres F β must be transverse to H α at H αβ . Writing ∂ α F β for the boundaries of these fibres at the corner, i.e. ∂ α F β := F β ∩H αβ , then H αβ is also the total space of a fibration with fibres ∂ α F β . Finally, we assume that the fibres F α at this corner are all contained in the fibres ∂ α F β , and in fact that each fibre ∂ α F β is the total space of a fibration with fibres F α . Two spaces X and X ′ with iterated fibration structures are isomorphic precisely when there exists a diffeomorphism Φ between these manifolds with corners which preserves all of the fibration structures at all boundary faces.
The index set A has a partial ordering: the ordered chains α 1 < . . . < α r in A are in bijective correspondence with the corners
H A ′ := H α1 ∩ . . . ∩ H αr , A ′ = {α 1 , . . . , α r }, where by definition α i < α j if dim F αi < dim F αj . In particular, α < β implies H α ∩ H β = ∅.
We say that H α has depth r if the longest chain β 1 < β 2 < . . . < β r in A with maximal element β r = α has length r. The depth of a manifold with corners with iteration fibration structure is the maximal depth of any of its boundary hypersurfaces, equivalently, the maximal codimension of any of its corners. The precise relationships between the induced fibrations on each corner are not easy to describe in general, but these do not play a role here.
Lemma 2.1. If α < β, then the boundary of each fibre F α ⊂ H α is disjoint from the interior of H αβ . Furthermore, the restriction of f α to H αβ has image lying within ∂B α , whereas the restriction of f β to H αβ has image intersecting the interior of B β . In particular, if α and β are, respectively, minimal and maximal elements in A, then the fibres F α and the base B β are closed manifolds without boundary.
Proof. Choose adapted local coordinates (x α , x β , y 1 , . . . , y n−2 ) in H αβ which simultaneously straighten out these fibrations. Thus (x β , y) are coordinates on H α = {x α = 0}, and there is a splitting y = (y ′ , y ′′ ) so that (x β , y ′ , y ′′ ) → (x β , y ′′ ) represents the fibration H α → B α . By part c) of the definition, since dim F α < dim F β , there is a further decomposition y ′′ = (y ′′ 1 , y ′′ 2 ) so that the fibration of H αβ with fibres ∂ α F β is represented by y → y ′′ 2 . Thus (x β , y ′′ ) and y ′ are local coordinates on B α and each F α , and y ′′ 2 and (x α , y ′ , y ′′ 1 ) are local coordinates on B β and each F β , respectively. All the assertions are direct consequences of this.
Unlike for smoothly stratified spaces, the structure of control data has not been incorporated into this definition of iterated fibration structures, because its existence and uniqueness can be inferred from standard facts in differential topology. Nonetheless, these data are still useful, and we discuss them now.
Definition 3. Let X be a manifold with corners with an iterated fibration structure. Then a control data set for X consists of a collection of triples { T H , , π H , ρ H }, one for each H ∈ H, where T H is a collar neighbourhood of the hypersurface H, ρ H is a defining function for H and π H is a diffeomorphism from each slice ρ H = const. to H. Thus the pair ( π H , ρ H ) gives a diffeomorphism T H → H × [0, 2), and hence an extension of the fibration of H to all of T H . These data are required to satisfy the following additional properties: for any hypersurface H ′ which intersects H with H ′ < H, the restriction of ρ H to H ′ ∩ T H is constant on the fibres of H ′ ; finally, near any corner H A ′ , A ′ = {α 1 , . . . , α r }, the extension of the set of fibrations of H A ′ induced by the product decomposition ( π Hα j , ρ Hα j )
αj ∈A ′ : r j=1 T Hα j ∼ = H A ′ × [0, 2) r
preserves all incidence and inclusion relationships between the various fibres.
The existence of control data for an iterated fibration structure on a manifold with corners X is discussed in [1, Proposition 3.7], so we make only a few remarks here. We can find some set of control data by successively choosing the maps π H and defining functions ρ H in order of increasing depth, at each step making sure to respect the compatibility relationships with all previous hypersurfaces. The uniqueness up to diffeomorphism can be established in much the same way, based on the fact that there is a unique product decomposition of a collar neighbourhood of any H up to diffeomorphism. Proposition 2.2. Let X be a manifold with corners with iterated fibration structure, and suppose that { π H , ρ H } and { π ′ H , ρ ′ H } are two sets of control data on it. Then there is a diffeomorphism f of X which preserves the iterated fibration structure, and which intertwines the two sets of control data.
The key idea in the proof is that we can pull back any set of control data on X to a 'universal' set of control data defined on the union of the inward pointing normal bundles to each boundary hypersurface which satisfies the obvious set of compatibility conditions. The fact that any two such sets of 'pre-control data' are equivalent can then be deduced inductively using standard results about uniqueness up to diffeomorphism of collar neighbourhoods of these boundary hypersurfaces.
Finally, note that if X has an iterated fibration structure, then any corner H A ′ inherits such a structure too (we forget about the fibration of its interior), with depth equal to k − codim H A ′ . Proposition 2.3. If X is a compact manifold with corners with an iterated fibration structure, then there is a smoothly stratified space X obtained from X by a process of successively blowing down the connected components of the fibres of each hypersurface boundary of X in order of increasing fibre dimension (or equivalently, of increasing depth). The corresponding blowdown map will be denoted β : X → X.
Proof. We warm up to the general case by first considering what happens when X is a manifold with boundary, so ∂ X is the total space of a fibration with fibre F and base space Y and both F and Y are closed manifolds. Choose a boundary defining function ρ and fix a product decomposition ∂ X × [0, 2) of the collar neighbourhood U = {ρ < 2}. This defines a retraction π : U → ∂X, as well as a fibration of U over ∂ X with fibre π −1 (F ) = F × [0, 2). Now collapse each fibre F at x = 0 to a point. This commutes with the restriction to each F ×[0, 2), so we obtain a bundle of cones C(F ) over Y . We call this space the blowdown of X along the fibration, and write it as X/F . Denote by T Y the image of U under this blowdown. The map π induces a retraction map π(U) = T Y → Y , and ρ also descends to T Y . Thus {T Y , π, ρ} are the control data for the singular stratum Y , and it is easy to check that these satisfy all of the axioms in §2.1, hence X/F is a smoothly stratified space. Now turn to the general case, which is proved by induction on the depth. As in the next subsection, where we we follow an argument from [7] and show how to blow up a smoothly stratified space, we use a 'doubling construction' to stay within the class of stratified pseudomanifolds while applying the inductive hypothesis to reduce the complexity of the problem. To set this up, beginning with X, a manifold with corners with iterated fibration structure of depth k, form a new manifold with corners and iterated fibration structure of depth k − 1 by simultaneously doubling X across all of its maximal depth hypersurfaces. In other words, consider
X ′ = ( X × −1) ⊔ ( X × +1) / ∼
where (p, −1) ∼ (q, +1) if and only if p = q ∈ H ∈ H where depth (H) = k. By standard arguments in differential topology, one can give X ′ the structure of a manifold with corners up to codimension k − 1. If H j ∈ H is any face with depth j < k which intersects a face H k of depth k, then as in Lemma 2.1, the boundaries of the fibres F j ⊂ H j only meet the corners H ij for i < j, and do not meet the interior of H jk . In terms of the local coordinates (x j , x k , y) in that Lemma, we simply let x k vary in (−ǫ, ǫ) rather than just [0, ǫ), and it is clear how to extend the fibrations accordingly.
The dimensional comparisons and inclusion relations at all other corners remain unchanged. Therefore, X ′ has an iterated fibration structure. This new space also carries a smooth involution which has fixed point set the union of all depth k faces, where the two copies of X are joined, as well as a function ρ k which is positive on one copy of X, negative on the other, and which vanishes simply on the interface between the two copies of X. For simplicity, assume that there is only one depth k face, H k . We can also choose ρ k so that it is constant on the fibres of all other boundary faces, and a retraction π k defined on the set | ρ k | < 2 onto H k . Now apply the inductive hypothesis to blow down the boundary hypersurfaces of X ′ in order of increasing fibre dimension to obtain a smoothly stratified space X ′ of depth k − 1. The function ρ k descends to a function (which we give the same name) on this space. Consider the open set X + := X ′ ∩ {ρ k > 0}, and also ∂ k X := X ′ ∩ {ρ k = 0}. Both of these are smoothly stratified spaces; for the former this is because (in the language of [7]) we are restricting to a 'saturated' open set of X ′ , though we do not need to appeal to this terminology since the assertion is clear, whereas for the latter it follows by induction since it is the blowdown of H k , which has depth less than k. This space ∂ k X, which we denote by H k is the total space of a fibration induced from the fibration of the face H k in X. By Lemma 2.1, since the H k are maximal, the base B k has no boundary, and the fibres F k are manifolds with corners with iterated fibration structures of depth less than k.
Hence after the blowdown, the base of the fibration of ∂ k X is still B k while the fibres are the blowdowns F k of the spaces F k , which are again well defined by induction. Finally, using the product decomposition of a neighbourhood of H k in X, collapsing the fibres of H k identifies the blowdown of this neighbourhood with the mapping cylinder for the fibration of ∂ k X. This produces the final space X.
It suffices to check that the stratification of X satisfies the axioms of a smoothly stratified space only near where this final blowdown takes place, since the inductive hypothesis guarantees that they hold elsewhere. These axioms are not difficult to verify from the local description of X in a product neighbourhood of H k .
Remark 2.4.
There is a subtlety in this result since there is typically more than one smoothly stratified space X which may be obtained by blowing down a manifold with corners X with iterated fibration structure. More specifically, there is a minimal blowdown, which associates to each connected hypersurface boundary of X a stratum of the blowdown X. However, it may occur that two strata of X of highest depth, for example, are diffeomorphic, and after identifying these strata we obtain a new smoothly stratified space. It may not be easy to quantify the full extent of nonuniqueness, but we do not attempt (nor need) this here.
2.3. The resolution of a smoothly stratified space. The other part of this description of the differential topology of smoothly stratified spaces is the resolution process: namely, conversely to the blowdown construction above, if X is any smoothly stratified space, one may resolve its singularities by successively blowing up its strata in order of decreasing depth to obtain a manifold with corners X with iterated fibration structure. Following Remark 2.4, two different smoothly stratified spaces X 1 , X 2 may resolve to the same manifold with corners X. Proposition 2.5. Let X be a smoothly stratified pseudomanifold. Then there exists a manifold with corners X with an iterated fibration structure, and a blowdown map β : X → X which has the following properties:
• there is a bijective correspondence Y ↔ X Y between the strata Y ∈ S of X and the (possibly disconnected) boundary hypersurfaces of X which blow down to these strata; • β is a diffeomorphism between the interior of X and the regular set of X;
we denote by X this open set, which is dense in either X or X; • β is also a smooth fibration of the interior of each boundary hypersurface X Y with base the corresponding stratum Y and fibre the regular part of the link of Y in X; moreover, there is a compactification of Y as a manifold with corners Y such that the extension of β to all of X Y is a fibration with base Y and fibre L Y ; finally, each fibre L Y ⊂ X Y is a manifold with corners with iterated fibration structure and the restriction of β to it is the blowdown onto the smoothly stratified space Y .
We sketch the proof, adapting the construction from [7], to which we refer for further details. The proof is inductive: if X has depth k and we simultaneously blow up the union of the depth k strata to obtain a space X 1 , then all the control data of the stratification on X lifts to give X 1 the structure of a smoothly stratified space of depth k − 1. Iterating this k times completes the proof. However, in order to stay within the category of smoothly stratified pseudomanifolds, which by definition have no codimension one boundaries, we proceed as in the proof of Proposition 2.3 (and as in [7]) and construct a space X ′ 1 which is the double across the boundary hypersurface of the blowup of X along its depth k strata, and show that X ′ 1 is a smoothly stratified pseudomanifold of depth k − 1. This space X ′ 1 is equipped with an involution τ 1 which interchanges the two copies of the double; the actual blowup is the closure of one component of the complement of the fixed point set of this involution. Iterating this k times, we obtain a smooth compact manifold X ′ k equipped with k commuting involutions {τ j } k j=1 ; the manifold with corners we seek is any one of the 2 k fundamental domains for this action.
Proof. To begin, fix a stratum Y which has maximal depth k; this is a smooth closed manifold. Recall the notation from §2.1, and in particular the stratified isomorphism F Y from the mapping cylinder of (S Y , π Y ) to T Y and the family of
local trivializations φ : π −1 Y (U) → U ×C(L Y ) for suitable U ⊂ Y . If u ∈ T Y ∩π −1 Y (U), we write φ(u) = (y, z, t) where y ∈ U, z ∈ L Y and t = ρ Y (u); by axiom v), there is a retraction R Y : T Y \ Y → S Y ,
given on any local trivialization by (y, z, t) → (y, z, 1) (which is well defined since t = 0).
To construct the first blowup, assume for simplicity that there is only one stratum Y of maximal depth k. Define
(2.1) X ′ 1 := ( X \ Y ) × {−1} ⊔ ( X \ Y ) × {+1} ⊔ S Y × (−2, 2) / ∼ where (if ǫ = ±1), (2.2) (p, ǫ) ∼ (R Y (p), ρ Y (p)) if p ∈ T Y \ Y and ǫt > 0. Let X ′ = ( X × {−1}) ⊔ ( X × {+1})/ ∼ where (u, ǫ) ∼ (u ′ , ǫ ′ ) if and only if u = u ′ ∈ Y . Note that X ′ 1 \ S Y × {0} is naturally identified with X ′ \ Y , so this construction replaces Y with S Y .
There is a blowdown map β 1 : X ′ 1 → X ′ given by
β 1 (u, ǫ) = (u, ǫ) if u / ∈ Y, β 1 (u, 0) = π Y (u).
Clearly
β 1 : X ′ 1 \ S Y × {0} → X ′ \ Y is an isomorphism of smoothly stratified spaces and (S Y × (−2, 2)) is a tubular neighbourhood of (β 1 ) −1 Y = S Y × {0} in X ′ 1 .
We shall prove that X ′ 1 is a smoothly stratified space of depth k − 1 equipped with an involution τ 1 which fixes S Y × {0} and interchanges the two components of the complement of this set in X ′ 1 , and which fixes all the control data of X ′ 1 . To do all of this, we must fix a stratification S 1 of X ′ 1 and define all of the corresponding control data and show that these satisfy properties i) -vi).
• Fix any stratum Z ∈ S of X with depth (Z) < k, and define
(2.3) Z ′ 1 := (Z × {±1}) ⊔ ((S Y ∩ Z) × (−2, 2)) / ∼,
where ∼ is the same equivalence relation as in (2.2). The easiest way to see that this is well-defined is to note that S Y ∩ Z is a stratum of the smoothly stratified space S Y and that the restriction
(2.4) F Y : Cyl (S Y ∩ Z, π Y ) −→ Z ∩ T Y
is an isomorphism. (This latter assertion follows from axiom ii).) As above, let Z ′ be the union of two copies of Z joined along Z ∩ Y .
• Now define the stratification
S 1 of X ′ 1 (2.5) S 1 := { Z ′ 1 : Z ∈ S \ Y }. We must now define the control data {T Z ′ 1 , π Z ′ 1 , ρ Z ′ 1 } Z ′ 1 ∈S1 associated to this strat- ification. • Following (2.3), set (2.6) T Z ′ 1 := T Z × {±1} ⊔ ((S Y ∩ T Z ) × (−2, 2)) / ∼, where (p, ǫ) ∼ (c, t) if tǫ > 0 and p = F Y (c, |t|)
. Extending (or 'thickening') (2.4), by axiom iii) we also have that F Y restricts to an isomorphism between Cyl (T Z ∩ S Y , π Y ) and T Z ∩ T Y . In turn, using axiom ii) again, within the smoothly stratified space S Y , F TY ∩Z is an isomorphism from Cyl (S Y ∩ S Z , π Z ) to the tubular neighbourhood of Z ∩ S Y in S Y , which is the same as T SY ∩Z . Using these representations, the fact that (2.6) is well-defined follows just as before.
Note that Y has been stretched out into S Y × {0}, and T Z ′
1 ∩ (S Y × {0}) is isomorphic to T Z ′ 1 ∩ (S Y × {t}) for any t ∈ (−2, 2). • The projection π Z ′ 1 is determined by π Z on each slice (S Y ∩ T Z ) × {t}, at least
when t = 0, and extends uniquely by continuity to the slice at t = 0 in X ′ 1 . A similar consideration yields the function ρ Z ′ 1 .
• One must check that the space X ′ 1 and this control data for its stratification satisfies axioms i) -vi). This is somewhat lengthy but straightforward, so details are left to the reader.
• Finally, this whole construction is symmetric with respect to the reflection τ 1 defined by t → −t in T Y and which extends outside of T Y as the interchange of the two components of
X ′ \ Y . The fixed point set of τ 1 is the slice S Y × {0}.
This establishes that the space X ′ 1 obtained by resolving the depth k smoothly stratified space X along its maximal depth strata via this doubling-blowup construction is a smoothly stratified space of depth k − 1, equipped with one extra piece of data, the involution τ 1 .
This process can now be iterated. After j iterations we obtain a smoothly strat-
ified space X ′ j of depth k − j which is equipped with j commuting involutions τ i , 1 ≤ i ≤ j.
In particular, the space X ′ k is a compact closed manifold. It is easy to check, e.g. using the local coordinate descriptions, that these involutions are 'independent' in the sense that for any point p which lies in the fixed point set of more than one of the τ i , the −1 eigenspaces of the dτ i are independent.
The complement of the union of fixed point sets of the involutions τ i has 2 k components, and X is the closure of any one of these components.
The construction is finished if we show that X carries the structure of a manifold with corners with iterated fibration structure. We proved already that X has the local structure of a manifold with corners, but we must check that the boundary faces are embedded. For this, first note that all faces of the resolution of X ′ 1 are embedded, and by its description in the resolution construction, H k is as well; finally, all corners of X which lie in H k are embedded since they are faces of the resolution of S Y where Y is the maximal depth stratum and we may apply the inductive hypothesis. This proves that X is a manifold with corners. Now examine the structure on the boundary faces inductively. The case k = 1 is obvious since then X is a manifold with boundary; ∂ X is the total space of a fibration and there are no compatibility conditions with other faces. Suppose we have proved the assertion for all spaces of depth less than k, and that X is a smoothly stratified space of depth k. Let Y be the union of all strata of depth k and consider the doubled-blowup space X ′ 1 . This is a stratified space of depth k − 1, so its resolution is a manifold with corners up to codimension k − 1 with iterated fibration structure. Since S Y is again a smoothly stratified space of depth k − 1, its resolution S Y is also a manifold with corners with iterated fibration structure. The blowdown of S Y along the fibres of all of its boundary hypersurfaces is a smoothly stratified space S Y and this is the boundary H k of X 1 , the 'upper half' of X ′ 1 . Once we have performed all other blowups, we know that the compatibility conditions are satisfied at every corner except those which lie in S Y . The images of the other boundaries of X 1 by blowdown into X 1 are the singular strata of this space. Furthermore, there is a neighbourhood of H ′ k in X 1 of the form S Y × [0, 2) (using the variable t in this initial blowup as the defining function ρ k ), so that near H k , X has the product decomposition S Y × [0, 2). From this it follows that each fibre F j of H j , j < k, lies in the corresponding corners H k ∩ H j ; it also follows that each fibre F k of H k is transverse to this corner, and has boundary ∂ j F k equal to a union of the fibres F j . This proves that conditions a) -c) of the iterated fibration structure are satisfied.
Smoothly stratified isomorphisms.
We now return to a closer discussion of a good definition of isomorphism between smoothly stratified spaces. Following Melrose, these isomorphisms are better understood through their lifts to the resolutions.
To begin, we state a result which is a straightforward consequence of the resolution and blowdown constructions above.
Proposition 2.6. Let X and X ′ be two smoothly stratified spaces and X, X ′ their resolutions, with blowdown maps β : X → X and β ′ : X ′ → X ′ . Suppose that f : X → X ′ is a stratified isomorphism as in [7], §2. Then there is a unique diffeomorphism of manifolds with corners f : X → X ′ which preserves the iterated fibration structures and which satisfies f
• β = β ′ • f .
Proof. If such a lift exists at all, it must be unique simply because it is defined by continuous extension from a map defined between the interiors of X and X ′ . Because of this uniqueness, it suffices to prove the existence of the lift in local coordinates, and this is done in [7], §2 Prop. 3.2 and Remark 4.2. Of course, since those authors are not using the notion of iterated fibration structures, they do not consider the issue of whether the lift preserves the fibrations at the boundaries; however, a cursory inspection of their proof shows that the map they construct does have this property.
The converse result is also true, up to a technical point concerning connectedness of the links. Proposition 2.7. Given X, X ′ , X and X ′ , as above, suppose that f : X → X ′ is a diffeomorphism of manifolds with corners which preserves the fibration structures at the boundaries. Suppose furthermore that X and X ′ are the minimal blowdowns of X, X ′ in the sense of Remark 2.4. Then there exists some choice of control data on the blown down spaces and a stratified isomorphism f :
X → X ′ such that f • β = β ′ • f . Proof.
As above, f is uniquely determined over the principal dense open stratum of X. The fact that f preserves the fibration structures means that f extends to a continuous map X → X ′ . However, this extension is not a stratified isomorphism unless we use the correct choices of control data on all these spaces. Thus fix control data on X; this may be pushed forward to control data on X ′ via f . Any set of control data for a manifold with corners with iterated fibration structure can be pushed down to a set of control data on its blowdown. Therefore we have now induced control data on X and X ′ , and it follows from this construction that the induced map f intertwines these sets of control data, as required.
Combined with Proposition 2.2, this gives another proof of the result from [7] that any two sets of control data on a smoothly stratified space X are equivalent by a smoothly stratified isomorphism.
This discussion motivates the following Definition 4. A smoothly stratified map f between smoothly stratified spaces X and X ′ is a continuous map f : X → X ′ sending the open strata of X smoothly into the open strata of X ′ and for which there exists a lift f : This definition has the advantage that it is not inductive (even though many of the arguments behind it are), and it provides a clear notion of the regularity of these isomorphisms on approach to the singular set.
X → X ′ , f • β = β ′ • f ,
Iterated edge metrics. Witt spaces.
In this section we first introduce the class of Riemannian metric on smoothly stratified spaces with which we shall work throughout this paper. These metrics are only defined on reg ( X), but the main point is their behaviour near the singular strata. These metrics were also considered by Brasselet-Legrand [8]; closely related metrics had been considered by Cheeger [13]; they are most easily described using adapted coordinate charts (see pp. 224-5 of [8]) or equivalently, on the resolution X. In the second part of the section we introduce the Witt condition and recall the fundamental theorem of Cheeger, asserting the isomorphism between intersection cohomology and Hodge cohomology on these spaces. In the following, we freely use notation from the last section.
Existence of iterated edge metrics.
We begin by constructing an open covering of reg ( X) by sets with an iterated conic structure. Let Y 1 be any stratum. By definition, for each q 1 ∈ Y 1 there exists a neighbourhood U 1 and a trivialization π −1 Y1 (U 1 ) ∼ = U 1 ×C(L Y1 ). Now fix any stratum Y 2 ⊂ L Y1 , and a point q 2 ∈ Y 2 . As before, there is a neighbourhood U 2 ⊂ Y 2 and a trivialization π −1 Y2 (U 2 ) ∼ = U 2 × C(L Y2 ). Continuing on in this way, the process must stop in no more than d = depth (Y 1 ) steps when q s lies in a stratum Y s of depth 0 in L Ys−1 ( which must, in particular, occur when L Ys−1 itself has depth 0). We obtain in this way an open set of the form
(3.1) U 1 × C U 2 × C(U 3 × . . . × C(U s ) ) · · · ,
where s ≤ d, which we denote by W = W q1,...,qs . Choose a local coordinate system y (j) on U j , and let r j be the radial coordinate in the cone C(L Yj ). Thus (y (1) , r 1 , y (2) , r 2 , . . . , y (s) ) is a full set of coordinates in W. Clearly we may cover all of X by a finite number of sets of this form. We next describe the class of admissible Riemannian metrics on reg ( X) by giving their structure on each set of this type.
Definition 5. We say that a Riemannian metric g defined on reg ( X) is an iterated edge metric if there is a covering by the interiors of sets of the form W q1,...,qs so that in each such set,
g = h 1 + dr 2 1 + r 2 1 (h 2 + dr 2 2 + r 2 2 (h 3 + dr 2 3 + r 2 3 (h 4 + . . . + r 2 s−1 h s ))),
with 0 < r j < ǫ for some ǫ > 0 and every j, and where h j is a metric on U j . We also assume that for every j = 1, . . . , s, h j depends only on y (1) , r 1 , y (2) , r 2 , . . . , y (j) , r j . If each h j is independent of the radial coordinates r 1 , . . . , r j , then we call g a rigid iterated edge metric. Note that this requires the choice of a horizontal lift of the tangent space of each stratum Y as a subbundle of the cone bundle T Y which is invariant under the scaling action of the radial variable on each conic fibre.
Proposition 3.1. Let X be a smoothly stratified pseudomanifold. Then there exists a rigid iterated edge metric g on reg ( X).
Proof. We prove this by induction. For spaces of depth 0, there is nothing to prove, so suppose that X is a smoothly stratified space of depth k ≥ 1, and assume that the result is true for all spaces with depth less than k.
Let Y be the union of strata of depth k, each component of which is necessarily a closed manifold; for convenience we assume that Y is connected. Consider the space X ′ 1 obtained in the first step of the resolution process in §2.3 by adjoining two copies of X along Y and replacing the double of the neighbourhood T Y by a cylinder S Y × (−2, 2). This is a space of depth k − 1, and hence possesses a rigid iterated edge metric g 1 . We may in fact assume that in the cylindrical region S Y × (−2, 2), g 1 has the form dt 2 + g SY , where g SY is a (rigid) iterated edge metric on S Y which is independent of t. Recalling that S Y is the total space of a fibration with fibre L Y , we can define a family of metrics g r SY on S Y by scaling the metric on each fibre by the factor r 2 . This leads to a rigid iterated edge metric g TY := dr 2 + g r SY on the tubular neighborhood T Y ⊂ X around Y , which by construction is also rigid. Now use the induction hypothesis to choose a rigid iterated edge metric g C on the complement C of the region {r < 1/2} ⊂ T Y . Finally, choose a smooth partition of unity {φ(r), ψ(r)} relative to the open cover [0, 2/3) ∪ (1/3, ∞) of R + ; the metric φg TY + ψg C on T Y extends to g C outside T Y , and satisfies our requirement. Proposition 3.2. 1) Any two iterated edge metrics on X are homotopic within the class of iterated edge metrics. 2) Any two rigid iterated edge metrics on X are homotopic within the class of rigid iterated edge metrics.
Proof. We proceed by induction. The result is obvious when the depth is 0, so assume it holds for all spaces of depth strictly less than k and consider a pseudomanifold of depth k with two iterated edge metrics g and g ′ .
To begin, then, fix a stratum Y which has maximal depth k. Then Y is a smooth closed manifold. Recall the notation from §2.1, and in particular the stratified isomorphism F Y from the mapping cylinder of (S Y , π Y ) to T Y and the family of
local trivializations φ : π −1 Y (U) → U ×C(L Y ) for suitable U ⊂ Y . If u ∈ T Y ∩π −1 Y (U), we write φ(u) = (y, z, t) where y ∈ U, z ∈ L Y and t = ρ Y (u); in particular, by axiom v), there is a retraction R Y : T Y \ Y → S Y ,
given on any local trivialization by (y, z, t) → (y, z, 1) (which is well defined since t = 0).
In any of these trivializations, the metric g has the form
(φ −1 ) * g = g U (y, t) + dt 2 + t 2 g LY (t, y, z)
and the homotopy
s → g U (y, s + (1 − s)t) + dt 2 + t 2 g LY (s + (1 − s)t, y, z)
removes the dependence of g U and g LY on t while remaining in the class of iterated edge metrics. Since the coordinate t = ρ Y (u) is part of the control data, this homotopy can be performed consistently across all of the local trivializations φ.
Without loss of generality we may assume that
(φ −1 ) * g = g U (y) + dt 2 + t 2 g LY (y, z), and (φ −1 ) * g ′ = g ′ U (y) + dt 2 + t 2 g ′ LY (y, z)
. The metrics g U and g ′ U are homotopic and, by inductive hypothesis, so are the metrics g LY and g ′ LY . Thus the metrics (φ −1 ) * g and (φ −1 ) * g ′ are homotopic within the class of iterated edge metrics on U × C(L Y ). Using consistency of the trivializations φ we can patch these homotopies together and see that g and g ′ are homotopic in a neighborhood of Y .
We can thus assume that g and g ′ coincide in a neighborhood of Y and, in this neighborhood, are independent of ρ Y . As in the proof of Proposition 2.5 we consider the space
X ′ 1 := ( X \ Y ) × {−1} ⊔ ( X \ Y ) × {+1} ⊔ S Y × (−2, 2) / ∼ Define the lift g of g to X ′ 1 by g on each copy of X \ Y and g U (y) + g LY (y, z) + dt 2 on each neighborhood of S Y ×(−2,
2) corresponding to the trivialization φ as above, and define g ′ similarly. Then g and g ′ are iterated edge metric on a space of depth k − 1 so by inductive hypothesis are homotopic. Moreover since they coincide in S Y × (−2, 2), the homotopy can be taken to be constant in a neighborhood of S Y , and hence the homotopy descends to a homotopy of g and g ′ . If g and g ′ are rigid, the homotopies above preserve this.
Cheeger also defines [12] (p. 127) a class of admissible metrics g on the regular part of a smoothly stratified pseudomanifold X. He uses a slightly different decomposition of X and assumes that on each 'handle' of the form (0, 1) n−i × C(N i−1 ), g induces a metric quasi-isometric to one of the form
(dy 1 ) 2 + . . . + (dy n−i ) 2 + (dr) 2 + r 2 g N i−1 ;
see [12] for the details. Using the proof of Proposition 3.1 as well as [12] (page 127), we obtain the following Proposition 3.3. 1) Any rigid iterated edge metric as in Definition 5) is admissible in the sense of Cheeger.
2) Any two admissible metrics are quasi-isometric.
Recall the manifold with corners with iterated fibration structure X, which is the resolution of X. Its interior is canonically identified with reg ( X), and we identify these without comment. Let x α be a global defining function for the boundary hypersurface H α of X (so H α = {x α = 0}); the total boundary defining function of X is, by definition,
ρ = α∈A x α .
If g is an iterated edge metric on reg ( X), then set
(3.2) g = ρ −2 g.
It is not hard to check that this metric is complete.
3.2. The Witt condition. Cheeger's Hodge theorem on Witt spaces.
In this paper, we consider only orientable Witt spaces, which are defined as follows.
Definition 6.
A pseudomanifold X is a Witt space if, for some (and hence any) stratification, all links of even dimension have vanishing lower middle perversity intersection homology in middle degree, i.e.,
Y ∈ S, dim L Y = f Y even =⇒ IH fY /2 m (L Y ) = 0.
It is a theorem that on a Witt space X the lower and upper middle perversity intersection homology groups are equal up to isomorphism: IH * m ( X) = IH * m ( X). A famous result concerning the L 2 cohomology of Witt spaces is due to Cheeger. (2) ( X) the L 2 maximal Hodge cohomology. Then
(3.3) H * (2) ( X) = H * (2) ( X) = IH * m ( X, C).
with m denoting either the upper or lower middle perversity.
In particular, if Y is a stratum with link L Y , and f Y = dim L Y is even, then (3.4) H fY /2 (2) (L Y ) = H fY /2 (2) (L Y ) = 0.
Iterated edge vector fields and operators
On a closed manifold, L 2 and Sobolev spaces are defined using a Riemannian metric but the spaces themselves are metric-independent. A differential operator induces a bounded map between suitable ones of these spaces, and ellipticity guarantees that this map is Fredholm. All of this fails when the manifold is not closed, and in this section we describe some of what is true for iterated edge metrics.
The space X := reg ( X) with complete metric g is an example of what is called a Riemannian manifold with bounded geometry. There are natural classes of L 2 and Sobolev spaces on any such space, as well as a class of 'uniform' differential operators, which induce bounded maps between these function spaces. There is also a calculus of uniform pseudo-differential operators which contains parametrices of uniform elliptic operators, and which can be used to prove certain uniform elliptic regularity results. Using that X compactifies to X, we can also define weighted L 2 and Sobolev spaces in this setting, and the uniform calculus gives some results for operators mapping between these as well. This uniform calculus does not establish that these mappings are Fredholm, and indeed, that requires more delicate arguments.
In this section we describe these ideas and explain how they can be applied to the de Rham operator of the edge iterated metric g. The uniform pseudodifferential calculus can also be used to obtain a parametrix even after twisting by a bundle of projective finitely generated modules over a C * -algebra.
Edge vector fields on X.
Associated to the complete metric g on X is the space of 'iterated edge' vector fields
(4.1) V ie = {V ∈ C ∞ ( X, T X) : X ∋ q → g q (V, V ) ∈ R + is bounded}.
In the notation of §3, on a neighbourhood of the form W q1,...,qs , this is locally spanned over C ∞ ( X) by vector fields of the form
r 1 . . . r s−1 ∂ r1 , r 1 . . . r s−1 ∂ y (1) , r 1 . . . r s−2 ∂ r2 , r 1 . . . r s−2 ∂ y (2) , . . . , ∂ y (s) .
It is easy to see that V ie forms a locally finitely generated, locally free Lie algebra with respect to the usual bracket on vector fields; furthermore, Swan's theorem shows that there is a vector bundle ie T X over X whose space of sections is V ie ,
(4.2) C ∞ ( X, ie T X) = V ie .
This bundle ie T X coincides with the usual tangent bundle T X over the interior of X and is isomorphic to T X, though there is no canonical isomorphism. It is easy to see that g defines a metric on ie T X. Proof. Recall the theorem of Gordon-de Rham-Borel, which states that a manifold is complete if and only if it admits a nonnegative, smooth, proper function with bounded gradient. For this metric g, such a function is − log(ρ), where ρ is the total boundary defining function. To prove that g has bounded geometry one must check that the curvature tensor of g, and its covariant derivatives, are bounded and that the injectivity radius of g has a positive lower bound. The former follows from the compactness of X, and the latter can be shown as in [2].
The set of ie-differential operators is the enveloping algebra of V ie ; i.e., it consists of linear combinations (over C ∞ ( X)) of finite products of elements of V ie . We denote by Diff k ie (X) the subset of differential operators that have local descriptions involving products of at most k elements of V ie . If E and F are vector bundles over X, then the space of ie-differential operators acting between sections of E and sections of F is defined similarly, by taking linear combinations over C ∞ ( X, Hom(E, F )).
We define Sobolev spaces for ie metrics by
H 0 ie (X) = L 2 ie (X) = L 2 (X, dvol( g)) H k ie (X) = {u ∈ L 2 ie (X) : Au ∈ L 2 ie (X), for every A ∈ Diff k ie (X)}, k ∈ N then define H t
ie (X) using Calderón interpolation for t ∈ R + and duality for t ∈ R − . Sobolev spaces for sections of bundles over X are defined similarly.
We will also allow for operators to act between sections of certain bundles of projective finitely generated modules over a C * -algebra; see [55] for the basic definitions. We assume that we have a continuous map r 0 : X → BΓ which extends continuously to r : X → BΓ where Γ is a countable, finitely generated, finitely presented group. This determines a Γ-covering, X ′ → X; and we will denote by C * r Γ the corresponding bundle, over X, of free left C * r Γ-modules of rank one:
(4.3) C * r Γ := C * r Γ × Γ X ′ .
Observe that this bundle induces, after pull back by the blowdown map X → X, a bundle on X (for which we keep the same notation). Given vector bundles E and F over X of rank k and ℓ, we define bundles E and F over X by tensoring E and F by C * r Γ; we obtain in this way bundles of projective finitely generated C * r Γ-modules of rank k and ℓ . We shall briefly refer to E and F as C * r Γ-bundles. An iterated edge differential operator acting between sections of E and F is defined as above, but allowing the coefficients to be C * r Γ-linear. The space of such operators will be denoted Diff * ie,Γ (X; E, F ). Finally, we denote by H t ie,Γ (X; E) the corresponding Sobolev C * r Γ-module, see [45].
Uniform pseudodifferential operators.
We showed above that ie metrics have bounded geometry. This allows us to use the calculus of uniform pseudo-differential operators as described in the work of Meladze-Shubin (see [41] and [31]). We single out the space BC ∞ (X) of functions which are uniformly bounded with uniformly bounded derivatives of all orders. Smooth functions on X are in BC ∞ (X), but the latter space is larger since general elements are not smooth at the boundary faces of X. A vector bundle over X is said to be a bundle of bounded geometry if it has trivializations whose transition functions are (matrices with entries) in BC ∞ (X). Vector bundles that extend smoothly to X have bounded geometry.
The spaces of operators Diff * B (X; E, F ) and, more generally, Diff * B,Γ (X; E, F ), are defined by requiring the coefficients to be in BC ∞ . These spaces contain Diff * ie (X; E, F ) and Diff * ie,Γ (X; E, F ), respectively. Next, the bounded geometry of (X, g) implies that it is possible to find a countable cover of X by open sets, each of which are normal coordinate charts for the complete metric g and which all have fixed radius ε > 0. Calling these charts U ε (ζ i ), then it is also possible to arrange that U 2ε (ζ i ) has uniformly bounded, finite multiplicity as a cover of X. We can then choose partitions of unity φ i , φ i subordinate to {U 2ε (ζ i )} and {U ε (ζ i )} respectively such that φ i , φ i have bounded derivatives uniformly in i, and such that φ i = 1 on supp φ i . These functions can be used to transplant constructions from R n to X.
We next recall how to transfer pseudodifferential operators from R n . Let E and F be vector bundles over X, and denote by d = d g the distance function associated to the complete metric g. An operator A :
C ∞ c (X; E) → C ∞ c (X; F ) is called a uniform pseudodifferential operator of order s ∈ R, A ∈ Ψ s B (X; E, F ), if its Schwartz kernel K A ∈ C −∞ (X 2 ;
Hom(E, F )) satisfies the following properties.
i) For some C A > 0,
K A (ζ, ζ ′ ) = 0 if d(ζ, ζ ′ ) > C A ,
ii) For every δ > 0, and any multi-indices α, β there is a constant C αβδ > 0 such that
|D α ζ D β ζ ′ K A (ζ, ζ ′ )| ≤ C αβδ , whenever d(ζ, ζ ′ ) > δ.
iii) For any i, and using the normal coordinate chart to identify U 2ε (ζ i ) with B 2ε (0) in R n , φ i Aφ i is a pseudodifferential operator of order s in B 2ε (0), whose full symbol σ satisfies the usual symbol estimates uniformly in i,
D α ζ D γ ξ σ( φ i Aφ i )(ζ, ξ) ≤ C αβγ (1 + |ξ| 2 g ) 1 2 (s−|γ|) ;
here |ξ| g is the norm of ξ ∈ T * ζ X with respect to g. We always assume that the symbols are (one-step) polyhomogeneous. Uniform pseudo-differential operators form an algebra. There is a well defined principal symbol map, with values in BC ∞ (S * X, hom(π * E, π * F )). Ellipticity is defined in a natural way (one requires the principal symbol to be uniformly invertible, i.e. invertible with inverse in BC ∞ ). The principal symbol σ(P ) of a uniform pseudodifferential operator P is a section of ie T * X (the bundle dual to ie T X) restricted to X. In general, σ(P ) does not extend to be a smooth section of ie T * X → X.
For a bundle of bounded geometry E and s ∈ R, define the B-Sobolev space
(4.4) H s B (X; E) = {u ∈ C −∞ (X; E) : φ i u ∈ H s (R n ; E) with norm bounded uniformly in i}.
The same definition holds for C * r Γ-bundles and we denote by H s B,Γ (X; E), the corresponding C * r Γ-module. Uniform pseudodifferential operators extend to bounded operators between B-Sobolev spaces.
If a map r : X → BΓ is given, then we can define uniform pseudo-differential operators between sections of E and sections of F by combining the above definition and the classic construction of Mishchenko and Fomenko; we denote by Ψ * B,Γ (X; E, F ) the corresponding algebra. Notice that the principal symbol is in this case a C * r Γ-linear map between the lifts of E and F to the cotangent bundle. The intersection over s ∈ R of the Ψ s B,Γ (X; E, F ) is denoted Ψ −∞ B,Γ (X; E, F ) and consists of smoothing operators whose integral kernel in X × X is in BC ∞ .
Elements of the uniform calculus also define bounded maps between weighted C * r Γ-Sobolev spaces. Let ρ be the total boundary defining function for X.
Lemma 4.2. If A ∈ Ψ s B,Γ (X; E, F ), then for any a, t ∈ R, A induces a bounded operator A : ρ a H t ie,Γ (X; E) → ρ a H t−s ie,Γ (X; F ). Proof. It is enough to check that ρ −a Aρ a ∈ A ∈ Ψ s B,Γ (X; E, F ) for any a. The integral kernel of ρ −a Aρ a is ρ(ζ) ρ(ζ ′ ) a K A (ζ, ζ ′ )
and the lemma follows by noting that if ρ(ζ)
ρ(ζ ′ )
a is a bounded smooth function on the support of K A .
An important property of the uniform pseudodifferential calculus is that it has a symbolic calculus. By standard constructions, this implies that any elliptic element in Diff k B,Γ (X; E, F ) has a symbolic parametrix, i.e. an inverse modulo smoothing operators. In particular, using the above Proposition, we see that an elliptic ie operator A ∈ Diff k ie,Γ (X; E, F ) has a symbolic parametrix
Q ∈ Ψ −k B,Γ (X; F , E) s.t. Id E −QP ∈ Ψ −∞ B,Γ (X; E), Id F −P Q ∈ Ψ −∞ B,Γ (X; F ).
The symbolic calculus also yields the standard characterization of Sobolev spaces. For instance, if N ∈ N, then
H N B (X) = {u ∈ C −∞ (X) : Au ∈ L 2 (X) for all A ∈ Diff N B (X)} = {u ∈ C −∞ (X) : Au ∈ L 2 (X) for some uniformly elliptic A ∈ Diff N B (X)}; in fact, if A ∈ Diff N B (X)
is uniformly elliptic, then H N B (X) equals the maximal domain of A as an unbounded operator on L 2 (X). This characterization, applied to an elliptic operator A ∈ Diff N ie (X), shows that H N ie (X) = H N B (X). Using Calderon interpolation and duality, we see that H t ie (X) = H t B (X) for all t ∈ R, and the same is true for sections of bundles of bounded geometry and the corresponding
C * r Γ-bundles. 4.3. Incomplete iterated edge operators. The set of incomplete iterated edge differential operators, Diff * iie,Γ (X; E, F ) is de- fined in terms of Diff * ie,Γ (X; E, F ) by Diff k iie,Γ (X; E, F ) = ρ −k Diff k ie,Γ (X; E, F ), where ρ = x 0 · · · x m−1 .
As an operator between weighted L 2 spaces with appropriate different weights, an operator A ∈ Diff k iie,Γ (X; E, F ) is unitarily equivalent to an iterated edge operator. Thus, for instance, for any a ∈ R, A defines an unbounded operator A : ρ a L 2 ie,Γ (X; E) → ρ a−k L 2 ie,Γ (X; F ) which has a unique closed extension whose domain is ρ α H k ie,Γ (X; E); moreover, A defines bounded operators ρ a H t ie,Γ (X; E) → ρ a−k H t−k ie,Γ (X; F ) for every a and t ∈ R. However, it is the more complicated behavior of A as an unbounded operator (4.5) A : ρ a L 2 ie,Γ (X; E) → ρ a L 2 ie,Γ (X; F ) that we will be concerned with. We point out that the operator (4.5) is unitarily equivalent to the unbounded operator A = ρ k/2 Aρ k/2 : ρ a−k/2 L 2 ie,Γ (X; E) → ρ a+k/2 L 2 ie,Γ (X; F ), Since A ∈ Diff * ie,Γ (X; E, F ), this shows that the study of incomplete iterated edge operators acting on a fixed Hilbert space is the same as the study of complete ie-operators acting between different Hilbert spaces.
We point out that the L 2 spaces of the incomplete iterated edge metric g and the associated complete ie metric g = ρ −2 g are related by L 2 ie,Γ (X, E) = ρ n/2 L 2 iie,Γ (X, E) with n equal to the dimension of X, so switching between them only involves a shift of the weight. Similarly, we introduce the spaces H t iie,Γ (X; E) for t ∈ R by H t ie,Γ (X; E) = ρ n/2 H t iie,Γ (X; E). Thus, for instance, if N ∈ N then H N iie,Γ (X, E) is the set of elements u ∈ L 2 iie,Γ (X, E) such that for any vector fields V 1 , . . . ,
V p ∈ V ie where p ≤ N, we have V 1 . . . V p u ∈ L 2
iie,Γ (X, E).
We say that A ∈ Diff k iie,Γ (X; E, F ) is elliptic if A = ρ k A is an elliptic ie operator. Elliptic ie operators always have a symbolic parametrix (see §4.2). A symbolic parametrix Q for A yields a symbolic parametrix Q = ρ k/2 Qρ k/2 for A. Recall that a continuous adjointable C * r Γ-linear operator K is called C * r Γ-compact if both K and K * are uniform limits of sequences of C * r Γ-linear operators whose ranges are finitely generated C * r Γ−modules. As is well-known, since smoothing operators are not necessarily C * r Γ-compact, a symbolic parametrix is generally not enough to determine when an operator is C * r Γ-Fredholm, so one also needs to know about the behavior at the boundary.
However, the uniform calculus does establish elliptic regularity in the sense that, whenever B ∈ Diff k ie,Γ (X; E, F ) is elliptic and a ∈ R, we have
(4.6) u ∈ ρ a L 2 iie,Γ (X, E), Bu ∈ ρ a L 2 iie,Γ (X, F ) =⇒ u ∈ ρ a H N iie,Γ (X, E).
The de Rham operator.
We are interested in analyzing the de Rham operator of an iie metric,
ð dR = d + δ : Ω * X → Ω * X.
As with the tangent bundle, it is convenient to replace the bundle of forms Ω * (X) = C ∞ (X, Λ * (T * X)) with the bundle of iie-forms,
iie Ω * (X) = C ∞ (X, Λ * ( iie T * X)),
where iie T * X → X is the rescaled bundle (cf. [42,Chapter 8]) defined by
C ∞ ( X, iie T * X) = ρC ∞ ( X, ie T * X).
We set iie Λ * X = Λ * ( iie T * X), and we have ð dR ∈ Diff 1 iie (X; iie Λ * (X), iie Λ * (X)) as we now explain.
First note that whether or not ð dR is an element of Diff 1 iie (X; iie Λ * (X), iie Λ * (X)) can be checked locally in coordinate charts. There is nothing to check in the interior of the manifold. Then, with the notations of §3, we consider a distinguished neighborhood W of a point of a stratum Y. Thus W is diffeomorphic to B × C(Z) where B is an open subset of Y which is diffeomorphic to a vector space and C(Z) is the cone whose base Z is a stratified space. The 'radial' coordinate of the cone will be denoted by x.
As in §3, the fibration over B extends to W ,
Z × [0, 1) x − W φ − → B,
and using x and a choice of connection for this fibration we can write
T * X W = dx ⊕ T * Y ⊕ T * Z.
With respect to this splitting the metric g restricted to W has the form g = dx 2 + φ * g Y + x 2 g Z and the differential forms on X can be decomposed as
(4.7) Λ * X = (Λ * Y ∧ Λ * Z) ⊕ dx ∧ (Λ * Y ∧ Λ * Z) iie Λ * X = (Λ * Y ∧ x N Λ * Z) ⊕ dx ∧ (Λ * Y ∧ x N Λ * Z)
where N is the 'vertical number operator', i.e., the map given by multiplication by k when restricted to forms of vertical degree k. This allows us to split the exterior derivative into
d = e dx ∂ x ⊕ d Y ⊕ d Z
where e dx denotes the exterior product by dx and correspondingly
δ = ⋆ −1 e dx ∂ x ⋆ ⊕ ⋆ −1 d Y ⋆ ⊕ ⋆ −1 d Z ⋆ = ⋆ −1 e dx ∂ x ⋆ ⊕δ Y x ⊕ δ Z x
where the x-dependence in δ Y x and δ Z x comes from the x-dependence of the Hodge star operator, ⋆. A straightforward computation shows that with respect to the splitting (4.7) of iie Λ * X, (and with f = dim Z),
(4.8) ð dR = 1 x (d Z + δ Z x ) + d Y + δ Y x − ⋆ −1 ∂ x ⋆ − 1 x (f − N) ∂ x + 1 x N − 1 x (d Z + δ Z x ) − d Y − δ Y x .
As in [26, (19)] one can write this in terms of operators related to the fibration, however for our purposes it is more important to point out that the leading order term with respect to x (as an iie operator) is given by
(4.9) ð dR ∼ 1 x ð Z dR + ð Y dR −∂ x − 1 x (f − N) ∂ x + 1 x N − 1 x ð Z dR − ð Y dR .
where f denotes the dimension of Z, ð Y dR and ð Z dR are the de Rham operators of φ * g Y x=0 and g Z x=0 , respectively. In effect, because of the weighting of the vertical forms, the Hodge star operator is asymptotically acting like the Hodge star operator of the product metric at {x = 0}.
By induction on the depth of the stratification and using (4.9) one proves without difficulties the following: We are also interested in the behaviour of ð dR after twisting to get C * -algebra coefficients. Thus we assume, as before, that we have a continuous map r : X → BΓ We compose r with the blow-down map β and we pull-back the universal bundle EΓ to X using f • β. We obtain a Galois Γ-covering X ′ over X and the associated bundle C * r Γ → X, with C * r Γ := C * r Γ × Γ X ′ . We restrict C * r Γ to X. Endowing C * r Γ × X ′ , as a trivial bundle over X ′ , with the trivial connection induces a (non-trivial) flat connection on the bundle C * r Γ → X; we also obtain a flat connection on the restriction of C * r Γ to X (and it is obvious that this connection will automatically extend to X). Using the latter connection we can define directly ð dR , the twisted de Rham operator on the sections of the vector bundle iie Λ * Γ (X) = iie Λ * X ⊗ C * r Γ. By construction ð dR ∈ Diff * iie,Γ , i.e. ρ ð dR is an element in Diff * ie,Γ .
Inductive analysis of the signature operator
In this section we analyze the behavior of the de Rham operator near the singular part of X. This is done inductively. The base case is that of a closed manifold, which is classical. Stratifications of depth one are analyzed in the work of Hunsicker and the third author [26], where the relationship between intersection cohomology and Hodge cohomology is treated in detail. Our results for depth one stratifications is implicitly contained in [26]. However, the treatment in [26] relies heavily on the edge calculus [40] which allows refined results, such as finding conormal representatives of cohomology classes. Though we cannot use the edge calculus directly, we proceed by adapting certain arguments from [40] to our context. More precisely, we define a model for this operator at each point of a singular stratum and then we establish that these model operators are invertible when acting on the appropriate Sobolev spaces. Taken together, ellipticity and this asymptotic invertibility are enough to establish the Fredholm properties we seek.
The main advantage of the de Rham operator over an arbitrary iie operator lies in (4.9). Indeed this shows that, at a given point q on the boundary, the leading order behavior of ð dR involves the fibre Z over q only through its de Rham operator ð Z dR . To take advantage of this structure we multiply this operator by a (symmetrically distributed) power of the radial distance x to the highest depth stratum Y . Since this is closely related to the de Rham operator for the metric x −2 g, which we regard as a 'partial completion' of g (i.e. we have made it complete near Y , but the link Z of the associated cone bundle with its induced metric remains incomplete. This allows us to set up an inductive scheme.
The partial completion of the de Rham operator.
Recall that (4.9) was written in a distinguished neighborhood W of a point of a stratum Y and that W is diffeomorphic to B × C(Z) where B is an open subset of Y diffeomorphic to a vector space and C(Z) is the cone with smoothly stratified link Z. The 'radial' coordinate of the cone is still denoted x, which we identify with one the boundary defining functions x j and thereby extend globally to X. To take advantage of the structure of the de Rham operator in W , as it appears in (4.9), we define the 'partial conformal completion' of the signature operator
D 0 = x 1/2 ð dR x 1/2 .
The advantage of using x 1/2 ð dR x 1/2 over, say, xð dR is that the former is symmetric as an operator
x −1/2 L 2 iie,Γ (X, iie Λ * Γ (X)) → x 1/2 L 2 iie,Γ (X, iie Λ * Γ (X))
(with respect to the natural pairing between the spaces on the right and left here), since ð dR is a symmetric operator on L 2 iie,Γ (X; iie Λ * Γ (X)) with core domain C ∞ c . To analyze ð dR it is useful to consider the operator it induces on various weighted L 2 spaces. For later use we point out first that ð dR satisfies
(5.1) ð dR (x a v) = [ð dR , x a ]v + x a ð dR v = x a [ae( dx x ) − ai( dx x ) + ð dR ]v,
where e and i denote exterior and interior product respectively, and, second, that we have a unitary equivalence of unbounded operators 1
ð dR : x a L 2 iie,Γ (X, iie Λ * Γ (X)) → x a L 2 iie,Γ (X, iie Λ * Γ (X)) ↔ D a = x 1/2−a ð dR x 1/2+a 0 : x −1/2 L 2 iie,Γ (X, iie Λ * Γ (X)) → x 1/2 L 2 iie,Γ (X, iie Λ * Γ (X)
). In order to adapt arguments from [40] it is more natural to work with the operator x 1/2−a ð dR x 1/2+a 0 as an unbounded operator from the space x −1/2 L 2 iie,Γ (X, iie Λ * Γ (X)) to itself. Thought of in this way, we denote it as P a ,
(5.2) P a : x −1/2 L 2 iie,Γ (X, iie Λ * Γ (X)) → x −1/2 L 2 iie,Γ (X, iie Λ * Γ (X)
). Our analysis of ð dR will proceed in two steps: in the first step we will analyze the behavior of P a by adapting two model operators from [40] -the normal operator and the indicial family. Then, in the second step, we will use the information gleaned about P a to analyze ð dR .
Remark. These two steps can be thought of in the following way. We first analyze x 1/2 ð dR x 1/2 as a partially complete edge operator on W ; complete in the (x, y) variables with values in iie-operators on Z. Then, as a second step, we analyze it as an incomplete edge operator in the (x, y) variables with values, again, in iieoperators on Z.
The normal operator of P a .
Recall that every point q ∈ Y has a neighborhood W which we identify with the product of U × C(Z), where U is a neighborhood of the origin in R b ∼ = T q Y . If this neighborhood is small enough that iie Λ * (X) W can be identified with the pull-back of some vector bundle over Z and similarly for iie Λ * Γ (X)| W , then we call W a basic neighborhood. In such a W , let us fix smooth nonnegative cutoff functions χ and χ, both independent of the Z variables, with supports in W and equaling one in a neighborhood of q, and such that χχ = χ. We refer to W , ψ, χ, χ as a basic setup at q ∈ Y .
We can identify a basic neighborhood W with a subset of the product of Z with T q Y + ∼ = R + s × R b u and use this identification to model the operator P a near q by an operator on Z × T q Y + , the normal operator of P a at q ∈ Y . Notice that the bundles iie Λ * (X) W , iie Λ * Γ (X)| W as pull-backs of bundles over Z, extend naturally to Z × T q Y + , and that the dilation maps R t : T q Y + → T q Y + for any t > 0 preserve the space of sections of these bundles. Definition 7. The normal operator N q (P a ) is the operator whose action on any
u ∈ C ∞ c (Z × T q Y + , iie Λ * Γ (Z × T q Y + )) is given by N q (P a )u = lim r→0 R * r (ψ −1 ) * χ P a ψ * χR * 1/r u.
Thus in local coordinates (s, y, z) the action of the normal operator of P a on a section u is obtained by evaluating u at (s/r, y/r, z), applying P a , dilating back by a factor of r, and then letting r → 0. It is easy to see that this procedure takes a vector field of the form a(s, y, z)(s∂ s ) + b(s, y, z)(s∂ y ) to the vector field 1 Note that in [26], for a stratification of depth one, Da denotes the de Rham operator of the complex (x a L 2 iie , d) while here Da denotes the de Rham operator of the complex (L 2 iie , d) as an operator on x a L 2 iie . a(0, 0, z)(s∂ s ) + b(0, 0, z)(s∂ y ), while for a vertical vector field V , this procedure returns V s=0,y=0 . In fact, it is easy to see that this procedure replaces the metric
g W = g U (x, y) + dx 2 + x 2 g Z (x, y, z)
which is a submersion metric with respect to the projection U × C(Z) → U, with the product of an iie metric on C(Z) and the flat metric on U,
g Z×TqY + = g U (0, 0) + ds 2 + s 2 g Z (0, 0, z).
It follows that any natural operator associated to g iie is taken by this procedure to the corresponding natural operator of g Z×TqY + -in particular this is true for ð dR .
Lemma 5.1. The normal operator of P a at q ∈ Y is equal to s 1/2−a ð dR s 1/2+a where ð dR is the de Rham operator of the metric g Z×TqY + . Thus in local coordinates,
(5.3) N q (P a ) = ð Z dR + sð R b dR −s∂ s − (f 0 − N + a + 1/2) s∂ s + N + a + 1/2 −ð Z dR − sð R b dR .
Remark. As explained above, this expression follows by naturality of the de Rham operator. Alternately, one can compute (5.3) directly from (4.9).
Localizing the maximal domain.
The following lemma will allow us to "localize the maximal domain" of ð dR near the singular locus.
Lemma 5.2. Let W , ψ, χ, χ be a basic setup at q ∈ Y . Let u ∈ x −1/2 L 2 iie,Γ (X; iie Λ * Γ (X)) be such that P a u ∈ x 1/2 L 2 iie,Γ (X; iie Λ * Γ (X)). Then χu ∈ x −1/2 L 2
iie,Γ (X; iie Λ * Γ (X)) and P a (χu) ∈ x 1/2 L 2 iie,Γ (X; iie Λ * Γ (X)). Proof. Clearly P a (χu) = χ(P a u) + [P a , χ]u, and, since χ is independent of the Zvariables, (4.9) allows us to see that [P a , χ] = σ(P a )(dχ) = xH where H is a multiplication operator by smooth bounded functions. Since u ∈ x −1/2 L 2 iie,Γ (X; iie Λ * Γ (X)) we see that [P a , χ]u ∈ x 1/2 L 2 iie,Γ (X; iie Λ * Γ (X)), which establishes the lemma.
Proposition 5.3. Let u ∈ x −1/2 L 2 iie (X; iie Λ * Γ (X)
) with compact support included in W and such that χ = 1 on supp u. Then P a u ∈ x 1/2 L 2 iie (X; iie Λ * Γ (X)) if and only
if N q (P a )(u • ψ −1 ) ∈ s 1/2 L 2 iie (Z × T q Y + , iie Λ * Γ (Z × T q Y + )
). Proof. We prove only one implication, the other one is similar. Since we work in the distinguished chart W , we may identify u with u • ψ −1 .
Let ρ denote a total boundary defining function. The operator ρ x P a is an elliptic ie differential operator, so elliptic regularity (4.6) yields u ∈ x −1/2 H 1 iie (X; iie Λ * Γ (X)). We observe that, in the expression (4.8),
x(d Y +δ Y x ) sends x −1/2 H 1 iie (X; iie Λ * Γ (X)) into x 1/2 L 2
iie (X; iie Λ * Γ (X)) and a similar observation is true for sð R b 0 dR , so using formulas (4.8) and (5.3), we get P a u − N q (P a )(u • ψ −1 ) ∈ s 1/2 L 2 iie , which proves the lemma. 5.4. Injectivity of N q (P a ). We take as an inductive hypothesis that the signature operator on Z is self adjoint with discrete spectrum. We make two further assumptions:
(5.4) a) Spec(ð Z dR ) ∩ (−1, 1) ⊆ {0}, b)
If k = f0 2 then H k (2) (Z) = 0.
By Theorem 3.4, b) is a topological condition on Z.
Proposition 5.4. 1) There exists a (rigid) iterated edge metric (cf Theorem 3.1) such that condition a) is satisfied on all links in X. Such a metric will be called adapted (rigid) iterated edge.
2) Any two adapted (rigid) iterated edge metrics are homotopic within the class of adapted (rigid) iterated edge metrics.
Proof. 1). Observe that condition a) can be arranged to hold along a given stratum by scaling the metric on Z. To check that this can be done coherently for all links in the Witt space X, one must retrace the proof of Theorem 3.1 concerning the existence of rigid iterated edge metrics. Following the inductive step there, we see that we can scale the metric on the link of the highest depth stratum so that a) is satisfied without disturbing the corresponding property for all the links of lower depth strata.
2). Retrace the proof of Proposition 3.2 along the lines of the previous proof.
Lemma 5.5. Let a ∈ (0, 1) and assume the conditions (5.4) and that Theorem 1.1 has been proven for Z. Then N (P a ) acting on
s −1/2 L 2 iie (Z × T q Y + , iie Λ * Γ (Z × T q Y + )) is injective on its maximal domain. Proof. Define R = s −1/2 N q (P 0 )s −1/2 (this is effectively N q (ð dR )), so that R = 1 s ð Z dR + ð R b dR −∂ s − 1 s (f 0 − N) ∂ s + 1 s N − 1 s ð Z dR − ð R b dR Since N q (P a ) = s 1/2−a Rs 1/2+a , if u solves N q (P a )u = 0 then v = s f 0 2 +a u solves Rs − f0−1 2 v = 0. Clearly u ∈ s −1/2 L 2 iie (T q ) precisely when we have v ∈ s f 0 −1 2
+a L 2 iie (T q ), so it suffices to solve
Rs − f0−1 2 v = 0, v ∈ s f 0 −1 2 +a L 2 iie (s f dsdydz, iie Ω) = s − 1 2 +a L 2 iie (dsdydz,
iie Ω). The advantage of this formulation is that v is also in the null space of (5.5)
s f0−1 2 (sR)s − f0−1 2 = ð Z dR + sð R b dR −s∂ s + N − f0 2 − 1 2 s∂ s + N − f0 2 + 1 2 −ð Z dR − sð R b dR and s f0−1 2 (s 2 R 2 )s − f0−1 2 = K 1 −2d Z −2δ Z K −1 , where K ℓ = ∆ Z + s 2 ∆ R b − (s∂ s ) 2 + (N − f0 2 + ℓ 2 ) 2 .
To analyze these systems, we point out that L 2 forms on Z satisfy a strong Kodaira decomposition, i.e., every L 2 form on Z can be written in a unique way as a sum of a form in the image of d Z , a form in the image of δ Z and a form in the joint kernel of d Z and δ Z . As explained in [26, §2] weak Kodaira decompositions are a general feature of Hilbert complexes. Inductively, we are assuming that d + δ is essentially self-adjoint and that its closed extension has closed range; this implies, see [26,Proposition 4.6], that d has a unique closed extension and that this extension has closed range (for instance, because d coincides with d + δ on (ker d) ⊥ ). Hence the weak Kodaira decomposition is a strong Kodaira decomposition.
The upshot is that if v = (α, β), then we can write
α = d Z α 1 + δ Z α 2 + α 3 , α 1 ∈ (ker d Z ) ⊥ , α 2 ∈ (ker δ Z ) ⊥ , α 3 ∈ ker d Z ∩ ker δ Z
and similarly for β. Inserting this decomposition into s
f 0 −1 2 (sR)s − f 0 −1 2 v = 0 and using d Z N = (N − 1)d Z , δ Z N = (N + 1)δ Z yields d Z (δ Z α 2 + sð R b dR α 1 − s∂ s β 1 + (N − f0 2 + 1 2 )β 1 ) +δ Z (d Z α 1 + sð R b dR α 2 − s∂ s β 2 + (N − f0 2 − 3 2 )β 2 ) +sð R b dR α 3 − s∂ s β 3 + (N − f0 2 − 1 2 )β 3 = 0 d Z (−δ Z β 2 − s∂ R b dR β 1 + s∂ s α 1 + (N − f0 2 + 3 2 )α 1 ) +δ Z (−d Z β 1 − sð R b dR β 2 + s∂ s α 2 + (N − f0 2 − 1 2 )α 2 ) −sð R b dR β 3 + s∂ s α 3 + (N − f0 2 + 1 2 )α 3 =
and hence another application of the Kodaira decomposition shows that
δ Z α 2 + sð R b dR α 1 − s∂ s β 1 + (N − f0 2 + 1 2 )β 1 = 0 d Z α 1 + sð R b dR α 2 − s∂ s β 2 + (N − f0 2 − 3 2 )β 2 = 0 −δ Z β 2 − s∂ R b dR β 1 + s∂ s α 1 + (N − f0 2 + 3 2 )α 1 = 0 −d Z β 1 − sð R b dR β 2 + s∂ s α 2 + (N − f0 2 − 1 2 )α 2 = 0 (5.6) sð R b dR α 3 − s∂ s β 3 + (N − f0 2 − 1 2 )β 3 = 0 −sð R b dR β 3 + s∂ s α 3 + (N − f0 2 + 1 2 )α 3 = 0. (5.7)
We also insert the Kodaira decomposition of v into s
f 0 −1 2 (s 2 R 2 )s − f 0 −1 2 v, and since d Z K ℓ = K ℓ−2 d Z , δ Z K ℓ = K ℓ+2 δ Z , this yields d Z (K 3 α 1 − 2δ Z β 2 ) + δ Z (K −1 α 2 ) + K 1 α 3 = 0, d Z (K 1 β 1 ) + δ Z (K −3 β 2 − 2d Z α 1 )K −1 β 3 = 0.
Once again another application of the Kodaira decomposition shows that
K 3 α 1 = 2δ Z β 2 (5.8) 2d Z α 1 = K −3 β 2 (5.9) K −1 α 2 = K 1 α 3 = K 1 β 1 = K −1 β 3 = 0. (5.10)
We are looking for solutions of (5.6)-(5.10) in s − 1 2 +a L 2 iie (dsdydz, iie Ω). Let us find the null space of K ℓ . Conjugating by the Fourier transform in R b (with dual variable η to y) and introducing the new variables t = s|η|, η = η |η| , takes K ℓ to
K ℓ = ∆ Z + t 2 − (t∂ t ) 2 + (N − f0 2 + ℓ 2 ) 2 .
By assumption ∆ Z has discrete spectrum and, since ∆ Z commutes with K ℓ , we can restrict to the λ eigenspace of ∆ Z ,
K ℓ,λ = λ + t 2 − (t∂ t ) 2 + (N − f0 2 + ℓ 2 ) 2 .
The null space of this operator can be described directly in terms of Bessel functions of an imaginary argument A I ν (t) + B K ν (t), ν = λ + (N − f0 2 + ℓ 2 ) 2 , t ∈ R + The functions I ν increase exponentially with t, so to stay in a (polynomially weighted) L 2 space, we must have A = 0. The functions K ν decrease exponentially with t as t → ∞, while
K ν (t) ∼ t −|ν| if ν = 0 − log t if ν = 0 as t → 0.
We are interested in avoiding K ν ∈ t a− 1 2 L 2 ( dt), which means we need to have
1 ≤ |ν| + a = a + λ + (N − f0 2 + ℓ 2 ) 2 , for all a > 0 hence 1 ≤ λ + (N − f0 2 + ℓ 2 ) 2
If λ = 0, then our assumption is that λ ≥ 1, so this is automatic. If λ = 0 then we are looking for elements in the null space of K ℓ that are also in the null space of ∆ Z , so this corresponds to α 3 and β 3 . From (5.7) we see that α 3 = 0 if and only if β 3 = 0 : indeed if α 3 = 0 then β 3 = s N− f0 2 − 1 2 F with F independent of s, but this is never in a polynomially weighted L 2 in s, and similarly if β 3 = 0. The same reasoning shows that ð R b dR α 3 = 0 if and only if β 3 = 0 and viceversa. Thus, since α 3 is in the null space of K 1 and β 3 is in the null space of K −1 , to avoid elements of the null space with λ = 0 we need to have either 1 ≤ |N − f0 2 + 1 2 | or 1 ≤ |N − f0 2 − 1 2 |. This is automatic unless N = f0 2 , but this case does not happen since by assumption there are no middle degree harmonic forms on Z.
This implies, from (5.10), that α 2 , α 3 , β 1 , and β 3 do not contribute to the null space of N q (P a ) for a > 0, and we only need to rule out α 1 and β 2 . First note that if α 1 = 0, then from (5.9) K −3 β 2 = 0, but since K −3 does not have non-zero null space in s − 1 2 +a L 2 iie (dsdydz, iie Ω), this implies β 2 = 0. Similarly β 2 = 0 implies α 1 = 0.
Next, substituting (5.9) into the second equation of (5.6) we have
K −3 β 2 + 2sð R b dR α 2 − 2s∂ s β 2 + 2(N − f0 2 − 3
2 )β 2 = 0 Applying K −1 s −1 kills the second term by (5.10), so
K −1 s −1 (K −3 − 2s∂ s + 2(N − f0 2 − 3 2 ))β 2 = 0, but K −3 − 2s∂ s + 2(N − f0 2 − 3 2 ) = ∆ Z + s 2 ∆ R b − (s∂ s ) 2 + (N − f0 2 − 3 2 ) 2 − 2s∂ s + 2(N − f0 2 − 3 2 ) = s −1 K −1 s,
so this says K −1 (s −2 K −1 s)β 2 = 0. Since we know that K −1 does not have non-zero null space in s − 1 2 +a L 2 iie (dsdydz, iie Ω), we must have s −2 K −1 sβ 2 = 0.
Similarly substituting (5.8) into the third equation of (5.6) and then applying K 1 s −1 yields K 1 (s −2 K 1 sα 1 ) = 0 and hence s −2 K 1 sα 1 = 0.
By the reasoning above, the projection onto the λ eigenspace of ∆ Z of β 2 is (after changing variables to t and η) of the form
(5.11) P λ β 2 = C |η| t K λ+(N− f0 2 − 1 2 ) 2 (t)
and the corresponding projection of α 1 is of the form
(5.12) P λ α 1 = C ′ |η| t K λ+(N− f0 2 + 1 2 ) 2 (t).
Thus to avoid elements of the null space we need to have either
1 ≤ 1 + a + λ + (N − f0 2 + 1 2 ) 2 or 1 ≤ 1 + a + λ + (N − f0 2 − 1
2 ) 2 and these are automatic for all a ≥ −1.
Indicial roots.
Another model operator of P a is its indicial family, defined using the action of P a on polyhomogeneous expansions. The indicial family is a one parameter family of operators on Y , I(P a ; ζ) defined by
P a (x ζ f ) = x ζ I(P a ; ζ)f x=0 + O(x ζ+1 ).
The base variables at the boundary enter into the indicial family as parameters, so we can speak of the indicial family at the point q ∈ Y by restricting not just to x = 0 but to the fibre over q. This refinement of the indicial family is denoted by I q (P a ; ζ); from (4.9) it is given by
I q (P a ; ζ) = ð Z dR −ζ − (f 0 − N + a + 1/2) ζ + N + a + 1/2 −ð Z dR ,
which coincides with the indicial family of the normal operator at q ∈ Y . The values of ζ for which I q (P a ; ζ) fails to be invertible (on L 2 iie (Z)) are known as the indicial roots of P a at q, or the boundary spectrum of P a at q, spec b (N q (P a )).
As we show below, this set depends on specð Z dR , and hence relies on the inductive hypothesis on Z.
An equivalent model of P a is the indicial operator:
I q (P a ) = ð Z dR −t∂ t − (f 0 − N + a + 1/2) t∂ t + N + a + 1/2 −ð Z dR .
It is related to the indicial family by the Mellin transform, M(I q (P a )u)(ζ) = I q (P a ; −iζ)M(u)(ζ).
Recall that this transform is defined, e.g., for u ∈ C ∞ c (R + ) by
(5.13) Mu(ζ) = ∞ 0 u(x)x iζ−1 dx,
and extends to an isomorphism between weighted spaces (5.14)
x α L 2 R + , dx x ∼ = − → L 2 ({η = α}; dξ)
where η = ℑζ and ξ = ℜζ. The inverse of the Mellin transform as a map (5.14) is given by
M −1 (v)(x) = 1 2π η=α v(ζ)x −iζ dξ.
Lemma 5.6. The indicial roots of P a are contained in the union of (6.16)
k = f0 2 − f0 2 − a ± k − f0 2 ± 1 2 , λ k =0 − f0 2 − a ± λ k + (k − f0 2 + ℓ 2 ) 2 , and λ k =0 1 − f0 2 − a ± λ k + (k − f0 2 + ℓ ′ 2 ) 2 where k ∈ {0, . . . , f 0 }, λ k is in the spectrum of ∆ Z acting on k-forms, ℓ ∈ {±1, ±3} and ℓ ′ ∈ {±1}.
The indicial operator of P a has a bounded inverse on the space t −1/2 L 2 iie (Z × R + t ) for all a ∈ (0, 1).
Proof. An analysis similar to -but simpler than -that above applies to the equation I q (P a )u = 0. Indeed, it suffices to replace (t∂ t ) 2 − t 2 in the 'equations to solve' by (t∂ t ) 2 . Since the solutions to (t∂ t ) 2 u = ν 2 u are linear combinations of t ν and t −ν , the solutions of I q (P a )u = 0 are obtained from the solutions to N q (P a )u = 0 by replacing each I v (t) by t ν and each K ν (t) by t −ν . Both of these contribute indicial roots, since for the indicial family we do not impose growth restrictions.
For the indicial operator, we are imposing growth restrictions and, as before, asking for solutions to be in t −1/2 L 2 (t f dt) excludes those involving t ν , hence conditions (a) and (b) show that there are no solutions involving t −ν for a > 0. Thus the proof of Lemma 5.5 shows that the indicial operator I q (P a ) is injective on t −1/2 L 2 (Z × R + ) as long as a > 0.
Similarly, the proof of Lemma 5.5 shows that if there is a non-zero solution to I q (P a ; ζ)u = 0 then ζ must be in one of the sets in (6.16). An advantage of the indicial family is that we can bring to bear our inductive hypotheses about ð Z dR . Indeed, decompose I q (P a ) as
I q (P a )(ζ) = ð Z dR −ζ − (f 0 − N + a + 1/2) ζ + N + a + 1/2 −ð Z dR = ð Z dR Id 0 0 − Id + 0 −ζ − (f 0 − N + a + 1/2) ζ + N + a + 1/2 0 = A + B.
Inductively we know that A is essentially self-adjoint, has closed range, and its domain, D(A), includes compactly into L 2 iie (Z). It follows that the operator B : D(A) → L 2 iie (Z) (where D(A) is endowed with the graph norm) is compact, i.e., B is relatively compact with respect to A, and so I q (P a ; ζ) has a unique closed extension, has closed range, and its domain is also D(A).
Since ð Z dR is essentially self-adjoint, the adjoint of I q (P a )(ζ) on L 2 (Z) is
I q (P a ; ζ) * = ð Z dR ζ + N + a + 1/2 −ζ − f + N − a − 1/2) −ð Z dR = I q (P a ; −(ζ + f + 2a + 1))
Notice that ζ is in one of the sets in (6.16) if and only if −(ζ + f + 2a + 1) is.
Thus we see that if ζ is not in one of these sets, then I q (P a ; ζ) is in fact invertible with bounded inverse. In fact, since the domain of I q (P a ; ζ) is D(A), its inverse is a compact operator. This proves that (6.16) contains the indicial roots of N q (P a ). Denote the inverse of I q (P a ; ζ) by
Q(ζ) : L 2 iie (Z) → D(A) ֒→ L 2 iie (Z)
. We obtain an inverse for I q (P a ) as an operator on t −1/2 L 2 iie (R + × Z) by applying the inverse Mellin transform to Q(ζ) along the line η = − f 2 − 1, which we can do as long as − f 2 − 1 is not an indicial root. If (a) and (b) hold, then this is true for all a ∈ (0, 1). 5.6. Bijectivity of N q (P a ). We now show that the normal operator N q (P a ) is a bijection between its maximal domain in s −1/2 L 2 iie (Z × T q Y + ) and s −1/2 L 2 iie (Z × T q Y + ) when 0 < a < 1. Observe first that by Lemma 5.5, assuming conditions a) and b), this mapping is injective for these values of a. Therefore, a simple duality argument shows that it suffices to show that it has closed range. Indeed, since ð dR is symmetric on L 2 iie (X; iie Λ * X), the operator D 0 : x −1/2 L 2 iie (X; iie Λ * X) → x 1/2 L 2 iie (X; iie Λ * X) coincides with its formal adjoint. It is then straightforward that the formal adjoint of P 0 is (P 0 ) * = x −1/2 ð dR x 3/2 : x −1/2 L 2 iie (X; iie Λ * X) → x −1/2 L 2 iie (X; iie Λ * X), and similarly,
(5.15) (P a ) * = (x 1/2−a ð dR x 1/2+a ) * = x −1/2+a ð dR x 3/2−a = P 1−a .
Lemma 5.7. The normal operator N q (P a ) is bijective as an operator on s −1/2 L 2 iie (Z× T q Y + ) acting on its maximal domain, for all a ∈ (0, 1).
Proof. For the duration of this section we write L 2 iie simply as L 2 and also omit the bundle iie Λ * (Z × T q Y + ) to simplify notation.
Following the proof of Lemma (5.5), we pass to the Fourier transform in the horizontal variables, introducing the variable η dual to y, and then rescale by setting t = s|η|, η = η/|η|. This leads to the family of operators
N q (P a , η) = ð Z dR + tcl ( η) −t∂ t − (f 0 − N + a + 1/2) t∂ t + N + a + 1/2 −ð Z dR − tcl ( η)
where cl ( η) = i η + e η is Clifford multiplication and η lies in the unit sphere S b−1 . Notice that (5.16) N q (P a , η) = I(P a ) + tA( η)
where A is a bounded matrix. These operations are all reversible, so it enough to study this simper family of operators, and in particular to show that it is a bijection from its maximal domain in t −1/2 L 2 to t −1/2 L 2 . We have already shown in Lemma 5.5 that this operator is injective, and by duality, i.e. using injectivity for N q (P a , η) * = N q (P 1−a , − η), it also has dense range. Thus it suffices to show that it has closed range, and to prove this we follow a standard procedure by constructing local parametrices for N q (P a , η) in the two regions (0, 2T ) × Z and (T, ∞) × Z for any fixed T . Notice that we only need to construct a right parametrix for N q (P a , η), since a left parametrix is obtained as the dual of a right parametrix for N q (P 1−a , − η). First consider the region t < 2T . We have indicated in §5.5 that I q (P a ) has an inverse H 0 = I q (P a ) −1 on t −1/2 L 2 , and hence
N q (P a , η) • H 0 = Id + tA( η)H 0
Since H 0 maps into the domain of I q (P a ), and the restriction of this domain to forms with bounded support in t includes compactly in t −1/2 L 2 , we see that the second term on the right is a compact operator on this subspace.
For forms supported in t > T , as in [40,Lemma 5.5], consider the partial symbol
σ( N q (P a , η) 2 ) = ∆ Z + t 2 + t 2 τ 2 0 0 −(∆ Z + t 2 + t 2 τ 2 )
where τ is the variable dual to ∂ t . Clearly,
| σ(N 2 q )u, u | ≥ t 2 (1 + τ 2 ) u ;
the inner product and norm are those of of t −1/2 L 2 . The operator norm of σ(N 2 q ) −1 is bounded by t −2 (1 + τ ) −2 , so that H ∞ (u) = e itτ σ(N 2 q ) −1 u dτ defines a parametrix for N 2 q ( η) in the large region. As before,
N 2 q • H ∞ = N q • ( N q • H ∞ ) = Id + B where B is compact, hence N q (P a , η) • H ∞ is the parametrix we seek.
Now choose a partition of unity {χ 0 , χ ∞ } relative to the open cover (0, 2T ) ∪ (T, ∞), and fix smooth functions χ j such that χ j = 1 on the support of χ j and which vanish outside a slightly larger neighbourhood. The right parametrix is then given by
H = χ 0 H 0 χ 0 + χ ∞ ( N q (P a , η) • H ∞ )χ ∞ .
The last thing we need to check is that N q (P a , η) • H = Id − Q, where Q is compact. However,
Q = [ N q (P a , η), χ 0 ]H 0 χ 0 + [ N q (P a , η), χ ∞ ]( N q (P a , η • H ∞ )χ ∞ + χ ∞ Bχ ∞ .
The two commutator terms are operators of order 0, i.e. multiplication operators, with compact support, and using the mapping properties of these two parametrices, we conclude that Q is compact, as claimed.
This proves that N q (P a , η) is Fredholm, which completes the argument.
5.7.
Integration by parts identity for N q (P a ).
In computing the indicial roots of P a , we have made strong use of the symmetries of the normal operator of P a , namely the translation invariance along horizontal directions (i.e., those tangent to Y ) and dilation invariance in T q Y + . In this section we exploit this invariance to establish an integration by parts identity, which will ultimately allow us to show that any 'extra' vanishing of N q (P a )u at x = 0 translates to some degree of vanishing of u at x = 0, the latter degree bounded by the indicial roots of N q (P a ). We will need the Sobolev spaces on Z × T q Y + analogous to those on X.
Definition 8. Let N ∈ N. We define H N pie (Z × T q Y + ; iie Λ * ) to be the set of u ∈ L 2 iie (Z × T q Y + ; iie Λ * )
such that for any positive integer p ≤ N,
X 1 . . . X p u ∈ L 2 iie (Z × T q Y + ; iie Λ * )
where the X j are vector fields which are either of the form s∂ s , s∂ uj (1 ≤ j ≤ b 0 ) or of the form X(z, s, u) = X(z) for each (z, s, u) ∈ Z × T q Y + , where X(z) is an edge vector field of the fibre Z = Z q . Notice that these vectors fields s∂ s , s∂ uj X(z) generate a Lie algebra.
As we have already used in §5.4, if a function in L 2 iie (X) is O(x γ ) near x = 0 then we must have 2γ + f 0 > −1. As the L 2 cut-off will be very important below we introduce the function
(5.17) δ 0 (γ) = γ − f 0 + 1 2 ,
thus a function in O(x γ ) is in x a L 2 (x f0 dx) precisely when γ > δ 0 (a). Briefly, let us abbreviate L 2 iie (Z × T q Y + , iie Λ * (Z × T q Y + )) by L 2 iie (q). Let C be a fixed number in [−1/2, 1] and ε ∈ (0, 1). Let now R be the unbounded operator induced by N q (P 0 ) on s C+ε L 2 iie (q) with domain C ∞ c ; with a small abuse of notation we denote also by R the operator induced by N q (P 0 ) on s C−ε L 2 iie (q) (acting distributionally). We consider the natural pairing ·, · : s C+ε L 2 iie (q) × s C−ε L 2 iie (q) → C between these two spaces 2 . Let R t be the formal transpose of R with respect to this pairing. R t is a differential operator and we let it act, distributionally, on s C−ε L 2 iie (q). We will establish that, if
u ∈ s C L 2 iie (q), v ∈ s C−ε L 2 iie (q),
and Ru, R t v ∈ s C+ε L 2 iie (q) then, with respect to the natural pairing ·, · above, we have v, Ru = u, R t v .
Notice that, although both pairings make sense, this is not an instance of the definition of R t , since both u and v are thought of as elements of s C−ε L 2 iie (q). Assume inductively that we have shown D max (ð dR ) = D min (ð dR ) for stratifications of depth at most m − 1 so that in particular
ð Z dR u, v = u, ð Z dR v for any two elements of D max (ð Z dR ).
On the one hand we know that, for u, v ∈ s C L 2 iie (q), the natural inner product is given by
u, v = s −2C u ∧ * v
and, on the other, the normal operator is given by
N q = N q (P a ) = ð Z dR + sð R b dR −s∂ s + N − f − (a + 1/2) s∂ s + N + (a + 1/2) −ð Z dR − sð R b dR ,
so as anticipated we only have to justify integrating by parts the s∂ s and sð R b dR . We can assume that we are working with sections compactly supported in a basic neighborhood W.
Our main tool is the Mellin transform (5.13). Using the inclusions x a L 2 ⊂ x b L 2 whenever b < a it follows that the Mellin transform of a function in x a L 2 (R + , dx) is holomorphic in the half-plane {η < a − 1/2}. The Mellin transform is very useful for studying asymptotics. For instance, if u is polyhomogeneous then Mu extends to a meromorphic function on the whole complex plane with poles at locations determined by the exponents occuring in the expansion of u. Switching from L 2 (R + ) to L 2 iie (X), assume that ω is supported in a basic neighborhood W of q ∈ Y , then we have
ω ∈ s α L 2 iie (X) ⇐⇒ Mω ∈ L 2 {η = δ 0 (α 0 )}, dξ; L 2 (dy dvol Z )
where M denotes Mellin transform in s (in the usual coordinates), dy denotes the Lebesgue measure of R b0 , and dvol Z denote the volume form associated to the edge iterated metric of Z. Notice that Mω extends to a holomorphic function on the half-plane {η < δ 0 (α 0 )} with values in L 2 (dy dvol Z ).
Elliptic regularity (via the symbolic calculus) tells us that elements in the null space of an elliptic edge-operator are in H ∞ iie (X; iie Λ * ), and hence smooth in the interior of the manifold. However, the derivatives of elements in H ∞ iie (X; E) will typically blow-up at the boundary, which is just to say that knowing ρ∂ y u ∈ L 2 iie (X; iie Λ * ) tells us that ∂ y u ∈ ρ −1 L 2 iie (X; iie Λ * ). Using the Mellin transform we can turn this around: if u is in the null space of an elliptic ie-operator, A, as a map A : ρ α L 2 iie (X; iie Λ * ) → ρ α L 2 iie (X; iie Λ * ) then, in the absence of indicial roots, we can view u as an element of a space with a stronger weight at the cost of giving up tangential regularity at the boundary. We shall concentrate directly on the normal operator of P a , even though much of what we prove could be extended to more general differential operators.
Lemma 5.8. Let W be a basic neighborhood for the point q ∈ Y . Set R = N q (P a ) and assume that, for some α ∈ R and ε ∈ (0, 1),
(5.18) {ℜζ + f 2 + 1 2 : ζ ∈ spec b (R)} ∩ [α − ε, α + ε] ⊆ {α}. (1) Assume v ∈ s α L 2 iie (Z×T q Y + ; iie Λ * ) is supported in W and Rv ∈ s α+ε L 2 iie (Z× T q Y + ; iie Λ * ) then v ∈ s α+ε L 2 (s f0 ds dvol Z , H −1 (dy) ⊗ iie Λ * ) = {s α+ε u : u ∈ Diff 1 (Y )L 2
iie (W ; iie Λ * )} Moreover, as a map into L 2 (dvol Z , H −1 (dy) ⊗ iie Λ * ) the Mellin transform of v is holomorphic in the half-plane {η < δ 0 (α + ε)}.
(2) Assume that u ∈ s α L 2 iie (Z×T q Y + ; iie Λ * ) and w ∈ s α−ε H 2 pie (Z×T q Y + ; iie Λ * ) (cf Definition 8) are such that
supp u ⊆ W Ru, R t w ∈ s α+ε L 2 iie (Z × T q Y + ; iie Λ * )
, then with respect to the natural pairing
·, · : s α−ε L 2 iie (Z × T q Y + ; iie Λ * ) × s α+ε L 2 iie (Z × T q Y + ; iie Λ * ) → C we have w, Ru = u, R t w .
Proof. (1): Since v is supported in a normal neighborhood of q ∈ Y , we can write
I q (R)v = Hv + h
where I q (R) is the indicial operator of R and H contains all of the 'higher order terms' at the boundary, e.g., s 2 ∂ s , s∂ u .
Passing to the Mellin transform, and using that I q (R; ζ) depends polynomially on ζ, we have an equality
(5.19) Mv(ζ) = I(R; iζ) −1 (M(Hv + h)(ζ)) as meromorphic functions {η < δ 0 (α)} → L 2 (dy dvol Z ; Λ * ),
of course since the left hand side is holomorphic on this half-plane so is the right hand side. On the other hand, M(h) is a holomorphic function into this space on the half plane {η < δ 0 (α + ε)}, and, reasoning as in [40], M(Hv) extends holomorphically to this half plane but we have to give up tangential regularity,
M(Hv) : {η < δ 0 (α + ε)} → L 2 (dvol Z ; H −1 (dy) ⊗ Λ * ) holomorphically.
This gives us an extension of (5.19) to as meromorphic functions {η < δ 0 (α + ε)} → L 2 (dvol Z ; H −1 (dy) ⊗ Λ * ).
The possible poles occur at indicial roots of R, so the first possibility would occur at ζ = δ 0 (α), and by hypothesis this is the only potential indicial root with real part less than or equal to δ 0 (α + ε). However we know that v(s, y, z) = 1 2π η=δ0(α) Mv(ξ, y, z)s −iζ dξ so in particular (as 1/ξ 2 is not integrable) Mv does not have any poles on this line. Hence
Mv(ζ) = I(R; iζ) −1 (M(Hv + h)(ζ)) as holomorphic functions {η < δ 0 (α + ε)} → L 2 (dvol Z ; H −1 (dy) ⊗ Λ * )
and we conclude that v ∈ s ε L 2 (s f0 ds dvol Z ; H −1 (dy) ⊗ Λ * ).
(2): This follows as in [40,Corollary 7.19] by analyzing the Mellin transform. Without loss of generality we can arrange, by conjugating R with an appropriate power of s, to work with the measure 1 s (dsdy dvol Z ). We will assume, for the duration of the proof, that this has been done without reflecting it in the notation. This has the advantage that the Parseval formula for the Mellin transform has the form 3
∞ 0 g 1 (s)g 2 (s) ds s = η=C Mg 1 (ζ) Mg 2 (−ζ) dξ
with C chosen so that the integral on the right makes sense. Notice that from knowing u, w ∈ s −ε L 2 iie and R(u), R t (w) ∈ s ε L 2 iie the respective Mellin transforms are defined on the half-planes
M(w)(ζ) on {η ≤ −ε}, M(Ru)(−ζ) on {η ≥ −ε} M(R t w)(ζ) on {η ≤ ε}, M(u)(−ζ) on {η ≥ ε}
so that a priori there is in each case only one choice for the constant C appearing in Parseval's formula. More precisely, C = −ε for the first pair and C = ε for the second pair.
Using part 1) of this Lemma we know we can extend
M(u)(−ζ) to {η ≥ −ε}
albeit with a loss in tangential regularity. Fortunately this loss in tangential regularity is compensated by a gain in tangential regularity in M(R t w) in this same region. Indeed, since w ∈ s −ε H 2 pie , we know that R t w ∈ s ε H 1 pie hence we have ∂ y R t w ∈ s −1+ε L 2 iie . It follows that the Mellin transform of ∂ y R t w is a holomorphic map from {η < −1 + ε} into L 2 (dy dvol Z ; Λ * ) and hence on this same halfplane M(R t w) maps holomorphically into L 2 (dvol Z , H 1 (dy) ⊗ Λ * ). Again applying Calderon's complex interpolation method, we conclude that
(5.21) M(R t w)(ζ) ∈ L 2 (dz, H ε−η ) for ε − 1 ≤ η ≤ ε.
The same reasoning applies to w. Thus if we start out with u, R t w which we can write as
End of induction: ð dR is essentially self-adjoint and Fredholm.
Our next task is to use the information gleaned in the previous section to show that elements of the maximal domain of ð dR as an operator on L 2 iie (X; iie Λ * X) are automatically in ρ ε L 2 iie,Γ (X; iie Λ * X). Proposition 5.9. Up to rescaling suitably the metric, the following is true. 1) Let u be in the maximal domain of ð dR as an operator on L 2 iie (X; iie Λ * X) then for any ε ∈ (0, 1), u ∈ ρ ε H 1 iie (X; iie Λ * X).
2) The maximal domain D max (ð dR ) is compactly embedded in L 2 iie . Proof. We can immediately localize and assume that u has support in a locally trivialized neighborhood U × C(Z) of the highest depth stratum.
We begin with the following intermediate result.
Proposition 5.10. Let u have compact support in U ×C(Z) and lie in the maximal domain of ð dR as an operator on L 2 iie (X; iie Λ * X). Then, for any ε ∈ (0, 1), u ∈ x ε L 2 iie (X; iie Λ * X). Proof. Fix ε 0 ∈ (0, 1) small enough that
{ℜζ + f 2 + 1 2 : ζ ∈ spec b (R)} ∩ [−1/2 − ε 0 , −1/2 + ε 0 ] ⊆ {−1/2}. Let u ∈ s −1/2 L 2 iie (Z × T q Y + ; iie Λ * ) satisfy N q (P 0 )(u) ∈ s 1/2 L 2 iie (Z × T q Y + ; iie Λ * ); by Proposition 5.3, we only need check that u ∈ s −1/2+ε0 L 2 iie (Z × T q Y + ; iie Λ * ).
Fix such a u, and write L 2 iie (Z × T q Y + ; iie Λ * ) in place of L 2 (q). Applying Lemmas 5.5 and 5.7, we know that R = N q (P 0 ) is injective and has closed range as a map from s −1/2+ε0 L 2 (q) to itself (on its maximal domain). It follows that R t is surjective from s −1/2−ε0 L 2 (q) to itself (on its minimal domain) .
Let G be the bounded generalized inverse of R t ; G is a bounded map from s −1/2−ε0 L 2 (q) to itself, with image contained in the domain of R t , and satisfies
R t G = Id s −1/2−ε 0 L 2 (q) .
Let φ be any element of
s −1/2+ε0 H 1 pie (Z × T q Y + ; iie Λ * ). Then v = Gφ satisfies v ∈ s −1/2−ε0 L 2 (a), R t v = R t Gφ = φ ∈ s −1/2+ε0 L 2 (q),
the latter statement and elliptic regularity allows us to strengthen the former to v ∈ s −1/2−ε0 H 2 pie (Z ×T q Y + ; iie Λ * ). On the other hand, we know that Ru ∈ s 1/2 L 2 (q) ⊂ s −1/2+ε0 L 2 (q), so by part 2) of Lemma 5.8 (with α = −1/2) we conclude that
Ru, v = R t v, u .
But then we also have
(5.22) Ru, v = Ru, Gφ = G t Ru, φ
where we recall that Ru ∈ s 1/2 L 2 (q) ⊂ s −1/2+ε0 L 2 (q), Gφ ∈ s −1/2−ε0 L 2 (q) and where G t denotes the functional analytic transpose of the bounded operator G; G t acts continuously on s −1/2+ε0 L 2 (q), so in fact G t Ru ∈ s −1/2+ε0 L 2 (q). Moreover, we have:
(5.23) Ru, v = R t v, u = R t Gφ, u = φ, u .
By comparing the last terms of (5.22) and (5.23) we see that u − G t Ru, φ = 0 and since φ was arbitrary we finally get: u = G t Ru. Therefore u ∈ s −1/2+ε0 L 2 (q).
Next, taking ε 1 ∈ (0, 1) small enough that
{ℜζ + f 2 + 1 2 : ζ ∈ spec b (R)} ∩ [−1/2 + ε 0 − ε 1 , −1/2 + ε 0 + ε 1 ] = ∅ ⊆ {−1/2 + ε 0 },
we can repeat the argument above and conclude u ∈ s −1/2+ε0+ε1 L 2 (q); continuing in this way we conclude that u ∈ s −1/2+ε L 2 (q) for any ε ∈ (0, 1) as required.
Proof of Proposition 5.9. 1) Proceed by induction on depth. For depth zero, there is nothing to prove. Let k > 0 and assume that the result is true for any Witt space of depth less than k. If u ∈ D max (ð dR ) has support in a locally trivialized neighbourhood U × C(Z) at the highest depth stratum, then Proposition 5.10 gives the stated decay and regularity in the final radial variable. Since the link Z has depth k − 1, we already know the result for it.
2) This follows since ρ ε H 1 iie (X; iie Λ * X) is compactly embedded in L 2
iie
We now know that elements of the maximal domain have some 'extra' degree of vanishing, and we can then apply an argument of Gil-Mendoza [18].
Proposition 5.11 (Gil-Mendoza). If D max (ð dR ) ⊆ ρ C L 2
iie (X; iie Λ * X) for some C > 0, then, as an operator on L 2 iie (X; iie Λ * X),
D max (ð dR ) ∩ ε>0 ρ 1−ε L 2 iie (X; iie Λ * X) ⊆ D min (ð dR )
Remark. Since we have actually shown not only that
D max (ð dR ) ⊆ ρ C H 1 iie (X; iie Λ * X) but in fact D max (ð dR ) ⊆ ε>0 ρ 1−ε H 1 iie (X; iie Λ * X),
this proposition implies D max (ð dR ) = D min (ð dR ).
Proof. We point out the following simple consequence of the formal self-adjointness of ð dR and the definitions of the minimal/maximal domains and weak derivatives:
Lemma 5.12. An element u ∈ D max (ð dR ) is in D min (ð dR ) if and only if (5.24) (ð dR u, v) = (u, ð dR v), for every v ∈ D max (ð dR ).
Proof. For any operator D with formal adjoint D * one has,
u ∈ D(D min ) ⇐⇒ u ∈ D ((D * ) max ) * ⇐⇒ Du, v = u, D * v for every v ∈ D((D * ) max )
If D is symmetric so that D * = D, then this is (5.24).
Let u ∈ D max (ð dR ) ∩ ε>0 ρ 1−ε L 2 iie (X; iie Λ * X), so u ∈ ε>0 ρ 1−ε H 1 iie (X; iie Λ * X).
Set u n = ρ 1/n u for n ∈ N, so that for each n, u n ⊆ ρH 1 iie (X; iie Λ * ), and, for every ε ∈ (0, 1), (5.25) u n → u in ρ 1−ε H 1 iie (X; iie Λ * ) and ð dR u n → ð dR u in ρ −ε L 2 iie (X; iie Λ * ).
Let ε ∈ (0, 1) so that D max (ð dR ) ⊆ ρ ε H 1 iie (X; iie Λ * ). Then, for any v ∈ D max (ð dR ), (5.25) implies
(ð dR u n , v) L 2 = (ρ ε ð dR u n , ρ −ε v) L 2 → (ρ ε ð dR u, ρ −ε v) L 2 = (ð dR u, v) L 2 , and (u n , ð dR v) → (u, ð dR v).
Moreover, by the previous Lemma, u n ∈ D min (ð dR ) implies (ð dR u n , v) = (u n , ð dR v).
It follows that (ð dR u, v) = (u, ð dR v) for every v ∈ D max (ð dR ) and hence u ∈ D min (ð dR ).
Altogether, we have now proved Theorem 1.1. We summarize for the benefit of the reader.
Proof. Parts 1) and 2) are direct consequences of the last Proposition. Let us show that ð dR is self-adjoint on its maximal domain. Denote by ð dR,max the operator ð dR on its maximal domain. If v is in the domain of ð dR,max then integration by parts, which is allowed because of the extra vanishing, implies that v is in the domain of (ð dR,max ) * and that ð dR,max v = (ð dR,max ) * v. Conversely, let v lie in the domain of (ð dR,max ) * . Observe that ∀u ∈ C ∞ c , ð dR u, v = u, ð dR v , with ð dR acting as a distribution on v. From the definition of adjointness we also know that ð dR u, v = u, (ð dR,max ) * v and since this is true for all u ∈ C ∞ c we infer that
ð dR v is in L 2 iie . Indeed, by definition, (ð dR,max ) * v ∈ L 2 iie .
Thus v is in the domain of ð dR,max and ð dR,max v = (ð dR,max ) * v. This proves that ð dR,max is self-adjoint.
To
prove 3), since ð dR is self-adjoint, (i Id + ð dR ) is invertible. Since D max (ð dR ) is compactly embedded into L 2
iie (X; iie Λ * X), (i Id + ð dR ) −1 defines a parametrix for ð dR acting on D max (ð dR ) with compact reminder.
Finally, for 4), since ð dR is Fredholm, there exists ǫ > 0 such that (ǫ Id + ð dR ) is invertible. Since the maximal domain is compactly embedded in L 2 iie , (ǫ Id + ð dR ) −1 is compact and self-adjoint. Thus, the spectrum of (ǫ Id+ ð dR ) −1 is discrete with finite multiplicity. Therefore, the spectrum of ð dR is discrete and has finite multiplicity.
The signature operator on Witt spaces
We now turn from the de Rham operator to the signature operator, first on forms with scalar coefficients and then with C * -algebra coefficients. We show first that these are Fredholm operators, but more importantly, that they define classes in the groups K * ( X) and K * (C * Γ), respectively. The index of these operators is independent of the choice of metric and defines a topological invariant. We will show later that this class enjoys even stronger properties: it is a Witt bordism invariant, a stratified homotopy invariant and it is equal, rationally, to a topologically defined invariant, the symmetric signature.
6.1. The signature operator ð sign . If X is even-dimensional, the Hodge star induces a natural involution on the differential forms on X, I : Ω * (X) → Ω * (X), I 2 = Id whose +1, −1 eigenspaces are known as the set of self-dual, respectively antiself dual, forms and are denoted Ω * + , Ω * − . The involution I extends naturally to
iie Ω * (X) and with respect to the splitting iie Ω * (X) = iie Ω * + ⊕ iie Ω * − , the de Rham operator decomposes
ð dR = 0 ð − sign ð + sign 0 where ð + sign = d + δ : iie Ω * + (X) → iie Ω * − (X), ð − sign = (ð + sign ) * .
If instead the manifold X is odd-dimensional, the signature operator of an (adapted) edge iterated metric is
ð sign = −i(dI + Id) = −iI(d − δ) = −i(d − δ)I.
We point out for later use that in either case, given a continuous map r : X → BΓ, we also obtain a twisted Mishchenko-Fomenko signature operator ð sign acting on sections of the bundle iie Λ * Γ (X).
Theorem 6.1. Up to rescaling suitably the metric the following is true. If X satisfies (5.4) for all strata, then the iterated incomplete edge signature operator ð sign is essentially self-adjoint with maximal domain contained in
ε>0 ρ 1−ε H 1 iie (X; iie Λ * X).
Its unique self-adjoint extension is Fredholm on its maximal domain endowed with the graph-norm; moreover it has discrete L 2 -spectrum of finite multiplicity.
Proof. If X is even-dimensional, it is immediate to see that
D min (ð + sign ) = D min (ð dR ) ∩ L 2 iie (X; iie Λ * + (X)), D max (ð + sign ) = D max (ð dR ) ∩ L 2 iie (X; iie Λ * + (X))
so the result follows from the corresponding results for ð dR . For X odd-dimensional, we point out that one can characterize the maximal domain of d − δ through the same analysis used for d + δ. Alternately, we can use the result for d+δ to deduce it for d−δ as follows. As explained above, a byproduct of our results is the existence of a strong Kodaira decomposition
L 2 iie Ω * = L 2 H ⊕ Image d ⊕ Image δ
where L 2 H is the intersection of the null spaces of d and δ. The de Rham operator d + δ decomposes into
(d : Image δ → Image d) ⊕ (δ : Image d → Image δ) ,
hence d and δ individually have closed range and
D max (ð dR ) ∩ Image δ = D max (d) ∩ Image δ D max (ð dR ) ∩ Image d = D max (δ) ∩ Image d hence i(d − δ)
has closed range with domain contained in (hence, by symmetry, equal to) D max (ð dR ). Applying Proposition 5.11 to i(d − δ) then shows that it too is essentially self-adjoint.
6.2. The K-homology class [ð sign ] ∈ K * ( X).
The results proved so far for the signature operator ð sign on a Witt space X allow one to define the K-homology class [ð sign ] ∈ K * ( X) = KK * (C( X), C). The Khomology signature class already appears in the work of Moscovici-Wu [46]; the definition there is based on the results of Cheeger.
Recall that an even unbounded Fredholm module for the C * -algebra C( X) is a pair (H, D) such that:
• H is a Hilbert space endowed with a unitary * -representation of C( X); D is a self-adjoint unbounded linear operator on H; • there is a dense * -subalgebra A ⊂ C( X) such that ∀a ∈ A the domain of D is invariant by a and [D, a] extends to a bounded operator on H; • (1 + D 2 ) −1 is a compact operator on H;
• H is equipped with a grading τ = τ * , τ 2 = I, such that τ f = f τ and τ D = −Dτ .
An odd unbounded Fredholm module is defined omitting the last condition. An unbounded Fredhom module defines a Kasparov (C( X), C)-bimodule and thus an element in KK * (C( X), C). We refer to [3] [6] for more on this foundational material.
Recall that adapted edge iterated metrics were defined in Proposition 5.4. The following Theorem already appears in [46], where it is proved using Cheeger's results. Here we give a proof using our approach.
Theorem 6.2. The signature operator ð sign associated to a Witt space X endowed with an adapted edge iterated metric g defines an unbounded Fredholm module for C( X) and thus a class [ð sign ] ∈ KK * (C( X), C), * ≡ dim X mod 2. Moreover, the class [ð sign ] does not depend on the choice of the adapted edge iterated metric on X.
Proof. We take H = L 2 iie (X; iie Λ * X), endowed with the natural representation of C( X) by multiplication operators. We take D as the unique closed self-adjoint extension of ð sign . These data depend of course on the choice of the adapted edge iterated metric. We take A equal to the space of Lipschitz functions on X with respect to g; A does not depend on the choice of g, since two adapted edge iterated metrics are quasi-isometric. Finally, in the even dimensional case we take the involution defined by I. All the conditions defining an unbounded Kasparov module are easily proved using the results of the previous section: indeed, if f is Lipschitz then it is elementary to see that multiplication by f sends the maximal domain of ð sign into itself; moreover [f, ð sign ] is Clifford multiplication by df which exists almost everywhere and is an element in L ∞ ( X); in particular [f, ð sign ] extends to a bounded operator on H; finally we know that (1 + D 2 ) −1 is a compact operator (indeed, we proved this is true for (±i + D) −1 ). Thus there is a well defined class KK * (C( X), C) which we denote simply by [ð sign ]; this class depends a priori on the choice of the metric g. Recall however that two adapted edge iterated metric g 0 and g 1 are joined by a path of adapted edge iterated metrics g t . Let ð 0 sign and ð 1 sign the corresponding signature operators, with domains in H 0 and H 1 . Proceeding as in the work of Hilsum on Lipschitz manifolds [22] one can prove that the 1-parameter family (H t , ð t sign ) defines an unbounded operatorial homotopy; using the homotopy invariance of KK-theory one obtains
[ð 0 sign ] = [ð 1 sign ] in KK * (C( X), C)
. We omit the details since they are a repetition of the ones given in [22].
6.3. The index class of the twisted signature operator ð sign . Let X be a Witt space endowed with an adapted edge iterated metric. Assume now that we are also given a continuous map r : X → BΓ and let Γ → X ′ → X the Galois Γ-cover induced by EΓ → BΓ. We consider the Mishchenko bundle
C * r Γ := C * r Γ × Γ X ′ .
and the signature operator with values in the restriction of C * r Γ to X, which we denote by ð sign . Proposition 6.3. The twisted signature operator ð sign is essentially self-adjoint as an operator on L 2 iie,Γ (X; iie Λ * Γ X), with maximal domain contained in ∩ ε>0 ρ 1−ε H 1 iie,Γ (X; iie Λ * Γ X) which is in turn C * r Γ-compactly included in the Hilbert C * r Γ-module L 2 iie,Γ (X; iie Λ * Γ X). Proof. We briefly point out how the proof given for ð dR and ð sign extends to the case of ð dR and ð sign . Recall that a C * r Γ−distribution on X = reg ( X) is a C−linear form T : C ∞ c (reg ( X), iie Λ * Γ X) → C * r Γ satisfying the following property. For any compact K ⊂ reg ( X), there exists a finite set S of elements of Diff * ie,Γ such that:
∀u ∈ C ∞ K (reg ( X), iie Λ * Γ X), T ; u C * r Γ ≤ sup Q∈S (Qu) L 2 iie,Γ .
Of course, any element of L 2 iie,Γ (X; iie Λ * Γ X) defines a C * r Γ−distribution on reg ( X). It is clear that ð dR sends L 2 iie,Γ (X; iie Λ * Γ X) into the space of C * r Γ−distributions. Therefore, the notion of maximal domain for ð dR is defined. The notion of minimal domain is also well defined (this is simply the closure of C ∞ c with respect to the norm u + ð dR u ). Notice that these two extensions are closed. Our first task is to show that these two extensions coincide. To this end we shall make use of the fundamental hypothesis that the reference map r : X → BΓ extends continously to the whole singular space X. Therefore, for any distinguished neighborhood W ≃ R b × C(Z), the induced Γ-coverings over W and over Z are trivial. This implies that for any q ∈ Y, N q ( ð dR ) is conjugate to N q (ð dR ) ⊗ Id C * r Γ . Once this has been observed we have, immediately, that Proposition 5.3 and Lemma 5.8 extend to the case of ð dR . Then, Proposition 5.10 also extends easily to the present case showing that the maximal domain of ð sign is included in ∩ ε>0 ρ 1−ε H 1 iie,Γ (X; iie Λ * Γ X). Once the extra vanishing is obtained, we can apply the argument give in the proof of Theorem 1.1 in order to show that the maximal extension is in fact self-adjoint. The argument of Gil-Mendoza can also be extended, showing the equality of the maximal and the minimal domain. The details of all this are easy and for the sake of brevity we omit them. Finally, proceeding as in [33], one can prove that ρ ε H 1 iie,Γ (X; iie Λ * Γ X) is C * r Γ-compactly included into L 2 iie,Γ (X; iie Λ * Γ X). The Proposition is proved for ð dR . The extra step needed for the signature operator is proved as in Theorem 6.1.
From now on we shall only consider the closed unbounded self-adjoint C * r Γoperator of Proposition 6.3 and with common abuse of notation we continue to denote it by ð sign .
We now proceed to show the following fundamental Proposition 6.4. The operator ð sign is a regular operator. Consequently (i ± ð sign ) and (1 + ð 2 sign ) are invertible. Proof. Recall that a closed unbounded self-adjoint operator D on a Hilbert C * r Γmodule is said to be regular if 1 + D 2 is surjective. One can show, see [32], that D is regular if and only if 1 + D 2 has dense image if and only if (i ± D) has dense image if and only if (i ± D) is surjective. Moreover, if D is regular then both (i ± D) and 1 + D 2 have an inverse. For a simple example of an unbounded self-ajoint operator on a Hilbert module such that (i + D) and (i − D) are not invertible see [23, page 415].
We shall prove that our operator is regular by employing unpublished ideas of George Skandalis, explained in detail in work of Rosenberg-Weinberger [51]. We have seen in the previous subsection that ð sign defines an unbounded Kasparov (C( X), C)-bimodule and thus a class [ð sign ] ∈ KK * (C( X), C). Consider now E := L 2 iie (X; iie Λ * X) ⊗ C C * r Γ ; tensoring ð sign with Id C * r Γ we obtain in an obvious way an unbounded Kasparov (C( X)⊗C * r Γ, C * r Γ)-bimodule that we will denote by (E, D). For later use we denote the corresponding KK-class as It is a non-trivial result ( [51]) that A is a dense *-subalgebra of A stable under holomorphic functional calculus. Consider now the Mishchenko bundle C * r Γ and its continuous sections C 0 ( X; C * r Γ) =: P . It is obvious that P is a finitely generated projective right A-module. The result cited above, together with Karoubi density theorem ( [27] exercice II.6.5), implies that there exists a finitely generated projective right A-module P such that P = P ⊗ A A. Consider for ξ ∈ P the operator T ξ : E → P ⊗ A E defined by T ξ (η) := ξ ⊗ η. T ξ is a bounded operator of C * r Γ Hilbert modules with adjoint T * ξ . Recall now, see [51], that a D-connection in the present context is a symmetric C * r Γ-linear operator D D :
P ⊗ A Dom(D) −→ P ⊗ A E
such that ∀ξ ∈ P the following commutator, defined initially on (Dom(D)) ⊕ P ⊗ A Dom(D), extends to a bounded operator on E ⊕ P ⊗ A E:
D 0 0 D , 0 T * ξ T ξ 0
Rosenberg and Weinberger have proved [51] that every D-connection is a self-adjoint regular operator. We can end the proof of the present Proposition as follows: first we observe that as C * r Γ Hilbert modules P ⊗ A E = L 2 iie,Γ (X; iie Λ * Γ X); next we consider ð sign and prove the following. Proof. It will suffice to prove the following. Let U be an open subset of X over which C * r Γ is trivial. Then the restriction of ξ ∈ P to U is a finite sum of terms of the form θ ⊗ u where θ is a flat section and u is a C 1 −function. So we shall assume that ξ = θ ⊗ u. Then for any η ∈ L 2 iie (U ; iie Λ * X |U ) ⊗ C * r Γ, one has:
( ð sign • T ξ − T ξ • D)(η) = θ ⊗ c(du)η + θ ⊗ u( ð sign − D)(η),
where cl (du) is Clifford multiplication. Recall that the restrictions to U of ð sign and D are differential operators of order one having the same principal symbol.
Therefore, ( ð sign • T ξ − T ξ • D) is bounded on L 2 iie (U ; iie Λ * X |U ) ⊗ C * r Γ.
One then gets immediately the Lemma by using a partition of unity.
Finally, we check easily that P ⊗ A Dom(D) ⊂ Dom max ( ð sign ). Since (i + ð sign ) has dense image with domain P ⊗ A Dom(D), we see that, a fortiori, the image of (i + ð sign ) with domain Dom max ( ð sign ) must also be dense.
These two Propositions yield at once the following Theorem 6.6. The twisted signature operator ð sign and the C * r Γ-Hilbert module L 2 iie,Γ (X; iie Λ * Γ X) define an unbounded Kasparov (C, C * r Γ)-bimodule and thus a class in KK * (C, C * r Γ) = K * (C * r Γ). We call this the index class associated to ð sign and denote it by Proof. We already know that ð sign is self-adjoint regular and Z 2 -graded in the even dimensional case. It remains to show that the inverse of (1+ ð 2 sign ) is a C * r Γ-compact operator. However, the domain of ð sign is compactly included in L 2 iie,Γ (X; iie Λ * Γ X); thus (i+ ð sign ) −1 and (−i+ ð sign ) −1 are both compacts. It follows that (1+ ð 2 sign ) −1 is compact. Thus ( ð sign , L 2 iie,Γ (X; iie Λ * Γ X)) defines an unbounded Kasparov (C, C * r Γ)bimodule as required. The equality Ind(
Ind( ð sign ) ∈ K * (C * r Γ). Moreover, if as in (6.1) we denote by [[ð sign ]] ∈ KK * (C( X) ⊗ C * r Γ, C * r Γ)ð sign ) = [ C * r Γ] ⊗ [[ð sign ]]
is in fact part of the theorem, attributed to Skandalis in [51], on D-connections. Finally, since we have proved that [ð sign ], and thus [[ð sign ]], is metric independent, and since [ C * r Γ] is obviously metric independent, we conclude that Ind( ð sign ) has this property too. The Theorem is proved. Corollary 6.7. Let β : K * (BΓ) → K * (C * r Γ) be the assembly map; let r * [ð sign ] ∈ K * (BΓ) the push-forward of the signature K-homology class. Then
(6.3) β(r * [ð sign ]) = Ind( ð sign ) in K * (C * r Γ) Proof. Since Ind( ð sign ) = [ C * r Γ] ⊗ [[ð sign ]]
, this follows immediately from the very definition of the assembly map. See [28] [29].
Witt bordism invariance
Let Y be an oriented odd dimensional Witt space with boundary ∂ Y = X. We assume that Y is a smoothly stratified space having a product structure near its boundary. We endow Y with an edge iterated metric having a product structure near ∂ Y = X and inducing an adapted edge iterated metric metric g (Proposition 5.4) on X. Consider a reference map r : Y → BΓ, its restriction to X and g induce a C * r Γ−linear signature operator on X. In this section only we shall be very precise and denote this operator by ð sign ( X).
Theorem 7.1. We have Ind ð sign ( X) = 0 in K 0 (C * r Γ) ⊗ Z Q. Proof.
We follow [37,Section 4.3] and Higson [21,Theorem 5.1]. Denote by Y ′ → Y and X ′ → X the two Γ−coverings associated to the reference map r : Y → BΓ.
The analysis of Section 7 shows that the operator ð sign ( X) induces a class [ ð sign ( X)] in the Kasparov group KK 0 (C 0 (∂ Y ), C * r Γ). In terms of the constant map π ∂ Y : ∂ Y → {pt}, one has:
Ind ð sign ( X) = π ∂ Y * ([ ð sign ( X)]) ∈ KK 0 (C, C * r Γ) ≃ K 0 (C * r Γ). Now let C ∂ Y ( Y ) ⊂ C( Y )
denote the ideal of continuous functions on Y vanishing on the boundary. Let i : ∂ Y → Y denote the inclusion and consider the long exact sequence in KK(·, C * r Γ) associated to the semisplit short exact sequence: Blackadar [6,page 197]). In particular, we have the exactness of
(7.1) 0 → C ∂ Y ( Y ) j → C( Y ) q → C(∂ Y ) → 0 (seeKK 1 (C ∂ Y ( Y ), C * r Γ) δ → KK 0 (C(∂ Y ), C * r Γ) i * → KK 0 (C( Y )
, C * r Γ) and thus i * • δ = 0. Recall that the conic iterated metric on Y (with product structure near ∂ Y = X) allows us to define a C * r Γ−linear twisted signature operator ð sign on Y with coefficients in the bundle Y ′ × Γ C * r Γ → Y . This twisted signature operator allows us to define a class [ ð sign ] ∈ KK 1 (C ∂ Y ( Y ), C * r Γ). Proof. We are using the proof of Theorem 5.1 of Higson [21]. We can replace Y by a collar neighborhood W (≃ [0, 1[×∂ Y ). Consider the differential operator d :
d = 0 −i d dx −i d dx 0
acting on [0, 1]. It defines a class in KK 1 (C 0 (0, 1), C * r Γ). Recall that the Kasparov product [d] ⊗ · induces an isomorphism:
[d] ⊗ · : KK 0 (C(∂ Y ), C * r Γ) → KK 1 (C ∂ Y (W )
, C * r Γ). As in [21], the connecting map δ :
KK 1 (C ∂ Y (W ), C * r Γ) δ → KK 0 (C(∂ Y )
, C * r Γ) is given by the inverse of [d] ⊗ ·. Denote by D W the restriction of ð sign to W and recall that X = ∂ Y is even dimensional. Then one checks (using [21] and [47, page 296
π ∂ Y * = π Y * • i * .
Since i * • δ = 0, the previous Lemma implies that:
2 Ind ð sign ( X) = π ∂ Y * ([2 ð sign ( X)]) = π ∂ Y * • δ([D W ]) = π Y * • i * • δ([D W ]) = 0. Therefore, Theorem 7.1 is proved.
We shall denote by Ω Witt,s * (BΓ) the bordism group in the category of smoothly stratified oriented Witt spaces. This group is generated by the elements of the form [ X, r : X → BΓ] where [ X, r : X → BΓ] is equivalent to the zero element if X is the boundary of a smoothly stratified Witt oriented space Y (as in Theorem 7.1) such that the map r extends continuously to Y . It follows that the index map
(7.2) Ω Witt,s * (BΓ) → K * (C * r Γ) ⊗ Q, sending [ X, r : X → BΓ]
∈ Ω Witt,s * (BΓ) to the higher index class Ind( ð sign ) (for the twisting bundle r * EΓ × Γ C * r Γ), is well defined. As in the closed case, see [52], it might be possible to refine this result and show that the index map actually defines a group homomorphism Ω Witt,s * (BΓ) → K * (C * r Γ) Recall that Siegel's Witt-bordism groups Ω Witt * (BΓ) are given in terms of equivalence classes of pairs ( X, u : X → BΓ), with X a Witt space which is not necessarily smoothly stratified.
We also recall that, working with PL spaces, Sullivan [56] has defined the notion of connected KO-Homology ko * (see also [54, page 1069]). Siegel [54,Chapter 4], building on work of Sullivan and Conner-Floyd, has shown that the natural map Ω SO * (BΓ) ⊗ Z Q → Ω Witt * (BΓ) ⊗ Z Q is surjective by showing that the natural map Ω SO * (BΓ) ⊗ Z Q → ko * (BΓ) ⊗ Z Q is surjective and the natural map ( [54]) Ω Witt * (BΓ) ⊗ Z Q → ko * (BΓ) ⊗ Z Q is an isomorphism. We need to extend these results for the corresponding groups associated with the category of smoothly stratified spaces.
Proposition 7.3. The natural map Ω SO * (BΓ) ⊗ Z Q → Ω Witt,s * (BΓ) ⊗ Z Q is surjec- tive.
Proof. Theorem 4.4 of [54] is still valid (by inspection) if one works in the category of smoothly stratified oriented Witt spaces. Namely, if X is an irreducible smoothly stratified Witt space of even dimension > 0 such that w( X) = 0, with w( X) ∈ W (Q), then X is Witt cobordant to zero in the category of smoothly stratified Witt spaces. The arguments of [54,Chapter 4] show that Siegel's map:
Ω Witt,s * (BΓ) ⊗ Z Q → ko * (BΓ) ⊗ Z Q
is an isomorphism and, using the surjectivity of the map Ω SO * (BΓ) ⊗ Z Q → ko * (BΓ) ⊗ Z Q, one gets the Proposition. 8. The homology L-class of a Witt space. Higher signatures.
The homology L-class L * ( X) ∈ H * ( X, Q) of a Witt space X was defined independently by Goresky and MacPherson [19], following ideas of Thom [57], and by Cheeger [13]. See also Siegel [54]. In this paper we shall adopt the approach of Goresky and MacPherson. We briefly recall the definition: if X has dimension n, k ∈ N is such that 2k − 1 > n, and N denotes the 'north pole' of S k , one can show that the map σ : π k ( X) → Z that associates to [f : X → S k ] the Witt-signature of f −1 (N ) is well defined and a group homomorphism. Now, by Serre's theorem, the Hurewicz map π k ( X) ⊗ Q → H k ( X, Q) is an isomorphism for 2k − 1 > n and we can thus view the above homomorphism, σ ⊗ Id Q , as a linear functional in Hom(H k ( X), Q) ≃ H k ( X, Q). This defines L k ( X) ∈ H k ( X, Q). The restriction 2k − 1 > n can be removed by crossing with a high dimensional sphere in the following way. Choose a positive integer ℓ such that 2(k + ℓ) − 1 > n + ℓ and k + ℓ > n. Then by the above construction, L k+ℓ ( X × S ℓ ) is well defined in H k+ℓ ( X × S ℓ , Q). Since k + ℓ > n, the Künneth Theorem shows that there is a natural isomorphism I : H k+ℓ ( X × S ℓ , Q) → H k ( X, Q). One then defines: L k ( X) := I(L k+l ( X × S ℓ )).
Once we have a homology L-class we can define the higher signatures as follows.
Definition 9. Let X be a Witt space and Γ := π 1 ( X). Let r : X → BΓ be a classifying map for the universal cover. The (Witt-Novikov) higher signatures of X are the collection of rational numbers:
(8.1) { α, r * L * ( X) , α ∈ H * (BΓ, Q)}
We set σ α ( X) := α, r * L * ( X) .
The Witt-Novikov higher signatures have already been studied, see for example [14]. If X is an oriented closed compact manifold and r : X → Bπ 1 (X) is the classifying map, it is not difficult to show that α, r * L * (X) = L(X) ∪ r * α, [X] ≡ L(X) ∪ r * α .
Thus the above definition is consistent with the usual definition of Novikov higher signatures in the closed case.
The Novikov conjecture in the closed case is the statement that all the higher signatures { L(X) ∪ r * α, [X] , α ∈ H * (BΓ, Q)} are homotopy invariants.
The Novikov conjecture in the Witt case is the statement that the Witt-Novikov higher signatures { α, r * L * ( X) , α ∈ H * (BΓ, Q)} are stratified homotopy invariants. Notice that intersection homology is not a homotopy invariant theory; however, it is a stratified homotopy-invariant theory, see [17].
We shall need to relate the homology L-class of Goresky-MacPherson to the signature class [ð sign ] ∈ K * ( X). This result is due to Cheeger, who proved it for piecewise flat metric of conic type, and to Moscovici-Wu, who gave an alternative argument valid also for any metric quasi-isometric to such a metric [13], [46]. It is worth pointing out here that our metrics do belong to the class considered in [46]. Notice that Moscovici-Wu prove that the straight Chern character of [ð sign ] Q ∈ K * ( X) ⊗ Q is equal to L * ( X) ∈ H * ( X, Q); the straight Chern character has values in Alexander-Spanier homology; the equality with L * ( X) ∈ H * ( X, Q) is obtained using the isomorphism between Alexander-Spanier and singular homology [46]. 9. Stratified homotopy invariance of the index class: the analytic approach
One key point in all the index theoretic proofs of the Novikov conjecture for closed oriented manifolds is the one stating the homotopy invariance of the signature index class in K * (C * r Γ). By this we mean that if r : X → BΓ as above, f : X ′ → X is a smooth homotopy equivalence and r ′ := r•f : X ′ → BΓ, then the index class, in K * (C * r Γ), associated to ð sign (i.e., associated to the signature operator on X, ð sign , twisted by r * EΓ × Γ C * r Γ) is equal to the one associated to ð ′ sign (i.e., associated to the signature operator on X ′ , ð ′ sign , twisted by (r ′ ) * EΓ × Γ C * r Γ). There are two approaches to this fundamental result:
(1) one proves analytically that Ind( ð sign ) = Ind( ð ′ sign ) in K * (C * r Γ); (2) one proves that the index class is equal to an a priori homotopy invariant, the Mishchenko (C * -algebraic) symmetric signature.
In this section we pursue the first of these approaches. We shall thus establish the stratified homotopy invariance of the index class on Witt spaces by following ideas from Hilsum-Skandalis [24], where this property is proved for closed compact manifolds. See also [49]. 9.1. Hilsum-Skandalis replacement of f . If X and Y are closed Riemannian manifolds, and f : X → Y is a homotopy equivalence, it need not be the case that pull-back by f induces a bounded operator in L 2 . Indeed, suppose f is an embedding and φ ε is a function which equals 1 on the ε tubular neighborhood of the image of X. The L 2 -norm of φ ε is bounded by Cε codim Y X and hence tends to zero, while f * φ ε ≡ 1 on X and so its L 2 norm is constant. Thus the closure of the graph of f * , say over piecewise constant functions, contains an element of the form (0, 1), and is not itself the graph of an operator.
On the other hand, if f is a submersion, and the metric on X is a submersion metric, then f * clearly does induce a bounded operator on L 2 . Since the latter property is a quasi-isometry invariant, and any two metrics on X are quasi-isometric, it follows that pull-back by a submersion always induces a bounded operator in L 2 .
As one is often presented with a homotopy equivalence f and interested in properties of L 2 spaces, it is useful to follow Hilsum and Skandalis [24] and replace pull-back by f by an operator that is bounded in L 2 . We refer to this operator as the Hilsum-Skandalis replacement of f * and denote it HS(f ).
Such a map is constructed as follows. Consider a disk bundle π Y : D Y → Y and the associated pulled back bundle f * D Y by the map f : X → Y. Denote by π X : f * D Y → X the induced projection. Then f admits a natural lift D(f ) such
that f * D Y D(f ) / / D Y X f / / Y commutes. Moreover, we consider a (smooth) map e : D Y → Y such that p = e • D(f ) : f * D Y → Y
is a submersion, and a choice of Thom form T for π X . The Hilsum-Skandalis replacement of f * is then the map
HS(f ) = HS T ,f * DY ,DY ,e (f ) : C ∞ (Y ; Λ * ) / / C ∞ (X; Λ * ) u / / (π X ) * (T ∧ p * u)
Notice that HS(f ) induces a bounded map in L 2 because p * = (e • D(f )) * does. For example, as in [24], one can start with an embedding j : Y → R N and a tubular neighborhood U of j(Y ) such that j(ζ)+D ⊆ U, and then take D X = X ×D, D Y = Y × D, D(f ) = f × id, and e(ζ, v) = τ (ζ + v) where τ : U → Y is the projection. Alternately, one can take D Y to be the unit ball subbundle of T Y and e(ζ, v) = exp f (ζ) (v). We will extend the latter approach to stratified manifolds.
In any case, one can show that HS(f ) is a suitable replacement for f * . Significantly, using HS(f ) we will see that the K-theory classes induced by the signature operators of homotopic stratified manifolds coincide.
Stratified homotopy equivalences.
Let X and Y denote stratified spaces, X and Y their regular parts, and S(X) and S(Y ) the corresponding sets of strata. Following [17] and [30, Def. 4.8.1 ff] we say that a map f : X → Y is stratum preserving if S ∈ S( Y ) =⇒ f −1 (S) is a union of strata of X and codimension preserving if also codimf −1 (S) = codimS.
We will say that a map is strongly stratum preserving if it is both stratum and codimension preserving.
In these references, a stratum-preserving homotopy equivalence between stratified spaces is a strongly stratum preserving map f : X → Y such that there exists another strongly stratum preserving map g : Y → X with both f •g and g •f homotopic to the appropriate identity maps through strongly stratum preserving maps. It is shown that stratum-preserving homotopy equivalences induce isomorphisms in intersection cohomology.
Notice that the existence of a homotopy equivalence between closed manifolds implies that the manifolds have the same dimension, so it is natural to impose a condition like strong stratum preserving on stratified homotopy equivalences. We shall also assume that f is a smooth strongly stratified map, see Definition 4, and that it is a smooth strongly stratified homotopy equivalence. (Once again, in the index-theoretic approach to the Novikov conjecture on closed manifolds, this additional hypothesis of smoothness is also made.)
We shall often omit the reference to the smoothness of f , given that our methods are obviously suited for these kind of maps only.
A smooth strongly stratified map lifts, according to Definition 4, to a smooth map between the resolutions of the stratified spaces f : X → Y preserving the iterated boundary fibration structures. In particular, f is a b-map and the differential of f sends tangent vectors to the boundary fibrations of X to tangent vector to the boundary fibrations of Y This implies that there exist linear maps
f * : C ∞ (Y ; ie Λ * ) → C ∞ (X; ie Λ * ), and f * : C ∞ (Y ; iie Λ * ) → C ∞ (X; iie Λ * ),
though, as on a closed manifold, these do not necessarily induce bounded maps in L 2 .
9.3. Hilsum-Skandalis replacement on complete edge manifolds. Suppose X and Y are both manifolds with boundary and boundary fibrations
φ X : ∂ X → H X , φ Y : ∂ Y → H Y .
Let X and Y denote the interiors of X and Y respectively. Endow Y with a complete edge metric g = ρ −2 g (3.2) such that g is adapted in the sense of Proposition 5.4. Let D Y ⊆ e T Y be the edge vector fields on Y with pointwise length bounded by one, and let exp : D Y → Y be the exponential map on Y with respect to the edge metric. The space D Y is itself an (open) edge manifold with boundary fibration φ DY : ∂D Y → ∂ Y → H Y . Notice that exp extends to a b-map that sends fibers of φ D Y to fibers of φ Y and hence induces a map exp * : e T D Y → e T Y which is seen to be surjective.
Let f : X → Y be a smooth b-map that sends fibers of φ X to fibers of φ Y . Pulling-back the bundle D Y → Y to X gives a commutative diagram
(9.1) f * D Y f / / πX D Y πY X f / / Y
which we use to construct the Hilsum-Skandalis replacement for pull-back by f . Namely, define e = exp : D Y → Y , let T the pull-back by f of a Thom form for D Y , and let
(9.2) HS(f ) = (π X ) * (T ∧ p * ) : C ∞ (Y ; ie Λ * ) → C ∞ (X; ie Λ * ) with p = e • D(f ).
Observe that p is a proper submersion and hence a fibration. Then, as above, HS(f ) induces a map between the corresponding L 2 spaces. The generalization to manifolds with corners and iterated fibrations structures is straightforward: we just replace the edge tangent bundle with the iterated edge tangent bundle. Indeed, it is immediate that if D Y ⊆ ie T Y is the set of iterated edge vector fields on Y with pointwise length bounded by one the exponential map exp : D Y → Y with respect to a (complete) iterated edge metric induces a map exp * : ie T D Y → ie T Y . That this map is surjective can be checked locally and follows by a simple induction. Then given a smooth b-map f : X → Y with the property that, whenever H ∈ M 1 ( X) is sent to K ∈ M 1 ( Y ), the fibers of the fibration on H are sent to the fibers of the fibration on K, we end up with a map
HS(f ) : C ∞ ( Y , ie Λ * ) → C ∞ ( X, ie Λ * )
that induces a bounded map between the corresponding L 2 ie spaces. Next, recall that
C ∞ ( Y ; iie Λ 1 ) = ρ Y C ∞ ( Y ; ie Λ 1 )
where ρ Y is a total boundary defining function for ∂ Y . Hence, if f :
X → Y induces f * : C ∞ ( Y ; ie Λ 1 ) → C ∞ ( X; ie Λ 1 ), it will also induce a map f * : C ∞ ( Y ; iie Λ 1 ) → C ∞ ( X; iie Λ 1 ) if f * (ρ Y ) is divisible by ρ X .
That is, we want f to map the boundary of X to the boundary of Y (a priori, it could map a boundary face of X onto all of Y ). For maps f coming from pre-stratified maps, this condition holds and hence the map HS(f ) induces a bounded map between iterated incomplete edge L 2 spaces. Of course, once f * induces a map on iie Λ 1 , it extends to a map on iie Λ * . 9.4. Stratified homotopy invariance of the analytic signature class. Suppose we have a stratum-preserving smooth homotopy equivalence between stratified spaces f : X → Y . Recall that X and Y denote the regular parts of X and Y , respectively. Recall the map r : Y → BΓ and the flat bundle V ′ of finitely generated C * r Γ-modules over Y :
V ′ = C * r Γ × Γ r * (EΓ).
Notice that using the blowdown map Y → Y , V ′ induces a flat bundle, still denoted V ′ on Y . Consider V = f * V ′ the corresponding flat bundle over X. We have a flat connection on V ′ , ∇ V ′ , over Y (and Y ) and associated differential d V ′ , and corresponding connection ∇ V and differential d V on X (and X). It is straightforward to see that the Hilsum-Skandalis replacement of f constructed above extends to
HS(f ) : C ∞ (Y ; iie Λ * ⊗ V ′ ) → C ∞ (X; iie Λ * ⊗ V)
and induces a bounded operator between the corresponding L 2 spaces. We now explain how the rest of the argument of Hilsum-Skandalis extends to this context. Suppose (f t ) 0≤t≤1 : X → Y is a homotopy of stratum-preserving smooth homotopy equivalences, let D Y be as above. Assume that (e s ) 0≤s≤1 : D Y → Y is a homotopy of smooth maps such that, for any s ∈ [0, 1], p s = e s • D(f s ) : f * s D Y → Y induces a surjective map on iie vector fields. Choose a smooth family of bundle isomorphisms (over X)
A s : f * s D Y −→ f * 0 D Y , (0 ≤ s ≤ 1), such that A 0 = Id . Set T s = A * s T 0 where T 0 is a Thom form for the bundle f * 0 D Y → X.
Consider ∇ a flat unitary connection on V ′ . It induces an exterior derivative d V ′ on the bundle
∧ * T * Y ⊗ V ′ . Choose a smooth family of C * r Γ−bundle isomorphism U s from the bundle (p s • A −1 s ) * V ′ → f * 0 D Y onto the bundle p * 0 V ′ → f * 0 D Y such that U 0 = Id.) * V ′ → f * 0 D Y .
Lemma 9.1. Under the above hypotheses and notation, there exists a bounded operator Υ :
L 2 iie (Y ; iie Λ * ⊗ V ′ ) → L 2 iie (f * 0 D Y ; iie Λ * ⊗ p * 0 V ′ ) such that (Id ⊗ U 1 ) • ( T 0 ∧ (p 1 • A −1 1 ) * ) − (T 0 ∧ p * 0 ) = p * 0 (d V ′ )Υ + Υd V ′ . Proof.
We follow Hilsum-Skandalis. Consider the map
H : f * 0 D Y × [0, 1] → Y (x, s) → H(x, s) = p s • A −1 s (x)
. Then the required map Υ is defined by, ∀ω ∈ L 2 iie (Y ; iie Λ * ⊗ V ′ ),
Υ(ω) = 1 0 i ∂ ∂t U t • (p t • A −1 t ) * F ⊗ (T 0 ∧ H * ω) dt.
We need to see how this construction handles composition. Recall that given f : X → Y we are taking D Y to be the ie vectors over Y with length bounded by one, D(f ) : f * D Y → D Y the natural map (9.1), e : D Y → Y the exponential map, p = e • D(f ), and T a Thom form on f * D Y , and then
HS(f )u = (π X ) * (T ∧ p * u).
Now suppose X, Y , and Z are manifolds with corners and iterated fibration structures, and
X h − → Y f − → Z
are smooth b-maps that send boundary hypersurfaces to boundary hypersurfaces and the fibers of boundary fibrations to the fibers of boundary fibrations. Assume that the map r : X → BΓ above is of the form r = r 1 • f for a suitable map r 1 : Z → BΓ. We then get a flat C * r Γ−bundle V ′′ over Z (and Z) such that V ′ = f * V ′′ . Denoting the various π · 's by τ 's, we have the following diagram
(f • p ′ ) * D Z τ1 p & & L L L L L L L L L L τ p ′′ h * D Y τ2
Proof. For simplicity, we give the proof only in the case Γ = {1}. Using the specific definitions of τ 1 , p, p ′ , τ 0 one checks easily that (τ 1 ) * p * = (p ′ ) * (τ 0 ) * . Therefore, ( p) * T τ0 is indeed a Thom form associated with τ 1 . Since p ′′ = p • p, one gets:
HS(f, h) = (τ 2 ) * (τ 1 ) * (T τ2 ∧ p * (T τ0 ∧ p * ))
Then replacing (τ 1 ) * p * by (p ′ ) * (τ 0 ) * one gets:
HS(f, g) = (τ 2 ) * (T τ2 ∧ (p ′ ) * ((τ 0 ) * (T τ0 ∧ (p) * ))) = HS(h) • HS(f ).
Next, notice that the maps
(t; ζ, ξ, η) → exp f (exp h(ζ) (tξ)) (η)
are a homotopy between p ′′ : (f • p ′ ) * D Z → Z and p : (f • h) * D Z → Z within submersions. Hence we can use the previous lemma to guarantee the existence of Υ.
Instead of the usual L 2 inner product, we will consider the quadratic form
Q X : C ∞ (X; iie Λ * ⊗ V) × C ∞ (X; iie Λ * ⊗ V) → C * r Γ Q X (u, v) = X u ∧ v *
and also the analogous Q Y , Q DY , Q f * DY . Recall that any element of C ∞ (X; iie Λ * ⊗ V) vanishes at the boundary of X so that Q X is indeed well defined. (We point out that the corresponding quadratic form in Hilsum-Skandalis [24, page 87] is given by i |u|(n−|u|) Q X (u, v).) We denote the adjoint of an operator T with respect to Q X (or Q Y ) by T ′ . Thus, for instance, d ′ V = −d V . From Theorem 6.6, we know that the signature data on X defines an element of K dim X (C * r Γ) and similarly for the data on Y . Hilsum and Skandalis gave a criterion for proving that two classes are the same which we now employ. Proposition 9.3. Consider a stratum-preserving homotopy equivalence f : X → Y , where dim X = n is even. Denote still by f the induced map X → Y . The bounded operator
HS(f ) : L 2 iie (Y ; iie Λ * ⊗ V ′ ) → L 2 iie (X; iie Λ * ⊗ V) satisfies the following properties: a) HS(f )d V ′ = d V HS(f ) and HS(f )(Dom d V ′ ) ⊆ Dom d V b) HS(f ) induces an isomorphism HS(f ) : ker d V ′ /ℑd V ′ → ker d V /ℑd V c)
There is a bounded operator Υ on a Hilbert module associated to Y such that
Υ(Dom d V ′ ) ⊆ Dom d V ′ and Id −HS(f ) ′ HS(f ) = d V ′ Υ + Υd V ′ d) There is a bounded self-adjoint involution ε on Y such that ε(Dom d V ′ ) ⊆ Dom d V ′ ,
which commutes with Id −HS(f ) ′ HS(f ) and anti-commutes with d V ′ . Hence the signature data on X and Y define the same element of K 0 (C * r Γ). Proof. The final sentence follows from (a)-(d) and Lemma 2.1 in Hilsum-Skandalis [24].
In Section 7 we showed that the signature operator has a unique closed extension, it follows that so do d V and d V ′ (see, e.g., [26,Proposition 11]). Since this domain is the minimal domain, as soon as we know that an operator is bounded in L 2 iie and commutes or anticommutes with these operators, we know that it preserves their domains. a) Since HS(f ) is made up of pull-back, push-forward, and exterior multiplication by a closed form,
HS(f )d V ′ = d V HS(f ).
b) From (a) we know that HS(f ) induces a map ker d V ′ /ℑd V ′ → ker d V /ℑd V . Let h denote a homotopy inverse of f and consider HS(h) : L 2 iie (X; iie Λ * ⊗ V) → L 2 iie (Y ; iie Λ * ⊗ V ′ ). We know from Lemma 9.2 that HS(f • h) and HS(h) • HS(f ) induce the same map in cohomology and, from Lemma 9.1 that HS(f • h) induces the same map as the identity. Since the same is true for HS(f • h) we conclude that HS(h) and HS(f ) are inverse maps in cohomology and hence each is an isomorphism. c) Recall that p : f * D Y → Y , being a proper submersion, is a fibration. Choose a Thom form T for the fibration π Y : D Y → Y so that D Y (f ) * T defines a Thom form for the fibration π X : f * D Y → X. These two facts allow us to carry out the following computation, where u ∈ C ∞ ( Y ; iie Λ * ⊗ V ′ ) and v ∈ C ∞ (X; iie Λ * ⊗ V).
Q X (HS(f )u, v) = Q X (π X ) * (D Y (f ) * T ∧ p * u), v = Q f * DY (D Y (f ) * T ∧ p * u, π * X v) = (−1) n(n−|v|) Q f * DY (p * u, D Y (f ) * T ∧ π * X v) = (−1) n(n−|v|) Q Y (u, p * (D Y (f ) * T ∧ π * X v)).
Since n is even this shows that HS(f ) ′ v = p * (D Y (f ) * T ∧ π * X v) and hence HS(f ) ′ HS(f )u = p * (D Y (f ) * T ∧ π * X (π X ) * ((D Y (f ) * T ∧ p * u))). Next one checks easily that, for any differential form ω on D Y , D Y (f ) * π * Y (π Y ) * ω = π * X (π X ) * D Y (f ) * ω. and so, from the identity p * = D Y (f ) * e * , HS(f ) ′ HS(f )u = (e • D Y (f )) * (D Y (f ) * ( T ∧ π * Y (π Y ) * ( T ∧ e * u))). Now observe that D Y (f ) : f * D Y → D Y , being a homotopy equivalence of manifolds with corners, sends the relative fundamental class of f * D Y to the relative fundamental class of D Y and so
Q f * DY (D Y (f ) * α, D Y (f ) * β) = Q DY (α, β).
From this identity, the previous equation, and the fact that e induces a fibration, one checks easily that Q Y (HS(f ) ′ HS(f )u, w) = Q Y (e * ( T ∧ π * Y (π Y ) * ( T ∧ e * u)), w) and hence HS(f ) ′ HS(f )u = e * ( T ∧ π * Y (π Y ) * ( T ∧ e * u)). Finally, e is homotopic to π Y , and since (π Y ) * ( T ∧ π * Y (π Y ) * ( T ∧ π * Y u)) = (π Y ) * ( T ∧ π * Y u) = u, Lemma 9.1, Id −HS(f ) ′ HS(f ) = d V ′ Υ + Υd V ′ as required.
d) It suffices to take εu = (−1) |u| u.
Remark. Consider now the case of an odd dimensional Witt space X endowed with an edge adapted iterated metric g and a reference map r : X → BΓ. We have defined in Section 7 the higher signature index class Ind ( ð sign ) ∈ KK 1 (C, C * r Γ) ≃ K 1 (C * r Γ) associated to the twisted signature operator defined by the data ( X, g, r). Recall that there is a suspension isomorphism Σ : K 1 (C * r Γ) ↔ K 0 (C * r Γ ⊗ C(S 1 )) which is induced by taking the Kasparov product with the Dirac operator of S 1 . Consider the even dimensional Witt space X × S 1 endowed with the obvious stratification and with the reference map r × Id S 1 : X × S 1 → B(Γ × Z) ≃ BΓ × S 1 .
As explained in [34, p. 624], [35, §3.2], the suspension of the odd index class Ind ( ð sign ) ∈ KK 1 (C, C * r Γ) ≃ K 1 (C * r Γ) is equal to the even signature index class associated to the data ( X × S 1 , g × (dθ) 2 , r × Id S 1 ). If now f : X → Y is a stratified homotopy equivalence of odd dimensional Witt spaces, then f induces a stratified homotopy equivalence from X × S 1 to Y × S 1 . By the previous Proposition the signature index classes of X×S 1 and Y ×S 1 are the same. Then using the suspension isomorphism Σ, we deduce finally that the odd signature index classes associated to X and Y are the same. Thus, the (smooth) stratified homotopy invariance of the signature index class is established for Witt spaces of arbitrary dimension.
Assembly map and stratified homotopy invariance of higher signatures
Consider the assembly map β : K * (BΓ) → K * (C * r Γ). The rationally injectivity of this map is known as the strong Novikov conjecture for Γ. In the closed case it implies that the Novikov higher signatures are oriented homotopy invariants. The rational injectivity of the assembly map is still unsettled in general, although it is known to hold for large classes of discrete groups; for closed manifolds having these fundamental groups the higher signatures are thus homotopy invariants. The following is the main topological result of this paper: Theorem 10.1. Let X be an oriented Witt space, r : X → Bπ 1 ( X) the classifying map for the universal cover, and let Γ := π 1 ( X). If the assembly map K * (BΓ) → K * (C * r Γ) is rationally injective, then the Witt-Novikov higher signatures { α, r * L * ( X) , α ∈ H * (BΓ, Q)} are stratified homotopy invariants.
Proof. The proof proceeds in four steps and is directly inspired by Kasparov's proof in the closed case, see for example [29] and the references therein:
(1) Consider ( X ′ , r ′ : X ′ → BΓ) and ( X, r : X → BΓ), with r = r ′ • f and f : X → X ′ a stratified homotopy equivalence between (smoothly stratified) oriented Witt spaces. Denote by ð ′ sign the twisted signature operator associated to ( X ′ , r ′ : X ′ → BΓ). We have proved that Ind( ð sign ) = Ind( ð ′ sign ) in K * (C * r Γ) ⊗ Q . (4) Since we know from Cheeger/Moscovici-Wu that Ch * (r * [ð sign ]) = r * (L * ( X)) in H * (BΓ, Q) we finally get that r * (L * ( X)) = (r ′ ) * (L * ( X ′ )) in H * (BΓ, Q)
which obviously implies the stratified homotopy invariance of the higher signatures {< α, r * L * ( X) >, α ∈ H * (BΓ, Q)}.
Examples of discrete groups for which the assembly map is rational injective include: amenable groups, discrete subgroups of Lie groups with a finite number of connected components, Gromov hyperbolic groups, discrete groups acting properly on bolic spaces, countable subgroups of GL(K) for K a field. 11. The symmetric signature on Witt spaces 11.1. The symmetric signature in the closed case. Let X be a closed orientable manifold and let r : X → BΓ be a classifying map for the universal cover. The symmetric signature of Mishchenko, σ(X, r), is a purely topological object [44]. In its most sophisticated presentation, it is an element in the L-theory groups L * (ZΓ). In general one can define the symmetric signature of any algebraic Poincaré complex, i.e., a cochain complex of finitely generated ZΓ-modules satisfying a kind of Poincaré duality. The Mishchenko symmetric signature corresponds to the choice of the Poincaré complex defined by the cochains on the universal cover. In the treatment of the Novikov conjecture one is in fact interested in a less sophisticated invariant, namely the image of σ(X, r) ∈ L * (ZΓ) under the natural map β Z : L * (ZΓ) → L * (C * r Γ). Recall also that there is a natural isomorphism ν : L * (C * r Γ) → K * (C * r Γ) (which is in fact valid for any C * -algebra). The C * -algebraic symmetric signature is, by definition, the element σ C * r Γ (X, r) := ν(β Z (σ(X, r)); thus σ C * r Γ (X, r) ∈ K * (C * r Γ). The following result, due to Mishchenko and Kasparov, generalizes the equality between the numeric index of the signature operator and the topological signature. With the usual notation: (11.1) Ind( ð sign ) = σ C * r Γ (X, r) ∈ K * (C * r Γ) As a corollary we see that the signature index class is a homotopy invariant; this is the topological approach to the homotopy invariance of the signature index class that we have mentioned in the introductory remarks in Section 9. The equality of the C * -algebraic symmetric signature with the signature index class (formula (11.1) above) can be restated as saying that the following diagram is commutative 11.2. The symmetric signature on Witt spaces. The middle perversity intersection homology groups of a Witt space do satisfy Poincaré duality over the rationals. Thus, it is natural to expect that for a Witt space X endowed with a reference map r : X → BΓ it should be possible to define a symmetric signature σ Witt QΓ (X, r) ∈ L * (QΓ). And indeed, the definition of symmetric signature in the Witt context, together with its expected properties, such as Witt bordism invariance, does appear in the literature, see for example [59], [10], [60]. However, no rigorous account of this definition was given in these references, which is unfortunate, given that things are certainly more complicated than in the smooth case and for diverse reasons that for the sake of brevity we shall not go into.
Fortunately, in a recent paper Markus Banagl [5] has given a rigorous definition of the symmetric signature on Witt spaces 4 using surgery techniques as well as previous results of Eppelmann [15]. Banagl's symmetric signature is an element σ Witt QΓ ( X, r) ∈ L * (QΓ); we refer directly to Banagl's interesting article for the definition and only point out that directly from his construction we can conclude that
• the symmetric signature σ Witt QΓ ( X, r) is equal to (the rational) Mishchenko's symmetric signature if X is a closed compact manifold;
• the Witt symmetric signature is a Witt bordism invariant; it defines a group homomorphism σ Witt QΓ : Ω Witt * (BΓ) → L * (QΓ). On the other hand, it is not known whether Banagl's symmetric signature σ Witt QΓ ( X, r) is a stratified homotopy invariant.
We define the C * -algebraic Witt symmetric signature as the image of σ Witt QΓ ( X, r) under the composite L * (QΓ) β Q −→ L * (C * r Γ) ν − → K * (C * r Γ) . We denote the C * -algebraic Witt symmetric signature by σ Witt C * r Γ (X, r). 11.3. Rational equality of the Witt symmetric signature and of the signature index class. Our most general goal would be to prove that there is a commutative diagram (11.3)
Ω Witt * (BΓ) Index − −−− → K i (C * r Γ) σ Witt QΓ ν −1 L * (QΓ) β Q − −−− → L * (C * r Γ). or, in formulae σ Witt C * r Γ (X, r) = Ind( ð sign ) in K i (C * r Γ)
with Ind( ð sign ) the signature index class decribed in the previous sections. We shall be happy with a little less, namely the rational equality.
Proposition 11.1. Let σ Witt C * r Γ (X, r) Q and Ind( ð sign ) Q be the rational classes, in the rationalized K-group K i (C * r Γ) ⊗ Q, defined by the Witt symmetric signature and by 4 Banagl actually concentrates on the more restrictive class of IP spaces, for which an integral symmetric signature, i.e. an element in L * (ZΓ), exists; it is easy to realize that his construction can be given for the larger class of Witt spaces, producing, however, an element in L * (QΓ).
the signature index class. Then (11.4) σ Witt C * r Γ (X, r) Q = Ind( ð sign ) Q in K i (C * r Γ) ⊗ Q Proof. We already know from [5] that the rationalized symmetric signature defines a homomorphism from (Ω Witt * (BΓ)) Q to K i (C * r Γ) ⊗ Q. However, it also clearly defines a homomorphism (Ω Witt,s * (BΓ)) Q → K i (C * r Γ) ⊗ Q, exactly as the signature index class. For notational convenience, let I : (Ω Witt,s * (BΓ)) Q → K i (C * r Γ) ⊗ Q be the (Witt) signature index morphism; let I ′ : (Ω Witt,s * (BΓ)) Q → K i (C * r Γ) ⊗ Q be the (Witt) symmetric signature morphism. We want to show that
I = I ′ .
We know from Proposition 7. with the first and third equality following from the above remark and the second equality obtained using the fundamental result of Kasparov and Mishchenko on closed manifolds. The proof is complete.
The above Proposition together with Proposition 9.3 implies at once the following result:
Corollary 11.2. The C * -algebraic symmetric signature defined by Banagl is a rational stratified homotopy invariant.
This Corollary does not seem to be obvious from a purely topological point of view. We add that very recently Friedman and McClure have given an alternative definition of symmetric signature on Witt spaces; while its relationship with Banagl's definition is for the time being unclear, we point out that the symmetric signature of Friedman and McClure is a stratified homotopy invariant; moreover, with the same proof given above, its image in K * (C * r Γ) is rationally equal to our signature index class.
Epilogue
Let X be an orientable Witt pseudomanifold with fundamental group Γ. We endow the regular part of X with an adapted iterated edge metric g (Proposition 5.4). Let X ′ be a Galois Γ-covering and r : X → BΓ a classifying map for X ′ . We now restate once more the signature package for the pair ( X, r : X → BΓ) indicating precisely where the individual items have been established in this paper.
(1) The signature operator defined by the edge (adapted) iterated metric g with values in the Mishchenko bundle r * EΓ × Γ C * r Γ defines a signature index class Ind( ð sign ) ∈ K * (C * r Γ), * ≡ dim X mod 2. Established in Theorem 6.6.
(2) The signature index class is a (smooth) Witt bordism invariant; more precisely it defines a group homomorphism Ω Witt,s * (BΓ) → K * (C * r Γ)⊗ Q. This is Theorem 7.1, together with (7.2).
(3) The signature index class is a stratified homotopy invariant. Proposition 9.3. (4) There is a K-homology signature class [ð sign ] ∈ K * (X) whose Chern character is, rationally, the homology L-Class of Goresky-MacPherson. Theorem 6.2 and Theorem 8.1. (5) The assembly map β : K * (BΓ) → K * (C * r Γ) sends the class r * [ð sign ] into Ind( ð sign ). Corollary 6.7. (6) If the assembly map is rationally injective one can deduce from the above results the homotopy invariance of the Witt-Novikov higher signatures. Theorem 10.1. (7) There is a topologically defined C * -algebraic symmetric signature σ Witt C * r Γ (X, r) ∈ K * (C * r Γ) which is equal to the analytic index class Ind( ð sign ) rationally. This is Banagl's construction together with our Proposition 11.1.
which is a b-map of manifolds with corners preserving the iterated fibration structures.
Theorem 3.4. (Cheeger) Let X be a Witt space endowed with an iterated edge metric g. Denote by H *(2) ( X) the cohomology of the L 2 de Rham complex with maximal domain; denote by H *
Proposition 4. 1 .
1(X, g) is a complete Riemannian manifold of bounded geometry.
Lemma 4. 3 .
3The operator ð dR is in Diff 1 iie , i.e., ρð dR is in Diff 1 ie
( 5 . 20 )
520Mv(ζ) = I(R; iζ) −1 (M(Hv + h)(ζ))
η=ε M(R t w)(ζ) M(u)(−ζ) dξ dy dvol Z , we can deform the contour from {η = ε} to {η = −ε} and throughout this deformation the integrand stays holomorphic with the loss in tangential regularity of M(u) exactly compensated by a gain in regularity by M(R t w), i.e. the integrand makes sense as a pairing throughout the deformation. Moreover the integrand is holomorphic in this region and so the value of the integral does not change during the deformation. Hence we can write u, R t w as η=−ε M(R t w)(ζ) M(u)(−ζ) dξ dy dvol Z .
Now integrating each term by parts we write this asη=−ε M(w)(ζ) M(Ru)(−ζ) dξ dy dvol Z ,which by another application of Parseval's formula we recognize as w, Ru .3 For the measure s f 0 ds the Parseval formula for the Mellin transform takes the form ∞ 0 g 1 (s)g 2 (s)s f 0 ds = η=C Mg 1 (ζ) Mg 2 (−(f 0 + 1)i − ζ) dξ
ð sign ]] ∈ KK * (C( X) ⊗ C * r Γ, C * r Γ). Consider A := C( X) ⊗ C * r Γ and set A := {a ∈ A : a(Dom D) ⊂ Dom D and [a, D] extends to an element of L(E)}.
Lemma 6 . 5 .
65The operator ð sign defines a D-connection.
the class obtained from [ð sign ] ∈ KK * (C( X), C) by tensoring with C * r Γ, then Ind( ð sign ) is equal to the Kasparov product of the class defined by Mishchenko bundle [ C * r Γ] ∈ KK 0 (C, C( X) ⊗ C * r Γ) with [[ð sign ]]: (6.2) Ind( ð sign ) = [ C * r Γ] ⊗ [[ð sign ]] In particular, the index class Ind( ð sign ) does not depend on the choice of the adapted edge iterated metric.
Lemma 7. 2 .
2One has δ[ ð sign ] = [2 ð sign ( X)].
Theorem 8.1. (Cheeger/Moscovici-Wu) The topological homology L-class L * ( X) ∈ H * ( X, Q) is the image, under the rationalized homology Chern character, of the signature K-homology class [ð sign ] Q ∈ K * ( X) ⊗ Q; in formulae (8.2) ch * [ð sign ] Q = L * ( X) in H * ( X, Q).
Implicit in the statement of the next Lemma is the fact that, for each s ∈ [0, 1], p s • A −1 s induces a morphism from the space of sections of the bundle V ′ → Y on the space of sections of the bundle (p s • A −1 s
( 2 )
2We know that the assembly map sends r * [ð sign ] ∈ K * (BΓ) to the Witt index class Ind( ð sign ). More explicitly:β(r * [ð sign ]) = Ind( ð sign ) in K * (C * r Γ) ⊗ Q (3)We deduce from the assumed rational injectivity of the assembly map thatr * [ð sign ] = (r ′ ) * [ð ′ sign ] in K * (BΓ) ⊗ Q.
−− → L * (C * r Γ).where i ≡ * mod 2.
3 that the natural map Ω SO * (BΓ) → Ω Witt,s * (BΓ) induces a rational surjection s : (Ω SO * (BΓ)) Q → (Ω Witt,s * (BΓ)) Q . In words, a smoothly stratified Witt space X with reference map r : X → BΓ is smoothly stratified Witt bordant to k-copies of a closed oriented compact manifold M with reference map ρ : M → BΓ. Moreover, we remark that the Witt index classes and the Witt symmetric signature of an oriented closed compact manifold coincide with the classic signature index class and the Mishchenko symmetric signature. Then I([X, r]) = I(k[M, ρ]) = I ′ (k[M, ρ]) = I ′ ([X, r])
]) that the KK−class [D W ] is equal to [d] ⊗ 2[ ð sign ( X)], and one finds that δ[D W ] = 2[ ð sign ( X)] which proves the result. Let π Y : Y → {pt} denote the constant map. By functoriality, one has:
Recall that this pairing is given by u, v := (u ′ , v ′ ) s C L 2 if u = s ǫ u ′ and v = s −ǫ v ′ .
Acknowledgements. The authors are happy to thank the referee for very helpful comments. P.A., R.M. and P.P. are grateful to MSRI for hospitality and financial support during the Fall Semester of 2008 when much of the analytic part of this paper was completed. P.A. was partly supported by NSF grant DMS-0635607002 and by an NSF Postdoctoral Fellowship, and thanks Stanford for support during visits; R.M. was supported by NSF grants DMS-0505709 and DMS-0805529, and enjoyed the hospitality and financial support of MIT, Sapienza Università di Roma and the Beijing International Center for Mathematical Research; P.P. wishes to thank the CNRS and Université Paris 6 for financial support during visits to Institut de Mathématiques de Jussieu in Paris. E.L was partially supported during visits to Sapienza Università di Roma by CNRS-INDAM (through the bilateral agreement GENCO (Non commutative Geometry)) and the Italian Ministero dell' Università e della Ricerca Scientifica (through the project "Spazi di moduli e teoria di Lie").The authors are particularly grateful to Richard Melrose for his help and encouragement at many stages of this project and for allowing them to include some of his unpublished ideas in section 2. They also thank: Markus Banagl for many interesting discussions on the notion of symmetric signature on Witt spaces; Shmuel Weinberger for suggesting the proof of Proposition 11.1; and Michel Hilsum for useful remarks on the original manuscript. Finally, they thank Thomas Krainer, Eugenie Hunsicker and Gerardo Mendoza for helpful correspondence.
. HS. C ∞ (Ziie Λ 1 ⊗ V ′′ ) → C ∞ (Yiie Λ 1 ⊗ V ′HS(f ) : C ∞ (Z; iie Λ 1 ⊗ V ′′ ) → C ∞ (Y ; iie Λ 1 ⊗ V ′ ),
. HS. C ∞ (YHS(h) : C ∞ (Y ;
. HS. C ∞ (ZHS(f, h) : C ∞ (Z;
HS(f )(u) = (τ 0 ) * (T τ0 ∧ p * u), HS(h)(u) = (τ 2 ) * (T τ2 ∧. p ′ ) * uHS(f )(u) = (τ 0 ) * (T τ0 ∧ p * u), HS(h)(u) = (τ 2 ) * (T τ2 ∧ (p ′ ) * u)
HS(f, h)(u) = τ * (T τ2 ∧ ( p) * T τ0 ∧. p ′′ ) * uHS(f, h)(u) = τ * (T τ2 ∧ ( p) * T τ0 ∧ (p ′′ ) * u)
HS(f, h) = HS(h)• HS(f ) and HS(f, h)− HS. Lemma 9. 2Lemma 9.2. HS(f, h) = HS(h)• HS(f ) and HS(f, h)− HS
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[] |
[
"Comparing the notions of optimality in CP-nets, strategic games and soft constraints",
"Comparing the notions of optimality in CP-nets, strategic games and soft constraints"
] |
[
"Krzysztof R Apt [email protected] \nCWI Amsterdam\nKruislaan 4131098 SJAmsterdamthe Netherlands\n\nUniversity of Amsterdam\nPlantage Muidergracht 24 1018 TVAmsterdamthe Netherlands\n",
"Francesca Rossi s:[email protected] \nDepartment of Pure and Applied Mathematics\nUniversity of Padova\nVia Trieste63 -35121PadovaItaly\n",
"Kristen Brent Venable [email protected] \nDepartment of Pure and Applied Mathematics\nUniversity of Padova\nVia Trieste63 -35121PadovaItaly\n"
] |
[
"CWI Amsterdam\nKruislaan 4131098 SJAmsterdamthe Netherlands",
"University of Amsterdam\nPlantage Muidergracht 24 1018 TVAmsterdamthe Netherlands",
"Department of Pure and Applied Mathematics\nUniversity of Padova\nVia Trieste63 -35121PadovaItaly",
"Department of Pure and Applied Mathematics\nUniversity of Padova\nVia Trieste63 -35121PadovaItaly"
] |
[] |
The notion of optimality naturally arises in many areas of applied mathematics and computer science concerned with decision making. Here we consider this notion in the context of three formalisms used for different purposes in reasoning about multi-agent systems: strategic games, CP-nets, and soft constraints. To relate the notions of optimality in these formalisms we introduce a natural qualitative modification of the notion of a strategic game. We show then that the optimal outcomes of a CP-net are exactly the Nash equilibria of such games. This allows us to use the techniques of game theory to search for optimal outcomes of CP-nets and vice-versa, to use techniques developed for CP-nets to search for Nash equilibria of the considered games. Then, we relate the notion of optimality used in the area of soft constraints to that used in a generalization of strategic games, called graphical games. In particular we prove that for a natural class of soft constraints that includes weighted constraints every optimal solution is both a Nash equilibrium and Pareto efficient joint strategy. For a natural mapping in the other direction we show that Pareto efficient joint strategies coincide with the optimal solutions of soft constraints.
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10.1007/s10472-008-9095-4
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[
"https://arxiv.org/pdf/0711.2909v2.pdf"
] | 3,204,000 |
0711.2909
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cf59136ab2452db02f005a955f6fa290b3440e66
|
Comparing the notions of optimality in CP-nets, strategic games and soft constraints
21 Apr 2008
Krzysztof R Apt [email protected]
CWI Amsterdam
Kruislaan 4131098 SJAmsterdamthe Netherlands
University of Amsterdam
Plantage Muidergracht 24 1018 TVAmsterdamthe Netherlands
Francesca Rossi s:[email protected]
Department of Pure and Applied Mathematics
University of Padova
Via Trieste63 -35121PadovaItaly
Kristen Brent Venable [email protected]
Department of Pure and Applied Mathematics
University of Padova
Via Trieste63 -35121PadovaItaly
Comparing the notions of optimality in CP-nets, strategic games and soft constraints
21 Apr 2008Strategic gamespure Nash equilibriapreferencesCP netssoft constraints AMS MOS Classification: 91B1091B5068T0168T30
The notion of optimality naturally arises in many areas of applied mathematics and computer science concerned with decision making. Here we consider this notion in the context of three formalisms used for different purposes in reasoning about multi-agent systems: strategic games, CP-nets, and soft constraints. To relate the notions of optimality in these formalisms we introduce a natural qualitative modification of the notion of a strategic game. We show then that the optimal outcomes of a CP-net are exactly the Nash equilibria of such games. This allows us to use the techniques of game theory to search for optimal outcomes of CP-nets and vice-versa, to use techniques developed for CP-nets to search for Nash equilibria of the considered games. Then, we relate the notion of optimality used in the area of soft constraints to that used in a generalization of strategic games, called graphical games. In particular we prove that for a natural class of soft constraints that includes weighted constraints every optimal solution is both a Nash equilibrium and Pareto efficient joint strategy. For a natural mapping in the other direction we show that Pareto efficient joint strategies coincide with the optimal solutions of soft constraints.
Introduction
The concept of optimality is prevalent in many areas of applied mathematics and computer science. It is of relevance whenever we need to choose among several alternatives that are not equally preferable. For example, in constraint optimization, each solution of a constraint satisfaction problem has a quality level associated with it and the aim is to choose an optimal solution, that is, a solution with an optimal quality level. In turn, in strategic games, two concepts of optimality have been commonly used: Nash equilibrium and Pareto efficient outcome.
Some formalisms proposed in AI employ 'their own' concept of an optimal outcome. The aim of this paper is to clarify the status of such notions of optimality used in CP-nets and soft constraints. To this end we use tools and techniques from game theory, more specifically theory of strategic games.
This allows us to gain new insights into the relationship between these formalisms which hopefully will lead to further cross-fertilization among these three different approaches to modelling optimality.
Background
Game theory, notably the theory of strategic games, forms one of the main tools in the area of multi-agent systems since they formalize in a simple and powerful way the idea that the agents interact with each other while pursuing their own interests. Each agent has a set of strategies and a payoff function on the set of joint strategies. The agents choose their strategies simultaneously with the aim of maximizing one's payoff.
The most commonly used concept of optimality is that of a Nash equilibrium. Intuitively, it is an outcome that is optimal for each player under the assumption that only he may reconsider his action. Another concept of optimality is that of Pareto efficient joint strategies, which are those in which no player can improve his payoff without decreasing the payoff of some other player. Sometimes it is useful to consider constrained Nash equilibria, that is, Nash equilibria that satisfy some additional requirements, see e.g. [12]. For example, Pareto efficient Nash equilibria are Nash equilibria which are also Pareto efficient among the Nash equilibria.
In turn, CP-nets (Conditional Preference nets) are an elegant formalism for representing conditional and qualitative preferences, see [6,5]. They model such preferences under a ceteris paribus (that is, 'all else being equal') assumption. A CP-net exploits the idea of conditional independence to provide a compact representation of preference problems. Preference elicitation in such a framework appears to be natural and intuitive.
Research on CP-nets has been focused on their modeling capabilities and on algorithms for solving various natural problems related to their use. Also, computational complexity of these problems was extensively studied. One of the fundamental problems is that of finding an optimal outcome, i.e., one that cannot be improved in the presence of the adopted preference statements. This is in general a complex problem since it was found that finding optimal outcomes and testing for their existence is in general NPhard, see [6,5]. In contrast, for so-called acyclic CP-nets this is an easy problem which can be solved by a linear time algorithm, see [6,5].
Finally, soft constraints, see e.g. [4], are a quantitative formalism which allow us to express constraints and preferences. While constraints state which combinations of variable values are acceptable, soft constraints allow for several levels of acceptance. An example are fuzzy constraints, see [8] and [21], where acceptance levels are between 0 and 1, and where the quality of a solution is the minimal level over all the constraints. An optimal solution is the one with the highest quality. The research in this area has dealt mainly with the algorithms for finding optimal solutions and with the relationship between modelling formalisms, see [19].
Main results
We consider the notions of optimality in two preference modelling frameworks, that is, CP-nets and soft constraints, and in strategic games. Although apparently there is no connection among these different ways of modelling preferences, we show that in fact there is a strong relationship. This is surprising and interesting on its own. Moreover, it might be exploited for a cross-fertilization among these three frameworks.
In particular, we start by considering the relationship between CP-nets and strategic games, and we show how game-theoretic techniques can be fruitfully used to study CP-nets. Our approach is based on the observation that the ceteris-paribus principle, typical of CP-nets, implies that an optimal outcome is worsened if a worsening change (to some variable) is made. This is exactly the idea behind Nash equilibria and the desired results easily follow once this observation is made formal by introducing an appropriate modification of strategic games. In this modification each player has at his disposal a preference relation on his set of strategies, parametrized by a joint strategy of his opponents. We call such games strategic games with parametrized preferences.
The cornerstone of our approach are two results closely relating CP-nets to such games. They show that the optimal outcomes of a CP-net are exactly the Nash equilibria of an appropriately defined strategic game with parametrized preferences. This allows us to transfer techniques of game theory to CP-nets, and vice-versa.
In strategic games techniques have been studied which iteratively reduce the game by eliminating some players' strategies, thus obtaining a smaller game while maintaining its Nash equilibria. In [11], for example, interesting results concerning the order in which such reductions are applied are described. We introduce two counterparts of such game-theoretic techniques that allow us to reduce a CP-net while maintaining its optimal outcomes. We also introduce a method of simplifying a CP-net by eliminating so-called redundant variables from the variables parent sets. Both techniques simplify the search for optimal outcomes in a CP-net.
In the other direction, we can use the techniques developed to reason about optimal outcomes of a CP-net to search for Nash equilibria of strategic games with parametrized preferences. We illustrate this point by introducing the notion of a hierarchical game with parametrized preferences and by explaining that such games have a unique Nash equilibrium that can be found in linear time.
In the final part of the paper we consider the relationship between strategic games and soft constraints, such as fuzzy, weighted and hard constraints. The appropriate notion of a strategic game is here that of a graphical game, see [13]. This is due to the fact that (soft) constraints usually involve only a small subset of the problem variables. This is in analogy with the fact that in a graphical game a player's payoff function depends only on a (usually small) number of other players.
We consider a natural mapping that associates with each soft constraint satisfaction problem (in short, a soft CSP or an SCSP) a graphical game. This mapping creates a direct corresponce between constraints and players' neighbourhoods. We show that, when using such a mapping, in general no relation exists between the notions of optimal solutions in soft CSPs and Nash equilibria in the corresponding games. On the other hand, for the class of strictly monotonic SCSPs (which includes in particular weighted constraints), every optimal solution corresponds to both a Nash equilibrium and Pareto efficient joint strategy. We also show that this mapping, when applied to a consistent CSP (that is, a satisfiable hard constraint satisfaction problem), defines a bijection between the solutions of the CSP and the set of joint strategies that are both Nash equilibria and Pareto efficient.
The latter holds in general, and not just for a subclass, if we consider a mapping from graphical games to soft CSPs which is independent of the constraint structure. This mapping, however, is less appealing from the computational complexity point of view since it requires that one considers all possible complete assignments, the number of which may be exponential in the size of the SCSP.
None of these two mappings are surjective, thus they cannot be used to pass from a generic graphical game to an SCSP. We also consider a mapping which goes in this direction. This mapping creates a soft constraint for each player, by looking at his neighbourhood. We show that this mapping defines a bijection between Pareto efficient joint strategies and optimal solutions of the SCSP.
The study of the relations among preference models coming from different fields such as AI and game theory has only recently gained attention. In [10] a mapping from the graphical games to hard CSPs has been defined, and it has been shown that the Nash equilibria of these games coincide with the solutions of the CSPs. We can use this mapping, together with our mapping from the graphical games to SCSPs, to identify the Pareto efficient Nash equilibria of the given game. In fact, these equilibria correspond to the optimal solutions of the SCSP obtained by joining the soft and hard constraints generated by the two mappings. The mapping of [10] leads to interesting results on the complexity of deciding whether a game has a pure Nash equilibrium or other kinds of desirable joint strategies.
In [14] a mapping from distributed constraint optimization problems (DCOPs) to graphical games is introduced, where the optimization criterion is to maximize the sum of utilities. By using this mapping, it is shown that the optimal solutions of the given DCOP are Nash equilibria of the generated game. This result is in line with our finding regarding strictly monotonic SCSPs, which include the class of problems considered in [14].
Organization of the paper
The paper is organized as follows. In Section 2 we introduce CP-nets, soft constraints, and strategic games. Next, in Section 3 we introduce a modification of the classical notion of strategic games considered in this paper. In Section 4 we show how to pass from CP-nets to so defined strategic games, while in Section 5 we deal with the opposite direction.
Then in Section 6 we show how to apply techniques developed in game theory to reason about CP-nets, while in Section 7 we study the other direction. Next, in Section 8 and 9 we study the relationship between soft CSPs and strategic games by relating optimal solutions of soft CSPs to Nash equilibria and Pareto efficient joint strategies. Finally, in Section 10 we summarize the main contributions of the paper.
Preliminary results of this research were reported in [2] and [3].
Preliminaries
In this section we recall the main notions regarding CP-nets, soft constraints, and strategic games.
CP-nets
CP-nets [6,5] (for Conditional Preference nets) are a graphical model for compactly representing conditional and qualitative preference relations. They exploit conditional preferential independence by decomposing an agent's preferences via the ceteris paribus assumption. Informally, CP-nets are sets of ceteris paribus (cp) preference statements. For instance, the statement "I prefer red wine to white wine if meat is served." asserts that, given two meals that differ only in the kind of wine served and both containing meat, the meal with a red wine is preferable to the meal with a white wine. On the other hand, this statement does not order two meals with a different main course. Many users' preferences appear to be of this type. CP-nets bear some similarity to Bayesian networks. Both utilize directed graphs where each node stands for a domain variable, and assume a set of features (variables) F = {X 1 , . . . , X n } with the corresponding finite domains D(X 1 ), . . . ,D(X n ). For each feature X i , a user specifies a (possibly empty) set of parent features P a(X i ) that can affect her preferences over the values of X i . This defines a directed graph, called dependency graph , in which each node X i has P a(X i ) as its immediate predecessors. A CP-net is said to be acyclic if its dependency graph does not contain cycles.
Given this structural information, the user explicitly specifies her preference over the values of X i for each complete assignment on P a(X i ). In this paper this preference is assumed to take the form of a linear order over D(X i ) [6,5]. 1 Each such specification is called below a preference statement for the variable X i . These conditional preferences over the values of X i are captured by a conditional preference table which is annotated with the node X i in the CP-net. An outcome is an assignment of values to the variables with each value taken from the corresponding domain.
As an example, consider a CP-net whose features are A, B, C and D, with binary domains containing f and f if F is the name of the feature, and with the following preference statements:
d : a ≻ a, d : a ≻ a, a : b ≻ b, a : b ≻ b, b : c ≻ c, b : c ≻ c, c : d ≻ d, c : d ≻ d.
Here the preference statement d : a ≻ a states that A = a is preferred to A = a, given that D = d. From the structure of these preference statements we see that P a(A) = {D}, P a(B) = {A}, P a(C) = {B}, P a(D) = {C} so the dependency graph is cyclic.
An acyclic CP-net is one in which the dependency graph is acyclic. As an example, consider a CP-net whose features and domains are as above and with the following preference statements:
a ≻ a, b ≻ b, (a ∧ b) ∨ (a ∧ b) : c ≻ c, (a ∧ b) ∨ (a ∧ b) : c ≻ c, c : d ≻ d, c : d ≻ d.
Here, the preference statement a ≻ a represents the unconditional preference for A = a over A = a. Also each preference statement for the variable C is a actually an abbreviated version of two preference statements. In this example we have P a(A) = ∅, P a(B) = ∅, P a(C) = {A, B}, P a(D) = {C}.
A worsening flip is a transition between two outcomes that consists of a change in the value of a single variable to one which is less preferred in the unique preference statement for that variable. By analogy we define an improving flip. For example, in the acyclic CP-net described in the previous paragraph, passing from abcd to abcd is a worsening flip since c is better than c given a and b. We say that an outcome α is better than the outcome β (or, equivalently, β is worse than α), written as α ≻ β, iff there is a chain of worsening flips from α to β. This definition induces a strict preorder over the outcomes. In the acyclic CP-net described in the previous paragraph, the outcome abcd is worse than abcd.
An optimal outcome is one for which no better outcome exists. So an outcome is optimal iff no improving flip from it exists. In general, a CPnet does not need to have an optimal outcome. As an example consider two features A and B with the respective domains {a, a} and {b, b} and the following preference statements:
a : b ≻ b, a : b ≻ b, b : a ≻ a, b : a ≻ a.
It is easy to see that then ab ≻ ab ≻ ab ≻ ab ≻ ab. Finding optimal outcomes and testing for optimality is known to be NPhard [6,5]. However, in acyclic CP-nets there is a unique optimal outcome and it can be found in linear time [6,5]. We simply sweep through the CP-net, following the arrows in the dependency graph, assigning at each step the most preferred value in the preference relation. For instance, in the CP-net above, we would choose A = a and B = b, then C = c and then D = d. The optimal outcome is therefore abcd.
Determining whether one outcome is better than another according to this order (a so-called dominance query) is also NP-hard even for acyclic CPnets, see [9]. Whilst tractable special cases exist, there are also acyclic CPnets in which there are exponentially long chains of worsening flips between two outcomes [9].
Hard constraints are enough to find optimal outcomes of a CP-net and to test whether a CP-net has an optimal outcome. In fact, given a CP-net one can define a set of hard constraints (called optimality constraints) such that their solutions are the optimal outcomes of the CP-net, see [7,20].
Indeed, take a CP-net N and consider a linear order ≻ over the elements of the domain of a variable X used in a preference statement for X. Let ϕ be the disjunction of the corresponding assignments used in the preference statements that use ≻. Then for each of such linear order ≻ the corresponding optimality constraint is ϕ → X = a j , where a j is the undominated element of ≻. The optimality constraints opt(N ) corresponding to N consist of the entire set of such optimality constraints, each for one such linear order ≻.
For example, the preference statements a ≻ a and (a ∧ b) ∨ (a ∧ b) : c ≻ c from the above CP-net map to the hard constraints A = a and (
A = a∧B = b) ∨ (A = a ∧ B = b) → C = c, respectively.
It has been shown that an outcome is optimal in the strict preorder over the outcomes induced by a CP-net N iff it is a satisfying assignment for opt(N ).
A CP-net is called eligible iff it has an optimal outcome. Even if the strict preorder induced by a CP-net has cycles, the CP-net may still be useful if it is eligible. All acyclic CP-nets are trivially eligible as they have a unique optimal outcome. We can thus test eligibility of any (even cyclic) CP-net by testing the consistency of the optimality constraints opt(N ). That is, a CP-net N is eligible iff opt(N ) is consistent.
Soft constraints
Soft constraints, see e.g. [4], allow us to express constraints and preferences. While constraints state which combinations of variable values are acceptable, soft constraints (also called preferences) allow for several levels of acceptance. A technical way to describe soft constraints is via the use of an algebraic structure called a c-semiring.
A c-semiring is a tuple A, +, ×, 0, 1 , where:
• A is a set, called the carrier of the semiring, and 0, 1 ∈ A;
• + is commutative, associative, idempotent, 0 is its unit element, and 1 is its absorbing element;
• × is associative, commutative, distributes over +, 1 is its unit element and 0 is its absorbing element.
Elements 0 and 1 represent, respectively, the highest and lowest preference. While the operator × is used to combine preferences, the operator + induces a partial order on the carrier A defined by
a ≤ b iff a + b = b.
Given a c-semiring S = A, +, ×, 0, 1 , and a set of variables V , each variable x with a domain D(x), a soft constraint is a pair def, con , where con ⊆ V and def : × y∈con D(y) → A. So a constraint specifies a set of variables (the ones in con), and assigns to each tuple of values from × y∈con D(y), the Cartesian product of the variable domains, an element of the semiring carrier A.
A soft constraint satisfaction problem (in short, a soft CSP or SCSP ) is a tuple C, V, D, S where V is a set of variables, with the corresponding set of domains D, C is a set of soft constraints over V and S is a c-semiring. Given an SCSP a solution is an instantiation of all the variables. The preference of a solution s is the combination by means of the × operator of all the preference levels given by the constraints to the corresponding subtuples of the solution, or more formally,
× c∈C def c (s ↓ conc ),
where × is the multiplicative operator of the semiring and def c (s ↓ conc ) is the preference associated by the constraint c to the projection of the solution s on the variables in con c .
A solution is called optimal if there is no other solution with a strictly higher preference.
Three widely used instances of SCSPs are:
• Classical CSPs (in short CSPs), based on the c-semiring {0, 1}, ∨, ∧, 0, 1 . They model the customary CSPs in which tuples are either allowed or not. So CSPs can be seen as a special case of SCSPs.
• Fuzzy CSPs, based on the fuzzy c-semiring [0, 1], max, min, 0, 1 .
In such problems, preferences are the values in [0, 1], combined by taking the minimum and the goal is to maximize the minimum preference.
• Weighted CSPs, based on the weighted c-semiring ℜ + , min, +, ∞ , 0 . Preferences are costs ranging over non-negative reals, which are aggregated using the sum. The goal is to minimize the total cost.
A simple example of a fuzzy CSP is the following one:
• three variables: x, y, and z, each with the domain {a, b};
• two constraints: C xy (over x and y) and C yz (over y and z) defined by: The unique optimal solution of this problem is bbb (an abbreviation for x = y = z = b). Its preference is 0.5.
Strategic games
Let us recall now the notion of a strategic game, see, e.g., [17]. A strategic game for a set N = {1, . . ., n} of n players (n > 1) is a tuple
(S 1 , . . ., S n , p 1 , . . ., p n ), where for each i ∈ [1..n]
• S i is the non-empty set of strategies available to player i,
• p i is the payoff function for the player i, so p i : S 1 × . . . × S n → R,
where R is the set of real numbers.
Given a sequence of non-empty sets S 1 , . . ., S n and s ∈ S 1 × . . . × S n we denote the ith element of s by s i , abbreviate N \ {i} to −i, and use the following standard notation of game theory, where i ∈ [1..n] and I := i 1 , . . ., i k is a subsequence of 1, . . ., n:
• s I := (s i 1 , . . ., s i k ), • (s ′ i , s −i ) := (s 1 , . . ., s i−1 , s ′ i , s i+1 , . . ., s n ), where we assume that s ′ i ∈ S i , • S I := S i 1 × . . . × S i k . A joint strategy s is called • a pure Nash equilibrium (from now on, simply Nash equilibrium) if p i (s) ≥ p i (s ′ i , s −i ) (1) for all i ∈ [1..n] and all s ′ i ∈ S i , • Pareto efficient if for no joint strategy s ′ , p i (s ′ ) ≥ p i (s) for all i ∈ [1..n] and p i (s ′ ) > p i (s) for some i ∈ [1..n].
Pareto efficiency can be alternatively defined by considering the following strict Pareto order < P on the n-tuples of reals:
(a 1 , . . ., a n )
< P (b 1 , . . ., b n ) iff ∀i ∈ [1..n] a i ≤ b i and ∃i ∈ [1..n] a i < b i .
Then a joint strategy s is Pareto efficient iff the n-tuple (p 1 (s), . . ., p n (s)) is a maximal element in the < P order on such n-tuples of reals.
To clarify these notions consider the classical Prisoner's Dilemma game represented by the following bimatrix representing the payoffs to both players:
C 2 N 2 C 1 3, 3 0, 4 N 1 4, 0 1, 1
Each player i represents a prisoner, who has two strategies, C i (cooperate) and N i (not cooperate). Table entries represent payoffs for the players (where the first component is the payoff of player 1 and the second one that of player 2).
The two prisoners gain when both cooperate (with a profit of 3 each). However, if only one of them cooperates, the other one will gain more (with a profit of 4). If none of them cooperates, both gain very little (a profit of 1 each), but more than the "cheated" prisoner whose cooperation is not returned (that is, 0).
Here the unique Nash equilibrium is (N 1 , N 2 ), while the other three joint strategies (C 1 , C 2 ), (C 1 , N 2 ) and (N 1 , C 2 ) are Pareto efficient.
A natural modification of the concept of strategic games, called graphical games, was proposed in [13]. These games stress the locality in taking decision. In a graphical game the payoff of each player depends only on the strategies of its neighbours in a given in advance graph structure over the set of players.
Formally, a graphical game for n players with the corresponding strategy sets S 1 , . . ., S n is defined by assuming a neighbour function neigh that given a player i yields its set of neighbours neigh(i). The payoff for player i is then a function p i from × j∈neigh(i)∪{i} S j to R.
By using the canonic extensions of these payoff functions to the Cartesian product of all strategy sets one can then extend the previously introduced concepts, notably that of a Nash equilibrium, to the graphical games. Further, when all pairs of players are neighbours, a graphical game reduces to a strategic game.
Strategic games with parametrized preferences
In game theory it is customary to study strategic games defined as above, in quantitative terms. A notable exception is [18] in which instead of payoff functions the linear quasi-orders on the sets of joint strategies are used.
For our purposes we need a different approach. To define it we first introduce the concept of a preference on a set A which in this paper denotes a strict linear order on A. We then assume that each player has to his disposal a preference relation ≻(s −i ) on his set of strategies parametrized by a joint strategy s −i of his opponents. So in our approach
• for each i ∈ [1..n] player i has a finite, non-empty, set S i of strategies available to him,
• for each i ∈ [1..n] and s −i ∈ S −i player i has a preference relation ≻(s −i ) on his set of strategies S i .
In what follows such a strategic game with parametrized preferences (in short a game with parametrized preferences, or just a game) for n players is represented by a tuple
(S 1 , . . ., S n , ≻(s −1 ), . . ., ≻(s −n )), where each s −i ranges over S −i .
It is straightforward to transfer to the case of games with parametrized preferences the basic notions concerning strategic games. In particular the following notions will be of importance for us (for the original definitions see, e.g., [18]), where G is a game with parametrized preferences specified as above:
• A strategy s i is a best response for player i to a joint strategy
s −i of his opponents if s i ≻(s −i ) s ′ i , for all s ′ i = s i . • A strategy s i is never a best response for player i if it is not a best
response to any joint strategy s −i of his opponents.
• A strategy s ′ i is strictly dominated by a strategy s i if s i ≻(s −i ) s ′ i , for all s −i ∈ S −i .
So according to this terminology a joint strategy s is a Nash equilibrium of G iff each s i is a best response to s −i . Note, however, that in our setup the underlying preferences are strict, so the above notions of a best response and Nash equilibrium correspond in the customary setting of strategic games to the notions of a unique best response and a strict Nash equilibrium. In particular, note that s is a Nash equilibrium of G iff for all i ∈ [1..n] and all s ′ i = s i s i ≻(s −i ) s ′ i , because to each joint strategy s −i a unique best response exists.
To clarify these definitions let us return to the above example of the strategic game that models the Prisoner's Dilemma. To view this game as a game with parametrized preferences we abstract from the numerical values and simply stipulate that
≻(C 2 ) := N 1 ≻ C 1 , ≻(N 2 ) := N 1 ≻ C 1 , ≻(C 1 ) := N 2 ≻ C 2 , ≻(N 1 ) := N 2 ≻ C 2 .
These orders reflect the fact that for each strategy of the opponent each player considers his 'not cooperate' strategy better than his 'cooperate' strategy.
It is easy to check that:
• for each player i the strategy C i is strictly dominated by N i (since N i ≻(C 3−i )C i and N i ≻(N 3−i )C i ),
• for each player i the strategy N i is a best response to the strategy N 3−i of his opponent,
• (as a result) (N 1 , N 2 ) is a unique Nash equilibrium of this game with parametrized preferences.
The framework of the games with parametrized preferences allows us to discuss only some aspects of the customary strategic games. In particular it does not allow us to introduce the notion of a mixed strategy, since the outcomes of playing different strategies by a player, given the joint strategy chosen by the opponents, cannot be aggregated. Also the notion of a Pareto efficient outcome does not have a counterpart in this framework because in general two joint strategies cannot be compared. For example, in the above modelling of the Prisoner's Dilemma game we cannot compare the joint strategies (N 1 , N 2 ) and (C 1 , C 2 ).
In the field of strategic games two techniques of reducing a game have been considered -by means of iterated elimination of strategies strictly dominated by a mixed strategy or of iterated elimination of never best responses to a mixed strategy (see, e.g., [18].) These techniques can be easily transferred to the games with parametrized preferences provided we limit ourselves to strict dominance by a pure strategy and never best responses to a pure strategy.
First, given such a game
G := (S 1 , . . ., S n , ≻(s −1 ), . . ., ≻(s −n )),
where each s −i ranges over S −i , and sets of strategies S ′ 1 , . . ., S ′ n such that
S ′ i ⊆ S i for i ∈ [1.
.n], we say that
G ′ := (S ′ 1 , . . ., S ′ n , ≻(s −1 ), . . ., ≻(s −n )),
where each s −i now ranges over S ′ −i , is a subgame of G, and identify in the context of G ′ each preference relation ≻(s −i ) with its restriction to S ′ i . We then introduce the following two notions of reduction between a game
G := (S 1 , . . ., S n , ≻(s −1 ), . . ., ≻(s −n )),
where each s −i ranges over S −i and its subgame
G ′ := (S ′ 1 , . . ., S ′ n , ≻(s −1 ), . . ., ≻(s −n )),
where each s −i ranges over S ′ −i :
• G → N BR G ′ when G = G ′ and for all i ∈ [1..n] each s i ∈ S i \ S ′ i is never a best response for player i in G, • G → S G ′ when G = G ′ and for all i ∈ [1..n] each s ′ i ∈ S i \S ′ i is strictly dominated in G by some s i ∈ S i .
In the literature it is customary to consider more specific reduction relations in which, respectively, all never best responses or all strictly dominated strategies are eliminated. The advantage of using the above versions is that we can prove the relevant property of both reductions by just one simple lemma, since by definition a strictly dominated strategy is never a best response and consequently
G → S G ′ implies G → N BR G ′ . Lemma 1 Suppose that G → N BR G ′ . Then s is a Nash equilibrium of G iff it is a Nash equilibrium of G ′ . Proof. ( ⇒ ) By definition each s i is a best response to s −i to G. So no s i is eliminated in the reduction of G to G ′ . ( ⇐ ) Suppose s is not a Nash equilibrium of G. So some s i is not a best response to s −i in G. Let s ′ i be a best response to s −i in G. (s ′ i exists since ≻(s −i ) is a linear order.)
So s ′ i is not eliminated in the reduction of G to G ′ and s ′ i is a best response to s −i in G ′ . But this contradicts the fact that s is a Nash equilibrium of G ′ .
2
Theorem 1 Suppose that G → * N BR G ′ , i.e.
, G ′ is obtained by an iterated elimination of never best responses from the game G.
(i) Then s is a Nash equilibrium of G iff it is a Nash equilibrium of G ′ .
(ii) If each player in G ′ has just one strategy, then the resulting joint strategy is a unique Nash equilibrium of G.
Proof.
(i) By the repeated application of Lemma 1.
(ii) It suffices to note that (s 1 , . . .s n ) is a unique Nash equilibrium of the game in which each player i has just one strategy, s i . 2
The above theorem allows us to reduce a game without affecting its (possibly empty) set of Nash equilibria or even, occasionally, to find its unique Nash equilibrium. In the latter case one says that the original game was solved by an iterated elimination of never best responses (or of strictly dominated strategies).
As an example let us return to the Prisoner's Dilemma game with parametrized preferences defined above. In this game each strategy C i is strictly dominated by N i , so the game can be solved by either reducing it in two steps (by removing in each step one C i strategy) or in one step (by removing both C i strategies) to a game in which each player i has exactly one strategy,
N i .
Finally, let us mention that [11] and [22] proved that all iterated eliminations of strictly dominated strategies yield the same final outcome. An analogous result for the iterated elimination of never best responses was established in [1]. Both results carry over to our framework of games with parametrized preferences by a direct modification of the proofs.
From CP-nets to strategic games
Consider now a CP-net with the set of variables {X 1 , . . ., X n } with the corresponding finite domains D(X 1 ), . . ., D(X n ). We write each preference statement for the variable X i as X I = a I : ≻ i , where for the subsequence I = i 1 , . . ., i k of 1, . . ., n:
• P a(X i ) = {X i 1 , . . ., X i k },
• X I = a I is an abbreviation for X i 1 = a i 1 ∧ . . . ∧ X i k = a i k ,
• ≻ i is a preference over D(X i ).
We also abbreviate D(X i 1 ) × . . . × D(X i k ) to D(X I ).
By definition, the preference statements for a variable X i are exactly all statements of the form X I = a I : ≻(a I ), where a I ranges over D(X I ) and ≻(a I ) is a preference on D(X i ) that depends on a I .
We now associate with each CP-net N a game G(N ) with parametrized preferences as follows:
• each variable X i corresponds to a player i,
• the strategies of player i are the elements of the domain D(X i ) of X i .
To define the parametrized preferences, consider a player i. Suppose P a(X i ) = {X i 1 , . . ., X i k } and let I := i 1 , . . ., i k . So I is a subsequence of 1, . . ., i − 1, i + 1, . . ., n and consequently each joint strategy a −i of the opponents of player i uniquely determines a sequence a I . Given now an arbitrary a −i we associate with it the preference relation ≻(a I ) on D(X i ) where X I = a I : ≻(a I ) is the unique preference statement for X i determined by a I .
In words, the preference of a player i over his strategies, assuming the joint strategy a −i of its opponents, coincides with the preference given by the CP-net over the domain of X i , assuming the assignment to its parents a I (which coincides with the projection of a −i over I). This completes the definition of G(N ).
As an example consider the first CP-net of Section 2. The corresponding game has four players A, B, C, D, each with two strategies indicated with f ,f for player F . The preference of each player on his strategies will depend only on the strategies chosen by the players which correspond to his parents in the CP-net. Consider for example player B. His preference over his strategies b andb, given the joint strategy of his opponents s −B = dac, is b ≻b. Notice that, for example, the same order holds for the opponents joint strategy s −B =dac, since the strategy chosen by the only player corresponding to his parent, A, has not changed.
We have then the following result.
Theorem 2 An outcome of a CP-net N is optimal iff it is a Nash equilibrium of the game G(N ).
From strategic games to CP-nets
We now associate with each game G with parametrized preferences a CP-net N (G) as follows:
• each variable X i corresponds to a player i,
• the domain D(X i ) of the variable X i consists of the set of strategies of player i,
• we stipulate that P a(X i ) = {X 1 , X i−1 , . . ., X i+1 , . . ., X n }, where n is the number of players in G.
Next, for each joint strategy s −i of the opponents of player i we take the preference statement
X −i = s −i : ≻(s −i ), where ≻(s −i ) is the preference relation on the set of strategies of player i associated with s −i .
This completes the definition of N (G). As an example of this construction let us return to the Prisoner's Dilemma game with parametrized preferences from Section 2.3. In the corresponding CP-net we have then two variables X 1 and X 2 corresponding to players 1 and 2, with the respective domains {C 1 , N 1 } and {C 2 , N 2 }. To explain how each parametrized preference translates to a preference statement take for example ≻(C 2 ) := N 1 ≻ C 1 . It translates to X 2 = C 2 : N 1 ≻ C 1 . We have now the following counterpart of Theorem 2.
Proposition 1 A joint strategy is a Nash equilibrium of the game G iff it is an optimal outcome of the CP-net N (G).
Proof. It suffices to notice that G(N (G)) = G and use Theorem 2.
2
The disadvantage of the above construction of the CP-net N (G) from a game G is that it always produces a CP-net in which all sets of parent features are of size n − 1 where n is the number of features of the CP-net. This can be rectified by reducing each set of parent features to a minimal one as follows.
Given a CP-net N , consider a variable X i with the parents P a(X i ), and take a variable Y ∈ P a(X i ). Suppose that for all assignments a to P a(X) − {Y } and any two values y 1 , y 2 ∈ D(Y ), the orders ≻(a, y 1 ) and ≻(a, y 2 ) on D(X i ) coincide.
We say then that Y is redundant in the set of parents of X i . It is easy to see that by removing all redundant variables from the set of parents of X i and by modifying the corresponding preference statements for X i accordingly, the strict preorder ≻ over the outcomes of the CP-nets is not changed.
Given a CP-net, if for all its variable X i the set P a(X i ) does not contain any redundant variable, we say that the CP-net is reduced .
By iterating the above construction every CP-net can be transformed to a reduced CP-net. As an example consider a CP-net with three features, X, Y and Z, with the respective domains {a 1 , a 2 }, {b 1 , b 2 } and {c 1 , c 2 }. Suppose now that P a(X) = P a(Y ) = ∅, P a(Z) = {X, Y } and that ≻(
a 1 , b 1 ) = ≻(a 2 , b 1 ), ≻(a 1 , b 2 ) = ≻(a 2 , b 2 ), ≻(a 1 , b 1 ) = ≻(a 1 , b 2 ), ≻(a 2 , b 1 ) = ≻(a 2 , b 2 )
. Then both X and Y are redundant in the set of parents of Z, so we can reduce the CP-net by reducing P a(Z) to ∅. Z becomes an independent variable in the reduced CP-net with the order over its domain which coincides with ≻(a 1 , b 1 ) (which is the same as the other three orders on the domain of Z).
In what follows for a CP-net N we denote by r(N ) the corresponding reduced CP-net. The following result, depicted in Figure 1, summarizes the relevant properties of r(N ) and relates it to the constructions of G(N ) and N (G). (iii) Each reduced CP-net N is a reduced CP-net corresponding to the game G(N ). Formally: N = r(N (G(N ))).
Proof.
Statements (i) and (ii) easily follow from the definition of function r and from from the construction of the game corresponding to a CP net. We will thus write explicitly only the proof of statement (iii). (iii) Given a reduced CP-net N , consider the CP-net N (G(N )). For each variable X i , P a(X i ) in N is a subset of P a(X i ) in N (G(N )), which is the set of all variables except X i . However, by the construction of the game corresponding to a CP-net and of the CP-net corresponding to a game, in each conditional preference table, if the assignments to the common parents are the same, the preference orders over X i are the same.
Let us now reduce N (G(N )) to obtain N ′ = r(N (G(N ))). Then P a(X i ) in N ′ coincides with P a(X i ) in N . Indeed, suppose there is a parent of X i in N which is not in N ′ . Since N is reduced, such a parent is not redundant in N . Thus the reduction r, when applied to N (G(N )), does not remove this parent since the orders in the conditional preference tables of N and N (G(N )) are the same.
Further, suppose there is a parent of X i in N ′ which is not in N . Since N ′ is reduced, such a parent is not redundant in N ′ . Thus it is also not redundant in N (G(N )). By the construction of N (G(N )), this parent is not redundant in N either.
2
Part (i) states that the reduction procedure r preserves the order over the outcomes. Part (ii) states that the construction of a game corresponding to a CP-net does not depend on the redundancy of the given CP-net. Finally, part (iii) states that the reduced CP-net N can be obtained 'back' from the game G(N ).
Game-theoretic techniques in CP-nets
Thanks to the established connections between CP-nets and games with parametrized preferences, we can now transfer to CP-nets the techniques of iterated elimination of strictly dominated strategies or of never best responses considered in Section 2.3. To introduce them in the context of CP-nets consider a CP-net N with the set of variables {X 1 , . . ., X n } with the corresponding finite domains D(X 1 ), . . ., D(X n ).
• We say that an element d i from the domain D(X i ) of the variable X i is a best response to a preference statement
X I = a I : ≻ i for X i if d i ≻ i d ′ i for all d ′ i ∈ D(X i ) such that d i = d ′ i .
• We say that an element d i from the domain of the variable X i is a never a best response if it is not a best response to any preference statement for X i .
• Given two elements
d i , d ′ i from the domain D(X i ) of the variable X i we say that d ′ i is strictly dominated by d i if for all preference statements X I = a I : ≻ i for X i we have d i ≻ i d ′ i .
By a subnet of a CP-net N we mean a CP-net obtained from N by removing some elements from some variable domains followed by the removal of all preference statements that refer to a removed element.
Then we introduce the following relation between a CP-net N and its subnet N ′ : N → N BR N ′ when N = N ′ and for each variable X i each removed element from the domain of X i is never a best response in N , and also introduce an analogous relation N → S N ′ for the case of strictly dominated elements. Since each strictly dominated element is never a best response, N → S N ′ implies N → N BR N ′ . The following counterpart of Theorem 1 then holds.
Theorem 3 Suppose that N → * N BR N ′ , i.e.,
the CP-net N ′ is obtained by an iterated elimination of never best responses from the CP-net N .
(i) Then s is an optimal outcome of N iff it is an optimal outcome of N ′ .
(ii) If each variable in N ′ has a singleton domain, then the resulting outcome is a unique optimal outcome of N .
2
To illustrate the use of this theorem reconsider the first CP-net from Section 2, i.e., the one with the preference statements d : a ≻ a, d : a ≻ a,
a : b ≻ b, a : b ≻ b, b : c ≻ c, b : c ≻ c, c : d ≻ d, c : d ≻ d. Denote it by N .
We can reason about it using the iterated elimination of strictly dominated strategies (which coincides here with the iterated elimination of never best responses, since each domain has exactly two elements).
We have the following chain of reductions: Indeed, in each step the removed element is strictly dominated in the considered CP-net. So using the iterated elimination of strictly dominated elements we reduced the original CP-net to one in which each variable has a singleton domain and consequently found a unique optimal outcome of the original CP-net N .
N → S N 1 → S N 2 → S N 3 → S N 4 ,
Finally, the following result shows that the introduced reduction relation on CP-nets is complete for acyclic CP-nets.
Theorem 4 For each acyclic CP-net N a subnet N ′ with the singleton domains exists such that N → * N BR N ′ . The outcome associated with N ′ is a unique optimal outcome of N and hence N ′ is unique.
Proof. First note that if N is an acyclic CP-net with some non-singleton domain, then N → N BR N ′ for some subnet N ′ of N . Indeed, suppose N is such a CP-net. By acyclicity a variable X exists with a non-singleton domain with no parent variable that has a non-singleton domain. So there exists in N exactly one preference statement for X, say X I = a I : ≻ i , where X I is the sequence of parent variables of X. Reduce the domain of X to the maximal element in ≻ i . Then for the resulting subnet N ′ we have N → N BR N ′ . Since N ′ is also acyclic and has one variable less with a non-singleton domain, by iterating this procedure we obtain a subnet N ′ with the singleton domains such that N → * N BR N ′ . The claim that the outcome associated with N ′ is a unique optimal outcome of N is a consequence of Theorem 3(ii).
2 The singleton domains obtained via the use of the → N BR reduction correspond to the unique optimal outcome of an acyclic CP-net, as defined in [6,5].
CP-net techniques in strategic games
The established relationship between CP-nets and strategic games with parametrized preferences also allows us to exploit the techniques developed for the CP-nets when studying such games.
One natural idea is to consider a counterpart of the notion of an acyclic CP-net. We call a game with parametrized preferences hierarchical if the CP-net r(N (G)) is acyclic.
We can introduce this notion directly, without using the CP-nets, by considering a partition of players 1, . . ., n in the game (S 1 , . . ., S n , ≻(s −1 ), . . ., ≻(s −n )), where each s −i ranges over S −i , into levels 1, . . ., k such that for each player i at level j and each s −i ∈ S −i the preference ≻(s −i ) depends only on the entries in s −i associated with the players from levels < j.
So a game is hierarchical if the players can be partitioned into levels 1, 2, . . . , k, such that each player at level j can express his preferences without taking into account the players at his level or higher levels (lower levels are more important).
We have then the following counterpart of Theorem 4.
Theorem 5 For each hierarchical game G a subgame G ′ with the singleton strategy sets exists such that G → * N BR G ′ . The resulting joint strategy associated with G ′ is a unique Nash equilibrium of G and hence G ′ is unique.
Proof. By an analogous argument as the one used in the proof of Theorem 4.
2
Given a hierarchical game G, by definition the CP-net r(N (G)) is acyclic. Thus we know that it has a unique optimal outcome which can be found in linear time. This means that the unique Nash equilibrium of G can be found in linear time by the usual CP-net techniques applied to r(N (G)).
Hierarchical games naturally represent multi-agent scenarios in which agents (that is, players of the game) can be partitioned into levels such that each agent can determine his preferences without consulting agents at his level or lower levels. Informally, agents at one level are 'more important' than agents at lower levels in the sense that they can take their decisions without consulting them.
A more general class of games is obtained by analogy to graphical games. We define a graphical game with parametrized preferences as follows. Given a neighbour function neigh we assume that for each player i and a joint strategy s i of his opponents, the preference ≻(s −i ) depends only on the entries in s −i associated with the players from neigh(i). Equivalently, we may just use the preference relations ≻ i s for each player i and each joint strategy s of the neighbours of i. Hierarchical games are then graphical games with parametrized preferences with acyclic neighbour graphs.
Given a CP-net N and the corresponding game G(N ), the dependency graph of N uniquely determines the neighbour function neigh between the players in G(N ). This allows us to associate with each CP-net N a graphical game with parametrized preferences. Conversely, each graphical game G with parametrized preferences uniquely determines a CP-net. It is obtained by proceeding as in Section 5 but by stipulating that the parent relation corresponds to the neighbour function neigh, that is, by putting
P a(X i ) := {X j | j ∈ neigh(i)}.
The counterparts of Theorems 1 and 2 then hold for CP-nets and graphical games with parametrized preferences.
Note that we arrived at the concept of a hierarchical game through the analogy with the acyclic CP-nets. To see a natural example of such games consider the problem of spreading a technology in a social network, inspired by the problems studied in [16] for the case of infinite number of players. We assume that the players (users) are connected in a network, which is a directed graph, and that there are k technologies (for example mobile telephone companies) t 1 , . . ., t k . Assume further that each user, given two technologies, prefers to use the one that is used by more of his neighbours in the network (for instance to cut down on the telephone costs).
We model this situation as a graphical game with parametrized preferences. We assume that each player i has k strategies, t 1 , . . ., t k , and for each joint strategy s of the neighbours of i we define the preference relation ≻ i s by putting t k ≻ i s t l iff |s(t k )| > |s(t l )| or (|s(t k )| = |s(t l )| and k < l),
where s(X) is the set of components of s that are equal to the strategy X. So we assume that in the case of a tie player i prefers a technology with the lower index.
We can now analyze the process of selecting a technology by exploiting the relation between hierarchical games and CP-nets. Namely, suppose that the above defined graphical game G with parametrized preferences is hierarchical. Then by virtue of Theorem 5 G → * N BR G ′ , where in G ′ each player has a single strategy, t 1 . The resulting joint strategy is then a unique Nash equilibrium of G. Additionally, by the corresponding order independence result mentioned at the end of Section 3, G ′ is a unique outcome of iterating the → N BR reduction.
This corresponds to an informal statement that when the neighbour function describes an acyclic graph, eventually technology t 1 is adopted by everybody. Because of the nature of the preference relations used above, this result actually holds for a larger class of graphical games with parametrized preferences.
They correspond to the following class of directed graphs. We call a directed graph well-structured if levels can be assigned to its nodes in such a way that each node has at least as many incoming edges from the nodes with strictly lower levels than from the other nodes. Of course, each directed acyclic graph is well-structured but other examples exist, see, e.g., Figure 2.
We have then the following result.
Theorem 6 Consider a graphical game G with parametrized preferences in which each player has k strategies, t 1 , . . ., t k , the preference relations ≻ i s are defined by (2), and the neighbour function describes a well-structured graph. Then G → * N BR G ′ , where in G ′ each player has a single strategy, t 1 , and the resulting joint strategy is a unique Nash equilibrium of G.
Proof. We prove by induction on the level m that Figure 2: A well-structured graph that is not acyclic where in G ′ each player of level ≤ m has a single strategy, t 1 . This yields then the desired conclusion about Nash equilibrium by Theorem 1. The claim holds for the lowest level, say 0, as then each player of level 0 has no neighbours and hence his strategies t 2 , . . ., t k can be eliminated as never best responses.
G → * N BR G ′ ,(3)
Suppose (3) holds for some level m. So we have G → * N BR G ′ , where in G ′ each player of level ≤ m has a single strategy, t 1 . Consider the players of level m + 1 in the game G ′ . Each of them has at least as many neigbours with the single strategy t 1 than with other sets of strategies. So each joint strategy of his neighbours has at least as many t 1 s as other strategies. Hence G ′ → * N BR G ′′ , where in G ′′ each player of level ≤ m + 1 has a single strategy, t 1 . Consequently G → * N BR G ′′ , which establishes the induction step. 2
The above example shows that graphical games with parametrized preferences can be used to provide a natural qualitative analysis of some problems studied in social networks. Expressing the process of selecting a technology using games with parametrized preferences, Nash equilibria and elimination of never best responses is more natural than using CP-nets. On the other hand we arrived at the relevant result about adoption of a single technology by searching for an analogue of Theorem 4 about acyclic CP-nets.
From SCSPs to graphical games
In this and the next section we relate the notion of optimality in soft constraints and graphical games. To obtain an appropriate match we assume that in graphical games payoffs are elements of a linearly ordered set A instead of the set of real numbers. (This precludes the use of mixed strategies but they are not needed here.) We denote then such games by (S 1 , . . . , S n , neigh, p 1 , . . . , p n , A),
where neigh is the given neighbour function.
In this section we define two mappings from SCSPs to a specific kind of graphical games. In what follows we focus on SCSPs based on c-semirings with the carrier linearly ordered by ≤ (e.g. fuzzy or weighted) and compare the concepts of optimal solutions in SCSPs with Nash equilibria and Pareto efficient joint strategies in the graphical games. In both mappings we identify the players with the variables. Since soft constraints link variables, the resulting game players are naturally connected, which explains why we use graphical games.
Local mapping
Given a SCSP P := C, V, D, S we define the corresponding graphical game for n = |V | players as follows:
• the players: one for each variable;
• the strategies of player i: all values in the domain of the corresponding variable x i ;
• the neighbourhood function: j ∈ neigh(i) iff the variables x i and x j appear together in some constraint from C;
• the payoff function of player i:
Let C i ⊆ C be the set of constraints involving x i and let X be the set of variables that appear together with x i in some constraint in C i (i.e., X = {x j | j ∈ neigh(i)}). Then given an assignment s to all variables in X ∪ {x i } the payoff of player i w.r.t. s is defined by:
p i (s) := Π c∈C i def c (s ↓ conc ).
We denote the resulting graphical game by L(P ) to emphasize the fact that the payoffs are obtained using local information about each variable, by looking only at the constraints in which it is involved.
One could think of a different mapping where players correspond to constraints. However, such a mapping can be obtained by applying the local mapping L to the hidden variable encoding [15] of the SCSP in input.
General case
In general, the concepts of optimal solutions of a SCSP P and the Nash equilibria of the derived game L(P ) are unrelated. Indeed, consider the fuzzy CSP defined at the end of Section 2.2. The corresponding game has:
• three players, x, y, and z;
• each player has two strategies, a and b;
• the neighbourhood function is defined by: where * stands for either a or b and where to facilitate the analysis we use the canonical extensions of the payoff functions p x and p z to the functions on {a, b} 3 . This game has two Nash equilibria: aaa and bbb. However, only bbb is an optimal solution of the fuzzy SCSP.
One could thus think that in general the set of Nash equilibria is a superset of the set of optimal solutions of the corresponding SCSP. However, this is not the case. Indeed, consider a fuzzy CSP with as before three variables, x, y and z, each with the domain {a, b}, but now with the constraints: Then aab, abb, bab and bbb are all optimal solutions but only aab and bbb are Nash equilibria of the corresponding graphical game.
SCSPs with strictly monotonic combination
Next, we consider the case when the multiplicative operator × is strictly monotonic. Recall that given a c-semiring A, +, ×, 0, 1 , the operator × is strictly monotonic if for any a, b, c ∈ A such that a < b we have c×a < c×b. (The symmetric condition is taken care of by the commutativity of ×.)
Note for example that in weighted CSP × is strictly monotonic, as a < b in the carrier means that b < a as reals, so for any c we have c + b < c + a, i.e., c × a < c × b in the carrier. In contrast, the fuzzy CSPs × are not strictly monotonic, as a < b does not imply that min(a, c) < min(b, c) for all c.
So consider now a c-semiring with a linearly ordered carrier and a strictly monotonic multiplicative operator. As in the previous case, given an SCSP P , it is possible that a Nash equilibrium of L(P ) is not an optimal solution of P . Consider for example a weighted SCSP P with
• two variables, x and y, each with the domain D = {a, b};
• one constraint C xy := {(aa, 3), (ab, 10), (ba, 10), (bb, 1)}.
The corresponding game L(P ) has:
• two players, x and y, who are neighbours of each other;
• each player has two strategies, a and b;
• the payoffs defined by: Notice that, in a weighted CSP we have a ≤ b in the carrier iff b ≤ a as reals, so when passing from the SCSP to the corresponding game, we have complemented the costs w.r.t. 10, when making them payoffs. In general, given a weighted CSP, we can define the payoffs (which must be maximized) from the costs (which must be minimized) by complementing the costs w.r.t. the greatest cost used in any constraint of the problem.
Here L(P ) has two Nash equilibria, aa and bb, but only bb is an optimal solution. Thus, as in the fuzzy case, we have that there can be a Nash equilibrium of L(P ) that is not an optimal solution of P . However, in contrast to the fuzzy case, the set of Nash equilibria of L(P ) is now a superset of the set of optimal solutions of P . In fact, a stronger result holds.
Theorem 7 Consider a SCSP P defined on a c-semiring A, +, ×, 0, 1 , where A is linearly ordered and × is strictly monotonic, and the corresponding graphical game L(P ). Then (i) Every optimal solution of P is a Nash equilibrium of L(P ).
(ii) Every optimal solution of P is a Pareto efficient joint strategy in L(P ).
Proof.
(i) We prove that if a joint strategy s is not a Nash equilibrium of game L(P ), then it is not an optimal solution of SCSP P .
Let a be the strategy of player x in s, and let s neigh(x) and s Y be, respectively, the joint strategy of the neighbours of x, and of all other players, in s. That is, V = {x} ∪ neigh(x) ∪ Y and we write s as (a, s neigh(x) , s Y ).
By assumption there is a strategy b for x such that the payoff p x (s ′ ) for the joint strategy s ′ := (b, s neigh(x) , s Y ) is higher than p x (s). (We use here the canonical extension of p x to the Cartesian product of all the strategy sets).
So by the definition of the mapping L
Π c∈Cx def c (s ↓ conc ) < Π c∈Cx def c (s ′ ↓ conc ),
where C x is the set of all the constraints involving x in SCSP P . But the preference of s and s ′ is the same on all the constraints not involving x and × is strictly monotonic, so we conclude that
Π c∈C def c (s ↓ conc ) < Π c∈C def c (s ′ ↓ conc ).
This means that s is not an optimal solution of P .
(ii) We prove that if a joint strategy s is not Pareto efficient in the game L(P ), then it is not an optimal solution of SCSP P . Since s is not Pareto efficient, there is a joint strategy s ′ such that p i (s) ≤ p i (s ′ ) for all i ∈ [1..n] and p i (s) < p i (s ′ ) for some i ∈ [1..n]. Let us denote with I = {i ∈ [1.
.n] such that p i (s) < p i (s ′ )}. By the definition of the mapping L, we have:
Π c∈C i def c (s ↓ conc ) < Π c∈C i def c (s ′ ↓ conc ),
for all i ∈ I and where C i is the set of all the constraints involving the variable corresponding to player i in SCSP P . Since the preference of s and s ′ is the same on all the constraints not involving any i ∈ I, and since × is strictly monotonic, we have:
Π c∈C def c (s ↓ conc ) < Π c∈C def c (s ′ ↓ conc ).
This means that s is not an optimal solution of P .
2
To see that there may be joint strategies that are both Nash equilibria and Pareto efficient but do not correspond to the optimal solutions, consider a weighted SCSP P with The corresponding game L(P ) has:
• two players, x and y, who are neighbours of each other;
• each player has two strategies: a and b; As above, when passing from an SCSP to the corresponding game, we have complemented the costs w.r.t. 10, when turning them to payoffs. L(P ) has two Nash equilibria: aa and bb. They are also both Pareto efficient. However, only aa is optimal in P .
Classical CSPs
Note that in the classical CSPs × is not strictly monotonic, as a < b implies that a = 0 and b = 1 but c ∧ a < c ∧ b does not hold then for c = 0. In fact, the above result does not hold for classical CSPs. Indeed, consider a CSP with:
• three variables: x, y, and z, each with the domain {a, b};
• two constraints: C xy (over x and y) and C yz (over y and z) defined by: This CSP has no solutions in the classical sense, i.e., each optimal solution, in particular baa, has preference 0. However, baa is not a Nash equilibrium of the resulting graphical game, since the payoff of player x increases when he switches to the strategy a.
C
On the other hand, if we restrict the domain of L to consistent CSPs, that is, CSPs with at least one solution with value 1, then it yields games in which the set of Nash equilibria that are also Pareto efficient joint strategies coincides with the set of solutions of the CSP.
Theorem 8 Consider a consistent CSP P and the corresponding graphical game L(P ). Then an instantation of the variables of P is a solution of P iff it is a Nash equilibrium and Pareto efficient joint strategy in L(P ).
Proof. Consider a solution s of P . In the resulting game L(P ) the payoff to each player is maximal, namely 1. So the joint strategy s is both a Nash equilibrium and Pareto efficient. Conversely, every Pareto efficient joint strategy in L(P ) corresponds to solution of P . 2
There are other ways to relate CSPs and games so that the CSP solutions and the Nash equilibria coincide. This is what is done in [10], where a mapping from the strategic games to CSPs is defined. Notice that our mapping goes in the opposite direction and it is not the reverse of the one in [10]. In fact, the mapping in [10] is not reversible.
Global mapping
The mapping L is in some sense 'local', since it considers the neighbourhood of each variable. An alternative 'global' mapping considers all constraints. More precisely, given a SCSP P = C, V, D, S , with a linearly ordered carrier A of S, we define the corresponding game on n = |V | players, GL(P ) = (S 1 , . . . , S n , p 1 , . . . , p n , A) by using the following payoff function p i for player i:
• given an assignment s to all variables in V p i (s) := Π c∈C def c (s ↓ conc ).
Notice that in the resulting game the payoff functions of all players are the same. Then the following result analogous to Theorem 8 holds.
Theorem 9 Consider an SCSP P over a linearly ordered carrier, and the corresponding graphical game GL(P ). Then an instantiation of the variables of P is an optimal solution of P iff it is a Nash equilibrium and Pareto efficient in GL(P ).
Proof. An optimal solution of P , say s, is a joint strategy for which all players have the same, highest, payoff. So no other joint strategy exists for which some player is better off and consequently s is both a Nash equilibrium and Pareto efficient. Conversely, every Pareto efficient joint strategy in GL(P ) has the highest payoff, so it corresponds to an optimal solution of P .
2
The global mapping GL has the advantage of providing a precise relationship between the optimal solutions and joint strategies that are both Nash equilibria and Pareto efficient. However, it has an obvious disadvantage from the computational point of view, since it requires to consider all the complete assignments of the SCSP.
From graphical games to SCSPs
Next, we define a mapping from graphical games to SCSPs. To define it we limit ourselves to SCSPs defined on c-semirings which are the Cartesian product of linearly ordered c-semirings (see Section 2.2).
The mapping
Given a graphical game G = (S 1 , . . . , S n , neigh, p 1 , . . . , p n , A) we define the corresponding SCSP L ′ (G) = C, V, D, S , as follows:
• each variable x i corresponds to a player i; • the c-semiring is A 1 × · · · × A n , (+ 1 , . . . , + n ), (× 1 , . . . , × n ), (0 1 , . . . , 0 n ), (1 1 , . . . , 1 n ) , the Cartesian product of n arbitrary linearly ordered semirings;
• soft constraints: for each variable x i , one constraint def, con such that:
-con = neigh(x i ) ∪ {x i };
def : Π y∈con D(y) → A 1 ×· · ·×A n such that for any s ∈ Π y∈con D(y), def(s) := (d 1 , . . . , d n ) with d j = 1 j for every j = i and d i = f (p i (s)), where f : A → A i is an order preserving mapping from payoffs to preferences (i.e., if r > r ′ then f (r) > f (r ′ ) in the c-semiring's ordering).
To illustrate it consider again the previously used Prisoner's Dilemma game:
C 2 N 2 C 1 3, 3 0, 4 N 1 4, 0 1, 1
Recall that in this game the only Nash equilibrium is (N 1 , N 2 ), while the other three joint strategies are Pareto efficient.
We shall now construct a corresponding SCSP based on the Cartesian product of two weighted semirings. This SCSP according to the mapping L ′ has: 2 • two variables: x 1 and x 2 , each with the domain {c, n};
• two constraints, both on x 1 and x 2 : The optimal solutions of this SCSPs are: cc, with preference 7, 7 , nc, with preference 10, 6 , cn, with preference 6, 10 . The remaining solution, nn, has a lower preference in the Pareto ordering. Indeed, its preference 9, 9 is dominated by 7, 7 , the preference of cc (since preferences are here costs and have to be minimized). Thus the optimal solutions coincide here with the Pareto efficient joint strategies of the given game. This is true in general.
Theorem 10 Consider a graphical game G and a corresponding SCSP L ′ (G). Then the optimal solutions of L ′ (G) coincide with the Pareto efficient joint strategies of G.
Proof. In the definition of the mapping L ′ we stipulated that the mapping f maintains the ordering from the payoffs to preferences. As a result each joint strategy s corresponds to the n-tuple of preferences (f (p 1 (s)), . . . , f (p n (s))) and the Pareto orderings on the n-tuples (p 1 (s), . . . , p n (s)) and (f (p 1 (s)), . . . , f (p n (s))) coincide. Consequently a sequence s is an optimal solution of the SCSP L ′ (G) iff (f (p 1 (s)), . . . , f (p n (s))) is a maximal element of the corresponding Pareto ordering.
2
We notice that L ′ is injective and, thus, can be reversed on its image. When such a reverse mapping is applied to these specific SCSPs, payoffs correspond to projecting of the players' valuations to a subcomponent.
Pareto efficient Nash equilibria
As mentioned earlier, in [10] a mapping is defined from the graphical games to CSPs such that Nash equilibria coincide with the solutions of CSP. Instead, our mapping is from the graphical games to SCSPs, and is such that Pareto efficient joint strategies and the optimal solutions coincide.
Since CSPs can be seen as a special instance of SCSPs, where only 1, 0, the top and bottom elements of the semiring, are used, it is possible to add to any SCSP a set of hard constraints. Therefore we can merge the results of the two mappings into a single SCSP, which contains the soft constraints generated by L ′ and also the hard constraints generated by the mapping in [10], Below we denote these hard constraints by H(G). We recall that each constraint in H(G) corresponds to a player, has the variables corresponding to the player and it neighbours and allows only tuples corresponding to the strategies in which the player has no so-called regrets. If we do this, then the optimal solutions of the new SCSP with preference higher than 0 are the Pareto efficient Nash equilibria of the given game, that is, those Nash equilibria which dominate or are incomparable with all other Nash equilibria according to the Pareto ordering. Formally, we have the following result.
Theorem 11 Consider a graphical game G and the SCSP L ′ (G) ∪ H(G). If the optimal solutions of L ′ (G) ∪ H(G) have global preference greater than 0, they correspond to the Pareto efficient Nash equilibria of G.
Proof. Given any solution s, let p be its preference in L ′ (G) and p ′ in L ′ (G) ∪ H(G). By the construction of the constraints H(G) we have that p ′ equals p if s is a Nash equilibrium and p ′ equals 0 otherwise. The remainder of the argument is as in the proof of Theorem 10.
2
For example, in the Prisoner's Dilemma game, the mapping in [10] would generate just one constraint on x 1 and x 2 with nn as the only allowed tuple. In our setting, when using as the linearly ordered c-semirings the weighted semirings, this would become a soft constraint with def(cc) := def(cn) := def(nc) = ∞, ∞ , def(nn) := 0, 0 .
With this new constraint, all solutions have the preference ∞, ∞ , except for nn which has the preference 9, 9 and thus is optimal. This solution corresponds to the joint strategy (N 1 , N 2 ) with the payoff (1, 1) (and thus preference (9,9)). This is the only Nash equilibrium and thus the only Pareto efficient Nash equilibrium.
This method allows us to identify among Nash equilibria the 'optimal' ones. One may also be interested in knowing whether there exist Nash equilibria which are also Pareto efficient joint strategies. For example, in the Prisoners' Dilemma example, there are no such Nash equilibria. To find any such joint strategies we can use the two mappings separately, to obtain, given a game G, both an SCSP L ′ (G) and a CSP H(G) (using the mapping in [10]). Then we should take the intersection of the set of optimal solutions of L ′ (G) and the set of solutions of H(G).
Conclusions
In this paper we related three formalisms that are commonly used to reason about optimal outcomes: strategic games, CP-nets and soft constraints. To this end we modified the concept of strategic games to games with parametrized preferences and showed that the optimal outcomes in CP-nets are exactly Nash equilibria of such games. This allowed us to exploit gametheoretic techniques in search for the optimal outcomes of CP-nets. In the other direction, we showed how the notion of an acyclic CP-net naturally leads to the concept of a hierarchical game. Such games have a unique Nash equilibrium.
We also considered the relation between graphical games and various classes of soft constraints. While for soft constraints there is only one notion of optimality, for graphical games there are at least two. In this paper we have considered Nash equilibria and Pareto efficient joint strategies. We showed that for a natural (local) mapping from soft CSPs to graphical games in general no relation exists between the notions of optimal solutions of soft CSPs and Nash equilibria. On the other hand, when in the SCSPs the preferences are combined using a strictly monotonic operator, the optimal solutions of the SCSP are included both in the Nash equilibria of the game and in the set of Pareto efficient joint strategies. In general the inclusions cannot be reversed. We have also exhibited a (global) mapping from the graphical games to a class of SCSPs such that the Pareto efficient joint strategies of the game coincide with the optimal solutions of the SCSP.
For the reverse direction we showed that for a natural mapping from the graphical games to a class of SCSPs the optimal solutions coincide with Pareto efficient joint strategies. Moreover, if we add suitable hard constraints to the soft constraints, optimal solutions coincide with Pareto efficient Nash equilibria.
The results of this paper clarify the relationship between various notions of optimality used in strategic games, CP-nets and soft constraints. These results can be used in a number of ways. One obvious way is to try to exploit computational results existing for one of these areas in another. This has been pursued already in [10] for games versus hard constraints. Using our results this can also be done for strategic games versus CP nets or soft constraints. For example, finding a Pareto efficient joint strategy involves mapping a game into a soft CSP and then solving it. Similar approach can also be applied to Pareto efficient Nash equilibria, which can be found by solving a suitable soft CSP.
C
xy := {(aa, 0.4), (ab, 0.1), (ba, 0.3), (bb, 0.5)}, C yz := {(aa, 0.4), (ab, 0.3), (ba, 0.1), (bb, 0.5)}.
Proof. ( ⇒ ) Take an optimal outcome o of N . Consider a player i in the game G(N ) and the corresponding variable X i of N . Suppose P a(X i ) = {X i 1 , . . ., X i k }. Let I := i 1 , . . ., i k , and let X I = o I : ≻(o I ) be the corresponding preference statement for X i . By definition there is no improving flip from o to another outcome, so o i is the maximal element in the order ≻(o I ).By the construction of the game G(N ), each outcome in N is a joint strategy in G(N ). Also, two outcomes are one flip away iff the corresponding joint strategies differ only in a strategy of one player. Given the joint strategy o considered above, we thus have that, if we modify the strategy of player i, while leaving the strategies of the other players unchanged, this change is worsening in ≻(o −i ), since ≻(o −i ) coincides with ≻(o I ). So by definition o is a Nash equilibrium of G(N ).( ⇐ ) Take a Nash equilibrium s of the game G(N ). Consider a variable X i of N . Suppose P a(X i ) = {X i 1 , . . ., X i k }. Let I := i 1 , . . ., i k , and let X I = s I : ≻(s I ) be the corresponding preference statement for X i .By definition for every strategy s ′ i = s i of player i, we have s i ≻(s −i ) s ′ i , so s i ≻(s I ) s ′ i since ≻(s −i ) coincides with ≻(s I ). So by definition s is an optimal outcome for N .2
Figure 1 :
1Relation between a CP-net N, its reduced form and corresponding games Proposition 2 (i) Each CP-net N and its reduced form r(N ) have the same order ≻ over the outcomes.(ii) For each CP-net N and its reduced form r(N ) we have G(N ) = G(r(N )).
•
N 1 results from N by removing a (from the domain of A) and the preference statements d : a ≻ a, d : a ≻ a, a : b ≻ b, • N 2 results from N 1 by removing b and the preference statements a : b ≻ b, b : c ≻ c, • N 3 results from N 2 by removing c and the preference statements b : c ≻ c c : d ≻ d, • N 4 results from N 3 by removing d from the domain of D and the preference statement c : d ≻ d.
neigh(x) := {y}, neigh(y) := {x, z}, neigh(z) := {y}; • the payoffs of the players are defined as follows: -for player x: p x (aa * ) := 0.4, p x (ab * ) := 0.1, p x (ba * ) := 0.3, p x (bb * ) := 0.5; -for player y: p y (aaa) := 0.4, p y (aab) := 0.3, p y (abb) := 0.1, p y (bbb) := 0.5, p y (bba) := 0.5, p y (baa) := 0.3, p y (bab) := 0.3, p y (aba) := 0.1; -for player z: p z ( * aa) := 0.4, p z ( * ab) := 0.3, p z ( * ba) := 0.1, p z ( * bb) := 0.5;
C
xy := {(aa, 0.9), (ab, 0.6), (ba, 0.6), (bb, 0.9)}, C yz := {(aa, 0.1), (ab, 0.2), (ba, 0.1), (bb, 0.2)}.
p x (aa) := p y (aa) := 7, p x (ab) := p y (ab) := 0, p x (ba) := p y (ba) := 0, p x (bb) := p y (bb) := 9.
•
two variables, x and y, each with domain D = {a, b}; • constraint C x := {(a, 2), (b, 1)}; • constraint C y := {(a, 4), (b, 7)}; • constraint C xy := {(aa, 0), (ab, 10), (ba, 10), (bb, 0)}.
•
the payoffs defined by: p x (aa) := 8, p y (aa) := 6, p x (ab) := p y (ab) := 0, p x (ba) := p y (ba) := 0, p x (bb) := 9, p y (bb) := 3.
xy := {(aa, 1), (ab, 0), (ba, 0), (bb, 0)}, C yz := {(aa, 0), (ab, 0), (ba, 1), (bb, 0)}.
•
the domain D(x i ) of the variable x i consists of the set of strategies of player i, i.e., D(x i ) := S i ;
-
constraint c 1 with def(cc) := 7, 0 , def(cn) := 10, 0 , def(nc) := 6, 0 , def(nn) := 9, 0 ; -constraint c 2 with def(cc) := 0, 7 , def(cn) := 0, 6 , def(nc) := 0, 10 , def(nn) := 0, 9 ;
In this we follow[5], where ties among values are initially allowed, (that is linear preorders are admitted) but in presentation only total orders are used. If ties are admitted, the notion of an optimal outcome of a CP-net has to be appropriately modified.
Recall that in the weighted semiring 1 equals 0.
AcknowledgementsWe thank the reviewers for helpful comments and suggestions.
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|
[] |
[
"The Matrix model and the non-commutative geometry of the supermembrane",
"The Matrix model and the non-commutative geometry of the supermembrane"
] |
[
"E G Floratos \nInstitute of Nuclear Physics\nNRCS Demokritos\nAthensGreece\n\nPhysics Department\nUniversity of Iraklion\nCreteGreece\n",
"G K Leontaris \nTheoretical Physics Division\nIoannina University\nGR-45110IoanninaGreece\n\nTheory Division\nCERN\n1211Geneva 23Switzerland\n"
] |
[
"Institute of Nuclear Physics\nNRCS Demokritos\nAthensGreece",
"Physics Department\nUniversity of Iraklion\nCreteGreece",
"Theoretical Physics Division\nIoannina University\nGR-45110IoanninaGreece",
"Theory Division\nCERN\n1211Geneva 23Switzerland"
] |
[] |
This is a short note on the relation of the Matrix model with the non-commutative geometry of the 11-dimensional supermembrane. We put forward the idea that Mtheory is described by the 't Hooft topological expansion of the Matrix model in the large N −limit where all topologies of membranes appear. This expansion can faithfully be represented by the Moyal Yang-Mills theory of membranes. We discuss this conjecture in the case of finite N , where the non-commutative geometry of the membrane is given be the finite quantum mechanics. The use of the finite dimensional representations of the Heisenberg group reveals the cellular structure of a toroidal supermembrane on which the Matrix model appears as a non-commutatutive Yang-Mills theory. The Moyal star product on the space of functions in the case of rational values of the Planck constant represents exactly this cellular structure. We also discuss the integrability of the instanton sector as well as the topological charge and the corresponding Bogomol'nyi bound.
|
10.1016/s0370-2693(99)01036-9
|
[
"https://arxiv.org/pdf/hep-th/9908106v2.pdf"
] | 7,774,859 |
hep-th/9908106
|
0f58fc1a72886bf615a645fd8923744bc69d4250
|
The Matrix model and the non-commutative geometry of the supermembrane
Aug 1999
E G Floratos
Institute of Nuclear Physics
NRCS Demokritos
AthensGreece
Physics Department
University of Iraklion
CreteGreece
G K Leontaris
Theoretical Physics Division
Ioannina University
GR-45110IoanninaGreece
Theory Division
CERN
1211Geneva 23Switzerland
The Matrix model and the non-commutative geometry of the supermembrane
Aug 1999arXiv:hep-th/9908106v2 22August 1999 * manolis@timaiosnucleardemokritosgr * * leonta@mailcernch -2-
This is a short note on the relation of the Matrix model with the non-commutative geometry of the 11-dimensional supermembrane. We put forward the idea that Mtheory is described by the 't Hooft topological expansion of the Matrix model in the large N −limit where all topologies of membranes appear. This expansion can faithfully be represented by the Moyal Yang-Mills theory of membranes. We discuss this conjecture in the case of finite N , where the non-commutative geometry of the membrane is given be the finite quantum mechanics. The use of the finite dimensional representations of the Heisenberg group reveals the cellular structure of a toroidal supermembrane on which the Matrix model appears as a non-commutatutive Yang-Mills theory. The Moyal star product on the space of functions in the case of rational values of the Planck constant represents exactly this cellular structure. We also discuss the integrability of the instanton sector as well as the topological charge and the corresponding Bogomol'nyi bound.
Introduction
One of the basic ingredients of M-threory [1,2] is the eleven dimensional (11-d) supermembrane for which some years ago [3] a consistent action has been written in a general background of 11-d supergravity. The supermembrane has a uniquely defined self-interaction, which comes in contrast to the superstring, from a infinite dimensional gauge symmetry apparent in the light-cone gauge as the area-preserving diffeomorphisms on the surface of the membrane.
Because of the absence of the dilaton field for the supermembrane, there is no topological expansion over all possible three manifolds analogous to the string case. The supermembrane, due to its unique self-interaction, is possible to break into other supermembranes so in a sense is already a second quantized theory but up to now there is no consistent perturbative expansion. In the light-cone gauge, and flat space-time, there are two classes of membrane vacua, points and tensionless strings, so a low-energy effective field theory of supermembrane massless excitations would be either eleven-dimensional supergravity or a field theory for tensionless strings. Hopefully, recent efforts for understanding the coupling of 11-d supergravity with the supermembrane will help to the construction of its effective low energy field theory [4].
In this letter, we present arguments that the Matrix model [5,6,7] describes the noncommutative geometry of the 11-d supermembrane, and M theory is the 't Hooft topological expansion of the Matrix model. We demostrate the existence of a topological charge and the corresponding Bogomol'nyi bound and we discuss the integrability of the instanton sector.
Non-commutative geometry of the membrane
It is a well known fact that the Matrix model [5,6,7] was one of the first ideas for the study of the dynamics of the bosonic membrane in the light-cone frame and in the approximation of finite number of oscillation modes [8,9]. The true dynamics would be determined by taking the limit of infinite number of modes. In the finite mode approximation the Hamiltonian of the membrane is exactly the same with SU(N) Yang-Mills (YM) classical mechanics and this system is known to possess interesting chaotic dynamics [10] and a discrete spectrum at the level of quantum mechanics (QM) [11]. Later on, Townsend et al [3,12] discovered the supermembrane Lagrangian in 11 dimensions and the finite mode truncation, as was expected, is described by the Hamiltonian of the supersymmetric SU(N) YM mechanics. It was found that the quantum mechanical spectrum of this model is continuous; at that time this was considered to be the end of the supermembrane as a fundamendal object replacing the superstring and producing all the low energy physics that could be useful for the unification of gauge and gravitational forces [13,14].
In ref. [15] the question of a deeper origin of the SU(N) YM classical mechanics as an approximation of the membrane dynamics was considered and it was found that SU(N) represents the Lie algebra of the finite Heisenberg group, which acts on a discretized membrane representing a toroidal discrete phase space. The membrane coordinates are approximated by N × N matrices (YM gauge fields), which represent collectively N 2 number of points in the target space. The large N-limit to reproduce the continuous surface of the membrane, should be such that all the positions of the SU(N) matrices are filled up in a continuous way and this limit has not been expressed, up to now, in a mathematically consistent way [16]. The non-commutative geometry of the discrete membrane is generated by the finite and discrete Heisenberg group and the space of functions on the surface of the membrane is the algebra of N × N complex matrices.
In modern language [17] the SU(N) YM classical mechanics is the YM theory on noncommutative 2-torus. It is interesting that the torus compactified Matrix model is equivalent with the M-theory compactification in a constant antisymmetric background gauge field. In this case, the Matrix model description becomes that of a gauge theory on a non-commutative torus [17,18,19,20,21].
The Heisenberg-Weyl group and the Moyal bracket
To start with, we introduce the irreducible representations of the finite Heisenberg group appropriate for the Matrix model non-commutative geometry of a toroidal membrane. The Hilbert space H Γ of the wave functions on the torus Γ = C/L of complex modulus τ = τ 1 + ıτ 2 , where L is the integer lattice, L = {m 1 + τ m 2 |(m 1 , m 2 ) ∈ Z × Z} is defined as the space of functions of complex argument z = x + ıy:
f (z) = n∈Z c n e ıπn 2 τ +2πınz (1) with norm ||f || 2 = e −2πy 2 /τ 2 |f (z)| 2 dxdy, τ 2 > 0.(2)
Consider the subspace H N (Γ) of H Γ with periodic Fourier coefficients {c n } n∈Z of period N:
c n = c n+N n ∈ Z, N ∈ N.(3)
The space H N (Γ) is N-dimensional and there is a discrete Heisenberg group, with generators S 1/N and T 1 acting as [27,28] (S 1/N f )(z) = n∈Z c n e 2πın/N e 2πınz+πın 2 τ 1 For a recent discussion see [23,24] and references therein. 2 For recent discussions see [26].
(
T 1 f )(z) = n∈Z c n−1 e 2πınz+πın 2 τ , c n ∈ C.(4)
On the N-dimensional subspace of vectors (c 1 , . . . , c N ) the two generators are represented by N × N matrices, Q, P
(S 1/N ) n 1 ,n 2 = Q n 1 ,n 2 = ω (n 1 −1) δ n 1 ,n 2 , (T 1 ) n 1 ,n 2 = P n 1 ,n 2 = δ n 1 −1,n 2 ,(5)
with ω = exp(2πı/N). They satisfy the Weyl relation QP = ωP Q.
The Heisenberg group elements are defined as
J r,s = ω r·s/2 P r Q s .(6)
These N × N matrices are unitary J † r,s = J −r,−s and periodic with period N, i.e. J N r,s = 1. They realize a projective representation of the discrete translation group Z N × Z N :
J r,s J r ′ ,s ′ = ω (r ′ s−rs ′ )/2 J r+r ′ ,s+s ′(7)
In ref [5] the finite N-Matrix model is considered as a non-commutative QM system (see also [15]), but the canonical commutation relations were not represented through the finite Heisenberg group basis J r,s . It is possible to define finite dimensional matricesp,q such that Q = e ıq and P = e −ıp
q ij = 2π N (s + 1 − i)δ ij ,p ij = −ı π N (−1) (i−j) sin π N (i−j)(8)
where N = 2s + 1 and s is an integer. Here we have shifted by s rows and columns of Q and P matrices defined in relations (5). These matrices satisfy new Heisenberg commutation relations, which have a very simple form [29] −
ı[q,p] ij = 2π N π N (i − j)(−1) (i−j) sin π N (i − j)(9)
when i = j and zero when i = j. The matrixq satisfies the torus compactification relations of the Matrix model, with corrections due to their finite size
P −1q P =q + 2π N I N − 2πI 0 ,(10)
where I N is the N × N identity matrix and I 0 the N × N diagonal matrix with diagonal elements {1, 0, . . . , 0}.
The bosonic part of the matrix model is the SU(N) YM classical mechanics and the gauge fields are linear combinations of the elements J r,s , i.e.,
A l (t) = N −1 r,s=0 A r,s l J r,s , l = 1, . . . , d − 1(11)
which can be considered as coherent states of the discrete and finite toroidal phase-space N × N lattice. The A l matrices are the non-commutative coordinates of the discrete membrane in d − 1 dimensions.
There is another representation of the standard quantum mechanics on the space of functions of the phase-space variables. This is the unique deformation of the Poisson bracket, the Moyal bracket [22,24] {{f,
g}} λ (u, v) = 1 λ sin (λ (∂ u ∂ v ′ − ∂ u ′ ∂ v )) f (u, v)g(u ′ , v ′ )| u=u ′ ,v=v ′(12)
Here, λ corresponds to the Planck constant and the Moyal bracket gives a structure of infinite dimensional algebra on the space of functions on the torus generated by
e r,s (u, v) = 1 2π e ı(ru+sv)(13)
where u, v ∈ [0, 2π] and r, s ∈ Z. This algebra is the trigonometric algebra of Fairlie Fletcher and Zachos [30]:
{{e r,s , e r ′ ,s ′ }} λ (u, v) = 1 2πλ sin (λ (rs ′ − r ′ s)) e r+r ′ ,s+s ′ (u, v)(14)
which also icludes the case λ = 2π N . This case gives the SU(N) algebra in the base J r,s :
[J r,s , J r ′ ,s ′ ] = −2ı sin 2π N (rs ′ − r ′ s) J r+r ′ ,s+s ′(15)
if the e r,s functions are identified with e r+kN,s+mN for k, m ∈ Z. The Heisenberg group matrices J r,s have been introduced by Weyl.
When λ → 0 (or N → ∞), we recover the Poisson algebra of the area preserving transformations of the torus [31] {e r,s , e r ′ ,
s ′ }(u, v) = (r ′ s − rs ′ ) 1 2π e r+r ′ ,s+s ′ .(16)
The Matrix model has various large N-limits. Up to now it is not known how to get the quantum mechanics of supermembrane starting from this model, even though, various compactifications indicate that it has membrane states as excitations. We believe that the appropriate limit is the 't Hooft topological expansion of the SU(N) YM-mechanics. To this end, we shall determine what happens to the Heisenberg group matrices J r,s in this limit. We observe that these matrices contain powers of the root of unity along two diagonals so we start with ω = e [32]. Since the limit → 0 replaces the Moyal bracket by Poisson, we get from Moyal YM theory the membrane. Higher order corrections to can be represented as membranes with attached handles on the initial membrane which is determined by the SU(N) chosen basis, in our case the torus.
In this limit, the light-cone gauge equations of motion for the membranë
X i = {X k , {X k , X i }}; i, k = 1, . . . , d − 1(17)
and the corresponding Gauss law {X i ,Ẋ i } = 0 are replaced bÿ
X i = {{X k , {{X k , X i }} }} (18) {{X i ,Ẋ i }} = 0, i, k = 1, . . . d − 1.(19)
When the space of functions on the toroidal membrane is replaced by the algebra of N ×N matrices, the coordinates of the membrane become the matrices A i (t), the velocity is the SU(N) electric field E i (t) =Ȧ i (t), and the magnetic field in three or seven dimensions is
B i (t) = 1 2 f ijk [A j , A k ]
where f ijk is the ǫ ijk totally antisymmetric symbol in three dimensions and Ψ ijk the octonionic multiplication table in seven dimensions [33].
The Moyal bracket generalizes both Poisson brackets and matrix commutators, so that one is tempted to consider a system where the Poisson bracket is replaced by the Moyal one [34]. The question of the appearance of Moyal bracket for physical reasons in the dynamics of membrane is up to now open. We know that there are other limits of the Matrix model, one leads to perturbative string field theory [35,36], and the Poisson limit in which the SU(N) symmetry becomes the area-preserving diffeomorphism group. We believe that the physical origin of the Moyal bracket is due to the presence of the antisymmetric background field C ijk in the light-cone gauge which gives a 'magnetic' flux (Hall effect), trasforming the surface of the membrane into a non-commutative phase-space [37]. This is true for open membranes where the topological term of the action receives a contribution from the boundary.
Topological charge, Bogomol'nyi bound and Integrability.
In order to explain the appearance of non-abelian electric-magnetic type of duality in the membrane theory, we recall that for YM-potentials independent of space coordinates the self-duality equation in the gauge A 0 = 0 iṡ
A i = 1 2 ǫ ijk [A j , A k ], i, j, k = 1, 2, 3(20)
According to ref [38] the only non-trivial higher dimensional YM self-duality equations exist in 8 space-time dimensions which, for the 7-space coordinate independent potentials, can be written (in the A 0 = 0 gauge) aṡ
A i = 1 2 Ψ ijk [A j , A k ], i, j, k = 1, · · · 7(21)
where Ψ ijk is the multiplication table of the seven imaginary octonionic units.
It is now tempting to take the large N-limit and replace the commutators by Poisson (Moyal) brackets to obtain the self-duality equations for membranes (non-commutative instantons for the Moyal case). In this limit we replace the gauge potentials A i by the membrane coordinates X i . Then, the 3-d system is [39,40],
X i = 1 2 ǫ ijk {X j , X k }, i, j, k = 1, 2, 3,(22)
while in seven space dimensions [34,41]
X i = 1 2 Ψ ijk {X j , X k }, i, j, k = 1, · · · , 7(23)
and correspondingly for the case of Moyal brackets in three dimensions [42,43] and in seven dimensions [24]. It is easy to see that the self-duality membrane equations, imply the second order Euclidean-time, equations of motion in the light-cone gauge as well as the Gauss law.
One striking feature of the self-duality membrane equations is their simple geometrical meaning [39,41]. These equations state that the normal vector at a point of the membrane surface and the velocity at the same point are parallel (self-dual) or anti-parallel (anti-selfdual). The possibility to write down self-duality equations based on the existence of vector cross-product comes from the existence of the quaternionic and octonionic algebras. Since these are the only existing division algebras the 3 and 7 dimensions are unique [33] 3 . The validity of this geometrical statement could be extended in a general curved space-time background as a definition of the self-dual membranes.
If one takes the limit where the commutator of matrices is replaced by commutator of operators or the Moyal bracket, then the self-duality equations become the Moyal Nahm or Moyal-Bogomol'nyi equations of [34].
The membrane instantons carry a topological charge density [45] which satisfies a Bogomol'nyi bound [46]:
Ω(X) = 1 3! ǫ abc f ijk X i a X j b X k c(24)
where X i a = ∂ ξa X i , a, b, c = 1, 2, 3 and f ijk = ǫ ijk when i, j, k = 1, 2, 3 and f ijk = Ψ ijk for i, j, k = 1, · · · , 7. This topological charge density defines the topological charge of the membrane
Q = 1 V 3 d 3 ξΩ(X)(25)
where V 3 is the volume of the integration region. The topological charge Q is an integer and represents the degree of the map from the membrane to its world volume. We display below the convenient representation of the topological charge which will help us demonstrate that it is a lower bound of the membrane action for topologically non-trivial membranes
Ω(X) = 1 2Ẋ i f ijk {X j , X k } = 1 2 {X j , X k } 2(26)
where the self-duality equations as well as the properties of f ijk in three and seven dimensions have been used. The topological charge of the membrane can now be written as
Q = 1 2V 3 M d 3 ξ{X j , X k } 2(27)
The minimum value of Q (Q = 1) is obtained for the membrane instanton compactified on a world-volume torus,
X 1 = √ 2σ 1 , X 2 = √ 2σ 2 and X 3 = 2t.
The Euclidean action can be written as
S = 1 V 3 d 3 ξ 1 2Ẋ 2 i + 1 4 {X j , X k } 2 (28) From the inequality (Ẋ i ± 1 2 f ijk {X j , X k }) 2 ≥ 0 we derive that, S ≥ Q(29)
and the equality holds only for the self-dual or anti-self-dual membranes. So the self-dual or anti-self-dual membranes are BPS Euclidean-time membrane world-volume solitons. As we have seen in ref [47], the 3−d and 7-d self-dual solutions preserve 8 and 1 supersymmetries respectively or 1/2 and 1/16 th of the supersymmetry of the light-cone supermembrane Hamiltonian. This is a direct consequence of the above Bogomol'nyi bound and the SO(3) and G 2 rotational space symmetry of the above cases.
The role of the membrane instantons is important in developing a perturbative expansion. Configurations of the membrane around instantons cannot collapse to points or strings, because they have different topological charge. The 3-index antisymmetric gauge field which is so crucial for the uniqueness of the supermembrane Lagrangian participates in the bosonic part through the Cern-Simons term. If its vacuum expectation value is non-zro and proportional to Ψ ijk (in the corresponding 7 dimensions), then the topological charge defined above, separates the functional integral into membrane topological sectors. Going now to the case of Moyal-Nahm equations, there is a corresponding topological charge without an obvious geometrical meaning and the Bogomol'nyi bound is valid in this case too. This bound is important for the stability of the corresponding Moyal-Nahm instantons. Recent discussions on the role of instantons in non-commutative YM theories (non-commutative instantons) imply that they can be considered as regularizations of small size instantons in standard YM theories (see e.g. [48]). The case of Moyal-Nahm equations could be considered as non-commutative membrane instantons which regularize the Poisson or membrane case.
We now make few remarks on the integrability of the self-dual equations. The 3-d selfduality system has a Lax pair and an infinite number of conservation laws [39,40]. In order to see this, we first rewrite the self-duality equations in the forṁ
X + = i{X 3 , X + },Ẋ − = i{X 3 , X − },Ẋ 3 = 1 2 i{X + , X − },(30)
where
X ± = X 1 ± iX 2(31)
The Lax pair equations can be written aṡ
ψ = L X 3 +λX − ψ,ψ = L 1 λ X + −X 3 ψ,(32)
where the differential operators L f are defined as
L f ≡ i ∂f ∂φ ∂ ∂ cos θ − ∂f ∂ cos θ ∂ ∂φ .(33)
The compatibility condition of (32) is
[∂ t − L X 3 +λX − , ∂ t − L 1 λ X + −X 3 ] = 0,(34)
from which, comparing the two sides for the coefficients of the powers 1 λ , λ 0 , λ 1 of the spectral parameter λ, we find (30). From the linear system (32), using the inverse-scattering method, one could in principle construct all solutions of the self-duality equations.
The infinite number of conservation laws are derived as follows: from the Cartesian formulation
dX i dt = 1 2 ǫ ijk {X j , X k }(35)
contracting with a complex 3-vector u i such that
u i = ǫ ijk u j v k ,(36)
where u i u i = 0, and v is another complex vector with v i v i = −1 and u i v i = 0, we find,
du · X dt = {u · X, v · X}(37)
The latter is a Lax pair type equation, which implies
d dt d 2 ξ(u · X) n = 0(38)
Applying the same method in seven dimensions with two complex 7-vectors u i , v i such that u i u i = 0, v i v i = −1 and u i v i = 0, leads to the equation
du · X dt = {u · X, v · X} + 1 2 φ jklm u j v k {X l , X m }(39)
The curvature tensor φ jklm is defined as the dual of Ψ ijk in seven dimensions. When equation (39) is restricted to three dimensions we recover (37). We observe that the presence of the curvature tensor is an obstacle for the integrability. At this point, we may look for an extended definition of integrability replacing the zero-curvature condition with the octonionic curvature one. We can restrict the above equation in particular subspaces of solutions where integrability appears. One possibility is the factorization of the time [41].
We conclude with a few remarks. In this note we have given arguments that the Matrix model describes a non-commutative YM theory for the supermembrane in the presence of background three-index antisymmetric gauge fields. If this conjecture is true, it implies that the excitations of this model in various compactifications are also physical excitations of the supermembrane. So the supermembrane should contain 11-d supergravity at least in weak coupling limits given by small radii of the compactification manifolds. It is tempting to calculate correlation functions of membrane observables using the Matrix model and then take the large N-limit as was discussed in section 3. On the other hand, perturbation theory for the supermembrane could be defined through the expansion in the parameter /N, with M/N → for M, N → ∞. In this expansion all the topologies of the membrane appear as splitting and joining interactions The other known large N-limit [35,36] gives the string perturbation theory as a QM sector of the supermembrane.
It is well known that the usual Quantum Mechanics can be represented on functions of the phase-space variables, with the Moyal bracket[22] 1 replacing the classical Poisson bracket. Recently the vertex operators of open strings in an external antisymmetric gauge field B µν were found to obey non-commutative relations of the Weyl type, which induces a Moyal bracket structure on the space of functions on the string momenta [25] 2 .
2πı M N , (M, N co-prime integers). The correct large N-limit for SU(N) is the inductive one, i.e., SU(N) → SU(N + 1) → SU(N + 2) . . . which we get if we let M, N → ∞ with M/N = constant. Note that the constant = 2π M N can be identified with the flux of the 3-index antisymmetric gauge field per unit membrane area. The Weyl relations become the Heisenberg group relations for an infinite phase-space lattice and in the Fourier transform space of both canonical variables the Matrix model describes a toroidal continuous membrane with Matrix commutators replaced by Moyal brackets
For other approach to self-duality see also[44].
As this work was written, we have been kindly informed that the Moyal limit of the Matrix model has been studied in connection with the higher derivative corrections to the Born-Infeld Lagrangians for the D2-brane[49]. For a very recent, interesting paper on D-branes in group manifolds, see[50]One of us (EGF) would like to thank prof. Albert Schwarz for a valuable discussion.
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|
[] |
[
"The Companion Model-a Canonical Model in Graph Signal Processing",
"The Companion Model-a Canonical Model in Graph Signal Processing"
] |
[
"Student Member, IEEEJohn Shi \nDepartment of Electrical and Computer Engineering\nCarnegie Mellon University\n15217PittsburghPAUSA\n",
"Fellow, IEEEJosé M F Moura \nDepartment of Electrical and Computer Engineering\nCarnegie Mellon University\n15217PittsburghPAUSA\n"
] |
[
"Department of Electrical and Computer Engineering\nCarnegie Mellon University\n15217PittsburghPAUSA",
"Department of Electrical and Computer Engineering\nCarnegie Mellon University\n15217PittsburghPAUSA"
] |
[] |
This paper introduces a canonical graph signal model defined by a canonical graph and a canonical shift, the companion graph and the companion shift. These are canonical because, under standard conditions, we show that any graph signal processing (GSP) model can be transformed into the canonical model. The transform that obtains this is the graph -transform (G T) that we introduce. The GSP canonical model comes closest to the discrete signal processing (DSP) time signal models: the structure of the companion shift decomposes into a line shift and a signal continuation just like the DSP shift and the GSP canonical graph is a directed line graph with a terminal condition reflecting the signal continuation condition. We further show that, surprisingly, in the canonical model, convolution of graph signals is fast convolution by the DSP FFT.
| null |
[
"https://arxiv.org/pdf/2203.13791v1.pdf"
] | 247,748,629 |
2203.13791
|
09a73419228e6b15d9069afe75c786ba5165e8b6
|
The Companion Model-a Canonical Model in Graph Signal Processing
Student Member, IEEEJohn Shi
Department of Electrical and Computer Engineering
Carnegie Mellon University
15217PittsburghPAUSA
Fellow, IEEEJosé M F Moura
Department of Electrical and Computer Engineering
Carnegie Mellon University
15217PittsburghPAUSA
The Companion Model-a Canonical Model in Graph Signal Processing
1Graph Signal ProcessingGSPGSP spSpectral ShiftSignal RepresentationsModulation
This paper introduces a canonical graph signal model defined by a canonical graph and a canonical shift, the companion graph and the companion shift. These are canonical because, under standard conditions, we show that any graph signal processing (GSP) model can be transformed into the canonical model. The transform that obtains this is the graph -transform (G T) that we introduce. The GSP canonical model comes closest to the discrete signal processing (DSP) time signal models: the structure of the companion shift decomposes into a line shift and a signal continuation just like the DSP shift and the GSP canonical graph is a directed line graph with a terminal condition reflecting the signal continuation condition. We further show that, surprisingly, in the canonical model, convolution of graph signals is fast convolution by the DSP FFT.
I. INTRODUCTION
In DSP, a sample of a real valued time signal like a sample of a segment of speech or of audio is indexed by the time instant at which the sample occurs. Similarly, the intensity or color of an image pixel is indexed by the location ( , ) of the pixel. Both indices, time and pixel ( , ), usually take values on regularly spaced one dimensional (1D) or two dimensional (2D) grids. 1 In contrast, the samples of graph signals like temperature readings in a network of meteorological stations or like voltages in a electric grid are indexed by the weather stations or by the buses of the power grid network, seemingly placed in arbitrary locations in space. This indexing structure is better described by a graph. Graph Signal Processing (GSP) [1]- [4] develops methods to analyze and study data indexed or defined on graphs, extending Discrete Signal Processing (DSP) to these signals [5]- [8].
Like with DSP, GSP commonly describes the graph signals by their values at the vertices = 0, · · · , − 1 of the indexing graph = ( , ), with vertex set and edge set . In GSP, the adjacency matrix of is taken as the shift [1]- [3]. The shift is the basic building block of linear shift invariant (LSI) filters that are polynomials ( ) of . Alternatively, the signals can be described by their graph spectrum = GFT , where GFT is the graph Fourier transform. The GFT is the inverse of the matrix of eigenvectors of and diagonalizes the shift. In [9], when studying graph sampling, we introduced a spectral graph shift . This led us to consider a new "spectral" graph signal model where the spectral LSI filters are polynomials ( ) in rather than in . The shift [9] replicates for GSP, with appropriate interpretation, the DSP property that shifting a signal in one domain, phase shifts the signal in the other domain, and viceversa.
The vertex signal and the graph spectrum , are two ways of describing the same graph signal but with respect to two different bases, the vertex standard Euclidean basis and the graph Fourier basis. Each of these signal descriptions has its own advantages. The vertex basis is the natural one, since the data is often collected at each node. The graph Fourier basis decomposes the signal model space into invariant subspaces (for diagonalizable shifts, these are the one-dimensional eigenvector spaces) and signals aligned with these invariant subspaces are invariant to linear graph filtering.
Contributions. This paper considers the following question: Is there, for arbitrary indexing graph , a signal model that replicates as closely as possible the DSP signal model. We interpret this question in terms of the action of the shift on the signal . In DSP, is a delayed signal, i.e., the signal samples are shifted downwards and then the signal is wrapped around so that the last signal sample −1 reappears at the top of . This is known as periodic signal extension. The answer to this question for generic graph turns out, surprisingly at first sight, to be yes. In other words, there is a graph signal representation that reproduces many of the characteristics of DSP starting with a GSP shift that moves downwards the graph signal (details in section V). For reasons that will become clear in sections IV and V, we call the signal transform that achieves this the graph -transform (G T), and we refer to the signal model as the graph companion model. Surprising facts associated with this signal model include: 1) the graph signal shift delays the graph signal; 2) like in DSP, the model is defined by the graph frequencies (eigenvalues of ), with no role played by the eigenvectors of ; 3) it conduces to a canonical (weighted) shift and a canonical (weighted) graph, regardless of the underlying graph ; and 4) it leads to a fast convolution of graph signals using the DSP FFT.
Brief review of the literature. The GSP literature is vast, by now covering many topics in processing graph signals. The approach in [1]- [3] identifies as basic building block the shift filter , building on the Algebraic Signal Processing in [10]- [14]. The approach in [4] departs from a variational operator, the graph Laplacian , motivated by for example earlier work from spectral graph theory [15]- [17], from work extending wavelets to data from irregularly placed sensors in sensor networks [18]- [21], and from research on sampling graph based data [22,23]. A comprehensive review covering both approaches and illustrating many different applications of GSP is [24].
Many additional topics have been considered in GSP. A sample of these include: alternative (unitary, but not local) shift operators [25,26]; approximating graph signals [27]; extensive work on sampling of graph signals, e.g., [28]- [33], see the recent review [34]; extending classical multirate signal processing to graphs [35,36]; an uncertainty principle for graph signals [37]; the study of graph diffusions [38]; graph signal recovery [39]; interpolation and reconstruction of graph signals [40,41]; stationarity of graph processes [42]; learning graphs from data [43]- [45]; or non-diagonalizable shifts and the graph Fourier transform [46].
All the references above describe graph signals by their standard (node or vertex) representation or their spectral representation , but practically none has discussed or studied other graph signal representations or the issues related to signal representations that we pursue here.
Summary of the paper. Section II casts DSP in the graph framework, provides background on GSP, and introduces graph impulses both in the vertex and the spectral domains. Section III introduces signal representations in general and then the two common ones, the vertex, standard, or Euclidean signal representation and the Fourier or spectrum representation. Section IV introduces the vertex impulsive representation and the graph -transform. Section V shows that the vertex impulsive representation leads to a canonical shift, the companion shift, and a canonical directed graph, the companion graph. These replicate the structure of the cyclic shift and the cyclic (time) graph, with appropriate boundary condition, given by the Cayley-Hamilton Theorem. The graph companion model comes closest to many DSP concepts. In particular, it requires only knowledge of the eigenfrequencies of , not its eigenvectors. Section VI extends the graph companion model for to its spectrum . Section VII summarizes the relations between the different GSP signal domains and shows how these GSP models coalesce into only two for DSP. Section VIII uses the companion signal model to introduce a fast convolution for graph signals using the DSP FFT. Finally, section IX presents concluding remarks.
II. GSP BACKGROUND
This section reviews briefly GSP following [1]- [3]. Let = ( , ) be a graph of order , i.e., with vertex or node set of cardinality | | = , and with edge set . The graph is arbitrary, possibly directed, undirected, or mixed with directed and undirected edges. The graph can be specified by an adjacency matrix , where = 1 if there is a directed edge from node to node or = 0 otherwise. 2 Remark 1 ( equivalence class). The adjacency matrix depends on the ordering of the nodes of . Different node orderings are related by permutations and the corresponding adjacency matrices are conjugated by , i.e., −1 = . In other words, adjacency matrices describing the same graph are an equivalence class under the symmetric group of per-2 Computer Science reverses this convention and the adjacency is .
mutations. We assume that a representative of this class has been chosen, by fixing the labeling order of the nodes in , which becomes now an ordered set, and identify graph with adjacency matrix rather than with the class of adjacency matrices. 3 A graph signal is an (ordered) -tuple = ( 0 , · · · , −1 ) that assigns to each node ∈ , = 0, · · · , − 1, the graph sample . In other words, graph signal samples are indexed by the nodes of the graph. In this paper, we consider the graph samples to be complex valued, ∈ C. The graph signal is then a vector in C , the -dimensional vector space over the complex field C.
A. DSP as GSP
To motivate GSP, we start by casting DSP in the context of GSP, see for example [1,3]. Consider the node directed cycle graph in figure 1 with adjacency matrix
= 0 0 . . . 0 1 1 0 . . . 0 0 . . . 1 . . . . . . . . . . . . . . . . . . . . . . . . 0 0 . . . 1 0 .(1)
The nodes of the cycle graph represent the time ticks and are naturally ordered. The time signal samples are indexed by the nodes of . Matrix is also the matrix representation of the shift −1 in DSP (assuming periodic boundary conditions [1,11,12])
0 1 · · · −1 = −1 0 · · · −2 .(2)
The eigenvalues (or a normalized version) and the eigenvectors of the cyclic in (1) are the discrete time frequencies and the discrete time harmonics, spectral components, or eigenmodes of time signals
= − 2 , , = 0, · · · , − 1 = 1 √ 1 2 · · · 2 ( −1)
.
The Discrete Fourier Transform, DFT, is obtained through the diagonalization of
= DFT Λ DFT, where Λ = diag [ ](3)= [ 0 · · · −1 ] = 1 − 2 · · · − 2 ( −1)(4)DFT = 1 √ 1 1 · · · 1 1 − 2 · · · − 2 ( −1)
. . . . . . . . .
1 − 2 ( −1) · · · − 2 ( −1) ( −1) (5) = 1 √ 0 · · · −1 (6) DFT = [ 0 · · · −1 ] .(7)
The matrix Λ in (3) is the diagonal matrix of the eigenvalues, i.e., its diagonal entries are the graph frequencies. Equation (4) defines the graph frequency vector and equation (6) uses the notation = · · · to represent times the Hadamard or entrywise product of the graph frequency vector . By (6) and (7), in DSP, the powers of the graph frequency vector are conjugates of the eigenvectors, 1 √ = * , and, by (7), the columns of the DFT are the eigenvectors of . The DFT is symmetric, DFT = DFT , and unitary, DFT −1 = DFT .
B. GSP basics
We now let be the adjacency matrix of an arbitrary (directed or undirected) graph of nodes and be a graph signal. As observed for DSP and following [1]- [3], in GSP, is the shift operator. It captures the local dependencies of the signal sample on the signal samples at the invertex neighbors ∈ of (given by the nonzero entries of row of ). The eigenvalues and eigenvectors of are the graph frequencies and graph spectral modes. Let graph frequency vector and matrix Λ be defined as before to collect the graph eigenvalues
= [ 0 , 1 , . . . , −1 ] , Λ = diag [ ] .
The following assumptions hold even when not stated. On occasion, we state them explicitly.
Assumption 1 (Strongly connected graph). The graph is strongly connected.
Under assumption 1, matrix has no zero column or row.
Assumption 2 (Distinct eigenfrequencies). The eigenvalues of are distinct.
Under assumption 2, 4 is diagonalizable and the Graph Fourier Transform (GFT) is found 5 by = GFT -1 Λ GFT.
If is symmetric, which is the case with undirected graphs, GFT is orthogonal GFT −1 = GFT , and if is normal then GFT is unitary GFT −1 = GFT . For general graphs, is neither symmetric nor normal, but GFT is full rank and invertible. 6 The spectral modes are the columns of GFT −1 . The graph Fourier transform of graph signal is = GFT . 4 Distinct eigenvalues are assumed for simplicity. The results can be proved in the more general setting of non-derogatory (equal minimum and characteristic polynomials (up to a factor ±1), or, equivalently, the geometric multiplicity of any eigenvalue to be 1 (single eigenvector)). 5 If assumption 2 does not hold, see [46] for further details on the GFT. 6 See [47] for numerically stable diagonalization of for directed graphs.
Remark 2 (Fixing the GFT). It is well known [48] that the diagonalization of a matrix is unique up to reordering of the eigenvalues and normalization of the eigenvectors. Paralleling remark 1, we assume the frequencies have been ordered and the eigenvectors appropriately normalized, see [9], fixing the GFT and Λ.
Cayley-Hamilton. Let the characteristic polynomial of be Δ( ) = 0 + 1 + · · · + −1 −1 + .
By the Cayley-Hamilton Theorem [48]- [50], satisfies its characteristic polynomial Δ( ) = 0 and so
= − 0 − 1 − · · · − −1 −1 ,(9)
and , ≥ , is reduced by modular arithmetic mod Δ( ) (·). Linear shift invariant (LSI) filtering. Under assumption 2, LSI filters in the vertex domain are polynomials ( ) in the shift. By Cayley-Hamilton, 7 ( ) is at most degree − 1.
The graph Fourier theorem [1] parallels DSP's theorem
( ) F − → (Λ) ,
and, in particular, the vertex shift relation
F − → Λ .(10)
Spectral shift . In [9,51,52], we introduce a spectral graph shift (see also [53] for a different definition) to shift a graph signal in the spectral domain preserving the dual of the shift invariance relation (10), i.e., such that
Λ * F − → . References [51, 52] show that = GFT Λ * GFT −1 ,
and LSI spectral filters are polynomials ( ).
C. Graph impulse
When studying graph signal representations, we need the concept of graph delta or graph impulse. In DSP, the impulse in the time domain and its Fourier transform are
,0 = 0 , F − → ,0 = DFT ,0 = 1 √ 1,
where 1 is the vector of ones. In DSP, the time impulse ,0 is impulsive in the vertex domain (nonzero only at 0) and flat in the frequency domain. Further, the delayed time impulses , = , are centered at and impulsive. Likewise, in DSP, the impulse in the frequency domain ,0 is impulsive now in frequency and flat in time. In other words, in DSP, the definition of impulse in time and frequency are symmetric-the time and frequency impulses are impulsive at = 0 and at = 0, respectively. In GSP, in general, we either get impulsivity in one domain or flatness in the other, but not both. We have then two possible definitions for the vertex impulse and two possible definitions for the spectral impulse. We choose to preserve flatness and define delta graph signals that are flat in one domain. We discuss next how to define 1) graph impulse signal in the vertex domain as the inverse GFT of a flat signal in the spectral domain; and 2) graph impulse signal in the spectral domain as the GFT of a flat signal in the vertex domain.
1) Vertex graph impulse
In the vertex domain, define the vertex impulse or delta 0 as the inverse GFT of a flat graph spectrum
0 F − → 0 = 1 √ 1 =⇒ 0 = Δ GFT −1 1 √ 1 . (11)
The shifted replicas of the vertex graph impulse 0 are
= 0 F − → = Λ 1 √ 1 = 1 √ .(12)
In GSP, the 's, delayed 0 by , are not impulsive.
2) Spectral graph impulse
We now consider the spectral graph impulse sp,0 in the spectral domain. We define it as the GFT of a flat signal in the vertex domain
sp,0 = 1 √ 1 F − → sp,0 =⇒ sp,0 = Δ GFT 1 √ 1 . (13)
The shifts of sp,0 in the spectral domain are obtained with the spectral shift . Replicating (12), get
sp, = Λ * 1 √ 1 = 1 √ * F − → sp, = sp,0 . (14)
Remark 3 (Notation on vertex and spectral quantities). When referring to quantities using the shift or the vertex impulse 0 we will often not qualify them with the word "vertex." In contrast, we will consistently qualify by "spectral" quantities related to the spectral shift or the spectral impulse sp, using the subscript 'sp' as a reminder.
Remark 4 (Vertex and spectral impulses). We emphasize that we have two graph impulses, the vertex impulse 0 (that is flat in the spectral domain, see (11)) and the spectral impulse sp, (that is flat in the vertex domain, see (13)). In general, in GSP, neither is actually "impulsive" in either domain.
III. VERTEX AND FOURIER SIGNAL REPRESENTATIONS
At an abstract level, graph signals are vectors in an dimensional graph signal vector space V over field F. Amplifying, attenuating, adding, filtering, or processing signals is simplified by first expressing them as linear combinations of basic signals. As a prelude to representations introduced in subsequent sections, here, we discuss first in subsection III-A a generic representation, and then in subsections III-B and III-C consider the vertex and spectral representations, respectively.
A. Graph signal representations
In the -dimensional signal vector space V over the field F, let = { 0 , · · · , −1 } be a basis. Recall that the basis vectors { } 0≤ ≤ −1 are all nonzero and linearly independent. Mathematically, for any ∈ V:
Graph signal representation = ( ) 0 0 + · · · + ( ) −1 −1(15)= [ 0 · · · −1 ] ( ) 0 . . . ( ) −1 .(16)
Remark 5 (V ≈ C ). In the paper, we assume the field F = C, so, ∈ C . By (15), V is isomorphic to C . In the sequel, we use this isomorphism and assume the signal space is the -dimensional vector space C over the field C.
Remark 6 (Ordered basis). For to be well defined, the basis is ordered. If we reorder the basis by a permutation , the coordinate vector is itself reshuffled by :
= .(17)
By equation (16), processing signals is computing with their coordinate vectors . Because of its significance, this coordinatization of signals receives a special designation. Choosing a signal representation corresponds to choosing a basis . There are infinitely many, with some particularly useful. DSP is essentially built around two representations, see section II. For GSP, we consider six representations and discuss their specific advantages.
B. Vertex, standard, or Euclidean representation
The graph signal is an indexed collection of samples = { } ∈ , one at each vertex of the graph. The vertex graph signal representation is the natural one where the thcomponent of the coordinate vector is the graph sample at indexing vertex ∈ of the graph. This representation corresponds to the standard or Euclidean ordered basis = { 0 , · · · , −1 }. Clearly, { ≠ 0} 0≤ ≤ −1 are linearly independent. For easy reference, we formally present the vertex or Euclidean representation.
Definition 2 (Vertex, standard, or Euclidean representation). The vertex, standard, or Euclidean representation of graph signal ∈ V ≈ C is the coordinate vector of with respect to the standard basis .
Vertex, standard, or Euclidean representation
= 0 0 + · · · + −1 −1 = [ 0 · · · −1 ] 0 . . . −1 = 0 . . . −1 = .
Component of the coordinate vector corresponds to ∈ and to ∈ . Ordering , orders nodes and is well defined. Because the matrix with columns is the identity, we usually omit the subindex and use the same symbol, e.g., , for the graph signal and its vertex representation .
Reordering or the vertices permutes as in (17). In DSP, time is ordered and this issue is taken for granted. In GSP, to process signals, the ordering should be fixed and shared.
C. Graph Fourier representation
Fourier analysis, frequency components, bandlimited, low pass come naturally from the spectral or Fourier transform domain description of the signal . This can also be interpreted as a representation of where the Fourier basis is:
Fourier = { 0 , · · · , −1 } ,
where the eigenvectors of are spectral modes and the columns of GFT −1 . We order the Fourier basis Fourier , ordering the spectral components and the graph frequencies . The graph Fourier representation is formally presented next, again, for easy reference.
Definition 3 (Graph Fourier representation). The graph Fourier representation of is its graph spectrum .
Graph Fourier representation
= 0 0 + · · · + −1 −1 = 0 · · · −1 GFT −1 0 · · · −1
IV. VERTEX IMPULSIVE REPRESENTATION AND GRAPH -TRANSFORM
In this section, we consider several representations for the graph signal: 1) as a linear combination of graph vertex impulses; 2) as the impulse response of a graph filter; 3) introduce the graph -transform (G T); and through the G T provide 4) a symbolic polynomial representation for graph signals.
A. Vertex Impulsive Representation
Consider the (ordered) set of the graph vertex impulse and its delayed replicas in (11) and (12):
imp = { 0 , 1 , · · · , −1 } = 0 , 0 , · · · , −1 0 .
To prove imp is a basis, introduce the vertex impulsive matrix imp with columns the vectors in imp :
imp = Δ [ 0 1 · · · −1 ] = 0 0 0 · · · −1 0 . We relate imp to a Vandermonde matrix V.
Result 1 (Vertex Impulsive and Vandermonde matrices).
imp F − → 1 √ V,(18)
where V is the Vandermonde matrix
V= 0 · · · −1 = 1 0 2 0 · · · −1 0 1 1 2 1 · · · −1 1 . . . . . . . . . . . . . . . 1 −1 2 −1 · · · −1 −1 .
Proof. This result follows by using (12) for in imp .
Result 2 (Full rank of vertex impulsive matrix). Under assumption 2, imp is full rank.
Proof. By result 1 and equation (18), imp is the GFT −1 of the Vandermonde matrix V. Under assumption 2, V is full rank [48]- [50]. Hence, imp is full rank.
Result 3 (Vertex impulsive basis). Under assumption 2, imp is a basis-the vertex impulsive basis.
Proof. The vectors in imp are the columns of imp , which by result 2 is full rank. Hence, imp is a basis.
Definition 4 (Vertex impulsive representation imp ). The vertex impulsive representation of graph signal is its coordinate vector imp with respect to basis imp :
Vertex impulsive representation
= 0 0 + 1 1 + · · · + −1 −1 (19) = 0 1 · · · −1 imp 0 · · · −1 imp(20)
Computing imp . To find the coordinate vector imp of with respect to imp , in general, we solve the linear system (20). In practice, a sparse approximation may suffice by minimizing imp imp − 2 2 + imp 1 .
B. Polynomial Transform Filter
Next, we interpret the vertex impulsive representation imp as the coefficients of a linear shift invariant (LSI) graph filter.
Result 4 ( as impulse response of ( )). Let assumption 2 hold. Then the graph signal is the impulse response = ( ) 0 (21) of the LSI polynomial filter
( ) = 0 + 1 + · · · + −1 −1 (22) iff the vector of coefficients coef of ( ) is the vertex impulsive representation imp in (20): coef = 0 1 · · · −1 = imp .(23)
Proof. The impulse response of the LSI ( ) is
( ) 0 = 0 + 1 + · · · + −1 −1 0(24)= 0 0 + 1 0 + · · · + −1 −1 0 = 0 1 · · · −1 imp coef .(25)
Under assumption 2, the impulse response of ( ) is the graph signal iff coef in (25) equals imp in (20). The polynomial transform filter ( ) in (22) is in powers of . We provide an alternative description.
Result 5 (Graph signal and ( )). Given F ← → , its LSI polynomial transform filter ( ) is alternatively given by
( ) = GFT −1 diag √ GFT(26)
Proof. From given as impulse response of ( ) in (24), using the diagonalization of ( ), it successively follows
( ) 0 = GFT −1 · (Λ) · GFT · GFT −1 · 1 √ 1 = =⇒ (Λ) 1 √ 1 = =⇒ 1 √ (Λ) = diag [ ] =⇒ ( ) = GFT −1 diag √ GFT,
where we used the definition of 0 = GFT −1 · 1 √ 1.
C. Graph -transform (G T)
The powers of the shift of the polynomial transform filter ( ) in (22) represent the GSP equivalent of the powers of the DSP shift −1 . This motivates the following definition.
Definition 6 (Graph -transform (G T)). The G T is
G T = −1 imp = 0 1 · · · −1 −1 .
In diagram form, the G T of and its reconstruction are: 1 and V). The G T −1 and V are GFT pairs:
G T − −−−−−− → imp = G T G T −1 − −−−−−− → Result 6 (G T −G T −1 GFT − −− → 1 √ V = GFT · G T −1 .
Proof. This follows from result 1 and definition 6.
G T maps vertex signals into -transformed signals imp . The G T of is the polynomial coefficient vector imp in (23) that defines ( ) in (22). To simplify notation, we introduce a symbolic polynomial representation ( ).
Definition 7 (Graph -transform representation ( )). The graph -transform representation of graph signal is its coordinate vector imp with respect to the monomial basis monomial = 1, , · · · , −1 :
Graph -transform representation G T ≈ ( ) = 0 + 1 + 2 2 + · · · + −1 −1 = 1 2 · · · −1 imp .
Remark 7 (Various meanings for imp ). We have multiple interpretations for imp : 1) as coordinate vector of with respect to basis imp ; 2) determining the polynomial transform filter ( ); 3) as G T of ; and 4) defining ( ). We will take advantage of these several understandings.
D. Spectral Impulsive Representation
Section III discusses the standard or Euclidean representation of graph signals (with respect to basis ), as well as the Fourier representation (with respect to basis Fourier ). Section IV presents the representation of graph signals with respect to the basis imp whose basis vectors are the vertex impulse 0 and its delayed replicas defined in section II-C1. In section II-C2 and equation (13), we define the graph impulse sp,0 , but now in the spectral domain. As (13) shows, sp,0 , the GFT −1 of the spectral graph impulse, is flat in the vertex domain. This section considers the representation of graph signal with respect to sp,0 and its delayed replicas.
Consider the set of sp,0 and its spectral shifts:
sp,imp = sp,0 , sp,1 , · · · , sp, −1 . Collect the vectors in sp,imp in the spectral impulse matrix 8 sp,imp = sp,0 sp,1 · · · sp, −1 .
Result 7 ( sp,imp and V * ). 8 Note that the columns of sp,imp are flat, not impulsive.
sp,imp = 1 √ V * = 1 √ 1 * · · · * −1 .(27)
Proof. Result follows from equation (14).
Equation (27) shows that the vectors of the set sp,imp are, apart a scaling, the columns of V * .
Result 8 (Spectral impulse basis sp,imp ). Under assumption 2,
sp,imp is a basis. Proof. By assumption 2, V * is full rank.
Definition 8 (Spectral impulsive representation). The spectral impulsive representation of graph signal is its coordinate vector sp,imp with respect to basis sp,imp :
Spectral impulsive representation = 0 sp,0 + 1 sp,1 + · · · + −1 sp, −1 (28)
= sp,imp 0 1 · · · −1 sp,imp = 1 √ V * sp,imp(29)
V. COMPANION MODEL-A CANONICAL GSP MODEL
In DSP, the DSP cyclic shift in (1) acts on graph signal as given in (2). Decompose the cyclic shift as
= 0 0 . . . 0 0 1 0 . . . 0 0 . . . 1 . . . . . . . . . . . . . . . . . . . . . . . . 0 0 . . . 1 0 ,line shift + 0 0 . . . ,periodic bc .(30)
Then the shifted time signal is delayed (moved downwards) by the line shift (left block in (30)) and the signal extension is determined by the periodic boundary condition (right block in (30)) [11]- [13] that wraps around the time signal so that sample −1 reappears as the first component of . In this section, we look for a GSP signal model where the graph shift acts in similar fashion to (30). We accomplish it with the impulsive GSP signal representation. The resulting GSP model leads to the companion shift and the companion graph. These are canonical shift and canonical graph representations to which, under assumptions 1 and 2, every other generic GSP model can be reduced to. We use these and the G T of section IV-C, to introduce a fast graph convolution in section VIII .
A. Canonical Companion Shift
To obtain the representation of with respect to imp , we apply the shift to each vector ∈ imp . Get 0 = 1 ,· · ·, = +1 0 = +1 ,· · ·, −2 = −1 . (31) We need a "signal extension" or "boundary condition" for
−1 = 0 .(32)
This boundary condition is embedded in the matrix and is obtained by reducing it by Cayley-Hamilton Theorem. Applying then this theorem through equation (9),
−1 = − 0 0 − 1 0 − 2 2 0 −· · ·− −1 −1 0 . (33)
The boundary condition in (33) for −1 is a linear combination of the basis vectors ∈ imp . The coefficients of the linear combination are the negative of the coefficients of the characteristic polynomial Δ ( ) of given in (8).
Putting together the equations (31)-(32) and using the boundary condition (33),
0 1 · · · −1 = = 0 1 · · · −1 0 0 · · · 0 − 0 1 0 · · · 0 − 1 0 1 . . . 0 − 2 . . . . . . . . . . . . . . . 0 0 · · · 1 − −1 comp .(34)
Equation (34) shows that the representation of the shift with respect to imp is the companion matrix comp . It is the companion matrix [48]- [50] of the characteristic polynomial Δ ( ) of the graph shift . We refer to comp as the companion shift. We can rewrite it as:
comp = 0 0 · · · 0 1 1 0 · · · 0 0 0 1 . . . 0 0 . . . . . . . . . . . . . . . 0 0 · · · 1 0 + −1 − 0 − 1 · · · − −1 [0 0 · · · 0 1] rank 1 . (35)
Equation (35) gives comp as the sum of a unitary matrix, the DSP cyclic shift , plus a rank one matrix. On the other hand, we may decompose comp as:
comp = 0 0 · · · 0 0 1 0 · · · 0 0 0 1 . . . 0 0 . . . . . . . . . . . . . . . 0 0 · · · 1 0 line shift + 0 0 · · · 0 − 0 0 0 · · · 0 − 1 0 0 · · · 0 − 2 . . . . . . . . . . . . . . . 0 0 · · · 0 − −1 linear bc .(36)
Equation (36) resolves comp as a 'line shift' line shift corrected by a 'boundary condition' linear bc . It replicates the structure of the DSP cyclic shift given in (30). Like ,line shift in (30), line shift moves the graph signal downwards, while linear bc retains the coefficients {− } 0≤ ≤ −1 of the boundary condition. This is a more general boundary condition than for the cyclic shift, since for ,periodic bc all = 0, except 0 = −1, see (30). This agrees with the characteristic polynomial of the DSP cyclic shift of for which Δ c ( ) = − 1. Since comp is determined by the characteristic polynomial Δ ( ), it only depends on the graph frequencies or eigenvalues of , not on the spectral modes or eigenvectors of . And this shows that, under diagonalization of , we can associate to arbitrary adjacency matrices a canonical weighted adjacency matrix, its companion shift.
Result 9 (Diagonalization of comp ). Under assumption 2, comp is diagonalized by the Vandermonde matrix
comp = V −1 Λ V.(37)
Proof. This is a well known result. It can be verified by direct substitution that 1 · · · −1 is a left eigenvector of comp for eigenvalue , from which the result follows.
Companion graph Fourier transform. Given (37), the Vandermonde matrix V is the graph Fourier transform for signals in impulsive representation, replicating the DSP result where the DFT is the Vandermonde matrix of the eigenfrequencies (apart a normalizing factor), see (5).
This shows that the impulsive representation replicates for GSP another dimension of DSP. In fact, just like for DSP, the eigenvalues (frequencies) provide the whole picture, since the companion graph Fourier transform is defined by the frequency vectors and its powers.
Next, we associate a weighted companion graph comp to comp . Both of these, comp and comp , are canonical graph representations connected with any GSP graph.
B. Canonical Companion Graph
The companion matrix comp defines the (weighted) companion graph comp = comp , comp displayed in figure 2. Under assumption 2, any directed or undirected signal graph has a corresponding weighted companion graph. The companion graph comp has a canonical structure: 1) its node set comp has nodes, node is associated with basis vector ∈ imp (or power ). In other words, these nodes are not the nodes of the original graph associated with ; 2) it is directed; 3) the edge set comp combines a directed path graph with possibly a self-loop at node − 1 and up to − 1 directed backward edges pointing from node − 1 to the previous nodes; 4) these directed edges are weighted by the negative of the coefficient of Δ ( ); 5) iff 0 ≠ 0, the companion graph is strongly connected. This is the case if zero is not an eigenvalue of . Figure 3 shows on top a "directed" ladder graph with 12 nodes and below it the corresponding canonical graph. The characteristic polynomial of the adjacency matrix of a ladder graph like shown in the figure but with 2 nodes is
C. Example
Δ ( ) = −1 − 2 − 4 − 8 − · · · − 2( −2) + 2 .
The polynomial Δ ( ) explains why the edge weights of the companion graph of the directed ladder graph are all ones (the coefficients of Δ ( ) are ≡ −1). The eigenfrequencies of As another example, consider the undirected node path. Its characteristic polynomial is the 3-term recursion
Δ ( ) = Δ −1 ( ) − Δ −2 ( ), Δ 0 ( ) = 1, Δ 1 ( ) = 2 .
This gives Δ ( ) = 2 where ( ) is the Chebyshev polynomial of the second kind [54]. For example, for = 8 Δ 8 ( ) = 8 − 7 6 + 15 4 − 10 2 + 1
The path and its companion graph are in figure 5.
VI. REPRESENTATIONS FOR THE SPECTRUM
Taking the GFT to both sides of a representation of , we obtain a representation for . We consider two representations for , derived from the vertex and the spectral impulsive representations given by (19) and (28), respectively.
A. Spectrum vertex impulse representation
Take the GFT to both sides of (19). Using result 1 and equation (12), obtain the representation of with respect to
= 1 √ 1, 1 √ , · · · , 1 √ −1 .
The set is a basis. In fact, its vectors, apart the scaling factor 1 √ , are the columns of the Vandermonde matrix V, and V is full rank under assumption 2.
Definition 9 (Spectrum vertex impulsive representation). The spectrum vertex impulsive representation of = GFT is the coordinate vector imp with respect to basis .
Spectrum vertex impulsive representation
= 0 0 + 1 1 + · · · + −1 −1 = 1 √ 0 1 √ · · · 1 √ −1 1 √ V 0 · · · −1 imp(38)
The spectrum vertex impulsive representation for has the same coordinate vector imp as the vertex impulsive representation for . It is the basis that is different. Now, it is the frequency vector and its powers that are the basis vectors for this representation of
B. Spectrum spectral impulse representation
Taking the GFT to both sides of equation (28), we obtain the representation of = GFT with respect to sp,imp = sp,0 , sp,1 , · · · , sp, −1
= 0 sp,0 , sp,0 , · · · , −1 sp,0 .
The set sp,imp is a basis, because its vectors are the GFT of the vectors of the basis sp,imp .
Definition 10 (Spectrum spectral impulsive representation). The spectrum spectral impulsive representation of = GFT is the coordinate vector sp,imp with respect to basis sp,imp .
Spectrum spectral impulsive representation
= 0 sp,0 + 1 sp,1 + · · · + −1 sp, −1 = sp,0 sp,1 · · · sp, −1 sp,imp 0 · · · −1 sp,imp(39)
The spectrum spectral impulsive representation for has the same coordinate vector sp,imp as the spectral impulsive representation for (see (39) and (29)). It is the basis that is different. Now, it is the spectral impulse sp,0 and its powers that are the basis vectors for this representation of .
Result 10 ( imp and sp,imp ).
sp,imp = (V * ) −1 GFT −1 V imp and imp = V −1 GFT V * sp,imp .
Proof. It follows from (38) and (39), using (27) in result 7.
We proceed to obtain results similar to sections IV-B and IV-C for sp,imp . We start by associating with it a spectral polynomial transfer filter ( ), now in the spectral shift .
Result 11 ( as impulse response of ( )). Let assumption 2 hold. Then is the impulse response of LSI filter ( )
= ( ) sp,0 ( ) = 0 + 1 + · · · + −1 −1 ,(40)
iff the vector of coefficients coef of ( ) is the spectral impulsive representation sp,imp in (39):
coef = 0 1 · · · −1 = sp,imp .
Filter
( ) is the spectral polynomial transform filter.
Note that equation (29) can be rewritten as
= (Λ * ) sp,0 = 0 + 1 Λ * +· · ·+ −1 (Λ * ) −1 1 √ 1.(41)
In other words, (41) interprets the original signal as the impulse response of the diagonal filter (Λ * ). From this, the next result follows.
Result 12 (Graph signal and
( )). The LSI polynomial transform filter ( ) is alternatively given by
( ) = GFT diag √ GFT −1
Results 11 and 12 parallel results 4 and 5.
C. Spectral graph -transform Like ( ) in (22) led us to the G T, we associate with ( ) a spectral graph -transform ( G T sp ).
Definition 11 (Spectral graph -transform ( G T sp )). Define
G T sp = −1 sp,imp = sp,0 sp,1 · · · sp, −1 −1 . Result 13 (Fourier pairs G T −1 sp and V * ). 1 √ V * GFT − −− → G T −1 sp = GFT 1 √ V * .
Proof. From (29), sp,imp = 1 √ V * , so, by definition 11
G T −1 sp = sp,imp = GFT sp,imp = GFT 1 √ V * .
G T sp maps into spectral -transformed signals sp,imp :
G T sp − −−−−−−− → sp,imp = G T sp G T sp −1 − −−−−−−− →
The G T sp of is the polynomial coefficient vector sp,imp in (39) that defines ( ) in (40) in result 11. To simplify notation, we use also a symbolic polynomial representation ( ) with coefficients given by sp,imp = G T sp G T sp ≈ ( ) = 1 2 · · · −1 sp,imp = 0 + 1 + 2 2 + · · · + −1 −1 .
The polynomial ( ) expresses the G T sp of in terms of the monomial basis monomial = 1, , · · · , −1 . Spectral companion model: Canonical companion matrix and graph. Since and are co-spectral (share the same spectrum), their characteristic polynomials Δ ( ) and Δ ( ) are equal. This means that we can associate with the spectral impulse representation the same companion matrix comp and the same companion graph comp as in sections V-A, equation (34), and V-B, respectively.
VII. SIGNAL REPRESENTATION DOMAINS
In sections III through VI, we discussed several signal representations. We summarize these in figure 6 that illustrates the corresponding signal domains and the transforms relating them, summarizing the main results from these sections. At the bottom, we have the standard Euclidean vertex domain signals with its shift and the spectral domain signals with its shift . The relation between these two domains is the GFT and its inverse GFT −1 . At the intermediate level, we have the two -transform domains corresponding to the two impulsive representations, the vertex impulsive -transformed signals imp and the spectral impulsive -transformed signals sp,imp . The graph -transform G T obtains imp from , while G T sp obtains sp,imp from . These two -transformed signal domains reflect a number of interesting and surprising facts. Their shift is the same, the companion matrix comp , to which we associate a "companion graph" comp , see figure 2. In these domains, the graph eigenvalues { } 0≤ ≤ −1 contain all needed information, since the eigenvectors derive from the graph frequency vector and its powers. At the top, we indicate that the Vandermonde matrix V and its conjugate relate the -transformed signal domains back to the spectral and vertex domains.
With DSP the picture is much simpler. Although not usually presented this way [5,6], by reinterpreting the above GSP representations in DSP, we cast four common DSP signal representations as vector coordinatizations of the signal ∈ C with respect to choices of basis in C .
1) Standard:
= { 0 , 1 , . . . , −1 } is the standard or Euclidean basis and the signal representation is the vector of signal samples = [ 0 , 1 , . . . , −1 ] 1 = 1 = 1 . 2) Impulsive: imp = { 0 , 1 , . . . , −1 } is the basis of the impulse and its delayed replicas. Since in this case, imp = , the signal representation is = [ 0 , 1 , · · · , −1 ] imp = imp , and imp = .
3) Spectral: Fourier = { 0 , 1 , . . . , −1 } is the basis of the eigenmodes or harmonics and the signal representation is the Fourier transform of the signal = [ 0 , 1 , . . . , −1 ] = . 4) Spectral impulsive: sp,imp = 1 √ * 0 , * 1 , · · · , * −1 . In this case, sp,imp = DFT , and the signal representation is = 1 √ * 0 , * 1 , · · · , * −1 sp,imp = sp,imp , and sp,imp = .
In DSP, the above four signal representations reduce to two distinct ones, see figure 7 that illustrates this for a = 4 signal. Since = , = 0, · · · , − 1, the standard and impulsive So, in DSP, the standard and impulsive representations can be used interchangeably as the time domain signal, , and, similarly, the eigenvalue and spectral representations can be used interchangeably as the frequency domain signal, . Also, in DSP, = [51,52]. Having this in mind, figure 6 is much simpler with DSP as illustrated in figure 8.
VIII. FAST GRAPH CONVOLUTION WITH THE FFT
Filtering in the vertex domain is defined in [1] as the product of matrix graph filter with graph vector signal . If the filter is linear shift invariant, it is a polynomial filter ( ). We now consider convolution of two graph signals.
Definition 12 (Convolution of vertex domain graph signals).
The (vertex domain) convolution of graph signals and is Figure 9 illustrates this convolution. Since polynomial Proof. Equation (43) follows from result 4 and equation (21). Equation (44) follows by taking the GFT of both sides of (42) and using the diagonalization of the transform filters in (26).
* = ( ) · ( ) 0 ,(42)
Equation (43) interprets convolution of and as filtering the graph signal by a filter whose impulse response is . In the spectral domain, equation (44) shows that convolution of the two signals is in the spectral domain the pointwise multiplication of the GFTs of the signals. This replicates the graph Fourier filtering theorem (see equation (27) in [1]).
A. Convolution of Graph Signals with the FFT
Equation (44) shows that, as in DSP, we can compute convolution by finding the two GFTs and of the two signals, then pointwise multiplying these, and finally taking the inverse GFT of the pointwise product. Even though this replicates the DSP result, GFTs and inverse GFTs are matrix vector products that are order 2 , not fast operations. We show that in "companion space," i.e., working with the impulsive representations, the G T of graph signals or their polynomial representation, and the polynomial transform filters of the signals, graph vertex convolution can be obtained by FFT.
Fast convolution. Consider the -transform representations ( ), ( ), and ( ) of , , and = * .
Result 15 (GSP convolution and linear convolution). We have ( ) = ( ( ) · ( )) mod Δ ( )
where Δ ( ) is the characteristic polynomial of , and = * is the vector of coefficients of ( ).
Proof. The product of polynomials (in or in ) is the polynomial whose coefficients are the linear convolution of the sequences of coefficients of the polynomials. Powers larger than − 1 are reduced by Cayley-Hamilton achieved by mod Δ ( ) reduction.
Remark 9. In DSP, = mod . This is the wrap-around effect, or "time-aliasing" in DSP. In DSP, the coefficient of the power of is added to the coefficient of the power of mod = 0, the coefficient of + 1 is added to the power of + 1 mod = 1, and so on. For a generic graph in GSP, there is also "vertex-aliasing," but it is not one-to-one like in DSP. The coefficient of the power of (and higher powers) is scaled differently and added to lower powers from 0 to − 1 as per Cayley-Hamilton. (45) are a fast convolution of the two graph signals when given their -transforms. The linear convolution of the sequences of coefficients of ( ) and ( ) is computed by fast Fourier transform (FFT). The mod reduction is computed by (fast) polynomial division, ( ) operations [55]. Result 15 is very pleasing. It evaluates GSP LSI convolution using the FFT, an intrinsically DSP algorithm.
Result 15 and equation
Remark 10. An interesting question is when are linear and circular (vertex) convolution equivalent in GSP. From (45), we see that, in GSP, if the degree of the product polynomial ( ) ( ) is not greater than − 1, then ( ) = ( ( ) · ( )) mod Δ ( ) = ( ) · ( ). Linear and circular convolution are equivalent and the reduction by mod Δ ( ) produces no effect. This is the same condition for when linear and circular convolution are equivalent in DSP. In practice, one may want to pad with zeros either or both of ( ) and ( ) to get faster processing.
An example of the convolution of graph signals using both polynomial filtering (figure 9) and the FFT for an expanded version (100 nodes) of the directed ladder graph in figure 3 is shown in figure 10. The steps in the figure illustrate the several representations and transforms introduced in sections V, III, VII. On top, we compute the convolution by equation (42) in definition 12. From left to right, we start with the vertex impulse 0 (obtained by GFT −1 of a flat impulse in the spectral domain), going through the polynomial transform filters ( ) and ( ) to get * . At the bottom, we illustrate the fast convolution in equation (45). We compute the G T of and to obtain imp and imp . These are linearly convolved and reduced by mod Δ ( ) to obtain imp . A final G T −1 gets the circular convolution * .
The polynomial coefficient vectors for signals and are: 1) the first 25 entries of imp are 1 and the remaining 75 entries are 0; and 2) the entries of imp were chosen as a triangle signal, with the first half going from 1 50 to 1 with a step of 1 50 and then the remaining to back down to 1 50 with a step of − 1 50 . Comparing the two plots on the right of figure 10), we conclude that different methods lead to the same result for * , with a maximum pointwise magnitude difference (due to roundoff errors) between them of 0.15 (with convolution result values of order 10 11 ).
IX. CONCLUSION
The paper introduces the companion graph signal model defined by a companion shift and a companion graph. These are canonical in the sense that every directed or undirected graph based signal model can be transformed into this canonical model by a graph -transform that we define in the paper. It is obtained from impulsive representations of graph signals. The companion graph signal model reflects many of the characteristics associated with the cyclic graph model of time signals and DSP. We show that, in the companion model, convolution of graph signals is fast convolution that is performed with the DSP FFT. Fig. 10. Example of circular convolution of and using both polynomial filtering (above) and the FFT (below) for the 100 node directed ladder graph in figure 3. Both methods produce the same * .
Fig. 1 .
1Directed cycle graph.
Definition 1 (
1Representation). The representation of with respect to the ordered basis is its coordinate vector . The th component ( ) of is the coefficient of the basis vector in the linear combination (15).
Definition 5 (
5Polynomial transform filter). The LSI polynomial filter ( ) in (22) is the polynomial transform filter of .
Fig. 2 .
2Companion graph. Unlabelled edges have weight 1. Other edges labeled by their weights. The structure of the companion graph in figure 2 extends the structure of the DSP cyclic graph in figure 1. The DSP cyclic graph follows form the companion graph of figure 2 by taking 0 = −1 and eliminating the self-loop and all the remaining backward pointing edges.
Fig. 3 .
3Directed ladder graph and its companion graph. this directed ladder graph are illustrated for k=4,6,8,10,12, and 14 nodes in figure 4. They distribute close to the unit circle.
Fig. 4 .
4Eigenfrequencies of directed ladder graphs with 4 (red), 6 (green), 8 (blue), 10 (brown), 12 (black), and 14 (orange) nodes.
Fig. 5 .
5Path graph and its canonical companion graph.
Remark 8 (
8DSP representation). In DSP, 0 and its delayed replicas are impulses (in the vertex domain) and sp,0 and its delayed replicas are impulses (in the frequency domain). So imp = and sp,imp = , from which (see(20)) = imp and (see (39)) = sp,imp .Further, for DSP, representation (29) is the Fourier representation of , i.e., decomposes the time signal in its harmonics (the powers * are the eigenvectors of ), while representation (38) decomposes in the powers of frequency vectors (that, in DSP, are the conjugate of the harmonics).
Fig. 6 .
6Graph signal domains and the transformations between them. For each domain, both the signal and shift are given.
Fig. 7 .
7DSP Signal Representations: = [1, 2, 3, 4] , = [5, −1+ , −1, −1− ] . The standard representation and impulsive representations coincide. The Fourier and spectral impulsive representations also coincide. bases and corresponding signal representations coincide, = imp and = imp = . Similarly, since = 1 √ * , the Fourier and spectral impulsive bases and corresponding signal representations coincide, Fourier = sp,imp and = sp,imp = DFT .
Fig. 8 .
8Figure 6for DSP. The three colored regions all have the same signals and shift. In DSP, = . The three colored regions are all identical to each other in DSP.
where ( ) and ( ) are the LSI polynomial transform filters for and . Definition 12 and equation (42) define convolution of graph signals and as the impulse response of the serial concatenation of the polynomial transform filters ( ) of and ( ) of .
Fig. 9 .
9Convolution of graphs signals and . filters commute, the convolution in (42) commutes. The next result provides alternative ways of computing the convolution. Result 14 (Vertex convolution). Consider graph signals and and their polynomial transform filters ( ) and ( ). Then * = ( ) · = (
Graphs that differ by permutations of are isomorphic.
Under assumption 2, the minimal polynomial of equals Δ( ).
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[] |
[
"Cooperative game theory and the Gaussian interference channel",
"Cooperative game theory and the Gaussian interference channel"
] |
[
"Senior member)Amir Leshem ",
"Senior member)Ephraim Zehavi "
] |
[] |
[] |
In this paper we discuss the use of cooperative game theory for analyzing interference channels. We extend our previous work, to games with N players as well as frequency selective channels and joint TDM/FDM strategies.We show that the Nash bargaining solution can be computed using convex optimization techniques. We also show that the same results are applicable to interference channels where only statistical knowledge of the channel is available. Moreover, for the special case of two players 2 × K frequency selective channel (with K frequency bins) we provide an O(K log 2 K) complexity algorithm for computing the Nash bargaining solution under mask constraint and using joint FDM/TDM strategies. Simulation results are also provided.
|
10.1109/jsac.2008.080906
|
[
"https://arxiv.org/pdf/0708.0846v1.pdf"
] | 1,559,119 |
0708.0846
|
8d0b81ee65196e3b577f0de31a5f5cb0da4f8e1c
|
Cooperative game theory and the Gaussian interference channel
6 Aug 2007
Senior member)Amir Leshem
Senior member)Ephraim Zehavi
Cooperative game theory and the Gaussian interference channel
6 Aug 2007arXiv:0708.0846v1 [cs.IT] 1Spectrum optimizationdistributed coordinationgame theoryNash bargaining solutioninterference channelmultiple access channel
In this paper we discuss the use of cooperative game theory for analyzing interference channels. We extend our previous work, to games with N players as well as frequency selective channels and joint TDM/FDM strategies.We show that the Nash bargaining solution can be computed using convex optimization techniques. We also show that the same results are applicable to interference channels where only statistical knowledge of the channel is available. Moreover, for the special case of two players 2 × K frequency selective channel (with K frequency bins) we provide an O(K log 2 K) complexity algorithm for computing the Nash bargaining solution under mask constraint and using joint FDM/TDM strategies. Simulation results are also provided.
I. INTRODUCTION
Computing the capacity region of the interference channel is an open problem in information theory [2]. A good overview of the results until 1985 is given by van der Meulen [3] and the references therein.
The capacity region of general interference case is not known yet. However, in the last forty five years of research some progress has been made. Ahslswede [4], derived a general formula for the capacity region of a discrete memoryless Interference Channel (IC) using a limiting expression which is computationally infeasible. Cheng, and Verdu [5] proved that the limiting expression cannot be written in general by a single-letter formula and the restriction to Gaussian inputs provides only an inner bound to the capacity region of the IC. The best known achievable region for the general interference channel is due to Han and Kobayashi [6]. However the computation of the Han and Kobayashi formula for a general discrete memoryless channel is in general too complex. Sason [7] describes certain improvement over the Han Kobayashi rate region in certain cases. A 2x2 Gaussian interference channel in standard form (after suitable School of Engineering, Bar-Ilan University, Ramat-Gan, 52900, Israel. Part of this work has been presented at ISIT 2006 [1]. This work was supported by Intel Corporation. e-mail: [email protected] . normalization) is given by:
x = Hs + n, H = 1 α β 1 (1)
where, s = [s 1 , s 2 ] T , and x = [x 1 , x 2 ] T are sampled values of the input and output signals, respectively.
The noise vector n represents the additive Gaussian noises with zero mean and unit variance. The powers of the input signals are constrained to be less than P 1 , P 2 respectively. The off-diagonal elements of H, α, β represent the degree of interference present. The capacity region of the Gaussain interference channel with very strong interference (i.e., α ≥ 1 + P 1 , β ≥ 1 + P 2 ) was found by Carleial given by
R i ≤ log 2 (1 + P i ), i = 1, 2.(2)
This surprising result shows that very strong interference dose not reduce the capacity. A Gaussian interference channel is said to have strong interference if min{α, β} > 1. Sato [8] derived an achievable capacity region (inner bound) of Gaussian interference channel as intersection of two multiple access gaussian capacity regions embedded in the interference channel. The achievable region is the intersection of the rate pair of the rectangular region of the very strong interference (2) and the region
R 1 + R 2 ≤ log 2 (min {1 + P 1 + αP 2 , 1 + P 2 + βP 1 }) .(3)
A recent progress for the case of Gaussian interference is described by Sason [7]. Sason derived an achievable rate region based on a modified time-(or frequency-) division multiplexing approach which was originated by Sato for the degraded Gaussian IC. The achievable rate region includes the rate region which is achieved by time/frequency division multiplexing (TDM/ FDM), and it also includes the rate region which is obtained by time sharing between the two rate pairs where one of the transmitters sends its data reliably at the maximal possible rate (i.e., the maximum rate it can achieve in the absence of interference), and the other transmitter decreases its data rate to the point where both receivers can reliably decode their messages.
While the two users fixed channel interference channel is a well studied problem, much less is known in the frequency selective case. An N × N frequency selective Gaussian interference channel is given by:
x k = H k s k + n k k = 1, ..., K
H k = h 11 (k) . . . h 1N (k) . . . . . . . . . h N 1 (k) . . . h N N (k) .(4)
where, s k , and x k are sampled values of the input and output signal vectors at frequency k, respectively.
The noise vector n k represents the additive Gaussian noises with zero mean and unit variance. The power spectral density (PSD) of the input signals are constrained to be less than p 1 (k), p 2 (k) respectively. The offdiagonal elements of H k , represent the degree of interference present at frequency k. The main difference between interference channel and a multiple access channel (MAC) is that in the interference channel, each component of s k is coded independently, and each receiver has access to a single element of x k .
Therefore iterative decoding schemes are much more limited, and typically impractical.
One of the simplest ways to deal with interference channel is through orthogonal signaling. Two extremely simple orthogonal schemes are using FDM or TDM strategies. For frequency selective channels (also known as ISI channels) we can combine both strategies by allowing time varying allocation of the frequency bins to the different users. In this paper we limit ourselves to joint FDM and TDM scheme
where an assignment of disjoint portions of the frequency band to the several transmitters is made at each time instance. This technique is widely used in practice because simple filtering can be used at the receivers to eliminate interference. In this paper we will assume a PSD mask limitation (peak power at each frequency) since this constraint is typically enforced by regulators.
While information theoretical considerations allow all points in the rate region, we argue that the interference channel is a conflict situation between the interfering links [1]. Each link is considered a player in a general interference game. As such it has been shown that non-cooperative solutions such as the iterative water-filling, which leads to good solutions for the multiple access channel (MAC) and the broadcast channel [9] can be highly suboptimal in interference channel scenarios [10], [11]. To solve this problem there are several possible approaches. One that has gained popularity in recent years is through the use of competitive strategies in repeated games [12]. Our approach is significantly different and is based on general bargaining theory originally developed by Nash [13]. Our approach is also different than that of [14] where Nash bargaining solution for interference channels is studied under the assumption of receiver cooperation. This translates the channel into a MAC, and is not relevant to distributed receiver topologies. In our analysis of the interference channel we claim that while all points on the boundary of the interference channel are achievable from the strict informational point of view, many of them will never be achieved since one of the players will refuse to use coding strategies leading to these points.
The rate vectors of interest are only rate vectors that dominate component-wise the rates that each user can achieve, independently of the other users coding strategy. The best rate pairs that can be achieved independently of the other users strategies form a Nash equilibrium [13]. This implies that not all the rates are indeed achievable from game theoretic prespective. Hence we define the game theoretic rate region.
Definition 1.1:
Let R be an achievable information theoretic rate region. The game theoretic rate region R G is given by
R G = {(R 1 , ..., R N ) ∈ R : R c i ≤ R i , for all i = 1, ..., N}(5)
where R c i is the rate achievable by user i in a non-cooperative interference game [11]. To see what are the pair rates that can be achieved by negotiation and cooperation of the users we resort to a well known solution termed the Nash bargaining solution. In his seminal papers, Nash proposed four axioms required that any solution to the bargaining problem should satisfy. He then proved that there exists a unique solution satisfying these axioms. We will analyze the application of Nash bargaining solution (NBS) to the interference game, and show that there exists a unique point on the boundary of the capacity region which is the solution to the bargaining problem as posed by Nash.
The fact that the Nash solution can be computed independently by users, using only channel state information, provides a good method for managing multi-user ad-hoc networks operating in an unregulated environment.
Application of Nash bargaining to OFDMA has been proposed by [15]. However in that paper the solution was used only as a measure of fairness. Therefore, R c i was not taken as the Nash equilibrium for the competitive game, but an arbitrary R min i . This can result in non-feasible problem, and the proposed algorithm might be unstable. The algorithm in [15] is suboptimal even in the two users case, and according to the authors can lead to an unstable situation, where the Nash bargaining solution is not achieved even when it exists. In contrast, in this paper we show that the NBS for the N palyers game can be computed using convex optimization techniques. We also provide detailed analysis of the two users case and provide an O(K log 2 K) complexity algorithm which provably achieves the joint FDM/TDM Nash bargaining solution. Our analysis provides ensured convergence for higher number of users and bounds the loss in applying OFDMA compared to joint FDM/TDM strategies. In the two users case we can show that the Nash bargaining solution requires TDM over no more than a single tone, so we can achieve a very good approximation to the optimal FDM based Nash bargaining solution. We also provide similar analysis for higher number of users, showing that for the Nash bargaining solution with N players, over a frequency selective channel with K frequency bins, only N 2 frequency bins has to be shared by TDM, while all other frequencies are allocated to a single user. When N 2 << K, this provides a near optimal solution to the game using FDM strategies, as well.
The structure of the paper is as follows: In section II we discuss competitive and cooperative solutions to interference games and provides an overview of the Nash bargaining theory. In section III we discuss the existence of the NBS for N players FDM cooperative game over slow, flat fading channels. In section IV we discuss the Nash bargaining over general frequency selective interference channel, with mask constraint. We show that computing the NBS under mask constraint and joint FDM/TDM strategies can be posed as a convex optimization problem. This shows that even for large number of palyers, computing the solution with many tones is feasible. We also show that in this case the N users will share only few frequencies, dividing all the others. In section V we specialize to the two players case, but with frequency selective channels. We provide an algorithm for computing the NBS in complexity O(K log 2 (K). Finally, we demonstrate in simulations the gains compared to to the competitive solution both in the flat fading and the frequency selective cases. We end up with some conclusions.
II. NASH EQUILIBRIUM VS. NASH BARGAINING SOLUTION
In this section we describe two solution concepts for N players games. The first notion is that of Nash equilibrium. The second is the Nash bargaining solution (NBS). In order to simplify the notation we specifically concentrate on the Gaussian interference game.
A. The Gaussian interference game
In this section we define the Gaussian interference game, and provide some simplifications for dealing with discrete frequencies. For a general background on non-cooperative games we refer the reader to [13].
The Gaussian interference game was defined in [16]. In this paper we use the discrete approximation game.
Let f 0 < · · · < f K be an increasing sequence of frequencies. Let I k be the closed interval be given by
I k = [f k−1 , f k ].
We now define the approximate Gaussian interference game denoted by GI {I 1 ,...,I K } .
Let the players 1, . . . , N operate over K parallel channels. Assume that the N channels have transfer functions h ij (k). Assume that user i'th is allowed to transmit a total power of P i . Each player can transmit a power vector p i = (p i (1), . . . , p i (K)) ∈ [0, P i ] K such that p i (k) is the power transmitted in the interval I k .
Therefore we have K k=1 p i (k) = P i . The equality follows from the fact that in non-cooperative scenario all users will use the maximal power they can use. This implies that the set of power distributions for all users is a closed convex subset of the cube N i=1 [0, P i ] K given by:
B = N i=1 B i(6)
where B i is the set of admissible power distributions for player i given by:
B i = [0, P i ] K ∩ (p(1), . . . , p(K)) : K k=1 p(k) = P i .(7)
Each player chooses a PSD p i = p i (k) : 1 ≤ k ≤ N ∈ B i . Let the payoff for user i be given by:
C i (p 1 , . . . , p N ) = K k=1 log 2 1 + |h i (k)| 2 p i (k) P |h ij (k)| 2 p j (k)+σ 2 i (k)(8)
where C i is the capacity available to player i given power distributions p 1 , . . . , p N , channel responses
h i (f ), crosstalk coupling functions h ij (k) and σ 2 i (k) > 0 is external noise present at the i'th receiver at frequency k. In cases where σ 2 i (k) = 0 capacities might become infinite using FDM strategies, however this is non-physical situation due to the receiver noise that is always present, even if small. Each C i is continuous on all variables.
Definition 2.1:
The Gaussian Interference game GI {I 1 ,...,I k } = {C, B} is the N players non-cooperative game with payoff vector C = C 1 , . . . , C N where C i are defined in (28) and B is the strategy set defined by (6).
The interference game is a special case of convex non-cooperative N-persons game.
B. Nash equilibrium in non-cooperative games
An important notion in game theory is that of a Nash equilibrium.
Definition 2.2:
An N-tuple of strategies p 1 , . . . , p N for players 1, . . . , N respectively is called a Nash equilibrium iff for all n and for all p (p a strategy for player n)
C n p 1 , ..., p n−1 , p, p n+1 , . . . , p N < C n (p 1 , ..., p N )
i.e., given that all other players i = n use strategies p i , player n best response is p n .
The proof of existence of Nash equilibrium in the general interference game follows from an easy adaptation of the proof of the this result for convex games [1]. A much harder problem is the uniqueness of Nash equilibrium points in the water-filling game. This is very important to the stability of the waterfilling strategies. A first result in this direction has been given in [17], [18]. A more general analysis of the convergence has been given in [19].
C. Nash bargaining solution for the interference game
Nash equilibria are inevitable whenever a non-cooperative zero sum game is played. However they can lead to substantial loss to all players, compared to a cooperative strategy in the non-zero sum case, where players can cooperate. Such a situation is called the prisoner's dilemma. The main issue in this case is how to achieve the cooperation in a stable manner and what rates can be achieved through cooperation.
In this section we present the Nash bargaining solution [13]. The underlying structure for a Nash bargaining in an N players game is a set of outcomes of the bargaining process S which is compact and convex. S can be considered as a set of possible joint strategies or states, a designated disagreement outcome d (which represents the agreement to disagree and solve the problem competitively) and a multiuser utility function U : S ∪ {d}→R N . The Nash bargaining is a function F which assigns to each pair (S ∪ {d}, U) as above an element of S ∪ {d}. Furthermore, the Nash solution is unique. In order to obtain the solution, Nash assumed four axioms:
Linearity. This means that if we perform the same linear transformation on the utilities of all players than the solution is transformed accordingly. Symmetry. If two players are identical than renaming them will not change the outcome and both will get the same utility.
Pareto optimality. If s is the outcome of the bargaining then no other state t exists such that U(s) < U(t) (coordinate wise).
A good discussion of these axioms can be found in [13]. Nash proved that there exists a unique solution to the bargaining problem satisfying these 4 axioms. The solution is obtained by maximizing
s = arg max s∈S∪{d} N n=1 (U n (s) − U n (d)) .(9)
Typically one assumes that there exist at least one feasible s ∈ S such that U(d) < U(s) coordinatewise, but otherwise we can assume that the bargaining solution is d.
We also define the Nash function F (s) :
S ∪ {d}→R F (s) = N n=1 (U n (s) − U n (d)) .(10)
The Nash bargaining solution is obtained by maximizing the Nash function over all possible states. Since the set of possible outcomes U (S ∪ {d}) is convex F (s) has a unique maximum on the boundary of
U (S ∪ {d}).
Whenever the disagreement situation can be decided by a competitive game, it is reasonable to assume that the disagreement state is given by a Nash equilibrium of the relevant competitive game. When the utility for user n is given by the rate R n , and U n (d) is the competitive Nash equilibrium, it is obtained by iterative waterfilling for general ISI channels. For the case of mask constraints the competitive solution is simply given by all users using the maximal PSD at all tones.
III. NASH BARGAINING SOLUTION FOR THE FLAT FADING N PLAYERS INTERFERENCE GAME
In this section we provide conditions for the existence of the Nash bargaining solution (NBS) for the N × N flat frequency interference game. In general, the rate region for the interference channel is unknown. However, by a simple time sharing argument we know that the rate region is always a convex set R, i.e.
R = {r : r = (R 1 , R 2 , ..., R N ) is in the rate region } .(11)
is a convex set. Typically we will use the utility defined by the rate, i.e., for every rate vector r = (R 1 , ..., R N ) T we have U n (r) = R n . Later we will show how the results can be generalized to other utility
functions such as U L n (t) = log (R n )
For some specific operational strategies one can define an achievable rate region explicitly. This allows for explicit determination of the strategies leading to the NBS. One such example is the use of FDM or TDM strategies in the interference channel. In the sequel we analyze the N players interference game, with FDM or TDM strategies. We provide conditions under which the bargaining solution exists, i.e., FDM strategies provide improvement over the competitive solution. This extends the work of [10] where we characterized when does FDM solution outperforms the competitive IWF solution for symmetric 2x2 interference game. We have shown there that indeed in certain conditions the competitive game is subject to the prisoner's dilemma where the competitive solution is suboptimal for both players. Let the utility of player n is given by U n = R n . The received signal vector x is given by
x = Hs + n(12)
where x = [x 1 , ..., x N ] T is the received signal, and H = {h ij }, 0 ≤ i, j ≤ N, is the interference coupling matrix, s = [s 1 , s 2 , ..., s N ] T is the vector of transmitted signals. We will assume that for all i, j |h ij | < 1.
Moreover, we will assume that the matrix H is invertible. This assumption is reasonable since typical wireless communication channels are random, and the probability of obtaining a singular channel is 0. Note that in our case both transmission and reception are performed independently, and the vector formulation is used for notational simplicity. First observe:
R c n = W 2 log 2 1 + |h nn | 2 P n W N 0 /2 + N j=1,j =n |h nj | 2 P ij(13)
Proof: To see that the flat power allocations form a Nash equilibrium for a flat channel, we first note that when all players j = n use flat power spectrum, the total interference plus noise spectrum is also flat.
Hence waterfilling by player n against flat power allocation results in flat power spectrum. This implies that the flat power spectrum is indeed a Nash equilibrium point. To obtain the uniqueness, assume that the total power limit of the users is given by p = [P 1 , ..., P N ] T and that the spectrum is divided into K identical bands. Assume that user n strategy at the equilibrium is given by ρ = [ρ n (1), ..., ρ n (K)] T . We note that the mutual waterfilling equations can be written for all k = k ′
HΛ k p + N 0 I = HΛ k ′ p + N 0 I(14)
where Λ k = diag{ρ 1 (k), . . . , ρ N (k)}. By our assumption H is invertible and Λ k is diagonal for each k so we must have for all n, k, ρ n (k) = ρ n (1), obtaining the uniqueness. Finally we note that when interference is very strong there are other Nash equilibrium points on the boundary of the strategy space, where not all frequencies are used by all users.
To simplify the expression for the competitive rates we divide the expression inside the log in (13) by the noise power W N 0 /2 obtaining:
R c n = W 2 log 2 1 + SNR n 1 + N j =n α nj SNR j (15) where SNR j = |h jj | 2 P j W N 0 /2 , α nj = |h nj | 2 |h jj | 2 .
Since the rates R c n are achieved by competitive strategy, player n would not cooperate unless he will obtain a rate higher than R c n . Therefore, the game theoretic rate region is defined by set of rates higher that R c n of equation (15). We are interested in FDM cooperative strategies. A strategy is a vector [ρ 1 , ..., ρ N ] T such that N n=1 ρ n ≤ 1. We assume that player n uses a fraction ρ n (0 ≤ ρ n ≤ 1) of the band (or equivalently uses the channel for a fraction ρ n of the time in the TDM case). The rate obtained by the n th player is given by
R n (ρ) = R n (ρ n ) = ρnW 2 log 2 1 + SNRn ρn .(16)
First we note that the FDM rate region
R F DM = {(R 1 , ..., R N )|R n = R n (ρ n )} is indeed convex. The
Pareto optimal points must satisfy N n=1 ρ n = 1, since by dividing the unused part of the band between users, all of them increase their utility. Also note that by strict monotonicity of R n (ρ) as a function of ρ each pareto optimal point is on the boundary of R F DM . It is achieved by a single strategy vector ρ.
Player n benefits from FDM cooperation as long as R c n < R n (ρ n ).
The Nash function is given by
F (ρ) = N n=1 (R n (ρ n ) − R c n ) .(18)
To better understand the gain in FDM strategies we define a function f (x, y) that is fundamental to the analysis.
Definition 3.1: For each 0 < x, y let f (x, y) be defined by f (x, y) = min ρ : 1 + x ρ ρ = 1 + x 1 + y .(19)
Claim 3.1: 1. f (x, y) is a well defined function for x, y ∈ R + .
2. For all x, y ∈ R + , 0 < f (x, y) < 1.
f (x, y) is monotonically decreasing in y.
Proof: Let g(x, y, ρ) be defined by:
g(x, y, ρ) = 1 + x ρ ρ − 1 − x 1 + y
For every x, y, g(x, y, ρ) is a continuous and monotonic function in ρ. Furthermore, for any 0 < x, y, g(x, y, 1) > 0, and lim ρ→0 g(x, y, ρ) < 0. Hence, there is a unique solution to (19). Furthermore, the value of f (x, y) is strictly between 0, 1. Finally f (x, y) is monotonically decreasing in y since g(x, y, ρ)
is increasing in y, so if we increase y we need to decreas ρ to maintain a fixed value.
Using the function f (x, y) we can completely characterize the cases where NBS is preferable to the Nash equilibrium.
Theorem 3.2: Nash bargaining solution exists if and only if the following inequality holds
N n=1 f SNR n , j =n α nj SNR j ≤ 1.(20)
Proof: In one direction, assume that a Nash bargaining solution exists. The next two conditions must hold 1. There is a partition of the band between the players such that player n gets a fraction ρ n > 0.
2. Each player gets by cooperation higher rate then the competitive rate, i.e, R n (ρ n ) ≥ R c n . Therefore, using equation (21) and inequality (17) we obtain that equation (20) must be satisfied. On the other direction by definition of f player n has at least the rate that it can get by competition if he can use a fraction ρ n , of the bandwidth. Since (20) implies that N n=1 ρ n ≤ 1, FDM is preferable to the competitive solution for the utility function U n = R n . By the convexity of the FDM rate region the Nash function has a unique maximum that is Pareto optimal and outperforms the competitive solution.
Interestingly, as long as the utility function U n (ρ) depends only on ρ n and U n (ρ) is monotonically increasing in ρ the same conclusion holds. This implies that the NBS when the utility is U L n (ρ) = log (R n (ρ n )) there is a unique frequency division vector ρ that achieves the NBS. Furthermore the optimization problem, of computing the optimal ρ is still convex.
We now examine the simple case of two players. Assume that player I uses a fraction ρ (0 ≤ ρ ≤ 1) of the band and user II uses a fraction 1 − ρ. The rates obtained by the two users are given by
R 1 (ρ) = ρW 2 log 2 1 + SNR 1 ρ R 2 (1 − ρ) = (1−ρ)W 2 log 2 1 + SNR 2 1−ρ(21)
The two users will benefit from FDM cooperation as long as
R c i ≤ R i (ρ i ), i = 1, 2 ρ 1 + ρ 2 ≤ 1(22)
Condition (20) can now be simplified:
f (SNR 1 , αSNR 2 ) + f (SNR 2 , βSNR 1 ) ≤ 1,(23)
where
SNR i = |h ii | 2 P i W N 0 /2 , α = |h 12 | 2 |h 22 | 2 , β = |h 21 | 2 |h 11 | 2 .
The NBS is given by solving the problem
ρ N BS = arg max ρ F (ρ)(24)
where the Nash function is now given by:
F (ρ) = (R 1 (ρ) − R c 1 ) (R 2 (1 − ρ) − R c 2 )(25)
and R i (ρ) are defined by (21). A special case can now be derived:
Claim 3.2: Assume that SNR 1 ≥ 1 2 (α 2 β 4 ) −1/3 and SNR 2 ≥ 1 2 (β 2 α 4 ) −1/3 . Then there is a Nash bargaining solution that is better than the competitive solution. When the channel is symmetric (α = β)
the solution exists as long as SNR ≥ 1 2α 2 . Proof: The proof of the claim follows directly by substituting solving the equation for ρ 1 = ρ 2 = 1/2.
Finally we note that as SNR i increases to infinity the NBS is always better than the NE.
1 + SNR 1 ρ ρ 1 + SNR 2 1 − ρ 1−ρ < 1 + SNR 1 1 + αSNR 2 1 + SNR 2 1 + βSNR 1 .(27)
Proof: The claim follows easily by applying the inequality x ρ y 1−ρ ≤ ρx + (1 − ρ) y on the left hand side of the above inequality and using the assumption.
The following example provides the intuition for the definitions of the game theoretic rate region, and the uniqueness of the NBS using FDM strategies. It also clearly demonstrates the relation between the competitive solution, the NBS and the game theoretic rate region R G . We have chosen SNR 1 = 20 dB, SNR 2 = 15 dB, and α = 0.4, β = 0.7. Figure 1 presents the FDM rate region, the Nash equilibrium point denoted by , and a contour plot of F (ρ). It can be seen that the concavity of NF (ρ) together with the convexity of the achievable rate region implies that at there is a unique contour tangent to the rate region. The tangent point is the Nash bargaining solution. We can see that the NBS achieves rates that are 1.6 and 4 times higher than the rates of the competitive Nash equilibrium rates for player I and player II respectively. The game theoretic rate region is the intersection of the information theoretic rate region with the quadrant above the dotted lines.
IV. BARGAINING OVER FREQUENCY SELECTIVE CHANNELS UNDER MASK CONSTRAINT
In this section we define a new cooperative game corresponding to the joint FDM/TDM achievable rate region for the frequency selective N users interference channel. We limit ourselves to the PSD mask constrained case since this case is actually the more practical one. In real applications, the regulator limits the PSD mask and not only the total power constraint. Let the K channel matrices at frequencies k = 1, ..., K be given by H k : k = 1, ..., K . Each player is allowed to transmit at maximum power p (k)
in the k'th frequency bin. In non-cooperative scenario, under mask constraint, all players transmit at the maximal power they can use. Thus, all players choose the PSD, p = p i (k) : 1 ≤ k ≤ K . The payoff for user i in the non-cooperative game is therefore given by:
R iC (p 1 ) = K k=1 log 2 1 + |h i (k)| 2 p i (k) j =i |h ij (k)| 2 p j (k) + σ 2 i (k) .(28)
Here, R iC is the capacity available to player i given a PSD mask constraint distributions p. σ 2 i (k) > 0 is the noise presents at the i'th receiver at frequency k. Note that without loss of generality, and in order to simplify notation, we assume that the width of each bin is normalized to 1. We know define the cooperative game G T F (N, K, p). 3) The utility of the i'th player is given by
R i = K k=1 R i (k) = K k=1 α ik log 2 1 + |h ii (k)| 2 p i (k) σ 2 i (k)(29)
Note that interference is avoided by time sharing at each frequency band, i.e only one player transmits at a given frequency bin at any time. Furthermore, since at each time instance each frequency is used by a single user, each user can transmit using maximal power.
The Nash bargaining can be posed as an optimization problem
max N n=1 (R i (α i ) − R iC ) subject to: N i=1 α i (k) = 1, ∀i, k α i (k) ≥ 0, ∀i R iC ≤ R i (α i ) ,(30)
where,
R i (α i ) = K k=1 α i (k) log 2 1 + |h i (k)| 2 P max (k) σ 2 i (k) = K k=1 α i (k)R i (k)) .(31)
This problem is convex and therefore can be solved efficiently using convex optimization techniques. To that end we explore the KKT conditions for the problem. The Lagrangian of the problem f (α) is given
by f (α) = − N i=1 log (R i (α i ) − R iC ) + K k=1 λ k N i=1 α i (k) − 1 − K k=1 N i=1 µ i (k)α i (k) − N i=1 δ i K k=1 α i (k) R i (k) − R iC .(32)
Taking the derivative with respect to the variable α i (k) and comparing the result to zero, we get
R i (k) R i (α i ) − R iC = λ k − µ i (k) − δ i(33)
with the constraints
N i=1 α i (k) = 1, δ i (R i (α i ) − R iC ) ≥ 0, µ i (k)α i (k) = 0, λ k ≥ 0.(34)
Based on (33, 34) one can easily come to the following conclusions:
1) If there is a feasible solution then for all i, δ i = 0.
2) Assume that a feasible solution exists. Then for all players sharing the frequency bin k (α i (k) > 0)
we have µ i (k) = 0, and
R i (k) R i (α i ) − R iC = λ k , ∀k satisfying α i (k) > 0.(35)
3) For all players that are not sharing the frequency bin k,(α i (k) = 0), µ i (k) ≥ 0. Therefore,
R i (k) R i (α i ) − R iC ≤ λ k , ∀k with α i (k) = 0.(36)
Clause (2) is very interesting. let L ij (k) = R i (k)/R j (k). Assume that for users i,j the values L ij (k) are all distinct. Then the two users can share at most a single frequency. To see this note that in this case
R i (k) R i (α i ) − R iC = R j (k) R j (α j ) − R jC(37)
and therefore
L ij (k) = R i (k) R j (k) = R i (α i ) − R iC R j (α j ) − R jC(38)
Since the right hand side is independent of the frequency k and L ij (k) are distinct, at most a single frequency can satisfy this condition. This proves the following theorem:
Theorem 4.1: Assume that for all i = j the values {L ij (k) : k = 1, ..., K} are all distinct. Then in the optimal solution at most N 2 frequencies are shared between different users. This theorem suggests, that when N 2 << K the optimal FDM NBS is very close to the joint FDM/TDM solution. It is obtained by allocating the common frequencies to one of the users.
While general convex optimization techniques are useful for computing the NBS, in the next section we will demonstrate that for the two players case the solution can be computed much more efficiently.
Furthermore, we will show that in the optimal solution only a single frequency is actually shared between the users even if the L ij (k) are not distinct.
Finally we comment on the applicability of the method to the case where only fading statistics is known. In this case the coding strategy will change, and the achievable rate in the competitive case and the cooperative case are given bỹ
R iC (p i ) = K k=1 E log 2 1 + |h i (k)| 2 p i (k) P j =i |h ij (k)| 2 p j (k)+σ 2 i (k) R i (α i ) = K k=1 α ik E log 2 1 + |h ii (k)| 2 p i (k) σ 2 i (k)(39)
respectively. All the rest of the discussion is unchanged, replacing R iC and
R i (α i ) byR iC ,R i (α i )
respectively.
V. COMPUTING THE NASH BARGAINING SOLUTION FOR TWO PLAYERS
For the two players case the optimization problem can be dramatically simplified. In this section we will provide an O(K log 2 K) complexity algorithm (in the number of tones) for computing the NBS optimal solution in a 2 users frequency selective channel. Furthermore, we will show that the two players will share at most a single frequency, no matter what the ratios between the users are. To that end let, α 1 (k) = α (k), and α 2 (k) = 1−α (k). We also define the surplus of players I and II when using Nash bargaining solution
as A = K m=1 α (m) R 1 (m) − R 1C and B = K ,=1 (1 − α (m)) R 2 (m) − R 2C
, respectively. The ratio, Γ = A/B is a threshold which is independent of the frequency and is set by the optimal assignment.
While Γ is a-priori unknown, it exists. Let L(k) = R 1 (k) /R 2 (k). Without loss of generality, assume that the rate ratios L(k), 1 ≤ k ≤ K are sorted in decreasing order i.e. L(k) ≥ L(k ′ ), ∀k ≤ k ′ . (This can be achieved by sorting the frequencies according to L(k).
We are now ready to define optimal assignment the α's. Let Γ k be a moving threshold defined by
Γ k = A k /B k . where A k = k m=1 R 1 (m) − R 1C , B k = K m=k+1 R 2 (m) − R 2C .(40)
A k is a monotonically increasing sequence, while B k is monotonically decreasing. Hence, Γ k is also monotonically increasing. A k is the surplus of user I respectively when frequencies 1, ..., k are allocated to user I. Similarly B k is the surplus of user II when frequencies k + 1, ..., K are allocated to user II. Let
k min = min k {k : A k ≥ 0} ; k max = min k {k : B k < 0} .(41)
Since we are interested in feasible NBS, we must have positive surplus for both users. Therefore, by the KKT equations, we obtain k min ≤ k max and L(k min ) ≤ Γ ≤ L(k max ). The sequence {Γ m : k min ≤ m ≤ k max − 1} is strictly increasing, and always positive. We first state two lemmas that are essential for finding the optimal partition.
Lemma 5.1: Assume that there is an NBS to the game. Then there is always a NBS satisfying that at most a single bin k s is partitioned between the players, and
α(k) = 1 k < k s 0 k > k s .(42)
Proof: By our assumption the sequence {L(k) : k = 1, ..., K} is monotonically decreasing (not necessarily strictly decreasing). If there is a k such that L(k − 1) < Γ < L(k) then the solution must be FDM type by the KKT equations and we finish. Otherwise assume that L(k) = Γ. Since Γ k is strictly increasing and L(k) is non-increasing there is at most a unique k such that Γ k−1 ≤ L(k) = Γ < Γ k . If no such k exists then the users can only share k max since for all k ≤ k max
A k B k ≤ Γ
and the only way to get something allocated to user II is by sharing k max . Otherwise such a k ≤ k max exists. By definition of Γ k we have
A k−1 B k−1 ≤ L(k) < A k B k .
Simple substitution yields
A k−1 B k−1 ≤ L(k) < A k−1 + R 1 (k) B k−1 − R 2 (k) = A k B k .
Since k min ≤ k < k max the denominator on the RHS is positive. Since for a, b, c, d > 0 the function a+xb c−xd is increasing with 0 ≤ x as long as the denominator is positive, we obtain that by continuity there is a unique ζ such that
L(k) = A k−1 + ζR 1 (k) B k−1 − ζR 2 (k) . But B k−1 − ζR 2 (k) = B k + (1 − ζ)R 2 (k) so that ζ satisfies Γ = L(k) = A k−1 + ζR 1 (k) B k + (1 − ζ)R 2 (k)
.
Setting α(m) = 1 for m < k, α(k) = ζ and α(m) = 0 for m > k we obtain a solution of the KKT equations. Note that when there are multiple values of k such that L(k) = Γ, we only showed that there is an NBS solution where a single frequency is shared.
While the threshold Γ is unknown, one can use the sequences Γ k and L(k).
If there is a Nash bargaining solution, let k s be the frequency bin that is shared by the players. Then, k min ≤ k s ≤ k max . Since, both players must have a positive gain in the game (A > A k min −1 ,B > B kmax ).
Let k s be the smallest integer such that L(k s ) < Γ ks , if such k s exists. Otherwise let k s = k max .
Lemma 5.2:
The following two statements provide the solution 1
If a Nash bargaining solution exists for k min ≤ k s < k max , then α (k s ) is given by α (k s ) = max{0, g}, where
g = 1 + B ks 2R 2 (k s ) 1 − Γ ks L(k s ) .(43)
2 If a Nash bargaining solution exists and there is no such k s , then k s = k max and α (k s ) = g.
Proof:
To prove 1 note that since Γ ks−1 ≤ L(k s ) ≤ Γ ks , α (k s ) is the solution to the equation
L(k s ) = A ks −(1−α(ks))R 1 (ks)
B ks +(1−α)R 2 (ks) . By simple mathematical manipulation, we get α (k s ) = g. Since, L(k) ≤ Γ ks , g ≤ 1. If g is negative, we set α (k s ) = 0, since k s is the smallest integer such that L(k s ) < Γ ks . Note, that in this case the Nash bargaining solution is given by pure FDM strategies.
To prove 2 note that since k s = k max and Γ k is increasing for k min ≤ k < k max , we must have that Γ kmax−1 ≤ Γ = L(k max ). Therefore, the only possibility that there is a solution is if k s = k max , and α (k s ) = g ≥ 0.
Based on the pervious lemmas the algorithm is described in table I. In the first stage the algorithm computes L(k) and sorts them in a non increasing order. Then k min , k max , A k , and B k are computed. In the second stage the algorithm computes k s and α. Figure 2 demonstrates the situation when SNR = 30dB and SIR is 10dB. In this case k max = 10 since B 11 becomes negative. Also Γ 8 < L(9) < Γ 9 . Therefore, only frequency 9 might be shared between the users. The algorithm computes a Nash bargaining solution if it exists, even in the case that L(k) is not a strictly decreasing sequence. However, reordering the bins with identical ratio may provides a different solution, with the same capacity gain for each player.
VI. SIMULATIONS
In this section we compare in simulations the Bargaining solution to the competitive solution for various situations with medium interference. The simulations are done both for flat slow fading and for frequency selective fading. First, we demonstrate the effect of the channel matrix and the signal to noise ratio on the gain of the NBS for flat fading channel. Then we performed extensive simulations that demonstrate the advantage of the NBS over the competitive approach for the frequency selective fading channel, as a function of the mean interference power.
A. Flat fading
We have tested the gain of the Nash bargaining solution relative to the Nash equilibrium competitive rate pair as a function of channel coefficients as well as signal to noise ratio for the flat fading channel.
To that end we define the minimum relative improvement describing the individual price of anarchy by:
∆ min = min R N BS 1 R c 1 , R N BS 2 R c 2(44)
and the usual price of anarchy [20], describing total loss due to lack of cooperation by
∆ sum = R N BS 1 + R N BS 2 R c 1 + R c 2 .(45)
In the first set of experiments we have fixed α, β and varied SNR 1 , SNR 2 from 0 to 40 dB in steps of 0.25dB. Figure 3 presents ∆ min for an interference channel with α = β = 0.7. We can see that for high SNR we obtain significant improvement. Figure 4 presents the relative sum rate improvement ∆ sum for the same channel. We can see that the achieved rates are 5.5 times those of the competitive solution. We have now studied the effect of the interference coefficients on the Nash Bargaining solution. We have set the signal to additive white Gaussian noise ratio for both users to 20 dB, and varied α and β between 0 and 1. Similarly to the previous case we present the minimal price of anarchy per user ∆ min and the sum rate price of anarchy ∆ sum . The results are shown in figures 5,6. We can clearly see that even with SINR of 10 dB we obtain 50 percent capacity gain per user.
B. Frequency selective Gaussian channel
In this experiment We demonstrate the advantage of the Nash bargaining solution over competitive approaches for a frequency selective interference channel. We assumed that two users having direct channels that are standard Rayleigh fading channels (σ 2 = 1), with SNR=30 dB, suffer from interference, with SINR of each user into the other channel (h ij ) was varied from 10 dB to 0 dB (σ h ij = 0.1, ...1).
We have used 32 frequency bins. At each pair of variances σ 2 1 = σ 2 h 21 , σ 2 2 = σ 2 h 12 we randomly picked 25 channels (each comprising of 32 2x2 matrices). The results of the minimal relative improvement (44) are depicted in figure 7. We can clearly see that the relative gain of the Nash bargaining solution over the competitive solution is 1.5 to 3.5 times, which clearly demonstrates the merrits of the method.
VII. CONCLUSIONS
In this paper we have defined the tic rate region for the interference channel. The region is a subset of the rate region of the interference channel. We have shown that a specific point in the rate region given by the Nash bargaining solution is better than other points in the context of bargaining theory. We have shown conditions for the existence of such a point in the case of the FDM rate region. We have shown that computing the Nash bargaining solution over a frequency selective channel can be described as a convex optimization problem. Moreover, we have provided a very simple algorithm for solving the problem in the 2xK case that is O(K log 2 K), where K is the number of tones. Finally, we have demonstrated through simulations the significant improvement of the cooperative solution over the competitive Nash equilibrium.
The adaptation of game theory approach for rate allocation in existing wireless and wireline system is very appealing. In many wireless LAN systems there is a central access point with full knowledge on the channel transfer functions. Moreover, it has been recognized by the 802.11 committee that radio resource management is importnat, especially when multiple networks are interfering with other. Knowledge of the transfer functions allows the access point to allocate the band for the subscribers on the uplink. Moreover, the results here can be extended to MIMO systems as well as for networks with multiple access points. If no such k exists, set ks = kmax and calculate g.
If g ≥ 0 set α ks = g, α(k) = 1, for k < kmax. Stop.
Else (g < 0)
There is no NBS. Use competitive solution.
End.
Independence of irrelevant alternatives. This axiom states that if the bargaining solution of a large game T ∪ {d} is obtained in a small set S. Then the bargaining solution assigns the same solution to the smaller game, i.e., The irrelevant alternatives in T \S do not affect the outcome of the bargaining.
Lemma 3. 1 :
1The competitive strategies in the Gaussian interference game are given by flat power allocation. The resulting rates are:
Claim 3 . 3 :
33If SNR 1 and SNR 2 are jointly increasing, while keeping the ratio SN R 1 SN R 2 = z fixed. Then, there is a constant g such that for SNR 1 > g, an FDM Nash Bargaining solution exists. the constant ratio x/y. The function h(x, z) is monotonically decreasing to zero as a function of x for any fixed value of z. Therefore, there is a constant g, such that for x > g the inequality,h x, z α + h x z , 1 βz < 1 is satisfied. Since by definition of f (x, y) we have h(x, z) > f (x, y), the equation f x, x αy + f y, y βx < 1 also holds for all x > g and y = zx.Claim 3.4: If SNR 1 + SNR 2 ≤ 1−α−β αβ there is no Nash bargaining solution. Proof: Nash Bargaining solution does not exists if
Definition 4 . 1 :
41The FDM/TDM game G T F (N, K, p) is a game between N players transmitting over K frequency bins under common PSD mask constraint. Each user has full knowledge of the channel matrices H k . The following conditions hold: 1) Player i transmits using a PSD limited by p i (k) : k = 1, ..., K satisfying p i (k) ≤ p(k).
2 )
2Strategies for player i are vectors α = [α i1 , ..., α iK ] T where α k is the proportion of time the player uses the k'th frequency channel. This is the TDM part of the strategy.
Define three sets: S 1 = {m : L(m) > Γ, A > 0, B > 0}, S 2 = {m : L(m) < Γ, A > 0, B > 0}, S c = {m : L(m) = Γ, A > 0, B > 0}. For all m ∈ S 1 α(m) = 1. For all m ∈ S 2 α(m) = 0. and for m ∈ S c 0 ≤ α(m) ≤ 1. Thus if the set S c is empty, pure FDM is a Nash bargaining solution.
Fig. 1 .
1FDM rate region (thick line), Nash equilibrium * , Nash bargaining solution and the contours of F (ρ). SNR1 = 20 dB, SNR2 = 15 dB, and α = 0.4, β = 0.7 TABLE I ALGORITHM FOR COMPUTING THE 2X2 FREQUENCY SELECTIVE NBS : Initialization: Sort the ratios L(k) in decreasing order. Calculate the values of A k , B k and Γ k , kmin, kmax, If kmin > kmax no NBS exists. Use competitive solution. Else For k = kmin to kmax − 1 if L(k) ≤ Γ k . Set ks = k and α ′ s according to the lemmas-This is NBS. Stop End End
Fig. 2 .
2Sorted L(k) and Γ k .
Fig. 3 . 23 Fig. 4 .
3234Per user price of anarchy (relative improvement of NBS sum rate over NE), as a function of SNR. α = β = 0.7. Price of anarchy, as a function of SNR. α = β = 0
Fig. 5 .
5Per user price of anarchy. SNR=20 dB.
Fig. 6 .
6Sum rate price of anarchy as a function of interference power. SNR=20 dB.
Fig. 7. Per user price of anarchy for frequency selective Rayleigh fading channel. SNR=30 dB.
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|
[] |
[
"Detecting Multi-Spin Interactions in the Inverse Ising Problem",
"Detecting Multi-Spin Interactions in the Inverse Ising Problem"
] |
[
"Joseph Albert \nPhysics Department\nCarnegie Mellon University\n15213PittsburghPennsylvaniaUSA\n",
"Robert H Swendsen \nPhysics Department\nCarnegie Mellon University\n15213PittsburghPennsylvaniaUSA\n"
] |
[
"Physics Department\nCarnegie Mellon University\n15213PittsburghPennsylvaniaUSA",
"Physics Department\nCarnegie Mellon University\n15213PittsburghPennsylvaniaUSA"
] |
[] |
While the usual goal in Monte Carlo (MC) simulations of Ising models is the efficient generation of spin configurations with Boltzmann probabilities, the inverse problem is to determine the coupling constants from a given set of spin configurations. Most recent work has been limited to local magnetic fields and pair-wise interactions. We have extended solutions to multi-spin interactions, using correlation function matching (CFM). A more serious limitation of previous work has been the uncertainty of whether a chosen set of interactions is capable of faithfully representing real data. We show how our confirmation testing method uses an additional MC simulation to detect significant interactions that might be missing in the assumed representation of the data.
|
10.1016/j.physa.2017.04.120
|
[
"https://arxiv.org/pdf/1703.01675v1.pdf"
] | 119,488,522 |
1703.01675
|
5edd2d07fd30a02c13424c5c9fa136b0e66d0c6a
|
Detecting Multi-Spin Interactions in the Inverse Ising Problem
(Dated: October 8, 2018)
Joseph Albert
Physics Department
Carnegie Mellon University
15213PittsburghPennsylvaniaUSA
Robert H Swendsen
Physics Department
Carnegie Mellon University
15213PittsburghPennsylvaniaUSA
Detecting Multi-Spin Interactions in the Inverse Ising Problem
(Dated: October 8, 2018)Inverse IsingInferenceCorrelation functions
While the usual goal in Monte Carlo (MC) simulations of Ising models is the efficient generation of spin configurations with Boltzmann probabilities, the inverse problem is to determine the coupling constants from a given set of spin configurations. Most recent work has been limited to local magnetic fields and pair-wise interactions. We have extended solutions to multi-spin interactions, using correlation function matching (CFM). A more serious limitation of previous work has been the uncertainty of whether a chosen set of interactions is capable of faithfully representing real data. We show how our confirmation testing method uses an additional MC simulation to detect significant interactions that might be missing in the assumed representation of the data.
I. INTRODUCTION
In many fields, such as biology, sociology, and neuroscience, obtaining information about underlying interactions between components of a system from observed correlations can clarify the structure of the system [1][2][3][4][5][6][7][8][9]. This reconstruction of cause from consequence is known as an inverse problem. Because of its relative simplicity, the inverse Ising model has become a standard test case for the development of methods to deal with intrinsically complex inverse problems. In 1984, a numerical solution to the problem was found by correlation function matching (CFM), using an identity due to Callen [10][11][12][13]. At the time, the solution was only applied to transitionally invariant problems, but as we show below, the modifications to remove this restriction are trivial.
Recently, equations originally found with CFM were rediscovered by Aurell and Ekeberg, starting from different principles (pseudo-likelihood), and successfully applied to the Sherrington-Kirkpatrick (SK) model [9]. Their approach has the advantage of exhibiting the relationship of the solution to a Bayesian probability distribution on the space of the coupling constants. The CFM approach, on the other hand, clarifies the relationship between extracting information from the configurations and making inferences about the original coupling constants.
Recent work on inverse problems has been largely restricted to pairwise interactions, as in the SK model. We have extended it to include multi-spin interactions [10][11][12][13], but there is still a question of whether a given set of real data can be faithfully represented by the chosen set of interactions. We answer this question by introducing a confirmation phase into our computations, using a new Monte Carlo (MC) simulation with the fitted coupling constants. By examining correlation functions that were not used in the inverse solution, we show that differences between the new MC simulation and the original * [email protected] † [email protected] data reveal neglected interactions. This "confirmation testing" provides a straightforward way of determining whether more interactions are needed for a faithful representation of the data, without having to perform a full computation of the coupling constants for the additional interactions. We will next describe the CFM equations that provide the basis for confirmation testing.
II. CFM EQUATIONS
To express a general, multi-spin Ising interaction, let α be a subset of m α spins, and define the product of all spins in α as
S α = mα j=1 σ j .(1)
For each operator S α , we will assign a corresponding dimensionless coupling constant, K α = βJ α , where k B is Boltzmann's constant, T is the temperature, and β = 1/k B T . The corresponding term in the dimensionless Hamiltonian H = −βH, (where H is the usual Hamiltonian) associated with α is K α S α . The full dimensionless Hamiltonian can then be written as a sum over the set of all spin products,
H = α K α S α ,(2)
where the set K = {K α }, are the true values of the coupling constants. We define an operator, H , that includes all terms in the Hamiltonian containing a specific spin, σ [10][11][12]. If σ ∈ α, we also define an operator S α, that omits the spin σ .
S α, = S α /σ(3)
If σ / ∈ α, S α, = 0. The sum of all terms in the Hamiltonian that contain the "central" spin σ is then
H = σ α K α S α, .(4)
arXiv:1703.01675v1 [cond-mat.dis-nn] 5 Mar 2017
The CFM method is based on fitting the correlation functions c α ≡ S α using an identity due to Callen [13].
c α (K) ≡ S α, tanh γ K γ S γ, = S α(5)
For comparison with earlier work, Eq. (5) corresponds to Eq. (14) in Ref. [10]. Aurell and Ekeberg considered P (σ j ) = (σ j + 1)/2 instead of σ j . Since the probability that a spin is positive is
P (+1) = (1 − exp [−2F ]) −1 = (tanh [F ] + 1) /2, (6) where F = γ K γ S γ, , Eq. (5) is equivalent to Eq. (4)
in Ref. [9]. Eq. (5) is exact for the correct values of the coupling constants and the exact correlation functions. Unfortunately, we never have the exact values of the correlation functions c α = S α . Data always come from a finite sample, which we will take to be N MC spin configurations. We denote the corresponding approximate correlations functions as c α = S α M C . We can then use a modification of Eq. (5) to find a set of approximate coupling parameters K = { K γ } to fit the MC correlation functions.
c α K ≡ S α, tanh γ K γ S γ, M C = S α M C (7)
The equality in Eq. (7) will only hold for specific values of the coupling parameters K. If trial values for the set K differ from the best-fitting value by δ K = {δK α }, an improved estimate can be obtained from a linearized approximation for the deviations from the best-fitting values.
c α K + δ K − S α M C ≈ β ∂ c α K ∂K β δ K β(8)
The derivatives in Eq. (8) are given by
∂ c α K ∂K β = S α, S β, sech 2 γ K γ S γ, M C(9)
Eq. (8) is iterated until convergence, which is quadratic in the absence of a degeneracy. Again for comparison with earlier work, Eqs. (9) corresponds to Eq. (15) in Ref. [10], and Eq. (7) in Ref. [9]. Note that for an interaction α with m α spins, this procedure produces m α values of K α , one for each choice of the "central" spin σ . Before going on to confirmation testing, we next describe the application of the CFM solution to the inverse Ising problem, along with limitations that not only CFM, but any inverse method, will have with respect to recovering the true coupling constants. A virtue of the CFM is that it exposes these limitations clearly.
III. THE LIMITED INFORMATION CONTAINED IN A SET OF CONFIGURATIONS
There is an important distinction between extracting information contained in the configurations and inferring the values of the true couplings. For example, if H = hσ, σ exact = tanh(h exact ) is the exact average value of a spin σ in a dimensionless field h exact . Given σ MC from an MC simulation, it is easy to find an effective magnetic field, h eff = tanh −1 σ MC , that reproduces it to arbitrary accuracy. Since σ MC is not exactly equal to σ exact , h eff will differ from h exact . The uncertainty in inferring h exact is given by δh ≈ cosh(h)/ √ N M C . For small values of h, the error is approximately equal to the minimum error, δh ≈ δh min = 1/ √ N M C . Our simulation results have shown that the errors in estimating coupling constants at high temperatures are very close to the minimum error, even for large numbers of spins.
This simple estimate of h eff breaks down for σ MC = 1 because it would imply that h eff = ∞. A simple Bayesian argument suggests replacing σ MC by 1 − 2/N MC , which gives a finite value for h eff . Although this is a coarse method, it is quite effective for improving results. Aurell and Ekeberg used a similar strategy, but took the factor to be 0.999 for all values of N MC [9].
The limited information contained in the configurations is illustrated by the Sherrington-Kirkpatrick (SK) model of a spin glass [14]. In this model, there are N spins, σ = {σ j = ±1|j = 1, 2, . . . , N }, and the Hamiltonian is
H(σ) = − j b j σ j − j>k J j,k σ j σ k ,(10)
where the couplings J j,k have a quenched Gaussian distribution of width J/ √ N . The local magnetic fields b j can either be set equal to zero or given independent quenched values. The corresponding dimensionless coupling constants are K j,k = βJ j,k and h j = βb j .
The SK model is known to have a rugged energy landscape and a spin-glass phase transition at k B T c = J. This makes it very difficult to generate independent configurations at low temperatures, which limits the information carried by the configurations. However, it does not affect our ability to extract whatever information there is. While we must be careful in interpreting our results, predictions cannot be made more accurate without additional information.
For large systems at high temperatures, a different limitation comes from the minimum error for the correlation functions. Since the magnitude of the couplings goes as 1/ √ N , the maximum temperature for which it is possible to determine the couplings to an accuracy of = δK ,j /K ,j is
T max J k B N M C N .(11)
To demonstrate that large coupling constants are not intrinsically difficult to determine -except for the fac-tor of cosh(h) in the errors -we've carried out simulations with optimal sampling, that is, choosing independent random values for the spins at neighboring sites of a central site . The values of σ are then chosen with the thermal probability for a trial set of K ,j 's. While the errors increase at low temperatures, good estimates of the original couplings can still be obtained for an SK distribution of quenched couplings down to T = 0.05 T c , as shown in Table I. TABLE I. Errors in estimates of original couplings from MC simulations for the toy model with random sampling. n = N − 1 = 49 neighboring spins were independently assigned the values ±1 with equal probability. The true couplings were generated randomly from a Gaussian distribution with the width βJ/ √ N . The width of the actual distribution is given as ∆K in the second column. For these simulations, NMC = 10 5 . The RMS error in the estimated couplings, given in the third column as δK ,j , was found from the results of ten independent trials. The errors in the K's were also compared to the minimum error δmin = 1/ √ NMC = 0.0032 in the last column.
T ∆K δK ,j δK ,j /K δK ,j /δmin The difficulties in extracting information from configurations generated below the critical temperature of the SK model are not due to any defect in the method of solution; the information is simply not available.
Because the low-temperature SK model has extremely long correlation times, MC simulations will usually only sample near a local free-energy minimum. For those states, it is common for some of the correlation functions to lock into values of ±1. When this happens, little information can be obtained about the corresponding interaction.
If we have the option to change the sampling method, restarting the simulations with different random initial conditions will generally improve results. Even though it is not possible to generate a complete sampling of a lowtemperature spin glass in this manner, it can improve estimates of coupling constants.
The twin features of having small coupling constants for large systems and high temperatures, and a phase transition that limits information at low temperatures, leave only a small range of parameters for testing. This led us to use short-range models with multi-spin interactions to illustrate confirmation testing, as discussed in the following section.
IV. CONFIRMATION TESTING
Any solution of the inverse Ising problem assumes a certain set of interactions that might be non-zero. When using confirmation testing, after fitting the coupling constants for those interactions, we perform an additional simulation using the fitted coupling constants to generate a new set of configurations. We then compare the correlation functions in the new set of configurations with those in the old set. If they match within the statistical errors discussed above, we have confirmed our assumptions. Deviations may reveal important interactions that were initially missing.
To illustrate confirmation testing, we have done MC simulations of a Hamiltonian with magnetic fields, nearest-neighbor pair interactions, and four-spin interactions on nearest-neighbor plaquettes on a 32 × 32 lattice, with N M C = 10 5 . Although it is not obvious from this plot, the errors in the four-spin coupling constants are smaller than those for the two-spin couplings, which are smaller than those for the local magnetic fields.
Next, we tried fitting our data with local magnetic fields and pairwise interactions, but omitting the fourspin couplings. Convergence was rapid, and the local magnetizations and the two-spin correlation functions were fit to better than 10 −10 . However, as can be seen in Fig. 2, the fitted values of the coupling constants deviated substantially from the true values.
We carried out a confirmation simulation, using the local fields and two-spin couplings (but no four-spin terms) shown in Fig. 2. As expected, we found good agree- ment with the local magnetizations and two-spin correlation functions from the full Hamiltonian (with four-spin terms). However, we found poor agreement for the fourspin correlations, as shown in Fig. 3. Although the twospin correlation functions and the local magnetizations agree to within the expected errors, there are significant differences in the four-spin correlations. For small values of the true four-spin coupling constants, the deviations are nearly linear, as expected. However, the linear approximation becomes worse as the magnitude of the true coupling constants increase. These systematic devi-ations demonstrate the existence of multi-spin couplings neglected in the initial assumptions.
V. CONCLUSIONS
We have shown that multi-spin coupling constants can be accurately obtained in the inverse Ising problem with the CFM approach [10][11][12]. We have introduced and demonstrated confirmation testing, which uses a new MC simulation to confirm (or deny) whether a given set of effective coupling constants provides a faithful representation of real data.
FIG. 1 .
1Plot of (well) estimated values of h f ield , Knn, and K f our vs. their true values using data from an MC simulation on a 32 × 32 lattice with NMC = 10 5 . The true values of all coupling constants were generated from a uniform distribution in the range [−0.5, 0.5].
Fig. 1
1shows that CFM accurately reproduces the values of the coupling constants for all types of interactions.
FIG. 2 .
2Plot of (badly) estimated coupling constants vs. the true values for the same MC simulation used in Fig. 1. While the estimates of the coupling constants only included nearestneighbor interactions, the Hamiltonian of the MC simulation also included four-spin interactions. Omitting the four-spin couplings severely distorted the estimated values of K f ield and Knn.
FIG. 3 .
3The two-spin coupling constants and the local magnetic fields shown in Fig. 2 were used in a new MC simulation on the same 32 × 32 lattice with NMC = 2.5 × 10 5 . This figure shows the differences in the four-spin correlation functions between the new simulation and the original data, plotted against the true values of the four-spin coupling constants.
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|
[] |
[
"VISUALIZING OVERTWISTED DISCS IN OPEN BOOKS",
"VISUALIZING OVERTWISTED DISCS IN OPEN BOOKS"
] |
[
"Tetsuya Ito ",
"Keiko Kawamuro "
] |
[] |
[] |
We give an alternative proof of a theorem of Honda-Kazez-Matić that every non-right-veering open book supports an overtwisted contact structure. We also study two types of examples that show how overtwisted discs are embedded relative to right-veering open books.
|
10.4171/prims/128
|
[
"https://arxiv.org/pdf/1310.6404v1.pdf"
] | 86,845,320 |
1310.6404
|
ed9eadca77894abbd66ebdab6ad7ec37702a2be6
|
VISUALIZING OVERTWISTED DISCS IN OPEN BOOKS
23 Oct 2013
Tetsuya Ito
Keiko Kawamuro
VISUALIZING OVERTWISTED DISCS IN OPEN BOOKS
23 Oct 2013arXiv:1310.6404v1 [math.GT]
We give an alternative proof of a theorem of Honda-Kazez-Matić that every non-right-veering open book supports an overtwisted contact structure. We also study two types of examples that show how overtwisted discs are embedded relative to right-veering open books.
Introduction
In [15], we have introduced open book foliations and their basic machinery by using that of braid foliations [2,3,4,5,6,7,8,9] and showed applications of open book foliations including a self-linking number formula of general closed braids. In [16] we study the geometric structure of a 3-manifold by using open book foliations. In this note we study more applications of open book foliations. We will assume the readers are familiar with the definition and basic machinery of open book foliations in [15].
One of the features of open book foliations is that one can visualize how surfaces are embedded with respect to general open books. In this paper we use this feature to illustrate overtwisted discs and give constructive methods to detect overtwisted contact structures.
We first give an alternative proof of a tightness criterion theorem by Honda, Kazez and Matić [14]: If an open book is not right-veering then it supports an overtwisted contact structure.
The converse does not hold: In fact, Honda, Kazez and Matić [14] show that if a contact structure ξ is supported by a non-right veering open book (S, φ), by applying positive stabilizations to (S, φ) one can find a right-veering open book ( S, φ) that supports ξ. We concretely visualize an overtwisted disc relative to the right-veering ( S, φ).
Lastly, we give an infinite family of open books that are right-veering and non-destabilizable but compatible with overtwisted contact structures. This negatively answers a question of Honda, Kazez and Matić [14]. Our family generalizes the previously known examples by Lekili [19] and Lisca [20], but our proof of overtwistedness is more direct.
Overtwisted discs in non-right-veering open books
Recall that an overtwisted disc is an embedded disc in a contact 3-manifold whose boundary is a limit cycle in the characteristic foliation of the disc. In particular, every overtwisted be an oriented disc whose boundary is positively braided (i.e., a transverse knot) with respect to the open book (S, φ). If the following are satisfied D is called a transverse overtwisted disc:
(1) G −− is a connected tree with no fake vertices, (2) G ++ is homeomorphic to S 1 .
(3) F ob (D) contains no c-circles, (Terminologies like G −− , G ++ , fake vertices and c-circles are defined in §2.1 of [15].)
In [15] we show that the manifold M (S,φ) contains a transverse overtwisted disc if and only if the contact manifold (M (S,φ) , ξ (S,φ) ) contains an overtwisted disc. Hence from now on, we may not distinguish a transverse overtwisted disc and a usual overtwisted disc, and often call a transverse overtwisted disc simply an overtwisted disc.
Next we review the notion of right-veering mapping classes then reprove Honda Kazez Matić's tightness criterion in Theorem 2.4.
Definition 2.2. [14] Let γ, γ ′ be oriented properly embedded arcs in the surface S that start from the same point * ∈ ∂S. Suppose, after some isotopy relative to the endpoints, γ and γ ′ realize the minimal geometric intersection number. We say that γ ′ lies strictly on the right side of γ if around the common starting point * the curve γ ′ strictly lies on the right side of γ. In such case we denote γ > γ ′ . Proof. If φ is not right-veering, then there exists a properly embedded oriented arc α ⊂ S such that φ(α) > α. By [14, Lemma 5.2], there exists a sequence of properly embedded oriented arcs α 0 , . . . , α k such that 2 (i) α 0 , . . . , α k have the same initial point, n ∈ ∂S, (ii) φ(α) = α 0 > · · · > α k = α, (iii) consecutive α i and α i+1 have disjoint interiors and distinct terminal points p i , p i+1 . Since φ(α k ) = α 0 and φ = id near the bindings we have p 0 = p k . We may assume that:
(iv) the terminal points p 0 , p 1 , . . . , p k−1 ∈ ∂S are mutually distinct. Let β i (resp.β i ) be a sub-arc of α i whose endpoints are p i and a point very close to p i (resp. n). See Figure 1. We orient α i against the parametrization, i.e., the positive
n p i β i α ǐ β i ∂S ∂S Figure 1. Arcs α i , β i , andβ i oriented from p i to n.
direction of α i is from p i to n. The orientation of α i induces those of β i andβ i . We define sets of oriented arcs for i = 0, . . . , k:
A i = β 0 ∪ · · · ∪ β i−1 ∪ α i ∪ β i+1 ∪ · · · ∪ β k . Let t i = i k ∈ [0, 1].
In the following, we construct an oriented surface
D i properly embedded in the product region S × [t i , t i+1 ], where i = 0, . . . , k − 1, such that D i ∩ S t i = −A i and D i ∩ S t i+1 = A i+1 .
The surface D i consists of k connected components; one non-product region and k − 1 product regions defined by
β j × [t i , t i+1 ] where j = i, i + 1 and 0 ≤ j ≤ k.
In the open book foliation of D i the point n becomes a negative elliptic point, the points p 0 , . . . , p k−1 become positive elliptic points, and the arc β j × {t} becomes an a-arc in the page S t . (See Prop. 2.2 of [15] for the definition of a-arcs.) The non-product component of D i is defined by the movie presentation as sketched in Figure 2. It is a saddle shape surface with a positive hyperbolic point h i . Now we glue D i and D i+1 along A i+1 ⊂ S t i+1 (i = 0, . . . , k − 2) and obtain a surface D 0 ∪ · · · ∪ D k−1 ⊂ S × [0, 1] whose oriented boundary is (−A 0 ) ∪ A k . Since arcs β 1 , . . . , β k are very close to ∂S and φ = id near ∂S, we have A 0 = φ(A k ). So in the manifold M (S,φ) we can identify A 0 and A k and obtain a surface which we denote by D.
The topological type of D is the disc and its open book foliation F ob (D) is depicted in Figure 3. Clearly our D is a transverse overtwisted disc.
Remark 2.6. Honda, Kazez and Matić's proof and our proof are based on the same combinatorial lemma [14,Lemma 5.2] in order to use the assumption of right-veeringness. Our proof is more elementary and different from the original one that is written in the language of convex surface theory:
In [14], they prove the existence of a bypass, half of an overtwisted disc, by applying [14, Lemma 5.2] and the right-to-life principal [
(on page S t i ) n p i p i+1 α i β i+1 n p i p i+1 h i (on page S t i+1 ) n p i p i+1 α i+1 β i n p i p i+1 α ǐ β i + n p i p i+1 β i α i+1n p 0 = p k h 0 p 1 h 1 p 2 p k−1 h k−1
Overtwisted discs in right-veering open books
The converse of Theorem 2.4 does not hold in general. Honda, Kazez and Matić [14] show that every contact structure is supported by a right-veering open book: 4
Their argument is the following: Given a contact structure (M, ξ) choose a compatible open book (S, φ). For a boundary component C of S take two boundary-parallel arcs a C and b C such that the geometric intersection number i(a C , b C ) = 2. Apply positive stabilizations to (S, φ) along a C and b C for all the boundary components C on which φ is non-right-veering. The new open book ( S, φ) is now right-veering, see [14,Prop.6.1], and supports the same contact structure ξ.
Suppose that we start from a non-right-veering open book (S, φ), hence ξ is overtwisted. In the following we concretely describe how is an overtwisted disc embedded with respect to the stabilized right-veering open book ( S, φ). [15]) that ends at the point n.
0 < t 1 < t 2 < · · · < t k < 1 2 < 1. For t ∈ [0, 1) let b t ∈ S t ∩ D be the b-arc (cf. Prop.2.2 of
Now we apply Honda-Kazez-Matić's stabilization to get a right-veering open book ( S, φ).
The monodromy φ satisfies φ = T β • T α • φ, where α and β are core circles of the annuli plumbed along a C and b C and T α , T β are positive Dehn twists along α, β.
We will construct an overtwisted disc D by giving a movie presentation relative to the open book ( S, φ). For sake of simplicity we assume that:
• φ is non-right-veering only along C.
• p k ∈ ∂S \ C.
In the general case a construction of D is similar but more complicated. It is obtained as an application of arguments in [17].
Choose stabilization arcs a C and b C such that i(α, b 1/2 ) = 1 and i(β, b 1/2 ) = 0 as shown in Figure 4 (a). Such arcs can always be found by the assumptions above.
To the region { S t | t ∈ [0, 1 2 ]} add a continuous family of b-arcs that are co-cores of the annuli plumbed along a C , see Figure 4 (a). We denote the positive and negative elliptic points which are the endpoints of the newly added b-arcs by p ′ and n ′ . Except this family of b-arcs, the movie presentation of D in the interval [0, 1 2 ] is the same as that of D.
Let b t := b t .
In the interval [ 1 2 , 1), D is described as in the passage of (b)→(c)→(d) of Figure 4. We form one negative and one positive hyperbolic points h − and h + as in (b) and (c), respectively. The describing arcs of h − and h + are parallel to α. Figure 4. A movie presentation of the overtwisted disc D. The shaded region is the union of a collar neighborhood of C and the two plumbed annuli. All the a-arcs are omitted for simplicity.
We have b 1 = T −1 α (b 1 ). Note that φ( b 1 ) = (T β • T α • φ)( b 1 ) = T β • T α (T −1 α b 0 ) = T β (b 0 ) = b 0 = b 0 , moreover φ( D ∩ S 1 ) = D ∩ S 0 so this movie presentation indeed defines an overtwisted disc in M ( S, φ) . 5 (a) (b) (c) (d) n new b-arc b t b 1 β α p ′ n ′ h − h + S t∈[0,1/2] S 1 C
The open book foliations of the overtwisted discs D and D are depicted in Figure 5, where F ob ( D) is obtained by inserting two bb-tiles of opposite signs into the shaded region of F ob (D) that is bounded by b 1 2 and b 1 .
Strictly speaking, the disc D is not a transverse overtwisted disc since the condition (3) of Definition 2.1 is not satisfied. However, applying the same technique we use in the alternative proof in [15] of the Bennequin-Eliashberg inequality [1,11], the condition (3) will be satisfied. Thus we can regard D as an overtwisted disc.
Generalization of Lekili and Lisca's examples
In [14, Question 6.2] Honda, Kazez and Matić ask whether a right-veering and nondestabilizable open book always supports a tight contact structure. Lekili [19] and Lisca [20] negatively answer the question by constructing examples. They study open book decompositions of 3-manifolds whose tight contact structures are well-studied and classified (in [19] Poincaré homology 3-spheres, and in [20] lens spaces). In both constructions the most technical points are showing that their open books indeed support overtwisted 6 Figure 6
b 1 2 b 1 b 1 2 b 1 n n p ′ n ′ h + h − p k p k h 2 p 1 p k−1 h k−1 h k. Let Φ h,i,k = T h a T i b T c T d T −k−1 e
where T x (x = a, b, c, d, e) denotes the right-handed Dehn twist along x. Then for all h, i, k ≥ 1, Φ h,i,k is right-veering and the open book (S, Φ h,i,k ) is non-destabilizable and supports an overtwisted contact structure. Proof. We owe [19] [20] the proof that Φ h,i,k is right-veering and non-destabilizable. Hence we only show that (S, Φ h,i,k ) supports an overtwisted contact structure. We define a transverse overtwisted disc D in the open book (S, Φ h,i,k ) by the movie presentation as in Sketches (1)-(7) of Figure 7. For example, Sketch (1) depicts the page S 0 with the set of 7 arcs D ∩ S 0 . On ∂S there are two negative elliptic points n 1 , n 2 , and (2k + 3) positive elliptic points p 1 , . . . , p 2k+3 . The movie presentation shows that F ob (D) contains two negative hyperbolic points and 2k + 3 positive hyperbolic points.
Note that Φ h,i,k (D ∩ S 1 ) = D ∩ S 0 .
The corresponding open book foliation F ob (D) is depicted in Figure 8. We can verify that D meets the conditions in Definition 2.1 of transverse overtwisted discs.
(1)
p 1 p 2 p 2k+3 p 3 p 2k+2 n 1 n 2 page S 0 −
• Leaves in the fiber S 0 consist of two b-arcs and (2k + 1) a-arcs. • The dashed describing arc corresponds to a negative hyperbolic point.
(2)
Acknowledgement
The first author was supported by JSPS Research Fellowships for Young Scientists. The second author was partially supported by NSF grants DMS-0806492 and DMS-1206770. • The describing arc joins the b-arc from n 1 to p 2k+2 and the a-arc emanating from p 1 and it represents a positive hyperbolic point. • The describing arc joins the b-arc from n 2 to p 2k+3 and the a-arc emanating from p 2 and it represents a positive hyperbolic point.
2k + 1
Definition 2.3. [14, Definition 2.1] Let C be a boundary component of S. We say that φ ∈ Aut(S, ∂S) is right-veering with respect to C if γ ≥ φ(γ) holds for any isotopy classes γ of properly embedded curves which start at a point on C. We say that the diffeomorphism φ (or the open book (S, φ)) is right-veering if φ is right-veering with respect to all the boundary components of S. In particular, the identify id ∈ Aut(S, ∂S) is right-veering. The following theorem gives a characterization of open books supporting tight contact structures. Theorem 2.4. [14, Theorem 1.1] If φ is not right-veering then (S, φ) supports an overtwisted contact structure. Remark 2.5. Conversely, Honda, Kazez and Matić also prove that given an overtwisted contact structure ξ there exists a non-right-veering open book (S, φ) supporting ξ in [14, p.444] where Eliashberg's classification of overtwisted contact structures [10] plays an important role. By Eliashberg's classification, an overtwisted contact structure admits an open book which is a negative stabilization of some open book. Clearly such an open book has a non-right-veering monodromy. Therefore, a contact structure ξ is tight if and only if every open book supporting ξ is right-veering.
Figure 2 .
2The non-product region of surface D i (movie presentation).
Figure 3 .
3The transverse overtwisted disc D.involves the Legendrian realization principle and Eliashberg's classification of tight contact structures on the 3-ball.On the other hand, we use [14, Lemma 5.2] to explicitly construct a chain of positive elliptic and hyperbolic points surrounding the center negative elliptic point of an overtwisted disc. Hence our proof concretely visualizes an overtwisted disc.
Example 3 . 1 (
31Overtwisted discs in Honda-Kazez-Matić's stabilizations). Let (S, φ) be a not-right-veering open book. By the proof of Theorem 2.4 one can construct an overtwisted disc, D, in M (S,φ) . The open book foliation of D has a unique negative elliptic point, say n, that lies on the binding component C ⊂ ∂S, and k positive elliptic points p 1 , . . . , p k and k positive hyperbolic points h 1 , . . . , h k . Let S t i be the singular fiber that contains h i . We may assume
Figure 5 .
5F ob (D) and F ob ( D) contact structures. Advanced tools such as Ozsváth-Szábo's Heegaard Floer invariants and properties of planar open books enable them to overcome the difficulty. We generalize Lekili and Lisca's examples in Theorem 4.1 below. Our proof of overtwistedness is direct and does not require any knowledge of classification of tight contact structures of ambient manifolds or Ozsváth-Szábo's invariants.
Theorem 4 . 1 .
41Let S be a 2-sphere with four holes. Let a, b, c, d, e be simple closed curves on S as shown in
Figure 6 .
6The surface S.
Remark 4. 2 .
2Lekili's examples [19, Theorem 1.2 and Remark 4.1] are Φ 2,i,1 (i ≤ 5), and Lisca's examples [20, Theorem 1.1] are Φ h,1,l (h, l > 0).
•
In Sketches (2), . . . ,(7), we omit some a-arcs emanating from p 3 , . . . , p 2k+2 if they are not involved in producing hyperbolic singularities.• The describing arc joining the b-arc emanating from n 2 and the a-arc from p 3 (or p 2i+1 in the i-th iteration) represents a positive hyperbolic point.
Remark 4. 3 .
3After the announcement of this example (on December 26, 2011), Kazez and Roberts[18] found more counterexamples to the conjecture of Honda, Kazez and Matić.
••
The describing arc joining the b-arc emanating from n 1 and the a-arc from p 4 (or, p 2i+2 in the i-th iteration) represents a positive hyperbolic point.• Iterate the Steps (The shaded boxes labeled h, 2k contain parallel h, 2k arcs. • The edges out of the shaded boxes are weighted as indicated. • The describing arc joins the b-arc from n 2 to p 2k+1 and the a-arc emanating from p 2k+3 and it represents a positive hyperbolic point.
•
The leaves in the page S 1 . It satis-fies Φ h,i,k (D ∩ S 1 ) = D ∩ S 0 .
Figure 7 .
7Movie presentation of a transverse overtwisted disc in (S, Φ h,i,k ).
Figure 8 .
8The open book foliation F ob (D).
Date: October 25, 2013. 2000 Mathematics Subject Classification. Primary 57M25, 57M27; Secondary 57M50. disc has Legendrian boundary. In the framework of open book foliations a transverse overtwisted disc plays a corresponding role: Definition 2.1. [15, Def 4.1] Let D ⊂ M (S,φ)
Entrelacements etéquations de Pfaff. D Bennequin, Astérisque. D. Bennequin, Entrelacements etéquations de Pfaff, Astérisque, 107-108, (1983) 87-161.
Studying links via closed braids. IV. Composite links and split links. J Birman, W Menasco, Invent. Math. 1021J. Birman, W. Menasco, Studying links via closed braids. IV. Composite links and split links, Invent. Math. 102 (1990), no. 1, 115-139.
Studying links via closed braids. II. On a theorem of Bennequin. J Birman, W Menasco, Topology Appl. 401J. Birman, W. Menasco, Studying links via closed braids. II. On a theorem of Bennequin, Topology Appl. 40 (1991), no. 1, 71-82.
Studying links via closed braids. V. The unlink. J Birman, W Menasco, Trans. Amer. Math. Soc. 3292J. Birman, W. Menasco, Studying links via closed braids. V. The unlink, Trans. Amer. Math. Soc. 329 (1992), no. 2, 585-606.
Studying links via closed braids. I. A finiteness theorem. J Birman, W Menasco, Pacific J. Math. 1541J. Birman, W. Menasco, Studying links via closed braids. I. A finiteness theorem, Pacific J. Math. 154 (1992), no. 1, 17-36.
Studying links via closed braids. VI. A nonfiniteness theorem. J Birman, W Menasco, Pacific J. Math. 1562J. Birman, W. Menasco, Studying links via closed braids. VI. A nonfiniteness theorem, Pacific J. Math. 156 (1992), no. 2, 265-285.
Studying links via closed braids. III. Classifying links which are closed 3-braids. J Birman, W Menasco, Pacific J. Math. 1611J. Birman, W. Menasco, Studying links via closed braids. III. Classifying links which are closed 3- braids, Pacific J. Math. 161 (1993), no. 1, 25-113.
Stabilization in the braid groups. J Birman, W Menasco, I. MTWS, Geom. Topol. 10J. Birman, W. Menasco, Stabilization in the braid groups. I. MTWS, Geom. Topol. 10 (2006), 413-540.
Stabilization in the braid groups. II. Transversal simplicity of knots. J Birman, W Menasco, Geom. Topol. 10J. Birman, W. Menasco, Stabilization in the braid groups. II. Transversal simplicity of knots, Geom. Topol. 10 (2006), 1425-1452.
Classification of overtwisted contact structures on 3-manifolds. Y Eliashberg, Invent. Math. 98Y. Eliashberg, Classification of overtwisted contact structures on 3-manifolds, Invent. Math. 98 (1989), 623-637.
Contact 3-manifolds twenty years since J. Martinet's work. Y Eliashberg, Ann. Inst. Fourier (Grenoble). 42Y. Eliashberg, Contact 3-manifolds twenty years since J. Martinet's work, Ann. Inst. Fourier (Greno- ble). 42 (1992), 165-192.
On the classification of tight contact structures I. K Honda, Geom. Topol. 4K. Honda, On the classification of tight contact structures I, Geom. Topol. 4 (2000), 309-368.
Tight contact structures on fibered hyperbolic 3-manifolds. K Honda, W Kazez, G Matić, J. Diff. Geom. 64K. Honda, W. Kazez, G. Matić, Tight contact structures on fibered hyperbolic 3-manifolds, J. Diff. Geom. 64 (2003), 305-358.
Right-veering diffeomorphisms of compact surfaces with boundary. K Honda, W Kazez, G Matić, Invent. math. 1692K. Honda, W. Kazez, G. Matić, Right-veering diffeomorphisms of compact surfaces with boundary, Invent. math. 169, No.2 (2007), 427-449.
Open book foliations, accepted by Geometry and Topology. T Ito, K Kawamuro, T. Ito, K. Kawamuro, Open book foliations, accepted by Geometry and Topology.
Essential open book foliation and fractional Dehn twist coefficient. T Ito, K Kawamuro, T. Ito, K. Kawamuro, Essential open book foliation and fractional Dehn twist coefficient, Available at http://arxiv.org/pdf/1208.1559.pdf and http://www.kurims.kyoto-u.ac.jp/∼tetitoh/
T Ito, K Kawamuro, arXiv:1309.4486Operations on open book foliations. T. Ito, K. Kawamuro, Operations on open book foliations, arXiv:1309.4486.
W Kazez, R Roberts, arXiv:1201.5290v1Fractional Dehn twists in knot theory and contact topology. W. Kazez, R. Roberts, Fractional Dehn twists in knot theory and contact topology arXiv:1201.5290v1.
Planer open books with four binding components. Y Lekili, Algebr. Geom. Topol. 11Y. Lekili, Planer open books with four binding components, Algebr. Geom. Topol. 11 (2011) 909-928.
Japan E-mail address: [email protected]. P Lisca, Pacific J. Math. 257Research Institute for Mathematical SciencesOn overtwisted, right-veering open booksP. Lisca, On overtwisted, right-veering open books, Pacific J. Math. 257 (2012) 219-225. Research Institute for Mathematical Sciences, Kyoto university Kyoto, 606-8502, Japan E-mail address: [email protected] URL: http://kurims.kyoto-u.ac.jp/~tetitoh/
USA E-mail address: kawamuro@iowa. Iowa City, IAMacLean Hall, Department of Mathematics, The University of Iowauiowa.eduMacLean Hall, Department of Mathematics, The University of Iowa, Iowa City, IA, 52242-1419, USA E-mail address: [email protected]
|
[] |
[
"Harnessing Natural Experiments to Quantify the Causal Effect of Badges",
"Harnessing Natural Experiments to Quantify the Causal Effect of Badges"
] |
[
"Tomasz Kusmierczyk [email protected] \nNorwegian University of Science and Technology\n\n",
"Manuel Gomez-Rodriguez [email protected] \nMax Planck Institute for Software Systems\n\n"
] |
[
"Norwegian University of Science and Technology\n",
"Max Planck Institute for Software Systems\n"
] |
[] |
A wide variety of online platforms use digital badges to encourage users to take certain types of desirable actions. However, despite their growing popularity, their causal effect on users' behavior is not well understood. This is partly due to the lack of counterfactual data and the myriad of complex factors that influence users' behavior over time. As a consequence, their design and deployment lacks general principles.In this paper, we focus on first-time badges, which are awarded after a user takes a particular type of action for the first time, and study their causal effect by harnessing the delayed introduction of several badges in a popular Q&A website. In doing so, we introduce a novel causal inference framework for badges whose main technical innovations are a robust survival-based hypothesis testing procedure, which controls for the utility heterogeneity across users, and a bootstrap difference-in-differences method, which controls for the random fluctuations in users' behavior over time. We find that first-time badges steer users' behavior if the utility a user obtains from taking the corresponding action is sufficiently low, otherwise, the badge does not have a significant effect. Moreover, for badges that successfully steered user behavior, we perform a counterfactual analysis and show that they significantly improved the functioning of the site at a community level. * This work was done during Tomasz Kusmierczyk's internship at the Max Planck Institute for Software Systems. 1 First-time badges are the simplest type of threshold badges, which are awarded after a user has taken an action a pre-specified number of times[4,18,25].
| null |
[
"https://arxiv.org/pdf/1707.08160v1.pdf"
] | 4,767,422 |
1707.08160
|
09f021684924ff7cce5c08329c4d9df2240ff60d
|
Harnessing Natural Experiments to Quantify the Causal Effect of Badges
Tomasz Kusmierczyk [email protected]
Norwegian University of Science and Technology
Manuel Gomez-Rodriguez [email protected]
Max Planck Institute for Software Systems
Harnessing Natural Experiments to Quantify the Causal Effect of Badges
A wide variety of online platforms use digital badges to encourage users to take certain types of desirable actions. However, despite their growing popularity, their causal effect on users' behavior is not well understood. This is partly due to the lack of counterfactual data and the myriad of complex factors that influence users' behavior over time. As a consequence, their design and deployment lacks general principles.In this paper, we focus on first-time badges, which are awarded after a user takes a particular type of action for the first time, and study their causal effect by harnessing the delayed introduction of several badges in a popular Q&A website. In doing so, we introduce a novel causal inference framework for badges whose main technical innovations are a robust survival-based hypothesis testing procedure, which controls for the utility heterogeneity across users, and a bootstrap difference-in-differences method, which controls for the random fluctuations in users' behavior over time. We find that first-time badges steer users' behavior if the utility a user obtains from taking the corresponding action is sufficiently low, otherwise, the badge does not have a significant effect. Moreover, for badges that successfully steered user behavior, we perform a counterfactual analysis and show that they significantly improved the functioning of the site at a community level. * This work was done during Tomasz Kusmierczyk's internship at the Max Planck Institute for Software Systems. 1 First-time badges are the simplest type of threshold badges, which are awarded after a user has taken an action a pre-specified number of times[4,18,25].
Introduction
In recent years, social media sites and online communities have increasingly relied on digital badges to reward their users for different types of online behavior. Similarly as their physical counterpart, digital badges have been used both as a reputation mechanism, summarizing the skills and accomplishments of the users who receive them, and as an incentive mechanism, encouraging users to take certain type of desirable actions. The promise of digital badges is that automated fine-grained monitoring and greater degree of control will help refine their design as incentive mechanisms, increasing users' engagement and improving the functioning of the corresponding online platform. However, to fulfill this promise, it is necessary to better understand their causal effect on the online behavior of the users who may receive them-identify when and why they are (not) able to steer their behavior.
In this paper, we focus on first-time badges, which are awarded after a user takes a particular type of action for the first time 1 , and study their causal effect by harnessing several natural experiments in Stack Overflow 2 , a popular Q&A website. Despite their simplicity, we need to tackle several challenges, which require careful reasoning:
-Measuring progress towards the badge: since first-time badges are awarded after performing just one single action, the action count does not provide a direct measure of progress towards the badge. This is in contrast with (non-binary) threshold badges, which were typically the focus of previous work [4,18,25].
-Utility heterogeneity: the utility each user obtains from taking an action differs wildly due to, e.g., user's intrinsic motivation, the target of the action, or other users' actions. As a consequence, the times users take to perform an action for the first time spans a large range of values.
-Random temporal changes: one can frequently observe random fluctuations in users' behavior over time due to many different complex factors. To assess the strength of the causal effect induced by a badge, it is necessary to control for these random fluctuations. We address the above mentioned challenges by developing a novel causal inference framework for firsttime badges, especially designed for our problem setting. Our framework avoids modeling the mechanisms underlying individual user actions and instead adopts a data-driven approach based on survival analysis and statistical hypothesis testing. At the heart of our approach there are two technical innovations: (i) a robust survival-based hypothesis testing procedure, inspired by the discrete choice literature on latent variable models [23,6], which allows us to account for the utility heterogeneity; and, (ii) a bootstrap difference-indifferences method, inspired by the economics literature on natural experiments [16,17,21], which allows us to control for the random fluctuations in users' behavior over time.
In contrast with recent empirical studies on threshold badges [4,18,19,25], which assume or conclude that badges (always) steer users' behavior, we do not find statistically significant causal evidence to back up this assumption (or conclusion) in all first-time badges. Instead, we provide strong empirical evidence of a more subtle picture. First-time badges steer users' behavior if the utility a user obtains from taking the corresponding action is sufficiently low, otherwise, the badge does not have a significant effect. Moreover, we hypothesize that this may be also the case for other types of badges, e.g., non-binary threshold badges, and thus argue that the user utilities should be carefully considered on the design and deployment of badges. Finally, for badges that successfully steered user behavior, we go a step further and, using a survival-based counterfactual analysis, show that they significantly improve the functioning of the site at a community level. Related work. Our work contributes to the growing literature on badges [2,4,7,13,25,18,19,11], which can be broadly divided into theoretical and empirical studies.
Theoretical studies on badges [13,25,11] analyze the effect of badges on users' behavior under stylized models of badges, which make strong assumptions, often without empirical support. Moreover, they typically ignore the (inherent) utility a user receives from taking the action the badge rewards-the action payoff and cost. In contrast, in our work, we avoid making strong assumptions about the mechanisms underlying individual user actions and instead adopt a data-driven approach, which enable us to account for the utility a user obtains from taking an action.
Empirical studies on badges [2,4,7,18,19] have mainly focused on threshold badges, where the action count provides a direct measure of progress towards the badge. In this context, several authors [18,4] have provided empirical evidence in favor of the goal-gradient hypothesis, which posits users increase their engagement as they get closer to earning a badge. However, most of these studies did not have access to control groups, which would have allowed them to assess users' behavior in the absence of a badge and control for random fluctuations in users' behavior over time. Therefore, they are unable to identify when or why different types of threshold badges are able to steer users' behavior. Two notable exceptions are by Abramovich et al. [2] and by Bornfeld et al. [7]. The former carried out a small controlled experiment in an educational setting and showed that the degree of success of a badge at steering a learner's behavior depends on her ability and motivation. The latter has been concurrently conducted with our work and it also leverages natural experiments in the context of badges. However, in contrast to our work, they rely on standard statistical tests on aggregated counts, account for the temporal fluctuations in community functioning in an ad-hoc manner, and ignore the utility heterogeneity across users. As a consequence, they are unable to conclude whether badges had a (significant) causal effect at a user's level and they do not shed light on when and why badges are (not) able to steer users' behavior.
In recent years, natural experiments [17,21], difference-in-difference designs [10,16] and propensity score matching [9,22] have been increasingly used to identify causal effects from observational data in online settings, e.g., social influence [5,3,8,15] or network formation [14,20]. However, together with Bornfeld et al. [7], the present work is the first that leverage natural experiments to quantify causal effects in the context of badges.
Data description
Our Stack Overflow dataset comprises of all individual timestamped actions performed by all users from the site's inception from July 31, 2008 to September 14, 2014, which allow us to track the complete sequence of actions users take. First-time badges: natural experiments. There are a great variety of badges, which reward users for different types of behaviors. In this work, we focus on first-time badges, which are awarded after a user takes a particular type of action for the first time, and identify those that were introduced some time after the site's inception. The delayed introduction of these badges can be thought of as natural experiments [17,21]. Figure 1 illustrates an example of such badge. More specifically, we select three first-time badges that reward actions whose utilities to the users are clearly different:
-Tag Editor badge: Stack Overflow users can include tags on questions (or answers) to concisely describe their content. In July 2010, Stack Overflow enabled the creation of tag wikis by the community, which aim to provide a description of all used tags. Shortly afterwards, it introduced a badge called Tag Editor, awarded after a user edits a tag wiki for the first time, to encourage users to edit tag wikis. To ensure the quality of the wiki tags, only users with at least a reputation level of 1,500 could (initially) edit a tag wiki 3 . Finally, note that a user obtains a low utility from editing a wiki tag-it requires some effort and she only receives the intangible reward of helping the community. Moreover, the more uncommon a tag is, the least this intangible reward may be.
-Promoter badge: When a Stack Overflow user does not receive a satisfactory answer to one of her questions, she can offer a bounty to reward, in the form of reputation points, the user who would provide such a satisfactory answer 4 . In July 2010, Stack Overflow introduced a badge called Promoter, awarded after a user offers a bounty for an answer to one of her questions for the first time, to encourage users to offer more bounties. Only users with at least a reputation of 75 points can offer a bounty. In contrast with editing a wiki tag, a user obtains a high utility from offering a bounty-it requires little effort and she may receive an answer to a question she is personally interested in, however, it entails a cost in terms of the reputation she transfers to the user providing the answer.
-Investor badge: Stack Overflow users can also offer bounties to receive a satisfactory answer to a question that has been asked by another user. In July 2010, Stack Overflow introduced a first-time badge called Investor to encourage users to offer more bounties for answers to other users' questions. Similarly as in the Promoter badge, only users with at least a reputation of 75 points can offer a bounty for an answer to a question asked by other user. However, in this case, a user may obtain a lower utility from offering a bounty for an answer to other user's question than her own-on the one hand, she may less interested in an answer since she did not originally ask the question and, on the other hand, the question may have already a (relatively) satisfactory answer when she found it.
Testing the effectiveness of badges
In this section, we first formalize the problem setting. Then, we introduce two survival-based hypothesis testing procedures of increasing statistical power, which we use in a novel bootstrap difference in differences method, especially designed for our problem setting. Finally, we evaluate the effectiveness of the overall framework using a variety of synthetic experiments. Problem setting. Given an action of interest a, we record the behavior of each user during an observation window [0, T ] as a tuple
e := ( start time ↓ s u , t u ↑ action time , utility ↓ v u ),(1)
which means that user u becomes eligible to perform the action at time s u , she performs the action at time t u , and obtains a utility v u , which is often intangible. If a user does not perform the action during the observation window [0, T ], we set the action time to t u = ∞, however, this does not imply she will never perform the action. Moreover, we assume a first-time badge b is introduced at time τ ∈ [0, T ] to incentivize users to take action a. That means, after time τ , a user receives badge b the first time she takes action a, potentially increasing its corresponding utility v u . Given the above setup, our goal is then to assess to which extent the introduction of the badge changes users' behavior, as measured by the time users take to perform the action for the first time, i.e., t u − s u . Next, we introduce two survival-based hypothesis testing procedures of increasing statistical power and then describe our bootstrap difference-in-difference method. Basic survival-based hypothesis testing. Given an action of interest a, we model the time t u when a user u takes action a using a survival process [1]. Following the literature on temporal point processes, we represent such survival process as a binary counting process N u (t) ∈ {0, 1}, which becomes one when the user performs the action for the first time. Then, we characterize this counting process using its corresponding intensity λ u (t), i.e., E[dN u (t)] = λ u (t)dt, which we define as follows:
λ u (t) = 0 if t < s u λ 0 if s u ≤ t < τ λ 1 otherwise (2)
where λ 0 and λ 1 are parameters shared across all users, which depend on the (intangible) utility users obtain from talking the action.
Under this model, the null hypothesis H 0 , i.e., the badge did not have an effect, corresponds to λ 0 = λ 1 ≥ 0 and the alternative hypothesis H 1 is λ 0 = λ 1 with λ 0 ≥ 0 and λ 1 ≥ 0. Moreover, given the behavior of n users, the maximum likelihood estimators of the model parameters,λ 0 andλ 1 , can be computed analytically. In particular, under the null hypothesis, they are readily given by:
λ 0 =λ 1 = u∈[n] I(t u ≤ T ) u∈[n] (min(t u , T ) − s u )
, and under the alternative hypothesis, they are given by:
λ 0 = u∈[n] I(t u ≤ τ ) u∈[n] (min(t u , τ ) − s u )I(s u < τ ) λ 1 = u∈[n] I(τ u < t u ≤ T ) u∈[n] (min(t u , T ) − τ )I(t u > τ ) ,
where I(·) is the indicator function and all the sums are over eligible users. Then, we can use a standard log-likelihood ratio (LLR) as test statistic [12]. Moreover, since the null model is nested in the alternative model, i.e., θ 0 ∈ {θ 0 , θ 1 }, using Wilks' theorem [24], it asymptotically holds that 2LLR ∼ χ 2 1 under the null hypothesis. Thus, we can easily find an approximate p-value, i.e., p ≈ 1 − χ 2 df (2LLR). Robust survival-based hypothesis testing. The survival-based hypothesis testing procedure described in the previous section assumes the model parameters are shared across all users and, by doing so, it ignores the utility heterogeneity across users. Inspired by the discrete choice literature on latent variable models [23,6], we account for the utility heterogeneity by considering different latent parameters per user, but sampled from the same distributions, i.e.,
λ u (t) = 0 if t < s u λ 0 (u) if s u ≤ t < τ λ 1 (u) otherwise λ 0 (u) ∼ Gamma(k 0 , r) λ 1 (u) ∼ Gamma(k 1 , r)(3)
where k 0 , k 1 are shape parameters and r is a rate parameter. Here, note that E[λ 0 ] = k 0 /r and E[λ 1 ] = k 1 /r. Then, we define the null and alternative hypothesis in terms of the shape parameters, i.e.,
H 0 : k 0 = k 1 ≥ 0 H 1 : k 0 = k 1 , k 0 ≥ 0, k 1 ≥ 0
Moreover, given the behavior of n users, we can estimate the shape parameters using maximum likelihood estimation, integrating out the latent parameters λ 0 (u) and λ 1 (u), and estimate the rate parameter by cross validation. More specifically, under the null hypothesis, the shape parameters are given by:
k 0 =k 1 = − u∈[n] I(t u ≤ T ) u∈[n] log r r+min(tu,T )−su
and under the alternative hypothesis, they are given by:
k 0 = − u∈[n] I(t u ≤ τ ) u∈[n] log r r+min(tu,τ )−su I(s u ≤ τ ) k 1 = − u∈[n] I(τ < t u ≤ T ) u∈[n] log r r+min(tu,T )−τ I(t u > τ )
Similarly as in the basic survival model, we can then use a standard log-likelihood ratio (LLR) as test statistic. However, in this case, the null model is not nested in the alternative model and, as a consequence, we cannot use Wilks' theorem to find a p-value. Instead, we will rely on the following bootstrap difference-indifference method to find a robust empirical estimate of the distribution of the test statistic under the null hypothesis. Bootstrap difference-in-differences method. Given an action of interest a, its corresponding first-time badge b with introduction time τ , the behavior of n users with respect to a, i.e., D a = {(s u , t u , v u )} u∈[n] , and a model-based hypothesis testing procedure, we design the following bootstrap difference-in-difference method to find a robust empirical estimate of the distribution of the corresponding test statistic under the null hypothesis, which accounts for the temporal fluctuations in users' behavior:
I. We select all users whose start time s u ∈ [τ − w/2, τ + w/2], where w ≥ 0 is a given parameter. Then, we run the model fitting procedure of choice on that subset of users, the treatment population, and obtain a test statistic, e.g., LLR τ . II. We introduce a set of virtual badges V at a times τ i ∈ [w/2, τ − w] ∪ [τ + w, T − w/2], picked uniformly at random (in practice, one can use a sliding window), where w ≥ 0 is the same given parameter as in the first step. Then, for each virtual badge i ∈ V, we select users whose start time s u ∈ [τ i − w/2, τ i + w/2]. Finally, we run the model fitting as in the previous step on each of these subsets of users, the control populations, and obtain a test statistic value per virtual badge, e.g., LLR τi . As a result, we can estimate an empirical cumulative density function (cdf) of the test statistic under the null hypothesis, F LLR (LLR), which is robust to temporal fluctuations in users' behavior. III. We measure the strength of the change induced by the badge by means of the probability that the test statistic of the control populations (for which the null hypothesis holds by design) is larger than the test statistic of the treatment population, p := F LLR (LLR τ ). The above bootstrap difference-in-differences method, which we also illustrate in Figure 2, equips us with a robust empirical estimate of the distribution of the test statistic under the null hypothesis F LLR (LLR) and a p-value p = F LLR (LLR τ ), which accounts for the temporal fluctuations in users' behavior and allows us to reject the null hypothesis with higher confidence. The main assumption needed for the above method to be valid (e.g., the empirical estimate of the test statistic distribution under the null hypothesis to be accurate) is that the treatment and control populations have similar characteristics. In other words, the process governing the exposures should resemble random assignment. Framework evaluation. In this section, we compare the effectiveness of the basic survival model with the theoretical distribution (χ 2 1 ) of the LLR under the null hypothesis ("basic theoretical") and the basic and robust survival models with the empirical distribution (F LLR ) of the LLR under the null hypothesis, as estimated by the proposed difference-in-differences bootstrap method ("basic bootstrap" and "robust bootstrap", respectively). More specifically, we proceed as follows.
First, we simulate the behavior of n = 10,000 users during a time interval [0, T ], where T = 360. For each user, we draw her starting times s u uniformly at random, s ∼ U [0, T ], and her action time t from an intensity λ u (t)(1 + at), where λ u (t) is given by Eq. 3 and a = 0.001. Moreover, in Eq. 3, we set the badge Figure 3: Performance of our causal inference framework on synthetic data. The left panel shows the average p-value against effect strength k 1 /k 0 , where lower (higher) is better for k 1 /k 0 > 1 (k 1 /k 0 = 1). The right panel shows the rejection probability of the null hypothesis H 0 at p = 0.05 against effect strength k 1 /k 0 , where higher (lower) is better for k 1 /k 0 > 1 (k 1 /k 0 = 1).
introduction time to τ = T /2, the rate parameter to r = 10, and consider different badge strength values, i.e., E(λ 1 (u))/E(λ 0 (u)) = k 1 /k 0 ∈ {1.0, 1.25, . . . , 10, 100}, where k 1 /k 0 = 1.0 is equivalent to not introducing a badge. Note that the term (1 + at) imposes a global linear trend, which is often observed in real data 5 . For each configuration, we run 100 independent simulations.
Then, we run the above methods ("basic theoretical", "basic bootstrap", and "robust bootstrap") on data from each of the independent simulations and measure their effectiveness in terms of two metrics: average p-value and rejection probability of the null hypothesis H 0 at p = 0.05. Figure 3 summarizes the results, which show that the robust bootstrap has a superior performance: it is more likely to reject H 0 when a badge is introduced (i.e., k 1 /k 0 > 1.0) while it is equally likely not to reject H 0 when a badge is not introduced (i.e., k 1 /k 0 = 1.0).
Do badges work?
In this section, we apply our causal inference framework to the three first-time badges described in Section 2. Figure 4 summarizes the results by means of:
(i) Test statistic over time for the basic and robust survival models, i.e., LLR τi and LLR τ against τ i and τ . (ii) Empirical distribution of the test statistic under H 0 and p-value for the robust survival model, i.e., F LLR (LLR) and F LLR (LLR τ ). (iii) Average intensities for first-time action under robust survival model, i.e.,k 0 before τ andk 1 after τ , using a sliding window of length w = 60 days. Overall, the results suggest that the Tag editor and Investor badges were successful -they had a significant causal effect on users' behavior (p = 0.004 and p = 0.017, respectively). In contrast, the Promoter badge was unsuccessful-it did not have a significant causal effect (p = 0.309). Moreover, a detailed analysis also reveals several interesting insights, which we will further expand in Sections 5 and 6.
First, the actions rewarded by the two successful badges were rare by the time the badges got introduced. For example, in the case of the Tag editor badge, only 100 tag wiki edits had been performed, however, there were ∼6,500 users who were eligible to perform edits. In the case of the Investor badge, only 40 bounties had been offered for an answer to other users' questions. In contrast, the action rewarded by the Promoter badge-offering a bounty for an answer to the users' own questions-was much more common by the time the badge got introduced. As a consequence, the average intensity for the Promoter badge was an order of magnitude higher than the intensities corresponding to the Tag editor or the Investor badge. Second, the introduction of the Tag editor and Investor badges was followed by an increase on the average intensity of the corresponding first-time action of more than 4×, fromk 0 ≤ 2 · 10 −4 tok 1 ≈ 8 · 10 −4 for Tag editor badge and fromk 0 ≤ 5 · 10 −5 tok 1 ≈ 2 · 10 −4 for Investor badge. Equivalently, the average time a user takes to perform the actions for the first time was reduced by 75%. Moreover, this change in user behavior did not vanish over time, as shown in the rightmost column.
Finally, in the case of the Investor badge, we find a transient increase on the average intensity of bounties to other users' questions around October-November 2010, which is statistically significant. Upon investigation, we notice that several users discovered ways of benefiting from offering bounties around that time, triggering subsequent first-time uses of bounties by other users 6 . Such discussions led to an increase on the minimum reputation one can transfer when offering a bounty.
Badges and utilities
In this section, we investigate the reasons why the Tag editor and Investor badges were successful at steering users' behavior while the Promoter badge was unsuccessful. To this aim, we resort to game-theoretic after,λ1(p) concepts such as user utilities, action payoffs and reservation values, and identify measurable proxies of some of these concepts for each of the above mentioned badges and actions. User utilities. In the game theory literature [11,13,25], the utility a user obtains from performing an action is defined as the difference between the action payoff p and the cost of effort c, i.e., v = p − c. Moreover, the fact that participation is a voluntary, strategic choice-users have a choice about whether or not to perform an action-is often modeled via a reservation value ω that the utility v must exceed in order for the user to perform the action. More specifically, if p − c < w, the user will decide not to perform the action and, otherwise, she will perform it. In this context, a badge b is assumed to increase the utility a user obtains from performing the action, i.e., v = p − c + v b , where v b is the badge value. Then, depending on the actual values of p, c, v b and ω, one can argue that a badge will induce users to perform an action that, in the absence of a badge, would not perform. However, in social media sites and online communities, the action payoffs, cost of effort, badge value and reservation values are typically intangible, hidden or ambiguously defined. As a consequence, our causal inference framework did not explicitly adopted the above model and instead used a data-driven approach based on survival analysis of items specific for particular badge, using only the observable temporal traces. In this section, however, we turn our attention towards the above stylized model, identify measurable proxies of the model parameters for each of the above mentioned badges and actions, and use them to investigate the reasons for the success or failure of badges at steering users' behavior, as concluded by our framework. Proxies to user utilities. We consider the following observable proxies for the utilities users obtain from editing a wiki tag and offering a bounty, respectively: (a) Tag popularity: the more popular a tag is, the greater the utility v a user may obtain from editing its wiki tag-the user needs to put less effort to create a wiki on a popular tag and she receives the satisfaction of helping a larger part of the community.
6 × 1 0 − 7 3 × 1 0 − 6 1 × 1 0 − 5 4 × 1 0 − 5 1 × 1 0 − 4 2 × 1 0 − 3 tag use probability
(b) Number of answers: the higher (lower) the number of answers a question receives after (before) offering a bounty, the greater the utility v a user obtains from offering the bounty. Moreover, users offering a bounty for an answer to other user's question may obtain less utility from the answers since they did not originally ask the question.
Given the above proxies, we proceeds as follows. In terms of tag wikis, we group tags by popularity (i.e., number of questions a tag was used on) and model the time the users take to create a wiki for a tag of a given popularity p as a survival process. Moreover, we characterize this process using an intensity λ p (t), which we define as follows: where λ 0 (p) and λ 0 (p) are parameters shared across all tags with popularity p, s is the time when the tag is first used in a question, and τ is the time when the Tag Editor badge is introduced. Then, by comparing the maximum likelihood estimators of the model parameters,λ 0 (p) andλ 1 (p), for different popularity levels p, we can assess the causal effect of the Tag Editor badge on tag wikis with different utility values.
λ p (t) = 0 if t < s λ 0 (p) if s ≤ t < τ λ 1 (p) otherwise
In terms of bounties, we first group questions by the number of answers they received in the first two days since they were asked and then compare the additional number of answers they received after those first two days if a bounty was (not) offered in the second day. Moreover, for questions that received a bounty, we estimate the distribution of the number of answers they received before the bounty was offered both before and after the badges Promoter and Investor were introduced. By controlling for the number of answers, we can assess the causal effect of both badges for bounties with different utility values. Results. Figure 5 summarizes the results for the Tag Editor badge, which shows the higher the popularity (utility) of a tag, the weaker the causal effect of the badge introduction. In other words, the introduction of the badge steered users to create tag wikis for less popular, low utility tags. Figure 6 summarizes the results for the Promoter and Investor badges, which let us better understand their failure and success, respectively: (i) the number of answers a bounty triggers (i.e., its utility) increases with the number of answers the question has received in its absence (left panel); (ii) the introduction of the Promoter badge did not significantly change the users' willingness to offer bounties to their own questions (right panel, top figure), in contrast, the Investor badge did change it for bounties offered to other users' questions (right panel, bottom figure). This change was especially pointed for questions with zero answers before offering the bounty, in which the utility of offering a bounty is the lowest (left panel).
Do badges improve the community functioning?
In this section, we use a survival-based counterfactual analysis, in which we investigate what would have happened if the Tag Editor and Investor badges had not been introduced, to assess to which extent the badges improved the site functioning at a community level. Counterfactual analysis. For the Tag editor, we assess the site functioning at a community level in terms of the number of new tag wikis over time and, for the Investor badges, we use the time to bounty and first answer across questions. More specifically, we proceed as follows.
-New tag wikis: we simulate the time the users take to create a new wiki for a tag of a given popularity p in the counterfactual world where the Tag editor badge is never introduced using the intensity defined by Eq. 4 with λ 0 (p) = λ 1 (p) =λ 0 (p), whereλ 0 (p) is the maximum likelihood estimate for λ 0 (p) in the true world. Time to first answer Then, we compare the number of new tag wikis over time as well as the popularity of their associated wikis in the true world and in the simulated counterfactual world.
-Time to bounty and first answer : for questions that received a bounty, we model the time that the users take to offer the bounty as a survival process with associated intensity λ b (t), which we define as follows:
λ b (t) = 0 if t < s λ 0 (b) if s ≤ t < τ λ 1 (b) otherwise(5)
where the parameter b ∈ {0, 1} denotes whether the badge is offered by the user asking the question or by another user, {λ i (b)} i,b∈{0,1} are (four) parameters shared across all questions, s is the time since two days after the question is asked 7 , and τ is the time when the Promoter and Investor badges are introduced. For all questions, we model the time that the users take to provide the first answer also as a survival process with an associated intensity defined similarly as in Eq. 5, however, in this case, the parameter b ∈ {−1, 0, 1} denotes whether the question received a badge and, if so, whether it was offered by the user asking the question or by another user, and thus the model has six parameters. Then, we compare the maximum likelihood estimators of the above parameters to assess what would have happened if the Investor badge had not been introduced. Results. Figure 7 summarizes the results for the Tag Editor badge. The badge introduction improved the site functioning at a community level in terms of the number of new tag wikis (left panel; the number of new tag wikis increased) and the average popularity (rank) of the associated (right panel; new tags wikis were created for less popular tags). However, shortly after its introduction, the rate of creation of new tag wikis decreased to its original value and matched the rate of creation in the simulated counterfactual world. A plausible hypothesis is that the badge lifted the utility value of badges over the reservation value for tags of certain popularity. As a consequence, after these tags had a tag wiki, the effect of the badge diminished. Figure 8 summarizes the results for the Investor badge, which show that the time to bounty and first answer for questions in which a bounty was offered by a user different than the user asking the question decreased (i.e., the intensities increased) after the Investor was introduced, improving the site functioning. In contrast, the time to first answer (and time to bounty) in questions without bounty (and with a bounty offered by the user asking the question) increased, suggesting that, in the counterfactual world where the Investor badge had not been introduced, the site functioning would have actually worsened.
Conclusions
Social media sites and online communities are dynamic environments where users change their behavior on a daily basis due to many complex factors. As a consequence, assessing the effectiveness of incentive mechanisms, which are ubiquitous among them, is challenging. In this work, we have focused on one of the simplest incentive mechanisms-first-time badges-and studied their effectiveness by developing a novel survival-based causal modeling framework, specially designed to harness the delayed introduction of several badges in a popular Q&A website.
Our work also opens up many interesting venues for future work. For example, it would be very useful to extend our framework to other types of badges, e.g., threshold badges. Badges are typically awarded to all users whose contributions exceed some predefined values, however, it would be interesting to also consider incentive mechanisms where users compete for a limited reward. Finally, in this work, we have focused on assessing the causal effect of (first-time) badges. A natural follow-up would be developing principled, effective methods to optimize their design.
Figure 1 :
1Time when users first edited a tag wiki (user action time, t) against time when they became eligible to edit tag wikis (user start time, s). The horizontal black line denotes the time when the Tag editor badge was introduced, which is awarded after a user edits a tag wiki for the first time.
Figure 2 :
2Our bootstrap difference-in-difference method. The treatment population (left) consists of users whose start time s lies in a window of size w around the time τ the badge is introduced. Each control population i (right) consists of users whose start time s lies in a window of size w around the time τ i a virtual badge is introduced. The method runs hypothesis testing on both the treatment population and all control populations and then compares the test statistic (e.g., p-value) of the treatment population with the empirical distribution of the test statistic of the control populations.
Figure 4 :
4Causal effect of three badges (Tag Editor, Promoter and Investor) on users' behavior. Panels in the left show the test statistic (log-likelihood ratio; LLR) over time for the basic and robust survival models, i.e., LLR τi and LLR τ against τ i and τ . Panels in the middle show the empirical distribution of the test statistic (LLR) under the null hypothesis H 0 and p-value for the robust survival model, i.e., F LLR (LLR) and p = F LLR (LLR τ ). Panels in the right show the average intensities for first-time action under robust survival model, i.e.,k 0 before τ andk 1 after τ , using a sliding window of length w = 60 days. The results suggest that the Tag editor and Investor badges were successful and the Promoter badge was unsuccessful.
Figure 5 :
5Causal effect of the Tag Editor badge for tags with different utility value, as estimated by their popularity level.
Figure 6 :
6Causal effect of the Promoter and Investor badges for bounties with different utility value, as estimated by the number of answers preceding the bounty offering.
Figure 7 :
7Tag wikis with and without the Tag Editor badge. Simulations means and 95%-CI shown.
Figure 8 :
8Time to bounty and first answer with and without the Promoter and Investor badges.
In February 2011, Stack Overflow lowered the minimum reputation level to 100 and thus the characteristics of the population that could earn the badge changed. Therefore, in our analysis, we only consider data up to January 2010.4 A user can also offer a bounty to a user after she has provided an answer, as a thank you gift, however, that usage is rarer.
We obtain quantitatively similar results in the absence of a linear trend, i.e., a = 0.
https://meta.stackexchange.com/questions/64824/clever-bounty-reputation-hack
A bounty can only be offered two days after the question has been asked.
AcknowledgementsWe thank Utkarsh Upadhyay and Isabel Valera for useful discussions.References
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|
[] |
[
"Top Quarks and Flavor Physics",
"Top Quarks and Flavor Physics"
] |
[
"Kenneth Lane \nDepartment of Physics\nBoston University\n590 Commonwealth Avenue02215BostonMA\n"
] |
[
"Department of Physics\nBoston University\n590 Commonwealth Avenue02215BostonMA"
] |
[] |
Because of the top quark's very large mass, about 175 GeV, it now provides the best window into flavor physics. Thus, pair-production of top quarks at the Tevatron Collider is the best probe of this physics until the Large Hadron Collider turns on in the next century. I discuss aspects of the mass and angular distributions that can be measured in tt production with the coming large data samples from the Tevatron and even larger ones from the LHC.1/95 †
|
10.1103/physrevd.52.1546
|
[
"https://arxiv.org/pdf/hep-ph/9501260v1.pdf"
] | 5,630,467 |
hep-ph/9501260
|
e2af0264f4f1dd38356a76bdd7877f2e0225e404
|
Top Quarks and Flavor Physics
arXiv:hep-ph/9501260v1 10 Jan 1995
Kenneth Lane
Department of Physics
Boston University
590 Commonwealth Avenue02215BostonMA
Top Quarks and Flavor Physics
arXiv:hep-ph/9501260v1 10 Jan 19951/95 † [email protected]
Because of the top quark's very large mass, about 175 GeV, it now provides the best window into flavor physics. Thus, pair-production of top quarks at the Tevatron Collider is the best probe of this physics until the Large Hadron Collider turns on in the next century. I discuss aspects of the mass and angular distributions that can be measured in tt production with the coming large data samples from the Tevatron and even larger ones from the LHC.1/95 †
Introduction
The CDF collaboration has reported evidence for top-quark production at the Tevatron Collider [1]. According to these papers, the top mass is m t = 174 ± 10 +13 −12 GeV. The CDF data are based on an integrated luminosity of 19.3 pb −1 . Taking into account detector efficiencies and acceptances, CDF reports the production cross section σ(pp → tt) = 13.9 +6.1 −4.8 pb at √ s = 1800 GeV. The predicted QCD cross section for m t = 174 GeV, including next-to-leading-log corrections [2], and soft-gluon resummation [3], is σ(tt) = 5.10 +0.73 −0.43 pb. This is 2.8 times smaller than the central value of the measured cross section. The uncertainty in α S increases the theoretical error in σ(tt) to at most 30% [4].
Very recently, the D / O Collaboration has also reported evidence for top-quark production [5]. A direct measurement of the top-quark mass and cross section was not made by D / O in this report. However, assuming that the excess of signal events over expected background is due to tt production and that m t = 180 GeV, D / O deduces the cross section σ(tt) = 8.2 ± 5.1 pb. This is consistent with the standard model and with CDF.
The experimental errors on the CDF and D / O measurements are large. Assuming the top mass is close to 175 GeV, the CDF cross section could be due to an up-fluctuation or to underestimated efficiencies (although the latter seems unlikely; see [1]). But, if it is confirmed by both experiments in their current higher-luminosity runs, the large of flavor physics that are more or less intimately connected to the large top mass and that lead to enhanced rates of the tt signals studied by the Tevatron experiments [6], [7], [8].
In this paper, we discuss two distributions that may be used to distinguish alternative models of tt production-including standard QCD [9]. These are the invariant mass distribution, dσ/dM tt , and the center-of-mass (c.m.) angular distribution of the top quark, dσ/d cos θ. The magnitude and shape of the invariant mass distribution (Section 2) can reveal whether tt production is standard or not, and whether resonances decaying to tt exist. We point out that, for standard QCD production, the mean and root-mean-square invariant masses are linear functions of the top-quark mass over the entire interesting range of m t . Thus, the M tt distribution can provide an independent determination of the top quark's mass. We apply this to the CDF data [1] and find quite good consistency with the directly measured mass. This analysis is made at the most elementary theoretical level. It needs to be carefully redone by the experimental collaborations themselves.
In Section 3, we apply the M tt analysis to examples of the three nonstandard mechanisms of tt production described in Refs. [6], [7] and [8]. The first involves resonant production of a 400-600 GeV color-octet vector meson ("coloron"), V 8 , which is associated with electroweak symmetry breaking via top-condensation [10] and which interferes with QCD production via the process qq → V 8 → tt. The second example invokes a coloroctet pseudoscalar, η T [11]. In multiscale models of walking technicolor [12], [13], the η T is produced strongly in gluon-gluon fusion and decays mainly to tt. The third model has additional production of the classic tt signals [1]. This occurs through pair-production of an electroweak-isoscalar, color-triplet quark, t s , which is approximately degenerate with the top quark and which, through mass-mixing, decays as t s → W + b. The agreement found in Section 2 between CDF's directly measured top-mass and that extracted from the M tt moments does not yet rule out these new mechanisms of top-quark production.
The M tt distributions from the current Tevatron run may do so. (Of course, finding the standard model cross section will also be powerful evidence against alternative production mechanisms.)
The angular distribution of top quarks (Section 4) also reflects the underlying production mechanism. Even though most of tt production is near threshold, the expectation that it is mainly s-wave can be overturned if there are large parity-violating components in the qq → tt process. We shall compare the angular distributions for standard and nonstandard tt production at the Tevatron and at the CERN Large Hadron Collider. We shall see that, because of the much larger τ =ŝ/s in top-pair production at the Tevatron, experiments there have an advantage over those at the LHC. 1 These angular tests require much larger 1 In this paper I do not discuss high-energy e + e − colliders such as the 500 GeV (or so) NLC.
Our focus is on distinguishing alternate mechanisms of tt production. Lepton machines cast no light on such strongly-coupled flavor physics aspects of tt production as the V 8 and η T . The higher rates possible at hadron machines also make them ideal for searches for new particles such as charged scalars in the decays of top quarks. data sets than will be available in the next year or two. To realize the full potential of this handle on flavor physics, it is essential that the Tevatron experiments be able to collect samples as large as 1-10 fb −1 .
Invariant Mass Distributions In QCD
Strictly speaking, the tt invariant mass M tt is not well-defined in QCD because of the emission of soft and collinear gluons from the t and t quarks. Nevertheless, the theoretical invariant mass is numerically not very different from a definition of M tt that allows for this gluon radiation. Furthermore, because the p T of the tt c.m. system typically is small compared to m t , it will be a good approximation for us to use the lowest-order QCD cross section dσ/dM tt to discuss the moments of the invariant mass distribution. This distribution is shown in Fig. 1 for the Tevatron Collider and for top-quark masses in the interesting range 100-220 GeV. 2 The mass distributions are seen to be sharply peaked at M max ≃ 2.1m t + 10 GeV. Consequently, low moments of the mass distribution, the mean and RMS, are nearly linear functions of the top-quark mass (also see Ref. [15]). mass. Indeed, in Ref. [15], it was shown that a quantity as indirect as the invariant mass M eµ of the isolated electron plus muon measured in tt → e ± + µ ∓ + jets is also a linear function of m t and may be used to determine it.
In Ref. [1], the top quark mass was determined from a sample of seven W → ℓν + 4 jets events by making an overall constrained best fit to the hypothesis pp → tt + X, followed by the standard top decays t → W + b with one W decaying leptonically and the other hadronically. At least one of the b-jets was tagged. The CDF paper provides the momentum 4-vectors of all particles in the event before and after the constrained fit.
From these, the central values of kinematic characteristics of the seven events may be determined. Table 1 lists the best-fit top-quark masses determined by CDF together with the invariant mass of the events before and after the constrained fit. 4 We used these M tt to compute the mean and RMS. Both sets of 4-momenta gave essentially identical results.
Using 4-momenta from the constrained fit, we found:
M tt = 439 ± 11 GeV =⇒ m t = 173 ± 5 GeV M 2 tt 1/2 = 443 ± 11 GeV =⇒ m t = 172 ± 5 GeV ∆M tt = 59.5 GeV .
(2.
2)
The errors in Eq. (2.2) were estimated by the "jacknife" method of computing the moments while omitting one of the seven events. They give some sense of the theoretical error in determining the mean and RMS invariant masses from the limited CDF sample. They are not to be interpreted as the true experimental errors; the CDF group must provide those.
However, we expect that the process of averaging the invariant mass will give moderately small experimental errors.
These results give some confidence that CDF's measured central value of the topquark mass, 174 GeV, is accurate. For example, if m t = 160 GeV (for which Ref. [3] predicts σ(tt) = 8.2 +1.3 −0.8 pb), we would expect M tt = 409 GeV and M 2 tt 1/2 = 415 GeV, well below the values determined above. Thus, if something is going to change in the CDF results from the current run, we expect it will be the cross section-which needs to become two to three times smaller to agree with the standard model. 4 Particle 4-vectors before the constrained fit do have various corrections-e.g., for the jet energy scale-made to them [1]. Only / E T is provided for the neutrino(s) in the before-fit 4vectors. The biggest change in the before and after momenta occurs in / E T . We used the W → ℓν 4-momenta determined from the constrained fit in both cases.
Nonstandard Mass Distributions
In this section we examine three nonstandard proposals [6], [7], [8] for tt production and the large cross section reported by CDF [1]. We shall find that they are not yet disfavored by the good agreement between the central values of the measured top mass and the top masses deduced in Eq. (2.2). We begin by quoting the differential cross sections for qq → tt and gg → tt in lowest-order QCD:
dσ(qq → tt) dz = πα 2 s β 9ŝ 2 − β 2 + β 2 z 2 , dσ(gg → tt) dz = πα 2 s β 6ŝ 1 + β 2 z 2 1 − β 2 z 2 − (1 − β 2 ) 2 (1 + β 2 z 2 ) (1 − β 2 z 2 ) 2 − 9 16 (1 + β 2 z 2 ) + 1 − β 2 1 − β 2 z 2 (1 − 1 8 β 2 + 3 8 β 2 z 2 ) ,(3.1)
where z = cos θ, θ is the c.m. scattering variable and β = 1 − 4m 2 t /ŝ. Forŝ ≫ 4m 2 t , these cross sections-especially the gluon fusion one-are forward-backward peaked. But, at the modestŝ at which QCD tt production is large, the cross sections are fairly isotropic.
For the "coloron" bosons of Ref. [6], we adopted a version of the model in which the gauge symmetry SU (3) 1 ⊗ SU (3) 2 breaks down to color SU (3), yielding eight massless gluons and equal-mass V 8 's. To study parity violation in the angular distributions in tt production (see Section 4), we made the theoretically inane assumption that the V 8 couples only to left-handed quarks with the amplitude
A(V a 8 (p, λ) → q(p 1 ) q(p 2 )) = g s ξ q ǫ µ (p, λ) u q (p 1 ) λ a 2 γ µ 1 − γ 5 2 v q (p 2 ) . (3.2)
Here, g s is the QCD coupling and, following Ref. [6], we took ξ t = ξ b = ±1/ξ q (q = u, d, c, s). For this chiral coupling, the qq → tt angular distribution in Eq. (3.1) is modified by the addition of
dσ(qq → V 8 → tt) dz = πα 2 s β 36ŝ (1 + βz) 2 1 + ξ q ξ tŝ s − M 2 V 8 + i √ŝ Γ(V 8 ) 2 − 1 , (3.3)
where, ignoring the mass of all quarks except the top's, the V 8 width is Fig. 3 for [6]) and in Fig. 4 for ξ t = ξ b = 1/ξ q = 40/3. The effect of interference with the QCD amplitude is obvious as is the tendency for the M tt distribution to be enhanced at lower (higher) masses for ξ t = −1/ξ q (+1/ξ q ). The theoretical width of Figures 5 and 6 show the mass distibutions for M V 8 = 475 GeV and ξ t = ∓ξ b = 40/3. Here, Γ(V 8 ) ∼ = 85 GeV and the mass distribution is significantly broader than in the case of M V 8 = 450 GeV. The characteristics of these mass distributions will be discussed below together with those of the other nonstandard production models we are considering. 5 Many technicolor models contain a color-octet pseudoscalar boson, η T . So long as the η T may be treated as a pseudo-Goldstone boson, its decay rates to gluons can be computed from the triangle anomaly [11]. We introduce a dimensionless factor C q in the Yukawa coupling of η T to qq [7]. While it is determined by the details of the underlying extended technicolor model, we expect C q = O(1). Then, the η T 's main decay modes are to two gluons and tt and they are given by
Γ(V 8 ) = α s M V 8 12 4ξ 2 q + ξ 2 t 1 + β t (1 − m 2 t /M 2 V 8 ) , (3.4) where β t = 1 − 4m 2 t /M 2 V 8 . The M tt distribution in the coloron model for M V 8 = 450 GeV is shown inξ t = ξ b = −1/ξ q = 40/3 (seethe V 8 in this example is Γ(V 8 ) ∼ = Γ(V 8 → bb)+Γ(V 8 → tt) = 40 GeV.Γ(η T → gg) = 5α 2 s N 2 T C M 3 η T 384 π 3 F 2 Q , Γ(η T → tt) = C 2 t m 2 t M η T β t 16πF 2 Q . (3.5)
The gluon fusion cross section for tt production is modified by the addition of
dσ(gg → η T → tt) dz = π 4 Γ(η T → gg) Γ(η T → tt) (ŝ − M 2 η T ) 2 +ŝ Γ 2 (η T ) + 5 √ 2 α 2 s N T C C t m 2 t β 768πF 2 Qŝ − M 2 η T (ŝ − M 2 η T ) 2 +ŝ Γ 2 (η T ) 1 − 2β 2 z 2 1 − β 2 z 2 . (3.6)
In these expressions, it is assumed that the η T is composed from a single doublet of techniquarks Q = (U, D) in the N TC representation of SU (N T C ); F Q is the decay constant of technipions in the QQ sector. The first term on the right is isotropic; the second 5 Note that this V 8 model predicts a strong resonance in qq → bb, providing another good way to to test it.
(interference) term is never very important, but we include it for completeness. In the narrow resonance approximation, the contribution of the η T to the pp ± → tt rate,
σ(pp ± → η T → tt) ≃ π 2 2s Γ(η T → gg) Γ(η T → tt) M η T Γ(η T ) × dη B f p g M η T √ s e η B f p g M η T √ s e −η B ,(3.7)
scales as N 2 T C /F 2 Q . Here, η B is the boost rapidity of the tt c.m. frame and f p g (x) is the gluon distribution function for the proton for mommentum fraction x and Q 2 = M 2 η T . As discussed in Ref. [7], the η T of the standard one-family technicolor model [11] has F Q = 123 GeV and, for N T C < ∼ 8, it cannot significantly increase the tt rate at the Tevatron. Thus, we were motivated to consider the η T arising in multiscale models [12] of walking technicolor [13]. Multiscale models are characterized by a small η T decay constant;
for the calculations presented here, we chose F Q = 30 GeV. The mass distribution for a model with N T C = 5 and C t = −1/3 is shown in Fig. 7 for M η T = 450 GeV. The width of the η T in this case is Γ(η T ) ∼ = Γ(η T → tt) + Γ(η T → gg) = 21 GeV + 11 GeV = 32 GeV. Before discussing features of these nonstandard M tt distributions, a comment on radiative corrections is in order. As noted above, we have multiplied all our lowest-order EHLQ1 cross sections by the factor 1.62. This is a composite of the radiative corrections at the Tevatron for the purely QCD processes gg → tt and qq → tt. For a 1.8 TeV pp collider, the qq process accounts for 90% of heavy tt production in the standard model. 6 On the other hand, the gluon fusion process receives the largest radiative correction [2], [3]. We do not know the radiative corrections appropriate to the resonant production processes qq → V 8 → tt and gg → η T → tt, but it seems likely that our multiplication by 1.62 overestimates the former and underestimates the latter process. Thus, the total Tevatron cross sections for these processes may be accurate to only about 30%. The total tt cross sections at the Tevatron and the characteristics extracted from the M tt distributions are displayed in Table 2 for the CDF data (see Eq. (2.2)) and for the three nonstandard production models described above. We note the following features:
1.) The CDF data is narrower (∆M tt = 60 GeV) than the QCD expectation (77 GeV).
While this ∆M tt is consistent with the resonant production models, the statistics are so low that we do not consider this significant. It is a feature worth watching for in future data samples.
2.) If ξ q ξ t = −1 in the coloron model (corresponding to the notation V − 8 in Table 2), the mass distribution is increased below the resonance and depressed above it; vice-versa for ξ q ξ t = +1 (V + 8 ). We see that, for a given M V 8 , this results in an extracted value of m t that is somewhat smaller than or significantly larger than the directly-measured one, depending on whether ξ q ξ t = −1 or +1.
3.) The η T 's we have considered are narrow enough to not interfere appreciably with the QCD gluon fusion process. Thus, the value of the extracted top mass depends mainly on M η T ; it tends to be larger for a larger M η T , but then the η T rate becomes smaller and the distortion of σ(tt) less important. Resonance masses in the range 400-500 GeV return a top mass close to the directly-measured value. 4.) It is easy to double the QCD value of σ(tt) in the isoscalar quark model: just choose m t s = m t . But, as could be foreseen, it is difficult for the isoscalar quark model to give both a 13.9 pb cross section and an extracted mass close to the directly-measured one.
To get a cross section ∼ 3 times as large as QCD requires choosing one of the masses significantly lower than 174 GeV, leading to too small an extracted value. This model could be the easiest to eliminate with data from the current Tevatron run.
Finally, we remark that subsystem invariant masses may be as interesting as the total invariant mass. For example, in multiscale technicolor, it is possible that a color-octet technirho is produced and decays as ρ T → W ∓ π ± T , with π + T → tb → W + bb, the same final state as in tt production [12]. Searches for processes such as these, using a constrained-fit procedure analogous to that employed by CDF for the tt hypothesis, should be carried out. All this will require a lot of data from the Tevatron, perhaps 1 fb −1 or more. At the expense of increasing backgrounds, larger data samples may be had by using appropriately selected events without a tagged b-jet. This was done in Ref. [1] and was found to give an excess of events with constrained-fit m t above 160 GeV.
Angular Distributions
The tt angular distribution of top quarks also provides information about their production mechanism. The distribution expected in lowest-order QCD was given in Eq. (3.1).
As we noted, the gg → tt process-10% of the QCD rate at the Tevatron and 90% of it at the LHC-is strongly forward-backward peaked atŝ ≫ 4 m 2 t , but fairly isotropic near threshold where most tt production occurs. Resonances such as the top-color V 8 and the technicolor η T may change the proportion of gg and qq-induced tt production and the top angular distribution at these colliders.
By Bose symmetry, the angular distribution in gg → tt processes is forward-backward symmetric. Although this is also true in lowest-order QCD for qq → tt, there is no reason it need be so for nonstandard production mechanisms. To illustrate the ability of hadron collider experiments to distinguish different angular distributions, we assume that the coloron V 8 couples only to left-handed quarks, implying the angular distribution
(1 + β cos θ) 2 .
The Tevatron has a distinct advantage in the study of top angular distributions. To determine θ in qq → tt processes, we need to know the direction of the incoming light quark as well as that of the outgoing top quark. 7 In pp → tt at √ s = 1800 GeV, the q-direction is the same as that of the proton practically all the time. Thus, if we denote by θ * the angle between the proton direction and the top-quark direction in the subprocess c.m., this angle is almost always the same as θ. As we shall see, the Tevatron's analyzing power would be significantly improved if the luminosity of the Tevatron were increased to 10 33 cm −2 s −1 or more and its detectors upgraded to handle this luminosity.
In pp collisions, the direction of the incoming quark can be inferred with confidence only for events with high boost rapidity, η B , or large fractional subprocess energy, τ =ŝ/s.
(For pp collisions, θ * will be defined as the angle between the direction of the boost and that of the top quark in the subprocess c.m.) For large τ , the quark direction tends to be the same as the boost of the c.m., even if η B is small [15]. However, τ is small for qq → tt at the LHC, making it hard to distinguish θ from π − θ. To make matters worse, tt production is dominated by gluon fusion, obscuring any interesting cos θ dependence.
Thus, angular information on top production is doubly difficult to come by at the LHC. 7 The distinction between t and t is based on the sign of the charged lepton in W -decay.
The cos θ * distributions we present below are integrals over tt invariant mass of dσ(pp ∓ → tt)/dM tt d cos θ * . The integration region is centered on the peak of the invariant mass distribution and is approximately the width of the resonance. For the η T , we used M η T = 450 GeV, N T C = 5, F Q = 30 GeV and C t = −1/3. Its width is 32 GeV. For the V 8 , we took M V 8 = 475 GeV and ξ t = ∓1/ξ q = 40/3. The V 8 width is 85 GeV.
The cos θ * distributions, defined as described above for pp (Tevatron) and pp (LHC) collisions, are shown for the η T and V 8 models in Figs. 11-14. Global features of these distributions are summarized in Table 3. The top quarks were required to have pseudorapidity |η| < 2, which we estimate to correspond to the average acceptance of the CDF and D / O detectors for leptons and jets from top decay. 8 We discuss them in turn: Figure 11 shows the qq → tt, gg → tt and gg → η T → tt components of the topproduction cos θ * distribution expected at the Tevatron. The M tt integration region is 430 to 470 GeV. The QCD contribution is flat, the forward-backward peaking diminished by the proximity of threshold. The η T contribution is also flat, of course, and makes up about 85% of the total cross section. The falloff above | cos θ * | = 0.90 is due to the rapidity cut, |η t,t | < 2.0. (We computed the cos θ * distribution of the seven tt candidate events reported by CDF [1]. The results, along with the top quark's c.m. velocity β, are listed in Table 1. They form a perfectly flat distribution.) Table 3 lists the total tt cross section as well as the cross sections σ F for cos θ * > 0 and σ B for cos θ * < 0. The forward-backward asymmetry is calculated as
A F B = N F − N B N F + N B = σ F − σ B σ F + σ B . (4.1)
The statistical error on A F B is
(∆A F B ) stat = 2 N F N B (N F + N B ) 3 = 2 σ F σ B (σ F + σ B ) 3 ǫ tt Ldt ,(4.2)
where ǫ tt is the overall efficiency, including branching ratios, for identifying and reconstructing tt events. For the CDF experiment at the Tevatron, we can infer from Ref. [1] that ǫ tt (CDF) ≃ 5-10 events/(19 pb −1 × 14 pb) = 2-4%. We use ǫ tt (TEV) = 3%. It is difficult to say what value of the efficiency is appropriate for LHC experiments; detailed 8 Our results do not change significantly if we require |η| < 1.5 for the Tevatron detectors and allow |η| < 2.5 for the LHC detectors.
simulations are needed (see e.g., Ref. [15]). We shall assume ǫ tt (LHC) = 5%, although it turns out not to matter in the examples we consider.
The components of the cos θ * distribution expected at the LHC are shown in Fig. 12.
Because of the small τ values involved, the roles of gluon fusion and qq annihilation are reversed, with gluon fusion making up about 90% of the QCD rate. The enormous η T → tt rate is due to the very large gg luminosity at small τ [14]. The slight central bowing of the cos θ * distribution is due to the top-rapidity cut. At the LHC energy, such large boost rapidities occur that events at large c.m. rapidity and cos θ * are depleted. The cos θ * distributions expected at the LHC for this V 8 are shown in Fig. 14. In this example, the contribution of the V 8 is about 20% of the total and it is polluted by the q ↔ q ambiguity, so that the rise in the cross section with cos θ * is invisible. The asymmetry is only 2%. This illustrates the dominance of gg processes and the uncertainty in determining the quark direction at small τ in a high-energy pp collider. Essentially similar results were obtained for the ξ t = 1/ξ q case (see Table 3). We found that there is nothing to be gained at the LHC by a looser η t cut, or by limiting the M tt integration region to a narrow band about M V 8 , or by selecting events produced at large boost rapidity.
Summary
Top quarks, of all known elementary particles, are most intimately connected to the physics of flavor and may provide keys to unlock its mysteries. Thus, top-quark production at the Tevatron provides our most incisive probe into flavor physics until the LHC turns on in the next century. The invariant mass distributions that can be formed in topquark production appear to be the best means for distinguishing between standard and nonstandard mechanisms.
The mean and RMS of the total invariant mass, M tt , provide an independent measure of m t which should agree with the directly-measured mass if production is governed by standard QCD. In QCD, the variance ∆M tt is expected to be about 75 GeV. The total invariant mass can reveal the presence of tt resonances such as the top-color vectors V 8 [6], [10] and the technihadron η T [7], [11]. Such resonances may easily double the tt rate. It is worth noting that, since the fraction of gluon-initiated processes rises fairly rapidly as the machine energy, an upgrade of the Tevatron to √ s = 2 TeV will lead to quite different changes in the V 8 → tt and η T → tt rates. 9 We find an increase of about 50% for qq → V 8 , but almost 100% for gg → η T . Subsystem invariant masses can be examined for alternative explanations of the top-production data and for unconventional top decays. In this regard, we emphasize that it may be dangerous to use the standard QCD tt production model to select top-candidate events. For example, a resonance in tt production may distort the summed scalar-E T and sphericity or aplanarity distributions of candidate events from their QCD expectation.
The angular dependence of top-production may also provide valuable information on the top-production mechanism. Although it is generally expected that, for production near threshold, the angular distribution will be isotropic, we have seen that chiral couplings can be detected if they are present and comparable to the QCD amplitudes. The dominance of qq annihilation in top-quark production processes at the Tevatron collider gives it an advantage over the LHC for studying angular distributions. Two distributions as different as those arising from the scalar-coupled η T and the chiral-coupled V 8 may be distinguished with a data sample of 1 fb −1 . However, it is clear that the resolving power of these distributions would benefit greatly from a significant upgrade of the collider and its detectors so that samples of O(10 fb −1 ) can be collected.
In conclusion, we emphasize that the studies done here have all been at the most naive parton level. We hope they will inspire the CDF and D / O collaborations to undertake more realistic, detector-specific simulations in the not-too-distant future. Table 3. Angular dependences of tt production in the η T and V 8 resonance models with parameters described in the text. Top quarks are produced with pseudorapidity |η| < 2.0 and cross sections (in pb) have been multiplied by 1.62.
Figure Captions
[1] The tt invariant mass distributions, in pp collisions at √ s = 1800 GeV, for m t = 100 − 220 GeV in 20 GeV increments. EHLQ1 distribution functions were used and the cross sections were multiplied by 1.62 as explained in the text. No rapidity cut is applied.
[2] The mean (solid) and root-mean-square (dashed) tt invariant mass, as a function of m t , for pp → tt at √ s = 1800 GeV. Lowest-order QCD cross sections (Fig. 1) were used.
[3] The tt invariant mass distribution in the presence of a V 8 , in pp collisions at √ s = 1800 GeV, for m t = 175 GeV and M V 8 = 450 GeV, ξ t = ξ b = −1/ξ q = 40/3. The QCD (dotted curve) and the total (solid) rates have been multiplied by 1.62 as explained in the text. No rapidity cut is applied to the top quarks.
[4] The tt invariant mass distribution in the presence of a V 8 , in pp collisions at √ s = 1800 GeV. The parameters and curves are as in Fig. 3 except that ξ t = ξ b = 1/ξ q = 40/3.
[5] The tt invariant mass distribution in the presence of a V 8 , in pp collisions at √ s = 1800 GeV, for m t = 175 GeV and M V 8 = 475 GeV, ξ t = ξ b = −1/ξ q = 40/3. The curves are labeled as in Fig. 3.
[6] The tt invariant mass distribution in the presence of a V 8 , in pp collisions at √ s = 1800 GeV. The parameters and curves are as in Fig. 5 except that ξ t = ξ b = 1/ξ q = 40/3.
[7] The tt invariant mass distribution in the presence of an η T , in pp collisions at √ s = 1800 GeV, for m t = 175 GeV and M η T = 450 GeV, F Q = 30 GeV and C t = −1/3. The QCD (dotted curve), η T → tt and its interference with the QCD amplitude (dashed), and total (solid) rates have been multiplied by 1.62 as explained in the text. No rapidity cut is applied to the top quarks.
[8] The tt invariant mass distribution in the presence of an η T , in pp collisions at √ s = 1800 GeV. The parameters and curves are as in Fiig. 7 except that M η T = 475 GeV.
[9] The effective tt mass distribution for pp → tt (dotted) and t s t s (dashed) at √ s = 1800 GeV; m t s = 160 GeV and m t = 175 GeV. The solid corve is the sum of the two mass distributions.
[10] The effective tt mass distribution for pp → tt and t s t s at √ s = 1800 GeV; m t s = 165 GeV and m t = 190 GeV. Curves are labeled as in Fig. 9.
[11] The cos θ * distribution for pp → tt at √ s = 1800 GeV in the presence of a 450 GeV η T with parameters as in Fig. 7; 430 < M tt < 470 GeV. The components are standard QCD gg → tt (dot-dash), qq → tt (long dashes), total QCD (dots), gg → η T → tt and interference with QCD (short dashes), and the total dσ/ cos θ * (solid). EHLQ1 distribution functions were used and all cross sections were multiplied by 1.62. The top quarks are required to have pseudorapidity |η| < 2.0.
[12] The cos θ * distribution for pp → tt at √ s = 15 TeV in the presence of a 450 GeV η T with parameters as in Fig. 11; 430 < M tt < 470 GeV. The curves are labeled as in Fig. 11.
[13] The cos θ * distribution for pp → tt at √ s = 1800 GeV in the presence of a 475 GeV V 8 with parameters as in Fig. 5; 400 < M tt < 500 GeV. The components are standard QCD gg → tt (dot-dash), qq → tt (long dashes), total QCD (dots), qq → V 8 → tt and interference with QCD (short dashes), and the total dσ/ cos θ * (solid). EHLQ1 distribution functions were used and all cross sections were multiplied by 1.62. The top quarks are required to have pseudorapidity |η| < 2.0.
[14] The cos θ * distribution for pp → tt at √ s = 15 TeV in the presence of a 475 GeV V 8 with parameters as in Fig. 13; 400 < M tt < 500 GeV. The curves labeled as in Fig. 13.
F
tt rate heralds the long-awaited collapse of the standard model. Even if the standard model result is found in the new data, however, it is clear that the top quark provides a wide-open window into the world of flavor physics. It is the heaviest elementary particle we know and, more to the point, the heaviest elementary fermion by a factor of 40as massive as tungsten. As a first example of flavor physics, we note that if the Higgs boson of the minimal one-doublet model exists, its coupling to the top quark, renormalized at m t = 174 GeV, is large: Γ t = 2 m t = 1.00. If there are charged scalars, members of Higgs-boson multiplets or technipions, they are expected to couple to top quarks with O(1) strength and to decay as H + → tb. Recently, several papers have discussed aspects
For 100 < 3 2
1003∼ m t < ∼ 200 GeV, the first two moments are well-fit by the formulae M tt = 50.range m t ≃ 140-200 GeV, the dispersion in M tt expected for standard QCD production is ∆M tt = 70-80 GeV. These plots and all other calculations in this paper were carried out using lowest-order QCD subprocess cross sections and the EHLQ Set 1 parton distribution functions[14]. To account for QCD radiative corrections, our tt cross sections have been multiplied by 1.62. This makes our QCD rates and the central values quoted in Ref.[3] agree to within one per cent over the entire interesting range of top masses. Our numerical results for the linear dependence of M tt and M 2 tt 1/2 on m t are accurate so long as the higher-order corrections are well-represented by a simple multiplicative factor.3 If there are experimental difficulties in measuring M tt that do not also affect the measurement of m t , one could instead use the mean value of the summed scalar-E T to extract the top-quark
Figure 8
8shows the cross section for M η T = 475 GeV; in this case, Γ(η T ) ∼ = 37 GeV. In both cases, the small decay constant results in a rate 2-3 times larger than QCD.The third model of enhanced top-production we considered is one in which an electroweak-isoscalar, charge 2 3 quark, t s , is approximately degenerate with the top-quark and mixes with it so that both have the same W b decay mode[8]. If m t s = m t = 174 GeV the expected rate for the top-quark signal is doubled to 10.2 pb. We illustrate the isoscalar quark model in Figs. 9 and 10 with two cases: m t s = 160 and m t = 175 GeV; m t s = 165 and m t = 190 GeV.
Figure 13 shows
13the components of the cos θ * distribution at the Tevatron for the 475 GeV V 8 coupling to left-handed quarks with relative strengths ξ t = −1/ξ q = 40/3.The M tt integration region is 400-500 GeV. The effect of the chiral coupling is evident, though somewhat diminished by the η t,t cut. The forward-backward asymmetry of 0.35 could be measured at the 5σ (statistical) level with an integrated luminosity of 1 fb −1 . For this luminosity, the statistical errors on dσ/d cos θ * in six bins 0.30 units wide would range from 20% down to 10%. This is one example of how useful it would be to upgrade the Tevatron luminosity to 10 33 cm −2 s −1 .
I am indebted to Alessandra Caner, Sekhar Chivukula, Estia Eichten, John Huth, Chris Quigg, Elizabeth Simmons, John Terning and Avi Yagil for helpful conversations. This research was supported in part by the Department of Energy under Grant No. DE-FG02-91ER40676. 9 I thank S. Parke for emphasizing this point to me.Table 1. Best fit top-quark masses and kinematic characteristics of the CDF experiment's tt candidate events (from Ref.[1]). Masses are in GeV. Transverse motion of the subprocess c.m. was neglected in determining the top-quark velocity β and scattering angle θ * .Table 2. pp → tt cross sections (in pb) at √ s = 1800 GeV and their kinematic characteristics for lowest-order QCD, CDF data[1], and the three nonstandard production models with parameters described in the text. Cross sections have been multiplied by 1.62.Run-Event
m t
M tt (before fit)
M tt (after fit)
β(after fit)
cos θ *
40758-44414
172 ± 11
523
526
0.757
0.404
43096-47223
166 ± 11
533
511
0.760
0.820
43351-266423
158 ± 18
440
460
0.727
0.512
45610-139604
180 ± 9
338
366
0.180
−0.0011
45705-54765
188 ± 19
440
431
0.489
−0.348
45879-123158
169 ± 10
411
412
0.572
−0.767
45880-31838
132 ± 8
384
365
0.691
−0.682
Model
σ(tt)
M tt
m t ( M tt )
M 2
tt
1/2
m t ( M 2
tt
1/2 )
∆M tt
LO-QCD (EHLQ1)
5.13
440
174
447
174
77
CDF data
13.9 +6.1
−4.8
439
173
443
172
60
M V −
8
= 450
13.3
431
170
433
168
46
M V +
= 450
11.0
465
185
469
184
58
M V −
8
= 475
14.9
440
174
444
173
53
M V +
8
= 475
10.8
482
193
487
192
67
M η T = 450
13.5
432
171
435
169
52
M η T = 475
11.4
442
175
446
174
55
t s (160) t(175)
13.2
421
166
428
166
77
t s (165) t(190)
10.0
437
173
444
173
77
Model
M tt range
Collider
σ(tt)
σ F
σ B
A F B
η T
430 − 470
TEV
4.82
2.41
2.41
0
η T
430 − 470
LHC
4360
2180
2180
0
V 8 −
400 − 500
TEV
8.14
5.48
2.66
0.35
V 8 −
400 − 500
LHC
285
145
140
0.017
V 8 +
425 − 525
TEV
5.93
4.21
1.72
0.42
V 8 +
425 − 525
LHC
255
130
125
0.021
This situation is reversed at the 15 TeV LHC pp collider, where gg → tt is 90% of the QCD cross section.
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The D / O Collaboration. S Abachi, hep-ex-9411001Search for High Mass Top Quark Production in pp Collisions at √ s = 1.8 TeV. submitted toS. Abachi, et al., The D / O Collaboration, "Search for High Mass Top Quark Production in pp Collisions at √ s = 1.8 TeV", hep-ex-9411001, submitted to Physical Review Letters, November 9, 1994.
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An abridged version of this work was given in K. Lane. BUHEP-94-25Proceedings of the 27th International Conference on High Energy Physics. the 27th International Conference on High Energy PhysicsBoston University PreprintTop-Quark Production and Flavor Physics-The Talk. to appear in theAn abridged version of this work was given in K. Lane, "Top-Quark Production and Flavor Physics-The Talk", to appear in the Proceedings of the 27th International Conference on High Energy Physics, Glasgow, 20-27th July 1994; Boston University Preprint BUHEP-94-25 (1994).
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Chapter 2; GEM TN-93-262, SSCL-SR-1219; Submitted by the GEM Collaboration to the Superconducting Super Collider Laboratory. GEM Technical DesignReportGEM Technical Design Report, Chapter 2; GEM TN-93-262, SSCL-SR-1219; Sub- mitted by the GEM Collaboration to the Superconducting Super Collider Laboratory (April 30, 1993);
K Lane, F Paige, T Skwarnicki, J Womersley, hep-ph-9412280Simulations of Supercollider Physics. submitted to Physics ReportsK. Lane, F. Paige, T. Skwarnicki and J. Womersley, Simulations of Supercollider Physics, hep-ph-9412280, submitted to Physics Reports.
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"Combinatorial constructions of modules for infinite-dimensional Lie algebras, II. Parafermionic space",
"Combinatorial constructions of modules for infinite-dimensional Lie algebras, II. Parafermionic space"
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"Galin Georgiev "
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The standard modules for an affine Lie algebraĝ have natural subquotients called parafermionic spaces -the underlying spaces for the so-called parafermionic conformal field theories associated withĝ.We study the caseĝ = sl(n + 1, C) for any positive integral level k ≥ 2. Generalizing the Z-algebra approach of Lepowsky, Wilson and Primc, we construct a combinatorial basis for the parafermionic spaces in terms of colored partitions. The parts of these partitions represent "Fourier coefficients" of generalized vertex operators (parafermionic currents) and can be interpreted as statistically interacting quasi-particles of color i, 1 ≤ i ≤ n, and charge s, 1 ≤ s ≤ k − 1. From a combinatorial point of view, these bases are essentially identical with the bases for level k−1 principal subspaces given in [GeI]. In the particular case of the vacuum module, the character (string function) associated with our basis is the formula of Kuniba, Nakanishi and Suzuki [KNS] conjectured in a Bethe Ansatz layout.New combinatorial characters are established for the whole standard vacuumĝmodules.
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Combinatorial constructions of modules for infinite-dimensional Lie algebras, II. Parafermionic space
Galin Georgiev
Combinatorial constructions of modules for infinite-dimensional Lie algebras, II. Parafermionic space
arXiv:q-alg/9504024v1 26 Apr 1995 q-alg/9504024
The standard modules for an affine Lie algebraĝ have natural subquotients called parafermionic spaces -the underlying spaces for the so-called parafermionic conformal field theories associated withĝ.We study the caseĝ = sl(n + 1, C) for any positive integral level k ≥ 2. Generalizing the Z-algebra approach of Lepowsky, Wilson and Primc, we construct a combinatorial basis for the parafermionic spaces in terms of colored partitions. The parts of these partitions represent "Fourier coefficients" of generalized vertex operators (parafermionic currents) and can be interpreted as statistically interacting quasi-particles of color i, 1 ≤ i ≤ n, and charge s, 1 ≤ s ≤ k − 1. From a combinatorial point of view, these bases are essentially identical with the bases for level k−1 principal subspaces given in [GeI]. In the particular case of the vacuum module, the character (string function) associated with our basis is the formula of Kuniba, Nakanishi and Suzuki [KNS] conjectured in a Bethe Ansatz layout.New combinatorial characters are established for the whole standard vacuumĝmodules.
0 Introduction 0.1
We begin with a short outline of some results from [GeI] which will be needed here.
Let g := sl(n + 1, C) with a triangular decomposition g = n − ⊕ h ⊕ n + and simple roots α i , 1 ≤ i ≤ n, where the indices reflect the roots' location on the Dynkin diagram (the considerations below can be carried out for any simple Lie algebra g over C of type A-D-E). Denote by x ±α i , 1 ≤ i ≤ n, the Chevalley generators of g. Let Q := n i=1 Zα i , P := n i=1 ZΛ i be the root and weight lattice respectively, where Λ i , 1 ≤ i ≤ n, are the fundamental weights of g. For some formal variable t, consider the untwisted affinizationŝ g := g ⊗ C[t, t −1 ] ⊕ Cc,n := n ⊕ C[t, t −1 ] with a scaling operator D := −td/dt. LetΛ i , 0 ≤ i ≤ n, be the fundamental weights ofĝ, so thatΛ i =Λ 0 + Λ i when 1 ≤ i ≤ n. For another formal variable z, denote by X ±α i (z) := m∈Z (x ±α i ⊗ t m )z −m−1 the vertex operators (bosonic currents) of charge one, corresponding to simple roots and their negatives. Define also currents of higher charge r ∈ Z + : X ±rα i (z) := X ±α i (z) r (cf. [GeI]). The operatorvalued ("Fourier ) coefficients of X rα i (z) are called quasi-particles of color i and charge r.
For a given positive integral level k (the eigenvalue of c), let L(kΛ 0 ) be the vacuum standardĝ-module, i.e., the highest weight integrable module with highest weight kΛ 0 (and highest weight vector v(kΛ 0 )). For the sake of simplicity, we shall consider in this Introduction only vacuum highest weights, although the results are proved for a larger class of highest weights. Denoting by U(·) universal enveloping algebra, recall from [FS] that W (kΛ 0 ) := U(n) · v(kΛ 0 ) is called principal subspace of the standard module. We constructed in [GeI] a basis for the principal subspace: It is built by quasi-particles of colors i, 1 ≤ i ≤ n, and charges s, 1 ≤ s ≤ k, acting on the highest weight vector v(kΛ 0 ). A straightforward counting of the basis resulted in the character formula announced by Feigin and Stoyanovsky [FS]: where (A lm ) n l,m=1 is the Cartan matrix of g, B st := min{s, t}, 1 ≤ s, t ≤ k, and for p ∈ Z + , (q) p := (1 − q)(1 − q 2 ) · · · (1 − q p ), (q) 0 := 1 (for the character of a principal subspace with more general highest weight, see formula (5.27) [GeI]). This formula was interpreted in the Introduction of [GeI] as a character for the Fock space of nk different free bosonic quasi-particles (n different colors and k different charges) with an additional twoparticle interaction A lm B st between a quasi-particle of color l and charge s and another quasi-particle of color m and charge t. This interaction can be interpreted as a statistical interaction in the sense of Haldane [H].
Tr q D W (kΛ 0 ) = p(1)
As indicated in [GeI], one can generate in a similar fashion a basis for the whole standard module, employing in addition quasi-particles corresponding to the negative simple roots. We shall see in Proposition 0.1 at the end this Introduction that it is sufficient to add only quasi-particles of charge k corresponding to the negative simple roots (we might also refer to the latter as to quasi-antiparticles of charge −k).
0.2
One of our objectives in this paper is the generalization (to higher rank affine Lie algebras) of the vertex operator construction of parafermionic spaces, given by Lepowsky and Primc [LP] forĝ = sl(2, C). Their construction was the untwisted version of the ground-breaking works on Z-algebras of Lepowsky and Wilson [LW].
Consider the so-calledĝ ⊃ĥ coset subspace L(kΛ 0 )ĥ + of L(kΛ 0 ) consisiting of all theĥ + := h ⊗ tC[t]-invariants, i.e., the vectors annihilated byĥ + . The parafermionic space L(kΛ 0 )ĥ + kQ is defined as the spase of kQ-coinvariants in L(kΛ 0 )ĥ + , i.e., L(kΛ 0 )ĥ + kQ := L(kΛ 0 )ĥ + /(ρ(kQ)−1)·L(kΛ 0 )ĥ + , where ρ is a natural action of the abelian group kQ ∼ = Q. There is a natural projection π L(kΛ 0 )ĥ + such that L(kΛ 0 )ĥ + = ρ(kQ)·π L(kΛ 0 )ĥ + ·W (kΛ 0 ) (cf.
(1.4) and (2.39) below) -this explains why structural results for the principal subspace are easily carried to theĝ ⊃ĥ subspace (and henceforth to the parafermionic space itself). For example, the bosonic current X rα i (z) has simply to be replaced by π L(kΛ 0 )ĥ + · X rα i (z) -the latter equals up to a nonzero constant and a power of z the familiar parafermionic current Ψ rα i (z) (called generalized vertex operator in the general setting of [DL]) which by analogy is said to have a color i and charge r. Roughly speaking, one could think of the parafermionic current Ψ as obtained from the bosonic current X by factoring a free bosonic field (cf. (2.20)). As expected, the quasi-particles of color i and charge r in the parafermionic setting will be the operator-valued ("Fourier") coefficients of the current Ψ rα i (z). Note by the way that the parafermionic counterpart of the simple relation X rα i (z) := X α i (z) r is more sophisticated and involves the familiar binomial correction terms:
Ψ rα i (z) = r l,p=1 l>p (z l − z p ) α i ,α i k Ψ α i (z r ) · · · Ψ α i (z 1 ) z r = · · · = z 1 = z , (0.2)
where the binomial terms are to be expanded in nonnegative inegral powers of the second variable (before the expression is restricted on the hyperplane z r = · · · = z 1 = z). There are other novelties in the parafermionic picture: The most important one is probably the new constraint
Ψ kα i (z) = const ρ(kα i ), const ∈ C × , 1 ≤ i ≤ n, (0.3)
which implies that the maximal allowed quasi-particle charge is reduced from k in the principal subspace picture to k − 1 in the parafermionic picture. Another subtlety is that the scaling operator D has to be shifted in the parafermionic setting by −Dĥ, where Dĥ is the rescaled 0th mode of the Virasoro algebra associated with the vertex operator algebra
U(ĥ) · v(kΛ 0 ),ĥ := h ⊗ C[t, t −1 ] ⊕ Cc of h (cf. (2.13)).
We build a quasi-particle basis for L(kΛ 0 )ĥ + kQ in section 4 (the corresponding basis for L(kΛ 0 )ĥ + is then obtained by multiplying with ρ(kα), α ∈ Q). As expected, it is essentially the same as the [GeI] quasi-particle basis for the level k − 1 principal subspace W ((k − 1)Λ 0 ). The only difference is that the two-particle interaction between a quasiparticle of color l and charge s and another quasi-particle of color m and charge t has a "parafermionic " shift −A lm st k and is therefore given by
A lm min{s, t} − A lm st k = A lm A (−1) st , 1 ≤ s, t ≤ k − 1, (0.4)
where (A lm ) n l,m=1 is again the Cartan matrix of g = sl(n + 1, C) and (A (−1) st ) k−1 s,t=1 is the inverse of the Cartan matrix of sl(k, C). In other words, the character formula associated with our basis is the one conjectured by Kuniba, Nakanishi and Suzuki [KNS]:
Tr q D−Dĥ L(kΛ 0 )ĥ + kQ = p (1) 1 ,...,p (k−1) 1 ≥0 ............ p (1) n ,...,p (k−1) n ≥0 q 1 2 s,t=1,...,k−1 l,m=1,...,n A lm A (−1) st p (s) l p (t) m n i=1 k−1 s=1 (q) p (s) i . (0.5)
Note that a dilogarithm proof of this formula has already been announced by Kirillov [Kir], so it might be appropriate to emphasize that our goal here is not simply to find another proof of this beautiful formula, but to reveal the underlying conceptual structure. This in-depth approach, although painful, will pay off as we shall see for example in the subsequent Proposition 0.1. Note also that our arguments work for more general dominant highest weight (cf. (4.1) below)-the associated character formula is given in (5.7) Section 5. After an appropriate normalization, one immediately obtaines from the above formulas combinatorial expressions for the corresponding string functions. Note that if we restrict our attention to the particular case g = sl(2, C), the above formula (0.5) is the celebrated Lepowsky-Primc character [LP]. The underlying basis in [LP] is nevertheless very different from ours: Their construction employs only quasiparticles of charge one, governed by the so-called "difference 2 at distance k−1" condition, while we work with quasi-particles of charge r, 1 ≤ r ≤ k − 1, governed by a "difference 2r at distance 1" condition. A straightforward generalization of the Lepowsky-Primc parafermionic basis for g = sl(3, C) was given in [P].
0.3
We continue with the higher level generalization of the new character formula for the level one standard module L(Λ 0 ) given in Proposition 0.1 [GeI] (for simplicity, we restrict ourselves to vacuum modules only).
As explained already, from the [GeI] basis for the principal subspace W (kΛ 0 ) we easily obtain a basis for theĝ ⊃ĥ subspace L(kΛ 0 )ĥ + . The latter immediately implies a basis for the whole standard module L(kΛ 0 ) ∼ = U(ĥ) · v(kΛ 0 ) ⊗ L(kΛ 0 )ĥ + : The associated character formula is (cf. (0.1)): cf. (5.15) for a more general highest weight). Note that the sum over Q incorporates the contribution of the factors ρ(kα), α ∈ Q, and is easily expressed in terms of the classical theta function of degree k:
Tr q D L(kΛ 0 ) = 1 (q) n ∞ p (1) 1 ,...,p (k−1) 1 ≥0 ............ p (1) n ,...,p (k−1) n ≥0 q 1 2 s,t=1,...,k−1 l,m=1,...,n A lm B st p (s) l p (t) m n i=1 k−1 s=1 (q) p (s) i · α∈Q q k 2 α,α + α, n i=1 r i α i , (0.6) where r i := k−1 s=1 sp (s) i (α∈Q q k 2 α,α + α,µ = q − µ,µ 2k Θ µ (q) = q − µ,µ 2k γ∈Q+ µ k q k 2 γ,γ . (0.7)
Although the presence of the theta function is somewhat intimidating, the above expression can still be interpreted as a trace along a combinatorial basis: This would be a semiinfinite monomial basis built up from quasi-particles corresponding to (positive) simple roots; cf. [FS] where the case g = sl(2, C) is discussed. But there is yet another -and probably the most natural -way to generate a basis for the whole standard module starting from the principal subspace: Simply add quasi-particles x −kα i (m) corresponding to negative simple roots (we shall call those antiquasiparticles of charge −k) and take into account the identity
(z 2 − z 1 ) r α i ,α i X −kα i (z 2 )X rα i (z 1 ) z 1 = z 2 = z = const X −(k−r)α i (z), const ∈ C × , (0.8)
for every simple root α i , 1 ≤ i ≤ n, and charge r, 1 ≤ r ≤ k. In particular, this identity implies that all the quasi-antiparticles of charge −r, 1 ≤ r < k, can be generated by antiquasiparticles of charge −k and usual quasi-particles. Moreover, when r = k, one gets an important new constraint between quasiparticles of charge k and anti-quasiparticles of charge −k (the right-hand side is just a constant). Following the layout of [GeI], it is now not difficult to generate a basis for the whole standard module and write down the character formula associated with it:
Proposition 0.1 One has the following q-character for the vacuum standard module at level k:
Tr q D L(kΛ 0 ) = p (1) ±1 ,...,p (k) ±1 ≥0 ............ p (1) ±n ,...,p (k) ±n ≥0 p (s) −i =0 ∀s<k, ∀i q 1 2 s,t=1,...,k l,m=1,...,n A lm B st (p (s) +l −p (s) −l )(p (t) +m −p (t) −m )+ n l=1 p (k) +l p (k) −l n i=1 k s=1 (q) p (s) +i (q) p (s) −i , (0.9)
where (A lm ) n l,m=1 is the Cartan matrix of g, B st := min{s, t}, 1 ≤ s, t ≤ k, and for p ∈ Z + , (q) p := (1 − q)(1 − q 2 ) · · · (1 − q p ), (q) 0 := 1. In complete analogy with the level one particular case (cf. [GeI] Proposition 0.1), the above formula follows from (0.6) and the "Durfee rectangle" combinatorial identity [A]
1 (q) ∞ := l≥0 (1 − q l ) −1 = a,b≥0 a−b=const q ab (q) a (q) b . (0.10)
The basis underlying the above expression will be discussed in details somewhere else.
In the simplest particular case n = k = 1, the right-hand side of the above character reduces to
p +1 ,p −1 ≥0 q p 2 +1 +p 2 −1 −p +1 p −1 (q) p +1 (q) p −1 ,
a formula which appeared first in [FS].
0.4
Since the above Proposition 0.1 is the higher level generalization of Proposition 0.1 [GeI], the natural question arises, what is the higher level generalization of Proposition 0.2 [GeI], i.e., how to build a basis from intertwining operators corresponding to fundamental weights and their negatives (this question was posed long time ago by J. Lepowsky)? This remains a difficult open problem although a promising breakthrough in the simplest particular caseĝ = sl(2, C) was recently made by Bouwknegt, Ludwig and Schoutens [BLS1] (cf. also [BPS], [I]): Using explicitly the easy to describe fusion algebra of the fusion category for sl(2, C)-modules, they built a basis of the above type (the so-called spinon basis) and derived the corresponding combinatorial character formula. As expected, the [BLS1] character is much more complicated than the particular sl(2, C)-case of formula (0.9). The reason is that (0.9) reflects a basis built up from usual (as opposed to intertwining!) vertex operators, namely, the ones corresponding to simple roots. Of course, the latter do not interchange different modules and for that matter, no knowledge of the fusion algebra is needed. Let us note that the [BLS1] basis in principal gradation was identified with the basis proposed in [FIJKMY] (cf. [BLS2]). Moreover, the [BLS1] character formula was later independently derived using crystaline spinon basis, i.e., spinon basis for standard modules over U q ( sl(2, C)) at q = 0 (cf. [NY], [ANOT]). No attempts have been made yet to crystalize the (q-deformation of the) basis underlying the above character formula (0.9).
Since the character (0.5) was originally conjectured by Kuniba, Nakanishi and Suzuki [KNS] from Thermodynamic Bethe Ansatz (TBA) considerations, it remains an open problem to understand the connection between our vertex operator construction and TBA. One possible approach, adopted by Melzer [M], Foda and Warnaar [FW], [W] in the context of Virasoro algebra modules, is through "finitization" of the above q-series, i.e., representing them as a limit of certain q-polynomials.
Finally, from the point of view of our approach, one of the most important and challenging unsolved problems is of course to find a generalization for arbitrary (nonintegral) levels and highest weights and also, to build vertex operator bases for other coset spaces. This will in particular provide vertex operator constructions for many other modules over W -algebras. It will also cast some light upon the connection with the approach of Berkovich and McCoy [BM] to Virasoro algebra modules from the minimal series and to branching functions associated with various other coset spaces considered in [KM], [KKMM], [DKKMM], [KMM], [WP], [BG].
0.5
In Section 1 we define theĝ ⊃ĥ coset subspace of a standardĝ-module (g = sl(n + 1, C)) at any positive integral level k ≥ 2. The parafermionic space is defined as its natural quotientspace. Section 2 introduces relative vertex operators (parafermionic currents). Section 3 explains how the notions of quasi-particle and quasi-particle momomial (from part I) have to be modified in the current setting. In Section 4 we build a quasi-particle monomial basis for the parafermionic space. Section 5 presents the corresponding character formulas for the parafermionic space (string function) and for the whole standard module. Tables 1 and 2 in the Appendix illustrate Examples 4.1 and 4.2, Section 4.
Theĝ ⊃ĥ coset subspace and its parafermionic quotientspace
Although the exposition below is selfcontained, we shall follow the framework of [GeI] and use many of its notations, definitions and results. We continue working withĝ = sl(n + 1, C) and a level k dominant integral highest weightΛ = k 0Λ0 + k jΛj for some j, 1 ≤ j ≤ n, and k 0 , k j ∈ N, k 0 + k j = k ≥ 2, where {Λ l } n l=0 are the fundamental weights ofĝ. In other words,Λ = kΛ 0 + Λ, where Λ = k j Λ j ∈ P and {Λ l } n l=1 are the fundamental weights of g. (All the objects introduced in Sections 1,2 and 3 have obvious generalizations for any dominant integral highest weight.)
Recall thatĥ := h ⊗ C[t −1 , t] ⊕ Cc ⊂ĝ is the affinization of the Cartan subalgebra h ⊂ g. Theĝ ⊃ĥ coset subspace L(Λ)ĥ + of the standardĝ-module L(Λ) is defined as the vacuum subspace forĥ + , i.e., the linear span of all the vectors {v ∈ L(Λ)|ĥ + ·v = 0}, wherê h + := h ⊗ tC [t]. A concrete realization of this space can be given in terms of the vertex operator construction of basic modules:
Set M(k) := U(ĥ) ⊗ U (h⊗C[t]⊕Cc) C, with h ⊗ C[t]
acting trivially on C and c acting as k (cf. [GeI] Preliminaries). Let V P := M(1) ⊗ C [P ] and recall that by the classical interpretation of V P as aĝ-module [FK], [S], one has V P ∼ = ⊕ n j=0 L(Λ j ) (cf. [GeI], Section 2). Therefore, the standard module L(Λ) can be explicitly generated by the universal eneveloping algebra U(ĝ) acting on the highest weight vector v(Λ) ⊂ V ⊗k P through the (k−1)-iterate ∆ k−1 of the standard coproduct ∆ (cf. [GeI], Sections 2, 5; no confusion can arise from the fact that we denote by the same letter ∆ the root system of g). Then theĝ ⊃ĥ coset subspace of L(Λ) is of course
L(Λ)ĥ + := span C {v ∈ L(Λ) = U(ĝ) · v(Λ)|ĥ + · v = 0} ⊂ V ⊗k P .
(1.1)
Note how easy it is to reconstruct the whole space L(Λ) from L(Λ)ĥ + because of the canonical isomorphisms of D-graded linear spaces [LW] U
(ĥ − ) ⊗ L(Λ)ĥ + ∼ → L(Λ) (1.2) h ⊗ u → h · u, and S(ĥ − ) ∼ = U(ĥ − ) ∼ = M(k), (1.3) whereĥ − :=ĥ ⊗ t −1 C[t −1 ], S(·)
is a symmetric algebra and M(k) was defined above. One can therefore consider the projection
π L(Λ)ĥ + : L(Λ) → L(Λ)ĥ + , (1.4)
given by the corresponding direct sum decomposition
L(Λ) = L(Λ)ĥ + ⊕ĥ − U(ĥ − ) · L(Λ)ĥ + . (1.5)
We can further reduce theĝ ⊃ĥ coset subspace using its natural structure of module for the root lattice Q: Observe that the map α → e α , α ∈ Q, defines an action of Q on V P = M(1)⊗C [P ] (cf. [GeI], Section 2) by restriction to the right factor. It thus commutes with the action ofĥ + on V P , which affects only the left factor. Define a diagonal action
of the sublattice kQ ⊂ Q (kQ ∼ = Q) on V ⊗k P : kα → ρ(kα) := e α ⊗ . . . ⊗ e α k f actors , α ∈ Q.
(1.6)
Note that it commutes with the action ∆ k−1 (ĥ + ) ofĥ + and hence preserves L(Λ)ĥ + ⊂ V ⊗k P . Following [DL], Ch. 4, we consider the space of kQ-coinvariants in the kQ-module L(Λ)ĥ + :
L(Λ)ĥ + kQ := L(Λ)ĥ + Span{(ρ(kα) − 1) · v|α ∈ Q, v ∈ L(Λ)ĥ + } . (1.7)
This quotientspace is called parafermionic space (of highest weightΛ) because it is a building block for the so-called parafermionic conformal field theories [ZF], [G], [LP], [DL]. We shall denote by
π L(Λ)ĥ + kQ : L(Λ) → L(Λ)ĥ + kQ (1.8)
the composition of π L(Λ)ĥ + from (1.4) and the obvious projection of L(Λ)ĥ + onto L(Λ)ĥ + kQ . Our goal here is to construct a quasi-particle basis for the parafermionic space L(Λ)ĥ + kQ , modifying appropriately the quasi-particle basis for the principal subspace W (Λ) ⊂ L(Λ) built in [GeI]. Observe that if we denote by L µ (Λ)ĥ + the weight subspace of L(Λ)ĥ + , corresponding to µ ∈ P and by L µ (Λ)ĥ + kQ its isomorphic (as graded linear space) image in the quotient L(Λ)ĥ + kQ , we have
ρ(kα) · L µ (Λ)ĥ + = L µ+kα (Λ)ĥ + , α ∈ Q, µ ∈ P,(1.9)
and
L(Λ)ĥ + kQ = µ∈Λ+Q/kQ L µ (Λ)ĥ + kQ ∼ = µ∈Λ+Q/kQ L µ (Λ)ĥ + .
(1.10) Therefore a basis for L(Λ)ĥ + kQ furnishes automatically a basis for the wholeĝ ⊃ĥ coset subspace L(Λ)ĥ + = µ∈Λ+Q L µ (Λ)ĥ + . (No need to say, the tensor product decomposition (1.2) then provides a basis for the whole standard module L(Λ).)
The trivial action ρ of kQ on the parafermionic space is very suggestive for considering a natural action (denoted by the same letter ρ) of the finite abelian group Q/kQ on L(Λ)ĥ + kQ (cf. [DL], Ch. 6): Set
ρ(α) · v := e 2πi α,µ k v, α ∈ Q/kQ, v ∈ L µ (Λ)ĥ + kQ .
(1.11)
The characters e 2πi ·,µ k , µ ∈ Λ + Q/kQ, are indeed the simple characters of the group Q/kQ, in other words, the decomposition (1.10) coincides with the character-space decomposition of L(Λ)ĥ + kQ under the above action of Q/kQ.
Generalized vertex operators (parafermionic currents)
In this Section, we shall largely use the methods of generalized vertex operator algebra theory developed by Dong and Lepowsky [DL] (cf. also [FLM]).
Recall that the main protagonist in the level k setting of [GeI] was the vertex operator (bosonic current)
X β (z) := ∆ k−1 (Y (e β , z)) = (2.1) = Y (e β , z) ⊗ 1 ⊗ · · · ⊗ 1 k f actors + 1 ⊗ Y (e β , z) ⊗ · · · ⊗ 1 k f actors + · · · + 1 ⊗ · · · ⊗ 1 ⊗ Y (e β , z) k f actors , where β ∈ ∆, Y (e β , z) := E − (−h β , z)E + (−h β , z) ⊗ e β z h β ε β , (2.2) E ± (h, z) := exp m≥1 h(±m) z ∓m ±m , h ∈ h,(2.3)
and ε β := ε(β, ·), ε : P × P → C × being a 2-cocycle on the weight lattice P (cf. [GeI] Sections 1 -3). The "Fourier coefficients" of this vertex operator are given by its expansion
X β (z) =: m∈Z x β (m)z −m−1 , (2.4)
on powers of the formal variable z. The action of the affine algebraĝ on the standard module
L(Λ) = U(ĝ) · v(Λ) is then given by x β ⊗ t m := x β (m), β ∈ ∆, m ∈ Z.
The Jacobi identity for the vertex operator algebra L(kΛ 0 ) generated by these currents, implies the usual formulas for the currents of higher charge r ∈ Z + :
X rβ (z) := X β (z) · · · X β (z) rf actors := Y (x β (−1) r · v(kΛ 0 ), z), (2.5)
where v(kΛ 0 ) is the vacuum highest weight vector at level k (cf. for example [DL] Proposition 13.16). Note that since the product X β (z 2 )X β (z 1 ) is not singular on the hyperplane z 2 = z 1 = z (cf. [GeI] (2.8)), one does not need to "regularize" it with powers of z 2 − z 1 before one sets z 2 = z 1 = z. This is not the case anymore in the parafermionic picture -see (2.19) below. We actually showed in [GeI] that the currents corresponding to the simple roots α i , 1 ≤ i ≤ n, are enough to build a basis of the principal subspace W (Λ) : their "Fourier coefficients" played the role of quasi-particles and the basis was generated by quasi-particle monomials (from an appropriate completion of the ordered product U := U(n αn ) · · · U(n α 1 ) acting on the highest weight vector v(Λ)). Switching now our attention from the principal subspace W (Λ) to the parafermionic space L(Λ)ĥ + kQ , we immediately observe that the above operators do not even preserve the vacuum subspace L(Λ)ĥ + , let alone its quotientspace L(Λ)ĥ + kQ ! Fortunately, one can perform a small cosmetic operation and fix this problem (cf. [LP], [ZF], [G], [DL]): For every β ∈ ∆, replace X β (z) by the relative vertex operator (called also parafermionic current) on V ⊗k
P Ψ β (z) := E − ( h β k , z) ⊗ · · · ⊗ E − ( h β k , z) k f actors X β (z) (2.6) z − h β k ε − 1 k β ⊗ · · · ⊗ z − h β k ε − 1 k β k f actors E + ( h β k , z) ⊗ · · · ⊗ E + ( h β k , z) k f actors .
Note that the coefficients of this operator lie in an appropriate completion of U(ĝ), i.e., they are infinite sums which are truncated when acting on modules from the category O. We do not have to specify here exactly which root of unity is to be taken in ε − 1 k β as long as it is the same on all tensor slots. The first and the last factor on right-hand side of the above definition, together with the k th root of formula (2.7) [GeI], ensure that
ĥ + , Ψ β (z) = ĥ − , Ψ β (z) = 0, (2.7)
i.e., the relative vertex operator indeed preserves the vacuum space L(Λ)ĥ + (recall thatĥ acts on V ⊗k P through the iterated coproduct ∆ k−1 ). But it is the term in the middle (not present in the Z-operators of Lepowsky and Primc [LP])
z − h β k ε − 1 k β ⊗ · · · ⊗ z − h β k ε − 1 k β k f actors , (2.8) which guarantees that [ρ(kα), Ψ β (z)] = 0, α, β ∈ Q, (2.9)
and therefore Ψ is well defined on the parafermionic space L(Λ)ĥ + kQ . It will be clear from the context whether Ψ acts on L(Λ)ĥ + or L(Λ)ĥ + kQ .
Remark 2.1 We should note that the nonzero numerical coset correction ε
− 1 k β ⊗ · · · ⊗ ε − 1 k β
is not present in the definition adopted in [DL]. As a result, the relative vertex operator in [DL] does not commute with ρ(kQ) but only with ρ(2kQ) and one is forced to consider a larger parafermionic space associated with the finite group Q/2kQ.
The components ("Fourier coefficients") of Ψ will be indexed in the very same fashion as the coefficients of X, but due to the term (2.8), their indices will typically be rational numbers rather than integers: For every β ∈ ∆, set
Ψ β (z) L µ (Λ)ĥ + =: m∈Z+ β,µ k ψ β (m)z −m−1 L µ (Λ)ĥ + , (2.10) where L µ (Λ)ĥ + is the µ-weight subspace of L(Λ)ĥ + , µ ∈ P (cf.
for example [DL] (6.52); note that the operator ψ β (m) is defined only on those µ-weight subspaces for which m ∈ Z + β,µ k ). This definition is designed so that the coefficients ψ β (m) can also be thought of as operators on the parafermionic space L(Λ)ĥ + kQ :
[ρ(kα), ψ β (m)] = 0, α, β ∈ Q.
(2.11)
We call −m − 1 k (rather than −m) a conformal energy of ψ β (m). This is because the Virasoro algebra generators are shifted on a coset space: In our parafermionic setting, we have to replace the grading operator D of the vertex operator algebra L(kΛ 0 ) (cf. [GeI],
Preliminaries) by D − Dĥ, where D − Dĥ, ψ β (m) = −(m + 1 k )ψ β (m), (2.12)
cf. e.g. [DL], (6.42) and (14.87). (Formula (2.12) is the parafermionic counterpart of the commutation relation [D,
x β (m)] = −mx β (m).) In other words, Dĥ L µ (Λ)ĥ + = Lĥ 0 − Λ, Λ 2k L µ (Λ)ĥ + = µ, µ 2k − Λ, Λ 2k L µ (Λ)ĥ + , (2.13)
where Lĥ 0 is the 0th mode of the Virasoro algebra associated with the vertex operator algebra U(ĥ) · v(kΛ 0 ) ∼ = M(k) (cf. [DL] (14.52) and [K] Remark 12.8 for example).
Pivotal for our arguments will be the observation that the "Fourier coefficients" of Ψ β (m) can be tied with the "Fourier coefficients" of X β (z) in a very simple way: For any given weight µ ∈ P, one has from the very definitions (2.4), (2.6) and (2.10)
π L(Λ)ĥ + · x β (m) L µ (Λ)ĥ + = const ψ β (m + β, µ k ) L µ (Λ)ĥ + ,
(2.14)
where m ∈ Z, const ∈ C × and the natural projection π L(Λ)ĥ + :
L(Λ) → L(Λ)ĥ + was introduced in (1.4), (1.5). Note that this identity does not make much sense if L(Λ)ĥ + is replaced by L(Λ)ĥ + kQ and if ψ is thought of as an operator on L µ (Λ)ĥ + kQ because π L(Λ)ĥ + kQ ·x β (m)
does not really act on the quotientspace L(Λ)ĥ + kQ . This is one of the reasons why we shall often use L(Λ)ĥ + as a mediator between the principal subspace and the parafermionic space.
Observe that one can reverse the definition (2.6) and thus "isolate" the part of the vertex operator X β (z) which acts on theĝ ⊃ĥ subspace L(Λ)ĥ + ⊂ L(Λ) :
X β (z) = E − (− h β k , z) ⊗ · · · ⊗ E − (− h β k , z) k f actors Ψ β (z) (2.15) z h β k ε 1 k β ⊗ · · · ⊗ z h β k ε 1 k β k f actors E + (− h β k , z) ⊗ · · · ⊗ E + (− h β k , z) k f actors .
Lemma 2.1 Theĝ ⊃ĥ subspace L(Λ)ĥ + (and therefore the parafermionic space L(Λ)ĥ + kQ ) is generated by the operators {ψ β (m)|β ∈ ∆} acting on the highest weight vector v(Λ), i.e.,
L(Λ)ĥ + = Span C ψ βr (m r ) · · · ψ β 1 (m 1 ) · v(Λ)|β l ∈ ∆, 1 ≤ l ≤ r . (2.16)
Proof Assume the opposite and using (2.15) and (1.2), arrive at a contradiction with the irreducibility of L(Λ) (cf. [DL] Proposition 14.9). ✷ Note by the way that for β ∈ ∆,
ρ(kβ) · v(Λ) = const x β (−1 − β, Λ )x β (−1) k−1 · v(Λ) = (2.17) = const ′ π L(Λ)ĥ + · x β (−1 − β, Λ )x β (−1) k−1 · v(Λ) = = const ′′ π L(Λ)ĥ + · x β (−1 − β, Λ ) π L(Λ)ĥ + · x β (−1) k−1 · v(Λ)
for some nonzero constants. Since v(Λ) is an eigenvector for D − Dĥ (cf. (2.13)), one can compute in a straightforward fashion from (2.12), (2.14) and (2.17) that D − Dĥ and ρ(kα), α ∈ Q, commute when acting on a highest weight vector. It follows immediately from (2.11), (2.12) and Lemma 2.1 that ρ(kα) and D − Dĥ commute on L(Λ)ĥ + . But ρ(kα) acts nontrivially only on the right factor of the decomposition (1.2), hence
[D − Dĥ, ρ(kα)] = 0, α ∈ Q.
(2.18)
Let us continue now with the parafermionic currents of higher charge (well known in the physics literature -cf. e.g. [ZF], [G]): For a given β ∈ ∆ and r ∈ Z + , we call a parafermionic current of charge r the generating function
Ψ rβ (z) := r l,p=1 l>p (z l − z p ) β,β k Ψ β (z r ) · · · Ψ β (z 1 ) z r = · · · = z 1 = z , (2.19)
where the binomial terms are to be expanded in nonnegative inegral powers of the second variable (before the expression is restricted on the hyperplane z r = · · · = z 1 = z). This generating function is well defined when acting on a highest weight module because the binomial terms cancel exactly the singularities related to the noncommutativity of the first and the last correction factors in the definition (2.6) of Ψ (cf. [GeI] (2.7), (2.8)). In other words, the above expression can be rewritten as
Ψ rβ (z) = const E − ( h β k , z) ⊗ · · · ⊗ E − ( h β k , z) k f actors r X rβ (z) (2.20) z − h β k ε − 1 k β ⊗ · · · ⊗ z − h β k ε − 1 k β k f actors r E + ( h β k , z) ⊗ · · · ⊗ E + ( h β k , z) k f actors r ,
where const ∈ C × and X rβ (z) = X β (z) r is the bosonic current of charge r from (2.5). It will be clear from the context whether Ψ acts on L(Λ)ĥ + or L(Λ)ĥ + kQ . The generating function Ψ rβ (z) is the parafermionic counterpart of X rβ (z) in the following sense: Recall that X rβ (z) is the vertex operator corresponding to the vector x β (−1) r · v(kΛ 0 ) in the vertex operator algebra L(kΛ 0 ) (cf. (2.5)). According to (2.14), the projection of this vector on the parafermionic space is
π L(Λ)ĥ + · x β (−1) r · v(kΛ 0 ) = const ψ β (−1 + β, (r − 1)β k ) · · · ψ β (−1)
r f actors ·v(kΛ 0 ), (2.21) (the nonzero const equals one if we assume without losing generality that ε is bimultiplicative and ε(α, α) = ε(α, α) 1 k = 1). But the Jacobi identity for the generalized (in the sense of [DL]) vertex operator algebra L(kΛ 0 )ĥ + kQ easily implies that Ψ rβ (z) is the vertex operator corresponding to this last vector, i.e.,
Ψ rβ (z) = Y (ψ β (−1 + β, (r − 1)β k ) · · · ψ β (−1) r f actors ·v(kΛ 0 ), z) (2.22)
(one can for example use repeatedly Proposition 14.29 [DL] which under the above assumptions for ε is still true even for our slightly modified Ψ, provided the notations are appropriately adjusted; be aware that the above Y (·, z) is not the same as Y (·, z) in (2.5) since these are vertex operators in two different vertex operator algebras).
In view of the correspondence between parafermionic and bosonic currents, the most natural generalization of the definition (2.10) of "Fourier coefficients" for higher-charge parafermionic currents is (cf. [GeI] (3.7))
Ψ rβ (z) L µ (Λ)ĥ + =: m∈Z+ rβ,µ k ψ rβ (m)z −m−r L µ (Λ)ĥ + , (2.23)
where β ∈ ∆, r ∈ Z + and L µ (Λ)ĥ + is as always the µ-weight subspace of L(Λ)ĥ + , µ ∈ P.
Note that ψ rβ (m) is a well defined operator on L µ (Λ)ĥ + kQ with µ, such that m ∈ Z + rβ,µ k , because ψ rβ (m) commutes with the action ρ of kQ. Moreover, by (2.12) and the definitions (2.19), (2.23), the conformal energy of ψ rβ (m) is
D − Dĥ, ψ rβ (m) = −(m + 1 k (r + β, β r 2 )ψ rβ (m) = (2.24) = −(m + r 2 k )ψ rβ (m).
Analogously to the charge-one situation (2.14), one can find for every operator x rβ (m) its parafermionic counterpart (acting on a given weight subspace of L(Λ)ĥ + ) by composing it with the projection π L(Λ)ĥ + :
π L(Λ)ĥ + · x rβ (m) L µ (Λ)ĥ + = const ψ rβ (m + rβ, µ k ) L µ (Λ)ĥ + ,(2.25)
where const ∈ C × .
We are now in position to formulate a key relation which -simple and beautiful as it is -explains both the similarity and the difference between the structure of the parafermionic space L(Λ)ĥ + kQ and the principal space W (Λ).
Proposition 2.1 For every β ∈ ∆ and r ∈ Z + , 1 ≤ r ≤ k, one has Ψ rβ (z) = const ρ(kβ)Ψ −(k−r)β (z),
(2.26)
const ∈ C × , (cf. (1.6)), i.e.,
Ψ rβ (z) = const Ψ −(k−r)β (z),(2.27)
const ∈ C × , as operators on L(Λ)ĥ + kQ .
Proof Follows from a direct computation employing the commutation relation [GeI] (2.7) (in complete analogy with [LP] Theorem 5.6 for example). ✷ This in particular implies that products of coefficients of parafermionic currents associated with positive roots are enough for generating the parafermionic space (when acting on the vacuum vector; cf. Lemma 2.1 and (2.19)). In other words, in complete analogy with the principal (sub)space
W (Λ) := Span C x βr (m r ) · · · x β 1 (m 1 ) · v(Λ)|β l ∈ ∆ + , 1 ≤ l ≤ r , (2.28) one has L(Λ)ĥ + = ρ(kQ) · Span C ψ βr (m r ) · · · ψ β 1 (m 1 ) · v(Λ)|β l ∈ ∆ + , 1 ≤ l ≤ r (2.29)
and hence L(Λ)ĥ + kQ = Span C ψ βr (m r ) · · · ψ β 1 (m 1 ) · v(Λ)|β l ∈ ∆ + , 1 ≤ l ≤ r .
(2.30)
Despite this similarity, there is an important difference between the parafermionic space and the principal (sub)space which is encoded in the particular case of Proposition 2.1 for r = k: Ψ kβ (z) = const ρ(kβ), (2.31)
const ∈ C × , β ∈ ∆, i.e., Ψ kβ (z) = const ∈ C × (2.32) when acting on the parafermionic space L(Λ)ĥ + kQ . In other words, one component of Ψ kβ (z) acts as a nonzero constant on the parafermionic space L(Λ)ĥ + kQ and all the other components vanish on it. Recall that on the principal subspace W (Λ), we have the constraint X (k+1)β (z) = 0, which of course implies Ψ (k+1)β (z) = 0. In contrast to this mutually shared constraint, the above constraint (2.32) is a purely parafermionic phenomenon with no analog in the principal subspace. It tells us that the maximal allowed charge of the (defined below) quasi-particles generating the parafermionic space is k − 1. This is also the maximal allowed charge of the quasi-particles generating a principal subspace at level k − 1. So, not surprisingly, the constructed below basis for a parafermionic space at level k will be combinatorially the same as a [GeI] basis for a principal space at level k − 1.
For the purpose of building a basis for L(Λ)ĥ + from the known already basis of W (Λ), it is very natural to employ not only the parafermionic counterpart Ψ rβ (z) of the bosonic current X rβ (z), but also the parafermionic counterpart of a whole product of bosonic currents (with different variables) which differs from the product of the respective parafermionic counterparts: For given roots β r , . . . , β 1 ∈ ∆ and corresponding sequence of charges n r , . . . , n 1 ⊂ Z + , set Ψ nrβr,...,n 1 β 1 (z r , . . . , z 1 ) := (2.33)
:= r l,p=1 l>p (z l − z p ) n l β l ,np βp k Ψ nrβr (z r ) · · · Ψ n 1 β 1 (z 1 ),
where the binomial terms are to be expanded as usual in nonnegative integral powers of the second variable. Just like in (2.19), they are inserted in order to ensure that the composition π L(Λ)ĥ + · X nrβr (z r ) · · · X n 1 β 1 (z 1 ) equals (up a nonzero constant) Ψ nrβr,...,n 1 β 1 (z r , . . . z 1 ). In other words, analogously to (2.20), one can show that Ψ nrβr,...,n 1 β 1 (z r , . . . z 1 ) = const
r l=1 E − ( h β l k , z) ⊗ · · · ⊗ E − ( h β l k , z) k f actors n l (2.34) X nrβr (z r ) · · · X n 1 β 1 (z 1 ) r l=1 z − h β l k l ε − 1 k β l ⊗ · · · ⊗ z − h β l k l ε − 1 k β l k f actors n l r l=1 E + ( h β l k , z) ⊗ · · · ⊗ E + ( h β l k , z) k f actors n l ,
where const ∈ C × . We call Ψ nrβr,...,n 1 β 1 (z r , . . . z 1 ) a normalized product of the parafermionic currents Ψ nrβr (z r ), . . . , Ψ n 1 β 1 (z 1 ). The last equality guarantees that a normalized product is invariant (up to a nonzero constant) under the permutation of two adjacent variables z l , z l+1 and the corresponding indices n l β l , n l+1 β l+1 as long as β l = β l+1 (this is not true for the usual product of parafermionic currents). Note that for any r ∈ Z + , one has Ψ β,...,β r entries (z, . . . , z) = Ψ rβ (z).
(2.35) Generalizing (2.23), one defines the "Fourier coefficients" of a normalized product as follows Ψ nrβr,...,n 1 β 1 (z r , . . . z 1 ) L µ (Λ)ĥ + =:
(2.36) =: mr∈Z+ nr βr ,µ k · · · m 1 ∈Z+ n 1 β 1 ,µ k ψ nrβr,...,n 1 β 1 (m r , . . . , m 1 )
L µ (Λ)ĥ + z −mr −nr r · · · z −m 1 −n 1 1 ,
where β ∈ ∆, r ∈ Z + and L µ (Λ)ĥ + is as always the µ-weight subspace of L(Λ)ĥ + , µ ∈ P. We call ψ nrβr,...,n 1 β 1 (m r , . . . , m 1 ) a normalized ψ-monomial or simply a normalized monomial, as opposed to the usual (ψ-) monomial ψ nrβr (m r ) · · · ψ n 1 β 1 (m 1 ). According to the very definition (2.33), a normalized monomial is a linear combination of usual monomials and vice versa (cf. (3.14), (3.15) below; no need to mention that our monomials are always acting on a designated space and the linear combinations in question are truncated, i.e., finite). The conformal energy of a normalized monomial is given by the following corollary of (2.24) and (2. (m l + n 2 l k + n l β l , p<l n p β p k )ψ nrβr,...,n 1 β 1 (m r , . . . , m 1 ).
As expected from the preceding discussion, the normalized monomials acting on L(Λ)ĥ + are indeed the parafermionic counterparts of monomials of type x nrβr (m 1 ) · · · x n 1 β 1 (m 1 ): One has from (2.34) and (2.36) that π L(Λ)ĥ + · x nrβr (m r ) · · · x n 1 β 1 (m 1 ) L µ (Λ)ĥ + = (2.38) = const ψ nrβr,...,n 1 β 1 (m r + n r β r , µ k , . . . , m 1 +
n 1 β 1 , µ k ) L µ (Λ)ĥ + ,
where const ∈ C × (cf. (2.25)). Since the usual ψ-monomials are linear combinations of normalized ψ-monomials, one can now conclude from (2.29) and (2.30) that (1.4) and (1.8)). This close relationship with the principal subspace will be instrumental in the subsequent arguments.
L(Λ)ĥ + = ρ(kQ) · π L(Λ)ĥ + · W (Λ). (2.39) and L(Λ)ĥ + kQ = π L(Λ)ĥ + kQ · W (Λ) (2.40) (cf.
Quasiparticles
Recall that in the context of the principal subspace [GeI], we restricted ourselves to vertex operators associated with the simple roots because those are perfectly enough to generate the whole space when acting on a highest weight vector v(Λ) (Lemma 3.1 [GeI]). This property is inherited in the parafermionic picture. At our convenience, we shall often be writing the ψ-monomials acting on L(Λ)ĥ + as parafermionic counterparts (cf. (2.25) or (2.38)):
Lemma 3.1 One has
L(Λ)ĥ + = Span C ρ(kα) · π L(Λ)ĥ + · x nrβr (m r ) · · · π L(Λ)ĥ + · x n 1 β 1 (m 1 ) · v(Λ) (3.1)
|x nrβr (m r ) · · · x n 1 β 1 (m 1 ) a monomial from U, α ∈ Q} .
Equivalently,
L(Λ)ĥ + kQ = Span C ψ nrβr (m r + n r β r , Λ + r−1 p=1 n p β p k ) · · · ψ n 1 β 1 (m 1 + n 1 β 1 , Λ k ) · v(Λ)
(3.2) |x nrβr (m r ) · · · x n 1 β 1 (m 1 ) a monomial from U := U(n αn ) · · · U(n α 1 )} .
Proof Follows immediately from Lemma 3.1 [GeI], the "surjectivity" (2.39) of the projection π L(Λ)ĥ + and the fact that normalized monomials are linear combinations of usual monomials of the same structure (cf. (2.25) and (2.38)). ✷
In other words, we are entitled again to dismiss all the nonsimple roots and fix an order in the set of simple roots.
Definition 3.1 For every simple root α i , 1 ≤ i ≤ n, and positive integer r, we shall say that the operator ψ rα i (m) from (2.23) represents a ψ-quasi-particle of color i, charge r and energy −m − r 2 k . If confusion with x-quasi-particles can not arise, we shall skip the prefix ψ and just talk about quasi-particles; it will be clear from the context whether ψ is an operator on L(Λ)ĥ + or L(Λ)ĥ + kQ . Abusing language, we shall say that ψ α i (m) is from π L(Λ)ĥ + · U(n α i ) and that the ψ-monomials (normalized or not) considered in Lemma 3.1 (3.1) are from π L(Λ)ĥ + · U. Be aware that the operator ψ rα i (m) is defined only on those µ-weight subspaces of L(Λ)ĥ + (resp., L(Λ)ĥ + ), for which m ∈ Z + rα i ,µ k .
Starting from here, we shall mostly work with (ψ-) monomials from π L(Λ)ĥ + kQ · U.
In complete analogy with the x-monomials from U ( [GeI] Section 3), we shall say that a ψ-monomial from π L(Λ)ĥ + · U ψ n r
(1) n ,n αn (m r (1) n ,n ) · · · ψ n 1,n αn (m 1,n ) · · · · · · ψ n r (1) 1 ,1 α 1 (m r (1) 1 ,1 ) · · · ψ n 1,1 α 1 (m 1,1 ) (3.3) and its normalized counterpart ψ n r (1) n ,n αn,...n 1,n αn;......;n r (1) 1 ,1 α 1 ,...,n 1,1 α 1 (m r (1) n ,n , . . . , m 1,n ; . . . . . . ; m r (1) 1 ,1 , . . . , m 1,1 ), (3.4) are of color-charge-type (n r (1) n ,n , . . . , n 1,n ; . . . ; n r (1) 1 ,1 , . . . , n 1,1 ), (3.5)
where 0 < n r (1) ,i ≤ · · · ≤ n 2,i ≤ n 1,i ≤ K,
r (1) i p=1 n p,i = r i , 1 ≤ i ≤ n,
of color-dual-charge-type (r (1) n , . . . , r (K) n ; . . . ; r
(1) 1 , . . . , r
(K) 1 ), (3.6) where r (1) i ≥ r (2) i ≥ · · · ≥ r (K) i ≥ 0, K t=1 r (t) i = r i , K ∈ Z + , 1 ≤ i ≤ n,
and of color-type (r n ; . . . ; r 1 ). We shall also say that the corresponding generating functions
Ψ n r (1) n ,n αn (z n r (1) n ,n ) · · · Ψ n 1,1 α 1 (z 1,1 ) (3.7)
and Ψ n r (1) n ,n αn,...,n 1,1 α 1 (z n r (1) n ,n , . . . , z 1,1 ) (3.8) are of the above color-charge-type, color-dual-charge-type and color-type. We would like to remind that no ψ-quasi-particles of charge greater than k − 1 will appear in our parafermionic space at level k: the constraint (2.31), (2.32) asserts in particular that for every color i, 1 ≤ i ≤ n, one has Ψ kα i (z) = const ρ(kα i ), const ∈ C × (3.9)
as an operator on L(Λ)ĥ + , i.e.,
Ψ kα i (z) = const ∈ C × (3.10)
as an operator on the parafermionic space L(Λ)ĥ + kQ . Note a curious corollary of this constraint: The inverse of the identity (2.20) implies that up to an invertible operator, every x-quasi-particle of charge k and color i acts on L(Λ) as an operator from U(ĥ), i.e.,
x kα i (m) = ρ(kα i ) · h, h ∈ U(ĥ),
(3.11) (recall that U(ĥ) acts through the iterated coproduct ∆ k−1 ). Since our ambition is to exploit results from [GeI], it is really more convenient to work with theĝ ⊃ĥ subspace L(Λ)ĥ + rather than with the parafermionic space L(Λ)ĥ + kQ : the ψ-monomial π L(Λ)ĥ + · x nrβr (m r ) · · · π L(Λ)ĥ + · x n 1 β 1 (m 1 ) and its normalized counterpart are of given color-charge-type, color-dual-charge-type and color-type if their x-counterpart x nrβr (m r ) · · · x n 1 β 1 (m 1 ) is of these types. (Strictly speaking, the ψ-monomials acting on L(Λ)ĥ + kQ do not have x-counterparts because π L(Λ)ĥ + kQ · x rβ (m) is not an operator on L(Λ)ĥ + kQ .) Note that given a ψ-monomial from π L(Λ)ĥ + · U, we do not exactly know the quasi-particle energies of its x-counterpart unless we specify the weight subspace on which the ψ-monomial acts (cf. (2.25) and (2.38)). If one discusses only the type of a monomial, this is not necessary.
In the same flow of thoughts, it is clear how to convey the linear ordering "<" and the partial ordering "≺" from the set of x-monomials of a given color-type (r n ; . . . ; r 1 ) (cf. [GeI] Section 3) to the set of ψ-monomials (acting on L(Λ)ĥ + ) of a given color-type: Set
π L(Λ)ĥ + ·x nrβr (m r ) · · · π L(Λ)ĥ + ·x n 1 β 1 (m 1 ) < π L(Λ)ĥ + ·x n ′ r βr (m ′ r ) · · · π L(Λ)ĥ + ·x n ′ 1 β 1 (m ′ 1 ) (3.12) and π L(Λ)ĥ + · x nrβr (m r ) · · · x n 1 β 1 (m 1 ) < π L(Λ)ĥ + · x n ′ r βr (m ′ r ) · · · x n ′ 1 β 1 (m ′ 1 ) (3.13) if
x nrβr (m r ) · · · x n 1 β 1 (m 1 ) < x n ′ r βr (m ′ r ) · · · x n ′ 1 β 1 (m ′ 1 ). Define analogously "≺" for ψ-monomials of a given color-type. These definitions can obviously be rewritten, replacing the projections with the corresponding ψ-operators according to (2.25) and (2.38). Moreover, we do not have to specify the weight subspace on which the ψ-monomials act: Recall from [GeI] Section 3 that the quassi-particle energies affect the ordering only if the color-charge-types (and for that matter, the color-dualcharge-types) are the same, in which case the shift of the indices of the corresponding x-monomials will be the same for the two compared ψ-monomials. Keep in mind that "≺" implies "<" but not vice versa.
We are now in position to explain why working with usual or with normalized ψmonomials is essentially the same. Roughly speaking, the transformation matrix between them is "upper triangular". More precisely, from the definitions (2.33), (2.23),(2.36) and the identities (2.25), (2.38), we obtain for a given (usual) monomial from π L(Λ)ĥ + · U that π L(Λ)ĥ + · x nrβr (m r ) · · · π L(Λ)ĥ + · x n 1 β 1 (m 1 ) = (3.14)
= const π L(Λ)ĥ + · x nrβr (m r ) · · · x n 1 β 1 (m 1 ) + a linear combination of normalized monomials of the same color-charge--type and greater in the ordering "≺",
where const ∈ C × . Conversely, for any given normalized monomial from π L(Λ)ĥ + · U, we have π L(Λ)ĥ + · x nrβr (m r ) · · · x n 1 β 1 (m 1 ) = (3.15) = const π L(Λ)ĥ + · x nrβr (m r ) · · · π L(Λ)ĥ + · x n 1 β 1 (m 1 ) + a linear combination of (usual) monomials of the same color--charge-type and greater in the ordering "≺", const ∈ C × .
Quasi-particle basis for the parafermionic space
Recall that we are working with level k ∈ Z + , k ≥ 2, and a highest weight
Λ := k 0Λ0 + k jΛj = kΛ 0 + Λ, where Λ := k j Λ j ,(4.1)
for some j, 1 ≤ j ≤ n; k 0 , k j ∈ N and k 0 + k j = k (cf. Section 1). Our level k spaces are all realized in the tensor product of level one modules given by the homogeneous vertex operator construction ( [GeI] Section 2). In order to treat the level one ingredients on equal footing, we introduced
j t := 0 for 0 < t ≤ k 0 j for k 0 < t ≤ k = k 0 + k j (4.2)
and then the highest weight vector of all the modules under consideration L(Λ), W (Λ), L(Λ)ĥ + and L(Λ)ĥ
+ kQ was defined as v(Λ) := v(Λ j k ) ⊗ · · · ⊗ v(Λ j 1 ) = (4.3) = v(Λ j ) ⊗ · · · ⊗ v(Λ j ) k j f actors ⊗ v(Λ 0 ) ⊗ · · · ⊗ v(Λ 0 ) k 0 f actors ,
where v(Λ jt ) is the highest weight vector of the level one module in the t th tensor slot (counted from right to left). The insightful reader has probably guessed already what is our basis-candidate for the level k parafermionic space L(Λ)ĥ + kQ -the most intuitive choice is of course the π L(Λ)ĥ + kQ projection of this particular subset of the basis for W (Λ) (from [GeI] Section 5), which contains only vectors generated by monomials with no quasi-particles of charge k. We shall rewrite for completeness the full definition of the prototype -the basis-generating set B W (Λ) from Definition 5.1 [GeI] -with the charge of the quasi-particles bounded by k − 1 rather than k, and call it B (k−1) W (Λ) . We shall not comment here on the origin and naturality of this incomprehensible at first sight affluence of parameters. The frustrated readers are referred to the Introduction of [GeI] for simple particular cases and to the Introduction and Section 5 here for much easier to grasp character formulas associated with these bases. We only emphasize that the basis-generating set of monomials is a disjoint union along color-charge-types (3.5) or, equivalently, along the corresponding color-dual-charge-types (3.6), with upper bound for the charges K := k − 1 (as opposed to K = k in the principal subspace picture):
Definition 4.1 Fix a highest weightΛ as in (4.1). Set
B (k−1) W (Λ) := 0≤n r
(1) n ,n ≤···≤n 1,n ≤k−1 ········· 0≤n r (1) 1 ,1
≤···≤n 1,1 ≤k−1 or, equivalently, r (1) n ≥···≥r (k−1) n ≥0 ········· r (1) 1 ≥···≥r (k−1) 1 ≥0 (4.4) x n r
(1) n ,n αn (m r (1) n ,n ) · · · x n 1,n αn (m 1,n ) · · · · · · x n r (1) 1 ,1 α 1 (m r (1) 1 ,1 ) · · · x n 1,1 α 1 (m 1,1 )
m p,i ∈ Z, 1 ≤ i ≤ n, 1 ≤ p ≤ r (1) i ; m p,i ≤ r (1) i−1 q=1 min {n p,i , n q,i−1 } − n p,i t=1 δ i,jt − p>p ′ >0 2min {n p,i , n p ′ ,i } − n p,i ; m p+1,i ≤ m p,i − 2n p,i for n p+1,i = n p,i ,
where r (1) 0 := 0 and j t was introduced in (4.2). Define B L(Λ)ĥ + := π L(Λ)ĥ + · x nrβr (m r ) · · · π L(Λ)ĥ + · x n 1 β 1 (m 1 ) (4.5)
x nrβr (m r ) · · · x n 1 β 1 (m 1 ) ∈ B (k−1) W (Λ) and B L(Λ)ĥ + kQ := ψ nrβr (m r + n r β r , Λ + r−1 p=1 n p β p k ) · · · ψ n 1 β 1 (m 1 + n 1 β 1 , Λ k ) (4.6)
x nrβr (m r ) · · · x n 1 β 1 (m 1 ) ∈ B Example 4.1 Consider g = sl(3), i.e., n = 2 and the vacuum highest weightΛ = 2Λ 0 at level k = 2. Denote for brevity the monomial
π L(2Λ 0 )ĥ + · x α 2 (s) · · · π L(2Λ 0 )ĥ + · x α 1 (t)
by (s α 2 . . . t α 1 ). For the first few energy levels (the eigenvalues of the scaling operator D − Dĥ under the adjoint action), Table 1 in the Appendix lists the elements of B L(2Λ 0 )ĥ + of color-types (1; 2) and (2; 2). This
∼ = B W (Λ 0 )
) with the entries in the column "energy" shifted according to (2.12) and (2.14).
Example 4.2 Let again g = sl(3) but consider the vacuum highest weightΛ = 3Λ 0 at level k = 3. Similarly to the previous example, denote the quasi-particle monomial
π L(3Λ 0 )ĥ + · x s ′ α 2 (s) · · · π L(3Λ 0 )ĥ + · x t ′ α 1 (t) by (s s ′ α 2 . . . t t ′ α 1 ).
For the first few energy levels, Table 2 in the Appendix lists the elements of B L(3Λ 0 )ĥ + of color-types (1; 2) and (2; 2). This Table is a copy of Table 2 [GeI], Appendix (listing the corresponding elements of B
(2) W (3Λ 0 ) ∼ = B W (2Λ 0 )
) with the entries in the column "energy" shifted according to (2.24) and (2.25).
We begin as usual with a proof of the spanning property of our basis-candidate.
Theorem 4.1 LetΛ be a highest weight as in (4.1) and v(Λ) be the corresponding highest weight vector (4.3). Then the set ρ
(kα) · b · v(Λ) b ∈ B L(Λ)ĥ + , α ∈ Q spans theĝ ⊃ĥ subspace L(Λ)ĥ + . Equivalently, the set b · v(Λ) b ∈ B L(Λ)ĥ + kQ spans the parafermionic space L(Λ)ĥ + kQ .
Proof We shall prove the statement for L(Λ)ĥ + (the statement for L(Λ)ĥ + kQ follows immediately from (2.25) and the definition (1.7)). In view of Lemma 3.1, it suffices to show that every vector b · v(Λ), b an usual ψ-monomial from π L(Λ)ĥ + · U, is a linear combination of vectors from the proposed set.
Suppose that b ∈ ρ(kQ) · B L(Λ)ĥ + . Due to the constraint (3.9), the (nonzero) quasiparticles of charge k in b (if any!) commute with all the other quasi-particles and can be moved ("exiled") all the way to the left. Using the "upper triangular" transformation (3.14), normalize the remaining ψ-monomial (which is by assumption ∈ B L(Λ)ĥ + ) and apply Remark 5.1 [GeI] -the strong form of the spanning Theorem 5.1 [GeI] -for the obtained x-monomials (new quasi-particles of charge k might be generated in the process!). Then switch back from normalized to usual ψ-monomials using the inverse transformation (3.15) and return the "exiled" quasi-particles of charge k to their old places. The newly obtained (usual) ψ-monomials from π L(Λ)ĥ + · U have the same color-type and index-sum as b and moreover, since b ∈ ρ(kQ) · B L(Λ)ĥ + , they are all greater than b in the ordering ′′ ≺" (by Remark 5.1 [GeI]). Since there are only finitely many such ψ-monomials which do not annihilate v(Λ), the statement follows by induction. ✷ Remark 4.1 The proof implies that the exact analog of Remark 5.1 [GeI] in the current setting is also true: Every vector b · v(Λ), b a usual ψ-monomial from π L(Λ)ĥ + · U, b ∈ ρ(kQ) · B L(Λ)ĥ + , is a linear combination of vectors of the form b ′ · v(Λ), b ′ ∈ ρ(kQ) · B L(Λ)ĥ + , b ′ ≻ b, with b ′ and b having the same color-type and total index-sum.
We proceed with the independence. For the familiar highest weightΛ = k t=1Λ jt (cf.
(4.1) and (4.2)), defineΛ
:= k−1 t=1Λ jt =Λ −Λ j k , (4.7)
i.e.,Λ = (k − 1)Λ 0 +Λ, whereΛ = Λ − Λ j k or equivalently,Λ = (k j − 1)Λ j ∈ P if k j > 0 andΛ = 0 otherwise. Note thatΛ is of the same type (4.1) asΛ (i.e., the results in [GeI] hold forΛ) and moreover, by the very definition (4.3), one has for the corresponding
highest weight vectors v(Λ) = v(Λ j k ) ⊗ v(Λ)
. Summoning the projection π U (ĥ − )·v(Λ j ) from [GeI] (2.12), one easily checks that
π U (ĥ − )·v(Λ j k ) ⊗ id ⊗ · · · ⊗ id k−1 f actors · B W (Λ) · v(Λ) = (4.8) = π U (ĥ − )·v(Λ j k ) ⊗ id ⊗ · · · ⊗ id k−1 f actors · B (k−1) W (Λ) · v(Λ) = v(Λ j k ) ⊗ B W (Λ) · v(Λ),
(cf. Definition 4.1 and [GeI] Definition 5.1). This is because any nontrivial x-quasi-particle action on the leftmost tensor factor vanishes under the above projection due to the term e β in the quasi-particle (cf. (2.1), (2.2)). But x-quasi-particles of charge k are zero unless they act on all the k tensor factors.
In the independence argument below, we shall employ the independence of the vectors from the set B W (Λ) · v(Λ) (rather than B W (Λ) · v(Λ)!) which was proven in Theorem 5.2 [GeI].
Theorem 4.2 LetΛ be again a highest weight as in (4.1) and v(Λ) be the corresponding highest weight vector (4.3). Then the set ρ(kα) · b · v(Λ) b ∈ B L(Λ)ĥ + , α ∈ Q is indeed a basis for theĝ ⊃ĥ subspace L(Λ)ĥ + . Equivalently, the set b · v(Λ) b ∈ B L(Λ)ĥ + kQ is a basis for the parafermionic space L(Λ)ĥ + kQ .
Proof It suffices to prove the independence of the vectors in the set ρ(kQ) · B L(Λ)ĥ + · v(Λ).
Let us first show that it would follow from the independence of the vectors in the set
v(Λ j k ) ⊗ B W (Λ) · v(Λ) over the ring U(ĥ − ) (recall that U(ĥ) acts through the (k − 1)-iterate ∆ k−1 of the standard coproduct ∆).
Suppose that there is a (nontrivial irreducible) linear relation among vectors from ρ(kQ) · B L(Λ)ĥ + · v(Λ). Without loss of generality, one can assume that the relation does not contain factors from ρ(kQ − ) (recall from [GeI] Preliminaries that Q − ⊂ Q is the semigroup generated by the negatives of the simple roots) and that at least one vector is from B L(Λ)ĥ + · v(Λ). An induction on the number of monomials involved shows that this can be easily achieved by multiplying the relation with appropriate invertible operators from ρ(kQ).
Normalize the ψ-monomials using the "upper triangular" relation (3.14): Observe that by (2.34), a vector π L(Λ)ĥ + · b · v(Λ), b a monomial from U, equals up to a nonzero constant b· v(Λ), plus a linear combination of vectors of the form h ′ · b ′ · v(Λ), where h ′ ∈ U(ĥ − ), b ′ is a monomial from U of the same color-type but of less index-sum than b, and in addition, b ′ ≻ b. Implement the strong form of the spanning Theorem 5.1 [GeI] (cf. Remark 5.1 [GeI]) for the obtained x-monomials and apply the projection
π U (ĥ − )·v(Λ j k ) ⊗ id ⊗ · · · ⊗ id k−1 f actors (4.9)
to the relation. (Note that it annihilates all the vectors containing a factor ρ(kα), α ∈ Q.) Due to (4.8), the result is a linear relation among vectors from v(Λ j k ) ⊗ B W (Λ) · v(Λ), with coefficients in U(ĥ − ). The relation is nontrivial: Among all the vectors from B L(Λ)ĥ + · v(Λ) in our initial relation (we have seen that without loss of generality, there is always at least one such vector), let b · v(Λ) be the one whose ψ-monomial b is smallest in the linear ordering "<". Then the projection (4.9) of the x-counterpart of b acting on v(Λ), is nonzero (by (4.8) and Theorem 5.2 [GeI]) and is present in the final relation, because "≺" implies "<".
In order to complete the proof, it remains to show that a linear relation among vectors from v(Λ j k ) ⊗ B W (Λ) · v(Λ), with coefficients in U(ĥ − ) is impossible. We shall reach a contradiction with the independence of the vectors from B W (Λ) · v(Λ) by restricting the relation to an appropriate homogeneous subspace where the action of the polynomial algebra U(ĥ − ) (given by ∆ k−1 ) is "squeezed" to the leftmost tensor slot: Suppose that m ∈ N is the maximal degree of the polynomials from U(ĥ − ) which are coefficients in our relation. Apply the projection
π U m (ĥ − )·v(Λ j k ) ⊗ id ⊗ · · · ⊗ id k−1 f actors ,
(4.10)
where π U m (ĥ − )·v(Λ j ) was introduced in (2.12) [GeI]. Since the vectors from the set v(Λ j k ) ⊗ B W (Λ) ·v(Λ) are nonzero (Theorem 5.2 [GeI]), the number m is maximal with the property that the projection (4.10) does not annihilate all the vectors in the relation. The reason we would like to apply the projection (4.10) to our relation is quite transparent: It is easily seen from the explicit form of the iterated coproduct ∆ k−1 that we shall thus obtain a nontrivial relation among vectors of the form h (Λ) . But this contradicts the independence of the vectors from the set B W (Λ) · v(Λ) which was proven in Theorem 5.2 [GeI]. ✷
· v(Λ j k ) ⊗ b · v(Λ), where h ∈ U m (ĥ − ) and b ∈ B W
Character formulas
We can finally reward ourselves and enjoy the entertaining world of characters associated with the above bases.
Consider an arbitrary monomial π L(Λ)ĥ + · x nrβr (m r ) · · · π L(Λ)ĥ + · x n 1 β 1 (m 1 ) (5.1) from π L(Λ)ĥ + · U of color-dual-charge-type (r (1) n , . . . , r (k−1) n ; . . . ; r
(1) 1 , . . . , r
(k−1) 1 ), (5.2) where r (1) i ≥ r (2) i ≥ · · · ≥ r (k−1) i ≥ 0, k−1 t=1 r (t) i = r i , 1 ≤ i ≤ n,
and hence, of color-type (r n ; . . . ; r 1 ) (cf. (3.6)). Set p (s) i to be the number of quasiparticles of color i and charge s in our monomial, i.e., p (s)
i := r (s) i − r (s+1) i , 1 ≤ s ≤ k − 2 and p (k−1) i := r (k−1) i , hence r i = k−1 s=1 sp (s) i , i = 1, .
. . , n. Then according to (2.37), (2.38) and (3.14), the D − Dĥ-eigenvalue of the vector
π L(Λ)ĥ + · x nrβr (m r ) · · · π L(Λ)ĥ + · x n 1 β 1 (m 1 ) · v(Λ) (5.3) equals − r l=1 m l − 1 2k n i=1 ( k−1 s=1 sp (s) i )α i , n i=1 ( k−1 s=1 sp (s) i )α i − 1 k n i=1 ( k−1 s=1 sp (s) i )α i , Λ = (5.4) = − r l=1 m l − 1 2k n i=1 r i α i , n i=1 r i α i − 1 k n i=1 r i α i , Λ = = − r l=1 m l − 1 2
l,m=1,...,n s,t=1,...,k−1
A lm C st p (s) l p (t) m − k j k k−1 s=1 sp (s) j = = − r l=1 m l − 1 k (r 2 1 + · · · + r 2 n − r 1 r 2 − · · · − r n−1 r n ) − k j k r j ,
where (A lm ) n l,m=1 is the Cartan matrix of g and C st := st k , 1 ≤ s, t ≤ k − 1. In view of Definition 4.1, (4.7) and Theorem 4.2, this formula provides the correction term needed to obtain Tr q D−Dĥ L(Λ)ĥ
+ kQ from Tr q D B (k−1) W (Λ) · v(Λ) = Tr q D W (Λ)
(cf. also (4.8)).
Recall from [GeI]
A lm B st p (s) l p (t) m n i=1 k−1 s=1 (q) p (s) i q k−1 s=k 0 +1 (s−k 0 )p (s) j
where B st := min{s, t}, 1 ≤ s, t ≤ k − 1. But one can immediately check that
B st − C st = A (−1) st , 1 ≤ s, t ≤ k − 1,(5.A lm A (−1) st p (s) l p (t) m n i=1 k−1 s=1 (q) p (s) i q k−1 s=k 0 +1 (s−k 0 )p (s) j − k j k k−1 s=1 sp (s) j .
If we restrict our attention to the vacuum module (Λ = kΛ 0 , that is k 0 = k and k j = 0), this formula is the sl(n + 1, C)-case of the Kuniba-Nakanishi-Suzuki conjecture [KNS] (a dilogarithm proof of this particular case was announced in [Kir]). Note that for a given weight µ ∈ P, such that µ − Λ ∈ Q, one obtains the (D − Dĥ)character of the weight subspace L µ (Λ)ĥ
+ kQ ∼ = L µ (Λ)ĥ + if the additional restriction µ = Λ + n i=1 r i α i mod kQ = Λ + n i=1 ( k−1 s=1 sp (s) i )α i mod kQ (5.8)
is imposed in the above formula (5.7). As far as the standard q-character is concerned, we know from (2.13) and (1.2) that for µ ∈ P,
Tr q D L µ (Λ) = q µ,µ 2k − Λ,Λ 2k (q) n ∞ Tr q D−Dĥ L µ (Λ)ĥ + kQ ,(5.9)
where (q) ∞ := l≥0 (1 − q l ). On the other hand, one defines the string function cΛ µ (q) as follows (cf. [KP] or [K] Section 12.7; departing from the tradition, we shall use subscript µ ∈ P, rather thanμ = kΛ 0 + µ):
cΛ µ (q) := q Λ+ρ,Λ+ρ 2(k+h ∨ ) − ρ,ρ 2h ∨ − µ,µ 2k
Tr q D L µ (Λ)
= (5.10) = q Λ+2ρ,Λ 2(k+h ∨ ) − 1 24 (dim g)k k+h ∨ − µ,µ 2k
Tr q D L µ (Λ)
, (the last identity follows from the strange formula of Freudenthal-de Vries). Hence, from (5.9),
cΛ µ (q) = q Λ+ρ,Λ+ρ 2(k+h ∨ ) − ρ,ρ 2h ∨ − Λ,Λ 2k (q) n ∞ Tr q D−Dĥ L µ (Λ)ĥ + kQ .
(5.11)
Remark 5.1 Observe that when the level k equals 2, formulas (5.11) and (5.9) (with the restriction (5.8)) provide combinatorial expression for every string function cΛ µ (q) corresponding to a generic dominant integral weightΛ =Λ i +Λ j , 0 ≤ i, j ≤ n: Due to the cyclic automorphism (of order n + 1) of the Dynkin diagram ofĝ, the generic string function equals (up to a power of q) a string function of the type cΛ 0 +Λ j µ (q), 0 ≤ j ≤ n, considered above.
Armed with an explicit expression for the string functions, we are only a step away from writing the character of the whole standard module: Since ρ(kα) acts nontrivially only on the right factor of the decomposition (1.2), one concludes from (2.13), (2.18) that where Θ µ (q, y) is the classical theta function of degree k (cf. e.g. [K] Chapters 12, 13):
Θ µ (q, y) := q µ,µ 2k α∈Q q k 2 α,α + α,µ n i=1 y Λ i ,kα+µ i = (5.14)
= γ∈Q+ µ k q k 2 γ,γ n i=1 y k Λ i ,γ i .
Formula (5.13) is of course the familiar expression for the normalized character of standard module in terms of string functions and theta functions (cf. [K] (12.7.12)). Note that the explicit combinatorial formula for cΛ µ is given by (5.11) and (5.7) with the additional restriction (5.8) imposed.
If we need only ch L(Λ), we can avoid any reference to D − Dĥ-characters and use directly the D-character (5.5) of the principal subspace W (Λ) (copied from [GeI] (5.27)) as well as the above theta function which incorporates as usual the contributions of the operators ρ(kα), α ∈ Q. This is because by its very definition (1.4), (1.5), the projection π L(Λ)ĥ + is D-invariant. In other words, if we setμ := µ − Λ j k , µ ∈ P, like in (4.7) and denote by Q (+) the monoid (with 0) generated by positive roots, we have from ( i . Recall thatΛ was defined in (4.1), (A lm ) n l,m=1 is the Cartan matrix of g and B st := min{s, t}, 1 ≤ s, t ≤ k − 1.
This last character formula corresponds also to a semiinfinite monomial basis of L(Λ) in the spirit of Feigin and Stoyanovsky (the caseĝ = sl(2, C) was described by them in the announcement [FS]). A proof of the particular case of (5.15) for the vacuum module (Λ = kΛ 0 , i.e., k 0 = k, k j = 0) was announced in [Kir].
Example 5.1 Let g = sl(3, C), k = 2. By (5.11) and (5.7) and (5.8) we have
c 2Λ 0 0 (q) = q − 2 15 (q) 2 ∞ p 1 ,p 2 ≥0
p 1 ,p 2 even q 1 2 (p 2 1 +p 2 2 −p 1 p 2 ) (q) p 1 (q) p 2 , (5.16)
c 2Λ 0 α 1 +α 2 (q) = q − 2 15 (q) 2 ∞ p 1 ,p 2 ≥0
p 1 ,p 2 odd q 1 2 (p 2 1 +p 2 2 −p 1 p 2 ) (q) p 1 (q) p 2 , (5.17)
cΛ 0 +Λ 1 Λ 1 (q) = q − 1 30 (q) 2 ∞ p 1 ,p 2 ≥0
p 1 ,p 2 even q 1 2 (p 2 1 +p 2 2 −p 1 p 2 −p 1 ) (q) p 1 (q) p 2 , (5.18)
cΛ 0 +Λ 1 Λ 1 +α 2 (q) = q − 1 30 (q) 2 ∞ p 1 ,p 2 ≥0
p 1 (p 2 ) even (odd) q 1 2 (p 2 1 +p 2 2 −p 1 p 2 −p 1 ) (q) p 1 (q) p 2 , (5.19)
Due to symmetries, the above string functions determine all the other string functions at level 2 (cf. for example [KP] Section 4.6; using the expressions of Kac and Peterson, the above formulas were verified on Maple up to O(50)). In particular, due to the cyclic automorphism of the affine Dynkin diagram, one has cΛ 0 +Λ 1 Λ 1 (q) = cΛ 0 +Λ 2 Λ 2 (q) = cΛ 1 +Λ 2 Λ 1 +Λ 2 (q). But notice thatΛ 1 +Λ 2 is not among the highest weights of type (4.1) for which our basis theorems and character formulas hold (cf. Remark 5.1). Thanks to the simplicity of the case, one can nevertheless find easily a combinatorial basis and write down the corresponding character formula:
Tr q D−Dĥ L(Λ 1 +Λ 2 )ĥ + 2Q = (5.20) = p 1 ,p 2 ≥0 q 1 2 (p 2 1 +p 2 2 −p 1 p 2 ) (q) p 1 (q) p 2 q p 1 − 1 2 (p 1 +p 2 ) + p 1 ,p 2 ≥0 q 1 2 [(p 1 +1) 2 +p 2 2 −(p 1 +1)p 2 ] (q) p 1 (q) p 2 q p 2 − 1 2 [(p 1 +1)+p 2 ] .
The first term counts only monomials which do not include a quasi-particle π L(Λ 1 +Λ 2 )ĥ + 2Q · x α 1 (−1), while the second term counts only monomials which contain such a quasi-particle.
6 Appendix color-energy basis -type (1; 2) 3/2 (1 α 2 − 3 α 1 − 1 α 1 ) 5/2
(1 α 2 − 4 α 1 − 1 α 1 ), (0 α 2 − 3 α 1 − 1 α 1 ) 7/2
(1 α 2 − 5 α 1 − 1 α 1 ), (1 α 2 − 4 α 1 − 2 α 1 ), (0 α 2 − 4 α 1 − 1 α 1 ), (1 α 2 − 3 α 1 − 1 α 1 ) (2; 2) 2 (−1 α 2 1 α 2 − 3 α 1 − 1 α 1 ) 3 (−2 α 2 1 α 2 − 3 α 1 − 1 α 1 ), (−1 α 2 1 α 2 − 4 α 1 − 1 α 1 ) 4 (−3 α 2 1 α 2 − 3 α 1 − 1 α 1 ), (−2 α 2 1 α 2 − 4 α 1 − 1 α 1 ), (−2 α 2 0 α 2 − 3 α 1 − 1 α 1 ), (−1 α 2 1 α 2 − 5 α 1 − 1 α 1 ), (−1 α 2 1 α 2 − 4 α 1 − 2 α 1 ) Table 1 color-energy color-basis -type -charge--type (1; 2) 1 (1; 2) (0 α 2 − 2 2α 1 ) 2
33): D − Dĥ, ψ nrβr,...,n 1 β 1 (m r , . . . , m
is the inverse of the Cartan matrix of sl(k, C).
,α + α,µ ρ(kα) L µ (Λ) , where ρ(kα) := e α ⊗ . . . ⊗ e α k f actors , α ∈ Q. (cf. (1.6)). Therefore (1.2), (1.9), (1.10) and (5.10) imply that ch L(Λ) = ch L(k 0Λ0 + k jΛj ) := Tr q cΛ µ (q)Θ µ (q, y),
L(Λ) = ch L(k 0Λ0 + k jΛj ) := Tr q
Table is
isof course a copy of Table 1 [GeI], Appendix
(listing the corresponding elements of B
(1)
W (2Λ 0 )
≥0 ............(5.27) that
Tr q D
W (Λ)
=
(5.5)
=
p
(1)
1 ,...,p
(k−1)
1
p
(1)
n ,...,p
(k−1)
n
≥0
q
1
2
s,t=1,...,k−1
l,m=1,...,n
Acknowledgments We are indebted to James Lepowsky whose advices, expertise and continuous encouragement made this work possible. Many thanks are due to Barry McCoy for the intense discussions, the constant support and interest toward this work. We would also like to thank Alexander Zamolodchikov.
α 2 − 3 2α 1 ), (−2 α 2 − 2 2α 1 ) 2 − 5 2α 1 ). 42 − 4 2α 1. −1 α 2 − 4 2α 1 ), (−2 α 2 − 3 2α 1. 2α 1 ), (−1 2α 2 − 3 2α 1 ), (−2 2α 2 − 2 2α 12 − 4 2α 1 ), (−1 α 2 − 3 2α 1 ), (−2 α 2 − 2 2α 1 ) 2 − 5 2α 1 ), (−1 α 2 − 4 2α 1 ), (−2 α 2 − 3 2α 1 ), 4 2α 1 ), (−1 2α 2 − 3 2α 1 ), (−2 2α 2 − 2 2α 1 )
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|
[] |
[
"NTUA-SLP at SemEval-2018 Task 3: Tracking Ironic Tweets using Ensembles of Word and Character Level Attentive RNNs",
"NTUA-SLP at SemEval-2018 Task 3: Tracking Ironic Tweets using Ensembles of Word and Character Level Attentive RNNs"
] |
[
"Christos Baziotis [email protected] \nSchool of ECE\nNational Technical University of Athens\nAthensGreece\n\nDepartment of Informatics\nAthens University of Economics and Business\nAthensGreece\n",
"Nikos Athanasiou \nSchool of ECE\nNational Technical University of Athens\nAthensGreece\n",
"Pinelopi Papalampidi \nSchool of ECE\nNational Technical University of Athens\nAthensGreece\n",
"Athanasia Kolovou [email protected]@central.ntua.gr \nSchool of ECE\nNational Technical University of Athens\nAthensGreece\n\nDepartment of Informatics\nUniversity of Athens\nAthensGreece\n",
"Georgios Paraskevopoulos \nSchool of ECE\nNational Technical University of Athens\nAthensGreece\n\nBehavioral Signal Technologies\nLos AngelesCA\n",
"Nikolaos Ellinas [email protected]@central.ntua.gr \nSchool of ECE\nNational Technical University of Athens\nAthensGreece\n",
"Alexandros Potamianos \nSchool of ECE\nNational Technical University of Athens\nAthensGreece\n\nBehavioral Signal Technologies\nLos AngelesCA\n"
] |
[
"School of ECE\nNational Technical University of Athens\nAthensGreece",
"Department of Informatics\nAthens University of Economics and Business\nAthensGreece",
"School of ECE\nNational Technical University of Athens\nAthensGreece",
"School of ECE\nNational Technical University of Athens\nAthensGreece",
"School of ECE\nNational Technical University of Athens\nAthensGreece",
"Department of Informatics\nUniversity of Athens\nAthensGreece",
"School of ECE\nNational Technical University of Athens\nAthensGreece",
"Behavioral Signal Technologies\nLos AngelesCA",
"School of ECE\nNational Technical University of Athens\nAthensGreece",
"School of ECE\nNational Technical University of Athens\nAthensGreece",
"Behavioral Signal Technologies\nLos AngelesCA"
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In this paper we present two deep-learning systems that competed at SemEval-2018 Task 3 "Irony detection in English tweets". We design and ensemble two independent models, based on recurrent neural networks (Bi-LSTM), which operate at the word and character level, in order to capture both the semantic and syntactic information in tweets. Our models are augmented with a self-attention mechanism, in order to identify the most informative words. The embedding layer of our wordlevel model is initialized with word2vec word embeddings, pretrained on a collection of 550 million English tweets. We did not utilize any handcrafted features, lexicons or external datasets as prior information and our models are trained end-to-end using back propagation on constrained data. Furthermore, we provide visualizations of tweets with annotations for the salient tokens of the attention layer that can help to interpret the inner workings of the proposed models. We ranked 2 nd out of 42 teams in Subtask A and 2 nd out of 31 teams in Subtask B. However, post-task-completion enhancements of our models achieve state-ofthe-art results ranking 1 st for both subtasks.
|
10.18653/v1/s18-1100
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[
"https://arxiv.org/pdf/1804.06659v1.pdf"
] | 4,942,082 |
1804.06659
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6ee45f56cc48e8483b750e60e508cac5dd9e6bd3
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NTUA-SLP at SemEval-2018 Task 3: Tracking Ironic Tweets using Ensembles of Word and Character Level Attentive RNNs
Christos Baziotis [email protected]
School of ECE
National Technical University of Athens
AthensGreece
Department of Informatics
Athens University of Economics and Business
AthensGreece
Nikos Athanasiou
School of ECE
National Technical University of Athens
AthensGreece
Pinelopi Papalampidi
School of ECE
National Technical University of Athens
AthensGreece
Athanasia Kolovou [email protected]@central.ntua.gr
School of ECE
National Technical University of Athens
AthensGreece
Department of Informatics
University of Athens
AthensGreece
Georgios Paraskevopoulos
School of ECE
National Technical University of Athens
AthensGreece
Behavioral Signal Technologies
Los AngelesCA
Nikolaos Ellinas [email protected]@central.ntua.gr
School of ECE
National Technical University of Athens
AthensGreece
Alexandros Potamianos
School of ECE
National Technical University of Athens
AthensGreece
Behavioral Signal Technologies
Los AngelesCA
NTUA-SLP at SemEval-2018 Task 3: Tracking Ironic Tweets using Ensembles of Word and Character Level Attentive RNNs
In this paper we present two deep-learning systems that competed at SemEval-2018 Task 3 "Irony detection in English tweets". We design and ensemble two independent models, based on recurrent neural networks (Bi-LSTM), which operate at the word and character level, in order to capture both the semantic and syntactic information in tweets. Our models are augmented with a self-attention mechanism, in order to identify the most informative words. The embedding layer of our wordlevel model is initialized with word2vec word embeddings, pretrained on a collection of 550 million English tweets. We did not utilize any handcrafted features, lexicons or external datasets as prior information and our models are trained end-to-end using back propagation on constrained data. Furthermore, we provide visualizations of tweets with annotations for the salient tokens of the attention layer that can help to interpret the inner workings of the proposed models. We ranked 2 nd out of 42 teams in Subtask A and 2 nd out of 31 teams in Subtask B. However, post-task-completion enhancements of our models achieve state-ofthe-art results ranking 1 st for both subtasks.
Introduction
Irony is a form of figurative language, considered as "saying the opposite of what you mean", where the opposition of literal and intended meanings is very clear (Barbieri and Saggion, 2014;Liebrecht et al., 2013). Traditional approaches in NLP (Tsur et al., 2010;Barbieri and Saggion, 2014;Karoui et al., 2015;Farías et al., 2016) model irony based on pattern-based features, such as the contrast between high and low frequent words, the punctuation used by the author, the level of ambiguity of The color intensity of each word / character, corresponds to its weight (importance), as given by the self-attention mechanism (Section 2.6). the words and the contrast between the sentiments. Also, (Joshi et al., 2016) recently added word embeddings statistics to the feature space and further boosted the performance in irony detection.
Modeling irony, especially in Twitter, is a challenging task, since in ironic comments literal meaning can be misguiding; irony is expressed in "secondary" meaning and fine nuances that are hard to model explicitly in machine learning algorithms. Tracking irony in social media posses the additional challenge of dealing with special language, social media markers and abbreviations. Despite the accuracy achieved in this task by handcrafted features, a laborious feature-engineering process and domain-specific knowledge are required; this type of prior knowledge must be continuously updated and investigated for each new domain. Moreover, the difficulty in parsing tweets (Gimpel et al., 2011) for feature extraction renders their precise semantic representation, which is key of determining their intended gist, much harder.
In recent years, the successful utilization of deep learning architectures in NLP led to alternative approaches for tracking irony in Twitter (Joshi et al., 2017;Ghosh and Veale, 2017). (Ghosh and Veale, 2016) proposed a Convolutional Neural Network (CNN) followed by a Long Short Term Memory (LSTM) architecture, outperforming the state-of-the-art. (Dhingra et al., 2016) utilized deep learning for representing tweets as a sequence of characters, instead of words and proved that such representations reveal information about the irony concealed in tweets.
In this work, we propose the combination of word-and character-level representations in order to exploit both semantic and syntactic information of each tweet for successfully predicting irony. For this purpose, we employ a deep LSTM architecture which models words and characters separately. We predict whether a tweet is ironic or not, as well as the type of irony in the ironic ones by ensembling the two separate models (late fusion). Furthermore, we add an attention layer to both models, to better weigh the contribution of each word and character towards irony prediction, as well as better interpret the descriptive power of our models. Attention weighting also better addresses the problem of supervising learning on deep learning architectures. The suggested model was trained only on constrained data, meaning that we did not utilize any external dataset for further tuning of the network weights.
The two deep-learning models submitted to SemEval-2018 Task 3 "Irony detection in English tweets" (Van Hee et al., 2018) are described in this paper with the following structure: in Section 2 an overview of the proposed models is presented, in Section 3 the models for tracking irony are depicted in detail, in Section 4 the experimental setup alongside with the respective results are demonstrated and finally, in Section 5 we discuss the performance of the proposed models.
Overview
Fig. 2 provides a high-level overview of our approach, which consists of three main steps: (1) the pre-training of word embeddings, where we train our own word embeddings on a big collection of unlabeled Twitter messages, (2) the independent training of our models: word-and char-level, (3) the ensembling, where we combine the predictions of each model.
Task definitions
The goal of Subtask A is tracking irony in tweets as a binary classification problem (ironic vs. nonironic). In Subtask B, we are also called to determine the type of irony, with three different classes of irony on top of the non-ironic one (four-class classification). The types of irony are:
(1) Verbal irony by means of a polarity contrast, which includes messages whose polarity (positive, negative) is inverted between the literal and the intended evaluation, such as "I really love this year's summer; weeks and weeks of awful weather", where the literal evaluation ("I really love this year's summer") is positive, while the intended one, which is implied in the context ("weeks and weeks of awful weather"), is negative.
(2) Other verbal irony, which refers to instances showing no polarity contrast, but are ironic such as "Yeah keeping cricket clean, that's what he wants #Sarcasm" and (3) situational irony which is present in messages that a present situation fails to meet some expectations, such as "Event technology session is having Internet problems. #irony #HSC2024" in which the expectation that a technology session should provide Internet connection is not met.
Data
Unlabeled Dataset. We collected a dataset of 550 million archived English Twitter messages, from Apr. 2014 to Jun. 2017. This dataset is used for (1) calculating word statistics needed in our text preprocessing pipeline (Section 2.4) and (2) train-ing word2vec word embeddings (Section 2.3).
Word Embeddings
Word embeddings are dense vector representations of words (Collobert and Weston, 2008;, capturing semantic their and syntactic information. We leverage our unlabeled dataset to train Twitter-specific word embeddings. We use the word2vec algorithm, with the skip-gram model, negative sampling of 5 and minimum word count of 20, utilizing Gensim's (Řehůřek and Sojka, 2010) implementation. The resulting vocabulary contains 800, 000 words. The pre-trained word embeddings are used for initializing the first layer (embedding layer) of our neural networks.
Preprocessing 1
We utilized the ekphrasis 2 (Baziotis et al., 2017) tool as a tweet preprocessor. The preprocessing steps included in ekphrasis are: Twitter-specific tokenization, spell correction, word normalization, word segmentation (for splitting hashtags) and word annotation. Tokenization. Tokenization is the first fundamental preprocessing step and since it is the basis for the other steps, it immediately affects the quality of the features learned by the network. Tokenization in Twitter is especially challenging, since there is large variation in the vocabulary and the used expressions. Part of the challenge is also the decision of whether to process an entire expression (e.g. anti-american) or its respective tokens. Ekphrasis overcomes this challenge by recognizing the Twitter markup, emoticons, emojis, expressions like dates (e.g. 07/11/2011, April 23rd), times (e.g. 4:30pm, 11:00 am), currencies (e.g. $10, 25mil, 50e), acronyms, censored words (e.g. s**t) and words with emphasis (e.g. *very*). Normalization. After the tokenization we apply a series of modifications on the extracted tokens, such as spell correction, word normalization and segmentation. We also decide which tokens to omit, normalize and surround or replace with special tags (e.g. URLs, emails and @user). For the tasks of spell correction (Jurafsky and James, 2000) and word segmentation (Segaran and Hammerbacher, 2009) we use the Viterbi algorithm. The prior probabilities are initialized using uni/bigram word statistics from the unlabeled dataset.
The benefits of the above procedure are the reduction of the vocabulary size, without removing any words, and the preservation of information that is usually lost during tokenization. Table 1 shows an example text snippet and the resulting preprocessed tokens.
Recurrent Neural Networks
We model the Twitter messages using Recurrent Neural Networks (RNN). RNNs process their inputs sequentially, performing the same operation, h t = f W (x t , h t−1 ), on every element in a sequence, where h t is the hidden state t the time step, and W the network weights. We can see that hidden state at each time step depends on previous hidden states, thus the order of elements (words) is important. This process also enables RNNs to handle inputs of variable length.
RNNs are difficult to train (Pascanu et al., 2013), because gradients may grow or decay exponentially over long sequences (Bengio et al., 1994;Hochreiter et al., 2001). A way to overcome these problems is to use more sophisticated variants of regular RNNs, like Long Short-Term Memory (LSTM) networks (Hochreiter and Schmidhuber, 1997) or Gated Recurrent Units (GRU) , which introduce a gating mechanism to ensure proper gradient flow through the network. In this work, we use LSTMs.
Self-Attention Mechanism
RNNs update their hidden state h i as they process a sequence and the final hidden state holds a summary of the information in the sequence. In order to amplify the contribution of important words in the final representation, a self-attention mechanism can be used original The *new* season of #TwinPeaks is coming on May 21, 2017. CANT WAIT \o/ !!! #tvseries #davidlynch :D processed the new <emphasis> season of <hashtag> twin peaks </hashtag> is coming on <date> . cant <allcaps> wait <allcaps> <happy> ! <repeated> <hashtag> tv series </hashtag> <hashtag> david lynch </hashtag> <laugh> (b) Attention RNN Figure 3: Comparison between the regular RNN and the RNN with attention. (Fig. 3). In normal RNNs, we use as representation r of the input sequence its final state h N . However, using an attention mechanism, we compute r as the convex combination of all h i . The weights a i are learned by the network and their magnitude signifies the importance of each hidden state in the final representation. Formally:
r = N i=1 a i h i , where N i=1 a i = 1, and a i > 0.
Models Description
We have designed two independent deep-learning models, with each one capturing different aspects of the tweet. The first model operates at the wordlevel, capturing the semantic information of the tweet and the second model at the character-level, capturing the syntactic information. Both models share the same architecture, and the only difference is in their embedding layers. We present both models in a unified manner.
Embedding Layer
Character-level. The input to the network is a Twitter message, treated as a sequence of characters. We use a character embedding layer to project the characters c 1 , c 2 , ..., c N to a lowdimensional vector space R C , where C the size of the embedding layer and N the number of characters in a tweet. We randomly initialize the weights of the embedding layer and learn the character embeddings from scratch. Word-level. The input to the network is a Twitter message, treated as a sequence of words. We use a word embedding layer to project the words w 1 , w 2 , ..., w N to a low-dimensional vector space R W , where W the size of the embedding layer and N the number of words in a tweet. We initialize the weights of the embedding layer with our pretrained word embeddings.
BiLSTM Layers
An LSTM takes as input the words (characters) of a tweet and produces the word (character) annotations h 1 , h 2 , ..., h N , where h i is the hidden state of the LSTM at time-step i, summarizing all the information of the sentence up to w i (c i ). We use bidirectional LSTM (BiLSTM) in order to get word (character) annotations that summarize the information from both directions. A bidirectional LSTM consists of a forward LSTM − → f that reads the sentence from w 1 to w N and a backward LSTM ← − f that reads the sentence from w N to w 1 . We obtain the final annotation for a given word w i (character c i ), by concatenating the annotations from both directions,
h i = − → h i ← − h i , h i ∈ R 2L
where denotes the concatenation operation and L the size of each LSTM. We stack two layers of BiLSTMs in order to learn more high-level (abstract) features.
Attention Layer
Not all words contribute equally to the meaning that is expressed in a message. We use an attention mechanism to find the relative contribution (importance) of each word. The attention mechanism assigns a weight a i to each word annotation h i . We compute the fixed representation r of the whole input message. as the weighted sum of all the word annotations.
e i = tanh(W h h i + b h ), e i ∈ [−1, 1]
(1)
a i = exp(e i ) T t=1 exp(e t ) , T i=1 a i = 1 (2) r = T i=1 a i h i , r ∈ R 2L(3)
where W h and b h are the attention layer's weights.
Character-level Interpretation. In the case of the character-level model, the attention mechanism operates in the same way as in the wordlevel model. However, we can interpret the weight given to each character annotation h i by the attention mechanism, as the importance of the information surrounding the given character.
Output Layer
We use the representation r as feature vector for classification and we feed it to a fully-connected softmax layer with L neurons, which outputs a probability distribution over all classes p c as described in Eq. 4:
p c = e W r+b i∈[1,L] (e W i r+b i )(4)
where W and b are the layer's weights and biases.
Regularization
In order to prevent overfitting of both models, we add Gaussian noise to the embedding layer, which can be interpreted as a random data augmentation technique, that makes models more robust to overfitting. In addition to that, we use dropout (Srivastava et al., 2014) and early-stopping.
Finally, we do not fine-tune the embedding layers of the word-level model. Words occurring in the training set, will be moved in the embedding space and the classifier will correlate certain regions (in embedding space) to certain meanings or types of irony. However, words in the test set and not in the training set, will remain at their initial position which may no longer reflect their "true" meaning, leading to miss-classifications.
Ensemble
A key factor to good ensembles, is to utilize diverse classifiers. To this end, we combine the predictions of our word and character level models. We employed two ensemble schemes, namely unweighted average and majority voting. Unweighted Average (UA). In this approach, the final prediction is estimated from the unweighted average of the posterior probabilities for all different models. Formally, the final prediction p for a training instance is estimated by:
p = arg max c 1 C M i=1 p i , p i ∈ IR C (5)
where C is the number of classes, M is the number of different models, c ∈ {1, ..., C} denotes one class and p i is the probability vector calculated by model i ∈ {1, ..., M } using softmax function. Majority Voting (MV). Majority voting approach counts the votes of all different models and chooses the class with most votes. Compared to unweighted averaging, MV is affected less by single-network decisions. However, this schema does not consider any information derived from the minority models. Formally, for a task with C classes and M different models, the prediction for a specific instance is estimated as follows:
v c = M i=1 F i (c) p = arg max c∈{1,...,C} v c(6)
where v c denotes the votes for class c from all different models, F i is the decision of the i th model, which is either 1 or 0 with respect to whether the model has classified the instance in class c or not, respectively, and p is the final prediction.
Experiments and Results
Experimental Setup
Class Weights. In order to deal with the problem of class imbalances in Subtask B, we apply class weights to the loss function of our models, penalizing more the misclassification of underrepresented classes. We weight each class by its inverse frequency in the training set.
Training We use Adam algorithm (Kingma and Ba, 2014) for optimizing our networks, with minibatches of size 32 and we clip the norm of the gradients (Pascanu et al., 2013) at 1, as an extra safety measure against exploding gradients. For developing our models we used PyTorch (Paszke et al., 2017) and Scikit-learn (Pedregosa et al., 2011).
Hyper-parameters. In order to find good hyperparameter values in a relative short time (compared to grid or random search), we adopt the Bayesian optimization (Bergstra et al., 2013) approach, performing a "smart" search in the high dimensional space of all the possible values. Table 2, shows the selected hyper-parameters.
Results and Discussion
Our official ranking is 2/43 in Subtask A and 2/29 in Subtask B as shown in Tables 3 and 4. Based on these rankings, the performance of the suggested model is competitive on both the binary and the multi-class classification problem. Except for its overall good performance, it also presents a stable behavior when moving from two to four classes. Additional experimentation following the official submission significantly improved the efficiency of our models. The results of this experimentation, tested on the same data set, are shown in Tables 5 and 6. The first baseline is a Bag of Words (BOW) model with TF-IDF weighting. The second baseline is a Neural Bag of Words (N-BOW) model where we retrieve the word2vec representations of the words in a tweet and compute the tweet representation as the centroid of the constituent word2vec representations. Both BOW and N-BOW features are then fed to a linear SVM classifier, with tuned C = 0.6. The best performance that we achieve, as shown in Tables 5 and 6 is 0.7856 and 0.5358 for Subtask A and B respectively 34 . In Subtask A the BOW and N-BOW models perform similarly with respect to f1 metric and word-level LSTM is the most competitive individual model. However, the best performance is achieved when the characterand the word-level LSTM models are combined via the unweighted average ensembling method, showing that the two suggested models indeed contain different types of information related to irony on tweets. Similar observations are derived for Subtask B, except that the character-level model in this case performs worse than the baseline models and contributes less to the final results.
Attention Visualizations
Our models' behavior can be interpreted by visualizing the distribution of the attention weights assigned to the words (characters) of the tweet. The weights signify the contribution of each word (character), to model's final classification decision. In Fig. 5, examples of the weights as- signed by the word level model to ironic tweets are presented. The salient keywords that capture the essence of irony or even polarity transitions (e.g. irony by clash) are correctly identified by the model. Moreover, in Fig. 6 we compare the behavior of the word and character models on the same tweets. In the first example, the character level model assigns larger weights to the most discriminative words whereas the weights assigned by the word level model seem uniform and insufficient in spotting the polarity transition. However, in the second example, the character level model does not attribute any weight to the words with positive polarity (e.g. "fun") compared to the word level model. Based on these observations, the two models indeed behave diversely and consequently contribute to the final outcome (see Section 3.6).
Conclusion
In this paper we present an ensemble of two different deep learning models: a word-and a character-level deep LSTM for capturing the semantic and syntactic information of tweets, respectively. We demonstrated that combining the predictions of the two models yields competitive results in both subtasks for irony prediction. Moreover, we proved that both types of informa-tion (semantic and syntactic) contribute to the final results with the word-level model, however, individually achieving more accurate irony prediction. Also, the best way of combining the outcomes of the separate models is by conducting majority voting over the respective posteriors. Finally, the proposed model successfully predicts the irony in tweets without exploiting any external information derived from hand-crafted features or lexicons. The performance reported in this paper could be further boosted by utilizing transfer learning methods from larger datasets. Moreover, the joint training of word-and character-level models can be tested for further improvement of the results. Finally, we make the source code of our models and our pretrained word embeddings available to the community 5 , in order to make our results easily reproducible and facilitate further experimentation.
Figure 1 :
1Attention heat-map visualization.
Figure 2 :
2High-level overview of our approach
Figure 4 :
4The word/character-level model.
Figure 5 :Figure 6 :
56Examples of the attention mechanism for identification of the type of irony in each sentence. Comparison of the behavior of the word and character level models.
Table 1 :
1Example of our text processor1
ℎ 1
2
ℎ 2
3
ℎ 3
…
(a) Regular RNN
1
1
ℎ 1
2
ℎ 2
3
ℎ 3
ℎ
…
2
3
Table 2 :
2Hyper-parameters of our models.
Table 3 :
3Competition results for Subtask A# Team Name
Acc
Prec
Rec
F1
1 Unknown
0.7321 0.5768 0.5044 0.5074
2 NTUA-SLP
0.6518 0.4959 0.5124 0.4959
3 THU_NGN
0.6046 0.4860 0.5414 0.4947
4 Unknown
0.6033 0.4660 0.5058 0.4743
5 NIHRIO, NCL 0.6594 0.5446 0.4475 0.4437
Table 4 :
4Competition results for Subtask B
Table 5 :
5Results of our models for Subtask Amodel
Acc
Prec
Rec
f1
BOW
0.5880 0.4460 0.4384 0.4371
N-BOW
0.6084 0.4649 0.4560 0.4520
LSTM-char
0.5726 0.4098 0.4102 0.3782
LSTM-word 0.6987 0.5394 0.5790 0.5315
Ens-MV
0.6888 0.5433 0.5442 0.5358
Ens-UA
0.6888 0.5361 0.4874 0.4959
Table 6 :
6Results of our models for Subtask B
Significant portions of the systems submitted to SemEval 2018 in Tasks 1, 2 and 3, by the NTUA-SLP team are shared, specifically the preprocessing and portions of the DNN architecture. Their description is repeated here for completeness. 2 github.com/cbaziotis/ekphrasis
The reported performance is boosted in comparison with the results presented in Tables 3 and 4 due to the utilization of unnormalized word vectors. Specifically, after further experimentation we found that normalization of word vectors provided to the LSTM is detrimental to performance, because semantic information is encoded by both the angle and length of the embedding vectors(Wilson and Schakel, 2015).4 For our DNNs, the results are computed by averaging 10 runs to account for the variability in training performance.
github.com/cbaziotis/ ntua-slp-semeval2018-task3
Acknowledgements. This work has been partially supported by the BabyRobot project supported by EU H2020 (grant #687831). Also, the authors would like to thank NVIDIA for supporting this work by donating a TitanX GPU.
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|
[] |
[
"EREMENKO'S CONJECTURE, WANDERING LAKES OF WADA, AND MAVERICK POINTS",
"EREMENKO'S CONJECTURE, WANDERING LAKES OF WADA, AND MAVERICK POINTS"
] |
[
"David Martí-Pete ",
"Lasse Rempe ",
"James Waterman "
] |
[] |
[] |
We develop a general technique for realising full closed subsets of the complex plane as wandering sets of entire functions. Using this construction, we solve a number of open problems.(1) We construct a counterexample to Eremenko's conjecture, a central problem in transcendental dynamics that asks whether every connected component of the set of escaping points of a transcendental entire function is unbounded.(2) We prove that there is a transcendental entire function for which infinitely many Fatou components share the same boundary. This resolves the long-standing problem whether Lakes of Wada continua can arise in complex dynamics, and answers the analogue of a question of Fatou from 1920 concerning Fatou components of rational functions. (3) We answer a question of Rippon concerning the existence of non-escaping points on the boundary of a bounded escaping wandering domain, that is, a wandering Fatou component contained in the escaping set. In fact we show that the set of such points can have positive Lebesgue measure. (4) We give the first example of an entire function having a simply connected Fatou component whose closure has a disconnected complement, answering a question of Boc Thaler. In view of (3), we introduce the concept of maverick points: points on the boundary of a wandering domain whose accumulation behaviour differs from that of internal points. We prove that the set of such points has harmonic measure zero, but that both escaping and oscillating wandering domains can contain large sets of maverick points.
| null |
[
"https://arxiv.org/pdf/2108.10256v2.pdf"
] | 246,608,281 |
2108.10256
|
83e0c26dce9ae7309b5cb37b737d93232b5e0218
|
EREMENKO'S CONJECTURE, WANDERING LAKES OF WADA, AND MAVERICK POINTS
David Martí-Pete
Lasse Rempe
James Waterman
EREMENKO'S CONJECTURE, WANDERING LAKES OF WADA, AND MAVERICK POINTS
We develop a general technique for realising full closed subsets of the complex plane as wandering sets of entire functions. Using this construction, we solve a number of open problems.(1) We construct a counterexample to Eremenko's conjecture, a central problem in transcendental dynamics that asks whether every connected component of the set of escaping points of a transcendental entire function is unbounded.(2) We prove that there is a transcendental entire function for which infinitely many Fatou components share the same boundary. This resolves the long-standing problem whether Lakes of Wada continua can arise in complex dynamics, and answers the analogue of a question of Fatou from 1920 concerning Fatou components of rational functions. (3) We answer a question of Rippon concerning the existence of non-escaping points on the boundary of a bounded escaping wandering domain, that is, a wandering Fatou component contained in the escaping set. In fact we show that the set of such points can have positive Lebesgue measure. (4) We give the first example of an entire function having a simply connected Fatou component whose closure has a disconnected complement, answering a question of Boc Thaler. In view of (3), we introduce the concept of maverick points: points on the boundary of a wandering domain whose accumulation behaviour differs from that of internal points. We prove that the set of such points has harmonic measure zero, but that both escaping and oscillating wandering domains can contain large sets of maverick points.
Introduction
The iteration of transcendental entire self-maps of the complex plane was initiated by Fatou in 1926 [Fat26] and has received much attention in recent years. As well as presenting beautiful and challenging problems in its own right, the techniques developed in transcendental dynamics have found applications in adjacent areas; compare, for example, the recent work of Dudko and Lyubich [DL18] on the local connectivity of the Mandelbrot set. Much work in the field in the last two decades has been motivated by the study of the escaping set of a transcendental entire function f , This set was introduced by Eremenko, who proved that the connected components of I(f ) are all unbounded [Ere89,Theorem 3]. He also states [Ere89,p. 343] that it is plausible that the set I(f ) itself has no bounded connected components; this is known as Eremenko's conjecture and has remained an open problem since then. [RS11], and that unbounded connected components are dense in I(f ) [RS05], but Eremenko's conjecture itself has remained open despite considerable research efforts. We resolve it in the negative. Recall that a compact set X ⊆ C is full if C \ X is connected. These two sets are named after Pierre Fatou and Gaston Julia, who independently laid the foundations of one-dimensional complex dynamics in the early 20th century. In his seminal memoir on the iteration of rational functions, Fatou posed the following question concerning the structure of the connected components of F (f ), which are today called the Fatou components of f . Question 1.3 (Fatou [Fat20,). If f has more than two Fatou components, can two of these components share the same boundary? 1 Fatou asked this question in the context of rational self-maps of the Riemann sphere, but it makes equal sense for transcendental entire functions, whose dynamical study Fatou initiated in 1926 [Fat26]. We give a positive answer to Question 1.3 in this setting.
Theorem 1.4 (Fatou components with a common boundary). There exists a transcendental entire function f and an infinite collection of Fatou components of f that all share the same boundary. In particular, the common boundary of these Fatou components is a Lakes of Wada continuum.
Here a Lakes of Wada continuum is a compact and connected subset of the plane that is the common boundary of three or more disjoint domains (see Figure 1). That such continua exist was first shown by Brouwer [Bro10,p. 427]; the name "Lakes of Wada" arises from a well-known construction that Yoneyama [Yon17,p. 60] attributed to his advisor, Takeo Wada; compare [HY61,p. 143], [HO95,Section 8] or [Ish17,Section 4]. In this construction, one begins with the closure of a bounded finitely connected domain, which we may think of as an island in the sea. We think of the complementary domains, which we assume to be Jordan domains, as bodies of water: the sea (the unbounded domain) and the lakes (the bounded domains). One then constructs successive canals, which ensure that the maximal distance from any point on the remaining piece of land to any body of water tends to zero, while keeping the island connected; see Figure 2. The land remaining at the end of this process is the boundary of each of its complementary domains; in particular, if there are at least three bodies of water, it is a Lakes of Wada continuum. One may even obtain infinitely many complementary domains, all sharing the same boundary, by introducing new lakes throughout the construction.
Lakes of Wada continua may appear pathological, but they occur naturally in the study of dynamical systems in two real dimensions; see [KY91]. For example, Figure 1 is (a projection of) the Plykin attractor [Ply74] (see also [Cou06]), which is the attractor of a perturbation of an Anosov diffeomorphism of the torus. Hubbard and Oberste-Vorth The island is black, the sea is white, and the two lakes are dark grey and light grey. The construction proceeds inductively, with the distance from any point in the island to any body of water tending to zero.
[HO95, Theorem 8.5] showed that, under certain circumstances, the basins of attraction of Hénon maps in R 2 form Lakes of Wada.
Theorem 1.4 provides the first example where such continua arise in one-dimensional complex dynamics, answering a long-standing open question. The second author learned of this problem for the first time in an introductory complex dynamics course by Bergweiler in Kiel during the academic year 1998-99; see [Ber13,p. 27 [Bak76] showed that transcendental entire functions can have wandering domains, while Sullivan [Sul85] proved in 1985 that they do not occur for rational functions. If f is a polynomial, a bounded periodic Fatou component U is either an immediate basin of a finite attracting or parabolic periodic point, or a Siegel disc, on which the dynamics is conjugate to an irrational rotation. In the former case, it is known that U is a Jordan domain [RY08]. Dudko and Lyubich have announced a proof that the boundary of a Siegel disc of a quadratic polynomial is also a Jordan curve. This would rule out Lakes of Wada boundaries, and provide a negative answer to Question 1.3 for maps in this family. Both questions remain open for polynomials of degree at least three, but it now seems reasonable to expect that, for polynomials and rational maps, the answer to Question 1.3 is negative. The problem whether the whole Julia set can be a Lakes of Wada continuum is related to Makienko's conjecture concerning completely invariant domains and buried points of rational functions; compare [SY03] and [CMMR09].
]. A Fatou component U is called periodic if f p (U ) ⊆ U for some p ∈ N, and U is preperiodic if there exists q ∈ N such that f q (U ) is
In contrast, our motivation for considering Question 1.3 comes from the study of wandering domains, which have been the focus of much recent research; compare [Bis15, BRS16, Bis18, MS20, BFE + 21a]. They have also recently been studied in higherdimensional complex dynamics; compare [ABD + 16, Boc21a, HP21, ABFP19]. A wandering domain U of an entire function f is called escaping if it is contained in the escaping set I(f ) and it is called oscillating if it is contained in the bungee set
BU (f ) . . = z ∈ C \ I(f ) : lim sup n→∞ |f n (z)| = +∞ .
We are interested in determining whether the behaviour of points on ∂U is determined by the interior dynamics of U . In all previously known examples of escaping wandering domains, the iterates f n in fact tend to infinity uniformly on U , and hence on ∂U . It is known [RS12, Theorem 1.2] that this must be the case whenever U is contained in the fast escaping set A(f ) introduced by Bergweiler and Hinkkanen [BH99]. This suggests the following question, which is Problem 2.94 in Hayman and Lingham's Research Problems in Function Theory. We answer this question in the negative.
Theorem 1.6 (Non-escaping points in the boundary of escaping wandering domains). There exists a transcendental entire function f with a bounded escaping wandering domain U such that ∂U \ I(f ) = ∅.
We prove Theorem 1.6 by showing that the function from Theorem 1.4 can be constructed such that at least one of these domains, U 1 , is an escaping wandering domain while another, U 2 , is an oscillating wandering domain. The set of bungee points in ∂U 1 = ∂U 2 has full harmonic measure seen from U 2 by a result of Osborne and Sixsmith [OS16, Theorem 1.3]; in particular, it is non-empty.
Both Theorem 1.4 and Theorem 1.6 are consequences of a much more general result, which concerns wandering compact sets of transcendental entire functions having arbitrary shapes.
Theorem 1.7 (Entire functions with wandering compacta). Let K ⊆ C be a full compact set. Let Z I , Z BU ⊆ K be disjoint finite or countably infinite sets such that no connected component of int(K) intersects both Z I and Z BU . Then there exists a transcendental entire function f such that
(i) ∂K ⊆ J(f ); (ii) f n (K) ∩ f m (K) = ∅ for n = m; (iii) every connected component of int(K) is a wandering domain of f ; (iv) Z I ⊆ I(f ) and Z BU ⊆ BU (f ).
Theorem 1.7 is inspired by a previous result of Boc Thaler [Boc21b]. Let U ⊆ C be a bounded domain such that C \ U is connected, and such that furthermore U is regular in the sense that U = int(U ). Using a version of Runge's approximation theorem, Boc Thaler proved that there is a transcendental entire function f for which U is a wandering domain on which the iterates of f tend to ∞ [Boc21b, Theorem 1]. He then asked the following. If U is a bounded domain such that ∂U is a Lakes of Wada continuum, then the complement of U has at least two connected components by definition. Hence the closure of the Fatou component from Theorem 1.4 has a disconnected complement, answering Question 1.8 in the negative.
In the case where Z BU = ∅ (which suffices for Theorem 1.4) we can prove Theorem 1.7 by a similar proof as Boc Thaler's, but applying a subtle change of point of view: Instead of beginning with a simply connected domain and approximating its closure, as in [Boc21b], we start the construction with the full compact set K ⊆ C and consider its interior components. (See Theorem 3.1.) We note that this construction can be used to provide new counterexamples to a stronger version of Eremenko's conjecture, which asked whether every escaping point can be connected to infinity by a curve consisting of escaping points. This was answered in the negative in [RRRS11, Theorem 1.1], but our construction is much simpler. (See Theorem 3.3.)
The iterates of the function f resulting from this argument converge to infinity uniformly on K, so additional ideas are needed to prove Theorems 1.6 and 1.7. A key new ingredient in the proof is to use Arakelyan's theorem (instead of Runge's theorem) to ensure that there is a sequence of unbounded domains that are mapped conformally over one another by any function involved in the construction (see Section 4 and Figure 6). The presence of these domains allows us to let the image of the compact set K be stretched (horizontally) at certain steps during the construction, ensuring that the spherical diameter of f n (K) does not tend to zero as n → ∞, and allowing K to contain points of both the escaping set and the bungee set.
In order to obtain a counterexample to Eremenko's conjecture, we develop the method even further, now applying it to an unbounded ray K that connects a finite endpoint to ∞. The unboundedness of K allows us to ensure that K is surrounded by connected components of an attracting basin, which means that any connected component of the escaping set intersecting K is also contained in K. By additionally ensuring that the finite endpoint of K escapes, while some other point is in BU (f ), we obtain a counterexample to Eremenko's conjecture. The stronger statement in Theorem 1.2 is obtained by a more careful application of similar ideas.
Recall that the example from Theorem 1.6 has an escaping wandering domain and an oscillating wandering domain, and their (shared) boundary contains both escaping points and bungee points. It seems interesting to investigate, more generally, those points on the boundary of a wandering domain whose orbits have different accumulation behaviour than the interior points. Definition 1.9 (Maverick points). Let f be a transcendental entire function and suppose that U is a wandering domain of f . We say that a point z ∈ ∂U is maverick if there is a sequence (n k ) such that f n k (z) → w ∈ C as k → ∞, but w is not a limit function of (f n k (U )).
Equivalently, a point z ∈ ∂U is maverick if for any (and hence all) w ∈ U , one has lim sup n→∞ dist # (f n (z), f n (w)) > 0, where dist # denotes spherical distance. (Compare Lemma 7.1.) If U is an escaping wandering domain, then z ∈ ∂U is maverick if and only if z / ∈ I(f ). However, if U is oscillating, then ∂U may contain maverick points that are also in the bungee set BU (f ), but with a different accumulation pattern. Indeed, the function f that is constructed in the proof of Theorem 1.7 satisfies dist # (f n (z), f n (w)) > 0 for any distinct z, w ∈ Z BU not belonging to the same interior component of K; see Remark 5.3. In particular, if z was chosen in some component U of int(K) and w ∈ ∂U , then w will be a maverick point of the wandering domain U .
As far as we are aware, Theorems 1.6 and 1.7 provide the first examples of wandering domains whose boundaries contain maverick points. The following result shows that, as suggested by their name, most points on ∂U are not maverick points.
Theorem 1.10 (Maverick points have harmonic measure zero). Let f be a transcendental entire function and suppose that U is a wandering domain of f . The set of maverick points in ∂U has harmonic measure zero with respect to U .
If U ⊆ I(f ), Theorem 1.10 reduces to [RS11, Theorem 1.1]. For non-escaping wandering domains, the theorem strengthens [OS16, Theorem 1.3], which states that, for points z ∈ ∂U from a set of full harmonic measure, the ω-limit set ω(z, f ) = ∞ n=1 {f k (z) : k > n} agrees with the ω-limit set of points in U . Observe that maverick points may have the same ω-limit set as points in U . Indeed, the function in Theorem 1.7 can be constructed so that all points in Z BU share the same ω-limit set, but have different accumulation behaviour; see Remark 5.8. So the set of maverick points may indeed be larger than the set considered by Osborne and Sixsmith.
Benini et al [BFE + 21b, Theorem 9.4] independently prove a result that implies the following weaker version of Theorem 1.10: if U is a wandering domain of a transcendental entire function, and E ⊆ ∂U is a set of maverick points such that additionally dist # (f n (z), f n (w)) → 0 as n → ∞ for all z, w ∈ E, then E has harmonic measure zero. However, their result applies in a setting (sequences of holomorphic maps between simply connected domains) in which the stronger conclusion of Theorem 1.10 becomes false in general; compare [ for the definition and a discussion of logarithmic capacity.) Question 1.11. Let U be a simply connected wandering domain of a transcendental entire function. Does the set of maverick points in ∂U have zero logarithmic capacity when seen from U ? That is, let ϕ : D → U be a conformal isomorphism between the unit disc D and U , and consider the set E ⊆ ∂D of points at which the radial limit of ϕ exists and is a maverick point. Does E have zero logarithmic capacity?
Our proof of Theorem 1.7 can be adapted to show that logarithmic capacity zero is the best one may hope for in the above question. Sets of zero harmonic measure (or zero logarithmic capacity) need not be small in an absolute geometric sense. Indeed, recall that the set of maverick points in our proof of Theorem 1.6 has full harmonic measure when seen from another complementary domain of the Lakes of Wada continuum, and hence has Hausdorff dimension at least 1 by a result of Makarov; see [GM05, Section VIII.2]. By a further application of our construction, we can strengthen the example as follows.
Theorem 1.13. There exists a transcendental entire function f with an escaping or oscillating wandering domain U such that the set of non-maverick points on ∂U has Hausdorff dimension 1, while the set of maverick points contains a continuum of positive Lebesgue measure.
Further questions. We conclude the introduction with several questions arising from our work. While Eremenko's conjecture is false in general, it is shown in [RRRS11, Theorem 1.6] that it holds for all functions of finite order in the Eremenko-Lyubich class B. (See [RRRS11] for definitions.) We may ask whether one of these two conditions can be omitted.
Question 1.14. Can I(f ) have a bounded connected component if f ∈ B has infinite order, or if f / ∈ B has finite order?
The following question is a modification of Question 1.8 which takes into account the new types of wandering domains provided by Theorem 1.7. Recall that if A ⊆ C is compact, then fill(A) denotes the fill of A, that is, the complement of the unbounded connected component of C \ A. Notation. For a set X ⊆ C, let ∂X, int(X), and X denote, respectively, the boundary, the interior, and the closure of X in C. We write dist, dist # and dist U for Euclidean distance, spherical distance, and hyperbolic distance in a domain U ⊆ C, respectively. The Euclidean disc of radius δ > 0 around z ∈ C is denoted D(z, δ), and the unit disc is denoted by D . . = D(0, 1). We use H . . = {z ∈ C : Im z > 0}.
Structure of the paper. In Section 2, we recall the results from approximation theory needed for our constructions, and prove a number of useful lemmas about behaviour that is preserved under approximation. In Section 3, we give a simplified proof of Theorem 1.7 in the special case that the wandering compact set K escapes uniformly, that is, Z BU = ∅ (see Theorem 3.1). We use this to obtain wandering domains that form Lakes of Wada (Theorem 1.4) and new counterexamples to the strong Eremenko conjecture (Theorem 3.3). The results of Section 3 are not strictly required for the proofs of the main results stated in the introduction (Theorem 1.4 also follows from Theorem 1.7), the proof of Theorem 3.1 already contains a number of ideas that are also present in the more involved constructions relating to Theorems 1.7 and 1.2.
In Section 4 we set up the basic structure of the examples constructed in both Theorem 1.7 and Theorem 1.2. The proof of Theorem 1.7 in full generality, and the deduction of Theorem 1.6 from it, is in Section 5. Our counterexamples to Eremenko's conjecture are constructed in Section 6. This construction is similar to that in Section 5, but can be read independently of it. Thus a reader primarily interested in the proof of Theorem 1.2 should study Sections 2, 4 and 6, but could choose to omit Sections 3 and 5.
Theorem 1.10 is proved in Section 7. We conclude the paper by indicating, in Section 8, how Theorems 1.12 and 1.13 can be obtained by modifications of the technique from Section 5.
Preliminary results on approximation
We recall two classical results of approximation theory. The classical theorem of Runge [Run85] (see also [Gai87, Theorem 2 in Chapter II §3]) concerns the approximation of functions on compact and full sets by polynomials. This is sufficient to prove Theorem 1.7 when Z BU = ∅, and hence Theorem 1.4. In contrast, Arakelyan's theorem [Ara64] (see also [Gai87,Theorem 3 in Chapter IV §2]) allows us to approximate functions defined on (potentially) unbounded sets by entire functions; this will be crucial in our remaining constructions, and in particular in the proof of Theorem 1.2.
Theorem 2.1 (Runge's theorem). Let A ⊆ C be a compact set such that C \ A is connected. Suppose that g : A → C is a continuous function that extends holomorphically to an open neighbourhood of A. Then for every ε > 0, there exists a polynomial f such that
|f (z) − g(z)| < ε for all z ∈ A.
Theorem 2.2 (Arakelyan's theorem). Let A ⊆ C be a closed set such that
(i) C \ A is connected;
(ii) C \ A is locally connected at ∞. Suppose that g : A → C is a continuous function that is holomorphic on int(A). Then for every ε > 0, there exists an entire function f such that
|f (z) − g(z)| < ε for all z ∈ A.
In each of our constructions, we use two simple facts concerning approximation. The first is essentially a uniform version of Hurwitz's theorem, while the second is an elementary exercise. For the reader's convenience, we include the proofs.
Lemma 2.3 (Approximation of univalent functions). Let U, V ⊆ C be open, and let ϕ : U → V be a conformal isomorphism. Let A ⊆ U be a closed set such that dist(A, ∂U ) > 0 and η . . = inf z∈A |ϕ (z)| > 0. Then there is ε > 0 with the following property: if f : U → C is holomorphic with |f (z) − ϕ(z)| ≤ ε for all z ∈ U then f is injective on A, with f (A) ⊆ V . Moreover, |f (z)| > η/2 for z ∈ A. If |ϕ (z)| is bounded from above on A, so is |f (z)|.
Remark 2.4. When A ⊆ U is compact, the hypotheses on A are automatically satisfied.
Proof. Let δ < dist(A, ∂U )/2. By Koebe's 1/4-theorem,
ϕ(D(z 0 , δ)) ⊇ D(ϕ(z 0 ), ρ) for all z 0 ∈ A, where ρ . . = δ · η/4. In particular, dist(ϕ(A), ∂V ) ≥ ρ > 0 (2.1) and, since ϕ is injective, |ϕ(z) − ϕ(z 0 )| ≥ ρ (2.2) for all z 0 ∈ A and z ∈ U with |z − z 0 | ≥ δ. Now let f be as in the statement of the lemma, where ε < ρ/2. Then f (z) = f (z 0 ) when |z − z 0 | ≥ δ, (2.3) by (2.2). Moreover, f (A) ⊆ V by (2.1).
We must show that also f (z) = f (z 0 ) for z = z 0 when z ∈ D . . = D(z 0 , δ). In other words, we claim that f (z) − f (z 0 ) = 0 has a unique solution z ∈ D. According to the argument principle, the number of such solutions is given by the winding number of f (∂D) around z 0 . By choice of ε and (2.2), the curves ϕ(∂D) − ϕ(z 0 ) and f (∂D) − f (z 0 ) are homotopic in C \ 0. Thus they have the same winding number around 0. As ϕ is injective, that winding number is 1, and the claim is proved. Together with 2.3, we see that f is injective on A.
Finally, by Cauchy's theorem we have
|f (z 0 ) − ϕ (z 0 )| = 1 2π ∂D(z 0 ,δ) f (ζ) − ϕ(ζ) (ζ − z 0 ) 2 dζ ≤ 2πδ 2π ε δ 2 = ε δ < η 8 (2.4)
for z 0 ∈ A. In particular, |f (z 0 )| > η/2. Moreover, if |ϕ | is bounded from above on A, then |f | is also bounded from above by (2.4). This proves the final statement of the lemma.
Lemma 2.5 (Approximation of iterates). Let U ⊆ C be open, and let g : U → C be continuous. Suppose that K ⊆ U is closed and n ∈ N is such that g k (K) is defined and a subset of U for all k < n. Suppose furthermore that K . . = n−1 k=0 g k (K) satisfies dist( K, ∂U ) > 0 and that g is uniformly continuous at every point of K, with respect to Euclidean distance.
Then for every ε > 0, there is δ > 0 with the following property.
If f : U → C is continuous with |f (z) − g(z)| < δ for all z ∈ U , then |f k (z) − g k (z)| < ε (2.5)
for all z ∈ K and all k ≤ n.
Remark 2.6. The assumptions on K are automatic when K is compact.
Proof. For every n ∈ N, we prove the existence of a function δ n such that δ = δ n (ε) has the desired property. The proof proceeds by induction on n; for n = 1 we may set δ 1 (ε) = ε. Suppose that the induction hypothesis holds for n, and let ε > 0. By uniform continuity, there is δ < dist(g n (K), ∂U ) such that |g(ζ) − g(ω)| < ε/2 whenever ω ∈ g n (K) and |ζ − ω| ≤ δ . Define δ n+1 (ε) = min δ n (ε), δ n (δ ), ε/2 and let z ∈ K. Then (2.5) holds for k ≤ n by the induction hypothesis. Setting ζ . . = f n (z) and ω . . = g n (z), we have |ζ − ω| ≤ δ by the induction hypothesis, and in particular ζ ∈ U . Thus,
|f n+1 (z) − g n+1 (z)| ≤ |f (ζ) − g(ζ)| + |g(ζ) − g(ω)| < ε,
as required.
Corollary 2.7 (Approximating univalent iterates). Let U ⊆ C be open and g : U → C be holomorphic. Suppose that G ⊆ U is open and K ⊆ G is closed with the following properties for some n ≥ 1.
(a) g n is defined and univalent on G.
(b) |g | is bounded from above and below by positive constants on U .
(c) dist(K, ∂G) > 0. Then for every ε > 0, there is δ > 0 with the following property. For any holomorphic f : U → C with |f (z) − g(z)| ≤ δ for all z ∈ U , f n is defined and injective on K, |f k (z) − g k (z)| ≤ ε on K for k ≤ n, and |f | is bounded from above and below by positive constants on K = n−1 k=0 f k (K). Remark 2.8. If K is compact, then the hypothesis on g can be omitted, since it is automatically satisfied for the restriction of f to a neighbourhood of K.
Proof. Note that g k is univalent on G for 1 ≤ k ≤ n. Furthermore, by Koebe's theorem and the assumption on g , g is uniformly continuous at every point of g k−1 (K) and dist(g k (K), ∂U ) > 0, for all k ≥ 1. So dist( K, ∂U ) > 0 and the hypotheses of Lemma 2.5 are satisfied. Thus there is
δ > 0 such that |f k (z) − g k (z)| ≤ ε for all z ∈ K and all k ≤ n if |f (z) − g(z)| ≤ δ on U .
If ε is chosen small enough, then for k ≤ n the hypotheses of Lemma 2.3 are satisfied for ϕ = g k on G. It follows that f k is injective on K with |(f k ) | bounded above and below by positive constants. This completes the proof.
In Sections 3 and 5, we also require the following fact about approximating compact and full sets from above by finite collections of Jordan domain; see Figure 3. This is a classical fact of plane topology, used already by Runge to prove Theorem 2.1 [Run85, Lemma 2.9 (Approximation by unions of Jordan domains). Let K ⊆ C be compact and full. Then there exists a sequence (K j ) ∞ j=0 of compact and full sets such that K j ⊆ int(K j−1 ) for all j ∈ N and ∞ j=0 K j = K.
Each K j may be chosen to be bounded by a finite disjoint union of closed Jordan domains.
Proof. DefineK
j := fill z ∈ C : dist(z, K) ≤ 1 j .
EachK j is compact and full by definition, and clearly K ⊆K j+1 ⊆ int(K j ) for all j. Any z ∈ C \ K can be connected to ∞ by a curve γ disjoint from K, and hence we have z / ∈K j for all sufficiently large j ∈ N. Now fix j and let V 1 , . . . , V m be the finitely many connected components of int(K j ) that intersect K. Then each V is a simply connected domain, and hence (say by the Riemann mapping theorem) there is a Jordan domain U ⊆ V with K ∩ V ⊆ U . The sets K j . . = m =1 U then have the required property.
Uniform escape
In this section, we prove Theorem 1.7 when Z BU = ∅, in which case we can choose f such that the iterates converge to infinity uniformly on all of K. (iv) f n | K → ∞ uniformly as n → ∞.
Let K ⊆ C be a full compact set as in Theorem 3.1, and choose a sequence (K j ) of approximating sets according to Lemma 2.9. By applying an affine transformation, we may assume without loss of generality that K 0 ⊆ D. For j ≥ 0, also choose a non-separating compact and 2 −j -dense subset
P j ⊆ ∂K j .
That is, dist(z, P j ) ≤ 2 −j for all z ∈ ∂K j ; in particular, ∂K is the Hausdorff limit of the sets P j . We shall construct a function f such that all P j are contained in a basin of attraction, while f n tends to infinity on K itself; this ensures that ∂K ⊆ J(f ). In most of our applications, we may choose the P j as finite sets, but for the proof of Theorem 3.3 below we shall choose larger sets P j .
For j ≥ −1, consider the discs
D j . . = D(3j, 1) = z ∈ C : |z − 3j| < 1 .
Our main goal now is to prove the following proposition, which implies Theorem 1.7.
Proposition 3.2. There exists a transcendental entire function f with the following properties:
(a) f (D −1 ) ⊆ D −1 . (b) f j+1 (P j ) ⊆ D −1 for all j ≥ 0. (c) f j is injective on K j for all j ≥ 0, with f j (K j ) ⊆ D j .
Proof of Theorem 3.1, using Proposition 3.2. Let j ≥ 0; then K ⊆ K j and hence we have f j (K) ⊆ D j by (c). In particular, f j | K → ∞ uniformly as j → ∞, and int(K) ⊆ F (f ) by Montel's theorem.
On the other hand, by (a) and (b), we have f k (P j ) ⊆ D −1 for k > j. Since every point z ∈ ∂K ⊆ K is the limit of a sequence of points p j ∈ P j , it follows that the family (f k ) ∞ k=1 is not equicontinuous at z, and thus z ∈ J(f ). In particular, every connected component of int(K) is a wandering Fatou component. This concludes the proof of Theorem 3.1.
Proof of Proposition 3.2. We construct f as the limit of a sequence of polynomials (f j ) ∞ j=0 , which are defined inductively using Runge's theorem. More precisely, for j ≥ 1, the function f j approximates a function g j , defined and holomorphic on a neighbourhood of a compact set A j ⊆ C, up to an error of at most ε j > 0. The function g j in turn is defined in terms of the previous function f j−1 .
Define ∆ j . . = D(−3, 1 + 3j) ⊇ D j−1 for j ≥ 0. The inductive construction ensures the following properties:
(i) For every j ≥ 0, f j j is injective on K j and f j j (K j ) ⊆ D j .
D −1 D 0 D 1 D 2 ∆ 2 ∆ 1 L 2 K 1 P 1 R 1 f 1 (L 2 ) f 1 g 2 g 2
Q 1 Figure 4. The construction of g j in the proof of Proposition 3.2, for j = 2.
(ii) For j ≥ 1, ∆ j−1 ⊆ A j ⊆ ∆ j . (iii) ε 1 < 1/4 and ε j ≤ ε j−1 /2 for j ≥ 2.
To anchor the induction, we set f 0 (z) . . = (z − 3)/2 for z ∈ C. Then (i) holds trivially for j = 0.
Let j ≥ 0 and suppose that f j has been defined, and that ε j has been defined if j ≥ 1. Applying Lemma 2.9 to K j+1 , we find a full compact set L j ⊆ K j with K j+1 ⊆ int(L j ). Set Q j . . = f j j (P j ). By (i), Q j ⊆ D j and Q j ∩ (∂D j ∪ f j j (L j )) = ∅. Let R j be a compact full neighbourhood of Q j disjoint from ∂D j and f j j (L j ), and set
A j+1 . . = ∆ j ∪ R j ∪ f j j (L j ).
We have R j ⊆ D j and f j j (L j ) ⊆ f j j (K j ) ⊆ D j by the inductive hypothesis (i). In particular, neither set intersects ∆ j . It follows that A j+1 satisfies (ii) and the hypotheses of Runge's theorem. We define Figure 4.) By definition, the function g j+1 extends analytically to a neighbourhood of A j+1 . Observe that g j+1 j+1 (P j ) = g j+1 (Q j ) ⊆ g j+1 (R j ) = {−3} ⊆ D −1 , and that g j+1 j+1 is defined and univalent on int(L j ) by (i). Now choose ε j+1 according to (iii) and sufficiently small that any entire function f with |f (z) − g j+1 (z)| ≤ 2ε j+1 on A j+1 satisfies:
g j+1 : A j+1 → C; z → f j (z), if z ∈ ∆ j , −3, if z ∈ R j , z + 3, if z ∈ f j j (L j ). (See
(1) f j+1 (P j ) ⊆ D −1 .
(2) f j+1 is injective on K j+1 ⊆ int(L j ).
(
3) f j+1 (K j+1 ) ⊆ D j+1 .
Here (1) is possible by Lemma 2.5, and (2) and (3) are possible by Corollary 2.7. We now let f j+1 : C → C be a polynomial approximating g j+1 up to an error of at most ε j+1 , according to Lemma 2.1. This completes the inductive construction.
Condition (iii) implies that (f j ) ∞ j=k forms a Cauchy sequence on every set A k , and by (ii), ∞ k=1 A k = C. So the functions f j converge locally uniformly to an entire function f . For 1 ≤ k ≤ j,
|f j (z) − g k (z)| ≤ ε k + · · · + ε j ≤ 2ε k for all z ∈ A k .
Hence the limit function f satisfies |f (z) − g k (z)| ≤ 2ε k for all z ∈ A k and k ≥ 1.
Since g 1 (D −1 ) = f 0 (D −1 ) = D(−3, 1/2), and 2ε 1 < 1/2, it follows that f (D −1 ) ⊆ D −1 . Moreover, f j+1 (P j ) ⊆ D −1 for j ≥ 0 by (1). Finally, by (2) and (3), f j is injective on K j and f j (K j ) ⊆ D j . This completes the proof of Proposition 3.2.
We now prove Theorem 1.4 regarding the existence of Fatou components with a common boundary.
Proof of Theorem 1.4. Let X ⊆ C be a continuum such that C \ X has infinitely many connected components and the boundary of every such component coincides with X. As mentioned in the introduction, such a continuum was first constructed by Brouwer [Bro10,p. 427], and can be obtained using the construction described by Yoneyama [Yon17,p. 60]. Let U be the unbounded connected component of C \ X.
The set K . . = fill(X) = C \ U satisfies ∂K = ∂U = X. Apply Theorem 1.7 to the full continuum K to obtain a transcendental entire function f for which X = ∂K ⊆ J(f ), and every connected component of int(K) ⊆ F (f ) is a simply connected wandering domain. Since there are infinitely many such components, each of which is bounded by X, Theorem 1.4 is proved.
As mentioned in the introduction, we can use the construction from this section to give new counterexamples to the strong version of Eremenko's conjecture. Proof. If X is a singleton, then we may consider a continuum X of which X is a pathconnected component and continue the proof with X instead of X. (For example, we can take X to be as shown in Figure 5.) From now on we therefore suppose that X is not a singleton.
Construct a function f as in Proposition 3.2, where the set P j ⊆ ∂K j is chosen such that no point of K is accessible from C \ (K ∪ ∞ j=0 P j ). For example, let z 0 , z 1 be distinct points of K, and for each j choose an open arc γ j ⊆ ∂K j in such a way that diam(γ j ) ≤ 2 −j for all j and such that, furthermore, γ 2j → z 0 and γ 2j+1 → z 1 as j → ∞. Then the sets P j . . = ∂K j \ γ j have the desired property. Now apply Proposition 3.2, to obtain a function f satisfying the conclusions of Theorem 3.1. Each P j belongs to an iterated preimage of the forward-invariant disc D −1 , and hence is disjoint from J(f ) ∪ I(f ). By choice of P j , this implies that there is no
Scaffolding
Both the function f constructed in Theorem 1.2 and that constructed in Theorem 1.7 will be constructed using a sequence of unbounded horizontal strips (S j ) ∞ j=0 and (T j ) ∞ j=0 , such that S j is mapped univalently over S j+1 and T j+1 . (See Figure 6.) Our wandering set K starts out in the strip T 0 , maps into S 0 and from there into T 1 , then back into S 0 and on to T 2 via S 1 , and so on. The crucial step happens at the time
N j . . = j =1 ( + 1) = j · (j + 3) 2 , (4.1)
when f N j (K) is in T j , and is mapped back inside S 0 by f . Similarly as in the proof of Theorem 3.1, f is the limit of a sequence of functions (f j ) ∞ j=0 constructed using approximation theory. We have to ensure that all these functions share the mapping behaviour on the strips (S j ) described above. This can be achieved using Lemma 2.3 together with Arakelyan's theorem.
To provide the details of this set-up, consider the affine map
Φ : C → C; z → 5z.
For j ≥ 0, set S j . . = {z ∈ C : 5 j+1 − 1 < 4 Im(z) < 5 j+1 + 3} and T j . . = S j + 2i = {z ∈ C : 5 j+1 + 7 < 4 Im(z) < 5 j+1 + 11}.
Observe that Φ(S j ) = {z ∈ C : 5 j+2 − 5 < 4 Im z < 5 j+2 + 15} ⊇ S j+1 ∪ T j+1 Then |Re f (z)| > 4|Re z| for all z ∈ S with |Re z| ≥ 1. Furthermore, for every j ≥ 1, there is a domain V j = V j (f ) ⊆ S 0 such that f j : V j → T j is a conformal isomorphism and f k (V j ) ⊆ S k for k = 0, . . . , j − 1. Moreover, Re z is unbounded from above and below in V j , and |f | is uniformly bounded from above and below by positive constants on
j−1 k=0 f k (V j ).
Proof. For k ≥ 0, definẽ S k . . = {z ∈ S k : dist(z, ∂S k ) ≥ 1/10)} = {z ∈ C : 5 k+1 − 3/5 ≤ 4 Im(z) ≤ 5 k+1 + 13/5} = Φ −1 ({z ∈ C : 5 k+2 − 3 ≤ 4 Im z ≤ 5 k+2 + 13}).
Set U . . = S, V . . = Φ(U ) and A . . = ∞ k=0S k . Then the hypotheses of Lemma 2.3 are satisfied. Let ε be as given in that lemma, chosen additionally such that ε < 1/2, and let f be ε-close to Φ on U . Then
|Re f (z)| ≥ |Re Φ(z)| − 1 2 = 5|Re z| − 1 2 > 4|Re z| for all z ∈ S with Re z ≥ 1. Moreover, f is injective on eachS k with f (S k ) ⊇ S k+1 ∪ T k+1 . So we may define inductively V j j = T j and, for k = 0, . . . , j − 1, V k j . . = (f |S k ) −1 (V k+1 j ).
Then V j . . = V 0 j has the desired properties. Finally, Φ (z) = 5 for all z ∈ C, and hence |f | is bounded from above and below on A by positive constants, according to Lemma 2.3.
Define
D . . = D 0, 1 2 and observe that D ∩S = ∅. The disc D will play the role of the disc D −1 from Section 3. We define A 0 . . = D ∪ S and
g 0 : A 0 → C; z → Φ(z), if z ∈ S, 0, if z ∈ D. Set ε 0 . . = min ε 2 , 1 5 ,
where ε is the number from Lemma 4.1. Apply Arakelyan's theorem to obtain an entire function f 0 such that |f 0 (z) − g 0 (z)| ≤ ε 0 for z ∈ A 0 . (4.3) Any entire function f that is ε 0 -close to f 0 (and hence ε-close to g 0 ) on A 0 satisfies the hypothesis and hence the conclusion of Lemma 4.1, and also
f (D) ⊆ D.
(4.4)
Entire functions with wandering compacta
We now prove Theorem 1.7. Let f 0 and ε 0 be as in the previous section, and suppose that K, Z I and Z BU are as in Theorem 1.7. We may assume that K ⊆ T 0 and that Z = Z I ∪ Z BU = ∅. We may also assume that no component of int(K) contains more than one point of Z (recall that any Fatou component that intersects I(f ) is contained in I(f ), and likewise for BU (f ).) Let (Z j ) ∞ j=0 be a sequence of finite subsets of Z with Z j ⊆ Z j+1 and such that Z = ∞ j=0 Z j .
Remark 5.1. We may always augment Z to be infinite. However, it turns out that the proof is slightly simpler in the case where #Z = 2, which is the case required for the proof of Theorem 1.6. So we allow Z to be either finite or infinite, and discuss those instances where the assumption #Z = 2 leads to simplifications.
Let (ζ j ) ∞ j=0 be a sequence in Z BU such that for every ζ ∈ Z BU there are infinitely many j ≥ 0 such that ζ j = ζ.
Again choose a sequence (K m ) ∞ m=0 of compact and full subsets as in Lemma 2.9, as well as non-separating and 2 −j -dense subsets P j ⊆ ∂K j . We may choose these sets such that K 0 ⊆ S 0 . Our goal is to prove the following version of Proposition 3.2. Recall that the domain V j+1 (f ) was defined in Lemma 4.1.
Proposition 5.2. Let N j be defined as in (4.1). There is a transcendental entire function f and increasing sequence (m j ) ∞ j=0 with the following properties:
(a) f (D) ⊆ D; (b) f N j +1 (P m j ) ⊆ D for all j ≥ 0; (c) f N j +1 is injective on K m j+1 for all j ≥ 0, with f N j +1 (K m j+1 ) ⊆ V j+1 (f ); (d) |Re f N j +1 (ζ j )| ≤ 1 for all j ≥ 0; (e) if N j + 1 ≤ n ≤ N j+1 and ζ ∈ Z j \ {ζ j }, |Re f n (ζ)| ≥ j.
Proof of Theorem 1.7, using Proposition 5.2. Similarly as in the proof of Theorem 3.1, all points in P m j have bounded orbits by (a) and (b), while by (c),
f N j+1 (K) = f N j +j+2 (K) ⊆ f j+1 (V j+1 (f )) = S j+1
for all j ≥ 0, and hence all points in K have unbounded orbits. So ∂K ⊆ J(f ), proving (i).
Furthermore, if N j + 1 ≤ n ≤ N j+1 , then
f n (K) ⊆ f n−N j −1 (V j+1 (f )) ⊆ S n−N j −1 ,
and in particular f n (K) ∩ S j+1 = ∅ for n < j + 1. Hence K is a wandering compactum, and (ii) holds. Together with (i), this also shows that every connected component of int(K) is a wandering domain. It remains to prove (iv). First, let ζ ∈ Z BU . As noted above, all points in K have unbounded orbits, On the other hand, there are infinitely many j ≥ 0 such that ζ j = ζ, and hence Re f N j +1 (ζ) ≤ 1 by (d). Furthermore, by (c), f N j +1 (ζ) ∈ V j+1 ⊆ S 0 and hence 1 ≤ Im f N j +1 (ζ) ≤ 2. So lim inf n→∞ |f n (ζ)| ≤ 3, and ζ ∈ BU (f ) as required.
On the other hand, let ζ ∈ Z I . Let j 0 ≥ 0 be large enough that ζ ∈ Z j for j ≥ j 0 . If n ≥ N j 0 + 1, let j ≥ j 0 be maximal with n > N j + 1. Then Re f n (ζ) ≥ j by (e). Hence Re f n (ζ) → +∞ as n → ∞, and ζ ∈ I(f ).
dist # (f N j +1 (ζ), f N j +1 (ζ )) > 0,
as noted in the introduction. (Recall that we assumed above that no interior component of K contains more than one point of Z.)
Proof of Theorem 1.6, using Theorem 1.7. Let X be a Lakes of Wada continuum with complementary components U 0 , U 1 , U 2 whose boundary agrees with X. We may assume that U 0 is the unbounded connected component of C \ X. Let K = C \ U 0 = fill(X). Choose ζ 1 ∈ U 1 and ζ 2 ∈ U 2 , and set Z I = {ζ 1 } and Z BU = {ζ 2 }. Now apply Theorem 1.7.
Then U 1 is an escaping wandering domain, and U 2 is an oscillating wandering domain. By Theorem 1.10 (or [RS11, Theorem 1.1] and [OS16, Theorem 1.3]),
I(f ) ∩ ∂U 1 = I(f ) ∩ X ⊆ X
has full harmonic measure when viewed from U 1 , while
BU (f ) ∩ ∂U 2 = BU (f ) ∩ X ⊆ X
has full harmonic measure when viewed from U 2 . In particular, these sets are nonempty.
We will require the following preliminary observation concerning the conformal geometry of the sets K m . For ζ ∈ Z and m ≥ 0, let U ζ m denote the connected component of int(K m ) containing ζ.
Observation 5.4. If ω ∈ K \ {ζ} does not belong to the same interior component as ζ, then the hyperbolic distance dist U ζ m (ζ, ω) → +∞ as m → ∞. Proof. Let δ m = max z∈Km dist(z, K). Then any curve connecting ζ and ω in K m must contain a point having Euclidean distance at most δ m from ∂K. This is trivial if one of the points belongs to ∂K, and otherwise it follows from the fact that ∂K separates ζ and ω. By the standard estimate on the hyperbolic metric in a simply connected domain, the hyperbolic density of U ζ m at a point z is bounded below by 1/(2 dist(z, ∂K m )). It follows that
dist U ζ m (ζ, ω) ≥ 1 2 ln 1 + |ζ − ω| δ m → +∞ as m → ∞.
Recall that U ζ m is a Jordan domain; let π ζ m : U ζ m → D be a conformal map with π ζ m (ζ) = 0. By Observation 5.4 and the assumption on the set Z, |π ζ m (ω)| → 1 as m → ∞ for every ω ∈ Z \ {ζ} that belong to the connected component of K containing ζ.
Passing to a subsequence of K m , we may assume that for every such pair ζ, ω ∈ Z, there is a point ξ ζ (ω) ∈ ∂D such that π ζ m (ω) → ξ ζ (ω) as m → ∞. Remark 5.5. Suppose that Z consists of exactly two points ζ and ω that both belong to the same connected component of K. Then we may normalise π ζ m and π ω m so that they map the other point to the positive real axis, and the above property is automatic with ξ ζ (ω) = ξ ω (ζ) = 1.
Recall from Proposition 5.2 that, at time N j +1, the point ζ j should map to a bounded part of the set V j+1 (f ), while other points of Z j should map sufficiently far to the right. In order to achieve this, we shall use the following fact about conformal mappings.
Proposition 5.6. Let Σ be a straight horizontal strip, let Ξ ⊆ ∂D be finite, let s ∈ Σ and let R > 0. Then there is a Jordan domain W ⊆ Σ with s ∈ W and a conformal map ϕ : D → W whose continuous extension to D satisfies ϕ(0) = s and Re ϕ(ξ) > R for all ξ ∈ Ξ.
Remark 5.7. Much more than Proposition 5.6 is true. Indeed, let Ξ ⊆ ∂D be any compact set of logarithmic capacity zero, and let g : Ξ → C be any continuous function such that preimages of different points are unlinked, i.e., belong to disjoint arcs of the unit circle. Then Bishop [Bis06] proved that there is a conformal map f from D to a simply connected domain Ω such that f extends continuously to ∂D and agrees with g on Ξ.
Proposition 5.6 is particularly simple when Ξ contains just a single element ξ. This is the case that arises in the proof of Proposition 5.2 when #Z = 2, and in particular is sufficient for the proof of Theorem 1.6. Here we may choose a Jordan domain W containing s and whose boundary contains a point z at real part greater than R, and choose the conformal map ϕ normalised so that ϕ(0) = s and ϕ(ξ) = z.
Proof of Proposition 5.6. We may assume without loss of generality that −1 / ∈ Ξ. Let H = {z ∈ C : Im z > 0} be the upper half-plane and let Σ = {x + iy : 0 < y < π}.
Define M to be the conformal isomorphism Let p 1 , . . . , p n be the elements of E, in increasing order. Define
M : D → H; z → i · 1 − z 1 + z and set E . . = M (Ξ) ⊆ R.ψ : H → Σ ; z → πi + −1 n · n j=1 Log(z − p j ), (5.1)
where Log is the principal branch of the logarithm. Then ψ maps H conformally to a domain Ω of the form
Ω = x + iy : 0 < y < π \ n−1 j=1 x + i · πj n : x ≥ x j ⊆ Σ ,
where the x j are real numbers. See [EY12, Section 2]. (In fact, this is a Schwarz-Christoffel mapping for the degenerate polygon Ω with angles 0 at ∞ and 2π at the points x j + πij/n.) The map ψ extends continuously to a map from the boundary R ∪ {∞} of H in C to the boundary ∂Ω ∪ {±∞} of Ω in C, with ψ(p j ) = +∞ for j = 1, . . . , n.
Let π : Σ → Σ be the conformal isomorphism with π(ψ(i)) = s and π(+∞) = +∞. Then ϕ 0 . . = π • ψ • M maps 0 to ζ 0 and all points of Ξ to +∞. For r sufficiently close to 1, the points ϕ 0 (r · ξ), with ξ ∈ Ξ, have real part greater than R. The map
ϕ : D → Σ; z → ϕ 0 (rz)
has the desired properties.
Proof of Proposition 5.2. The construction follows a similar pattern as the proof of Proposition 3.2. We inductively construct a sequence (f j ) ∞ j=0 of entire functions, where f j approximates a function g j -continuous on a closed (but unbounded) set A j and holomorphic on its interior -up to a uniform error of at most ε j . For j ≥ 0, define Σ j . . = S ∪ z ∈ C : |4 Im z| ≤ 5 j+1 + 3 .
Then Σ j ∩ T j = ∅, T ⊆ Σ j for < j and ∞ j=0 Σ j = C. The inductive construction ensures the following inductive hypotheses.
(i) For every
j ≥ 0, f N j j is injective on K m j , and f N j j (K m j ) ⊆ T j . (ii) For every j ≥ 1, Σ j−1 ⊆ A j ⊆ Σ j . (iii) ε j ≤ ε j−1 /2 for j ≥ 1.
Recall that f 0 and ε 0 are already defined; we set m 0 . . = 0. The inductive hypothesis (i) holds trivially for j = 0.
Suppose that f j and m j have been constructed. Let R j ⊆ T j be a compact full
neighbourhood of Q j . . = f N j j (P m j ) disjoint from f N j j (K m j +1 ). Let Z j consist of those ω ∈ Z j \ {ζ j } that belong to the same connected component of K as ζ j . Define Ξ . . = {ξ ζ j (ω) : ω ∈ Z j } ⊆ ∂D.
The inductive hypothesis implies, in particular, that f j is 2ε 0 -close to g 0 on A 0 , so the domains V j+1 = V j+1 (f j ) from Lemma 4.1 are defined. Consider the strip Σ = T j+1 . Let s 0 ∈ V j+1 with Re s 0 = 0, and set s .
. = f j+1 j (s 0 ) ∈ Σ. Also set R . . = max Re f j+1 j (z) : Re z ≤ j .
We apply Proposition 5.6 to obtain a domain W j ⊆ Σ and a conformal map ϕ : D → W j , extending continuously to ∂D, such that ϕ(0) = s and Re ϕ(ξ) > R for all ξ ∈ Ξ. Now choose m j+1 ≥ m j + 2 sufficiently large that
Re ϕ(π ζ j m j+1 −1 (ω)) > R for all ω ∈ Z j , and such that ω / ∈ U m j+1 −1 (ζ j ) for ω ∈ Z j \ (Z j ∪ {ζ j }). (Recall that π ζ j m j+1 −1 (ω) → ξ ζ j (ω) ∈ Ξ for ω ∈ Z j .) We set L j . . = K m j+1 −1 . Let α = α j+1 be an affine map such that
α(f N j j (L j )) ⊆ T j+1 \ W j . Set A j+1 . . = Σ j ∪ R j ∪ f N j j (L j )
, then A j+1 satisfies the hypotheses of Arakelyan's theorem. Set B j . . = f N j j (U m j+1 −1 (ζ j )) and define
g j+1 : A j → C; z → f j (z), if z ∈ Σ j , 0, if z ∈ R j , (f j+1 j | V j+1 (f j ) ) −1 (ϕ(π ζ j m j+1 −1 ((f N j j | L j ) −1 (z)))), if z ∈ B j , (f j+1 j | V j+1 (f j ) ) −1 (α j+1 (z)), otherwise.
Then g j+1 is continuous on A j+1 and holomorphic on its interior. Now choose ε j+1 according to (iii) and sufficiently small that any entire function f with |f (z) − g j+1 (z)| ≤ 2ε j+1 on A j+1 satisfies:
(1) f N j +1 (P m j ) ⊆ D.
(2) f N j+1 is injective on K m j+1 ⊆ int(L j ).
(3) f N j + (K m j+1 ) ⊆ S −1 for = 1, . . . , j + 1.
(4) f N j+1 (K m j+1 ) ⊆ T j+1 .
(5) Re f N j +1 (ζ j ) < 1.
(6) Re f N j +1 (ω) > j for ω ∈ Z j . Note that g j+1 itself satisfies these properties, and they are preserved under sufficiently close approximation by Lemma 2.5 and Corollary 2.7.
The inductive construction is complete; let f be the limit function of the f j , which exists by (ii). We have |f (z) − g(z)| < ε j on A j for j ≥ 0. It remains to verify that f satisfies the desired properties. Claim (a) holds by (4.4). Claim (b) holds by (1). Claim (c) holds by (2) and (3). Claim (d) follows from (5). Finally, (e) follows from (6) together with Lemma 4.1.
Remark 5.8. With a slight modifiction of the proof, we may achieve that all points ζ ∈ Z BU have the same ω-limit set. Indeed, for any map f as in Lemma 4.1, the set of points with f n (z) ∈ S n for all n ≥ 0 forms a curve γ f connecting −∞ and +∞, which is the limit of the sets V j . Moreover, iff is uniformly close to f , then γf is uniformly close to γ f .
In choosing the point s 0 = s 0 (j + 1) in the definition of the map g j+1 , we could have chosen any point in V j+1 , as long as for every ζ ∈ Z BU , the points {s 0 (j) : ζ = ζ j } do not tend to infinity. It is not difficult to see that we may choose this sequence in such a way that, for the limit function, the corresponding orbit points accumulate everywhere on γ f , which implies that the ω-limit set of every ζ ∈ Z BU consists of γ f , its forward iterates, and ∞. We omit the details.
Counterexamples to Eremenko's conjecture
We prove Theorem 1.2 first in the case where X = {0}, and then indicate how to modify the proof in order to obtain the more general case. Proof. We shall prove the theorem with [0, ∞) replaced by K . . = [0, ∞) + 7i/2; the result then follows by conjugating with a translation. Let the strips (S j ), (T j ), the disc D, the map f 0 and the numbers N j and ε 0 be as in Section 4. Also recall the definition of the domains V j (f ) from Lemma 4.1. Observe that K ⊆ T 0 .
Similarly as in Proposition 5.2, we construct an entire function f and a sequence (K j ) ∞ j=0 with the following properties for all j ≥ 0, where ζ . . = 7i/2 and t j . . = j + 2: (a) K j is a closed horizontal half-strip with K ⊆ K j+1 ⊆ int(K j ) ⊆ T 0 for all j ≥ 0, and ∞ j=0 K j = K;
(b) f (D) ⊆ D; (c) f N j +1 (∂K j ) ⊆ D; (d) f N j +1 is injective on K j+1 , with f N j +1 (K j+1 ) ⊆ V j+1 (f ); (e) if z = t + 7i/2 with 1/t j ≤ t ≤ t j , then |Re f N j +1 (z)| ≤ 1; (f) if N j + 1 ≤ n ≤ N j+1 , then |Re f n (ζ)| ≥ j.
Observe that this proves the theorem. Indeed, let z ∈ K \ {ζ}. Then by (d) and (e), for j ≥ 0, |Re f N j +1 (z)| ≤ 1 and 1 ≤ Im f N j +1 (z) ≤ 2, while f N j+1 (z) ∈ T j+1 , and hence f N j+1 (z) → ∞ as j → ∞. So z ∈ BU (f ). On the other hand, ζ ∈ I(f ) by (f). Furthermore, by (b) and (c), ∂K j belongs to F (f ) \ I(f ). So, by (a), ∞ j=0 ∂K j separates K from every other point of I(f ) ∪ J(f ).
To construct the function f , we proceed similarly as in the proof of Proposition 5.2, but will need to take some care because now the sets K and K j are unbounded. Thus we once more construct a sequence of entire functions f j , each of which approximates a function g j (defined inductively in terms of f j−1 ) on a set A j up to an error ε j , where A j satisfies the hypotheses of Arakelyan's theorem. Again define Σ j . . = S ∪ z ∈ C : |4 Im z| ≤ 5 j+1 + 3 .
Along with f j , we construct a half-strip K j as in (a), in such a way that the following properties are satisfied:
(
i) f N j j is injective on K j , f N j j (K j ) ⊆ T j and dist(f N j j (K j ), ∂T j ) > 0; (ii) Re f N j j (z) → +∞ as z → ∞ in K j ;
(iii) |f j | is bounded from above and below by positive constants on
N j −1 k=0 f k j (K j ); (iv) Σ j−1 ⊆ A j ⊆ Σ j if j ≥ 1; (v) ε j ≤ ε j−1 /2 if j ≥ 1.
We choose K 0 ⊆ int(T 0 ) to be an arbitrary closed half-strip whose interior contains K. Suppose that f j and K j have been constructed. LetL j be a closed horizontal halfstrip contained in int(K j ) such that ζ ∈ ∂L j and K \ {ζ} ⊆ int(L j ). Let δ j denote the hyperbolic distance, in int(L j ), between 1/t j + 7i/2 and t j + 7i/2. By (i) and (ii), Q j . . = f N j j (∂K j ) is a Jordan arc with both ends at infinity and dist(Q j , ∂T j ) > 0. By (iii), also dist(Q j , f N j j (L j )) > 0. Let R j ⊆ T j be a closed neighbourhood of Q j homeomorphic to a bi-infinite closed strip such that dist(Q j , ∂R j ) > 0 and dist(R j , f N j j (L j )) > 0. Now let W j ⊆ T j+1 be a horizontal strip. If W j is chosen sufficiently thin, then any hyperbolic ball of radius δ j has Euclidean diameter less than 1. Let ϕ : int(L j ) → W j be a conformal isomorphism with Re ϕ(1 + 7i/2) = 0 whose continuous extension to the boundary satisfies ϕ(ζ) = −∞ and ϕ(+∞) = +∞. Then |Re ϕ(t + 7i/2)| < 1 for
1/t j ≤ t ≤ t j . Observe that ϕ(z) is finite for all z ∈ ∂L j \ {ζ}.
Let η > 0 be small (see below); define L j . . =L j − η/2 and ϕ j (z) . . = ϕ(z + η). Then ϕ j : L j → C is continuous. Set
A j+1 . . = Σ j ∪ R j ∪ f N j j (L j ) and g j+1 : A j+1 → C; z → f j (z), if z ∈ Σ j , 0, if z ∈ R j , f j+1 j | V j+1 (f j ) −1 ϕ j (f N j j | L j ) −1 (z) ,
otherwise. If η is chosen sufficiently small, then
• L j ⊆ K j and f N j j (L j ) is disjoint from R j ; • |Re g N j +1 j+1 (ζ)| = Re f j+1 j | V j+1 (f j )
−1 (ϕ j (ζ)) > j;
• |Re g N j+1 j+1 (t + 7i/2)| = |Re ϕ(t + η + 7i/2)| < 1 for 1/t j ≤ t ≤ t j . Let K j+1 be any closed half-strip contained in the interior of L j and containing K, chosen sufficiently small such that
max z∈∂K j+1 dist(z, K) ≤ 1 j + 1 . (6.1)
The set A j+1 consists of a sequence of pairwise disjoint topological closed strips, tending uniformly to ∞, and therefore satisfies the hypotheses of Arakelyan's theorem. Observe that g N j+1 j+1 is injective on L j . By the inductive hypothesis, |g j+1 | is bounded from above and below by positive constants on N k −1 k=0 g k j+1 (L j ). The same is true on S by Lemma 4.1. We claim that it also holds on f N j j (L j ). To see this, it is enough to show that ϕ j is bounded from above and below on L j .
Indeed, any conformal isomorphism between a half-stripΣ and a bi-infinite strip Σ that takes ∞ to ∞ can be explicitly obtained as a composition of the following elementary operations:
• an affine change of variable mappingΣ to the half-strip {a+ib : a < 0, 0 < b < π};
• the exponential map, which maps the latter half-strip to the upper half-disc and takes −∞ to 0; • the transformation z → ((1 + z)/(1 − z)) 2 , which maps the half-disc to the upper half-plane, with 0 → 1; • a Möbius transformation of the upper half-plane that maps 1 to 0;
• the principal branch of the logarithm, which maps the upper half-plane to the strip {a + ib : 0 < b < π}; with 0 → −∞; • an affine change of variable mapping the latter strip to Σ.
From this, it follows easily that ϕ (z) converges to a constant as Re z → +∞ inL j , from which the claim follows.
We now choose ε j+1 ≤ ε j /2 so small that any entire function f with |f (z) − g j+1 (z)| ≤ 2ε j+1 on A j+1 satisfies:
(1) f N j +1 (∂K j ) ⊆ D;
(2) f N j+1 is injective on K j+1 , with |f | bounded above and below by positive constants on
N j+1 −1 k=0 (f k (K j+1 )); (3) f N j +1 (K j+1 ) ⊆ V j+1 (f ); (4) |Re f N j +1 (ζ)| > j; (5) |Re f N j+1 (t + 7i/2)| < 1 for 1/t j ≤ t ≤ t j .
Since g j+1 itself satisfies these properties, such an ε j+1 exists by Lemma 2.5 and Corollary 2.7. Apply Arakelyan's theorem to obtain the desired function f j+1 , with |f j+1 (z) − g j+1 (z)| ≤ ε j+1 on A j+1 . Clearly (i) to (v) hold for f j+1 ; this concludes the inductive construction.
By (iv), we have A j+1 ⊇ A j and ∞ j=1 A j = C. By (v), the functions (f j ) form a Cauchy sequence on every A j , and thus converge locally uniformly to an entire function f with |f (z) − g j (z)| ≤ 2ε j for all j ≥ 0 and all z ∈ A j . Properties (a) to (d) follow directly from the construction. Indeed, Property (a) holds by definition of K j and property (b) by choice of ε 0 . Claim (c) is a consequence of (1), while (d) follows from (2) and (3).
Finally, property (e) follows from (5) by the statement on real parts in Lemma 4.1; likewise, (f) follows from (4).
We now prove Theorem 1.2 which is a more general version of Theorem 6.1.
Proof of Theorem 1.2. Let X ⊆ C be a full plane continuum. As in the proof of Theorem 6.1, we may assume that X ⊆ T 0 . Let ξ ∈ X be a point with maximal real part, and let K be the union of X with the horizontal ray γ . . = ξ + [0, +∞).
We proceed with the recursive construction as before, but must adjust the definition of L j and K j+1 . While these can no longer be chosen to be straight horizontal strips, the intersection of each of these sets with a suitable right half-plane will be such a strip. Let K 0 to be an arbitrary topological half-strip with this property, such that K 0 ⊆ T 0 and K ⊆ int(K 0 ).
In the recursive step of the construction (when K j has been defined), we first let L j ⊆ int(K j ) to be a closed horizontal half-strip with ξ + ε ∈ ∂L j and ξ + (ε, +∞) ⊆ int(L j ), where ε < 1/t j . Exactly as in the proof of Theorem 6.1, we find a conformal map ϕ : int(L j ) → W j , where W j ⊆ T j+1 is a horizontal strip such that ϕ(+∞) = +∞, ϕ(ξ + ε) = −∞, and |Re ϕ(ξ + t)| < 1 for 1/t j ≤ t ≤ t j . Now letL j ⊇L j be a closed topological half-strip obtained fromL j by adding a small neighbourhood of X toL j . Letφ : int(L j ) → W j be a conformal isomorphism witĥ ϕ(ξ + 1) = ϕ(ξ + 1) andφ(+∞) = +∞. We suppose that int(L j ) is sufficiently close to int(L j ) in the sense of Carathéodory kernel convergence (seen from ξ + 1); compare [Pom92, Section 1.4]. Thenφ still maps ξ + [1/t j , t j ] to points at real parts less than 1, while |Re(f j+1 j | V j+1 (f j ) ) −1 (φ(z))| > j for all z ∈ K.
We now let K j+1 ⊆ L j ⊆L j be slightly smaller topological half-strips and let ϕ j+1 be the restriction ofφ to L j . It is easy to see that |ϕ j+1 | is bounded from above and below on L j , and the remainder of the proof proceeds as before.
Maverick points
In this section, we prove Theorem 1.10. We begin with the following observation.
Lemma 7.1 (Maverick points and spherical diameter). Let f be a transcendental entire function and suppose that U is a wandering domain of f . Let ζ ∈ U and z ∈ ∂U . Then z is maverick if and only if dist # (f n (ζ), f n (z)) → 0 as n → ∞. In particular, if diam # (f n (U )) → 0 as n → ∞, then there are no maverick points for U .
Proof. The first claim is a simple restatement of the definition of a maverick point. Indeed, if f n k (z) → w ∈ C and f n k (ζ) → w as k → ∞, then dist # (f n k (z), f n k (ζ)) → 0 as k → ∞. Conversely, if (n k ) is a subsequence such that dist # (f n k (z), f n k (ζ)) ≥ ε > 0 for all k ∈ N, then we may pass to a further subsequence f n k j such that f n k j (z) and f n k j (ζ) converge to distinct points on the sphere.
The second claim follows immediately from the first.
Our proof of Theorem 1.10 is similar to that of [OS16, Theorem 1.5]. In fact, it turns out that the proof is simplified when we use our more general, and perhaps more natural, definition of maverick points, despite giving a stronger result. As with [OS16, Theorem 1.5], we use the following, lemma which is [OS16, Lemma 4.1], based in turn on [RS11, Theorem 1.1].
Lemma 7.2. Let (G n ) be a sequence of pairwise disjoint simply connected domains in C. Suppose that, for each n ∈ N, g n : G n−1 → G n is analytic in G n−1 , continuous in G n−1 , and satisfies g n (∂G n−1 ) ⊆ ∂G n . Let h n = g n • · · · • g 2 • g 1 for n ∈ N.
Let ξ ∈ C, δ ∈ (0, 1), c > 1, and z 0 ∈ G 0 . Then H = {z ∈ ∂G 0 : dist # (h n (z), ξ) ≥ cδ and dist # (h n (z 0 ), ξ) < δ for infinitely many n} has harmonic measure zero relative to G 0 .
Proof of Theorem 1.10. Let f be a transcendental entire function with a wandering domain U . As mentioned in the introduction, if U is multiply connected, then U ⊆ I(f ); see [RS05,Theorem 2]. Hence, we assume that U is simply connected and fix some z 0 ∈ U .
For 0 < δ < 1 and ξ ∈Ĉ define H(ξ, δ) to consist of all points z ∈ ∂U for which there is an incrasing sequence n k such that dist # (f n k (z 0 ), ξ) < δ/2 and dist # (f n k (z), ξ) > δ for all k.
Set G 0 = U , g n = f n k −n k−1 , and let G n be the Fatou component of f that contains the image of the wandering domain f n k (U ). Then, each G n is a disjoint simply connected wandering domain and satisfies g n (∂G n−1 ) ⊆ ∂G n . We observe that by Lemma 7.2 the harmonic measure of H(ξ, δ) is zero relative to U . Now, we consider a covering of the plane by discs with rational center and rational radius, and consider limit points of U in these discs. By Lemma 7.1, if z ∈ ∂U is a maverick point, then there exists a subsequence n k j , δ ∈ Q with 1/2 > δ > 0, and ξ ∈ Λ(U, f, (n k j )) such that dist # (f n k j (z 0 ), ξ ) < δ/4 and dist # (f n k j (z), ξ ) > 2δ. Further, by the density of the rational numbers, there exists ξ ∈ Q(i) such that dist # (f n k j (z 0 ), ξ) < δ and dist # (f n k j (z), ξ) > 3δ/2. Hence, z ∈ H(ξ, δ). Thus, {z ∈ ∂U : z is a maverick point} ⊆ ξ∈Q(i) δ∈Q∩(0,1/2) H(ξ, δ).
Since H(ξ, δ) has zero harmonic measure relative to U and by countable additivity of the harmonic measure, the set of maverick points has zero harmonic measure relative to U . Remark 7.3. Note that the proof does not require us to distinguish whether U is escaping, oscillating or orbitally bounded. (The latter means that every point in U has bounded orbit; it is unknown whether orbitally bounded wandering domains exist.)
Large sets of maverick points
We now return to the construction of Section 5 and indicate how we may modify the proof of Proposition 5.2 to obtain examples with further properties, including Theorems 1.12 and 1.13. Recall that Theorem 1.12 says that in the case that U = D, we may obtain any compact subset of ∂D of logarithmic capacity zero as the set of maverick points. On the other hand, Theorem 1.13 shows that for some domains, the set of maverick points may have positive Lebesgue measure.
We begin with a general observation on the proof of Proposition 5.2. For simplicity, assume that K is connected, so that U ζ m = int(K m ) for all m and ζ ∈ K. Suppose that, instead of the sets Z I and Z BU , we instead are given a sequence ζ j ∈ K and sets Z j ,Z j ⊆ K. Assume furthermore that the conformal maps π ζ j m can be chosen such that π ζ j m (Z j ) converges to a compact set Ξ j ⊆ ∂D of zero logarithmic capacity, and similarly π ζ j m (Z j ) converges to a compact setΞ j of zero logarithmic capacity, disjoint from Ξ j .
By the results of [Bis06] mentioned above, we may choose the domain W j and the conformal map ϕ in such a way that Re ϕ(ξ) > R for all ξ ∈ Ξ, such that |Re ϕ(ξ)| < 1 for all ξ ∈Ξ, and such that either |Re ϕ(0)| < 1 or |Re ϕ(0)| > R. In particular, for the resulting function f , any point that belongs to infinitely many Z j will be in BU (f ), and any point that is inZ j for all sufficiently large j belongs to I(f ).
(We remark that [Bis06] does not explicitly mention the image of the point 0, but it follows from the construction that we may choose this to be arbitrarily close to any given point of the complex plane. Moreover, for our two applications, which we discuss in greater detail below, the main results of [Bis06] are not required.)
Proof of Theorem 1.12. To construct the function f BU , we follow the proof of Proposition 5.2, using K . . = D(3i/2, 1/4) and a sequence K m = D(3i/2, r m ) of concentric discs shrinking to K. We take ζ j = 0 for all j ≥ 0 and define Z . . = 3i 2 + ξ 4 : ξ ∈ Ξ .
Proposition 5.6 also holds for any compact set Ξ of zero logarithmic capacity, with the same proof except that the map ψ on the upper half-plane is now given by ψ = u + iv, where
u(z) = R log 1 |z − x| dµ(x),
with µ a suitable probability measure supported on Ξ; see [Bis06, Proof of Theorem 6].
The proof then proceeds exactly as before.
To construct f I , we note that we may choose the map ϕ in Proposition 5.6 in such a way that Re ϕ(0) > R and |Re ϕ(ξ)| < 1 for all ξ ∈ Ξ. Indeed, we can postcompose the map ψ in the proof of the proposition by a conformal isomorphism mapping the strip Σ to a the positive half-strip {z ∈ Σ : Re z > 0} in such a way that +∞ maps to a point with real part 0 and s = ϕ(0) maps to a point at real part greater than R. Using this map in the construction of Proposition 5.2 yields a function with the desired properties.
Proof of Theorem 1.13. Let Z ⊆ C be a Jordan arc of positive Lebesgue measure, let γ : (0, 1) → C be an injective analytic curve that accumulates everywhere on Z in both directions, and let U be the bounded connected component of C \ (Z ∪ γ). Then U is a regular simply connected domain with ∂U = Z ∪ γ; we set K . . = U . Then Z is the impression of a single prime end of U , while every other prime end corresponds to a point on γ. Let ζ 0 ∈ U and let (K m ) be a nested sequence of closed Jordan domains shrinking down to K. Set U m = int(K m ) and let π m : U m → D be a conformal isomorphism with π m (ζ 0 ) > 0. Then it follows that π m (Z) converges to a single point ξ ∈ ∂D. We may now proceed with the construction as before (with ζ j = ζ 0 for all j ≥ 0 and Ξ = {ξ} at every step) to obtain a transcendental entire function f for which U is an oscillating wandering domain and Z ⊆ I(f ). Since BU (f ) ∩ ∂U ⊆ γ has Hausdorff dimension 1, the function has the desired properties.
To obtain an escaping wandering domain, we proceed as above ensuring that ζ 0 ∈ I(f ) and Z ⊆ BU (f ).
Conjecture 1. 1 (
1Eremenko's conjecture). Every connected component of the escaping set of a transcendental entire function is unbounded. The conjecture has been confirmed in a number of cases; see [SZ03, RS05, Rem07, RS13, ORS19, NRS21] and [RRRS11, Theorem 1.2]. On the other hand, stronger versions have been disproved; see [RRRS11, Theorem 1.1] and [Rem16, Theorem 1.6]. It is known that I(f ) ∪ {∞} is always connected
Theorem 1 . 2 (
12Counterexamples to Eremenko's conjecture). Let X ⊆ C be a non-empty full and connected compact set. Then there exists a transcendental entire function f such that X is a connected component of I(f ). (A bounded connected component of I(f ) need not be compact, but any compact connected component is necessarily full.) The examples in Theorem 1.2 are constructed using approximation theory; more precisely, we use a famous theorem of Arakelyan on approximation by entire functions (see Theorem 2.2). The use of approximation theory to construct examples in transccendental dynamics has a long history, going back to work of Eremenko and Lyubich from 1987 [EL87]. Our Theorem 1.2 is in fact an application of a new general method for prescribing the dynamical behaviour of an entire function on certain closed subsets of the complex plane. We use this procedure to answer a number of further open problems in transcendental dynamics, which we describe in the following. For a transcendental entire function or a rational self-map of the Riemann sphere C . . = C ∪ {∞}, the Fatou set F (f ) is the set of those values whose orbits under f remain stable under small perturbations. More formally, F (f ) is the largest open set on which the iterates of f are equicontinuous with respect to spherical distance; equivalently, it is the largest forward-invariant open set that omits at least three points of the Riemann sphere [Ber93, Section 2.1]. The complement J(f ) . . = C \ F (f ) is called the Julia set.
Figure 1 .
1A Lakes of Wada continuum bounding four domains.
Figure 2 .
2First steps in the construction of a Lakes of Wada continuum.
Question 1 . 5 (
15Rippon [HL19, p. 61]). If U is a bounded escaping wandering domain of a transcendental entire function f , is ∂U ⊆ I(f )?
Question 1 . 8 (
18Boc Thaler[Boc21b, p. 3]). Is it true that the closure of any bounded simply connected Fatou component of an entire function has a connected complement?
Theorem 1. 12 .
12Let E ⊆ ∂D be a compact set of zero logarithmic capacity. Then there exist transcendental entire functions f I and f BU such that (i) D is an escaping wandering domain of f I ; (ii) D is an oscillating wandering domain of f BU ; (iii) every point in E is maverick for both f I and f BU .
Question 1 . 15 .
115Suppose that U is a bounded simply connected Fatou component of a transcendental entire function, and let K = fill(U ). Is it true that ∂U = ∂K?A positive answer to Question 1.15 would imply that Theorem 1.7 gives a complete description of the bounded simply connected wandering domains of entire functions.Our results imply the existence of both escaping and oscillating simply connected wandering domains with Lakes of Wada boundaries. In contrast, it appears much more difficult to construct invariant Fatou components with Lakes of Wada boundaries, if this is at all possible.Question 1.16. Let f be a transcendental entire function and suppose that U is an invariant Fatou component of f . Must every connected component of C\U intersect J(f )?A positive answer to Question 1.16 would imply, in particular, that a transcendental entire function has at most one completely invariant Fatou component; compare[RS19]. The above-mentioned partial results in the polynomial case motivate the following strengthening of Question 1.16 for bounded U .
Question 1 . 17 .
117Let f be an entire function and suppose that U is a bounded invariant Fatou component of f . Must ∂U be a simple closed curve?
Figure 3 .
3Nested sequences of compacta (K j ) shrinking down to the filled-in Julia set K = K(p) of p(z) = z 2 + c with c ≈ −0.12 + 0.74i. pp. 230-231] (see also [Wal69, Theorem 2 on pp. 7-8]). For the reader's convenience, we provide a simple proof.
Theorem 3. 1 (
1Wandering compacta with uniform escape). Let K ⊆ C be a full compact set. Then there exists a transcendental entire function f such that (i) ∂K ⊆ J(f ); (ii) f n (K) ∩ f m (K) = ∅ when n = m; (iii) every connected component of int(K) is a wandering domain of f ;
Theorem 3 . 3 (
33Counterexamples to the strong Eremenko conjecture). Let X ⊆ C be a full continuum. Then there exists a transcendental entire function f such that every pathconnected component of X is a path-connected component of the escaping set I(f ), and every path-connected component of ∂X is a connected component of J(f ). In particular, no point of X can be connected to ∞ by a curve in I(f ).
Figure 5 .
5A continuum X having a singleton path-connected component. X is obtained as a countable union of progressively smaller copies of the sin(1/x) continuum, accumulating on a single point (the left-most point in thefigure).curve in J(f ) ∪ I(f ) connecting a point of C \ K to a point of K. Since K ⊆ I(f ), every path-connected component of K is a path-connected component of I(f ); similarly, since ∂K = K ∩ J(f ), every path-connected component of ∂K is a path-connected component of J(f ).
Figure 6 .
6≥ 0. (Compare Figure 6.) Define S . . = ∞ j=0 S j . Lemma 4.1. There is an ε > 0 with the following property. Suppose that f : S → C is analytic and |f (z) − Φ(z)| ≤ ε for all z ∈ S. The strips (S k ) and (T k ) used in the construction.
Remark 5 . 3 .
53Let ζ ∈ Z BU , and ζ ∈ Z \ {ζ}. Then (d) and (e) show that lim sup j→∞
Theorem 6 . 1 .
61There is a transcendental entire function f such that(a) [0, ∞) ⊆ J(f ); (b) [0, ∞) is a connected component of J(f ) ∪ I(f ); (c) 0 ∈ I(f ); (d) (0, ∞) ⊆ BU (f ).In particular, {0} is a connected component of I(f ).
contained in a periodic Fatou component; otherwise U is called a wandering domain of f . The work of Fatou and Julia left open the question whether wandering domains exist. In 1976, Baker
"S'il y a plus de deux régions contiguës distinctes (et par suite une infinité), deux régions contiguës peuvent-elles avoir la même frontière, sansêtre identiques?"
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The iterated minimum modulus and conjectures of Baker and Eremenko. J W Osborne, P J Rippon, G M Stallard, J. Anal. Math. 1392J. W. Osborne, P. J. Rippon, and G. M. Stallard, The iterated minimum modulus and conjectures of Baker and Eremenko, J. Anal. Math. 139 (2019), no. 2, 521-558.
On the set where the iterates of an entire function are neither escaping nor bounded. J W Osborne, D J Sixsmith, Ann. Acad. Sci. Fenn. Math. 41J. W. Osborne and D. J. Sixsmith, On the set where the iterates of an entire function are neither escaping nor bounded, Ann. Acad. Sci. Fenn. Math. 41 (2016), 561-578.
Sources and sinks of A-diffeomorphisms of surfaces. R V Plykin, Mat. Sb. (N.S.). 94136R. V. Plykin, Sources and sinks of A-diffeomorphisms of surfaces, Mat. Sb. (N.S.) 94(136) (1974), 243-264, 336.
Boundary behaviour of conformal maps, Grundlehren der mathematischen Wissenschaften. Ch, Pommerenke, Springer-Verlag299BerlinFundamental Principles of Mathematical SciencesCh. Pommerenke, Boundary behaviour of conformal maps, Grundlehren der mathematis- chen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 299, Springer- Verlag, Berlin, 1992.
On a question of Eremenko concerning escaping components of entire functions. L Rempe, Bull. Lond. Math. Soc. 394L. Rempe, On a question of Eremenko concerning escaping components of entire functions, Bull. Lond. Math. Soc. 39 (2007), no. 4, 661-666.
Arc-like continua, Julia sets of entire functions, and Eremenko's conjecture. L Rempe-Gillen, L. Rempe-Gillen, Arc-like continua, Julia sets of entire functions, and Eremenko's conjec- ture, 2016.
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Fast escaping points of entire functions. P J Rippon, G M Stallard, Proc. Lond. Math. Soc. 3P. J. Rippon and G. M. Stallard, Fast escaping points of entire functions, Proc. Lond. Math. Soc. (3) 105 (2012), no. 4, 787-820.
Baker's conjecture and Eremenko's conjecture for functions with negative zeros. P J Rippon, G M Stallard, J. Anal. Math. 120P. J. Rippon and G. M. Stallard, Baker's conjecture and Eremenko's conjecture for functions with negative zeros, J. Anal. Math. 120 (2013), 291-309.
On connected preimages of simply-connected domains under entire functions. L Rempe-Gillen, D Sixsmith, Geom. Funct. Anal. 295L. Rempe-Gillen and D. Sixsmith, On connected preimages of simply-connected domains under entire functions, Geom. Funct. Anal. 29 (2019), no. 5, 1579-1615.
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Escaping points of exponential maps. D Schleicher, J Zimmer, J. London Math. Soc. 2D. Schleicher and J. Zimmer, Escaping points of exponential maps, J. London Math. Soc. (2) 67 (2003), no. 2, 380-400.
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Theory of continuous set of points. K Yoneyama, Tôhoku Math. J. 12K. Yoneyama, Theory of continuous set of points, Tôhoku Math. J. 12 (1917), 43-158.
|
[] |
[
"Dynamical gluon mass in the instanton vacuum model",
"Dynamical gluon mass in the instanton vacuum model"
] |
[
"M Musakhanov [email protected] \nTheoretical Physics Department\nNational University of Uzbekistan\n100174TashkentUzbekistan\n",
"O Egamberdiev \nTheoretical Physics Department\nNational University of Uzbekistan\n100174TashkentUzbekistan\n"
] |
[
"Theoretical Physics Department\nNational University of Uzbekistan\n100174TashkentUzbekistan",
"Theoretical Physics Department\nNational University of Uzbekistan\n100174TashkentUzbekistan"
] |
[] |
We consider the modifications of gluon properties in the instanton liquid model (ILM) for the QCD vacuum. Rescattering of gluons on instantons generates the dynamical momentum-dependent gluon mass M g (q). First, we consider the case of a scalar gluon, no zero-mode problem occurs and its dynamical mass M s (q) can be found. Using the typical phenomenological values of the average instanton size ρ = 1/3 f m and average inter-instanton distance R = 1 f m we get M s (0) = 256 M eV . We then extend this approach to the real vector gluon with zero-modes carefully considered. We obtain the following expression M 2 g (q) = 2M 2 s (q). This modification of the gluon in the instanton media will shed light on nonperturbative aspect on heavy quarkonium physics.
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10.1016/j.physletb.2018.01.080
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[
"https://arxiv.org/pdf/1706.06270v3.pdf"
] | 118,961,323 |
1706.06270
|
e14dc1b3263b611699404e6581bb4ea61d9155ed
|
Dynamical gluon mass in the instanton vacuum model
31 Jan 2018
M Musakhanov [email protected]
Theoretical Physics Department
National University of Uzbekistan
100174TashkentUzbekistan
O Egamberdiev
Theoretical Physics Department
National University of Uzbekistan
100174TashkentUzbekistan
Dynamical gluon mass in the instanton vacuum model
31 Jan 2018PACS numbers: * Electronic address: 1
We consider the modifications of gluon properties in the instanton liquid model (ILM) for the QCD vacuum. Rescattering of gluons on instantons generates the dynamical momentum-dependent gluon mass M g (q). First, we consider the case of a scalar gluon, no zero-mode problem occurs and its dynamical mass M s (q) can be found. Using the typical phenomenological values of the average instanton size ρ = 1/3 f m and average inter-instanton distance R = 1 f m we get M s (0) = 256 M eV . We then extend this approach to the real vector gluon with zero-modes carefully considered. We obtain the following expression M 2 g (q) = 2M 2 s (q). This modification of the gluon in the instanton media will shed light on nonperturbative aspect on heavy quarkonium physics.
I. INTRODUCTION
Without any doubt instantons represent a very important topologically nontrivial component of the QCD vacuum. The main parameters of the QCD instanton vacuum developed in the instanton liquid model (ILM) are the average instanton size ρ and inter-instanton distance R (see, for example, following reviews [1,2]). They were phenomenologically estimated as ρ = 1/3 fm, R = 1 fm and confirmed by theoretical variational calculations [1,2] and recent lattice simulations of the QCD vacuum [3][4][5][6][7]. In particular, the spontaneous breakdown of chiral symmetry is realized very well via the ILM [8]. Hence, instantons play a pivotal and significant role in describing the lightest hadrons and their interactions.
In the ILM the instanton induces strong interactions between light quarks and produce a large dynamical mass M , which was initially almost massless. Consequently, light quarks are bound and the pions as a pseudo-Goldstone boson appear as a result of spontaneous breakdown of chiral symmetry (SBCS).
On the other hand, the instantons from the QCD vacuum also interact with heavy quarks and are responsible for the generation of the heavy-heavy and heavy-light quark interactions with trace of the SBCS [9][10][11]. It is important to note that the packing parameter is given as ρ 4 /R 4 and it become very small (∼ 0.01) with the phenomenological values of ρ and R used. Then, the dynamical quark mass M is expressed as (packing parameter) 1/2 ρ −1 ∼ 365 MeV [1] while the instanton contribution to the heavy quark mass is given as ∆M ∼ (packing parameter)ρ −1 ∼ 70 MeV [12]. We see that these specific packing parameter dependencies explain the values of M and ∆M . These factors define the coupling between the light-light, heavy-light and heavy-heavy quarks induced by the instantons from the QCD vacuum.
The direct instanton effects mainly contribute to the intermediate region characterized by the instanton size (ρ 0.33 fm), as was studied in Ref. [13] in which the instanton effects are marginal but still important to be considered for a quantitative description of the heavy quarkonium spectra. One-gluon exchange is dominant at smaller distances. On the other hand, a size of the heavy quarkonium is small [14]. It means that heavy quarkonium properties might be sensitive to a modification of gluon properties in instanton media (ILM)
induced by rescattering of gluons on the instantons.
Previously the dynamical gluon mass M g within ILM was estimated in [15] as M g ∼ 400 M eV with phenomenological values of ρ and R. However, this estimation was obtained by ignoring the gluon zero-modes problem and some SU (N c ) factors.
In this work, we aim at investigating the dynamical gluon mass within the ILM, extending the method developed in Ref. [16], where the formulae for the quark correlators were derive.
II. SCALAR "GLUON" PROPAGATOR
We start from the scalar massless field φ belonging to the adjoint representation as a real gluon. We have to find its propagator in the external classical gluon field in the ILM The action is defined as
A µ = I A I µ (γ I ), where A I µ (γ I )S φ = (φ + P 2 φ) where P µ = p µ + A µ (in the coordinate represen- tation p µ = i∂ µ ).
The scalar gluon-like propagator is given by
∆ = (p + A) −2 = (p 2 + i ({p, A i } + A 2 i ) + i =j A i A j ) −1 , ∆ 0 = p −2 ,(1)∆ = (p 2 + i ({p, A i } + A 2 i )) −1 , ∆ i = P −2 i = (p 2 + {p, A i } + A 2 i ) −1 .
There are no zero modes in ∆ −1 i = P 2 i and ∆ −1 = P 2 , which means the existence of the inverse operators ∆ i and ∆. Our aim is to find the propagator averaged over instanton collective coordinates∆ ≡< ∆ >= Dγ ∆. However, we start first from∆. Expanding∆
over ({p, A i } + A 2
i ) carrying out further resummation, we obtain the multi-scattering series
∆ = ∆ 0 + i (∆ i − ∆ 0 ) + i =j (∆ i − ∆ 0 )∆ −1 0 (∆ j − ∆ 0 ) + ...(2)
As in Ref. [16], the main contribution to the∆ can be summed up by the following equatioñ
∆ − ∆ 0 = i < {∆ −1 (∆ −1 0 − ∆ −1 i ) −1 ∆ −1 i − (∆ −1 0 −∆ −1 )} −1 >(3)
Rewriting this equation, we havē
∆ −1 − ∆ −1 0 = i < {∆ + (∆ −1 i − ∆ −1 0 ) −1 } −1 >(4)
We can derive the solution of Eqs. (3) and (4)
∆ −1 − ∆ −1 0 = < i {∆ 0 + (∆ −1 i − ∆ −1 0 ) −1 } −1 > = − < i ∆ −1 0 (∆ i − ∆ 0 )∆ −1 0 >= N ∆ −1 0 (∆ I − ∆ 0 )∆ −1 0 ,(5)where∆ I = dγ I ∆ I . Now we compare ∆ with∆. Expanding ∆ with respect to A i A j , we get ∆ =∆ −∆ i =j A i A j ∆ =∆ −∆ i =j A i A j∆ + ...(6)
It means immediately∆ −∆ = O(N 2 ) which is negligible. We have to take a well-known results for ∆ I from Ref. [17]:
∆ ab I = 1 2 tr τ a F (x, y)τ b F (y, x) 4π 2 (x − y) 2 Π(x)Π(y) , Π(x) = x 2 + ρ 2 x 2 ,(7)τ µ = ( τ , i), τ + µ = ( τ , −i), τ µ τ + ν = δ µν + iη aµν τ a ,(8)
F (x, y) = 1 + ρ 2 (τ x)(τ + y) x 2 y 2 = 1 + ρ 2 (xy)
x 2 y 2 + ρ 2 iη aµν τ a x µ y ν x 2 y 2 ,(9)
whereη aµν = −η aνµ is the 'tHooft symbol. We assume that the position of the instanton z = 0 and the orientation U = 1. It is clear to see from Eq. (5) that the gluon-like scalar dynamical mass operator is given by
M 2 s δ ab =< i p 2 (∆ ab i − ∆ ab 0 )p 2 >= N (p 2∆ab I p 2 − δ ab p 2 ).(10)
In order to average over the position z, we have to change x → x − z, y → y − z and perform integration d 4 z. Similarly, we average over the color orientation U . Introducing the orientation factor O ab = tr(U + t a U τ b ), where t a are SU (N c )-matrices, we change ∆ ab I to be O ab O a b ∆ bb I , and carry out integration dO. Here
dOO ab O ab = δ bb , dOO ab O a b = (N 2 c − 1) −1 δ aa δ bb . Also, dOO abη bµν O a b η b µ ν = (N 2 c − 1) −1 δ aa (δ µµ δ νν − δ µν δ νµ ). In coordinate space, we find ∆ aa I (x, y) − ∆ aa 0 (x, y) = d 4 zdOO ac O a c (∆ cc I (x , y ) − ∆ cc 0 (x , y )) (x ≡ x − z, y ≡ y − z), = δ aa d 4 z[ 3ρ 2 4π 2 (N 2 c − 1) f 1 (x )f 1 (y ) + 2ρ 4 N 2 c − 1 f 2 (x )g(x − y )f 2 (y )],(11)f 1 (x) = 1 (x 2 + ρ 2 ) , f 2 (x) = (x µ x ν , ix 2 ) x 2 (x 2 + ρ 2 ) , g(x − y) = 1 4π 2 (x − y) 2 .
In momentum space, we find the contribution from the first term in Eq. (11) as
M 1,s (q) = [ 3ρ 2 (N 2 c − 1)R 4 4π 2 ] 1/2 qρK 1 (qρ)(12)
where the form factor qρK 1 (qρ). K 1 denotes the modified Bessel function. In Fig. 1
III. REAL GLUON PROPAGATOR
The total gluon field in the ILM is A + a, where A = i A i (γ i ). The number of all collective coordinates γ i is equal to 4N c N . We have to decompose the so-called zero modes φ i µ from the total fluctuation a, which are the fluctuations along the collective coordinates γ i in the functional space. Consider first the single instanton case, based on Refs. [18,19]. All of the fluctuations will be taken with gauge fixing condition P I µ a µ = 0 imposed. Then, the quadratic part of the effective action is given as (a µ M I µν a ν ), where M I µν = P I 2 δ µν + 2iG I µν − (1 − 1/ξ)P I µ P I ν and G I µν = −i[P I µ , P I ν ]. Here ξ stands for the gauge fixing parameter. The zero modes are the solutions of the following equation
M I µν φ i ν = 0.(13)
In some sense the zero modes can be considered as derivatives with respect to collective coordinates of the instanton field together with the additional longitudinal term dictated by the gauge fixing condition. The projection operator to the instanton zero-modes space is defined as P I µν = i φ i µ φ i+ ν , while that to the nonzero modes space is defined as Q I µν = δ µν − P I µν . The gluon propagator S I µν is defined by the following equation
M I µν S I νρ = Q I µρ(14)
The explicit solution of this equation was already derived in Ref. [17]. To generalize the formulae in Ref. [16], we introduce an artificial gluon mass m,which will be taken zero at the end of calculation. So, we define g I m,µν and take the limit of lim m→0 g I m,µν = S I µν , where
(M I µρ + m 2 δ µρ )g I m,ρν = Q I µν .
We also introduce G I m,ρν , satisfying (M I µρ + m 2 δ µρ )G I m,ρν = δ µν .
As was shown in Ref. [19], we find G I m,ρν = g I m,ρν + 1 m 2 P I ρν .
It is clear to see that
G I −1 m,µρ = (M I µρ + m 2 δ µρ ).
Now we may repeat the the same method with which we are able to obtain the averaged ILM "scalar" gluon propagator∆ given in Eqs. (3,4). First, we introduce the ILM inverse massive gluon propagator
G −1 m,µρ = P 2 δ µν + 2iG µν + m 2 δ µρ =M µν + m 2 δ µρ + i =j (A i A j δ µν − i[A i µ , A j ν ]). andG −1 m,µν =M µν + m 2 δ µν = p 2 + i (({p, A i } + A i 2 )δ µν + 2iG i µν ) + m 2 δ µν .
Following the way we have derived Eqs.
We finally seeḠ m,ρν =S m,ρν + 1 m 2Pρν , whereP ρν = NP I ρν . The ILM non-zero modes propagatorS m,ρν in the limit of m → 0 limit becomeS ρν and is given bȳ
S ρν − S 0 ρν = N (S I ρν − S 0 ρν ),(17)
where S 0 µν = (δ µν − (1 − ξ)p µ p ν /p 2 )/p 2 is the free gluon propagator. It is obvious to see that S 0 −1 µν = δ µν p 2 − (1 − 1/ξ)p µ p ν . We expectS ρν = (δ µν − (1 − ξ)p µ p ν /p 2 )/(p 2 + M 2 g ). Thus, we are able to rewrite Eq.(17) in another equivalent form
is a generic notation for the QCD (anti-) instanton in the singular gauge. γ I stands for all the relevant collective coordinates: the position in Euclid 4D space z I , the size ρ I and the SU (N c ) color orientation U I . The number of the collective coordinates is 4N c .
in the ILM by expanding with respect to the instanton density N/V = 1/R 4 , since the actual dimensionless expansion parameter is in fact the packing parameter ρ 4 /R 4 . The solution of Eq. (4) in the first-order expansion with respect to the density comes from the iteration if replaces the right-hand side of this equation by∆ → ∆ 0 . Then we havē
FIG. 1 :
1, we draw the form factor. The estimation of the contribution from the second term in Eq. Scalar "gluon" dynamical mass form-factor as a function of x = qρ.leads to M 2,s (0) = 0, while in general we know M 2,s (q → ∞) → 0. It means that we may neglect this contribution at all and obtain M s (q) = M 1,s (q). Using the phenomenological values of ρ and R, we obtain M s (0) = 256 M eV .
(3,4) and neglecting O(ρ 8 /R 8 ) terms, we can immediately find the averagedḠ m,µν as follows G m,ρν − G 0
M 2 g δ ρν = N S 0 −1 ρσ (S I σµ − S 0 σµ )S 0 −1 µα (δ αν − (1 − ξ)p α p ν /p 2 ).
This equation defines the gauge-invariant (ξ-independent) dynamical gluon mass M g .Here the single instanton gluon propagator[17]is given aswhere q I µνρσ = δ µν δ ρσ + δ µρ δ νσ − δ µσ δ νρ + µνρσ . From Eq.(11)we know that in coordinate space the most slowly decreasing part part of the ∆ I in the limit of x → ∞ and similarly y → ∞ is given by the first term (∼ f 1 (x−z)f 1 (y− z)) of Eq.(11). Only this term gave a contribution to the M s . A similar analysis can be done for M g . We expect that the most slowly decreasing part part S I νµ − S 0 νµ will only contribute to M g . In coordinate space we find. Comparing the effects from i∂ µ with A I µ , we conclude from Eq. (19) that the the most slowly decreasing part part of the S I νµ − S 0 νµ in Eq. (18) comes from p ρ (the most slowly decreasing part of (∆ I − ∆ 0 )∆ 0 + ∆ 0 (∆ I − ∆ 0 ))p σ and only this term will contribute to M g . Comparing it with Eq. (12), we conclude that M 2 g (q) = 2M 2 s (q), where q dependence is represented byFig.1. Using the phenomenological values of ρ and R, we obtain M g (0) = 362 M eV .IV. CONCLUSIONThe strength of the gluon-instanton interaction is given by dynamical gluon mass M g and it is large. It depends on the parameters ρ and R as in the case of the dynamical light quark mass:This modification of the gluon propagator from the instanton vacuum will provide the Yukawa-type potential in addition to all other pieces with instanton effects recently reported in Ref.[13]. Since the distance that is most sensitive to this modification is approximately around r g ≈ M −1 g = 0.55 f m, we conclude that it may still give some effects on charmonium properties. Further investigation is under way.
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|
[] |
[
"Influence of broken flavor and C and P symmetry on the quark propagator",
"Influence of broken flavor and C and P symmetry on the quark propagator"
] |
[
"Axel Maas \nInstitute of Physics\nNAWI Graz\nUniversity of Graz\nUniversitätsplatz 5A-8010GrazAustria\n",
"Walid Ahmed Mian \nInstitute of Physics\nNAWI Graz\nUniversity of Graz\nUniversitätsplatz 5A-8010GrazAustria\n"
] |
[
"Institute of Physics\nNAWI Graz\nUniversity of Graz\nUniversitätsplatz 5A-8010GrazAustria",
"Institute of Physics\nNAWI Graz\nUniversity of Graz\nUniversitätsplatz 5A-8010GrazAustria"
] |
[] |
Embedding QCD into the standard model breaks various symmetries of QCD explicitly, especially C and P. While these effects are usually perturbatively small, they can be amplified in extreme environments like merging neutron stars or by the interplay with new physics. To correctly treat these cases requires fully backcoupled calculations. To pave the way for later investigations of hadronic physics, we study the QCD quark propagator coupled to an explicit breaking. This substantially increases the tensor structure even for this simplest correlation function. To cope with the symmetry structure, and covering all possible quark masses, from the top quark mass to the chiral limit, we employ Dyson-Schwinger equations. While at weak breaking the qualitative effects have similar trends as in perturbation theory, even moderately strong breakings lead to qualitatively different effects, non-linearly amplified by the strong interactions.
|
10.1140/epja/i2017-12211-0
|
[
"https://arxiv.org/pdf/1611.08130v1.pdf"
] | 119,447,086 |
1611.08130
|
61fb73012c5eaabd9ba9870cb99659d12ea82638
|
Influence of broken flavor and C and P symmetry on the quark propagator
Axel Maas
Institute of Physics
NAWI Graz
University of Graz
Universitätsplatz 5A-8010GrazAustria
Walid Ahmed Mian
Institute of Physics
NAWI Graz
University of Graz
Universitätsplatz 5A-8010GrazAustria
Influence of broken flavor and C and P symmetry on the quark propagator
(Dated: November 28, 2016)
Embedding QCD into the standard model breaks various symmetries of QCD explicitly, especially C and P. While these effects are usually perturbatively small, they can be amplified in extreme environments like merging neutron stars or by the interplay with new physics. To correctly treat these cases requires fully backcoupled calculations. To pave the way for later investigations of hadronic physics, we study the QCD quark propagator coupled to an explicit breaking. This substantially increases the tensor structure even for this simplest correlation function. To cope with the symmetry structure, and covering all possible quark masses, from the top quark mass to the chiral limit, we employ Dyson-Schwinger equations. While at weak breaking the qualitative effects have similar trends as in perturbation theory, even moderately strong breakings lead to qualitatively different effects, non-linearly amplified by the strong interactions.
I. INTRODUCTION
With the detection of gravitational waves [1] a whole new era of astronomy has begun. Eventually, this will allow to investigate neutron star mergers. In such dense environments, weak interactions become so prevalent that the dynamical backcoupling between the weak and the strong interaction becomes relevant, see e. g. [2][3][4][5][6][7][8][9]. This requires therefore a fully coupled, and necessarily nonperturbative, description. This is so far only possible at the level of comparatively simple effective models, but not yet in an ab-initio calculation. This is not the only reason to consider this problem. Beyond the known standard model (hidden) sectors may exist in which strongly interacting parity-conserving and parity-breaking interactions both exist.
Both these considerations motivate to understand such backcouplings better. The main hallmark of both scenarios is the appearance of explicit C and P symmetry breaking, as well as, to a lesser extent, flavor breaking. The presence of additional, or non-negligible, symmetry breaking effects implies always a more involved tensor structure of correlation functions. Therefore, we focus here on the simplest object, which exhibits the full additional complexity, the quark propagator (QP). Since we are mainly interested in how the strong interaction may amplify, or modify, the symmetry breaking effects, we will consider only an explicit source of the symmetry breaking, as will be discussed in greater detail in section II. For the weak interactions, which, due to explicit masses of the W and Z bosons do not show a strong momentum dependence at low energies, this is a sufficient approximation.
However, the inclusion of such a breaking, and the large differences in relevant energies, already limits the possible choices of methods. Especially, lattice gauge theory is not suitable. This is on the one hand due to the immense computational costs for the vastly different energy levels involved. On the other hand, there is no fully proven way yet to upgrade the static breaking considered here to the full weak interactions using lattice methods [10].
As an alternative, here functional methods in the form of Dyson-Schwinger equations (DSEs) will be employed. These have been used very successfully to determine the quark propagator in QCD in various levels of sophistication [11][12][13][14][15]. In the more exploratory investigations here, the most important features are the dynamical mass generation as well as the correct implementation of chiral symmetry. These features are particularly well implemented in the so-called rainbow-ladder truncation [11,12,14]. This completes our setup, which we describe in much more detail in section II as well as in the appendices A and B. Especially in appendix A, we will discuss the tensor structure of the quark propagator, which has now four matrix-valued, rather than two real-valued, dressing functions, demonstrating the much higher complexity compared to QCD alone.
The most important results will be discussed in section III. Probably the most relevant insight gained, aside from the necessary technology to deal with the increase of complexity, is that the fully coupled system behaves as expected from perturbation theory only at very weak breaking. Already at moderately small breaking, the amplification by the strong interaction can lead to qualitatively different behaviors for various dressing functions of the quark propagator than perturbatively expected. Of course, only future investigations of observable quantities will tell what the implications for physics are, but the present results mandates caution with extrapolation of perturbative notions. In this section, we will also present information on the analytic structure of the quark propagator using its Schwinger function, an issue which is already in QCD alone highly non-trivial [11,12,14].
Finally, we list many further results in appendix C, providing a complete picture of the quark propagator in this setup for a wide range of parameters and quark masses. All of the insights and results are finally summarized in section IV. Some preliminary results have already been reported in [16].
II. BASICS AND METHODS
A. Ansatz
The breaking of the symmetries will be realized by including an explicitly symmetry-violating term in the Lagrangian. The breaking is thus generated already at treelevel. Since the weak interactions motivate our study, the term will break C, P, and flavor within generations. Thus, our QP becomes matrix-valued in flavor space, with the off-diagonal elements mediating flavor changes.
Flavor violation within a generation is actually not possible without involving further particles, due to electric charge conservation. In the standard model, this is ensured by the emission of a lepton and a (anti)neutrino. To avoid this additional complexity, here the leptons are modeled as an external background field. Given that our ultimate interest is in neutron star mergers, where such a reservoir is readily available, this appears like a reasonable approximation.
In the following the subscript u and d denotes up-like quarks and down-like quarks and the superscript L and R left-handed and right-handed quarks, respectively. The strength of the weak interaction, or more precisely the coupling to the symmetry breaking external field, will be denoted by g w , the effective weak strength. We will vary the value of g w from small values to large values to turn on the effects smoothly. Finally, the quark fields are denoted by ψ.
All together leads to the Lagrangian L =L QCD + L Effective ,
L QCD =ψ u − / ∂ + m u ψ u + ψ d − / ∂ + m d ψ d + g s ψ / A i T i ψ + L Rest , L Effective = − 2g w ψ L u / ∂ψ L d + ψ L d / ∂ψ L u ,(1)ψ L = 1 2 (1 1 − γ 5 )ψ,
where A i are the gluon fields and T i are the generators of SU (3). m u and m d are the masses for up-like and down-like quarks. L Rest is the remainder of the QCD Lagrangian, which includes the gluon self interaction and the gluon-ghost part, and does not play an explicit role in the following. It also contains the gauge-fixing terms of our choice of (minimal) Landau gauge. For brevity, we also suppressed the renormalization constants.
B. Quark Propagator
In the following, P AB is the propagator from flavor A to B with A, B ∈ {u, d} for up-like and down-like quarks.
Its inverse will be denoted by P −1 AB . Because of parity violation the QPs have in addition to the usual vectorand scalar channels also non-vanishing axial-and pseudoscalar channels. The standard notation for the vectorand scalar channel dressing functions of the inverse Propagator in the literature is A and B. We will keep this and denote the dressing functions for the axial-and pseudoscalar channels of the inverse propagator with C and D. The corresponding dressing functions of the propagator are denoted by a tilde.
This leads to the form
P AB (p 2 ) =Ã AB (p 2 ) i / p +B AB (p 2 )1 1 +C AB (p 2 ) i / pγ 5 +D AB (p 2 )γ 5 , P −1 AB (p 2 ) = −A AB (p 2 ) i / p + B AB (p 2 )1 1 + C AB (p 2 ) i / pγ 5 + D AB (p 2 )γ 5 .(2)
The dressing functions of the propagator and its inverse are related with each other. In general the dressing functions of the propagator depends on all the dressing functions of the inverse propagator in a complicated way, see for details appendix A, and generically denoted as
A AB =Ã AB (A CD , B CD , C CD , D CD ).(3)
Instead of splitting the Lorenz channels into the vectorand axial channels, it can also be split into left-handed L(p 2 ) and right-handedR(p 2 ) components, leading to
P AB (p 2 ) =L AB (p 2 ) √ 2 i / p(1 1 − γ 5 ) +R AB (p 2 ) √ 2 i / p(1 1 + γ 5 ) +B AB (p 2 )1 1 +D AB (p 2 )γ 5 ,(4)
with the relations
L AB (p 2 ) = 1 √ 2 Ã AB (p 2 ) −C AB (p 2 ) , R AB (p 2 ) = 1 √ 2 Ã AB (p 2 ) +C AB (p 2 ) .(5)
At tree-level the propagator reads
P 0,uu (p 2 ) = 1 N (p 2 ) (m 2 d + (1 − 2g 2 w )p 2 ) i / p +m u (m 2 d + p 2 )1 1 + 2g 2 w p 2 i / pγ 5 , P 0,dd (p 2 ) = 1 N (p 2 ) (m 2 u + (1 − 2g 2 w )p 2 ) i / p +m d (m 2 u + p 2 )1 1 + 2g 2 w p 2 i / pγ 5 ,(6)P 0,ab,a =b (p 2 ) = g w N (p 2 ) (m u m d − p 2 ) i / p −(m u + m d )p 2 1 1 − (m u m d + p 2 ) i / pγ 5 −σ ab (m u − m d )p 2 γ 5 ,
where the common denominator is given by
N (p 2 ) =m 2 d m 2 u + (m 2 u + m 2 d )p 2 + (1 − 4g 2 w )p 4 =(1 − 4g 2 w )(p 2 + M 2 l )(p 2 + M 2 h ), M l = m 2 u + m 2 d − (m 2 u − m 2 d ) 2 + 16g 2 w m 2 u m 2 d 2(1 − 4g 2 w ) ,(7)M h = m 2 u + m 2 d + (m 2 u − m 2 d ) 2 + 16g 2 w m 2 u m 2 d 2(1 − 4g 2 w )
.
The quantity σ ab is 1 for a = u and b = d and −1 in the other case. The tree-level propagator already reveals the major contributions for the QPs. They are separated in the different channels at tree-level, but will mix in the full case. Consider the denominator. In the second line of equation (7) we have factorized the denominator to see both poles of the tree-level propagator. For g w → 0 M l goes to the mass of the lighter quark and M h to the mass of the heavier quark. By increasing g w , the value of M l is decreased and M h is increased and thus increases the effect of the mass splitting.
Note that M l and M h diverge at g w = 0.5. At this point the poles turn imaginary, indicating a breakdown of the trivial vacuum around which the perturbative expansion is performed. This feature will actually not be lifted in the full non-perturbative treatment, and we are only able to find solutions as long as g w 0.4. Since this is a very large breaking, probably far too strong for the setting of neutron star mergers guiding this work, we did not endeavor to find out what happens beyond this point, and restrict ourselves to breaking strengths below this value.
The explicit chiral symmetry breaking by the tree-level masses manifests itself in the scalar channel of the pure and mixed flavors. In the chiral limit the scalar channel vanishes at tree-level, but due to dynamical chiral breaking the scalar channel does not vanish for the full propagator.
The vector channel of the mixed propagator is proportional to g w (m u m d − p 2 ), which changes its sign for p 2 > m u m d . The influence of this is a contribution in different direction for large and low momenta. For heavier bare quark masses this contribution is shifted to higher momenta.
The most remarkable contribution, a difference of both quark masses, appears in the pseudo-scalar channel of the mixed propagator. This will have a significant impact on the full propagator. It is remarkable that this contribution appears with opposite sign for the propagator from up-like quarks to down-like quarks and the other way around: Although we have taken the same strength for the propagation of both mixed QP in our ansatz, we get a difference, if the quarks have different masses.
C. DSEs
We can write the QP in a matrix form, where the diagonal elements are the QPs for pure flavor and the off-diagonal elements are the QPs for mixed flavors, see for details appendix A. A graphical representation of the DSEs is given in figure 1. The equations look similar to those of QCD, where the QP can also be given in a matrix form, but with vanishing off-diagonal elements. It is only the appearance of the off-diagonal tree-level elements, which gives rise to all differences.
The full quark-gluon-vertex appears in the self energy graph, which is determined by a separate DSEs involving even higher-order correlation functions.
To avoid this complication, the DSEs are truncated at this level, which is known as the rainbow truncation [11,12,14]. The required inputs of the full gluon propagator and the full quark-gluon-vertex are then replaced by a bare gluon propagator and a quark-gluon vertex given by the tree-level tensor structure, but dressed with an effective running coupling α. Denoting the tree-level inverse propagator with P −1 0,AB , the final DSE reads
P −1 AB (p 2 , µ 2 ) = Z 2,A (µ 2 , Λ 2 )Z 2,B (µ 2 , Λ 2 )P −1 0,AB + Z 2,A (µ 2 , Λ 2 )Z 2,B (µ 2 , Λ 2 ) 3π 3 × × Λ d 4 q α(k 2 ) k 2 δ νρ − k ν k ρ k 2 γ ν P BA (q 2 , µ 2 )γ ρ ,(8)
where Z 2,A and Z 2,B are the quark wave functionrenormalization constants for flavor A and B. Λ represents a translationally-invariant regularization with the UV-cutoff Λ. µ is the renormalization point and k = q−p.
For the dressing function α we choose the Maris-Tandy coupling [17] α(q 2 ) = π ω 6 Dq 4 e − q 2
ω 2 + 2πγ m [1 − exp (− q 2 m 2 t )] ln[e 2 −1 + (1 + q 2 Λ 2 QCD ) 2 ] , .(9)
Here the parameters are adapted to describe pions in the vacuum adequately. In the literature these parameters are fitted for degenerate masses of up and down quarks [18]. For a detailed analyses of different parameter sets see, e. g., [19,20]. To allow for comparison, we choose here one such set, namely Λ QCD = 0.234 GeV, GeV. In addition, we will also consider the chiral limit as well degenerate cases.
Further details can be found in appendix B.
−1 = −1 + FIG.
1: Diagrammatic representation of the DSEs for the QPs. The solid lines with an arrow and x represents tree-level QPs for pure and mixed flavors, respectively. Analog the solid lines with filled and empty blob represents the full QPs for pure and mixed flavors, respectively. The wiggly lines represents the propagation of gluons. A small and big filled blob at a vertex indicates a bare and a full vertex, respectively.
D. Schwinger function and Masses
To obtain information on the analytic structure, we also determine the Schwinger function [21,22]. It is defined as
∆ AB (t) = 1 π ∞ 0 d p 4 cos(tp 4 )σ AB (p 2 4 ).(10)
σ AB (p 2 4 ) is one of the dressing functions from the propagator evaluated at zero spatial momenta ( p = 0).
The actual analytic form of the propagator is yet unknown, but poles and/or cuts appear likely [11,14,21]. If there would be only an ordinary mass pole, the Schwinger function would show an exponential decay [22] ∆(t) ∼ e −mt (11) and m would be the mass. However, investigation of pure QCD in the rainbow truncation yielded rather a structure with complex conjugated poles [11,14,21], which is e. g. expected in the Gribov-Stingl scenario [23][24][25]. Note, however, that this may be a truncation artifact. In this case the Schwinger function is roughly given by
∆(t) ∼ e −at cos(bt + δ).(12)
The decay rate is given by the real part and the oscillation frequency by the imaginary part of the mass pole.
III. RESULTS
For each quark generation, we have 2 dressing functions for pure flavor and 2 dressing functions for mixed flavor and each has 4 Lorentz channels, resulting in 16 dressing functions. We will consider 6 cases in total, the chiral limit and three physical quark generations and two cases of degenerate masses. Therefore we have numerical results for 192 dressing functions and each of them as a function of the weak strength. In addition to that, we also have the Schwinger function for each dressing function. To avoid cluttering up the main text, most of these results are relegated to appendix C. Here, only the qualitatively most remarkable results will be analyzed. The results in the appendix do not add any conceptual new to this section.
The result for the QP in QCD are usually [11,14,21] given in terms of the wave function renormalization Z(p 2 ) = 1/A(p 2 ) and mass function M (p 2 ) = B(p 2 )/A(p 2 ). For ease of comparison, the case with g w = 0 will serve as reference. Therefore in section III B the results for Z andà AA will be explored. Afterwards, the mass function and the relatedB AA will be discussed in section III C. In section III D we will analyze the results for the axial channelC AA , and study the impact of parity violation. The results for the Schwinger function will be discussed in section III E. But first, it is necessary to discuss the involved scales, as the problem is now a multi-scale one. Note that in the chiral limit there is no difference between the up-like quark and down-like quark, and thus the flavor-diagonal elements coincide. In these cases, always the up-type one will be shown.
A. Relative scales g w is dimensionless. Thus, a comparison of the strength of breaking with the strong interaction scale Λ QCD is not directly possible. However, in the scalar channel, the interaction is found to be transmuted into a momentum scale. This is particularly true for the flavor off-diagonal-element, which is zero without breaking. It is shown for various quark masses in figure 2 It is seen that in the IR this dressing function is approximately proportional to the weak strength for g w ≤ 0.1. Also, for small values of the weak strength, the scale generated is small compared to Λ QCD . At the largest values of g w the generated scale becomes of the same order as Λ QCD , which therefore substantially deviates from nature. This also justifies our choice to restrict to not too large values of g w . However, such effects may play a role charm-strange top-bottom in theories with strongly-interacting chiral sectors. Note that the generated scale depends on the quark masses. For both light generations the difference to the chiral limit is small. For top and bottom quark, the generated scale is one order of magnitude bigger at the same g w . Thus, there is a linking of the different involved scales.
g w =10 -6 g w =10 -5 g w =10 -4 g w =10 -3 g w =10 -2 g w =10 -1 g w =0.4g w =10 -6 g w =10 -5 g w =10 -4 g w =10 -3 g w =10 -2 g w =10 -1 g w =0.4
B. Wave function renormalization
From equation (B3) follows that A uu andà uu are directly linked with each other, and thus Z = 1/A uu is also directly related toà uu . These dressing functions are shown in figure 3 for different values of g w in the chiral limit. For values of g w 0.01 no appreciable effect is seen. At larger valuesà uu slightly increases in the UV when increasing g w . In the mid momenta regime it is slightly decreased and in the IR it is significantly increased. A consequence of this is that Z also increases in the IR. This can be understood from (B3), as A uu is obtained from integratingà uu multiplied with a kernel over all momenta. Becauseà uu is increased in the UV very little and more decreased in the mid range, Z is slightly decreased in the UV range due to the integration. For the same reason Z is not increased as much asà uu is increased in the IR.
For other quark masses the same behavior is seen, as shown in figure 4. The graph shows thatà is increased in the IR for all quark flavors and thus Z is also increased.
The effect comes from different sources. One is from g w and the other from a combination of g w and the bare quark masses. Especially, A and Z are increased for the up quarks more than in the chiral limit. Also, in general the value for up-like quarks is increased more than for down-like quarks. This is also seen in figure 15 in appendix C 1 in more detail.
This can be understood from equation (6) for the treelevel case. One of the contributions arises from the bare quark masses and another from mass splitting with differ- ent signs. This creates the cross-talks leading to the observed effects. We will return to this later in section III D.
In addition, the absolute value is decreased for higher bare quark masses, as anticipated because the masses of the quarks enter in the denominator of the QP. This can bee seen already for the tree-level propagator in equation (7).
C. Mass function
The relations between the dressing functions of the QP and its inverse are more involved as in QCD, see also appendix A. In QCD, the relation is given bỹ
A AA (p 2 ) = A AA (p 2 ) A 2 AA (p 2 )p 2 + B 2 AA (p 2 ) = Z AA (p 2 ) p 2 + M 2 AA (p 2 ) , B AA (p 2 ) = B AA (p 2 ) A 2 AA (p 2 )p 2 + B 2 AA (p 2 ) = Z AA (p 2 )M AA (p 2 ) p 2 + M 2 AA (p 2 ) .(13)
To be able to compare, we therefore choose to define a (pseudo) mass function as
M AA (p 2 ) = B AA (p 2 ) A AA (p 2 ) ,(14)
which by construction coincides with the usual one in the QCD case. Of course, neither in QCD nor here this function needs to coincide with the actual mass. Any such statement requires the Schwinger function in section III E. Nonetheless, we will stick here with the usual convention and call this quantity mass function. The dependence on g w of this mass function is shown in figure 5. As in QCD, the mass function is non-zero, indicative of chiral symmetry breaking. The mass function starts to change appreciably for g w 0.01, like the wave function renormalization. The same is true forB uu , which is also shown in figure 5. Since the connection between B uu andB uu , due to equation (B3), is similar as for A uu andà uu , the same analysis as before applies, and the response to g w follows the same pattern.
At non-zero masses, the picture changes. This is shown in figure 6 for up and down quarks. The mass function M of the up quark is decreased by increasing g w for g w 0.3. For larger values it increases again, but here our approximations start to break down. This replicates the result of the tree-level propagator in section II B: The mass of the heavier quark in a generation is increased and the mass of the lighter quark is decreased.
The other flavors are shown in figure 7. The second generation follows the pattern of the first, but not the third. In the latter case the mass function of both quarks increases. This implies different contributions to the mass functions. One increases the mass function of the heavier quark and decreases the one of the lighter quark. The other contribution increases with the mass of both quarks. An indication of this is already seen in the second generation, albeit not creating a qualitative change.
D. Parity Violation
In the following the handiness of the quarks is investigated, using the definition (5). This requires the axial channel, shown in figure 8 for the chiral limit. The corresponding dressing functionC is for the flavor-diagonal elements found to be positive for higher momenta and negative in the IR. At the same time, for increasing g w the absolute value ofC also increases. Of course, at large momenta the dressing function goes to its tree-level part, which from equation (6) is
C 0,AA = 2g 2 w p 2 N (p 2 ) ,(15)
which is positive, actually for all momenta. Therefore the backcoupling to QCD forces it to be negative in the IR. This happens at a transition scale of approximately 1 GeV 2 , which is the typical QCD scale.
To assess the consequences of this for the left-handed and right-handed contributions, it is useful to define their relative ratio as
r AB (p 2 ) =L AB (p 2 ) −R AB (p 2 ) L AB (p 2 ) +R AB (p 2 ) = −C AB (p 2 ) A AB (p 2 ) .(16)
Sinceà is always positive, the sign ofr is given by the sign ofC. This already entails a change of sign, and that the left-handed part is larger in the infrared. This is also shown in figure 8. The effect increases non-linearly with g w : For g w 0.1 the absolute value |r| in the UV and IR is increased by two order of magnitudes, when g w is increased by one order of magnitude.
The flavor-off-diagonalC is always negative in the chiral limit, butà changes its sign, see figures 17 and 22 in appendix C. At the same transition scale of approximately 1 GeV 2 as for the flavor-diagonal case, and in the chiral limit,à has a zero crossing andC not. This leads to a divergingr ud at this scale. Thus also in this case there is a transition from right-handed in the UV to left-handed in the IR. This is shown in figure 9.
Increasing the mass, the situation for up and down quarks is shown in figure 10. The absolute value ofC is different for up quark and down quark, but for g w = 10 −5 the behavior for up and down quark is as in the chiral case. Slightly increasing g w to 5 · 10 −5 entails a drastic qualitative change. For the up quarkC is still positive in the UV and negative in the IR, but for the down quark it remains positive for all momenta. Therefore the up quark still flips its chirality at long distances, but the down quark does not do so.
To understand the origin of this effect, it is helpful to study the degenerate mass case, also shown in figure 10. There is no (numerically detectable) difference between up and down quark 1 , andC changes again sign, as in the chiral limit. For higher values of g w the absolute value of C just increases in the IR. This implies, that the different behavior ofC for the non-degenerate case is due to the mass splitting of the quarks. Since at tree-level only in the pseudoscalar channel a contribution proportional to the mass splitting occurs, this effect must have been propagated by the QCD interaction to the axial channel. Moreover, the effect becomes already important and the, compared to QCD, very small mass splitting and a very small breaking scale of g w ≈ 5 · 10 −5 . This can only happen if there is a strong non-linear amplification mechanism is at work. This implies that the QCD medium strongly affects the helicity at long ranges, but only for non-degenerate quark masses. That was certainly not expected. The same is true for the other quark generations, see for the second generation figure 9. The effect is still there for the third generation, with its very large mass splitting, see figure 11, and there occurs already at an even smaller breaking strength of g w = 10 −6 . Thus the absolute value of the involved mass scales amplifies the non-linear backcoupling, such that it occurs at weaker breaking strength.
The corresponding relative ratios are shown in figure 25 in appendix C 5. These graphs support the existence of a transition scale, where the left-handed and right-handed contribution change their relative contribution.
For the mixed flavor case the effect is solely driven by the absolute value of the mass splitting, and the breaking strength plays only a minor role. This is shown in figure 12. Whether there is a mass-splitting or not plays only a role if the mass splitting is large enough, i. e. in the third generation. Only then the behavior with or without mass splitting differ qualitatively in the infrared. In fact, already the second generation is sufficient for this, as can be seen in figure 9.
E. Schwinger function
As noted, the analytic structure is accessible through the Schwinger function (10). Let us now consider the Schwinger function in the chiral limit for the flavordiagonal dressing function B. The other flavor-diagonal dressing functions do not lead to qualitatively new results, and are even quantitatively similar, so these will be skipped here.
The results are shown in figure 13. The Schwinger function shows an oscillatory behavior, consistent with the form (12), and thus complex conjugate poles, as in pure QCD for the rainbow-ladder truncation [21]. In fact, a fit using a more detailed ansatz, see appendix D, of this type works very well, as is also shown in figure 13. The values of the fit parameters are listed, for completeness, in appendix D.
However, the oscillation period starts to substantially increase for g w > 0.01, up to a point where at large g w the first zero crossing has moved to a time which we can no longer numerically resolve reliably. Thus, the imaginary part shrinks with increasing breaking. The curvature at short distances is still not quite right for a physical particle. A similar behavior, though with a suppression of oscillations for decreasing interaction strength, has already been observed for adjoint scalar particles [26,27]. This strongly suggests that the interaction strength plays a crucial role for the scale at which negative norm contributions become relevant, even though the coupling does not differentiate between positive-norm and negative-norm states. At the same time the steepness decreases, making the real part smaller. Thus, the increase in g w moves in total the poles closer to the origin, as both real and imaginary part decrease.
The flavor-off-diagonal Schwinger function, again only the scalar part as the others are very similar, is shown in figure 14. In principle, it shows a very similar behavior as for the flavor-diagonal part, except that it always retains a first zero crossing, relatively independent of g w , at very short times. It is still possible to fit it using the same fit form. The results for the fit are also listed in appendix D. It is found that the zero crossing at small momenta comes from the phase shift δ in (12). It is very close to π/2 and causes the sign change for the Schwinger function at small t. Still, the position of the pole also moves towards the origin with increasing breaking strength. The Schwinger function for the first two generations show the same behavior as in the chiral limit, see figure 26 in appendix C 6. For the third generation the fall-off was too fast, due to the large real part, as that any unambiguous statements could be drawn before numerical noise drowns out the signal.
It is a quite interesting result that the breaking pushes the poles closer to the origin. As we expect a change of physics when crossing the threshold g W 0.4, this could be a first indication of a drastic change at strong breaking. However, this is probably not of relevance to neutron star physics. On the downside, the decreasing distance to the origin will create additional problems in any mesonic correlators in rainbow-ladder calculations [11,12,14]. In these cases more elaborate schemes will be necessary than a tree-level breaking, which we are currently developing.
IV. CONCLUSION
We have calculated the quark propagator in the presence of explicit flavor, C and P symmetry breaking. Moreover, we took into account the non-linear backcoupling from QCD in the rainbow-ladder truncation. The latter lead to qualitative effects, even for relatively small explicit breaking strengths, at long (hadronic) distances. They also couple in a highly non-trivial way the various dressing functions to each other. This was particularly visible in the way how effects from mass splitting and mass averages surfaced in various dressing functions. The non-linear amplification also surfaced in other ways. This is a very important insight: External perturbations can, even in rainbow-ladder truncation, be substantially amplified by the strong interactions. This must be regarded as a warning that even small effects can play a non-perturbatively large role when QCD is involved.
From the point of view of physics, another interesting insight is obtained when considering how left-handed and right-handed particle propagation changes. Under particular conditions, flips between handedness can be amplified at long distances by the strong interaction. This can deplete or enlarge the available particles in some handedness. As the weak interactions only couple to a particular handedness, this can increases or decrease the reservoir of particles which are weakly interacting in a system. If this pertains to the full system, this can influence the dynamics in forming or merging neutron stars, as this could alter, e. g., the opacity for neutrinos. This is even more important as the typical range where this occurs is only of the size of a hadron.
Concluding, this investigations showed that weak interactions effects, even if themselves small, can be amplified by the strong interactions, and this backcoupling can have qualitative impact. Keeping this in mind will be important in the next step, when relaxing the assumption of a reservoir, and taking the weak interactions explicitly into account, including the emitted neutrinos and electrons. This will require to work on a hadronic level, which is our next aim. In the following a number of useful relations between the dressing functions of the quark propagator and its inverse will be collected. Combining the flavor elements in a matrix, e. g. for the vectorial channel as
A = A uu A ud A du A dd ,(A1)
allows for a compact notation. Writing the propagator and its inverse in terms of these matrices yields
P (p 2 ) =Ã(p 2 ) i / p +B(p 2 )1 1 +C(p 2 ) i / pγ 5 +D(p 2 )γ 5 , P −1 (p 2 ) = −A(p 2 ) i / p + B(p 2 )1 1 + C(p 2 ) i / pγ 5 + D(p 2 )γ 5 .(A2)
The propagator satisfies the condition
P −1 P = 1 1. (A3)
yielding relations between the matrix-valued dressing functions
AÃp 2 + BB + CCp 2 + DD = 1 1 −AB + BÃ + CD − DC = 0 −AD − DÃ + BC + CB = 0 (A4) ACp 2 + CÃp 2 + BD + DB = 0
While this system is a system of linear equations for either the matrix elements of the propagator or its inverse, an explicit solution is of little use. The expressions become extremely lengthy, and therefore prohibitively expensive to evaluate during numerical calculations. Therefore, in our investigations we always solved such equations numerically at double precision.
Appendix B: DSEs for Quark Propagators
To derive the DSEs for the different dressing functions, insert in equation (8) the quark propagator of equation (2) and project out the different channels, by taking suitable traces. We define the following two kernels
K 1 (p, q, k) = 12πC F α(k 2 ) k 2 , K 2 (p, q, k) = 4πC F α(k 2 ) k 2 p 2 (p · q) + 2 (p · k)(q · k) k 2 ,(B1)
where C F = (N 2 c −1)/2N c and N c is the number of colors, i. e. N c = 3.
We further define the functional Π i for i = 1, 2 as
Π i,A (f, p 2 ) = Z 2,A (µ 2 , Λ 2 ) Λ d 4 q (2π) 4 K i (p, q, k)f (q 2 , µ 2 ) . (B2)
This yields the DSEs of the the different dressing functions for flavor-diagonal elements A ∈ {u, d}
A AA (p 2 , µ 2 ) = Z 2,A (µ 2 , Λ 2 ) 1 + Π 2,A (Ã AA , p 2 ) , B AA (p 2 , µ 2 ) = Z 2,A (µ 2 , Λ 2 ) m A + Π 1,A (B AA , p 2 ) , C AA (p 2 , µ 2 ) = Z 2,A (µ 2 , Λ 2 )Π 2,A (C AA , p 2 ),(B3)D AA (p 2 , µ 2 ) = Z 2,A (µ 2 , Λ 2 )Π 1,A (D AA , p 2 ).
and for mixed flavors, A, B ∈ {u, d} , A = B,
A AB (p 2 , µ 2 ) = Z 2,A (µ 2 , Λ 2 )Z 2,B (µ 2 , Λ 2 )g w + Z 2,B (µ 2 , Λ 2 )Π 2,A (Ã BA , p 2 ), B AB (p 2 , µ 2 ) = Z 2,B (µ 2 , Λ 2 )Π 1,A (B BA , p 2 ), C AB (p 2 , µ 2 ) = Z 2,A (µ 2 , Λ 2 )Z 2,B (µ 2 , Λ 2 )g w + Z 2,B (µ 2 , Λ 2 )Π 2,A (C BA , p 2 ),(B4)
D AB (p 2 , µ 2 ) = Z 2,B (µ 2 , Λ 2 )Π 1,A (D BA , p 2 ). This system of equation is technically very similar to the ordinary rainbow-ladder truncation. Therefore a numerical solution using standard fixed-point iteration schemes is possible, and was done here. Only the the quark propagator was numerically inverted at every step, due to the involved structure, as discussed in appendix A. There are, however, a few more subtle numerical issues to be mentioned. To perform the integral for Π i,A the dressing functionsÃ,B,C andD are evaluated for various momenta using interpolation. We performed our calculations using linear and cubic interpolation. If we use a precision of 5×10 −5 , which is considered to be sufficient in the standard fixed-point iteration scheme, for A, B, C and D then we get different numerical solutions for linear and cubic interpolation. By increasing the precision the solution from the linear interpolation approaches the solution from the cubic interpolation for all dressing functions except for D AA andD AA . Especially, using linear interpolation we get different solutions for D AA and D AA by using different start values for the iteration. In contrast to this, we get the same solutions for different start values using the cubic interpolation. Thus, we consider the solutions from the cubic interpolation as stable, and used them throughout this work.
In addition, we used a precision of 5 × 10 −7 , instead of the standard value, for A, B, C and D and 2 8 = 256 grid points. In this case the difference for the solutions from the linear and cubic interpolation were at most in the third significant digit, and thus lead essentially to the same results.
Let us note that our precision for the iteration procedure is for the dressing functions of the inverse propagator (A, B, C and D). The dressing functions of the propagator (Ã,B,C andD) are calculated by a numerical inversion, with a precision of roughly 10 −20 . Appendix C: Numerical Results
Vector Channel
The flavor-diagonal vector dressing functions for the different generations and values of the breaking strength are shown in figure 15. A detailed discussion is given in section III B.
A comparison between the different generations, the chiral limit, and tree-level for the flavor-off-diagonal vector dressing function is shown in figure 16 for a fixed value g w = 0.2. The tree-level value is given by, see equation
(6),Ã 0,ud = g w (m u m d − p 2 ) N (p 2 ) .(C1)
This demonstrates that at tree-level the dressing function is negative for momenta p 2 ≥ m u m d and positive for lower momenta. The full dressing function exhibits the same behavior, but the absolute value is substantially larger. Note especially that in the chiral limit the treelevel propagator is proportional to −p 2 and thus negative for all momenta. The full dressing function is, however, positive in the IR. As the masses of up and down quark are comparatively small, the result for them is close to the one in the chiral limit.
The dependence on the generation and g w is shown in figure 17. In every generationà ud increases with g w . The zero ofà ud shifts for higher momenta for heavier mass. This is already the case for the tree-level propagator, where the zero is determined by the condition p 2 = m u m d .
Scalar Channel
Complementing the results in section III C the flavordiagonal scalar dressing functions for the different quarks are shown in figure 18.
A comparison to the tree-level case of the third generation flavor-off-diagonal scalar dressing function is shown Thus, at tree-levelB 0,ud is negative for all momenta, in particular in the UV. The latter is also seen in the full case. But it switches to a positive value in the IR. This qualitative behavior is also seen for the other generations, as is also plotted in figure 19, but at differing absolute values. This persists even in the chiral limit. The value of this quantity is found to also increase when increasing g w , which is shown in figure 20.
Axial Channel
The flavor-off-diagonal axial dressing function is shown for the various generations and the chiral limit in com-parison to tree-level in figure 21 at fixed g w = 0.2. The tree-level propagator takes the form, see equation (6),
C 0,ud = − g w (m u m d + p 2 ) N (p 2 ) . (C3)
Therefore the full dressing function becomes negative in the UV. In the IR it is positive for the third and second generation, as is visible from figure 21, but remains negative for the first generation and the chiral limit. The change of the relative ratio between the left-handed and right-handed contribution is related to the sign ofC, see equation (16). For the second and third generation the contribution from the mass splitting is high enough to makeC positive in the IR. A detailed discussion is given in section III D.
In figure 22 the dependence on g w is shown for the different generation and the chiral limit. The higher the value of g w , the larger the dressing function. But the qualitative behavior is unchanged, and therefore entirely controlled by the masses.
Pseudo-scalar Channel
At tree-level the flavor-diagonal pseudo-scalar dressing function vanishes. The full dressing function has a finite value, which is depicted in figure 23 for all quarks at g w = 0.2.
The flavor-off-diagonal dressing function at tree-level is zero for degenerate quark masses,
D 0,ud = −D 0,du = − g w (m u − m d )p 2 N (p 2 ) ,(C4)
as can be obtained from equation (6). This is also true for the full case, also shown in figure 23, and especially so for the chiral limit. For non-degenerate quark masses the dressing function does no longer vanish. This is in as far remarkable as non-trivial effects due to (non-)degeneracy propagate in other dressing functions, as discussed in the main text. We also find that the difference between upto-down and down-to-up dressing function, as discussed in section II B, also persists in the full case, and both differ by a sign. This effect does not propagate to other dressing functions, where we do not find any difference. The dependence of the flavor-off-diagonal pseudoscalar dressing function on generation and g w , also in comparison to tree-level, is shown in figure 24. Here also the sign switch between up-to-down and down-to-up is shown. Note however that the sign in the infrared is again different for the first generation and the second and third generation. This is because of the switch of relative sign in the mass difference from the first to the other generation, as is already the case at tree-level in (C4). The size, but not the qualitative features, increase again with g w . The size of the dressing function also decreases with increasing quark mass.
Relative Ratio
In figure 25 the relative ratio (16) for the flavordiagonal elements are shown, see section III D for a detailed discussion.
Schwinger function
In figure 26 the Schwinger function for the scalar dressing function for the first and second generation, both for flavor-diagonal and flavor-off-diagonal elements, are shown. The third generation's large tree-level mass leads to a too quick drowning in numerical noise to provide any reasonable results. For a detailed discussion see section section III E. (3.0 ± 0.5) × 10 −5 (9.5 ± 0.5) × 10 −4 (3.5 ± 0.5) × 10 −3 (1.2 ± 0.5) × 10 −2 (3.5 ± 0.5) × 10 −2 0.11 ± 0.05 0.25 ± 0.05 0.13 ± 0.05 0.16 ± 0.05 0.10 ± 0.05 (−2.5 ± 0.5) × 10 −6 (−2.5 ± 0.5) × 10 −5 (−3.0 ± 0.5) × 10 −4 (−2.8 ± 0.5) × 10 −3 (−3.0 ± 0.5) × 10 −2 −0. 15
m t = 1.0 GeV, ω = 0.4 GeV, and D = 0.93 GeV. γ m = 12/(11N c − 2N f ) is the anomalous dimension of the quark propagator. Because we consider each quark generation on its own, we choose N f = 2 and N c = 3. For the bare quark masses we take m up = 2.3 MeV, m down = 4.8 MeV, m strange = 95 MeV, m charm = 1.275 GeV, m bottom = 4.18 GeV and m top = 160 GeV, always at the renormalization point of µ = 10 6
FIG. 2 :
2The scalar channel for the inverse mixed propagator in the chiral limit and for all three quark generations. For gw ≤ 0.1 the mass in the IR is approximately proportional to gw.
FIG. 3 :
3The wave function renormalization Z(p 2 ) (top panel) and the vector channel (bottom panel) for different values of gw in the chiral limit.
FIG. 4 :
4The flavor-diagonal vector channel without (left panels) and with (right panel) explicit breaking. The lower panels show the same for Z(p 2 ).
FIG. 5 :
5The mass function M (p 2 ) (top panel) and the flavordiagonal scalar channel (bottom panel) for different values of gw in the chiral limit.
FIG. 6 :
6The mass function M (p 2 ) for different values of gw for the up quark (top panel) and down quark (bottom panel).
FIG. 7 :
7The mass function for different values of gw and different quarks masses.
FIG. 8 :
8Flavor-diagonal axial channel (top panel) and the ratio (16) (bottom panel) in the chiral limit for different values of gw.
FIG. 9 :
9The flavor-off-diagonal ratio(15) in the chiral case (top panel) and for the second generation (bottom panel) for different values of gw.
FIG. 10 :
10The axial dressing function (top panel) and the ratio (16) (bottom panel) for the up quark and down quark at two different breaking strengths strength. The left panels show the physical mass splitting while in the right panel both masses are degenerate.
FIG. 11 :
11The axial channel for the top and bottom quark (top panel) and heavy degenerate quarks (bottom panel) at gw = 10 −6 (top panel) . Appendix A: Structure of the Quark Propagator
FIG. 12 :
12The flavor-off-diagonal ratio(16) for different gw for the first generation (top panels) and third generation (bottom panels) for physical mass splittings (first panel) and degenerate masses (bottom panels).
FIG. 13 :
13Flavor-diagonal Schwinger function in the chiral limit for different gw (top panel) and with fits (bottom panel).
FIG. 14 :
14Flavor-off-diagonal Schwinger function in the chiral limit for different gw (top panel) and with fits (bottom panel).
FIG. 15 :FIG. 16 :
1516The flavor-diagonal vector dressing function for different values of gw. From top to bottom generations one to three are shown, and the left-hand panels show up-type quarks, and the right-hand panels down-type quarks.infigure 19. Note that the tree-level result is, see equa-tion (6),B0,ud = − g w (m u + m d )p 2 N (p 2 ). The flavor-off-diagonal vector dressing function for gw = 0.2 in comparison to tree-level (left panel) and for different generations (right panel).
FIG. 17 :
17The flavor-off-diagonal vector dressing function for the chiral limit and the different generations for different gw.
FIG. 18 :
18The flavor-diagonal scalar dressing function for different gw for. From top to bottom generations one to three are shown, and the left-hand panels show up-type quarks, and the right-hand panels down-type quarks.
FIG. 19 :FIG. 20 :
1920The flavor-off-diagonal vector dressing function for the third generation at gw = 0.2 in comparison to tree-level (left panel) and the flavor-off-diagonal scalar dressing function for different generations and the chiral limit at gw = 0.2 (right panel). The flavor-off-diagonal scalar dressing function for different values gw and the different quark generations and the chiral limit.
FIG. 21 :FIG. 22 :
2122The flavor-off-diagonal axial dressing function at gw = 0.2 for the third generation compared to tree-level (left panel) and for the different generations (right panel). The dependence of the flavor-off-diagonal axial dressing function on gw for the different generations and the chiral limit.
FIG. 24 :
24The flavor-off-diagonal pseudo-scalar dressing function in comparison to tree-level for the first (top-left panel) and third (top-right panel) generation, as well as a function of gw for the first generation (bottom-left panel) and for the different generations at fixed gw = 0.2 (bottom-right panel).
FIG. 25 :
25Ratio (16) for the left-handed and right-handed flavor-diagonal quark propagator for different values of gw. From top to bottom generations one to three are shown, and the left-hand panels show up-type quarks, and the right-hand panels down-type quarks.
FIG. 26 :
26The Schwinger function for different values of gw. Top and middle panels show generations one and two and the lefthand panels show up-type quarks, and the right-hand panels down-type quarks. The bottom panels show the flavor-off-diagonal elements for the first (left panel) and second (right panel) generation.
FIG. 23:The left-hand panel shows the flavor-diagonal pseudo-scalar dressing function for the different quarks at gw = 0.2. In the right-hand panel the flavor-off-diagonal pseudo-scalar dressing function is shown for degenerate (here: up and down) quarks masses.g w =0.2
chiral
up
down
strange
charm
bottom
top
-5x10
-19
-4x10
-19
-3x10
-19
-2x10
-19
-1x10
-19
0
1x10
-19
2x10 -19
10
-6
10
-4
10
-2
1
10
2
10
4
10
6
10
8
D
ud (p
2
) [1/GeV]
p
2 [GeV
2 ]
up and down degenerate masses
0
0.01
0.1
0.2
0.3
0.4
gw
a
b
δ
e
f
m
φ
chiral
0
0.01
0.05
0.1
0.2
0.3
0.4
0.54 ± 0.05
0.54 ± 0.05
0.54 ± 0.05
0.47 ± 0.05
0.42 ± 0.05
0.38 ± 0.05
0.31 ± 0.05
0.30 ± 0.01
0.30 ± 0.01
0.26 ± 0.01
0.21 ± 0.01
0.17 ± 0.01
0.05 ± 0.01
10 −7 ± 0.01
−1.13 ± 0.05
−1.13 ± 0.05
−0.90 ± 0.05
−0.66 ± 0.05
−0.14 ± 0.05
0 ± 0.05
0 ± 0.05
0.44 ± 0.05
0.44 ± 0.05
0.48 ± 0.05
0.25 ± 0.05
0.16 ± 0.05
0.26 ± 0.05
0.15 ± 0.05
0.043 ± 0.005
0.043 ± 0.005
0.14 ± 0.05
0.12 ± 0.05
−0.04 ± 0.05
0.46 ± 0.05
0.25 ± 0.05
0.61 ± 0.05
0.61 ± 0.05
0.60 ± 0.05
0.51 ± 0.05
0.45 ± 0.05
0.38 ± 0.05
0.31 ± 0.05
0.51 ± 0.05
0.51 ± 0.05
0.46 ± 0.05
0.41 ± 0.05
0.38 ± 0.05
0.14 ± 0.05
10 −9 ± 0.05
up
0
0.01
0.05
0.1
0.2
0.3
0.4
0.54 ± 0.05
0.54 ± 0.05
0.54 ± 0.05
0.49 ± 0.05
0.45 ± 0.05
0.39 ± 0.05
0.34 ± 0.05
0.31 ± 0.01
0.31 ± 0.01
0.27 ± 0.01
0.22 ± 0.01
0.17 ± 0.01
0.06 ± 0.01
10 −7 ± 0.01
−1.13 ± 0.05
−1.13 ± 0.05
−0.92 ± 0.05
−0.82 ± 0.05
−1.15 ± 0.05
0 ± 0.05
0 ± 0.05
0.46 ± 0.05
0.46 ± 0.05
0.45 ± 0.05
0.30 ± 0.05
0.25 ± 0.05
0.27 ± 0.05
0.22 ± 0.05
−0.045 ± 0.005
−0.045 ± 0.005
0.10 ± 0.05
0.11 ± 0.05
−0.02 ± 0.05
0.47 ± 0.05
0.44 ± 0.05
0.63 ± 0.05
0.63 ± 0.05
0.60 ± 0.05
0.53 ± 0.05
0.48 ± 0.05
0.39 ± 0.05
0.34 ± 0.05
0.51 ± 0.05
0.51 ± 0.05
0.47 ± 0.05
0.43 ± 0.05
0.36 ± 0.05
0.14 ± 0.05
10 −9 ± 0.05
down
0
0.01
0.05
0.1
0.2
0.3
0.4
0.56 ± 0.05
0.56 ± 0.05
0.56 ± 0.05
0.50 ± 0.05
0.46 ± 0.05
0.39 ± 0.05
0.34 ± 0.05
0.32 ± 0.01
0.32 ± 0.01
0.28 ± 0.01
0.20 ± 0.01
0.19 ± 0.01
0.06 ± 0.01
10 −7 ± 0.01
−1.13 ± 0.05
−1.13 ± 0.05
−0.86 ± 0.05
−0.40 ± 0.05
−1.45 ± 0.05
0 ± 0.05
0 ± 0.05
0.47 ± 0.05
0.47 ± 0.05
0.46 ± 0.05
0.24 ± 0.05
0.17 ± 0.05
0.22 ± 0.05
0.17 ± 0.05
−0.046 ± 0.005
−0.046 ± 0.005
0.15 ± 0.05
0.15 ± 0.05
−0.06 ± 0.05
0.30 ± 0.05
0.25 ± 0.05
0.64 ± 0.05
0.64 ± 0.05
0.63 ± 0.05
0.54 ± 0.05
0.49 ± 0.05
0.40 ± 0.05
0.34 ± 0.05
0.51 ± 0.05
0.51 ± 0.05
0.46 ± 0.05
0.39 ± 0.05
0.39 ± 0.05
0.15 ± 0.05
10 −9 ± 0.05
TABLE I :
IFit values for the flavor-diagonal Schwinger functions.
TABLE II :
IIFit values for the flavor-off-diagonal Schwinger functions.
Which is not trivial, as even at tree-level the flavor propagation is not symmetric.
AcknowledgementsWe are grateful to Helios Sanchis-Alepuz, Jordi Paris-Lopez and Adrian Lorenz Blum for helpful discussions. W. M. has been supported by the FWF doctoral school W1203-N16.Appendix D: Fit parameters for the Schwinger functionThe values for the fit-function[22]∆(t) = e 2 2m 3 sin (2φ) e −tm cos(φ)for the Schwinger function in section III E are listed for the various cases studied in tables I and II for the flavordiagonal and flavor-off-diagonal elements, respectively. The fit form(12)in section II D captures the essential features of this form. See[22]for a more detailed discussion. The errors are an estimate of how strongly the fit can be distorted without introducing any substantial deviations from the data.
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|
[] |
[
"ESSENTIAL SURFACES IN HIGHLY TWISTED LINK COMPLEMENTS",
"ESSENTIAL SURFACES IN HIGHLY TWISTED LINK COMPLEMENTS"
] |
[
"Ryan Blair ",
"ANDDavid Futer ",
"Maggy Tomova "
] |
[] |
[] |
We prove that in the complement of a highly twisted link, all closed, essential, meridionally incompressible surfaces must have high genus. The genus bound is proportional to the number of crossings per twist region. A similar result holds for surfaces with meridional boundary: such a surface either has large negative Euler characteristic, or is an n-punctured sphere visible in the diagram.
|
10.2140/agt.2015.15.1501
|
[
"https://arxiv.org/pdf/1312.5016v2.pdf"
] | 56,033,305 |
1312.5016
|
8ebb02b1738b32debb2a1153b715b10acdd3ffb4
|
ESSENTIAL SURFACES IN HIGHLY TWISTED LINK COMPLEMENTS
11 Oct 2014
Ryan Blair
ANDDavid Futer
Maggy Tomova
ESSENTIAL SURFACES IN HIGHLY TWISTED LINK COMPLEMENTS
11 Oct 2014arXiv:1312.5016v2 [math.GT]
We prove that in the complement of a highly twisted link, all closed, essential, meridionally incompressible surfaces must have high genus. The genus bound is proportional to the number of crossings per twist region. A similar result holds for surfaces with meridional boundary: such a surface either has large negative Euler characteristic, or is an n-punctured sphere visible in the diagram.
Introduction
Links in S 3 are most easily visualized via a projection diagram. However, obtaining topological and geometric information directly from link diagrams has proved to be a difficult task. Historically, alternating links are one of the few classes of links for which this information has been accessible. For instance, links with prime alternating diagrams contain no incompressible tori [12], and have minimal-genus Seifert surfaces constructible directly from the diagram [4,13]. The goal of this paper is to extend results in this vein to diagrams with a high degree of twisting. To state our results, we must define what this means.
A bigon in a link diagram D(K) is a disk in the projection plane, whose boundary consists of two arcs in the projection of K. Define an equivalence relation on crossings in a diagram, in which two crossings are considered equivalent if they are connected by a string of one or more consecutive bigons. Then, a twist region of a diagram is an equivalence class of crossings. The minimal number of crossings in a twist region of D(K) is called the height of D, denoted h(D), and the number of twist regions of D(K) is called the twist number, denoted t(D).
The height and twist number of a diagram turn out to be deeply related to the geometric structure of the link it depicts. Lackenby showed that given a prime alternating diagram, the hyperbolic volume of the link complement is bounded both above and below by linear functions of the twist number t(D) [11]. Futer, Kalfagianni, and Purcell extended these volume estimates to non-alternating diagrams for which h(D) ≥ 7; that is, diagrams where every twist region contains at least 7 crossings [6]. Additionally, the results of Futer and Purcell [8] imply that when h(D) is large, there is a close connection between the link diagram and any generalized Heegaard decomposition for the exterior of K.
In this paper, we show that h(D) provides a linear lower bound on the genus of essential surfaces in a link complement. Stating our results precisely requires several definitions.
A link diagram is prime if every simple closed curve in the projection plane P that meets D(K) transversely in two points in the interior of edges bounds a disk in P that is disjoint from all crossings of the diagram. A diagram is called twist-reduced if, for every simple closed curve in P that meets D(K) in exactly two crossings, those two crossings belong to the same twist region. (See Figure 1, left.) We will implicitly assume that the diagram Left: in a twist-reduced diagram, these crossings must belong to the same twist region. Right: in a twist-reduced diagram with alternating twist region, this configuration cannot occur. D(K) is connected and alternating within each twist region (so the configuration of Figure 1, right cannot occur). It is easy to verify that every prime link K has a prime, twist-reduced diagram, with alternating twist regions. This can be achieved by first applying a maximal number of type II Reidermeister moves that eliminate crossings, followed by applying flypes to consolidate crossings into a minimal number of twist regions.
A surface embedded in S 3 is n-punctured if it meets K transversely in exactly n points. Two n-punctured surfaces are equivalent if they are transversely isotopic with respect to K. A surface F embedded in S 3 is c-incompressible if every disk or 1-punctured disk D embedded in S 3 such that D ∩ F = ∂D is transversely isotopic to a disk or 1-punctured disk contained in F while fixing the boundary. Although c-incompressibility is a strictly stronger condition than incompressibility, it is often better behaved than incompressibility and more natural to use when studying surfaces in link exteriors. We can now state the main theorem. Theorem 1.1. Let K ⊂ S 3 be a link with a connected, prime, twist-reduced diagram D(K). Suppose D(K) has at least 2 twist regions and h(D) ≥ 6. Let F ⊂ S 3 K be a closed, essential, c-incompressible surface in the link complement. Then
χ(F ) ≤ 5 − h(D).
Furthermore, if K is a knot, then χ(F ) ≤ 10 − 2h(D).
A special case of Theorem 1.1 was proved by Futer and Purcell [7,Theorem 1.4]: if h(D) ≥ 6, then χ(F ) < 0, which implies that F cannot be a sphere or torus.
There is an analogous statement for surfaces with meridional boundary.
Theorem 1.2. Let K ⊂ S 3 be a link with a connected, prime, twist-reduced diagram D(K). Suppose D(K) has at least 2 twist regions and h(D) ≥ 6. Let F ⊂ S 3 K be a connected, essential, c-incompressible surface in S 3 K, whose boundary consists of meridians of K. Then one of two conclusions holds:
(1) F is a sphere with n punctures, which intersects the projection plane in a single closed curve that meets the link n times and is disjoint from all twist regions.
(2) χ(F ) ≤ 5 − h(D).
In other words: either F is "visible in the projection plane", or we obtain the same Euler characteristic estimate as in Theorem 1.1.
There is an interesting analogue between several results involving the height h(D) and results involving distance of bridge surfaces. Distance is an integer measure of complexity for a bridge surface for a knot that has deep implications for the underlying topology and geometry of the knot exterior. The distance of a bridge surface bounds below the genus of certain essential surfaces in the knot exterior [2], while Theorem 1.1 and Theorem 1.2 demonstrate an analogous property for height. It is known that both diagrams with large height and bridge surfaces with large distance produce knots with no exceptional surgeries [3,7]. Additionally, both height and bridge distance give strong restrictions on the Heegaard surfaces for the knot exterior [8,15].
The analogous results about height and bridge distance are all the more striking given that the two notions are in some ways orthogonal. For instance, for 2-bridge knots, distance is essentially equal to the number of twist regions t(D) in a minimal diagram [16], while the height h(D) is the minimal number of crossings per twist region. It would be interesting to know whether the analogous results are indicative of some deeper underlying structure.
Here is a brief outline of the proofs of Theorems 1.1 and 1.2. We begin by adding a number of extra link components to K, so that there is a link component encircling each twist region. (See Figure 2.) In Section 2, we review the construction of this augmented link L, and show that F can be moved by isotopy into a favorable position with respect to the added link components. In Section 3, we describe a decomposition of the augmented link complement into right-angled ideal polyhedra, and again isotope F into a favorable position with respect to these polyhedra.
Sections 4 and 5 constitute the heart of the paper. Here, we use the combinatorics of the ideal polyhedra to estimate the number of times that the surface F must intersect the extra link components that we added to construct L. Each of these intersections will make a definite contribution to the Euler characteristic of F , implying the estimates of Theorems 1.1 and 1.2.
Augmented links and crossing disks
In the arguments that follow, we will assume that D(K) satisfies the hypotheses of Theorems 1.1 and 1.2. Specifically, D(K) is a connected, prime, twist-reduced diagram with at least 2 twist regions and h(D) ≥ 6. By [7,Theorem 1.4], these hypotheses on D(K) imply that K is prime and S 3 K is irreducible.
The proof of Theorems 1.1 and 1.2 relies on the geometric study of augmented links. Let us recap the definitions, while pointing the reader to Purcell's survey paper [14] for more details.
For every twist region of D(K), we add an extra link component, called a crossing circle, that wraps around the two strands of the twist region. The result is a new link J. (See Figure 2.) Now, the manifold S 3 J is homeomorphic to S 3 L, where L is obtained by removing all full twists (pairs of crossings) from the twist regions of J. This link J is called the augmented link corresponding to D(K). By [7, Theorem 2.4], both J and L are prime and S 3 L ∼ = S 3 J is irreducible.
Every crossing circle C i bounds a crossing disk D i that is punctured twice by strands of K. These twice-punctured disks play a particularly significant role in the hyperbolic geometry of S 3 L. Note that S 3 K can be recovered from S 3 L by 1/n i Dehn filling on C i , where |n i | is the number of full twists that we removed from the corresponding twist region.
A key goal in proving Theorems 1.1 and 1.2 is to place the surface F into a particularly nice position with respect to the crossing circles and crossing disks. This will be done in two steps. First, we move F by isotopy through S 3 K into a position that minimizes the intersections with the crossing disks. Then, in the next section, we drill out the crossing circles and place the remnant surface F • ⊂ F into normal form with respect to a polyhedral decomposition. Proof. The first step of the proof is to rule out closed curves of intersection. Since D i is a twice-punctured disk, every closed curve in D i is either trivial or parallel to one of the boundary components. Isotope F to intersect the union of the D i minimally.
Since F is incompressible and S 3 K is irreducible, no curve of intersection can bound a disk in D i since we could eliminate such a curve of intersection via an isotopy of F . Similarly, since F is c-incompressible and K is prime, no closed curve of intersection can be parallel to a meridian of K. Thus all closed curves of F ∩ D i are parallel to C i . We may then move F by isotopy in S 3 K, past the crossing circle C i , and remove all remaining curves of intersection F ∩ D i , contradicting our minimality assumption. See Figure 3. Figure 3. If F intersects a crossing disk D i in a closed curve, this closed curve must be parallel to crossing circle C i , and can be removed by isotopy. Now that we have ruled out closed curves of F ∩ D i , all components of intersection must be arcs. An arc with an endpoint on K cannot occur, because F is a meridional surface and after an small perturbation of F we can assume that F is disjoint from the points of intersection between K and the crossing disks. Therefore, every component of F ∩ D i is an arc from C i to C i . If any of these arcs are inessential in D i , then an outermost such arc α can be removed via an isotopy of F supported in a neighborhood of the subdisk of D i cobounded by α and an arc in C i . Thus, every component of intersection between F and a crossing disk D i is an essential arc in D i with endpoints in C i .
F F ⇒ D i D i C i
Corollary 2.2. Suppose that F is moved by isotopy into a position that minimizes the number of components of F ∩ i D i , as in Lemma 2.1. This position also minimizes the number of points of intersection between F and the crossing circles C i .
Proof. By Lemma 2.1, every component of F ∩ D i is an arc from C i back to C i . This arc has two endpoints on C i . Suppose F * is an isotopic copy of F that minimizes the number of points of intersection between F and the crossing circles C i . As described in the proof of Lemma 2.1, c-incompressibility of F * and primeness of K implies we can eliminate loops of intersection between F * and any D i that bound disks or 1-punctured disks in D i via an isotopy the fixes F * ∩ ∪ i C i . Similarly, we can remove loops of intersection between F * and any D i that are isotopic to C i in D i via an isotopy of F * that fixes F * ∩ ∪ i C i . Hence, we can assume that the points of intersection between F * and the C i are in two-to-one correspondence with the components of intersection of F * and the D i . Thus, minimizing the number of components of F ∩ i D i also minimizes the number of points of intersection between F and the crossing circles C i . Our next step is to drill out the crossing circles C i . Suppose, following Corollary 2.2, that F intersects i D i , and thus i C i , minimally. Let F • = F i C i be the remnant of F after removing the crossing circles. Lemma 2.3. Let L be the augmented link, as in Figure 2. Then, after isotoping F to minimize the number of components of
F ∩ i D i , F • = F i C i is an essential c-incompressible surface in S 3 L. Proof. Suppose that F • is compressible in S 3 L. Let γ be an essential curve in F • that bounds a compressing disk D in S 3 L.
If γ is essential in F , then we contradict the incompressibility of F . Hence γ bounds a disk E in F that is disjoint from K but meets i C i in a nontrivial number of points. Since E ∪ D is a 2-sphere bounding a 3-ball, there is an isotopy of F taking E to D that strictly reduces the number of components of F ∩ i D i , a contradiction.
Suppose that F • is incompressible and c-compressible in S 3 L. Let γ be an essential curve in F • that bounds a 1-punctured disk D in S 3 L. If γ is essential in F , then we contradict the c-incompressibility of F . Hence, γ bounds a punctured disk E in F that meets i C i in at least two points and meets K in at most one point. If E meets K in exactly one point, then, since K is prime, E ∪ D is a 2-sphere bounding a 3-ball that meets K in a single unknotted arc. Thus, there is an isotopy of F transverse to K taking E to D that strictly reduces the number of components of F ∩ i D i , a contradiction. If E is disjoint from K, then the isotopy of F taking E to D is supported in a 3-ball disjoint from K and again strictly reduces the number of components of F ∩ i D i , a contradiction.
Suppose that F • is boundary parallel in the exterior of L. Then F • is isotopic to the boundary of a regular neighborhood of a component of L or F • is boundary compressible. If F • is boundary compressible, then, since the exterior of L has all torus boundary components, F • is compressible in S 3 L, which is a contradiction as previously demonstrated, or F • is a boundary parallel annulus. If F • is a boundary parallel annulus, then F is not essential in S 3 K, a contradiction. If F • is isotopic to the boundary of a regular neighborhood of a component of L, then F is not essential in S 3 K, a contradiction.
The polyhedral decomposition
In this section, we consider the intersection between the punctured surface F • and a certain polyhedral decomposition of of S 3 L. For the purposes of this paper, a right-angled A A C C Figure 4. Decomposing S 3 L into ideal polyhedra. First, slice along the projection plane, then split remaining halves of two-punctured disks. This produces the polygon on the right. Figured borrowed from [7].
ideal polyhedron is a convex polyhedron in hyperbolic 3-space, all of whose vertices lie on the sphere at infinity, and all of whose dihedral angles are π/2. A right-angled polyhedral decomposition of a 3-manifold M is an expression of M as the union of finitely many rightangled ideal polyhedra, glued by isometries along their faces. Note that a right-angled polyhedral decomposition endows M with a complete hyperbolic metric. In our setting, where M is the augmented link complement S 3 L, there is a well-studied way to decompose M into two identical right-angled ideal polyhedra, first considered by Adams [1] and later popularized by Agol and Thurston [10, Appendix]. Purcell's survey article [14] describes the polyhedral decomposition in great detail. For our purposes, the salient features are summarized in the following theorem, and illustrated in Figure 4.
Theorem 3.1. Let D(K) be a prime, twist-reduced diagram of a link K with at least 2 twist regions. Let L be the augmented link constructed from D(K). Then the augmented link complement S 3 L is hyperbolic, and there is a decomposition of S 3 L into two identical totally geodesic polyhedra P and P ′ . In addition, these polyhedra have the following properties.
(1) The faces of P and P ′ can be checkerboard colored, with shaded faces all triangles corresponding to portions of crossing disks, and white faces corresponding to regions into which L cuts the projection plane. Our goal is to place F • in normal form with respect to this polyhedral decomposition. Our convention is that the ideal vertices of the polyhedra are truncated to form boundary faces that tile the boundary tori of S 3 L. Then, ∂F intersects the boundary faces in a union of arcs.
Definition 3.2. Let P be a truncated ideal polyhedron. An embedded disk D ⊂ P is called normal if its boundary curve γ = ∂D satisfies the following conditions:
(1) γ is transverse to the edges of P , (2) γ doesn't lie entirely in a face of P , (3) no arc of γ in a face of P has endpoints on the same edge, or on a boundary face and an adjacent edge, (4) γ intersects each edge at most once, and (5) γ intersects each boundary face at most once.
If M is a 3-manifold subdivided into ideal polyhedra, a surface S is called normal if its intersection with each polyhedron is a disjoint union of normal disks.
It is a well-known fact, originally due to Haken [9], that every essential surface in an irreducible 3-manifold can be isotoped into normal form. However, in our context, we would like to make F • normal while preserving the conclusion of Lemma 2.1. This requires carefully managing the complexity of the surface.
(S) = #(S ∩ f 1 ), . . . , #(S ∩ f n ) .
Here, # denotes the number of components. Given two surfaces S and S ′ , we say that c(S) ≤ c(S ′ ) if the inequality holds in each coordinate. We say that c(S) < c(S ′ ) if c(S) ≤ c(S ′ ) and there is a strict inequality in at least one coordinate. This argument is adapted from Futer and Guéritaud [5, Theorem 2.8], and the figures are drawn from that paper.
Proof. We need to ensure that S satisfies the conditions of Definition 3.2. By hypothesis, S is transverse to the polyhedra. This transversality implies that for every polyhedron P , each component of S ∩ ∂P is a simple closed curve, and gives condition (1). Additionally, since S is incompressible, we can assume that we have isotoped S to meet each polyhedron in a collection of properly embedded disks. Now, whenever some component of S ∩ ∂P violates one of the conditions (2)-(5), we will describe a move that reduces the complexity c(S). That is, for each face σ of the polyhedra, the intersection number #(S ∩ σ) will either remain constant or decrease, with a strict decrease for at least one face.
Suppose that γ is a closed curve, violating (2). Without loss of generality, we may assume that γ is innermost on the face σ. Then γ bounds a disk D ⊂ σ, whose interior is disjoint from S. But since S is incompressible, γ also bounds a disk D ′ ⊂ S. Furthermore, since M is irreducible, the sphere D ∪ γ D ′ must bound a ball. Thus, we may isotope S through this ball, moving D ′ past D. This isotopy removes the curve γ from the intersection between S and σ. In addition, the isotopy will remove the intersections between D ′ and any other faces of P .
Next, suppose that γ runs from an edge e back to e, violating the first half of condition (3). Then γ and a sub arc of e co-bound a disk D ⊂ σ, and we can assume γ is innermost (i.e. S does not meet D again). We can use this disk D to guide an isotopy of S past the edge e, as in the left panel of Figure 5. This isotopy removes γ from the intersection between S and σ. Some intersection components between S and the interiors other faces adjacent to e will also merge. Hence, #(S ∩ σ) stays constant or decreases for each face.
Suppose that an arc γ runs from a boundary face to an adjacent interior edge in a face σ, violating the second half of condition (3). Then γ has endpoints in adjacent edges of ∂σ, and we may assume without loss of generality that it is outermost in σ. Thus γ once again cuts off a disk D from σ. By isotoping S along this disk, as in the right panel of Figure 5, we remove γ from S ∩ σ and alter the intersection of S with any other face by an isotopy of arcs in that face.
Suppose a component γ ′ of S ∩ ∂P intersects an edge e twice, violating (4). Let γ be the closure of a component of γ ′ − e such that γ together with a subarc of e cobound a disk D. By passing to an outermost arc of intersection between S and D, we can assume that D ∩ S = ∂D ∩ S = γ. If γ is contained in a face of ∂P , then we violate (3). Hence, we can assume that γ meets the face of ∂P that contains a neighborhood of ∂γ in at least two components. While fixing ∂D ∩ e isotope the rest of D slightly into the interior of P . If S meets the interior of D it does so in simple closed curves. Since S meets P in a collection of properly embedded disks, then we can eliminate all components of intersection between S and the interior of D via a isotopy of S that is supported in the interior of P and fixes c(S). After this isotopy, D is a boundary compressing disk for the component of S ∩ P that contains γ ′ in its boundary. As in Figure 5, left, we may use D to guide an isotopy of S past edge e. Since γ meets the face of ∂P that contains a neighborhood of ∂γ in at least two components, this isotopy will strictly reduce #(S ∩ σ) for that face and will not increase #(S ∩ σ) for every other face σ that meets e.
Finally, suppose that γ meets a boundary face twice, violating (5). Then the polyhedron P contains a boundary compression disk D for S such that ∂D ∩ ∂M is contained in the boundary face. Since S is boundary-incompressible, γ must also cut off a disk D ′ ⊂ S, as in Figure 6. Since S 3 L is irreducible, it follows that the disk D ∪ γ D ′ is boundaryparallel. Thus we may isotope S through a boundary-parallel ball, moving D ′ past D, which eliminates all components of intersection between D ′ and ∂P . Since ∂D ′ meets the edge of a boundary face, this isotopy strictly lowers c(S). Since each of the above moves reduces the complexity c(S), a minimum-complexity position will be normal.
As a consequence, we get the following structural statement. Proof. Recall that in the construction of F • , we have assumed that we have isotoped F to minimize the number of components of F ∩ i D i . Hence, by Lemma 2.1, conclusion (1) holds before we begin the normalization procedure. Additionally, there is an isotopy of F • supported in a neighborhood of the D i such that after this isotopy any component of F • ∩ D i meets any shaded face of the polyhedra in at most one arc. Since each arc of intersection of F • ∩ D i is essential in D i , then for each shaded face σ of the polyhedra, F • ∩ σ is an arc from an ideal vertex at a crossing circle to the opposite edge. Hence, we can assume both conclusion (1) and conclusion (2) hold before we begin the normalization procedure. We claim that before the normalization procedure, the total number of arcs of F • in shaded faces is 2
i # F ∩ D i = i # F ∩ C i .
This is because each component of F • ∩ D i runs from C i to C i , and consists of one arc in each of the two shaded faces comprising D i . Each such arc runs from an ideal vertex at C i to the opposite edge, as in Figure 7. Now, consider what happens during the normalization procedure of Lemma 3.4. That procedure monotonically reduces the complexity c(F ). In other words, for every face σ, #(F ∩ σ) either stays constant or goes down. But by Corollary 2.2, the quantity 2 i # F ∩ D i is already minimal before normalization. Since this quantity is the total number of intersections between F • and the shaded faces, it follows that #(F ∩ σ) stays constant for every shaded face σ. This means that the intersections between F and the shaded faces remain as in Figure 7, and conclusions (1) and (2) remain true throughout the normalization process. Proof. Recall from [7, Lemma 2.6] that the cusp torus T i corresponding to crossing circle C i is cut by the polyhedra into two rectangular boundary faces, one in each polyhedron. In the universal cover of T i , we have a rectangular lattice spanned by s and w, where s is a step parallel to a shaded face (horizontal in Figure 8) and w is a step parallel to a white face (vertical in Figure 8). In order to recover S 3 K from S 3 L, we need to fill the torus T i along a slope corresponding to the meridian of C i in S 3 J. By [7,Theorem 2.7], this Dehn filling slope is homologous to w + n i s.
In S 3 J, the punctured surface F • meets the neighborhood of each crossing circle in a meridian. By the above paragraph, each component of ∂F • on T i has homological intersection ±1 with the shaded faces. On the other hand, by Lemma 3.5, each puncture of F • at C i gives rise to a single arc in the the shaded disk. Thus, each curve of ∂F • on T i only has geometric intersection number 1 with the shaded faces. The only way to do this while staying in the homology class w + n i s is to take (n i − 1) segments parallel to s, along with two diagonal segments whose sum is w + s.
In the following section we will need a vocabulary that allows us to translate combinatorial statements regarding normal loops in the boundary of P into combinatorial statements regarding the knot diagram D(K). The following remark provides this translation.
Remark 3.7. The homeomorphism from S 3 J to S 3 L can be taken to be the identity outside of a neighborhood of the union of the crossing disks in S 3 . We can view this fact diagrammatically by shrinking the twist regions in the diagram of K until each is contained in the regular neighborhood of the arc of intersection between the corresponding crossing disk and the plane of projection for K. By Theorem 3.1, the white faces of the polyhedral decomposition of the complement of L meet the complement of the neighborhood of the union of the crossing disks in S 3 exactly in the plane of projection for L. Equivalently, the white faces of the polyhedral decomposition of the complement of J meet the complement of a regular neighborhood of the twist regions of K exactly in the plane of projection for K. In this way, arcs and loops in the white faces of the polyhedral decomposition are arcs and loops in the complement of the twist regions in the plane of projection for K. Additionally, in light of Lemma 3.6, we know exactly how a normal surface meets the faces of the boundary of a polyhedron that correspond to cusp tori. In particular, if a normal loop in the boundary of a polyhedron meets only white faces and cusp tori faces, then each component of intersection with the cusp tori faces is a segment in the s direction. Hence, if a normal surface meets the boundary of a polyhedron in a loop that is disjoint from the shaded faces and this loop meets the collection of cusp tori faces in n components, then there is a curve in the plane of projection for K that cuts through twist regions n times and meets K in exactly 2n points. See Figure 9.
Intersections with crossing circles
In this section, we bound from below the number of times that a c-incompressible surface F must meet the crossing circles. We note that some, but not all, of the subsequent lemmas carry the hypothesis that F is closed. This will allow us maximum flexibility in proving Theorems 1.1 and 1.2.
Lemma 4.1. Suppose that F ⊂ S 3 K is a closed, c-incompressible surface. Then F must intersect a crossing circle.
Proof. Suppose that F is disjoint from every C i . Then F = F • , and by Lemma 3.5, we can assume F is normal and disjoint from the crossing disks. By Theorem 3.1 the shaded faces of P ∪ P ′ glue to form the crossing disks. Thus F ∩ (∂P ∪ ∂P ′ ) is entirely contained in the white faces.
Since P and P ′ are checkerboard colored, every side of every white face borders on a shaded face. But F is disjoint from the shaded faces, hence it cannot meet any edge of the white faces. Thus any intersection of F with a white face must be a simple closed curve, contradicting the normality of F = F • . Lemma 4.2. Suppose that F ⊂ S 3 K is a c-incompressible surface, either meridional or closed. Let ∆ ⊂ F • be a normal disk that meets exactly one crossing circle cusp. Then ∆ must also meet a cusp corresponding to K.
In particular, if F is closed, some normal disk ∆ ⊂ F • must meet at least two crossing circles.
Proof. Let γ ⊂ ∂∆ be the unique arc of ∂∆ in a boundary face T i corresponding to a crossing circle C i . By Lemma 3.6, γ is either a segment in the s direction, parallel to a shaded face, Figure 10. The two cases of Lemma 4.2. Left: ∂∆ must intersect a cusp of K, or else the dotted arc lies in a single white face, violating primeness. Right: ∂∆ must intersect a cusp of K, because the two endpoints of the dotted arc are separated by an odd number of knot strands.
K K K K ω ω ∂∆ ∂∆ γ γ σ
or else a diagonal segment that runs from a white face to a shaded face. We will consider these possibilities in turn. Case 1: γ runs parallel to the shaded faces, from a white face ω ′ to another white face ω. Consider where ∂∆ can go next. If ∂∆ crosses an edge of a polyhedron into a shaded face σ, Lemma 3.5 implies that it must next run into a boundary face T j corresponding to some crossing circle C j . But by hypothesis, ∂∆ meets only one boundary face, hence T i = T j . Thus, ∂∆ must meet T i both in a segment parallel to a shaded face and in a diagonal segment that runs from a white face to a shaded face, contradicting normality.
If ∂D runs from ω directly into the boundary face T j of a crossing circle C j , then again we must have T i = T j , which means that ω = ω ′ and ∂∆ contains only two segments. But then, as Figure 10 shows, we can use Remark 3.7 to find a loop in D(K) corresponding to γ that intersects K twice with non-trivial regions on each side. This contradicts the primeness of the diagram D(K).
The remaining possibility, if γ is a segment in the s direction, is that ∂∆ runs through ω to the a truncated ideal vertex corresponding to K. This is our desired conclusion.
Case 2: γ is a diagonal segment that runs from a shaded face σ to a white face ω. Then, observe that the two ends of γ are separated by an odd number of knot strands. (See Figure 10.) Thus, to form a closed curve, ∂∆ must either cross a strand of K, which is our desired conclusion, or cross through another shaded face σ ′ . But then, as above, ∂∆ would have to run through σ ′ to a boundary face T j of some crossing circle C j , which contradicts either
normality (if T i = T j ) or the hypotheses (if T i = T j ).
Thus, in all cases, ∂D must meet a cusp corresponding to K. Proof. Suppose F meets strictly fewer than three crossing circles. By Lemmas 4.1 and 4.2, F must meet exactly two distinct crossing circles, C i and C j . Recall that Lemma 3.6 implies that each component of intersection between ∂F • and the cusp torus T i (resp. T j ) contains n i − 1 ≥ 5 (resp. n j − 1 ≥ 5) segments parallel to s, and only two other segments. By Lemma 4.2, a normal disk ∆ ⊂ F cannot intersect T i only. It follows that some disk ∆ ⊂ F intersects each of T i and T j in a segment parallel to s. Figure 11. Lemma 4.3: if ∂∆ meets exactly two crossing circle cusps, in segments parallel to s, then the diagram D(K) cannot be twist-reduced.
K K K K ∂∆
Consider how the curve ∂∆ can close up. This curve cannot meet the ideal vertices corresponding to K, and it also cannot meet any additional shaded faces (otherwise Lemma 3.5 would force ∆ to run into an additional crossing circle C k ). The only remaining possibility is that ∂∆ runs through a white face from T i to T j , and then through another white face back to T i . Now, Figure 11 shows that we can use Remark 3.7 to find a loop in the projection plane corresponding to ∂∆ that will intersect K four times, with two intersections adjacent to the twist region of C i and the remaining intersections adjacent to the twist region of C j . This violates the hypothesis that D(K) is twist-reduced. Proof. The first conclusion is an immediate consequence of the fact that F is separating.
For the second conclusion, suppose that F is a closed surface and K is a knot. Since F is closed, it must separate S 3 into two components. The knot K must lie in one of these components. But every arc of F ∩ D i separates the two strands of K that puncture the disk D i . Therefore, since K lies on one side of F , the arcs of F ∩ D i must come in pairs. Hence, if F intersects a crossing circle C i at all, it must meet it at least 4 times.
Combinatorial length
The lemmas in the previous section give us a lot of control over the number of times that F • meets the cusps T i corresponding to the crossing circles. To prove Theorems 1.1 and 1.2, we need to show that each component of ∂F • ∩ T i makes a substantial contribution to the Euler characteristic of F • , hence to that of F as well.
This can be done in one of two ways: either by estimating the geometric length of each component of ∂F • on a maximal cusp corresponding to the crossing circle C i , or to estimate its combinatorial length in the sense of bounding the complexity of normal disks comprising F • . The paper of Futer and Purcell contains readily applicable estimates on both combinatorial length and geometric length [7, Theorem 3.10 and Proposition 5.13], and either result would suffice for Theorem 1.1. We choose to pursue the combinatorial approach, because this is the approach that will generalize to meridional surfaces in Theorem 1.2.
The notion of combinatorial length was developed by Lackenby, as part of his study of Dehn surgeries on alternating knots [10]. The main idea is that the Euler characteristic of a surface can be controlled by understanding the intersections between that surface and the truncated ideal vertices in an ideal polyhedral decomposition.
Let us recap the key definitions. Every normal disk ∆ ⊂ F • has a well-defined combinatorial area, computed using the dihedral angles of the polyhedra in a manner that mimics the area formula for hyperbolic polygons.
Definition 5.1. Let D be a normal disk in a right-angled ideal polyhedron P , with the boundary faces of P lying on ∂M . Let n be the number of interior edges of P crossed by ∂D. Then the combinatorial area of D is defined to be
area(D) = π 2 n + π|∂D ∩ ∂M | − 2π.
Furthermore, the combinatorial area of a normal surface H is defined to be the sum of the combinatorial areas of all of its constituent normal disks and is denoted area(H). Specializing to the case where M = S 3 L, we have a way to "see" combinatorial area from the crossing circles. Definition 5.3. Let ∆ be a normal disk with respect to the polyhedral decomposition of S 3 L. Let γ 1 , . . . , γ n be the segments of ∂∆ that lie in boundary faces corresponding to crossing circles, and suppose that n ≥ 1. Then, for each γ i , we define ℓ(γ i , ∆) = area(∆)/n.
In other words, the area of ∆ is distributed evenly among its intersections with the crossing circle cusps.
It is worth remarking that our definition of ℓ(γ i , ∆) differs slightly from the corresponding definition in Futer and Purcell [7,Definition 4.9]. The difference is that the latter definition divides the area of ∆ among all the segments of ∆ in boundary faces, not just those corresponding to crossing circles. Definition 5.3 is designed to give stronger versions of some of the following estimates.
area(S) ≥ i ℓ(γ i , ∆),
where the sum is taken over all normal disks ∆ ⊂ S and all segments of ∂∆ in crossing circle cusps.
Proof. This is immediate, since Definition 5.3 ensures that the area of each disk is counted with the appropriate weight. Note that the inequality might be strict, because there may be normal disks in S that have positive area but do not meet any crossing circle cusps.
In the case where S is the meridional, c-incompressible surface F • , we have a lot of control over the areas of disks and the corresponding combinatorial areas. Lemma 5.6. Let ∆ ⊂ F • be a normal disk that meets n crossing circle cusps, where n ≥ 1.
Then, for each segment γ i of ∂∆ in a crossing circle cusp,
ℓ(γ i , ∆) ≥ max{π/n, π/3}.
Proof. Let m be the number of segments of ∆ in all boundary faces (belonging either to K or to a crossing circle). By Lemma 4.2, m ≥ 2. Furthermore, by Lemma 3.5, ∆ cannot be a bigon (because the boundary of a bigon runs between two consecutive ideal vertices). If ∆ is an ideal triangle, then area(∆) = π. Thus, by Proposition 5.5, area(∆) ≥ π in all cases. By Definition 5.3, it follows that ℓ(γ i , ∆) ≥ π/n.
It remains to show that ℓ(γ i , ∆) ≥ π/3. If n ≤ 3, we are done by the previous paragraph. Alternately, if n > 3, Proposition 5.5 gives area(D) ≥ mπ 2 , where m ≥ n. Thus
ℓ(γ i , ∆) = area(∆) n ≥ mπ 2n ≥ π 2 .
We can now complete the proof of Theorem 1.1.
Proof of Theorem 1.1. Let F ⊂ S 3 K be a closed, c-incompressible surface. Isotope F into a position that minimizes the intersection number with the crossing disks D i . After drilling out the crossing circles, we obtain a surface F • = F ∩S 3 L, which can be placed into normal form via the procedure of Lemma 3.4. Let b be the number of boundary components of F • on the crossing circle cusps of S 3 L. By Corollary 4.5, we have b ≥ 6, with b ≥ 12 in case K is a knot. Furthermore, by Lemma 3.6, each of these b components consists of (n i + 1) segments in boundary faces, where n i ≥ h(D). Thus, by Lemma 5.6, each component of ∂F • contributes at least (h(D) + 1)π/3 to the area of F • . Now, we may compute:
(1) −2π(χ(F ) − b) = −2πχ(F • ) by the construction of F • = area(F • )
by the Gauss-Bonnet formula ≥ i ℓ(γ i , ∆) by Lemma 5.4 ≥ π/3 · b · (h(D) + 1) by Lemma 5.6.
We may compare the first and last terms to get
− 2π(χ(F ) − b) ≥ π/3 · b · (h(D) + 1) −6χ(F ) + 6b ≥ b h(D) + b −6χ(F ) ≥ b (h(D) − 5). χ(F ) ≤ b/6 (5 − h(D)).(2)
Substituting b ≥ 6 for links and b ≥ 12 for knots gives the desired result.
The same ideas, with one added ingredient, also prove Theorem 1.2.
Proof of Theorem 1.2. Let F ⊂ S 3 K be a compact, connected, meridional, c-incompressible surface. Isotope F into a position that minimizes the intersection number with the crossing disks D i . Drill out the crossing circles, and normalize F • in S 3 L.
Unlike the setting of closed surfaces (that is, unlike Lemma 4.1), it may happen that F • is disjoint from the crossing circle cusps, i.e. F • = F . Then, by Lemma 3.5, each normal disk ∆ ⊂ F • must be disjoint from the shaded faces. In other words, ∂∆ is a closed curve in the white projection plane, which intersects the cusps of K some number of times. This closed curve bounds a disk ∆ in polyhedron P , unique up to isotopy. Recall that P is glued to P ′ along all its white faces, and the gluing map is the identity on white faces. Thus we have an identical normal curve in P ′ , which again bounds a normal disk ∆ ′ that is unique up to isotopy. Since ∆ and ∆ ′ are glued to each other along all their edges, we conclude that F = F • is a sphere punctured some number of times by K, and, by Remark 3.7, that it meets the projection plane for K along the single closed curve ∂∆ = ∂∆ ′ .
Next, consider what happens if ∂F • contains b components along the crossing circle cusps, where b > 0. By Lemma 4.4, b is even. If b ≥ 6, then we argue exactly as in the proof of Theorem 1.1. The computations (1) and (2) in that proof produce the same estimate as for closed surfaces, namely χ(F ) ≤ 5 − h(D).
If b = 2, then ∂F • must intersect only one crossing circle C i , and in particular every normal disk of F • has at most one segment along a crossing circle cusp. Thus Lemma 5.6 tells us that ℓ(γ, ∆) ≥ π for every segment γ along T i , hence each component of ∂F • along T i contributes at least (h(D) + 1)π to the area of F • . As a consequence, the calculation of (1) gives −2π(χ(F ) − b) ≥ π · b · (h(D) + 1).
After substituting b = 2, this simplifies to
χ(F ) ≤ 1 − h(D).
Similarly, if b = 4, then ∂F • intersects either one or two crossing circles. Consequently, every normal disk of F • has at most two segments along a crossing circle cusp. Thus Lemma 5.6 (with n ≤ 2) tells us that ℓ(γ, ∆) ≥ π/2 for every segment γ along along a crossing circle cusp T i . Hence, each component of ∂F • along T i contributes at least (h(D) + 1)π/2 to the area of F • , and the calculation of (1) gives −2π(χ(F ) − b) ≥ π/2 · b · (h(D) + 1).
After substituting b = 4, this simplifies to
χ(F ) ≤ 3 − h(D).
Futer is supported in part by NSF grants DMS-1007221 and DMS-1408682. Tomova is supported in part by NSF grant DMS-1054450.October 14, 2014.
Figure 1 .
1Figure 1. Left: in a twist-reduced diagram, these crossings must belong to the same twist region. Right: in a twist-reduced diagram with alternating twist region, this configuration cannot occur.
Figure 2 .
2An augmented link L is constructed by adding a crossing circle around each twist region of D(K), then removing full twists. The crossing circles are shown in red. Figure borrowed from [7]. Lemma 2.1. Let F be a c-incompressible surface in S 3 K, whose boundary (if any) consists of meridians. Move F by isotopy into a position that minimizes the number of components of intersection with the crossing disks. Then every component of intersection between F and a crossing disk D i is an essential arc in D i with endpoints in C i .
( 2 )
2All ideal vertices are 4-valent. (3) The dihedral angle at each edge of P and P ′ is π 2 . Proof. The hyperbolicity of S 3 L is a theorem of Adams [1]; compare [7, Theorem 2.2]. The remaining assertions are proved in [14, Proposition 2.2].
Definition 3. 3 .
3Let M be a 3-manifold with a prescribed polyhedral decomposition. Let S ⊂ M be a properly embedded surface, transverse to the edges and faces of the polyhedra. Order the faces of the polyhedral decomposition: f 1 , . . . , f n . Then the complexity of S is the ordered n-tuple c
Lemma 3. 4 .
4Let M be an irreducible 3-manifold with incompressible boundary, and with a prescribed polyhedral decomposition. Let S ⊂ M be a properly embedded essential surface, transverse to the edges and faces of the polyhedra. Then S can be isotoped to a normal surface by a sequence of moves that monotonically reduces the complexity c(S).
Figure 5 .Figure 6 .
56When a surface violates condition (3) of normality, then an isotopy in the direction of the arrow removes intersections between S and all the faces that meet edge e. When a surface violates condition (5) of normality, isotoping disk D ′ past D removes intersections between S and the faces.
Figure 7 .
7Left: normal disk D i is subdivided by the projection plane of L into two shaded faces, one in each polyhedron. By Lemma 3.5, each component of F • ∩ D i is a collection of arcs from C i to C i . Right: the picture in a single shaded face σ.
Lemma 3 . 5 .
35Let F • = F i C i be as inLemma 2.3. Suppose that F • has been isotoped into normal form via the procedure of Lemma 3.4. Then the following hold.(1) For each crossing disk D i , each component of F • ∩ D i is an essential arc in D i with endpoints in C i . (2)For each shaded face σ of the polyhedra, F • ∩ σ is an arc from an ideal vertex at a crossing circle to the opposite edge. SeeFigure 7.
Figure 8 .
8The cusp torus T i of a crossing circle C i is subdivided into two boundary rectangles. There are two combinatorial possibilities, depending on whether the number of crossings n i in the twist region is even (shown on the left) or odd (shown on right). The normal curve in T i representing a component of ∂F • must be as shown in red.
Lemma 3 . 6 .
36Assume that F • is in normal form. For each cusp torus T i corresponding to crossing circle C i , each component of ∂F • ∩ T i consists of (n i − 1) segments parallel to shaded faces and 2 diagonal segments that have one endpoint on a white face and one endpoint on a shaded face. Here, n i is the number of crossings in the twist region of C i . SeeFigure 8.
Figure 9 .
9Curves in ∂P that are contained in the union of the white faces and the boundary faces give rise to curves in the plane of projection.
Lemma 4. 3 .
3Suppose that F ⊂ S 3 K is a closed, c-incompressible surface. Then F must intersect at least 3 crossing circles.
Lemma 4 . 4 .
44Every cusp T i of a crossing circle C i contains an even number of components of ∂F • . Furthermore, if K is a knot and F is a closed surface, then every cusp T i met by F • contains at least 4 components of ∂F • .
Corollary 4 . 5 .
45If F is closed, the punctured surface F • must meet the crossing circle cusps at least b times, where b ≥ 12 if K is a knot, and b ≥ 6 if K is a link. Proof. Immediate from Lemmas 4.3 and 4.4.
Proposition 5. 2 (
2Gauss-Bonnet Theorem). Let H ⊂ M be a normal surface in a 3manifold with a right-angled polyhedral decomposition. Then area(H) = −2πχ(H).
Lemma 5. 4 .
4Let S be any normal surface in the polyhedral decomposition of S 3 L. Then
Let D be a normal disk in a polyhedron P of a right-angled polyhedral decomposition of M , such that ∂D passes through at least one boundary face. Let m = |∂D ∩ ∂M |. If D is not a bigon or an ideal triangle, then area(D) ≥ mπ 2 .
Augmented alternating link complements are hyperbolic, Low-dimensional topology and Kleinian groups (Coventry/Durham, 1984). Colin C Adams, London Math. Soc. Lecture Note Ser. 112Cambridge Univ. PressColin C. Adams, Augmented alternating link complements are hyperbolic, Low-dimensional topology and Kleinian groups (Coventry/Durham, 1984), London Math. Soc. Lecture Note Ser., vol. 112, Cambridge Univ. Press, Cambridge, 1986, pp. 115-130.
Distance and bridge position. David Bachman, Saul Schleimer, Pacific J. Math. 2192David Bachman and Saul Schleimer, Distance and bridge position, Pacific J. Math. 219 (2005), no. 2, 221-235.
Ryan Blair, Marion Campisi, Jesse Johnson, Scott Taylor, Maggy Tomova, arXiv:1209.0197Exceptional and cosmetic surgeries on knots. Ryan Blair, Marion Campisi, Jesse Johnson, Scott Taylor, and Maggy Tomova, Exceptional and cosmetic surgeries on knots, arXiv:1209.0197.
Genus of alternating link types. Richard Crowell, Ann. of Math. 2Richard Crowell, Genus of alternating link types, Ann. of Math. (2) 69 (1959), 258-275.
Angled decompositions of arborescent link complements. David Futer, François Guéritaud, Proc. London Math. Soc. 982David Futer and François Guéritaud, Angled decompositions of arborescent link complements, Proc. Lon- don Math. Soc. 98 (2009), no. 2, 325-364.
Dehn filling, volume, and the Jones polynomial. David Futer, Efstratia Kalfagianni, Jessica S Purcell, J. Differential Geom. 783David Futer, Efstratia Kalfagianni, and Jessica S. Purcell, Dehn filling, volume, and the Jones polynomial, J. Differential Geom. 78 (2008), no. 3, 429-464.
Links with no exceptional surgeries. David Futer, Jessica S Purcell, Comment. Math. Helv. 823David Futer and Jessica S. Purcell, Links with no exceptional surgeries, Comment. Math. Helv. 82 (2007), no. 3, 629-664.
Explicit Dehn filling and Heegaard splittings. Comm. Anal. Geom. 213, Explicit Dehn filling and Heegaard splittings, Comm. Anal. Geom. 21 (2013), no. 3, 625-650.
Theorie der Normalflächen. Wolfgang Haken, Acta Math. 105Wolfgang Haken, Theorie der Normalflächen, Acta Math. 105 (1961), 245-375.
Word hyperbolic Dehn surgery. Marc Lackenby, Invent. Math. 1402Marc Lackenby, Word hyperbolic Dehn surgery, Invent. Math. 140 (2000), no. 2, 243-282.
The volume of hyperbolic alternating link complements. Proc. London Math. Soc. Ian Agol and Dylan Thurston3, The volume of hyperbolic alternating link complements, Proc. London Math. Soc. (3) 88 (2004), no. 1, 204-224, With an appendix by Ian Agol and Dylan Thurston.
Closed incompressible surfaces in alternating knot and link complements. William Menasco, Topology. 231William Menasco, Closed incompressible surfaces in alternating knot and link complements, Topology 23 (1984), no. 1, 37-44.
On the genus of the alternating knot. I, II. Kunio Murasugi, J. Math. Soc. Japan. 10Kunio Murasugi, On the genus of the alternating knot. I, II, J. Math. Soc. Japan 10 (1958), 94-105, 235-248.
An introduction to fully augmented links, Interactions between hyperbolic geometry, quantum topology and number theory. Jessica S Purcell, Contemp. Math. 541Amer. Math. SocJessica S. Purcell, An introduction to fully augmented links, Interactions between hyperbolic geometry, quantum topology and number theory, Contemp. Math., vol. 541, Amer. Math. Soc., Providence, RI, 2011, pp. 205-220.
Multiple bridge surfaces restrict knot distance. Maggy Tomova, Algebr. Geom. Topol. 7Maggy Tomova, Multiple bridge surfaces restrict knot distance, Algebr. Geom. Topol. 7 (2007), 957-1006.
Bridge and pants complexities of knots. Alexander Zupan, J. Lond. Math. Soc. 2Alexander Zupan, Bridge and pants complexities of knots, J. Lond. Math. Soc. (2) 87 (2013), no. 1, 43-68.
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"; Beck \nUniversity of Copenhagen\nCERN\n1211Geneva 23Switzerland\n",
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"Morten Jørgensen \nUniversity of Copenhagen\nCERN\n1211Geneva 23Switzerland\n",
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Search for metastable heavy charged particles with large ionisation energy loss in pp collisions at root s=8 TeV using the ATLAS experiment Aad, G.; Abbott, B.; Abdallah, J.; Abdinov, O.; Aben, R.; Abolins, M.; AbouZeid, O.S.; Abramowicz, H.; Abreu, H.; Abreu, R.; Dam, Mogens;Abstract Many extensions of the Standard Model predict the existence of charged heavy long-lived particles, such as R-hadrons or charginos. These particles, if produced at the Large Hadron Collider, should be moving non-relativistically and are therefore identifiable through the measurement of an anomalously large specific energy loss in the ATLAS pixel detector. Measuring heavy long-lived particles through their track parameters in the vicinity of the interaction vertex provides sensitivity to metastable particles with lifetimes from 0.6 ns to 30 ns. A search for such particles with the ATLAS detector at the Large Hadron Collider is presented, based on a data sample corresponding to an integrated luminosity of 18.4 fb −1 of pp collisions at √ s = 8 TeV. No significant deviation from the Standard Model background expectation is observed, and lifetime-dependent upper limits on Rhadrons and chargino production are set. Gluino R-hadrons with 10 ns lifetime and masses up to 1185 GeV are excluded at 95 % confidence level, and so are charginos with 15 ns lifetime and masses up to 482 GeV. ation of the LHC, as well as the support staff from our institutions without whom ATLAS could not be operated efficiently. We acknowledge the support of ANPCyT,
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1506.05332
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01a5275dc11aa138476679fabbc636095fc9564a
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Search for metastable heavy charged particles with large ionisation energy loss in pp collisions at √ s = 8 TeV using the ATLAS experiment ATLAS Collaboration
2015
Jørn Hansen
University of Copenhagen
CERN
1211Geneva 23Switzerland
; Dines
University of Copenhagen
CERN
1211Geneva 23Switzerland
Jørgen Hansen
University of Copenhagen
CERN
1211Geneva 23Switzerland
; Beck
University of Copenhagen
CERN
1211Geneva 23Switzerland
Stefania ; Xella
University of Copenhagen
CERN
1211Geneva 23Switzerland
Peter Hansen
University of Copenhagen
CERN
1211Geneva 23Switzerland
; Henrik
University of Copenhagen
CERN
1211Geneva 23Switzerland
Troels Petersen
University of Copenhagen
CERN
1211Geneva 23Switzerland
; Christian
University of Copenhagen
CERN
1211Geneva 23Switzerland
Lotte Thomsen
University of Copenhagen
CERN
1211Geneva 23Switzerland
; Ansgaard
University of Copenhagen
CERN
1211Geneva 23Switzerland
Sascha ; Mehlhase
University of Copenhagen
CERN
1211Geneva 23Switzerland
Morten Jørgensen
University of Copenhagen
CERN
1211Geneva 23Switzerland
; Dam
University of Copenhagen
CERN
1211Geneva 23Switzerland
AlmutMaria ; Pingel
University of Copenhagen
CERN
1211Geneva 23Switzerland
AskEmil ; Løvschall-Jensen
University of Copenhagen
CERN
1211Geneva 23Switzerland
Alonso Diaz
University of Copenhagen
CERN
1211Geneva 23Switzerland
Alejandro ; Monk
University of Copenhagen
CERN
1211Geneva 23Switzerland
James William
University of Copenhagen
CERN
1211Geneva 23Switzerland
; Pedersen
University of Copenhagen
CERN
1211Geneva 23Switzerland
Lars Egholm
University of Copenhagen
CERN
1211Geneva 23Switzerland
; Wiglesworth
University of Copenhagen
CERN
1211Geneva 23Switzerland
Graig ; Galster
University of Copenhagen
CERN
1211Geneva 23Switzerland
Gorm Aske
University of Copenhagen
CERN
1211Geneva 23Switzerland
Gram Krohn
University of Copenhagen
CERN
1211Geneva 23Switzerland
G Aad
University of Copenhagen
CERN
1211Geneva 23Switzerland
AbdallahB Abbott
University of Copenhagen
CERN
1211Geneva 23Switzerland
J Abdinov
University of Copenhagen
CERN
1211Geneva 23Switzerland
O Aben
University of Copenhagen
CERN
1211Geneva 23Switzerland
Abolins, M., .R
University of Copenhagen
CERN
1211Geneva 23Switzerland
Search for metastable heavy charged particles with large ionisation energy loss in pp collisions at √ s = 8 TeV using the ATLAS experiment ATLAS Collaboration
European Physical Journal C. Particles and Fields European Physical Journal C. Particles and Fields
7594072015Publication date: 2015 Received: 18 June 2015 / Accepted: 7 August 2015 / Published online: 3 September 2015Published in: Document Version Publisher's PDF, also known as Version of record Citation for published version (APA):.. Galster, G. A. G. K. (2015). Search for metastable heavy charged particles with large ionisation energy loss in pp collisions at root s=8 TeV using the ATLAS experiment. Regular Article -Experimental Physics
Search for metastable heavy charged particles with large ionisation energy loss in pp collisions at root s=8 TeV using the ATLAS experiment Aad, G.; Abbott, B.; Abdallah, J.; Abdinov, O.; Aben, R.; Abolins, M.; AbouZeid, O.S.; Abramowicz, H.; Abreu, H.; Abreu, R.; Dam, Mogens;Abstract Many extensions of the Standard Model predict the existence of charged heavy long-lived particles, such as R-hadrons or charginos. These particles, if produced at the Large Hadron Collider, should be moving non-relativistically and are therefore identifiable through the measurement of an anomalously large specific energy loss in the ATLAS pixel detector. Measuring heavy long-lived particles through their track parameters in the vicinity of the interaction vertex provides sensitivity to metastable particles with lifetimes from 0.6 ns to 30 ns. A search for such particles with the ATLAS detector at the Large Hadron Collider is presented, based on a data sample corresponding to an integrated luminosity of 18.4 fb −1 of pp collisions at √ s = 8 TeV. No significant deviation from the Standard Model background expectation is observed, and lifetime-dependent upper limits on Rhadrons and chargino production are set. Gluino R-hadrons with 10 ns lifetime and masses up to 1185 GeV are excluded at 95 % confidence level, and so are charginos with 15 ns lifetime and masses up to 482 GeV. ation of the LHC, as well as the support staff from our institutions without whom ATLAS could not be operated efficiently. We acknowledge the support of ANPCyT,
Introduction
The main motivation for heavy long-lived particle (LLP) searches at the Large Hadron Collider (LHC) arises from proposed solutions to the gauge hierarchy problem [1], which typically involve previously unseen particles at the TeV mass scale. Hadronising LLPs are anticipated in a wide range of physics models that extend the Standard Model (SM). For example, these particles appear in both R-parity-conserving [2][3][4][5][6][7][8][9] and R-parity-violating [10][11][12] supersymmetry (SUSY) and in universal extra dimensions theories [13,14]. e-mail: [email protected] These particles can be stable, or metastable; 1 the lifetime of these metastable particles may depend on the mass splitting with the lightest SUSY particle or on the size of any R-parity-violating coupling [15].
LLPs produced at the LHC are expected to be slow (β significantly below 1) and, therefore, should have specific ionisation higher than any SM particle at high momenta. The ATLAS detector [16] has a number of subsystems able to measure the velocity of charged particles. The pixel detector [17] provides measurements of ionisation energy loss (dE/dx) whereas the calorimeters and the muon spectrometer give a direct measurement of the time of flight for particles traversing them. A search for stable LLPs has been performed with the ATLAS detector [18] with 4.7 fb −1 of √ s = 7 TeV proton-proton ( pp) collisions using both the full detector information as well as the pixel detector information alone, and has been recently updated with the entire √ s = 8 TeV dataset [19], but without a pixel-only analysis. The CMS Collaboration has recently published [20] an analysis searching for stable LLPs based on the measurement of dE/dx, β, and on muon identification. In CMS, a search for metastable LLPs has been carried out by looking for secondary vertices [21], or disappearing tracks [22]. A displaced vertex search performed by the ATLAS Collaboration [23] sets limits on metastable particles in a number of scenarios. Limits on chargino production, from an analysis searching for disappearing tracks, have also been published by the ATLAS Collaboration [24].
The analysis described in this article has sensitivity to metastable particles if they have unit charge and their track length before decay is more than 45 cm in the radial direction, so that they can be measured in the first few layers of the ATLAS tracker. This measurement does not depend on the way the LLP interacts in the dense calorimeter material nor, to a first approximation, on the LLP decay mode. It can therefore address many different models of New Physics, especially those predicting the production of metastable heavy particles with O(ns) lifetime at LHC energies, such as mini-split SUSY [25,26] or anomaly-mediated supersymmetry breaking (AMSB) models [27,28]. A metastable gluino with a mass of 1 TeV would be compatible with the measured Higgs boson mass according to mini-split SUSY models, which also predict squark masses of 10 3 -10 5 TeV, therefore making the gluino the only observable strongly produced SUSY particle at LHC energies. In AMSB models, SUSY breaking is caused by loop effects and the lightest chargino can be only slightly heavier than the lightest neutralino, resulting in a heavy charged particle that can be measured before decaying into very low energy SM particles and a neutralino.
Results are presented in the context of SUSY models assuming the existence of R-hadrons [29] formed from a long-lived coloured sparticle (squark or gluino) and light SM quarks or gluons, and in AMSB models for the case of longlived charginos.
The paper is organised as follows. After a brief description of the experiment (Sect. 2) and of the measurement strategy (Sect. 3), the simulation of the signal processes is described (Sect. 4). The triggering strategy is then summarised and the trigger efficiency is calculated for R-hadrons and charginos (Sect. 5.1), after which the event selection is defined and motivated (Sect. 5). The data-driven background estimation is then described (Sect. 6) and the systematic uncertainties are presented and discussed (Sect. 7). Finally, after a brief description of the statistical method used to extract lifetimedependent limits on R-hadron and chargino production cross sections and masses, the results are reported in Sect. 8.
ATLAS detector and pixel dE/dx measurement
The ATLAS detector 2 consists of a tracker surrounded by a solenoid magnet for measuring the trajectories of charged particles, followed by calorimeters for measuring the energy of particles that have electromagnetic or strong interactions with matter, and a muon spectrometer. The muon spectrometer is immersed in a toroidal magnetic field and provides tracking for muons, which have typically passed through the calorimeters. The detector is hermetic within its η acceptance and can therefore measure the missing transverse momentum (whose magnitude is denoted by E miss T ) associated with each 2 ATLAS uses a right-handed coordinate system with its origin at the nominal interaction point in the centre of the detector and the z-axis coinciding with the axis of the beam pipe. The x-axis points from the interaction point to the centre of the LHC ring, and the y-axis points upward. Cylindrical coordinates (r , φ) are used in the transverse plane, φ being the azimuthal angle around the beam pipe. The pseudorapidity is defined in terms of the polar angle θ as η = − ln tan(θ/2). event. A complete description of the ATLAS detector can be found elsewhere [16]. The tracker is made of three detector systems. Starting from the solenoid magnet and moving toward the beam collision region one finds a ≈400-thousandchannel transition radiation tracker [30] followed by a ≈6million-channel silicon microstrip detector [31], and finally a ≈80-million-channel pixel detector. The pixel detector is crucial for this measurement and is described in more detail below.
As the innermost sub-detector in ATLAS, the silicon pixel detector provides at least three precision measurements for each track in the region |η| < 2.5 at radial distances of 5 to 13 cm from the LHC beam line. At normal incidence, the average charge released by a minimum-ionising particle (MIP) in a pixel sensor is ≈20000 e − and the charge threshold is set to 3500 ± 40 e − for each pixel. Signals above this threshold are time-stamped within one beam crossing; the hit efficiency under these conditions exceeds 99 %. When detector data are read out, the time over threshold (ToT), i.e. the length of time for which the signal is above the threshold, is digitised with 8 bits. The ToT is proportional to the ionisation charge [32] and its maximum value corresponds to 8.5 times the average charge released by a MIP track normal to the silicon detectors and leaving all of its ionisation charge on a single pixel. If this value is exceeded, the signal is lost.
The charge released by a track crossing a layer of the pixel detector is rarely contained within just one pixel. Neighbouring pixels are thus joined together to form clusters and the charge of a cluster is calculated by summing up the charges of all pixels after calibration corrections. The specific energy loss (dE/dx) is defined as an average of the individual cluster ionisation measurements (charge collected in the cluster, corrected for the track length in the sensor), for the clusters associated with the track. To reduce the Landau tails, the average is evaluated after having removed the highest dE/dx cluster and amounts to 1.24 ± 0.19 MeV/g cm 2 for a MIP [33]. The minimum measurable βγ with the dE/dx method is ≈0.3 for particles with unit charge and is determined by the ToT overflow in any of the pixels in a cluster.
Measurement strategy
Charged massive LLPs are expected to interact with matter following the Bethe-Bloch distribution according to their βγ . The mass of the LLPs can be obtained by fitting their specific energy loss and momentum to an empirical Bethe-Bloch distribution in the range 0.3 < βγ < 1.5. This range overlaps with the expected average βγ of LLPs produced at the LHC, which decreases from 2.0 to 0.5 for an increase in particle mass from 100 GeV to 1600 GeV. The parametric function describing the relationship between the most probable value of the energy loss (dE/dx MPV ) and βγ is [33]: Fig. 1 Ratio of the reconstructed mass, computed as the most probable value of a fit to a Landau distribution convolved with a Gaussian, to the generated mass, as a function of the generated mass for stable gluino R-hadrons. The yellow band is the half-width at half maximum of the reconstructed mass distribution normalised to the generated mass
dE/dx MPV (βγ ) = p 1 β p 3 ln[1 + (| p 2 |βγ ) p 5 ] − p 4 .(1)
The p i calibration constants have been measured [33] using low-momentum pions, kaons and protons reconstructed in ATLAS and their values are monitored by checking the stability of the proton mass measurement as a function of time.
The calibration is found to be stable at the 1 % level for all data-taking conditions and detector settings. Given a measured value of dE/dx, the mass estimate m is obtained from Eq. (1) by numerically solving the equation dE/dx MPV ( p/m) = dE/dx for the unknown m. Figure 1 shows the ratio of the reconstructed mass to the generated mass for simulated R-hadrons with masses up to 1600 GeV. The overestimation of the reconstructed mass for heavy particles is due to the pion scattering model assumed in the track reconstruction and momentum measurement. A ≈3 % rescaling of the measured mass is then applied in the analysis. The half-width at half maximum of the reconstructed mass distribution increases with the mass value. This is due to the momentum measurement uncertainty dominating the mass resolution above masses of 200 GeV.
The measurement strategy consists of looking for an excess of events, compatible with the expected measurement resolution, in the mass distribution for particles that are selected as LLP candidates. These should appear as high transverse momentum ( p T ) isolated particles with large dE/dx. Since there is no trigger based on these observables, events of interest were selected using the lowest-threshold unprescaled calorimetric E miss T trigger. For signal events, the E miss T originates from jets from QCD initial-state radiation (ISR) and, whenever relevant, by the LLP decays to unde-tected neutralinos. The trigger efficiency for all the LLPs searched for with this analysis is shown in Sect. 5.1.
Simulation of signal
A number of Monte Carlo (MC) simulated signal samples are used in this analysis to determine the expected efficiencies and to estimate the systematic uncertainties. A description of the simulation techniques is presented below for both stable and metastable R-hadrons and charginos. All simulated events are processed through the Geant4 [34] standard ATLAS simulation [35] and digitisation, followed by event reconstruction. This includes a realistic description of additional pp interactions in the same or neighbouring bunch crossings (pile-up). In order to take into account residual discrepancies with data, the simulated pile-up distribution is scaled to that observed during the 2012 data-taking period.
Stable R-hadrons
Pair production of gluinos with masses between 100 and 1700 GeV is simulated in Pythia 6.4.27 [36] with the AUET2B [37] set of MC tuneable parameters and the CTEQ6L1 [38] parton distribution function (PDF) set, incorporating dedicated hadronisation routines [39] to produce final states containing R-hadrons. Additional samples of gluinos with some representative mass values are generated using MadGraph5 [40] 1.3.33. Since the MadGraph samples are generated with an additional outgoing parton in the matrix element they provide a more accurate description of ISR and thus a more accurate distribution of the transverse momentum of the gluino-gluino system. The gluino samples simulated with Pythia6 are reweighted to match this gluino-gluino system p T distribution obtained from the MadGraph samples. The cross sections are calculated to next-to-leading order in the strong coupling constant (NLO), including the resummation of soft-gluon emission at nextto-leading-logarithmic accuracy (NLO+ NLL) [41][42][43][44][45]. The nominal cross section is calculated assuming a squark mass of 10 TeV. The uncertainty is taken from an envelope of crosssection predictions using different PDF sets and factorisation and renormalisation scales, as described in Ref. [46].
Simulated R-hadron events are passed through a full detector simulation, where interactions with matter are handled by dedicated Geant4 routines based on different scattering models. The model described in Refs. [39,47], and hereafter referred to as the generic model, imposes few constraints on the allowed stable states. This is the only model where doubly charged R-hadrons are predicted, with a production probability of 0.1 %. Hadronic scattering is described through a purely phase-space-driven approach. A second model, referred to as the Regge model in the following, employs a triple-Regge formalism to describe hadronic scattering and describes R-hadrons containing gluinos according to Ref. [48]. More recent models for the hadronic scattering of gluino R-hadrons predict that the majority would be electrically neutral after just a few hadronic interactions. The third scenario considered here belongs to this family and is based on bag-model calculations presented in Ref. [49]. This is referred to as the intermediate model. The probability for a gluino to form a gluon-gluino bound state, based on a colouroctet model, is assumed to be 10 % [2]. Results are presented for the generic (Regge) model for gluino (squark) R-hadrons. Variations resulting from the use of a model different from the nominal one are taken into account as systematic uncertainties on the signal efficiency.
Metastable R-hadrons
The simulation of metastable gluino-based R-hadron samples is performed in a similar way to that of the stable R-hadrons, as described in Sect. 4.1. The gluinos within R-hadrons are required to decay via the radiative process g → gχ 0 1 org → qqχ 0 1 , using Pythia6. Decays involving tt pairs are treated separately. Gluino masses between 400 GeV and 1400 GeV are simulated, with the neutralino mass either fixed to 100 GeV or set to m(g) − 100 GeV (or m(g) − 480 GeV for tt channels). Samples with τg = 0.1, 1 and 10 ns are generated for different mass points.
Results of this search are presented as a function of the Rhadron lifetime. Signal MC samples with different lifetimes are obtained, starting from those simulated with a fixed value of the lifetime, by applying event weights, such that the distribution of the proper lifetime in the modified sample corresponds to the chosen new value of the mean lifetime. The event weight w is given by:
w(τ R H ) = n R H i τ 0 τ R H exp − t i 1 τ R H − 1 τ 0 ,(2)
where n R H , τ 0 and t i are the number of R-hadrons in the event, the R-hadron mean lifetime as set by the simulation, and the proper lifetime of the ith R-hadron, respectively. The modified mean lifetime obtained after the reweighting procedure is indicated by τ R H .
Stable charginos
A chargino may be stable in a simplified scenario where the mass difference between the chargino and neutralino is less than 160 MeV so that the chargino decay to a pion and a neutralino is kinematically suppressed. Samples with longlived charginos are generated using Herwig++ 2.6.3 [50] along with the UEEE3 [51] tune and the CTEQ6L1 PDF set.
The chargino mass is varied between 100 GeV and 800 GeV. The chargino is forced to remain stable and the other particles in the model are set to be too heavy to be produced in √ s = 8 TeV pp collisions. Signal cross sections are calculated to NLO using PROSPINO2 [52]. They are in agreement with the NLO+NLL calculations within ∼2 % [53][54][55]. The total cross section is dominated by direct production ofχ 0 1χ ± 1 pairs (∼67 %), and byχ + 1χ − 1 pairs (∼30 %). The relative proportion of these two production mechanisms was checked and found to be constant at the 1 % level over the considered chargino mass range.
Metastable charginos
Samples with metastable charginos are produced similarly to the samples of stable charginos. The mean lifetime of the chargino is set to a given value (τχ± 1 ), and charginos are forced to decay intoχ 0 1 + π ± in the Geant4 simulation following an exponential decay with lifetime τχ± 1 in the chargino rest frame. The chargino-neutralino mass splitting is set to 160 MeV. Samples with τχ± 1 = 1, 5, 15 and 30 ns are generated for different mass points. To reduce the use of simulation resources, all the samples were generated with a jet filter requiring at least one generator-level jet with p T > 70 GeV and |η| < 5. This choice was optimised for the first metastable chargino search [24]. The present analysis does not make any explicit requirement on the energy of the jets, while it requires missing transverse momentum in the event. There is a strong correlation between these two requirements. The residual bias due to the jet filter is evaluated and assigned as a systematic uncertainty.
Candidate selection
Trigger
Events are selected by the E miss T > 80 GeV trigger, the lowest-threshold E miss T trigger that remained unprescaled throughout the 2012 data taking. This is based uniquely on the energy deposited in the calorimeters. Figure 2 shows the efficiency of this trigger as a function of the R-hadron or chargino mass.
The decay of each LLP to jets and a neutralino occurring within the ATLAS active volume contributes to the transverse momentum imbalance and leads to higher trigger efficiency in the metastable cases. In the case of metastable Rhadrons, the E miss T also depends on the mass of the neutralino produced in the decay itself. The two R-hadrons tend to be emitted back-to-back and the same holds for the two heavy neutralinos, which therefore approximately balance E miss T . If neutralinos are light, highp T jets coming from R-hadron
χ ∼ ) = 10 ns, m( g ( τ , 0 1 χ ∼ q /q 0 1 χ ∼ g → g ) = 100 GeV 0 1 χ ∼ ) = 10 ns, m( g ( τ , 0 1 χ ∼ q /q 0 1 χ ∼ g → g ) = 100 GeV 0 1 χ ∼ ) = 1.0 ns, m( g ( τ , 0 1 χ ∼ q /q 0 1 χ ∼ g → g stable g stable ± 1 χ ∼ Fig. 2
Efficiency for the calorimetric E miss T > 80 GeV trigger as a function of the R-hadron mass or of the chargino mass. Separate curves are also shown for the stable and metastable cases. Only statistical uncertainties are shown, which are too small to be visible on these graphs decay also contribute to the transverse momentum imbalance. For this reason, the calorimetric E miss T is significantly larger when the neutralino is light. The E miss T also depends on the lifetime of the parent particle as this defines the fraction that decay before the calorimeter and therefore affects E miss T . If the lifetime is shorter than 1 ns, the decay happens very close to the primary vertex and the calorimetric E miss T does not depend very much on the lifetime. On the other hand, if the lifetime is long enough, the decay may happen beyond the calorimeter region and, therefore, the calorimetric E miss T is close to the stable case. The 10 ns lifetime is an intermediate case, as decays happen mainly, but not exclusively, in the tracker region.
Offline selection
This search is based on a sample of well-measured highp T isolated tracks in events with large missing transverse momentum. The data sample considered in this analysis was collected with tracking detectors, calorimeters, muon chambers and magnets fully operational and corresponds to a total integrated luminosity of 18.4 fb −1 with an uncertainty of ±2.8 % measured using beam separation scans following the technique described in Ref. [56].
The first step in the selection is the confirmation that the event has sufficient E miss T . The E miss T variable computed using the offline reconstruction [57,58], which uses refined calorimetric information and includes the contributions of the energy of the muons, must exceed 100 GeV. Candidate events are then required to have at least one primary vertex with a minimum of five tracks with p T > 0.4 GeV. There must be 3 with at least three pixel hits, measured over 45 cm in the radial direction, and with transverse momentum p T > 80 GeV and |η| ≤ 2.5. The set of requirements described above, including the trigger requirement, defines the preselection entry in Tables 1 and 2.
The following additional requirements must be satisfied by at least one of the preselected candidate tracks in order to select the event.
The track must be isolated. A track is considered isolated if its distance ΔR = (Δη) 2 + (Δφ) 2 to any other track associated with the primary vertex and with p T ≥ 1 GeV is greater than 0.25. About 70 % of these isolated tracks are highp T leptons originating from W boson production.
The track must not be identified as an electron [59] as LLPs can very rarely (<1 %) be identified as electrons. The selected tracks are required not to match any reconstructed electron within ΔR ≤ 0.01.
The track must have momentum p > 150 GeV and the relative uncertainty on the momentum σ p / p < 50 %. The first requirement improves the signal-to-background ratio, while the second ensures good mass resolution.
The track must not be a muon originating from a W boson decay. Muons cannot be simply identified and rejected at this stage, as hypothetical very long-lived particles would often be mis-identified as muons in the detector. Therefore, to reject muons from a W boson decay, a requirement on transverse mass 4 (m T > 130 GeV) is applied. According to simulation, this requirement reduces the fraction of W boson events in the data sample to ∼40 %.
The selected track is required to have specific ionisation measured by the pixel detector larger than 1.800−0.034|η|+ 0.101η 2 − 0.029|η| 3 MeV/g cm 2 . This requirement corrects the slight |η| dependence [60] of the dE/dx variable and selects ∼1.3 % of the tracks in the data independently of the pseudorapidity region. The selection cut chosen is the lowest with mass-discriminating power (below this, the dE/dx values of all particles are too close to the MIP value, irrespective of their masses).
The above requirements complete the selection for the stable particle search. One additional requirement is applied to improve the sensitivity for the metastable case. The highly ionising particles can be matched with reconstructed jets [61] or muons [62]. Out of 85 candidates, 57 are geometrically matched to muons (ΔR ≤ 0.01) and 26 are ΔR ≤ 0.07 from a jet. The other two candidates have no signals in the calorimeters or muon system in the vicinity of the LLP. If the LLPs are stable, they are usually reconstructed as muons. If the heavy particles are not stable, the matching with muons becomes much more rare, in particular for particles with a lifetime of O(ns). In the search for metastable particles a muon veto is applied, and tracks that are matched with a muon are rejected.
Finally, if more than one track per event passes all requirements, the highestp T candidate is chosen, in order not to bias the distribution of the variables and to allow for proper normalisation in the background estimate. Table 1 shows the number of events in data and for an example gluino R-hadron signal for the different selection criteria. In data 85 events are selected before and 28 after the muon veto. None of the events has more than one selected track per event. Table 2 shows the yields for the same event selection as in Table 1, but applied to simulated signal events with 1000 GeV gluino R-hadrons that are either stable, or otherwise decay to g/qq plus a light neutralino of mass m(χ 0 1 ) = 100 GeV, and with a 1 ns lifetime. If the R-hadron lifetime is shorter, while the event efficiency is rather unaffected, the efficiency of reconstructing good quality, highp T , isolated tracks falls dramatically. Figure 3 shows the overall signal efficiencies for a representative set of simulated signal samples to which the full selection procedure is applied.
4 m T = 2 p T E miss T (1 − cos(Δφ(E miss T , track)).
When the LLPs decay inside the ATLAS active volume, E miss T increases and trigger and offline E miss T selection becomes more efficient than for the stable case. However, as the lifetime decreases, the probability to reconstruct a track segment in the silicon detectors decreases dramatically. At a mass of 1000 GeV, these two effects give a total efficiency of ≈15 % for the 10 ns lifetime samples and ≈1 % for the 1 ns samples, while for stable particles the efficiency has intermediate values of ≈7 %.
Background estimation
In order to estimate the background, a data-driven approach is used. The method uses data to fit the distributions of key variables, taking into account their inter-dependence, and then to generate a large random sample of background events based on the same distributions. The choice of the control samples takes into account the measured correlations between the variables used: p, dE/dx and η. The dE/dx dependence on the path length in the sensor is not linear [60], but depends on η, increasing by ∼10 % from central to high |η|. The dE/dx also depends on the particle βγ via the Bethe-Bloch formula and therefore on its momentum, until the Fermi plateau is reached. Finally p and η depend on event kinematics, as high-momentum tracks are more likely to be produced at high |η| values. Total background systematic uncertainty:
Stable particle search 11
Metastable particle search 10 Two samples are constructed to describe the distributions of the key variables. Both selections use the full data sample, but with requirements minimising the possible contamination by signal events. These control region samples are the same for the searches for stable and metastable particles, except for the rejection of track candidates geometrically matched with spectrometer muons in the latter case.
A first sample (CR1) is selected by applying all the selections described in Sect. 5 except for the high ionisation requirement, which is instead inverted to ensure orthogonality with the search sample. Otherwise, the kinematic prop- The expected background is shown with its total uncertainty (sum in quadrature of statistical, normalisation and systematic errors). The signal distributions are stacked on the expected background, and a narrower binning is used for them to allow the signal shape to be seen more clearly. The number of signal events is that expected according to the theoretical cross sections. For both distributions, the bin-per-bin ratio of data to expected background is also shown subset with the muon veto applied is shown in Fig. 4. In their bulk the distributions are very similar, thus the larger CR2 sample is also used for the metastable search with the muon veto. Nevertheless a larger tail, likely due to e + e − pair production, is seen at high dE/dx for the tracks with muon veto, and this difference is accounted for in the systematic uncertainty evaluation. A large background sample consisting of five million { p, η, dE/dx} triplets is randomly generated according to the following procedure: the momentum is generated according to a binned function based on selected tracks in CR1; the pseudorapidity is generated according to the η-binned (where η depends on p) functions based on tracks in CR1 and the ionisation is generated according to dE/dx-binned (where dE/dx depends on η) functions based on all tracks in CR2.
χ ∼ q g/q → g )=10ns g ( τ )=100 GeV, 0 1 χ ∼ m( )=350 GeV ± 1 χ ∼ , m( 0 1 χ ∼ ± π → ± 1 χ ∼ )=1nsCharginoχ 0 1 + π ± m(χ ± 1 ) − 0.14 1.0 239 Charginoχ 0 1 + π ± m(χ ± 1 ) − 0.14 15 482
For the ∼50000 random combinations in which dE/dx is larger than the selection requirement, both in the stable and the metastable scenario, the particle mass m is obtained given the {dE/dx, p} generated values, using the technique explained in Sect. 3. The normalisation of the generated background distribution to the selected data is obtained by scaling the background distribution to the data in the low-mass region of the mass distribution (40 < m < 160 GeV) where a possible signal has already been excluded [18,24]. The normalisation is performed on the samples before the ionisation requirement, and its uncertainty is dominated by the statistical uncertainty in the data. The complete procedure, from the key variables description, to the random generation, to the normalisation, is tested on signal-depleted regions. These regions are the same as CR1 and CR2 except for requiring tracks with 100 < p < 150 GeV instead of p > 150 GeV. Applied to these validation samples and the stable scenario, the procedure described above yields a predicted background of 48.9±0.2 (stat.) ± 0.6 (norm.) events, while the number of events in the data sample is 49, for m > 160 GeV. The same procedure applied to the metastable scenario yields a predicted background of 16.9 ± 0.1 (stat.) ± 0.4 (norm.) events, compared to 20 observed in the data, for m > 160 GeV.
Systematic uncertainties
Systematic uncertainties from several sources affecting the background estimate and the signal yield have been evaluated. The uncertainties are quoted as the maximum deviation from the nominal expectation for the background or for the Fig. 6 Upper limits on the production cross section as a function of mass for metastable gluino R-hadrons, with lifetime τ = 10 ns, decaying into g/qq plus a light neutralino of mass m(χ 0 1 ) = 100 GeV (top) or a heavy neutralino of mass m(χ 0 1 ) = m(g) − 100 GeV (bottom). Theoretical values for the cross section are shown with their uncertainty. The expected upper limit in the background-only case is shown as a solid black line, with its ±1σ and ±2σ bands, green and yellow, respectively. The observed 95 % CL upper limit is shown as a solid red line signal in the probed mass range. The actual systematic uncertainties are calculated and assigned per mass bin.
χ ∼ q g/q → g )=10 ns g ( τ ) -100 GeV, g )=m( 1 0 χ ∼ m( g Prediction Observed 95% CL limit ) exp σ 1 ± Expected 95% CL limit ( ) exp σ 2 ± Expected 95% CL limit ( ATLAS
The uncertainties in the background estimation can be divided into three categories: those related to the particular choices made for the binning, intervals, and fitting functions; those related to the different description of the key variables for control samples with different compositions than the search region sample; and those related to the stability as a function of pile-up. The uncertainty on the background estimate due to each of these sources is evaluated by changing the description of the key variables, repeating the entire generation procedure, and comparing the resulting mass distribution with the nominal one. The uncertainties are estimated separately for the searches for stable and metastable particles. In each iteration, five million events are generated Fig. 7 The excluded range of lifetimes as a function of gluino mass for gluino R-hadrons decaying into g/qq plus a light neutralino of mass m(χ 0 1 ) = 100 GeV (top) or a heavy neutralino of mass m(χ 0 1 ) = m(g)− 100 GeV (bottom). The expected exclusion, with its experimental ±1σ band, is given with respect to the nominal theoretical cross section as explained in Sect. 6. The resulting uncertainties are summarised in Table 3.
σ 1 ± Observed ) exp σ 1 ± Expected 95% CL exclusion ( )-100 GeV g )=m( 0 χ ∼ , m( 0 χ ∼ q g/q → g ATLAS
The uncertainties on the signal yield are summarised in Table 4 and can be divided into three categories: those on the phenomenological modelling of the signal process with Monte Carlo generators; those on the modelling of the detector efficiency or calibration; and those affecting the overall signal yield.
The uncertainty on QCD radiation is evaluated for stable and metastable R-hadrons as the difference in efficiency between the Madgraph and Pythia6 samples. This uncertainty is large for the benchmark channels in which the E miss T is dominated by the ISR contribution, such as the stable Rhadrons or the gluino R-hadrons decaying into g/qq plus a heavy neutralino of mass m(χ 0 1 ) = m(g) − 100 GeV. The Fig. 8 Upper limits on the production cross section as a function of mass for metastable gluino R-hadrons, with lifetime τ = 10 ns, decaying into tt plus a light neutralino of mass m(χ 0 1 ) = 100 GeV (top) or a heavy neutralino of mass m(χ 0 1 ) = m(g) − 480 GeV (bottom). Theoretical values for the cross section are shown with their uncertainty. The expected upper limit in the background-only case is shown as a solid black line, with its ±1σ and ±2σ bands, green and yellow, respectively. The observed 95 % CL upper limit is shown as a solid red line uncertainty on QCD radiation is evaluated for metastable charginos using the same procedure as described in Ref.
χ ∼ t t → g )=10 ns g ( τ ) -480 GeV, g )=m( 1 0 χ ∼ m( g Prediction Observed 95% CL limit ) exp σ 1 ± Expected 95% CL limit ( ) exp σ 2 ± Expected 95% CL limit ( ATLAS
[24], while for stable charginos the same uncertainty used for the stable R-hadrons is used, as the E miss T distributions are very similar. For the R-hadrons there is an uncertainty on how they interact with the detector material, and this is evaluated by comparing the efficiencies obtained from generating events according to the three scattering models described in Sect. 4.1. Efficiencies for chargino events generated with and without the jet filter (see Sect. 4.4) are compared, and their difference is accounted for as a systematic uncertainty. A systematic uncertainty is assigned to the lifetime reweighting procedure, and is estimated as the discrepancy between the efficiencies obtained for the same reweighted lifetime starting from different samples. Fig. 9 The excluded range of lifetimes as a function of gluino mass for gluino R-hadrons decaying into tt plus a light neutralino of mass m(χ 0 1 ) = 100 GeV (top) or a heavy neutralino of mass m(χ 0 1 ) = m(g)− 480 GeV (bottom). The expected exclusion, with its experimental ±1σ band, is given with respect to the nominal theoretical cross section
σ 1 ± Observed ) exp σ 1 ± Expected 95% CL exclusion ( )-480 GeV g )=m( 0 χ ∼ , m( 0 χ ∼ t t → g ATLAS
The systematic uncertainties on the detector modelling are dominated by the trigger efficiency modelling, by the E miss T scale, and by the parameterisation of the ionisation. Systematic uncertainties related to the trigger are evaluated by varying the threshold and resolution parameters of the calorimetric E miss T trigger efficiency modelling curve and then looking at the efficiency difference between data and MC simulated Z → μ + μ − events. Systematic uncertainties of the E miss T measurement are evaluated with the methods described in Refs. [57,63] and are propagated to the uncertainty of the efficiency. Since the pile-up distribution is different in data and MC simulation, the simulated samples are reweighted to match the data, and a systematic uncertainty is calculated by varying the weighting factors. The systematic uncertainty on the pixel ionisation is evaluated by compar- Fig. 10 Upper limits on the production cross section as a function of mass for metastable charginos, with lifetime τ = 1.0 ns, decaying intõ χ 0 1 + π ± . Theoretical values for the cross section are shown with their uncertainty. The expected upper limit in the background-only case is shown as a solid black line, with its ±1σ and ±2σ bands, green and yellow, respectively. The observed 95 % CL upper limit is shown as a solid red line ing the ionisation of simulated and real tracks. These tracks are compatible with MIPs, selected with the same requirements as those used for the search. Other smaller uncertainties are due to momentum [18] and track efficiency [64] parameterisation, and electron [65] and muon [66] identification.
ATLAS
The uncertainty on the signal cross section is calculated as described in Sect. 4. The uncertainty ranges from 15 % (at 100 GeV) to 56 % (at 1700 GeV) for R-hadrons, and is ≈8.5 % for the charginos, slightly dependent on the mass.
Results
The mass distribution is shown in Fig. 5 for the selection of stable and metastable particles. The data, 85 events for the stable selection and 28 events for the metastable selection, are compared to the background estimate. In addition, mass distributions for examples of gluino and chargino signals are shown.
No evidence of a signal above the expected background is observed. The largest deviation has a local p-value of 4.3 % and occurs at a mass of 700 GeV for metastable gluino Rhadrons with lifetime τ = 10 ns.
Upper limits at 95 % confidence level (CL) on R-hadron or chargino production cross sections (and, therefore, given a model cross section, a lower limit on their masses) are extracted using the CL S method [67] by scanning the signal strength over a suitable range and using the profile likelihood ratio as a test statistic. In the procedure for setting the limit, the systematic uncertainties on the signal and back- Fig. 11 The excluded range of lifetimes as a function of chargino mass for charginos decaying toχ 0 1 + π ± . The expected exclusion, with its experimental ±1σ band, is given with respect to the nominal theoretical cross section ground yields, as evaluated in Sect. 7, are treated as Gaussiandistributed nuisance parameters. The statistical uncertainty on the background distribution also takes into account the uncertainty due to the normalisation. Signal, background and data events are counted in a mass window of ±1.4σ around the signal peak, where the signal peak and the width are estimated by a Gaussian fit to the mass distribution in simulated signal MC samples. Lower limits on the mass are derived by comparing the measured cross-section limits to the lower edge of the ±1σ band around the theoretically predicted cross section for each process. The resulting lower limits set on the mass of the stable and metastable particles are summarised in Table 5 for the different searches. Figure 6 shows the upper limits on the production cross section for the gluino R-hadron with a lifetime of 10 ns decaying into g/qq plus a light neutralino of mass m(χ 0 1 ) = 100 GeV, or a heavy neutralino of mass m(χ 0 1 ) = m(g) − 100 GeV. Figure 7 shows the excluded range of lifetimes for the same R-hadron decays versus the particle mass. Figure 8 shows the upper limits on the production cross section for the gluino R-hadron with a lifetime of 10 ns decaying into tt plus a light neutralino of mass m(χ 0 1 ) = 100 GeV, or a heavy neutralino of mass m(χ 0 1 ) = m(g) − 480 GeV. Figure 9 shows the excluded range of lifetimes for the same R-hadron decays as a function of the particle mass. As shown in Figs. 7 and 9 the sensitivity of the measurement for R-hadrons is maximal for lifetimes around 10 ns. Figure 10 shows the upper limits on the production cross section for charginos with lifetime of 1 ns decaying intoχ 0 1 + π ± and Fig. 11 shows the excluded range of lifetimes as a function of the particle mass for the same chargino decay.
σ 1 ± Observed ) exp σ 1 ± Expected 95% CL exclusion ( )-160 MeV ± 1 χ ∼ )=m( 0 χ ∼ , m( ± π 0 1 χ ∼ → ± 1 χ ∼ ATLAS
Summary
A search has been performed for stable and metastable nonrelativistic long-lived charged particles identified through their anomalous specific ionisation energy loss in the ATLAS pixel detector. The search uses 18.4 fb −1 of pp collision data at √ s = 8 TeV collected by the ATLAS detector at the LHC. In the scenario considered, stable charginos with masses smaller than 534 GeV are excluded at 95 % confidence level, and so are stable R-hadrons with masses smaller than 1115 GeV for gluinos, 751 GeV for bottom squarks and 766 GeV for top squarks. For metastable particles the maximum sensitivity is reached at 10 ns lifetime for R-hadrons, where masses below 1185 GeV are excluded, and at 15 ns lifetime for charginos, where masses up to 482 GeV are excluded. Compared to previous searches this search provides the best sensitivity for gluinos with lifetimes between 3 and 20 ns. 15
Fig. 3
3Total selection efficiencies for some MC samples as a function of the R-hadron mass or of the chargino mass. Both stable and some metastable cases are shown. Only statistical uncertainties are included
Fig. 4
4Ionisation distribution of all the CR2 tracks (filled circles), and those not matched to a reconstructed muon (open squares). The two distributions are normalised to their total number of entries
GeV), while keeping all the other selection requirements unchanged. This procedure ensures a negligible signal content in the sample (signal contamination of less than 0.02 %), and that the selected tracks are in the same kinematic ranges of momentum and pseudorapidity as the search sample. With this selection, ∼440000 background events are kept in CR2, the majority (≈96 %) being matched with muons. The ionisation of all the CR2 tracks and of the
Fig. 5
5Distribution of the mass of selected candidates, derived from the specific ionisation loss, for data, background, and examples of gluino Rhadron and chargino signals, for searches for stable (top) and metastable (bottom) particles.
Table 1
1Observed data event yields at different steps of the selection procedure compared with the expected number of events for 1000 GeV R-hadrons decaying, with a 10 ns lifetime, to g/qq plus a light neutralino of mass m(χ 0 1 ) = 100 GeV. The simulated yields are normalised to a total integrated luminosity of 18.4 fb −1 and their statistical uncertainty is also shown. See text for detailsRequirement
Selected events
(data)
Expected events
(τ = 10 ns)
Preselection
543692
112 ± 3
Isolation
88431
102 ± 3
Electron veto
60450
102 ± 3
High-p
35684
91 ± 3
High-m T
6589
75 ± 2
Ionisation
85
68 ± 2
Muon veto
28
62 ± 2
Table 2
2Expected number of events at different steps of the selection procedure for 1000 GeV gluino R-hadrons decaying, with a 1 ns lifetime, to g/qq plus a light neutralino of mass m(χ 0 1 ) = 100 GeV, and for stable R-hadrons. The simulated yields are normalised to a total integrated luminosity of 18.4 fb −1 and their statistical uncertainty is also shown. See text for detailsRequirement
Expected events
(τ = 1 ns)
Expected events
(stable)
Preselection
19.5 ± 1.1
6 2 ± 2
Isolation
9.3 ± 0.7
5 5 ± 2
Electron veto
9.3 ± 0.7
5 4 ± 2
High-p
4.4 ± 0.5
5 2 ± 2
High-m T
3.9 ± 0.4
4 3 ± 1
Ionisation
3.0 ± 0.4
3 7 ± 1
Muon veto
2.9 ± 0.4
-
at least one track associated with this primary vertex,
Table 3
3Summary table of the systematic uncertainties that affect the background estimations. All the sources are common to searches for stable and metastable particles, unless explicitly indicated. The uncertainties depend on the mass, and the maximum values are reported. In the limit calculations the actual value of the uncertainty for a given mass is usedSource of uncertainty:
[%]
Modification of:
-Binning in p
5
-Binning in η
5
-Momentum intervals in η
2
-Binning in dE/dx
4
-Analytical description of dE/dx
2
Different fractions of:
-Non-leptons in CR2 (metastable only)
2
-Tracks with more than 3 pixel clusters in CR2
1.5
-Non-leptons in CR1 (stable only)
4
-W decays in CR1 and CR2
3
Pile-up dependence
4
Table 4
4Summary table for the sources of systematic uncertainty considered for R-hadrons and charginos. The values are separately indicated for metastable and stable cases when these are different. The uncertainty depends on the mass and on the decay model, and the maximum negative and positive values are reported. In the limit calculations the actual value of the uncertainty for a given mass is usedSource of uncertainty
−[%]
+[%]
R-hadron
QCD radiation modelling (stable)
−28
28
QCD radiation modelling (metastable)
−12
12
Scattering models
−9.9
6 .6
Lifetime reweighting (metastable)
−10
10
Chargino
QCD radiation modelling (stable)
−27
27
QCD radiation modelling (metastable)
−21
30
Generator jet filter
−8.3
0
R-hadron and Chargino
Lifetime reweighting (metastable)
−10
10
Trigger efficiency modelling
−4.5
4 .5
E miss
T
scale
−3.8
3 .5
Pile-up
−1.7
1 .7
Ionisation parameterisation
−5.8
0
Momentum parameterisation (stable)
−1.0
1 .0
Momentum parameterisation (metastable)
−2.0
2 .0
Track Efficiency parameterisation
−2.0
2 .0
Electron identification
−1.0
1 .0
Muon identification (metastable only)
−1.0
1 .0
Total systematic uncertainty
Stable R-hadron
−30
29
Metastable R-hadron
−18
15
Stable chargino
−30
28
Metastable chargino
−24
31
Uncertainties on signal yield
Luminosity
−2.8
2 .8
Cross section uncertainty (R-hadron)
−56
56
Cross section uncertainty (chargino)
−8.5
8 .5
erties and overall event characteristics are expected to be
similar to the signal region. With this selection, ∼6000 back-
ground events are kept in CR1 with signal contamination of
less than 0.6 %.
A second sample (CR2) is used to obtain the dE/dx tem-
plates and is selected by inverting the E miss
Table 5
5The 95 % CL lower limit on the relevant LLP mass for the different models considered. Other relevant parameters (decay mode, neutralino mass, lifetime if metastable) are also shownParticle
Decay
m(χ 0
1 ) [GeV]
τ [ns]
m > [GeV]
g R-hadron
Stable
-
-
1115
b R-hadron
Stable
-
-
751
t R-hadron
Stable
-
-
766
Chargino
Stable
-
-
534
g R-hadron
g/qq
100
10
1185
g R-hadron
g/qq
m (g) − 100
10
1099
g R-hadron
tt
100
10
1182
g R-hadron
tt
m (g) − 480
10
1157
g R-hadron
g/qq
100
1.0
869
g R-hadron
g/qq
m (g) − 100
1.0
821
g R-hadron
tt
100
1.0
836
g R-hadron
tt
m (g) − 480
1.0
836
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In this article, particles that do not decay in the active detector volume are considered stable, and those that decay inside this volume are considered metastable.
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Fakultät für Physik und Astronomie, Julius-Maximilians-Universität, Würzburg, Germany 175 Fachbereich C Physik, Bergische Universität Wuppertal, Wuppertal, Germany
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Tam; Adelaide; Albany, NY; Edmonton, AB, Canada; Istanbul, Turkey; Gaziantep, Turkey175University of Adelaide ; Australia 2 Physics Department, SUNY Albany ; USA 3 Department of Physics, University of Alberta ; Department of Physics, Bogazici University, Istanbul, Turkey; (b) Department of Physics, Dogus University ; c) Department of Physics Engineering, Gaziantep UniversityJ. Veatch. Veloso 126a,126c , T. Velz 21 , S. Veneziano 132a , A. Ventura 73a,73b , D. Ventura 86 , M. Venturi 169 , N. Venturi 158 , A. Venturini 23 , V. Vercesi 121a , M. Verducci 132a,132b , W. Verkerke 107 , J. C. Vermeulen 107 , A. Vest 44 , M. C. Vetterli 142,d , O. Viazlo 81 , I. Vichou 165 , T. Vickey 139 , O. E. Vickey Boeriu 139 , G. H. A. Viehhauser 120 , S. Viel 15 , R. Vigne 62 , M. Villa 20a,20b , M. Villaplana Perez 91a,91b , E. Vilucchi 47 , M. G. Vincter 29 , V. B. Vinogradov 65 , I. Vivarelli 149 , F. Vives Vaque 3 , S. Vlachos 10 , D. Vladoiu 100 , M. Vlasak 128 , M. Vogel 32a , P. Vokac 128 , G. Volpi 124a,124b , M. Volpi 88 , H. von der Schmitt 101 , H. von Radziewski 48 , E. von Toerne 21 , V. Vorobel 129 , K. Vorobev 98 , M. Vos 167 , R. Voss 30 , J. H. Vossebeld 74 , N. Vranjes 13 , M. Vranjes Milosavljevic 13 , V. Vrba 127 , M. Vreeswijk 107 , R. Vuillermet 30 , I. Vukotic 31 , Z. Vykydal 128 , P. Wagner 21 , W. Wagner 175 , H. Wahlberg 71 , S. Wahrmund 44 , J. Wakabayashi 103 , J. Walder 72 , R. Walker 100 , W. Walkowiak 141 , C. Wang 33c , F. Wang 173 , H. Wang 15 , H. Wang 40 , J. Wang 42 , J. Wang 33a , K. Wang 87 , R. Wang 6 , S. M. Wang 151 , T. Wang 21 , X. Wang 176 , C. Wanotayaroj 116 , A. Warburton 87 , C. P. Ward 28 , D. R. Wardrope 78 , A. Washbrook 46 , C. Wasicki 42 , P. M. Watkins 18 , A. T. Watson 18 , I. J. Watson 150 , M. F. Watson 18 , G. Watts 138 , S. Watts 84 , B. M. Waugh 78 , S. Webb 84 , M. S. Weber 17 , S. W. Weber 174 , J. S. Webster 31 , A. R. Weidberg 120 , B. Weinert 61 , J. Weingarten 54 , C. Weiser 48 , H. Weits 107 , P. S. Wells 30 , T. Wenaus 25 , T. Wengler 30 , S. Wenig 30 , N. Wermes 21 , M. Werner 48 , P. Werner 30 , M. Wessels 58a , J. Wetter 161 , K. Whalen 116 , A. M. Wharton 72 , A. White 8 , M. J. White 1 , R. White 32b , S. White 124a,124b , D. Whiteson 163 , F. J. Wickens 131 , W. Wiedenmann 173 , M. Wielers 131 , P. Wienemann 21 , C. Wiglesworth 36 , L. A. M. Wiik-Fuchs 21 , A. Wildauer 101 , H. G. Wilkens 30 , H. H. Williams 122 , S. Williams 107 , C. Willis 90 , S. Willocq 86 , A. Wilson 89 , J. A. Wilson 18 , I. Wingerter-Seez 5 , F. Winklmeier 116 , B. T. Winter 21 , M. Wittgen 143 , J. Wittkowski 100 , S. J. Wollstadt 83 , M. W. Wolter 39 , H. Wolters 126a,126c , B. K. Wosiek 39 , J. Wotschack 30 , M. J. Woudstra 84 , K. W. Wozniak 39 , M. Wu 55 , M. Wu 31 , S. L. Wu 173 , X. Wu 49 , Y. Wu 89 , T. R. Wyatt 84 , B. M. Wynne 46 , S. Xella 36 , D. Xu 33a , L. Xu 33b,aj , B. Yabsley 150 , S. Yacoob 145a , R. Yakabe 67 , M. Yamada 66 , Y. Yamaguchi 118 , A. Yamamoto 66 , S. Yamamoto 155 , T. Yamanaka 155 , K. Yamauchi 103 , Y. Yamazaki 67 , Z. Yan 22 , H. Yang 33e , H. Yang 173 , Y. Yang 151 , W-M. Yao 15 , Y. Yasu 66 , E. Yatsenko 5 , K. H. Yau Wong 21 , J. Ye 40 , S. Ye 25 , I. Yeletskikh 65 , A. L. Yen 57 , E. Yildirim 42 , K. Yorita 171 , R. Yoshida 6 , K. Yoshihara 122 , C. Young 143 , C. J. S. Young 30 , S. Youssef 22 , D. R. Yu 15 , J. Yu 8 , J. M. Yu 89 , J. Yu 114 , L. Yuan 67 , S. P. Y. Yuen 21 , A. Yurkewicz 108 , I. Yusuff 28,ak , B. Zabinski 39 , R. Zaidan 63 , A. M. Zaitsev 130,aa , J. Zalieckas 14 , A. Zaman 148 , S. Zambito 57 , L. Zanello 132a,132b , D. Zanzi 88 , C. Zeitnitz 175 , M. Zeman 128 , A. Zemla 38a , K. Zengel 23 , O. Zenin 130 , T. Ženiš 144a , D. Zerwas 117 , D. Zhang 89 , F. Zhang 173 , H. Zhang 33c , J. Zhang 6 , L. Zhang 48 , R. Zhang 33b , X. Zhang 33d , Z. Zhang 117 , X. Zhao 40 , Y. Zhao 33d,117 , Z. Zhao 33b , A. Zhemchugov 65 , J. Zhong 120 , B. Zhou 89 , C. Zhou 45 , L. Zhou 35 , L. Zhou 40 , N. Zhou 33f , C. G. Zhu 33d , H. Zhu 33a , J. Zhu 89 , Y. Zhu 33b , X. Zhuang 33a , K. Zhukov 96 , A. Zibell 174 , D. Zieminska 61 , N. I. Zimine 65 , C. Zimmermann 83 , S. Zimmermann 48 , Z. Zinonos 54 , M. Zinser 83 , M. Ziolkowski 141 , L. Živković 13 , G. Zobernig 173 , A. Zoccoli 20a,20b , M. zur Nedden 16 , G. Zurzolo 104a,104b , L. Zwalinski 30 1 Department of PhysicsG. Aad 85 , B. Abbott 113 , J. Abdallah 151 , O. Abdinov 11 , R. Aben 107 , M. Abolins 90 , O. S. AbouZeid 158 , H. Abramowicz 153 , H. Abreu 152 , R. Abreu 30 , Y. Abulaiti 146a,146b , B. S. Acharya 164a,164b,a , L. Adamczyk 38a , D. L. Adams 25 , J. Adelman 108 , S. Adomeit 100 , T. Adye 131 , A. A. Affolder 74 , T. Agatonovic-Jovin 13 , J. Agricola 54 , J. A. Aguilar-Saavedra 126a,126f , S. P. Ahlen 22 , F. Ahmadov 65,b , G. Aielli 133a,133b , H. Akerstedt 146a,146b , T. P. A. Åkesson 81 , A. V. Akimov 96 , G. L. Alberghi 20a,20b , J. Albert 169 , S. Albrand 55 , M. J. Alconada Verzini 71 , M. Aleksa 30 , I. N. Aleksandrov 65 , C. Alexa 26a , G. Alexander 153 , T. Alexopoulos 10 , M. Alhroob 113 , G. Alimonti 91a , L. Alio 85 , J. Alison 31 , S. P. Alkire 35 , B. M. M. Allbrooke 149 , P. P. Allport 74 , A. Aloisio 104a,104b , A. Alonso 36 , F. Alonso 71 , C. Alpigiani 76 , A. Altheimer 35 , B. Alvarez Gonzalez 30 , D. Álvarez Piqueras 167 , M. G. Alviggi 104a,104b , B. T. Amadio 15 , K. Amako 66 , Y. Amaral Coutinho 24a , C. Amelung 23 , D. Amidei 89 , S. P. Amor Dos Santos 126a,126c , A. Amorim 126a,126b , S. Amoroso 48 , N. Amram 153 , G. Amundsen 23 , C. Anastopoulos 139 , L. S. Ancu 49 , N. Andari 108 , T. Andeen 35 , C. F. Anders 58b , G. Anders 30 , J. K. Anders 74 , K. J. Anderson 31 , A. Andreazza 91a,91b , V. Andrei 58a , S. Angelidakis 9 , I. Angelozzi 107 , P. Anger 44 , A. Angerami 35 , F. Anghinolfi 30 , A. V. Anisenkov 109,c , N. Anjos 12 , A. Annovi 124a,124b , M. Antonelli 47 , A. Antonov 98 , J. Antos 144b , F. Anulli 132a , M. Aoki 66 , L. Aperio Bella 18 , G. Arabidze 90 , Y. Arai 66 , J. P. Araque 126a , A. T. H. Arce 45 , F. A. Arduh 71 , J-F. Arguin 95 , S. Argyropoulos 42 , M. Arik 19a , A. J. Armbruster 30 , O. Arnaez 30 , V. Arnal 82 , H. Arnold 48 , M. Arratia 28 , O. Arslan 21 , A. Artamonov 97 , G. Artoni 23 , S. Asai 155 , N. Asbah 42 , A. Ashkenazi 153 , B. Åsman 146a,146b , L. Asquith 149 , K. Assamagan 25 , R. Astalos 144a , M. Atkinson 165 , N. B. Atlay 141 , K. Augsten 128 , M. Aurousseau 145b , G. Avolio 30 , B. Axen 15 , M. K. Ayoub 117 , G. Azuelos 95,d , M. A. Baak 30 , A. E. Baas 58a , M. J. Baca 18 , C. Bacci 134a,134b , H. Bachacou 136 , K. Bachas 154 , M. Backes 30 , M. Backhaus 30 , P. Bagiacchi 132a,132b , P. Bagnaia 132a,132b , Y. Bai 33a , T. Bain 35 , J. T. Baines 131 , O. K. Baker 176 , E. M. Baldin 109,c , P. Balek 129 , T. Balestri 148 , F. Balli 84 , E. Banas 39 , Sw. Banerjee 173 , A. A. E. Bannoura 175 , H. S. Bansil 18 , L. Barak 30 , E. L. Barberio 88 , D. Barberis 50a,50b , M. Barbero 85 , T. Barillari 101 , M. Barisonzi 164a,164b , T. Barklow 143 , N. Barlow 28 , S. L. Barnes 84 , B. M. Barnett 131 , R. M. Barnett 15 , Z. Barnovska 5 , A. Baroncelli 134a , G. Barone 23 , A. J. Barr 120 , F. Barreiro 82 , J. Barreiro Guimarães da Costa 57 , R. Bartoldus 143 , A. E. Barton 72 , P. Bartos 144a , A. Basalaev 123 , A. Bassalat 117 , A. Basye 165 , R. L. Bates 53 , S. J. Batista 158 , J. R. Batley 28 , M. Battaglia 137 , M. Bauce 132a,132b , F. Bauer 136 , H. S. Bawa 143,e , J. B. Beacham 111 , M. D. Beattie 72 , T. Beau 80 , P. H. Beauchemin 161 , R. Beccherle 124a,124b , P. Bechtle 21 , H. P. Beck 17,f , K. Becker 120 , M. Becker 83 , S. Becker 100 , M. Beckingham 170 , C. Becot 117 , A. J. Beddall 19c , A. Beddall 19c , V. A. Bednyakov 65 , C. P. Bee 148 , L. J. Beemster 107 , T. A. Beermann 175 , M. Begel 25 , J. K. Behr 120 , C. Belanger-Champagne 87 , W. H. Bell 49 , G. Bella 153 , L. Bellagamba 20a , A. Bellerive 29 , M. Bellomo 86 , K. Belotskiy 98 , O. Beltramello 30 , O. Benary 153 , D. Benchekroun 135a , M. Bender 100 , K. Bendtz 146a,146b , N. Benekos 10 , Y. Benhammou 153 , E. Benhar Noccioli 49 , J. A. Benitez Garcia 159b , D. P. Benjamin 45 , J. R. Bensinger 23 , S. Bentvelsen 107 , L. Beresford 120 , M. Beretta 47 , D. Berge 107 , E. Bergeaas Kuutmann 166 , N. Berger 5 , F. Berghaus 169 , J. Beringer 15 , C. Bernard 22 , N. R. Bernard 86 , C. Bernius 110 , F. U. Bernlochner 21 , T. Berry 77 , P. Berta 129 , C. Bertella 83 , G. Bertoli 146a,146b , F. Bertolucci 124a,124b , C. Bertsche 113 , D. Bertsche 113 , M. I. Besana 91a , G. J. Besjes 36 , O. Bessidskaia Bylund 146a,146b , M. Bessner 42 , N. Besson 136 , C. Betancourt 48 , S. Bethke 101 , A. J. Bevan 76 , W. Bhimji 46 , R. M. Bianchi 125 , L. Bianchini 23 , M. Bianco 30 , O. Biebel 100 , D. Biedermann 16 , S. P. Bieniek 78 , M. Biglietti 134a , J. Bilbao De Mendizabal 49 , H. Bilokon 47 , M. Bindi 54 , S. Binet 117 , A. Bingul 19c , C. Bini 132a,132b , S. Biondi 20a,20b , C. W. Black 150 , J. E. Black 143 , K. M. Black 22 , D. Blackburn 138 , R. E. Blair 6 , J.-B. Blanchard 136 , J. E. Blanco 77 , T. Blazek 144a , I. Bloch 42 , C. Blocker 23 , W. Blum 83,* , U. Blumenschein 54 , G. J. Bobbink 107 , V. S. Bobrovnikov 109,c , S. S. Bocchetta 81 , A. Bocci 45 , C. Bock 100 , M. Boehler 48 , J. A. Bogaerts 30 , D. Bogavac 13 , A. G. Bogdanchikov 109 , C. Bohm 146a , V. Boisvert 77 , T. Bold 38a , V. Boldea 26a , A. S. Boldyrev 99 , M. Bomben 80 , M. Bona 76 , M. Boonekamp 136 , A. Borisov 130 , G. Borissov 72 , S. Borroni 42 , J. Bortfeldt 100 , V. Bortolotto 60a,60b,60c , K. Bos 107 , D. Boscherini 20a , M. Bosman 12 , J. Boudreau 125 , J. Bouffard 2 , E. V. Bouhova-Thacker 72 , D. Boumediene 34 , C. Bourdarios 117 , N. Bousson 114 , A. Boveia 30 , J. Boyd 30 , I. R. Boyko 65 , I. Bozic 13 , J. Bracinik 18 , A. Brandt 8 , G. Brandt 54 , O. Brandt 58a , U. Bratzler 156 , B. Brau 86 , J. E. Brau 116 , H. M. Braun 175,* , S. F. Brazzale 164a,164c , W. D. Breaden Madden 53 , K. Brendlinger 122 , A. J. Brennan 88 , L. Brenner 107 , R. Brenner 166 , S. Bressler 172 , K. Bristow 145c , T. M. Bristow 46 , D. Britton 53 , D. Britzger 42 , F. M. Brochu 28 , I. Brock 21 , R. Brock 90 , J. Bronner 101 , G. Brooijmans 35 , T. Brooks 77 , W. K. Brooks 32b , J. Brosamer 15 , E. Brost 116 , J. Brown 55 , P. A. Bruckman de Renstrom 39 , D. Bruncko 144b , R. Bruneliere 48 , A. Bruni 20a , G. Bruni 20a , M. Bruschi 20a , N. Bruscino 21 , L. Bryngemark 81 , T. Buanes 14 , Q. Buat 142 , P. Buchholz 141 , A. G. Buckley 53 , S. I. Buda 26a , I. A. Budagov 65 , F. Buehrer 48 , L. Bugge 119 , M. K. Bugge 119 , O. Bulekov 98 , D. Bullock 8 , H. Burckhart 30 , S. Burdin 74 , C. D. Burgard 48 , B. Burghgrave 108 , S. Burke 131 , I. Burmeister 43 , E. Busato 34 , D. Büscher 48 , V. Büscher 83 , P. Bussey 53 , J. M. Butler 22 , A. I. Butt 3 , C. M. Buttar 53 , J. M. Butterworth 78 , P. Butti 107 , W. Buttinger 25 , A. Buzatu 53 , A. R. Buzykaev 109,c , S. Cabrera Urbán 167 , D. Caforio 128 , V. M. Cairo 37a,37b , O. Cakir 4a , N. Calace 49 , P. Calafiura 15 , A. Calandri 136 , G. Calderini 80 , P. Calfayan 100 , L. P. Caloba 24a , D. Calvet 34 , S. Calvet 34 , R. Camacho Toro 31 , S. Camarda 42 , P. Camarri 133a,133b , D. Cameron 119 , R. Caminal Armadans 165 , S. Campana 30 , M. Campanelli 78 , A. Campoverde 148 , V. Canale 104a,104b , A. Canepa 159a , M. Cano Bret 33e , J. Cantero 82 , R. Cantrill 126a , T. Cao 40 , M. D. M. Capeans Garrido 30 , I. Caprini 26a , M. Caprini 26a , M. Capua 37a,37b , R. Caputo 83 , R. Cardarelli 133a , F. Cardillo 48 , T. Carli 30 , G. Carlino 104a , L. Carminati 91a,91b , S. Caron 106 , E. Carquin 32a , G. D. Carrillo-Montoya 8 , J. R. Carter 28 , J. Carvalho 126a,126c , D. Casadei 78 , M. P. Casado 12 , M. Casolino 12 , E. Castaneda-Miranda 145b , A. Castelli 107 , V. Castillo Gimenez 167 , N. F. Castro 126a,g , P. Catastini 57 , A. Catinaccio 30 , J. R. Catmore 119 , A. Cattai 30 , J. Caudron 83 , V. Cavaliere 165 , D. Cavalli 91a , M. Cavalli-Sforza 12 , V. Cavasinni 124a,124b , F. Ceradini 134a,134b , B. C. Cerio 45 , K. Cerny 129 , A. S. Cerqueira 24b , A. Cerri 149 , L. Cerrito 76 , F. Cerutti 15 , M. Cerv 30 , A. Cervelli 17 , S. A. Cetin 19b , A. Chafaq 135a , D. Chakraborty 108 , I. Chalupkova 129 , P. Chang 165 , J. D. Chapman 28 , D. G. Charlton 18 , C. C. Chau 158 , C. A. Chavez Barajas 149 , S. Cheatham 152 , A. Chegwidden 90 , S. Chekanov 6 , S. V. Chekulaev 159a , G. A. Chelkov 65,h , M. A. Chelstowska 89 , C. Chen 64 , H. Chen 25 , K. Chen 148 , L. Chen 33d,i , S. Chen 33c , X. Chen 33f , Y. Chen 67 , H. C. Cheng 89 , Y. Cheng 31 , A. Cheplakov 65 , E. Cheremushkina 130 , R. Cherkaoui El Moursli 135e , V. Chernyatin 25,* , E. Cheu 7 , L. Chevalier 136 , V. Chiarella 47 , G. Chiarelli 124a,124b , J. T. Childers 6 , G. Chiodini 73a , A. S. Chisholm 18 , R. T. Chislett 78 , A. Chitan 26a , M. V. Chizhov 65 , K. Choi 61 , S. Chouridou 9 , B. K. B. Chow 100 , V. Christodoulou 78 , D. Chromek-Burckhart 30 , J. Chudoba 127 , A. J. Chuinard 87 , J. J. Chwastowski 39 , L. Chytka 115 , G. Ciapetti 132a,132b , A. K. Ciftci 4a , D. Cinca 53 , V. Cindro 75 , I. A. Cioara 21 , A. Ciocio 15 , Z. H. Citron 172 , M. Ciubancan 26a , A. Clark 49 , B. L. Clark 57 , P. J. Clark 46 , R. N. Clarke 15 , W. Cleland 125 , C. Clement 146a,146b , Y. Coadou 85 , M. Cobal 164a,164c , A. Coccaro 138 , J. Cochran 64 , L. Coffey 23 , J. G. Cogan 143 , L. Colasurdo 106 , B. Cole 35 , S. Cole 108 , A. P. Colijn 107 , J. Collot 55 , T. Colombo 58c , G. Compostella 101 , P. Conde Muiño 126a,126b , E. Coniavitis 48 , S. H. Connell 145b , I. A. Connelly 77 , S. M. Consonni 91a,91b , V. Consorti 48 , S. Constantinescu 26a , C. Conta 121a,121b , G. Conti 30 , F. Conventi 104a,j , M. Cooke 15 , B. D. Cooper 78 , A. M. Cooper-Sarkar 120 , T. Cornelissen 175 , M. Corradi 20a , F. Corriveau 87,k , A. Corso-Radu 163 , A. Cortes-Gonzalez 12 , G. Cortiana 101 , G. Costa 91a , M. J. Costa 167 , D. Costanzo 139 , D. Côté 8 , G. Cottin 28 , G. Cowan 77 , B. E. Cox 84 , K. Cranmer 110 , G. Cree 29 , S. Crépé-Renaudin 55 , F. Crescioli 80 , W. A. Cribbs 146a,146b , M. Crispin Ortuzar 120 , M. Cristinziani 21 , V. Croft 106 , G. Crosetti 37a,37b , T. Cuhadar Donszelmann 139 , J. Cummings 176 , M. Curatolo 47 , C. Cuthbert 150 , H. Czirr 141 , P. Czodrowski 3 , S. D'Auria 53 , M. D'Onofrio 74 , M. J. Da Cunha Sargedas De Sousa 126a,126b , C. Da Via 84 , W. Dabrowski 38a , A. Dafinca 120 , T. Dai 89 , O. Dale 14 , F. Dallaire 95 , C. Dallapiccola 86 , M. Dam 36 , J. R. Dandoy 31 , N. P. Dang 48 , A. C. Daniells 18 , M. Danninger 168 , M. Dano Hoffmann 136 , V. Dao 48 , G. Darbo 50a , S. Darmora 8 , J. Dassoulas 3 , A. Dattagupta 61 , W. Davey 21 , C. David 169 , T. Davidek 129 , E. Davies 120,l , M. Davies 153 , P. Davison 78 , Y. Davygora 58a , E. Dawe 88 , I. Dawson 139 , R. K. Daya-Ishmukhametova 86 , K. De 8 , R. de Asmundis 104a , A. De Benedetti 113 , S. De Castro 20a,20b , S. De Cecco 80 , N. De Groot 106 , P. de Jong 107 , H. De la Torre 82 , F. De Lorenzi 64 , L. De Nooij 107 , D. De Pedis 132a , A. De Salvo 132a , U. De Sanctis 149 , A. De Santo 149 , J. B. De Vivie De Regie 117 , W. J. Dearnaley 72 , R. Debbe 25 , C. Debenedetti 137 , D. V. Dedovich 65 , I. Deigaard 107 , J. Del Peso 82 , T. Del Prete 124a,124b , D. Delgove 117 , F. Deliot 136 , C. M. Delitzsch 49 , M. Deliyergiyev 75 , A. Dell'Acqua 30 , L. Dell'Asta 22 , M. Dell'Orso 124a,124b , M. Della Pietra 104a,j , D. della Volpe 49 , M.Delmastro 5 , P.A.Delsart 55 , C. Deluca 107 , D. A. DeMarco 158 , S. Demers 176 , M. Demichev 65 , A. Demilly 80 , S. P. Denisov 130 , D. Derendarz 39 , J. E. Derkaoui 135d , F. Derue 80 , P. Dervan 74 , K. Desch 21 , C. Deterre 42 , P. O. Deviveiros 30 , A. Dewhurst 131 , S. Dhaliwal 23 , A. Di Ciaccio 133a,133b , L. Di Ciaccio 5 , A. Di Domenico 132a,132b , C. Di Donato 104a,104b , A. Di Girolamo 30 , B. Di Girolamo 30 , A. Di Mattia 152 , B. Di Micco 134a,134b , R. Di Nardo 47 , A. Di Simone 48 , R. Di Sipio 158 , D. Di Valentino 29 , C. Diaconu 85 , M. Diamond 158 , F. A. Dias 46 , M. A. Diaz 32a , E. B. Diehl 89 , J. Dietrich 16 , S. Diglio 85 , A. Dimitrievska 13 , J. Dingfelder 21 , P. Dita 26a , S. Dita 26a , F. Dittus 30 , F. Djama 85 , T. Djobava 51b , J. I. Djuvsland 58a , M. A. B. do Vale 24c , D. Dobos 30 , M. Dobre 26a , C. Doglioni 81 , T. Dohmae 155 , J. Dolejsi 129 , Z. Dolezal 129 , B. A. Dolgoshein 98,* , M. Donadelli 24d , S. Donati 124a,124b , P. Dondero 121a,121b , J. Donini 34 , J. Dopke 131 , A. Doria 104a , M. T. Dova 71 , A. T. Doyle 53 , E. Drechsler 54 , M. Dris 10 , E. Dubreuil 34 , E. Duchovni 172 , G. Duckeck 100 , O. A. Ducu 26a,85 , D. Duda 107 , A. Dudarev 30 , L. Duflot 117 , L. Duguid 77 , M. Dührssen 30 , M. Dunford 58a , H. Duran Yildiz 4a , M. Düren 52 , A. Durglishvili 51b , D. Duschinger 44 , M. Dyndal 38a , C. Eckardt 42 , K. M. Ecker 101 , R. C. Edgar 89 , W. Edson 2 , N. C. Edwards 46 , W. Ehrenfeld 21 , T. Eifert 30 , G. Eigen 14 , K. Einsweiler 15 , T. Ekelof 166 , M. El Kacimi 135c , M. Ellert 166 , S. Elles 5 , F. Ellinghaus 175 , A. A. Elliot 169 , N. Ellis 30 , J. Elmsheuser 100 , M. Elsing 30 , D. Emeliyanov 131 , Y. Enari 155 , O. C. Endner 83 , M. Endo 118 , J. Erdmann 43 , A. Ereditato 17 , G. Ernis 175 , J. Ernst 2 , M. Ernst 25 , S. Errede 165 , E. Ertel 83 , M. Escalier 117 , H. Esch 43 , C. Escobar 125 , B. Esposito 47 , A. I. Etienvre 136 , E. Etzion 153 , H. Evans 61 , A. Ezhilov 123 , L. Fabbri 20a,20b , G. Facini 31 , R. M. Fakhrutdinov 130 , S. Falciano 132a , R. J. Falla 78 , J. Faltova 129 , Y. Fang 33a , M. Fanti 91a,91b , A. Farbin 8 , A. Farilla 134a , T. Farooque 12 , S. Farrell 15 , S. M. Farrington 170 , P. Farthouat 30 , F. Fassi 135e , P. Fassnacht 30 , D. Fassouliotis 9 , M. Faucci Giannelli 77 , A. Favareto 50a,50b , L. Fayard 117 , P. Federic 144a , O. L. Fedin 123,m , W. Fedorko 168 , S. Feigl 30 , L. Feligioni 85 , C. Feng 33d , E. J. Feng 6 , H. Feng 89 , A. B. Fenyuk 130 , L. Feremenga 8 , P. Fernandez Martinez 167 , S. Fernandez Perez 30 , J. Ferrando 53 , A. Ferrari 166 , P. Ferrari 107 , R. Ferrari 121a , D. E. Ferreira de Lima 53 , A. Ferrer 167 , D. Ferrere 49 , C. Ferretti 89 , A. Ferretto Parodi 50a,50b , M. Fiascaris 31 , F. Fiedler 83 , A. Filipčič 75 , M. Filipuzzi 42 , F. Filthaut 106 , M. Fincke-Keeler 169 , K. D. Finelli 150 , M. C. N. Fiolhais 126a,126c , L. Fiorini 167 , A. Firan 40 , A. Fischer 2 , C. Fischer 12 , J. Fischer 175 , W. C. Fisher 90 , E. A. Fitzgerald 23 , N. Flaschel 42 , I. Fleck 141 , P. Fleischmann 89 , S. Fleischmann 175 , G. T. Fletcher 139 , G. Fletcher 76 , R. R. M. Fletcher 122 , T. Flick 175 , A. Floderus 81 , L. R. Flores Castillo 60a , M. J. Flowerdew 101 , A. Formica 136 , A. Forti 84 , D. Fournier 117 , H. Fox 72 , S. Fracchia 12 , P. Francavilla 80 , M. Franchini 20a,20b , D. Francis 30 , L. Franconi 119 , M. Franklin 57 , M. Frate 163 , M. Fraternali 121a,121b , D. Freeborn 78 , S. T. French 28 , F. Friedrich 44 , D. Froidevaux 30 , J. A. Frost 120 , C. Fukunaga 156 , E. Fullana Torregrosa 83 , B. G. Fulsom 143 , T. Fusayasu 102 , J. Fuster 167 , C. Gabaldon 55 , O. Gabizon 175 , A. Gabrielli 20a,20b , A. Gabrielli 132a,132b , G. P. Gach 38a , S. Gadatsch 107 , S. Gadomski 49 , G. Gagliardi 50a,50b , P. Gagnon 61 , C. Galea 106 , B. Galhardo 126a,126c , E. J. Gallas 120 , B. J. Gallop 131 , P. Gallus 128 , G. Galster 36 , K. K. Gan 111 , J. Gao 33b,85 , Y. Gao 46 , Y. S. Gao 143,e , F. M. Garay Walls 46 , F. Garberson 176 , C. García 167 , J. E. García Navarro 167 , M. Garcia-Sciveres 15 , R. W. Gardner 31 , N. Garelli 143 , V. Garonne 119 , C. Gatti 47 , A. Gaudiello 50a,50b , G. Gaudio 121a , B. Gaur 141 , L. Gauthier 95 , P. Gauzzi 132a,132b , I. L. Gavrilenko 96 , C. Gay 168 , G. Gaycken 21 , E. N. Gazis 10 , P. Ge 33d , Z. Gecse 168 , C. N. P. Gee 131 , D. A. A. Geerts 107 , Ch. Geich-Gimbel 21 , M. P. Geisler 58a , C. Gemme 50a , M. H. Genest 55 , S. Gentile 132a,132b , M. George 54 , S. George 77 , D. Gerbaudo 163 , A. Gershon 153 , S. Ghasemi 141 , H. Ghazlane 135b , B. Giacobbe 20a , S. Giagu 132a,132b , V. Giangiobbe 12 , P. Giannetti 124a,124b , B. Gibbard 25 , S. M. Gibson 77 , M. Gilchriese 15 , T. P. S. Gillam 28 , D. Gillberg 30 , G. Gilles 34 , D. M. Gingrich 3,d , N. Giokaris 9 , M. P. Giordani 164a,164c , F. M. Giorgi 20a , F. M. Giorgi 16 , P. F. Giraud 136 , P. Giromini 47 , D. Giugni 91a , C. Giuliani 48 , M. Giulini 58b , B. K. Gjelsten 119 , S. Gkaitatzis 154 , I. Gkialas 154 , E. L. Gkougkousis 117 , L. K. Gladilin 99 , C. Glasman 82 , J. Glatzer 30 , P. C. F. Glaysher 46 , A. Glazov 42 , M. Goblirsch-Kolb 101 , J. R. Goddard 76 , J. Godlewski 39 , S. Goldfarb 89 , T. Golling 49 , D. Golubkov 130 , A. Gomes 126a,126b,126d , R. Gonçalo 126a , J. Goncalves Pinto Firmino Da Costa 136 , L. Gonella 21 , S. González de la Hoz 167 , G. Gonzalez Parra 12 , S. Gonzalez-Sevilla 49 , L. Goossens 30 , P. A. Gorbounov 97 , H. A. Gordon 25 , I. Gorelov 105 , B. Gorini 30 , E. Gorini 73a,73b , A. Gorišek 75 , E. Gornicki 39 , A. T. Goshaw 45 , C. Gössling 43 , M. I. Gostkin 65 , D. Goujdami 135c , A. G. Goussiou 138 , N. Govender 145b , E. Gozani 152 , H. M. X. Grabas 137 , L. Graber 54 , I. Grabowska-Bold 38a , P. O. J. Gradin 166 , P. Grafström 20a,20b , K-J. Grahn 42 , J. Gramling 49 , E. Gramstad 119 , S. Grancagnolo 16 , V. Gratchev 123 , H. M. Gray 30 , E. Graziani 134a , Z. D. Greenwood 79,n , K. Gregersen 78 , I. M. Gregor 42 , P. Grenier 143 , J. Griffiths 8 , A. A. Grillo 137 , K. Grimm 72 , S. Grinstein 12,o , Ph. Gris 34 , J.-F. Grivaz 117 , J. P. Grohs 44 , A. Grohsjean 42 , E. Gross 172 , J. Grosse-Knetter 54 , G. C. Grossi 79 , Z. J. Grout 149 , L. Guan 89 , J. Guenther 128 , F. Guescini 49 , D. Guest 176 , O. Gueta 153 , E. Guido 50a,50b , T. Guillemin 117 , S. Guindon 2 , U. Gul 53 , C. Gumpert 44 , J. Guo 33e , Y. Guo 33b , S. Gupta 120 , G. Gustavino 132a,132b , P. Gutierrez 113 , N. G. Gutierrez Ortiz 78 , C. Gutschow 44 , C. Guyot 136 , C. Gwenlan 120 , C. B. Gwilliam 74 , A. Haas 110 , C. Haber 15 , H. K. Hadavand 8 , N. Haddad 135e , P. Haefner 21 , S. Hageböck 21 , Z. Hajduk 39 , H. Hakobyan 177 , M. Haleem 42 , J. Haley 114 , D. Hall 120 , G. Halladjian 90 , G. D. Hallewell 85 , K. Hamacher 175 , P. Hamal 115 , K. Hamano 169 , A. Hamilton 145a , G. N. Hamity 145c , P. G. Hamnett 42 , L. Han 33b , K. Hanagaki 118 , K. Hanawa 155 , M. Hance 15 , P. Hanke 58a , R. Hanna 136 , J. B. Hansen 36 , J. D. Hansen 36 , M. C. Hansen 21 , P. H. Hansen 36 , K. Hara 160 , A. S. Hard 173 , T. Harenberg 175 , F. Hariri 117 , S. Harkusha 92 , R. D. Harrington 46 , P. F. Harrison 170 , F. Hartjes 107 , M. Hasegawa 67 , S. Hasegawa 103 , Y. Hasegawa 140 , A. Hasib 113 , S. Hassani 136 , S. Haug 17 , R. Hauser 90 , L. Hauswald 44 , M. Havranek 127 , C. M. Hawkes 18 , R. J. Hawkings 30 , A. D. Hawkins 81 , T. Hayashi 160 , D. Hayden 90 , C. P. Hays 120 , J. M. Hays 76 , H. S. Hayward 74 , S. J. Haywood 131 , S. J. Head 18 , T. Heck 83 , V. Hedberg 81 , L. Heelan 8 , S. Heim 122 , T. Heim 175 , B. Heinemann 15 , L. Heinrich 110 , J. Hejbal 127 , L. Helary 22 , S. Hellman 146a,146b , D. Hellmich 21 , C. Helsens 12 , J. Henderson 120 , R. C. W. Henderson 72 , Y. Heng 173 , C. Hengler 42 , A. Henrichs 176 , A. M. Henriques Correia 30 , S. Henrot-Versille 117 , G. H. Herbert 16 , Y. Hernández Jiménez 167 , R. Herrberg-Schubert 16 , G. Herten 48 , R. Hertenberger 100 , L. Hervas 30 , G. G. Hesketh 78 , N. P. Hessey 107 , J. W. Hetherly 40 , R. Hickling 76 , E. Higón-Rodriguez 167 , E. Hill 169 , J. C. Hill 28 , K. H. Hiller 42 , S. J. Hillier 18 , I. Hinchliffe 15 , E. Hines 122 , R. R. Hinman 15 , M. Hirose 157 , D. Hirschbuehl 175 , J. Hobbs 148 , N. Hod 107 , M. C. Hodgkinson 139 , P. Hodgson 139 , A. Hoecker 30 , M. R. Hoeferkamp 105 , F. Hoenig 100 , M. Hohlfeld 83 , D. Hohn 21 , T. R. Holmes 15 , M. Homann 43 , T. M. Hong 125 , L. Hooft van Huysduynen 110 , W. H. Hopkins 116 , Y. Horii 103 , A. J. Horton 142 , J-Y. Hostachy 55 , S. Hou 151 , A. Hoummada 135a , J. Howard 120 , J. Howarth 42 , M. Hrabovsky 115 , I. Hristova 16 , J. Hrivnac 117 , T. Hryn'ova 5 , A. Hrynevich 93 , C. Hsu 145c , P. J. Hsu 151,p , S.-C. Hsu 138 , D. Hu 35 , Q. Hu 33b , X. Hu 89 , Y. Huang 42 , Z. Hubacek 128 , F. Hubaut 85 , F. Huegging 21 , T. B. Huffman 120 , E. W. Hughes 35 , G. Hughes 72 , M. Huhtinen 30 , T. A. Hülsing 83 , N. Huseynov 65,b , J. Huston 90 , J. Huth 57 , G. Iacobucci 49 , G. Iakovidis 25 , I. Ibragimov 141 , L. Iconomidou-Fayard 117 , E. Ideal 176 , Z. Idrissi 135e , P. Iengo 30 , O. Igonkina 107 , T. Iizawa 171 , Y. Ikegami 66 , K. Ikematsu 141 , M. Ikeno 66 , Y. Ilchenko 31,q , D. Iliadis 154 , N. Ilic 143 , T. Ince 101 , G. Introzzi 121a,121b , P. Ioannou 9 , M. Iodice 134a , K. Iordanidou 35 , V. Ippolito 57 , A. Irles Quiles 167 , C. Isaksson 166 , M. Ishino 68 , M. Ishitsuka 157 , R. Ishmukhametov 111 , C. Issever 120 , S. Istin 19a , J. M. Iturbe Ponce 84 , R. Iuppa 133a,133b , J. Ivarsson 81 , W. Iwanski 39 , H. Iwasaki 66 , J. M. Izen 41 , V. Izzo 104a , S. Jabbar 3 , B. Jackson 122 , M. Jackson 74 , P. Jackson 1 , M. R. Jaekel 30 , V. Jain 2 , K. Jakobs 48 , S. Jakobsen 30 , T. Jakoubek 127 , J. Jakubek 128 , D. O. Jamin 114 , D. K. Jana 79 , E. Jansen 78 , R. W. Jansky 62 , J. Janssen 21 , M. Janus 54 , G. Jarlskog 81 , N. Javadov 65,b , T. Javůrek 48 , L. Jeanty 15 , J. Jejelava 51a,r , G.-Y. Jeng 150 , D. Jennens 88 , P. Jenni 48,s , J. Jentzsch 43 , C. Jeske 170 , S. Jézéquel 5 , H. Ji 173 , J. Jia 148 , Y. Jiang 33b , S. Jiggins 78 , J. Jimenez Pena 167 , S. Jin 33a , A. Jinaru 26a , O. Jinnouchi 157 , M. D. Joergensen 36 , P. Johansson 139 , K. A. Johns 7 , K. Jon-And 146a,146b , G. Jones 170 , R. W. L. Jones 72 , T. J. Jones 74 , J. Jongmanns 58a , P. M. Jorge 126a,126b , K. D. Joshi 84 , J. Jovicevic 159a , X. Ju 173 , C. A. Jung 43 , P. Jussel 62 , A. Juste Rozas 12,o , M. Kaci 167 , A. Kaczmarska 39 , M. Kado 117 , H. Kagan 111 , M. Kagan 143 , S. J. Kahn 85 , E. Kajomovitz 45 , C. W. Kalderon 120 , S. Kama 40 , A. Kamenshchikov 130 , N. Kanaya 155 , S. Kaneti 28 , V. A. Kantserov 98 , J. Kanzaki 66 , B. Kaplan 110 , L. S. Kaplan 173 , A. Kapliy 31 , D. Kar 145c , K. Karakostas 10 , A. Karamaoun 3 , N. Karastathis 10,107 , M. J. Kareem 54 , M. Karnevskiy 83 , S. N. Karpov 65 , Z. M. Karpova 65 , K. Karthik 110 , V. Kartvelishvili 72 , A. N. Karyukhin 130 , L. Kashif 173 , R. D. Kass 111 , A. Kastanas 14 , Y. Kataoka 155 , C. Kato 155 , A. Katre 49 , J. Katzy 42 , K. Kawagoe 70 , T. Kawamoto 155 , G. Kawamura 54 , S. Kazama 155 , V. F. Kazanin 109,c , R. Keeler 169 , R. Kehoe 40 , J. S. Keller 42 , J. J. Kempster 77 , H. Keoshkerian 84 , O. Kepka 127 , B. P. Kerševan 75 , S. Kersten 175 , R. A. Keyes 87 , F. Khalil-zada 11 , H. Khandanyan 146a,146b , A. Khanov 114 , A. G. Kharlamov 109,c , T. J. Khoo 28 , V. Khovanskiy 97 , E. Khramov 65 , J. Khubua 51b,t , S. Kido 67 , H. Y. Kim 8 , S. H. Kim 160 , Y. Kim 31 , N. Kimura 154 , O. M. Kind 16 , B. T. King 74 , M. King 167 , S. B. King 168 , J. Kirk 131 , A. E. Kiryunin 101 , T. Kishimoto 67 , D. Kisielewska 38a , F. Kiss 48 , K. Kiuchi 160 , O. Kivernyk 136 , E. Kladiva 144b , M. H. Klein 35 , M. Klein 74 , U. Klein 74 , K. Kleinknecht 83 , P. Klimek 146a,146b , A. Klimentov 25 , R. Klingenberg 43 , J. A. Klinger 139 , T. Klioutchnikova 30 , E.-E. Kluge 58a , P. Kluit 107 , S. Kluth 101 , J. Knapik 39 , E. Kneringer 62 , E. B. F. G. Knoops 85 , A. Knue 53 , A. Kobayashi 155 , D. Kobayashi 157 , T. Kobayashi 155 , M. Kobel 44 , M. Kocian 143 , P. Kodys 129 , T. Koffas 29 , E. Koffeman 107 , L. A. Kogan 120 , S. Kohlmann 175 , Z. Kohout 128 , T. Kohriki 66 , T. Koi 143 , H. Kolanoski 16 , I. Koletsou 5 , A. A. Komar 96,* , Y. Komori 155 , T. Kondo 66 , N. Kondrashova 42 , K. Köneke 48 , A. C. König 106 , T. Kono 66 , R. Konoplich 110,u , N. Konstantinidis 78 , R. Kopeliansky 152 , S. Koperny 38a , L. Köpke 83 , A. K. Kopp 48 , K. Korcyl 39 , K. Kordas 154 , A. Korn 78 , A. A. Korol 109,c , I. Korolkov 12 , E. V. Korolkova 139 , O. Kortner 101 , S. Kortner 101 , T. Kosek 129 , V. V. Kostyukhin 21 , V. M. Kotov 65 , A. Kotwal 45 , A. Kourkoumeli-Charalampidi 154 , C. Kourkoumelis 9 , V. Kouskoura 25 , A. Koutsman 159a , R. Kowalewski 169 , T. Z. Kowalski 38a , W. Kozanecki 136 , A. S. Kozhin 130 , V. A. Kramarenko 99 , G. Kramberger 75 , D. Krasnopevtsev 98 , M. W. Krasny 80 , A. Krasznahorkay 30 , J. K. Kraus 21 , A. Kravchenko 25 , S. Kreiss 110 , M. Kretz 58c , J. Kretzschmar 74 , K. Kreutzfeldt 52 , P. Krieger 158 , K. Krizka 31 , K. Kroeninger 43 , H. Kroha 101 , J. Kroll 122 , J. Kroseberg 21 , J. Krstic 13 , U. Kruchonak 65 , H. Krüger 21 , N. Krumnack 64 , A. Kruse 173 , M. C. Kruse 45 , M. Kruskal 22 , T. Kubota 88 , H. Kucuk 78 , S. Kuday 4b , S. Kuehn 48 , A. Kugel 58c , F. Kuger 174 , A. Kuhl 137 , T. Kuhl 42 , V. Kukhtin 65 , Y. Kulchitsky 92 , S. Kuleshov 32b , M. Kuna 132a,132b , T. Kunigo 68 , A. Kupco 127 , H. Kurashige 67 , Y. A. Kurochkin 92 , V. Kus 127 , E. S. Kuwertz 169 , M. Kuze 157 , J. Kvita 115 , T. Kwan 169 , D. Kyriazopoulos 139 , A. La Rosa 137 , J. L. La Rosa Navarro 24d , L. La Rotonda 37a,37b , C. Lacasta 167 , F. Lacava 132a,132b , J. Lacey 29 , H. Lacker 16 , D. Lacour 80 , V. R. Lacuesta 167 , E. Ladygin 65 , R. Lafaye 5 , B. Laforge 80 , T. Lagouri 176 , S. Lai 54 , L. Lambourne 78 , S. Lammers 61 , C. L. Lampen 7 , W. Lampl 7 , E. Lançon 136 , U. Landgraf 48 , M. P. J. Landon 76 , V. S. Lang 58a , J. C. Lange 12 , A. J. Lankford 163 , F. Lanni 25 , K. Lantzsch 30 , A. Lanza 121a , S. Laplace 80 , C. Lapoire 30 , J. F. Laporte 136 , T. Lari 91a , F. Lasagni Manghi 20a,20b , M. Lassnig 30 , P. Laurelli 47 , W. Lavrijsen 15 , A. T. Law 137 , P. Laycock 74 , T. Lazovich 57 , O. Le Dortz 80 , E. Le Guirriec 85 , E. Le Menedeu 12 , M. LeBlanc 169 , T. LeCompte 6 , F. Ledroit-Guillon 55 , C. A. Lee 145b , S. C. Lee 151 , L. Lee 1 , G. Lefebvre 80 , M. Lefebvre 169 , F. Legger 100 , C. Leggett 15 , A. Lehan 74 , G. Lehmann Miotto 30 , X. Lei 7 , W. A. Leight 29 , A. Leisos 154,v , A. G. Leister 176 , M. A. L. Leite 24d , R. Leitner 129 , D. Lellouch 172 , B. Lemmer 54 , K. J. C. Leney 78 , T. Lenz 21 , B. Lenzi 30 , R. Leone 7 , S. Leone 124a,124b , C. Leonidopoulos 46 , S. Leontsinis 10 , C. Leroy 95 , C. G. Lester 28 , M. Levchenko 123 , J. Levêque 5 , D. Levin 89 , L. J. Levinson 172 , M. Levy 18 , A. Lewis 120 , A. M. Leyko 21 , M. Leyton 41 , B. Li 33b,w , H. Li 148 , H. L. Li 31 , L. Li 45 , L. Li 33e , S. Li 45 , Y. Li 33c,x , Z. Liang 137 , H. Liao 34 , B. Liberti 133a , A. Liblong 158 , P. Lichard 30 , K. Lie 165 , J. Liebal 21 , W. Liebig 14 , C. Limbach 21 , A. Limosani 150 , S. C. Lin 151,y , T. H. Lin 83 , F. Linde 107 , B. E. Lindquist 148 , J. T. Linnemann 90 , E. Lipeles 122 , A. Lipniacka 14 , M. Lisovyi 58b , T. M. Liss 165 , D. Lissauer 25 , A. Lister 168 , A. M. Litke 137 , B. Liu 151,z , D. Liu 151 , H. Liu 89 , J. Liu 85 , J. B. Liu 33b , K. Liu 85 , L. Liu 165 , M. Liu 45 , M. Liu 33b , Y. Liu 33b , M. Livan 121a,121b , A. Lleres 55 , J. Llorente Merino 82 , S. L. Lloyd 76 , F. Lo Sterzo 151 , E. Lobodzinska 42 , P. Loch 7 , W. S. Lockman 137 , F. K. Loebinger 84 , A. E. Loevschall-Jensen 36 , A. Loginov 176 , T. Lohse 16 , K. Lohwasser 42 , M. Lokajicek 127 , B. A. Long 22 , J. D. Long 89 , R. E. Long 72 , K. A. Looper 111 , L. Lopes 126a , D. Lopez Mateos 57 , B. Lopez Paredes 139 , I. Lopez Paz 12 , J. Lorenz 100 , N. Lorenzo Martinez 61 , M. Losada 162 , P. Loscutoff 15 , P. J. Lösel 100 , X. Lou 33a , A. Lounis 117 , J. Love 6 , P. A. Love 72 , N. Lu 89 , H. J. Lubatti 138 , C. Luci 132a,132b , A. Lucotte 55 , F. Luehring 61 , W. Lukas 62 , L. Luminari 132a , O. Lundberg 146a,146b , B. Lund-Jensen 147 , D. Lynn 25 , R. Lysak 127 , E. Lytken 81 , H. Ma 25 , L. L. Ma 33d , G. Maccarrone 47 , A. Macchiolo 101 , C. M. Macdonald 139 , B. Maček 75 , J. Machado Miguens 122,126b , D. Macina 30 , D. Madaffari 85 , R. Madar 34 , H. J. Maddocks 72 , W. F. Mader 44 , A. Madsen 166 , J. Maeda 67 , S. Maeland 14 , T. Maeno 25 , A. Maevskiy 99 , E. Magradze 54 , K. Mahboubi 48 , J. Mahlstedt 107 , C. Maiani 136 , C. Maidantchik 24a , A. A. Maier 101 , T. Maier 100 , A. Maio 126a,126b,126d , S. Majewski 116 , Y. Makida 66 , N. Makovec 117 , B. Malaescu 80 , Pa. Malecki 39 , V. P. Maleev 123 , F. Malek 55 , U. Mallik 63 , D. Malon 6 , C. Malone 143 , S. Maltezos 10 , V. M. Malyshev 109 , S. Malyukov 30 , J. Mamuzic 42 , G. Mancini 47 , B. Mandelli 30 , L. Mandelli 91a , I. Mandić 75 , R. Mandrysch 63 , J. Maneira 126a,126b , A. Manfredini 101 , L. Manhaes de Andrade Filho 24b , J. Manjarres Ramos 159b , A. Mann 100 , A. Manousakis-Katsikakis 9 , B. Mansoulie 136 , R. Mantifel 87 , M. Mantoani 54 , L. Mapelli 30 , L. March 145c , G. Marchiori 80 , M. Marcisovsky 127 , C. P. Marino 169 , M. Marjanovic 13 , D. E. Marley 89 , F. Marroquim 24a , S. P. Marsden 84 , Z. Marshall 15 , L. F. Marti 17 , S. Marti-Garcia 167 , B. Martin 90 , T. A. Martin 170 , V. J. Martin 46 , B. Martin dit Latour 14 , M. Martinez 12,o , S. Martin-Haugh 131 , V. S. Martoiu 26a , A. C. Martyniuk 78 , M. Marx 138 , F. Marzano 132a , A. Marzin 30 , L. Masetti 83 , T. Mashimo 155 , R. Mashinistov 96 , J. Masik 84 , A. L. Maslennikov 109,c , I. Massa 20a,20b , L. Massa 20a,20b , N. Massol 5 , P. Mastrandrea 148 , A. Mastroberardino 37a,37b , T. Masubuchi 155 , P. Mättig 175 , J. Mattmann 83 , J. Maurer 26a , S. J. Maxfield 74 , D. A. Maximov 109,c , R. Mazini 151 , S. M. Mazza 91a,91b , L. Mazzaferro 133a,133b , G. Mc Goldrick 158 , S. P. Mc Kee 89 , A. McCarn 89 , R. L. McCarthy 148 , T. G. McCarthy 29 , N. A. McCubbin 131 , K. W. McFarlane 56,* , J. A. Mcfayden 78 , G. Mchedlidze 54 , S. J. McMahon 131 , R. A. McPherson 169,k , M. Medinnis 42 , S. Meehan 145a , S. Mehlhase 100 , A. Mehta 74 , K. Meier 58a , C. Meineck 100 , B. Meirose 41 , B. R. Mellado Garcia 145c , F. Meloni 17 , A. Mengarelli 20a,20b , S. Menke 101 , E. Meoni 161 , K. M. Mercurio 57 , S. Mergelmeyer 21 , P. Mermod 49 , L. Merola 104a,104b , C. Meroni 91a , F. S. Merritt 31 , A. Messina 132a,132b , J. Metcalfe 25 , A. S. Mete 163 , C. Meyer 83 , C. Meyer 122 , J-P. Meyer 136 , J. Meyer 107 , H. Meyer Zu Theenhausen 58a , R. P. Middleton 131 , S. Miglior anzi 164a,164c , L. Mijović 21 , G. Mikenberg 172 , M. Mikestikova 127 , M. Mikuž 75 , M. Milesi 88 , A. Milic 30 , D. W. Miller 31 , C. Mills 46 , A. Milov 172 , D. A. Milstead 146a,146b , A. A. Minaenko 130 , Y. Minami 155 , I. A. Minashvili 65 , A. I. Mincer 110 , B. Mindur 38a , M. Mineev 65 , Y. Ming 173 , L. M. Mir 12 , T. Mitani 171 , J. Mitrevski 100 , V. A. Mitsou 167 , A. Miucci 49 , P. S. Miyagawa 139 , J. U. Mjörnmark 81 , T. Moa 146a,146b , K. Mochizuki 85 , S. Mohapatra 35 , W. Mohr 48 , S. Molander 146a,146b , R. Moles-Valls 21 , K. Mönig 42 , C. Monini 55 , J. Monk 36 , E. Monnier 85 , J. Montejo Berlingen 12 , F. Monticelli 71 , S. Monzani 132a,132b , R. W. Moore 3 , N. Morange 117 , D. Moreno 162 , M. Moreno Llácer 54 , P. Morettini 50a , M. Morgenstern 44 , D. Mori 142 , M. Morii 57 , M. Morinaga 155 , V. Morisbak 119 , S. Moritz 83 , A. K. Morley 150 , G. Mornacchi 30 , J. D. Morris 76 , S. S. Mortensen 36 , A. Morton 53 , L. Morvaj 103 , M. Mosidze 51b , J. Moss 111 , K. Motohashi 157 , R. Mount 143 , E. Mountricha 25 , S. V. Mouraviev 96,* , E. J. W. Moyse 86 , S. Muanza 85 , R. D. Mudd 18 , F. Mueller 101 , J. Mueller 125 , R. S. P. Mueller 100 , T. Mueller 28 , D. Muenstermann 49 , P. Mullen 53 , G. A. Mullier 17 , J. A. Murillo Quijada 18 , W. J. Murray 170,131 , H. Musheghyan 54 , E. Musto 152 , A. G. Myagkov 130,aa , M. Myska 128 , B. P. Nachman 143 , O. Nackenhorst 54 , J. Nadal 54 , K. Nagai 120 , R. Nagai 157 , Y. Nagai 85 , K. Nagano 66 , A. Nagarkar 111 , Y. Nagasaka 59 , K. Nagata 160 , M. Nagel 101 , E. Nagy 85 , A. M. Nairz 30 , Y. Nakahama 30 , K. Nakamura 66 , T. Nakamura 155 , I. Nakano 112 , H. Namasivayam 41 , R. F. Naranjo Garcia 42 , R. Narayan 31 , T. Naumann 42 , G. Navarro 162 , R. Nayyar 7 , H. A. Neal 89 , P. Yu. Nechaeva 96 , T. J. Neep 84 , P. D. Nef 143 , A. Negri 121a,121b , M. Negrini 20a , S. Nektarijevic 106 , C. Nellist 117 , A. Nelson 163 , S. Nemecek 127 , P. Nemethy 110 , A. A. Nepomuceno 24a , M. Nessi 30,ab , M. S. Neubauer 165 , M. Neumann 175 , R. M. Neves 110 , P. Nevski 25 , P. R. Newman 18 , D. H. Nguyen 6 , R. B. Nickerson 120 , R. Nicolaidou 136 , B. Nicquevert 30 , J. Nielsen 137 , N. Nikiforou 35 , A. Nikiforov 16 , V. Nikolaenko 130,aa , I. Nikolic-Audit 80 , K. Nikolopoulos 18 , J. K. Nilsen 119 , P. Nilsson 25 , Y. Ninomiya 155 , A. Nisati 132a , R. Nisius 101 , T. Nobe 155 , M. Nomachi 118 , I. Nomidis 29 , T. Nooney 76 , S. Norberg 113 , M. Nordberg 30 , O. Novgorodova 44 , S. Nowak 101 , M. Nozaki 66 , L. Nozka 115 , K. Ntekas 10 , G. Nunes Hanninger 88 , T. Nunnemann 100 , E. Nurse 78 , F. Nuti 88 , B. J. O'Brien 46 , F. O'grady 7 , D. C. O'Neil 142 , V. O'Shea 53 , F. G. Oakham 29,d , H. Oberlack 101 , T. Obermann 21 , J. Ocariz 80 , A. Ochi 67 , I. Ochoa 78 , J. P. Ochoa-Ricoux 32a , S. Oda 70 , S. Odaka 66 , H. Ogren 61 , A. Oh 84 , S. H. Oh 45 , C. C. Ohm 15 , H. Ohman 166 , H. Oide 30 , W. Okamura 118 , H. Okawa 160 , Y. Okumura 31 , T. Okuyama 66 , A. Olariu 26a , S. A. Olivares Pino 46 , D. Oliveira Damazio 25 , E. Oliver Garcia 167 , A. Olszewski 39 , J. Olszowska 39 , A. Onofre 126a,126e , P. U. E. Onyisi 31,q , C. J. Oram 159a , M. J. Oreglia 31 , Y. Oren 153 , D. Orestano 134a,134b , N. Orlando 154 , C. Oropeza Barrera 53 , R. S. Orr 158 , B. Osculati 50a,50b , R. Ospanov 84 , G. Otero y Garzon 27 , H. Otono 70 , M. Ouchrif 135d , E. A. Ouellette 169 , F. Ould-Saada 119 , A. Ouraou 136 , K. P. Oussoren 107 , Q. Ouyang 33a , A. Ovcharova 15 , M. Owen 53 , R. E. Owen 18 , V. E. Ozcan 19a , N. Ozturk 8 , K. Pachal 142 , A. Pacheco Pages 12 , C. Padilla Aranda 12 , M. Pagáčová 48 , S. Pagan Griso 15 , E. Paganis 139 , F. Paige 25 , P. Pais 86 , K. Pajchel 119 , G. Palacino 159b , S. Palestini 30 , M. Palka 38b , D. Pallin 34 , A. Palma 126a,126b , Y. B. Pan 173 , E. Panagiotopoulou 10 , C. E. Pandini 80 , J. G. Panduro Vazquez 77 , P. Pani 146a,146b , S. Panitkin 25 , D. Pantea 26a , L. Paolozzi 49 , Th. D. Papadopoulou 10 , K. Papageorgiou 154 , A. Paramonov 6 , D. Paredes Hernandez 154 , M. A. Parker 28 , K. A. Parker 139 , F. Parodi 50a,50b , J. A. Parsons 35 , U. Parzefall 48 , E. Pasqualucci 132a , S. Passaggio 50a , F. Pastore 134a,134b,* , Fr. Pastore 77 , G. Pásztor 29 , S. Pataraia 175 , N. D. Patel 150 , J. R. Pater 84 , T. Pauly 30 , J. Pearce 169 , B. Pearson 113 , L. E. Pedersen 36 , M. Pedersen 119 , S. Pedraza Lopez 167 , R. Pedro 126a,126b , S. V. Peleganchuk 109,c , D. Pelikan 166 , O. Penc 127 , C. Peng 33a , H. Peng 33b , B. Penning 31 , J. Penwell 61 , D. V. Perepelitsa 25 , E. Perez Codina 159a , M. T. Pérez García-Estañ 167 , L. Perini 91a,91b , H. Pernegger 30 , S. Perrella 104a,104b , R. Peschke 42 , V. D. Peshekhonov 65 , K. Peters 30 , R. F. Y. Peters 84 , B. A. Petersen 30 , T. C. Petersen 36 , E. Petit 42 , A. Petridis 146a,146b , C. Petridou 154 , P. Petroff 117 , E. Petrolo 132a , F. Petrucci 134a,134b , N. E. Pettersson 157 , R. Pezoa 32b , P. W. Phillips 131 , G. Piacquadio 143 , E. Pianori 170 , A. Picazio 49 , E. Piccaro 76 , M. Piccinini 20a,20b , M. A. Pickering 120 , R. Piegaia 27 , D. T. Pignotti 111 , J. E. Pilcher 31 , A. D. Pilkington 84 , J. Pina 126a,126b,126d , M. Pinamonti 164a,164c,ac , J. L. Pinfold 3 , A. Pingel 36 , B. Pinto 126a , S. Pires 80 , H. Pirumov 42 , M. Pitt 172 , C. Pizio 91a,91b , L. Plazak 144a , M.-A. Pleier 25 , V. Pleskot 129 , E. Plotnikova 65 , P. Plucinski 146a,146b , D. Pluth 64 , R. Poettgen 146a,146b , L. Poggioli 117 , D. Pohl 21 , G. Polesello 121a , A. Poley 42 , A. Policicchio 37a,37b , R. Polifka 158 , A. Polini 20a , C. S. Pollard 53 , V. Polychronakos 25 , K. Pommès 30 , L. Pontecorvo 132a , B. G. Pope 90 , G. A. Popeneciu 26b , D. S. Popovic 13 , A. Poppleton 30 , S. Pospisil 128 , K. Potamianos 15 , I. N. Potrap 65 , C. J. Potter 149 , C. T. Potter 116 , G. Poulard 30 , J. Poveda 30 , V. Pozdnyakov 65 , P. Pralavorio 85 , A. Pranko 15 , S. Prasad 30 , S. Prell 64 , D. Price 84 , L. E. Price 6 , M. Primavera 73a , S. Prince 87 , M. Proissl 46 , K. Prokofiev 60c , F. Prokoshin 32b , E. Protopapadaki 136 , S. Protopopescu 25 , J. Proudfoot 6 , M. Przybycien 38a , E. Ptacek 116 , D. Puddu 134a,134b , E. Pueschel 86 , D. Puldon 148 , M. Purohit 25,ad , P. Puzo 117 , J. Qian 89 , G. Qin 53 , Y. Qin 84 , A. Quadt 54 , D. R. Quarrie 15 , W. B. Quayle 164a,164b , M. Queitsch-Maitland 84 , D. Quilty 53 , S. Raddum 119 , V. Radeka 25 , V. Radescu 42 , S. K. Radhakrishnan 148 , P. Radloff 116 , P. Rados 88 , F. Ragusa 91a,91b , G. Rahal 178 , S. Rajagopalan 25 , M. Rammensee 30 , C. Rangel-Smith 166 , F. Rauscher 100 , S. Rave 83 , T. Ravenscroft 53 , M. Raymond 30 , A. L. Read 119 , N. P. Readioff 74 , D. M. Rebuzzi 121a,121b , A. Redelbach 174 , G. Redlinger 25 , R. Reece 137 , K. Reeves 41 , L. Rehnisch 16 , H. Reisin 27 , M. Relich 163 , C. Rembser 30 , H. Ren 33a , A. Renaud 117 , M. Rescigno 132a , S. Resconi 91a , O. L. Rezanova 109,c , P. Reznicek 129 , R. Rezvani 95 , R. Richter 101 , S. Richter 78 , E. Richter-Was 38b , O. Ricken 21 , M. Ridel 80 , P. Rieck 16 , C. J. Riegel 175 , J. Rieger 54 , M. Rijssenbeek 148 , A. Rimoldi 121a,121b , L. Rinaldi 20a , B. Ristić 49 , E. Ritsch 30 , I. Riu 12 , F. Rizatdinova 114 , E. Rizvi 76 , S. H. Robertson 87,k , A. Robichaud-Veronneau 87 , D. Robinson 28 , J. E. M. Robinson 42 , A. Robson 53 , C. Roda 124a,124b , S. Roe 30 , O. Røhne 119 , S. Rolli 161 , A. Romaniouk 98 , M. Romano 20a,20b , S. M. Romano Saez 34 , E. Romero Adam 167 , N. Rompotis 138 , M. Ronzani 48 , L. Roos 80 , E. Ros 167 , S. Rosati 132a , K. Rosbach 48 , P. Rose 137 , P. L. Rosendahl 14 , O. Rosenthal 141 , V. Rossetti 146a,146b , E. Rossi 104a,104b , L. P. Rossi 50a , R. Rosten 138 , M. Rotaru 26a , I. Roth 172 , J. Rothberg 138 , D. Rousseau 117 , C. R. Royon 136 , A. Rozanov 85 , Y. Rozen 152 , X. Ruan 145c , F. Rubbo 143 , I. Rubinskiy 42 , V. I. Rud 99 , C. Rudolph 44 , M. S. Rudolph 158 , F. Rühr 48 , A. Ruiz-Martinez 30 , Z. Rurikova 48 , N. A. Rusakovich 65 , A. Ruschke 100 , H. L. Russell 138 , J. P. Rutherfoord 7 , N. Ruthmann 48 , Y. F. Ryabov 123 , M. Rybar 165 , G. Rybkin 117 , N. C. Ryder 120 , A. F. Saavedra 150 , G. Sabato 107 , S. Sacerdoti 27 , A. Saddique 3 , H. F-W. Sadrozinski 137 , R. Sadykov 65 , F. Safai Tehrani 132a , M. Sahinsoy 19a , M. Saimpert 136 , T. Saito 155 , H. Sakamoto 155 , Y. Sakurai 171 , G. Salamanna 134a,134b , A. Salamon 133a , M. Saleem 113 , D. Salek 107 , P. H. Sales De Bruin 138 , D. Salihagic 101 , A. Salnikov 143 , J. Salt 167 , D. Salvatore 37a,37b , F. Salvatore 149 , A. Salvucci 60a , A. Salzburger 30 , D. Sammel 48 , D. Sampsonidis 154 , A. Sanchez 104a,104b , J. Sánchez 167 , V. Sanchez Martinez 167 , H. Sandaker 14 , R. L. Sandbach 76 , H. G. Sander 83 , M. P. Sanders 100 , M. Sandhoff 175 , C. Sandoval 162 , R. Sandstroem 101 , D. P. C. Sankey 131 , M. Sannino 50a,50b , A. Sansoni 47 , C. Santoni 34 , R. Santonico 133a,133b , H. Santos 126a , I. Santoyo Castillo 149 , K. Sapp 125 , A. Sapronov 65 , J. G. Saraiva 126a,126d , B. Sarrazin 21 , O. Sasaki 66 , Y. Sasaki 155 , K. Sato 160 , G. Sauvage 5,* , E. Sauvan 5 , G. Savage 77 , P. Savard 158,d , C. Sawyer 131 , L. Sawyer 79,n , J. Saxon 31 , C. Sbarra 20a , A. Sbrizzi 20a,20b , T. Scanlon 78 , D. A. Scannicchio 163 , M. Scarcella 150 , V. Scarfone 37a,37b , J. Schaarschmidt 172 , P. Schacht 101 , D. Schaefer 30 , R. Schaefer 42 , J. Schaeffer 83 , S. Schaepe 21 , S. Schaetzel 58b , U. Schäfer 83 , A. C. Schaffer 117 , D. Schaile 100 , R. D. Schamberger 148 , V. Scharf 58a , V. A. Schegelsky 123 , D. Scheirich 129 , M. Schernau 163 , C. Schiavi 50a,50b , C. Schillo 48 , M. Schioppa 37a,37b , S. Schlenker 30 , E. Schmidt 48 , K. Schmieden 30 , C. Schmitt 83 , S. Schmitt 58b , S. Schmitt 42 , B. Schneider 159a , Y. J. Schnellbach 74 , U. Schnoor 44 , L. Schoeffel 136 , A. Schoening 58b , B. D. Schoenrock 90 , E. Schopf 21 , A. L. S. Schorlemmer 54 , M. Schott 83 , D. Schouten 159a , J. Schovancova 8 , S. Schramm 49 , M. Schreyer 174 , C. Schroeder 83 , N. Schuh 83 , M. J. Schultens 21 , H.-C. Schultz-Coulon 58a , H. Schulz 16 , M. Schumacher 48 , B. A. Schumm 137 , Ph. Schune 136 , C. Schwanenberger 84 , A. Schwartzman 143 , T. A. Schwarz 89 , Ph. Schwegler 101 , H. Schweiger 84 , Ph. Schwemling 136 , R. Schwienhorst 90 , J. Schwindling 136 , T. Schwindt 21 , F. G. Sciacca 17 , E. Scifo 117 , G. Sciolla 23 , F. Scuri 124a,124b , F. Scutti 21 , J. Searcy 89 , G. Sedov 42 , E. Sedykh 123 , P. Seema 21 , S. C. Seidel 105 , A. Seiden 137 , F. Seifert 128 , J. M. Seixas 24a , G. Sekhniaidze 104a , K. Sekhon 89 , S. J. Sekula 40 , D. M. Seliverstov 123,* , N. Semprini-Cesari 20a,20b , C. Serfon 30 , L. Serin 117 , L. Serkin 164a,164b , T. Serre 85 , M. Sessa 134a,134b , R. Seuster 159a , H. Severini 113 , T. Sfiligoj 75 , F. Sforza 30 , A. Sfyrla 30 , E. Shabalina 54 , M. Shamim 116 , L. Y. Shan 33a , R. Shang 165 , J. T. Shank 22 , M. Shapiro 15 , P. B. Shatalov 97 , K. Shaw 164a,164b , S. M. Shaw 84 , A. Shcherbakova 146a,146b , C. Y. Shehu 149 , P. Sherwood 78 , L. Shi 151,ae , S. Shimizu 67 , C. O. Shimmin 163 , M. Shimojima 102 , M. Shiyakova 65 , A. Shmeleva 96 , D. Shoaleh Saadi 95 , M. J. Shochet 31 , S. Shojaii 91a,91b , S. Shrestha 111 , E. Shulga 98 , M. A. Shupe 7 , S. Shushkevich 42 , P. Sicho 127 , P. E. Sidebo 147 , O. Sidiropoulou 174 , D. Sidorov 114 , A. Sidoti 20a,20b , F. Siegert 44 , Dj. Sijacki 13 , J. Silva 126a,126d , Y. Silver 153 , S. B. Silverstein 146a , V. Simak 128 , O. Simard 5 , Lj. Simic 13 , S. Simion 117 , E. Simioni 83 , B. Simmons 78 , D. Simon 34 , R. Simoniello 91a,91b , P. Sinervo 158 , N. B. Sinev 116 , G. Siragusa 174 , A. N. Sisakyan 65,* , S. Yu. Sivoklokov 99 , J. Sjölin 146a,146b , T. B. Sjursen 14 , M. B. Skinner 72 , H. P. Skottowe 57 , P. Skubic 113 , M. Slater 18 , T. Slavicek 128 , M. Slawinska 107 , K. Sliwa 161 , V. Smakhtin 172 , B. H. Smart 46 , L. Smestad 14 , S. Yu. Smirnov 98 , Y. Smirnov 98 , L. N. Smirnova 99,af , O. Smirnova 81 , M. N. K. Smith 35 , R. W. Smith 35 , M. Smizanska 72 , K. Smolek 128 , A. A. Snesarev 96 , G. Snidero 76 , S. Snyder 25 , R. Sobie 169,k , F. Socher 44 , A. Soffer 153 , D. A. Soh 151,ae , C. A. Solans 30 , M. Solar 128 , J. Solc 128 , E. Yu. Soldatov 98 , U. Soldevila 167 , A. A. Solodkov 130 , A. Soloshenko 65 , O. V. Solovyanov 130 , V. Solovyev 123 , P. Sommer 48 , H. Y. Song 33b , N. Soni 1 , A. Sood 15 , A. Sopczak 128 , B. Sopko 128 , V. Sopko 128 , V. Sorin 12 , D. Sosa 58b , M. Sosebee 8 , C. L. Sotiropoulou 124a,124b , R. Soualah 164a,164c , A. M. Soukharev 109,c , D. South 42 , B. C. Sowden 77 , S. Spagnolo 73a,73b , M. Spalla 124a,124b , F. Spanò 77 , W. R. Spearman 57 , D. Sperlich 16 , F. Spettel 101 , R. Spighi 20a , G. Spigo 30 , L. A. Spiller 88 , M. Spousta 129 , T. Spreitzer 158 , R. D. St. Denis 53,* , S. Staerz 44 , J. Stahlman 122 , R. Stamen 58a , S. Stamm 16 , E. Stanecka 39 , C. Stanescu 134a , M. Stanescu-Bellu 42 , M. M. Stanitzki 42 , S. Stapnes 119 , E. A. Starchenko 130 , J. Stark 55 , P. Staroba 127 , P. Starovoitov 58a , R. Staszewski 39 , P. Stavina 144a,* , P. Steinberg 25 , B. Stelzer 142 , H. J. Stelzer 30 , O. Stelzer-Chilton 159a , H. Stenzel 52 , G. A. Stewart 53 , J. A. Stillings 21 , M. C. Stockton 87 , M. Stoebe 87 , G. Stoicea 26a , P. Stolte 54 , S. Stonjek 101 , A. R. Stradling 8 , A. Straessner 44 , M. E. Stramaglia 17 , J. Strandberg 147 , S. Strandberg 146a,146b , A. Strandlie 119 , E. Strauss 143 , M. Strauss 113 , P. Strizenec 144b , R. Ströhmer 174 , D. M. Strom 116 , R. Stroynowski 40 , A. Strubig 106 , S. A. Stucci 17 , B. Stugu 14 , N. A. Styles 42 , D. Su 143 , J. Su 125 , R. Subramaniam 79 , A. Succurro 12 , Y. Sugaya 118 , C. Suhr 108 , M. Suk 128 , V. V. Sulin 96 , S. Sultansoy 4c , T. Sumida 68 , S. Sun 57 , X. Sun 33a , J. E. Sundermann 48 , K. Suruliz 149 , G. Susinno 37a,37b , M. R. Sutton 149 , S. Suzuki 66 , M. Svatos 127 , M. Swiatlowski 143 , I. Sykora 144a , T. Sykora 129 , D. Ta 90 , C. Taccini 134a,134b , K. Tackmann 42 , J. Taenzer 158 , A. Taffard 163 , R. Tafirout 159a , N. Taiblum 153 , H. Takai 25 , R. Takashima 69 , H. Takeda 67 , T. Takeshita 140 , Y. Takubo 66 , M. Talby 85 , A. A. Talyshev 109,c , J. Y. C. Tam 174 , K. G. Tan 88 , J. Tanaka 155 , R. Tanaka 117 , S. Tanaka 66 , B. B. Tannenwald 111 , N. Tannoury 21 , S. Tapprogge 83 , S. Tarem 152 , F. Tarrade 29 , G. F. Tartarelli 91a , P. Tas 129 , M. Tasevsky 127 , T. Tashiro 68 , E. Tassi 37a,37b , A. Tavares Delgado 126a,126b , Y. Tayalati 135d , F. E. Taylor 94 , G. N. Taylor 88 , W. Taylor 159b , F. A. Teischinger 30 , M. Teixeira Dias Castanheira 76 , P. Teixeira-Dias 77 , K. K. Temming 48 , H. Ten Kate 30 , P. K. Teng 151 , J. J. Teoh 118 , F. Tepel 175 , S. Terada 66 , K. Terashi 155 , J. Terron 82 , S. Terzo 101 , M. Testa 47 , R. J. Teuscher 158,k , T. Theveneaux-Pelzer 34 , J. P. Thomas 18 , J. Thomas-Wilsker 77 , E. N. Thompson 35 , P. D. Thompson 18 , R. J. Thompson 84 , A. S. Thompson 53 , L. A. Thomsen 176 , E. Thomson 122 , M. Thomson 28 , R. P. Thun 89,* , M. J. Tibbetts 15 , R. E. Ticse Torres 85 , V. O. Tikhomirov 96,ag , Yu. A. Tikhonov 109,c , S. Timoshenko 98 , E. Tiouchichine 85 , P. Tipton 176 , S. Tisserant 85 , K. Todome 157 , T. Todorov 5,* , S. Todorova-Nova 129 , J. Tojo 70 , S. Tokár 144a , K. Tokushuku 66 , K. Tollefson 90 , E. Tolley 57 , L. Tomlinson 84 , M. Tomoto 103 , L. Tompkins 143,ah , K. Toms 105 , E. Torrence 116 , H. Torres 142 , E. Torró Pastor 167 , J. Toth 85,ai , F. Touchard 85 , D. R. Tovey 139 , T. Trefzger 174 , L. Tremblet 30 , A. Tricoli 30 , I. M. Trigger 159a , S. Trincaz-Duvoid 80 , M. F. Tripiana 12 , W. Trischuk 158 , B. Trocmé 55 , C. Troncon 91a , M. Trottier-McDonald 15 , M. Trovatelli 169 , P. True 90 , L. Truong 164a,164c , M. Trzebinski 39 , A. Trzupek 39 , C. Tsarouchas 30 , J. C-L. Tseng 120 , P. V. Tsiareshka 92 , D. Tsionou 154 , G. Tsipolitis 10 , N. Tsirintanis 9 , S. Tsiskaridze 12 , V. Tsiskaridze 48 , E. G. Tskhadadze 51a , I. I. Tsukerman 97 , V. Tsulaia 15 , S. Tsuno 66 , D. Tsybychev 148 , A. Tudorache 26a , V. Tudorache 26a , A. N. Tuna 122 , S. A. Tupputi 20a,20b , S. Turchikhin 99,af , D. Turecek 128 , R. Turra 91a,91b , A. J. Turvey 40 , P. M. Tuts 35 , A. Tykhonov 49 , M. Tylmad 146a,146b , M. Tyndel 131 , I. Ueda 155 , R. Ueno 29 , M. Ughetto 146a,146b , M. Ugland 14 , F. Ukegawa 160 , G. Unal 30 , A. Undrus 25 , G. Unel 163 , F. C. Ungaro 48 , Y. Unno 66 , C. Unverdorben 100 , J. Urban 144b , P. Urquijo 88 , P. Urrejola 83 , G. Usai 8 , A. Usanova 62 , L. Vacavant 85 , V. Vacek 128 , B. Vachon 87 , C. Valderanis 83 , N. Valencic 107 , S. Valentinetti 20a,20b , A. Valero 167 , L. Valery 12 , S. Valkar 129 , E. Valladolid Gallego 167 , S. Vallecorsa 49 , J. A. Valls Ferrer 167 , W. Van Den Wollenberg 107 , P. C. Van Der Deijl 107 , R. van der Geer 107 , H. van der Graaf 107 , R. Van Der Leeuw 107 , N. van Eldik 152 , P. van Gemmeren 6 , J. Van Nieuwkoop 142 , I. van Vulpen 107 , M. C. van Woerden 30 , M. Vanadia 132a,132b , W. Vandelli 30 , R. Vanguri 122 , A. Vaniachine 6 , F. Vannucci 80 , G. Vardanyan 177 , R. Vari 132a , E. W. Varnes 7 , T. Varol 40 , D. Varouchas 80 , A. Vartapetian 8 , K. E. Varvell 150 , F. Vazeille 34 , T. Vazquez Schroeder 87 , J. Veatch 7 , L. M. Veloce 158 , F. Veloso 126a,126c , T. Velz 21 , S. Veneziano 132a , A. Ventura 73a,73b , D. Ventura 86 , M. Venturi 169 , N. Venturi 158 , A. Venturini 23 , V. Vercesi 121a , M. Verducci 132a,132b , W. Verkerke 107 , J. C. Vermeulen 107 , A. Vest 44 , M. C. Vetterli 142,d , O. Viazlo 81 , I. Vichou 165 , T. Vickey 139 , O. E. Vickey Boeriu 139 , G. H. A. Viehhauser 120 , S. Viel 15 , R. Vigne 62 , M. Villa 20a,20b , M. Villaplana Perez 91a,91b , E. Vilucchi 47 , M. G. Vincter 29 , V. B. Vinogradov 65 , I. Vivarelli 149 , F. Vives Vaque 3 , S. Vlachos 10 , D. Vladoiu 100 , M. Vlasak 128 , M. Vogel 32a , P. Vokac 128 , G. Volpi 124a,124b , M. Volpi 88 , H. von der Schmitt 101 , H. von Radziewski 48 , E. von Toerne 21 , V. Vorobel 129 , K. Vorobev 98 , M. Vos 167 , R. Voss 30 , J. H. Vossebeld 74 , N. Vranjes 13 , M. Vranjes Milosavljevic 13 , V. Vrba 127 , M. Vreeswijk 107 , R. Vuillermet 30 , I. Vukotic 31 , Z. Vykydal 128 , P. Wagner 21 , W. Wagner 175 , H. Wahlberg 71 , S. Wahrmund 44 , J. Wakabayashi 103 , J. Walder 72 , R. Walker 100 , W. Walkowiak 141 , C. Wang 33c , F. Wang 173 , H. 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Zaidan 63 , A. M. Zaitsev 130,aa , J. Zalieckas 14 , A. Zaman 148 , S. Zambito 57 , L. Zanello 132a,132b , D. Zanzi 88 , C. Zeitnitz 175 , M. Zeman 128 , A. Zemla 38a , K. Zengel 23 , O. Zenin 130 , T. Ženiš 144a , D. Zerwas 117 , D. Zhang 89 , F. Zhang 173 , H. Zhang 33c , J. Zhang 6 , L. Zhang 48 , R. Zhang 33b , X. Zhang 33d , Z. Zhang 117 , X. Zhao 40 , Y. Zhao 33d,117 , Z. Zhao 33b , A. Zhemchugov 65 , J. Zhong 120 , B. Zhou 89 , C. Zhou 45 , L. Zhou 35 , L. Zhou 40 , N. Zhou 33f , C. G. Zhu 33d , H. Zhu 33a , J. Zhu 89 , Y. Zhu 33b , X. Zhuang 33a , K. Zhukov 96 , A. Zibell 174 , D. Zieminska 61 , N. I. Zimine 65 , C. Zimmermann 83 , S. Zimmermann 48 , Z. Zinonos 54 , M. Zinser 83 , M. Ziolkowski 141 , L. Živković 13 , G. Zobernig 173 , A. Zoccoli 20a,20b , M. zur Nedden 16 , G. Zurzolo 104a,104b , L. Zwalinski 30 1 Department of Physics, University of Adelaide, Adelaide, Australia 2 Physics Department, SUNY Albany, Albany, NY, USA 3 Department of Physics, University of Alberta, Edmonton, AB, Canada 19 (a) Department of Physics, Bogazici University, Istanbul, Turkey; (b) Department of Physics, Dogus University, Istanbul, Turkey; (c) Department of Physics Engineering, Gaziantep University, Gaziantep, Turkey
Infn Sezione Di Bologna, Dipartimento di Fisica e Astronomia. Bologna, Italy; Bologna, Italy; Bonn, Germany; Boston, MA, USA; Waltham, MA, USAUniversità di Bologna ; 21 Physikalisches Institut, University of Bonn ; 22 Department of Physics, Boston University ; 23 Department of Physics, Brandeis UniversityINFN Sezione di Bologna, Bologna, Italy; (b) Dipartimento di Fisica e Astronomia, Università di Bologna, Bologna, Italy 21 Physikalisches Institut, University of Bonn, Bonn, Germany 22 Department of Physics, Boston University, Boston, MA, USA 23 Department of Physics, Brandeis University, Waltham, MA, USA
b) Physics Department, National Institute for Research and Development of Isotopic and Molecular Technologies. 26São Paulo; Upton, NY, USA; Bucharest, Romania; Cluj Napoca, Romania; Bucharest, Romania; Timisoara, RomaniaUniversidade Federal do Rio De Janeiro COPPE/EE/IF, Rio de Janeiro, Brazil; (b) Electrical Circuits Department, Federal University of Juiz de Fora (UFJF), Juiz de Fora, Brazil; (c) Federal University of Sao Joao del Rei (UFSJ) ; Universidade de Sao Paulo ; Brazil 25 Physics Department, Brookhaven National Laboratory ; National Institute of Physics and Nuclear Engineering ; c) University Politehnica Bucharest ; West University in TimisoaraSao Joao del Rei, Brazil; (d) Instituto de Fisica. dUniversidade Federal do Rio De Janeiro COPPE/EE/IF, Rio de Janeiro, Brazil; (b) Electrical Circuits Department, Federal University of Juiz de Fora (UFJF), Juiz de Fora, Brazil; (c) Federal University of Sao Joao del Rei (UFSJ), Sao Joao del Rei, Brazil; (d) Instituto de Fisica, Universidade de Sao Paulo, São Paulo, Brazil 25 Physics Department, Brookhaven National Laboratory, Upton, NY, USA 26 (a) National Institute of Physics and Nuclear Engineering, Bucharest, Romania; (b) Physics Department, National Institute for Research and Development of Isotopic and Molecular Technologies, Cluj Napoca, Romania; (c) University Politehnica Bucharest, Bucharest, Romania; (d) West University in Timisoara, Timisoara, Romania
. Física Departamento De, Canada. 30Universidad de Buenos Aires ; University of Cambridge ; 29 Department of Physics, Carleton University ; CERNDepartamento de Física, Universidad de Buenos Aires, Buenos Aires, Argentina 28 Cavendish Laboratory, University of Cambridge, Cambridge, UK 29 Department of Physics, Carleton University, Ottawa, ON, Canada 30 CERN, Geneva, Switzerland
. Física Departamento De, Chinese Academy of Sciences. 33Pontificia Universidad Católica de Chile ; b) Departamento de Física, Universidad Técnica Federico Santa María ; Institute of High Energy Physics ; b) Department of Modern Physics, University of Science and Technology of China ; c) Department of Physics, Nanjing University ; Shandong University ; Shanghai Key Laboratory for Particle Physics and Cosmology, Department of Physics and Astronomy, Shanghai Jiao Tong University ; f) Physics Department, Tsinghua UniversityPhysicsDepartamento de Física, Pontificia Universidad Católica de Chile, Santiago, Chile; (b) Departamento de Física, Universidad Técnica Federico Santa María, Valparaiso, Chile 33 (a) Institute of High Energy Physics, Chinese Academy of Sciences, Beijing, China; (b) Department of Modern Physics, University of Science and Technology of China, Hefei, Anhui, China; (c) Department of Physics, Nanjing University, Nanjing, Jiangsu, China; (d) School of Physics, Shandong University, Shandong, China; (e) Shanghai Key Laboratory for Particle Physics and Cosmology, Department of Physics and Astronomy, Shanghai Jiao Tong University, Shanghai, China; (f) Physics Department, Tsinghua University, Beijing 100084, China
b) Dipartimento di Fisica. Laboratori Infn Gruppo Collegato Di Cosenza, Nazionali Di Frascati, Frascati, Italy; Rende, ItalyUniversità della CalabriaINFN Gruppo Collegato di Cosenza, Laboratori Nazionali di Frascati, Frascati, Italy; (b) Dipartimento di Fisica, Università della Calabria, Rende, Italy
. Faculty of Physics and Applied Computer Science. AGH University of Science and TechnologyAGH University of Science and Technology, Faculty of Physics and Applied Computer Science, Kraków, Poland;
. Eur. Phys. J. C. 75407Marian Smoluchowski Institute of Physics, Jagiellonian University, Kraków, Poland 39 Institute of Nuclear Physics, Polish Academy of Sciences, Kraków, Poland 40 Physics Department, Southern Methodist UniversityMarian Smoluchowski Institute of Physics, Jagiellonian University, Kraków, Poland 39 Institute of Nuclear Physics, Polish Academy of Sciences, Kraków, Poland 40 Physics Department, Southern Methodist University, Dallas, TX, USA Eur. Phys. J. C (2015) 75 :407
Germany 49 Section de Physique. T X Richardson, Hamburg Desy, Zeuthen ; Infn Sezione Di Genova, 48 Fakultät für Mathematik und Physik. b) Dipartimento di Fisica, Università di GenovaDurham, NC, USA; Frascati, Italy; Freiburg; Geneva, Switzerland; Genoa, Italy; Genoa, Italy; Tbilisi, Georgia; Tbilisi, Georgia 52 II Physikalisches Institut; Giessen, Germany; Glasgow, UK 54 II Physikalisches Institut; Grenoble; Hampton, VA; Cambridge, MA, USA46Physics Department, University of Texas at Dallas ; Germany 43 Institut für Experimentelle Physik IV, Technische Universität Dortmund, Dortmund, Germany 44 Institut für Kern-und Teilchenphysik, Technische Universität Dresden, Dresden, Germany 45 Department of Physics, Duke University ; University of Edinburgh, Edinburgh, UK 47 INFN Laboratori Nazionali di Frascati ; Albert-Ludwigs-Universität ; Université de Genève ; Andronikashvili Institute of Physics, Iv. Javakhishvili Tbilisi State University ; High Energy Physics Institute, Tbilisi State University ; Justus-Liebig-Universität Giessen ; University of Glasgow ; Georg-August-Universität, Göttingen, Germany 55 Laboratoire de Physique Subatomique et de Cosmologie, Université Grenoble-Alpes ; CNRS/IN2P3 ; France 56 Department of Physics, Hampton University ; USA 57 Laboratory for Particle Physics and Cosmology, Harvard UniversityPhysics and AstronomyPhysics Department, University of Texas at Dallas, Richardson, TX, USA 42 DESY, Hamburg and Zeuthen, Germany 43 Institut für Experimentelle Physik IV, Technische Universität Dortmund, Dortmund, Germany 44 Institut für Kern-und Teilchenphysik, Technische Universität Dresden, Dresden, Germany 45 Department of Physics, Duke University, Durham, NC, USA 46 SUPA-School of Physics and Astronomy, University of Edinburgh, Edinburgh, UK 47 INFN Laboratori Nazionali di Frascati, Frascati, Italy 48 Fakultät für Mathematik und Physik, Albert-Ludwigs-Universität, Freiburg, Germany 49 Section de Physique, Université de Genève, Geneva, Switzerland 50 (a) INFN Sezione di Genova, Genoa, Italy; (b) Dipartimento di Fisica, Università di Genova, Genoa, Italy 51 (a) E. Andronikashvili Institute of Physics, Iv. Javakhishvili Tbilisi State University, Tbilisi, Georgia; (b) High Energy Physics Institute, Tbilisi State University, Tbilisi, Georgia 52 II Physikalisches Institut, Justus-Liebig-Universität Giessen, Giessen, Germany 53 SUPA-School of Physics and Astronomy, University of Glasgow, Glasgow, UK 54 II Physikalisches Institut, Georg-August-Universität, Göttingen, Germany 55 Laboratoire de Physique Subatomique et de Cosmologie, Université Grenoble-Alpes, CNRS/IN2P3, Grenoble, France 56 Department of Physics, Hampton University, Hampton, VA, USA 57 Laboratory for Particle Physics and Cosmology, Harvard University, Cambridge, MA, USA
Kirchhoff-Institut Für Physik, 59 Faculty of Applied Information Science. Heidelberg, Germany; Heidelberg, Germany; Mannheim, Germany; Hiroshima, Japan; Shatin, NT, Hong Kong; Pok Fu Lam, Hong Kong; Kowloon, Hong Kong, China60Ruprecht-Karls-Universität Heidelberg ; Ruprecht-Karls-Universität Heidelberg ; Ruprecht-Karls-Universität Heidelberg ; Hiroshima Institute of Technology ; Department of Physics, The Chinese University of Hong Kong ; Department of Physics, The University of Hong Kong ; c) Department of Physics, The Hong Kong University of Science and Technology, Clear Water BayZITI Institut für technische InformatikKirchhoff-Institut für Physik, Ruprecht-Karls-Universität Heidelberg, Heidelberg, Germany; (b) Physikalisches Institut, Ruprecht-Karls-Universität Heidelberg, Heidelberg, Germany; (c) ZITI Institut für technische Informatik, Ruprecht-Karls-Universität Heidelberg, Mannheim, Germany 59 Faculty of Applied Information Science, Hiroshima Institute of Technology, Hiroshima, Japan 60 (a) Department of Physics, The Chinese University of Hong Kong, Shatin, NT, Hong Kong; (b) Department of Physics, The University of Hong Kong, Pok Fu Lam, Hong Kong; (c) Department of Physics, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong, China
(b) Dipartimento di Fisica. Infn Sezione Di Roma Tor, Vergata, Rome, Italy; Rome, ItalyUniversità di Roma Tor VergataINFN Sezione di Roma Tor Vergata, Rome, Italy (b) Dipartimento di Fisica, Università di Roma Tor Vergata, Rome, Italy
b) Centre National de l'Energie des Sciences Techniques Nucleaires. Tre Infn Sezione Di Roma, ) Faculté des Sciences. b) Dipartimento di Matematica e Fisica, Università Roma TreRome, Italy; Rome, Italy; Casablanca, Morocco; Rabat, Morocco; Marrakech, Morocco; Oujda, Morocco; Rabat, MoroccoRéseau Universitaire de Physique des Hautes Energies-Université Hassan II ; Université Cadi Ayyad, LPHEA-Marrakech ; Université Mohammed V-Agdalc) Faculté des Sciences Semlalia135 (a) Faculté des Sciences Ain Chock. d. (e) Faculté des SciencesINFN Sezione di Roma Tre, Rome, Italy; (b) Dipartimento di Matematica e Fisica, Università Roma Tre, Rome, Italy 135 (a) Faculté des Sciences Ain Chock, Réseau Universitaire de Physique des Hautes Energies-Université Hassan II, Casablanca, Morocco; (b) Centre National de l'Energie des Sciences Techniques Nucleaires, Rabat, Morocco; (c) Faculté des Sciences Semlalia, Université Cadi Ayyad, LPHEA-Marrakech, Marrakech, Morocco; (d) Faculté des Sciences, Université Mohamed Premier and LPTPM, Oujda, Morocco; (e) Faculté des Sciences, Université Mohammed V-Agdal, Rabat, Morocco
Dsm/Irfu, CEA Saclay (Commissariat à l'Energie Atomique et aux Energies Alternatives). Gif-sur-Yvette, FranceInstitut de Recherches sur les Lois Fondamentales de l'UniversDSM/IRFU (Institut de Recherches sur les Lois Fondamentales de l'Univers), CEA Saclay (Commissariat à l'Energie Atomique et aux Energies Alternatives), Gif-sur-Yvette, France
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[] |
[
"EIGENVALUES ON SPHERICALLY SYMMETRIC MANIFOLDS",
"EIGENVALUES ON SPHERICALLY SYMMETRIC MANIFOLDS"
] |
[
"Marie Stine ",
"Berge "
] |
[] |
[] |
In this article we will explore Dirichlet Laplace eigenvalues on balls on spherically symmetric manifolds. We will compare any Dirichlet Laplace eigenvalue with the corresponding Dirichlet Laplace eigenvalue on balls in Euclidean space with the same radius. As a special case we will show that the Dirichlet Laplace eigenvalues on balls with small radius on the sphere are smaller than the corresponding eigenvalues on the Euclidean ball with the same radius. While the opposite is true for the Dirichlet Laplace eigenvalues of hyperbolic spaces.
| null |
[
"https://arxiv.org/pdf/2203.11911v1.pdf"
] | 247,596,983 |
2203.11911
|
b83751d7f73d8bdf7b878c6d79a4c7e801faf77c
|
EIGENVALUES ON SPHERICALLY SYMMETRIC MANIFOLDS
22 Mar 2022
Marie Stine
Berge
EIGENVALUES ON SPHERICALLY SYMMETRIC MANIFOLDS
22 Mar 2022
In this article we will explore Dirichlet Laplace eigenvalues on balls on spherically symmetric manifolds. We will compare any Dirichlet Laplace eigenvalue with the corresponding Dirichlet Laplace eigenvalue on balls in Euclidean space with the same radius. As a special case we will show that the Dirichlet Laplace eigenvalues on balls with small radius on the sphere are smaller than the corresponding eigenvalues on the Euclidean ball with the same radius. While the opposite is true for the Dirichlet Laplace eigenvalues of hyperbolic spaces.
Introduction
It is well know that the Dirichlet Laplace eigenvalues of the ball B r0 (0) ⊂ R n with radius r 0 are on the form One of the tools used in the proof are comparison theorems for Sturm-Liouville equations on the radial part of the eigenfunction. More recently, the first eigenvalue of a ball on a spherically symmetric, (also called rotationally symmetric) manifold was compared to the first eigenvalue of a ball in Euclidean space in [4,Lem. 3.1]. When applied to hyperbolic space and the sphere the result gives a generalization of (1.1) for the first eigenvalue found in [4,Thm. 3.3]. As a general consequence of this result, Borisov and Freitas showed that the first Dirichlet Laplace eigenvalue on balls on hyperbolic space are larger than the first eigenvalue on Euclidean space. Prior, a lower bound in the case of hyperbolic space for the first Dirichlet Laplace eigenvalue on balls has been shown by Artamoshin, see [1]. In the same article, the author also computed all the eigenvalues corresponding to radial eigenfunctions in 3-dimensional hyperbolic space.
In this article, we will present two different proofs of similar inequalities to (1.1) for all eigenvalues for both the spheres and hyperbolic spaces. Both the proofs will heavily dependent on the existence of certain continuous families of eigenfunctions and corresponding eigenvalues. The construction of these continuous families will for constant curvature spaces be outlined in Section 2, and generalized to spherically symmetric manifolds in Section 5. This parametrized family of Dirichlet eigenvalues on the geodesic ball B r (p) with radius r centered at p will be denoted by λ m,l (r) and satisfies lim r→0 r 2 λ m,l (r) = (j l m+n/2−1 ) 2 . We will show that these families of eigenvalues satisfies a lower and upper bound similar to (1.1). As a corollary, we will show that for every eigenvalue there exists a continuous family of Dirichlet Laplace eigenvalues λ m,l (r) on hyperbolic space with curvature K on the geodesic ball B r (p), such that
lim r→∞ λ m,l (r) = − n − 1 2 2 K.
For the first eigenvalue corresponding to the case m = 0 and l = 1 this was previously shown by Randol, see [5, p. 46]. The outline of the paper is as follows: In Section 2 we will prove a version of (1.1) for all eigenvalues using Sturm-Liouville theory as outlined in [2,3]. We will go through the definition and some theory about spherical symmetric manifolds in Section 3. Of special importance for this paper, is representing eigenfunctions of balls on spherically symmetric manifolds using spherical harmonics. The goal of Section 4 will be to express the eigenvalues as zeros of radial functions satisfying a Sturm-Liouville equation.
In the last section, we will write the an family of eigenvalue as an integral of the corresponding eigenfunction. To be more precise, let u t be the normalized continuous family of Dirichlet Laplace eigenfunction satisfying ∆u t = λ m,l (t)u t on B t (p) in a spherically symmetric manifold of dimension n. Using this notation we the eigenvalue λ m,l (r 1 ) can be written as the integral
(1.2) λ m,l (r 1 ) = j l m+n/2−1 r 1 2 + 1 2r 2 1 r1 0 t Bt(p) (u t ) 2 (x)g(r(x)) dvol dt,
where g is a radial function only depending on the geometry. For the eigenvalue λ 0,1 (r) this result can be found in [4,Lem. 3.1]. As an application of formula (1.2) we give another proof of the theorem presented in Section 2 for eigenvalues on spheres and hyperbolic space.
Dirichlet Laplace Eigenvalues on Model Spaces
In this section we will work on the model space (M K , g K ) with constant sectional curvature K and dimension n. Denote by sin K (r) the function
sin K (r) = sin(r √ K)/ √ K for K > 0 r for K = 0 sinh(r √ −K)/ √ −K for K < 0,
and cos K (r) = sin K (r) ′ . For a fixed point p ∈ M K denote by r p (x) = dist(p, x) the radial distance function and let S n−1 1 ⊂ R n denote the (n − 1)-sphere of radius 1. To represent points in M K we will use geodesic spherical coordinates (r, θ) where θ ∈ S n−1
1
. Geodesic coordinates are defined by using the exponential map, see [6, Sec. III. 1.]. Define B r0 (p) = r −1 p ([0, r 0 )) and S n−1 r0 (p) = ∂B r0 (p).
In Section 3 we will show that the solutions to the Dirichlet Laplace eigenvalue problem (2.1) ∆u r0 + λ(r 0 )u r0 = 0 in B r0 (p) for p ∈ M K , u r0 = 0 at S n−1 r0 (p) that are given on the form u r0 (r, θ) = R r0 m (r)Θ m (θ) form a basis for L 2 (B r0 (p)). Using separation of variables we can show that Θ m is a spherical harmonic function solving
∆ S n−1 1 Θ m + m(m + n − 2)Θ m = 0,
where m ∈ N. The radial part R r0 m on the other hand solves the equation
(2.2) R r0 m (r) ′′ + (n − 1) cos K (r) sin K (r) R r0 m (r) ′ − m(m + n − 2) sin 2 K (r) R r0 m (r) = −λ(r 0 )R r0 m (r)
with the condition R r0 m (r 0 ) = 0 and R r0 m (0) is bounded. Notice that (2.2) makes R r0 m to an eigenfunction of the second order differential operator d 2 dr 2 + (n − 1)
cos K (r) sin K (r) d dr − m(m + n − 2) sin 2 K (r)
.
Denote by R r0 m,l the eigenfunction to (2.2) corresponding to the l'th eigenvalue when ordered by size. The corresponding eigenvalue will be denoted by λ m,l (r 0 ). It is known, see e.g. [5, p. 318
], that lim r0→0 r 2 0 λ m,l (r 0 ) = (j l m+n/2−1 ) 2 ,
where j l m+n/2−1 is the l'th positive zero of the Bessel function J m+n/2−1 . We are now ready to state the main theorem: Theorem 2.1. Let λ m,l (r 0 ) be defined as above. Furthermore, when K > 0 we will assume that r 0 < π/ √ K. In the case that either m > 0 or n > 2 we have that
j l m+n/2−1 r 0 2 + 2m 2 + 2m(n − 2) − n(n − 1) 6 K ≤ λ m,l (r 0 ) ≤ j l m+n/2−1 r 0 2 − n − 1 2 2 K + n − 2 2 + m 2 − 1 4 1 sin 2 K (r 0 ) − 1 r 2 0 .
When m = 0 and n = 2 we get
j l 0 r 0 2 − 1 4 K + 1 sin 2 K (r 0 ) − 1 r 2 0 ≤ λ 0,l (r 0 ) ≤ j l 0 r 0 2 − 1 3 K.
Remark 2.2.
• In the case of the sphere S 2 this inequality is known from [2,3]. Additionally, for the first eigenvalue the theorem is known from [4, Thm. 3.3]. • When K < 0 we have that 1
sin 2 K (r) − 1 r 2 ≤ 0 and lim r→∞ 1 sin 2 K (r) − 1 r 2 = 0.
When K > 0 we have that
lim r→ π √ K 1 sin 2 K (r) − 1 r 2 = ∞.
For both positive and negative curvature
lim r→0 1 sin 2 K (r) − 1 r 2 = K 3 .
• For n = 3 and m = 0 Theorem 2.1 simplifies to
λ 0,l (r 0 ) = j l 1/2 r 0 2 − K.
For hyperbolic space this equality was shown in [1].
To compare the eigenfunctions of (2.2) to the Bessel equation we are going to use the Sturm-Picone comparison theorem.
(p 1 (x)y ′ 1 (x)) ′ + q 1 (x)y 1 (x) = 0, (p 2 (x)y ′ 2 (x)) ′ + q 2 (x)y 2 (x) = 0, on the interval [a, b]
. Assume that 0 < p 2 ≤ p 1 and q 1 ≤ q 2 , and let z 1 and z 2 be two consecutive zeros of y 1 . Then either y 2 has a zero in the interval (z 1 , z 2 ), or y 1 = y 2 .
Proof Thm. 2.1. In this section we will assume that for K > 0 we have
r 0 < π 2 √ |K| .
For the case when r 0 ≥ π 2 √ |K| see the proof in Section 6.
To get (2.2) on Sturm-Liouville form we will define v K (r) := sin n−1 2 K (r)R r0 m,l (r), and letm := n−2 2 + m. Then we have that
(2.3) 0 = v ′′ K (r) + n − 1 2 2 K + λ m,l (r 0 ) + 1 − 4m 2 4 sin 2 K (r) v K (r).
Let Jm be a Bessel function of orderm'th, and define u C (r) := √ rJm(Cr) to be the solution to
(2.4) 0 = u ′′ C (r) + C 2 + 1 − 4m 2 4r 2 u C (r).
Our goal is to apply Sturm-Picone comparison theorem (Thm. 2.3) to (2.3) and (2.4). We will use Sturm-Picone comparison twice with the C in (2.4) being two different constants which we will denote by C 1 and C 2 . Notice that
1 sin 2 K (r) − 1 r 2 is increasing with the lower limit lim r→0 + 1 sin 2 K (r) − 1 r 2 = K 3 .
Denote by
am := 4m 2 − 1 4
1 sin 2 K (r 0 ) − 1 r 2 0 , bm := K(4m 2 − 1) 12 .
For the next part we will need a lower bound for the eigenvalue. We will show that for K > 0 and m > 0 we have that
(2.5) λ m,l (r 0 ) > m(m + n − 2) sin 2 K (r 0 )
.
When m > 0 we have that the first zero r 0 of R r0 m,l is occurs after the first extremal point r max < r 0 . Without loss of generality, we can assume that the first extremal point is a maximum. At the point r max we have that R r0 m,l
′′ (r max ) < 0 and R r0 m,l ′ (r max ) = 0, which by using (2.2) implies (λ m,l (r 0 ) sin 2 K (r 0 ) − m(m + n − 2))R r0 m,l (r) > (λ m,l (r 0 ) sin 2 K (r) − m(m + n − 2))R r0 m,l (r) > 0, hence (2.5) follows.
For the negatively curved spaces we will use that
− (n − 1) 2 4 K < λ m,l (r 0 )
which can be found e.g. [5, p. 46]. When either n > 2 or m > 0 we have that 1 − 4m 2 ≤ 0, and we will set
C 1 = λ m,l (r 0 ) + n − 1 2 2 K − am and C 2 = λ m,l (r 0 ) + n − 1 2 2 K − bm.
In this case we have that
C 2 1 + 1 − 4m 2 4r 2 ≤ n − 1 2 2 K + λ m,l (r 0 ) + 1 − 4m 2 4 sin 2 K (r) ≤ C 2 2 + 1 − 4m 2 4r 2 .
Using Theorem 2.3 for solutions to the equations (2.3) and (2.4) leads to the estimate of r 0 by j lm
C 2 ≤ r 0 ≤ j lm C 1 .
Solving for λ m,l (r 0 ) we get
j lm r 0 2 + 2m 2 + 2m(n − 2) − n(n − 1) 6 K ≤ λ m,l (r 0 ) ≤ j lm r 0 2 − n − 1 2 2 K + 4m 2 − 1 4 1 sin 2 K (r 0 ) − 1 r 2 0 .
When m = 0 and n = 2 we have that
C 2 2 + 1 − 4m 2 4r 2 ≤ n − 1 2 2 K + λ 0,l (r 0 ) + 1 − 4m 2 4 sin 2 K (r) ≤ C 2 1 + 1 − 4m 2 4r 2 .
Hence we obtain
j l 0 r 0 2 − 1 4 K + 1 sin 2 K (r 0 ) − 1 r 2 0 ≤ λ 0,l (r 0 ) ≤ j l 0 r 0 2 − 1 3 K.
A corollary of Theorem 2.1 is as follows:
Corollary 2.4. Assume that K < 0. Then in the notation of Theorem 2.1 we have that
(2.6) lim r0→∞ λ m,l (r 0 ) = − n − 1 2 2 K.
Proof. By [5, p. 46] we know that the smallest eigenvalue λ 0,1 (r 0 ) satisfies
λ 0,1 (r 0 ) ≥ − n − 1 2 2 K.
Using this lower bound together with taking the limit as r 0 goes towards infinity of the upper bound in Theorem 2.1 gives the result for the case when either m > 0 or n > 2. Hence we are only left with the case when n = 2 and m = 0. Let us assume that n = 2 and consider the operators
(L 0 φ)(r) = (sin K (r)φ ′ (r)) ′ sin K (r) and (L 1 φ)(r) = (sin K (r)φ ′ (r)) ′ sin K (r) − φ(r) sin 2 K (r)
.
Then both L 0 and L 1 are negative and symmetric operators on the L 2 -space
X r0 = φ ∈ H 2 loc (0, r 0 ) : r0 0 sin K (r)φ(r) 2 dr < ∞, φ(r 0 ) = φ(0)φ ′ (0) = 0 , with the norm φ, ξ sin K = r0 0 φ(r)ξ(r) sin K (r) dr.
Then for φ ∈ X r0 one have the inequality
L 0 φ, φ sinK ≤ L 1 φ, φ sinK .
This implies that the eigenvalues satisfy
λ k (B r (x)) ≤ (n − 1) 2 4 κ + c n k 2 r 2 . This gives the inequality lim r0→∞ λ k (r 0 ) ≤ (n − 1) 2 4 ,
for H n with constant curvature −1.
Eigenvalues on Spherically Symmetric Manifolds
Let (M, g) be a Riemannian manifold of dimension n. Fix a point p ∈ M and use the notations r p (x) = dist(x, p) and S n−1 r0 = r −1 p (r 0 ). In this section we will work with geodesic spherical coordinates (r, θ) with respect to the point p, see [6, Sec. III. 1.]. Furthermore, we will assume that (M, g) is spherically symmetric with respect to the point p. This means that the metric g can be written as
g = dr ⊗ dr + f 2 (r)g S n−1
in geodesic spherical coordinates as long as r p < ρ for some fixed ρ. By
∆ = ∂ 2 ∂r 2 + (n − 1) f ′ (r) f (r) ∂ ∂r + 1 f 2 (r) ∆ S n−1 1 .
We refer the reader to [12,Sec. 4.2.3] for more information about spherically symmetric manifolds.
Using separation of variables we will study families of Dirichlet eigenvalues on the ball B t (p). We will refer to the parametrized family R t (r) as the radial part of the solution and Θ(θ) as the spherical part. We will show that there exists an L 2 (B t (p))basis on the form R t (r)Θ(θ) consisting of Dirichlet Laplace eigenfunctions on the ball B t (p). The parametrized family R t will always be assumed to be continuous in t. In the case for the sphere this was shown in [5, Chap. II.5] with a similar approach. The radial part R t solves the equation
(3.1) (f n−1 (r)(R t ) ′ (r)) ′ − f n−3 (r)m(m + n − 2)R t (r) = −λ(t)f n−1 (r)R t (r),
with R t (t) = 0 and where R t (0) is bounded. Introduce the norm
φ 2 f = t 0 φ(r) 2 f n−1 (r) dr.
The operator
(L m φ)(r) = f 1−n (r) (f n−1 (r)φ ′ (r)) ′ − f n−3 (r)m(m + n − 2)φ(r)
is unbounded and symmetric on L 2 ((0, t), f n−1 (r)dr) defined on the subspace
X t = φ ∈ H 2 loc (0, t) : t 0 f n−1 (r)φ(r) 2 dr < ∞, φ(t) = φ(0)φ ′ (0) = 0 .
Recall that H 2 loc (0, t) is the space where the second weak derivatives are locally in L 2 (0, t). The eigenvalues of L m are simple, since if u, v are two eigenfunctions with the same eigenvalue λ we have that
0 = f n−1 (r)(uL m v − vL m u) = u(f n−1 (r)v ′ ) ′ − v(f n−1 (r)u ′ ) ′ = (f n−1 (r)(uv ′ − vu ′ )) ′ .
This means that f n−1 (r)(uv ′ − vu ′ ) is constant. Using that u(t) = v(t) = 0 we get that the Wronskian uv ′ − vu ′ is zero. Hence u and v are linearly dependent.
We will let λ m,l (t) be the l'th eigenvalue of the problem (3.1) with the corresponding eigenfunction R t m,l . Proposition 3.1. Let (M, g) be a spherically symmetric manifold with respect to the point p and a constant 0 < ρ and consider the ball B t (p) ⊂ M with t < ρ. Then there exists an L 2 -basis on B t (p) which consists of Dirichlet Laplace eigenfunctions on the form u t (r, θ) = R t m,l (r)Θ m (θ), where ∆ S n−1 Θ m + m(m + n − 2)Θ m = 0, and R t m,l is the eigenfunction corresponding to the l'th eigenvalue of (3.1). Proof. Let u(r, θ) = R(r)Θ(θ) be an eigenfunction written in geodesic spherical coordinates with eigenvalue λ(t). Then we have
∆u(r, θ) = R ′′ (r)Θ(θ) + (n − 1) f ′ (r) f (r) R ′ (r)Θ(θ) + R(r) 1 f 2 (r) ∆ S n−1 1 Θ(θ) = −λ(t)R(r)Θ(θ).
The above expression simplifies to
(3.2) f 2 (r)R ′′ (r) + (n − 1)f ′ (r)f (r)R ′ (r) + λ(t)f 2 (r)R(r) R(r) = − ∆ S n−1 1 Θ(θ) Θ(θ) .
Since the left hand-side of (3.2) is independent of θ we have that Θ is a spherical harmonic function. Using that the eigenvalues on the (n − 1)-sphere have the form m(m + n − 2) we get that R satisfies (3.1). Hence if R(r) is a solution to (2.2) satisfying R(t) = 0 and R(0) is bounded, then u is a solution to the Dirichlet Laplace eigenvalue problem. The only thing left to show is that each eigenfunction can be written as a sum of eigenfunctions on the form R(r)Θ(θ). Since the spherical harmonics are the eigenfunctions of S n−1 , we know that the spherical harmonics form an orthonormal basis for L 2 (S n−1 ). This means that an arbitrary eigenfunction u(r, θ) with eigenvalue λ(t) can be written as
u(r, θ) = ∞ i=1 a i (r)Θ i (θ),
where Θ i is the i'th spherical harmonic function with eigenvalue m i (m i + n − 2). Using the Laplacian written out in spherical coordinates gives
−m i (m i + n − 2)a i (r) = S n−1 1 u(r, θ)∆ S n−1 1 Θ i (θ) dS = S n−1 1 ∆ S n−1 1 u(r, θ)Θ i (θ) dS = −λ(t)f 2 (r)a i (r) − f 2 (r)a ′′ i (r) − (n − 1)f ′ (r)f (r)a ′ i (r)
. In particular, the function a i (r)Θ i (θ) is an eigenfunction for all i. By orthogonality of eigenfunctions we get that u can be written on the form
u(r, θ) = m i=j a i (r)Θ i (θ),
where a i (r)Θ i (θ) has eigenvalue λ(t).
Eigenvalues on Balls
Let (M, g) be a Riemannian manifold (not necessarily spherically symmetric) and consider the ball B t (p) ⊂ M for p ∈ M and t < Inj(p). The notation Inj(p) denotes the injectivity radius at p. Again we will look at the Dirichlet problem
(4.1) ∆u t + λ (t) u t = 0 on B t (p) u t = 0 on S n−1 t (p) .
We will assume that Bt(p) (u t ) 2 dvol = 1, in which case
λ(t) = Bt(p) | grad u t | 2 dvol .
By using the Hadamard formula presented in [8, Cor. 2.1] one has, assuming that λ(t) and u t are differentiable with respect to t, that
λ ′ (t) = − S n−1 t (p) u t n 2 dS .
For solutions to the Dirichlet problem we have the following result.
Proposition 4.1. Let u t be a parametrized family of solution to (4.1) which is normalized in L 2 . Denote by r the radial distance from the point p and let r 1 < Inj(p). Assume that both λ(t) and u t are differentiable in t for r 0 ≤ t ≤ r 1 . Then
r 2 1 λ(r 1 ) = lim t→r0 t 2 λ(t)+ r1 r0 t Bt(p) (u t ) 2 ∆ (r p ∆r p ) 2 +2 grad S n−1 rp (p) u t 2 −2r p ∇ 2 r p grad u t , grad u t dvol dt.
The above proposition was proved for the first eigenvalue in [4, Lem. 3.1]. The proof was based on variational methods. We will give another proof for general eigenvalues. We first develop the following lemma of independent interest. Lemma 4.2. Let u be a solution to ∆u + λu = 0 on the ball B t (p) where t ≤ Inj(p). Let ϕ : (0, ∞) → (0, ∞) be such that grad rp ϕ(rp(x)) is a smooth vector field. Then
S n−1 t (p) | grad S n−1 t (p) u| 2 − u 2 n dS = ϕ (t) Bt(p) | grad u| 2 ϕ (r p (x)) ∆r p − ϕ ′ (r p (x)) ϕ 2 (r p (x)) dvol − 2ϕ (t) Bt(p) ∇ 2 r p (grad u, grad u) − λu r u ϕ (r p (x)) − ϕ ′ (r p (x)) u 2 r ϕ 2 (r p (x)) dvol .
Proof. We will use the notation X = grad rp φ(rp(x)) and V = 2X(u) grad u − | grad u| 2 X.
Taking the divergence of V , one obtains div(V ) = 2 grad X(u), grad u − 2λuX(u) − grad | grad u| 2 , X − | grad u| 2 div(X) = 2 ∇ grad u X, grad u + 2 X, ∇ grad u grad u − 2 ∇ X grad u, grad u − 2λuX(u) − | grad u| 2 div(X)
= 2 ∇ grad u X, grad u + ∇ 2 u(grad u, X) − 2∇ 2 u(X, grad u) − 2λuX(u) − | grad u| 2 div(X) = 2 ∇ grad u X, grad u − 2λuX(u) − | grad u| 2 div(X).
Expanding the term
∇ grad u grad r p φ(r p (x)) , grad u = φ(r p (x))∇ 2 r p (grad u, grad u) − φ ′ (r p (x))u 2 r φ(r p (x)) 2 we get div(V ) = 2(φ(r p (x))∇ 2 r p (grad u, grad u) − φ ′ (r p (x))u 2 r ) − 2λφ(r p (x))uu r φ(r p (x)) 2 + | grad u| 2 (φ(r p (x))∆r p − φ ′ (r p (x))) φ(r p (x)) 2 .
Using the divergence theorem with div(V ) together with
V, grad r p = | grad S n−1 t (p) u| 2 − (u r ) 2 φ(t) ,
gives the result.
Proof of Prop. 4.1. Lemma 4.2 with ϕ(r p (x)) = 1 rp(x) implies that tλ ′ (t) = Bt(p)
grad u t 2 (r p ∆r p + 1)
− 2 r p ∇ 2 r p grad u t , grad u t + (u t ) 2 n − λ (t) r p (u t ) n u t dvol .
Using the equation
Bt(p) (u t ) n u t r p dvol = − 1 2 Bt(p) (u t ) 2 dvol − 1 2 Bt(p) r p ∆r p (u t ) 2 dvol gives tλ ′ (t) = Bt(p) grad u t 2 (r p ∆r p + 1) − 2 r p ∇ 2 r p grad u t , grad u t + (u t ) 2 n dvol − λ (t) − λ (t) Bt(p) r p ∆r p (u t ) 2 dvol . (4.2)
The equation (4.2) together with (t 2 λ(t)) ′ = t(2λ(t) + tλ ′ (t)) implies that
t 2 λ (t) ′ = t Bt(p) grad u t 2 (r p ∆r p + 2) − 2r p ∇ 2 r p grad u t , grad u t dvol − 2t Bt(p) (u t ) 2 n dvol −tλ (t) Bt(p) r p ∆r p (u t ) 2 dvol .
Simplifying the expression further gives us that
(t 2 λ (t)) ′ = t Bt(p) 1 2 ∆((u t ) 2 ) + λ(t)(u t ) 2 r p ∆r p dvol + 2t Bt(p) grad S n−1 r (p) u t 2 − r p ∇ 2 r p grad u t , grad u t dvol − tλ (t) Bt(p) r p ∆r p (u t ) 2 dvol = −tλ (t) Bt(p) 1 2 grad(u t ) 2 , grad (r p ∆r p ) dvol + 2t Bt(p) grad S n−1 r (p) u t 2 − r p ∇ 2 r p grad u t , grad u t dvol = t Bt(p) (u t ) 2 ∆ (r p ∆r p ) 2 + 2 grad S n−1 r (p) u t 2 dvol − 2t Bt(p) r p ∇ 2 r p grad u t , grad u t dvol .
Finally, integrating the identity above gives the result.
Remark 4.3. Let (M, g) be an analytic Riemannian manifold and let φ s : B t (p) → B s+t (p) be the flow of the vector field grad r p which is analytic for r + t < Inj(p). Then by [8,Lem. 3.1] we can find a differentiable family λ(t) and u t for t ∈ (r − ǫ, r + ǫ).
Spherical Symmetric Manifolds
In this section we will assume that the n-dimensional Riemannian manifold (M, g) is spherically symmetric with respect to the point p ∈ M on the ball B ρ (p). We will use the notation introduced in Section 3. Before stating the main result, we will show that the following Hadamard formula holds:
(t) dt = −f (t) n−1 (R t m,l (t) ′ ) 2 ,
a.e. in t ∈ (0, ρ).
To show Theorem 5.1 we need the following result:
dU h (r) dr = A h (r)U h (r) for r ∈ (a, b), U h (x 0 ) = y h , has a unique solution U h (x) that is continuous in (−c, c) × (a, b).
Proof. For completeness, we repeat and slightly modify the proof from [15,Thm. 2.1 p. 23].
Fix h and let [x 0 − η, x 0 + η] be such that
x0+η x0−η |A h | dt ≤ q < 1.
Define the operator B h :
C([x 0 − η, x 0 + η]) → C([x 0 − η, x 0 + η]) by B h u(x) = y h + x x0 A h (t)u(t) dt, where C([x 0 − η, x 0 + η])
is the space of continuous functions with the supremum norm · ∞ . Notice that B h is a contraction since
B h u − B h v ∞ ≤ q u − v ∞ .
By the Banach fixed point theorem there exists a unique fixed point, which we will denote by U h , giving the existence and uniqueness at the interval [x 0 − η, x 0 + η]. Choosing a compact set K inside (a, b) and a finite covering
{(x i − η, x i + η)} i∈J of K where xi+η xi−η |A h | dt ≤ q < 1,
gives the existence and uniqueness of the solution on K. A compact sweeping of the interval (a, b) gives the uniqueness and existence result on (a, b).
We are only left to show that U h is continuous. Fix a value h ′ ∈ (−c, c) and let
I = [x 0 − η, x 0 + η] ⊂ (a, b) be such that I |A h ′ (t)| dt ≤ 1 2 .
Then for y ∈ I we obtain
sup x∈I |U h ′ (y) − U h (x)| = sup x∈I |B h ′ U h ′ (y) − B h U h (x)| ≤ |y h ′ − y h | + |x − y| sup x∈I |A h ′ (x)| sup x∈I |U h ′ (x)| + sup x∈I |U h ′ (y) − U h (x)| I |A h (t)| dt + sup x∈I |U h ′ (y) − U h ′ (x)| I |A h (t)| dt + sup x∈I |U h ′ (x)| I |A h (t) − A h ′ (t)||U h ′ (x)| sup h∈[h ′ −ξ,h ′ +ξ] I |A h (t) − A h ′ (t)| dt ≤ ǫ 8 , sup x∈I |U h ′ (y) − U h ′ (x)| I |A h (t)| dt ≤ ǫ 8 , |x − y| sup x∈I |A h ′ (x)| sup x∈I |U h ′ (x)| < ǫ 8 , and sup h∈[h ′ −ξ,h ′ +ξ] |y h − y h ′ | ≤ ǫ 8 ,
showing continuity at (h ′ , y). Hence U h (x) is continuous. The solution is assumed to satisfy t 0 f n−1 (r)(R t m,l (r)) 2 dr = 1.
We will start by showing that the family R t m,l is uniformly continuous on compact subsets of (0, t] . The differential equation
(f n−1 (r)(R t m,l (r)) ′ ) ′ + f n−1 (r) λ m,l (t) − m(m + n − 2) f 2 (r) R t m,l (r) = 0
has two linearly independent solutions on the interval (0, t], one of which is bounded at 0 and the other is unbounded at 0. By existence and uniqueness of second order ordinary differential equations, see e.g. [13, Thm. A p. 488], the two solutions can not have common zeroes. Additionally, we will use that the eigenvalues are continuous in the variable t, see e.g. [14,Thm. 2.10]. Let 0 ≤ h < ǫ 1 , where t + ǫ 1 < ρ, and ρ is the largest radius where the manifold is radially symmetric. Define the matrix
A h (r) = t + h t 0 f 1−n ( t+h t r) f n−1 ( t+h t r) m(m+n−2) f 2 ( t+h t r) − λ m,l (t + h) 0 .
Then we can consider the problem on the interval (0, t]. This means that the components of U h given by U h,1 (r) = t t+h R t+h m,l ( t+h t r) f n−1 (t + h)(R t+h m,l ) ′ (t + h) and U h,2 (r) = t t+h f n−1 ( t+h t r)(R t+h m,l ) ′ ( t+h t r) f n−1 (t + h)(R t+h m,l ) ′ (t + h) are continuous.
(5.1) U ′ h (r) = A h (r)U h (r) on (0, tρ/(t + ǫ 1 )),
We will now turn to finding the derivative of λ m,l (t). Denote by N (t, h) = t/(t + h) f n−1 (t + h)(R t+h m,l ) ′ (t + h)f n−1 (t)(R t m,l ) ′ (t)
.
Using integration by parts shows that U h,1 t 2 t + h U 0,2 (t) = N (t, h) Taking the limit as h → 0 and using the continuity of U h,2 gives the result.
Using Proposition 4.1 we get the following corollary.
the l'th zero of the Bessel function J m+n/2−1 . In[2,3] Baginski compared all the Dirichlet Laplace eigenvalues on the spherical cap B r0 (p) ⊂ S 2 to the Dirichlet Laplace eigenvalues of the ball in the plane with the same radius. In particular, the author showed that the first eigenvalue λ 1 (r 0 ) on B r0 (p)
Theorem 2 . 3 (
23Sturm-Picone Comparison Theorem, [9, Theorem B]). Let y 1 and y 2 be non-zero solutions to
λ 0 ,l (r 0 )( 1 )
001≤ λ 1,l (r 0 ).Taking the limit as r 0 For the smallest Dirichlet eigenvalue on hyperbolic space, i.e. m = 0 and l = 1, Corollary 2.4 is well known, see e.g.[5, p. 46]. (2) There exist similar inequalities for eigenvalues for more general manifolds.One example of similar inequalities is given in e.g.[7, Cor. 2.3]. As noted in [10, Re. 1.3], given a complete simply connected Riemannian manifold (M, g) with Ricci curvature satisfying Ric ≥ (n − 1)(−κ) for κ > 0 we have that
[ 12 ,
12Prop. 1.4.7] the function f : [0, ρ) → [0, ∞) satisfies f (0) = 0, f ′ (0) = 1, f ′′ (0) = 0. Many harmonic manifolds and surfaces of revolution are examples of spherically symmetric manifolds. Writing the Laplacian in geodesic spherical coordinates gives
Theorem 5 . 1 .
51Let u t (r, θ) = R t m,l (r)Θ(θ) be an L 2 -normalized solution to (4.1) on the ball B t (p) where ∆ S n−1 1 Θ = −m(m + n − 2)Θ, and R t m,l satisfies (3.1) with R t m,l (t) = 0 and R t m,l (0) being bounded. Then we have that dλ m,l
Theorem 5.2 ([15, Thm. 2.1 p. 23]). Let A : (−c, c) × (a, b) → Mat(R n ) and y : (−c, c) → R n be continuous functions. Fix a point x 0 ∈ (a, b). Then the problem
Proof of Thm. 5.1. The outline of the proof is inspired by[11, Thm. 3.1], where the author shows several similar results for Sturm-Liouville equations.Recall that R t m,l satisfies the equation(f n−1 (r)(R t m,l (r)) ′ ) ′ + f n−1 (r) λ m,l (t) − m(m+n−2) f 2 (r)R t m,l (r) = 0, R t m,l (0)(R t m,l ) ′ (0) = R t m,l (t) = 0.
≤
m,l (r)f n−1 (r)(R t m,l ) ′ (r)) ′ dr = N (t, h) t 0 R t+h m,l (r)(f n−1 (r)(R t m,l ) ′ (r)) ′ dr (l ) ′ (r) − v(t, 0)f n−1 (r)(R t+h m,l ) ′ (r) f n−1 (r)dr .
Corollary 5. 3 .F
3Let u t (r, θ) = R t m,l (r)Θ(θ) be a solution to (4.1) on the ball B t (r) = n − 1 f 3 (r) (3 − n)rf ′ (r) 3 + (rf ′′′ (r) + 2f ′′ (r))f 2 (r) + ((n − 4)rf ′′ (r) + (n − 3)f ′ (r))f ′ (r)f (r) .
dt .
dtOne can find a box [h ′ − ξ, h ′ + ξ] × [y − δ, y + δ] such that for all x ∈ [y − δ, y + δ] and all h ∈ [h ′ − ξ, h ′ + ξ] we have that supx∈I
h t+h t 1 f n−1 (r) (v(t, 0)f n−1 (r)(R t m,l ) ′ (r)) dr
− N (t, h) t 0 ((R t+h m,l ) ′ (r)f n−1 (r)) ′ R t m,l (r) dr.Using the differential equation for R t+h m,l and R t m,l gives By the continuity of U h (x) we immediately have for 0 < ǫ < t the limitwe will show that the integrand f n−1 (r)R t+h m,l (r)R t m,l (r) is uniformly bounded. In the case that m > 1 we have that R t+h m,l (r), f n−1 (r), and R t m,l (r) are increasing functions for small r. To be more precise, fix r 0 such that for all r < r 0 and all h ∈ (−c, c) we have that f ′ (r) > 0 and min h∈(−c,c)To get this result, notice that (3.1) implies that (R t+h m,l ) ′ (r) can not be zero as long asby the second derivative test. Choosing ǫ < r 0 we get by the dominated convergence theorem thatIn the case that m = 0, we fix r 0 such that for all r < r 0 we have that R t 0,l (r) > 0 and f ′ (r) > 0. ThenUsing integration by parts and the Cauchy-Schwarz inequality we obtainHence we can use the dominated convergence theorem once more and obtain (5.(v(t, 0)f n−1 (r)(R t m,l ) ′ (r)) dr.If we insert the above equation into (5.2) and divide by h we getThen for 0 < r 0 < r 1 < ρ we haves 2 λ m,l (s).In particular taking the limit of r 0 → 0 one has that Proof. Using Proposition 4.1 we need to compute ∆(r p ∆r p ). Notice first thatMoreover, the radial part of the Laplacian is ∂ 2Using the divergence theorem with s < t we get thatwhere we have used that ∆ S n−1 s = 1 f 2 (s) ∆ S n−11. Thus the result follows.6. Second Proof of Theorem 2.1As stated in the introduction of the article, we can use Corollary 5.3 to give a new proof of Theorem 2.1.Proof of Thm. 2.1. In the case that (M K , g K ) is a model space we have that f (r) = sin K (r). Hence the function F in Corollary 5.3 simplifies to.When K > 0 we will assume that r 1 < π √ K . The function G(r) = sin K (r) − r cos K (r) sin 3 K (r) is increasing and satisfies lim r→0 G(r) = −K 3 . Thus for all r ∈ (0, t] we have.We will assume that λ m,l (t) is the l'th eigenvalue of R t m,l and that u t (r, θ) = R t m,l (r)Θ m (θ) where u t is normalized in L 2 (B t (p)) and R t m,l satisfies (3.1) and Θ m is a spherical harmonic function with eigenvalue m(m + n − 2). When n ≥ 3 or m ≥ 1 we get thatThis is the same inequality as in Theorem 2.1. Using the lower bound of G(t) we obtainFor n = 2 and m = 0 we have thatIn this case, we have that the eigenvalue λ 0,l (r 1 ) satisfies λ 0,l (r 1 ) = 1 2r
Lower bounds for the first Dirichlet eigenvalue of the Laplacian for domains in hyperbolic space. S Artamoshin, Mathematical Proceedings of the Cambridge Philosophical Society. 1602S. Artamoshin. Lower bounds for the first Dirichlet eigenvalue of the Laplacian for domains in hyperbolic space. Mathematical Proceedings of the Cambridge Philosophical Society, 160(2), 2016.
Upper and lower bounds for eigenvalues of the Laplacian on a spherical cap. F E Baginski, Quarterly of Applied Mathematics. 483F. E. Baginski. Upper and lower bounds for eigenvalues of the Laplacian on a spherical cap. Quarterly of Applied Mathematics, 48(3), 1990.
Errata: Upper and lower bounds for eigenvalues of the Laplacian on a spherical cap. F E Baginski, Quarterly of Applied Mathematics. 492F. E. Baginski. Errata: Upper and lower bounds for eigenvalues of the Laplacian on a spherical cap. Quarterly of Applied Mathematics, 49(2), 1991.
The spectrum of geodesic balls on spherically symmetric manifolds. D Borisov, P Freitas, Communications in Analysis and Geometry. 253D. Borisov and P. Freitas. The spectrum of geodesic balls on spherically symmetric manifolds. Communications in Analysis and Geometry, 25(3), 2017.
Including a chapter by B. Randol, With an appendix by. I Chavel, J. DodziukAcademic PressEigenvalues in Riemannian geometryI. Chavel. Eigenvalues in Riemannian geometry. Academic Press, 1984. Including a chapter by B. Randol, With an appendix by J. Dodziuk.
Riemannian geometry. I Chavel, Cambridge University Presssecond editionI. Chavel. Riemannian geometry. Cambridge University Press, second edition, 2006.
Eigenvalue comparison theorems and its geometric applications. S Y Cheng, Mathematische Zeitschrift. 1433S. Y. Cheng. Eigenvalue comparison theorems and its geometric applications. Mathematische Zeitschrift, 143(3), 1975.
Domain deformations and eigenvalues of the Dirichlet Laplacian in a Riemannian manifold. A El Soufi, S Ilias, Illinois Journal of Mathematics. 512A. El Soufi and S. Ilias. Domain deformations and eigenvalues of the Dirichlet Laplacian in a Riemannian manifold. Illinois Journal of Mathematics, 51(2), 2007.
Sturm's 1836 oscillation results evolution of the theory. D Hinton, Sturm-Liouville theory. Birkhäuser, BaselD. Hinton. Sturm's 1836 oscillation results evolution of the theory. In Sturm-Liouville theory, pages 1-27. Birkhäuser, Basel, 2005.
A note on eigenvalue bounds for non-compact manifolds. M Keller, S Liu, N Peyerimhoff, Mathematische Nachrichten. 29462021M. Keller, S. Liu, and N. Peyerimhoff. A note on eigenvalue bounds for non-compact mani- folds. Mathematische Nachrichten, 294(6), 2021.
Dependence of eigenvalues of Sturm-Liouville problems on the boundary. Q Kong, A Zettl, Journal of Differential Equations. 1262Q. Kong and A. Zettl. Dependence of eigenvalues of Sturm-Liouville problems on the bound- ary. Journal of Differential Equations, 126(2), 1996.
Riemannian geometry. P Petersen, SpringerChamthird editionP. Petersen. Riemannian geometry. Springer, Cham, third edition, 2016.
G F Simmons, Differential equations with applications and historical notes. Textbooks in Mathematics. CRC PressThird editionG. F. Simmons. Differential equations with applications and historical notes. Textbooks in Mathematics. CRC Press, 2017. Third edition.
Spectral analysis and differential geometry of the Laplacian. H Urakawa, World Scientific Publishing CoSpectral geometry of the LaplacianH. Urakawa. Spectral geometry of the Laplacian. World Scientific Publishing Co., 2017. Spec- tral analysis and differential geometry of the Laplacian.
Spectral theory of ordinary differential operators. J Weidmann, Welfengarten. 130167Springer-VerlagInstitute for Analysis, Leibniz University HannoverHannoverJ. Weidmann. Spectral theory of ordinary differential operators. Springer-Verlag, Berlin, 1987. Institute for Analysis, Leibniz University Hannover, Welfengarten 1 30167 Han- nover, Germany Email address: [email protected]
|
[] |
[
"The case for three-body decaying dark matter",
"The case for three-body decaying dark matter"
] |
[
"Hsin-Chia Cheng \nDepartment of Physics\nUniversity of California\n95616DavisCA\n",
"Wei-Chih Huang \nSISSA\nINFN-sezione di Trieste\nVia Bonomea 26534136TriesteItaly\n",
"Ian Low \nHigh Energy Physics Division\nArgonne National Laboratory\n60439ArgonneIL\n\nDepartment of Physics and Astronomy\nNorthwestern University\n60208EvanstonIL\n",
"Gabe Shaughnessy \nHigh Energy Physics Division\nArgonne National Laboratory\n60439ArgonneIL\n\nDepartment of Physics and Astronomy\nNorthwestern University\n60208EvanstonIL\n\nDepartment of Physics\nUniversity of Wisconsin\n53706MadisonWI\n"
] |
[
"Department of Physics\nUniversity of California\n95616DavisCA",
"SISSA\nINFN-sezione di Trieste\nVia Bonomea 26534136TriesteItaly",
"High Energy Physics Division\nArgonne National Laboratory\n60439ArgonneIL",
"Department of Physics and Astronomy\nNorthwestern University\n60208EvanstonIL",
"High Energy Physics Division\nArgonne National Laboratory\n60439ArgonneIL",
"Department of Physics and Astronomy\nNorthwestern University\n60208EvanstonIL",
"Department of Physics\nUniversity of Wisconsin\n53706MadisonWI"
] |
[] |
Fermi-LAT has confirmed the excess in cosmic positron fraction observed by PAMELA, which could be explained by dark matter annihilating or decaying in the center of the galaxy. Most existing models postulate that the dark matter annihilates or decays into final states with two or four leptons, which would produce diffuse gamma ray emissions that are in tension with data measured by Fermi-LAT. We point out that the tension could be alleviated if the dark matter decays into three-body final states with a pair of leptons and a missing particle. Using the goldstino decay in a certain class of supersymmetric theories as a prime example, we demonstrate that simultaneous fits to the total e + + e − and the fractional e + /e − fluxes from Fermi-LAT and PAMELA could be achieved for a 2 TeV parent particle and a 1 TeV missing particle, without being constrained by gamma-ray measurements. By studying different effective operators giving rise to the dark matter decay, we show that this feature is generic for three-body decaying dark matter containing a missing particle. Constraints on the hadronic decay widths from the cosmic anti-proton spectra are also discussed.
|
10.1088/1475-7516/2013/01/033
|
[
"https://arxiv.org/pdf/1205.5270v2.pdf"
] | 118,746,184 |
1205.5270
|
80a224869c531b266375236e9b0e190c3c6b4e91
|
The case for three-body decaying dark matter
28 Jun 2012
Hsin-Chia Cheng
Department of Physics
University of California
95616DavisCA
Wei-Chih Huang
SISSA
INFN-sezione di Trieste
Via Bonomea 26534136TriesteItaly
Ian Low
High Energy Physics Division
Argonne National Laboratory
60439ArgonneIL
Department of Physics and Astronomy
Northwestern University
60208EvanstonIL
Gabe Shaughnessy
High Energy Physics Division
Argonne National Laboratory
60439ArgonneIL
Department of Physics and Astronomy
Northwestern University
60208EvanstonIL
Department of Physics
University of Wisconsin
53706MadisonWI
The case for three-body decaying dark matter
28 Jun 20121
Fermi-LAT has confirmed the excess in cosmic positron fraction observed by PAMELA, which could be explained by dark matter annihilating or decaying in the center of the galaxy. Most existing models postulate that the dark matter annihilates or decays into final states with two or four leptons, which would produce diffuse gamma ray emissions that are in tension with data measured by Fermi-LAT. We point out that the tension could be alleviated if the dark matter decays into three-body final states with a pair of leptons and a missing particle. Using the goldstino decay in a certain class of supersymmetric theories as a prime example, we demonstrate that simultaneous fits to the total e + + e − and the fractional e + /e − fluxes from Fermi-LAT and PAMELA could be achieved for a 2 TeV parent particle and a 1 TeV missing particle, without being constrained by gamma-ray measurements. By studying different effective operators giving rise to the dark matter decay, we show that this feature is generic for three-body decaying dark matter containing a missing particle. Constraints on the hadronic decay widths from the cosmic anti-proton spectra are also discussed.
I. INTRODUCTION
The intriguing positron excess in PAMELA [1] has been recently confirmed by Fermi-LAT [2] with the help of the Earth's magnetic field to distinguish positrons from electrons due to the lack of the magnetic field on board. Fermi-LAT also extends the positron excess range up to a higher energy scale, around 200 GeV. In addition, measurements in the total e + + e − by Fermi-LAT also exhibit some interesting feature well beyond the 100 GeV region [3]. These results imply the existence of new sources of primary positrons. The new source could be astrophysical, for example, nearby pulsars [4]. The more interesting possibility is that they may be a sign of the dark matter (DM), due to either the dark matter annihilation in the galaxy halo [5], or the decaying dark matter [6]. In the case of the dark matter annihilation, it requires a very large boost factor due to the fact that both the resulting positron flux and the relic abundance are determined by the same annihilation channel. Some of the possible solutions include the Sommerfeld enhancement [7] [8] [9] and Breit-Wigner Enhancement [10] [11]. On the other hand, the decaying dark matter has no such a limitation, but a mechanism responsible for a long lifetime has to be realized in this case.
The possibility that the excesses in the PAMELA positron fraction and Fermi-LAT e + +e − spectrum come from the dark matter annihilation or decay and the corresponding constraints from the antiproton and gamma ray spectra has been the subject of intensive investigations.
Most of the studies in the literature focused on the two-body and four-body final states of the standard model (SM) particles [6], with the latter comes from an intermediate stage of two light "portal" particles [8] [12] [13] [14] [15]. The general lesson from these studies is that final states have to be dominantly µ's or τ 's, since a significant fraction of the electron final states would produce a sharp feature in the total e + + e − flux which is not seen in the Fermi-LAT data. However, these charged final states would produce diffuse gamma ray emissions through inverse Compton scatterings that are excluded by the Fermi-LAT data [16] [17] [18] [19]. There also exist constraints on decays into final state producing hadrons from the anti-proton measurements made by PAMELA and on prompt decays into photons from Fermi-LAT [20], HESS [21], and VERITAS [22], especially for annihilating dark matter.
In a previous paper [23], we proposed a novel scenario of the decaying dark matter where the final states include two standard model particles and a heavy missing particle. 1 In a certain class of supersymmetric theories where supersymmetry (SUSY) is spontaneously broken in multiple sequestered sectors, there will be a goldstino in each SUSY breaking sector.
Only one linear combination of them is eaten and becomes the longitudinal component of the gravitino. The others are dubbed as goldstini in Ref. [28]. If the lightest supersymmetric particle (LSP) is the gravitino and the next lightest supersymmetric particle (NLSP) is a goldstino, and R-parity is conserved, the goldstino decays through dimension-8 operators to three-body final states containing the gravitino plus a pair of standard model (SM) particles. As a result, the goldstino could have a long lifetime and be cosmologically stable. For suitable SUSY breaking scales in the hidden sectors [23], the lifetime can naturally be in the range of 10 26 − 10 27 sec, which is necessary for the decaying dark matter interpretation of the observed e + /e − excess. Unlike the gravitino, the goldstino interactions with the SM fields are not universal and it is easy to come up with scenarios where the goldstino decays dominantly to leptons [23], thereby producing the observed excesses. A distinct feature of this scenario is that the decay of the dark matter is three-body, with the gravitino escaping detection and carrying off part of the energy. The SM particle pair in the final states has a smooth and soft injection energy spectrum, and consequently, a good fit to both the PAMELA e + /e − and Fermi-LAT e + + e − data can be achieved with a universal coupling to all three generations of leptons, which is a welcoming feature from the model-building point of view. This is in contrast with the case of two-body or four-body final states where decays into e + e − pair are disfavored.
In a more general context, one can imagine other models where the dark matter decays into three-body final states containing a missing particle. Given the very different energy spectra from the well-studied cases of two-body and four-body decays, the scenario of threebody decays obviously deserves a detailed investigation to map out the parameter space allowed by various cosmic ray measurements such as the antiproton and gamma ray data. This is the main purpose of this work. While we use the goldstini scenario as the prime illustrative example, more general setup can be included by studying various effective operators giving rise to the decay of the dark matter. We will see that, for the mass range that could fit both the e + /e − and e + + e − excesses, the injection energy spectra are quite insensitive to the type of operators mediating the decay as well as the spin of the dark matter.
This work is organized as follows. In the next section, we first perform fits to PAMELA and Fermi-LAT e + /e − and Fermi-LAT e + + e − data for a wide range of the dark matter and missing particle masses, using the goldstino decay as an example. Then, we consider different types of higher-dimensional operators that lead to three-body decays and compare the spectra of the decay products from different decay operators. We also compare their fits to the PAMELA and Fermi-LAT e + /e − data. There are small variations but they in general give similar results. In section III, we study astrophysical constraints on three-body decaying dark matter using the gamma-ray and anti-proton data. Our conclusions are drawn in section IV.
II. THREE-BODY DARK MATTER DECAY FOR THE POSITRON EXCESS
In the scenario of goldstino dark matter [23], the goldstino decays to the gravitino through a three-body process. It was shown in Ref. [23] that the leptons pair produced in the decay has a smooth spectrum and can give a good fit to the PAMELA positron excess and the Fermi-LAT e + + e − spectrum, thereby providing a possible explanation of the observed anomalies. In this section we perform a general analysis of the three-body decaying dark matter explanation of the positron excesses in PAMELA and Fermi-LAT data by including the new Fermi-LAT result of the positron excess up to ∼ 200 GeV in our fitting.
In particular, we vary both the dark matter and missing particle masses to obtain a best fit and go beyond the goldstino scenario by studying various effective operators mediating the three-body decay.
A. Goldstini
Goldstini arise when there are multiple sequestered sectors which break SUSY. For simplicity, let us consider that SUSY is spontaneously broken in two hidden sectors, then there is a goldstino in each sector. If the superpartners of the SM particles receive SUSY breaking masses from both sectors, there will be couplings of the SM particles and their superpartners to each goldstino. If the SM superpartners are heavier than the goldstini, upon integrating out the superpartners, we will obtain dimension-8 operators between a pair of goldstini and a pair of SM particles. 2 One linear combination of the goldstini is eaten and becomes the longitudinal mode of the gravitino. The other obtains a mass due to supergravity effects [28]. If there is a hierarchy in the SUSY breaking scales of the two hidden sectors, the eaten goldstino mostly comes from the sector with a larger SUSY breaking scale. The uneaten goldstino, which we assume to be the dark matter, is made mostly of the goldstino in the sector with a smaller SUSY breaking scale. After going to the mass eigenstates, the dimension-8 operators contains interactions which allow the uneaten goldstino decays to a gravitino and a pair of SM particles. If R-parity is conserved, the lifetime is naturally longer than the age of the universe and, with suitable choices of SUSY breaking scales of the hidden sectors, can be the required time scale to explain the e + /e − excess observed by PAMELA and Fermi-LAT.
The goldstino decay operators were derived in the previous paper [23]. We have for SM fermions,
L (1) 2f = − 1 f 2 ef f m 2 1 tan θ − m 2 2 cot θ m 2 q ∂ µ (ζq)∂ µ ( G L q) + h. c. ,(1)
where ζ is the goldstion, G L is the logitudinal mode of the gravitino, q is the SM fermion,
f ef f = f 2 1 + f 2 2
is the effective total SUSY breaking scale with f 1,2 being the SUSY breaking F -terms of the two SUSY breaking sectors, tan θ = f 2 /f 1 , m 2 1,2 are the soft SUSY breaking mass of the superpartner of q coming from the two SUSY breaking sectors respectively, and
m 2 q = m 2 1 + m 2 2 . For gauge bosons, L(1)2γ = −i f 2 ef f m 1 tan θ − m 2 cot θ m λ G L F σ · ∂ (F ζ) ,(2)
where Higgs field, there are terms from both Kähler potential and superpotential,
F (F ) ≡ F µν σ µν (F µν σ µν )L (0) 2h = − 1 µf 2 eff G L ζ m 2 Hu + |µ| 2 φ † u − Bµφ d δm 2 d φ † d − δBµφ u +u ↔ d + h.c. ,(3)L (1) 2h = 1 µ 2 f 2 eff ∂ µ m 2 Hu + |µ| 2 φ u − Bµφ † d G L iσ µ δm 2 u φ † u − δBµφ d ζ +∂ µ δm 2 u φ u − δBµφ † d ζ iσ µ m 2 Hu + |µ| 2 φ † u − Bµφ d G L +u ↔ d + h.c. ,(4)
where
m 2 Hα = i m 2 iα , δm 2 α = m 2 1α tan θ − m 2 2α cot θ , α = u, d ,(5)B = i B i , δB = B 1 tan θ − B 2 cot θ ,(6)
with i = 1, 2 indicating the SUSY breaking sector where the soft SUSY breaking parameters come from. The operators with the Higgs fields can actually induce two-body decay ζ → G L h after substituting the Higgs vacuum expectation values. The two-body decay mode is suppressed by a factor (v/m ζ ) 2 . Compared with the phase space suppression of the threebody decay, the two-body decay rate is expected to be of the same order as the three-body decay rate but may be somewhat larger depending on the model parameters.
Unlike the gravitino, couplings of the goldstino to SM fields are not universal, but rather depend on the fractions of the soft SUSY breaking parameters of the corresponding superpartners coming from the two SUSY breaking sectors. To explain the cosmic e + /e − excess while satisfying other cosmic ray constraints, the decays should dominantly go to leptons.
The constraints on the branching fractions to other particles will be studied in Sec. III.
To the leading order, the goldstino acquires a mass which is twice that of gravitino due to supergravity effects [28]. However, the relation can be modified significantly beyond the leading order [29]. In this subsection, we extend the analysis to explore a wider range of the goldstino mass m ζ and the gravitino masse m G L by relaxing the mass relation of m ζ = 2m G L . We follow exactly the same procedure as in Ref. [23]. To recap, we make use of the Bessel function method of Ref. [30] to obtain the positron flux detected on the Earth due to the DM decay by assuming the MED model parameters. For the background fluxes we employ the "model" presented by the Fermi-LAT collaboration [31], which can be analytically parameterized as shown in Ref. [32]. The Moore profile in Ref. [33] is used for shows the contour plot of m ζ versus m ζ − m G L , from which we see that χ 2 ∼ 1 occurrs when m ζ − m G L ∼ 1 TeV and m ζ 2 TeV. Consequently, we will use m ζ = 2 TeV as a benchmark value for later discussion.
The underlying reason for these features is the hardening feature around 300 ∼ 500
GeV in the e + + e − flux from Fermi-LAT, which dictates m ζ − m G L to be around 1 TeV.
This number comes about because we assume universal couplings to all three lepton flavors and the resulting softer energy spectra due to the µ and τ components drives the value of m ζ −m G L to be larger than twice the energy scale at which the hardening spectrum appears:
m ζ − m G L 600 − 1000 GeV.
On the other hand, the goldstino mass m ζ , which sets the overall energy scale, need to be heavier than 2 TeV since a smaller mass will produce a feature in the total e + + e − flux at around 500 − 1000 GeV, which worsens the fit with the In addition to varying masses of the goldstino and gravitino, we also consider the best fit lifetime τ of the decaying goldstino, which is determined by the strength of the excess in the PAMELA e + /e − flux in the energy regime of 10 − 200 GeV. We present a contour plot corresponding to 90%, 95% and 99% C.L. of fits to PAMELA and Fermi-LAT data in the m G L versus τ plane in Fig. 3. It shows that for m ζ = 2 TeV, the best fit corresponds to m G L ∼ 900 GeV. As m G L increases (decreases), it results in the softer (harder) injection spectrum, which can be compensated by a shorter (longer) lifetime.
Before switching to three-body kinematics, we would like to comment that for m ζ ≥ 2 TeV, the superpartners of the SM particles need to be heavier than a few TeV which implies some fine-tuning of the electroweak breaking scale. However, the absence of the SUSY signal at the LHC so far indicates that the superpartners may indeed be heavier than expected if SUSY exists at the TeV scale. The hint of a Higgs boson around 125 GeV is also consistent with a heavy SUSY spectrum.
B. Three-Body Dark Matter Decay Kinematics
If a dark matter particle decays to two-body final states, the energies of the decay products in the rest frame of the DM particle are fixed. In contrast, the three-body decay will produce softer and broader spectra as we saw in the goldstino example. One can imagine there are other models where DM particles have similar three-body decays into a missing particle and two SM particles. The spectra of the decay products depend on the properties of DM particles and their interactions, but do share some generic features. By parametrizing different three-body decay mechanisms by effective higher dimensional operators, we will examine the similarities and differences of various three-body decay spectra in this subsection.
We start by comparing the injection energy spectrum of the SM fermions from decays via dimension-six four-fermi interactions with that from the benchmark scenario of goldstino decays, which occur through the dimension-eight operator in Eq. (1). More specifically, we consider the following Dirac structures [27]: (pseudo-)scalar, (pseudo-)vector and tensor interactions, which are written as
λ 1 Λ 2 Ψ X Ψ DM Ψ q Ψ q + h. c., λ 2 Λ 2 Ψ X γ 5 Ψ DM Ψ q γ 5 Ψ q + h. c., λ 3 Λ 2 Ψ X γ µ Ψ DM Ψ q γ µ Ψ q + h. c.,(7)λ 4 Λ 2 Ψ X γ µ γ 5 Ψ DM Ψ q γ µ γ 5 Ψ q + h. c., λ 5 Λ 2 Ψ X σ µν Ψ DM Ψ q σ µν Ψ q + h. c.,
where we assume both the dark matter (DM) and the missing particle in the decay product (X) are Dirac fermions, while q refers to the SM fermions. The decay process induced by these operators is Fig. 4(a) shows the comparison of energy spectra of SM fermions from the three-body decaying dark matter, where the total width is normalized to unity and the masses are fixed to m DM = 2m X = 2 TeV. The goldstino decay results in the hardest spectrum due to fact that goldstino is derivatively coupled, which gives more weight to the higher energy modes. However, one sees that all spectra exhibit similar qualitative features, peaking in the 400 − 600 GeV region. Therefore, we expect all of them to give rise to similar fits to the Fermi-LAT and PAMELA data points.
DM → X + q + q .(8)
In addition to decays to SM fermions, we also compare various decay operators into SM gauge bosons, as they will be constrained by the anti-proton data from PAMELA as well as the gamma ray data from Fermi-LAT. Here we list three types of different operators to be compared with the benchmark scenario of goldstino decay in Eq. (2):
λ h v 2 Λ 3 Ψ X Ψ DM W a µ W aµ + h. c., λ s Λ 3 Ψ X Ψ DM F a µν F aµν + h. c.,(9)λ v Λ 3 Ψ X γ µν γ ρσ Ψ DM F a µν F a ρσ + h. c.,
where F a µν can be any of the SU (3), SU (2), and U (1) SM gauge bosons. The suppression of v 2 in front of the first operator signals its origin from the gauge invariant operator
(λ h /Λ 3 )Ψ X Ψ DM D µ H † D µ H.
We see in Fig. 4(b) that, again, various operators have very similar qualitative features, even more so than the energy spectra of the SM fermions.
dΓ/dE DM=2 TeV X=1 TeV Goldstino decaȳ ΨX ΨDMΨq Ψq/Λ 2 ΨX γ µ ΨDMΨq γµΨq/Λ 2 ΨX γ µ γ 5 ΨDMΨq γµγ 5 Ψq/Λ 2 ΨX γ 5 ΨDMΨq γ 5 Ψq/Λ 2 ΨX σ µν ΨDMΨq σµν Ψq/Λ 2 (a)dΓ/dE DM=2 TeV X=1 TeV Goldstino decaȳ ΨX ΨDM W aµ W a µ v 2 /Λ 3 ΨX ΨDM F a µν F aµν /Λ 3 ΨX γ µν γ ρσ ΨDM F a µν F a ρσ /Λ 3 (b)dΓ/dE DM=2 TeV X=0 TeV Goldstino decaȳ ΨX ΨDMΨq Ψq/Λ 2 ΨX γ µ ΨDMΨq γµΨq/Λ 2 ΦX ΦDMΨq Ψq/Λ ΦX ∂ µ ΦDMΨq γµΨq/Λ 2 (b) FIG. 5:
The comparison between spin-0 and spin-1/2 dark matter with several decay mechanisms.
We assume m DM = 2m X = 2 TeV.
For the higher dimensional operators involving a pair of Higgs fields, if the Higgs fields are not both derivatively coupled, substituting in the Higgs VEV will induce a two-body decay DM → h X. The Higgs boson will have a delta-function spectrum independent of the exact operators. We shall see that this two-body decay gives comparable constraints to those from the corresponding three-body decay in section III.
We conclude this subsection by comparing energy spectra of three-body decaying scalar dark matter with those of fermionic dark matter. From Fig. 5(a) it is notable that, for m X = m DM /2, spectra of scalar dark matter follows closely the counterparts of fermionic dark matter. However this behavior disappears in the limit of massless missing particle mass, m X = 0, as shown in Fig. 5 Energy (GeV) We first examine the fits for two-body decays of dark matter to a pair of leptons with universal couplings to all three generations. As explained before, the observed excess of positron fraction from both PAMELA and Fermi-LAT can be accounted for by varying the lifetime of the decaying dark matter as long as the injection energy spectrum is not too soft. However, in order to explain the hardening feature in the total e − + e + flux observed by Fermi-LAT, the injection spectrum must peak around O(400) GeV. As a consequence, the two-body decay has difficulty fitting both the positron fraction and total flux, since the injection energy is a delta-function peak at half the mass of the dark matter in the centre-of-mass frame, which is demonstrated in the upper panels of Fig. 6. Note that one could use only the PAMELA and Fermi-LAT e + /(e − + e + ) data in computing the χ 2 /d.o.f., then the positron excess can be well described with a shorter lifetime compared to those of three-body decay cases. But the resulting total (e − + e + ) flux is in severe tension with the Fermi-LAT data, as can be seen in the lower panel of Fig. 6.
E 3 dΦ dE (GeV −2 cm 2 s sr) −1 DM → Xl + l − (ΨXγ µ ΨDMΨqγµΨq) DM
The best fits for several three-body decay mechanisms are displayed in Fig. 7, again assuming universal couplings to all three lepton flavors. We see the overall fits to both PAMELA and the Fermi-LAT are better than the case of two-body decay. Among the different decay mechanisms, the goldstino case has largest missing particle mass, which softens the harder energy spectrum resulting from its derivative couplings. Similarly, the pseudo-scalar operator needs a larger m X than the vector one due to a harder injection spectrum, but a shorter lifetime, driven mostly by the excess of e + /(e − + e + ), to compensate for the flatness in the spectrum shown in Fig. 4(a).
So far we have assumed universal couplings to all three lepton flavors. However, the (pseudo-)scalar operators break the chiral symmetry and one may expect that the coefficients are proportional to the fermion masses, which implies that the decay to τ leptons will be dominant. In this case, these two operators can not explain both PAMELA and Fermi-LAT data. The spectrum of e + (e − ) from τ decays is quite soft and would populate mostly the low-energy region. In order to explain the excess in PAMELA data, the lifetime has to be increased significantly and the missing particle mass m X has to be reduced, which results in a much harder spectrum than the Fermi-LAT total e − + e + flux data as shown in Fig. 8.
III. ASTROPHYSICAL CONSTRAINTS
In this section we consider astrophysical constraints on three-body decaying dark matter. There are three main categories: 1) the diffuse gamma-ray due to inverse Compton scatterings (ICS) and final state radiation (FSR), 2) prompt photons that are direct decay product of the dark matter, and 3) anti-proton flux measurements which constrain decays into hadronic final states.
A. Diffuse γ
It is known that the leptonic final states from dark matter decays yield photons via FSR as well as ICS when the produced leptons interact with background photons. Both are subject to the constraints from the Fermi-LAT gamma ray data. In this subsection, we study such constraints by assuming the e + /e − excess in PAMELA and Fermi-LAT is the consequence of dark matter decays. Since we have shown that different three-body decay operators have similar best-fits to e + /e − data for 2 TeV dark matter mass, it is sufficient to focus on the goldstino case only.
We make use of the Fermi-LAT gamma ray data in Ref. [34], which provides information of diffuse Galactic emission (DGE) and extragalactic gamma ray background (EGB). For DGE we fit to the "Galactic diffuse (fit)" data from Table I in Ref. [34], which is DGE averaged over the Galactic latitude range |b| ≥ 10 • as measured by Fermi-LAT. The DM signal results from the sum of ICS and FSR produced by charged leptons, the products of the goldstino decay. The averaged photon flux from FSR over a solid angle of interest from Table 3 in Ref. [35] and a numerical code for ICS in Ref.
[36] are utilized for calculating the photon flux from the Galactic dark matter decay. For the fitting procedure, we vary the normalization of the background, whose shape is taken from the "Galactic diffuse (model)"
numbers in Table 1 of Ref. [34], to minimize the χ 2 for various goldstino decay widths. In addition, for the goldstino decay, the gravitino mass is extracted from the best fits to the e + /e − and e + + e − data.
Before presenting the results, we would like to comment on the effect of different electron We present results in Fig. 9 corresponding to the 90% and 95% C.L. upper (lower) limit on the decay width (or equivalently lifetime) of the goldstino, normalized to Γ best from best fits to e + and e − data, based on the Fermi-LAT DGE data. Again, the photon flux resulted from the goldstino decay consists of both ICS and FSR contributions. The goldstino decay can satisfy the diffuse gamma ray constraints. In contrast, for two-body decays with the MAX or MED propagation model, the decaying dark matter is ruled out as an explanation to the excess of e + and e − . In the case of MIN, both two-and three-body decay can avoid the gamma ray constraint as explained before. Note that FSR contribution is numerically subdominant compared to those of ICS on both two-and three-body cases. The reason is that the τ decay channel, which produces relatively more photons via showering than e and µ channel, gets diluted by the assumption of universal leptonic couplings. Therefore, if the dark matter decays into τ only, FSR would be far more stringent than ICS as demonstrated in Refs. [17,18].
On the other hand, the Fermi-LAT EBG data provide constraints on the isotropic diffuse gamma ray. From Ref. [19], the DM decay contributes to the isotropic flux in two ways, which can be expressed in terms of the differential flux,
dΦ Isotropic dE γ = dΦ ExGal dE γ + 4π dΦ Gal dE γ dΩ minimum ,(10)
where the first term is an isotropic extragalactic cosmological flux from the decays at all past redshifts and the second term is the residual contribution from the Galactic DM halo. Note that the latter is of course not isotropic but the minimum will be the irreducible component of the isotropic gamma ray flux. We employ the same assumption as in Ref. [19] that the minimum of Galactic contribution is located at the anti-Galactic Center. With the help of the code, EGgammaFluxDec in Ref. [36], which is based on the two-body decay and includes both FSR and ICS from the primary charge leptons [37], we properly convolute it with our three-body injection spectra to obtain the isotropic cosmological flux. For the fitting procedure, we again use the EGB data from Table I In this subsection, we explore the situation where photons are direct products of threebody dark matter decays, i.e., primary photons. 3 In this situation, we have two independent parameters: the decay widths of goldstino into leptons and into photons. Moreover, the results are obviously independent of the electron propagation models like FSR and the cosmological diffuse flux, and the constraints would be model-dependent. As the injection spectra of different operators from II B are similar, we expect those four operators to have the similar results. Here, we also involve both DGE and EBG data with the same fitting procedure described before. From Fig. 11, the constraint mainly comes from EGB data and the goldstino decay can not be ruled out even if the goldstino decays 80% into photons and 20% into leptons, which accounts for the e + /e − excess observed by PAMELA and MED pass the test of the isotropic diffuse gamma ray flux. Like in Fig. 9, we see the dip appears especially for the goldstino decay, but here the power-law index of the background is being varied and therefore the shape of the background changes for the different DM lifetime. In addition, the change on the power-law index is also the underlying reason for the local minimum around ∆χ 2 ∼ 5 in the case of the two-body decay in MAX and MED.
Fermi-LAT. The underlying reason is, for 2 TeV goldstino, the prompt photons are mainly located beyond 100 GeV and consequently the DGE data yield a very loose bound. In these cases, Air Cherenkov Telescopes such as Veritas are well poised for DM decay to photons GeV goldstino is severely constrained by Fermi gamma ray data.
of O(TeV) energy. On the other hand, the cosmological diffuse photons will get redshifted and constrained by the EGB data. For comparison, 500 GeV goldstino would be severely constrained by both of DGE and EGB data as shown in Fig. 11.
C.p
In addition to measurements of the e + spectra made by PAMELA and Fermi-LAT, thep spectra has also been well measured by PAMELA. While the e + species show an excess at high energies, thep measurement shows a spectra consistent with astrophysical sources [39].
The most recent PAMELA anti-proton measurement from 60 MeV to 180 GeV has further confirmed this result [40]. This agreement with the expected background can serve to limit the total annihilation rate of dark matter to hadronic final states [41] as well as hadronic interactions of the dark matter, which could have important implications for direct detection experiments [42].
We study the restrictions of hadronic decay modes of the dark matter by the potential contribution from the decay to qq, gg, W + W − , ZZ and hh modes, with m h = 125 GeV. In the left panel of Fig. 12 we show the fragmentation function of the above final states into an anti-proton. Since these final states may also produce electrons upon further decays, it is important to study whether the partial width allowed by the anti-proton spectra could result in additional noticeable contributions in the positron measurements that were not included in our fits in the previous section. Thus in the right panel of Fig. 12 we also show the fragmentation function of all possible final states into the positron. If the dark matter decays through two-body kinematics, then Fig. 12 gives precisely the injection energy spectra of the anti-proton and the positron. For three body decay, these energy distributions are convolved with the intermediate state's energy distribution. It is clear that due to the nature of fragmentation and hadronization, all non-leptonic modes provide a soft anti-proton spectra.
Once produced, the anti-protons propagate through the galaxy under the effects of diffusion due to the galactic magnetic field, the convective wind away from the plane of the galaxy and annihilations with interstellar protons; we model the anti-proton propagation according to Ref. [43]. As with the modeling of the e + propagation, we primarily adopt the MED model and assume the dark matter is distributed according to the Moore profile.
In addition, once the anti-protons approach Earth, solar modulation effects alter their low energy spectra. We include this effect for anti-protons since the energy range for solar modulation is well positioned within the PAMELA data. To this end, we adopt a Fisk potential of φ = 500 MV [44].
To illustrate how consistent the two-body and three-body decay scenarios are with the anti-proton data, we fit the prediction of each model to the PAMELAp/p data [40]. We take the best-fits to the electron data in Sect. II C and simultaneously vary the dark matter signal normalization and totalp background normalization. We show in detail the fits for the qq and W + W − mode in the MED propagation model. We later summarize the results for other decay modes and propagation models.
The two-body decay of dark matter yields two particles with well defined energy. If they are massless, the resulting anti-proton spectra appears identical to Fig. 12 with E parent = m DM /2. The energy of injected anti-protons from massive parents are altered with twobody kinematics. In Fig. 13, we show the two-body dark matter decay to quark modes (left panels) and W boson modes (right panels) for m DM = 2 TeV. The comparison with the PAMELA anti-proton data (top panels) shows that a considerable anti-proton flux is possible in energies beyond the present data. The ∆χ 2 fits (middle panels) show no preferred value for the partial width, as expected since no excess is observed in this data. Had there been a preferred decay rate, the position of the minimum χ 2 would not be asymptotically approaching zero. Finally, we show in the bottom panels the components of the proton and anti-proton flux based on the 95% C.L. fit. Overall, we see that decay rate to the quark mode at 95% C.L. must be smaller than ≈ 1/(2.4 × 10 27 s) and the rate for the W -boson mode must be smaller than ≈ 1/(1.3 × 10 27 s). One could also compare these bounds on the hadronic decay width with the best fit decay widths into the lepton modes from fitting the electron/positron data. If we take the best-fit lifetime from fitting twobody decays to both the positron fraction and the total e + + e − flux, which is shown in the upper panels in Fig. 6, the hadronic partial widths fall around O(10%) of the lepton modes.
The upper limit on the two body decay of dark matter in the W -boson mode is roughly lower limits from the PAMELAp data on the decay lifetime of the two body decay modes to qq, gg, W + W − , ZZ, and hh are also summarized. In parentheses are the respective ratios between the leptonic lifetime fit and the hadronic lifetime limit. We see that obviously the most constrained modes are the purely hadronic qq and gg modes. Generally, within the MAX propagation model, more suppression into the decay of hadronic final states is required to maintain agreement with data. Within the MIN propagation model, the hadronic states have to be suppressed no more than an order of magnitude.
Γ DM →W + W − < 0.17 × Γ DM → + − from
Generically, the three-body decays through various operators of dark matter to quarks and W -bosons create similarp spectra. We therefore concentrate on the analysis of thep flux in the goldstino decay scenario where it decays through the dimension-8 operators. Since the injected q and W in the three-body decay is of lower energy, the resulting anti-protons are softer than the two-body case. As a result, the constraint on the allowed decay width in every hadronic channel is weaker than the corresponding two-body case, which can be seen by comparing Table II with Table I. The decay rate to the quark mode at 95% C.L. must be smaller than ≈ 1/(1.3 × 10 27 s) while the rate for the W -boson mode must be smaller than ≈ 1/(1.1 × 10 27 s). In terms of the ratio of the hadronic decay rates over the best-fit leptonic decay rates, the 95% C.L. upper limit on the three-body decay of dark matter in the W -boson mode is roughly Γ DM →W + W − < 0.14 × Γ DM → + − from the PAMELA anti-proton data. Likewise the limit on the q mode is Γ DM →qq < 0.11×Γ DM → + − . Similar to the Table I for the two-body decays, the propagation model dependence for three-body decays is shown in Table II. The bounds is much looser than ones from antiproton data which justify our fitting procedure.
ζ → G L + − ζ → G L qq ζ → G L gḡ ζ → G L W + W − ζ → G L ZZ ζ → G L hh ζ → G L h MAX
It is worthwhile to mention that, for the three-body decay, in principle, we should simultaneously fit both of the positron (electron) and the antiproton data by including all kinds of decay products such as leptons, quarks and gauge bosons, etc. Instead, we fit positron data first with universal couplings to leptons only and obtain m G L or m X , and then fit antiproton data with the extracted m G L (or m X ) to see how much the hadronic final state is allowed. In this procedure, first it is easier to pin down the sufficient conditions for good fits to positron data and second it is numerically faster on searching for minima. Nevertheless, one can argue that including other final states might change best fits to positron data significantly since these final states yield positrons (electrons) as well, and therefore modify the whole picture. Fig. 15 clearly shows that inclusion of hadron final states will have a modest effect in terms of χ 2 on fits to PAMELA and Fermi-LAT positron data even if the quarks or W -bosons comprise more than 25% of decay products, which is much higher than the upper bounds from the antiproton data consideration. Similar conclusions hold for the effect on the gravitino mass and the goldstino lifetime. As a result, our way of fitting is properly justified.
IV. CONCLUSIONS
In this work we have attempted to make a strong case for the three-body decaying dark matter, by studying how the three-body kinematics could allow simultaneous fits to both the PAMELA and Fermi-LAT positron excesses without being excluded by other astrophysical measurements such as the gamma-ray and anti-proton data. As a contrast, conventional decaying dark matter models have trouble achieving the above goal due to the restrictive two-body decay kinematics.
Using the goldstino as a prime example of the three-body decaying DM, which arises in a certain class of supersymmetric theories where SUSY is broken by multiple sectors, we found that the goldstino decay with universal leptonic couplings could fit PAMELA and Fermi-LAT positron data well, if the goldstino mass is at around 2 TeV and the gravitino, which escapes detection and results in missing energy, has a mass around 1 TeV. The mass difference is driven by the hardening feature around 300 ∼ 500 GeV in the Fermi-LAT e + + e − data. A slightly smaller (larger) mass difference than 1 TeV can be counterbalanced with a shorter (longer) lifetime.
We further demonstrated that these features of goldstino decays persist in other types of three-body decay mechanisms, by studying the injection energy spectra of several different types of four-fermi interactions with universal leptonic couplings, as well as scalar dark matter decays. We found that these other mechanisms all have softer injection spectra and consequently smaller missing particle mass for best fits to the positron data. However, the best fit χ 2 are all similar to the goldstino case. The only exception is when certain operators are associated with chiral symmetry breaking and might have couplings proportional to the masses of SM fermions. Then the τ decay channel would be the dominant one and in this situation the resulting soft injection spectrum can not fit e + /(e − + e + ) and e − + e + at the same time, while satisfying the gamma ray constraints. Interestingly, the scalar dark matter have the very similar injection spectrum for the mass region of interest and in turn the similar best fits. These observations suggest that it would be difficult to distinguish different decay mechanisms using data considered in this work.
One important advantage of three-body decaying dark matter over other conventional models is the ability to avoid null searches in cosmic gamma-ray and anti-proton data for dark matter. Due to the softer injection energy spectra of the three-body kinematics, we showed that the diffuse gamma-ray measurements are compatible with the three-body decays, while at the same time maintaining the fits to the PAMELA and Fermi-LAT positron data. The allowed hadronic decay widths from the anti-proton data are also larger in the three-body decay scenario than in the two-body case, due to the softness of the decay energy spectra.
In the end, we hope it is clear that there is a strong case for three-body decaying dark matter, if the excesses in the positron measurements by PAMELA and Fermi-LAT are believed to be due to dark matter. It would therefore be important to explore ways to definitively determine the decay mechanism of the dark matter in other types of measurements, which will be a subject for future studies.
FIG. 1 :
1The best fits to the e + /(e − + e + ) ratios observed by PAMELA and Fermi-LAT, and the Fermi-LAT e − + e + spectrum for several goldstino masses.the DM halo. With the assumption that the dark matter density is ρ = 0.3 GeV/cm 3 , we perform combined fits to both PAMELA and Fermi-LAT data, including the new Fermi-LAT positron fraction data[2], by varying the decaying DM lifetime and the overall normalization of the primary e − component of the background flux, as described in Ref.[32]. Because the e + /e − flux with energies below 10 GeV measured at the top of the atmosphere is significantly influenced by the solar modulation effect, we only include data points above 10 GeV from PAMELA for the total χ 2 . We assume universal couplings to all three lepton flavors which give a better fit than couplings to any single flavor.The fits to the PAMELA and Fermi-LAT data for several sample goldstino masses are shown inFig. 1. The χ 2 per degree of freedom (d.o.f) of the fits as functions of the goldstino mass m ζ and the gravitino mass m G L are shown in Fig. 2. The solid line in Fig. 2(a) is the best fit χ 2 /d.o.f. for a given m ζ by varying m G L freely, only subject to m G L ≤ m ζ . We see that χ 2 /d.o.f. asymptotically approaches 1 when m ζ 2 TeV. The dashed line, however,assumes m ζ = 2m G L , in which case the minimum of χ 2 occurs around m ζ ∼ 2 TeV.Fig. 2(b)
FIG. 2 :
2These plots display relations between χ 2 /d.o.f. of best fits to both PAMELA and Fermi-LAT data and the goldstino mass m ζ . On the left plot, we vary m ζ and m G L independently (solid line) as well as dependently by assuming the mass relation: m ζ = 2m G L . The right plot shows contours on the m ζ and m ζ − m G L plane.
FIG. 3 :
3The contour plot represents 90%, 95% and 99% C.L. for fits to PAMELA and Fermi-LAT data by varying the m G L and the lifetime τ of ζ.
FIG. 4 :
4The comparison among different DM decay mechanisms into fermions and W W , by normalizing the total decay width Γ = 1. We assume the mass of the dark matter is 2 TeV and 1 TeV for X and G L .
FIG. 6 :
6Fits to PAMELA and Fermi-LAT using two-body decaying dark matter. For the upper panel we include data on both positron fraction and total e − + e + flux in the fit, while in the lower panel we only include the positron fraction data in computing the χ 2 .
DMFIG. 7 :
7→ Xl + l − (ΨXγ 5 ΨDMΨqγ 5 Ψq) The fits to PAMELA and Fermi-LAT between different operators. Three-body decays have the similar χ 2 and lifetime. m G L controls the hardness of injection energy spectra; therefore, goldstino has the largest m G L to balance the hardest spectrum.
FIG. 8 :
8Best fits for the pseudo-scalar operator to PAMELA and Fermi-LAT combined, assuming couplings proportional to the fermion mass. The PAMELA(Fermi-LAT) excess can be accounted for properly at the cost of a longer lifetime and m X = 0, which yields an unwanted bump in the Fermi-LAT e − + e + data.
propagation models, dubbed MIN, MED, and MAX[30] [38]. For FSR, secondary photons are produced near where primary leptons are produced and once produced, photons propagate directly toward the Earth; therefore, they are independent of the propagationmodels of electrons through the galaxy. On the other hand, ICS photons are produced from electrons scattering off background photons while the electrons propagate and lose energy. Hence, it has dependence on the propagation models of electrons. For the MIN model, the low/intermediate energy electron ( 200 GeV) will be more easily stopped and fail to produce ICS photons. Fermi-LAT gamma ray data ranging from 0.2 to 100 GeV are influenced by this effect. Therefore, for both two-and three-body decay, decay scenarios adopting the MIN propagation model are less constrained compared to MAX and MED. MAX has an opposite feature, i.e., producing more low/intermediate ICS photons but the effect is less dramatic than MIN. In other words, MAX is similar to MED.
ζζ
DM→ l + l − (MED) DM(2 TeV)→ l + l − 90% C.L. τ > 3.50 × 10 26 sec 95% C.L. τ > 3.12 × 10 26 sec τ best = 2.13 × 10 26 sec → GLl + l − (MED) ζ(2 TeV)→ GLl + l − 90% C.L. τ > 1.15 × 10 26 sec 95% C.L. τ > 1.02 × 10 26 sec τ best = 1.44 × 10 26 sec DM→ l + l − (MIN) DM(2 TeV)→ l + l − 90% C.L. τ > 8.10 × 10 25 sec 95% C.L. τ > 7.21 × 10 25 sec τ best = 2.01 × 10 26 sec → GLl + l − (MIN) ζ(2 TeV)→ GLl + l − 90% C.L. τ > 1.55 × 10 25 sec 95% C.L. τ > 1.37 × 10 25 sec τ best = 8.77 × 10 25 sec FIG. 9: The goldstino decay can satisfy the Fermi-LAT DGE gamma ray constraints. In contrast, for the two-body decay, only MIN is allowed by virtue of the reason mentioned in the text. Note that for the certain range of the DM lifetime, the existence of the DM signal compensates the difference between the data and background and in turn results in dips in χ 2 fits.
of Ref.[34], which can be described very well by a featureless power law with index γ = 2.41 ± 0.05, and hence a power law background. The normalization of the background and the index of the power law are being varied to find the minimum of χ 2 for different goldstino lifetimes. The results are presented inFig. 10. Based on the numerical results, the cosmological flux, which is independent of the propagation model, is dominant over that of the DGE minimum and clearly the bounds on the goldstino lifetime are similar for different propagation models. Therefore, the EGB data impose more stringent constraints than those of DGE, i.e.,Fig. 9, subject to uncertainties on the propagation models. We can see that all of two-body decay are excluded and even the goldstino decay in the case of MIN is also disfavored, since the lifetime (decay width) of the goldstino is shorter (larger) in MIN to account for fewer electrons and positrons reaching the Earth.B. Prompt γ
ζζ
DM→ l + l − (MED) DM(2 TeV)→ l + l − 90% C.L. τ > 4.19 × 10 26 sec 95% C.L. τ > 3.58 × 10 26 sec τ best = 2.13 × 10 26 sec → GLl + l − (MED) ζ(2 TeV)→ GLl + l − 90% C.L. τ > 1.16 × 10 26 sec 95% C.L. τ > 1.03 × 10 26 sec τ best = 1.44 × 10 26 sec DM→ l + l − (MIN) DM(2 TeV)→ l + l − 90% C.L. τ > 3.87 × 10 26 sec 95% C.L. τ > 3.33 × 10 26 sec τ best = 2.01 × 10 26 sec → GLl + l − (MIN) ζ(2 TeV)→ GLl + l − 90% C.L. τ > 1.21 × 10 26 sec 95% C.L. τ > 1.08 × 10 25 sec τ best = 8.77 × 10 25 sec FIG. 10: The constraints from the Fermi-LAT EGB data. Only the goldstino decay with MAX and
TeV) → GLγγ 90% C.L. τ > 2.76 × 10 25 sec 95% C.L. τ > 2.45 × 10 25 sec τ best = 1.44 × 10 26 sec GeV) → GL(250 GeV)γγ ζ(500 GeV) → GL(250 GeV)γγ 90% C.L. τ > 1.47 × 10 27 sec 95% C.L. τ > 1.30 × 10 27 sec τ best = 1.44 × 10 26 sec FIG. 11: The Fermi-LAT DGE constraints (upper panels) and EGB constraints (lower panels). The 2 TeV goldstino decay to photons could easily pass the test of the Fermi-LAT gamma ray data because the injection spectrum of photons peaks well beyond 100 GeV. On the other hand, a 500
FIG. 12 :
12Antiproton and electron energy distribution functions of µ, τ -leptons, u-quarks, gluons, W, Z and Higgs boson decay. The hardness of the electron in the lepton and W/Z decays result from prompt production, whereas the quark/gluon decays are very soft, producing electrons only after QCD showering and hadronization. Similarly, the antiproton energies are very soft and are similar for quarks and W bosons, with a slightly harder component for W decays. For two body decay, the resulting positron and antiproton spectra are similar to these distributions. The corresponding three body decay convolves the intermediate state energy with these distributions.
the PAMELA anti-proton data. Likewise the limit on the q mode is Γ DM →qq < 0.09 × Γ DM → + − . However, these ratios are a factor of 2 − 3 smaller if one only includes the positron fraction in the fit for the leptonic widths, which was demonstrated in the lower panels ofFig. 6.The quark mode is more constrained by roughly a factor of two relative to the W W mode since the anti-protons originate from quark hadronization. In comparison, the W -boson can decay to ν, giving a reduction in the overallp rate. Moreover, as the W -boson decays to pairs of quarks, the resulting anti-protons are softer compared with the dark matter decay to quarks. However, below Tp = m p E inj /m W , thep spectra is cut off. This can be understood since thep's are predominantly relativistic, meaning they have a kinetic energy of at least O(m p ) in the W boson rest frame. Boosting to the dark matter rest frame places the energy cutoff near Tp = m p E inj /m W . Overall, the flux from lower energy anti-protons becomes buried in the increasing backgroundp flux, thus providing a lowerp/p contribution. This is clearly seen in the position of the peak of the DM signal in the lower panels. The propagation model dependence is shown inTable I. The decay lifetime fit to the lepton modes from the PAMELA data are shown in units of 10 26 s. The subsequent 95% C.L.
FIG. 13 :
13Contribution to thep/p ratio from dark matter 2-body decay for the W and q decay modes. Since the W -boson tends to produce fewerp than the q-mode, the decay rate allowed by the PAMELAp data is larger by nearly a factor of two. Dark matter decay rates for W -bosons (q) as high as 0.17 (0.09) × Γ DM → + − can be accommodated by the PAMELAp data.
FIG. 14 :
14Similar toFig. 13, but for dark matter three-body decay via the Dim-8 goldstino decay operators. Decay rate for W -bosons (q) as high as 0.14 (0.11) × Γ DM → G L + − can be accommodated by the PAMELAp data.
FIG. 15 :
15The effect on fits to PAMELA and Fermi-LAT from inclusion of hardonic final states.
is the gauge field strength tensor, m 1,2 are the SUSY breaking gaugino masses coming from the two SUSY breaking sectors, and m λ = m 1 + m 2 . For the The operators involving the Higgs fields actually have a lower dimension. However, it is suppressed by the same SUSY breaking scale with the mass dimension made up by the Higgs mass parameters in the numerator.2
(b). Therefore, in terms of fits to PAMELA and Fermi-LAT, we expect to have similar χ 2 /d.o.f. for both scalar and fermionic dark matter.C. e + and e − Having established that the injection energy spectra of three-body decaying dark matter do not depend sensitively on the dynamics giving rise to the decay in the mass range of interest, we now illustrate how three-body decaying dark matter could give better fits to both PAMELA and Fermi-LAT e + /e − data than the conventional model of two-bdoy decaying dark matter.
TABLE I :
IDark matter two-body decay mode lifetime dependence on the propagation model for fitsto the PAMELA/Fermi e + , Fermi e + /(e + + e − ), and PAMELAp data with m ζ = 2 TeV. The decay to the + − mode is the best lifetime fit provided by the Fermi-LAT and PAMELA electron data, while the other decay modes are 95% C.L. lower limits on the lifetime and the ratio with respect to the best fit lepton lifetime in parenthesis.10 26 s ζ → + − ζ → qq ζ → gḡ ζ → W + W − ζ → ZZ ζ → hhτ
MAX
2.17
49.7 (22.9) 80.9 (37.3) 29.6 (13.7) 31.4 (14.5) 47.5 (21.9)
MED
2.13
23.8 (11.2) 39.2 (18.4) 12.8 (6.01) 13.7 (6.42) 21.0 (9.86)
MIN
2.01
5.63 (2.80) 9.30 (4.62) 2.89 (1.44) 3.09 (1.54) 4.78 (2.38)
TABLE II :
IIGoldstino decay mode lifetime dependence on the propagation model for fits to the PAMELA/Fermi e + , Fermi e + /(e + + e − ), and PAMELAp data with m ζ = 2 TeV. The decay to the G L + − mode is the best lifetime fit provided by the Fermi and PAMELA electron data, while the other decay modes are 95% C.L. lower limits on the lifetime and the ratio with respect to the best fit lepton lifetime in parenthesis.τ
10 26 s
Other possibilities of three-body DM decays were considered in Ref.[24] ,[25] ,[26] ,[27] and by A. Ibarra et al in Ref.[6].
Note that the signal of the two-body prompt decay is situated beyond the reach of the Fermi-LAT gamma ray data for 2 TeV dark matter[34].
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|
[] |
[
"Black Hole Mass Estimates Based on C iv are Consistent with Those Based on the Balmer Lines †",
"Black Hole Mass Estimates Based on C iv are Consistent with Those Based on the Balmer Lines †"
] |
[
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"K D Denney \nDepartment of Astronomy\nThe Ohio State University\n140 W. 18th Ave43210ColumbusOHUSA [\n",
"C S Kochanek \nDepartment of Astronomy\nThe Ohio State University\n140 W. 18th Ave43210ColumbusOHUSA [\n\nThe Center for Cosmology and Astroparticle Physics\nThe Ohio State University\n191 West Woodruff Avenue43210ColumbusOHUSA\n",
"B M Peterson \nDepartment of Astronomy\nThe Ohio State University\n140 W. 18th Ave43210ColumbusOHUSA [\n\nThe Center for Cosmology and Astroparticle Physics\nThe Ohio State University\n191 West Woodruff Avenue43210ColumbusOHUSA\n",
"S Koz Lowski \nDepartment of Astronomy\nThe Ohio State University\n140 W. 18th Ave43210ColumbusOHUSA [\n",
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"P Buschkamp \nMax-Planck-Institut fuer Extraterrestrische Physik, Giessenbachstr\nD-85748GarchingGermany\n",
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"E Falco \nHarvard-Smithsonian Center for Astrophysics\n60 Garden Street02138CambridgeMAUSA\n",
"C Feiz \nLandessternwarte, ZAH\n10 Astron. Institut\nRuhr Univ. Bochum\nKoenigstuhl 12, Universitaetsstr. 150D-69117, D-44780Heidelberg, BochumGermany, Germany\n",
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"R Hofmann \nMax-Planck-Institut fuer Extraterrestrische Physik, Giessenbachstr\nD-85748GarchingGermany\n",
"M Juette ",
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"V Knierim ",
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"A Quirrenbach \nLandessternwarte, ZAH\n10 Astron. Institut\nRuhr Univ. Bochum\nKoenigstuhl 12, Universitaetsstr. 150D-69117, D-44780Heidelberg, BochumGermany, Germany\n",
"W Seifert \nLandessternwarte, ZAH\n10 Astron. Institut\nRuhr Univ. Bochum\nKoenigstuhl 12, Universitaetsstr. 150D-69117, D-44780Heidelberg, BochumGermany, Germany\n",
"D Stern \nJet Propulsion Laboratory\nCalifornia Institute of Technology\nMS 169-530, 4800 Oak Grove Drive91109PasadenaUSA\n",
"B Shappee \nDepartment of Astronomy\nThe Ohio State University\n140 W. 18th Ave43210ColumbusOHUSA [\n",
"C Storz \nMax-Planck-Institut fuer Astronomie\nKoenigstuhl 17D-69117HeidelbergGermany\n",
"J Van Saders \nDepartment of Astronomy\nThe Ohio State University\n140 W. 18th Ave43210ColumbusOHUSA [\n",
"P Weiser \nFachhochschule fuer Technik und Gestaltung\nThe University of Arizona on behalf of the Arizona university system; Istituto Nazionale di Astrofisica, Italy; LBT Beteiligungsgesellschaft the Astrophysical Institute Potsdam, and Heidelberg University; and The Research Corporation, on behalf of The University of Notre Dame, University of Minnesota and University of Virginia\nThe Ohio State University\nWindeckstr. 110D-68163MannheimGermany, Germany\n",
"D Zhang \nDepartment of Astronomy\nThe Ohio State University\n140 W. 18th Ave43210ColumbusOHUSA [\n"
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"Department of Astronomy\nThe Ohio State University\n140 W. 18th Ave43210ColumbusOHUSA [",
"Max-Planck-Institut fuer Astronomie\nKoenigstuhl 17D-69117HeidelbergGermany",
"Department of Astronomy\nThe Ohio State University\n140 W. 18th Ave43210ColumbusOHUSA [",
"Fachhochschule fuer Technik und Gestaltung\nThe University of Arizona on behalf of the Arizona university system; Istituto Nazionale di Astrofisica, Italy; LBT Beteiligungsgesellschaft the Astrophysical Institute Potsdam, and Heidelberg University; and The Research Corporation, on behalf of The University of Notre Dame, University of Minnesota and University of Virginia\nThe Ohio State University\nWindeckstr. 110D-68163MannheimGermany, Germany",
"Department of Astronomy\nThe Ohio State University\n140 W. 18th Ave43210ColumbusOHUSA ["
] |
[] |
-Using a sample of high-redshift lensed quasars from the CASTLES project with observed-frame ultraviolet or optical and near-infrared spectra, we have searched for possible biases between supermassive black hole (BH) mass estimates based on the C iv, Hα and Hβ broad emission lines. Our sample is based upon that of Greene, Peng & Ludwig, expanded with new near-IR spectroscopic observations, consistently analyzed high S/N optical spectra, and consistent continuum luminosity estimates at 5100Å. We find that BH mass estimates based on the FWHM of C iv show a systematic offset with respect to those obtained from the line dispersion, σ l , of the same emission line, but not with those obtained from the FWHM of Hα and Hβ. The magnitude of the offset depends on the treatment of the He ii and Fe ii emission blended with C iv, but there is little scatter for any fixed measurement prescription. While we otherwise find no systematic offsets between C iv and Balmer line mass estimates, we do find that the residuals between them are strongly correlated with the ratio of the UV and optical continuum luminosities. This means that much of the dispersion in previous comparisons of C iv and Hβ BH mass estimates are due to the continuum luminosities rather than any properties of the lines. Removing this dependency reduces the scatter between the UV-and optical-based BH mass estimates by a factor of approximately 2, from roughly 0.35 to 0.18 dex. The dispersion is smallest when comparing the C iv σ l mass estimate, after removing the offset from the FWHM estimates, and either Balmer line mass estimate. The correlation with the continuum slope is likely due to a combination of reddening, host contamination and object-dependent SED shapes. When we add additional heterogeneous measurements from the literature, the results are unchanged. Moreover, in a trial observation of a remaining outlier, the origin of the deviation is clearly due to unrecognized absorption in a low S/N spectrum. This not only highlights the importance of the quality of the observations, but also raises the question if whether cases like this one are common in the literature, further biasing comparisons between C iv and other broad emission lines.
|
10.1088/0004-637x/742/2/93
|
[
"https://arxiv.org/pdf/1009.1145v3.pdf"
] | 41,187,697 |
1009.1145
|
c332289550e357845b1b8e72e54892d6e7a20644
|
Black Hole Mass Estimates Based on C iv are Consistent with Those Based on the Balmer Lines †
30 Aug 2011
R J Assef [email protected]]
Department of Astronomy
The Ohio State University
140 W. 18th Ave43210ColumbusOHUSA [
Jet Propulsion Laboratory
California Institute of Technology
MS 169-530, 4800 Oak Grove Drive91109PasadenaUSA
Dark Cosmology Centre
Niels Bohr Institute
NASA Postdoctoral
Program Fellow 4 DARK Fellow
University of Copenhagen
Juliane Maries Vej 302100CopenhagenDenmark
K D Denney
Department of Astronomy
The Ohio State University
140 W. 18th Ave43210ColumbusOHUSA [
C S Kochanek
Department of Astronomy
The Ohio State University
140 W. 18th Ave43210ColumbusOHUSA [
The Center for Cosmology and Astroparticle Physics
The Ohio State University
191 West Woodruff Avenue43210ColumbusOHUSA
B M Peterson
Department of Astronomy
The Ohio State University
140 W. 18th Ave43210ColumbusOHUSA [
The Center for Cosmology and Astroparticle Physics
The Ohio State University
191 West Woodruff Avenue43210ColumbusOHUSA
S Koz Lowski
Department of Astronomy
The Ohio State University
140 W. 18th Ave43210ColumbusOHUSA [
N Ageorges
Max-Planck-Institut fuer Extraterrestrische Physik, Giessenbachstr
D-85748GarchingGermany
R S Barrows
Arkansas Center for Space and Planetary Sciences
University of Arkansas
72701FayettevilleAR
P Buschkamp
Max-Planck-Institut fuer Extraterrestrische Physik, Giessenbachstr
D-85748GarchingGermany
M Dietrich
Department of Astronomy
The Ohio State University
140 W. 18th Ave43210ColumbusOHUSA [
E Falco
Harvard-Smithsonian Center for Astrophysics
60 Garden Street02138CambridgeMAUSA
C Feiz
Landessternwarte, ZAH
10 Astron. Institut
Ruhr Univ. Bochum
Koenigstuhl 12, Universitaetsstr. 150D-69117, D-44780Heidelberg, BochumGermany, Germany
H Gemperlein
Max-Planck-Institut fuer Extraterrestrische Physik, Giessenbachstr
D-85748GarchingGermany
A Germeroth
Landessternwarte, ZAH
10 Astron. Institut
Ruhr Univ. Bochum
Koenigstuhl 12, Universitaetsstr. 150D-69117, D-44780Heidelberg, BochumGermany, Germany
C J Grier
Department of Astronomy
The Ohio State University
140 W. 18th Ave43210ColumbusOHUSA [
R Hofmann
Max-Planck-Institut fuer Extraterrestrische Physik, Giessenbachstr
D-85748GarchingGermany
M Juette
R Khan
Department of Astronomy
The Ohio State University
140 W. 18th Ave43210ColumbusOHUSA [
M Kilic
Harvard-Smithsonian Center for Astrophysics
60 Garden Street02138CambridgeMAUSA
V Knierim
W Laun
Max-Planck-Institut fuer Astronomie
Koenigstuhl 17D-69117HeidelbergGermany
R Lederer
Max-Planck-Institut fuer Extraterrestrische Physik, Giessenbachstr
D-85748GarchingGermany
M Lehmitz
Max-Planck-Institut fuer Astronomie
Koenigstuhl 17D-69117HeidelbergGermany
R Lenzen
Max-Planck-Institut fuer Astronomie
Koenigstuhl 17D-69117HeidelbergGermany
U Mall
Max-Planck-Institut fuer Astronomie
Koenigstuhl 17D-69117HeidelbergGermany
K K Madsen
California Institute of Technology
1200 E. California Blvd91125PasadenaCAUSA
H Mandel
Landessternwarte, ZAH
10 Astron. Institut
Ruhr Univ. Bochum
Koenigstuhl 12, Universitaetsstr. 150D-69117, D-44780Heidelberg, BochumGermany, Germany
P Martini
Department of Astronomy
The Ohio State University
140 W. 18th Ave43210ColumbusOHUSA [
The Center for Cosmology and Astroparticle Physics
The Ohio State University
191 West Woodruff Avenue43210ColumbusOHUSA
S Mathur
Department of Astronomy
The Ohio State University
140 W. 18th Ave43210ColumbusOHUSA [
The Center for Cosmology and Astroparticle Physics
The Ohio State University
191 West Woodruff Avenue43210ColumbusOHUSA
K Mogren
Department of Astronomy
The Ohio State University
140 W. 18th Ave43210ColumbusOHUSA [
P Mueller
Landessternwarte, ZAH
10 Astron. Institut
Ruhr Univ. Bochum
Koenigstuhl 12, Universitaetsstr. 150D-69117, D-44780Heidelberg, BochumGermany, Germany
V Naranjo
Max-Planck-Institut fuer Astronomie
Koenigstuhl 17D-69117HeidelbergGermany
A Pasquali
Max-Planck-Institut fuer Astronomie
Koenigstuhl 17D-69117HeidelbergGermany
K Polsterer
R W Pogge
Department of Astronomy
The Ohio State University
140 W. 18th Ave43210ColumbusOHUSA [
The Center for Cosmology and Astroparticle Physics
The Ohio State University
191 West Woodruff Avenue43210ColumbusOHUSA
A Quirrenbach
Landessternwarte, ZAH
10 Astron. Institut
Ruhr Univ. Bochum
Koenigstuhl 12, Universitaetsstr. 150D-69117, D-44780Heidelberg, BochumGermany, Germany
W Seifert
Landessternwarte, ZAH
10 Astron. Institut
Ruhr Univ. Bochum
Koenigstuhl 12, Universitaetsstr. 150D-69117, D-44780Heidelberg, BochumGermany, Germany
D Stern
Jet Propulsion Laboratory
California Institute of Technology
MS 169-530, 4800 Oak Grove Drive91109PasadenaUSA
B Shappee
Department of Astronomy
The Ohio State University
140 W. 18th Ave43210ColumbusOHUSA [
C Storz
Max-Planck-Institut fuer Astronomie
Koenigstuhl 17D-69117HeidelbergGermany
J Van Saders
Department of Astronomy
The Ohio State University
140 W. 18th Ave43210ColumbusOHUSA [
P Weiser
Fachhochschule fuer Technik und Gestaltung
The University of Arizona on behalf of the Arizona university system; Istituto Nazionale di Astrofisica, Italy; LBT Beteiligungsgesellschaft the Astrophysical Institute Potsdam, and Heidelberg University; and The Research Corporation, on behalf of The University of Notre Dame, University of Minnesota and University of Virginia
The Ohio State University
Windeckstr. 110D-68163MannheimGermany, Germany
D Zhang
Department of Astronomy
The Ohio State University
140 W. 18th Ave43210ColumbusOHUSA [
Black Hole Mass Estimates Based on C iv are Consistent with Those Based on the Balmer Lines †
30 Aug 2011ABSTRACT † This works relies partly on observations the Large Binocular Telescope. The LBT is an international collaboration among institutions in the United States, Italy and Germany. LBT Corporation partners are:Subject headings: gravitational lensing -galaxies: active -quasars: emission lines
-Using a sample of high-redshift lensed quasars from the CASTLES project with observed-frame ultraviolet or optical and near-infrared spectra, we have searched for possible biases between supermassive black hole (BH) mass estimates based on the C iv, Hα and Hβ broad emission lines. Our sample is based upon that of Greene, Peng & Ludwig, expanded with new near-IR spectroscopic observations, consistently analyzed high S/N optical spectra, and consistent continuum luminosity estimates at 5100Å. We find that BH mass estimates based on the FWHM of C iv show a systematic offset with respect to those obtained from the line dispersion, σ l , of the same emission line, but not with those obtained from the FWHM of Hα and Hβ. The magnitude of the offset depends on the treatment of the He ii and Fe ii emission blended with C iv, but there is little scatter for any fixed measurement prescription. While we otherwise find no systematic offsets between C iv and Balmer line mass estimates, we do find that the residuals between them are strongly correlated with the ratio of the UV and optical continuum luminosities. This means that much of the dispersion in previous comparisons of C iv and Hβ BH mass estimates are due to the continuum luminosities rather than any properties of the lines. Removing this dependency reduces the scatter between the UV-and optical-based BH mass estimates by a factor of approximately 2, from roughly 0.35 to 0.18 dex. The dispersion is smallest when comparing the C iv σ l mass estimate, after removing the offset from the FWHM estimates, and either Balmer line mass estimate. The correlation with the continuum slope is likely due to a combination of reddening, host contamination and object-dependent SED shapes. When we add additional heterogeneous measurements from the literature, the results are unchanged. Moreover, in a trial observation of a remaining outlier, the origin of the deviation is clearly due to unrecognized absorption in a low S/N spectrum. This not only highlights the importance of the quality of the observations, but also raises the question if whether cases like this one are common in the literature, further biasing comparisons between C iv and other broad emission lines.
Using a sample of high-redshift lensed quasars from the CASTLES project with observed-frame ultraviolet or optical and near-infrared spectra, we have searched for possible biases between supermassive black hole (BH) mass estimates based on the C iv, Hα and Hβ broad emission lines. Our sample is based upon that of Greene, Peng & Ludwig, expanded with new near-IR spectroscopic observations, consistently analyzed high S/N optical spectra, and consistent continuum luminosity estimates at 5100Å. We find that BH mass estimates based on the FWHM of C iv show a systematic offset with respect to those obtained from the line dispersion, σ l , of the same emission line, but not with those obtained from the FWHM of Hα and Hβ. The magnitude of the offset depends on the treatment of the He ii and Fe ii emission blended with C iv, but there is little scatter for any fixed measurement prescription. While we otherwise find no systematic offsets between C iv and Balmer line mass estimates, we do find that the residuals between them are strongly correlated with the ratio of the UV and optical continuum luminosities. This means that much of the dispersion in previous comparisons of C iv and Hβ BH mass estimates are due to the continuum luminosities rather than any properties of the lines. Removing this dependency reduces the scatter between the UV-and optical-based BH mass estimates by a factor of approximately 2, from roughly 0.35 to 0.18 dex. The dispersion is smallest when comparing the C iv σ l mass estimate, after removing the offset from the FWHM estimates, and either Balmer line mass estimate. The correlation with the continuum slope is likely due to a combination of reddening, host contamination and object-dependent SED shapes. When we add additional heterogeneous measurements from the literature, the results are unchanged. Moreover, in a trial observation of a remaining outlier, the origin of the deviation is clearly due to unrecognized absorption in a low S/N spectrum. This not only highlights the importance of the quality of the observations, but also raises the question if whether cases like this one are common in the literature, further biasing comparisons between C iv and other broad emission lines.
Subject headings: gravitational lensing -galaxies: active -quasars: emission lines
Introduction
It is thought that every massive galaxy has a supermassive black hole (BH) at its center, and some physical properties of the BH appear to be tightly correlated with those of the galaxy. In particular, the mass of the central BH correlates well with the luminosity of the spheroidal component of the host (see, e.g., Marconi & Hunt 2003;Graham 2007) and with its velocity dispersion (see, e.g., Ferrarese & Merritt 2000;Gebhardt et al. 2000;Tremaine et al. 2002;Gültekin et al. 2009;Graham et al. 2011). Both of these properties of galaxies have physical scales a few orders of magnitude larger than the sphere of influence of the BH, so mechanisms linking their properties are not immediately apparent. Theoretical models try to account for the correlation through co-evolution of the galaxy and its BH, in which accretion induced by galaxy mergers regulates the BH's growth, and feedback from the accretion regulates the growth of the galaxy by quenching star formation and removing cold gas (e.g., Granato et al. 2004;Hopkins et al. 2005Hopkins et al. , 2006Hopkins et al. , 2008Somerville et al. 2008;Shankar et al. 2009). However, the existence of these correlations does not necessarily imply co-evolutionary mechanisms, as some authors argue that they can be a simple consequence of mergers and the central limit theorem (Peng 2007(Peng , 2010Jahnke & Maccio 2010).
Direct measurements of BH masses in inactive galaxies are only possible for a small number of nearby objects because it is necessary, or at least desirable (see, e.g., Merritt & Ferrarese 2001;Gültekin et al. 2009), to resolve the BH's sphere of influence in order to determine the BH mass from the kinematics of the stars and gas closest to it. Galaxies with active nuclei (AGNs) offer a completely different means of estimating BH masses at any distance. In particular, Type 1 AGNs show bright broad emission lines in their spectra produced by gas in the broad line region (BLR), which is close to the central black hole but outside the hot accretion disk. The large line-widths are thought to arise from the Doppler broadening due to the orbital velocity of the gas around the BH. Thus, measuring the mass of the central BH from the width of the broad lines is possible if the distance of the BLR from the BH is known.
This distance can be directly measured with the reverberation mapping (RM) technique (Blandford & McKee 1982;Peterson 1993). This technique works by measuring the light travel time between the continuum and the broad-line emitting regions, which is derived from the time lag between changes in their respective luminosities. Unfortunately the timescale over which appreciable variability is observed in AGNs increases with BH mass (e.g., Vanden Berk et al. 2004;Wilhite et al. 2008;Kelly et al. 2009;MacLeod et al. 2010), making it difficult (i.e., more time intensive) to apply RM to the luminous QSOs that possess the most massive BHs. For example, MacLeod et al. (2010) find that for a typical quasar with M BH = 10 8 M ⊙ (typical magnitude of M i ≈ −23 mag), the rest-frame timescale, ∆t, required to reach an r.m.s. variability amplitude of 0.1 mag is approximately 45 days, while for a quasar with M BH = 10 9 M ⊙ (M i ≈ −25.5 mag), ∆t is approximately 125 days. This is further complicated by the time dilation due to the higher redshift of these rare objects. It has been shown, however, that the distance from the BH to the BLR correlates well with the continuum luminosity of the AGN (see, e.g., Kaspi et al. 2000Kaspi et al. , 2005Bentz et al. 2006Bentz et al. , 2009Zu et al. 2010). Given this correlation, BH masses can be estimated for distant broad-line quasars for which the RM technique is not reasonably applicable. Masses estimated in this way are usually referred to as single epoch (SE) BH mass estimates.
Because it is generally easier to obtain optical rather than UV or IR spectra, SE BH masses are typically estimated from the Hβ and Hα broad emission lines and the continuum luminosity at 5100Å at low redshifts (z 0.7). The overlap with RM targets has allowed for very accurate calibration of these SE mass estimators (Collin et al. 2006;Vestergaard & Peterson 2006, VP06;McGill et al. 2008). At high redshift, however, these emission lines are shifted into the IR, and most mass estimates are then based upon the UV Mg ii λ2798 and C iv λ1549 broad emission lines and the continuum luminosities at 3000Å for Mg ii and 1450Å or 1350Å for C iv. Unlike the Balmer emission lines, these UV lines lack large local calibration samples because of the difficulty of obtaining UV-based RM measurements. Onken & Kollmeier (2008) have argued that the Mg ii line can provide accurate mass estimates, but that there is a small, but significant, dependence on the Eddington ratio of the AGN. C iv, on the other hand, is not thought to have this bias, and VP06 have calibrated a C iv-based mass estimator based on local RM AGNs using space-based UV spectra. However, there are still concerns about whether the C iv velocity widths are attributed solely to gravity or if there are bulk flows due to winds of ejected material, and the impact of these effects on the accuracy of C iv-based BH mass estimates is still debated. For example, C iv is typically slightly displaced in wavelength (usually blueshifted) with respect to the rest of the quasar emission lines (see, e.g., Gaskell 1982;Tytler & Fan 1992;Richards et al. 2002), and frequently shows broad absorption features (e.g., Weymann et al. 1981) and strong line asymmetries correlated with quasar properties (e.g., Wilkes 1984;Richards et al. 2002;Leighly & Moore 2004).
The simplest approach to test the reliability of C iv mass estimates is to systematically compare them to Balmer line estimates for the same sources (see, e.g., Dietrich et al. 2009). High redshift lensed quasars are some of the best targets for such tests. Generally, the problem is that the high redshift makes it easy to observe the C iv line, but the better calibrated Hα and Hβ lines lie in the near-IR, where it is difficult to observe them. Magnification increases the apparent brightness of the lensed quasars, and also, because their observed brightness is not uniquely determined by their intrinsic luminosity and distance, it helps to mitigate any Eddington biases in the sample or, in other words, it makes objects in the sample unlikely to be preferentially brighter than average for their BH mass.
In a recent work, Greene, Peng & Ludwig (2010, GPL10) presented near-IR spectral observations for a sub-sample of lensed quasars from the CfA-Arizona Space Telescope LEns Survey (CASTLES) of gravitational lenses (Falco et al. 2001) whose C iv or Mg ii BH masses had been estimated in a previous work by Peng et al. (2006). GPL10 measured, whenever possible, the full width at half maximum (FWHM) of the Hβ and Hα emission lines of these objects and found no systematic biases between BH masses estimated from these lines and those estimated from C iv. Their sample, however, did not cover a large enough range in BH mass to decide whether there was a mass dependent slope to the relation between the masses. This comparison also suffered from the fact that Peng et al. (2006) lacked access to the original UV/optical spectra for many targets and frequently had to rely on the printed spectra in published papers to measure line widths.
In this work we start from the sample of GPL10 and attempt to improve on both of these issues. First, we add Balmer line based BH mass estimates for the lens SDSS1138+0314 and make revised estimates based on new, higher S/N, spectra of HS0810+2554 and SBS0909+532. We obtained near-IR observations for SDSS1138+0314 and HS0810+2554 using the newly commissioned Large Binocular Telescope (LBT) NIR Spectrograph Utility Camera and Integral Field Unit (LUCIFER; Seifert et al. 2003;Ageorges et al. 2010), while for SBS0909+532 we use the UV through IR observations of Mediavilla et al. (2010). Second, we made consistent C iv BH mass estimates from high S/N spectra using the original observations analyzed by Peng et al. (2006), other published or unpublished spectra, or new spectra for all targets in the sample. Finally, we obtained continuum luminosities at 5100Å for all objects in the sample in a consistent manner. This allows us to include the lenses SDSS0246-0825, HS0810+2554 and Q2237+030, which were excluded by GPL10. With these additions we expand the sample of GPL10 with both C iv and Balmer lines mass estimates from 7 to 12 quasars and the mass range covered by approximately 0.5 dex. In §2 we describe the sample of gravitationally lensed quasars we use in this study as well as our observations. In §3 we describe the methods we use to measure emission line velocity widths and their uncertainties, the continuum luminosities of the quasars and the SE BH masses. In §4 we compare the different mass estimates we have derived and determine the possible biases we measure between them while in §5 we expand our results using a heterogeneous sample of measurements from other studies. In §6 we summarize the conclusions. In an appendix we discuss individual objects in detail. We use a standard ΛCDM cosmology with Ω M = 0.3, Ω Λ = 0.7 and H 0 = 73 km s −1 Mpc −1 throughout the paper.
The Sample of Lensed QSOs
We selected 12 lensed quasars from the CASTLES survey with high quality UV/optical, typically ground-based, spectra of C iv and either published near-IR spectra of the Balmer lines or IR magnitudes bright enough to obtain such spectra. The targets are listed in Table 1. All 12 objects have been observed by CASTLES with HST in the V (F555W), I (F814W) and H (F160W) bands, except for B1422+231, which was not observed in I.
We start from the sample of GPL10, who observed most of these lensed quasars in the near-IR with the Triplespec spectrograph at the Apache Point Observatory. The wavelength range of these spectra is 0.95-2.46 µm with R = 3500, and either the Hβ or Hα (or both) emission line is observable in one of the atmospheric windows. Although GPL10 considered objects with a large span of redshift and reddening, we limit our sample to objects with sufficiently high redshift and small enough reddening for C iv emission to be observable in ground based UV/optical spectra 1 . GPL10 presented FWHM velocity width measurements for all the objects in their sample but did not present BH mass estimates for three of them. For these three lensed QSOs, SDSS0246-0825, HS0810+2554 and Q2237+030, we have measured the continuum luminosity and estimated BH masses so we can include them in our sample.
We obtained near-IR spectra in the H and K band for SDSS1138+0314 ( Fig. 1) and in the J-band for HS0810+2554 ( Fig. 2) with the LBT LUCIFER spectrograph. The first was obtained as part of the LUCIFER science demonstration time and is discussed here, while the second was a target of a separate project to be presented by Mogren et al. (in prep.). We also analyzed the near-IR J-and H-band observations of SBS0909+532 presented by Mediavilla et al. (2010), shown in Figure 3.
LUCIFER Observations of SDSS1138+0314
We obtained a near-infrared spectrum of SDSS1138+0314 using the new LUCIFER instrument at the LBT during its science demonstration time. LUCIFER is a near-infrared spectrograph and imager with an overall wavelength range of 0.85 -2.5 µm. We observed SDSS1138+0314 in the longslit mode with the OrderSep filter, a 0. ′′ 5 slit, the 200_H+K grating and the N1.8 camera for a total integration time of 840s over 7 dithered exposures during the night of UTC 2010-01-04. This configuration gives an effective wavelength range of 1.49 -2.4 µm, which includes both the H and K bands, with a resolving power of 1880 at H and 2570 at K. The slit was oriented to include images A and C of the lensed quasar, as well as part of the lens galaxy. No emission from the lens galaxy is detected in our data. The B9V star HIP 33350 was observed with the same configuration, except for a change in slit width from 0. ′′ 5 to 1 ′′ , and was used to correct the spectrum of SDSS1138+0314 for telluric absorption features. The difference in resolution caused by the different slit widths degrades our telluric corrections, but has little consequence for measuring the width of broad emission lines. We estimated the seeing was ∼ 0. ′′ 8 during the observations. We reduced the data using standard IRAF packages in combination with the IDL task xtellcor_general of Vacca et al. (2003) for the telluric absorption corrections. We performed a 2-D wavelength calibration on each of the 7 exposures using the sky emission lines and built a sky frame by median combining them. The sky frame was then used to remove the sky from each exposure before extracting the spectrum. We also did an alternate sky subtraction of the spectra using a version of the COSMOS software modified to work on LUCIFER data. This software, designed for reduction of spectral observations with IMACS (Dressler et al. 2006) and LDSS-3 (upgraded from LDSS-2, Allington-Smith et al. 1994) on the Magellan telescopes, follows the procedures of Kelson (2003). It produces an accurate model of the sky emission by creating a sub-pixel resolution map of the sky line profiles using the full extent of the lines in the spectrum coupled with a model of the optical distortions. Both extractions of the spectra yield equivalent results, and both are shown in Figure 1. While in principle we could use the telluric standard to perform an absolute flux calibration, it is hard to model the slit losses, especially considering the difference in the slit widths. Instead, the flux calibration was performed by convolving the spectrum corrected for telluric absorption with the NICMOS F160W filter curve and matching it to the estimated de-magnified absolute magnitude of the quasar from the CASTLES HST imaging of this lens (see §3.3 for details on the lens magnification). The blue edge of the LUCIFER SDSS1138+0314 spectrum is somewhat redder than the blue edge of the F160W band, so we extended the observed spectrum using the AGN SED template of Assef et al. (2010) assuming no reddening. Note that the HST NICMOS observations were obtained on UTC 2003-11-06, approximately 6 years before the LUCIFER observations, so we attempt to correct for the intrinsic variability of the quasar. However, this is typically not an important correction (see §3.3). We use the R−band light curves obtained with the SMARTS 1.3 m telescope for a gravitational lens monitoring project (see Morgan et al. 2010). These data show that the quasar intrinsically brightened by 56 ± 17% between UTC 2004-02-03 and UTC 2010-01-09. We assume that no significant variability occurred between the HST NIC-MOS and the first SMARTS observations and between the LUCIFER and the last SMARTS observations.
From an optical spectrum of SDSS1138+0314, Eigenbrod et al. (2006) estimated a redshift of z = 2.438 for the quasar, while SDSS provides z = 2.4427 ± 0.0014. Using the narrow component of Hα and the [OIII] λλ 4959, 5007 emission lines, we obtained z = 2.4417, consistent with SDSS. We did not use the [NII] lines or the narrow component of Hβ as they could not be centroided accurately because of blending with the broad Hα and Hβ profiles, respectively.
UV/Optical Spectra
For most of the GPL10 sample, as well as for SDSS1138+0314, we found suitable high S/N optical spectroscopic observations in the literature that the owners kindly made available for this study (see Table 1 for the references, where applicable, and Appendix A for details on each object). When needed, we performed an absolute flux calibration using photometry from several different sources, as this was not always required for the science goals of the original project. All the UV/optical spectra compiled from the literature are shown in Figure 4.
We could not locate suitable optical spectra for HS0810+2554 and FQB1633+3134. Both objects were observed by the SDSS spectroscopic survey, but these spectra did not have high enough S/N to provide accurate line-width measurements with good continuum subtraction. We obtained new optical spectra of these objects using the MDM observatory 2.4m Hiltner telescope with the Boller & Chivens CCD Spectrograph 2 (CCDS). HS0810+2554 was observed on UTC 2010-02-24 with a grating center of 5300Å and was flux calibrated using the standard star Feige 34. FBQ1633+3134 was observed on UTC 2010-03-21 and UTC 2010-03-22 with a grating center of 4700Å and was flux calibrated using the standard star Feige 98. Absolute fluxes were obtained for both objects by performing a cross-calibration between SDSS g-band photometry of other objects in the field and g-band photometric observations with the RETROCAM instrument (Morgan et al. 2005) obtained on UTC 2010-03-06 and UTC 2010-03-22 for HS0810+2554 and FBQ1633+3134, respectively. The reduced spectra are shown in Figure 4.
Models and Measurements
In this section, we briefly discuss the methods we use to measure the line widths and estimate the black hole masses from the optical and near-IR spectra.
Line-width Measurements
There is no standard prescription for measuring the line-width characterizations of the broad C iv emission line in QSOs. While for other emission lines this may not be a significant source of uncertainties, there is a shelf-like emission feature redward of C iv that blends with the line profile and is created by a combination of broad He II λ1640, OIII] λ1663, and a feature of unknown origin at 1600Å usually referred to as the λ1600 feature (Laor et al. 1994;Marziani et al. 1996;Fine et al. 2010). While the λ1600 feature is commonly thought to correspond to Fe ii, Fine et al. (2010) argue that this cannot account for all the observed flux, yet it is also unlikely that C iv can reach large enough velocities to produce the feature. Different prescriptions for modeling the blended emission can have significant effects on line width estimates (Denney et al. 2009;Fine et al. 2010), so it is important to explore how these affect our results. Fine et al. (2010) explored three different and widely used approaches and their effects on the C iv width measurements. The three prescriptions are: (1) to assume that the λ1600 feature corresponds to C iv emission and therefore remove only the He II λ1640 and OIII] λ1663 contributions; (2) to assume that the λ1600 feature belongs to a different species from C iv and so removing its contribution along with that of the other two components on the shelf; and (3) to fit the λ1600 feature as part of the continuum (see Fine et al. 2010, for details on each prescription). While Fine et al. (2010) selects prescription (2) as their preferred method, in large part because it produces symmetric C iv profiles, it is hard to apply this approach to low S/N data (see Fine et al. 2010, for details). Moreover, it is not guaranteed to produce more accurate BH masses than the other two prescriptions. The simple prescription of (3) produces line-width characterizations that are systematically smaller than prescription (2) but with very low dispersion between individual measurements, while (1) produces estimates with a larger scatter relative to (2) but without a systematic offset. The differences between the prescriptions is smallest for FWHM and largest for the line dispersion, σ l .
Based, in part, on these issues, we considered two different prescriptions for removing the continuum and blended emission from the C iv emission line profile. Both prescriptions are amenable to large scale automated use. The first prescription, which we will refer to as prescription A, is very similar to that used by VP06, where the shelf feature redward of C iv is considered part of the C iv line profile, but only the region within ±10, 000 km s −1 of the peak is considered. The continuum is fit by linearly interpolating between the two continuum windows in the wavelength ranges 1425-1470 and 1680-1705Å. When these continuum windows were affected by absorption, we slightly shifted them as detailed in Table 2. Our continuum fitting is in principle different from that of VP06, who considered 5 different continuum windows and then fit a power-law to them, but the differences of the measured line-widths are not significant and our approach requires a much smaller wavelength range for the spectra. In this prescription, He ii and O iii] emission is not explicitly removed, but this has negligible effects due to the limit on the velocity range, making it analogous to prescription (1) of Fine et al. (2010). The second prescription, B, is analogous to prescription (3) of Fine et al. (2010), as we fit the λ1600 feature as part of the continuum. It only differs in that the red continuum region is chosen to match the minimum between C iv and the λ1600 feature. In general, prescription A will lead to broader estimates of the C iv line width than prescription B.
The observed wavelength continuum windows for each object and prescription are listed in Table 2. The C iv emission line flux was then measured above the fit continuum and between the emission line wavelength regions listed in Table 2. In addition, for objects that showed mild absorption features, bad pixels, and/or significant night sky line residuals, we used a low-order polynomial (i.e., first, second or third order depending on the size and location of the feature) to interpolate across the feature before measuring the line widths. Details for the individual targets are given in Appendix A. We did not attempt to remove any narrow-line emission from C iv λ1549, since this line is typically very weak and cannot be reliably isolated (Wills et al. 1993, although see Sulentic et al. 2007), and the separate lines of the C iv doublet are unresolved in AGN spectra (see VP06, and references therein for further discussion). We characterized the line width by both its FWHM and line dispersion (σ l , the second moment of the line profile). The widths were measured directly from the actual or interpolated spectrum (except where noted below and in Appendix A) following the procedures described by Peterson et al. (2004).
We also fit the original or interpolated line profiles with a sixth-order Gauss-Hermite (GH) polynomial, because making functional fits to emission-line profiles is a common way of mitigating the effects of low S/N on line-width measurements (see, e.g., Woo et al. 2007;McGill et al. 2008, for similar approaches). The Gauss-Hermite polynomials we fit utilize the normalization of van der Marel & Franx (1993) and the functional forms of Cappellari et al. (2002). We then use a Levenberg-Marquardt least-squares fitting procedure to determine the best-fitting coefficients. We measured the widths of these line profile models using the same software as was used to measure widths directly from the data (see Peterson et al. 2004). Ultimately we only used the results from the line profile models for PG1115+080 (see Appendix A). Instead, these fits were primarily used to determine uncertainties in our width measurements as described in §3.2. The continuum and the Gauss-Hermite fits to the C iv line profiles are shown in Figure 5 for both prescriptions. In the cases of SDSS1138+0314 and SBS0909+532, reasonable fits could not be achieved because of the extremely high S/N and peculiar shape of these line profiles (a very narrow peak with broad base; see Appendix A). No fits are shown for these objects.
Both the FWHM and line dispersion, σ l , measurements of the C iv λ1549 emission line are listed in Table 3 for all objects in our sample for both prescriptions. We have corrected the widths for spectral resolution effects following Peterson et al. (2004), when possible, using the resolutions given in Table 2. Except for PG1115+080, we utilize the line widths measured directly from the data (interpolated across gaps where noted) for the subsequent black hole mass calculations. For objects with multiple spectra of the individually lensed images we averaged their line widths. Our C iv λ1549 widths are smaller than those given by GPL10 for the objects in which we both used the SDSS spectra (Q0142-100, SDSS0246-0825, PG1115+080, and H1413+117). The likely origin of the discrepancy is that GPL10 fit a narrow line component as part of the C iv profile, which would naturally yield larger FWHM values. We note, however, that GPL10 do not use their SDSS line-width measurements to estimate BH masses in their analysis, but always use those determined by Peng et al. (2006). The lens HE1104-1805 is the only object in the sample for which we use the same optical spectrum as Peng et al. (2006), that of Wisotzki et al. (1995), and we find a FWHM that is smaller by 260 km s −1 , compared to our measured uncertainty of 50 km s −1 . Although Peng et al. (2006) do not quote errors in their line width measurements, the disagreement (∼ 5Å in the observed-frame) is likely within their uncertainties.
Line widths of the Hβ and Hα broad-emission lines are given in Table 3, while the continuum and broad line spectral wavelength regions used are given in Table 4. We measured them from the near-IR spectra following a similar procedure to the C iv line-widths except that (1) the best Gauss-Hermite polynomial fit was used for all line-width measurements, with the exception of Hα for SBS0909+532, because the S/N of the near-IR data was typically too poor to justify measurement directly from the data, (2) blended emissionline components were removed from each spectrum before the line width was measured, as described in Appendix A, and (3) a power-law, instead of a linear, continuum was fit to the Hβ spectrum of HS0810+2554 because it was fit simultaneously with additional blended emission-line components over a larger wavelength range.
For the objects where we lack the near-IR spectroscopic observations, we rely on the published Hα and Hβ line widths of GPL10. These measurements were done using somewhat different methods than ours. While we consider most of the GPL10 FWHM estimates to be reliable, there are some that we believe are suspect because (1) they were measured from very low S/N spectra, (2) the lines were not fully contained in the wavelength range of the spectrum, and/or (3) we do not agree with the narrow-line component models subtracted before the line width was measured. In the relevant Figures and Tables, we differentiate between the Balmer-line velocity widths we think are reliable (group I, solid symbols) and those we believe are affected by any of these issues (group II, open symbols). Individual objects can be in both groups because these issues may affect only one of the Balmer lines. We also include in group I the Hα and Hβ line-width measurements from our new IR spectra. The decision to split our sample is a conservative choice, and our conclusions are not significantly modified when the group II line widths are included.
Line-Width Measurement Uncertainties
Line-width measurements can be affected by sources of error that are difficult to model, as they depend not only on the overall S/N ratio, but also on the line profile and the presence of sky emission and absorption lines, with the latter being of particular importance in the near-IR. We use a Monte Carlo approach to determine the uncertainties in our line-width measurements. Using the flux uncertainty per pixel in each spectrum and the best fit Gauss-Hermite line profile (with the exception of the optical SDSS1138+0314 and SBS0909+532 spectra, see Appendix A), we produced 1000 resampled spectra by adding random Gaussian deviates based on the error spectrum to the flux in each pixel of the GH model spectrum and then re-measured the line width using the methods described in the previous section. For the UV/optical spectra from the literature without an error spectrum, we estimated one by propagating the measured S/N of a small continuum window near the C iv λ1549 emission line to the overall spectrum. In this case, δF λ , the flux error in a pixel of wavelength λ with flux F λ , is given by
δF λ = λ c λ F λc F λ S N −1 c ,(1)
where λ c and F λc are the average wavelength and flux per unit wavelength of the continuum window chosen, and (S/N) c is the signal-to-noise ratio per pixel in the chosen continuum window. This equation is constructed by assuming that the only source of error is Poisson fluctuations, and that the number of detected photons is proportional to F λ (hc/λ) −1 , where the proportionality constant is empirically determined in the continuum window from (S/N) c , λ c and F λc . This approach neglects the sky background and the presence of strong absorption or emission sky lines, which is reasonable for the UV/optical spectra. It also neglects changes in the instrument sensitivity as a function of wavelength and assumes a constant pixel wavelength-width, both of which are reasonable because the continuum S/N is measured in close proximity to the emission line of interest. While the parametric fits are not exact representations of each line, this still provides a reasonable estimate of the fractional uncertainties.
Luminosity Measurements
We estimated the continuum luminosities at 5100Å by fitting the AGN SED template of Assef et al. (2010) to the unmagnified quasar magnitudes obtained from the CASTLES project HST NICMOS imaging. To correct the observed quasar fluxes for the lens magnification, we modeled each system using the astrometry and lens galaxy photometry from the CASTLES HST WFPC2 and NICMOS observations following the procedures of Lehár et al. (2000). The image is decomposed into a set of point sources for the quasars, de Vaucouleurs models for the lens galaxy and, if necessary, a lensed host component, convolved with model or empirical PSFs. The resulting component positions and image fluxes were modeled using lensmodel (Keeton 2001). The lens was modeled as a singular isothermal ellipsoid in an external shear with the ellipsoid's orientation and ellipticity constrained by those of the light of the lens galaxy and a weak prior on the external shear. The models were not tightly constrained to match the observed fluxes due to systematic errors in image flux ratios such as source variability and microlensing. Aside from substructure, the dominant uncertainty in the magnifications is the radial mass distribution of the lens (see Kochanek, Schneider & Wambsganss 2004), and this is less than a factor of two even if we allow the full range of models between a flat rotation curve and a constant M/L model. Since we have extensive evidence that lenses have mass distributions corresponding to flat rotation curves on these scales (e.g., Rusin et al. 2003;Jiang & Kochanek 2007;Koopmans et al. 2009), the model uncertainties are considerably less than this factor, and the uncertainties are dominated by the systematic uncertainties in the image fluxes. Table 1 lists the magnifications used for each object in the sample. The only object for which a different model was used is Q0957+561, where we used the magnifications determined by Fadely et al. (2010).
We did not apply reddening corrections other than removing Galactic foreground extinction (see below), as the requirement that C iv is observable in the UV/optical severely limits the presence of dust absorption, especially at rest-frame 5100Å. For all four-image lenses, we estimated the true source flux for all images, rejected the highest and lowest estimates and averaged the remaining two to limit the effects of microlensing. For two-image lenses we simply averaged the two estimates. Table 1 shows the estimated unmagnified H-band magnitude of each quasar. Note that in general we did not apply a correction for variability. Although there is a 5 to 10 year time difference between the CASTLES and the GPL10 Triplespec observations, the typical uncertainty introduced falls well below the systematic uncertainties in the SE BH mass estimates. An estimate of the typical variability of a quasar can be obtained from measurements of their structure function. Using the power-law fit of Vanden Berk et al. (2004) to the i-band structure function of SDSS quasars, we find that the typical quasar would experience a change in magnitude of approximately 0.2 mag for a rest-frame time-lag of 1500 days (approximately 10 years in the observer's frame for our lowest redshift quasar). A change of 0.2 magnitudes results in a change to the BH mass estimate of 0.04 dex, well below their typical error bar of 0.3 dex, and we would expect the H−band variability to be still smaller, as the average variability amplitude decreases with increasing wavelength (see, e.g., Vanden Berk et al. 2004;MacLeod et al. 2010).
For SDSS1138+0314, HS0810+2554 and SBS0909+532, we performed an absolute flux calibration of the near-IR spectra and measured the 5100Å continuum luminosity directly. The calibration for the first object is discussed in detail in §2.1. For HS0810+2554 we fit a power-law to the continuum of our MDM CCDS spectrum (see §2.2) and extrapolated it to rest-frame 5100Å. For SBS0909+532 we calibrated the spectrum using the HST NICMOS H-band photometry, as the object did not show significant flux variations between the two relevant epochs (J. Muñoz, private communication).
To obtain the rest-frame continuum UV luminosities at 1350Å and 1450Å, we flux calibrated the spectra whenever it was necessary and measured the flux by fitting a straight line to the region between rest-frame 1349Å and 1355Å for the estimate at 1350Å and to the region between 1440Å and 1460Å for the estimate at 1450Å. We corrected these luminosities for foreground Galactic extinctions obtained through the NASA/IPAC Extragalactic Database 3 from the dust maps of Schlegel et al. (1998). Errors in the continuum luminosity will be dominated by the uncertainties in the magnification models, which are hard to quantify. We assume a conservative error of 20% in each continuum luminosity estimate.
GPL10 obtained continuum luminosities at 5100Å for their sample of objects by following a similar approach. They fit a power-law to the unmagnified HST photometry from the CASTLES survey, using the lensing models of Peng et al. (2006). In comparison to GPL10 we observe that our luminosity estimates are, on average, 0.20 ± 0.05 dex smaller. The offset is likely caused by a combination of the differences in the lensing models, in the prescription used to deal with the flux ratio anomalies, and in the use of the AGN SED template of Assef et al. (2010) instead of the power-law fits of Peng et al. (2006). We note that this offset translates to 0.1 dex in BH mass, well below the uncertainties we estimate for our SE mass measurements in the next section. We also note that our conclusions are unaltered if we replace our 5100Å continuum luminosity estimates with those of GPL10 for all objects where this is possible.
Black Hole Mass Estimates
The width of a given broad emission line in a Type 1 AGN is primarily caused by the gravitational attraction of the supermassive black hole on the gas in the broad line region (BLR). Hence, the mass of the black hole, M BH , can be estimated from virial assumptions by
M BH = f R BLR (∆v) 2 G ,(2)
were ∆v is the velocity dispersion of the BLR gas, estimated from the width of the broad emission line, G is the gravitational constant and R BLR is the distance from the black hole to the BLR. The factor f is a scale factor of order unity that depends on the structure, kinematics and inclination of the BLR (see, e.g., Collin et al. 2006, and references therein). The term R BLR (∆v) 2 /G is usually referred to as the virial product (VP) and encapsulates all the observable quantities for a single object. The radius of the BLR can only be measured through reverberation mapping (see, e.g., Peterson et al. 2004), but has been shown to correlate well with the continuum luminosity (see, e.g., Kaspi et al. 2005;Bentz et al. 2006Bentz et al. , 2009Zu et al. 2010).
For the broad hydrogen emission lines we estimate the BLR radius using the R BLR − λL λ (5100Å) relation of Bentz et al. (2009), which was calibrated using a large sample of RM AGNs. The f factor of equation (2) depends on the characterization of the line width, generally either the FWHM or the line dispersion, σ l , as well as on the emission line being used. For estimating M BH from the width of the Hβ broad line, we use the f factor calibrations of Collin et al. (2006) for the FWHM and for σ l . While for σ l a unique f factor of 3.85 for all AGNs suffices, Collin et al. (2006) argued that f is strongly dependent on the line profile shape for FWHM-based estimates, where the shape was quantified as the ratio between the FWHM and σ l . We choose, however, to use the best-fit fixed f factor of 1.17 for FWHM instead of the line-shape dependent calibrations because Denney et al. (2009) have shown that σ l is affected by blending with other emission lines, making the correlation found by Collin et al. (2006) hard to interpret. For Hα there is no equivalent calibration of the f -factor, so we cannot directly estimate the black hole masses. Instead, we use the relation determined by Greene & Ho (2005) between the FWHM of Hα and Hβ,
FWHM Hβ = (1.07 ± 0.07) × 10 3 FWHM Hα 10 3 km s −1 (1.03±0.03) km s −1 ,(3)
to estimate the Hβ FWHM and then estimate M BH (Hα) using the same f −factor and R BLR − L relation as for M BH (Hβ). Unfortunately, there is no equivalent transformation for σ l , so we cannot use this measurement to estimate the mass of the black hole from Hα.
Combining equations (2) and (3)
M ⊙ ,(5)
where in equation (4) ∆v Hβ can be either the line dispersion or the FWHM. Because equation (5) is fully dependent upon the scaling relations for Hβ, the f factor in it is the same as for FWHM Hβ in equation (4). Table 5 shows our BH mass estimates based on Hα and Hβ for all objects in the sample.
For the UV/optical spectra we use the empirical M BH calibrations of VP06 for the C iv broad emission line, given by
M BH (C IV) = 10 κ ∆v C IV 10 3 km s −1 2 λL λ (1350Å) 10 44 erg s −1 0.53 M ⊙ ,(6)
where ∆v is either FWHM or σ l , and κ = 6.66 ± 0.01 or 6.73 ± 0.01, respectively, for these line-width characterizations. The constant κ implicitly contains the f factor, which is assumed to be a constant for all objects. Whenever possible, we use the observed 1350Å flux to determine the continuum luminosity. Unfortunately 1350Å is not within the observed wavelength range of all the UV/optical spectra we use. In these cases we estimate the continuum luminosity at 1350Å using the observed flux at 1450Å, as VP06 have shown L λ at these wavelengths to be equivalent. We list our C iv BH mass estimates in Table 5 for both prescriptions used to measure the widths of C iv. As expected, masses determined from the FWHM are highly consistent for both prescriptions, with a mean difference of 0.04 dex and a scatter of 0.02 dex, with the average prescription B based mass estimates being smaller. The agreement is much worse for σ l , with a mean difference of 0.23 dex, in the sense that B is smaller, and a scatter of 0.18 dex.
We estimate the uncertainties in our BH mass estimates by propagating the errors in the velocity widths and in the continuum luminosities. For masses based on the width of the broad Hydrogen lines, we also propagate the uncertainties in the f -factor and in R BLR . Collin et al. (2006) determined that the uncertainty in f when using σ l is 30%, while that in FWHM is 43%. For R BLR we assume the intrinsic scatter of 0.11 dex estimated by Peterson (2010) for the radius-luminosity relation. Adding the uncertainties in f and R BLR is not possible for the C iv estimates of the BH masses. Instead, we add the measurement errors and the intrinsic scatter between C iv and RM BH mass estimates in quadrature. Using the sample of VP06, we estimate intrinsic scatters of 0.32 dex and 0.28 dex for FWHM and σ l respectively. VP06 found that the total scatter, including measurement errors, was 0.32 dex for both line-width characterizations of C iv, showing that the intrinsic scatter dominates over measurement errors, especially for FWHM estimates.
Biases in C iv Black Hole Mass Estimates
In this section we use the sample described in §2 to study biases in the C iv black hole mass estimates. We first compare how the mass depends on the characterization of the C iv line-width, and then we proceed to compare these rest-frame UV estimates to those based on the Hα and Hβ emission lines. In the next section we will compare our results with those of other studies on the relations between C iv and Hβ BH estimated masses.
Comparison of FWHM and σ l Derived Masses
Given that we have measured both FWHM and σ l for C iv in all our objects, the simplest test we can perform is to determine if there are any biases between them as BH mass estimators. Both measurements have advantages, and some contention exists in the literature as to which constitutes a more reliable mass estimator (see Peterson et al. 2004, and references therein). Figure 6 compares the C iv-based BH masses determined for both line-width estimates and for the two continuum and line blending prescriptions A and B, respectively. A clear bias is observed for both prescriptions, where most objects have a lower estimated BH mass if we use σ l instead of the FWHM. The bias for prescription A (B) width measurements seems to be well represented by a constant offset of K = 0.13 ± 0.06 dex (0.24 ± 0.07 dex) or, equivalently, a factor of 1.3 (1.7). We fit for K while simultaneously fitting for the intrinsic scatter between the two mass estimators by adding a scatter S in quadrature to the error of each logarithmic mass difference. Note that the logarithmic mass difference does not depend on the continuum luminosity or the intrinsic scatter with respect to the RM estimates. In practice we maximize the likelihood
L = σ 2 + S 2 −1/2 N i=1 σ 2 i + S 2 −1/2 e −χ 2 (S)/2 ,(7)
where σ 2 i is the variance due to measurement errors in the logarithmic mass difference of object i and σ 2 is its average over all objects. We exclude objects for which we consider the C iv-based BH mass estimates to be lower bounds due to absorption. The leading factor in equation (7) is a logarithmic prior on the overall dispersion. The best fit scatter is similar for both prescriptions, with case A line-widths producing S = 0.16 dex while case B ones have S = 0.19 dex. Since the logarithmic mass difference only depends on the line-widths and not on the continuum luminosities, the constant BH mass offsets K can also be expressed as an offset between the line-width characterizations. As such, these values imply an offset of 0.10 ± 0.03 dex (0.16 ± 0.04 dex) between the FWHM and σ l line-width characterizations of C iv for prescription A (B).
It is not surprising that prescription A provides a smaller offset between BH masses obtained from the FWHM and σ l of C iv, as this prescription is modeled after that used by VP06, who used their measurements to determine equation (6). However, given the similarity, the presence of a non-zero offset for prescription A is somewhat puzzling. If we examine the sample of VP06, the scatter is larger, 0.2 dex, and there is no offset (−0.02 ± 0.03 dex), although the lack of an offset is by definition small since both mass estimators were calibrated against the same RM data set.
The large overlap in the mass and continuum luminosity ranges of our sample and that of VP06 suggest that dependence on a secondary parameter is unlikely. Furthermore, we do not see any correlation of this bias with BH mass, continuum luminosity or Eddington ratio. There is also no correlation with redshift, suggesting that it is unlikely to be an evolutionary trend. The only other major difference between the samples is lensing by foreground galaxies. This, however, is very unlikely to cause such an effect, as quasars are quite compact and strong lensing affects the whole object. Microlensing by the stars in the foreground galaxy could in principle distort the shape of the C iv broad emission lines due to the spatial dependence of their velocity structure, but this is very unlikely for two reasons. First, the width of C iv is typically well below 10,000 km/s, constraining the location of the gas to a distance greater than 10 3 Schwarzschild radii (R S ) from the black hole, while microlensing is only observed to have significant effects on scales below 100 R S (Morgan et al. 2010). Second, the gas moving at the highest velocities is expected to be closest to the black hole, so microlensing would tend to magnify the wings of the line more than the core, and hence producing the inverse of the effect we see by making σ l too large rather than too small compared to the FWHM. While microlensing can also produce regions of demagnification in the source plane, these are of very large spatial extent, and so it is unlikely to see significant magnification variations across the BLR.
It is likely then that other minor differences in the method we use to measure σ l as compared to VP06 give rise to the remaining bias. Denney et al. (2009) showed that estimates of σ l depend on the exact prescription used for the line-width measurement and the segregation of blended emission for Hβ. Our investigation shows that this may be the case for C iv as well (see also Fine et al. 2010). However, the remarkably low scatter in Figure 6 suggests that if σ l is measured in a self-consistent manner it can be as accurate as the FWHM for estimating BH masses, but the calibration will depend on the exact prescription. In the next section we will explore the reliability of the C iv FWHM and σ l BH mass estimates by comparing them to those based on Hα and Hβ.
C iv compared to Hα and Hβ
Figures 7 and 8 compare the mass estimates based on the Hα and Hβ lines to those based on the width of C iv. We only show here (and for the rest of the figures) UV BH masses based on the prescription B width measurements of C iv. The FWHM based BH masses are almost equal for prescriptions A and B (see §3.4), but they show a systematic offset for the σ l estimates (see §4.1). We adopt the prescription B masses for the rest of this section, but our conclusions are unaltered if we instead use prescription A measurements. We have made the assumption that the C iv FWHM mass estimates are unbiased, and so those obtained from the prescription B σ l measurement of C iv have been shifted by the systematic offset of 0.24 dex derived in the previous section.
We measure no significant offset between the C iv-based and Hα-or Hβ-based masses when using only objects with C iv line-widths that are not lower bounds and have reliable Balmer line widths (group I). We find best fit offsets of −0.12±0.15, −0.11±0.16, −0.15±0.16 and −0.19 ± 0.18 dex for panels a), b), c) and d), respectively, of Figure 8, with residual scatter of 0.30, 0.23, 0.46 and 0.38 dex. Including the objects with group II Hα and Hβ line-width estimates does not change this conclusion, with best fit offsets of −0.05 ± 0.14, −0.13 ± 0.13, −0.07 ± 0.15 and −0.15 ± 0.14 dex, respectively, with measured scatters of 0.36, 0.33, 0.46 and 0.41 dex. The lack of offsets confirms our assumption that C iv FWHM BH masses are unbiased and that only those based on σ l need to be corrected. The constant offset fits yield χ 2 per degree of freedom (χ 2 ν ) values of 0.6, 0.5, 1.4 and 1.1 for panels a), b), c) and d) of Figure 7 when using only the solid symbols. The scatter in each panel of Figure 7 is largely consistent with the estimated uncertainties, although the errors in the C iv σ l masses may be slightly overestimated. We find no evidence based on the χ 2 statistic that a slope different from unity is required to describe the relation between the logarithms of the BH masses (Figure 7), independent of whether we include the group II Balmer line-width measurements.
We next investigate if the residuals between the C iv and Balmer line masses are correlated with any other observables. Figures 9 -15 show the residuals as a function of the 1350Å and 5100Å continuum luminosities, redshift, Eddington ratio, blueshift of the C iv line, asymmetry of C iv (parametrized by the ratio of the widths red and blue of the centroid), and the ratio of the UV and optical continuum luminosities. Table 6 summarizes the significance of the correlations based on their Spearman rank-order coefficients. Only the correlation with the ratio of the rest-frame optical and UV continuum luminosities is significant ( Figure 15). Figure 16 compares the C iv and Balmer line derived BH masses after rescaling the C iv masses using the best fit correlation determined from the corresponding panel in Figure 15. We applied corrections of the form
log M Corr BH (C IV) = log M VP06 BH (C IV) − b − a log λL λ (1350Å) λL λ (5100Å) .(8)
where the coefficients a and b are listed in Table 7. For completeness, this Table also shows the coefficients obtained when using the prescription A line-widths of C iv, which are of similar magnitude and significance. Note that the uncertainties given for these coefficients have been determined after rescaling the errors such that the best fit has χ 2 ν ≡ 1. The agreement between the rest-frame UV and optical BH mass estimates after applying this correction is remarkable, and the scatter of objects with group I and non-lower bound line-widths has decreased from 0.30 to 0.11, 0.23 to 0.10, 0.46 to 0.25 and 0.38 to 0.22 dex for panels a-d of Figures 7 and 16, respectively. We find that the lowest scatter is between the BH masses estimated from the σ l of C iv σ l and the FWHM of either Balmer line. This supports our conclusion in the previous section that σ l C iv BH masses have small random errors, even if their systematic errors may be much larger than those of the FWHM estimates due to blending of emission lines. Such a small scatter places strong constraints on the strength of a possible correlation between the mass residuals and any tertiary parameter. We find again that a slope different from unity is not required to describe the relation between the logarithm of the C iv and Balmer line BH masses.
Since BH mass estimates generally scale as ∆v 2 L 1/2 (eqns.
[4] and [6]), a naive interpretation of the reduced scatter is that we have simply shifted from showing L 1/2 1350Å vs. L 1/2 5100Å to L 1/2 5100Å vs. L 1/2 5100Å . The best fit correction is statistically different from simply replacing L 1350Å by L 5100Å by 1-2σ, so it is not simply swapping the luminosities. More importantly, even if the slope was exactly α = 1/2, it reveals the crucial point that a significant fraction of any problems in reconciling C iv and Balmer line estimates of BH masses is due to the estimates of the continuum luminosities rather than any properties of either line.
There are 3 potential causes for a correlation of the mass ratio with the ratio of the continuum luminosities: i) obscuration, ii) host contamination and iii) non-universal AGN SEDs. Unfortunately, our analysis does not allow us to determine which BH mass estimate is more accurate. Extinction will reduce the rest-frame UV continuum luminosity while having little effect on the rest-frame optical luminosity. Conversely, host contamination will raise the optical luminosity while leaving the UV unchanged, as galaxies are typically brighter in the optical than in the UV. With respect to case iii), the radius of the BLR is really determined by the flux of the ionizing continuum (λ < 912Å). The R BLR − L relations used to construct equations (4), (5) and (6) implicitly assume a universal SED for all quasars, as they imply that the ionizing continuum can be uniquely predicted from the continuum luminosity at longer wavelengths. This approximation is likely to be better for the restframe UV continuum than for the optical. All three cases discussed would produce a slope of a ≃ 0.5 in equation (8), simply representing the luminosity power indices in equations (4), (5) and (6). This is generally shallower than the observed slope but within 2σ of the best-fit relations. A larger sample is needed to fully determine if the slope of this correlation is statistically different from α ≃ 0.5. We note that in order to create a slope larger than 0.5, it would be necessary for the velocity widths of the quasar broad lines to be dependent on the ratio of the continuum luminosities. There is some evidence that the inclination angle of the accretion disk with respect to the line of sight may correlate with both the SED of the continuum (Gallagher et al. 2005, and references therein) and the FWHM of the broad Hβ line (Wills & Browne 1986;Wills & Brotherton 1995;Jarvis & McLure 2006), although no such correlation is observed for the FWHM of C iv (Vestergaard et al. 2000, but see Decarli et al. 2008). Accretion disk inclination corrections, however, would act in the opposite sense to the observed correlation and hence cannot be responsible for a slope in excess of 0.5 -disks with higher inclination angles (closer to edge-on) would appear to have higher FWHM of Hβ and bluer continua for a fixed "true" BH mass (i.e. not estimated from spectral features; see Gallagher et al. 2005, and references therein).
Our sample is likely representative of observations of the general quasar population in terms of reddening and host contamination. It could, in principle, have a larger typical reddening due to additional obscuration by dust associated with the lens, but this is unlikely to be important for our sample. Reddening by the lens galaxy will typically vary between quasar images. Falco et al. (1999) studied most of the objects in our sample and found that only two of them showed significant differential reddening: SBS0909+523 (∆E(B − V ) = 0.2 mag for image B with respect to A, see also Appendix A) and Q2237+0305 (∆E(B − V ) = 0.18 and 0.17 mag for images C and D with respect to A). Small but nonzero differential reddening was also detected for three other lenses (HE1104-1805, H1413+117 and B1422+231). The lensed quasars SDSS0246-0825, HS0810+2554, FBQ1633+3134 and SDSS1138+0314 were not part of the sample studied by Falco et al. (1999). We studied the latter object in §2.1 and concluded images B and C did not show evidence for differential reddening between them, but there is no information in this regard for the other three quasars. Lensing can also alter host contamination in the quasar observations as compared to an unlensed case. The exact amount of host contamination depends on the size of the PSF and aperture used, the morphology of the lens and the surface brightness profile of the quasar's host galaxy (see, e.g., Kochanek et al. 2001;Ross et al. 2009), however the zeroth order effect is to not alter the amount of host contamination compared to an unlensed quasar.
While we have shown that the dominant source of scatter in the comparison between the BH mass estimates based on C iv and the Balmer lines is due to the continuum luminosities, we still wish to assess the relation between the widths of the different emission lines used. Figure 17 shows the comparison between the C iv and Balmer line widths. Note that we do not show measurements for which we only have lower bounds on the C iv width due to absorption. The best agreement is between σ l of C iv and FWHM of Hβ, which is expected given that these measurements also give the lowest scatter in the BH mass estimates, however a generally good agreement is also observed in all panels. We remind the reader, however, that the corrections we found between the BH mass estimates residuals and the ratio of the continuum luminosities did not have a slope of 0.5. This implies that the ratio of the linewidths may have a dependence on the luminosity ratio, with a power given by the excess of the slope from 0.5. This could be a source of additional scatter in Figure 17, and so, instead of comparing the line-widths directly, we also compare them after applying a correction based on the continuum luminosity estimates, of the form log ∆v(Hβ or Hα) corr = log ∆v(Hβ or Hα) + (a − 0.53) 2 log λL λ (1350Å) λL λ (5100Å) + 5×10 −3 log λL λ (5100Å) 10 44 erg s −1 , (9) as shown in Figure 18. Note that since we don't know the origin of the corrections, applying it to Hβ rather than C iv is a completely arbitrary decision made for display purposes. The agreement is now better and a correlation between the measurements is clear, suggesting that the widths of both lines are equally good tracers of BH mass. We have quantified the correlation between the different line-widths in Table 8 using the Spearman rank-order coefficient, as we did for the BH mass residuals. After applying the correction from equation (9), we find positive correlations which are typically statistically significant between the C iv and Balmer line-widths. A weak anti-correlation, however, is measured between the FWHMs of C iv and Hβ, but it is not statically significant.
Comparison with Other Studies
In §4.2 we used a sample of lensed quasars to compare BH masses based on observations of the C iv emission line and of the Balmer lines Hα and Hβ. We found that the agreement between the rest-frame UV and rest-frame optical based BH masses is reasonably good.
We also found that this agreement is even better once we apply an empirically determined correction based on the ratio of the 1350Å and 5100Å continuum luminosities. There have been a number of previous studies that have explored the relative accuracy of BH masses based on C iv and Hβ, and they have reached both similar and opposite conclusions.
The studies of VP06, which we have discussed previously, and Dietrich & Hamann (2004) found that C iv derived BH masses are consistent with those obtained from the width of Hβ, and hence constitute a valid replacement as a mass estimator. Shemmer et al. (2004, Sulentic et al. (2007) and Dietrich et al. (2009), however, reached opposite conclusions. Shemmer et al. (2004) concluded that BH masses derived from C iv were poorly matched to those obtained from Hβ and could be systematically different. They showed that for a sample of narrow-line Seyfert 1 galaxies, C iv based BH masses are larger by an average factor of ∼ 3 with respect to those obtained from Hβ. Dietrich et al. (2009) also found a large disagreement between the two estimates of the BH mass, but they found that using C iv tends to underestimate the BH masses by a factor of ∼ 1.7, although the significance of this result is limited by the small number of objects (9) in their sample. While Sulentic et al. (2007) also found significant disagreement, they argue that the magnitude of the offset depends on the spectroscopic characteristics of the quasar. Netzer et al. (2007), on the other hand, found no significant offset between the mass estimates, but argued there was also no discernible correlation between them. It is likely that many of the differences between the results of these studies are due to the use of different R BLR − L calibrations, different f -factors, different prescriptions for measuring line widths, limited mass ranges and data quality.
Here we take their measurements, where possible, and make estimates of the BH masses using equations (4), (5) and (6). We caution the reader, however, that we are not redoing the line-width and continuum luminosity measurements in a consistent manner and that this may be a significant source of additional scatter. We used all 21, 15 and 9 sources from VP06, Netzer et al. (2007) and Dietrich et al. (2009) with Hβ and C iv line FWHM and continuum luminosity measurements. We could not use 10, 29 and 1 sources from these studies that lack either or both line widths, or any of the sources from Dietrich & Hamann (2004), which lack measurements of the 5100Å continuum luminosity. We also could not use the sources of Sulentic et al. (2007), as they only report narrow-component subtracted C iv widths, which are not compatible with the rest of the measurements we discuss. We note that the 29 sources we could not use from the study of Netzer et al. (2007) also belong to the sample of Shemmer et al. (2004), for which the C iv line widths and UV continuum luminosities are not reported.
The left panel of Figure 19 compares the C iv and Hβ BH masses derived for all these objects along with those in our sample. A clear correlation is observed for the complete ensemble of objects, albeit with a considerable scatter of 0.41 dex. The scatter is comparable with the 0.46 dex we find for our sample of C iv and Hβ FWHM-based BH masses (see §4.2). A Spearman rank-order coefficient analysis returns r s = 0.79 with a probability that both mass estimates are uncorrelated of P ran = 2 × 10 −12 . A linear fit to the left panel of Figure 19 of the form
log M BH (C IV) 10 8 M ⊙ = m log M BH (Hβ) 10 8 M ⊙ + n(10)
returns a best-fit slope of m = 0.89±0.08 and intercept of n = −0.09±0.08 (the measurement uncertainties were scaled to make χ 2 ν = 1 before determining the uncertainties in the fit parameters). If we plot the residuals between the two BH mass estimates we find, just as in §4.2, that a significant correlation is observed with the ratio of the UV and continuum luminosities ( Figure 20), but not with BH mass, redshift, Eddington ratio or the continuum luminosity at 5100Å (all shown in Figure 21), or with the continuum luminosity at 1350Å (not shown). The best-fit linear relation to the correlation between BH mass residuals and the ratio of the continuum luminosities, shown in Table 7, has a slope of a = 0.82 ± 0.18 and an intercept of b = −0.40 ± 0.07. While the slope is consistent with the value obtained for our sample alone (a = 0.86 ± 0.25, b = −0.23 ± 0.12), the intercept differs by approximately 0.2 dex (approximately 1.5σ). The offset is likely produced by the different prescriptions used to measure the width of the broad emission lines. Figure 19 also compares the C iv and Hβ derived masses after correcting for this correlation by applying equation (8) (see 4.2 for details). While the strength of the correlation has not increased substantially (r s = 0.80, P ran = 6.4 × 10 −13 ), the scatter has decreased from 0.41 dex to 0.34 dex. This change is significantly more modest than that found for our sample of lensed quasars, but this is likely due to the inhomogeneous prescriptions used to measure the width of the emission lines. A linear fit of the form of equation (10) to the relation between the BH mass estimates after applying the correction returns very similar parameters as before, with a best-fit slope of m = 0.88 ± 0.07 and intercept of n = 0.06 ± 0.07. Note that the measurement errors have again been scaled to make χ 2 ν = 1 before estimating the uncertainties in the fit parameters. Given the inhomogeneity of the measurements used, the relatively small number statistics of the sample, their typically large error bars and the likely intrinsic dispersion, however, we cannot currently determine whether the deviation from a slope of unity is significant or not.
As argued before, the inhomogeneity of the measurements can be a very significant source of scatter in the comparisons discussed above. We have shown in the previous section that with homogeneously analyzed, high S/N spectra, the difference between the line widths is not the dominant source of scatter in the comparison between C iv and Balmer line-based BH mass estimates. We would still like to assess whether the C iv and Hβ line widths are correlated in this combined sample. Figure 22 shows this comparison for all the objects used in this section with and without applying the correction based on equation (9). While the scatter in Figure 22 is large, there is still a statistically significant (99%) correlation between the measured line-widths (see Table 8). Most of the scatter is due to the sample of Netzer et al. (2007). Upon inspection of the SDSS spectra used for that study, we find that almost all the outliers correspond to low S/N spectra. Given that the C iv line is typically very complex, this can be a major source of uncertainty.
As an experiment, we obtained higher S/N spectra for one of the outliers, SDSS1151+0340. It has the third most discrepant line-width ratio in the sense that the C iv line is too narrow compared to the Balmer lines. We obtained two independent spectra, one with OSMOS (Martini et al. 2011) at the MDM 2.4m telescope and one with the Double Spectrograph (Oke & Gunn 1982) at the Palomar 200-inch telescope. Due to poor weather conditions and aperture size, only the Double Spectrograph observations yielded a higher S/N spectrum than that of SDSS. All three spectra of SDSS1151+0340 are shown in Figure 23. The spectrum obtained with Double Spectrograph reveals that there is significant absorption near the C iv line, with two clear absorption troughs. These can be seen in the lower S/N spectra, but are difficult to distinguish from the noise. Due to the very substantial absorption, it is not possible to reliably measure the width of the C iv line, and the width measurement of Netzer et al. (2007) should only be considered as a lower bound. While the SDSS spectrum of this source has the lowest continuum S/N in their sample (S/N = 1.5), it is comparable to many of their other sources. The average continuum S/N of the SDSS spectra is only 6.7, with all objects having a lower S/N than any optical spectra in our lensed quasar sample. In particular, the second largest outlier in their sample also has the second lowest S/N of 4.1.
While our example comes from Netzer et al. (2007), low S/N spectra are also present in all the additional samples we consider. If we eliminate objects with continuum S/N < 10 in the vicinity of C iv, the statistical correlation between Hβ and C iv line widths increases dramatically. The bottom panels of Figure 22 show the comparison of the line-widths in the absence of these objects, and a clear correlation is observed between the C iv and Hβ FWHM measurements, regardless of whether we apply the continuum luminosity based correction of equation (9). These correlations are more statistically significant by about two orders of magnitude than when including the low S/N spectra, with a probability of not being real of ∼ 5 × 10 −4 (see Table 8 for details). This suggests that the width of C iv is as good a tracer of BH mass as the width of Hβ, with the caveat that high S/N spectra of the rest-frame UV region are fundamental to accurately model the structure of the C iv emission line.
Conclusions
Using a sample of high-redshift gravitationally lensed quasars observed spectroscopically in the UV/optical and NIR, we have studied the agreement between single-epoch BH mass estimators based on the C iv, Hβ and Hα broad emission lines. Our sample consists of 12 lensed quasars observed with HST by the CASTLES project. In particular, we have used the sample of NIR spectroscopic observations by GPL10 as a starting point and improved on it by (i) adding new NIR observations for 3 objects (SDSS1138+0314, SBS0909+253 and HS0810+2554), (ii) adding high S/N, uniformly analyzed, optical spectroscopic observations for all targets, and (iii) adding the missing rest-frame λL λ (5100Å) luminosity estimates for SDSS0246-0852, HS0810+2554 and Q2237+030.
We described in detail all the methods we used to measure velocity widths and their uncertainties, rest-frame continuum luminosities and to estimate the BH mass of each quasar using the Hβ, Hα and C iv emission lines. We first compared the C iv BH mass estimates based on the FWHM and σ l line-width characterizations and the calibration of VP06 and found that, for our sample, the σ l based BH masses are systematically underestimated with respect to the FWHM-based ones by 0.13 ± 0.06 dex if using prescription A and 0.24 ± 0.07 dex if using prescription B. A similar offset is not observed in the VP06 data set. The difference probably arises from our treatment of the blending of the broad C iv emission line with the nearby broad HeII λ1640 and FeII emission redward of C iv, which is partly confirmed by the lower difference found for the prescription A measurements. This adds to the arguments in Denney et al. (2009) that σ l is not universally reliable for SE mass estimates in the presence of blending, as the results obtained are highly dependent on the exact prescription used for the line characterization. When comparing with BH masses derived from the Hα and Hβ broad emission lines, we find that C iv FWHM based BH masses are not biased, reinforcing the conclusion that the bias is in the σ l estimates. We note, however, that the scatter between C iv FWHM and σ l derived masses is relatively small, suggesting that if a consistent prescription for measuring σ l is applied, σ l would be at least as accurate as FWHM. This is important because σ l measurements are significantly more reliable for complex line profile shapes and in the presence of narrow-line component residuals (Peterson et al. 2004;Denney et al. 2009).
We then compared the C iv and Balmer line BH mass estimates. After offsetting the C iv σ l masses to agree with the FWHM estimates, we find there is no significant offset between C iv and either Balmer line BH mass estimates. Averaged over the 4 possible C iv/Balmer line mass comparisons (see, for example, Figure 7), the offset is −0.15 ± 0.17 dex and the scatter is 0.35 dex. Note that the error in the mean offset corresponds to the average of the errors of the four estimates, which is representative given that the estimates are not truly independent. The scatter of 0.35 dex is very close to the scatter of 0.34 dex found by Shen et al. (2008) between Mg ii and C iv FWHM based BH mass estimates, and significantly larger than the scatter of 0.22 dex they found between Mg ii and Hβ FWHM based BH masses.
We find that the residuals between the C iv and Hβ and Hα based mass estimates are not strongly correlated with the UV or optical continuum luminosities, redshift or Eddington ratio, but we find a strong dependence on the ratio of the UV to optical continuum luminosities. If we correct for this color dependence, the agreement between the C iv and Balmer line estimates is remarkably good, with an average scatter of 0.18 dex, almost a factor of 2 smaller. We find the scatter is smallest -approximately 0.1 dex -when using the Hβ line and the σ l characterization of C iv rather than its FWHM. This observed correlation could be caused by i) reddening, ii) host contamination, or iii) an object-dependent SED shape. The slope we observe is somewhat steeper than that expected in any of these cases, and may suggest a luminosity component to the line-width characterization of the broad emission lines. A larger sample is needed to accurately determine the slope of this correlation and determine its nature with certainty. More generally, the comparison shows that many of the problems in comparing C iv and Balmer line BH mass estimates are associated with the continuum luminosities rather than any potential physical complexities with the C iv lines. When we compare the line-widths directly instead of the BH masses, we find that the width of C iv is well correlated with those of the Balmer lines once the correction based on the ratio of the continuum luminosities is applied.
Our conclusions are unchanged if we add 45 additional, but heterogeneously analyzed, C iv and Hβ estimates from VP06, Netzer et al. (2007) and Dietrich et al. (2009). We used the published FWHM of both emission lines and rest-frame UV and optical continuum luminosities of these sources, but the mass calibrations used for our sample. There is a clear linear correlation between the BH mass estimates, and the residuals are again correlated with the ratio of the continuum luminosities. The residuals are not correlated with either continuum luminosity alone, redshift, BH mass or Eddington ratio. We also find for this heterogeneous sample that the width of C iv is well correlated with that of Hβ, particularly after we eliminate the objects with low S/N C iv spectra. Relatively high S/N spectra are essential to obtaining accurate line widths.
In summary, our results show that C iv is a good BH mass estimator but with small prescription-dependent offsets. The correlation of the mass residuals with the continuum slope could be a bias in either or both of the estimators. Determining the "blame" would require an independent mass estimate, but its existence should not be a surprise given that quasar SEDs are not universal (e.g., Yip et al. 2004). More generally, unless we are to believe that all properties of AGN are determined by a single quantity, the black hole mass, both single-epoch mass estimates and reverberation-mapping radius estimates must depend on additional parameters. That the black hole mass seems to dominate is convenient, but the excess scatter in mass and radius estimates beyond the measurement uncertainties requires either that the error estimates are incorrect or is evidence for additional parameters. One possibility is that radiation pressure plays a significant role (Marconi et al. 2008) and it could easily affect different lines in different ways. While there has been considerable recent effort to expand the range of black hole masses included in these studies (e.g., Kaspi et al. 2007;Bentz et al. 2009;Botti et al. 2010), it is equally important to expand the range in other physical parameters such as spectral shape and Eddington ratio in order to better search for these additional correlations.
A. Notes on Individual Objects
In this section we discuss some details of our line-width and continuum measurements of individual objects. All UV/optical spectra, as well as the continuum and line-profile fits, are shown in Figure 4. LUCIFER spectra of SDSS1138+0314 and HS0810+2554 are shown in Figures 1 and 2, while the LIRIS spectra of SBS0909+532 are shown in Figure 3.
HS0810+2554 -The C iv λ1549 profile of HS0810 shows a small amount of absorption near the peak of the line. We interpolate over this region before making the line-width measurements and fitting the GH polynomial. Our results are consistent with or without the interpolation, as the absorption is weak and only seen near the very peak of the line. To fit the continuum and emission-line features that blended with the Hβ emission of HS0810+2554, a power-law continuum and Fe ii broad-emission line template were fit to the spectrum based on the continuum regions listed in Table 4 and the rest-frame optical Fe ii template of Boroson & Green (1992) from observations of I Zw1 (see Wills et al. 1985;Dietrich et al. 2002Dietrich et al. , 2005, for more details). Narrow [O iii] λλ4959, 5007 emission was then removed by creating a template from a two-component Gaussian fit to the [O iii] λ5007 narrow line and then scaling it to [O iii] λ4959 based on standard emission line ratios. We could not remove narrow Hβ emission because such a component was not obvious in the spectrum 4 . After subtracting these components, the remaining broad Hβ emission was fit with a Gauss-Hermite polynomial, and the FWHM and line dispersion were measured from this fit as described in §3.1. The deblended spectrum of HS0810+2554, showing each component including the GH fit, is shown in Figure 2. Our Hβ FWHM measured from the LUCIFER spectrum of Mogren et al. (in prep.) is consistent with that of GPL10. SBS0909+532 -We use the combined UV-optical-NIR spectrum of Mediavilla et al. (2010) of images A and B of this object, based on a combination of HST STIS and WHT INTEGRAL and LIRIS observations. The UV section of the spectrum is shown in Figure 4 while the NIR section is shown in Figure 3. The C iv profile of SBS0909+532 showed a small absorption trough near observed frame 3600Å and we interpolated over this region before measuring widths. The SBS0909+532 C iv profile shape is 'peaky' with broader wings at the base, and our GH fitting procedure was unable to satisfactorily fit this line profile, so we estimate errors based on the original spectra instead of a GH polynomial fit. For this object we only measure the UV continuum luminosity on image A, as image B shows clear differential reddening with respect to A. We note that the Peng et al. (2006) mass quoted by GPL10 is based on MgII, so using the C iv line-width measurements given in Table 3 provides the first estimate of a C iv-based black hole mass for this object. For the IR spectra of Mediavilla et al. (2010), shown in Figure 3, we removed narrow-line components from the IR spectra using the [O iii] λ5007 line as a template and scaling it to the other narrow lines using standard emission-line ratios between lines of the same atomic species and basing the strength of the Balmer narrow lines on the ratio of [O iii] λ5007/Hβ determined by inspection. We are not as confident in our narrow-line subtraction for this object as for the others because (1) we see residuals near the peak of Hβ, and (2) the exact strength of Hα is uncertain because narrow-line emission remains present after subtraction. The exact level of the residuals for Hα is unclear, since increasing the fraction of emission by as much as a factor of 2 does not result in an obviously improved subtraction. In the case of Hβ, the residuals are not larger than expected based on the S/N of the images, but for Hα we report uncertainties determined from difference between the widths determined with or without the narrow-line subtraction. This results in an Hα FWHM uncertainty several times larger than would be estimated by our Monte Carlo simulations. Comparable σ l uncertainties are measured using both methods, because the line dispersion is far less dependent on the presence of a narrow-line component (see Denney et al. 2009). We measure the Hβ line-widths from the GH fits to the profile and the Hα line-widths directly from the data because the GH polynomials did not accurately fit the line profile. Image B may have a residual sky line peak just blueward of the Hβ narrow-line component. The presence of this emission has little effect on our fits, however, since we measure consistent line widths if we interpolate under this emission to remove it. Our Hα and Hβ widths are consistent with those reported by GPL10.
Q0957+561 -We use the HST STIS UV spectrum of both images obtained by Goicoechea et al. (2005). The rather strange C iv λ1549 line profiles in this object may indicate that there is absorption and/or that the profile shapes in individual images are affected by microlensing from the lens galaxy. However, since there was no definite source of uncertainty to correct for, we simply measured the observed line widths from each spectrum.
HE1104-1805 -We use the EFOSC1 ESO 3.6m telescope UV/optical spectrum of Wisotzki et al. (1995). The C iv λ1549 profile shows a small amount of absorption near the peak of the line, similar to that of the HS0810+2554 C iv λ1549 profile. We therefore apply the same treatment to this line as to the HS0810+2554 profile, and find similarly consistent results with or without interpolation.
PG1115+080 -Due to the severe absorption, both narrow and broad, in the C iv λ1549 line profile in this object, we could not measure the C iv λ1549 line-width directly from the data. However, by masking out the absorption regions, we made a reasonable GH fit to this line profile, from which we measured the line widths given in Table 3.
SDSS1138+0314 -To measure the width of C iv and the UV continuum luminosity we use the FORS1 VLT spectra of images B and C obtained by Eigenbrod et al. (2006). The C iv λ1549 line profile not only shows absorption in the blue side of the line, but is also particularly narrow and 'peaky' with a broad base, similar to SBS0909+523. We were unable to reasonably approximate the profile shape with a sixth-order Gauss-Hermite polynomial. However, since the S/N of this spectrum was very high (see Table 2), we interpolated over the absorption with a 2nd order polynomial, measured the line width directly from the interpolated data, and used this interpolated spectrum and the error spectrum formed with equation (1) to derive uncertainties in the C iv λ1549 width measurement. Because of the combined effects of absorption and the narrow line profile (i.e., where the absorption could be masking the true width), we treat our C iv widths as lower limits. At rest-frame optical wavelengths, the difficulty in removing the blended narrow-line components of Hα and [N ii] λλ6548, 6583 combined with our attempt to accurately fit the emission-line peak (often underestimated with line profile fits) led to an overestimate of the flux between the Hα and N ii λ6583 narrow lines. This overestimate does not significantly affect our width measurements. This object was not part of the GPL10 sample.
H1413+117 -This object is a BAL QSO and therefore a large portion of the C iv λ1549 line profile is completely absorbed on the blue side. Hence, we adopt the C iv λ1549 width measured from only the red side of the line, and we consider this to be a lower limit on the width. B1422+231 -We use the LRIS Keck II UV/optical spectrum of Tonry (1998). We interpolated over the two small absorption troughs near ∼6875Å and ∼7020Å before measuring the C iv λ1549 widths directly from the data. Our treatment of these regions did not affect the resulting GH fit to the data. From the GPL10 data, we cannot assess the reliability of their fit to the Hβ profile, because they plotted the Hβ spectrum of HE1104-1805 in place of the spectrum of B1422+231. In order to be conservative, we therefore flag B1422+231 as one of the objects with possible problems in the sample of GPL10.
FBQ1633+3134 -There is evidence for absorption in the blue side of C iv, however, it is not clear that a reliable interpolation could be made across this possible absorption. We measure the line width as is, and treat this measurement as a lower limit.
Q2237+030 -We use the FORS1 VLT UV/optical spectra of images C and D obtained by Eigenbrod et al. (2008). We follow the same prescription as for SDSS1138+0314 and Q0957+561 and use an average of the C iv line-widths of each image to estimate M BH . This object shows C iv λ1549 absorption in the red side of the line. We interpolate over this absorption with a third order polynomial before measuring the line width and fitting the GH polynomial to the data. The interpolation creates a peak slightly higher than that observed in the original spectrum, but makes for a much more symmetric line profile, which is more typical of the core of C iv λ1549 line profiles, than a linear or quadratic interpolation. This increase in the assumed line peak decreases our line-width measurements, but not significantly. a Rest frame wavelengths are used here because line boundaries were chosen after the deblending procedure that transfers the spectrum into the rest frame. Note. -The fits discussed in §4.2 correspond those performed using the prescription B C iv line-widths. Fits obtained using prescription A measurements are shown for completeness. The fits to the combined sample discussed in §5 are also reported here. Note. -The table shows the correlation strength of the C iv and Balmer line-width estimates for both C iv line-width measurement prescription, quantified by the Spearman rank order coefficient, r s . Results are shown for group I and the combination of groups I and II measurements. In each case, N indicates the number of lenses used to estimate the correlation strength and P ran indicates the probability of observing such a correlation by chance if the variables are uncorrelated. Mediavilla et al. (2010). The top panel shows the complete spectrum while the bottom four panels show the spectral regions around Hα and Hβ of each quasar image. Overlaid on top are the best fit continuum and narrow-and broad-line components, as well as their overall sum, using the same line-styles as in Figure 1. each UV/optical spectrum around C iv along with the best fit continuum (dotted line) around the line, the best fit line profile (long dashed) and the addition of both (short dashed). For each spectrum, the fits obtained using prescription A are shown in the left panel while those obtained following prescription B are shown in the right panel. We could not obtain good fits for SDSS1138+0314 and SBS0909+532 C iv lines (see Appendix A for details). -Comparison of C iv BH masses derived from the FWHM and σ l velocity widths for all objects in our sample using the relations of VP06. The left (right) panel compares the BH mass estimates based on the prescription A (B) line-width measurements of C iv. Masses are equal along the dashed line and the dotted line correspond to the best fit offset. The objects with arrows correspond to those for which we believe our C iv line-width measurements to be lower bounds. Figure 8, but as a function of the estimated Eddington ratio. We used a factor of 11.91 to convert between λL λ (5100Å) and L Bol , as determined from the AGN SED of Assef et al. (2010). Figure 8, but as a function of the asymmetry of the C iv line, parametrized as the ratio of the widths red and blue of the line centroid. We do not show lower bounds on the C iv line-width as those object do not have a well defined blue side width due to the presence of absorption. Figure 8, but as a function of the ratio of the UV to optical continuum luminosities. The solid line shows the best fit linear relation to all objects with group I Balmer-line width estimates (solid symbols) and the dashed line shows the linear relation obtained when also including object with group II estimates (open symbols). Note that we do not include objects with lower bound C iv line widths on the fits. Figure 7, but after correcting the C iv BH masses for the dependence on the ratio of the UV to optical continuum luminosities observed in Figure 15. Figure 19. The solid line shows the best-fit linear relation for our data while the dashed line shows the best-fit to the combined sample. Error-bars are not shown in order to make the plot more legible. Fig. 19. The large gray hexagon shows SDSS1151+0340. Top panels show the complete literature sample while bottom panels only show objects with spectra that have continuum S/N > 10 in the vicinity of C iv. . The arrows mark the probable absorption troughs near the C iv line. The spectra have been resampled to a common resolution, and the continuum S/N shown in the upper left corner of each panel has been calculated in the same way as for all other objects in our sample.
with the R BLR − λL λ (5100Å) relation ofBentz et al.
Note. -Literature UV/optical spectra obtained from the following references: a) SDSS DR7 (Abazajian et al. 2009), b) Mediavilla et al. (2010), c) Goicoechea et al. (2005), d) Wisotzki et al. (1995), e) Eigenbrod et al. (2006), f ) Tonry (1998), g) Eigenbrod et al. (2008). † Unmagnified quasar H-band magnitudes.
Fig. 1 .
1-(Top) LUCIFER H-and K-band spectra of SDSS1138+0314. The black solid line shows the spectrum obtained by performing the sky subtraction with the median combination of the sky frames while the gray line shows that obtained by using the modified version of the COSMOS software described in the text. (Bottom Left) Spectral region around Hβ. Overlaid on top are the best fit continuum (black dotted line) and narrow (black short-dashed line) and broad line components (black long-dashed line), as well as their sum (black solid line) and the error spectrum (thin gray solid line). (Bottom Right) Same as bottom left but for Hα.
Fig. 2 .
2-LUCIFER J-band spectrum of HS0810+2554 (gray solid line). Overlaid are the best fit continuum and FeII emission (black dotted line), narrow line emission (black shortdashed line), broad line component (black long-dashed line) and their sum (black solid line), as well as the error spectrum (thin gray solid line).
Fig. 3 .
3-LIRIS near-IR spectra of images A and B of SBS0909+532 obtained by
Fig. 4 .
4-UV/optical spectra of all objects used in this study. For each object the panel shows the full spectrum, the source from which it was obtained and its redshift.
Fig. 5 .
5-The Figure shows the region of
Fig. 6 .
6Fig. 6.-Comparison of C iv BH masses derived from the FWHM and σ l velocity widths for all objects in our sample using the relations of VP06. The left (right) panel compares the BH mass estimates based on the prescription A (B) line-width measurements of C iv. Masses are equal along the dashed line and the dotted line correspond to the best fit offset. The objects with arrows correspond to those for which we believe our C iv line-width measurements to be lower bounds.
Fig. 7 .
7-Comparison between BH masses estimated from the prescription B σ l and FWHM of C iv and from the FWHM of Hα and Hβ. For the estimates based on the line dispersion of C iv we have added the systematic offset of 0.24 dex described in §4.1. Solid symbols correspond to the objects with group I Hα or Hβ line-width estimates, while open symbols correspond to those with group II estimates. Six-pointed stars mark the objects not considered in the analysis of GPL10. The dotted line shows where the BH masses are equal.
Fig. 8 .
8-Ratio between the C iv and Hβ/Hα mass estimates as a function of the corresponding hydrogen line mass estimate. Symbols and lines have the same definitions as inFigure 7.
Fig. 9 .
9-Same asFigure 8, but as a function of the UV continuum luminosity.
Fig. 10 .
10-Same asFigure 8, but as a function of the 5100Å continuum luminosity.
Fig. 11 .
11-Same asFigure 8, but as a function of redshift.
Fig. 12 .
12-Same as
Fig. 13 .
13-Same asFigure 8, but as a function of the blueshift of the C iv line.
Fig. 14 .
14-Same as
Fig. 15 .
15-Same as
Fig. 16 .
16-Same as
17.-Comparison between the C iv and Balmer lines measured line-widths. Points and lines have the same meaning as inFigure 7. We do not show lower bounds for clarity.
Fig. 18 .
18-Comparison between the C iv and Balmer lines measured line-widths after applying correction from equation (9). Points and lines have the same meaning as inFigure 7. We do not show lower bounds for clarity.
Fig. 19 .
19-Panel a) shows the BH masses estimated using eqns. (4), (5) and (6) for the objects in the samples of VP06 (open squares), Netzer et al. (2007) (solid gray pentagons) and Dietrich et al. (2009) (solid black triangles) for which this was possible (see §5 for details). Objects in our sample are shown by the solid and open six-pointed stars and circles, keeping the point style conventions used in previous plots. The dotted line shows where the masses are equal. Error-bars are not shown in order to make the plot more legible. Panel b) shows the results after applying the continuum slope correction fromTable 7.
Fig. 20 .
20-Residuals between BH masses estimated from the FWHM of the C iv and Hβ broad emission lines as a function of the logarithm of ratio of the continuum luminosities at 1350Å and 5100Å for the samples of VP06 (open squares), Netzer et al. (2007) (solid gray pentagons) and Dietrich et al. (2009) (solid black triangles) as well as our sample (solid and open six-pointed stars and circles). Point-styles have the same definitions as in
Fig. 21 .
21-Residuals between BH masses estimated from the FWHM of the C iv and Hβ broad emission lines as a function of the logarithm of Hβ-based BH mass (top left), redshift (top right), Eddington ratio (bottom left) and continuum luminosity at 5100Å(bottom right). Objects belong to the samples of VP06 (open squares), Netzer et al. (2007) (solid gray pentagons) and Dietrich et al. (2009) (solid black triangles) as well as our sample (solid and open six-pointed stars and circles). Point-styles have the same definitions and inFigure 19. The dotted line shows where the masses are equal. Error-bars are not shown in order to make the plot more legible.
Fig. 22 .
22-Comparison of the C iv and Hβ line-widths for the sample we have compiled from the literature. Point types and line styles are the same as for
Fig. 23 .
23-Spectra of the QSO SDSS1151+0340 obtained by SDSS (top), with MDM/OSMOS (middle) and with Palomar/Double Spectrograph(bottom)
Table 1 .
1Lens Magnifications and Continuum LuminositiesObject
Ref.
z
Image Magnification
m H
†
log λL λ / erg s −1
A
B
C
D
(mag)
1350Å 1450Å 5100Å
Q0142-100
a
2.72
3.3 0.4 · · ·
· · ·
16.56
46.83 46.76 46.27
SDSS0246-0825
a
1.69
26.9 8.9 · · ·
· · ·
20.39
· · ·
44.53 44.59
HS0810+2554
· · · 1.51
47.2 51.1 13.5 7.7
18.72
· · ·
44.44 44.84
SBS0909+532
b
1.38
1.7 1.5 · · ·
· · ·
15.18
46.08 46.05 46.31
Q0957+561
c
1.41
3.1 1.7 · · ·
· · ·
16.51
46.31 46.25 45.79
HE1104-1805
d
2.32
16.2 2.3 · · ·
· · ·
18.52
46.15 46.09 45.38
PG1115+080
a
1.72
19.6 18.7
3.2 4.9
19.13
· · ·
45.47 44.93
SDSS1138+0314
e
2.44
7.3 3.7
5.2 6.9
20.65
44.83 44.77 44.81
H1413+117
a
2.55
8.2 6.8
6.8 3.4
18.05
45.73 45.78 45.63
B1422+231
f
3.62
6.6 8.2
4.3 0.3
16.55
46.83 46.74 46.42
FBQ1633+3134
· · · 1.52
2.7 0.7 · · ·
· · ·
16.85
45.65 45.64 45.72
Q2237+030
g
1.69
4.9 4.3
2.2 4.1
16.83
· · ·
45.53 45.98
Table 2 .
2C iv Emission Line and Continuum Region Boundaries Prescription A Prescription B Blue Cont. Red Cont. Broad Line Blue Cont. Red Cont. The S/N quoted is the S/N per pixel averaged over all continuum regions listed for each object.Broad Line Res
Table 4 .
4NIR Emission Line and Continuum Region BoundariesEmission
Blue Cont.
Red Cont.
Broad Line Res
Object
Line
S/N a
(Å)
(Å)
(Å)
(Å)
HS0810+2554 a
Hβ
36
4680-4710
5080-5120
4710-4960
8.0
SBS0909+532-A
Hβ
68 11205-11310 11840-11855 11380-11840 · · ·
SBS0909+532-B
Hβ
22 11205-11310 11840-11855 11380-11840 · · ·
SBS0909+532-A
Hα
25 14210-14440 17500-17700 14865-16650 · · ·
SBS0909+532-B
Hα
18 14210-14440 17500-17700 14865-16650 · · ·
SDSS1138+0314
Hβ
12 16150-16300 17485-17623 16300-17100 8.0
SDSS1138+0314
Hα
8
21780-21850 23270-23310 22135-23040 8.0
Table 6 .
6Correlations of BH Mass ResidualsNote. -The table shows the correlation strength of the BH mass residuals as a function of each different variable, quantified by the Spearman rank order coefficient, rs. Results are shown for group I and the combination of groups I and II measurements. In each case, N indicates the number of QSOs used to estimate the correlation strength and Pran indicates the probability of observing such a correlation by chance if the variables are uncorrelated.Variable
C iv
Balmer
Group I Estimates
Group I & II Estimates
Width
Line
rs
N
Pran
rs
N
Pran
M BH Balmer lines
σ l
Hβ
-0.214
7
0.644
-0.183
9
0.637
Hα
0.100
5
0.873
-0.048
8
0.911
FWHM Hβ
-0.500
7
0.253
-0.333
9
0.381
Hα
-0.400
5
0.505
-0.143
8
0.736
λL λ (1350Å)
σ l
Hβ
0.357
7
0.432
0.467
9
0.205
Hα
0.500
5
0.391
0.571
8
0.139
FWHM Hβ
0.143
7
0.760
0.367
9
0.332
Hα
0.100
5
0.873
0.429
8
0.289
λL λ (5100Å)
σ l
Hβ
-0.214
7
0.644
-0.067
9
0.865
Hα
-0.300
5
0.624
-0.048
8
0.911
FWHM Hβ
-0.500
7
0.253
-0.233
9
0.546
Hα
-0.700
5
0.188
-0.214
8
0.610
Redshift
σ l
Hβ
0.607
7
0.148
0.517
9
0.154
Hα
0.500
5
0.391
0.500
8
0.207
FWHM Hβ
0.750
7
0.052
0.583
9
0.099
Hα
0.600
5
0.285
0.619
8
0.102
L/L Edd
σ l
Hβ
-0.036
7
0.939
0.200
9
0.606
Hα
-0.300
5
0.624
0.024
8
0.955
FWHM Hβ
-0.286
7
0.534
0.067
9
0.865
Hα
-0.200
5
0.747
-0.071
8
0.867
C iv Blueshift
σ l
Hβ
0.536
7
0.215
0.033
9
0.932
Hα
0.100
5
0.873
-0.214
8
0.610
FWHM Hβ
0.679
7
0.094
0.133
9
0.732
Hα
0.600
5
0.285
0.143
8
0.736
C iv assymetry
σ l
Hβ
0.429
7
0.337
0.117
9
0.765
Hα
0.300
5
0.624
-0.048
8
0.911
FWHM Hβ
-0.393
7
0.383
-0.333
9
0.381
Hα
0.300
5
0.624
-0.476
8
0.233
λL λ (1350Å)/λL λ (5100Å)
σ l
Hβ
0.929
7
0.003
0.883
9
0.002
Hα
1.000
5
0.000
0.809
8
0.015
FWHM Hβ
0.750
7
0.052
0.767
9
0.016
Hα
0.700
5
0.188
0.857
8
0.007
Table 7 .
7Linear Fits to Correlations of BH Mass Residuals with AGN colorC iv
Balmer
Group I Estimates
Group I & II Estimates
Width
Line
a
b
a
b
Prescription A
σ l
Hβ
0.64 ± 0.13 -0.13 ± 0.06
0.68 ± 0.16 -0.13 ± 0.08
Hα
0.58 ± 0.15 -0.11 ± 0.06
0.57 ± 0.11 -0.17 ± 0.06
FWHM
Hβ
0.89 ± 0.25 -0.20 ± 0.12
0.95 ± 0.22 -0.19 ± 0.11
Hα
0.75 ± 0.30 -0.20 ± 0.13
0.79 ± 0.15 -0.23 ± 0.08
Prescription B
σ l
Hβ
0.60 ± 0.11 -0.18 ± 0.05
0.68 ± 0.17 -0.16 ± 0.08
Hα
0.51 ± 0.14 -0.14 ± 0.06
0.58 ± 0.16 -0.22 ± 0.08
FWHM
Hβ
0.86 ± 0.25 -0.23 ± 0.12
0.91 ± 0.22 -0.22 ± 0.10
Hα
0.72 ± 0.30 -0.23 ± 0.13
0.76 ± 0.16 -0.27 ± 0.08
Combined Sample
FWHM
Hβ
0.82 ± 0.18 -0.40 ± 0.07
0.85± 0.18 -0.40± 0.07
Table 8 .
8Correlations of Line-Width Estimates Combined Sample (S/N > 10) FWHM Hβ 0.551 31 1.3 × 10 −3 0.539 33 1.2 × 10 −3C iv Prescription
C iv
Balmer
Group I Estimates
Group I & II Estimates
Width
Line
r s
N
P ran
r s
N
P ran
Without color correction
Prescription A
σ l
Hβ
0.214 7
0.644
0.300 9
0.433
Hα
0.700 5
0.188
0.786 8
0.021
FWHM Hβ
-0.143 7
0.760
0.000 9
1.000
Hα
0.300 5
0.624
0.500 8
0.207
Prescription B
σ l
Hβ
0.679 7
0.094
0.333 9
0.381
Hα
0.900 5
0.037
0.524 8
0.183
FWHM Hβ
-0.143 7
0.760
0.000 9
1.000
Hα
0.300 5
0.624
0.500 8
0.207
Combined Sample
FWHM Hβ
0.346 52
0.012
0.342 54
0.011
With color correction
Prescription A
σ l
Hβ
0.571 7
0.180
0.550 9
0.125
Hα
0.900 5
0.037
0.905 8
0.002
FWHM Hβ
0.357 7
0.432
0.383 9
0.308
Hα
0.400 5
0.505
0.738 8
0.037
Prescription B
σ l
Hβ
0.929 7
0.003
0.467 9
0.205
Hα
0.900 5
0.037
0.452 8
0.260
FWHM Hβ
0.357 7
0.432
0.383 9
0.308
Hα
0.400 5
0.505
0.667 8
0.071
Combined Sample
FWHM Hβ
0.326 52
0.018
0.323 54
0.017
Combined Sample (S/N > 10) FWHM Hβ
0.602 31 3.4 × 10 −4
0.587 33 3.3 × 10 −4
http://www.astronomy.ohio-state.edu/MDM/CCDS/
http://nedwww.ipac.caltech.edu/
This preprint was prepared with the AAS L A T E X macros v5.2.
GPL10 are similarly unable to isolate a narrow component in their observations of HS0810+2554
* Group II line-widths. See §3.1 for details.
Fig. 5.-Continued
We would like to thank Jenny E. Greene, Christopher Onken, Chien Y. Peng, Kristen Sellgren, Marianne Vestergaard and Linda Watson for their help and suggestions that improved our work. We thank F. Courbin, E. Mediavilla, V. Motta, L.J. Goicoechea, S. Sluse, J.L. Tonry, L. Wisotzki and J. Muñoz for sending us their optical spectra of Q2237+030, SDSS1138+0314, Q0957+561, HE1104-1805, B1422+231, and SBS0909+532. We thank F. Harrison for helping us obtain an optical spectrum of SDSS1151+0340. We would also like to thank all the people in the LUCIFER science demonstration time team that did not participate directly in this work. We thank the anonymous referee for suggestions that help improve our work. R.J.A. was supported in part by an appointment to the NASA Postdoctoral Program at the Jet Propulsion Laboratory, administered by Oak Ridge Associated Universities through a contract with NASA. C.S.K. is supported by NSF grants AST-0708082 and AST-1009756. B.M.P., M.D. and R.W.P. are supported by NSF grant AST-1008882. P.M. is supported by NSF grant AST-0705170. This research has made use of the NASA/IPAC Extragalactic Database (NED) which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration.3.96 ± 0.18 3.78 ± 0.12 3.49 ± 0.07 2.51 ± 0.05 3.80 ± 1.40 · · · 4.80 ± 0.60 * · · · † All Hα and Hβ line width measurements correspond to those inTable 1of GPL10, except for SDSS1138+0314, SBS0909+523 and the Hβ widths of HS0810+2554. 9.69 ± 0.32 9.65 ± 0.32 9.37 ± 0.29 9.27 ± 0.29 9.72 ± 0.38 * · · · · · · FBQ1633+3134 8.88 ± 0.32 ‡ 8.82 ± 0.32 ‡ 8.77 ± 0.29 ‡ 8.29 ± 0.29 ‡ 9.11 ± 0.28 * · · · 9.11 ± 0.27 Q2237+030 8.67 ± 0.33 8.63 ± 0.32 8.63 ± 0.29 8.34 ± 0.29 9.08 ± 0.39 · · · 9.38 ± 0.25 * Note. -All BH masses correspond to those obtained from eqns. (4),(5)and(6). None of the corrections discussed in § §4.1 and 4.2 have been applied. * Based on group II line-width. See §3.1 for details. ‡ Should be considered as lower bound. See Appendix A for details.
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|
[] |
[
"Discrete geodesics and cellular automata",
"Discrete geodesics and cellular automata"
] |
[
"Pablo Arrighi [email protected] \nAix-Marseille Univ\nLIF\nF-13288Marseille Cedex 9France\n",
"Gilles Dowek [email protected] \nInria\n23 avenue d'Italie81321, 75214Paris Cedex 13CSFrance\n"
] |
[
"Aix-Marseille Univ\nLIF\nF-13288Marseille Cedex 9France",
"Inria\n23 avenue d'Italie81321, 75214Paris Cedex 13CSFrance"
] |
[] |
This paper proposes a dynamical notion of discrete geodesics, understood as straightest trajectories in discretized curved spacetime. The notion is generic, as it is formulated in terms of a general deviation function, but readily specializes to metric spaces such as discretized pseudo-riemannian manifolds. It is effective: an algorithm for computing these geodesics naturally follows, which allows numerical validation-as shown by computing the perihelion shift of a Mercury-like planet. It is consistent, in the continuum limit, with the standard notion of timelike geodesics in a pseudo-riemannian manifold. Whether the algorithm fits within the framework of cellular automata is discussed at length.
|
10.1007/978-3-319-26841-5_11
|
[
"https://arxiv.org/pdf/1507.06836v1.pdf"
] | 6,574,957 |
1507.06836
|
705ff7afa9b5648854268104ac4a67004624c253
|
Discrete geodesics and cellular automata
Pablo Arrighi [email protected]
Aix-Marseille Univ
LIF
F-13288Marseille Cedex 9France
Gilles Dowek [email protected]
Inria
23 avenue d'Italie81321, 75214Paris Cedex 13CSFrance
Discrete geodesics and cellular automata
Discrete connectionparallel transportgeneral relativityRegge calculus
This paper proposes a dynamical notion of discrete geodesics, understood as straightest trajectories in discretized curved spacetime. The notion is generic, as it is formulated in terms of a general deviation function, but readily specializes to metric spaces such as discretized pseudo-riemannian manifolds. It is effective: an algorithm for computing these geodesics naturally follows, which allows numerical validation-as shown by computing the perihelion shift of a Mercury-like planet. It is consistent, in the continuum limit, with the standard notion of timelike geodesics in a pseudo-riemannian manifold. Whether the algorithm fits within the framework of cellular automata is discussed at length.
Introduction
Three reasonable hypotheses-bounded velocity of propagation of information, homogeneity in time and space, and bounded density of information-lead to the thesis that natural phenomena can be described and simulated by cellular automata. This implication has in fact been formalized into a theorem both in the classical [9] and the quantum case [1], albeit in flat space. Further evaluating this thesis leads to the project of selecting specific physical phenomena, such as gravitation, and attempting to describe them as cellular automata. A first step in this direction is to build discrete models of the phenomena. In the case of gravitation, this leads to the question we address in this paper: what is a discrete geodesics?
Geodesics generalize the flat space notion of line, to curved spaces. A line is both the shortest, and the straightest path between two points, but in curved space the two criteria do not coincide [12]. In computer graphics and discrete geometry, discrete geodesics as shortest path between two given point have been studied extensively [13,14]. This is not the case of geodesics as straightest path given an initial point and velocity-with the noticeable exception of [15], in the framework of simplicial complexes. Yet, it is this criterion that one must adopt in order to describe and simulate the timelike geodesics trajectories of particles. In this paper we adopt the dynamical, spacetime view on geodesics which is typical of numerical relativity [16]. But instead of discretizing geodesics defined by partial differential equations in a continuous spacetime, we seek to discrete geodesics as a native notion of a discretized spacetime, for instance of a grid endowed with a metric.
More precisely, the paper proposes both a notion of discrete-spacetime geodesics and a notion of discrete-time continuous-space geodesics (Section 2). Both are generic, that is formulated in terms of a general deviation function, but readily specialize for metric spaces (Section 4). They are effective: an algorithm for computing timelike geodesics naturally follows (Section 3), which allows us to validate the notions numerically, by computing the perihelion shift of a Mercury-like planet (Section 5). They are consistent with one another: the former is clearly a discretization the the latter. Moreover, the latter is proven to have the standard notion of continuous-spacetime geodesics in a Riemanian space as its limit, which validates both notions as legitimate discrete counterparts (Section 9). Whether the algorithm fits within the framework of cellular automata is discussed at length, as well as how this impacts on precision (Sections 6-8).
These results apply to natively discrete formulations of General Relativity such as Regge calculus [17,4]. For instance, our method yields perihelion shift computations of the right order, an issue in [17] pointed out in [4]. We discuss how our approach differs from [15] and why it fixes this issue. Finally, an often underestimated contribution is the pedagogical: the simple discrete model summarized in Figure 1 has continuum limit the complicated, well-known equations of (9) and (10).
Besides assessing this "digital physics" program, we believe that these results can be applied in any inherently discrete geometrical setting, in order to compute geodesics without the need to interpolate a continuous surface. Such applications may arise in computer vision and graphics [14] including computer anatomy [10].
Discrete geodesics
Consider a discrete-time continuous-space spacetime Z × R n where Z is the discrete timeline and R n a continuous space. Consider a deviation function w from (Z × R n ) 3 to R + , the number w(E, F, G) measuring how the path E, F, G deviates from "going straight ahead". In this setting, a geodesic is a sequence of points in Z × R n (E i ) i such that for any i, w(E i−1 , E i , E i+1 ) = 0. Such a property can be read as a condition on E i+1 : if the points E i−1 and E i are given, the geodesics must continue with a point E i+1 such that w(
E i−1 , E i , E i+1 ) = 0.
Consider now a discrete spacetime M = Z × Z n where Z is the discrete timeline and Z n a discrete space and a deviation function w from M 3 to R + which, as above, measures how the path E, F, G deviates from going straight ahead. In this setting we cannot demand w(E i−1 , E i , E i+1 ) to be exactly zero, but we demand that it be a minimum with respect to spatial local variations of E i+1 .
Spatial local variations can be defined as follows. Let us write x 0 , x 1 , ..., x n , the coordinates of a point E in M , where x 0 is the time coordinate and x 1 , ..., x n the space coordinates. Two points of M , x 0 , x 1 , ..., x n and x 0 , x 1 , ..., x n are said to be spatial neighbors if x 0 = x 0 and for all i ≥ 1, |x i − x i | ≤ 1.
Thus, a discrete geodesics in M can be defined as a sequence of points in M , (E i ) i such that for any i, the deviation w(E i−1 , E i , E i+1 ) is a local minimum with respect to spatial local variations of E i+1 , that is for any spatial neighbor G of E i+1 , we have
w(E i−1 , E i , G) ≥ w(E i−1 , E i , E i+1 )(1)
Notice how this condition may be understood as a discrete counterpart of the Euler-Lagrange equation, in the spirit of [11].
An algorithm to compute a geodesic
We now give a gradient descent-like algorithm to compute a discrete geodesic, t 0 , A 0 , t 1 , A 1 , t 2 , A 2 given a deviation function w, a timeline t 0 , t 1 , t 2 , ..., and two starting points A 0 and A 1 .
Assume, A i−1 and A i are computed. To compute A i+1 start with a point t i+1 , C . Compute w( t i−1 , A i−1 , t i , A i , t i+1 , C ) for all 3 n spatial neighbour C of C. If they are all larger than w( t i−1 , A i−1 , t i , A i , t i+1 , C ) take C for A i+1 . Otherwise chose a C which minimizes w( t i−1 , A i−1 , t i , A i , t i+1 , C )
and iterate, starting from this C .
Whether this iteration will eventually end depends, in general, on w(., ., .). For instance, say that w( t i−1 , A i−1 , t i , A i , t i+1 , C ) increases as soon as A i C > t i+1 − t i . Then A i+1 will have to lie within distance t i+1 − t i of A i , thereby imposing a bounded velocity c = 1, as well as enforcing termination.
Distance induced deviation function
Most of the times, the idea of deviating from going straight is induced from a notion of distance. Here is how. Suppose a distance function d, and define the three point distance function
l(E, F, G) = d(E, F ) + d(F, G).
Intuitively, FG is understood to deviate from EF if it "leans" in some spatial direction FF , as witnessed by the fact that
l(E, F , G) < l(E, F, G) for F some neighbour of F .
In a continuous-space discrete-time setting this would be formalized by letting w(E, x 0 , . . . , x n , G) be
∂ 0 l(E, x 0 , . . . , x n , G)) 2 + . . . + (∂ n l(E, x 0 , . . . , x n , G)) 2 . (2) E F F G Fig. 1.
Discrete geodesics seek to find G such that FG minimizes its deviation relative to EF. In the case of metric spaces, FG is understood to "deviate towards FF relative to EF", whenever l(E, F , G) < l(E, F, G)-such deviations must be minimized.
Thus, for continuous-space discrete-time geodesics
(E i ) i each point E i is a local extremum for l(E i−1 , E i , E i+1 ).
In the discrete spacetime case, w(E, x 0 , . . . , x n , G) is simply obtained by replacing partial derivatives with finite differences in Equation (2):
∂ µ l(E, x 0 , . . . , x n , G) becomes (l(E, x 0 , . . . , x µ − 1, . . . , x n , G) − l(E, x 0 , . . . , x µ + 1, . . . x n , G))/2.(3)
And, for discrete spacetime geodesics
(E i ) i each point E i minimizes the possibly non-zero w(E i−1 , E i , E i+1 ).
Discrete Schwarzschild spacetime
In this section, we give an example of discrete spacetime, which is a discretization of the Schwarzschild spacetime of General Relativity. Discretize spacetime down to ∆ = 1cm. Consider a star of mass M = 2.10 30 kg-alike the Sun. Its Schwarzschild radius is m = 2GM/c 2 = 3km = 3.10 5 cm. In order to evaluate distances, consider the metric tensor
g( t, x, y ) = 1 − m r 0 0 0 − x 2 r(r−m) − y 2 r 2 − mxy r 2 (r−m) 0 − mxy r 2 (r−m) − x 2 r 2 − y 2 r(r−m)
where r = x 2 + y 2 , and let the distance function d be defined by
d(E, F ) = EF † g(E)EF.
We study the geodesics trajectory of a planet, with respect to a timeline 0, r∆, 2r∆, 3r∆, ... with r = 10 7 . Thus r∆ = 10 7 cm = 3.33.10 −4 s. The fake planet has parameters chosen so as to maximize relativistic effects: its first point is E = x 0 = 0, x 1 = 10 8 cm = 1000km, x 2 = 0 , and its initial velocity is vx = 0, vy = 2.10 −2 c = 6000km.s −1 .
We compute the geodesics with respect to the w(., ., .) induced by d(., .) as in Section 4, and following the algorithm of Section 3. Recall that in this algorithm at iteration i the point A i+1 is found by gradient descent starting from some point C. In the context of planetary movement, a good guess for C is obtained as follows. Define velocity
S i = A i − A i−1 and acceleration R i = S i − S i−1 ,
and make the guess that acceleration will remain constant, that is
R i+1 = R i . This would entail that A i+1 = A i + S i + R i , thus take C = A i + S i + R i
as the first guess and start exploring for the real A i+1 . Within reasonable ranges other heuristics-for instance, C = A i + S i -lead to the same trajectories, but may require longer computation times.
A run of the simulation is shown in Figures 2 and 3. Computation time is a few seconds. The code is available in [2]. The code is easily augmented to detect aphelion, typically a = 1000km, and perihelion, typically p = 150km.
The perihelion shift is visible on Figure 2. A well-known formula [8] states that perihelion shift in radians per revolution should be
σ = 24π 3 L 2 T 2 c 2 (1 − e 2 ) = 6πGM c 2 L(1 − e 2 ) = 3πm P
where T is the revolution period of the planet, L is the semi-major axis of the trajectory of the planet, e its eccentricity, and P = L(1 − e 2 ) its parameterrecall that, by Kepler's third law, T 2 = 4π 2 L 3 /(GM ), and m = 2GM/c 2 . Then an easy geometrical relation is
P = 2 1/a + 1/p hence σ = (3/2)πm(1/a + 1/p)
which typically is 6.17 deg. The observed shift is around 6.27 deg.
Cellular Automata in Mechanics
As suggested in the Introduction, one motivation for discretizing General Relativity is to describe the motion of a planet in a cellular automaton.
Recall that, in the cellular automata vocabulary, a configuration σ is a function which associates, to each cell C of the grid Z n , some internal state σ(C) taken in the set Σ. A cellular automaton is a function F from configurations to configurations, which has the following physics-inspired symmetries:
bounded velocity of propagation of information; homogeneity time and space; bounded density of information, that is Σ is finite.
The state of a cell can be used to express the presence or the absence of a particle in this region of space. This way cellular automata can describe particle motions. For instance the simplest n-dimensional cellular automata-2 states, radius 1-can describe one particle motion among 3 n , as in each dimension, it could have velocity −1, 0, or 1.
To describe more complex motions, we must increase the number of states. For instance in a 1-dimensional automaton with radius 1, we can describe the motion of a particle that goes to the right at velocity 1/2, by alternating states s 1 -stay still-and s 2 -step-, but also the motion of a particle that goes to the right at velocity 1, staying in the state s 3 .
Another option is to increase the radius of the automaton. For instance, in a 1-dimensional automaton, with radius 2 we can describe the motion of a particle that goes on the right at velocity 1 staying in a state s 1 or at velocity 2 staying in a state s 2 . Notice that modulo changing the units, the former behaviour can be obtained from the latter just by cell grouping.
We want to address the following question: to what extent is the algorithm of Section 3 just a cellular automaton?
Geodesics as Cellular Automata
Whether the algorithm of Section 3 enforces a bounded velocity of propagation of information c = 1 depends, in general, on the properties of w(., ., .). If such a velocity bound is enforced, then the motion of the body can be described in a cellular automaton of radius r. It is well-known that the velocity of a particle in a continuous Schwarzschild spacetime is bounded by c = 1. We conjecture that this is also the case for the discretized Schwarztchild spacetime.
If w(., ., .) does not depend on space and time, then the algorithm clearly acts the same everywhere and everywhen, so that homogeneity is also enforced. In the important case where it depends upon a space-dependent metric, then this metric field has to be carried by the internal state of the cells, even if it does not contain a particle, so that homogeneity is still enforced.
Let us evaluate whether bounded density of information holds. Even when w(., ., .) does not depend on space and time, it is still the case that if a particle is at E i , we need its velocity E i−1 E i to compute its next position E i+1 . But thanks to bounded velocity of propagation, and the fact that positions are discrete, the number of possible velocities is bounded above by b = (2r +1) n , so that bounded density of information is preserved. In the important case of a space-dependent metric carried by the internal state of the cells, whether bounded density holds depends upon whether we can assume that the metric field can be given with bounded precision. Even if this is not the case, notice that for a given cell, all that matters is to distinguish, for each input velocity of the particle, between b output candidate target cells. This map is a discrete counterpart to the connection associated to the metric. It contains just the finite amount of information that needs to be attached to the cell in order to compute geodesics. It could in fact be pre-compiled into each cell, thereby yielding a cellular automaton with b + 1 internal states to code for presence and velocity, times b b to code for the discrete connection.
Time versus space, precision
Geodesics have been popularized by General Relativity. General Relativity likes to put space and time on an equal footing. Numerical schemes for General Relativity ought to pursue that path, in particular it would be nice if the timeline of the computed geodesic were just 0, ∆, 2∆, 3∆, ... that is if r was equal to 1. In the scheme of Section 3, this choice leads to a cellular automaton of radius 1, which is appealing, but it also restricts to b = 3 n the number of possible velocities. As we discussed in Section 6, this severely limits the number of motions that can be described. In the quantum setting, superpositions of basic velocities may compensate for this [6,7,3]. Classically, this is dramatic loss in precision. This is why, in Section 5, we took r = 10 7 .
However, we also saw that a radius of r = 1 can be obtained from a cellular automaton of arbitrary radius r simply by grouping each r n hypercube of cells into one supercell. Each supercell now has an internal state in Σ = Σ r n . Notice that keeping the position of the single particle within the hypercube is crucial. Otherwise, all the velocities of norm less than one supercell are rounded up to the center of the supercell-and so the increased precision in the velocities is not much use. Hence, Σ is really just coding for a velocity amongst b possibilities, which is appealing. . . but also for the position of the single particle within the hypercube, which perhaps is not so satisfactory. After all, what this space grouping has done is really just to hide the discrepancy between the disctetization step ∆ and the computed geodesics timeline step r∆, by hiding some of spatial precision within the internal space of the supercells.
Hence, r 1 appears to be fundamental requirement for precision. Notice that large values for r are better obtained by diminishing the discretization step ∆ rather than augmenting the timeline step r∆, as we cannot hope to achieve a pseudo-elliptic trajectory with just a handful of velocity changes per revolution. Running the simulations, it was indeed observed that r large, for instance r = 10 7 , yields increased stability. But only to some extent: after a while the number of possible velocities b = (2r + 1) n exceeds those which can be stored as a vector of machine-sized integers.
It also helps to fine-grain the discretization step ∆, keeping r constant. Running the simulations, it was indeed observed that this yields increased stability and convergence-at the expense of (reasonably) longer computation times. At some point, however, the finite-differences of (3) can become unstable, due to very small differences between l(E, F, G) and l(E, F , G) when F F = 1, again hitting bounded machine floating point-arithmetic precision-but this can easily be fixed by evaluating these derivatives with F F a fraction of l(E, F, G) independent of ∆.
Recovering continuous spacetime geodesics
The algorithm of Section 3 is successful in computing geodesics in discrete time and space Z × Z n , in a way which is consistent with continuous-space discretetime geodesics. We now explain how continuous-space discrete-time geodesics are themselves consistent with the standard geodesics of the fully continuous setting. For this question to make sense, we place ourselves in the case of Section 4: a distance-induced deviation function.
As in Figure 4, consider three points E, F , G. Say that the distance EF is measured according to g(E), and that the distance F G is measured according to g(F ). We said that trajectory EF G is a continuous-space discrete-time geodesics if and only if it minimizes the distance EF + F G, with respect to infinitesimal changes of F into F . Let us take EF = εv with v normalized with respect to g(E), and FG = εv with v normalized with respect to g(F ). Let us take F F = δd with d normalized with respect to g(F ). The distance EF + F G is given by:
(εv + δd) † g(E)(εv + δd) + (εv − δd) † g(F )(εv − δd)
Consider the first term. Its derivative with respect to δ is d † g(E)(εv + δd) + (εv + δd) † g(E)d 2 (εv + δd) † g(E)(εv + δd)
Taken at δ = 0 and using the symmetry of g(E) we get:
d † g(E)(εv) (εv) † g(E)(εv) = d † g(E)v
Consider the second term. If g(F ) was just g(F ), the same process would yield
−d † g(F )v . We would then just have d † g(E)v − d † g(F )v = 0, yielding v = g(F ) −1 g(E)v.
This is the equation derived in [17], and is in the same spirit as that obtained [15] in the framework of simplicial complexes. Unfortunately it does not yield accurate predictions for perihelion shift, as pointed out in [17,4] and confirmed by our simulations. This is because one has to take into account that the variations of g(F ) around F yield a third term:
(εv ) † (∂g(F ).d)(εv ) 2 (εv ) † g(F )(εv ) = ε 2 v † (∂g(F ).d)v
Let us emphasize that straightest geodesics on simplicial complexes [15] do not see this term either: quite simply because a path EF G between two adjacent simplices sees the geometry of the first simplex-that is the term g(E)-and the geometry of the second simplex-that is the term g(F )-, but ignores the variations of the geometry in some arbitrary direction F F . In other words, simplices are usually thought of as polyhedrons of constant metric-but in order to be consistent with the continuum they must be interpreted as surfaces of constant metric derivatives. Altogether, we get that trajectory EF G is a geodesics if and only if
d † (g(F )v − g(E)v) = ε 2 v † (∂g(F ).d)v(4)
in every directions d. Unfortunately, this is still inconvenient to solve for v . At this stage the traditional, continuous approach to geodesics follows two simplifying steps, which in the discrete setting translate into two approximations. The first step is to evaluate the condition for d only along the coordinate directions:
(g(F ) λ. v − g(E) λ. v) = ε 2 v † (g(F ) ,λ )v(5)
for all λ. The second step is to neglect the difference between v and v on the right-hand side, as it yields only second order terms. We get:
g(F ) λ. v = g(E) λ. v + ε 2 v † (g(F ) ,λ )v. (6) v = g(F ) −1 ·λ g(E) λ· v + ε 2 v † g(F ) ,λ v(7)
Notice that this provides another explicit discrete scheme for geodesics: it suffices to evaluate each g ,λ as (g(F (λ)) − g(F ))/ε, where F (λ) is point F moved by ε along coordinate λ. Another useful form is obtained letting g = g(F ) and using g(E) = g −ε∂g.v. We get:
g λ· v = g λ· v − εg λ·,µ v µ v + ε 2 v † g ,λ v g λ· (v − v) = ε 2 v † g ,λ v − 2g λν,µ v µ v ν g λ· (v − v) = ε 2 (g µν,λ v µ v ν − g λν,µ v µ v ν − g λµ,ν v µ v ν ) (v − v) = −εΓ µν v µ v ν(8)
where Γ µν = g −1 ·λ (g λν,µ + g λµ,ν − g µν,λ ) /2
Let us study the continuum limit ε → 0. We haveẍ = lim ε→0 (v − v)/ε,ẋ = lim ε→0 v, and henceẍ = −Γ µνẋ µẋν (10) which is your traditional geodesics equation.
Conclusion
Summarizing, we have introduced a generic notion of discrete geodesics as straightest trajectories in discretized spacetime, which are such that any three successive points E, F, G must minimize the deviation function w(E, F, G). Given E and F , G is implicitly determined: this can be viewed as a dynamical system and computed via a gradient descent algorithm. For a metric space, the canonical choice for w(., ., .) measures how the length EF +F G varies with small variation of F . This was validated numerically, by computing the trajectory of a planet in discretized Schwarzschild spacetime, and recovering a perihelion shift of the right order. This was also validated by taking the continuum limit and recovering the standard geodesics equations on pseudo-riemannian manifolds. Part of our motivations were to evaluate the strength and limits of cellular automata. Recall that three well-accepted postulates about physics-bounded velocity of propagation of information, homogeneity in time and space, and bounded density of information-necessarily imply that physics may be cast in the framework of cellular automata-both in the classical and quantum settings [9,1]. Both theorems, however, rely on the implicit hypothesis of a flat spacetime. To which extent can cellular automata account for relativistic trajectories, that is geodesics? This paper shows that discrete geodesics can be cast in the framework of cellular automata, provided that a few extra assumptions are met: that the metric can be given with bounded precision, and that it has the property of fixing a velocity limit. These extra assumptions do not contradict the three postulates: they are but instances of them.
Yet, this paper shows that a large discrepancy between the time discretization step the space discretization step is necessary in order to maintain a good precision on the velocity of particles. Namely, the number of particle velocities varies in (2r + 1) n , with n the dimension of space and the radius r of the cellular automaton, which is therefore inherently large. Thus, computing discrete geodesics-and straight lines in euclidean space, for that matter-is local. . . but not that local. This may come as a surprise, and suggest that geodesics equations are better-behaved in the continuum. An alternative is to live with imprecise velocities. A planet is a collection of particles, and so it may be the average of their imprecise velocities which grants it a precise averaged velocity. In fact, a single particle is itself quantum, and may thus be in a superposition of these imprecise velocities, yielding a precise averaged velocity-as is made formal in the eikonal approximation [5]. This is in fact precisely what happens in quantum cellular automata models of quantum particles in curved spacetime, as shown in [6,7,3]. All of these considerations suggest that nature's way of working out timelike geodesics trajectories may in fact be emergent, from the simpler and more local behaviour of spinning particles.
Hence, for future work, it may be interesting to look for discrete models based on spinning particles, oscillating along a few, cardinal, light-like directions. These may in fact be closer to mimicking the real behaviour of fermions in curved spacetime, with the hope to recover the Mathisson-Papapetrou-Dixon equation-a generalization of the geodesics equation to spatially extended massive spinning bodies-as emergent, in analogy with the continuum [5]. Such discrete models may be more local. Another approach is to work directly in terms of a discrete connection [10]. In the continuum, the Levi-Civita connection is axiomatized as being the unique metric-compatible and torsion-free connection. That given in Equation (9) is exactly torsion-free, but interpreted as a discrete connection, as in Equation (8), it is metric-compatible only to first order. One can ask for both properties to be met exactly even in the discrete setting, this specifies the intersection of two ellipses. In 2-dimensions the number of solutions is finite, but this is not even the case in higher-dimensions: the axiomatization suggested by the continuum breaks down and demands fixing.
Fig. 2 .
2The computed trajectory of a planet.
----------------------------------------
Fig. 3 .
3Numerics of the computed trajectory of a planet.
Fig. 4 .
4The continuum limit is obtained for ε and δ tending to zero.
AcknowledgementsThis work has been funded by the ANR-12-BS02-007-01 TARMAC grant, the ANR-10-JCJC-0208 CausaQ grant, and the John Templeton Foundation, grant ID 15619. Pablo Arrighi benefited from a visitor status at the IXXI institute of Lyon.
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|
[] |
[
"CONQ: CONtinuous Quantile Treatment Effects for Large-Scale Online Controlled Experiments",
"CONQ: CONtinuous Quantile Treatment Effects for Large-Scale Online Controlled Experiments"
] |
[
"Weinan Wang [email protected] \nSnap Inc. Santa Monica\nCaliforniaUSA\n",
"Xi Zhang [email protected] \nSnap Inc. Santa Monica\nCaliforniaUSA\n"
] |
[
"Snap Inc. Santa Monica\nCaliforniaUSA",
"Snap Inc. Santa Monica\nCaliforniaUSA"
] |
[] |
In many industry settings, online controlled experimentation (A/B test) has been broadly adopted as the gold standard to measure product or feature impacts. Most research has primarily focused on user engagement type metrics, specifically measuring treatment effects at mean (average treatment effects, ATE), and only a few have been focusing on performance metrics (e.g. latency), where treatment effects are measured at quantiles. Measuring quantile treatment effects (QTE) is challenging due to the myriad difficulties such as dependency introduced by clustered samples, scalability issues, density bandwidth choices, etc. In addition, previous literature has mainly focused on QTE at some pre-defined locations, such as P50 or P90, which doesn't always convey the full picture. In this paper, we propose a novel scalable non-parametric solution, which can provide a continuous range of QTE with point-wise confidence intervals while circumventing the density estimation altogether. Numerical results show high consistency with traditional methods utilizing asymptotic normality. An end-to-end pipeline has been implemented at Snap Inc., providing daily insights on key performance metrics at a distributional level.
|
10.1145/3437963.3441779
|
[
"https://arxiv.org/pdf/2010.15326v1.pdf"
] | 225,103,170 |
2010.15326
|
0ba168cc28c7a168654af0468b4859140dc85f6d
|
CONQ: CONtinuous Quantile Treatment Effects for Large-Scale Online Controlled Experiments
Weinan Wang [email protected]
Snap Inc. Santa Monica
CaliforniaUSA
Xi Zhang [email protected]
Snap Inc. Santa Monica
CaliforniaUSA
CONQ: CONtinuous Quantile Treatment Effects for Large-Scale Online Controlled Experiments
10.1145/1122445.1122456ACM Reference Format: Weinan Wang and Xi Zhang. 2020. CONQ: CONtinuous Quantile Treatment Effects for Large-Scale Online Controlled Experiments. In Proceedings of WSDM '21: ACM Symposium on Neural Gaze Detection (WSDM '21). ACM, New York, NY, USA, 9 pages. https://controlled experimentsquantile treatment effectscalabilitynon- parametric estimationBahadur representationcontinuous
In many industry settings, online controlled experimentation (A/B test) has been broadly adopted as the gold standard to measure product or feature impacts. Most research has primarily focused on user engagement type metrics, specifically measuring treatment effects at mean (average treatment effects, ATE), and only a few have been focusing on performance metrics (e.g. latency), where treatment effects are measured at quantiles. Measuring quantile treatment effects (QTE) is challenging due to the myriad difficulties such as dependency introduced by clustered samples, scalability issues, density bandwidth choices, etc. In addition, previous literature has mainly focused on QTE at some pre-defined locations, such as P50 or P90, which doesn't always convey the full picture. In this paper, we propose a novel scalable non-parametric solution, which can provide a continuous range of QTE with point-wise confidence intervals while circumventing the density estimation altogether. Numerical results show high consistency with traditional methods utilizing asymptotic normality. An end-to-end pipeline has been implemented at Snap Inc., providing daily insights on key performance metrics at a distributional level.
are conducted to judge if the treatment effects are indeed statistically significant. These metrics can be roughly divided into two types: engagement vs. performance. For engagement type metrics, typically two sample -tests are utilized to determine if the average treatment effects (ATE) are truly nonzero [14]. However for performance metrics (e.g., camera transcoding latency, page load latency, etc.), evaluating treatment effects at mean is typically ill-advised [23], and the industry standard is to measure QTE (quantile treatment effects), most notably at P50 and P90, whereas P50 is aimed at a summary statistic for overall performance and P90 for tail area performance.
However, it is not straightforward to get such QTE with valid -values and confidence intervals as events can no longer be considered independent (each randomization unit, typically users, can contribute multiple events, which in itself is another random variable measuring engagement), therefore delta-methods and user-level correlation calculations are required [23]. In addition, while the method proposed in [23] is statistically valid, it lacks a principled and data-driven way on choosing the bandwidth for its kernel density estimation, which affects the result greatly since the density's square appears as the denominator of the variance term. Furthermore, QTE at the median and the 90-th quantile are sometimes not enough to give experimenters the whole picture, especially when the significance or directions do not agree (example in Figure 1), or when heterogeneous treatment effect (HTE) is present for different devices with high and low overall performances. This motivates us to come up with a robust, statistically valid, and scalable method which can provide continuous QTE with point-wise -values and confidence intervals. In this paper, we propose a novel method named CONQ (continuous quantile treatment effect), which utilizes Bag of Little Bootstrap (BLB) [20] and the Bahadur Representation [30] to construct point-wise asymptotically valid confidence intervals and -values, which are consistent with the normal approximation method in [23]. To further improve Monte Carlo efficiency, balanced bootstrap is implemented for variance reduction. We also show the added benefit of log-transformation on the original metric for calculating percentage change QTE due to quantile's invariance to arXiv:2010.15326v1 [stat.ME] 29 Oct 2020 monotone transformations. Extensive evaluation on Snapchat data shows high consistency of our method with method in [23] at P50 and P90, and stable performance across a range of quantile locations bench-marked by known A/A tests. A daily pipeline has been implemented using Spark [33] and Airflow, generating continuous QTE on core performance metrics for all A/B tests at Snap Inc.
The key contributions of our paper include:
• A scalable and theoretically sound method that can provide QTE with percentage change -values and confidence intervals at arbitrary range of quantile locations simultaneously. • Circumvents the issue of density estimation and bandwidth choice altogether, improve accuracy of delta-method by logtransformation for percentage changes. • Validation of the method on real experiments at Snap Inc., which shows consistency with existing P50 and P90 results, and stable performance across various quantile locations.
In Section 2, we describe the background and literature on QTE in the A/B setting and statistical inference for quantiles. In Section 3, we detail the motivation and theoretical justification of the CONQ procedure and its implementation. In Section 4, we evaluate CONQ using a large set of A/B experiments at Snap Inc., showing its comparable performance with delta-method based approach in [6,23], and shows its stable performance using a set of known A/A tests as validation corpus. In Section 5, we discuss the possible extension and summarize the paper.
BACKGROUND AND RELATED WORK 2.1 Quantile Treatment Effects
Although in many tech firms, the vast majority of metrics are measuring user engagement activities, another important category of metrics aim to gauge app or web performance, i.e. performance metrics. For such performance metrics, not only do we care about the average users' experience, we are also interested in the cohort of users that are suffering from the worst app or web performances. Bearing these concerns in mind, it's important to evaluate these metrics at at least multiple quantile locations in A/B tests, especially in the tail area. At Snap Inc., P50 and P90 are the typical objectives various teams optimize for in A/B experiments. [23] proposed a statistically valid and scalable method that can calculate QTE with -values and confidence intervals by appealing to quantile estimator's asymptotic normality [5], however the varying bandwidths choice mentioned is rather ad-hoc and lacks theoretically guarantee. [6] introduced an approach that appeals to delta-method and the outer CI, circumventing the density estimation which still involves user-level correlation calculation for the variance term. Furthermore, these approaches still focus on treatment effects at a number of pre-determined quantile locations (.e.g., P50 or P90), where generalizations to a continuous range of quantile locations cannot be easily obtained without point-wise recalculation.
QTE at various quantile locations has been receiving increased attention these days in the industry, as it provides a fuller picture of distributional changes between control and treatment, and better captures heterogeneity [24]. At Uber, the approach they adopted is quantile regression, citing the advantage of relying on existing abundant literature [13,21]. However, the methodology proposed still requires an optimization algorithm, which is not suitable for the myriad of performance metrics and experiments at Snap Inc. at scale.
Statistical Inference for Quantiles
In the Statistics literature, the relationship between quantiles and CDF (cumulative distribution function) has been investigated extensively. Most notably, for 1 , 2 , · · · , that are i.i.d. random variables from an unknown non-parametric with density , for 0 < < 1, denote by = inf { : ( ) ≥ } the -th quantile of , , the -th sample quantile of , [1] coined the famed Bahadur representation:
, = + − ( ) ( ) + . . [ −3/4 (log ) 1/2 (log log ) 1/4 ].
To put it simply, it stated that an asymptotic relationship between quantile and CDF can be established through its density function . Refinements of Bahadur's result in the i.i.d. setting were further proposed by Kiefer [17][18][19]. [30] further extended such representation for a wide range of dependent cases. [29] proposed the Woodruff confidence interval, which essentially established a one to one relationship between point-wise confidence intervals of the CDF and quantiles.
Another line of research focused on a L ∞ -norm deviation of the empirical CDF from the truth (uniform bound), instead of at a specific quantile location (point-wise). The celebrated Dvoretsky-Kiefer-Wolfowitz-Massart (DKWM) inequality [7,25] provides a tight bound on the CDF for i.i.d. samples:
P sup ∈R ( ( ) − ( )) > ≤ exp −2 2 , ∀ ≥ √︂ 1 2 ln 2
, which by its expression, is indifferent to quantile locations. [12] further proposed two methods for point-wise quantile confidence intervals and two methods for L ∞ -norm type confidence bounds, which are proven to be adaptive in a sequential setting, resonating with the always-valid -values line of research [15,16]. However, coming from a QTE perspective, we are still most interested in point-wise statistical inference, and [12]'s method requires either parameter tuning or numeric root finding, and generalization to the dependent case is non-trivial.
In this paper, we focus on the end-horizon time setting which is the common-case at Snap Inc., instead of a sequential test setting as considered by [12]. We propose an easy to interpret and implement data-driven method for a range of quantile level treatment effects for percentage changes.
CONTINUOUS POINT-WISE STATISTICAL INFERENCE ON QTE 3.1 Sample Quantiles and Bahadur Representation
At Snap Inc., hundreds of metrics are evaluated in A/B experiments simultaneously, aiming to not only gauge user telemetry data, but also on app-performance. As a social network company with a camera focus, myriad facets of in-app performances are the direct goal of optimization for many teams. There's almost always a dual performance metric for an engagement metric. For example, as we measure the number of app opens, we also measure app open latency. This brings about another layer of complication in A/B tests. As a typical practice, randomization units are users; for engagement type metrics, each user contributes exactly one point, and all data-points can be considered i.i.d. observations, hence statistical inference can be by and large accomplished by appealing to the Central Limit Theorem. However, the dual performance metrics are not as straight-forward, as each user contributes multiple data-points (exactly the engagement count number of data-points), therefore we have a clustered set of observations which are no longer independent. Using the same notation in 2.2, for i.i.d. samples 1 , · · · , , the sample -th quantile , is asymptotically normal:
√ , − → N 0, (1 − ) 2 .
With performance metrics, typically we need delta method to approximate the numerator in the variance term [6,23]. However, this is only for a fixed quantile location , and it's hard to generalize to other locations without recalculation. Furthermore, kernel density estimation is typically required forˆ( ), in which its bandwidth is hard to choose. Although some ad-hoc rules could work well in practice [23], small deviations in the bandwidth affect the result greatly.
To demonstrate such issue, here is an example on a typical performance metric measuring the latency on starting the Discover section inside the Snapchat app. The study improved this latency at P50 from 911.991 to 908.173, with a -value of 0.044 using the method mentioned in [23] with the recommended bandwidth choice. However, if we were to vary the bandwidth using the below choices (with normal reference rule [26] chosen as ℎ = 1.06 min{ , IQR/1.34}/ 1/5 (IQR iinter-quantile range, is sample standard deviation and is sample size), we get drastically different -values demonstrated in Table 1. Using 0.05 as a threshold, we can see that -values vary from being very significant to not significant at all as the bandwidth varies. However, we lack a robust and data-driven approach on choosing such bandwidth which can guarantee good estimation quality for inference across all quantiles. As noted in sub-Section 2.2, the famed Bahadur representation establishes the asymptotic relationship between sample quantiles and the empirical CDF, especially in various dependent cases [30].
Such Bahadur representation has a nice geometric interpretation as illustrated by 2. At the neighborhood of point ( , ), to convert from the confidence interval of empirical CDF to quantiles, we can simply divide by the tangent of the curve ( ), which is ′ ( ) = ( ). Therefore, for i.i.d. samples, we have
− → N 0, (1 − )
which no longer involves the density estimation. For clustered sample's case, simply replacing the variance term (1 − )/ with would suffice, which can be empirically estimated using the Bag of Little Bootstrap (BLB) technique in the next section. Take one step further, we have the result that:
Letˆ= inf : ( ) ≥ − /2 1/2 [ ( )] , = inf : ( ) ≥ + /2 1/2 [ ( )] , then P ˆ≤ ≤ˆ ≈ 1 − .
Woodruff [29] first proposed this interval empirically, which ingeniously circumvents the density estimation altogether, while still provides valid statistical inference results.
Below we discuss how we extend such CI to a continuous setting and clustered sample's case, suitable for the large-scale A/B testing need at Snap Inc.
Balanced Bag-of-Little-Bootstrap for CDF
In the previous section, we discuss the relationship between empirical CDF and quantiles through the Bahadur representation. In this section, we discuss how the variance term can be efficiently calculated for at a continuous range of quantile locations using one set of bootstrap results, and how Woodruff type confidence intervals can be applied.
As mentioned in [6], 'ntiles' operation (query quantile from set of observations) is expensive and should be avoided if possible. Furthermore, bearing in mind the one to one relationship between CI for quantiles and CI for CDF, in order to simultaneously get quantile level inference for all possible locations, we can first tackle the variance of CDF, and then appeal to Woodruff CI for transforming back to quantiles. In this paper, we use the Bag of Little Bootstrap in [20] with the balanced method [10] for resampling to further reduce Monte Carlo variance, achieving higher bootstrap efficiency.
Suppose there are users, for user = 1, · · · , , there are total of events, denoted as ,1 , · · · , , , while =1 = . We first take the logarithm of these metrics, then preserve a pre-defined digits of precision (we chose 2): round(log , , 2), = 1, · · · , . Note this step is to reduce the total number of unique log-scaled values for efficiency. Furthermore, due to quantile's invariance to monotone transformation [4], we have
log ( ) ( ) = (log ) ( ), ∀0 < < 1.
This ensures we can get back the QTE on the original metric by simply taking the exponent. Furthermore, by delta method, we have Var(%diff between control and treatment at p-th quantile)
=Var ( ) ( ) − ( ) ( ) ( ) ( ) = Var ( ) ( ) ( ) ( ) =Var exp log ( ) ( ) − log ( ) ( ) =Var exp (log ) ( ) − (log ) ( ) ≈Var (log ) ( ) − (log ) ( ) ( ) ( ) ( ) ( ) 2 = Var (log ) ( ) + Var (log ) ( ) ( ) ( ) ( ) ( ) 2 ,
where the first term in the last expression can be bootstrapped on log-transformed data separately for control and treatment. This way of applying the delta-method is empirically more accurate than on the original scale metric, especially when percentage change is large, or when ( ) ( ) is small. Then we split users together with their corresponding events into subsets ( = 100 for Snap) for control and treatment separately, all events for any given user shall be preserved in the same subset. For each subset, generate a data sketch with unique rounded logscaled metric values (denoted by˜, = 1, · · · , , here is the total number of unique values in the original dataset) with its corresponding count , = 1, · · · , , = 1, · · · , : · · · · · · · · · · · · · · · 1, 2,
· · · ,
Let be the number of bootstraps, to achieve balanced bootstrapping, i.e. where all samples appear exactly times, randomly permute the long vector =
1, · · · , 1 B repetition , 2, · · · , 2 B repetition , · · · · · · , , · · · , B repetition
and then split into vectors of length , and treat each vector , = 1, · · · , as one bootstrap sample on the buckets. For each bootstrap sample vector , we can get a counter of how many times 1, · · · , each appears, and weight corresponding , by how many times -th bucket appears in . For each bootstrap sample, we can get the empirical cdf at all unique log-scaled values , = 1, · · · , . Using all bootstrap samples, we can get an estimate of the standard deviation of empirical cdf at all˜(denoted as F , (˜), = 1, · · · , ) as well, which would approximate at F (˜):
Var F (˜) ≈ −1 ∑︁ =1 F , (˜) − −1 ∑︁ =1 F , (˜) 2 :=ˆ[F (˜)].
Bootstrapping on the empirical cdf avoids the 'ntiles' operations, and enables us to use one set of bootstrap to get all potentially interested QTE by interpolating the estimated variance via a general continuity assumption. Below is a typical example of the interpolated bootstrapped standard error across all quantiles: Since the original Woodruff method does not always produce a symmetric CI, empirically, we propose the following conservative estimate for each unique log-scaled metric value˜for control and treatment separately:
LetˆF (˜) = inf : F ( ) ≥ F (˜) −ˆ[F (˜)] , (1) F (˜) = inf : F ( ) ≥ F (˜) +ˆ[F (˜)] , (2) then let SE(˜) = max ˜−ˆF (˜) ,ˆF (˜) −˜ .(3)
Furthermore, using these results joined with the original observations' empirical CDF, we can then linearly interpolate these results on a fixed grid of quantiles (ensuring common quantile locations between control and treatment for QTE evaluation on such grid). At Snap, we chose a range of P20 to P99. The same example referenced in Figure 3 has the interpolated standard error, shown in Figure 4. Once we have such common quantile locations and estimated log-scaled metrics' SE for control and treatment, we have all the ingredients for traditional point-wise -test based statistical inference, including percentage change -values, confidence intervals, etc.
The following section provides a detailed summary of the aforementioned steps constituting the algorithm named CONQ (Continuous QTE for large-scaled experiments).
CONQ Algorithm and Implementations
Here we summarize the end-to-end CONQ algorithm in Algorithm 1, to adopt earlier notations, superscripts and denote data associated with the control and the treatment group respectively, being the total number of events, being the total number of users. As can be seen, it only requires several hyper-parameters including for log-scaling precision, for number of buckets, for balanced bootstrap iterations, a pre-determined grid on quantile locations for evaluation, and 1 − for confidence level. As for the bag of little bootstrap part, it can be simply construed as re-weighting observations in buckets based on samples on bucket indices. This greatly reduces computational cost in comparison to the method in [6] as the number of unique observations reduce from to , and here can be further adjusted by the log-scaling precision .
One key step that enables us to get a range of continuous QTE evaluation is the evaluation of F * , , on all˜for each bootstrap sample and ∈ { , }. This is straightforward after we do a simple sort on˜and a weighted cumulative sum on the re-weighted count , . This avoids 'ntiles' operation and improves efficiency. We implemented the CONQ algorithm at Snap's A/B experimentation pipeline, the flowchart illustrated in Figure 5 demonstrates the overall workflow orchestrated by Spark [34] and Airflow.
Recall our earlier example in Figure 1 where we see significant but directionally different results at P50 and P90 QTE, here using CONQ, in Figure 6, we can see that the regression mainly happens at P20 to P80, i.e. lower quantiles, while improvements mainly happen after P90, thus indicating presence of heterogeneity in treatment effects, especially for devices with varying performance. Further breakdown results by varying device performance clusters (2 to 7, with lower number indicating devices with general poorer performance, while higher number indicating higher performance) in Figure 7, the results corroborate with our speculation that low regressions mainly occur for high end devices, whereas improvements occur for low end devices.
EVALUATION
In this section we evaluate CONQ using real A/B experiments measuring various performance type metrics at Snap. Specifically we aim to address the following questions:
(1) Comparing with density estimation based methods [6,23], can CONQ achieve similar results empirically at P50 and P90? (2) Under an known A/A test setting (null case), can CONQ be robust across all quantile locations when it comes to false positives after multiple testing adjustment, thus indicating stability?
Evaluation on Snap's A/B Experiments
For real experiments comparison, we use method in [6] (denoted by DELTA) with kernel density bandwidth choice ℎ chosen by the ad-hoc rule proposed in [23], and compare Δ % -value results at P50 and P90 with CONQ. A total of 15 performance metrics on 696
Algorithm 1: CONQ: Continuous QTE on performance type metrics for large-scale online controlled experiments. Input and data-processing:
• Users and events : , , = 1, · · · , , = 1, · · · , , =1 = ; , , = 1, · · · , , = 1, · · · , , =1 = .
• Digits of precision for log-scaling: , 2 at Snap.
• Unique log-scaled observations and their counts:˜, = 1, · · · , ;˜, = 1, · · · , . Further let , = 1, · · · , ; , = 1, · · · , be their corresponding count in the dataset. • # of user buckets: = = , 100 at Snap. • # of bootstrap iterations: = = , 200 at Snap. • Quantile grid locations for QTE evaluation: grid ∈ (0, 1), = 1, · · · , grid ; e.g., 20% to 99% with step size 1% chosen for Snap. • Confidence level: 1 − , = 0.05 at Snap.
Output: Quantile Treatment Effects-QTE
• Δ% and -value evaluated at grid: Δ %, , = 1, · · · , grid . • Δ% confidence interval at level 1 − evaluated at grid: CI := [CI , CI ], = 1, · · · , grid . for control and treatment identifier ∈ { , }, do
• evaluate the overall empirical CDF F (˜), = 1, · · · , .
• split users into buckets of equal size ⌊ / ⌋; for all users in bucket , put all events in bucket , get corresponding log-scaled event's count˜, = 1, · · · , as , .
• permute the long vector =
1, · · · , 1 B repetition , 2, · · · , 2 B repetition , · · · · · · , , · · · , B repetition
randomly and split into small vectors of equal length .
for each bootstrap sample = ( 1 , · · · , ), define F * , • Use linear interpolation on F (˜), SE(˜) , = 1, · · · , , and evaluate on grid , = 1, · · · , grid → grid , SE F ,−1 (grid ) .
, ( ) := =1 =1 , I(˜≤ ) / =1 =1 , ; -evaluate F * , , ( ) at =˜1 ,˜2 , · · · ,˜; For each˜, = 1, · · · , , estimate Var F (˜) ≈ −1 =1 F * , , (˜) − −1 =1 F * , ,(˜)
QTE:
• Δ% at grid: Δ % = exp F ,−1 (grid ) − exp F ,−1 (grid ) /exp F ,−1 (grid ) × 100%;
• SE for Δ% at grid:
SE(Δ %) = √︂ SE F ,−1 (grid ) 2 + SE F ,−1 (grid ) 2 × exp F ,−1 (grid ) exp F ,−1 (grid )
;
• Δ% -value at grid:
= 2 × Φ − Δ % SE(Δ %) ; • Δ% confidence interval at level 1 − at grid: CI = [Δ % − /2 SE(Δ %), Δ % + /2 SE(Δ %)].
treatment and control pairs are evaluated (total of 10440 -values for P50 and P90). The performance metrics included measure a wide array of latencies and crashes for the Snapchat app, which itself should be representative enough for industry A/B experiments. We compare the resulting -values from DELTA and CONQ first using a scatter-plot. As can be seen in Figure 8, these two methods align quite well and mostly adhere a linear pattern as expected, indicating stable performance of CONQ across various settings and -value regions. Then we compare the proportions of significance discoveries with a static threshold on -values ranging from 0.0001 to 0.2, DELTA and CONQ also perform quite similarly at both P50 and P90 levels, with CONQ resulting in slightly more discoveries across all thresholds.
Evaluation on Snap's A/A Tests
In this section, we evaluate CONQ on a subset of known A/A tests at Snap where no QTE is expected across all quantile locations. Then for any given quantile location, the set of -values associated QTE across different treatment control pairs and various performance metrics should have no significance after multiple testing Figure 1 evaluated by CONQ at P20 to P99, as can be seen, regression (indicated by red) happens mainly at P20 to P80, whereas improvements (indicated by green) mainly happen after P90, indicating presence of heterogeneity in treatment effects. Purple dotted line is plotted at 0.05, y-axis is log-scaled. The threshold for -value here is chosen as 0.05. Figure 1 evaluated by CONQ at P20 to P99 at device cluster breakdown level, as can be seen, regressions mainly occur at high end devices, while improvements mainly occur at low end devices. adjustment using the Benjamini-Hochberg procedure [3]. The reason for using BH here instead of testing for a uniform distributed -value, is the positive dependency relationships between many performance metrics, and the robustness of BH under such setting. Here we choose the nominal FDR level as = 0.05, 0.1, 0.2. We evaluate a total of 15 performance metrics across 54 treatment and control pair combinations. The percentile grid for evaluation is chosen as P20 to P95 with 5% increment. BH is applied conditioning on a fixed percentile. Since we know that all scenarios considered are A/A tests, any discoveries by BH are false discoveries. Table 3 summarizes the percentile and their corresponding count of false discoveries and its percentage (total of 810 -value per quantile). As can be seen, CONQ performs stable across various quantile locations under a realistic FDR nominal thresholds of either 0.05 or 0.1. With a larger threshold at 0.2, CONQ still performs robust with slightly elevated number of false positives in the tail area. P20 0 (0%) 0 (0%) 0 (0%) P25 0 (0%) 0 (0%) 0 (0%) P30 0 (0%) 0 (0%) 0 (0%) P35 0 (0%) 0 (0%) 0 (0%) P40 0 (0%) 0 (0%) 0 (0%) P45 0 (0%) 0 (0%) 0 (0%) P50 0 (0%) 0 (0%) 0 (0%) P55 0 (0%) 0 (0%) 0 (0%) P60 0 (0%) 0 (0%) 0 (0%) P65 0 (0%) 0 (0%) 0 (0%) P70 0 (0%) 0 (0%) 0 (0%) P75 0 (0%) 0 (0%) 0 (0%) P80 0 (0%) 0 (0%) 1 (0.12%) P85
1 (0.12%) 2 (0.25%) 3 (0.37%) P90 0 (0%) 0 (0%) 5 (0.62%) P95
2 (0.25%) 2 (0.25%) 8 (0.99%) Table 3: : number of false positives discovered and its percentage using various nominal FDR thresholds for BH adjustment out of 810 -values, evaluated at different percentiles.
CONCLUDING REMARKS AND POSSIBLE EXTENSION
In this paper, We propose a non-parametric method for point-wise statistical inference on a continuous range of QTE in A/B experiments, which is density estimation free and scalable for the largescale setting at industry. It achieves stable performance across all ranges and performs similarly to existing delta-method based approaches in [6,23]. Note the confidence interval provided is conservative to ensure symmetry and -value calculation, for tail areas, the direct application of Woodruff type confidence intervals may have better coverage rate as investigated in [27]. Furthermore, we can indirectly use the estimated SE on quantile and empirical CDF to get an estimate of the density, which can be used for other tasks, avoiding kernel bandwidth choice.
we mainly focus on point-wise statistical inference for QTE, namely for any given ∈ (0, 1), we produce 1 − confidence intervals CI s.t. P( ∈ CI) ≥ 1 − . We could also be interested in making a confidence interval for all quantiles simultaneously, namely P(∃ ∈ (0, 1), s.t.
∈ CI) ≥ 1 − , which is in line with the DKWM inequality mentioned above. The method CONQ described in this paper can be further extended to produce L ∞norm type bound as well, we can use the bootstrapped variance to get an estimate of the so-called "effective degrees of freedom", and use it to substitute in the DKWM inequality or the inequality produced in [12]. This approach is justified by the central limit theory on clustered samples, specifically in [11]. We leave this for future research.
Figure 1 :
1A sample experiment where a performance metric has statistically significant QTE at both P50 and P90, however the direction differs.
Figure 2 :
2Geometric interpretation of the Bahadur representation between the empirical CDF and metric quantiles.
Figure 3 :
3A typical example of bootstrapped SE for the empirical CDF across all quantile locations for control and treatment.
Figure 4 :
4Estimated SE on a grid of quantile locations using Woodruff type estimation mentioned above for control and treatment.
Figure 5 :
5Illustration of CONQ engineering pipeline at Snap, orchestrated through Airflow and Spark.
=
inf : F ( ) ≥ F (˜) −ˆ[F (˜)] , F (˜) = inf : F ( ) ≥ F (˜) +ˆ[F (˜)] , then let SE(˜) = max ˜−ˆF
Figure 6 :
6Example in
Figure 7 :
7Example in
Figure 8 :
8Comparison of DELTA and CONQ on p-values and significance discoveries at various threshold levels.
Table 1: How bandwidth choice affects kernel density estimation, resulting in different percentage change -values at P50.kernel density estimator's bandwidth Δ% -value
0.002
1.08e-34
0.01
0.0125
0.02
0.049
normal reference rule
0.352
Table 2 :
2Disjoint subsets splitting the original data, events belong to the same user are in the same bucket.1
2
· · ·
s
1
1,1
2,1
· · ·
,1
2
1,2
2,2
· · ·
,2
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[
"Low-scale SUSY breaking and the (s)goldstino physics",
"Low-scale SUSY breaking and the (s)goldstino physics"
] |
[
"I Antoniadis [email protected]. \nCERN -Theory Division\nCH-1211Geneva 23Switzerland\n",
"D M Ghilencea \nCERN -Theory Division\nCH-1211Geneva 23Switzerland\n\nTheoretical Physics Department\nUMR CNRS 7644) Ecole Polytechnique\nNational Institute of Physics and Nuclear Engineering (IFIN-HH\nMG-6 077125, F-91128Bucharest, PalaiseauRomania., France\n"
] |
[
"CERN -Theory Division\nCH-1211Geneva 23Switzerland",
"CERN -Theory Division\nCH-1211Geneva 23Switzerland",
"Theoretical Physics Department\nUMR CNRS 7644) Ecole Polytechnique\nNational Institute of Physics and Nuclear Engineering (IFIN-HH\nMG-6 077125, F-91128Bucharest, PalaiseauRomania., France"
] |
[] |
For a 4D N=1 supersymmetric model with a low SUSY breaking scale (f ) and general Kahler potential K(Φ i , Φ † j ) and superpotential W (Φ i ) we study, in an effective theory approach, the relation of the goldstino superfield to the (Ferrara-Zumino) superconformal symmetry breaking chiral superfield X. In the presence of more sources of supersymmetry breaking, we verify the conjecture that the goldstino superfield is the (infrared) limit of X for zero-momentum and Λ → ∞ (Λ is the effective cut-off scale). We then study the constraint X 2 = 0, which in the one-field case is known to decouple a massive sgoldstino and thus provide an effective superfield description of the Akulov-Volkov action for the goldstino. In the presence of additional fields that contribute to SUSY breaking we identify conditions for which X 2 = 0 remains valid, in the effective theory below a large but finite sgoldstino mass. The conditions ensure that the effective expansion (in 1/Λ) of the initial Lagrangian is not in conflict with the decoupling limit of the sgoldstino (1/m sgoldstino ∼ Λ/f , f < Λ 2 ). † on leave from CPHT (
|
10.1016/j.nuclphysb.2013.01.015
|
[
"https://arxiv.org/pdf/1210.8336v2.pdf"
] | 118,694,910 |
1210.8336
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75b00075269a35485ee37bc1ae8383fd8adc4c7c
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Low-scale SUSY breaking and the (s)goldstino physics
24 Jan 2013 May 3, 2014
I Antoniadis [email protected].
CERN -Theory Division
CH-1211Geneva 23Switzerland
D M Ghilencea
CERN -Theory Division
CH-1211Geneva 23Switzerland
Theoretical Physics Department
UMR CNRS 7644) Ecole Polytechnique
National Institute of Physics and Nuclear Engineering (IFIN-HH
MG-6 077125, F-91128Bucharest, PalaiseauRomania., France
Low-scale SUSY breaking and the (s)goldstino physics
24 Jan 2013 May 3, 2014
For a 4D N=1 supersymmetric model with a low SUSY breaking scale (f ) and general Kahler potential K(Φ i , Φ † j ) and superpotential W (Φ i ) we study, in an effective theory approach, the relation of the goldstino superfield to the (Ferrara-Zumino) superconformal symmetry breaking chiral superfield X. In the presence of more sources of supersymmetry breaking, we verify the conjecture that the goldstino superfield is the (infrared) limit of X for zero-momentum and Λ → ∞ (Λ is the effective cut-off scale). We then study the constraint X 2 = 0, which in the one-field case is known to decouple a massive sgoldstino and thus provide an effective superfield description of the Akulov-Volkov action for the goldstino. In the presence of additional fields that contribute to SUSY breaking we identify conditions for which X 2 = 0 remains valid, in the effective theory below a large but finite sgoldstino mass. The conditions ensure that the effective expansion (in 1/Λ) of the initial Lagrangian is not in conflict with the decoupling limit of the sgoldstino (1/m sgoldstino ∼ Λ/f , f < Λ 2 ). † on leave from CPHT (
Introduction
Supersymmetry, if realised in Nature, must be broken at some high scale. In this work we consider the case of SUSY breaking at a (low) scale √ f ≪ M P lanck in the hidden sector, an example of which is gauge mediation. In this case the transverse gravitino couplings (∼ 1/M P lanck ) can be neglected relative to their longitudinal counterparts (∼ 1/ √ f ) that are due to its goldstino component. If so, one can then work in the gravity decoupled limit, with a massless goldstino. The auxiliary field of the goldstino superfield breaks SUSY spontaneously, while the goldstino scalar superpartner (sgoldstino) can acquire mass and decouple at low energy, similar to SM superpartners, to leave a non-linear SUSY realisation. To describe this regime one can work with component fields and integrate out explicitly the sgoldstino and other superpartners too (if massive), to obtain the effective Lagrangian. Alternatively one can use a less known but elegant superfield formalism endowed with constraints, see [1] for a review. If applied to matter, gauge and goldstino superfields, these constraints project out the massive superpartners, giving a superfield action for the light states. Such constraint may be applied only to the goldstino superfield which can be coupled to the linear multiplets of the model (e.g. MSSM), to parametrize SUSY breaking.
In this work we study SUSY breaking in the hidden sector and its relation to the goldstino superfield in the presence of more sources of SUSY breaking and the connection of the goldstino superfield to the superconformal symmetry breaking chiral superfield X [2].
It was noticed long ago [3] (see also [1]) that a Lagrangian, function of Φ 1 = (φ 1 , ψ 1 , F 1 )
L = d 4 θ Φ † 1 Φ 1 + d 2 θ f Φ 1 + h.c. , with a constraint: (Φ 1 ) 2 = 0,(1)
provides an onshell superfield description of the Akulov-Volkov action for the goldstino field [4]. Indeed, the constraint (which generates interactions) has a solution Φ 1 = ψ 1 ψ 1 /(2F 1 )+ √ 2θψ 1 + θθF 1 , which "projects out" the sgoldstino φ 1 . When this Φ 1 is used back in (1), one obtains the onshell-SUSY Lagrangian for a massless goldstino ψ 1 . If more fields are present and for general K, W , the situation is more complicated and was little studied.
Further, it was only recently conjectured [1] that the goldstino superfield is the infrared (i.e. zero-momentum) limit of the superconformal symmetry breaking chiral superfield X, that breaks the conservation of the Ferrara-Zumino current [2]. In the light of the above discussion, one then also expects that in such limit X 2 ∼ (Φ 1 ) 2 = 0, and this conjecture was verified in very simple examples. We address these two problems, more exactly: a) the convergence of the field X to the goldstino superfield, in the limit of vanishing momentum. We note that one must also take Λ → ∞. b) the validity and implications of the constraint X 2 = 0 [1,3]. This is supposed to decouple (project out) the sgoldstino. In particular this can mean an infinite sgoldstino mass [5], however on dimensional grounds this is actually proportional to f /Λ. It is difficult to satisfy a) and b) simultaneously in general cases, because of opposite limits, of large Λ and large sgoldstino mass, while f < Λ 2 . This situation is further complicated by the presence of more fields, some of which can also contribute to (spontaneous) SUSY breaking.
The above problems were studied for simple superpotentials, like linear superpotentials [1,6] or with only one field breaking SUSY [5], with many phenomenological applications studied in [7]. We investigate below problems a), b) for general K and W , with more fields contributing to SUSY breaking in the hidden sector. This helps a better understanding of SUSY breaking and its transmission to the visible sector via the coupling [1] of the X field to models like the Minimal Supersymmetric Standard Model (MSSM).
2 (s)Goldstino and its relation to the chiral superfield X.
Goldstino and sgoldstino eigenstates for arbitrary K, W.
The starting point is the general Lagrangian
L = d 4 θ K(Φ i , Φ † j ) + d 2 θ W (Φ i ) + d 2 θ W † (Φ † i ) = K j i ∂ µ φ i ∂ µ φ † j + i 2 ψ i σ µ D µ ψ j − D µ ψ i σ µ ψ j + F i F † j + 1 4 K kl ij ψ i ψ j ψ k ψ l + W k − 1 2 K ij k ψ i ψ j F k − 1 2 W ij ψ i ψ j + h.c.(2)
where we ignored a (−1/4)✷K in the rhs.
Here K i ≡ ∂K/∂φ i , K n ≡ ∂K/∂φ † n , K n i ≡ ∂ 2 K/(∂φ i ∂φ † n ), W j = ∂W/∂φ j , W j = (W j ) † , etc, with W = W (φ i ), K = K(φ i , φ † j )
. Terms with more than two derivatives of K are suppressed by powers of Λ which is the UV cutoff of the model, K k ij ∼ 1/Λ, K ij km ∼ 1/Λ 2 , etc. We also used the notation
D µ ψ l ≡ ∂ µ ψ l − Γ l jk (∂ µ φ j ) ψ k , Γ l jk = (K −1 ) l m K m jk D µ ψ l ≡ ∂ µ ψ l − Γ jk l (∂ µ φ † j ) ψ k , Γ jk l = (K −1 ) m l K jk m(3)
Eq. (2) is the offshell form of the Lagrangian. The eqs of motion for auxiliary fields
F † m = −(K −1 ) i m W i + (1/2) Γ lj m ψ l ψ j F m = −(K −1 ) m i W i + (1/2) Γ m lj ψ l ψ j(4)
can be used to obtain the onshell form of L:
L = K j i ∂ µ φ i ∂ µ φ † j + i 2 ψ i σ µ D µ ψ j − D µ ψ i σ µ ψ j − W k (K −1 ) i k W i − 1 2 W ij − Γ m ij W m ψ i ψ j + h.c. + 1 4 R kl ij ψ i ψ j ψ k ψ l , R kl ij = K kl ij − K n ij Γ kl n(5)
Here R kl ij is the curvature tensor and the potential of the model is
V = W i (K −1 ) i j W j(6)
The derivatives of K, W are scalar fields-dependent. In the following we always work in normal coordinates, in which case
k i j = δ j i , k i jk... = k jk.... i = 0, where k i... j.... are the values of K i....
j... evaluated on the ground state (denoted φ k , F k , ψ k = 0). We denote the field fluctuations by δφ i = φ i − φ i . In normal coordinates k kl ij used below is actually k kl ij = R kl ij . From the eqs of motion for F i , φ i , after taking the vev's, then
k j i F † j + f i = 0, k j im F i F † j + f km F m = 0(7)
We denote by f i , f ik , f ijk , the values of corresponding, field dependent W i , W ik , W ijk , evaluated on the ground state, so
f i = W i ( φ m ), f ij = W ij ( φ m ), f ijk = W ijk ( φ m ), f i = W i ( φ m ), etc.(8)
Eq.(7) then becomes
F † j = −f j , f km F m = 0.(9)
To break supersymmetry, a non-vanishing vev of an auxiliary field is needed, which requires det f ij = 0. The goldstino mass matrix is (M F ) ij = W ij − Γ m ij W m evaluated on the ground state, giving (M F ) ij = f ij in normal coordinates. A consequence of the last eq in (9) is that the goldstino eigenvector (normalised to unity) is
ψ 1 = − F † m ψ m F † i F i 1/2 = f m ψ m f i f i 1/2 , mψ 1 = 0.(10)
Further, regarding the scalar sector, the mass matrix has the form
M 2 b = (V ) k l (V ) kl (V ) kl (V ) l k = f ik f il − k jk il f i f j f jkl f j f jkl f j f il f ik − k jl ik f i f j ,(11)where V k l = ∂ 2 V /(∂φ l ∂φ † k ), V kl = ∂ 2 V /(∂φ l ∂φ k )
, etc, is evaluated on the ground state. The two real components of the complex sgoldstino are mass degenerate only if in (11) the off-diagonal (holomorphic or anti-holomorphic) blocks vanish. For simplicity we assume that this is indeed the case. This restricts the generality of our superpotential by the condition
f ijk f k = 0,(12)
that we assume to be valid in this paper 1 . In this case, the block (V ) k l determines the mass spectrum and eigenstates. The mass of (complex) sgoldstino obtained from this block must involve Kahler terms (their derivatives), it cannot acquire corrections from f ij and it must be proportional to SUSY breaking, thus it depends on f i , f i . The only possibility in normal coordinates is to contract the only non-trivial, non-vanishing tensor k ij kl = R ij kl with f i , f i and ensure the correct mass dimension and sign. The result for this (mass) 2 is given in the equation below and a discussion can be found in [8]. Finally since SUSY is broken spontaneously, the sgoldstino mass eigenvector is expected to have a form similar to that of goldstino itself in eq.(10). Indeed, in the limit of ignoring the Kahler part of (V ) k l which is sub-dominant, of order O(1/Λ 2 ), the sgoldstino is the (massless) eigenvector of f ik f il matrix, and has the form:
φ 1 = f m δφ m [f i f i ] 1/2 + O(1/Λ 2 ), m 2 φ 1 = − k ij kl f i f j f k f l f m f m (13)
where δφ m = φ m − φ m is the field fluctuation about the ground state. The mass of sgoldstinoφ 1 comes from D-terms, which are O(1/Λ 2 ). Spontaneous SUSY breaking suggests the auxiliary of goldstino superfield should have a similar structure:
F 1 = f m F m [f i f i ] 1/2(14)
This is verified onshell, when
F † i = −W i + O(1/Λ 2 ) is expanded about the ground state to linear order fluctuations F † i = −f i − f im δφ m + O(1/Λ 2 )
. One then finds from (14) 2
F 1 = f m (−f m ) [f i f i ] 1/2 + O(1/Λ 2 )(15)
For illustration let us now consider in detail the case of only two fields present in Lagrangian (2), and also present the expression of the second mass eigenvector. For the fermions, the mass eigenvectors (normalised to unity) are given below, withψ 1 the goldstino field:
ψ 1 = 1 [f i f i ] 1/2 f 1 ψ 1 + f 2 ψ 2 , m 2 ψ 1 = 0. ψ 2 = 1 [f i f i ] 1/2 − (f 1 /ρ) ψ 1 + f 2 ρ ψ 2 , m 2 ψ 2 = f ij f ij , ρ = |f 1 | |f 2 | .(16)
1 An example when such condition is respected is for a superpotential of the type W = f1Φ 1 + λ/6 (Φ 2 ) 3 , with Φ 1 breaking SUSY and Φ 2 a matter field. This example will be considered later. 2 Also at minimum, V should be just | F 1 | 2 which is respected.
For the scalars sector, we find after some algebra the mass eigenstates 3 of (M 2 b ) = V k l :
φ 1 = 1 [f i f i ] 1/2 f 1 (1 + ξk 11 ) δφ 1 + f 2 (1 + ξk 12 ) δφ 2 , φ 2 = 1 [f i f i ] 1/2 − (f 1 /ρ) (1 + ξk 21 ) δφ 1 + f 2 ρ (1 + ξk 22 ) δφ 2 ,(17)
with the notatioñ
k 11 = k ij kl ρ k i f i f j f k f l ,k 12 = k ij kl ν k i f i f j f k f l , ξ ≡ [2(f k f k ) (f ij f ij )] −1 k 21 = k ij kl σ k i f i f j f k f l ,k 22 = k ij kl δ k i f i f j f k f l ,(18)
where
ρ 1 1 = 2 /ρ 2 , ρ 2 1 = 1 + ρ 2 + 2/ρ 2 , ρ 1 2 = −3 − ρ 2 , ρ 2 2 = −2, ν 1 1 = −2, ν 2 1 = −(1 − ρ 2 ), ν 1 2 = −(1 − ρ 2 ), ν 2 2 = 2 ρ 2 , σ 1 1 = −2, σ 2 1 = 1 + 1/ρ 2 + 2ρ 2 , σ 1 2 = −3 − 1/ρ 2 , σ 2 2 = 2 ρ 2 , δ 1 1 = 2 /ρ 2 , δ 2 1 = −(1 − 1/ρ 2 ), δ 1 2 = −(1 − 1/ρ 2 ), δ 2 2 = −2.(19)
In our normal coordinates k kl ij = R kl ij . We also find the masses:
m 2 φ 1 = − k ij kl f i f j f k f l f m f m , m 2 φ 2 = f ij f ij − k jk ik f i f j + k ij kl f i f j f k f l f m f m .(20)
We identifyφ 1 of (17) as the sgoldstino, since its mass should not receive corrections from f ij , in the limit of ignoring the curvature tensor corrections in (17), and it has a form similar to that of goldstino eigenstate (10), (16). Regarding the auxiliary fields, one can show thatF 2 = O(1/Λ 2 ) and thatF 1 is that in (15). We conclude that the goldstino superfield has the onshell SUSY form
Φ 1 on−shell = f k δΦ k [f i f i ] 1/2 on−shell + O(1/Λ 2 ) δΦ k on−shell ≡ δφ k + √ 2 θ ψ k + θθ (−f k ).(21)
where we used that k ij kl = O(1/Λ 2 ) and that auxiliary fields are on-shell. Eqs.(17) to (21) are valid under the assumption that corrections suppressed by powers of Λ are sub-leading to the superpotential SUSY corrections, proportional to f ij , see also (11). Let us introduce a parameter ζ equal to the ratio of the Kahler curvature tensor contracted by the SUSY breaking scale(s) f i to the SUSY "mass term" (f ij ):
ζ = ξk ij ∼ k kl ij f i f j f k f l (f p f p )(f mn f mn ) ∼ m 2 sgoldstino f mn f mn ≤ 1.(22)
If ζ ≤ 1 the results of this section such as (17) and (21) are valid and terms suppressed by high powers of Λ can be neglected, as we actually did. For ζ ∼ 1 the eigenvectors have a more complicated form (easily obtained) and is not presented here. The limit ζ ≫ 1 corresponds to decoupling a massive sgoldstino and is discussed in Section 2. 4 We shall compare eq.(21) to the chiral superfield X that breaks superconformal symmetry, conjectured in [1] to be equal, in the infrared limit to the goldstino superfieldΦ 1 .
2.2
The chiral superfield X and its low-energy limit.
Let us explore the properties of the superconformal symmetry breaking chiral superfield X and examine its relation to the goldstino superfield found earlier. The definition of X is
DαJ αα = D α X, X ≡ (φ X , ψ X , F X )(23)
where J is the Ferrara-Zumino current [2]. For a review of this topic, see for example section 2.1 in [1]. ψ X is related to the supersymmetry current and F X to the energymomentum tensor. For the general, non-normalizable action in (2), this equation has a solution [9]
X = 4 W − 1 3 D 2 K − 1 2 D 2 Y † (Φ † )(24)
We find the component fields of X to be (ignoring the improvement term D 2 Y † (Φ † )):
φ X = 4 W (φ i ) + 4 3 K j F † j − 1 2 K ij ψ i ψ j ψ X = ψ k ∂φ X ∂φ k − 4 i 3 σ µ K j ∂ µ ψ j + K ij ψ j ∂ µ φ † i F X = F i ∂φ X ∂φ i − 1 2 ψ i ψ j ∂ 2 φ X ∂φ i ∂φ j + 4 3 K j i ∂ µ φ i ∂ µ φ † j + i 2 ψ i σ µ D µ ψ j −D µ ψ i σ µ ψ j − ∂ µ K j ∂ µ φ † j − i 2 K j i ψ i σ µ ψ j(25)
In these relations all derivatives are scalar-fields dependent quantities. As a side-remark, one also notices that the integer powers n ≥ 1 of these components have a nice compact structure:
φ X n = (φ X ) n , (n ≥ 1) ψ X n = n (φ X ) n−1 ψ X = ψ j ∂φ X n ∂φ j + O(∂ µ ) F X n = n (φ X ) n−2 φ X F X − n − 1 2 ψ X ψ X = F j ∂φ X n ∂φ j − 1 2 ψ i ψ j ∂ 2 φ X n ∂φ i ∂φ j + O(∂ µ ),(26)
where the terms O(∂ µ ) vanish in the infrared limit of zero momenta. Notice that in the leading (zero-th) order in 1/Λ, the only dependence of these components on the Kahler comes through φ X via its term K j F † j , with additional contributions, fermionic dependent being 4 O(1/Λ).
From (25) we expand X about the ground state and denote w = W ( φ k ). Keeping linear fluctuations in fields, one obtains from eq.(25) that
φ X = 4 w + 8 3 f j δφ j + O(1/Λ), ψ X = 8 3 f k ψ k + O(1/Λ), F X = 8 3 f k (−f k ) − 4 f k f km δφ m − 4 3 f k δF † k + 8 3 f k δF k + O(1/Λ).(27)
Up to a constant we can write, using eq.(21), that onshell-SUSY:
X on−shell = 8 3 f k δφ k + √ 2 θ ψ k + θθ (−f k ) + O(1/Λ) = 8 3 f k δΦ k on−shell + O(1/Λ)(28)
Comparing this result against that for the goldstino superfield of (21), one has
X on−shell = 8 3 f i f iΦ 1 on−shell + O(1/Λ).(29)
Note that the X field goes to the (onshell) goldstino field in the limit of vanishing momentum and in addition Λ → ∞ when higher dimensional terms in the Kahler potential decouple. This clarifies the relation between the goldstino and the superconformal symmetry breaking superfields for general K and W , in the presence of more sources of SUSY breaking, and is one of the results of this work. All directions of supersymmetry breaking contribute to the relation between these two superfields.
Further properties of the field X.
Let us compute the onshell form of X by eliminating the auxiliary fields F k in (25)
φ X = σ + σ mn ψ m ψ n , ψ X = 8 3 ψ k W k , F X = β + β mn ψ m ψ n + β kl mn (ψ m ψ n ) (ψ k ψ l ),(30)
where
σ = 4 W − (4/3) K l (K −1 ) k l W k , σ mn = (2/3) K l Γ mn l − K mn = O(1/Λ), β = −(8/3) W m (K −1 ) m k W k , β mn = 2 W k Γ k mn − W mn = −2 W mn + O(1/Λ), β kl mn = (1/3) K kl mn − K j mn Γ kl i ≡ (1/3) R kl mn = O(1/Λ 2 ).(31)
Here we made explicit the terms which are suppressed by powers of the cutoff scale. In [1,3] it was used that the constraint X 2 = 0 projects out the sgoldstino field 5 . In a strict sense this constraint is valid only in the limit of an infinite sgoldstino mass. So the problem is that one has an expansion in 1/Λ of the initial Lagrangian which can conflict with an expansion in the inverse sgoldstino mass, 1/m 2 φ 1 ∼ 1/(f 2 i /Λ 2 ) = Λ 2 /f 2 i , that decouples the sgoldstino. The effective Kahler terms must give a mass to sgoldstino (which would otherwise be massless at tree level in spontaneous Susy breaking), and must simultaneously be large enough for the sgoldstino to decouple at low energy. The two expansions may have only a very small overlap region of simultaneous convergence.
To have X 2 = 0 it is necessary and sufficient to have F X 2 = 0. This can be checked directly. If for example F X 2 = 0 then one immediately shows that φ X ∼ ψ X ψ X so X 2 = 0. Let us compute the value of F X 2 in general, using (30). One has
F X 2 = α+α mn (ψ m ψ n )+λ mn (ψ m ψ n )+ν kl mn (ψ m ψ n )(ψ k ψ l )+ξ mn (ψ m ψ n )(ψ 1 ψ 1 )(ψ 2 ψ 2 ) = α + α mn (ψ m ψ n ) + O(1/Λ)(32)with α = 2σβ, α mn = 2σβ mn − 64 9 W m W n ,(33)
while the remaining coefficients are suppressed, of order O(1/Λ) or higher 6 and vanish at large Λ. In this limit only, expanding (32) about the ground state (or using (27)) we find
F X 2 = − 64 9 (2 f k f k f l δφ l + f k f l ψ k ψ l ) + O(1/Λ)(34)
up to a constant (∝ w). To see if F X 2 and thus X 2 vanish (hereafter this is considered up to O(1/Λ) terms) after decoupling massive scalar fields, one should integrate out δφ k , k = 1, 2..., via the eqs of motion. With δφ k expressed in terms of the light fermionic and other scalar degrees of freedom, one checks in this way if X 2 = 0, without the need of computing the mass eigenstates 7 . In general, upon integrating out the sgoldstino and additional massive scalars, X 2 necessarily contains terms suppressed by the sgoldstino mass. In most cases X 2 does not vanish anymore if this mass is finite, except in specific cases, due to additional simplifying assumptions (symmetries, etc) for the terms (e.g. k ij mn ) of the Lagrangian. In these cases the convergence problem mentioned earlier is not an issue. We discuss such a case in the next two sections.
Decoupling all scalar fields, for vanishing SUSY mass terms.
Let us consider the special case of a vanishing SUSY term, i.e. f ij = 0 or assume it is much smaller than the Kahler terms in the mass matrix of eq.(11). We consider only two fields present in the Lagrangian (2), both of which can contribute to the SUSY breaking. This can be generalised to more fields. For f ij = 0 we have two massless fermions. As a result, both scalar fields, which are massive (via Kahler terms), can be integrated out and expressed in terms of these massless fermions, without special restrictions for scales present. We shall do this and then examine under what conditions X 2 vanishes after decoupling.
Regarding the relation of X to the goldstino superfield, in this case it is difficult to define the latter, as both fields contribute to SUSY breaking and they can also mix. In previous sections, see eq.(22), (29) the superpotential (f ij ) terms were dominant and Kahler terms were a small correction (∝ 1/Λ 2 ) to the scalars mass matrix (11), while here the situation is reversed. Eq.(11) with vanishing off-diagonal blocks and vanishing f ik f il gives a mass matrix
M 2 b = (V ) k l = −k jk il f i f j in basis δφ 1,2 with eigenvalues m 2 φ 1,2 = 1 2 − k mj mk f k f j ± √ ∆ + O(f ij f ij ), ∆ = (k mj mk f k f j ) 2 − 4 det(k pj mk f k f j )(35)
where all indices take values 1 and 2, and the determinant is over the free indices. This is the counterpart to the result in (20). The eigenvectors can also be found 8 . How do we 6 The expressions of these terms are λ mn = 2 β σ mn = O(1/Λ), ν kl mn = 2 σ β kl mn + βmnσ kl = O(1/Λ), ξmn = 2 σ 11 β 22 mn + σ 22 β 11 mn − 2σ 12 β 12 mn = O(1/Λ) . 7 For one field breaking SUSY, f 1 = 0, (f j =1 = 0) and if w = 0, F X 2 = 0 if δφ 1 = ψ 1 ψ 1 /(−2 f 1 ) see [1]. 8 They areφ 1,2
= (1/|φ 1,2 |){(2k 2k 1j f k f j ) −1 (k 1n 1m − k 2n 2m )f m fn ± √ ∆ δφ 1 +δφ 2 } + O(f ij fij ) (|φ 1,2 |: norm).
identify the sgoldstino? The term f ij f jk which defined in (11) the leading contribution to the mass matrix and eigenvectors, is vanishing, so it cannot be used. One can identify the sgoldstino from a transformation that ensures that only one linear combination of auxiliary fields breaks supersymmetry. The scalar in the same supermultiplet is then the sgoldstino; further, if no mixing is induced by Kahler curvature terms (this mixing is controlled by k ij mn and is therefore UV and model-dependent) then this state is also a mass eigenstate.
To this end define new superfields
Φ 1 = 1 [f k f k ] 1/2 (f 1 δΦ 1 + f 2 δΦ 2 ), Φ 2 = 1 [f k f k ] 1/2 (−(f 1 /ρ) δΦ 1 + f 2 ρ δΦ 2 ), ρ = |f 1 |/|f 2 |.(36)
where δΦ j = (δφ j , ψ j , F j ).Φ 1 is inferred from the auxiliary fields combination andΦ 2 was determined by unitarity arguments. One can apply this transformation to the original Lagrangian, then if scalar components are not mixing,φ 1 is also a mass eigenstate 9 .
Let us now discuss the decoupling of the scalars and check under what conditions X 2 can vanish, without demanding an infinite sgoldstino mass (which would bring convergence problems). We integrate the scalars, so in the low energy they are combinations of the light/massless fermions. To this purpose, we do not need to identify the sgoldstino. From (5), the eq of motion of scalar field φ † l , at zero-momentum, is
W kl (K −1 ) i k W i + W k (K −1 ) il k W i + 1 2 W ijl − ∂ l (Γ ij m W m ))ψ i ψ j − 1 2 ∂ l Γ m ij W m ψ i ψ j = 0. (37)
We expand this about the ground state, in normal coordinates and use our simplifying assumptions
f ij = 0, f ijl f l = 0, and f ijlm = 0.(38)
The result is
k il kj δφ j f k f i + 1 2 k lm ij f m ψ i ψ j − 1 2 f ijl ψ i ψ j + O(1/Λ 3 ) = 0, i, j, k, l, m = 1, 2.(39)
Taking l = 1, 2, we solve this system for δφ 1,2 to find
δφ 1 = 1 2 det(k kn lm f n f m ) A ij ψ i ψ j + B ij ψ i ψ j + O(1/Λ) δφ 2 = 1 2 det(k kn lm f n f m ) C ij ψ i ψ j + D ij ψ i ψ j ) + O(1/Λ)(40)
with
A ij = k 2p ij k 1r 2s − k 1p ij k 2r 2s f r f s f p , B ij = −f ij2 k mr 2s f s f m f r (f 1 ) −1 , C ij = k 1p ij k 2r 1s − k 2p ij k 1r 1s f r f s f p , D ij = −f ij1 k mr 1s f s f m f r (f 2 ) −1 .(41)
The fields are suppressed by the mass of sgoldstino since the determinant in the denominator is a product of scalar masses, see (35); but due to interactions (f ijl ), by counting the mass dimensions, the expansion can also be regarded as proportional to Λ 2 /f 2 i ! Indeed:
δφ 1 ∝ 1 m 2 sgoldstino A ij ψ i ψ j + B ij ψ i ψ j , (similar for δφ 2 ).(42)
This is the sgoldstino decoupling limit, opposite to that considered in eq.(22).
To check if X 2 vanishes for finite sgoldstino/scalars masses, we use the result of (34). From (40) one finds that this happens if
(f k f k ) f 1 A ij + f 2 C ij + det(k m n ) f i f j = 0, f 1 B ij + f 2 D ij = 0(43)
withk m n ≡ k mr ns f r f s . The last two equations can be re-written as
f p k 2p ij (f 1k 1 2 − f 2k 1 1 ) − k 1p ij (f 1k 2 2 − f 2k 2 1 ) + det(k m n ) f i f j f k f k = 0 k mr ls f ijl f s f m f r = 0,(44)
where i, j are fixed to any value, 1,2. If these relations are respected one has in the model X 2 = 0 for a finite sgoldstino mass and trilinear interactions in the superpotential. These relations ultimately imply some constraints for the curvature tensor and thus for the UV regime. The first relation in (44) simplifies further in specific cases, for example ifφ 1 of (36) is also a mass eigenstate which happens fork 1 2 =k 2 1 = 0. Conditions (44) can be generalised to more fields and should be verified in those applications in which the constraint X 2 = 0 was used. These conditions would also be recovered with our definition of the goldstino superfield in (36). With this definition, the above relations are obtained by demanding (onshell) (Φ 1 ) 2 = 0, or equivalently
F (Φ 1 ) 2 = 2F 1φ1 −ψ 1ψ1 = 0.
To illustrate some implications, let us take in eq.(40) the limit of only one field breaking supersymmetry, i.e. assume f 2 = 0. One finds
δφ 1 = − ψ 1 ψ 1 2 f 1 + det(k 1i 2j ) det(k 1m 1n ) ψ 2 ψ 2 2 f 1 − k 11 21 f ij2 det(k 1m 1n ) ψ i ψ j 2 |f 1 | 2 + O(1/Λ) δφ 2 = − ψ 1 ψ 2 f 1 + k 12 11 k 11 22 − k 11 11 k 12 22 det(k 1m 1n ) ψ 2 ψ 2 2 f 1 + k 11 11 f ij2 det(k 1m 1n ) ψ i ψ j 2 |f 1 | 2 + O(1/Λ)(45)
This is the general result for the scalars as functions of the massless fermionic fields, when superpotential interactions are present 10 . This result recovers eqs.(33), (37) in [6] but have additional corrections due to superpotential couplings. The terms proportional to f ij2 in both δφ 1,2 are actually dominant, since they grow like Λ 2 , as it can be seen from the mass dimensions of the k ij lm . The other terms, coefficients of ψ 1 ψ 1 and ψ 2 ψ 2 are actually independent of Λ, although for ψ 2 ψ 2 they involve UV details 11 .
From (34) one obtains F X 2 ∝ (2 f 1 δφ 1 + ψ 1 ψ 1 ) [1], which we demand to vanish. Using δφ 1 of (45) or directly from the two equations in (44), one finds that X 2 = 0 if det(k 1i 2j ) = 0, and f ij2 k 11 12 = 0.
These constraints are a particular case of the general conditions in (44). If the Lagrangian respects these conditions, one can have 12 X 2 = 0 in the presence of trilinear interactions, with one field breaking SUSY and finite mass sgoldstino. Finally, let us add that eq.(45) and conditions (46) simplify further if one demandsφ 1 of (36) be also a mass eigenstate which only happens under a special, additional UV assumption: k 11 12 = k 12 11 = 0. Then condition (46) reduces to k 11 22 = 0. This is however a particular case, not considered further.
Decoupling the sgoldstino in the presence of a light matter field.
There are situations when the sgoldstino is significantly heavier than other scalar (matter) fields and is the first or the sole field to decouple at low energy. If so, under what conditions is X 2 = 0? To examine this briefly, consider the case of the previous section, of two fields Φ 1,2 in the Lagrangian, with a simple superpotential
W = f 1 Φ 1 + λ 3! (Φ 2 ) 3 ,(47)
So Φ 1 breaks supersymmetry and we also assume that its scalar component (sgoldstino) is much heavier than the second scalar (matter) field belonging to Φ 2 . One can ensure such mass hierarchy by assuming that det(k 1n 1m ) is small enough, see (35). Although we do not consider here the extreme case when it actually vanishes, in that case one has (if k 11 11 + k 12 12 < 0) that h 11 11 k 12 12 − k 11 12 k 12 11 ≈ 0, m 2 φ 1 ≈ −(k 11 11 + k 12 12 ) f 2 1 , m 2 φ 2 ≈ 0. Let us then integrate out the sgoldstino. Its eq of motion, from Lagrangian (5) or (39), is
δφ 1 = − 1 f 1 k 11 11 (1/2) k 11 mn ψ m ψ n + f 1 δφ 2 k 11 12 + O(1/Λ),(48)
which is a function of the light scalar and massless fermions. Using (48) one finds 10 As usual, in the normal coordinates used here k mn ij = R mn ij . 11 Similar effects were discussed in [5,6]. 12 up to O(1/Λ) corrections.
F X 2 = 64 9 (f 1 ) 2 k 11 11 2 k 11 12 (f 1 δφ 2 + ψ 1 ψ 2 ) + k 11 22 ψ 2 ψ 2 + O(1/Λ)(49)
For any value of the scalar matter field (φ 2 ), with sgoldstino decoupled at finite mass, one can thus have F X 2 = X 2 = 0 only if k 11 12 = 0, k 11 22 = 0 and for a large Λ. These conditions can be compared to those when both scalars are decoupled shown in (46) (with f ij2 → λ). Therefore the action for which the formalism of [1] applies with X 2 = 0, has K given by
K = Φ † 1 Φ 1 + Φ † 1 Φ 1 + k 11 11 (Φ † 1 Φ 1 ) 2 + k 22 22 (Φ † 2 Φ 2 ) 2 + k 21 22 (Φ † 2 Φ 2 )(Φ 2 Φ † 1 ) + h.c. + k 12 12 (Φ † 1 Φ 1 )(Φ † 2 Φ 2 ) + O(1/Λ 3 )(50)
with a nontrivial superpotential as in (47) and a finite sgoldstino mass.
In the Lagrangian obtained after decoupling δφ 1 one can now also integrate out δφ 2 and obtain a solution for it as in (45) but with the replacement f ij2 ψ i ψ j → λψ 2 ψ 2 . This solution, if used in (48), brings δφ 1 to the form shown in (45), as expected. With this δφ 2 one then easily verifies that F X 2 of (49) becomes
F X 2 = 64 9 −(f 1 ) 2 det(k 1m 1n ) det(k 1i 2j ) ψ 2 ψ 2 − λ k 11 12 f 1 ψ 2 ψ 2(51)
and
X 2 = − F X 2 f 1 −1 2 f 1 ψ 1 ψ 1 + 9 128 F X 2 (f 1 ) 2 + √ 2 θ ψ 1 + θθ (−f 1 ) + O(1/Λ).(52)
Therefore F X 2 vanishes and so does X 2 provided that det(k 1l 2m ) = 0 and λ k 11 12 = 0, and this recovers the result in eq.(46) when both scalars are decoupled.
Higher powers of X can vanish with weaker restrictions. This is actually expected from the properties of the Grassmann variables. Indeed, one shows that in onshell-SUSY case after decoupling only the sgoldstino (δφ 1 ) then:
X 3 ∝ k 11 12 k 11 11 f 1 × (function of δφ 2 , ψ 1,2 ).(53)
This vanishes for any δφ 2 and finite sgoldstino mass, provided that k 11 12 = 0 which is a weaker constraint than that found for X 2 . Higher powers of X show that k 11 12 = 0 is still needed for X 4 to vanish for any light matter field, because in (48) δφ 2 is multiplied by k 11 12 . Recall however that k 11 12 vanishes if there is no scalars mixing induced by Kahler curvature terms i.e. if φ 1 of (47) is also a mass eigenstate.
Conclusions.
In this work we considered the relation of the superconformal symmetry breaking chiral superfield X and the goldstino superfield, in effective models with low scale of SUSY breaking, when transverse gravitino couplings are negligible relative to their longitudinal counterparts of its goldstino component. The models considered have a general Kahler (K) and superpotential (W ) with more sources of supersymmetry breaking.
In this case we verified the conjecture that the superfield X becomes the goldstino superfield in the limit of zero-momentum and, in addition, Λ → ∞, where Λ is the UV cutoff. This happens when the higher dimensional Kahler terms are sub-leading to the supersymmetric mass terms in the scalar mass matrix. For vanishing SUSY mass terms, but otherwise rather general K and W we also investigated the decoupling of the massive scalars simultaneously or separately. In this case we identified the conditions for which the sgoldstino decoupling condition X 2 = 0 is still satisfied in the presence of additional fields, for a finite sgoldstino mass. This is important to ensure that the effective expansion (∝ 1/Λ) of the Lagrangian does not conflict with the sgoldstino decoupling limit (of small ∝ 1/m 2 sgoldstino ∼ Λ 2 /f 2 i where f i is the SUSY breaking scale). The above conditions are lifted in the formal limit of very large sgoldstino mass (or when all scalar and fermion fields other than the Goldstino fermion have all non-zero masses and are integrated out); then, in the far infrared (i.e. far below any of these mass scales and at zero momentum) one recovers the relation X 2 = 0 of the Akulov-Volkov action for the goldstino.
One can reverse the above arguments and conclude that the use of the constraint X 2 = 0, although appealing and apparently UV independent, is of somewhat restricted applicability in the case of general K, W (with massless fields present, additional SUSY breaking fields and interactions, etc); ultimately it implicitly makes assumptions about UV details, difficult to justify without additional input (symmetry, etc). The situation can improve in models where the UV details are under control, such as in renormalizable models of supersymmetry breaking (O'Raifeartaigh, etc), not considered here (where in the sgoldstino decoupling limit Λ is replaced by an appropriate SUSY mass scale).
What does this mean for model building? When parametrizing SUSY breaking in models like the MSSM one commonly uses a spurion field that is a limit of the goldstino superfield with the dynamics integrated out. The above observation regarding UV assumptions suggests that it may be preferable, when studying the details of a low-scale SUSY breaking case, to couple (offshell!) the goldstino superfield to the MSSM, as a linear superfield 13 rather than as a non-linear representation that is a solution of the constraint X 2 = 0. One can then eventually decouple the sgoldstino explicitly, via the eqs of motion.
and then examine under what conditions X 2 could vanish. The results below are valid up to O(∂ µ ), where all terms K j , W k ... etc are actually scalar-fields dependent. One has
eqs.(16) and (17) are multiplied in the rhs by |f2|/f2 which is set to unity by phase rescaling Φ2.
The bracket in FXn is SUSY invariant for n = 2, and then F X 2 = 0 is invariant.
Additional constraints for matter superfields can be used to decouple superpartners at low energy.
This requires a diagonal mass matrix in δφ 1,2 initial basis, i.e. k 1k2m f k f m = k 2k1m f k f m = 0 (for a vanishing f2 this means k 11 12 = k 12 11 = 0). In this case the masses are m 2 φ 1 = −k 1m 1n fmf n , m 2 φ 2 = −k 2m 2n fmf n .
identified as in the text, in the case of more sources of SUSY breaking.
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|
[] |
[
"Artificial Intelligence and Location Verification in Vehicular Networks",
"Artificial Intelligence and Location Verification in Vehicular Networks"
] |
[
"Ullah Ihsan \nSchool of Electrical Engineering & Telecommunications\nThe University of New South Wales\n2052SydneyNSWAustralia\n",
"Ziqing Wang \nSchool of Electrical Engineering & Telecommunications\nThe University of New South Wales\n2052SydneyNSWAustralia\n",
"Robert Malaney \nSchool of Electrical Engineering & Telecommunications\nThe University of New South Wales\n2052SydneyNSWAustralia\n",
"Andrew Dempster \nSchool of Electrical Engineering & Telecommunications\nThe University of New South Wales\n2052SydneyNSWAustralia\n",
"Shihao Yan \nSchool of Engineering\nMacquarie University\n2109SydneyNSWAustralia\n"
] |
[
"School of Electrical Engineering & Telecommunications\nThe University of New South Wales\n2052SydneyNSWAustralia",
"School of Electrical Engineering & Telecommunications\nThe University of New South Wales\n2052SydneyNSWAustralia",
"School of Electrical Engineering & Telecommunications\nThe University of New South Wales\n2052SydneyNSWAustralia",
"School of Electrical Engineering & Telecommunications\nThe University of New South Wales\n2052SydneyNSWAustralia",
"School of Engineering\nMacquarie University\n2109SydneyNSWAustralia"
] |
[] |
Location information claimed by devices will play an ever-increasing role in future wireless networks such as 5G, the Internet of Things (IoT), and Intelligent Transportation Systems (ITS). Against this background, the verification of such claimed location information will be an issue of growing importance. A formal information-theoretic Location Verification System (LVS) can address this issue to some extent, but such a system usually operates within the limits of idealistic assumptions on a-priori information on the proportion of genuine user-vehicles in the field. In this work we address this critical limitation by the use of a Deep Neural Network (DNN) showing how such a DNN based LVS is capable of efficiently functioning even when the proportion of genuine user-vehicles is completely unknown a-priori. We demonstrate the improved performance of this new form of LVS based on Time of Arrival (ToA) measurements from multiple verifying base stations within the context of vehicular networks, quantifying how our DNN-based LVS outperforms the stand-alone information-theoretic LVS in a range of anticipated real-world conditions.
|
10.1109/globecom38437.2019.9014171
|
[
"https://arxiv.org/pdf/1901.03001v1.pdf"
] | 57,761,148 |
1901.03001
|
3766f8112d10afe834e9198ea09c0edcb2709909
|
Artificial Intelligence and Location Verification in Vehicular Networks
10 Jan 2019
Ullah Ihsan
School of Electrical Engineering & Telecommunications
The University of New South Wales
2052SydneyNSWAustralia
Ziqing Wang
School of Electrical Engineering & Telecommunications
The University of New South Wales
2052SydneyNSWAustralia
Robert Malaney
School of Electrical Engineering & Telecommunications
The University of New South Wales
2052SydneyNSWAustralia
Andrew Dempster
School of Electrical Engineering & Telecommunications
The University of New South Wales
2052SydneyNSWAustralia
Shihao Yan
School of Engineering
Macquarie University
2109SydneyNSWAustralia
Artificial Intelligence and Location Verification in Vehicular Networks
10 Jan 2019
Location information claimed by devices will play an ever-increasing role in future wireless networks such as 5G, the Internet of Things (IoT), and Intelligent Transportation Systems (ITS). Against this background, the verification of such claimed location information will be an issue of growing importance. A formal information-theoretic Location Verification System (LVS) can address this issue to some extent, but such a system usually operates within the limits of idealistic assumptions on a-priori information on the proportion of genuine user-vehicles in the field. In this work we address this critical limitation by the use of a Deep Neural Network (DNN) showing how such a DNN based LVS is capable of efficiently functioning even when the proportion of genuine user-vehicles is completely unknown a-priori. We demonstrate the improved performance of this new form of LVS based on Time of Arrival (ToA) measurements from multiple verifying base stations within the context of vehicular networks, quantifying how our DNN-based LVS outperforms the stand-alone information-theoretic LVS in a range of anticipated real-world conditions.
I. INTRODUCTION
We are at the verge of new wireless networks that aim to bring communication revolutions in homes, hospitals, education, transportation, and other aspects of society. Emerging Intelligent Transportation Systems (ITS) are particularly exciting due to their potential to save many lives. The success of these new technologies in general, and ITS in particular, find their roots in the true location information of the clients (devices, users, vehicles) involved. In many scenarios it is anticipated that clients can directly obtain their location information [1], [2] through the Global Navigation Satellite System (GNSS). This location information is then usually provided by the clients to other clients or to some central Processing Center (PC), for verification and other network functionality purposes. But what if a client provides incorrect information about his true location intentionally in an attempt to obtain some advantage over other users [3], [4]? Such circumstances could also occur unintentionally due to difficulty in recording the GNSS location information or faulty hardware issues.
For the application that forms the focus of this work, namely ITS, if a malicious user provides inaccurate location information the possible aftermath could range from suboptimal traffic routing all the way through to life-threatening collisions [5], [6]. Verification of the reported location infor-mation is hence critical for successful operation in ITS [5], [7]- [10]. Due to this, Location Verification System (LVS) performance has been a research focus in ITS for well over a decade. Recently, several information-theoretic LVSs have been devised [11]- [13]. These LVSs operate under a set of well-defined rules and conditions. Additionally, they have limitations in addressing various anomalies since they usually assume idealized channel conditions [11]. As such, information-theoretic LVSs usually possess performance limitations in real-world situations. One of the most important of these limitations is the a-priori lack of knowledge on the proportion (fraction) of vehicles in the field that will be malicious (alternatively, the fraction that will be genuine).
Artificial intelligence has recently brought breakthroughs into many aspects of modern society. Web mining [14], content filtering [15], image recognition [16], [17], speech processing [18], language identification [19], speaker verification [20], object detection [21], [22], advanced genomics [23], and drug discovery [24] are just a few of the fields impacted. Many of these breakthroughs are achieved through the development of new machine-learning algorithms [25].
Artificial intelligence also lays the foundation for many aspects of the self-driving car paradigm [26]. In this work, we show that unlike the information theoretic LVS which assumes an a-priori knowledge about the proportion of malicious user-vehicles in the field, our Deep Neural Network (DNN)-LVS works satisfactorily in the complete absence of this knowledge. For focus we consider an LVS based on Time of Arrival (ToA) measurements [27] under the influence of Non-Line-of-Sight (NLoS) biases. As we shall see, the DNN-LVS is also able to compensate for unknown NLoS conditions significantly better than an information-theoretic LVS.
Recent advancements in digital signal processing and hardware design now provide us with very accurate physical-layer timing information for wireless networks [28], [29]. These developments provide us with the clock synchronization that enables the LVS we study here. As such, we suggest our new DNN-based LVS can offer a viable and pragmatic solution to the important task of vehicle location verification under real-world conditions and uncertainties.
II. SYSTEM MODEL We outline the system model and assumptions considered for our framework:
1) The framework consists of N trusted Base Stations (BSs) as verifiers with publicly known locations that are assumed to be in the range of the prover (the vehicle whose claimed location is to be authenticated).
The location of the i-th BS is
X i = [x i , y i ] where i = 1, 2, ..., N .
2) The true location from a genuine or malicious uservehicle (the prover) is denoted by X t = [x t , y t ] and is assumed to possess zero localization error. 3) We refer to the announced location from a legitimate or malicious user-vehicle as claimed location and denote it by X c = [x c , y c ]. For a legitimate user-vehicle, the claimed location is exactly the same as his true location. On the other hand, a malicious user-vehicle spoofs his true location to the BSs (to potentially obtain an advantage over other user-vehicles or to disrupt the system performance). The true location of a malicious user-vehicle is unknown to the wider network. 4) One of the N BSs is chosen as the PC. Measurements from all BSs are collected at the PC before being processed into a a binary decision on the integrity of a user's claimed location. 5) Under the null hypothesis H o , the framework assumes a user-vehicle to be legitimate i.e.,
H o : X c = X t .(1)
6) Under the alternate hypothesis H 1 , the framework considers a user-vehicle to be malicious i.e.,
H 1 : X c = X t .(2)
Under H o , the ToA value measured by the i-th BS from a legitimate user-vehicle is given by,
Y i = U i + X i , i = 1, 2, . . . , N,(3)
where X i , the BS's receiver thermal noise, is a zero-mean normal random variable with variance σ 2 T . U i is the ToA and is given by,
U i = d c i c ,(4)
where d c i is the Euclidian distance of i-th BS to a legitimate user-vehicle's true location, and is given by
d c i = (x c − x i ) 2 + (y c − y i ) 2 , with c as the speed of light.
We assume the measurements made by the N BSs to be independent of each other. Under H o , they collectively form a vector Y = [Y 1 , Y 2 , . . . , Y N ] T . Vector Y follows a multivariate normal distribution given as,
Y|H o ∼ N (U , R),(5)
where U = [U 1 , U 2 , . . . , U N ] T is the mean vector under the null hypothesis, and R = σ 2 T I N is the variance matrix.
Under H 1 , a malicious user-vehicle claims to be at a location removed from his true location. In a real-world scenario, we can think of this as if the malicious user pretends to be on the road when he actually is placed off the road in a street or in a building. The ToA value measured by the i-th verifier from a malicious user-vehicle is given by,
Y i = T x + W i + X i , i = 1, 2, . . . , N,(6)
where T x is a time bias potentially added by malicious uservehicle which impacts the overall ToA value. W i is given by,
W i = d t i c ,(7)
where d t i is the Euclidian distance of i-th BS to a malicious user's true location, and is given by
d t i = (x t − x i ) 2 + (y t − y i ) 2 .
We assume the measurements made by N BSs to be independent of each other. Under H 1 , they collectively form a vector
Y = [Y 1 , Y 2 , . . . , Y N ] T .
Vector Y follows a multi-variate normal distribution given as,
Y|H 1 ∼ N (W + T x 1, R),(8)
where
W + T x 1 = [W 1 + T x , W 2 + T x , . . . , W N + T x ] T and
1 is a vector equal to the length of the number of BSs N with all its elements set to 1. For later convenience we set V = W + T x 1 and rewrite (8) as,
Y|H 1 ∼ N (V , R).(9)
III. PERFORMANCE ANALYSIS
The outcome of an LVS is a binary result i.e. true or false, yes or no, legitimate or malicious. This is different from a localization system where the output is an estimated location. We measure the performance of our LVS using two methodologies; through information theoretic analysis followed in [27] and, through the newly designed DNN method which makes use of the machine-learning techniques. In both cases, a Bayes average cost function is chosen as the performance metric for LVS in terms of the 'Total Error'. The information theoretic method is based on the a-priori assumption that the proportion of malicious user-vehicles is known. Usually this is set to 0.5 in the absence of any other information. On the other hand, the DNN-LVS calculates the Total Error irrespective of any such a-priori assumption and it can function with any proportion of malicious user-vehicles in the field. The Total Error is given by
ǫ = p(Y|H o )p(H o ) + p(Y|H 1 )p(H 1 ),(10)
where p(H o ) and p(H 1 ) are the a priori probabilities of occurrences of H o (i.e. legitimate user) and H 1 (i.e. malicious user), respectively. We set the a priori probabilities to 0.5 and represent p(Y|H o ) as the False Positive Rate α (the rate of a legitimate user being detected incorrectly) and p(Y|H 1 ) as the Detection Rate β (the rate of a malicious user being detected correctly). Equation (10) then takes the form,
ǫ = 0.5×α + 0.5× (1−β) .(11)
A. LVS Performance Analysis Using Information Theory
The Likelihood Ratio Test (LRT) is used for performance measurement of the LVS. It has been proven earlier that the LRT achieves the optimum detection results for a given false positive rate [30]. This leads to the conclusion that the LRT minimizes the Total Error and maximizes the mutual information between input and output of LVS [31]. We follow the decision rule below for the LRT,
Λ (Y ) p (Y | H 1 ) p (Y | H o ) D1 ≥ < D0 λ,(12)
where Λ (Y ) is the likelihood ratio, λ is the decision threshold, and D 1 and D 0 are the binary decision values (i.e. whether the user is legitimate or malicious). Given the multi-variate normal form of the observations, (12) can be reformulated as
Λ (Y ) = e − 1 2 (Y −V ) T R −1 (Y −V ) e − 1 2 (Y −U ) T R −1 (Y −U ) D1 ≥ < D0 λ.(13)
B. LVS Performance Analysis Using Artificial Intelligence
This section highlights the novel approach used to design a classification framework for the verification of a user's claimed location through supervised machine-learning techniques. The framework uses a multi-layer feed-forward DNN for the binary classification of a user-vehicle as either legitimate or malicious.
For uniformity, the framework considers the same inputs as considered for the information theoretic method. These inputs include U (the ToA of the signal based on the user's claimed location) and Y (the observation vector influenced by the thermal noise X i ). Based on a series of trials with changing architectures for the DNN-LVS, we finalised a framework that has an input vector, a hidden layer (with 10 neurons), and a binary output layer as shown in Fig. 1. The DNN LVS achieved optimum performance through the use of a Hyperbolic tangent sigmoid and linear transfer functions in the hidden and output layers respectively.
IV. NUMERICAL RESULTS
We now present some numerical results based on our analysis from the information-theoretic and DNN-based LVS. In carrying out these simulations, BSs are located in a 1000 meters by 500 meters area at fixed publicly known locations. This area closely resembles a small district of a city and corresponds to the context of VANET where the BSs are trusted verifiers located on the roadside or in the nearby parking lots. The claimant vehicle (the prover whose location is yet to be verified) resides in a 500 X 500 meters area in between the BSs. In order to simulate the attacking scenario and thus study the performance of both the LVS, we assume there are two claimants that are within the communication range, namely a legitimate uservehicle which is reporting his true location to the BSs and a malicious user-vehicle which is performing the locationspoofing attack. Both the user-vehicles can overhear the communication between the BSs and thus both acquire the locations of the BSs. If malicious, the user-vehicle can also overhear the communication between legitimate user-vehicles and the BSs so that it can forge his claimed location to that of the legitimate user's true location.
The malicious user-vehicle sets his true location at a faroff point so that its transmitted signal (with the appropriate timing offset) has equal ToA at all the BSs (in the limit of the true location of a malicious vehicle being much greater that any other scale all NLoS biases at all BSs are the same). Under this approximation the mean ToA at the BSs is just the mean of the timings anticipated from a vehicle at the claimed location. The resultant alteration in ToA due to the receiver's thermal noise is extracted from a Gaussian random distribution with fixed standard deviation. The values of standard deviation considered in our simulation is set to 300 nanoseconds.
We use simulated ToA data in our numerical experiments. The claimed location for genuine and malicious user-vehicles in equal proportion is generated randomly in the specified area. The ToA from the claimed locations at the 4 BSs is calculated using equation (4). The receivers in the BSs are under the influence of independent thermal noise X i and thus the ToA measurements they make have a certain degree of variation. We extract this variation (in nanoseconds) from a Gaussian random function that has a fixed standard deviation. The area around the user-vehicles are blocked so their transmitted signal cannot reach the BSs directly and hence their ToA have an additional NLoS bias φ i in them. To mimic reality, we extract φ i from an exponential distribution as given below,
f (φ i ) = ρ i e −ρiφi ,(14)
where ρ i is the scale parameter. For the information-theoretic LVS, we calculate the Total Error, the false positive rate, and the detection rate using equations (11) and (13), respectively.
The data considered for the information-theoretic LVS analysis is used to also train the DNN-LVS. We call this data the training data. 1 In the training phase, we feed the DNN-LVS with random user-vehicles data at a speed of one uservehicle data per second. During each second the DNN-LVS is trained with the available training data. The backpropagation algorithm has a set of internal parameters to terminate the training phase for the DNN-LVS. We observe that in most of the cases the maximum validation failures; which is the maximum number of iterations in a row during which the DNN-LVS's performance fails to improve or remains the same, terminates the training phase. We set this parameter to 6. The weights and biases are considered as optimised once the training phase has concluded. The DNN-LVS afterwards can be used to classify a user-vehicle as genuine or malicious in the test data 2 .
The DNN-LVS is trained during the 1 st second with an input (the ToA and claimed location) training data from a single random user-vehicle. At the end of 1 st second, we subject the DNN-LVS (with its weights and biases optimized) to calculate a Total Error for the test data. In the 2 nd second, 1 By training data we mean ToA data received from user-vehicles who we know a priori to be legitimate or malicious. Use of such data in order to set the neural network parameters, prior to its use on 'unlabeled' data (i.e., data from user-vehicles who we do not know a priori to be legitimate or malicious), is known as the training phase. 2 The test data is simulated under a different realization with same settings as training data. Further, test data has no labels. we add another random user-vehicle training data to the previously available single user-vehicle training data. The combined data forms a new training data set which is used to retrain the DNN-LVS (from 1 st second). After a re-training, the DNN-LVS is used to calculate a new Total Error for the test data. We add yet another random user-vehicle training data to the previously available training data in the 3 rd second and use the updated data set to once again train the DNN-LVS. At the end of the third second a revised Total Error is calculated for the test data. This process of updating the training data set, retraining the DNN-LVS and recalculating a new Total Error for the test data continues in the following seconds. The Total Error keeps on decreasing with the passage of time. In Fig. 3 we initially calculate the Total Error for a data set that has genuine and malicious user-vehicles in equal proportions (Po= 0.5). The standard deviation for X i is 300 nanoseconds while the standard deviation for NLoS bias is indicated by the different curves. The number of BSs used is 4. The LRT (i.e. the Total Error arising from the information theoretic LVS) corresponding to each NLoS curve is indicated by the dashed lines. We can see that performance for the information theoretic LVS deteriorates as the NLoS bias increases while the performance for the DNN-LVS improves with an increase in the NLoS bias in the ToA data. It is clear that the DNN-LVS is able to accommodate the NLoS conditions significantly better than an information theoretic LVS.
Next, we train a DNN-LVS through similar procedures as described earlier but change the proportion of malicious user-vehicles in the test data. In one of the experiments, we fix the standard deviation for BSs receiver's thermal noise and the NLoS bias to 300 nanoseconds. The number of BSs are 4. We can see in Fig. 4 that DNN-LVS performs consistently even when Po (the proportion of malicious uservehicles) is different in the test data. We can see that DNN-LVS performance is satisfactory even when the test data has 99.95% genuine user-vehicles and 0.05% malicious uservehicles. The red line in the Fig. 4 shows the Total Error for the information theoretic LVS when the genuine and malicious user-vehicles are in equal proportions in the data. Our study shows that unlike the information theoretic LVS whose performance is conditioned to the a-priori knowledge of Po, the DNN-LVS's performance is largely independent of Po. In Fig. 4, we change the standard deviation for NLoS bias to 500 nanoseconds. X i still is extracted from a Gaussian random distribution with a standard deviation of 300 nanoseconds. We can observe that DNN-LVS's performance is independent of the Po value.
In Fig. 6, we change number of BSs to 6 while the standard deviation for NLoS bias and X i are kept the same as in Fig. 5 describe the real-world channels. We also plan to compliment ToA systems by adding Received Signal Strength (RSS) and Angle of Arrival (AoA) measurements thus enhancing the reliability of DNN-based LVSs. A combination of ToA, RSS and AoA will make the DNN LVS even more reliable and efficient in its location verification. We also plan to extend the DNN framework to more complex channel fading models such as Rician fading channels. Estimating channel parameters through artificial intelligence will also result in an extended performance for the DNN LVS. In this case the neural network architecture will need to be extended so as to accommodate the additional unknowns that must be learned.
VI. CONCLUSION
Information-theoretic LVS frameworks, due to their operating limitations, are not practical in many real-world scenarios. To address this gap, we have proposed the use of a machine learning approach to location verification. This approach is particulary useful when we consider that one of the key inputs to any LVS is knowledge on the proportion of vehicles anticipated to be malicious -and input usually unknown. Using simulated ToA data, we have shown how a DNN-based LVS outperforms a state-of-the-art information theoretic LVS. Unlike the information theoretic LVS, the working of the DNN-LVS is shown to be largely independent of the proportion of malicious user-vehicles in the area. Unknown channel conditions, such as NLoS bias effects, were also shown to be better accommodated by the DNN-LVS approach. More salient real-world channel features, will be considered in our future work to further improve the overall system performance. This will help us develop a more robust state-of-the-art artificially intelligent LVS, an LVS which will be wholly practical in terms of its location verification performance in a wide range of future wireless networks beyond the ITS we have studied here. We believe the novel approach for enhancing the performance of realworld LVSs that we have developed here potentially forms the foundation for all future works in this important area.
Fig. 1 .
1The architecture of the Deep Neural Network (DNN) used for Location Verification in this work. This architecture arose from many trials of biased timings using different architectures with varying numbers of layers.
Fig. 2 .
2Schematic of the LVS architecture studied in this paper. Other trusted vehicles (previously verified) or BSs are used as the verifiers when authenticating claimed location of a prover vehicle. The diagram shows a malicious user spoofing a claimed location. In this work we assume only BSs are used as the verifiers.
Fig. 3 .
3Total Error performance of the DNN-LVS as it is being trained under changing NLoS bias conditions. The training and the testing data has genuine and malicious user-vehicles in equal proportions. X i is extracted from a Gaussian random distribution with a standard deviation of 300 nanoseconds. The NLoS bias (represented by different colour of the curves) is extracted from an exponential distribution with a fixed standard deviation. The DNN-LVS indicates an improved performance with a Total Error of 0.10 (NLoS 300ns), 0.06 (NLoS 500ns) and 0.04 (NLoS 700ns) as compared to the LRT method of[27] which gives a Total Error of 0.16 (NLoS 300ns), 0.18 (NLoS 500ns) and 0.21 (NLoS 700ns). Higher NLoS leads to easier discrimination between a genuine vehicle and a malicious one placed far from the BSs.
Fig. 4 .
4Total Error performance of the DNN-LVS with 4 BSs. The test data has different proportions for genuine and malicious user-vehicles as highlighted by the different colour of curves. X i is extracted from a Gaussian random distribution with a standard deviation of 300 nanoseconds. The NLoS bias is extracted from an exponential distribution with a fixed standard deviation of 300 nanoseconds. The red line shows the Total Error for the information theoretic LVS (based on LRT method) for a data (realised under same settings of standard deviation for thermal noise and NLoS bias) which has a Po equal to 0.5. We can see that the DNN-LVS performs consistently with different Po(s) in the test data.
. Still we can see a promising performance from the DNN-LVS under different Po(s).V. FUTURE WORKWe aim to add more prominent features related to the channel environment so as to further investigate gains achieved by DNN-based LVSs. These additional features will betterFig. 5. Total Error performance of the DNN-LVS with 4 BSs.The test data has different proportions for genuine and malicious user-vehicles as highlighted by the different colour of curves. X i is extracted from a Gaussian random distribution with a standard deviation of 300 nanoseconds. The NLoS bias is extracted from an exponential distribution with a fixed standard deviation of 500 nanoseconds. The red line shows the Total Error for the information theoretic LVS (based on LRT method) for a data (realised under same settings of standard deviation for thermal noise and NLoS bias) which has a Po equal to 0.5. We can see that the DNN-LVS performs consistently with different Po(s) in the test data.Fig. 6. Total Error performance of the DNN-LVS with 6 BSs. The test data has different proportions for genuine and malicious user-vehicles as highlighted by the different colour of curves. X i is extracted from a Gaussian random distribution with a standard deviation of 300 nanoseconds. The NLoS bias is extracted from an exponential distribution with a fixed standard deviation of 500 nanoseconds. The red line shows the Total Error for the information theoretic LVS (based on LRT method) for a data (realised under same settings of standard deviation for thermal noise and NLoS bias) which has a Po equal to 0.5. We can see that the DNN-LVS performs consistently with different Po(s) in the test data.0
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|
[] |
[
"Limiting curves for polynomial adic systems. *",
"Limiting curves for polynomial adic systems. *"
] |
[
"A R Minabutdinov "
] |
[] |
[] |
We prove the existence and describe limiting curves resulting from deviations in partial sums in the ergodic theorem for cylindrical functions and polynomial adic systems. For a general ergodic measurepreserving transformation and a summable function we give a necessary condition for a limiting curve to exist. Our work generalizes results by É. Janvresse, T. de la Rue and Y. Velenik and answers several questions from their work.
|
10.1007/s10958-017-3415-3
|
[
"https://arxiv.org/pdf/1701.07617v1.pdf"
] | 119,623,047 |
1701.07617
|
b00c778994b14ce3e0e04d50cea26b6d0c9d6082
|
Limiting curves for polynomial adic systems. *
November 14, 2018
A R Minabutdinov
Limiting curves for polynomial adic systems. *
November 14, 2018Polynomial adic systemsergodic theoremdeviations in er- godic theorem MSC: 37A3028A80
We prove the existence and describe limiting curves resulting from deviations in partial sums in the ergodic theorem for cylindrical functions and polynomial adic systems. For a general ergodic measurepreserving transformation and a summable function we give a necessary condition for a limiting curve to exist. Our work generalizes results by É. Janvresse, T. de la Rue and Y. Velenik and answers several questions from their work.
Introduction
In this paper we develop the notion of a limiting curve introduced by É. Janvresse, T. de la Rue and Y. Velenik in [16]. Limiting curves were studied for the Pascal adic in [16] and [11]. In this paper we study it for a wider class of adic transformations.
Let T be a measure preserving transformation defined on a Lebesgue probability space (X, B, µ) with an invariant ergodic probability measure µ. Let g denote a function in L 1 (X, µ). Following [16] for a point x ∈ X and a positive integer j we denote the partial sum j−1 k=0 g T k x by S g x (j). We extend the function S g x (j) to a real valued argument by a linear interpolation and denote extended function by F g x (j) or simply F (j), j ≥ 0.
Let (l n ) ∞ n=1 be a sequence of positive integers. We consider continuous on [0, 1] functions ϕ n (t) = F (t·ln(x))−t·F (ln) Rn ≡ ϕ g x,ln (t) , where the normalizing coefficient R n is canonically defined to be equal to the maximum in t ∈ [0, 1] of |F (t · l n (x)) − t · F (l n )|. Definition 1. If there is a sequence l g n (x) ∈ N such that functions ϕ g
x,l g n (x)
converge to a (continuous) function ϕ g x in sup-metric on [0, 1], then the graph of the limiting function ϕ = ϕ g x is called a limiting curve, sequence l n = l g n (x) is called a stabilizing sequence and the sequence R n = R g x,l g n (x) is called a normalizing sequence. The quadruple x, l n ∞ n=1 , R n ∞ n=1 , ϕ is called a limiting bridge.
Heuristically, the limiting curve describes small fluctuations (of certainly renormalized) ergodic sums 1 l F (l), l ∈ (l n ), along the forward trajectory x, T (x), T 2 (x) . . . . More specifically, for l ∈ (l n ) it holds F (t · l) = tF (l) + R l ϕ(t) + o(R l ), where t ∈ [0, 1].
In this paper we will always assume that T is an adic transformation. Adic transformations were introduced into ergodic theory by A. M. Vershik in [1] and were extensively studied since that time. The following important theorem shows that adicity assumption is not restrictive at all:
Theorem. (A. M. Vershik, [2]). Any ergodic measure preserving transformation on a Lebesgue space is isomorphic to some adic transformation. Moreover, one can find such an isomorphism that any given countable dense invariant subalgebra of measurable sets goes over into the algebra of cylinder sets.
In [2,3,5] authors encouraged studying different approaches to combinatorics of Markov's compacts (sets of paths in Bratteli diagrams). In particular, it is interesting to find a natural class of adic transformations such that the limiting bridges exist for cylindric functions. Moreover, it is interesting to study joint growth rates of stabilizing and normalizing sequences.
In this paper we give necessary condition for a limiting curve to exist. Next we find necessary and sufficient conditions for almost sure (in x) existence of limiting curves for a class of self-similar adic transformations and cylindric functions. These transformations (in a slightly less generality) were considered by X. Mela and S. Bailey in [19] and [12]. Our work extends [16] and answers several questions from this research.
Limiting curves and cohomologous to a constant functions
In this section we show that a necessary condition for limiting curves to exist is unbounded growth of the normalizing coefficient R n . Contrariwise we show that normalizing coefficients are bounded if and only if function g is cohomologous to a constant. In particular this implies that there are no limiting curves for cylindric functions for an ordinary odometer.
Notions and definitions
Let B = B(V, E) denote a Bratteli diagram defined by the set of vertices V and the set of edges E. Vertices at the level n are numbered k = 0 through L(n). We associate to a Bratteli diagram B the space X = X(B) of infinite edge paths beginning at the vertex v 0 = (0, 0). Following fundamental paper [1] we assume that there is a linear order ≤ n,k defined on the set of edges with a terminate vertex (n, k), 0 ≤ k ≤ L(n). These linear orders define a lexicographical order on the set of edges paths in X that belong to the same class of the tail partition. We denote by corresponding partial order on X. The set of maximal (minimal) paths is defined by X max (correspondingly, X min ).
Definition 2.
Adic transformation T is defined on X \ X max ∪ X min by setting T x, x ∈ X, equal to the successor of x, that is, the smallest y that satisfies y x.
Let ω be a path in X. We denote by (n, k n (ω)) a vertex through which ω passes at level n. For a finite path c = (c 1 , . . . , c n ) we denote k n (c) simply by k(c). A cylinder set C = [c 1 c 2 . . . c n ] = {ω ∈ X|ω 1 = c 1 , ω 2 = c 2 , . . . , ω n = c n } of a rank n is totally defined by a finite path from the vertex (0, 0) to the vertex (n, k) = (n, k(c)). Sets π n,k of lexicographically ordered finite paths c = (c 0 , c 1 , . . . , c n−1 ), k(c) = n, are in one to one correspondence with towers τ n,k made up of corresponding cylinder sets C j = τ n,k (j), 1 j dim(n, k). The dimension dim(n, k) of the vertex (n, k) is the total number of such finite paths (rungs of the tower).
We denote by Num(c) the number of finite paths in lexicographically ordered set π n,k . Evidently, 1 ≤ Num(c) ≤ dim(n, k). For a given level n the set of towers {τ n,k } 0≤k≤L(n) defines approximation of transformation T , see [1], [3].
We can consider a vertex (n, k) of Bratteli diagram B as an origin in a new diagram B n,k = (V , E ). The set of vertices V , edges E and edges paths X(B n,k ) are naturally defined. As above partial order on X(B ) is induced by linear orders ≤ n ,k , n > n.
Definition 3. Ordered Bratteli diagram (B, ) is self-similar if ordered diagrams (B, ) and (B n,k , n,k ) are isomorphic n ∈ N, 0 k L(n).
Let F denote the set of all functions f : X → R. We denote by F N the space of cylindric functions of rank N (i.e. functions that are constant on cylinders of rank N ).
Let g ∈ F N , N < n. We denote by F g n,k linearly interpolated partial sums S g x∈τ k n (1) . Assume that self-similar Bratteli diagram B has L + 1 vertices at level N and let ω ∈ π n,k , 0 k L(n), be a finite path such that its initial segment ω = (ω 1 , ω 2 , . . . , ω N ) is a maximal path, i.e. Num(ω ) = dim(N, k(ω )). Let E N,l n,k denote the number of paths from (0, 0) to (n, k) passing through the vertex (N, l), 0 ≤ l ≤ L, and not exceeding path ω. We denote by ∂ N,l n,k (ω) the ratio of E N,l n,k to dim(N, l). It is not hard to see that a partial sum F g n,k evaluated at j = Num(ω) has the following expression:
F g n,k (j) = L l=0 h g N,l ∂ N,l n,k (ω),(1)
where coefficients h g N,l are equal to F g N,l (H N,l ), 0 l L. Expression (1) is a generalization of Vandermonde's convolution formula.
A necessary condition for existence of limiting curves
Let (X, T ) be an ergodic measure-preserving transformation with invariant measure µ. Let g be a summable function and a point x ∈ X. We consider a sequence of functions ϕ g x,ln and normalizing coefficients R g x,ln given by the identity
ϕ g x,ln(x) (t) = S g x ([t · l n (x)]) − t · S g x (l n (x)) R g x,ln(x) ,
where R g x,ln(x) equals maximum of absolute value of the numerator. Without loss of generality, we assume that the limit g * (x) = lim n→∞ 1 n S g x exists at the point x. The following theorem generalizes Lemma 2.1 from [16] for an arbitrary summable function. Theorem 1. If a continuous limiting curve ϕ g x = lim n ϕ g x,ln exists for µ-a.e. x, then the normalizing coefficients R g x,ln are unbounded in n.
Proof. Assume the contrary that |R g x,ln | K. For simplicity we introduce the following notation: S = S g x , ϕ n = ϕ g x,ln , R n = R g x,ln and ϕ = ϕ x . Since ϕ = 0, there is j ∈ N such that 1 j S(j) = g * . This in turn implies lim inf n ϕ n ( j ln ) = lim inf n 1 Rn S(j) − jS(ln) (g −g * )•T j of a cohomologous function are µ-a.e. bounded, therefore normalizing coefficients R g x,ln are µ-a.e. bounded too. The proof of the converse statement exploits the result by A. G. Kachurovskiy from [7]. Assume that the normalizing coefficients R g x,ln are bounded. Then for µ-a.e. point x ∈ X and for any j ∈ N the following inequality holds |S g x (j) − j ln S g x (l n )| C. Going to the limit in n, we see that | [7] (see also G. Halasz, [14]), inequality |S f x | C, is equivalent to existence of a function h ∈ L ∞ , such that f = h • T − h. Therefore g equals to h • T − h + g * .
ln 1 K |S(j) − jg * | = j K 1 j S(i) − g * > 0,j i=1 f • T i (x)| C, where f = g − g * . Theorem 19 from
Definition 5. Let B be a Bratteli diagram such that there is only one vertex at each level, and let the edge ordering be such that the edges increase from left to right. This transformation is called an odometer. A stationary odometer is an odometer for which the number of edges connecting consecutive levels is constant.
Theorem 3. Let (X, T ) be an odometer. Any cylindric function g ∈ F N is cohomologous to a constant. Therefore there is no limiting curve for a cylindric function.
Proof. There is only one vertex (n, 0) at each level n. Expression (1) for the partial sum F g n,0 (i) is evidently valid for any odometer (even without assumption of self-similarity). Moreover, expression (1) is defined by the only coefficient h g N,0 and therefore is proportional to H N = dim(N, 0). We can subtract such constant C to the function g that equality h g−C N,0 = 0 holds. But this is equivalent to the following: The function function g − C belongs to the linear space spanned by the functions f j − f j • T, 1 j H N , where f j is the indicator-function of the j-th rung in the tower τ N,0 . Therefore function g − C is cohomologous to zero.
Existence of limiting curves for polynomial adic systems
In this part we will show that any not cohomologous to a constant cylindric function in a polynomial adic system has a limiting curve. These generalizes Theorem 2.4. from [16].
Polynomial adic systems
Let p(x) = a 0 + a 1 x · · · + a d x d be an integer polynomial of degree d ∈ N with positive integer coefficients
a i , 0 ≤ i ≤ d. Bratteli diagram B p = (V, E) p associated to polynomial p(x)
is defined as follows:
1. Number of vertices grows linearly:
|V 0 | = 1 è |V n | = |V n−1 | + d = nd + 1, n ∈ N.
2. If 0 j d vertices (n, k) and (n + 1, k + j) are connected by a j edges.
Polynomial p(x) is called a generating polynomial of the diagram B p , see paper [12] by S. Bailey. Since the number of edges into vertex (n, k) is exactly p(1) = a 0 + a 1 + · · · + a d it is natural to use the alphabet A = {0, 1, . . . , a 0 + a 1 + · · · + a d − 1} for edges labeling. We call a lexicographical order defined in [12] a canonical order. It is defined as follows: Edges connecting (0, 0) with (1, d) are labeled through 0 to a d − 1 (from left to right); edges connecting (0, 0) and (1, d − 1), are indexed by a d to a d + a d−1 − 1, etc. Edges connecting (0, 0) and (1, 0), are indexed through a 0 + a 1 + · · · + a d−1 to a 0 + a 1 + · · · + a d .
Infinite paths are totally defined by this labeling and may be considered as one sided infinite sequences in A N . We denote the path space by X p . We denote by T p the adic transformation associated with the canonical ordering.
Remark. Any self-similar Bratteli diagram is either a diagram of a stationary odometer or is associated to some polynomial p(x). Any noncanonical ordering is obtained from canonical by some substitution σ.
Everywhere below we stick to the canonical order. Case of general order needs several straightforward changes that are left to the reader.
Dimension of the vertex (n, k) from diagram B p equals to the coefficient of x k in the polynomial (p(x)) n and is called generalized binomial coefficient. We denote it by C p (n, k). For n > 1 coefficients C p (n, k) can be evaluated by a recursive expression C p (n, k) = d j=0 a j C d (n − 1, k − j). In [19] and [12] X. Méla and S. Bailey showed that the fully supported invariant ergodic measures of the system (X p , T p ) are the one-parameter family of Bernoulli measures:
Theorem 4. (S. Bailey, [12], X. Méla, [19]) 1. Let q ∈ (0, 1
a 0 ) and t q is the unique solution in (0, 1) to the equation
a 0 q d + a 1 q d−1 t + · · · + a d t d − q d−1 = 0,
then the invariant, fully supported, ergodic probability measures for the adic transformation T p are the one-parameter family of Bernoulli measures µ q , q ∈ (0, 1 a 0 ),
µ q = ∞ 0 q, . . . , q a 0 , t q , . . . , t q a 1 , t 2 q q , . . . , t 2 q q a 2 , . . . , t d q q d−1 , . . . , t d q q d−1 a d .
2.Invariant measures that are not fully supported are
∞ 0 1 a 0 , . . . , 1 a 0 a 0 , 0, . . . , 0 and ∞ 0 0, . . . , 0, 1 a d , . . . , 1 a d a d
, .
Definition 6. Polynomial adic system associated with polynomial p(x), is a triple (X p , T p , µ q ), q ∈ (0, 1 a 0 ). In particular, if p(x) = 1 + x system (X p , T p , µ q ), q ∈ (0, 1), is the wellknown Pascal adic transformation. Transformation was defined in [2] by A. M. Vershik 1 and was studied in many works [20,15,4,6], see more complete list in the last two papers. For the Pascal adic space X p is an infinite dimensional unit cube I = {0, 1} ∞ , while measures µ q are dyadic Bernoulli measures ∞ 1 (q, 1 − q). Transformation T p = P is defined by the following formula (see [2]) 2 :
x → P x; P (0 m−l 1 l 10 . . . ) = 1 l 0 m−l 01 . . .(2)
(that is only first m+2 coordinates of x are being changed). De-Finetti's theorem and Hewitt-Savage 0-1 law imply that all P -invariant ergodic measures are the Bernoulli measures µ p = ∞ 1 (p, 1 − p), where 0 < p < 1. Below we enlist several known properties of the polynomial systems:
1. Polynomial systems are weakly bernoulli (the proof essentially follows [15] and is performed in [19] and [12]).
2. Complexity function has polynomial growth rate (for the Pascal adic first term of asymptotic expansion is known to be equal to n 3 6 , see [20]). 3. Polynomial system (X p , T p , µ q ) defined by a polynomial p(x) = a 0 +a 1 x with a 0 a 1 > 1 has a non-empty set of non-constant eigenfunctions.
Authors of [16] studied limiting curves for the Pascal adic transformation (I, P, µ q ), q ∈ (0, 1).
Theorem 5. ( [16], Theorem 2.4.) Let P be the Pascal adic transformation defined on Lebesgue probability space (I, B, µ q ), q ∈ (0, 1), and g be a cylindric function from F N . Then for µ q -a.e. x limiting curve ϕ g x ∈ C[0, 1] exists if and only if g is not cohomologous to a constant.
For the Pascal adic limiting curves can be described by nowhere differentiable functions, that generalizes Takagi curve.
Theorem 6. ( [11], Theorem 1.) Let P be the Pascal adic transformation defined on the Lebesgue space (I, B, µ q ), N ∈ N and g ∈ F N be a not cohomologous to a constant cylindric function. Then for µ q -a.e. x there is a stabilizing sequence l n (x) such that the limiting function is α g,x T 1 q , where α g,x ∈ {−1, 1}, and T 1 q is given by the identity
T 1 q (x) = ∂F µq ∂q • F −1 µq (x), x ∈ [0, 1],
where F µq is the distribution function 3 of µ q .
The graph of 1 2 T 1 1/2 is the famous Takagi curve, see [22]. For a function g correlated with the indicator functions of i-th coordinate
1 {x i =0} , x = (x j ) ∞ j=1 ∈ X Theorem 6 was proved in [16].
Combinatorics of finite paths in the polynomial adic systems
In this section we'll specify representation (1) for the polynomial adic systems. 3 More precisely Fµ q is distribution function of measureμq, that is image of µq under
canonical mapping φ : I → [0, 1], φ(x) = ∞ i=1 x i 2 i .
For a finite path ω = (ω 1 , ω 2 , . . . , ω n ) we set k 1 (ω) equal to nd − k(ω). Using self-similarity of the diagram B p we can inductively prove the following explicit expression for Num(ω):
Proposition 1. Index Num(ω) of a finite path ω = (ω j ) n j=1
in lexicographically ordered set π n,k(ω) is defined by equality:
Num(ω) = r j=2 ωa j −1 i=0 C P (a j − 1; k 1 (ω) − k 1 (i) − m j ) + Num(ω 1 ),(3)where m j = r−1 t=j k 1 (ω at ), 2 j r −1, m r = 0, and polynomial P (x) is given by the identity P (x) = x d p(x −1 ).
Remark. If initial segment (ω 1 , ω 2 , . . . , ω N ) of ω ∈ π n,k is a maximal path to some vertex (N, l), then (3) can be rewritten as follows:
Num(ω) = r j=N +1 ωa j −1 i=0 C P (a j − 1; k 1 (ω) − k 1 (i) − m j ) + C P (a N ; k 1 (ω) − m l ).
(4) Let N and l, 0 l N d, be positive integers and ω ∈ π n,k be a finite path. Function ∂ N,l k 1 :
Z + → Z + , k 1 = nd − k, is defined by the identity ∂ N,l k 1 ω = r j=N +1 ωa j −1 i=0 C P (a j − 1 − N ; k 1 − k 1 (i) − m j − l),(5)
where positive integers a j , k 1 (i), m j , are defined as in (4). Parameters N and l correspond to shifting the origin vertex (0, 0) to the vertex (N, l). Therefore value of the function ∂ N,l k 1 ω, k 1 = k 1 (ω), equals to the number of paths from the vertex (0, 0) going through the vertex (N, l) to the vertex (n, k), k = nd − k 1 , and non-exceding path ω divided by dim(N, l).
Let K M,n,k , 1 M n, denote indexes (in lexicographical order) of those paths ω = (ω 1 , . . . , ω n ) ∈ π n,k , such that their initial segment (ω 1 , ω 2 , . . . , ω M ) is maximal (as a path from (0, 0) to some vertex (M, l)).
Let g ∈ F N . FunctionF g,M n,k : K M,n,k → R (where M, N M n is a positive integer) is defined by the identitỹ
F g,M n,k (j) = N d l=0 h g M,l ∂ M,l nd−k ω,(6)
where Num(ω) = j, j ∈ K M,n,k , ω ∈ π n,k . We extend domain of the functioñ F g,M n,k to the whole interval [1, H n,k ] using linear interpolation. Expression (1) implies that for j ∈ K M,n,k the identityF g,M n,k (j) = F g n,k (j) holds. Non strictly speaking, higher values of parameter M, M > N, makes functions F g,M n,k to be more and more rough approximation of function F g n,k and points from K M,n,k correspond to nodes of this approximation. Lemma 1. Let 1 j H n,k and g ∈ F N . There exists a constant C = C(g), such that the following inequality holds for all n, k:
|F g,N n,k (j) − F g n,k (j)| C.
Remark. If the function g equals 1 and Num(ω) = dim(n, k) ≡ C P (n, k 1 (ω)), then expression (6) (as well as (1)) reduces to:
C P (n, k) = N d l=0 C P (N, l)C P (n − N, k − l),
that is Vandermonde's convolution formula for generalized binomial coefficients.
A generalized r-adic number system on [0, 1]
Let parameter q ∈ (0, 1/a 0 ) and number t q ∈ (0, 1/a 1 ) be defined as in Theorem 4. We denote by r = p(1) number of letters in the alphabet A.
Let ω = (ω i ) ∞ i=1 ∈ X p , ω i ∈ A, be an infinite path. It is also natural to consider ω as a path in an infinite perfectly balanced tree M r .
Bys n = (s 0 n , . . . s r−1 n ) T we denote r-dimensional vector with j-th, 0 j r − 1, component equal to number of occurrences of letter j among (ω 1 , ω 2 , . . . , ω n ).
Letā i , 0 i r − 1, denote r-dimensional vector (0, 0, . . . , 0 i−1 j=0 a j , 1, 1, . . . , 1 a i , 0, 0, . . . , 0 d j=i+1 a j ),
Let u · v denote scalar product of r-dimensional vectors u and v. We define mapping θ q : X → [0, 1] by the following identity:
x = ∞ j=1 I q (ω j )q j t q q ā 1 ·s j +2ā 2 ·s j +···+dā d ·s j ,(7)
where I q (w) = a 0
q h+1 t h+1 q + a 1 tqq h t h+1 q + · · · + a h q tq + s with w = h i=0 a i + s, 0 s < a h+1 , 0 h < d.
Let X 0 denote the set of stationary paths.Function θ 1/r is a canonical bijection φ = θ 1/r : X \ X 0 → [0, 1] \ G, G = φ(X 0 ). Function φ maps measure µ q , q ∈ (0, 1), defined on X to measureμ q on [0, 1], the family of towers {τ n,k } nd k=0 to the family {τ n,k } nd k=0 of disjunctive intervals. That defines isomorphic realizationT p on [0, 1] \ G of polynomial adic transformation T p .
As shown by A. M. Vershik any adic transformation has a cutting and stacking realization on the subset of a full measure of [0, 1] interval. However, nice explicit expression (7) needs some regularity from the Bratteli diagram.
Conversely, any point x ∈ [0, 1] could be represented by series (7). We call this representation q-r-adic representation associated to the polynomial p(x). (If r = 2 representation (7) for q = 1/2 is a usual dyadic representation of x ∈ (0, 1).) Let G m q denote the set (vector) of all stationary numbers of rang m, m ∈ N, i.e. numbers with a finite representation
x = m j=1 I(ω j ) r−1 i=0 p s i j i ,
and let G q = ∪ m G m q be the set of all q-r-stationary numbers. Let l ∈ N and x be a path in X p . We consider r l -dimensional vectors K n = K n−l,n,kn(x) and renormalization mappings D n,k : [1, C p (n, k)] → [0, 1] defined by D n,k (j) = j Cp(n,k) . Using ergodic theorem it is straightforward to show that for µ q -a.e. x it holds lim n→∞ R n,kn(x) (K n ) = G l q (where convergence is the componentwise convergence of vectors).
Existence of limiting curves for polynomial adic systems
In this part we generalize Theorem 2.4 from [16] for polynomial adic systems (X p , T p ) associated with positive integer polynomial p.
First we prove a combinatorial variant of the theorem. Let, as above, x ∈ X p be an infinite path going through vertices (n, k n (x)) ∈ B p . Below we write vertex (n, k n (x)) as (n, k n ) or simply as (n, k). To simplify notation, the dimension dim(n, k) = C p (n, k) is denoted by H n,k .
We define function ϕ g n,k = ϕ g x∈τ n,k (1),H n,k : [0, 1] → R by identity ϕ g n,k (t) = F g n,k (tH n,k ) − tF g n,k (H n,k ) R g n,k . Let F be a function defined on [1, H n,k ]. Define function ψ F,n,k on [0, 1] by
ψ F (t) = F (tH n,k ) − tF (H n,k ) R n,k ,
where R n,k is a canonically defined normalization coefficient. Then the following identity holds ψ F g n,k = ϕ g n,k . Let g ∈ F N be not cohomologous to a constant cylindric function. Theorem 2 implies that normalization sequence R g n,kn n 1 monotonically increases. Lemma 1 shows that
ψ F g n,k − ψ F g,N n,k ∞ −−−→ n→∞ 0.
We want to show that there is a sequence (n j ) j 1 and a continuous function ϕ(t), t ∈ [0, 1], such that
lim j→∞ ψ F g,N n j ,kn j − ϕ ∞ = 0.
Following [16], we consider an auxiliary object: a family of polygonal
functions ψ M n = ψ F g,n−M +N n,k , N + 1 M n. Graph of each function ψ M n is defined by (2r) M -dimensional array (x M i (n), y M i (n)) r M i=1 , such that ψ M n (x M i (n)) = y M i (n). Results from Section 3.3 show that vector (x M i (n)) r M i=1
converges pointwise to q-r-stationary numbers G M q of rank M, given by polynomial p(x).
Let l and M be positive integers, such that N + 1 l < M < n. Functions F g,n−M n,k and F g,n−l n,k coincide at each point from K n−l,n,kn , therefore functions ψ M n and ψ l n also coincide at (x l i (n)) r l i=1 . Moreover, Proposition 2 (it generalizes Proposition 3.1 from [16]) provides the following estimate:
ψ M n j − ψ l n j ∞ C 1 e −C 2 (M −l) , with C 1 , C 2 > 0.
For a fixed M we can extract a subsequence (n j ) such that polygonal functions ψ M n j converge to a polygonal function ϕ M in sup-metric. Then, as in [16], using a standard diagonalization procedure we can find subsequence (that again will be denoted by (n j ) j ) such that convergence to some continuous on [0, 1] function holds for any M :
lim M →∞ lim sup j→∞ ψ M n j − ϕ ∞ = 0.
Auxiliary functions ϕ M are polygonal approximations to the function ϕ. Therefore we have proved the following claim, generalizing Theorem 5:
Theorem 7. Let (X, T, µ q ), q ∈ (0, 1 a 0 ), be a polynomial adic transformation defined on Lebesgue probability space (I, B, µ q ), q ∈ (0, 1), and g be a not cohomologous to a constant cylindric function from F N . Then for µ q -a.e.
x passing through vertices (n, k n (x)) we can extract a subsequence (n j ) such that ϕ g n j ,kn j (x) converges in sup-metric to a continuous function on [0, 1].
Each limiting curve ϕ is a limit in j of polygonal curves ψ m n j , m 1, with nodes at stationary points G q ⊂ [0, 1]. Therefore, its values ϕ(t) can be obtained as limits lim with Num(ω) ∈ K n j −m,n j ,kn j . Self-similar structure of towers simplifies this task. We write simply n for n j (x), F for F g n,k and R n = R g n,k . The following lemma in fact generalizes results from Section 3.1. of [16]:
Lemma 2.
Limiting curve is totally defined by the following limits n → ∞:
lim n→∞ 1 R n F (L m,i,n,k ) − L m,i,n,k H n,k F (H n,k ) , where L m,i,n,k = m j=0 a j H n−i,k(ω)+j−di , 0 m d.
Proof. We may assume that δ < kn n(x) < δd for some δ > 0. First, we suppose that some typical n = n(x) >> m and k n are taken. We consider a set of ingoing finite paths of length m to the vertex (n, k), d ≤ k ≤ d(n − 1), n >> m. Self-similarity of B p implies that these paths can be considered as paths going from the origin to some vertex (m, j), 0 ≤ j ≤ md, of B p , see Fig. 4. As shown in Section 3.3 above, each such path correspond to a point from G m q , which in its turn correspond to q-r-adic interval of rank m. Let x m,j , 0 j md denote length of such interval and y m,j , denote increment of the function Stochastic version of Theorem 7 is obtained from the following claim: for any ε > 0 for µ q -a.e. x there exists subsequence n j (x) such that Num(w j ), w j = (x 1 , . . . x n j ), satisfies the following condition Num(w j ) H n j ,kn j < ε. In fact, even more strong result holds. It follows from the recurrence property of onedimensional random walk and was first proved by É. Janvresse and T. de la Rue in [15] to show that the Pascal adic transformation is loosely Bernoulli. Later it was generalized in [19,12] for the polynomial adic systems.
Lemma 3. For any ε > 0 and µ q × µ q -a.e. pair of paths (x, y) ∈ X × X there is a subsequence n j such that k n j (x) = k n j (y) and indices Num(ω x ), Num(ω y ) of paths ω x = (x 1 , x 2 , . . . x n j ) and ω y = (y 1 , y 2 , . . . y n j ) satisfy the following inequlity Num(ω z )/H n j ,kn j (x) < ε, z ∈ {x, y}, for each j ∈ N. Proof. Follows from Lemma 3, Theorem 7 and Theorem 2.
Remark Lemma 3 implies that appropriate choice of stabilizing sequence l n (x) can provide the same limiting curve ϕ g x , lim j→∞ ||ϕ g x,l j (x) − ϕ g x || = 0, for µ q -a.e. x.
Finally we prove Proposition 2 used above. It generalizes Proposition 3.1 from [16]. However, its proof needs an additional statement due to the non unimodality of generalized binomial coefficients C p (n, k): Cp(n,k+1) } C 1 n for n > n 1 .
Lemma 4. Let p(x) = a 0 + a 1 x + · · · + a d x2. C p (n − 1, k − i) 1 a i max{ k n , 1 − k n }C p (n, k), 0 i d.
Proof. 1. Let X be a discrete random variable on {0, 1, . . . , d} with distribution associated to the polynomial p(x), that is Prob(X = k) = a k /p(1), 0 k d. Distribution of a sum Y n = X 1 + X 2 + . . . X n of i.i.d. random variables X k , 0 k n, with distributions associated to the polynomial p(x), is associated to the polynomial p n (x), i.e. Prob(Y n = k) = C p (n, k)/p n (1), 0 k nd. A. Oldyzko and L. Richmond showed in [21] that the function f n (k) ≡ Prob(Y n = k) is asymptotically unimodal, i.e. for n ≥ n 1 , coefficients C p (n, k), 0 k nd, n n 1 , first increase (in k) and decrease then.
We denote by C and c the maximum and the minimum values of the coefficients {C p (n 1 , k)} n 1 d k=0 of the polynomial p n 1 (x). Let also a max denote the maximum of the coefficients {a 0 , . . . , a d } of the polynomial p(x). We will use induction in n to prove that Now assume that we have already shown
Cp(n−1,k) Cp(n−1,k−1) Cdamax c (n − 1),
where 1 k d(n − 1) and n ≥ n 1 . We need to show that Cp(n,k) Cp(n,k−1) Cdamax c n, 1 k dn.
C p (n, k + 1) C p (n, k) = d i=0 a i C p (n − 1, k + 1 − i) d i=0 a i C p (n − 1, k − i) C p (n − 1, k) a 0 + a 1 + · · · + a d−1 + da max a d C c (n − 1) a d C p (n − 1, k) a 0 + a 1 + · · · + a d−1 − da max a d + a max
Cdn c a max Cd c n. (8) 2. The statement follows directly from the following identity for the generalized binomial coefficients:
d i=1 C p (n − 1, k − i)a i i = k n C p (n, k).(9)
To show it we differentiate identity p n (x) = k≥0 C p (n, k)x k resulting np n−1 (x)p (x) = k≥0 kC p (n, k)x k−1 , p (x) = a 1 + 2a 2 x + · · · + da d x d−1 . It remains to equate exponents from the two sides.
The following proposition generalizes Propositon 3.1 from [16]. We preserved the original notation where it was possible. Proposition 2. Let N 1 be a positive integer and δ ∈ (0, 1 4 ) be a small parameter. Let A = A(n,k) ∈ B p be a vertex with coordinates n,k satisfying 2δn k (d − 2δ)n and 2δn nd − k (d − 2δ)n. Let α l , 0 l N d, be real numbers, such that N d l=0 α 2 l > 0. Let n, N n n, and B(n, k) = n, k be a vertex with coordinates satisfying 0 k k , 0 n − k n −k. Define
γ n,k = 1 R N d l=0 α l C d (n − N, k − l),(10)
where R = R(A, B, δ) is a renormalization constant such that |γ n,k | are uniformly in n and k from 0 k k , 0 n − k n −k, N n n bounded by 2. Then there exist a constant C = C(δ, N ), such that, providedn is large enough, the following inequality holds for all n, k:
|γ n,k | 3e −C(n−n) .
Conditions on the vertex A δ-separate it from "boundary" vertices (n, 0) and (n, dn). Conditions on the vertex B = B(n, k) provides it can be considered as a vertex in a "flipped" graph and that it can be connected with the vertex A, see Fig. 5. Proof. We can assume thatn > 2n 1 , where n 1 = n 1 (a 0 , a 1 , . . . , a d ), is defined in the proof of Lemma 4. Let l 0 , 0 l 0 N d, be such that coefficient α l 0 is nonzero. We can rewrite the right hand side of (10) as follows:
Rγ n,k = C d (n − N, k − l 0 )P (n, k, l 0 ), N n n, 0 k nd, where P (n, k, l 0 ) is defined by N l=0 α l C d (n−N,k−l) C d (n−N,k−l 0 ) . Let α denote the maximum of |α l |, 0 l N d. We want to show that there is a polynomial Q(x) of degree deg(Q) ≤ N d such that |P (n, k, l 0 ) − P (n,k, l 0 )| Q(n).(11)
It is enough to show that there is
c 1 > 0 such that | C d (n−N,k−l) C d (n−N,k−l 0 ) | c 1 n N d , 0 l N d, N n n.
The latter inequality follows from N d fold application of part 1 of Lemma 4. Define functionQ byQ = P (n, k, l 0 ) − P (n,k, l 0 ). We can write
γ n,k = 1 R C d (n − N, k − l 0 )P (n, k, l 0 ) = = C d (n − N, k − l 0 ) C d (n − N,k − l 0 ) C d (n − N,k − l 0 )P (n, k, l 0 ) R = = C d (n − N, k − l 0 ) C d (n − N,k − l 0 ) C d (n − N,k − l 0 )(P (n,k, l 0 ) +Q) R .(12)
By the assumption we have |γn ,k | = | 1 R P (n,k, l 0 )C d (n − N,k − l 0 )| 2. Therefore inequality (11) can be written as |Q| Q. We get
|γ n,k | 3 Q(n)C d (n − N, k − l 0 ) C d (n − N,k − l 0 ) .
Applying the estimate from part 2 of Lemma 4 (n − n) times and using assumptions on the vertices A and B, we obtain that C d (n−N,k−l 0 )
C d (n−N,k−l 0 )
3e −C(δ)(n−n) for someC(δ) > 0. Finally we get (an independent of the initial choice of l 0 ) estimate:
|γ n,k | 3 Q(n)C d (n − N, k − l 0 ) C d (n − N,k − l 0 ) 3e −C(δ)(n−n)
for some C(δ) > 0.
Examples of limiting curves
Let q 1 and q 2 be two numbers (parameters) from (0, 1). We consider the function S p q 1 ,q 2 : [0, 1] → [0, 1] that maps a number x with q 1 -r-adic repre-
sentation x = ∞ j=1 I q 1 (ω j )q j 1 tq 1 q 1 ā 1 ·s j +2ā 2 ·s j +···+dā d ·s j to S p q 1 ,q 2 (x) = ∞ j=1 I q 2 (ω j )q j 2 t q 2 q 2 ā 1 ·s j +2ā 2 ·s j +···+dā d ·s j .(13)
For any q 1 -r-stationary point x 0 = m j=1 I q 1 (ω j )q j 1 tq 1 q 1 ā 1 ·s j +2ā 2 ·s j +···+dā d ·s j
and any x ∈ [0, 1] the function S p q 1 ,q 2 satisfies the following self-affinity property:
S p q 1 ,q 2 x 0 + r q 1 x = S p q 1 ,q 2 (x 0 ) + r q 2 S p q 1 ,q 2 (x),(14)
where (14) means that the graph of S p q 1 ,q 2 considered on the q-r-adic interval [x 0 , x 0 + r q 1 ] coincides after renormalization with the graph of S p q 1 ,q 2 on the whole interval [0, 1]. Also for q 1 = 1/r function S p 1/r,q 2 is the distribution function of the measureμ q 2 .
r q i = q m i tq i q i ā 1 ·sm+2ā 2 ·sm+···+dā d ·sm , i = 1, 2. Expression
Functions S p q 1 ,q 2 (·) allow us to define new functions
T k p,q 1 := ∂ k S p q 1 ,q 2 ∂q k 2 q 2 =q 1 , k ∈ N.
If k = 0 we will assume that T 0 p,q (x) = x. For q = 1/2 and k = 1 function 1 2 T 1 1+x,1/2 is the Takagi function, see [22]. The function T k p,q 1 on the interval [x 0 , x 0 + r q 1 ] can be expressed by a linear combination of the functions T j p,q 1 , 0 j k. (Expression can be easily obtained by differentiating identity (14) with respect to parameter q 2 and defining q 2 equal to q 1 .)
Theorem 9. Functions T k p,q , q ∈ (0, 1/a 0 ), k ≥ 1, are continuous functions on [0, 1].
Proof. The proof is based on the fact that any two points x and y from the same q-r-adic interval of rank m have the same coordinates (ω 1 , ω 2 , . . . , ω m ) in q-r-adic expansion. This provides a straightforward estimate for the difference |T 1 p,q (x) − T 1 p,q (y)|. Let b = b q denote the ratio t q /q. As shown in Section 3.3 above any x in (0, 1) can be coded by a path ω = (ω i ) ∞ i=1 , ω i ∈ {0, 1 . . . , r − 1} = A, in Figure 6: Graph of the funciton T 1 p,q , defined by polynomial p(x) = 1 + x + 2x 2 with parameter q equal to 1 4 . r-adic (perfectly balanced) tree M r . The function T 1 p,q maps x = x q ∈ [0, 1] with q-r adic series representation
x = ∞ j=1 ω j −1 i=0 b i q j bā 1 ·s j +2ā 2 ·s j +···+dā d ·s j ,
to z = ∂ ∂q x q . Lets j denote the sumā 1 ·s j + 2ā 2 ·s j + · · · + dā d ·s j . Derivative ∂ ∂q (q j b l ) equals to q j−1 b l−1 [(j − l)b + lt q ], where l =s j − ω j + i. Using implicit function theorem we find that
t q = − a 0 dq d−1 + a 1 (d − 1)q d−2 t q + · · · + a d−1 t d−1 q − (d − 1)q d−2 a 1 q d−1 + 2a 2 q d−2 t q + . . . a d t d−1 q d .(15)
Let also a max denote the maximum of the coefficients {a 0 , . . . , a d } of the polynomial p(x). We have |t q | ≤ a max 2d 2 q . Let p max ∈ (0, 1) denote the maximum of {q, t q ,
t 2 q q , . . . , t d q q d−1 }.
Assume y is the left boundary of some q-r-adic interval of rank m, containing point x. Then the following inequality holds (we simply write T for T 1 p,q ):
|T (y) − T (x)| ≤ ∞ j=m ω j −1 i=0 ∂ ∂q q j bs j +i .
Using estimate |q j bs j +i | ≤ (p max ) j , 0 ≤ i ≤ r−1, we see that the absolute value of ∂ ∂q (q j b l ) for j > 2 is estimated by expression P (j, q)(p max ) j−2 , where P (j, q) is some polynomial. Define ε to be equal to 0.99. Then for m large enough it holds:
|T (y) − T (x)| ≤ ∞ j=m ω j −1 i=0 P (j, q)(p max ) j−2 ≤ C(p max ) mε ,(16)
where C is some constant.
In general case we can assume that points x and x + δ are from some q-r-adic interval of rank m = m(δ), lim δ→0 m(δ) = +∞, and let y be the left boundary point of this interval. Then
|T (x + δ) − T (x)| ≤ |T (y) − T (x)| + |T (y) − T (x + δ)| ≤ 2Cp mε max
For k > 1 we can use a similar argument based on the following estimate for the k-th derivative:
| ∂ k ∂q k q j b l | ≤ P k (j, q)(p max ) j−k−1 ,
where j > k and P k (j, q) is some polynomial.
Proposition 3. For a cylindrical function g = − d j=0 ja j 1 {k 1 (x 1 )=j} ∈ F 1
and for µ q -a.e. x there is a stabilizing sequence l n (x) such that the limiting function is T 1 p,q .
Proof. For simplicity we will present the proof for p(x) = 1 + x + x 2 . The general case follows the same steps. Theorem 7 implies that we can find the limiting function ϕ(x) as lim n→∞ ϕ n,k , where (by the law of large numbers) kn n → E µq k 1 . Lemma 2 imply that it is sufficient to show that the function ϕ(x) coincide with T 1 p,q at x = q j and x = q j−1 (q + t q ), where j ∈ N. The function T 1 p,q 1 maps point x = q j to ∂ ∂q q j = jq j−1 and point x = q j−1 (q + t q ) to q j−2 jq + (j − 1)t q + t q q . Using expression (15) we see that
t q = 1−(2q+tq)
2tq+q . Identity (9) implies that h g n,k = k n H n,k . We need to find the following limits for i ∈ N, n → ∞ and kn n → E µq k 1 = 2q + t q (we write F for F n,k ):
1. lim 1 Rn F (H n−i,k−2i ) − H n−i,k−2i H n,k F (H n,k ) 2. lim 1 Rn F (H n−i,k−2i + H n−i,k−2i+1 ) − H n−i,k−2i +H n−i,k−2i+1 H n,k F (H n,k )
We define the normalizing coefficient R n by R n = qH n,k n (2 − E µq k 1 ). After some computations we see that the first limit equals iq i−1 , and the second to q i−2 iq + (i − 1)t q + t q q . These shows that that the limiting function ϕ coincides with the function T 1 p,q 1 on a dense set of q-2-stationary points. Therefore, by Theorem 9 these functions coincide.
Numerical simulations show that limiting functions T k p,q , k 1, and their linear combinations arise as limiting functions lim n→∞ ϕ g n,kn for a general cylindrical function g ∈ F N . We do not have any proof of this statement except for the case of the Pascal adic, see Theorem 6 above. Expression (3) shows that for a cylindrical function g ∈ F N the partial sum F g n,k is defined by the coefficients h g N,k , 0 ≤ k ≤ N d. Its seems to be useful to define h gm N,k by the generating function h gm
N,k = coeff[v m ] (h 0 + h 1 v + · · · + h d v d ) k p(v) N d−k ,
where functions g m forms an orthogonal basis. (For the Pascal adic the function (1 − av) k (1 + v) n−k , a = 1−q q , is the generating function of the Krawtchouk polynomials and the basis g m is the basis of Walsh functions, see [11]).
Limit of limiting curves
In this section we answer the question by É. Janvresse, T. de la Rue and Y. Velenik from [16], page 20, Section 4.3.1. Let q ∈ (0, 1) and t q ∈ (0, 1) be the unique solution in (0, 1) of the equation q d + q d−1 t + · · · + t d = q d−1 .
As above, we denote by b = b q the ratio t q /q. Any x in (0, 1) has an almost unique (d + 1)-adic representation:
x = ∞ j=1 ω j −1 i=0 b i q j b s 1 j +2s 2 j +...ds d j −ω j ,(17)
where ω = (ω i ) ∞ i=1 , ω i ∈ {0, 1 . . . , d} = A, is a path in (d + 1)-adic (perfectly balanced) tree M d+1 and s k j is the number of occurrences of letter k among (ω 1 , ω 2 , . . . , ω j ).
We denote by S q (x) the (anaclitic in parameter q) function defined by (a uniformly summable in x) series (17). We put q * equal to 1/(d + 1) (this is so called symmetric case t q * = q * ). If d = 1 representation (17) for q * = 1/2 is a usual dyadic representation of x ∈ (0, 1).
The authors of [16] were interested in the limiting behavior of the graph of the functionT
d : x → 1 d + 1 ∂S q ∂q q=q *
for large values of d (we also introduced vertical normalization by d + 1, if d = 1 the graph of 1 2 T 1 is the Takagi curve). On the basis of a series of numerical simulations they noticed that limiting curves for d → ∞ seem to converge to a smooth curve. Below we will show that the limiting curve for d = ∞ is actually a parabola, see Fig. 7.
We are going split the unit interval into d+1 subintervals I i = ( i d+1 ; i+1 d+1 ), 0 ≤ i ≤ d, of equal length and evaluate the functionT d at each of the (left) boundary points of these intervals. We also want to show that the functionT d is uniformly in d bounded at these intervals. After that we go to the limit in d.
Symmetry assumption q = t q = q * and implicit function theorem (see (15)) imply that t q * = − 2−d d . In its turn this implies b q = t q − bq q q=q * = − 2(d+1) d . Finally we find that ∂ ∂q (q j b r ) = jq j−1 b r +rb r−1 q j b q q=q * = (q * ) j−1 (j+ rq * (− 2(d+1) d )) = (q * ) j−1 (j − 2r d )). Note that the left boundary point a d of I ad , a ∈ [0, 1], ad ≡ [ad], ([ · ] is an integer part) equals a d = ad d+1 and is coded by the stationary path ω = (ω j ) ∞ j=1 ∈ M d+1 with ω 1 = ad and ω j ≡ 0, j ≥ 2. We havẽ
T d (a d ) = 1 d + 1 da−1 i=0 j− 2(da(j − 1) + i) d = a d (1−a d ) d + 1 d −−−→ d→∞ a d (1−a d ).
This shows that the smooth curve (if exists) should be a parabola.
To complete the proof of the theorem it only remains to show thatT d (x) is uniformly bounded in d at the intervals I ad . Analogously to (16) we see that for x ∈ I ad it holds
|T d (x)−T d (a d )| = 1 d+1 ∞ j=2 q j−1 * ω j −1 i=0 j− 2(s 1 j +2s 2 j +···+ds d j −ω j +i) d ≤ 100d d+1 ∞ j=2 jq j−1 * ≤
contradicting continuity of the limiting curve ϕ at the origin.
Definition 4 .
4A function g ∈ L ∞ (X, µ) (µ-a.e.) of the form g = c+h•T −h for some c ∈ R and h ∈ L ∞ (X, µ) is called cohomologous to a constant in L ∞ .Theorem 2. Normalizing sequence R gx,ln is bounded if and only if function g is cohomologous to a constant.
Figure 1 :
1A Bratteli diagram of an odometer.
Figure 2 :
2Bratteli diagram associated to polynomial 1 + x + 3x 2 .
Figure 3 :
3Labeling of the polynomial system associated to 1 + x + 3x 2 .
ψ M n on the interval (m, j). Values (x m,j , y m,j ) may be defined inductively: For m = 0 by x 0,0 = 1, y 0,0 = 0, and for m > 0 and indices j such that(m − 1)d < j md by x m,j = j i=0 a i Cp(n−m,nd−k+j−di) H n,k , y m,j = ψ M n (x m,j ); for other values of j by recursive expression x m−1,i = d j=0 a j x m,i−j , y m−1,i = d j=0a j y m,i−j . Therefore function ψ M n j is totally defined by its values at x m,j , 1 m M, (m−1)d < j md. Going to the limit we obtain the claim.
Figure 4 :
4For Bratteli diagram B p and graph of ingoing paths.
Theorem 8 .
8(Stochastic variant of Theorem 7.) Let (X, T, µ q ), q ∈ (0, 1 a 0 ), and g be a cylindric function from F N . Then for µ q -a.e. x limiting curve ϕ g x ∈ C[0, 1] exists if and only if function g is not cohomologous to a constant.
d be a positive integer polynomial. Then the following holds:1. There exist n 1 ∈ N and C 1 > 0, depending only on {a 0 , . . . , a d },
,
can be proved in the same way). We start now with the base case: For n = n 1 it obviously holds that 0 k n, hence we have shown the base case.
Figure 5 :
5Vertices A and B in the graph B p .
Figure 7 :
7Limiting curves observed for the polynomial adic transformations (from left to right): d + 1 = 2, 3, 8, 32.
However earlier isomorphic transformation was used by[13] and[17]. 2 P k (x), k ∈ Z, is defined for all x except eventually diagonal, i.e., except those x for which there exists n ∈ N such that either x k = 0 for all k ≥ n or x k = 1 for all k ≥ n
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A M Odlyzko, L B Richmond, On the Unimodality of High Convolutions of Discrete Distributions. 13A. M. Odlyzko, L. B. Richmond, On the Unimodality of High Convolu- tions of Discrete Distributions, Ann. Probab., 13 (1985), 299-306.
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"(k + 1)(k + 2)(2k + 3)M 2 k 3 n−1−k = n(n + 1)(n + 2)M n M n−1",
"(k + 1)(k + 2)(2k + 3)M 2 k 3 n−1−k = n(n + 1)(n + 2)M n M n−1"
] |
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"Zhi-Wei Sun [email protected] \nDepartment of Mathematics\nNanjing University\n210093NanjingPeople's Republic of China\n"
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"Department of Mathematics\nNanjing University\n210093NanjingPeople's Republic of China"
] |
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"Adv. Appl. Math"
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The Motzkin numbers M n =
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10.1016/j.aam.2021.102319
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"https://arxiv.org/pdf/1801.08905v3.pdf"
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1801.08905
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d6224402f7337417550ace43f28d6603be1b3772
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(k + 1)(k + 2)(2k + 3)M 2 k 3 n−1−k = n(n + 1)(n + 2)M n M n−1
1 Feb 2022. 2022
Zhi-Wei Sun [email protected]
Department of Mathematics
Nanjing University
210093NanjingPeople's Republic of China
(k + 1)(k + 2)(2k + 3)M 2 k 3 n−1−k = n(n + 1)(n + 2)M n M n−1
Adv. Appl. Math
1361 Feb 2022. 2022ON MOTZKIN NUMBERS AND CENTRAL TRINOMIAL COEFFICIENTS
The Motzkin numbers M n =
Introduction
In combinatorics, the Motzkin number M n with n ∈ N = {0, 1, 2, . . . } is the number of lattice paths from the point (0, 0) to the point (n, 0) which never dip below the line y = 0 and are made up only of the allowed steps (1, 0) (east), (1, 1) (northeast) and (1, −1) (southeast). It is well known that
M n = ⌊n/2⌋ k=0 n 2k C k
where C k denotes the Catalan number 2k k − 2k k+1 = 2k k /(k + 1).
For n ∈ N, the central trinomial coefficient T n is the constant term in the expansion of (1 + x + x −1 ) n . By the multi-nomial theorem, we see that
T n = ⌊n/2⌋ k=0 n 2k 2k k = n k=0 n k n − k k .
It is known that T n coincides with the number of lattice paths from the point (0, 0) to (n, 0) with only allowed steps (1, 0) (east), (1, 1) (northeast) and (1, −1) (southeast). The Motzkin numbers, the Catalan numbers and the central trinomial coefficients arise naturally in enumerative combinatorics. As the Fibonacci numbers arising from combinatorics have rich number-theoretic properties, we think that important combinatorial quantities like M n and T n with n ∈ N should also have nice arithmetic properties. For example, in [S14a] we conjectured that for any n ∈ Z + = {1, 2, 3, . . . } the arithmetic mean of the n numbers (8k + 5)T 2 k (k = 0, . . . , n − 1) is always an integer, and this was later confirmed by Y.-P. Mu and the author [MS] via symbolic computation. Motivated by congruence properties of such numbers, we found in [S14b, S20] many series for 1/π involving central trinomial coefficients or their extensions. For example, in [S20, Section 10] we conjectured the combinatorial identity where p is a prime greater than 3 and (−) is the Legendre symbol. Thus it is interesting to investigate congruence properties of combinatorial quantities like M n and T n with n ∈ N, and the study in turn may stimulate us to find some new combinatorial identities. Let p > 3 be a prime. In [S14a, Conjecture 1.1(ii)] we conjectured
p−1 k=0 M 2 k ≡ (2 − 6p) p 3 (mod p 2 ), p−1 k=0 kM 2 k ≡ (9p − 1) p 3 (mod p 2 ), and p−1 k=0 T k M k ≡ 4 3 p 3 + p 6 1 − 9 p 3 (mod p 2 ).
The three supercongruences look curious and challenging. Motivated by the above conjectures, we establish the following new results.
n 2 (n 2 − 1) 6 n−1 k=0 k(k + 1)(8k + 9)T k T k+1 . (1.3)
Remark 1.2. If we define t(n) := 6 n 2 (n 2 − 1) n−1 k=0 k(k + 1)(8k + 9)T k T k+1 (n = 2, 3, . . . ), then the values of t(2), t(3), . . . , t(10) are as follows: 51, 271, 1398, 8505, 54387, 367551, 2570931, 18510739, 136282347. Let b, c ∈ Z and n ∈ N. The generalized central trinomial coefficient T n (b, c) denotes the coefficient of x n in the expansion of (x 2 + bx + c) n (cf. [S14a] and [S14b]). By the multi-nomial theorem, we see that
T n (b, c) = ⌊n/2⌋ k=0 n 2k 2k k b n−2k c k .
The generalized Motzkin number M n (b, c) introduced in [S14a] is given by
M n (b, c) = ⌊n/2⌋ k=0 n 2k C k b n−2k c k .
Note that T n (1, 1) = T n , M n (1, 1) = M n , T n (2, 1) = 2n n and M n (2, 1) = C n+1 . Also, T n (3, 2) coincides with the (central) Delannoy number
D n = n k=0 n k n + k k = n k=0 n + k 2k 2k k ,
which counts lattice paths from (0, 0) to (n, n) in which only east (1, 0), north (0, 1), and northeast (1, 1) steps are allowed (cf. R. P. Stanley [St99,p. 185]). And M n (3, 2) equals the little Schröder number
s n+1 = n+1 k=1 N (n + 1, k)2 n+1−k with the Narayana number N (m, k) (m k 1) given by N (m, k) := 1 m m k m k − 1 ∈ Z.
The little Schröder numbers and the Narayana numbers also have many combinatorial interpretations (cf. [St97] and [Gr,). See also [S11, S18b] for some congruences involving the Delannoy numbers or the little Schröder numbers.
Theorem 1.3. Let b, c ∈ Z with b = 0 and d = b 2 − 4c = 0, and let n ∈ Z + . Then b n(n + 1) 2 n k=1 kT k (b, c)T k−1 (b, c)d n−k (1.4) and b n 2 (n + 1) 2 4 3 n k=1 k 3 T k (b, c)T k−1 (b, c)d n−k .
(1.5)
Also,
(2, n) n(n + 1)(n + 2)
n−1 k=0 (k + 1)(k + 2)(2k + 3)M k (b, c) 2 d n−1−k ∈ Z
(1.6) and n−1 k=0 (k + 1)(k + 2)(2k + 3) n(n + 1)(n + 2)
M k (b, c) 2 (−d) n−1−k = M n (b, c)M n−1 (b, c) b ∈ Z, (1.7)
where (m, n) denotes the greatest common divisor of two integers m and n.
Remark 1.3. For each n ∈ Z + , (1.7) with b = c = 1 gives the curious identity n−1 k=0 (k + 1)(k + 2)(2k + 3)M 2 k 3 n−1−k = n(n + 1)(n + 2)M n M n−1 .
(1.8)
In the case b = 3 and c = 2, Theorem 1.3 yields the following consequence.
Corollary 1.1. For any n ∈ Z + we have 3 n(n + 1) 2 n k=1 kD k D k−1 , n 2 (n + 1) 2 4 n k=1 k 3 D k D k−1 , (1.9) n(n + 1)(n + 2) (2, n) n k=1 k(k + 1)(2k + 1)s 2 k , (1.10) and 1 n(n + 1)(n + 2) n k=1 k(k + 1)(2k + 1)(−1) n−k s 2 k = s n s n+1 3 ∈ Z.
(1.11) Theorems 1.1-1.3 are quite sophisticated and their proofs need various techniques. We will prove Theorems 1.1-1.3 in Sections 2-4 respectively. In Section 5 we are going to pose some related conjectures for further research.
Proof of Theorem 1.1
For n ∈ Z + , in [S18b] we introduced the polynomial s n (x) := n k=1 N (n, k)x k−1 (x + 1) n−k (2.1) for which s n (1) is just the little Schróder number s n . For n ∈ N, define S n (x) = n k=0 n k n + k k
x k k + 1 = n k=0 n + k 2k C k x k . (2.2)
Then S n (1) equals the large Schröder number S n which counts the lattice paths from the point (0, 0) to (n, n) with steps (1, 0), (0, 1) and (1, 1) that never rise above the line y = x. As proved in [S18b], we have
S n (x) = (x + 1)s n (x) for all n ∈ Z + . (2.3)
Lemma 2.1. (i) For any n ∈ Z + we have
n(n + 1)s n (x) 2 = n k=1 n + k 2k 2k k 2k k + 1 (x(x + 1)) k−1 . (2.4) (ii) Let b, c ∈ Z with d = b 2 − 4c = 0. For any n ∈ N we have M n (b, c) = ( √ d) n s n+1 b/ √ d − 1 2 .
(2.5)
Proof. As (x + 1)s n (x) = S n (x) by (2.3), the identity (2.4) has the equivalent version
n(n + 1)S n (x) 2 = n k=1 n + k 2k 2k k 2k k + 1 x k−1 (x + 1) k+1
which appeared as [S12a, (2.1)]. So (2.4) holds. The identity (2.5) was proved in [S18b, Lemma 3.1].
Remark 2.1. For n ∈ N and b, c ∈ Z with b 2 = 4c, by combining the two parts of Lemma 2.1 we obtain that
M n (b, c) 2 = 1 (n + 1)(n + 2) n+1 k=1 n + k + 1 2k 2k k 2k k + 1 c k−1 (b 2 − 4c) n+1−k .
(2.6)
Lemma 2.2. For any n ∈ Z + we have n k=1 (2k + 1)M 2 k = n+1 k=0 (4n − 2k + 3)(n + k + 2) n + 2 n + k + 1 2k 2k k 2k + 1 k (−3) n+1−k .
(2.7)
Proof. In view of (2.6), we have
n k=0 (2k + 1)M 2 k = n k=0 2k + 1 (k + 1)(k + 2) k+1 j=1 k + j + 1 2j 2j j 2j j + 1 (−3) k+1−j = n k=0 2k + 1 (k + 1)(k + 2) k l=0 k + l + 2 2l + 2 2l + 2 l + 1 2l + 2 l (−3) k−l = n k=0 n l=0 F (k, l), where F (k, l) := 2k + 1 (k + 1)(k + 2) k + l + 2 2l + 2 2l + 2 l + 1 2l + 2 l (−3) k−l .
By the telescoping method developed by Chen, Hou and Mu [CHM] and applied by Mu and Sun [MS], the double sum can be reduced to a single sum:
n k=0 n l=0 F (k, l) = 1+(4n+3)(−3) n+1 + n j=0 (−3) n−j (4n − 2j + 1)(n + j + 3)!(2j + 3)! (n + 2)(n − j)!(j + 2)(j + 1)! 4 . (2.8) Therefore n k=1 (2k + 1)M 2 k = n j=−1 (−3) n−j (4n − 2j + 1)(n + j + 3)!(2j + 3)! (n + 2)(n − j)!(j + 2)(j + 1)! 4 = n+1 k=0 (−3) n+1−k (4n − 2k + 3)(n + k + 2)!(2k + 1)! (n + 2)(n + 1 − k)!(k + 1)k! 4 = n+1 k=0 (4n − 2k + 3)(n + k + 2) n + 2 n + k + 1 2k 2k k 2k + 1 k (−3) n+1−k
and this concludes the proof.
For each integer n we set
[n] q = 1 − q n 1 − q ,
which is the usual q-analogue of n. For any n ∈ Z, we define
n 0 q = 1 and n k q = k−1 j=0 [n − j] q k j=1 [j] q for k = 1, 2, 3, . . . .
Obviously lim q→1 n k q = n k for all k ∈ N and n ∈ Z. It is easy to see that
n k q = q k n − 1 k q + n − 1 k − 1 q for all k, n = 1, 2, 3, . . . .
By this recursion, n k q ∈ Z[q] for all k, n ∈ N. For any integers a, b and n > 0, clearly
a ≡ b (mod n) =⇒ [a] q ≡ [b] q (mod [n] q ).
Let n be a positive integer. The cyclotomic polynomial
Φ n (q) := n a=1 (a,n)=1 q − e 2πia/n ∈ Z[q]
is irreducible in the ring Z[q]. It is well-known that
q n − 1 = d|n Φ d (q). Note that Φ 1 (q) = q − 1.
Lemma 2.3. For any a, b ∈ N and n ∈ Z + , we have
n−1 k=0 n + 1 k a q n + k k b q 2k k q [k + 2] q (−[3] q ) n−1−k ≡ 0 (mod [n] q ). (2.9)
Proof. (2.9) is trivial in the case n = 1. Below we assume n > 1. As
[n] q = 1<d|n Φ d (q)
and Φ 2 (q), Φ 3 (q), . . . are pairwise coprime, it suffices to show that the sum in (2.9) is divisible by Φ d (q) for any given divisor d > 1 of n.
A well-known q-Lucas theorem (see, e.g., [O]) states that if a, b, d, s, t ∈ N with s < d and t < d then
ad + s bd + t q ≡ a b s t q (mod Φ d (q)).
Let S denote the sum in (2.9) and write n = dm with m ∈ Z + . Then
S = m−1 j=0 d−1 r=0 md + 1 jd + r a q md + jd + r jd + r b q 2jd + 2r jd + r q [jd + r + 2] q (−[3] q ) md−1−(jd+r) ≡ m−1 j=0 d−1 r=0 m j a 1 r a q m + j j b r r b q 2jd + 2r jd + r q [r + 2] q (−[3] q ) (m−j)d−(r+1) ≡ m−1 j=0 m j a m + j j b 1 r=0 2jd + 2r jd + r q [r + 2] q (−[3] q ) (m−j)d−(r+1) ≡ m−1 j=0 m j a m + j j b 2j j 0 0 q [2] q (−[3] q ) (m−j)d−1 + m−1 j=0 m j a m + j j b [1 + 2] q (−[3] q ) (m−j)d−2 × 2j+1 j 0 1 q if d = 2, 2j j 2 1 q if d > 2, ≡0 (mod Φ d (q)).
(Note that [2] q = 1 + q = Φ 2 (q).) This concludes the proof.
Lemma 2.4. For any prime p > 3 we have
p−1 k=1 2k k k3 k ≡ 3 p−1 − 1 p (mod p).
(2.10)
Proof. Let u n = ( n 3 ) for n ∈ N. Then u 0 = 0, u 1 = 1 and u n+1 = −u n − u n−1 for all n = 1, 2, 3, . . . . Applying [S12b, Lemma 3.5] with m = 1, we obtain
p−1 k=1 2k k k3 k ≡ (−3) p−1 − 1 p − 1 2 −3 p u p−( −3 p ) p (mod p). Note that u p−( −3 p ) = 0 since p ≡ ( −3 p ) (mod 3). So (2.10) holds. Proof of Theorem 1.1. (i) Observe that 4 n + 2 ≡ 4/2 = 2 (mod n) if 2 ∤ n, 2/(n/2 + 1) ≡ 2 (mod n) if 2 | n.
Thus, for each k ∈ {1, . . . , n + 1}, we have
2 × 2k k n + 2 = 4 n + 2 2k − 1 k ≡ 2 2k − 1 k = 2k k (mod n).
Combining this with (2.7) we see that
2 n k=1 (2k + 1)M 2 k ≡2(4n + 3)(−3) n+1 + n+1 k=1 (4n − 2k + 3)(n + k + 2) n + k + 1 2k 2k k 2k + 1 k (−3) n+1−k ≡ − n+1 k=0 (2k − 3)(k + 2) n + k + 1 n + 1 n + 1 k 2k + 1 k (−3) n+1−k ≡ − n+1 k=0 (2k − 3)(k + 2) n + k + 1 n + 1 n + k k n + 1 k (2k + 1)C k (−3) n+1−k ≡ − n+1 k=0 (2k − 3)(k + 2)(k + 1) n + k k n + 1 k (2k + 1)C k (−3) n+1−k (mod n).
For each k = 0, . . . , n + 1, clearly k(k − 1) n + 1 k = n(n + 1) n − 1 n + 1 − k ≡ 0 (mod n).
Since (2k − 3)(2k + 1) = 4k(k − 1) − 3, by the above we have
2 n k=1 (2k + 1)M 2 k ≡ − n+1 k=0 n + 1 k n + k k 2k k (k + 2)(−3) n+2−k (mod n).
Note that n+1 k=n n + 1 k
n + k k 2k k (k + 2)(−3) n+2−k = n + 1 n 2n n 2 (n + 2)(−3) 2 + 2n + 1 n + 1 2n + 2 n + 1 (n + 3)(−3) ≡18 2n n 2 − 18 2n + 1 n + 1 2n n 2 ≡ 0 (mod n). Therefore 2 n k=1 (2k + 1)M 2 k ≡ 27 n−1 k=0 n + 1 k n + k k 2k k (k + 2)(−3) n−1−k (mod n).
(2.11) By (2.9) with a = b = 1 and q = 1, we have
n−1 k=0 n + 1 k n + k k 2k k (k + 2)(−3) n−1−k ≡ 0 (mod n).
Combining this with (2.11) we immediately obtain the desired (1.1).
(ii) Applying (2.7) with n = p − 1, we get
p−1 k=1 (2k + 1)M 2 k = p k=0 (4p − 2k − 1)(p + k + 1) p + 1 p + k 2k 2k k 2k + 1 k (−3) p−k = p−1 k=1 (4p − 2k − 1)(p + k + 1) p + 1 p k p + k k 2k + 1 k + 1 2k k (−3) p−k + (4p − 1)(−3) p + (2p − 1)(2p + 1) p + 1 2p p 2p + 1 p + 1 2p p ≡3 p−1 k=1 p k p − 1 k − 1 (2k + 1) 2 2k k (−3) k + (3 − 12p)3 p−1 − 2 2p − 1 p − 1 2 ≡ − 3p p−1 k=1 4k + 4 + 1 k 2k k 3 k + 3 p − 12p − 4 (mod p 2 )
with the aid of Wolstenholme's congruence 2p−1 p−1 ≡ 1 (mod p 3 ) (cf. [W]). Com-bining this with (2.10) and noting that −1/2
k = 2k k /(−4) k for k ∈ N, we obtain − 1 12p p−1 k=0 (2k + 1)M 2 k ≡1 + p−1 k=1 (k + 1) −1/2 k (−4) k 3 k ≡ (p−1)/2 k=0 (p − 1)/2 k − 4 3 k + (p−1)/2 k=1 k (p − 1)/2 k − 4 3 k ≡ 1 − 4 3 (p−1)/2 − 4 3 · p − 1 2 (p−1)/2 k=1 (p − 3)/2 k − 1 − 4 3 k−1 ≡ −3 p + 2 3 1 − 4 3 (p−3)/2 ≡ −3 p − 2 −3 p = − p 3 (mod p).
This proves (1.2). The proof of Theorem 1.1 is now complete. and
T k (b, c) 2 = k j=0 k + j 2j 2j j 2 c j d k−j for all k ∈ N. (3.2)
Remark 3.1. For (3.1) and (3.2), see [S14a, (1.19) and (4.1)].
Lemma 3.2. For any n ∈ Z + , we have n−1 k=0 k(k + 1)(8k + 9)T k T k+1 = (−1) n n 6 n−1 k=0 n − 1 k −n − 1 k C k 3 n−1−k a(n, k),
(3.3) where a(n, k) = 4k 2 n 2 − 8kn 3 − 14k 2 n − 14kn 2 − 4n 3 + 13k 2 − 11kn − 26n 2 + 39k + 4n + 26.
Proof. In light of (3.1) with b = c = 1,
n−1 k=0 k(k + 1)(8k + 9)T k T k+1 = n k=1 k(k − 1)(8k + 1)T k T k−1 = n k=1 (k − 1)(8k + 1) k−1 j=0 (2j + 1)T 2 j 3 k−1−j = n−1 j=0 (2j + 1)T 2 j n k=j+1 (k − 1)(8k + 1)3 k−1−j . By induction, for each j ∈ N we have m k=j+1 (k − 1)(8k + 1)3 k−1−j = 1 4 3 m−j (16m 2 − 30m + 21) − (16j 2 − 30j + 21)
for all m = j + 1, j + 2, . . . . Thus, in view of the above and (3.2) with b = c = 1, we get 4 n−1 k=0 k(k + 1)(8k + 9)T k T k+1 = n k=0 (2k + 1)T 2 k 3 n−k (16n 2 − 30n + 21) − (16k 2 − 30k + 21) = n k=0 n l=0 F (k, l),
where F (k, l) denotes (2k + 1) k + l 2l 2l l 2 (−3) k−l 3 n−k (16n 2 − 30n + 21) − (16k 2 − 30k + 21) .
Via the telescoping method stated in [CHM, MS], the double sum can be reduced to a single sum: Lemma 3.3. For any n ∈ Z + , we have
n 2 − 1 n−1 k=0 n − 1 k −n − 1 k C k 3 n−1−k a(n, k) (3.5)
with a(n, k) given in Lemma 3.2.
Proof. It suffices to show that n 2 − 1 divides n−1 k −n−1 k a(n, k) for any fixed k ∈ {0, . . . , n − 1}. Clearly, a(n, k) ≡4k 2 − 8kn − 14k 2 n − 14k − 4n + 13k 2 − 11kn − 26 + 39k + 4n + 26 =k 2 (17 − 14n) + k(25 − 19n) (mod n 2 − 1),
and (±n − 1) | k ±n−1 k since k ±n−1 k = (±n − 1) ±n−2 k−1 if k > 0. So n − 1 k −n − 1 k a(n, k) ≡ n − 1 k −n − 1 k k(25 − 19n) (mod n 2 − 1).
If 2 ∤ n, then n ± 1 and 25 − 19n are all even, hence both 2(n − 1) and 2(n + 1) divide n−1 k −n−1 k a(n, k). If n is even, then (n − 1, n + 1) = (n − 1, 2) = 1 and hence n 2 − 1 coincides with the least common multiple [n − 1, n + 1] of n − 1 and n + 1. Note that when n is odd we have (2, n − 1) = 2 and [2(n − 1), 2(n + 1)] = 2(n − 1)2(n + 1) (2(n − 1), 2(n + 1)) = 4(n 2 − 1) 2(n − 1, 2) = n 2 − 1.
Therefore n 2 − 1 | n−1 k −n−1 k a(n, k) no matter n is odd or even. This concludes the proof.
Lemma 3.4. Let a, b ∈ N with a + b even, and let n ∈ Z + . Then
2n n−1 k=0 n − 1 k a −n − 1 k b 2k k (k + 2)3 n−1−k . (3.6)
Proof. Let f (k) = k 2k−1 k 3 n−k for k = 0, . . . , n. For each k = 0, . . . , n − 1, we clearly have
∆f (k) =f (k + 1) − f (k) = (k + 1) 2k + 1 k + 1 3 n−k−1 − k 2k − 1 k 3 n−k =(2k + 1) 2k k 3 n−k−1 − 3k 2k − 1 k 3 n−1−k = k + 2 2 2k k 3 n−1−k .
Thus, by [S18a,Theorem 4.1] we get
n−1 k=0 n − 1 k a −n − 1 k b k + 2 2 2k k 3 n−1−k = n−1 k=0 n − 1 k a −n − 1 k b ∆f (k) ≡ 0 (mod n)
and hence (3.6) holds.
Proof of Theorem 1.2. Since (n, n 2 − 1) = 1, by Lemmas 3.2 and 3.3 it suffices to show that
n−1 k=0 n − 1 k −n − 1 k C k 3 n−1−k a(n, k) ≡ 0 (mod n).
For each k = 0, . . . , n − 1, clearly a(n, k) ≡ 13k 2 + 39k + 26 = 13(k + 1)(k + 2) (mod n).
So n−1 k=0 n − 1 k −n − 1 k C k 3 n−1−k a(n, k) ≡13 n−1 k=0 n − 1 k −n − 1 k 2k k (k + 2)3 n−1−k ≡ 0 (mod n).
with the help of Lemma 3.4. This completes the proof.
Proof of Theorem 1.3
Lemma 4.1. Let b, c ∈ Z and d = b 2 − 4c. For any n ∈ Z + we have
nT n (b, c)T n−1 (b, c) = b n−1 j=0 (n − j) n + j 2j 2j j 2 c j d n−1−j . (4.1)
Proof. In view of Lemma 3.1,
nT n (b, c)T n−1 (b, c) =b n−1 k=0 (2k + 1) k j=0 k + j 2j 2j j 2 c j d k−j (−d) n−1−k =b n−1 j=0 2j j 2 c j d n−1−j n−1 k=j (−1) n−1−k (2k + 1) k + j 2j .
For each j ∈ N, by induction we have
m−1 k=j (−1) m−1−k (2k + 1) k + j 2j = (m − j) m + j 2j
for all m = j + 1, j + 2, . . . .
(4.2) Thus
nT n (b, c)T n−1 (b, c) = b n−1 j=0 2j j 2 c j d n−1−j (n − j) n + j 2j
and hence (4.1) holds.
Lemma 4.2. For any k, n ∈ Z + with k n, we have n(n + 1)(n + 2) (2, n) (n + k + 1) n + k k
n + 1 k + 1 2k k + 1 . (4.3)
Proof. Clearly,
(n + k + 1) n + k k n + 1 k + 1 2k k + 1 =(n + k + 1) n + k k n + 1 k + 1 n k kC k =(n + 1) n + k + 1 k + 1 n n − 1 n − k C k ,
and also
(n + k + 1) n + k k 2k k + 1 ≡ (k − 1)(−1) k −n − 1 k kC k ≡ 0 (mod n + 2) since k(k − 1) −n − 1 k = (−n − 1)(−n − 2) −n − 3 k − 2 if k > 1.
Thus
[n(n + 1), n + 2] (n + k + 1) n + k k n + 1 k + 1 2k k + 1 .
Note that
[n(n + 1), n + 2] = n(n + 1)(n + 2) (n(n + 1), n + 2) = n(n + 1)(n + 2) (2, n) .
So we have (4.3).
Lemma 4.3. For any n ∈ N we have 6 2n n ≡ 0 (mod n + 2).
(4.4)
Proof. Observe that 2n + 2 n + 1 = 2 2n + 1 n = 2(2n + 1) n + 1 2n n and hence 2(2n + 1) 2n n = (n + 1) 2n + 2 n + 1 = (n + 1)(n + 2)C n+1 .
Thus n + 2 (n + 2, 2n + 1) 2n + 1 (n + 2, 2n + 1) 2 2n n and hence n + 2 (n + 2, 2n + 1) 2 2n n .
(4.5)
Since (n + 2, 2n + 1) = (n + 2, 2(n + 2) − 3) = (n + 2, 3) divides 3, we obtain (4.4) from (4.5).
As in [S18b], for n ∈ Z + we define
w n (x) := n k=1 w(n, k)x k−1 with w(n, k) = 1 k n − 1 k − 1 n + k k − 1 ∈ Z.
Lemma 4.4. For any integers n k 1, we have
w(n, k) = k j=1 n − j k − j N (n, j) (4.6)
and
N (n, k) = k j=1 n − j k − j (−1) k−j w(n, j). (4.7)
Proof. We first prove (4.7). Observe that
k j=1 n − j k − 1 (−1) k−j w(n, j) = k j=1 n − j k − j (−1) k−j n n j n + j j − 1 = (−1) k−1 n n k k j=1 k k − j −n − 2 j − 1 .
Thus, with the help of the Chu-Vandermonde identity (cf. [G, (3
.1)]), we get k j=1 n − j k − 1 (−1) k−j w(n, j) = (−1) k−1 n n k k − n − 2 k − 1 = N (n, k).
This proves (4.7). In view of (4.7), we have
k j=1 n − j k − j N (n, j) = k j=1 n − j k − j j i=1 n − i j − i (−1) j−i w(n, i) = k i=1 w(n, i) n − i k − i k j=i k − i j − i (−1) j−i = w(n, k).
So (4.6) also holds. This ends the proof.
Lemma 4.5. For any n ∈ Z + we have w n (x) = s n (x).
(4.8)
Proof. With the aid of (4.7), we get
s n (x) = n k=1 N (n, k)x k−1 (x + 1) n−k = n k=1 k j=1 n − j k − j (−1) k−j w(n, j)x k−1 (x + 1) n−k = n j=1 w(n, j)x n−1 n k=j n − j k − j (−1) k−j 1 + 1 x n−j−(k−j) = n j=1 w(n, j)x n−1 1 + 1 x − 1 n−j = w n (x).
This concludes the proof.
Lemma 4.6. For any n ∈ Z + we have the new identity (2x + 1) n k=1 k(k + 1)(2k + 1)(−1) n−k w k (x) 2 = n(n + 1)(n + 2)w n (x)w n+1 (x). (4.9)
Proof. In the case n = 1, both sides of (4.9) are equal to 6(2x + 1). Now assume that (4.9) holds for a fixed positive integer n. Applying the Zeilberger algorithm (cf. [PWZ,) via Mathematica 9 we find that (n + 3)w n+2 (x) = (2x + 1)(2n + 3)w n+1 (x) − nw n (x). Thus (2x + 1) n+1 k=1 k(k + 1)(2k + 1)(−1) n+1−k w k (x) 2 =(2x + 1)(n + 1)(n + 2)(2n + 3)w n+1 (x) 2 − (2x + 1) n k=1 k(k + 1)(2k + 1)(−1) n−k w k (x) 2 =(2x + 1)(n + 1)(n + 2)(2n + 3)w n+1 (x) 2 − n(n + 1)(n + 2)w n (x)w n+1 (x) =(n + 1)(n + 2)w n+1 (x)((2x + 1)(2n + 3)w n+1 (x) − nw n (x)) =(n + 1)(n + 2)(n + 3)w n+1 (x)w n+2 (x).
In view of the above, by induction, (4.9) holds for each n ∈ Z + .
Proof of Theorem 1.3. (i) Let δ ∈ {0, 1}. In light of Lemma 4.1,
n k=1 k 2δ+1 T k (b, c)T k−1 (b, c)d n−k = n k=1 k 2δ b k−1 j=0 (k − j) k + j 2j 2j j 2 c j d k−1−j d n−k =b n−1 j=0 2j j 2 c j d n−1−j n k=j+1 k 2δ (k − j) k + j 2j . By induction, for each j ∈ N, we have m k=j+1 k 2δ (k − j) k + j 2j = m δ (m + 1) δ 2 · (m − j)(m + j + 1) j + δ + 1 m + j 2j (4.10)
for every m = j + 1, j + 2, . . . . Therefore,
n k=1 k 2δ+1 T k (b, c)T k−1 (b, c)d n−k =b n δ (n + 1) δ 2 n−1 j=0
2j j 2 c j d n−1−j (n − j)(n + j + 1) j + δ + 1 n + j 2j = b 2 (n(n + 1)) δ n−1 j=0 2j j j + δ + 1 c j d n−1−j (n − j)(n + j + 1) n j n + j j and hence n k=1
k 2δ+1 T k (b, c)T k−1 (b, c)d n−k = b 2 (n(n + 1)) δ+1 n−1 j=0 n − 1 j n + j + 1 j 2j j j + δ + 1 c j d n−1−j .
(4.11)
In the case δ = 0, (4.11) yields (1.4) since 2j j /(j + 1) = C j ∈ Z. By Lemma 4.3 and (4.11) with δ = 1, we immediately obtain (1.5).
(ii) By induction, for each j ∈ N we have m k=j (2k + 1) k + j 2j = (m + 1)(m + j + 1) j + 1 m + j 2j for all m = j, j + 1, . . . .
(4.12)
In view of this and (2.4), we have
n k=1 k(k + 1)(2k + 1)s k (x) 2 = n k=1 (2k + 1) k j=1 k + j 2j 2j j 2j j + 1 (x(x + 1)) j−1 = n j=1 2j j 2j j + 1 (x(x + 1)) j−1 n k=j (2k + 1) k + j 2j = n j=1
2j j 2j j + 1 (x(x + 1)) j−1 (n + 1)(n + j + 1) j + 1 n + j 2j = n j=1 2j j + 1 (x(x + 1)) j−1 (n + 1)(n + j + 1) j + 1 n j n + j j and hence n k=1 k(k + 1)(2k + 1)s k (x) 2 = n k=1 (n + k + 1) n + 1 k + 1 n + k k 2k k + 1 (x(x + 1)) k−1 .
(4.13)
Let x = (b/ √ d − 1)/2
. Then x(x + 1) = c/d. In view of Lemma 2.1(ii) and (4.13), we have
n−1 k=0 (k + 1)(k + 2)(2k + 3)M k (b, c) 2 d n−1−k = n−1 k=0 (k + 1)(k + 2)(2k + 3)d k s k+1 (x) 2 d n−1−k =d n−1 n k=1 k(k + 1)(2k + 1)s k (x) 2 = n k=1 (n + k + 1) n + 1 k + 1 n + k k 2k k + 1 c k−1 d n−k .
Combining this with Lemma 4.2, we get the desired (1.6).
In light of Lemma 2.1(ii) and Lemmas 4.5-4.6, we have
n−1 k=0 (k + 1)(k + 2)(2k + 3)M k (b, c) 2 (−d) n−1−k = n−1 k=0 (k + 1)(k + 2)(2k + 3)d k s k+1 (x) 2 (−d) n−1−k =d n−1 n k=1 k(k + 1)(2k + 1)(−1) n−k w k (x) 2 =n(n + 1)(n + 2)d n−1 s n (x)s n+1 (x) 2x + 1 =n(n + 1)(n + 2)d n−1 M n−1 (b, c) √ d n−1 · M n (b, c) √ d n · √ d b =n(n + 1)(n + 2) M n (b, c)M n−1 (b, c) b . If 2 ∤ n then b | M n (b, c); if 2 | n then 2 ∤ n − 1 and b | M n−1 (b, c). So b divides M n (b, c)M n−1 (b, c)
. Therefore (1.7) holds. The proof of Theorem 1.3 is now complete.
Some open problems
Clearly,
2k k 2k − 1 = 2 2k − 1 2k − 1 k = 2 k 2k − 2 k − 1 = 2C k−1 for k ∈ Z + ,
and thus 2k − 1 | 2k k for all k ∈ N. Motivated by this we introduce a new kind of numbers (ii) For any prime p > 3 and positive integer n, the number
pn−1 k=0 W 2 k − 2( n−1 k=0 T k ) 2 pn
is always a p-adic integer.
Remark 5.1. We also guess that the sequence (W n+1 /W n ) n 5 is strictly increasing to the limit 3 and the sequence ( n+1 W n+1 / n √ W n ) n 9 is strictly decreasing to the limit 1.
For h, n ∈ Z + , we define w (h) n (x) := n k=1 w(n, k) h x k−1 .
Conjecture 5.2. Let h, m, n ∈ Z + . Then
(2, n) n(n + 1)(n + 2) n k=1 k(k + 1)(2k + 1)w
(h) k (x) m ∈ Z[x].
(5.4) Also, (2, m − 1, n) n(n + 1)(n + 2) n k=1 (−1) k k(k + 1)(2k + 1)w k (x) m ∈ Z[x], (5.5) and 1 n(n + 1)(n + 2) n k=1 (−1) k k(k + 1)(2k + 1)w
(h) k (x) m ∈ Z[x] for h > 1.
(5.6)
Remark 5.2. Fix n ∈ Z + . By combining (4.13) with Lemma 4.2, we obtain (2, n) n(n + 1)(n + 2) n k=1 k(k + 1)(2k + 1)s k (x) 2 ∈ Z[x(x + 1)].
(5.7)
As s k (x) = w k (x) for all k ∈ Z + (by Lemma 4.5), this implies (5.4) with h = 1 and m = 2. Since w 2j (x)/(2x + 1) ∈ Z[x] for all j ∈ Z + (cf. [S18b, Section 4]), (5.5) with m = 2 follows from (4.9).
For h ∈ Z + and n ∈ N, we define Note that S
(1) n (x) = S n (x) for all n ∈ N.
Conjecture 5.3. Let h, m, n ∈ Z + . (i) We have
(2, n) n(n + 1)(n + 2) n k=1 k(k + 1)(2k + 1)S
(h) k (x) m ∈ Z[x]
(5.8)
and
(2, m − 1, n) n(n + 1)(n + 2) n k=1 (−1) k k(k + 1)(2k + 1)S
(h) k (x) m ∈ Z[x].
(5.9) (ii) We have
(2, n) n(n + 1)(n + 2) n k=1 k(k + 1)(2k + 1)D
(h) k (x) m ∈ Z[x]
and
(2, hm − 1, n) n(n + 1)(n + 2) n k=1 (−1) k k(k + 1)(2k + 1)D
(h) k (x) m ∈ Z[x].
Remark 5.3. Fix n ∈ Z + . As S k (x) = (x + 1)s k (x) = (x + 1)w k (x) for all k ∈ Z + , (5.8) and (5.9) with h = 1 and m = 2 do hold in view of Remark 5.2. We also conjecture that 2 3n(n + 1) n k=1 (−1) n−k k 2 D k D k−1 and 1 n n k=1 (−1) n−k (4k 2 + 2k − 1)D k−1 s k are positive odd integers.
Conjecture 5.4. (i) For any h, m, n ∈ Z + we have 2(2, n) n(n + 1)(n + 2) n k=1 k(k + 1)(k + 2)(w
(h) k (x)w (h) k+1 (x)) m ∈ Z[x]
(5.10) (ii) For any m, n ∈ Z + we have 2(2, n) n(n + 1)(n + 2)(2x + 1) m n k=1 k(k + 1)(k + 2)(w k (x)w k+1 (x)) m ∈ Z[x]. (5.11) If n ∈ Z + is even, then 4 n(n + 1)(n + 2)(2x + 1) 3 n k=1 k(k + 1)(k + 2)w k (x)w k+1 (x) ∈ Z[x].
(5.12)
Remark 5.4. Recall that w 2j (x)/(2x + 1) ∈ Z[x] for all j ∈ Z + (by [S18b, Section 4]).
. 1 .
1Let b, c ∈ Z and d = b 2 − 4c. T k (b, c) 2 (−d) n−1−k = nT n (b, c)T n−1 (b, c)for any n ∈ Z + , (3.1)
3 n−1−k a(n, k) and hence (3.3) holds.
k and S (h) n (x) := n k=0 n + k 2k h C h k x k .
Theorem 1.1. (i) For any n ∈ Z + , we have Theorem 1.2. For any integer n 2, we haves(n) :=
2
n
n
k=1
(2k + 1)M 2
k ∈ Z.
(1.1)
(ii) For any prime p > 3, we have
p−1
k=0
(2k + 1)M 2
k ≡ 12p
p
3
(mod p 2 ).
(1.2)
Remark 1.1. The values of s(1), . . . , s(10) are as follows:
6, 23, 90, 432, 2286, 13176, 80418, 513764, 3400518, 23167311.
W n := which are analogues of the Motzkin numbers. The values of W 0 , W 1 , . . . , W 12 are as follows:Applying the Zeilberger algorithm (cf.[PWZ,) via Mathematica 9, we obtain the recurrence (n + 3)W n+3 = (3n + 7)W n+2 + (n − 5)W n+1 − 3(n + 1)W n (n = 0, 1, 2, . . . ).For this new kind of numbers, we have the following conjecture similar to Theorem 1.1.Conjecture 5.1. (i) For any n ∈ Z + we have Also, for any odd prime p we have⌊n/2⌋
k=0
n
2k
2k
k
2k − 1
(n = 0, 1, 2, . . . )
(5.1)
−1, −1, 1, 5, 13, 29, 63, 139, 317, 749, 1827, 4575, 11699.
n−1
k=0
(8k + 9)W 2
k ≡ n (mod 2n).
(5.2)
1
p
p−1
k=0
(8k + 9)W 2
k ≡ 24 + 10
−1
p
− 9
p
3
− 18
3
p
(mod p).
(5.3)
Acknowledgments. The author would like to thank Prof. Qing-Hu Hou and the anonymous referee for their helpful comments.
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"Learning Visual Representations with Optimum-Path Forest and its Applications to Barrett's Esophagus and Adenocarcinoma Diagnosis",
"Learning Visual Representations with Optimum-Path Forest and its Applications to Barrett's Esophagus and Adenocarcinoma Diagnosis"
] |
[
"Luis A · De Souza [email protected] \nDepartment of Computing\nMedizinische Klinik III\nDepartment of Computing\nRegensburg Medical Image Computing (ReMIC), Ostbay-erische Technische Hochschule Regensburg -OTH Regens-burg\nFederal University of São Carlos -UFScar\nSão Paulo State University -UN-ESPBrazil, Klinikum Augsburg -Germany, Germany, Brazil\n",
"C S Luis \nDepartment of Computing\nMedizinische Klinik III\nDepartment of Computing\nRegensburg Medical Image Computing (ReMIC), Ostbay-erische Technische Hochschule Regensburg -OTH Regens-burg\nFederal University of São Carlos -UFScar\nSão Paulo State University -UN-ESPBrazil, Klinikum Augsburg -Germany, Germany, Brazil\n",
"Afonso \nDepartment of Computing\nMedizinische Klinik III\nDepartment of Computing\nRegensburg Medical Image Computing (ReMIC), Ostbay-erische Technische Hochschule Regensburg -OTH Regens-burg\nFederal University of São Carlos -UFScar\nSão Paulo State University -UN-ESPBrazil, Klinikum Augsburg -Germany, Germany, Brazil\n",
"Alanna Ebigbo [email protected] \nDepartment of Computing\nMedizinische Klinik III\nDepartment of Computing\nRegensburg Medical Image Computing (ReMIC), Ostbay-erische Technische Hochschule Regensburg -OTH Regens-burg\nFederal University of São Carlos -UFScar\nSão Paulo State University -UN-ESPBrazil, Klinikum Augsburg -Germany, Germany, Brazil\n",
"Andreas Probst [email protected] \nDepartment of Computing\nMedizinische Klinik III\nDepartment of Computing\nRegensburg Medical Image Computing (ReMIC), Ostbay-erische Technische Hochschule Regensburg -OTH Regens-burg\nFederal University of São Carlos -UFScar\nSão Paulo State University -UN-ESPBrazil, Klinikum Augsburg -Germany, Germany, Brazil\n",
"Helmut Messmann [email protected] \nDepartment of Computing\nMedizinische Klinik III\nDepartment of Computing\nRegensburg Medical Image Computing (ReMIC), Ostbay-erische Technische Hochschule Regensburg -OTH Regens-burg\nFederal University of São Carlos -UFScar\nSão Paulo State University -UN-ESPBrazil, Klinikum Augsburg -Germany, Germany, Brazil\n",
"· Robert \nDepartment of Computing\nMedizinische Klinik III\nDepartment of Computing\nRegensburg Medical Image Computing (ReMIC), Ostbay-erische Technische Hochschule Regensburg -OTH Regens-burg\nFederal University of São Carlos -UFScar\nSão Paulo State University -UN-ESPBrazil, Klinikum Augsburg -Germany, Germany, Brazil\n",
"Mendel \nDepartment of Computing\nMedizinische Klinik III\nDepartment of Computing\nRegensburg Medical Image Computing (ReMIC), Ostbay-erische Technische Hochschule Regensburg -OTH Regens-burg\nFederal University of São Carlos -UFScar\nSão Paulo State University -UN-ESPBrazil, Klinikum Augsburg -Germany, Germany, Brazil\n",
"Christian Hook [email protected] \nDepartment of Computing\nMedizinische Klinik III\nDepartment of Computing\nRegensburg Medical Image Computing (ReMIC), Ostbay-erische Technische Hochschule Regensburg -OTH Regens-burg\nFederal University of São Carlos -UFScar\nSão Paulo State University -UN-ESPBrazil, Klinikum Augsburg -Germany, Germany, Brazil\n",
"Christoph Palm [email protected] \nDepartment of Computing\nMedizinische Klinik III\nDepartment of Computing\nRegensburg Medical Image Computing (ReMIC), Ostbay-erische Technische Hochschule Regensburg -OTH Regens-burg\nFederal University of São Carlos -UFScar\nSão Paulo State University -UN-ESPBrazil, Klinikum Augsburg -Germany, Germany, Brazil\n",
"João P Papa \nDepartment of Computing\nMedizinische Klinik III\nDepartment of Computing\nRegensburg Medical Image Computing (ReMIC), Ostbay-erische Technische Hochschule Regensburg -OTH Regens-burg\nFederal University of São Carlos -UFScar\nSão Paulo State University -UN-ESPBrazil, Klinikum Augsburg -Germany, Germany, Brazil\n",
"Luis A De Souza Jr\nDepartment of Computing\nMedizinische Klinik III\nDepartment of Computing\nRegensburg Medical Image Computing (ReMIC), Ostbay-erische Technische Hochschule Regensburg -OTH Regens-burg\nFederal University of São Carlos -UFScar\nSão Paulo State University -UN-ESPBrazil, Klinikum Augsburg -Germany, Germany, Brazil\n",
"Luis C S Afonso [email protected] \nDepartment of Computing\nMedizinische Klinik III\nDepartment of Computing\nRegensburg Medical Image Computing (ReMIC), Ostbay-erische Technische Hochschule Regensburg -OTH Regens-burg\nFederal University of São Carlos -UFScar\nSão Paulo State University -UN-ESPBrazil, Klinikum Augsburg -Germany, Germany, Brazil\n",
"Alanna Ebigbo \nDepartment of Computing\nMedizinische Klinik III\nDepartment of Computing\nRegensburg Medical Image Computing (ReMIC), Ostbay-erische Technische Hochschule Regensburg -OTH Regens-burg\nFederal University of São Carlos -UFScar\nSão Paulo State University -UN-ESPBrazil, Klinikum Augsburg -Germany, Germany, Brazil\n",
"Andreas Probst \nDepartment of Computing\nMedizinische Klinik III\nDepartment of Computing\nRegensburg Medical Image Computing (ReMIC), Ostbay-erische Technische Hochschule Regensburg -OTH Regens-burg\nFederal University of São Carlos -UFScar\nSão Paulo State University -UN-ESPBrazil, Klinikum Augsburg -Germany, Germany, Brazil\n",
"Helmut Messmann \nDepartment of Computing\nMedizinische Klinik III\nDepartment of Computing\nRegensburg Medical Image Computing (ReMIC), Ostbay-erische Technische Hochschule Regensburg -OTH Regens-burg\nFederal University of São Carlos -UFScar\nSão Paulo State University -UN-ESPBrazil, Klinikum Augsburg -Germany, Germany, Brazil\n",
"Robert Mendel [email protected] \nDepartment of Computing\nMedizinische Klinik III\nDepartment of Computing\nRegensburg Medical Image Computing (ReMIC), Ostbay-erische Technische Hochschule Regensburg -OTH Regens-burg\nFederal University of São Carlos -UFScar\nSão Paulo State University -UN-ESPBrazil, Klinikum Augsburg -Germany, Germany, Brazil\n",
"Christian Hook \nDepartment of Computing\nMedizinische Klinik III\nDepartment of Computing\nRegensburg Medical Image Computing (ReMIC), Ostbay-erische Technische Hochschule Regensburg -OTH Regens-burg\nFederal University of São Carlos -UFScar\nSão Paulo State University -UN-ESPBrazil, Klinikum Augsburg -Germany, Germany, Brazil\n",
"Christoph Palm \nDepartment of Computing\nMedizinische Klinik III\nDepartment of Computing\nRegensburg Medical Image Computing (ReMIC), Ostbay-erische Technische Hochschule Regensburg -OTH Regens-burg\nFederal University of São Carlos -UFScar\nSão Paulo State University -UN-ESPBrazil, Klinikum Augsburg -Germany, Germany, Brazil\n",
"João P Papa \nDepartment of Computing\nMedizinische Klinik III\nDepartment of Computing\nRegensburg Medical Image Computing (ReMIC), Ostbay-erische Technische Hochschule Regensburg -OTH Regens-burg\nFederal University of São Carlos -UFScar\nSão Paulo State University -UN-ESPBrazil, Klinikum Augsburg -Germany, Germany, Brazil\n"
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Considering the rose number of the Barret's esophagus (BE) number in the last decade, and its expectation of continue increasing, methods that can provide an early diagnosis of dysplasia in BE diagnosed patients may provide a high probability of cancer remission. The limitations related to traditional methods of BE detection and management encourage the creation of computer-aided tools to assist in this problem. In this work, we introduce the unsupervised Optimum-Path Forest (OPF) classifier for learning visual dictionaries in the context of Barrett's esophagus (BE) and automatic adenocarcinoma diagnosis. The proposed approach was validated in two datasets (MICCAI 2015 and Augsburg) using three different feature extractors (SIFT, SURF, and the not yet applied to the BE context A-KAZE), as well as five supervised classifiers, including two variants of the OPF, Support Vector Machines with Radial Basis Function and Linear kernels,
|
10.1007/s00521-018-03982-0
|
[
"https://arxiv.org/pdf/2101.07209v2.pdf"
] | 57,617,057 |
2101.07209
|
476f76ff88d5a589502a58e6db46cfa36a20d9af
|
Learning Visual Representations with Optimum-Path Forest and its Applications to Barrett's Esophagus and Adenocarcinoma Diagnosis
Luis A · De Souza [email protected]
Department of Computing
Medizinische Klinik III
Department of Computing
Regensburg Medical Image Computing (ReMIC), Ostbay-erische Technische Hochschule Regensburg -OTH Regens-burg
Federal University of São Carlos -UFScar
São Paulo State University -UN-ESPBrazil, Klinikum Augsburg -Germany, Germany, Brazil
C S Luis
Department of Computing
Medizinische Klinik III
Department of Computing
Regensburg Medical Image Computing (ReMIC), Ostbay-erische Technische Hochschule Regensburg -OTH Regens-burg
Federal University of São Carlos -UFScar
São Paulo State University -UN-ESPBrazil, Klinikum Augsburg -Germany, Germany, Brazil
Afonso
Department of Computing
Medizinische Klinik III
Department of Computing
Regensburg Medical Image Computing (ReMIC), Ostbay-erische Technische Hochschule Regensburg -OTH Regens-burg
Federal University of São Carlos -UFScar
São Paulo State University -UN-ESPBrazil, Klinikum Augsburg -Germany, Germany, Brazil
Alanna Ebigbo [email protected]
Department of Computing
Medizinische Klinik III
Department of Computing
Regensburg Medical Image Computing (ReMIC), Ostbay-erische Technische Hochschule Regensburg -OTH Regens-burg
Federal University of São Carlos -UFScar
São Paulo State University -UN-ESPBrazil, Klinikum Augsburg -Germany, Germany, Brazil
Andreas Probst [email protected]
Department of Computing
Medizinische Klinik III
Department of Computing
Regensburg Medical Image Computing (ReMIC), Ostbay-erische Technische Hochschule Regensburg -OTH Regens-burg
Federal University of São Carlos -UFScar
São Paulo State University -UN-ESPBrazil, Klinikum Augsburg -Germany, Germany, Brazil
Helmut Messmann [email protected]
Department of Computing
Medizinische Klinik III
Department of Computing
Regensburg Medical Image Computing (ReMIC), Ostbay-erische Technische Hochschule Regensburg -OTH Regens-burg
Federal University of São Carlos -UFScar
São Paulo State University -UN-ESPBrazil, Klinikum Augsburg -Germany, Germany, Brazil
· Robert
Department of Computing
Medizinische Klinik III
Department of Computing
Regensburg Medical Image Computing (ReMIC), Ostbay-erische Technische Hochschule Regensburg -OTH Regens-burg
Federal University of São Carlos -UFScar
São Paulo State University -UN-ESPBrazil, Klinikum Augsburg -Germany, Germany, Brazil
Mendel
Department of Computing
Medizinische Klinik III
Department of Computing
Regensburg Medical Image Computing (ReMIC), Ostbay-erische Technische Hochschule Regensburg -OTH Regens-burg
Federal University of São Carlos -UFScar
São Paulo State University -UN-ESPBrazil, Klinikum Augsburg -Germany, Germany, Brazil
Christian Hook [email protected]
Department of Computing
Medizinische Klinik III
Department of Computing
Regensburg Medical Image Computing (ReMIC), Ostbay-erische Technische Hochschule Regensburg -OTH Regens-burg
Federal University of São Carlos -UFScar
São Paulo State University -UN-ESPBrazil, Klinikum Augsburg -Germany, Germany, Brazil
Christoph Palm [email protected]
Department of Computing
Medizinische Klinik III
Department of Computing
Regensburg Medical Image Computing (ReMIC), Ostbay-erische Technische Hochschule Regensburg -OTH Regens-burg
Federal University of São Carlos -UFScar
São Paulo State University -UN-ESPBrazil, Klinikum Augsburg -Germany, Germany, Brazil
João P Papa
Department of Computing
Medizinische Klinik III
Department of Computing
Regensburg Medical Image Computing (ReMIC), Ostbay-erische Technische Hochschule Regensburg -OTH Regens-burg
Federal University of São Carlos -UFScar
São Paulo State University -UN-ESPBrazil, Klinikum Augsburg -Germany, Germany, Brazil
Luis A De Souza Jr
Department of Computing
Medizinische Klinik III
Department of Computing
Regensburg Medical Image Computing (ReMIC), Ostbay-erische Technische Hochschule Regensburg -OTH Regens-burg
Federal University of São Carlos -UFScar
São Paulo State University -UN-ESPBrazil, Klinikum Augsburg -Germany, Germany, Brazil
Luis C S Afonso [email protected]
Department of Computing
Medizinische Klinik III
Department of Computing
Regensburg Medical Image Computing (ReMIC), Ostbay-erische Technische Hochschule Regensburg -OTH Regens-burg
Federal University of São Carlos -UFScar
São Paulo State University -UN-ESPBrazil, Klinikum Augsburg -Germany, Germany, Brazil
Alanna Ebigbo
Department of Computing
Medizinische Klinik III
Department of Computing
Regensburg Medical Image Computing (ReMIC), Ostbay-erische Technische Hochschule Regensburg -OTH Regens-burg
Federal University of São Carlos -UFScar
São Paulo State University -UN-ESPBrazil, Klinikum Augsburg -Germany, Germany, Brazil
Andreas Probst
Department of Computing
Medizinische Klinik III
Department of Computing
Regensburg Medical Image Computing (ReMIC), Ostbay-erische Technische Hochschule Regensburg -OTH Regens-burg
Federal University of São Carlos -UFScar
São Paulo State University -UN-ESPBrazil, Klinikum Augsburg -Germany, Germany, Brazil
Helmut Messmann
Department of Computing
Medizinische Klinik III
Department of Computing
Regensburg Medical Image Computing (ReMIC), Ostbay-erische Technische Hochschule Regensburg -OTH Regens-burg
Federal University of São Carlos -UFScar
São Paulo State University -UN-ESPBrazil, Klinikum Augsburg -Germany, Germany, Brazil
Robert Mendel [email protected]
Department of Computing
Medizinische Klinik III
Department of Computing
Regensburg Medical Image Computing (ReMIC), Ostbay-erische Technische Hochschule Regensburg -OTH Regens-burg
Federal University of São Carlos -UFScar
São Paulo State University -UN-ESPBrazil, Klinikum Augsburg -Germany, Germany, Brazil
Christian Hook
Department of Computing
Medizinische Klinik III
Department of Computing
Regensburg Medical Image Computing (ReMIC), Ostbay-erische Technische Hochschule Regensburg -OTH Regens-burg
Federal University of São Carlos -UFScar
São Paulo State University -UN-ESPBrazil, Klinikum Augsburg -Germany, Germany, Brazil
Christoph Palm
Department of Computing
Medizinische Klinik III
Department of Computing
Regensburg Medical Image Computing (ReMIC), Ostbay-erische Technische Hochschule Regensburg -OTH Regens-burg
Federal University of São Carlos -UFScar
São Paulo State University -UN-ESPBrazil, Klinikum Augsburg -Germany, Germany, Brazil
João P Papa
Department of Computing
Medizinische Klinik III
Department of Computing
Regensburg Medical Image Computing (ReMIC), Ostbay-erische Technische Hochschule Regensburg -OTH Regens-burg
Federal University of São Carlos -UFScar
São Paulo State University -UN-ESPBrazil, Klinikum Augsburg -Germany, Germany, Brazil
Learning Visual Representations with Optimum-Path Forest and its Applications to Barrett's Esophagus and Adenocarcinoma Diagnosis
Received: date / Accepted: dateNeural Computing and Applications manuscript No. (will be inserted by the editor)
Considering the rose number of the Barret's esophagus (BE) number in the last decade, and its expectation of continue increasing, methods that can provide an early diagnosis of dysplasia in BE diagnosed patients may provide a high probability of cancer remission. The limitations related to traditional methods of BE detection and management encourage the creation of computer-aided tools to assist in this problem. In this work, we introduce the unsupervised Optimum-Path Forest (OPF) classifier for learning visual dictionaries in the context of Barrett's esophagus (BE) and automatic adenocarcinoma diagnosis. The proposed approach was validated in two datasets (MICCAI 2015 and Augsburg) using three different feature extractors (SIFT, SURF, and the not yet applied to the BE context A-KAZE), as well as five supervised classifiers, including two variants of the OPF, Support Vector Machines with Radial Basis Function and Linear kernels,
and a Bayesian classifier. Concerning MICCAI 2015 dataset, the best results were obtained using unsupervised OPF for dictionary generation using supervised OPF for classification purposes and using SURF feature extractor with accuracy nearly to 78% for distinguishing BE patients from adenocarcinoma ones. Regarding the Augsburg dataset, the most accurate results were also obtained using both OPF classifiers but with A-KAZE as the feature extractor with accuracy close to 73%. The combination of feature extraction and bagof-visual-words techniques showed results that outperformed others obtained recently in the literature, as well as we highlight new advances in the related research area. Reinforcing the significance of this work, to the best of our knowledge, this is the first one that aimed at addressing computer-aided BE identification using bag-of-visual-words and OPF classifiers, being this application of unsupervised technique in the BE feature calculation the major contribution of this work. It is also proposed a new BE and adenocarcinoma description using the A-KAZE features, not yet applied in the literature.
Keywords Barrett's esophagus · optimum-path forest · machine learning · adenocarcinoma · image processing
Introduction
Pattern classification has been paramount in the last decades, mainly due to the increasing number of applications that require some intelligent-decision-making mechanism. The standard pipeline adopted for so many years follows a robust but straightforward workflow: (i) feature extraction, (ii) model learning, and (iii) classification outcomes. The former step can be performed us-ing handcrafted features or information learned through deep learning approaches. In this latter case, one may not know what kind of information the model is learning, since the set of outcome values that minimizes some loss function is the one employed in the model learning step. Handcrafted features require a more knowledgeable personnel, which is usually in charge of selecting and extracting features that matter when performing pattern classification.
Describing images using their most important information, the so-called "points of interest" (PoIs) or key points, has been an active area of interest by many researchers worldwide. Notable approaches have been proposed in the literature to compute those points, which somehow aim at capturing subtle information that is less variant to geometric transformations such as rotation, and translation, among others. Scale-Invariant Feature Transform (SIFT) [17], Speeded-Up-Robust-Features (SURF) [4], and Accelerated-KAZE features (A-KAZE) [2] are some examples.
However, the main problem related to the mentioned approaches concern the final feature vector. Since the number of PoIs may vary from one image to another, the feature vectors used to represent the images shall have different dimensions. To overcome this issue, an additional step called "quantization" is required (some works refer to this step as the "codebook generation" [9]). In a nutshell, given a training set composed of PoIs extracted from all training images, we can build a "bag" (i.e., a visual dictionary) with the most representative PoIs (from now on called "visual words"). Further, for each training image and each of its PoIs, we can find the "closest" visual word in the bag and build up a histogram that stores the number of times a visual word is nearest to each PoI from the training images. Therefore, the final feature vector of each training image will be that histogram with a dimensionality that corresponds to the number of visual words (i.e., the size of the dictionary or bag). Essentially, that is the main reason such approaches are usually referred to as "bag-of-visual-words" (BoVW) [6].
Bag-of-visual-words have been widely used in the literature for a number of different purposes, such as video-based action recognition [25], retinal health diagnosis [14], and perivascular spaces categorization in brain data [10], among others. Nonetheless, one still has two problems to face regarding the BoVW approach: (i) how to find out the most representative visual words, and (ii) how to establish a proper bag size, i.e., the number of visual words. Notice that both issues are pretty much crucial since they are in charge of the feature vector composition and dimensionality.
To cope with the first issue, i.e., finding out the most representative visual words, two approaches are commonly used: (i) random sampling and (ii) clustering. The former randomly selects a given number of visual words to compose the bag. On the other hand, clustering-based approaches make use of some unsupervised learning algorithm (usually k-means) to group the visual words, and the most representative ones (i.e., centroids) are elected to compose the dictionary [1]. However, randomly choosing visual words does not lead to good results, and the usage of certain unsupervised learning algorithms turns out to be a problem since most of them require the number of clusters (i.e., the bag size) beforehand.
Therefore, clustering techniques that do not require a priori information about the data are usually preferred. Among several techniques that have been proposed in the literature, one is gaining attention daily due to its effectiveness and efficiency in different research areas. The Optimum-Path Forest (OPF) is a framework for the design of pattern classifiers based on graph partition. In short, OPF-based classifiers work on a reward-competition process, in which previously selected samples called "prototypes" try to conquer other samples by offering them optimum-path costs. Once a sample is conquered by another one, it receives its label and a "mark" (i.e., a predecessor map) that reveals its conqueror. The Optimum-Path Forest framework comprises supervised [21,20,22], unsupervised [28], and semi-supervised [3] versions that have been widely employed in a number of applications, from remote sensing [27,19] to human intestinal parasites identification [38], just to cite a few.
One particular strength of unsupervised OPF concerns the fact it does not require the number of clusters beforehand, i.e., it finds clusters on-the-fly. Such feature is quite interesting in the context of BoVW generation since we skip the problem of choosing suitable bag sizes. As far as we are concerned, only two works attempted at using OPF in the context of BoVW: (i) Papa and Rocha [23] evaluated the supervised OPF for image categorization using visual words, and further (ii) Afonso et al. [1] studied the impact of using unsupervised OPF for learning proper visual dictionaries.
We are particularly interested in the application of such technique for the recognition of Barrett's esophagus (BE), which happens to be a side effect of some reflux diseases. BE comprises a very severe and growing disease in the last decades, and since BE is often not identified properly at the early stages, it may evolve to a more aggressive version, and even to cancer. However, the early diagnosis of dysplastic tissue in BE diagnosed patients may provide very high rates of remission after the treatment [7,31,26]. There are several endoscopic techniques for the BE diagnosis and detection, such as chromoendoscopy and narrow-band imaging, but the human screening for the injured region definition is still often misclassified by endoscopists, once the region does not present enough goblet cells in biopsy or the experts refuse to use the recommended procedure for extensive biopsies [32]. Moreover, computer-assisted diagnosis may bring precision and accuracy to the BE screening and evaluation, once this task can be very influenced by the human factor [35,37,36]. To the best of our knowledge, only one very recent work coped with BE identification using OPF. Souza et al. [35] introduced the supervised OPF for Barrett's esophagus automatic identification using features based on BoVW. The authors considered both random-and k-means-based sampling strategies to build the visual dictionaries and then used OPF for classification purposes.
Some works that dealt with endoscopic image analysis can be referred to as well, but that is still an emerging area of research [37]. Seibel et al. [30] developed a low-cost but high-performance technology to assist the diagnosis of BE and esophageal cancer. However, their primary contributions rely on hardware advances rather than software ones. The work presented by van der Sommen [34] aimed at using machine learning techniques to detect early neoplasia in Barrett's esophagus, and Swager et al. [39] addressed the very same context mentioned above but using volumetric laser endoscopy images.
Klomp et al. [13] proposed new features for computer-aided Barrett's esophagus identification, and Hassan and Haque [11] used endoscopy videos obtained from wireless capsules to assess gastrointestinal hemorrhages. Later on, Seguí et al. [29] used the same source of images (i.e., wireless capsules) together with Deep Convolutional Neural Networks for intestine motility characterization. Mendel et al. [18] started the study of deep learning application to the BE and adenocarcinoma evaluation problem.
The major problems around the computer-assisted systems developed for the BE and adenocarcinoma evaluation are related to the type of technique to provide a correct description of the injured areas and which classification techniques should be designed for the problem. These problems are related to all proposed works, and considering the high potential in this research area, new ways to describe the injured areas (that are very similar), and the evaluation of different classifiers can deliver important and substantial improvements to the precision and correct differentiation of both.
As one can observe, Barrett's esophagus automatic identification using machine learning techniques presents a growing interest in the last years. Therefore, there is plenty of room for new works that employ techniques that were not considered in such a context. In this work, we extended and outperformed the approach proposed by Souza et al. [35] by learning proper visual dictionaries using unsupervised OPF, as well as we introduced a variant of supervised OPF (OPF knn ) proposed by Papa et al. [22] in the context of BE identification. The OPF was never applied to such problem in the visual learning step, and this could deliver, besides the novelty in the feature vector calculation, a new way to evaluate the key points provided by the feature extraction techniques. Last but not least, we introduce the A-KAZE feature extraction technique for the calculation of the key points, for comparison with SURF and SIFT, previously adopted for the BE and adenocarcinoma differentiation context [37]. The results presented in this paper are close to some state-of-the-art recognition rates [18], and it features recent advances to BE automatic identification by means of machine learning and computer vision.Therefore, the main contributions of this paper are five-fold:
to extend and outperform the recent results obtained by Souza et al. [35] in which the evaluation of BE and adenocarcinoma context were performed using: (i) SURF and SIFT techniques for key points calculation, (ii) k-means and random techniques for the bag-of-visual-words calculation, and (iii) OPF and SVM classifiers for the classification task; to introduce OPF knn [22] for BE and adenocarcinoma automatic diagnosis, considering that Souza et al. [35] employed only the complete graph version of OPF classifier for the classification task; to introduce A-KAZE features for the aforementioned context, once such technique has been largely applied in the literature for image description and retrieving; to extend the work by Afonso et al. [1] with a more robust evaluation of the unsupervised OPF for learning visual dictionaries; to introduce a new representation of feature extraction techniques (such as SURF and SIFT) based on their most representative words in the feature space using the OPF clustering technique.
The remainder of this paper is organized as follows. Sections 2 to 4 present a theoretical background of unsupervised OPF and the methodology adopted in this work, respectively. Section 5 discusses the experiments, and Section 6 states conclusions and future works.
Unsupervised Learning with Optimum-Path Forest
In this section, we briefly present the theoretical background related to unsupervised OPF, which is used to learn proper visual dictionaries.
Let D = {x 1 , x 2 , . . . , x m } be an unlabeled dataset such that x i ∈ n stands for a feature vector extracted from some sample (i.e., images in our case) related to the problem to be addressed. Additionally, let G = (D, A k ) be a graph derived from that dataset, which means D denotes the set of graph nodes (i.e., vertices) and A k stands for a k-nearest neighbors adjacency relation.
In a nutshell, the OPF working mechanism is based on a reward-competition problem, where some samples called "prototypes" employ a competitive process among themselves to conquer the other samples from the dataset D. Such competition ends up partitioning D into optimum-path trees (OPTs), which are rooted at each prototype node. It is worth mentioning that a sample that belongs to a given OPT is more "strongly connected" to the root and samples of that tree than to any other in the forest (i.e., a collection of all trees in the graph).
At a glance, the whole process can be summarized in the following steps:
1. To establish a proper neighborhood size and build up A k (i.e., to find out "suitable" k values); 2. To elect the prototypes and Learning Visual Representations with Optimum-Path Forest and its Applications to Barrett's esophagus and Adenocarcinoma Diagnosis 3. To start the competition process.
Concerning step 1), a number of different approaches to cope with the task could be considered. Rocha et al. [28] proposed to compute the best value of k (i.e., the neighborhood size), say that k * , as the one that minimizes the normalized graph cut, which is a measure that considers both the dissimilarity between clusters as well as the similarity within the groups of samples [33].
Soon after computing k * , the next move concerns finding the prototypes (i.e., step 2), also known as the "roots of the trees". Such essential samples are in charge of ruling the competition process that ends up partitioning the graph into OPTs (i.e., clusters). Those samples will be used as the visual words to compose the final dictionary, as further discussed.
The supervised OPF proposed by Papa et al. [21] elects the prototypes as the nearest samples from different classes, which can be accomplished by computing a Minimum Spanning Tree (MST) over the training graph. Then, the samples from different classes that are connected in the MST are marked as prototypes. However, unsupervised OPF does not make use of labeled datasets, which motivated Rocha et al. [28] to elect the prototypes as the samples that are located at the center of the clusters. Such samples can be computed by assigning a density score ρ(x i ) for each dataset sample x i ∈ D. That score is computed using a probability density function (pdf) given by a Gaussian distribution considered in the neighborhood of each sample as follows:
ρ(x i ) = 1 √ 2πσ 2 k ∀xj ∈A k (xi) exp −d(x i , x j ) 2σ 2 ,(1)
where i = j and σ = d max /3. In this case, d max stands for the maximum arc-weight in G. Using such formulation, ρ(x i ) considers all adjacent nodes for the probability computation purposes since a Gaussian function covers 99.7% of the samples within d(
x i , x j ) ∈ [0, 3σ].
After computing Equation 1 for all nodes, the competition process among samples can take place. Each density value will be used to populate a priority queue, where the idea of the unsupervised OPF algorithm is to end up maximizing the cost of each sample, and thus partitioning the graph.
The definition of "cost" is based on paths on graphs, i.e., a sequence of adjacent samples with no cycles. Let π xi be a path with terminus at sample x i and starting from some root R(x i ), where R stands for the set of prototype samples. Additionally, let π xi = x i be a trivial path (i.e., a path composed of a single sample) and π xi · x i , x j the concatenation of π xi and the arc (
x i , x j ) such that i = j.
The OPF algorithm assigns to each path π xi a value f (π xi ) given by a connectivity function f : X → . In this context, a path π xi is considered optimum if f (π xi ) ≥ f (τ xi ) for any other path τ xi . Such sort of functions are known as "smooth functions", and they figure important constraints that ensure the theoretic correctness of the OPF algorithm [8].
Among different path-cost functions that have been proposed in the literature, unsupervised OPF employs the following formulation for ∀x i , x j ∈ D such that i = j:
f ( x i ) = ρ(x i ) if x i ∈ R ρ(x i ) − δ otherwise,(2)
and
f (π xi · x i , x j ) = min{f (π xi ), ρ(x j )},(3)
where δ = min ∀(xi,xj )∈A k |ρ(t) =ρ(s) |ρ(t) − ρ(s)|. In a nutshell, δ stands for the smallest quantity required to avoid plateaus in the regions nearby the prototypes (i.e., areas with the highest density). Among all possible paths π xi from the maxima of the pdf, the method assigns to sample x i a final path whose minimum density value along it is maximum. Such final path value is represented by a cost map C, as follows:
C(x i ) = max ∀πx j ∈(D,A k ),i =j {f (π xj · x j , x i )}.(4)
The OPF algorithm maximizes the connectivity map C(x i ), ∀x i ∈ D, by computing an optimum-path forest over the dataset. Such forest is encoded as a predecessor map P with no cycles that assigns to each sample x i / ∈ R its predecessor P(x i ) in the optimum path from R, or a marker nil when x i ∈ R. Figures 1 to 3 depict a toy example concerning the unsupervised OPF working mechanism. Figures 1a and 1b illustrate an unlabeled dataset and its 3-nearest neighbors graph, respectively (we assume k = 3 to explain step 1). For the sake of visualization purposes, we assigned the same color to each graph node and the arcs corresponding to its 3-nearest neighbors. Notice the arcs are also weighted by the distance (e.g., Euclidean distance) among their corresponding nodes. One can observe that some arcs and their weights are double-colored, which means their corresponding nodes share the very same 3-neighborhood. Figure 2 illustrates the density computation step to further elect the prototypes (i.e., step 2). Therefore, given the arc-weights depicted in Figure 1b, we can use Equation 1 to compute ρ(x i ), ∀x i ∈ D. Notice the density values are computed over the adjacency relation encoded by A k . One can realize that the samples located at the center of the clusters tend to be the ones with the highest value of ρ since they are connected by smaller arc-weights. The density values are then stored in a priority queue (i.e., a max-heap) that pops out the sample x i with the highest ρ(x i ). Concerning the toy example depicted in Figure 2, the first sample to come out of the queue is either 'H' or 'A' since both have the highest densities. Suppose 'H' has been added first to the queue. Since it has no predecessor, it is added to the set R and assigned f (H) = ρ(H) = 0.66 according to Equation 2.
Further, the competition process (i.e., step 3) takes place. In short, sample 'H' evaluates its neighbors 'I', 'J', and 'K' to offer better costs to them (i.e., costs that are greater than the ones they have already). Therefore, one has f (H· H,I ) = \min{0.66, 0.63} = 0.63, f (H· H,J ) = \min{0.66, 0.65} = 0.65, and f (H· H,K ) = \min{0.66, 0.65} = 0.65. Since the costs offered by 'H' are greater or equal than the costs of its neighbors, they are conquered by sample 'H'. Such process is encoded by the aforementioned predecessor map P, i.e., after this first move of sample 'H', one has that P(I) = H, P(J) = H, and P(K) = H.
The next sample to start the competition process is sample 'A', and the very same process mentioned earlier is repeated until all samples have played in the competition process. The resulting optimum-path forest is depicted in Figure 3. Notice one can obtain a different number of clusters based on the value of k max . In this toy example, we obtained two clusters, which are labeled with the same color of its prototype/root of the tree (i.e., the dashed nodes 'A' and 'H'). The unsupervised OPF algorithm finds the number of the clusters on-the-fly, which means there is no need to have such information beforehand. The only parameter that needs to be set is the k max , which constraints the search for suitable neighborhood sizes. One can observe that the knowledge required to set k max is considerably lower than the one needed to set the number of clusters used by k-means, for instance. Such skill makes OPF pretty much attractive to the application addressed in this paper, as discussed in the next section.
Barrett's Esophagus
The BE disease is known as the replacement of squamous cells by columnar cells in the esophagus. This process is a result of a complication of gastroesophageal reflux disease, being able to progress into esophageal cancer [7,31].
The incidence of BE and Barrett's adenocarcinoma in the western population of the world has risen significantly in the past decade. Their close association with the metabolic syndrome suggest growth in the next years [15,7,16]. The early diagnosis of Esophageal adenocarcinoma in BE diagnosed patients is critical for remission and justifies the necessity of robust surveillance, detection, and characterization. However, the detection of dysplastic tissues and their characterization of abnormalities within BE-diagnosed patients can be challenging, especially for manual evaluation made by endoscopists. Even considering the dangerousness of the disease, when detected at the early stages, the disease can be treated with very high rates of remission (93% after 10 years) [7,31,26].
The esophagus mucosa is composed of squamous cells (similar to the skin or mouth cells), with a whitishpink color surface, while the gastric mucosa goes sharply from salmon-pink to red [7,31]. The point in which the stomach and the stomach meet is called squamo-columnar junction or "Z-line". BE's mucosa may extend upward in a continuous pattern, changing the Zline position [7,31,26]. Figure 4 shows the two cases in which patients can present long-segment of BE and short-segment of BE in a Z-line variation.
Methodology and Proposed Approach
In this section, we present the proposed approach and the methodology adopted to cope with the problem of Barrett's esophagus automatic identification using bag-of-visual-words. First, the proposed method is defined, followed by the datasets used for the experiments, adopted classifiers and experimental delineation.
Proposed Method
As mentioned earlier, one of the leading contributions of this work is to evaluate the robustness of the OPF clustering for learning visual dictionaries. To fulfill that purpose, we considered three distinct feature descriptors based on key point extraction from images: (i) SIFT, (ii) SURF, and (iii) A-KAZE. Although any other approaches could be used, we opted to employ these mainly because they are well known and widely considered in the literature of bag-of-visual-words for both image classification and retrieval, but any other techniques could be applied considering the generalization of the learning visual dictionaries.
For the inicial step of the proposed model, given a set of training images, it is needed to build a bag of key points extracted from them. In hands of a feature extraction technique for the description of an image (being SURF, SIFT or A-KAZE, in this specific case), the model aims to provide the most discriminative key points in the feature dimension based on the entire feature domain. Therefore, taking into account the key points of the entire dataset, a clustering algorithm can be used to group the key points into clusters that share similar properties for choosing the "best key point" from each cluster and use it as the representative of that group. Such samples will compose the final bagof-visual-words. The main contribution of this work is the calculation of such most representative key points from clusters (as we use to call "prototypes") by using the OPF clustering technique. After obtaining the bagof-visual-words, the last step of the model is known as "quantization" and computes the new representation for both training and testing images. For each image, it is computed the frequency of each visual word from the bag in the given image by finding the most similar visual word to each key point based on a distance metric. The outcome of that process is a histogram (feature vector) where each bin has the number of key points that are similar to its corresponding visual word. Notice that the representation of both training and testing images are computed based on the same bag. Finally, in hands of the feature vectors, each image of the evaluated dataset shows the exact same number of features for its description, but with the calculation based on the entire feature space domain. The training and testing may be conducted as the final step of the model.
In this work, we propose to cluster the dataset of key points using the OPF technique presented in the previous section and then use the prototypes to compose the bag-of-visual-words. As aforementioned, the prototypes are located in the regions of highest densities, which means they are pretty much suitable to describe the clusters. Another decisive point about OPF concerning other optimization-based clustering techniques relates the fact of not being attracted to local optima, such as k-means or k-medoids, for instance, which are widely used for learning dictionaries due to their simplicity and low computational cost.
As mentioned earlier, OPF finds the clusters onthe-fly, i.e., the clustering process is dynamic, and the forest configuration can change until the last sample finishes the conquering process. Instead of varying the size of the dictionary, one can change the value of k max and then may find the different number of clusters. The cluster calculation comprises one of the most important steps of the proposed method. The prototype computation is performed in an unsupervised process, turning the calculation of the feature vectors based only on the key points themselves, and providing a high generalization for this task. The problem of a different number of key points for each image can be solved using this bag-of-visual-words approach, proposing a consistent way of regular description for images evaluated by feature extraction techniques. Figure 5 depicts the pipeline adopted in this work.
Since the images are colored, a gray-scale normalization is applied to the images so that the key points can be extracted. Later, such PoIs are then mapped onto a feature space for clustering purposes. An example of the outcome of the clustering process is depicted at the bottom of Figure 5. Each color stands for a different group and the dashed nodes represent the prototypes selected by OPF to be part of the visual dictionary. As aforementioned, a histogram is built upon the training PoIs and the visual words for the further design of the final set of handcrafted features. For the selected visual words, an evaluation of their appearance is performed in the PoIs of each dataset image aiming to calculate the final cumulative histogram that represents each feature vector, with dimension depending on the number of visual words generated in the clustering calculation of the bag.
Datasets
An in-depth analysis concerning the robustness of the proposed approach is provided through two datasets. The first dataset comprises a set of images from a benchmark dataset provided at the "MICCAI 2015 EndoVis Challenge" 1 was considered, hereinafter called "MIC-CAI 2015" dataset, which aimed at differentiating Barrett's esophagus from cancerous images. Such dataset is composed of 100 endoscopic pictures of the lower esophagus captured from 39 individuals, 22 of them being diagnosed with early-stage Barrett's, and 17 showing signs of esophageal adenocarcinoma. Each patient has several endoscopic images available, ranging from one to a maximum of eight. The database comprises a total of 50 images displaying cancerous tissue areas as well as 50 images showing dysplasia without signs of cancer. Suspicious lesions observed in the cancerous images had been delineated individually by five endoscopy experts.
Additionally, a dataset provided by the Augsburg Klinikum, Medizinische Klinik III was also used for the experiments. Such dataset is composed of 76 endoscopic images (esophagus) captured from different patients with adenocarcinoma (34 samples) and BE (42 samples). The images were annotated (manual segmentation of the adenocarcinoma's and Barrett's area, respectively) by an expert from the Augsburg Klinikum, and the diagnosis was provided using biopsy. Since we are dealing with a classification problem, the annota- tions provided by the experts were not considered in our work. Figure 6 depicts some examples of the MICCAI 2015 dataset positive for cancer (i.e., negative for BE) and their respective delineations performed by five experts. However, we are not working with the delineation information since we compute the PoIs for the whole image. One could use the information about the delineated regions to extract PoIs from that areas only, which could guarantee that pure adenocarcinoma PoIs are computed, but the problem still concerns the fact that delineations are not available to all real-world images. Figure 7 displays some images positive for cancer from Augsburg dataset. In this case, we have only one delineation per image. Once again, such information is not used in this work since we are interested mostly in the differentiation of Barrett's esophagus and adenocarcinoma rather than its segmentation.
Adopted Classifiers
We considered different supervised pattern recognition techniques to assess the robustness of unsupervised OPF for learning visual dictionaries:
-OPF cpl : supervised OPF with complete graph proposed by Papa et al. [21,20]; -OPF knn : supervised OPF with k-nn graph proposed by Papa et al. [22]; -SVM-RBF: Support Vector Machines with Radial Basis Function kernel and parameters optimized by cross-validation [5]; -SVM-Linear: Support Vector Machines with Linear kernel and parameters optimized by crossvalidation [5]; -Bayes: standard Bayesian classifier.
Regarding the OPF-based classifiers, we used the Li-bOPF [24], which is an open-source library that implements both the supervised as well as the unsupervised versions of the OPF used in this work. With respect to the Bayesian classifier, we employed our own implementation.
Experimental Delineation
To compose the set of experiments, we considered three different sizes for the dictionaries: 100, 500, and 1, 000. The main idea is to evaluate the robustness of the techniques used in this work under different scenarios. As we shall discuss later, the usage of dictionaries with 500 visual words seemed to achieve better results, as stated in a previous work [35], which motivated us to set k max = 500 for this one. However, this not implies in constraining OPF to find exactly 500 clusters, just to limit the size of the neighborhood of each sample to be 500. Regarding OPF knn , its parameter k is fine-tuned within the range [1,500], and the value that maximized the accuracy over the training set was used.
Regarding the experimental validation, it was considered a cross-validation approach with 20 runs and using 70% of the dataset for training purposes, as well as the remaining 30% for classification. Moreover, the experimental results were assessed using a statistical analysis using the Wilcoxon signed-rank test with confidence as of 5% [40]. All experiments were conducted on an 8GB-memory computer equipped with an Intel Core i5 -2.30 GHz processor. Additionally, we employed the OpenCV [12] implementation for feature extraction using SIFT, SURF, and A-KAZE.
Experimental Results
In this section, we present the experiments used to evaluate the proposed approach. Five supervised classifiers were considered to discriminate between samples positive and negative to adenocarcinoma: OPF cpl , OPF knn , SVM-RBF, SVM-Linear, and Bayesian classifier (hereinafter called Bayes). For all the classifiers adopted for such evaluation, there was no need for setting any parameter, as long as they were used in the default set. The same experimental protocol was applied to all techniques using cross-validation, i.e., three distinct feature representations were considered (SURF, SIFT, and A-KAZE, with the metric threshold of all sets is default), and with different bag sizes (i.e., 100, 500 and 1, 000 visual words). The results are presented and discussed considering each dataset individually.
A statistical evaluation using the signed-rank Wilcoxon test [40] was used for comparison purposes as follows:
1. For each dictionary generation approach (i.e., clustering by k-means, random or unsupervised OPF), it was verified the classification results and the best ones were highlighted in bold. Statistically similar results were highlighted in bold. 2. For each feature extractor (i.e., A-KAZE, SIFT, and SURF), the best statistical results were underlined. 3. Additionally, the best results among all configurations were marked with a ' ' symbol.
This very same procedure was adopted to both datasets. In this work, we used the following accuracy rate:
A = T P + T N T P + T N + F P + F N · 100,(5)
where T P and T N stand for the true positives and true negatives, respectively, and F N and F P denote the false negatives and false positives, respectively. In a nutshell, the above equation computes the ratio between the number of correct classifications (i.e., T P + T N ) and the size of the dataset (i.e., all correct and wrong classifications).
MICCAI 2015 Dataset
Tables 1, 2, and 3 present the results related to A-KAZE, SURF, and SIFT descriptors, respectively, concerning MICCAI 2015 dataset. Regarding the A-KAZE results presented in Table 1, one can draw the following conclusions: (i) OPF cpl obtained the best results for all dictionary generation techniques, and (ii) OPF clustering achieved the best results (77.6% of recognition rate with 1, 000 visual words) for BE recognition among all configurations, although being statistically similar to kmeans with OPF cpl with 500 and 1, 000 visual words as well.
The average number of PoIs used for training and test sets concerning A-KAZE feature extractor were 16, 024 and 6, 868, respectively, taking an average computational load of 4.05 minutes. A training set composed of around 16, 000 visual words is enough to support the sizes of the dictionaries we used to build the feature vector of each image, i.e., 100, 500, 1, 000. Larger dictionaries may not be interesting since there will be numerous small-sized clusters, which means less spatial information about the visual words is captured. Table 2 presents the results concerning the SURF feature extractor. Once again, OPF cpl achieved the best classification results regarding all dictionary generation approaches, and OPF clustering allowed the best results among all, i.e., it could learn better dictionaries for image representation. In this context, a dictionary of size 500 computed by k-means also achieved the best recognition rates according to the statistical test. The average number of PoIs used for training and test sets concerning SURF feature extractor were 14, 411 and 6, 189, respectively, taking an average computational load of 13.77 minutes. One can observe that SVM did not obtain proper recognition rates in both situations, i.e., A-KAZE and SURF feature extractors. One possible reason concerns the number of training samples, which is usually lower than the number of features. Therefore, SVM will map samples to a lower-dimensionality feature space instead of a higher one, thus neglecting the assumption of linearity in higher-dimensionality spaces. Table 3 presents the results considering the SIFT feature extractor. Once again, OPF cpl achieved the best results so far, with OPF knn and SVM-RBF being statistically similar for k-means with 1, 000 words and a random generation of dictionaries with 1, 000 words. However, the best global results were achieved using OPF clustering with OPF cpl with 500 and 1, 000 visual words, outperforming by far the other results with SIFT feature extractor. The average number of PoIs used for training and test sets concerning SIFT feature extractor were 28, 137 and 12, 059, respectively, taking an average computational load of 5.95 minutes. Last but not least, the best results among all three feature extractors (i.e., the ones marked with ' ') were obtained using OPF clustering for dictionary generation and OPF cpl for classification with 500 and 1, 000 visual words considering SURF and SIFT, and the same pair (i.e., OPF clustering and OPF cpl ) regarding A-KAZE with 1, 000 visual words, and finally k-means and OPF cpl with 500 words. Notice the best absolute result was obtained using OPF clustering for visual words generation and OPF cpl for classification purposes with SIFT-based features (i.e., 78.9%).
Augsburg Dataset
Tables 4, 5, and 6 present the results related to A-KAZE, SURF, and SIFT descriptors, respectively, concerning Augsburg dataset. Starting with the A-KAZE feature extractor, one can observe the best results were mostly obtained by both OPF cpl and OPF knn . The best global results were achieved by OPF clustering, Random and k-means, but the most accurate one (i.e., absolute results) was OPF clustering for visual dictionary generation and OPF cpl for classification purposes with accuracy of 72.6%. Such result is slightly less accurate than the same feature extractor considering MICCAI 2015 dataset since Augsburg dataset is more challenging due to different levels of adenocarcinoma. The average number of PoIs used for training and test sets concerning A-KAZE feature extractor were 40, 064 and 17, 170, respectively, taking an average computational load of 4.18 minutes. Table 5 presents the results concerning the SURF feature extractor. Once again, OPF-based classifiers obtained the best results in most of the scenarios, being OPF clustering and k-means the best approaches for visual dictionary generation. The best absolute classification results were obtained by OPF cpl and Bayes with accuracies nearly to 68%. The average number of PoIs used for training and test sets concerning SURF feature extractor were 14, 251 and 6, 108, respectively, taking an average computational load of 9.23 minutes.
The Augsburg dataset figured out as being more challenging than MICCAI 2015 dataset due to the considerably low results achieved ( Table 2). SVM-RBF presented better results with higher-dimensionality bags (i.e., 65.4% with 1, 000 words with OPF clustering), and the same behavior can be observed regarding SVM-Linear. Table 6 presents the results considering the SIFT feature extractor. In this case, OPF-based classifiers and SVM-RBF figured as the most accurate techniques and OPF clustering as the best one for visual dictionary generation (absolute results). A comparison against A-
Discussion
In this section, we aim at providing a more in-depth discussion about the experiments, as well as insightful conclusions regarding the usage of bag-of-visual words in the context of computer-aided differentiation between Barrett's esophagus and adenocarcinoma. Table 7 presents a summary with the best results obtained in the previous two sections concerning the number of visual words and feature extractor. Concerning both datasets, OPF cpl figured as the more accurate classification technique, meanwhile OPF clustering appears as the best dictionary generation approach. The results support the primary contributions stated previously, which are related to the robustness of OPFbased classifiers for both supervised and unsupervised learning in the context of automatic adenocarcinoma identification. Additionally, the number of visual words strongly affects the results, but we believe a trade-off between the size of the dictionary and the information it carries on shall be established beforehand. Table 8 summarizes the mean sensitivity and specificity results of both datasets with the best configuration of the number of visual words, dictionary generation approach, feature extractor, and classification technique mentioned above. Sensitivity stands for the classification rate considering adenocarcinoma identification, i.e., those positive to Barrett's esophagus and to adenocarcinoma, and specificity denotes the accuracy regarding those negative to adenocarcinoma, i.e., positive only to BE. Considering such sensitivity and specificity results, some conclusions can be drawn: (i) for the MICCAI 2015 dataset, the sensitivity results presented higher values than the specificity ones, suggesting a very good generalization in the positive adenocarcinoma identification. Even with lower results, the specificity still showed a convincing value, and the misclassification can be justified by two factors: the fuzzy region (region in which the experts disagree in the annotation) and lack of enough key points in the noncancerous regions during the feature vector calculation. For the Augsburg results of sensitivity and specificity, a better trade-off between the correct classification of positive and non-positive adenocarcinoma samples could be found, but still with lower results when compared to the MICCAI 2015 dataset ones. The Augsburg dataset presents images with different behavior and acquisition technology when compared to the MICCAI 2015 ones, thus justifying the worse results.
To provide more insightful comments and to better understand the working mechanism of visual words in the context of computer-assisted BE identification, we performed some additional experiments with cancerous images that were classified either as cancer or as Barrett's esophagus since we have their delineated regions. In a nutshell, the main idea is to compute the percentage of PoIs located inside those regions with respect to the remaining ones (i..e, those located outside cancerous areas). This information allows us to compare whether the number of PoIs placed inside the delineated regions are enough or not to provide accurate classifications. Table 9 presents the mean percentage of PoIs located inside the cancerous area for the whole dataset, as well as the average percentage of PoI inside the cancerous area concerning the misclassified images (i.e., cancerous images that were classified as BE). Since we conducted a cross-validation approach with 20 runs, the average percentages concerning the misclassified images (i.e., Cancer→BE) were computed to each run, for the further computation of the average value of all. Additionally, since MICCAI 2015 dataset comprises delineations from five experts, we took the intersection of them all as the final delineated area to compute the percentage of PoIs into account.
One can observe that the percentage of PoIs inside the cancerous images were more significant than the values obtained from the misclassified images. Such assumption is pretty interesting since we can conclude that the number of PoIs inside the delineated regions are essential to achieve accurate results and to avoid misclassifications. The only exception stands for the Augsburg dataset with A-KAZE features, where the number of PoIs were slightly higher for the misclassified images. Note that the percentage of PoIs inside the cancer region is in general higher for the Augsburg databaset than for the MICCAI 2015 dataset. This can be explained because the Augsburg images use the nearfocal imaging technique, in which the suspicious region is displayed larger. Figure 8 were calculated using SIFT and belong to the MICCAI 2015 dataset, and their percentage of incidence is 21.72%, which is slightly lower considering the average percentage presented in Table 9 (23, 04%).
One can observe a considerable amount of PoIs located at the left-middle portion of Figure 8b, mainly due to some air bubbles and foam. Problems with light (upper part of the image) also contribute to placing PoIs outside the delineated area.
The PoIs showed in Figure 9 were calculated using A-KAZE on an image from the Augsburg dataset. Their percentage of incidence is 7.5%, which is quite low considering the average percentage presented in Table 9 (53.77%). In this case, the main reason for placing PoIs outside the delineated area concerns illumination problems (brighter areas).
Conclusions and Future Works
In this paper, we dealt with the problem of computerassisted Barrett's esophagus identification by means of bag-of-visual-words calculated using the OPF clustering technique. Such technique showed promising results, outperforming the previous handcrafted feature results in the same context. This suggests the generalization relevance of such technique, which can improve previous results in the same field not only for the BE context but for other in which the image representation configures the context to be evaluated. BE stands for an illness that is likely to be confused with adenocarcinoma, and its early detection and prevention is of great concern.
We observed that only a very few works attempted at coping with the problem of automatic BE identification using computer vision and machine learning techniques to date. In this work, we fostered the research towards such area by introducing a supervised variant of the Optimum-Path Forest classifier for automatic BE recognition, as well as we showed how to build proper visual dictionaries using unsupervised OPF learning, outperforming the results obtained in some recent works in which the same database and protocol were applied [35,36]. Considering some previous works [35,37,36], the use of handcrafted features were based on the SURF and SIFT PoIs, but without the use of the OPF clustering as a way of dimensional reduction of the problem. Moreover, considering the improvements of the results, the use of the OPF clustering provides a new and promising way of BE and adenocarcinoma problem evaluation based on extracted key points. The presented results showed the relevance of such technique addressed to the BE and adenocarcinoma evaluation and description, contributing to the context literature and influencing the evaluation and description of other tissue diseases. Comparing the proposed method with others already published, we can ensure that with the use of the OPF for the BoVW step, improvements could be achieved considering the higher results obtained. Also, such technique provides advantages in the dimension reduction of the feature vector calculation, once even with a different number of key points per image, a standard method of feature calculation is established. Again, the OPF clustering may provide flexibility and time saving for such task.
The experimental results were considered over two datasets: (i) MICCAI 2015, and (ii) Augsburg. For both scenarios, we evaluated five classification techniques and three unsupervised learning approaches to build the visual dictionaries. Also, we considered dictionaries with three distinct sizes and even three different feature extractors.
The experiments pointed out that bag-of-visualwords techniques are suitable to handle BE automatic identification, and there must be a trade-off between the number of visual words and the amount of informa-tion they can encode (i.e., size of the clusters). Additionally, both supervised and unsupervised OPF-based classifiers achieved the most accurate results, thus supporting the main contributions of this paper.
In the following, a bullet list of trends based on the achieved results is presented:
the OPF classifier presented the highest results of accuracy in all experiments, and may be highly recommended considering the high generalization that provided for such a context, even for different description scenarios; the representation of BE and adenocarcinoma by means of image description techniques may provide encouraging results, and with less computation processing cost as needed in more sophisticated techniques; the A-KAZE features showed the very best results for BE and adenocarcinoma description in the Augsburg dataset evaluation, suggesting to be a very important technique for the description of such diseases; the use of handcrafted features still has potential to evaluated for BE and adenocarcinoma problem, considering the several number of techniques, such as fisher vectors and sparse coding; the use of OPF clustering improved the current results and can be applied to a large number of cases of image description of the BE and adenocarcinoma regions; the way of improving the selection of key points in each region (cancerous and non-cancerous) still shows potential considering the influence of the number of key points in each region for the correct classification result.
Regarding future works, we aim at considering deep learning and post-processing techniques after the construction of the bags, such as feature selection (i.e., visual word selection). Additionally, this post-processing can be performed using the large number of machine learning techniques, such as SVM, OPF or even Convolution Neural Networks, providing intermediate learning for the dictionaries calculation. More techniques for image description are also considered to be evaluated using the bag-of-visual-words provided by the OPF clustering.
Fig. 1
1Toy example: (a) unlabeled dataset and its (b) 3nearest neighbors graph.
Fig. 2
2Computing the densities of each graph node according to its 3-neighborhood. The values under/over the nodes stand for their density values computed using Equation 1.
Fig. 3
3Resulting optimum-path forest with two clusters and prototypes highlighted.
Fig. 4
4BE's short-segment (a) and BE's long-segment (b), with their respective endoscopic views (extracted from[37]).
Fig. 5
5Pipeline adopted in this work for Barrett's esophagus identification.
Fig. 6
6Some examples of images positive for cancer and their respective delineations (MICCAI 2015 dataset).
Fig. 7
7Some examples of images positive for cancer and their respective delineations (Augsburg dataset).
Fig. 8
8Misclassified image (patient 31) from MICCAI 2015 dataset: (a) gray-scale, (b) PoIs (SIFT), and (c) RGB version with delineations.
Fig. 9
9Misclassified image (patient 39) from Augsburg dataset: (a) gray-scale, (b) PoIs (A-KAZE), and (c) RGB version with delineation.
Table 1
1Mean accuracy results using A-KAZE features with 100, 500, and 1, 000 visual words.Dictionary
100
500
1000
k-means
OPF cpl
73.6%
76.3%
77.2%
OPF knn
59.2%
61.7%
68.1%
SVM-RBF
60.7%
65.9%
66.1%
SVM-Linear 58.5%
63.0%
67.4%
Bayes
56.8%
60.0%
60.9%
Random
OPF cpl
59.5%
63.9%
70.3%
OPF knn
58.3%
61.7%
62.3%
SVM-RBF
62.1%
65.6%
63.7%
SVM-Linear 55.3%
59.0%
59.1%
Bayes
55.5%
62.2%
61.1%
OPF clustering
OPF cpl
72.2%
73.1%
77.6%
OPF knn
62.3%
60.1%
66.2%
SVM-RBF
61.9%
65.1%
70.9%
SVM-Linear 55.8%
60.5%
66.8%
Bayes
55.8%
58.0%
61.3%
Table 2
2Mean accuracy results using SURF Features and 100, 500, and 1, 000 visual words.Dictionary
100
500
1000
k-means
OPF cpl
70.0%
74.8%
73.6%
OPF knn
64.1%
66.0%
65.1%
SVM-RBF
63.6%
64.8%
62.6%
SVM-Linear
62.0%
58.6%
62.8%
Bayes
56.4%
56.9%
57.4%
Random
OPF cpl
69.7%
70.2%
66.1%
OPF knn
58.0%
58.4%
61.8%
SVM-RBF
61.0%
63.4%
62.1%
SVM-Linear
51.7%
57.6%
56.5%
Bayes
50.5%
53.5%
56.9%
OPF clustering
OPF cpl
69.4%
78.4%
77.1%
OPF knn
63.6%
69.6%
71.6%
SVM-RBF
67.5%
71.8%
70.9%
SVM-Linear
65.1%
66.9%
66.7%
Bayes
53.3%
56.8%
57.1%
Table 3
3Mean accuracy results using SIFT Features and 100, 500, and 1, 000 visual words.Dictionary
100
500
1000
k-means
OPF cpl
68.3%
72.3%
71.4%
OPF knn
67.0%
71.8%
72.1%
SVM-RBF
67.3%
71.4%
71.9%
SVM-Linear
55.2%
56.8%
67.3%
Bayes
53.5%
60.0%
60.7%
Random
OPF cpl
66.4%
70.7%
71.2%
OPF knn
58.1%
63.9%
66.1%
SVM-RBF
62.1%
65.6%
63.7%
SVM-Linear
53.2%
54.5%
52.7%
Bayes
50.2%
53.0%
54.4%
OPF clustering
OPF cpl
71.2%
77.7%
78.9%
OPF knn
63.9%
71.3%
75.7%
SVM-RBF
68.0%
70.2%
69.7%
SVM-Linear
61.3%
64.7%
64.4%
Bayes
50.2%
53.0%
54.4%
Table 4 Mean
4accuracy results using A-KAZE Features and
100, 500, and 1, 000 visual words.
Dictionary
100
500
1000
k-means
OPF cpl
60.7%
69.4%
65.6%
OPF knn
61.9%
66.1%
70.1%
SVM-RBF
60.4%
63.5%
63.1%
SVM-Linear 55.1%
60.4%
62.1%
Bayes
56.9%
60.1%
61.3%
Random
OPF cpl
59.4%
68.4%
69.9%
OPF knn
59.9%
61.4%
62.4%
SVM-RBF
57.9%
62.2%
63.2%
SVM-Linear 55.3%
58.8%
58.9%
Bayes
56.5%
57.1%
61.0%
OPF clustering
OPF cpl
68.4%
68.7%
72.6%
OPF knn
67.4%
69.3%
70.3%
SVM-RBF
59.4%
63.0%
69.8%
SVM-Linear 57.7%
57.3%
62.7%
Bayes
62.4%
60.7%
63.1%
Table 5
5Mean accuracy results using SURF Features and 100, 500, and 1, 000 words.Dictionary
100
500
1000
k-means
OPF cpl
66.3%
67.9%
61.5%
OPF knn
62.8%
63.2%
65.4%
SVM-RBF
57.1%
61.1%
62.9%
SVM-Linear
56.7%
57.1%
59.4%
Bayes
60.8%
59.9%
61.1%
Random
OPF cpl
60.0%
62.2%
63.5%
OPF knn
54.2%
58.1%
60.8%
SVM-RBF
61.3%
61.9%
62.0%
SVM-Linear
57.1%
55.4%
56.4%
Bayes
51.9%
59.0%
59.1%
OPF clustering
OPF cpl
59.2%
62.1%
66.1%
OPF knn
61.1%
63.9%
64.5%
SVM-RBF
58.5%
62.0%
65.4%
SVM-Linear
53.5%
60.8%
64.6%
Bayes
59.8%
67.0%
67.9%
KAZE and SURF showed these to be quite more ac-
curate than SIFT, an opposite situation that occurred
over MICCAI 2015 dataset, where SIFT achieved the
best recognition rates. Additionally, the average num-
ber of PoIs used for training and test sets concerning
SIFT feature extractor were 89, 514 and 38, 363, respec-
tively, taking an average computational load of 8.71
minutes.
Table 6 Mean
6accuracy results using SIFT Features and 100,
500, and 1, 000 visual words.
Dictionary
100
500
1000
k-means
OPF cpl
60.3% 60.5%
59.3%
OPF knn
58.9%
60.6% 62.0%
SVM-RBF
60.8%
61.8%
59.8%
SVM-Linear
55.5%
57.1%
59.9%
Bayes
53.1%
54.8%
58.7%
Random
OPF cpl
59.2%
60.5%
61.6%
OPF knn
57.0%
58.4%
60.5%
SVM-RBF
57.8%
62.6% 62.1%
SVM-Linear
54.4%
55.6%
61.5%
Bayes
51.9%
57.0%
59.0%
OPF clustering
OPF cpl
60.4%
62.8% 62.1%
OPF knn
58.1%
61.6%
63.9%
SVM-RBF
57.0%
60.5%
62.1%
SVM-Linear
58.8%
58.9%
58.7%
Bayes
61.1%
62.2%
61.8%
Table 7
7Summarization of the results.Dataset
Accuracy Feature Extractor #visual words
MICCAI 2015
78.9%
SIFT
1, 000
Augsburg
72.6%
A-KAZE
1, 000
Table 8
8Mean sensitivity (i.e., positive to BE) and specificity (i.e., negative to BE) results.Dataset
Sensitivity Specificity
MICCAI 2015
81.7%
76.4%
Augsburg
70.9%
74.9%
Table 9
9Percentage of PoIs inside the delineated (cancerous) ares. For visualization purposes, Figures 8 to 9 depict some cancer patients that were misclassified as BE from both datasets. The PoIs showed inDataset
Feature
Cancer Cancer→BE
Extractor
PoI %
PoI %
MICCAI 2015
A-KAZE
30.34%
21.69%
MICCAI 2015
SURF
25.58%
23.05%
MICCAI 2015
SIFT
30.73%
23.04%
Augsburg
A-KAZE
53.77%
55.70%
Augsburg
SURF
42.97%
39.54%
Augsburg
SIFT
48.34%
44.06%
https://endovissub-barrett.grand-challenge.org/ home/
AckonowledgementsThe authors are grateful to DFG grant PA 1595/3-
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|
[] |
[
"Surface plasmon in 2D Anderson insulator with interactions Typeset using REVT E X 1",
"Surface plasmon in 2D Anderson insulator with interactions Typeset using REVT E X 1"
] |
[
"T V Shahbazyan \nDepartment of Physics\nUniversity of Utah\n84112Salt Lake CityUT\n",
"M E Raikh \nDepartment of Physics\nUniversity of Utah\n84112Salt Lake CityUT\n"
] |
[
"Department of Physics\nUniversity of Utah\n84112Salt Lake CityUT",
"Department of Physics\nUniversity of Utah\n84112Salt Lake CityUT"
] |
[] |
We study the effect of interactions on the zero-temperature a.c. conductivity of 2D Anderson insulator at low frequencies. We show that the enhancement of the real part of conductivity due to the Coulomb correlations in the occupation numbers of localized states results in the change of the sign of imaginary part within a certain frequency range. As a result, the propagation of a surface plasmon in a localized system becomes possible. We analize the dispersion law of the plasmon for the two cases: unscreened Coulomb interactions and the interactions screened by a gate electrode spaced by some distance from the electron plane.
|
10.1103/physrevb.53.7299
|
[
"https://export.arxiv.org/pdf/cond-mat/9508009v2.pdf"
] | 15,726,439 |
cond-mat/9508009
|
f14d9b5b00a4e2545d056c8f9c9a12c7d04207eb
|
Surface plasmon in 2D Anderson insulator with interactions Typeset using REVT E X 1
arXiv:cond-mat/9508009v2 2 Aug 1995
T V Shahbazyan
Department of Physics
University of Utah
84112Salt Lake CityUT
M E Raikh
Department of Physics
University of Utah
84112Salt Lake CityUT
Surface plasmon in 2D Anderson insulator with interactions Typeset using REVT E X 1
arXiv:cond-mat/9508009v2 2 Aug 1995
We study the effect of interactions on the zero-temperature a.c. conductivity of 2D Anderson insulator at low frequencies. We show that the enhancement of the real part of conductivity due to the Coulomb correlations in the occupation numbers of localized states results in the change of the sign of imaginary part within a certain frequency range. As a result, the propagation of a surface plasmon in a localized system becomes possible. We analize the dispersion law of the plasmon for the two cases: unscreened Coulomb interactions and the interactions screened by a gate electrode spaced by some distance from the electron plane.
I. INTRODUCTION
It is believed that in two-dimensions (2D) even small disorder localizes all electron states. 1 First hint for the absence of the mobility threshold in 2D came from the calculation of the weak localization correction, δσ(ω), to the conductivity at finite frequency ω. It was shown 2 that in the limit k F l ≫ 1 one has δσ(ω)/σ 0 = 1 k F l ln |ωτ |, where k F is the Fermi momentum, l is the mean free path, and τ is the elastic scattering time; σ 0 = (e 2 /h)k F l is the Drude conductivity. The logarithmic singularity in δσ(ω) indicates that the zerotemperature (T = 0) conductivity is metallic only if ω ≫ ω 0 , where
ω 0 ∼ 1 τ e −πk F l(1)
is the characteristic frequency which marks the crossover to the exponentially localized regime. 1 From the finite size, L, correction to the zero-frequency conductivity, which is of the order of 1 k F l ln(L/l), one can estimate the localization length ξ as 1 ξ = l exp π 2 k F l .
Therefore, starting from metal, one cannot describe the T = 0 behavior of the conductivity at frequencies ω ∼ ω 0 . The adequate language for this region would be the language of the localized states. Within this language the conductivity originates from the transitions between the localized states induced by an external a.c. field.
Let us briefly remind the corresponding derivation of σ(ω), carried out by Mott 3 in the strongly localized regime. The Hamiltonian for strongly localized electrons reads
H = i ǫ i c † i c i + ij I ij (c † i c j + c † j c i ),(3)
where ǫ i is the energy of a localized state centered at r = r i and I ij is the overlap integral which falls off exponentially with distance: I ij = I 0 exp(−|r i − r j |/a), where a is the size of the wave-function of the localized electron. The general expression for the conductivity is given by the Kubo formula σ(ω) = ie 2 ω hA ij | i|x|j | 2 n i − n j ω + ω ij + i0 (4) where i|x|j is the matrix element of x calculated from the exact eigenstates |i and |j of the Hamiltonian (3) with energies E i and E j ,hω ij = E i −E j and n i is the occupation number of the state |i .
The dissipative conductivity, Reσ(ω), is determined by the pairs of states with ω ij = ω.
Athω ≪ I 0 the spatial separation between bare states |i and |j is much larger than a.
Then in the calculation of eigenstates of a resonant pair one should take into account the overlap I ij within this pair only and neglect the overlap with all the other localized states.
This gives
|i = ǫ i − ǫ j Γ |i + 2I ij Γ |j , |j = 2I ij Γ |i + ǫ j − ǫ i Γ |j .(5)
The corresponging energies are
E i,j = (ǫ i + ǫ j )/2 ± Γ/2, Γ = [(ǫ i − ǫ j ) 2 + 4I 2 ij ] 1/2 =hω ij .(6)
Using (5) the matrix element in (4) takes the form
i|x|j = (x i − x j )I ij /Γ.(7)
The contribution to Reσ(ω) from pairs with shoulder r can be presented as
Reσ(ω, r) = e 2 h πx 2 I 2 (r) hω F (hω, r),(8)
where F (hω, r) is the density of pairs with shoulder r and excitation energyhω [here I(r) =
I 0 e −r/a ].
The density F (hω, r) is determined by the condition that the pair is singly occupied (with energies on the opposite sides from the Fermi level). Then we have
F (hω, r) = g 2 dǫ 1 dǫ 2 θ hω 2 + ǫ 1 + ǫ 2 2 θ hω 2 − ǫ 1 + ǫ 2 2 δ (ǫ 1 − ǫ 2 ) 2 + 4I 2 (r) −hω = 2g 2 (hω) 2 (hω) 2 − 4I 2 (r) ,(9)
where g is the density of states and θ(x) is the step-function. Substituting (9) into (8) and
integrating over r we get
Reσ(ω) = e 2 h (2π 2 g 2h ω) ∞ 0 drr 3 I 2 (r) (hω) 2 − 4I 2 (r) .(10)
Forhω ≪ I 0 the main contribution to the integral comes from r ∼ r ω = a ln(2I 0 /hω) ≫ a and one obtains
Reσ(ω) = √ 2π 2 e 2 h (g 2 ah 2 ω 2 r 3 ω ).(11)
Although we cannot show it explicitly, we argue below that the Mott expression (11) for Reσ(ω) is valid also for the Anderson insulator with k F l ≫ 1. When applying (11) to the Anderson insulator one should replace a by ξ from (2) Note that in 1D case a similar argument leads to I 0 ∼ 1/g 1 ξ, g 1 being the 1D density of states. Important is that in 1D the Kubo formula can be evaluated exactly 6 resulting in the 1D version of the Mott formula, from which one can recover the above estimate for I 0 .
Such a mapping was first established by Shklovskii and Efros. 7 Using Eq. (10) one can formally evaluate Reσ(ω) forhω > 2I 0 . In this case the main contribution to the integral comes from r ∼ a and we get
Reσ(ω) = 3π 2 4 e 2 h (I 2 0 g 2 a 4 ).(12)
Certainly, the presentation of I(r) in the form I 0 e −r/a makes sense only for r ≫ a This means that the numerical coefficient in Eq. (12) is not reliable.
Clearly, at large frequencies ω ≫ ω 0 the conductivity of the Anderson insulator should have the Drude form. The fact that Eq. (12), calculated for strongly localized electrons, is also frequency independent allows us to assume that the description of a.c. transport in the Anderson insulator based on Hamiltonian (3) is accurate within a numerical coefficient. In other words, we assume that despite the complex structure of the electron wave-functions in the Anderson insulator the energy dependence of the matrix elements calculated between these functions is still given by (7).
Eq. (12) provides yet another way to estimate I 0 in the limit k F l ≫ 1. Namely, it matches σ 0 if we take I 0 ∼ (k F l) 1/2 /gξ 2 . We see that the dependence of I 0 on ξ in both estimates is the same; the extra factor (k F l) 1/2 presumably can be accounted for the ln(k F l) corrections to the exponent of ξ.
There is also another argument in favor of the above estimate for I 0 . The frequency dependence of Reσ(ω) in the Anderson insulator becomes strong for ω ≪ ω 0 , whereas in the picture of strongly localized electrons the demarcation frequency is ω ∼ I 0 /h. Equating I 0 tohω 0 we get I 0 = k F l/gξ 2 , with another extra factor k F l.
The simplified description of the Anderson insulator based on the Hamiltonian (3) allows one to include into consideration the Coulomb correlations (i.e., the correlations in the occupation numbers of the localized states caused by electron-electron interactions) using the ideas first spelled out in Refs. 8,7,9. This is the main goal of the present paper. We study the effect of Coulomb correlations on both Reσ(ω) and Imσ(ω). The most drastic conclusion we come to is that due to modification of Imσ(ω) by the Coulomb correlations a system of localized electrons can support surface plasmons within a certain frequency range.
We also show that these plasmons cause an additional structure in the behavior of Reσ(ω)
at ω > ω 0 ∼ I 0 /h.
The paper is organized as follows. In Section II we analyze the polarizability of the localized system in the absence of Coulomb correlations. In Section III we introduce the Coulomb correlations and find the dispersion law for the surface plasmons. In Section III we study the corrections to the dispersion law due to the resonant scattering of plasmons by pairs of localized states. In Section IV we calculate the plasmon contribution to Reσ(ω).
Section V concludes the paper.
II. POLARIZABILITY IN THE ABSENCE OF COULOMB CORRELATIONS
In this section we demonstrate that without Coulomb correlations the 2D Anderson insulator cannot support surface plasmon.
In the framework of the linear response theory the dispersion law of a plasmon, ω(q), is determined from the condition 10
1 = v(q)ReP(ω, q),(13)
where v(q) is the Fourier component of the electron-electron interaction v(r) and P(ω, q)
is the polarization operator. Within a standard approach P(ω, q) is calculated for noninteracting electrons described by Hamiltonian (3):
P (ω, q) = 1 A ij | i|e iqr |j | 2 n i − n j hω + E i − E j + i0 .(14)
In the absence of disorder the eigenstates |i and |j are the plain waves so that the matrix element in (14) reduces to the delta-function δ(i − j − q). Then evaluating P (ω, q) and
substituting it into (13) together with 2D Coulomb interaction v(q) = 2πe 2 /κq (κ is the dielectric constant) yields the surface plasmon with the well-known dispersion law
ω(q) = 2πne 2 q mκ 1/2 ,(15)
where n is the 2D concentration of electrons. The plasmon mode is undamped if q < ω/v F where v F = (4πn) 1/2h /m is the Fermi velocity (for larger q the Landau damping leads to a finite ImP ). The latter condition can be rewritten as q < 1/2a B where a B =h 2 κ/me 2 is the Bohr radius.
In the case of a strong disorder the eigenstates i and j in (14) are the localized states.
The polarization operator (14) can be evaluated in a way similar to that employed in the Introduction for calculation of a.c. conductivity. For small q the matrix element in (14) can be evaluated using (7):
i|e iqr |j = iqrI(r)/Γ.(16)
Then the contribution to P (ω, q) from the pairs with the shoulder r [density P(ω, q, r) of the polarization operator] can be presented as
ReP(ω, q, r) = (qr) 2 I 2 (r) ∞ 2I(r) dΓ Γ F (Γ, r) (hω) 2 − Γ 2 ,(17)
where Γ and F (Γ, r) is given by Eqs. (6) and (9), respectively (the integral is understood as principal part). Substituting F (Γ, r) into (17) and integrating over Γ we obtain
ReP(ω, q, r) = 2g 2 (qr) 2 I 2 (r) ∞ 2I(r) dΓ Γ Γ 2 − 4I 2 (r)[(hω) 2 − Γ 2 ] = − πg 2 (qr) 2 I 2 (r) 4I 2 (r) − (hω) 2 , forhω < 2I(r), = 0, forhω > 2I(r).(18)
We see that ReP(ω, q, r) is either negative or zero, so that Eq. (13) cannot be satisfied.
In fact, the result (18) is almost obvious. Indeed, the imaginary part of polarization operator density, ImP(ω, q, r), at small q differs from Reσ(ω, r) in (8) by a factor e 2 ω/q 2 , so it follows from (8) and (9) that
ImP(ω, q, r) = − πg 2 (qr) 2 I 2 (r) (hω) 2 − 4I 2 (r) , forhω > 2I(r), = 0, forhω < 2I(r).(19)
Since ReP(ω, q, r) and ImP(ω, q, r) are connected via the Kramers-Kronig relation, the form (19) immediately follows from (18).
III. COULOMB CORRELATIONS AND SURFACE PLASMON
In the previous section the polarization operator was evaluated using the pair density (9) which was derived for non-interacting electrons. As it was first pointed out by Efros 8 , interactions modify strongly the density of singly-occupied pairs. The underlying physics is the following. A pair can be singly-occupied even if both energy states reside below the Fermi level. The right condition for the pair to be singly-occupied is that the addition of a second electron (which interacts with the first one) is energetically unfavorable. Such a Coulomb correlations effectively enhance the density of "soft" pairs (i.e., the pairs with small excitation energy Γ).
Our goal is to apply the latter argument, which was presented for strongly localized system, to the Anderson insulator with large ξ. In order to do so we will adopt two assumptions:
(i) The interactions do not change the localization radius ξ.
(ii) The estimate for the overlap integral, I 0 ∼ 1/gξ 2 , is unchanged in the presence of interactions.
In other words, we assume that switching on the interactions leads to the Coulomb shifts of the eigenenergies but does not affect the wave functions. Note that assumptions (i) and
(ii) contradict those made in Refs. 11 and 12, respectively.
As we will see below, the relevant pairs would be those with shoulder r ∼ ξ. In other words, the relevant transitions shift the position of electron by ∼ ξ. To establish the form of the density of singly occupied pairs F (Γ, r) with such a shoulder one can argue as follows.
An isolated region of a size ξ can be viewed as a small metallic granule. The transfer of an additional electron into this granule leads to the charging energy U = e 2 /2C, where C is the capacitance of a granule. In other words, the levels in the granule get shifted 13 by an amount ∼ U. Consider now two neighboring granules and assume that U ≫ I 0 . Due to aforementioned charging effect the highest occupied levels in the two granules typically differ by ∼ U. Let for concreteness the highest occupied level in the first granule be higher by U than in the second one. Then the sought singly occupied pair with frequency Γ can be composed from the top occupied states in the first granule (these states should belong to the energy interval Γ + U measured from the highest occupied level) and the empty states in the second granule. Then the density of pairs F (Γ, r) (with r ∼ ξ) can be estimated as
g 2 (Γ + U)
. With the energy splitting taken into account it can be written as
F (Γ, r) = 2g 2 Γ(Γ + U) Γ 2 − 4I 2 (r) .(20)
Then at U = 0 we return to (9). Certainly, our consideration, based on artificial arranging the localized states into the granules, provides only the order of magnitude estimate of F (Γ, r). In particular, the numerical coefficient in (20) cannot be found from such a consideration. Our choice of numerical coefficient in (20) provides matching with a similar expression for strongly localized regime. 7,9,14 With the pair density (20) we can now easily evaluate ReP(ω, q, r). Calculating the integral over Γ in (17) yields
ReP(ω, q, r) = − πg 2 (qr) 2 I 2 (r) 4I 2 (r) − (hω) 2 1 + 2 π Ū hω arctan hω 4I 2 (r) − (hω) 2 , forhω < 2I(r), = Ū hω 2g 2 (qr) 2 I 2 (r) (hω) 2 − 4I 2 (r) ln (hω) 2 − 4I 2 (r) +hω 2I(r) , forhω > 2I(r).(21)
The expression (21) for ReP(ω, q, r) can be also obtained, using the Kramers-Kronig relations, from ImP(ω, q, r) which has a simple form
ImP(ω, q, r) = − π(qr) 2 I 2 (r) (hω) 2 F (hω, r) = − πg 2 (qr) 2 I 2 (r) (hω) 2 − 4I 2 (r) 1 + Ū hω , forhω > 2I(r), = 0, forhω < 2I(r).(22)
Note that as in (19), ImP(ω, q, r) = 0 only forhω > 2I(r).
We see that the enhancement in the density of pairs with small Γ leads to a positive sign of ReP(ω, q, r) forhω > 2I(r). This is our main observation.
In order to obtain ReP (ω, q) one should integrate ReP(ω, q, r) over r. Forhω > 2I 0 the main contribution to this integral comes from r ∼ ξ [like in derivation of Eq. (12)] and we get ReP (ω, q) = 3π 4
q 2 UI 2 0 g 2 ξ 4 (hω) 2 ln hω I 0 .(23)
For generality we will assume that there is also a gate at a distance d from the plane of 2D
electrons. In this case the Fourier component of electron-electron interaction has the form
v(q) = 2πe 2 κq 1 − e −2qd .(24)
If the gate is close to the electron plane, that is qd ≪ 1, we can expand the exponent in (24) and get v(q) = 4πe 2 d/κ. If d ≪ ξ then the capacitance C reduces to the capacitance of two disks with area S = ξ 2 separated by a distance d, so that C = κS/4πd and consequently U = 4πe 2 d/κξ 2 . With these v(q) and U, after substituting (23) into the plasmon equation (13), we obtain
1 = 3π 4 (qξ) 2 Ū hω 2 (I 0 gξ 2 ) 2 ln hω I 0 .(25)
Certainly, the numerical coefficient in (25) should not be taken seriously. According to the assumtion (ii), I 0 ∼ gξ 2 . Then Eq. (25) yields the following dispersion law for the surface plasmon q(ω) = 1 ξ
hω U ln −1/2 hω I 0 .(26)
We see that the dispersion law is close to acoustic.
Let us establish the frequency range for the surface plasmon with dispersion law (26).
The validity of expansion (16), qξ ≪ 1, implies that U ≫hω ln −1/2 (hω/I 0 ). On the other hand,hω > 2I 0 . Then the frequency range for plasmon is 2I 0 < ∼h ω < ∼ U. The nesessary condition for this range to be wide is U ≫ I 0 . The ratio U/I 0 = 4πe 2 d/ξ 2 I 0 with I 0 ∼ 1/gξ 2 can be presented as 8π 2 d/a B , where a B =h 2 κ/me 2 is the Bohr raduis. Thus, the condition d ≫ a B insures that the plasmon equation (13) has a solution within a wide frequency range.
If d < a B , the screening of Coulomb interaction by the gate is strong and the number of soft pairs is not sufficient to change the sign of ReP (ω, q).
A similar condition can be obtained from the analysis of ImP (ω, q), which can be derived by integration of Eq. (22) over r ImP (ω, q) = 3π 2 8
q 2 I 2 0 g 2 ξ 4 (hω) 2 (hω + U) ∼ q 2 (hω) 2 (hω + U).(27)
The origin of ImP (ω, q) is the interaction of a plasmon with "resonant" pairs having excitation energy ω. In fact, ImP (ω, q) describes the resonant scattering of a plasmon by a pair of localized states. One can introduce a mean free path, l, associated with such a scattering and obtain ql ∼ ln 1/2 (hω/I 0 )/(1 +hω/U). Thus, the condition 2I 0 < ∼h ω < ∼ U reduces to the condition ql > ∼ 1.
In the absense of gate [qd ≫ 1 in Eq. (24)] one should take U = e 2 /κξ, so that U/I 0 ∼ ξ/a B ≫ 1. Then after a simple algebra we obtain
q(ω) = 1 ξ hω U 2 ln −1 hω I 0 .(28)
It can be shown that the above analysis of the validity applies in this case as well and leads to the same frequency range 2I 0 < ∼h ω < ∼ U. Within this range we again have qξ ≪ 1.
Thus, one should use Eq. (26) for qd < 1 and Eq. (28) for qd > 1. On the other hand, the magnitude of U depends on the ratio d/ξ. Note that for d > ξ one still can have qd < 1.
In this case the dispersion law is given by Eq. (26) with U = e 2 /κξ.
IV. RENORMALIZATION OF THE PLASMON DISPERSION LAW
In the previous section, when calculating the polarization operator, we took into account the Coulomb correlations within a pair, but neglected the effect of polarization of surrounding pairs on a given pair. On the other hand, by averaging of the polarization operator (14) over frequencies of pairs Γ and their shoulders r we have effectively replaced the localized system by a medium. The average polarization of this medium gave rise to a plasmon mode.
Within this procedure the "feedback" from surrounding pairs reduces to the interaction of a given pair with plasmons. In the present section we study the renormalization of the plasmon spectrum due to this effect.
Generally, the plasmon excitation is defined as a pole in the density-density correlation function, Π(ω, q, q ′ ), which is related to the polarization operator P(ω, q, q ′ ) by the Dyson equation Π(ω, q, q ′ ) = P(ω, q, q ′ ) + dq 1 (2π) 2 P(ω, q, q 1 )v(q 1 )Π(ω, q 1 , q ′ ).
Before averaging, both P(ω, q, q ′ ) and Π(ω, q, q ′ ) depend on two momentum variables q and q ′ . The approximation we made above reduces to replacing of P(ω, q, q ′ ) by its average P(ω, q), so that the solution of (29) takes the form
Π(ω, q) = P(ω, q) 1 − v(q)P(ω, q) .(30)
Then the pole of Π(ω, q) is determined by the plasmon equation (13).
As a next step, we took for P(ω, q) its expression (14) for non-interacting electrons, which represents a sum of polarizations of pairs,
P (ω, q) = 1 A ij | i|e iqr |j | 2 P ij (ω), P ij (ω) = n i − n j hω + E i − E j + i0 ,(31)
and performed the summation neglecting correlations between the pairs but with the Coulomb correlations within a pair included.
The renormalized P ij (ω) for a given pair can be obtained from the following procedure.
The function Π(ω, q) has a diagrammatic presentation in a form of a series of bubbles (ij), corresponding to P ij , connected by the Coulomb interaction lines (see Fig. 1a). First, we arrange into a single block the sum over all combinations of bubbles which appear between two bubbles (ij) (see Fig. 1b). Then we replace this block by its average, so that the result can be presented as two bubbles (ij) connected by a plasmon propagator Π(ω, q) (see Fig. 1b) (there is also an extra factor v(q) in each vertex). The renormalized bubble (ij) can be then obtained by summing up the series, consisting from the bubbles (ij), connected by plasmon lines (see Fig. 1c). The resulting expression for P ij (ω) reads
P ij (ω) = P ij (ω, ) 1 − P ij (ω)R ij (ω) ,(32)
with
R ij (ω) = dq (2π) 2 | i|e iqr |j | 2 v 2 (q)Π(ω, q).(33)
Finally, replacing P ij (ω) in (31) by P ij (ω) we obtain
P(ω, q) = 1 A ij | i|e iqr |j | 2 n i − n j hω + E i − E j − (n i − n j )R ij (ω)
.
The equations (30), (33), and (34) form a closed system which determines Π(ω, q) and, correspondingly, the renormalized dispersion law of a plasmon in a self-consistent way. The approximation made in order to get the closed system [replacement of a block by a sought function Π(ω, q)] is known as the effective-medium approximation.
The analysis of the system (30,33,34) reveals that the renormalization of the plasmon dispersion law is weak. Namely, the appearence of the term R ij (ω) in the denominator of (34) has the physical meaning that a pair (ij) acquires a finite life-time τ ij due to the interaction with plasmons. We will show that this life-time is long, i.e., 1/τ ij ≪ ω ij . It can be readily seen that the difference between the renormalized polarization ReP(ω, q) and
ReP (ω, q) originates from resonant pairs with (ω −ω ij ) ∼ 1/τ ij and r ∼ ξ (note that the main term, ReP (ω, q), is determined by the entire interval ω ij ∼ ω). If we neglect the dependence of the matrix element in (34) on ω ij then the renormalization correction to ReP (ω, q) would be identically zero. A finite correction results from a slight asymmetry of the matrix element within the narrow interval (ω − ω ij ) ∼ 1/τ ij . Then the relative magnitude of the correction is of the order of 1/ωτ ij with τ ij calculated for a pair with ω ij = ω. Expecting τ ij to be long, we can calculate it by substituting the nonrenormalized dispersion law of a plasmon into Eq. (33). Performing the integration we obtain
R ij (ω) =h τ ij ∼hω hω U I 0 U 2 ln −2 hω I 0 , with gate, ∼hω hω U 3 I 0 U 2 ln −3 hω I 0 , without gate.(35)
Since both ratios,hω/U and I 0 /U are small, the correction to the dispersion law,
δq(ω)/q(ω) ∼ 1/ωτ ij , is negligible.
In the next section we will see that the corresponding renormalization of Reσ(ω) is much larger than the renormalization of the dispersion law.
V. RENORMALIZATION OF THE REAL PART OF THE CONDUCTIVITY
As we have seen in the previous section, the renormalization of the polarization operator results in an appearance of ih/τ ij in the denominator of Eq. (34). Correspondingly, the renormalized expression (4) for Reσ(ω) can now be rewritten as
Reσ(ω) = e 2 ω hA Im ij I ij (x i − x j ) hω ij 2 1 ω + ω ij + i/τ ij ,(36)
where the summation is performed over the singly occupied pairs ij and in this way the Coulomb correlations are taken into account. In the limit τ ij → ∞ and for ω > 2I 0 one should use the pair density F (Γ, r), given by Eq. (20), in order to perform the summation.
The result is determined by resonant pairs with ω = ω ij = Γ/h:
Reσ(ω) = 3π 2 4 e 2 h (I 0 gξ 2 ) 2 1 + Ū hω = σ 0 1 + Ū hω .(37)
We see that within the frequency interval 2I 0 <hω < U the real part of the conductivity exceeds the Drude value due to Coulomb correlations.
When calculating the correction to Reσ(ω) caused by the finite value of τ ij it is important to realize thath/τ ij is maximal for soft pairs with small ω ij . This is because the matrix element i|e iqr |j is proportional to 1/ω ij . In the previous Section this was not important since the correction to ReP (ω, q) came from the resonant pairs only. Here, however, we have Im(ω + ω ij + i/τ ij ) −1 ∝ 1/τ ij ∝ 1/ω 2 ij , so that the soft pairs give the main contribution to the correction δReσ(ω). Assuming ω ij ≪ ω we can present this correction in the form
Reσ(ω) = e 2 hωA ij [I ij (x i − x j )] 2 (hω ij ) 2 τ ij = e 2 hωA ij I ij (x i − x j ) hω ij 4 dq (2π) 2 q 2 v 2 (q)ImΠ(ω, q),(38)
where we have substitutedh/τ ij = R ij from Eq. (33). The sum over pairs is again evaluated with the pair density (20)
1 A ij I ij (x i − x j ) hω ij 4 = 1 4 drr 4 I 4 (r) ∞ 2I(r) dΓ F (Γ, r) Γ 4 .(39)
The main contribution to the integral over Γ comes from the lower limit Γ ∼ I(r). Then the integral over r is again determined by r ∼ ξ, so that the sum (39) appears to be ∼ I 0 ξ 6 g 2 U.
As in the previous section, the value of the integral in (39) takes different values in the presence and in the absence of a gate. Finally we obtain
δReσ(ω) σ 0 ∼ hω U 3 hω I 0 ln −2 hω I 0 , with gate, ∼ hω U 5 hω I 0 ln −3 hω I 0 , without gate.(40)
Comparing (35) to (40) we see that both corrections are of the same order (I 0 /U) 2 athω ∼ I 0 .
However the correction to Reσ is much bigger athω ≫ I 0 . This reveals a new mechanism of absorption of a.c. field: by resonant excitation of plasmons. More precisely, the field polarizes the soft pairs (with ω ij ∼ I 0 /h) and the induced polarization excites the plasmon waves. Therefore, the energy of a.c. field is effectively absorbed by plasmons. The rapid increase of δσ with ω is caused by the number (phase volume) of plasmons which absorb the field.
Note that with correction δσ(ω) the total conductivity Reσ(ω) exhibits a rather complicated behavior. Forhω < 2I 0 the conductivity increases with ω. Then it passes through a maximum at ω ∼ I 0 /h and falls off with ω according to Eq. (37). However athω ∼ U(I 0 /U) 1/4 (with gate) andhω ∼ U(I 0 /U) 1/6 (without gate) we have δReσ(ω) ∼ Reσ(ω) and the conductivity starts rising again. On the other hand, the expression for δReσ was derived assuming that it is small. Therefore in the region δReσ > Reσ(ω) the renormalization of σ(ω) by plasmons is strong. In this case one cannot calculate τ ij using the bare polarization operator. The full analysis of the system (30,33,34) in this frequency range is out of the scope of the present paper.
VI. CONCLUSION
In the present paper we argue that the wave of electric field can propagate along the surface of the 2D Anderson insulator. The field originates from the density fluctuations of localized electrons. One should distinguish this wave from the usual plasmon in an ideal 2D gas with the dispersion law given by Eq. (15): (i) the derivation of (15) implies that ω ≫ 1/τ while we predict the existence of a plasmon at much lower frequencies ω > ∼ I 0 /h.
The minimal frequencies 1/τ and I 0 /h differ by a factor exp(πk F l); (ii) the plasmon (15) results from the solution of Eq. (13) with polarization operator calculated for free electrons.
Within this approximation Eq. (13) has no solutions for localized electrons. The solution appears only if one takes into account the Coulomb correlations in the occupation numbers of the localized states.
The obvious consequence of the existence of a plasmon excitation is that localized electrons "feel" a fluctuating electric field E(ω, q) with the spectral density
|E(ω, q)| 2 = κ 2 2πe 2 q 2 v 2 (q)ImP(ω, q) [1 − v(q)ReP(ω, q)] 2 + [v(q)ImP(ω, q)] 2 ,(41)
where P(ω, q) is the polarization operator (34). The spectral density (41) has a peak at ω = ω(q) corresponding to the dispersion law of a plasmon, which is different with and without gate.
The basic assumption of our theory is that in the presence of interactions the Anderson insulator is characterized by two energy scales: I 0 -the spacing between energy levels in the area ξ 2 and the charging energy U which is the Coulomb interaction of two localized electrons separated by a distance ∼ ξ. This energy modifies the pair density F (ω, r). Such a modification results in an enhancement [7][8][9]14 of the dissipative conductivity Reσ(ω). If we denote as ρ(ω) the matrix element of r calculated for a pair with frequency ω then
Reσ(ω) ∝ ωρ 2 (ω)F (ω, r ω ).(42)
As it was discussed in the Introduction, it is plausible to assume that for the Anderson insulator in the absence of interactions ρ(ω) behaves as ρ(ω) ∼ r ω for ω < ∼ I 0 /h and ρ(ω) ∼ ξI 0 /ω for ω > ∼ I 0 /h. Since the pair density in the absence of interactions is proportional to ω we reproduce Eqs. (11) and (12):
Reσ(ω) ∝ ω 2 , for ω < ∼ I 0 /h,(43a)
Reσ(ω) = const,
for ω > ∼ I 0 /h,(43b)
Where we neglected a logarithmic factor in (43a). We have assumed that the frequency dependence of ρ(ω) remains the same in the presence of interactions. 15 At the same time, F (ω, ξ) is changed drastically by interactions: F (ω, ξ) = const for ω < ∼ U/h and F (ω, ξ) ∝ ω
for ω > ∼ U/h. As a result, we get the following frequency dependence of Reσ(ω) in the presence of interactions [see also (37)]:
Reσ(ω) ∝ ω, for ω < ∼ I 0 /h,(44a)Reσ(ω) ∝ 1/ω, for I 0 /h < ∼ ω < ∼ U/h,(44b)Reσ(ω) = const, for ω > ∼ U/h,(44c)
This simple analysis shows that Reσ(ω) exhibits a maximum at ω ∼ I 0 /h (see also the end of the previous section). In fact, this maximum is intimately related to the existence of a plasmon. Indeed, for Eq. (13) to have solutions we need a positive sign of ReP(ω, q), which is equivalent to a positive Imσ(ω). At the same time, Imσ(ω) can be obtained from Reσ(ω)
using the Kramers-Kronig relation. In order to trace how the modification of Reσ(ω) by
Coulomb correlations leads to the change of sign of Imσ(ω), we can interpolate the frequency dependence of ρ(ω) as ρ(ω) ∝ [(hω) 2 + 4I 2 0 ] −1/2 and the frequency dependence of F (ω, ξ) as F (ω, ξ) ∝ (hω + U). Then we have
ReG(ω) =h ω(hω + U) (hω) 2 + 4I 2 0 , ω > 0,(45)
where G = σ e 2 /h is the conductance. It can be easily seen that Eq. (45) reproduces correctly all the limiting cases (44) and (44) both with and without charging effect. The imaginary part of the conductance calculated from (45) has a simple form:
ImG(ω) = −h ω[2I 0 − (2U/π) ln(hω/2I 0 )] (hω) 2 + 4I 2 0 , ω > 0,(46)
In Fig. 2 we have plotted ReG(ω) and ImG(ω) for different ratios U/2I 0 . We see that in the absence of Coulomb correlations (U = 0), ImG(ω) is strictly negative (in our calculation in Section II it turns to zero for ω > I 0 /h). However, at finite U the change of sign occurs at ω = (2I 0 /h) exp(πI 0 /U). For U ≫ I 0 this frequency is just 2I 0 /h.
As a final remark, let us outline the difference between our approach and that of Ref. 16.
In Ref. 16 the authors addressed electron-electron interactions in the strongly localized regime when the localization radius is much smaller than the interpair separation. In this case singly-occupied pairs can be considered as point-like dipoles. The dipole moment p k induced by an external field E 0 e −iωt can be written as
p µ k = α µν k [E ν 0 + l =k E (l)ν k ],(47)
where E (l)ν k is a component ν of the electric field caused by polarization of a dipole l which acts on the dipole k (summation over repeating indices ν is implied). The polarizability α µν k of a dipole k has the form
α µν k = 2e 2 h ρ µ k ρ ν k ω k ω 2 − ω 2 k = e 2 h ρ µ k ρ ν k ij P ij ,(48)
where ρ µ k and ω k are correspondingly the matrix element and the frequency of the dipole k. Here P ij is given by Eq. (31) with the states i and j making up the dipole k. To keep the discussion simple we will assume that the polarizability is isotropic, i.e., α µν k = α k δ µν . The field E (l)ν k can, in turn, be expressed through p k as
E (l)µ k = − ∂ ∂R µ p l R R 3 R=R k −R l ,(49)
where R k is the position of the dipole k.
Upon substituting (49) into (47) we obtain an infinite system of linear equations. It can be easily shown that iterating this system leads to the renormalization of polarizabilities of dipoles α k
α k = e 2 h ρ 2 k ij P ij 1 − P ij Σ k = 2e 2 h ρ 2 k (ω k + Σ k ) ω 2 − (ω k + Σ k ) 2 ,(50a)Σ k = − l α l ∂ ∂R µ R ν R 3 ∂ ∂R ν R µ R 3 R=R k −R l + · · · .(50b)
Here Σ k is the self-energy. Fleishman and Anderson 16 anylized this self-energy using the arguments similar to those put forward by Anderson 17 when he demonstrated the existence of localization transition for eigenfunctions of the Schroedinger equation with disorder. They argued that ImΣ takes a finite value with non-zero probability. This means that in the absence of an external field the system
p k + α k ∇ l =k p l R R 3 R=R k −R l = 0,(51)
has delocalized solutions. 18 In other words, by analogy to the Schoedinger equation in the tight-binding approximation, the eigenstates {p k } of the system (51) extend throughout the entire volume.
Fleishman and Anderson 16 considered a three-dimensional system and neglected the Coulomb correlations. Note that if one rewrites the system (51) in the momentum representation and averages it by factorizing the average of the product αp (mean-field) then one arrives at the plasmon equation (13). As it was demonstrated in Section II, this equation has no propagating solutions in the absence of Coulomb correlations. Our central point is that with the Coulomb correlations the delocalized solution exists even at the mean-field level. This solution, that is surface plasmon, is specific for the two-dimensional system and is characterized by the dispersion law ω(q).
An interesting question that could be addressed within the same approach is how the Coulomb correlations modify the a.c. Hall conductivity σ xy (ω) of the Anderson insulator. 5,19
and substitute g = m/2πh 2 (m is the electron mass). However, the question remains: what is the magnitude of I 0 in this case? A plausible estimate can be obtained in the spirit of the Thouless picture of localization. 4 By definition, I 0 represents the splitting of energy levels of two neighboring localized states (with centers at distance ∼ ξ). The estimate for I 0 emerges if one equates this splitting to the mean energy spacing for localized states centered within the area ∼ ξ 2 ,so that I 0 ∼ 1/gξ 2 (see also Ref. 5).
ACKNOWLEDGMENTSThe authors are indebted to B. I. Shklovskii for numerous discussions. In fact, Sections IV-VI have emerged from his hot contesting of our basic assumptions.
. P A Lee, T V Ramakrishnan, Rev. Mod. Phys. 57287P. A. Lee and T. V. Ramakrishnan, Rev. Mod. Phys. 57, 287 (1985).
. L P Gor'kov, A I Larkin, D E Khmelnitskii, Pis'ma Zh. Eksp. Teor. Fiz. 30228JETP Lett.L. P. Gor'kov A. I. Larkin and D. E. Khmelnitskii, Pis'ma Zh. Eksp. Teor. Fiz. 30, 248 (1979) [JETP Lett. 30, 228 (1979)].
. N F Mott, Phil. Mag. 227N. F. Mott, Phil. Mag. 22, 7 (1970).
. D J Thouless, Phys. Rev. Lett. 391167D. J. Thouless, Phys. Rev. Lett. 39, 1167 (1977).
A similar estimate for the overlap integral in the case of the Anderson insulator was used by Y. Imry. Phys. Rev. Lett. 711868in calculation of the Hall conductivity at finite frequenciesA similar estimate for the overlap integral in the case of the Anderson insulator was used by Y. Imry, Phys. Rev. Lett. 71, 1868 (1993), in calculation of the Hall conductivity at finite frequencies.
. V L Berezinskii, Zh. Eksp. Teor. Fiz. 65Sov. Phys. JETPV. L. Berezinskii, Zh. Eksp. Teor. Fiz. 65, 125, (1973) [Sov. Phys. JETP 38, 620 (1974)].
. B I Shklovskii, A L Efros, Zh. Eksp. Teor. Fiz. 81Sov. Phys. JETPB. I. Shklovskii and A. L. Efros, Zh. Eksp. Teor. Fiz. 81, 406, (1981) [Sov. Phys. JETP 54, 218 (1981)].
. A L Efros, Phil. Mag. 43829A. L. Efros, Phil. Mag. B43, 829 (1981).
A L Efros, B I Shklovskii, Electron-Electron Interactions in Disordered Systems. A. L. Efros, and M. PollakAmsterdamNorth-HollandA. L. Efros and B. I. Shklovskii, in Electron-Electron Interactions in Disordered Systems, edited by A. L. Efros, and M. Pollak (North-Holland, Amsterdam, 1985).
G D Mahan, Many-Particle Physics. New YorkPlenumG. D. Mahan, Many-Particle Physics, (Plenum, New York, 1990).
. I I Aleiner, B I Shklovskii, Phys. Rev. B. 49I. I. Aleiner and B. I. Shklovskii, Phys. Rev. B 49, 13 721 (1994).
. D G Polyakov, B I Shklovskii, Phys. Rev. B. 48167D. G. Polyakov and B. I. Shklovskii, Phys. Rev. B 48, 11 167 (1993);
. B I Shklovskii, Pis'ma Zh. Eksp. Teor. Fiz. 36352JETP Lett.B. I. Shklovskii, Pis'ma Zh. Eksp. Teor. Fiz. 36, 287 (1982) [JETP Lett. 36, 352 (1983)].
. A L Efros, Zh. Eksp. Teor. Fiz. 891057Sov. Phys. JETPA. L. Efros, Zh. Eksp. Teor. Fiz. 89, 1834, (1985) [Sov. Phys. JETP 62, 1057 (1985)].
Note that in Ref. 12 it was assumed that interactions result in the enhancement of ρ(ω) by a factor U/I 0. Note that in Ref. 12 it was assumed that interactions result in the enhancement of ρ(ω) by a factor U/I 0 .
. L Fleishman, D C Licciardello, P W Anderson, Phys. Rev. Lett. 401340L. Fleishman, D. C. Licciardello, and P. W. Anderson, Phys. Rev. Lett. 40, 1340 (1978);
. L Fleishman, P W Anderson, Phys. Rev. B. 212366L. Fleishman and P. W. Anderson, Phys. Rev. B 21, 2366 (1980).
. P W Anderson, Phys. Rev. 1091492P. W. Anderson, Phys. Rev. 109, 1492 (1958).
See also the renormalization group analysis of this system in L. S. Levitov. Phys. Rev. Lett. 64547See also the renormalization group analysis of this system in L. S. Levitov, Phys. Rev. Lett. 64, 547 (1990).
. S.-C Zhang, S Kivelson, D.-H Lee, Phys. Rev. Lett. 691252S.-C. Zhang, S. Kivelson, and D.-H. Lee, Phys. Rev. Lett. 69, 1252 (1992). FIGURES
Each vertex contains matrix element i|e iqr |j . (b) The block connecting two bubbles (ij) is replaced by a plasmon line. (c) The renormalization of the bubble (ij) caused by interaction with plasmons. FIG. 2. The real (a) and imaginary (b) parts of conductance is plotted as a function of dimensionless frequencyhω/2I 0 for U/2I 0 = 0. U/2I 0 = 3); bubble (ij) stands for P ij (ω) while dashed line corresponds to the Coulomb interaction v(q). solid line. and U/2I 0 = 8 (dashed lineFIG. 1. (a) Diagrammatic presentation of the density-density correlation function Π(ω, q); bub- ble (ij) stands for P ij (ω) while dashed line corresponds to the Coulomb interaction v(q). Each vertex contains matrix element i|e iqr |j . (b) The block connecting two bubbles (ij) is replaced by a plasmon line. (c) The renormalization of the bubble (ij) caused by interaction with plasmons. FIG. 2. The real (a) and imaginary (b) parts of conductance is plotted as a function of dimen- sionless frequencyhω/2I 0 for U/2I 0 = 0 (solid line), U/2I 0 = 3 (long-dashed line), and U/2I 0 = 8 (dashed line).
|
[] |
[
"Charge regulation of patchy charged particles Anomalous Multipole Expansion: Charge Regulation of Patchy, Inhomogeneously Charged Spherical Particles",
"Charge regulation of patchy charged particles Anomalous Multipole Expansion: Charge Regulation of Patchy, Inhomogeneously Charged Spherical Particles"
] |
[
"Anže Lošdorfer Božič \nDepartment of Theoretical Physics\nJožef Stefan Institute\nSI-1000LjubljanaSlovenia\n",
"Rudolf Podgornik \nDepartment of Theoretical Physics\nJožef Stefan Institute\nSI-1000LjubljanaSlovenia\n\nDepartment of Physics\nFaculty of Mathematics and Physics\nUniversity of Ljubljana\nSI-1000LjubljanaSlovenia\n\nSchool of Physical Sciences and Kavli Institute for Theoretical Sciences\nUniversity of Chinese Academy of Sciences\n100049BeijingChina\n\nInstitute of Physics\n) CAS Key Laboratory of Soft Matter Physics\nChinese Academy of Sciences\n100190BeijingChina\n"
] |
[
"Department of Theoretical Physics\nJožef Stefan Institute\nSI-1000LjubljanaSlovenia",
"Department of Theoretical Physics\nJožef Stefan Institute\nSI-1000LjubljanaSlovenia",
"Department of Physics\nFaculty of Mathematics and Physics\nUniversity of Ljubljana\nSI-1000LjubljanaSlovenia",
"School of Physical Sciences and Kavli Institute for Theoretical Sciences\nUniversity of Chinese Academy of Sciences\n100049BeijingChina",
"Institute of Physics\n) CAS Key Laboratory of Soft Matter Physics\nChinese Academy of Sciences\n100190BeijingChina"
] |
[] |
Charge regulation is an important aspect of electrostatics in biological and colloidal systems, where the charges are generally not fixed, but depend on the environmental variables. Here, we analyze the charge regulation mechanism in patchy, inhomogeneously charged spherical particles, such as globular proteins, colloids, or viruses. Together with the multipole expansion of inhomogeneously charged spherical surfaces, the charge regulation mechanism on the level of linear approximation is shown to lead to a mixing between different multipole moments depending on their capacitance -the response function of the charge distribution with respect to the electrostatic potential. This presents an additional anomalous feature of molecular electrostatics in the presence of ionic screening. We demonstrate the influence of charge regulation on several examples of inhomogeneously charged spherical particles, showing that it leads to significant changes in their multipole moments.
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10.1063/1.5037044
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[
"https://arxiv.org/pdf/1804.07596v2.pdf"
] | 51,690,761 |
1804.07596
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5edc52b0cfbe03040b87deeaf52efef818f95126
|
Charge regulation of patchy charged particles Anomalous Multipole Expansion: Charge Regulation of Patchy, Inhomogeneously Charged Spherical Particles
Anže Lošdorfer Božič
Department of Theoretical Physics
Jožef Stefan Institute
SI-1000LjubljanaSlovenia
Rudolf Podgornik
Department of Theoretical Physics
Jožef Stefan Institute
SI-1000LjubljanaSlovenia
Department of Physics
Faculty of Mathematics and Physics
University of Ljubljana
SI-1000LjubljanaSlovenia
School of Physical Sciences and Kavli Institute for Theoretical Sciences
University of Chinese Academy of Sciences
100049BeijingChina
Institute of Physics
) CAS Key Laboratory of Soft Matter Physics
Chinese Academy of Sciences
100190BeijingChina
Charge regulation of patchy charged particles Anomalous Multipole Expansion: Charge Regulation of Patchy, Inhomogeneously Charged Spherical Particles
(Dated: 1 June 2018)
Charge regulation is an important aspect of electrostatics in biological and colloidal systems, where the charges are generally not fixed, but depend on the environmental variables. Here, we analyze the charge regulation mechanism in patchy, inhomogeneously charged spherical particles, such as globular proteins, colloids, or viruses. Together with the multipole expansion of inhomogeneously charged spherical surfaces, the charge regulation mechanism on the level of linear approximation is shown to lead to a mixing between different multipole moments depending on their capacitance -the response function of the charge distribution with respect to the electrostatic potential. This presents an additional anomalous feature of molecular electrostatics in the presence of ionic screening. We demonstrate the influence of charge regulation on several examples of inhomogeneously charged spherical particles, showing that it leads to significant changes in their multipole moments.
I. INTRODUCTION
In general, charges in biological and colloidal systems are not fixed, but result from some charge separation mechanism stemming from dissociation/association equilibria. Typical examples are the acid/base equilibria of dissolved salts and minerals, dissociation of protonating/deprotonating titrating groups in membrane lipids or protein amino acids, and adsorption of solution ions onto solid substrates in aqueous solutions 1 . These charge separation mechanisms all involve a specific chemical component, which can be described both by a chemical equilibrium constant, arising from detailed quantum chemical considerations, as well as by a more coarse-grained electrostatic component, originating in the dissociated charges. The latter component was first considered in the context of "protein ionization" by Linderstrøm-Lang in the 1920s 2 . The conceptual basis of these acid/base equilibria was developed by Marcus 3 , who considered the effect of solution electrostatic interactions on the fractional charging and titration curves of general polyelectrolytes in aqueous electrolyte solutions as a function of the ionic strength of the bathing solution. A similar approach for the description of protein titration curves within a dielectric continuum model was developed by Tanford and Kirkwood 4,5 . The framework of charge separation mechanisms was substantially widened when Ninham and Parsegian 6 coupled the charge dissociation processes with the interactions between dissociable surfaces within the paradigm of charge regulation (CR), which found application in various domains of the theory of the stability of colloids 7,8 . In more recent decades, CR was a) Electronic mail: [email protected] formalized for surface binding either by using the law of mass action 9,10 , or equivalently, by using a model-specific surface free energy [11][12][13][14][15][16][17][18][19][20][21][22] . CR is also closely related to the binding of ions to small molecules, proteins, polymers, colloid particles, and membranes, as well as to proton binding to these substrates, and has been reviewed in detail 1, 23 .
Electrostatic interactions dominate many aspects of protein behavior, and cannot be properly understood without paying due attention to the protonation and deprotonation of their constituent ionizable amino acid residues 24,25 , with the implied CR of the protein-specific distribution of dissociable charges 26,27 . In order to quantify the electrostatic interactions in proteins, one needs to encode not only the magnitudes of their charges, but also the anisotropic part of their distribution along their solvent-exposed molecular surface 28,29 . This naturally leads to a multipole expansion of the protein surface charge density 30 . Multipole moments of high order are known to influence the phase equilibria of concentrated protein solutions 31 , orientationally steer protein complexes into place 32 , and can be used as a versatile phenomenological tool to dress up bare spheres in the first step towards the consideration of more complicated details of both protein charge distributions as well as patchy charge distributions in general 33,34 . Each multipole in this expansion series describes a particular charge motif, starting from the simplest monopolar, sphericallysymmetric distribution 35 . The most straightforward representation of the multipole expansion is obtained by mapping the charge distribution on the original solventaccessible protein surface onto a sphere circumscribed to the protein 30,[36][37][38] . Such a multipole expansion provides a mapping between a coarse-grained collective description of the charge density and the underlying detailed microscopic charge site distribution, so that any level of detail can be reached if enough multipole orders of the expansion are taken into account.
While a small number of multipole moments can often be used as a proxy to characterize the microscopic details of charge distributions in charged macromolecules 28,29 , the details of the standard (Coulomb) multipole expansion 39 are substantially modified when the macromolecules are placed in a screening environment of aqueous bathing solutions. In fact, contrary to the standard multipole expansion, the screened electrostatic potential retains the full directional dependence for all multipole moments, so that the effects of charge anisotropy and high-order multipole moments extend all the way to the far-field region [40][41][42][43] . This entails an admixture of higher multipole moments to all the low-order multipoles -including the monopole -so that the usual argument that at large separations between charge distributions only the monopole moment matters is not substantiated. While this anomalous property of screened multipole electrostatics has been derived some time ago, it is often overlooked, e.g., in the modelling of protein-protein interactions.
In what follows, we will show that the CR mechanism in conjunction with the multipole moment expansion of inhomogeneously charged spherical surfaces leads to an additional anomalous feature of molecular electrostatics, similar to the full directional dependence of all multipole moments in a screened environment, yet differing in relevant details. CR mechanism will be shown to imply a mixing between different multipoles depending on their capacitance, i.e., the response function of the charge distribution with respect to the electrostatic potential. We will base our approach on a reformulation of the CR problem, akin to the framework set forth by Marcus 3 : We will start with the multipole expansion of the Debye-Hückel (DH) electrostatic energy for general surface charge distributions, weaving them afterwards into the CR theory through the corresponding CR free energy terms. The nature of the CR of dissociable groups will immediately yield an equation coupling a given multipole moment to other multipole moments. After deriving a linearized form of the CR theory, fully consistent with the DH approximation, we will provide a comprehensive discussion of these anomalous CR effects on the electrostatic multipoles of patchy spherical colloids and globular proteins with inhomogeneous surface charge distributions.
II. SPHERICAL SURFACE CHARGE DENSITY IN PRESENCE OF CHARGE REGULATION
A. Model of surface charge density While our derivation of the CR mechanism can be generalized to arbitrary inhomogeneously charged surfaces (with 33 or without 30 any symmetry), our main concern will be spherical particles, such as patchy colloids and globular proteins (Fig. 1a). To model their surface charge densities, we will assume that all association/dissociation sites η k are located on a sphere with radius R. The charge on each of these sites, q k , will be modelled by a normal distribution on the sphere, characterized by its mean direction Ω k = (ϑ k , ϕ k ) and the spread around this direction λ k (the "patchiness" of the charge) 44 . In addition, the surface charge distribution can consist of two different types of moieties: The first one can acquire a positive charge by protonation, and we write q + k = eη k , where η k ∈ [0, 1]. The second one acquires a negative charge by deprotonation, and we write q − k = e(η k − 1), where again η k ∈ [0, 1].
The total surface charge distribution on the sphere is then given by 44
σ(Ω) = 1 4πR 2 N k=1 q ± k λ k sinh λ k exp(λ k cos γ k ),(1)
where cos γ k is the great circle distance between Ω and Ω k . When λ → 0, the distribution becomes uniform on the sphere, whereas in the opposite limit of λ → ∞, we obtain a distribution composed of point charges described by Dirac delta functions. Details of this model are given in Ref. 44. Figure 1b shows an example of the surface charge distribution of a globular protein with λ = 20 for all the constituent charges, mapped from a sphere onto a plane using the Mollweide projection 30,42 .
Writing the contribution of an individual charge as σ (k) (Ω), we expand it in terms of multipole moments as
σ (k) (Ω) = 1 4πR 2 l,m σ (k) (lm) Y lm (Ω).(2)
The corresponding multipole expansion coefficients are 44
σ (k) (lm) = 4π q k g l (λ k ) Y * lm (Ω k ),(3)
where we have introduced the function
g l (λ) = λ sinh λ i l (λ),(4)
and i l (x) are the modified spherical Bessel functions of the first kind. The multipole coefficients of the total surface charge density are then simply the sum of the contributions of the coefficients of individual charges, σ(lm) = k σ (k) (lm).
B. Charge regulation
The surface charge density on the spheres should not, however, be assumed a priori, but should follow from the minimization of the relevant total thermodynamic potential, yielding the equilibrium state in terms of equilibrium charge densities on the surface without any additional assumptions. This CR mechanism can be formalized either by invoking the chemical dissociation equilibrium of the surface binding sites with the corresponding law of mass action, or equivalently by adding a model surface free energy to the Poisson-Boltzmann (PB) or DH bulk free energy. The latter approach then leads to the same basic self-consistent CR boundary conditions for surface dissociation equilibrium through a minimization procedure, but without an explicit connection with the law of mass 30 . Panel (b) shows the "bare" surface charge distribution (in absence of CR) of 2BLG projected onto a plane using the Mollweide projection 30,42 . The distribution is shown at pH = 10, n0 = 100 mM, εp = 4, and λ = 20; these parameters are described in more detail in the main text. Grey stars show the positions of the Cartesian coordinate axes. (c) Linearized CR mechanism, shown on the examples of a positive charge with pKa = 10 and a negative charge with pKa = 3. Linearized CR contribution to each charge consists of its (pH-dependent) bare charge q (0) and bare capacitance c (0) [Eqs. (18) and (19)]. This mechanism leads to deviations from the bare charge distribution in absence of CR. Panel (d) shows the difference between the surface charge distributions of 2BLG at pH = 10 in presence and absence of CR. Other parameters of the distributions are the same as in panel (b).
action. We will use this approach which will allow us to derive some useful approximations.
We first write down the total free energy of the system:
F = F DH [ψ] + F CR [η k ].(5)
The first term, F DH [ψ], is the DH free energy, dependent on the electrostatic potential ψ of the system arising from the charge density σ(Ω) on the surface of the sphere. The DH free energy has the standard form
F DH [ψ(r)] = − 1 2 εε 0 V d 3 r (∇ψ) 2 + κ 2 ψ 2 + S dS σ(Ω) ψ(R, Ω),(6)
where κ 2 = 2e 2 n 0 /(εε 0 k B T ) is the square of the inverse Debye screening length. Here, T is the temperature, k B the Boltzmann constant, and n 0 the bulk univalent salt concentration. The surface integral in Eq. (6) runs over the entire surface of the sphere. A full PB free energy could also be used here; however, its use would preclude all analytical calculations, which can however illuminate the problem in spite of their approximate nature, as we will show later on. The second, CR term of the free energy in Eq. (5) can be written as 45
F CR [η k ] = k α k η k + 1 β k η k ln η k + (1 − η k ) ln(1 − η k ) ,(7)
and corresponds to the Langmuir-Davies isotherm, or indeed to the Ninham-Parsegian CR condition 6 . Here, α k represents the non-electrostatic dissociation free energy penalty of the site k. Other models of course exist, leading to different surface charging isotherms 1 . Next, by minimizing the total free energy [Eq. (5)] with respect to ψ, solving for ψ, and then evaluating the first two terms in Eq. 5, we remain with
F = F ES [σ(Ω)] + F CR [η k ] = F [η k ].(8)
Here, F ES [σ(Ω)] is now the equilibrium DH electrostatic free energy for a surface charge distribution given by σ(Ω), in absence of CR. Using furthermore the multipole expansion of the surface charge density, the free energy can be written as
F ES [σ(Ω)] = 1 2 S dS σ(Ω) ψ(R, Ω) = 1 8π dΩ ψ(R, Ω) l,m σ(lm) Y lm (Ω). (9)
The last equality in Eq. (9) follows from the definition of the multipole expansion [Eq. (2)], additionally implying
F ES [σ(Ω)] → F ES [η k ].
A full PB electrostatic free energy would differ from the above expression only in the substitution
1 2 σ(Ω) ψ(R, Ω) → σ(Ω) 0 ψ(R, Ω) dσ(Ω),(10)
corresponding to the Casimir charging process 46 , with everything else remaining unchanged.
Minimizing now further the expression for the free energy in Eq. (8) with respect to η i , we obtain the CR condition
∂F ES [η k ] ∂η i + α i + 1 β ln η i 1 − η i = 0.(11)
In absence of electrostatics (F ES = 0), the CR condition yields
η i = 1 1 + e βαi ,(12)
where we can identify βα i = ln 10(pH − pK (i) a ) 30 . Here, pK (i) a is the association/dissociation constant of the i-th charged group. Equation (11) then gives the full electrostatic modification of this result:
η i = 1 + exp ln 10(pH − pK (i) a ) + β ∂F ES [η k ] ∂η i −1 .(13)q ± i = ±e 1 + exp ± ln 10(pH − pK (i) a ) ± β ∂F ES [η k ] ∂ηi .(14)
To evaluate the derivative of F ES [σ(Ω)] = F ES [η k ], we can rewrite it in terms of σ(lm), thus obtaining
∂F ES [σ(Ω)] ∂η i = 4πe l,m ∂F ES [σ(lm)] ∂σ(lm) g l (λ i ) Y * lm (Ω i ),(15)
where we have taken into account the multipole expansion of the surface charge density, Eq. (3). This expression can be inserted back into Eq. (14) to obtain a set of non-linear equations for the charges q k . In addition, one can also evaluate the equilibrium free energy after the values for the CR charges have been obtained. Inserting Eq. (11) into Eq. (8), we are left with
F [η k ] = F ES [η k ] − i ∂F ES [η k ] ∂η i η i − 1 β i ln 1 + exp −βα i − β ∂F ES [η k ] ∂η i .(16)
In this way, we have expressed the total free energy of the system through its electrostatic part and its derivatives. The above equation is of course valid only for the Langmuir-Davies dissociation isotherm, and different forms would be obtained for different models of CR.
C. Linearized CR approximation
Since the free energy we have used in our derivation is based on the DH approximation, valid for small electrostatic potentials and thus necessarily for small electrostatic interactions, an expansion of Eq. (14) in terms of the electrostatic contribution is in order. This is the logic behind the linear CR approximation previously used to tame the non-linearity of the CR theory 47,48 , and we use the same argumentation also in our case. We thus proceed with the linear order of the CR condition that leads to the following simplified form:
q i = q (0) i − βe c (0) i ∂F ES [η k ] ∂η i ,(17)
where q (0) i is the bare regulated charge of the i-th moiety in absence of electrostatics [Eq. (12)]:
q (0) i = ±e 1 + exp ± ln 10(pH − pK (i) a ) ,(18)
and c
= 1 e ln 10 ∂ q
(0) i ∂ pH ,
The last line explicitly shows that this is indeed a capacitance associated with the i-th moiety 26 . In this context, we use the term "bare" to denote that the electrostatic contribution to these quantities has not been taken into account. Figure 1c shows the pH dependence of the bare charge and bare capacitance of a single positive and negative charge with pK a = 10 and pK a = 4, respectively. We see that, first of all, when the bare charges are "fully" charged, q (0) ± = ±e, their bare capacitance is negligible. The latter attains a maximum when the charges reach their mid-point, q (0) ± = ±e/2, where c (0) ± = 1/4. This occurs when pH is equal to the pK a value of the charge. When the bare charge goes to zero, so does the bare capacitance, with their ratio becoming equal to 1. With this in mind, we can already predict that CR should have the biggest influence close to the pK a value of an individual charge, and gradually lose in importance as the pH value moves further away from the pK a .
D. Linearized CR approximation and the multipole expansion
The linearized form of CR can also be written on the level of the multipole coefficients of the surface charge density σ(lm). Inserting Eq. (17) into Eq. (3), and at the same time taking into account Eq. (15), we are left with
σ(lm) = σ (0) (lm) − β(4πe) 2 k c (0) k × p,q ∂F ES ∂σ(pq) g l (λ k ) g p (λ k ) Y * lm (Ω k ) Y * pq (Ω k ) ,(20)
which is completely equivalent to the local relation of Eq. (17). The expression for σ (0) (lm) is identical to the one for σ(lm) with q k → q (0) k . In order to stress the physical content of the condition in Eq. (20), we can use the well-known formula for the product of two spherical harmonics 49 and write
σ(lm) = σ (0) (lm) − β(4πe) 2 L,M p,q T (lm|pq|LM ) ∂F ES [σ(pq)] ∂σ(pq) C (0) (LM |l|p),(21)
where T (lm|pq|LM ) = (2l + 1)(2p + 1)(2L + 1) 4π
× l p L m q M l p L 0 0 0(22)
connects the product of two different spherical harmonic functions to the sum over a single spherical harmonic function, all at the same solid angle. The expressions in parentheses denote the Wigner 3-j symbols. In Eq. (21) we have also introduced the collective multipole capacitance of the complete charge distribution, obtained by summing over all the sites:
C (0) (LM |l|p) = k c (0) k g l (λ k ) g p (λ k ) Y * LM (Ω k ).(23)
This expression connects the bare capacitance of the k-th site c Equation (21) thus connects only the collective variables: the multipole moments σ(lm) and the multipole capacitance C (0) (LM |l|p). Since the derivative ∂F ES [σ(pq)]/∂σ(pq) is a linear function of σ(pq) in the DH approximation, it follows that the CR condition mixes different spherical harmonic coefficients in such a way that M = m + q and | l − p| L | l + p|. This admixture of different multipole moments is a simple consequence of the fact that the association/dissociation equilibrium giving rise to CR condition [Eq. (14)] selfconsistently couples the local charge at each site with the value of the global electrostatic potential resulting from the charges of all the other sites. The mixing of different multipoles then follows straightforwardly from the connection between the local site variables and the collective multipole moments.
III. SINGLE SPHERE WITH INHOMOGENEOUS SURFACE CHARGE DISTRIBUTION
The derived analytical expressions for the charges and multipole moments in presence of CR can be used in any system where the form of the electrostatic free energy is known. As an example, we will apply them to spherical shells with the same bathing solution on both sides of a thin, proteinaceous surface, such as viral capsids and virus-like particles, as well as to globular proteins or patchy colloids with a dielectric core impermeable to the bathing solution.
In general, the electrostatic free energy of an inhomogeneously charged sphere with radius R can be written in the DH limit as 42,50
F ES [σ(lm)] = 1 32π 2 ε w ε 0 R l C(l, κR) m |σ(lm)| 2 .(24)
The only difference between the the case of an ionpermeable shell and the case of an impermeable, dielectric sphere lies in the function C(l, κR), which is defined as
C P (l, x) = I l+1/2 (x) K l+1/2 (x)(25)
for the ion-permeable shell, and
C D (l, x) = 1 x (ε − 1) l x − K l+3/2 (x) K l+1/2 (x) −1(26)
for the impermeable dielectric sphere. Here, I l and K l are the modified Bessel functions of the first and second kind, respectively, ε w is the dielectric constant of water, and we have defined for the dielectric sphere the ratio ε = ε p /ε w , where ε p is the dielectric constant of the sphere. From Eq. (24) we can immediately write down the derivative of the free energy with respect to the chargeable moieties:
∂F ES [η k ] ∂η i = e 4πε w ε 0 R l (2l + 1) g l (λ i ) C(l, κR) × k q k g l (λ k ) P l (cos γ ik ),(27)
where we have applied the addition theorem for spherical harmonics 49 . (Alternatively, one could use here also the expression for the derivative of the free energy with respect to the multipole coefficients.) Introducing the Bjerrum length in water, B = βe 2 /4πε w ε 0 , we next define
ξ ik = B R l (2l + 1) g l (λ i ) g l (λ k ) C(l, κR) P l (cos γ ik ) (28) and thus obtain βe ∂F ES [η k ] ∂η i = k ξ ik q k .(29)
We also note that ξ ik = ξ ki . Taking into account the linearized CR approximation [Eq. (17)], we are left with a linear system of equations:
k δ ik + c (0) i ξ ik q k = q (0) i .(30)
To solve this system of equations, we need the knowledge of the bare charges (q i ) at a given pH, as well as the corresponding coefficients ξ ik . From Eq. (30) we can again clearly observe the admixing of different charging sites proportional to the capacitance of a given site.
A. Examples
Once we obtain the CR-corrected values of charge, we can immediately obtain also the corresponding coefficients of the multipole expansion σ(lm), wherefrom we can calculate the multipole magnitudes of the surface charge distribution as 44
S l = 4π 2l + 1 m |σ(lm)| 2 .(31)
We will use the magnitudes of the monopole ( = 0), dipole ( = 1), and quadrupole ( = 2) moment to explore the effects of CR on several different spherical surface charge distributions. With S (0) l we will denote the multipole moments in absence of CR, whereas S l will be used to refer to multipole moments with CR taken into account. We will keep the temperature of the system fixed (T = 300 K), while we will mainly vary the salt concentration n 0 , the radius of the sphere R, and the solution pH. In addition, we will consider distributions with different numbers of charges N , their dissociation constants pK a , and "patchiness" λ (which will be, for simplicity, assumed identical for all charges in a distribution).
As a first example, we consider a model dipolar patchy distribution on a dielectric, impermeable sphere with ε p = 4. To generate such a distribution, we pick the positions of the chargeable moieties using Mitchell's best candidate algorithm, which randomly places charges on the sphere while preserving some minimal distance between them (for details, see Ref. 44). Then, to obtain a dipolar distribution, charges on the northern hemisphere (ϑ k π/2) are assigned a positive sign (q k = +e), whereas the charges on the southern hemisphere are assigned a negative sign. There are thus N/2 positive and N/2 negative charges in the distribution. For simplicity, we also assign a pK a = 10 to all positive charges and pK a = 4 to all negative charges.
The effects of CR on the first three multipole moments of such a distribution with N = 40 and λ = 20 are shown in Fig. 2 for the entire range of pH values, where we plot the difference of the multipole magnitudes in the presence and absence of CR, S l − S (0) l . To give a sense of the scale of the CR correction, the insets in Fig. 2 show only the bare multipole magnitudes in absence of CR. We can immediately observe that CR has a significant effect on all of the multipole magnitudes. The effects of CR are particularly large in the vicinity of the pK a values of the constituent charges, as can be expected from Fig. 1c and the fact that the admixing of different higher order multipoles is proportional to the multipole capacitance. Lower salt concentrations, and consequently smaller κR, lead to larger effects of CR. As can be inferred from Eq. (28), the radius and salt concentration have to be treated as separate parameters, and we can observe that the effects of CR become less pronounced when the radius is increased, a change more drastic than when we vary the salt concentration. Figure 2 does not show the influence of other parameters, such as N or λ, on the effects of CR, as they turn out to be mostly quantitative, influencing to an extent -but not much -the magnitude of the CR correction, but not its overall qualitative behaviour. In addition, when the sphere is considered to be permeable to ions, the overall behaviour retains the significance of the CR effects, which could also be expected from the general similarity of the functions in Eqs. (25) and (26). Lastly, we note that we observe similar behaviour also when constructing model quadrupolar distributions, where the two polar caps are covered with charges of the same type and the equator is covered with the same number of opposite charges.
Next, as an example of more complicated charge distributions, we consider the surface charge distributions of two globular proteins, lysozyme (PDB ID: 2LYZ) and β-lactoglobulin (PDB ID: 2BLG), whose multipole moments and their pH dependence have already been studied previously in Ref. 30 numerous charged amino acids of different types (N = 28 for 2LYZ and N = 52 for 2BLG), whose coordinates were obtained from PDB 52 and whose pK a values we determined using PROPKA3.1 software 51 . Due to a more complex surface charge distribution, the resulting pH behaviour of the multipole magnitudes becomes more complex than in the previously considered model dipolar and quadrupolar distributions 30 . The two proteins are considered to be impermeable to salt ions with a dielectric constant of ε p = 4, and the patchiness of their constituent charges is taken to be λ = 20 (cf. Fig. 1b).
The pH dependence of the CR correction to the first three multipole magnitudes is shown in Figs. 3 and 4 for 2LYZ and 2BLG, respectively. We see immediately that the scale of CR effects is again significant and comparable to the magnitudes of the bare multipole moments, while their pH dependence is more complicated than in the case of a simple dipolar distribution. We can also observe two rapid shifts in the CR correction of the monopole moments in panels 3a and 4a, which are related to the isoelectric points of the two proteins (located at approximately pI ∼ 11 and pI ∼ 4.5 for 2LYZ and 2BLG, respec- tively). As before, the effect of CR is reduced with an increase in salt concentration, yet the CR corrections cannot be neglected even in the limit of high-salt, n 0 ∼ 1000 mM. By changing the patchiness of individual charges λ, the scale of CR corrections is modified, with its effects increasing slightly with an increasing λ, yet the qualitative behaviour shown in Figs. 3 and 4 persists.
The CR effects on the multipole moments of general inhomogeneous charge distributions described in the examples above are thus robust, qualitative, and generally cannot be ignored. Even pronounced electrolyte screening is apparently not enough to completely wipe out the multipole mixing, unless in the highly unlikely case when the screening length would become smaller than the separation between the chargeable moieties on the sphere. This is a simple consequence of the collective effect of CR, since at each dissociation site the collective electrostatic field of all the other sites contributes to the dissociation reaction.
IV. CONCLUSIONS
We have derived the expressions for the full CR of inhomogeneous spherical surface charge densities and their multipole moments, and solved them explicitly on the level of linear approximation. As a result, we have shown that the CR mechanism leads to a mixing between different multipoles depending on their capacitance (as well as other system parameters), and thus presents an additional anomalous feature of molecular electrostatics. Such effects need to be considered especially in the context of proteins, where multipoles are often assumed to be fixed by structural details.
By considering several examples of inhomogeneously charged spherical particles, we demonstrated that CR can lead to significant changes in the magnitude of their multipole moments even when the system is in the linearized, DH regime. The effects of CR consequently cannot be neglected, and should thus be considered in, e.g., the calculation of the protein-protein interactions, which is often based on multipole expansion. Based on our analysis and the observed effects in terms of multipole mixing, a caveat is also appropriate for computer simulations, where it is obviously quite crucial to incorporate the dissociation equilibrium of the chargeable amino acid moieties in the case of proteins 34 or weakly acidic or basic monomers in the case of general polyelectrolytes 53 .
While the linearized version of the CR condition derived here cannot be readily applied when the electrostatic effects are large, the consistent behaviour of the CR effects upon changes in different system parameters shows that the CR effects should nonetheless persist even in the non-linear electrostatic regime, retaining a considerable effect on the multipole moments. Together with the full directional dependence of the electrostatic interaction in the presence of ionic screening, which involves involving all multipole moments even in the farfield regime 40-43 , we have therefore identified another anomalous feature of the multipolar expansion in the case of CR in charged macromolecules.
FIG. 1 .
1(a) An example of a globular protein, β-lactoglobulin (PDB: 2BLG), consisting of 52 charged amino acids with different pKa values. Coloured in red are the amino acids which can become positively charged, and in blue the amino acids which can become negatively charged. The charge distribution of the protein can be mapped onto its circumscribed sphere with radius R = 1.06 nm
It is clear by comparison with the results from Ref. 30 that the last term in the exponent in Eq. (13), ∂F ES /∂η i , plays the role of the local electrostatic potential at site i, ψ i . From Eq. (13) we can also immediately obtain the values of individual charges:
Eq. 19] and the measure of the angular size of the k-th site g l (λ k ) [Eq. 4].
FIG. 2 .
2Difference between the (a) monopole ( = 0), (b) dipole ( = 1), and (c) quadrupole ( = 2) moments in presence and absence of CR, S l − S (0) l , as a function of pH. The plots are shown for a model patchy dipolar distribution with N = 40 charges, consisting of 20 positive charges with pKa = 10 and 20 negative charges with pKa = 4 charges, placed on the opposite hemispheres of a sphere using Mitchell's algorithm 44 . Solid curves represent spheres with radius R = 1 nm and dot-dashed curves spheres with radius R = 10 nm. Different curves then correspond to different values of the univalent bulk salt concentration (n0 = 1, 10, 100, 1000 mM). Other parameters of the system are λ = 20 and εp = 4. To give a sense of the scale of the CR correction, the dashed curves in the insets show the pH dependence of the bare multipole moments S (0) l in absence of CR.
FIG. 3 .
3Difference between the (a) monopole ( = 0), (b) dipole ( = 1), and (c) quadrupole ( = 2) moments in presence and absence of CR, S l − S (0) l , as a function of pH. The plots are shown for the charge distribution of lysozyme (PDB: 2LYZ), consisting of 28 positively-and negativelycharged amino acids 30 , whose pKa values were predicted by PROPKA3.1 51 . The circumscribed radius of the protein is R = 2.24 nm, and different curves correspond to different values of salt concentration (n0 = 1, 10, 50, 100, 200, 500 mM). Other parameters of the system are λ = 20 and εp = 4. To give a sense of the scale of the CR correction, the dashed curves in the insets show the pH dependence of the bare multipole moments S (0) l in absence of CR.
FIG. 4 .
4Difference between the (a) monopole ( = 0), (b) dipole ( = 1), and (c) quadrupole ( = 2) moments in presence and absence of CR, S l − S (0) l , as a function of pH. The plots are shown for the charge distribution of β-lactoglobulin (PDB: 2BLG), consisting of 52 positively-and negativelycharged amino acids 30 , whose pKa values were predicted by PROPKA3.1 51 . The circumscribed radius of the protein is R = 2.28 nm, and different curves correspond to different values of salt concentration (n0 = 1, 10, 50, 100, 200, 500 mM). Other parameters of the system are λ = 20 and εp = 4. To give a sense of the scale of the CR correction, the dashed curves in the insets show the pH dependence of the bare multipole moments S (0) l in absence of CR.
M. Borkovec, B. Jönsson, and G. J. Koper, in Surface and colloid science (Springer, 2001) pp. 99-339.
ACKNOWLEDGMENTSALB and RP acknowledge the financial support from the Slovenian Research Agency (research core funding No.(P1-0055)). RP also acknowledges the support of the 1000-Talents Program of the Chinese Foreign Experts Bureau, and the hospitality of the University of the Chinese Academy of Sciences.
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[
"THE BETTI TABLE OF A HIGH DEGREE CURVE IS ASYMPTOTICALLY PURE",
"THE BETTI TABLE OF A HIGH DEGREE CURVE IS ASYMPTOTICALLY PURE"
] |
[
"Daniel Erman "
] |
[] |
[] |
We prove that asymptotically in the degree, the main term of the Boij-Söderberg decomposition of a high degree curve is a single pure diagram that only depends on the genus of the curve. This answers a question of Ein and Lazarsfeld in the case of curves.Dedicated to Rob Lazarsfeld on the occasion of his sixtieth birthday.
|
10.1017/cbo9781107416000.012
|
[
"https://arxiv.org/pdf/1308.4661v2.pdf"
] | 119,328,054 |
1308.4661
|
0174d55d76d93027fdf6e0414ecee185b2a0c35a
|
THE BETTI TABLE OF A HIGH DEGREE CURVE IS ASYMPTOTICALLY PURE
8 Jan 2014
Daniel Erman
THE BETTI TABLE OF A HIGH DEGREE CURVE IS ASYMPTOTICALLY PURE
8 Jan 2014
We prove that asymptotically in the degree, the main term of the Boij-Söderberg decomposition of a high degree curve is a single pure diagram that only depends on the genus of the curve. This answers a question of Ein and Lazarsfeld in the case of curves.Dedicated to Rob Lazarsfeld on the occasion of his sixtieth birthday.
Introduction
Syzygies can encode subtle geometric information about an algebraic variety, with the most famous examples coming from the study of smooth algebraic curves. Though little is known about the syzygies of higher dimensional varieties, Ein and Lazarsfeld have shown that at least the asymptotic behavior is uniform [2]. More precisely, given a projective variety X ⊆ P n embedded by the very ample bundle A, Ein and Lazarsfeld ask: which graded Betti numbers are nonzero for X reembedded by dA? They prove that, asymptotically in d, the answer (or at least the main term of the answer) only depends on the dimension of X.
Boij-Söderberg theory [5] provides refined invariants of a graded Betti table, and it is natural to ask about the asymptotic behavior of these Boij-Söderberg decompositions. In fact, this problem is explicitly posed by Ein and Lazarsfeld [2,Problem 7.4], and we answer their question for smooth curves in Theorem 2.3.
Fix a smooth curve C and a sequence {A d } of increasingly positive divisors on C. We show that, as d −→ ∞, the Boij-Söderberg decomposition of the Betti table of C embedded by |A d | is increasingly dominated by a single pure diagram that depends only on the genus of the curve. The proof combines an explicit computation about the numerics of pure diagrams with known facts about when an embedded curve satisfies Mark Green's N p -condition.
Setup
We work over an arbitrary field k. Throughout, we will fix a smooth curve C of genus g and a sequence {A d } of line bundles of increasing degree. Since we are interested in asymptotics, we assume that for all d,
deg A d ≥ 2g + 1. Let r d := dim H 0 (C, A d ) − 1 = deg A d − g so that the complete linear series |A d | embeds C ⊆ P r d .
For each d, we consider the homogeneous coordinate ring R(C, A d ) := ⊕ e≥0 H 0 (C, eA d ) of this embedding. We may then consider R(C, A d ) as a graded module over the polynomial ring Sym(H 0 (C, A d )). If F = [F 0 ← F 1 ← · · · ← F n ← 0] is a minimal graded free resolution of R(C, A d ), then we will use β i,j (O C , A d ) to denote the number of minimal generators of F i of degree j. Equivalently, we have
β i,j (O C , A d ) = dim k Tor Sym(H 0 (C,A d )) i (R(C, A d ), k) j .
We define the graded Betti table β(O C , A d ) as the vector with coordinates
β i,j (O C , A d ) in the vector space V = n i=0 j∈Z Q.
We use the standard Macaulay2 notation for displaying Betti tables, where
β = β 0,0 β 1,1 β 2,2 . . . β 0,1 β 1,2 β 2,3 . . . β 0,2 β 1,3 β 2,4 . . . . . . . . . . . . . . . .
Boij-Söderberg theory focuses on the rational cone spanned by all graded Betti tables in V. The extremal rays of this cone correspond to certain pure diagrams, and hence every graded Betti table can be written as a positive rational sum of pure diagrams; this decomposition is known as a Boij-Söderberg decomposition. For a good introduction to the theory, see either [6] or [7]. We introduce only the notation and results that we need.
For a given d and some i ∈ [0, g], we define the (degree) sequence e = e(i, d) :
= (0, 2, 3, 4, . . . , r d − i, r d − i + 2, r d − i + 3, . . . , r d + 1) ∈ Z r d −1 , and we define the pure diagram π i,d ∈ V by the formula: (1.1) β p,q (π i,d ) = (r d − 1)! · ℓ =p 1 |e ℓ −ep| if p ∈ [0, r d − 1] and q = e p 0
else.
Note that the shape of π i,d is the following, where * indicates a nonzero entry:
π i,d = 0 1 . . . r d − i − 1 r d − i . . . r d − 1 * 0 . . . 0 0 . . . 0 0 * . . . * 0 . . . 0 0 0 . . . 0 * . . . * .
It turns out that these are the only pure diagrams that appear in the Boij-Södergberg decomposition of the Betti tables β(C, A d ) (see Lemma 2.1 below). We next recall the notion of a (reduced) Hilbert numerator, which will be central to our proof. If S = k[x 0 , . . . , x n ] is a polynomial ring, and M is a graded S-module, then the Hilbert series of a finitely generated, graded module M is the power series HS M (t) :=
i∈Z dim k M i · t i ∈ Q[[t]]
. The Hilbert series can be written uniquely as a rational function of the form As is standard in Boij-Söderberg theory, we allow formal rescaling of Betti tables by rational numbers. Note the Hilbert numerator is invariant under modding out by a regular linear form or adjoining an extra variable; also, the Hilbert numerator is computable entirely in terms of the graded Betti table (see [3, §1]). Similar statements hold for the codimension of a module. Thus we may and do formally extend the notions of Hilbert numerator, codimension, and multiplicity to all elements of the vector space V. Lemma 1.2. For any i, d, the diagram π i,d has multiplicity 1.
HS M (t) = HN M (t) (1 − t) dimProof. By (1.1) we have β 0,0 (π i,d ) = (r d −1)! 2·3···(r d −i)·(r d −i+2)···(r d +1) .β 0,0 (π i,d ) · 2 · 3 · · · (r d − i) · (r d − i + 2) · · · (r d + 1) (r d − 1)! = β 0,0 (π i,d ) · (β 0,0 (π i,d )) −1 = 1.
Main result and proof
To make sensible comparisons between the graded Betti tables β(O C , A d ) for different values of d, we will rescale by the degree of the curve so that we are always considering Betti tables of (formal) multiplicity equal to 1. Namely, we define
β(O C , A d ) := 1 deg A d · β(O C , A d ).(2.2) β(O C , A d ) = g i=0 c i,d · π i,d ,
where c i,d ∈ Q ≥0 and i c i,d = 1.
The above lemma shows that the number of potential pure diagrams appearing in the decomposition of β(C, A d ) is at most g + 1. Note that the precise number of summands with a nonzero coefficient is closely related to Green and Lazarsfeld's Gonality Conjecture [9,Conjecture 3.7], and hence will vary even among curves of the same genus. However our main result, which we now state, shows that this variance plays a minor role in the asymptotics:
Theorem 2.3. The Betti table β(O C , A d )
converges to the pure diagram π g,d in the sense that
c i,d −→ 0 i = g 1 i = g as d −→ ∞.
1 Strictly speaking, Huneke and Miller's computation is for graded algebras. But by including a β 0,0 factor, the argument goes through unchanged for a graded Cohen-Macaulay module generated in degree 0 and with a pure resolution.
In particular, the limiting pure diagram only depends on the genus of the curve. A nearly equivalent statement of the theorem is: asymptotically in d, the main term of the Boij-Söderberg decomposition of the (unscaled) Betti table β(C, A d ) is the π g,d summand.
Proof of Lemma 2.1. Since the homogeneous coordinate ring of C ⊆ P r d is Cohen-Macaulay (see [3, §8A] for a proof and the history of this fact), it follows from [5, Theorem 0.2] that β(O C , A d ) can be written as a positive rational sum of pure diagrams of codimension r d − 1.
Since C ⊆ P r d satisfies the N p condition for p = r d − g − 1 by [8, Theorem 4.a.1], it follows that the shape of β(O C , A d ) is:
0 1 2 . . . r d − g − 1 r d − g . . . r d − 1 * − . . . − − − . . . − − * * . . . * * . . . * − − − . . . − * . . . * .
Thus the pure diagrams π i,d for i = 0, 1, . . . , g are the only diagrams that can appear in the Boij-Söderberg decomposition of β(C, A d ), and so we may write
β(O C , A d ) = g i=0 c i,d · π i,d with c i,d ∈ Q ≥0
. The (formal) multiplicity of β(C, A d ) is 1 by construction, and the same holds for the π i,d by Lemma 1.2, so it follows that c i,d = 1.
Lemma 2.4. The Hilbert numerator of the pure diagram π i,d is
HN π i,d (t) = r d − i + 1 r d (r d + 1) t 0 + (r d − 1)(r d − i + 1) r d (r d + 1) t 1 + i r d + 1 t 2 .
The Hilbert numerator of the rescaled Betti table β(O C , A d ) is
1 r d + g t 0 + r d − 1 r d + g t 1 + g r d + g t 2 .
Proof. We prove the first statement by direct computation. Since π i,d represents, up to scalar multiple, the Betti table of a Cohen-Macaulay module M, we may assume by Artinian reduction that the module M has finite length. For a finite length module, the Hilbert numerator equals the Hilbert series. Since the Betti table π i,d has 2 rows, it follows that the Castelnuovo-Mumford regularity of M equals 2 (except for π 0,d which has regularity 1). The coefficient of t 0 is thus the value of the Hilbert function in degree 0, which is the 0th Betti number of the pure diagram π i,d . By (1.1), this equals
β 0,0 (π i,d ) = (r d − 1)! 2 · 3 · · · (r d − i) · (r d − i + 2) · · · (r d + 1) = r d + 1 − i r d (r d + 1) .
Similarly, the coefficient of t 2 is given by the bottom-right Betti number of π i,d which is
β r d −1,r d +1 (π i,d ) = i r d + 1 .
Remark 2.5. If X is a variety with dim X > 1, then our argument fails in several important ways. To begin with, Ein and Lazarsfeld's nonvanishing syzygy results from [2] show that the number of potential pure diagrams for the Boij-Söderberg decomposition of β(X, A d ) is unbounded. Moreover, in the case of curves, the Hilbert numerator of the embedded curves converged to the Hilbert numerator of one of the potential pure diagrams; the N p condition then implied that this pure diagram had the largest degree sequence of any potential pure diagram. Our result then followed by the semicontinuous behavior of the Hilbert numerators of pure diagrams (for a related semicontinuity phenomenon see [4, Monotonicity Principle, p. 758]). Ein and Lazarsfeld's asymptotic nonvanishing results imply that, even for P 2 , the limit of the Hilbert numerator will fail to correspond to an extremal potential pure diagram, and so the semicontinuity does not obviously help.
Date: May 11, 2014. Research of the author partially supported by the Simons Foundation.
M and we define the Hilbert numerator of M as the polynomial HN M (t). The multiplicity of M is HN M (1).
The Boij-Söderberg decomposition of β(O C , A d ) has a relatively simple form. Lemma 2.1. For any d, the Boij-Söderberg decomposition of β(O C , A d ) has the form
Up to a positive scalar multiple, the diagram π i,d equals the graded Betti table of a Cohen-Macaulay module by [5, Theorem 0.1]. Then by Huneke and Miller's multiplicity computation for Cohen-Macaulay modules with a pure resolution 1 [10, Proof of Theorem 1.2], it follows that the multiplicity of π i,d equals
AcknowledgmentsThe questions considered in this paper arose in conversations with Rob Lazarsfeld, and in addition I learned a tremendous amount about these topics from him; it is a great pleasure to thank Rob Lazarsfeld for his influence and his superb mentoring. I also thank Lawrence Ein and David Eisenbud for helpful insights and conversations related to this paper. I thank Christine Berkesch, Frank-Olaf Schreyer, and the referee for comments that improved this paper.Finally, since π i,d has multiplicity 1 by Lemma 1.2, it follows that HN π i,d (1) = 1 and hence the coefficient of t 1 equals 1 minus the coefficients of t 0 and t 2 :For the Hilbert numerator of β(O C , A d ) statement, we note that deg A d = r d + g, yielding. As above, we can compute the t 0 and t 2 coefficients via the first and last entries in the Betti table, and these are thus 1 r d +g and g r d +g respectively (see[3, §8A], for instance). Since β(O C , A d ) has multiplicity 1, the t 1 coefficient is again 1 minus the t 0 and t 2 coefficients, and the statement follows immediately.Proof of Theorem 2.3. Note that r d −→ ∞ as d −→ ∞. We rewrite the Hilbert numerator of π i,d aswhere r d ǫ j,i,d −→ 0 as d −→ ∞ for all j = 0, 1, 2 and i = 0, . . . , g. For instance.We may similarly rewrite the Hilbert numerator of β(O C , A d ) asSince the Hilbert numerator is additive with respect to the Betti table decomposition of (2.2), combining the above computations with our Boij-Söderberg decomposition from (2.2), we see that the t 2 coefficient of the Hilbert numerator of β(O C , A d ) may be written asWe multiply through by r d and take the limit as d −→ ∞. Since r d δ j,d and r d ǫ j,i,d both go to 0 as d −→ ∞, this yields:But c i,d ≥ 0 and i c i,d = 1. Hence, as d −→ ∞, we obtain c i,d −→ 0 for all i = g and c g,d −→ 1.
L Ein, D Erman, R Lazarsfeld, arXiv:1207.5467Asymptotics of random Betti tables. L. Ein, D. Erman, and R. Lazarsfeld, Asymptotics of random Betti tables, (2012), arXiv:1207.5467.
L Ein, R Lazarsfeld, arXiv:1103.0483Asymptotic syzygies of algebraic varieties. 16L. Ein and R. Lazarsfeld, Asymptotic syzygies of algebraic varieties (2011), arXiv:1103.0483. 1, 6
The geometry of syzygies. D Eisenbud, Graduate Texts in Mathematics. 2295Springer-VerlagD. Eisenbud, The geometry of syzygies, Graduate Texts in Mathematics, vol. 229, Springer-Verlag, New York 2005. 3, 4, 5
Filtering free resolutions. D Eisenbud, D Erman, F O Schreyer, Compositio Math. 1496D. Eisenbud , D. Erman, and F.O. Schreyer, Filtering free resolutions, Compositio Math. 149 (2013), 754-772. 6
Betti numbers of graded modules and cohomology of vector bundles. D Eisenbud, F O Schreyer, J. Amer. Math. Soc. 2234D. Eisenbud and F.O. Schreyer, Betti numbers of graded modules and cohomology of vector bundles, J. Amer. Math. Soc., 22 (2009), no. 3, 859-888. 1, 3, 4
Betti numbers of syzygies and cohomology of coherent sheaves. D Eisenbud, F O Schreyer, Proceedings of the International Congress of Mathematicians. the International Congress of MathematiciansHyderabad, IndiaD. Eisenbud and F.O. Schreyer, Betti numbers of syzygies and cohomology of coherent sheaves, Pro- ceedings of the International Congress of Mathematicians (2010), Hyderabad, India. 2
Boij-Söderberg theory: Introduction and survey. G Fløystad, Progress in commutative algebra 1. Berlinde Gruyter154G. Fløystad, Boij-Söderberg theory: Introduction and survey, Progress in commutative algebra 1, 154, de Gruyter, Berlin, 2012. 2
Koszul cohomology and the geometry of projective varieties. M L Green, J. Differential Geom. 191M.L. Green, Koszul cohomology and the geometry of projective varieties, J. Differential Geom. 19 (1984), no. 1, 125-171. 4
On the projective normality of complete linear series on an algebraic curve. M Green, R Lazarsfeld, Invent. Math. 831M. Green and R. Lazarsfeld, On the projective normality of complete linear series on an algebraic curve, Invent. Math. 83 (1985), no. 1, 73-90. 3
A note on the multiplicity of Cohen-Macaulay algebras with pure resolutions. C Huneke, M Miller, Canad. J. Math. 376C. Huneke and M. Miller, A note on the multiplicity of Cohen-Macaulay algebras with pure resolutions, Canad. J. Math. 37 (1985), no. 6, 1149-1162. 3
E-mail address: derman@math. wisc.eduE-mail address: [email protected]
|
[] |
[
"The strong convergence of operator-splitting methods for the Langevin dynamics model",
"The strong convergence of operator-splitting methods for the Langevin dynamics model"
] |
[
"Adam Telatovich \nDepartment of Mathematics\nPennsylvania State University\n16802University ParkPennsylvaniaUSA\n",
"Xiantao Li \nDepartment of Mathematics\nPennsylvania State University\n16802University ParkPennsylvaniaUSA\n"
] |
[
"Department of Mathematics\nPennsylvania State University\n16802University ParkPennsylvaniaUSA",
"Department of Mathematics\nPennsylvania State University\n16802University ParkPennsylvaniaUSA"
] |
[] |
We study the strong convergence of some operator-splitting methods for the Langevin dynamics model with additive noise. It will be shown that a direct splitting of deterministic and random terms, including the symmetric splitting methods, only offers strong convergence of order 1. To improve the order of strong convergence, a new class of operator-splitting methods based on Kunita's solution representation [1] are proposed. We present stochastic algorithms with strong orders up to 3. Both mathematical analysis and numerical evidence are provided to verify the desired order of accuracy.
| null |
[
"https://arxiv.org/pdf/1706.04237v1.pdf"
] | 119,301,943 |
1706.04237
|
3a605887f7f3d28fa5f35b787c8f64afaa4b8fcd
|
The strong convergence of operator-splitting methods for the Langevin dynamics model
Adam Telatovich
Department of Mathematics
Pennsylvania State University
16802University ParkPennsylvaniaUSA
Xiantao Li
Department of Mathematics
Pennsylvania State University
16802University ParkPennsylvaniaUSA
The strong convergence of operator-splitting methods for the Langevin dynamics model
Taylor approximation 2010 MSC: 60H35, 65C30Langevin equationBrownian motionstrong convergenceoperator splitting methodsItô
We study the strong convergence of some operator-splitting methods for the Langevin dynamics model with additive noise. It will be shown that a direct splitting of deterministic and random terms, including the symmetric splitting methods, only offers strong convergence of order 1. To improve the order of strong convergence, a new class of operator-splitting methods based on Kunita's solution representation [1] are proposed. We present stochastic algorithms with strong orders up to 3. Both mathematical analysis and numerical evidence are provided to verify the desired order of accuracy.
Introduction
The Langevin dynamics (LD) equation plays a fundamental role in the modeling of many complex dynamical systems subject to random noise. In its simplest form, it can be expressed as the Newton's equations of motion with added frictional and random forces, which are usually posed to satisfy the fluctuation-dissipation theorem.
As a system of stochastic differential equations (SDE), there are various classical methods for approximating the solutions [2]. However, low order methods, such as the Euler-Maruyama method, often do not have sufficient accuracy. On the other hand, higher order methods that are constructed based on direct expansions of solutions (Itô-Taylor expansion) usually involve high order derivatives of the drift and diffusion coefficients, which makes the implementation rather difficult. For instance, for bio-molecular $ Research supported by the National Science Foundation DMS-1522617 and DMS-1619661. models [3], this implies that one has to compute the derivatives of the inter-molecular forces, which typically is not plausible. As a result, these methods have been largely neglected in the molecular simulation community. Instead, operator splitting methods have been more widely used. Such algorithms, especially with applications to molecular dynamics simulations, have been treated extensively in [4,5], where many theoretical and practical aspects have been discussed. The idea is to separate out terms on the right hand side and form two or more SDEs that can be solved explicitly. This is denoted by an [ABO] notation in [4]. Many existing methods can be recast into this form, e.g., [6,7,8,9,10,11]. One particular advantage of the splitting methods is that they are very easy to implement, since each substep can be carried out exactly. The splitting methods can also be designed to better sample the equilibrium averages. Another approach is based on solving the coordinate and momentum equations consecutively. For example, one can start by assuming the coordinates remain constant, and integrate out the momentum equation exactly. Then using this solution for the momentum, one can integrate the first equation and obtain an updated coordinate for the next step. A further correction can be made by assuming the force is linear in time, constructed using the coordinates at the current and next steps. This led to the stochastic velocity Verlet method (SVV) [12,13,14], which has been implemented in simulation packages, e.g., TINKER [15]. Other integration methods can also be found in the literature, e.g., [16,17,18,10,19].
Part of this paper is concerned with the numerical accuracy of the Langevin integrators. This fundamental issue has been discussed in [4] as well. In particular, the weak convergence of the numerical solution has been rigorously proved in [20]. Such analysis is crucial when the approximation methods are used to sample the corresponding equilibrium statistics. This is particularly useful when the averages of certain quantities are of interest. On the other hand, to the best of our knowledge, the strong convergence has not been fully studied. Strong convergence ensures the accuracy in terms of individual realizations and solutions at transient stages. Strong convergence usually implies weak convergence, but not vice versa. Typically, strong convergence can be examined by comparing to the Itô-Taylor expansion.
Therefore, the fact that the splitting methods discussed in the literature often do not involve multiple Itô integrals of order 2 or higher is already an indication that those methods are only of strong order 1 or less.
In general, each improvement of the strong convergence will only increase the order by 0.5 [2].
The purpose of this paper is to present some mathematical analysis of the strong convergence properties for some existing numerical algorithms for the Langevin dynamics model. In addition, we present a new formalism for constructing algorithms that are robust and easy to implement. Our starting point is the solution representation by Kunita [1]. Written formally as an operator exponential form, the differ-ential operator is expressed in terms of the commutators involving the differential operators associated with the drift and diffusion coefficients, along with multiple Itô integrals. Intuitively, we can make truncations at various levels, to obtain approximation methods of increasing order. Such truncation schemes have been used in [21] as a starting point to construct robust algorithms for scalar SDEs with multiplicative noise. It was demonstrated that such algorithms can preserve the non-negativity of the solution. For the applications of these truncations to the Langevin dynamics, we provide the mathematical analysis of these approximations, and examine the strong order of the approximations. The strong convergence is in the L 1 sense following the notations in [2].
With the truncations of the solution operator, we obtain approximate solutions that can be written as solutions of ODEs, for which many efficient methods exist. We choose the well established operatorsplitting methods. With the various truncation schemes, together with an appropriate operator splitting for the resulting ODEs, we found methods of higher strong order. More specifically, we present the explicit forms of the methods with strong order 2 and 3.
The rest of the paper is organized as follows. Section 2 presents the analysis of some existing numerical methods. In section 3, we introduce the new class of operator splitting methods based on the truncations of the Kunita's solution operator and examined the strong order of accuracy. Section 4 contains numerical tests that will demonstrate the expected order of convergence.
The basic theory
Let us start with the Langevin dynamics model with n space dimension,
x =v, v = f (x) − Γv + σW (t ),(1)
eq: lgv where x = (x 1 , . . . , x n ), v = (v 1 , . . . , v n ) ∈ R n can be interpreted as position and velocity components respectively, W (t ) = (W 1 (t ), . . . ,W n (t )) ∈ R n is the standard n−dimensional Brownian motion and 0 ≤ t ≤ T . Assume the function f = f (x) : R n → R n , representing the conservative force (for example, the Morse potential), has bounded second derivatives. Here we will consider the case where σ and Γ are constant n × n matrices. In particular, the noise is additive, and as a consequence, (1) may be interpreted as either an Itô or Stratonovich SDE: in differential form, •dW t = dW t = W (t )d t . For particle dynamics, typically n = 3N , with N being the total number of particles.
Itô-Taylor expansion of the solution.
For numerical solutions of the Langevin dynamics model (1), the consistency is defined via a comparison with the Itô-Taylor expansion of the exact solution. Let us recall [2, eq. 5.5.3] that in general, for a multi-dimensional SDE,
d X t = a(X t )d t + bdW t , (2) eq: SDEs where a = a(x) : R d → R d , b ∈ R dX k+1 = X k + α∈A γ f α (X k )I α + α∈B γ I α [ f α (X t )] t n ,t n+1 ,(3)
eq: IT0
where we have denoted t k = k∆t , X (k∆t ) = X k , with k being the time step with step size ∆t , and final time step n T satisfying n T ∆t = T . In addition, A γ and B γ are sets of multi-indices, f α = f α (x) are coefficient functions, and I α are multiple Itô integrals. This idea generalizes the Taylor expansion of a deterministic function. Specifically the set A γ is defined as,
A γ = {α : l (α) + n(α) ≤ 2γ or l (α) = n(α) = γ + 0.5},(4)
where l (α) := l is the length of the multi-index α = ( j 1 , j 2 , . . . , j l ), with entries j k ∈ [0, 1, · · · , m], n(α) is the number of zero entries in α, and the remainder set B γ is defined by
B γ = {α = ( j 1 , j 2 , . . . , j l ) ∉ A γ : ( j 2 , j 3 , . . . , j l ) ∈ A γ }.(5)
The coefficient functions f α are given by
f α (x) = L j 1 . . . L j l −1 L j l x(6)
where the L j are differential operators
L 0 = a · ∇ x + 1 2 (b · ∇ x ) 2 , L j = b · ∇ x = d k=1 b k ∂ ∂x k for 1 ≤ j ≤ m(7)
and ∇ x = ∂ ∂x 1 , . . . , ∂ ∂x n . Finally, the multiple Itô integrals I α = I α,t k ,t k+1 are given by
I α,t k ,t k+1 = t k+1 t k s l −1 t k · · · s 2 t k dW j 1 s 1 dW j 2 s 2 . . . dW j l s l ,(8)
where by convention dW 0 t := d t , and more generally,
I α [ f α (X t )] t k ,t k+1 = t k+1 t k s l −1 t k · · · s 2 t k f α (X s 1 )dW j 1 s 1 dW j 2 s 2 . . . dW j l s l .(9)
This reduces to the usual Taylor expansion when d = 1, a = 1, and b = 0. We refer the readers to [2] for the detailed explanation of the notations.
One-step numerical schemes can be directly obtained from the Itô-Taylor expansion (3). For example, to find the strong order γ = 0.5 Itô-Taylor approximation of X n , first notice
A 0.5 = {(0), ( j ) : j = 1, . . . , m}, then calculate f 0 = a and f j = b j ( j = 1, . . . , m). Next, calculate I 0 = ∆t , I j = W j ∆t =: ∆W j , so that Y n+1 = Y n + a(Y n )∆t + m j =1 b j (Y n )∆W j .(10)
Here we have used Y n to denote the numerical solution at step n. This is the Euler-Maruyama method. In simulations, the increments ∆W j ∈ N (0, ∆t ) are implemented as ∆t ξ j where ξ j ∈ N (0, 1) are standard normal random variables. The strong order γ = 1 Itô Taylor approximation can be found by simply adding more multi-indices α = ( j 1 , j 2 ), with j 1 , j 2 = 1, 2, · · · , m, and computing f ( j 1 , j 2 ) = bb , so that
Y n+1 = Y n + a(Y n )∆t + m j =1 b j (Y n )∆W j + l j 1 , j 2 =1 L j 1 b j 2 I ( j 1 , j 2 ) ,(11)E(|X (T ) − Y (T )|) ≤ C ∆t γ for all 0 < ∆t < ∆.(12)
Theorem 2. [2,Thm. 11.5.1] The strong Itô-Taylor approximation (3) X t of order γ converges strongly to the solution of (2) with order γ. Furthermore, any discrete approximation
Y k+1 = Y k + α∈A γ g α,k I α + R k (k = 0, 1, 2, . . . )(13)
with continuous functions g α,k satisfying the following two conditions for some constants K 1 , K 2 > 0, converges strongly to the exact solution X of (2) with order γ. In this case we call Y a strong Itô scheme of order γ.
E max 0≤k≤n T |g α,k − f α (Y k )| 2 ≤ K 1 ∆t 2γ−φ(α) (k = 0, 1, 2 . . . ) φ(α) := 2l (α) − 2 if l (α) = n(α) l (α) + n(α) − 1 if l (α) = n(α) ,(14)
Our main result in this paper is that we have developed strong Itô schemes of orders 1, 2 and 3 for the Langevin equation with additive noise. These schemes are stochastic operator splitting methods, which are natural generalizations of well-known operator splitting methods for ODEs. These methods will be presented in section 3. To analyze the strong order of accuracy, we first show the strong Itô-Taylor approximations of order 1,2 and 3 for the Langevin equation. In particular, for the Langevin equation (1),
we have X t = (x(t )
, v(t )) ∈ R 2n so that d = 2n. In addition, the drift and diffusion coefficients are given by,
a = v f (x) − Γv and b = 0 σ ,(16)
so that m = n.
x(∆t ) v(∆t ) = x v + v f (x) − Γv ∆t + n j =1 0 σ j ∆W j , x(∆t ) v(∆t ) = x v + v f (x) − Γv ∆t + n j =1 0 σ j ∆W j + f (x) − Γv D f (x)v − Γ f (x) + Γ 2 v ∆t 2 2 + n j =1 σ j −Γσ j I ( j ,0) (∆t ), x(∆t ) v(∆t ) = x v + v f (x) − Γv ∆t + n j =1 0 σ j ∆W j + f (x) − Γv D f (x)v − Γ f (x) + Γ 2 v ∆t 2 2 + n j =1 σ j −Γσ j I ( j ,0) (∆t ) + n j =1 D f (x)v − Γ f (x) + Γ 2 v D[D f (x)v]v − ΓD f (x)v + D f (x)( f (x) − Γv) + Γ 2 ( f (x) − Γv) ∆t 3 3! + n j =1 −Γσ j D f (x)σ j + Γ 2 σ j I ( j ,0,0) (∆t ) (17) eq: ito-lgv where σ j = σ(column j ), and D f (x) is the Jacobian of f at x, (D f (x)) i j = ∂ f i (x)
∂x j . Furthermore, the strong order 0.5, 1.5 and 2.5 Itô-Taylor approximations are the same as the strong order 1, 2 and 3 approximations, respectively, due to the fact that the noise is additive.
Proof. We will just verify the strong order 1 Itô-Taylor approximation here. The derivations of the higher order approximations are very similar, yet more tedious (see (84),(88), and (92) in the appendix).
First observe that the multi-index set is
A 1 = {α : l (α) + n(α) ≤ 2} = {(0), ( j 1 ), ( j 1 , j 2 ) : j 1 , j 2 = 1, 2, · · · , n}.(18)
Next observe that the differential operators L j are given by
L 0 = a · ∇ x = v f (x) − Γv · ∇ x ∇ v = v · ∇ x + ( f (x) − Γv) · ∇ v , L j = b j · ∇ x = 0 σ j · ∇ x ∇ v = σ j · ∇ v .(19)
Then we can calculate the coefficient functions:
f (0) x v = L 0 x L 0 v = v f (x) − Γv f ( j ) x v = L j x L j v = D v (x)σ j D v (v)σ j = 0 n×1 σ j j = 1, 2, · · · , n f ( j 1 , j 2 ) x v = L j 2 L j 2 x L j 1 L j 2 v = L j 1 0 n×1 L j 1 σ j 2 = 0 2n×1 for j 1 , j 2 = 1, 2, · · · , n since the noise is additive.(20)
We do not need to compute I ( j 1 , j 2 ) (∆t ) because the corresponding coefficient functions f ( j 1 , j 2 ) above are identically zero. We just need to observe that
I (0) (∆t ) = ∆t 0 d t = ∆t I ( j ) (∆t ) = ∆t 0 dW j (t ) = W j (∆t ) − W j (0) =: ∆W j 1 ≤ j ≤ n.(21)
Therefore, the strong order 1 Itô-Taylor approximation of (1) is
x(∆t ) v(∆t ) = x v + α∈A 1 f α x v I α (∆t ) = x v + v f (x) − Γv ∆t + n j =1 0 σ j ∆W j ,(22)
as desired. To see why the strong order 0.5 approximation is the same, simply observe that the multiindex set A 0.5 satisfies A 0.5 = {(0), ( j 1 ) : j 1 = 1, 2, · · · , n} = A 1 − {α ∈ A 1 : f α = 0}.
Analysis of some existing methods
The Itô-Taylor approximations (17) revealed the leading terms in the Itô-Taylor expansion, and they serve as an important reference to study the strong convergence, as indicated by Theorem 2. On the other hand, a direct implementation of these approximation may not be practical, especially because the formulas (17) contains the derivatives of f (x) which are not easy to compute in practice. Here we consider some existing methods and examine their strong order of accuracy.
Direct operator splitting methods
A natural approximation of (17) can be obtained by splitting the equation into several subproblems, each of which can be solved exactly. A wide variety of splitting methods have been discussed in [4]. For example, we may consider to split the Langevin equation as follows,
x =v v =0,(23)
eq: A
and x =0 v = f (x) − Γv + σW (t ).
(24) eq: B
Both of these equations have explicit solutions. The solution steps can be denoted by A and B , respectively, and approximations can be obtained by following the operations, e.g., AB , A 2 B A 2 , etc [4]. The proof will be postponed to the next section.
The stochastic velocity-Verlet's method
Here we analyze another widely implemented scheme -the stochastic velocity-Verlet's (SVV) method [14]. This method starts with an assumption that x(t ) remains as a constant, and integrates the second
equation in (1) exactly, giving rise to, v(t ) = e −Γt v(0) + Γ −1 (I − e −Γt ) f (x(0)) + t 0 e −Γ(t −s) σdW s ,(25)
where I denotes the n × n identity matrix.
With this approximation of v(t ), one can now turn to the first equation, and integrate. This gives,
x(∆t ) = x(0) + c 1 v(0)∆t + c 2 ∆t 2 f (x(0)) + ∆t 0 t 0 e −Γ(t −s) σdW s d t . (26) eq: svv-x
Here the coefficients are given by,
c 0 = e −Γ∆t , c 1 = (Γ∆t ) −1 (I − e −Γ∆t ), c 2 = 1 ∆t 2 ∆t 0 Γ −1 (I − e −Γt )d t .(27)
One might stop here and accept the position and velocity values. Or one can use the updated position value and approximate the function f by a linear function,
f (x(t )) ≈ f (x(0)) + f (x(∆t )) − f (x(0)) t .(28)
With this approximation, one can integrate the velocity equation again. One finds that,
v(∆t ) = c 0 v(0) + c 1 f (x(0))∆t + c 2 f (x(∆t )) − f (x(0)) ∆t + ∆t 0 e −Γ(∆t −s) σdW s . (29) eq: svv-v
Equations (26) and (29) form the basis for the SVV method. The formulas can be repeated, and at each step, the function f is evaluated only once at each step, which is typically considered as an considerable advantage.
Theorem 5.
The SVV algorithm has strong order 2.
Proof. The details are in the appendix. To sketch the proof, we compare SVV to the strong order γ = 2
Ito Taylor approximation, finding that the remainder term R k , k = 0, 1, · · · , n T −1 for the position after the kth step is a discrete martingale [2, pg. 195]
R k = n j =1 (k+1)∆t k∆t t k∆t e −Γ(t −s) − I σ j dW j s d t ,(30)
which satisfies convergence estimate (15) by using Doob's lemma.
For the velocity components, the strategy is similar, with
R k = n j =1 (k+1)∆t k∆t e −Γ(∆t −s) − I + Γ(∆t − s) σ j dW j s , k = 0, . . . , n T − 1.(31)
New operator-splitting algorithm with higher order strong convergence
Here we will present new splitting algorithms. Our starting point is the Kunita's solution operator [1].
In particular, for the standard SDE (2) we define the differential operators,
X 0 = a · ∇ x , X j = b j · ∇ x .(32)
These are none other than the familiar operators L 0 and L j . Then the exact solution of the SDE can be formally expressed as,
X t = exp(D t )X 0 ,(33)
where, by [21, eq. (2.5)],
D ∆t = ∆t X 0 + n j =1 ∆W j X j + 1 2 n j =1 [∆t , ∆W j ][X 0 , X j ] + 1 18 n j =1 [[∆t , ∆W j ], ∆t ][[X 0 , X j ], X 0 ] + . . .(34)
and [X 0 , X j ] denotes the commutator bracket
X 0 X j − X j X 0 ,(35)
which is a differential operator, so that [[X 0 ,
X j ], X 0 ] = [X 0 , X j ]•X 0 −X 0 •[X 0 , X j ]. The terms [∆t , ∆W j ]
are given by
[∆t , ∆W j ] := ∆t 0 t dW j t − ∆t 0 W j t d t ,(36)
In particular, we will use exp(D ∆t ) to define our numerical solution.
First-order truncation
We first make a truncation and keep the first two terms [21, eq. 3.18]:
D I ∆t = ∆t X 0 + n j =1 ∆W j X j = ∆t v · ∇ x + ∆t ( f (x) − Γv) + n j =1 σ j ∆W j · ∇ v .(37)
Once the Brownian motion ∆W has been sampled (and realized), the operator exp(D I ∆t ) corresponds to the solution operator of the following ODE system,
x = ∆t v v = ∆t ( f (x) − Γv) + n j =1 σ j ∆W j .
(38) eq: ODE-I This approximation by the solution of the above ODE system will be referred to as truncation I.
At this point, we can prove the strong order convergence. While the first order truncation is incapable of competing with the SVV algorithm, it serves as an alternative to understand operator splitting methods surveyed in [4, Sec. 7.3.1].
Lemma 1. By (100) in the appendix,
exp(D I ∆t ) x v = x v + v f (x) − Γv ∆t + f (x) − Γv D f (x)v − Γ( f (x) − Γv) ∆t 2 2 + n j =1 0 σ j ∆W j + n j =1 σ j −Γσ j I (0, j ) + I ( j ,0) + higher order terms (39)
where the higher order terms do not involve I ( j ,0) .
Theorem 6.
For the Langevin equation with additive noise, truncation I given by X ∆t = exp(D I ∆t )(X 0 ) is precisely a strong order 1 approximation.
Proof. Use the lemma above and recall that the strong order 1.5 and 2 Itô Taylor method is
x(∆t ) v(∆t ) = x v + v f (x) − Γv ∆t + f (x) − Γv D f (x)v − Γ( f (x) − Γv) ∆t 2 2 + n j =1 0 σ j ∆W j + n j =1 σ j −Γσ j I ( j ,0) .(40)
We can match terms exactly for all α ∈ A 2 (see (81)) except for the term α = (0, j ), and
I (0, j ) (∆t ) ∈ N (0, ∆t 3 /3).
Criterion (14) is satisfied for order γ = 1 and γ = 2, since (14) is satisfied with powers 3 on the left hand side and 2 on the right hand side. As for criterion (15), the remainder term
g α = f α for all α ∈ A 1 ∩ A 2 except for α = (0, j ). For α = (0, j ) and γ = 2, we have 2γ−φ(α) = 2 and g 0 j − f 0 j = (σ j , −Γσ j )I (0, j ) ∈ N (0, ∆t 3 ). Thus,R k after k steps (k = 0, . . . , n T − 1) is R k = n j =1 σ j −Γσ j I (0, j ) ∈ N 0 2n×n ,O ∆t 3(41)
and satisfies
E max 1≤m≤n T m−1 k=0 R k 2 ≤ 4E n T −1 k=0 R k 2 by Doob's lemma ≤ 4E n T −1 k=0 R 2 k since E(R k R l ) = δ kl E R 2 k = 4 n T −1 k=0 E R 2 k = 4 n T −1 k=0 O(∆t 3 ) = 4n T O(∆t 3 ) = O(∆t 2 ) since n T ∆t = T.(42)
Then, letting Y k+1 = Y k + α∈A γ f α (X n )I α and Z k+1 = Z k + α∈A γ g α,k I α + R k denote the strong order γ ∈ {1, 2} Itô Taylor approximation and truncation 1 method, respectively, we have that
E(|Y (n T ) − Z (n T )|) ≤ E |Y (n T ) − Z (n T )| 2 by Jensen's inequality ≤ E max 1≤m≤n T |Y m − Z m | 2 = E max 1≤m≤n T m−1 k=0 R k 2 ≤ O(∆t 2 ) = O(∆t ).(43)
Then, since Y has strong order γ ∈ {1, 2}, by the triangle inequality Z converges strongly at time T = n T ∆t with order 1 to the exact solution, X :
E (|X (n T ) − Z (n T )|) ≤ E (|X (n T ) − Y (n T )|) + E (|Z (n T ) − Y (n T )|) = O(∆t γ ) + O(∆t ) = O(∆t ).(44)
For the symmetric and non-symmetric operator splitting methods, consider
D I = A + B where A = v∆t · ∇ x and B = ( f (x) − Γv)∆t + σ∆W ∇ v .(45)
Lemma 2. We have
I + A + B + 1 2 [A, B ] + 1 2 (A 2 + AB + B A + B 2 ) x v = x(∆t ) v(∆t ) + 0 n×1 D x f (x)v − (1/2)Γ f (x) + (1/2)Γ 2 v ∆t 2 + n j =1 0 n×1 −Γσ j ∆t ∆W 2 ,(46)
and
I + A + B + 1 2 (A 2 + AB + B A + B 2 ) x v = x(∆t ) v(∆t ) + f (x) − Γv D x f (x)v − Γ f (x) + Γ 2 v ∆t 2 2 + n j =1 σ j −Γσ j ∆t ∆W 2 ,(47)
where (x(∆t ), v(∆t )) denotes the strong order γ = 1 Ito Taylor approximation.
Proof. See appendix.
Theorem 7. The non-symmetric splitting scheme
exp D I ≈ exp(A) exp(B )(48)
and symmetric splitting scheme
exp D I ≈ exp(A/2) exp(B ) exp(A/2)(49)
both yield approximations with strong order γ = 1.
Proof. Use the previous lemma and the fact that
exp(A) exp(B ) x v ≈ I + A + B + 1 2 [A, B ] + 1 2 (A 2 + AB + B A + B 2 ) x v (50) and exp(A/2) exp(B ) exp(A/2) x v ≈ I + A + B + 1 2 (A 2 + AB + B A + B 2 ) x v .(51)
In our numerical tests (see Figures 1 and 2), we see that the (naive and symmetric) operator splitting methods applied to truncation D I ∆t both converge with order 1.
Second-order truncation
Now we consider the truncation of D which includes the first order bracket [21, eq. 3.22]:
D II ∆t = ∆t X 0 + n j =1 ∆W j X j + 1 2 n j =1 [∆t , ∆W j ][X 0 , X j ] = ∆t v − 1 2 n j =1 σ j [∆t , ∆W j ] · ∇ x + ∆t f (x) − Γv + 1 2 n j =1 Γσ j [∆t , ∆W j ] · ∇ v(52)
Here, [∆t , ∆W j ] is given by [21, pg.169]:
[∆t , ∆W j ] = I (0, j ) − I ( j ,0) = ∆t 0 t dW j t − ∆t 0 W j t d t = 2I (0, j ) − ∆t ∆W j .
In the computation, the integral I (0, j ) can be sampled as normal random variables. In [2], the following variable was introduced,γ
j = 3 ∆t 3/2 2I (0, j ) − ∆t ∆W j .
Using the fact that 〈I 2 (0, j ) 〉 = ∆t 3 /3 and 〈I (0, j ) ∆W j 〉 = ∆t 2 /2, one finds thatγ j ∼ N (0, 1), for 1 ≤ j ≤ n. Here for the analysis we choose to define ∆U = (∆U 1 , . . . , ∆U n ), where
∆U j = 1 2 [∆t , ∆W j ] = I (0, j ) − 1 2 ∆t ∆W j , j = 1, 2, · · · , n.(53)
We now write our operator as follows, with ∆W := (∆W 1 , . . . , ∆W n ):
D II ∆t = (∆t v − σ∆U ) · ∇ x + ∆t ( f (x) − Γv) + σ∆W + Γσ∆U · ∇ v(54)
Once ∆W and ∆U are realized, the solution corresponds to that of the following ODEs at time t = 1,
x = v∆t − σ∆U , v = f (x)∆t − Γv∆t + σ∆W + Γσ∆U . Lemma 3. We have exp D II x v = x v + v f (x) − Γv ∆t + f (x) − Γv D x f (x) − Γ f (x) + Γ 2 v ∆t 2 2 + n j =1 0 n×1 σ j ∆W j + n j =1 σ j −Γσ j I ( j ,0) (∆t ) + n j =1 Γσ j −D x f (x)σ j − Γ 2 σ j ∆t ∆U j 2 + 1 3! (D II ) 3 x v + higher order terms.(55)
Proof. See appendix, (110).
Theorem 8. The operator exp D II ∆t generates a solution with strong order 2.
Proof. By the previous lemma, we see that
exp(D II ) x v = x(∆t ) v(∆t ) + n j =1 Γσ j −D x f (x)σ j − Γ 2 σ j ∆t ∆U j 2 + 1 3! (D II ) 3 x v + higher order terms,(56)
where (x(∆t ), v(∆t )) denotes the strong order γ = 2 Ito Taylor approximation. With g α,n = f α for all α ∈ A 2 (see appendix for A 2 ), we observe that the convergence criterion (14) is trivially satisfied:
E(max 1≤m≤n T |g α,m − f α (X m )| 2 ) = 0.
The multi-index α in the remainder set B 2 whose corresponding multiple Itô integral
I α has minimal variance is α = ( j , 0, 0) where j = 1, 2, · · · , n, since B 2 = {( j , 0, 0), (0, 0, 0) : j = 1, 2, · · · , n}
and I ( j ,0,0) ∈ N (0, ∆t 5 ) has lower order variance than I (0,0,0) = ∆t 3 /3!. Specifically, by [2, Lemma 5.7.3],
I ( j ,0,0) (∆t ) has variance bound E(I 2 ( j ,0,0) ) ≤ 4∆t 4 (k+1)∆t k∆t 1d t = 4∆t 5 .(57)
Observe that R k = n j =1 c j I ( j ,0,0) (k∆t , (k + 1)∆t ) + O(∆t 3 ) for some constants c j , which are bounded be-cause f has bounded derivatives. Then
E max 1≤m≤n T n T −1 k=0 R k 2 ≤ E n T −1 k=0 R k 2 by Doob's lemma, since R k is a Martingale ≤ n T −1 k=0 E R 2 k since E(R k R l ) = δ kl E R 2 k ≤ n T −1 k=0 n j =1 c 2 j E(I 2 ( j ,0,0) ) since I ( j ,0,0) are independent ≤ n T −1 k=0 4∆t 5 · n j =1 c 2 j = 4K n T ∆t 5 = O(∆t 4 ) for K := n j =1 c j << n T and n T ∆t = T.(58)
Taking square roots and using Jensen's inequality, we get that the method converges with order γ = 2.
For the numerical implementation, we use the follow splitting,
D II ∆t = A + B , where, A = ∆t v + −σ∆U · ∇ x B = f (x)∆t − Γv∆t + σ∆W + σΓ∆U · ∇ v .(59)
Theorem 9. The naive splitting scheme
exp(D II ∆t ) ≈ exp(A) exp(B )
yields an approximation with strong order 1. The symmetric splitting scheme,
exp(D II ∆t ) ≈ exp( A 2 ) exp(B ) exp( A 2 )
gives strong order 2.
Proof. We use the Baker Campbell Hausdorff formula [22] for both splitting schemes: The symmetric splitting will give us a higher order method because the bracket [A, B ] does not show up.
exp A exp B = exp{A + B + 1 2 [A, B ] +
Fortunately, we do not have to compute the difficult third order brackets in order to see this. It can be shown (see (123)) that
exp A exp B x v ≈ I + A + B + 1 2 [A, B ] + 1 2 A 2 + AB + B A + B 2 x v = x(∆t ) v(∆t ) + 0 n×1 2D x f (x)v − Γ f (x) + Γ 2 v ∆t 2 2 + n j =1 −σ j Γσ j ∆U j + n j =1 0 n×1 −Γσ j ∆t ∆W j 2 + n j =1 0 n×1 −2D x f (x)σ j − Γ 2 σ j ∆t ∆U j 2 ,(61)
where (x(∆t ), v(∆t )) denotes the strong order γ = 1 Ito Taylor approximation (84). We see that g α = f α for all α ∈ A 1 , but g α = f α for α = (0, 0), ( j , 0) ∈ A 2 . Therefore the naive splitting exp A exp B of truncation 2
D II ∆t = A + B
gives a strong order γ = 1 method but not a strong order γ = 2 method. Turning now to the symmetric splitting, one can show that (see (124)) that
exp(A/2) exp B exp(A/2) x v ≈ I + A + B + 1 2 (A 2 + AB + B A + B 2 ) x v = x(∆t ) v(∆t ) + n j =1 Γσ j −D x f (x)σ j − Γ 2 σ j ∆t ∆U j 2 ,(62)
where (x(∆t ), v(∆t )) denotes the strong order γ = 2 Ito Taylor approximation (88). Then g α = f α for all α ∈ A 2 , and the remainder term ∆t ∆U j /2 has variance O(∆t 5 ). Therefore the symmetric splitting of truncation 2 has strong order 2.
The numerical tests in Figures 1 and 2 confirmed the convergence orders for truncation II with the naive and symmetric splittings methods can be found in the next section.
Third-order truncation
Finally, we turn to the next truncation,
D III ∆t = ∆t + n j =1 ∆W j X j + n j =1 ∆U j [X 0 , X j ] + n j =1 ∆V j [X 0 , X j , X 0 ] ∆U j = 1 2 [∆t , ∆W j ] = 1 2 (I (0, j ) − I ( j ,0) ) = I (0, j ) − ∆t ∆W j /2 ∆U = (∆U 1 , . . . , ∆U n ) ∆V j = [∆t , ∆W j , ∆t ] = 1 18 (2I (0, j ,0) − 2I ( j ,0,0) + ∆t I ( j ,0) − ∆t I (0, j ) ) = 1 9 I (0, j ,0) − I ( j ,0,0) − ∆t ∆U j ∆V = (∆V 1 , . . . , ∆V n ).(63)
This comes from [21, eq. 2.15], using the fact that [[X 0 , X i ], X j ] = 0 when i , j = 0 for additive noise. Since
X 0 = v · ∇ x + ( f (x) − Γv) · ∇ v X j = σ j · ∇ v [X 0 , X j ] = −σ j · ∇ x + Γσ j · ∇ v [[X 0 , X j ], X 0 ] = Γσ j · ∇ x − (D f (x)σ j + Γ 2 σ j ) · ∇ v ,(64)
we can rewrite D III ∆t as
D III ∆t = (v∆t − σ∆U + Γσ∆V ) · ∇ x + ( f (x) − Γv)∆t + σ∆W + Γσ∆U − (D f (x)σ + Γ 2 σ)∆V · ∇ v ,(65)
which gives us the ODEs
x = v∆t − σ∆U + Γσ∆V v = ( f (x) − Γv)∆t + σ∆W + Γσ∆U − (D f (x)σ + Γ 2 σ)∆V.
(66) eq: III
We choose not to pursue a proof of the strong convergence order of the method X ∆t = exp(D III )X 0 due to the lengthy calculations, but rather rely on the numerical results.
For the numerical implementation, we use the splitting D III ∆t = A + B where,
A = v∆t − σ∆U + Γσ∆V B = ( f (x) − Γv)∆t + σ∆W + Γσ∆U − (D f (x)σ + Γ 2 σ)∆V,(67)
eq: AB-III and the two corresponding ODEs are given by
x = v∆t − σ∆U + Γσ∆V v = 0(68)
and
x = 0 v = ( f (x) − Γv)∆t + σ∆W + Γσ∆U − (D f (x)σ + Γ 2 σ)∆V.(69)
They have explicit solutions
x(t ) = x + t v − σ∆U ∆t + Γσ∆V ∆t(70)
and
v(t ) = c 0 v + c 1 ∆t f (x)∆t + σ∆W + Γσ∆U − (D f (x)σ + Γ 2 σ)∆V(71)
where c 0 = exp(−Γt ) and c 1 = Γ −1 (I − c 0 ).
In principle, the symmetric splitting methods applied to ODEs have order 2. In order to achieve higher order of accuracy, we solve the ODEs (66) using the Neri's splitting method [22], which consists of alternating the operators in (67) three times:
exp(D III ) ≈ exp(c 1 A) exp(d 1 B ) exp(c 2 A) exp(d 2 B ) exp(c 3 A) exp(d 3 B ) exp(c 4 A), where c 1 = 1 2(2 − 3 2) , c 2 = 1 2 − c 1 , c 3 = c 2 , c 4 = c 1 , and d 1 = 1 2 − 3 2 , d 2 = 1 − 2d 1 , d 3 = d 1 .(72)
In the implementation, the term D f (x)σ will be approximated by a finite-difference formula,
D f (x)σ∆V ≈ f (x + εσ∆V ) − f (x) ε .(73)
For practical implementations, the joint covariances of ∆W, ∆U , and ∆V are needed in order to sample these mean-zero Gaussian random variables. Note that E(∆W i ∆W j ) = ∆t δ i j for i , j = 1, 2, · · · , n, since the components of the Brownian motion are independent with mean 0 and variance ∆t . Therefore (75) eq: cov
the matrix E(∆W 2 ) i j := E(∆W i ∆W j ) satisfies E(∆W 2 ) = ∆t I n×n .E(∆W i ∆U j ) = 0, E(∆W i ∆V j ) = 0, E(∆U i ∆U j ) = ∆t 3 12 δ i j , E(∆U i ∆V j ) = −∆t 4 216 δ i j E(∆V i ∆V j ) = ∆t 5 2430 δ i j (74) so that E(∆W 2 ) E(∆W ∆U ) E(∆W ∆V ) E(∆U ∆W ) E(∆U 2 ) E(∆U ∆V ) E(∆V ∆W ) E(∆V ∆U ) E(∆V 2 ) = ∆t I
Numerical tests
A one-dimensional pendulum model
In the first sets of experiments, we consider the one-dimensional pendulum model:
f (x) = − sin(x)(76)
when x ∈ R. To examine the strong order, we generate the 'exact' solution using the Euler-Maruyama method with very small step size δt = 2 −18 . In order to verify strong convergence, we used 100 realizations. Furthermore, in order to follow the same realization in the implementation of each algorithms, we first generate the Brownian motions with small step size, and then the multiple stochastic integrals are evaluated using a numerical quadrature (Simpson's rule). Notice that this is only for the purpose of examining the strong order of accuracy. In practice, one can sample the integrals using the covariance matrix (75). As can be seen from Figure 1, the order of accuracy is as expected.
Lennard-Jones cluster
The Lennard Jones model is given by The matrix Γ = 10I 21×21 , and σ = 2k B T Γ, where k B T = 0.3, and T is the temperature, not to be confused with the final time T . Initially, the atoms are arranged at the vertices of a hexagon and its center. The side length corresponds to the minimum of the Lennard-Jones potential, 2 1/6 . Notice that for this model, the function f (x) does not have bounded derivatives unless a cut-off is introduced. Nevertheless, as shown in Figure 2, the strong order of accuracy is still consistent with the results of the analysis.
f i (x) = n j =i , j =1 12 1 r i j 13 − 6 1 r i j 7 r i j r i j , where r i j = x i − x j ∈ R 3 and r i j = r i j 2 ,(77)
Summary and discussion
In this paper, we presented the analysis of strong convergence of some numerical schemes for the Langevin equation with additive noise. This type of convergence is important for predicting the transient stage of the stochastic processes. In addition, we presented several new operator-splitting schemes based on Kunitas solution representation. In particular, we obtained algorithms with strong order up to order 3.
There are several remaining challenges in simulating algorithms for Langevin-type of equations. First, there might be multiple scales involved in the force term f (x) [23]. In this case, a more appropriate splitting is between the fast and slow forces, e.g., [24]. Secondly, the damping and diffusion coefficients can be position-dependent. Such models arise, for instance, in the dissipative-particle dynamics (DPD) [25,26].
Finally, there are Langevin equations with strong stiffness, e.g., large damping coefficients. In this case, implicit algorithm are needed. These issues will be addressed in separate works.
Appendix
Due to the lengthy calculations in the analysis, we have included some parts of the proofs in the appendix. These details are provided here also for the interested readers who intend to implement the new algorithms.
Coefficient function computations
The coefficient function f α for a multi-index α = ( j 1 , j 2 , . . . , j l ) is given by
f α (x) = L j 1 L j 2 . . . L j l x, L 0 := d k=1 a k ∂ ∂x k + 1 2 d k,l =1 m j =1 b k j b l j ∂ 2 ∂x k ∂x l L j := d k=1 b k j ∂ ∂x k , j = 1, 2, . . . , m,(78)
Observe that for the Langevin equation with additive noise,
L 0 = v · ∇ x + ( f (x) − Γv) · ∇ v L j = σ j · ∇ v f (0) (x, v) = L 0 (x, v) = L 0 x, L 0 v = v f (x) − Γv , f ( j ) (x, v) = L j (x, v) = (L j x, L j v) = 0 σ j , j = 1, 2, · · · , n, f (0,0) (x, v) = L 0 v, L 0 ( f (x) − Γv) = f (x) − Γv D f (x)v − Γ( f (x) − Γv) , where D f (x) is the Jacobian matrix f ( j 1 , j 2 ) (x, v) = L j 1 (0, σ j 2 ) = (0, L j 1 σ j 2 ) = 0 for additive noise f (0, j ) (x, v) = L 0 (0, σ j ) = 0 f ( j ,0) (x, v) = L j (v, f (x) − Γv) = σ j −Γσ j f ( j 1 , j 2 , j 3 ) = 0 if j 1 , j 2 = 0 and j 3 ∈ {0, 1, . . . , n} f ( j 1 , j 2 , j 3 , j 4 ) = 0 if j 1 , j 2 , j 3 , j 4 = 0 f ( j ,0,0) = −Γσ j D f (x)σ j + Γ 2 σ j for all j = 1, 2, · · · , n where Γ 2 means Γ squared f (0,0,0) = D f (x)v − Γ f (x) + Γ 2 v D[D f (x)v]v − D[Γ f (x)]v + D f (x)( f (x) − Γv) + Γ 2 ( f (x) − Γv)
f ( j 1 , j 2 ,0,0) = 0 for all j 1 , j 2 = 1, 2, · · · , n. (79) eq: coeffs
These are all the coefficient functions we'll need for methods of strong order ≤ 3.
Hierarchical set computations
Throughout this paper, α will denote a multi-index α = ( j 1 , j 2 , . . . , j l ) with entries j k ∈ {0, 1, 2, . . . , n}, length l (α) := l , and n(α) will denote the number of zero entries in α. In this section we focus on hierarchical sets [2, eq. 10.
6.2]
A γ := {α : l (α) + n(α) ≤ 2γ or l (α) = n(α) = γ + 0.5}
where γ ∈ {0.5, 1.0, 1.5, 2.0, 2.5, 3}, and f α refers to the coefficient functions for the Itô Taylor truncation corresponding to the Langevin equation with additive noise. To find A γ − {α : f α = 0} we will use the coefficient function calculations from the previous section. All of the following calculations were done by hand, are easily verified in the case n = 1, and generalize immediately to the case n ≥ 1: ( j 1 , 0), (0, 0), (0, 0, 0), ( j 1 , 0, 0) : j 1 = 1, 2, · · · , n}.
A 0.5 = {α : l (α) + n(α) ≤ 1 or l (α) = n(α) = 1} = {( j 1 ), (0) : j 1 = 1, 2, · · · , n} A 0.5 − {α : f α = 0} = A 0.5 A 1 = {α : l (α) + n(α) ≤ 2} = {( j 1 ), (0), ( j 1 , j 2 ) : j 1 , j 2 = 1, 2, · · · , n} A 1 − {α : f α = 0} = {( j 1 ), (0) : j 1 = 1, 2, · · · , n} A 1.5 = {α : l (α) + n(α) ≤ 3 or l (α) = n(α) = 2} = {( j 1 ), (0), ( j 1 , j 2 ), (0, j 1 ), ( j 1 , 0), ( j 1 , j 2 , j 3 ), (0, 0) : j k = 1, 2, · · · , n} A 1.5 − {α : f α = 0} = {( j 1 ), (0), ( j 1 , 0), (0, 0) : j 1 = 1, 2, · · · , n} A 2 = {α : l (α) + n(α) ≤ 4} = {( j 1 ), (0), ( j 1 , j 2 ), (0, j 1 ), ( j 1 , 0)} ∪ {( j 1 , j 2 ,A 3 = A 2.5 ∪ {α : l (α) + n(α) = 6} = A 2.5 ∪ {( j 1 , j 2 , 0, 0), ( j 1 , 0, j 2 , 0), ( j 1 , 0, 0, j 2 )} ∪ {(0, j 1 , j 2 , 0), (0, j 1 , 0, j 2 ), (0, 0, j 1 , j 2 )} ∪ {(0, j 1 , j 2 , j 3 , j 4 ), ( j 1 , 0, j 2 , j 3 , j 4 ), ( j 1 , j 2 , 0, j 3 , j 4 ), ( j 1 , j 2 , j 3 , 0, j 4 )} ∪ {( j 1 , j 2 , j 3 , j 4 , 0), ( j 1 , . . . , j 6 ) : j k = 1, 2, · · · , n} A 3 − {α : f α = 0} = A 2.5 − {α : f α = 0} = {( j 1 ), (0),
(81) eq: indices
The large index sets A γ are possibly necessary for multiplicative noise, but for additive noise what we really need are the smaller index sets A γ − {α : f α = 0}: ( j 1 , 0), (0, 0), ( j 1 , 0, 0), (0, 0, 0) : j 1 = 1, 2, · · · , n}.
A 1 − {α : f α = 0} = A 0.5 − {α : f α = 0} = {( j 1 ), (0) : j 1 = 1, 2, · · · , n} A 2 − {α : f α = 0} = A 1.5 − {α : f α = 0} = {( j 1 ), (0), ( j 1 , 0), (0, 0) : j 1 = 1, 2, · · · , n} A 3 − {α : f α = 0} = A 2.5 − {α : f α = 0} = {( j 1 ), (0),
(82)
Derivation of the strong order 1 Taylor approximation for the Langevin equation with additive noise
The strong order 1 (and 0.5) Itô Taylor truncation is
X n+1 = X n + α∈A 1 : f α =0 f α (X n )I α ,(83)which gives us x(∆t ) v(∆t ) = x v + v f (x) − Γv ∆t + n j =1 0 σ j ∆W j .
(84) eq: it1
Strong order 2 Itô Taylor approximation for the Langevin equation with additive noise
The strong order 2 (and 1.5) Itô Taylor (discrete time) approximation for d X t = a(X t )d t + n j =1 b j dW j is then
X n+1 = X n + α∈A 2 −{α: f α =0} f α (X n )I α .
where the coefficient functions are given in (79) and the multiple Itô integrals I α are given by
I α = ∆t 0 t l 0 · · · t 2 0 dW j 1 t 1 dW j 2 t 2 . . . dW j l t l .(85)
Recall from (81) that
A 2 − {α : f α = 0} = {( j 1 ), (0), ( j 1 , 0), (0, 0) : j 1 = 1, 2, · · · , n}(86)
and the relevant coefficient functions are (79)
f (0) = a = v f (x) − Γv , f ( j ) = b j = 0 σ j , j = 1, . . . , n, f (0,0) = f (x) − Γv D f (x)v − Γ( f (x) − Γv) , f ( j ,0) = σ j −Γσ j (87)
Then the strong order 2 Itô Taylor approximation is
x(∆t ) v(∆t ) = x v + v f (x) − Γv ∆t + f (x) − Γv D f (x)v − Γ( f (x) − Γv) ∆t 2 2 + n j =1 0 σ j ∆W j + n j =1 σ j −Γσ j I ( j ,0) .
(88) eq: it2
Strong order 3 Itô Taylor approximation for the Langevin equation with additive noise
Building off of the previous section, the strong order 3 (and 2.5) Itô Taylor truncation (not given in [2])
requires
For this method, we just have to compute the additional coefficient functions f (0,0,0) and f ( j ,0,0) , and multiple Itô integrals I (0,0,0) and I (1,0,0) . The easiest of these four things to compute is the Riemann integral
I (0,0,0) = ∆t 0 t 0 s 0 d ud sd t = ∆t 3 3! .(90)
For the coefficient functions, recall from (79) that
f ( j ,0,0) (x, v) = −Γσ j D f (x)σ j + Γ 2 σ j for all j = 1, 2, · · · , n f (0,0,0) (x, v) = vD f (x) − Γ( f (x) − Γv) D(D f (x)v)v − D(Γ f (x))v + D f (x)( f (x) − Γv) + Γ 2 ( f (x) − Γv) .(91)
Then, with ∆V j := I ( j ,0,0) , the strong order 3 Itô Taylor approximation is
x(∆t ) v(∆t ) = x v + v f (x) − Γv ∆t + f (x) − Γv D f (x)v − Γ( f (x) − Γv) ∆t 2 2 + n j =1 0 σ j ∆W j + n j =1 σ −Γσ j ∆Z j + n j =1 vD f (x) − Γ( f (x) − Γv) D(D f (x)v)v − D(Γ f (x))v + D f (x)( f (x) − Γv) + Γ 2 ( f (x) − Γv) ∆t 3 3! + n j =1 −Γσ D f (x)σ j + Γ 2 σ j ∆V j .
(92) eq: it3
The analysis of the stochastic velocity Verlet method
Here we give a detailed proof of the following:
Theorem 10.
The SVV algorithm has strong order 2.
Proof. We start with the displacement component. We compare
x(∆t ) = x + v∆t + ( f (x) − Γv) ∆t 2 2 + n j =1 σ j I ( j ,0) (∆t )
i.e., the Taylor approximation with strong order 2, with the SVV method,
x(∆t ) = x + v∆t + ( f (x) − Γv) ∆t 2 2 + O(∆t 3 ) + n j =1 σ j I ( j ,0) (∆t ) + n j =1 ∆t 0 t 0 e −Γ(t −s) − I σ j dW j s d t .
(93)
Setting f 0 = g 0 = v, f j = g j = 0, f 00 = g 00 = f (x) − Γv, f j 0 = g j 0 = σ j , f 0 j = g 0 j = 0, f j 1 j 2 = 0,
and it remains to show that R k satisfies convergence condition (15) with γ = 2. To that end, first notice that E(R k R l ) = δ kl E(R 2 k ) (where δ kl = 1 if k = l and 0 otherwise), since the increments of the Wiener process are independent. Next, since R k is a martingale, we may apply the discrete version of Doob's lemma with p = 2[2, eq. 2.3.7] to obtain an estimate for (15):
E max 1≤m≤n T m−1 k=0 R k 2 ≤ 4E n T −1 k=0 R k 2 by Doob's lemma ≤ 4 n T −1 k=0 E(R 2 k ) since E(R k R l ) = δ kl E(R 2 k ) = 4n T E(R 2 k ).(95)
Notice e −Γ(t −s) − I n = O(∆t )ones n×n for k∆t ≤ s ≤ t ≤ (k + 1)∆t , and recall that E(I 2 j 0 ) = O(∆t 3 ) for all j = 1, 2, · · · , n [2, pg. 172, exercise 5.2.7]. As E(I ( j 1 ,0) I ( j 2 ,0) ) = 0 for distinct j 1 , j 2 = 1, 2, · · · , n (W j 1 ,W j 2 are independent, see [2, p. 223, eq. 5.12.7] for more details), we have (15) is
E(R 2 k ) = E (O(∆t 2 ) n j =1 σ j I ( j ,0) 2 = O(∆t 2 )E( n j =1 σ 2 j I 2 ( j ,0) ) = O(∆t 2 ) n j =1 σ 2 j E(I 2 ( j ,0) ) = O(∆t 2 )O(∆t 3 ) since n << n T = O(∆t 5 ).(96)Therefore, E max 1≤m≤n T m−1 k=0 R k 2 ≤ 4n T O(∆t 5 ) = O(∆t 4 ) = O(∆t 2γ ), so convergence criterionsatisfied for γ = 2.
For the velocity components, with c 0 = e −Γ∆t , c 1 = (Γ∆t ) −1 (I − c 0 ), and c 2 = 1
∆t 2 ∆t 0 Γ −1 (I − e −Γt )d t ,
where I is the n ×n identity matrix, we compare the Itô Taylor approximation with strong order 2 and the
SVV algorithm, v(∆t ) = v + ( f (x) − Γv)∆t + (D f (x)v − Γ( f (x) − Γv)) ∆t 2 2 + n j =1 σ j ∆W j − n j =1 Γσ j I ( j ,0) (∆t ) v(∆t ) = c 0 v + c 1 f (x)∆t + c 2 ( f (x(∆t )) − f (x))∆t + n j =1 ∆t 0 e −Γ(∆t −s) σ j dW j s = v + ( f (x) − Γv)∆t + [D f (x)v − Γ( f (x) − Γv)] ∆t 2 2 + n j =1 σ j ∆W j − Γσ j I ( j ,0) + n j =1 ∆t 0 e −Γ(∆t −s) − I + Γ(∆t − s) σ j dW j s ,(97)
where the last equality is non-trivial and holds from Taylor expanding the c's and f (x(∆t )) − f (x). Setting f α = g α for all α ∈ A 2 , we see from the calculation above that convergence criterion (14) is trivially satisfied. Next, set
R k = n j =1 (k+1)∆t k∆t e −Γ(∆t −s) − I + Γ(∆t − s) σ j dW j s , k = 0, . . . , n T − 1,(98)
and notice that R k is a discrete martingale. Once again, E(R k R l ) = δ kl E(R 2 k ) since Brownian increments are independent and the Itô integrals are taken over disjoint intervals. Then we see that
E R 2 k = ∆t 0 (e −Γ(= K 5 ∆t 5 ⇒ E max 1≤m≤n T m−1 k=0 R k 2 ≤ 4E n T −1 k=0 R 2 k by discrete Doob's lemma ≤ 4E n T −1 k=0 |R k | 2 since E(R k , R l ) = δ kl E R 2 k = 4K 5 n T −1 k=0 E R 2 k = 4K 5 n T O(∆t 5 ) = O(∆t 4 ) since n T ∆t = T.(99)
Therefore convergence criterion (15) is satisfied with γ = 2.
Expansion of exp D I ∆t (x, v)
Here we derive the following formula to show the strong order convergence of the first order truncation:
exp(D I ∆t ) x v = x v + v f (x) − Γv ∆t + f (x) − Γv D f (x)v − Γ( f (x) − Γv) ∆t 2 2 + n j =1 0 σ j ∆W j + n j =1 σ −Γσ j ∆t ∆W j 2 + higher order terms (100) eq: trunc1
where the higher order terms do not involve I ( j ,0) .
Proof. Recall that D I ∆t is given by
D I ∆t = ∆t X 0 + n j =1 ∆W j X j (101) where X 0 (·) = D x (·)v + D v (·)( f (x) − Γv) and X j (·) = D v (·)σ j . Observe that X 0 (x) = D x (x)v + D v (x)( f (x) − Γv) = I n v = v X j (x) = D v (x)σ j = 0 n×n σ j = 0 n×1 ⇒ D I ∆t (x) = ∆t v X 0 (v) = D x (v)v + D v (v)( f (x) − Γv) = f (x) − Γv X j (v) = D v (v)σ j = σ j ⇒ D I ∆t (v) = ∆t X 0 (v) + n j =1 ∆W j X j (v) = ∆t ( f (x) − Γv) + n j =1 σ j ∆W j ⇒ D I ∆t x v = D I ∆t (x) D I ∆t (v) = v f (x) − Γv ∆t + n j =1 0 n×1 σ j ∆W j (102) X 0 (D I ∆t x) = X 0 (v∆t ) = ∆t X 0 (v) = ∆t ( f (x) − Γv) X j (D I ∆t x) = ∆t X j (v) = σ j ∆t ⇒ (D I ∆t ) 2 x = ∆t X 0 (D I ∆t (x)) + n j =1 ∆W j X j (D I ∆t (x)) = ( f (x) − Γv)∆t 2 + n j =1 σ j ∆t ∆W j X 0 (D I ∆t v) = X 0 ∆t ( f (x) − Γv) + n j =1 σ j ∆W j = ∆t X 0 ( f (x) − Γv) + n j =1 ∆W j X 0 (σ j ) = ∆t D x ( f (x) − Γv)v + D v ( f (x) − Γv)( f (x) − Γv) + 0 n×1 = ∆t D x f (x)v − ∆t Γ f (x) + ∆t Γ 2 v X j (D I ∆t v) = ∆t X j ( f (x) − Γv) + n k=1 ∆W k X j (σ k ) = ∆t D v ( f (x) − Γv)σ j = −∆t Γσ j ⇒ (D I ∆t ) 2 v = ∆t X 0 (D I ∆t v) + n j =1 ∆W j X j (D I ∆t v) = ∆t ∆t D x f (x)v − ∆t Γ f (x) + ∆t Γ 2 v + n j =1 ∆W j (−∆t Γσ j ) = ∆t 2 D x f (x)v − Γ f (x) + Γ 2 v + n j =1 (−Γσ j )∆t ∆W j ⇒ (D I ∆t ) 2 x v = (D I ∆t ) 2 x (D I ∆t ) 2 v = f (x) − Γv D x f (x)v − Γ f (x) + Γ 2 v ∆t 2 + n j =1 σ j −Γσ j ∆t ∆W j ⇒ exp(D I ∆t ) x v = I + D I ∆t + 1 2 (D I ∆t ) 2 + . . . x v = x v + v f (x) − Γv ∆t + n j =1 0 σ j ∆W j + f (x) − Γv D x f (x)v − Γ f (x) + Γ 2 v ∆t 2 2 + n j =1 σ j −Γσ j ∆t ∆W j 2 + higher order terms.
(103)
Derivation of exp D II ∆t (x, v)
We define truncation 2 by the truncation of the Kunita solution operator D ∆t after the second order brackets [X i , X j ]:
D II ∆t := ∆t X 0 + n j =1 ∆W j X j + 1 2 i < j [∆W i , ∆W j ][X i , X j ](104)
where ∆W 0 := ∆t , are identically zero: indeed for i , j = 1, 2, · · · , n,
[∆W i , ∆W j ] = ∆t 0 W i t dW j t − ∆t 0 W j t dW i t ,(105)X i X j x = X i (0 n×1 ) = 0 n×1 , X i X j v = X i σ j = 0 n×1 , [X 0 , X j ]x = X 0 0 n×1 − X j v = 0 n×1 − σ j = −σ j , [X 0 , X j ]v = X 0 σ j − X j ( f (x) − Γv) = −D v ( f (x) − Γv)σ j = Γσ j .(106)
Therefore, for additive noise, with ∆U j := 1 2 [∆t , ∆W j ], we have
D II ∆t x = v∆t − n j =1 σ j ∆U j , D II ∆t v = ( f (x) − Γv)∆t + n j =1 σ j ∆W j + n j =1 Γσ j ∆U j , ⇒ D II ∆t x v = D II ∆t x D II ∆t v = v f (x) − Γv ∆t + n j =1 0 n×1 σ j ∆W j + n j =1 −σ j Γσ j ∆U j (107) X 0 (D II x) = X 0 v∆t − n j =1 σ j ∆U j = ∆t X 0 (v) = ( f (x) − Γv)∆t X j (D II x) = ∆t X j (v) = σ j ∆t [X 0 , X j ](D II x) = ∆t [X 0 , X j ](v) = ∆t Γσ j ⇒ (D II ) 2 x = ∆t X 0 (D II x) + n j =1 ∆W j X j (D II x) + n j =1 ∆U j [X 0 , X j ](D II x) = ( f (x) − Γv)∆t 2 + n j =1 σ j ∆t ∆W j + n j =1 Γσ j ∆t ∆U j (108) X 0 (D II v) = ∆t X 0 ( f (x) − Γv) = ∆t D x ( f (x) − Γv)v + D v ( f (x) − Γv)( f (x) − Γv) = ∆t D x f (x)v − Γ f (x) + Γ 2 v X j (D II v) = ∆t X j ( f (x) − Γv) = ∆t D v ( f (x) − Γv)σ j = −∆t Γσ j [X 0 , X j ](D II v) = ∆t [X 0 , X j ]( f (x) − Γv) = ∆t [X 0 X j ( f (x) − Γv) − X j X 0 ( f (x) − Γv)] = ∆t [X 0 (−Γσ j ) − X j (D x f (x)v − Γ f (x) + Γ 2 v)] = ∆t [0 n×1 − D v (D x f (x)v)σ j + 0 n×1 − D v (Γ 2 v)σ j ] = ∆t [−D x f (x)σ j − Γ 2 σ j ] ⇒ (D II ) 2 v = ∆t X 0 (D II v) + n j =1 ∆W j X j (D II v) + n j =1 ∆U j [X 0 , X j ](D II v) = ∆t 2 [D x f (x)v − Γ f (x) + Γ 2 v] − n j =1 Γσ j ∆t ∆W j − n j =1 (D x f (x)σ j + Γ 2 σ j )∆t ∆U j ⇒ (D II ) 2 x v = (D II ) 2 x (D II ) 2 v = f (x) − Γv D x f (x)v − Γ f (x) + Γ 2 v ∆t 2 + n j =1 σ j −Γσ j ∆t ∆W j + n j =1 Γσ j −D x f (x)σ j − Γ 2 σ j ∆t ∆U j (109) ∴ exp D II x v = I + D II + 1 2 (D II ) 2 + . . . x v = x v + v f (x) − Γv ∆t + f (x) − Γv D x f (x) − Γ f (x) + Γ 2 v ∆t 2 2 + n j =1 0 n×1 σ j ∆W j + n j =1 −σ j Γσ j ∆U j + n j =1 σ j −Γσ j ∆t ∆W j 2 + n j =1 Γσ j −D x f (x)σ j − Γ 2 σ j ∆t ∆U j 2 +
and a second application of this formula yields the following useful formula [22, eq. (3.2)]:
exp(A/2) exp B exp(A/2) = exp{A + B + 1 12 [B, B, A] − 1 24 [A, A, B ] + . . . }.(112)
This symmetric product causes the bracket [A, B ] to vanish, and in terms of numerics this observation will give us a higher order method.
Convergence of naive and symmetric splittings for truncation 1
Consider the first order truncation D I split into
D I = A + B with A = v∆t ∇ x and B = ( f (x) − Γv)∆t + σ∆W ∇ v .(113)
We compute the following:
Ax = D x (x)(v∆t ) = I (v∆t ) = v∆t Av = D x (v)(v∆t ) = 0 n×n (v∆t ) = 0 n×1 B v = ( f (x) − Γv)∆t + σ∆W B x = D v (x)B v = 0 n×n B v = 0 n×1
In general:
A(·) = D x (·)Ax and B (·) = D v (·)B v(114)(A + B ) x v = v f (x) − Γv ∆t + n j =1 0 n×1 σ j ∆W j (115) A 2 x = A(Ax) = D x (Ax)Ax = D x (v∆t )Ax = 0 n×n Ax = 0 n×1 A 2 v = A(Av) = A0 n×1 = 0 n×1 AB x = A0 n×1 = 0 n×1 AB v = D x (B v)Ax = D x ( f (x)∆t )Ax = ∆t 2 D x f (x)v B 2 x = B (B x) = B 0 n×1 = 0 n×1 B 2 v = B (B v) = D v (B v)B v = D v (−Γv∆t )B v = −∆t ΓB v = (−Γ f (x) + Γ 2 v)∆t 2 − Γσ∆t ∆W (116) 1 2 [A, B ] + A 2 + AB + B A + B 2 x v = 1 2 2AB + B 2 x v since A 2 (x, v) = 0 = AB + 1 2 B 2 x v = 0 n×1 D x f (x)v − (1/2)Γ f (x) + (1/2)Γ 2 v ∆t 2 + n j =1 0 n×1 −Γσ j ∆t ∆W 2(117)
By the BCH formula,
exp(A) exp(B ) x v ≈ I + A + B + 1 2 [A, B ] + 1 2 (A 2 + AB + B A + B 2 ) x v = x v + v f (x) − Γv ∆t + n j =1 0 n×1 σ j ∆W j + 0 n×1 D x f (x)v − (1/2)Γ f (x) + (1/2)Γ 2 v ∆t 2 + n j =1 0 n×1 −Γσ j ∆t ∆W 2 .(118)
We see by comparison with the Ito Taylor approximations that the non-symmetric method above has strong order γ = 1 by matching terms and noticing that the multiple stochastic integral I ( j ,0) does not show up.
As for the symmetric splitting, we first compute
B Ax = D v (Ax)B v = D v (v∆t )B v = ∆t B v = ( f (x) − Γv)∆t 2 + σ∆t ∆W B Av = B 0 n×1 = 0 n×1 ⇒ B A x v = f (x) − Γv 0 n×1 ∆t 2 + n j =1 σ j 0 n×1 (119)
and use the computations for A 2 , AB, B 2 given above to obtain
(A 2 + AB + B A + B 2 ) x v = 0 2n×1 + 0 n×1 D x f (x)v ∆t 2 + n j =1 σ j 0 n×1 ∆t ∆W j + 0 n×1 −Γ f (x) + Γ 2 v ∆t 2 + n j =1 0 n×1 −Γσ j ∆t ∆W j = f (x) − Γv D x f (x)v − Γ f (x) + Γ 2 v ∆t 2 + n j =1 σ j −Γσ j ∆t ∆W j .(120)
Then using the BCH formula, we get the approximation
exp(A/2) exp(B ) exp(A/2) x v ≈ I + A + B + 1 2 (A 2 + AB + B A + B 2 ) x v = x v + v f (x) − Γv ∆t + n j =1 0 n×1 σ j ∆W j + f (x) − Γv D x f (x)v − Γ f (x) + Γ 2 v ∆t 2 + n j =1 σ j −Γσ j ∆t ∆W j .(121)
By comparing again to the Ito Taylor approximations, we see that this approximation is much closer to attaining strong order γ = 2 than the non-symmetric splitting, but still falls short of strong order γ = 2, only because the multiple Ito integral I ( j ,0) is not present.
Convergence of naive and symmetric splittings for truncation 2
Here we consider the second order truncation D II ∆t split into D II ∆t = A + B . By the BCH formula in the previous section, it seems necessary to compute the brackets Observe that with A = (v∆t − σ∆U ) · ∇ x and B = (( f (x) − Γv)∆t + σ∆W + Γσ∆U ) · ∇ v , we have
Ax = v∆t − σ∆U , Av = 0 B v = ( f (x) − Γv)∆t + σ∆W + Γσ∆U A(·) = D x (·)Ax B (·) = D v (·)B v where D x (·) = ∂(·) i ∂x j and D v (·) = ∂(·) i ∂v j (i , j = 1, 2, · · · , n) Av = D x (v)Ax = 0 n×n Ax = 0 n×1 B x = D v (x)B v = 0 n×n B v = 0 n×1 AB x = B Av = 0 n×1 B Ax = B (Ax) = D v (Ax)B v = ∆t I n×n B v = ∆t B v AB v = A(B v) = D x (B v)Ax = ∆t D x f (x)Ax ⇒ AB x v = 0 n×1 D x f (x)v ∆t 2 + n j =1 0 n×1 −D x f (x)σ j ∆t ∆U j B A x v = f (x) − Γv 0 n×1 ∆t 2 + n j =1 σ j 0 n×1 ∆t ∆W j + n j =1 Γσ j 0 n×1 ∆t ∆U j A 2 x v = 0 2n×1 B 2 x v = 0 n×1 −∆t ΓB v = 0 n×1 −Γ f (x) + Γ 2 v ∆t 2 + n j =1 0 n×1 −Γσ j ∆t ∆W j + n j =1 0 n×1 −Γ 2 σ j ∆t ∆U j (A 2 + AB + B A + B 2 ) x v = f (x) − Γv D x f (x)v − Γ f (x) + Γ 2 v ∆t 2 + n j =1 σ j −Γσ j ∆t ∆W j + n j =1 Γσ j −D x f (x)σ j − Γ 2 σ j ∆t ∆U j [A, B ] x v = [A, B ]x [A, B ]v = AB x − B Ax AB v − B Av = −B Ax AB v = −∆t B v ∆t D x f (x)Ax = − f (x) + Γv D x f (x)v ∆t 2 + n j =1 −σ j 0 n×1 ∆t ∆W j + n j =1 −Γσ j −D x f (x)σ j ∆t ∆U j(122)
Observe that the naive splitting gives
exp A exp B x v ≈ I + A + B + 1 2 [A, B ] + A 2 + AB + B A + B 2 x v = x v + v f (x) − Γv ∆t + n j =1 0 n×1 σ j ∆W j + 0 n×1 2D x f (x)v − Γ f (x) + Γ 2 v ∆t 2 2 + n j =1 −σ j Γσ j ∆U j + n j =1 0 n×1 −Γσ j ∆t ∆W j 2 + n j =1 0 n×1 −2D x f (x)σ j − Γ 2 σ j ∆t ∆U j 2 ⇒ exp A exp B x v − x(∆t ) v(∆t ) = 0 n×1 2D x f (x)v − Γ f (x) + Γ 2 v ∆t 2 2 + n j =1 −σ j Γσ j ∆U j + n j =1 0 n×1 −Γσ j ∆t ∆W j 2 + n j =1 0 n×1 −2D x f (x)σ j − Γ 2 σ j ∆t ∆U j 2 (123) eq: naive
where we let (x(∆t ), v(∆t )) denote the strong order γ = 1 Ito Taylor approximation. We see that g α = f α for all α ∈ A 1 , but g α = f α for α = (0, 0), ( j , 0) ∈ A 2 . Therefore the naive splitting exp A exp B of truncation 2 D II ∆t = A + B gives a strong order γ = 1 method but not a strong order γ = 2 method. As for the symmetric splitting, observe that
exp(A/2) exp B exp(A/2) x v ≈ I + A + B + 1 2 (A 2 + AB + B A + B 2 ) x v = x v + v f (x) − Γv ∆t + n j =1 0 n×1 σ j ∆W j + n j =1 −σ j Γσ j ∆U j + n j =1 σ j −Γσ j ∆t ∆W j 2 + n j =1 Γσ j −D x f (x)σ j − Γ 2 σ j ∆t ∆U j 2
(124) eq: sym and ∆t ∆W j 2 − ∆U j = I ( j ,0) (∆t ). Therefore,
exp(A/2) exp B exp(A/2) x v − x(∆t ) v(∆t ) = n j =1 Γσ j −D x f (x)σ j − Γ 2 σ j ∆t ∆U j 2 ,(125)
where (x(∆t ), v(∆t )) denotes the strong order γ = 2 Ito Taylor approximation (88). That is, the strong order γ = 2 Ito Taylor approximation agrees with this method up to the terms ∆t ∆U j
Covariances
Here we derive the covariances of ∆W, ∆U , ∆V in the multi-dimensional case, where
∆U i = I (0,i ) − ∆t ∆W i 2 ∆V i = 1 9 I (0,i ,0) − I (i ,0,0) − ∆t ∆U i .(126)
Notice E(∆W i ∆W j ) = E(W i ∆t W j ∆t ) = ∆t δ i j , so that E(∆W 2 ) = ∆t I n×n . To find E(∆W ∆U ), we need to know E(∆W i I (0, j ) ), since
E(∆W i ∆U j ) = E(∆W i I (0, j ) ) − ∆t 2 E(∆W i ∆W j ) = E(∆W i I (0, j ) ) − ∆t 2 2 δ i j .(127)
So, we look on [2, p. 223] and find that E(∆W i I (0, j ) ) = ∆t 2 2 δ i j , or verify this fact:
E(∆W i I (0, j ) ) = E W i ∆t ∆t 0 t dW j t by definition = E W i ∆t ∆tW j ∆t − ∆t 0 W j t d t integration by parts = ∆t E W i ∆t W j ∆t − ∆t 0 E(W i ∆t W j t )d t = ∆t 2 δ i j − ∆t 0 E W i ∆t − W i t W j t + E W i t W j t d t = ∆t 2 δ i j −
Therefore, E(∆W i ∆U j ) = E(∆W i I (0, j ) ) − ∆t 2 E(∆W i ∆W j ) = ∆t 2 2 δ i j − ∆t 2 2 δ i j = 0 for all i , j = 1, 2, · · · , n.
Then also E(∆U i ∆W j ) = 0, so that E(∆W ∆U ) = E(∆U ∆W ) = 0 n×n . Next, to find E(∆W ∆V ), we first need to know E(∆W i I (0, j ,0) ) and E(∆W i I ( j ,0,0) ), because
E(∆W i ∆V j ) = 1 9 E(∆W i I (0, j ,0) ) − E(∆W i I ( j ,0,0) ) − ∆t E(∆W i ∆U j )(130)
and we already know that E(∆W i ∆U j ) = 0. To that end, we go to [2, p. 223] again and find E(∆W i I (0, j ,0) ) = E(∆W i I ( j ,0,0) ) = ∆t 3 6 δ i j , or verify ourselves:
E(∆W i I ( j ,I (0, j ) (t )d t = ∆t 0 E W i ∆t − W i t I (0, j ) (t ) + E W i t I (0, j ) (t ) d t = ∆t 0 t 2 2 δ i j d t = ∆t 3 3! δ i j .(131)
Therefore, E(∆W i ∆V j ) = E(∆V i ∆W j ) = 0, i.e. E(∆W ∆V ) = E(∆V ∆W ) = 0 n×n .
We now seek E(∆U 2 ), which will require the covariances E(I (0,i ) I (0, j ) ), yet to be determined, because we can write E(∆U i ∆U j ) = E(I (0,i ) I (0, j ) ) − ∆t E(∆W i I (0, j ) ) +
To find E(I (0,i ) I (0, j ) ), use the Ito isometry:
E(I (0,i ) I (0, j ) ) = E
Therefore 9E(∆U i ∆V j ) = −∆t 4 4! δ i j , that is,
E(∆U ∆V ) = E(∆V ∆U ) = −∆t 4 216 δ i j .(139)
For E(∆V 2 ), we need E(I (0,i ,0) I ( j ,0,0) ), E(I (0,i ,0) I (0, j ,0) ), and E(I (i ,0,0) I ( j ,0,0) ). These higher order calculations cannot be found in [2], but we can use [2, Cor. 5.12.3] to find them:
Theorem 4 .
4The splitting methods have strong order at most 1.
1 12 (
12[A, A, B ] + [B, B, A]) + . . . } exp(A/2) exp B exp(A/2) = exp{A + B + 1 12 [B, B, A] − 1 24 [A, A, B ] + . . . } (60) where [A, B ] := AB − B A is the commmutator bracket from Lie algebras, and [A, A, B ] := [A, [A, B ]], etc.
Figure 1 :
1The error plot versus the time step for the pendulum model on the log-log scale. From top to bottom: Truncations I, II and III. fig: pend and f (x) = −∇E (x) where E is the Lennard Jones potential. For our simulations, we used 7 particles in R 3 .
Figure 2 :
2The error plot for the LJ-7 cluster on a log-log scale. From top to bottom: Truncations I, II and III. fig: LJ
j 3 )
3, (0, 0), (0, j 1 , j 2 ), ( j 1 , 0, j 2 ), ( j 1 , j 2 , 0), ( j 1 , j 2 , j 3 , j 4 ) : j k = 1, 2, · · · , n}A 2 − {α : f α = 0} = {( j 1 ), (0), ( j 1 , 0), (0, 0) : j 1 = 1, 2, · · · , n} A 2.5 = A 2 ∪ {l (α) + n(α) = 5 or l (α) = n(α) = 3}= A 2 ∪ {(0, 0, 0), (0, 0, j 1 ), (0, j 1 , 0), ( j 1 , 0, 0)} ∪ {( j 1 , j 2 , j 3 , 0), ( j 1 , j 2 , 0, j 3 ), ( j 1 , 0, j 2 , j 3 ), (0, j 1 , j 2 , j 3 ), ( j 1 , j 2 , j 3 , j 4 ) : j k = 1, 2, · · · , n} A 2.5 − {α : f α = 0} = {( j 1 ), (0), ( j 1 , 0), (0, 0), (0, 0, 0), ( j 1 , 0, 0) : j 1 = 1, 2, · · · , n}
computing coefficient functions and multiple Itô integrals for multi-indices in the hierarchical set (see (81)) A 3 − {α : f α = 0} = {( j ), (0), ( j , 0), (0, 0), ( j , 0, 0), (0, 0, 0) : j = 1, 2, · · · , n}.
and g j 1 , j 2 = O(∆t 3 ) for all j 1 , j 2 = 1, 2, · · · , n, we see that convergence condition(14) is clearly satisfied for γ = 2 and all α ∈ A 2 = {(0), ( j ), (0, 0), ( j , 0), (0, j ), ( j 1 , j 2 )}.(For α = ( j 1 , j 2 ), the left hand side of(14) is O(∆t 6 ) and the right hand side is O(∆t 3 )). The remainder term R k , k = 0, . . . , n T − 1 after the kth step is
∆t −s) − I + Γ(∆t − s)) 2 d s by the Itô isometry [27, Cor.
the vector fields X i , i = 0 : n are defined as in the previous section, and [A, B ] = AB − B A is the commutator bracket from Lie algebras. For additive noise, that is, constant σ, most of the commutators [X i , X j ]
[A, B ], [A, [A, B ]] and [B, [B, A]] in order to find the strong convergence order for the naive splitting and the symmetric splitting below.
∆t 2 4=
2E(∆W i ∆W j ) = E(I (0,i ) I (0, j ) ) E(I (0,i ) I (0, j ) ) − ∆t 3 4 δ i j .
to E(∆U ∆V ), it is straightforward, using our previous analyses, to show 9E(∆U i ∆V j ) = E(I (0,i ) I (0, j ,0) ) − E(I (0,i ) I ( j ,
III (i , 0 ) I ( j ,0, 0 ) d t by [ 2 ,
002(0,i ) I (0, j ,0) d t by Cor. (0,i ) I ( j ,0,0) d t + E ∆t 0I (0,i ,0) I ( j ,0) Cor∆V j = I (0,i ,0) I (0, j ,0) − I (0,i ,0) I ( j ,0,0) − ∆t ∆U j I (0,i ,0) − I (i ,0,0) I (0, j ,0) + I (i ,0,0) I ( j ,0,0) + ∆t ∆U j I (i ,0,0) − ∆t ∆U i I (0, j ,0) + ∆t ∆U j I ( j ,0,0) + ∆t 2 ∆U i ∆U j 81E(∆V i ∆V j ) 2∆t E(∆U i I (0, j ,0) ) + 2∆t E(∆U i I ( j ,2∆t E(∆U i I (0, j ,0) ) + 2∆t E(∆U i I ( j ,0,0) ) + ∆t 5 12 all times δ i j E(∆U i I (0, j ,0) ) = E(I (0,i ) I (0, j ,0) ) − ∆t 2 E(∆W i I (0, j ,∆U i I ( j ,0,0) ) = E(I (0,i ) I ( j ,0,0) ) − ∆t 2 E(∆W i I ( j ,
×m and W t is the standard m−dimensional Brownian motion, the strongorder γ ∈ {0.5, 1, 1.5, 2, 2.5, 3, ...} Itô-Taylor expansion is given by
also known as the Milstein method[2, eq. 10.3.3] Definition 1. Let γ ∈ {0.5, 1, 1.5, 2, 2.5, 3, ...}. A discrete time approximation Y of X with uniform step size
∆t converges with strong order γ at time T [2, eq. 10.6.3] provided that there are constants C and ∆ > 0
such that
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|
[] |
[
"Visual Search at eBay",
"Visual Search at eBay"
] |
[
"Fan Yang [email protected] \neBay Inc\n\n",
"Ajinkya Kale [email protected] \neBay Inc\n\n",
"Yury Bubnov [email protected] \neBay Inc\n\n",
"Leon Stein [email protected] \neBay Inc\n\n",
"Qiaosong Wang [email protected] \neBay Inc\n\n",
"Hadi Kiapour [email protected] \neBay Inc\n\n",
"Robinson Piramuthu [email protected] \neBay Inc\n\n"
] |
[
"eBay Inc\n",
"eBay Inc\n",
"eBay Inc\n",
"eBay Inc\n",
"eBay Inc\n",
"eBay Inc\n",
"eBay Inc\n"
] |
[] |
In this paper, we propose a novel end-to-end approach for scalable visual search infrastructure. We discuss the challenges we faced for a massive volatile inventory like at eBay and present our solution to overcome those 1 . We harness the availability of large image collection of eBay listings and state-of-the-art deep learning techniques to perform visual search at scale. Supervised approach for optimized search limited to top predicted categories and also for compact binary signature are key to scale up without compromising accuracy and precision. Both use a common deep neural network requiring only a single forward inference. e system architecture is presented with in-depth discussions of its basic components and optimizations for a trade-o between search relevance and latency.is solution is currently deployed in a distributed cloud infrastructure and fuels visual search in eBay ShopBot and Close5. We show benchmark on ImageNet dataset on which our approach is faster and more accurate than several unsupervised baselines. We share our learnings with the hope that visual search becomes a rst class citizen for all large scale search engines rather than an a erthought.
|
10.1145/3097983.3098162
|
[
"https://arxiv.org/pdf/1706.03154v2.pdf"
] | 22,367,645 |
1706.03154
|
c2777072fb18b9f8071f1c2d3c9839e8a537cde4
|
Visual Search at eBay
Fan Yang [email protected]
eBay Inc
Ajinkya Kale [email protected]
eBay Inc
Yury Bubnov [email protected]
eBay Inc
Leon Stein [email protected]
eBay Inc
Qiaosong Wang [email protected]
eBay Inc
Hadi Kiapour [email protected]
eBay Inc
Robinson Piramuthu [email protected]
eBay Inc
Visual Search at eBay
10.1145/3097983.3098162Visual SearchSearch Enginee-CommerceDeep LearningSeman- tics
In this paper, we propose a novel end-to-end approach for scalable visual search infrastructure. We discuss the challenges we faced for a massive volatile inventory like at eBay and present our solution to overcome those 1 . We harness the availability of large image collection of eBay listings and state-of-the-art deep learning techniques to perform visual search at scale. Supervised approach for optimized search limited to top predicted categories and also for compact binary signature are key to scale up without compromising accuracy and precision. Both use a common deep neural network requiring only a single forward inference. e system architecture is presented with in-depth discussions of its basic components and optimizations for a trade-o between search relevance and latency.is solution is currently deployed in a distributed cloud infrastructure and fuels visual search in eBay ShopBot and Close5. We show benchmark on ImageNet dataset on which our approach is faster and more accurate than several unsupervised baselines. We share our learnings with the hope that visual search becomes a rst class citizen for all large scale search engines rather than an a erthought.
INTRODUCTION
With the exponential rise in online photos and openly available image datasets, visual search or content-based image retrieval (CIBR) has a racted a lot of interest lately. Although many successful commercial systems have been running visual search, there are very few publications describing the end-to-end system in detail, including algorithms, architecture, challenges and optimization of deploying it at scale in production [10,11,19].
Visual Search is an extremely challenging problem for a marketplace like eBay for 4 major reasons: Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for pro t or commercial advantage and that copies bear this notice and the full citation on the rst page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permi ed. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior speci c permission and/or a fee. Request permissions from [email protected]. Accepts novel images as query. Note the quality of query images taken at store. Exact product was retrieved in these cases. Price comparison by taking a picture instead of bar code scanning or typing will become a common trend.
• Volatile Inventory: Unlike the scenario for standard search engines, in a dynamic marketplace like eBay, numerous items are listed and sold every minute. us, listings are short-lived. • Scale: Most solutions for visual search work on small to mid scale datasets but fail to operate at eBay scale. In addition, eBay inventory covers numerous ne-grained categories that are dicult to classify. We need a distributed architecture to handle our massive inventory with high search relevance and low latency. • Data ality: Image quality is diverse in eBay inventory since it is a platform that enables both high volume and occasional sellers. Compare this with catalog quality images in various popular commerce sites. Some listings may have incorrect or missing labels, which adds more challenges to model learning. • ality of ery Image: eBay ShopBot allows users to upload query images taken from any source and not just limited to an existing image in the inventory (see Figure 1). We describe how we address the challenges above. We present our approach, tips and tricks for building and running a computationally e cient visual search system at scale. We will touch on the architecture of our system and how we leverage deep neural networks for precision and speed. Details of training a uni ed deep neural network (DNN) for category classi cation and binary signature extraction, along with aspect prediction are discussed. We experiment with a public image dataset (ImageNet [5]) to quantitatively evaluate the e ectiveness of our network. We also showcase Figure 2: Overview of our visual search system. e top part shows image ingestion and indexing. e bottom part illustrates the ow during inference. Our core service accepts a query image from the client, performs category/aspect prediction and hash generation in a single pass, and then sends the results to image ranking service (Section 4.2). e image ranking service takes the top predicted categories and hash of the query image to match against hash in live index to produce an initial recall set of retrieval results. ese are further re ned by aspect-based re-ranking for better semantic relevance (Section 3.2).
how visual search at eBay is deployed in a recently released product -eBay ShopBot, a chatbot integrated into Facebook Messenger platform 3 , and an eBay-owned location-based buying and selling mobile application -Close5 4 , capable of discovering similar products from eBay inventory for users. Rest of the paper is organized as follows: Section 2 reviews recent literature on visual search algorithms based on deep learning and how our approach di ers from existing commercial applications.
is is followed by our approach of category/aspect prediction and binary hash extraction in Section 3. We present our cloud-based distributed image search architecture in Section 4. Next, we show experimental results with quantitative analysis on ImageNet [5] containing 1.2M images to prove the e ectiveness of our model. Finally, results by eBay ShopBot and Close5 are presented in Section 6 and Section 7, followed by conclusion in Section 8.
RELATED WORK
Finding similar items using a seed image is a well-studied problem [4], but it remains a challenging one as images can be similar on multiple levels. Recently, with the rise of deep convolutional neural networks, visual search has a racted lot of interest [2,9,10,21]. Deep learning has proven extremely powerful for semantic feature representation and multi-class image classi cation on large image datasets such as ImageNet [12].
Krizhevsky et al. [12] demonstrate strong visual retrieval results on ImageNet, where they use the last hidden layer to encode images and compute the distance between images in Euclidean space. is is infeasible for large scale visual search. Various approaches utilize deep convolutional neural networks (CNNs) to learn a compact representation of images to achieve fast and accurate retrieval, while reducing the storage overhead. Most approaches [1,7,13,14,20,22] focus on learning hash functions in a supervised manner using pairwise similarity of similar and dissimilar images so that the learned binary hashes capture similarity of images in the original Euclidean space. PCA and discriminative dimensionality reduction proposed by Babenko et al. [1] provide short codes that give state-of-the-art accuracy, but require a large set of image pairs to minimize the distances between them. Xia et al. [20] propose a two-stage framework to learn hash functions as well as feature representations, but it requires expensive similarity matrix decomposition. Lai et al. [13] directly learn hash functions with a triplet ranking loss while Zhu et al. [22] incorporate a pairwise cross-entropy loss and a pairwise quantization loss, both under a deep CNN framework. However, for a volatile inventory like eBay, where new products surface frequently, it is computationally ine cient and infeasible to collect a huge amount of image pairs across all categories, especially when the category tree is ne-grained and overlapping at times.
Alternatively, some works directly learn hash functions using point-wise labels without resorting to pairwise similarities of images [16,17]. Lin et al. [16] suggest a supervised learning technique to avoid pairwise inputs and learn binary hash in a point-wise manner that makes it enticing for large-scale datasets. An unsupervised learning approach [15] is also proposed to learn binary hashes by optimizing three types of objectives without utilizing image annotations. Despite of the success of these works, there are still challenges and unanswered questions about how to convert these research works into real production. Previous algorithms have been only evaluated on datasets containing at most few millions of images. However, given eBay scale datasets, it is challenging and non-trivial to perform hundreds of millions of hamming distance computations with low latency, while discovering the most relevant items at the same time. Figure 3: Network Topology. We modi ed ResNet-50 [8] with split topology for category prediction (top stream) and binary hash extraction (bottom stream). Sigmoid activations from the bottom stream are binarized during inference to get hash. Last layers in both streams were trained to predict category. is supervision helps with infusing semantic information in binary hash. Details in Section 3.
With these realistic challenges in mind, we propose a hybrid scalable and resource e cient visual search engine. Our rst contribution is a hybrid technique which uses supervised learning in three stages, rst to classify an item image into the right category bucket to signi cantly cut down on the search space, followed by a supervised binary hashing technique proposed by Lin et al. [16] and end with a re-ranking based on visual aspect (brand, color, etc.) recognition to ensure we show similar items not just based on the class or category but also across similar aspects. e rst 2 tasks are performed using a single DNN at inference. e second contribution is to demonstrate the distributed architecture we deployed in eBay ShopBot to serve Visual Search in a highly available and scalable cloud-based solution. Some commercial visual search engines are heavily fashion-oriented [11,19], where supported categories are limited to clothing products, while our goal is generic visual search and object discovery with signi cantly wider category coverage. We also learn binary image representations from deep CNN with full supervision instead of using low-level visual features [19] or expensive oating point deep features [10]. In addition, we allow users to freely take photos in an unconstrained environment, which di erentiates us from existing visual search systems that only accept clean, high-quality catalog images already in their inventory. Figure 2 illustrates the overall system architecture of our visual search system. As in [10,11], our approach is based on a DNN. However, instead of directly extracting features from the DNN and performing exhaustive search over the entire database, we search only among top predicted categories and then use semantic binary hash with Hamming distance for fast ranking. For speed and low memory footprint, we use shared topology for both category prediction and binary hash extraction, where we use just a single pass for inference.
APPROACH
Category Recognition
Rather than classifying objects into generic categories, such as persons, cats and clothes, etc., we aim to recognize much ner object categories of products. For example, we would like to distinguish men's athletic shoes from women's athletic shoes, maternity dresses from dancing dresses, or even coins from di erent countries. We refer to these ne-grained categories as leaf categories. We use state-of-the-art ResNet-50 network [8] for a good trade-o between accuracy and complexity. e network was trained from scratch based on a large set of images from diverse eBay product categories including fashion, toys, electronics, sporting goods, etc. We remove any duplicate images and split the dataset into a training set and a validation set, where every category has images in both training and validation sets. Each training image is resized to 256 × 256 pixels and the 227 × 227 center crop and its mirrored version are fed into the network as input. We used the standard multinomial logistic loss for classi cation tasks to train the network. To fully exploit the capacity of the deep network, we ne-tune the network with various combinations of learning parameters. Speci cally, we change learning parameters a er training for several epochs, and repeat this process several times until validation accuracy saturates.
To further improve the network's ability to handle object variations, we also include on-the-y data augmentation 5 during training to enrich training data, which includes random geometric transformations, image variations and lighting adjustments. Table 1 shows the absolute improvement by data augmentation against various image rotations. All images used in this experiment are from the validation set, so they cover all the categories. All learning parameters are the same expect for the additional data augmentation module. Overall, data augmentation brings 2% absolute improvement to top-1 accuracy in category prediction.
Aspect Prediction
Product images are associated with rich information apart from the leaf category labels, such as color, brand, style, material, etc. Such properties, called aspects ( Figure 4), add semantic information to each product. Our aspect classi ers are built on top of the shared category recognition network (Section 3.1). Speci cally, in order to save computation time and storage, all aspect models share the parameters with the main DNN model, up to the nal pool layer. Next, we create a separate branch for each aspect type. Our aspects cover a wide range of visual a ributes including color, brand, style, etc. Note that while some aspects such as color appear under multiple leaf categories, other aspects are speci c to certain categories. e.g., precious stones are relevant to jewelry, while sleeve length is relevant to tops & blouses. erefore, we embed the image representation from pool5 layer with a one-hot encoded vector representation of leaf category. is integrates visual appearance and categorical information into one representation. is is particularly useful for our choice of multi-class classi ers. We use gradient boosted machines [6] for their speed and exibility in supporting both categorical and non-categorical input. Since the idea behind boosting is combining a series of weak learners, our model creates 5 Figure 4: Simpli ed eBay Listing. Each listing has aspects (item speci cs). e associated category is part of a category tree. We use leaf nodes for categories.
several splits of the visual space, according to the leaf categories. We use XGBoost [3] to train the aspect models. It allows for fast inference with minimal resources using CPU only. We train a model for each aspect that can be inferred from an image.
Deep Semantic Binary Hash
Scalability is a key factor when designing a real world large-scale visual search system. Although we can directly extract and index features from some of the convolutional layers or fully-connected layer, it is far from optimal and does not scale well. It is both burdensome and costly to store real-valued feature vectors. In addition, computing pairwise Euclidean distance is ine cient, which signi cantly increases latency and deteriorates user experience. To address such challenges, we represent images as binary signatures instead of real values in order to greatly reduce storage requirement and computation overhead. We will show in Section 5.2.1 that we do not lose much in accuracy in going from real valued to binary feature vector. Following [16], we append an additional fully-connected layer to the last shared layer (Figure 3), and use sigmoid function as the activation function. e sigmoid activation function limits the activations to bounded values between 0 and 1. erefore, it is a natural choice to binarize these activations by a simple threshold of 0.5 during inference. Although more sophisticated binarization algorithms can be applied, we nd that using 0.5 as the threshold already achieves balanced bits and leads to satisfactory performance in our production. is bo om stream of split topology in network is trained independently from the top stream, but by xing the shared layer and learning weights for the later layers with the same classi cation loss that we use for category recognition. is supervised approach helps encode semantic information in the binary hash. We will show in Section 5.2.2 that our approach gives huge gains when compared to a popular unsupervised approach.
us, we use a single DNN to predict category as well as to extract binary semantic hash. We also use this network to extract feature vector for aspect prediction. All of these operations are performed in a single pass. e complete model is presented in Figure 3. We use 4096 bits in our binary semantic hash, which corresponds to the number of neurons of the newly added fully-connected layer from the deep semantic hash branch in Figure 3. Using 4096dimensional binary feature vectors substantially reduces the storage requirement. Speci cally, one such binary vector occupies only 512 bytes, making the total storage space under 100GB for 200M images. In contrast, if extracting features from the last shared layer (pool5 layer), we obtain an 8192-dimensional oating point vector (from 2 × 2 feature maps of 2048 convolutional lters), which requires 32KB space (using 32-bit oating point), resulting in an enormous storage of 6.1TB for the same amount of images. is gives over 90% storage reduction. Moreover, it is far more e cient to use binary representation as Hamming distance is much faster than Euclidean distance to compute.
To further improve speed, we only search for the most similar images from the top predicted leaf categories, so that we can greatly reduce the overhead in an exhaustive search over the entire database covering all categories. Top matches returned from the top categories are merged and ordered again to generate the nal ranked list.
Aspect-based Image Re-ranking
Our initial results are obtained by only comparing binary signatures of images. However, we can further improve search relevance by utilizing semantic information from aspect prediction. Suppose our model generates n aspects (a q i ) for a query image q. Each matched item in the inventory has a set of m aspects (a j ) and values as populated by the seller during listing. We check whether the predicted aspects match such ground-truth aspects and assign a "reward point" w i to each predicted aspect a q i that has an exact match. e nal score, de ned as aspect matching score S aspect =
1 n i =1 w i n i=1 m j=1 w i I(a q i = a j )
, is obtained by accumulating all scores of matched aspects. Here I is an indicator function that equals 1 only when the predicted aspect and ground-truth aspect are the same. For simplicity, we assigned hard-coded reward points for all aspects, although they can be learned from data. In addition, rather than treating all aspects equally, we assign di erent reward points to them considering that some aspects are more important for users to make a purchase in e-commerce scenarios. In our system, we assign larger points to the aspects size, brand and price while equal importance for all other aspects.
A er calculating the aspect matching score, we blend it with the visual appearance score S appear ance (normalized Hamming distance) from image ranking to obtain the nal visual search score to re-rank the initial ranked list of product images, i.e., e ective similarity score S = λS appear ance + (1 − λ)S aspect . Linear combination allows fast computation without performance degradation. e combination weight λ is xed (0.75) in our current solution but is also con gurable dynamically to adapt to changes over time. We give more importance to appearance scores in order to be less sensitive to possible noise in aspect labels created by inexperienced sellers. Figure 5 shows some examples where relevance is improved by aspect re-ranking. For the blue denim skirt, before re-ranking, the rst and fourth retrieved images do not match the query in color. We observe that deep features, trained for category classi cation, might sometimes deprioritize color information. Color matches a er re-ranking by aspect value for color. e second example contains a handbag in poor lighting conditions. Due to image quality, we could not get exact match of product. Instead, we get similar bags. Without aspect prediction, the second image is not restricted to the brand in query. A er re-ranking by aspects, brand in top retrieved images match with query. In the third example, straps of the wedding dress get overlooked in the hash representations. Re-ranking by aspects such as shape of the neckline re nes the result to match ne-grained properties while preserving the overall similarity, even though we did not retrieve the exact product. To detect duplicates, we compare MD5 hashes over image bits. Initially, we computed hash over bits of color channels a er resizing and converting image to RGB model, however, it turned out that using exact matching, i.e., computing hash directly on image bits without resizing gives close duplicate detection rates, while being signi cantly cheaper computationally.
As images from new listings arrive, we compute image hashes for the main listing image in micro-batches against the batch hash extraction service (Figure 2), which is a cluster of GPU servers For indexing, we generate daily image hash extracts from Bigtable for all available listings in the supported categories. e batch extraction process runs as a parallel Spark job in cloud Dataproc using HBase Bigtable API. e extraction process is driven by scanning a table with currently available listing identi ers along with their category IDs, and ltering listings in the supported categories. Filtered identi ers are then used to query listings from catalog table in micro-batches. For each returned listing, we extract the image identi er, and then lookup corresponding image hashes in micro-batches. e image hashes preceded by listing identi er are appended to a binary le. Both listing identi er and image hash are wri en with xed length (8 bytes for listing identi er and 512 bytes for image hash). We write a separate le for each category for each job partition, and store these intermediate extracts in cloud storage. A er all job partitions are complete, we download intermediate extracts for each category and concatenate them across all job partitions. Concatenated extracts are uploaded back to the cloud storage.
We update our DNN models frequently. To handle frequent updates, we have a separate parallel job that scans all active listings in batches, and recomputes image hashes from stored images. We keep up to 2 image hashes in Bigtable for each image corresponding to the older and the newer DNN model versions, so the older image hash version can be still used in extracts while hash re-computation is running.
Image Ranking
To build a robust and scalable solution for nding similar items across all items in the supported categories, we create an image ranking service and deploy it in a Kubernetes 6 cluster. Given the huge amount of data, we have to split image hashes for all the images across the cluster containing multiple nodes, rather than storing them on a single machine. is allows us to provide fault tolerance in case any single node becomes unavailable, and makes it possible to easily scale according to user tra c. In this scenario, each instance of the application will be responsible for a subset of the data, but collectively the cluster performs search across all the data. As our goal is to provide close-to-linear scalability, each node in the cluster should have knowledge about other nodes in order to decide which part of the data to serve. We use Hazelcast 7 (an open source in-memory data grid) for cluster awareness.
When a node participates in the Hazelcast cluster, it receives noti cations if other nodes are leaving or joining the cluster. Once the application starts, a "cluster change" event is received by every node. en, each node checks current set of nodes. If there is a change, the data redistribution procedure is kicked o . In this procedure, we split the data for each category into as many partitions as the number of nodes in the cluster. Each partition is assigned to a node in round robin fashion using a list of nodes sorted according to the ID assigned by Hazelcast. Starting node is determined by Ketama consistent hash 8 from node ID. us, each node can generate the same distribution of data across the cluster and identify the part it is responsible for. To guarantee that all nodes have the same data, we leverage Kubernetes to share single disk, in read-only mode, across multiple pods.
For initial discovery, during cluster startup, we leverage Kubernetes service with type "Cluster IP". When node performs DNS resolution of the service, it receives information about all nodes already in the Hazelcast cluster. Each node also periodically pulls information about other nodes from the DNS record to prevent cluster separation.
is design allows us to scale out to any number of nodes to satisfy di erent requirements, such as supporting increasing user tra c, decreasing search latencies by making each partition smaller and providing fault tolerance, etc. If any single node becomes unavailable at exact moment when user sends request, search will be performed on live ones. Since data for each leaf category spreads across cluster, service availability can be guaranteed.
When our core vision service (Figure 7) sends incoming search requests to the image ranking service, the requests could be served be any node in the cluster, referred to as "serving node". Since each node is aware of the state of the cluster, it proxies incoming requests to all other nodes including itself. Each node further looks through partitions assigned to it and nds the closest N listings for a given image hash if it has categories mentioned in the request (Figure 8). We use Hamming distance as the distance metric to discover the most similar listings. Data for each category is represented as a continuous array of listing IDs and corresponding image hashes. Each node divides category partition into a set of sub-partitions of the same number of available CPU cores and executes search in parallel to nd the nearest N items. Once the search is done in each sub-partition for each leaf category in the request, results are merged and the closest overall N listings are returned. When 6 search is completed on all data nodes, serving node performs similar merging procedure and sends back the result.
EXPERIMENTS ON IMAGENET
In this section, we conduct extensive experiments on the Ima-geNet [5] dataset for reproducible proof of concept. We take the ResNet-50 [8] model for category prediction and follow the same learning protocol in Section 3.3 by ne-tuning a newly added hashing branch on ImageNet. Our classi cation branch (Figure 3) achieves top-1 validation error 25.7% and top-5 validation error 7.9%, which are only slightly higher than the accuracy of hashing branch: 24.7% and 7.8%, implying the discriminative power of the learned hash functions and capture of semantic content. We will con rm this quantitatively in the series of experiments to follow. Figure 9 illustrates the embedding based on 4096-bit binary semantic hash for 5 synsets from ImageNet. is strengthens our claim qualitatively that the binary hash preserves semantic information and also local neighborhood. It is important to encode semantic information in hash to mitigate the undesirable e ects of collision, since the items in collision will then be semantically similar.
Bit Distribution
One of the critical properties of learning good binary hash functions is to have balanced bits, i.e., forcing bits to be equally distributed over the full data. Given a speci c bit of the binary hash, ideally it activates on half of the images in the training set while equals 0 on the remaining half. Such balanced distribution generates discriminative bits and reduces collision, as well as maintaining the largest entropy in the perspective of information theory, thus leading to more accurate matching for visual search.
In Figure 10, we show the percentage of images from the training set where a bit is activated for each of the 4096 bits. While the curve is not perfectly at at 50% as in ideal case, it mostly lies around 50%, within the range from 45% to 55%, which indicates that the bits learned from our model are roughly equally distributed. Speci cally, as many as 3445 out of 4096 bits (84.1% of bits) activate on 45% to 55% images from the training set.
is veri es the encoding e ciency of our binary semantic hash. is experiment is performed on 5 synsets, each containing 1300 images. We extract binary hashes for all images and apply t-SNE to obtain the image coordinates. Images belonging to the same category are marked by the same color.
is illustrates that the binary hash preserves semantic information as well as similarity in the local neighborhood. Actual bit dsitribution Ideal bit distribution Figure 10: Bit distribution on ImageNet [5] training set. 84.1% of bits are activate on 45% to 55% images. Ideal distribution for maximum entropy is uniform. Deviation from ideal is partly due to non-uniform diversity of data set and capture of semantic information. Details in Section 5.1.
antitative Comparison
We conduct further experiments to evaluate the performance of our approach against a popular unsupervised visual search, and show that our approach results in be er retrieval results in terms of various evaluation metrics. We use the validation set of Ima-geNet [5] as the query set to search for similar images from the training set. As a baseline, we perform k-means clustering (k = 2 n where n = 4, 6, 8, 10) on binary hashes of images from the training set. Instead of rst predicting N class labels for each query image, we nd the top N nearest clustering centers rst, and then search for M most similar images within each cluster. In this way, we obtain N × M initial retrieved images, from which K images are returned as the nal search results.
Category prediction.
Using predicted categories during search, our supervised approach produces more accurate results. We use precision@K and accuracy@K by varying the number of top K returned results to evaluate the performance of baselines and our approach. ese metrics measure the number of relevant results, i.e., from the same class as the query, and the classi cation accuracy, i.e., the query is considered to be correctly classi ed if there is at least one returned image belonging to the same class among the top K retrieved results. For all experiments, we set M = 50 so that we only search for 50 similar images within each predicted category. We have 2 avors for our approach. First is to look at absolute top N categories. e second is to use predictions up to a maximum of N until we get a cumulative so max con dence of 0.95. Our results show that the former is be er. In Figure 11, accuracy of our approach rapidly improves as K increases when K ≤ 100 while precision only drops slightly, which clearly shows that our approach does not introduce many irrelevant images into the top search results. Compared to di erent variants of the baseline, we are able to achieve be er performance when retrieving as few as only 20 images. On the other hand, when we search within more predicted categories, the accuracy signi cantly improves as we include more candidates and outperforms all baseline variants.
Similarity search.
We further compare the performance of our approach and baseline in terms of similarity search, where we consider the results of exact K nearest neighbors of a query as ground-truth and evaluate how the compared methods approximate the ground-truth. erefore, we use normalized discounted cumulative gain (NDCG) to measure the quality (relevance) of the ranked list considering the ranks, and again precision@K. By taking into account the category information of images, we have another ground-truth to evaluate these methods by looking at the relevance at category-level. Our approach again outperforms baseline variants under both of the two scenarios (see Figure 12).
Timing.
We also compare speed (excluding network forward pass) of our approach and several variations of the baseline. e top and bottom rows use a di erent ground-truth for nearest neighbor (NN). e former uses NNs in the entire data set and the latter is restricted to NNs from ground truth category of query. Note high relevance from our approach even though search is limited to only the top predicted categories. Absolute top N is again better than top N based on cumulative so max con dence. See Section 5.2.2 for details.
We run experiments with di erent N , i.e., top predictions or nearest k-means centroids and present the results in Table 2. Our method is much more e cient than the baseline since we reduce the search space signi cantly by only searching the top N predicted categories. Speci cally, our method is 1.14× faster than the baseline with a large k = 1024 when N = 5, while being more accurate. Note that all experiments are conducted in Python with single-thread processing on a desktop, while the production environment (for live eBay inventory) has been extensively optimized (Section 4.2) to greatly reduce latency (Section 6.4).
APPLICATION: EBAY SHOPBOT
Our visual search was recently deployed in eBay ShopBot. Since then, visual search has become an indispensable part of users' shopping journey and has served numerous customers. People can nd the best deals from eBay's 1 billion-plus live listings by uploading a photo. Early user data from ShopBot has shown that the number of shopping missions that started with a photo search has doubled since the launch of eBay ShopBot beta. In the following, we summarize two main scenarios where visual search is used in ShopBot.
User query
eBay ShopBot allows users to freely take a photo (from camera or photo album) and nd similar products in eBay's massive inventory. is is invaluable speci cally when it is hard to precisely describe a product solely in words. We do not set any restrictions on how the photos are taken, and can support a wide range of queries from professional quality images to low quality user photos. Figure 1 shows examples of user interface on ShopBot. Our visual search successfully nds exactly matching products despite noisy background and low quality query images. ery is from active eBay listing (not user-uploaded). Images on the right were shown when scrolled. See Section 6.2 for details.
Anchor search
e second use case of visual search in eBay ShopBot is anchor search. Every time a user is presented with a list of products (even if she used text to initiate the search), she can click on the "more like this" bu on on any of the returned products to re ne or broaden initial searches, and to nd visually similar items. Here, the photos from the selected item serve as the anchor to initialize a new visual search. Since the anchor is actively listed, we take advantage of its meta information, such as category and aspects, to guide visual search even further. In this way, we provide users with a non-linear search and browsing experience, which allows more exibility and freedom. is feature has been a very popular feature with users with engagement increasing by 65% since it became available. Figure 13 shows an example where the user is looking for inspirations by nding visually similar handbags.
alitative results
Since we could not share eBay dataset due to proprietary information, we present only qualitative results in Figure 14. Our visual search engine successfully discovers visually similar images from the massive and dynamic inventory of eBay despite the variety of categories, diverse composition and illumination. Note that all retrieved images are from active listings of eBay at the time of writing.
Latency
We report the latency of several main components. By extensive optimization and leveraging the computational power of cloud, the batch hash generation service takes 34ms per image on a single GPU (Tesla K80). In ShopBot scenario, given a user query, the deep network takes 125ms on average to predict the category, recognize aspects and generate image hash. e image ranking service only takes 25ms and 70ms to return 50 and 1000 items, respectively, depending on the size of each category. e aspect re-ranking only takes 10ms to re-rank as many as 1000 results. erefore, the total latency is only a couple of hundreds milliseconds, plus miscellaneous overhead, which provides users with a fast and enjoyable shopping experience. alitative results of our visual search. ery images (not taken from eBay site) have red border and are followed by top 4 ranked images from active eBay listings. Note the diversity in categories and image quality. When exact product is not found, retrieved results still share common semantics with query.
APPLICATION: CLOSE5
Close5 is an eBay-owned platform for buying and selling locally. Users can freely take photos of and add descriptions to their products they want to sell to create a listing on Close5, which can be viewed by nearby users. Our visual search solution has been deployed and integrated with Close5 native application support millions of Close5 users. It has been extensively used in the following two scenarios. Figure 15: Screenshot of a listing from Close5. "Similar items on eBay" are provided by our visual search. Note that we nd an exact match listed by a di erent seller at ebay.com, which has di erent inventory than close5.com.
Auto categorization
While most listings have user-created descriptions, some only have photos taken by users as title and description are not mandatory. erefore, these listings without descriptions are not search-able by text, resulting in 200× less chance of selling than others. In this case, our visual search predicts the category of the product based on images to populate category-related textual information, so that these listings can be search-able. Early statistics show that the average view of a listing increased by 21% compared to those without descriptions and no auto categorization.
Similar items on eBay
To enable product discovery from eBay inventory for Close5 users, we integrate the visual feature functionality as a feature called "Similar items on eBay". When a user browses a listing, visual search automatically triggers with the rst image from this Close5 listing as query. Results are presented at the bo om of the listing page with links directing users to eBay website. As shown in Figure 15, our visual search successfully nds the same oral dress, as well as other similar dresses. With this feature, eBay inventory is exposed to millions of Close5 users, which provides both Close5 users and eBay sellers with improved buying and selling experience. Click through rate doubled since we rolled out this user using visual search.
CONCLUSION
We presented a scalable visual search infrastructure that leverages the power of deep neural network and cloud-based platform for e cient product discovery given a massive and volatile inventory like eBay. Highlights include searching only among images from top predicted categories, single DNN with split topology to predict category as well as extract compact and e cient binary semantic hash, aspect-based re-ranking to reinforce semantic similarity. We have also presented the system architecture and discussed several optimizations for a trade-o between search relevance and latency. Extensive experiments on a large public dataset veri es the discriminative ability of our learned model. Additionally, we show that our visual search solution has been deployed successfully to the recently launched eBay ShopBot, and integrated into eBay-owned Close5 native application.
KDD'17, August 13-17, 2017, Halifax, NS, Canada. © 2017 ACM. ISBN 978-1-4503-4887-4/17/08. . . $15.00 DOI: h p://dx.doi.org/10.1145/3097983.3098162
Figure 1 :
1Visual Search in eBay ShopBot 2 .
Figure 5 :
5Re ning search results by aspects. Predicted aspect values are shown. For each query, the 1 st and 2 nd rows present search results before & a er aspect-based reranking (Section 3.4), respectively. Red cross is shown on images with unmatched aspects as query.
eBay sellers worldwide generate numerous inventory image updates per second. Each listing may contain multiple images, and also multiple variations with multiple images. Our ingestion pipeline (Figure 6) detects image updates in near-real-time and maintains them in cloud storage. To reduce storage requirements, duplicate images (about a third) across listings are detected and cross-linked.
Figure 6 :Figure 7 :
67Image Image ranking deployment running our pre-trained DNN models. Image hashes are stored in a distributed database (we use Google Bigtable), keyed by the image identi er.
Figure 9 :
9t-SNE visualization[18] of 4096-bit binary semantic hash on ImageNet vehicle categories (best viewed electronically and in color).
Figure 11 :
11Category prediction based on top K images. Performance of our approach and baseline by varying number of retrieved images K and top N category predictions. Here, k denotes number of centroids from k-means. Absolute top N is better than top N based on cumulative so max condence. Best accuracy for N > 3. See Section 5.2.1 for details.
Figure 12 :
12Search relevance.
Figure 13 :
13Anchor search.
Figure 14 :
14Figure 14:
Table 1 :
1Robustness of data augmentation against image rotation. Numbers are absolute improvement of top-1 accuracy to predict category. Rotation angles are clockwise (in degree).Rotation angle
0
90
180
270
Mean
Improvement 1.73% 2.04% 2.22% 1.98% 2.00%
h ps://github.com/kevinlin311tw/ca e-augmentationClothing, Shoes & Accessories
All
…
…
Women's Shoes
…
…
Flats & Oxfords
Heels
Sandals & Flip Flops
Slippers
Boots
Athletic Shoes
Item Specifics
Condition:
New with box
Brand:
Michael Kors
Heel Type:
Kitten
Style:
Kitten Heels
Heel Height:
Med (1 ¾ in. to 2 3/4 in.)
US Shoe Size (Women's):
7
Material:
Haircalf
Width:
Medium (B, M)
Pattern:
Houndstooth
Color:
Multi-Colored
h ps://en.wikipedia.org/wiki/Kubernetes 7 h ps://en.wikipedia.org/wiki/Hazelcast 8 h ps://en.wikipedia.org/wiki/Consistent hashingCoreVS
CoreVS
Image Search Request
w/ Hash & Categories
Return Top N Results
Serving Node
Serving Node
Data Nodes
Data Nodes
Category Partition
Get Top N Categories
Return Top N Results
Get Top N Results
Top N Results
Merge and Get Top N
Merge and Get Top N
Category Partition
Figure 8: Image ranking request sequence. CoreVS := Core
Vision Service that predicts top categories, aspects and ex-
tracts binary semantic hash.
Table 2 :
2Comparison of search ranking time per query (in
ms), averaged over all queries, by our approach and base-
lines with various number k of k-means centroids.
Method Ours k = 16 k = 64 k = 256 k = 1024
N = 1
33.5
2073.4
605.9
124.7
47.1
N = 5
150.6 8293.0
2246.4
720.9
172.3
N = 10 323.7 16252.1 4190.0
1056.1
325.3
A demonstration video can be found at h ps://youtu.be/iYtjs32vh4g. 2 h ps://shopbot.ebay.com
h ps://shopbot.ebay.com/ 4 h ps://www.close5.com
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|
[] |
[
"INTELLIGENT COMPUTATIONAL MODEL FOR THE CLASSIFICATION OF COVID-19 WITH CHEST RADIOGRAPHY COMPARED TO OTHER RESPIRATORY DISEASES",
"INTELLIGENT COMPUTATIONAL MODEL FOR THE CLASSIFICATION OF COVID-19 WITH CHEST RADIOGRAPHY COMPARED TO OTHER RESPIRATORY DISEASES"
] |
[
"C David ",
"Wyld "
] |
[] |
[] |
Lung X-ray images, if processed using statistical and computational methods, can distinguish pneumonia from COVID-19. The present work shows that it is possible to extract lung X-ray characteristics to improve the methods of examining and diagnosing patients with suspected COVID-19, distinguishing them from malaria, dengue, H1N1, tuberculosis, and Streptococcus pneumonia. More precisely, an intelligent computational model was developed to process lung X-ray images and classify whether the image is of a patient with COVID-19. The images were processed and extracted their characteristics. These characteristics were the input data for an unsupervised statistical learning method, PCA, and clustering, which identified specific attributes of X-ray images with Covid-19. The introduction of statistical models allowed a fast algorithm, which used the X-means clustering method associated with the Bayesian Information Criterion (CIB). The developed algorithm efficiently distinguished each pulmonary pathology from X-ray images. The method exhibited excellent sensitivity. The average recognition accuracy of COVID-19 was 0.93 ± 0.051.
|
10.5121/csit.2021.111011
|
[
"https://arxiv.org/pdf/2108.05536v1.pdf"
] | 236,770,227 |
2108.05536
|
37d0e0cde1f6989ac3f533404c1d8e155103533a
|
INTELLIGENT COMPUTATIONAL MODEL FOR THE CLASSIFICATION OF COVID-19 WITH CHEST RADIOGRAPHY COMPARED TO OTHER RESPIRATORY DISEASES
2021
C David
Wyld
INTELLIGENT COMPUTATIONAL MODEL FOR THE CLASSIFICATION OF COVID-19 WITH CHEST RADIOGRAPHY COMPARED TO OTHER RESPIRATORY DISEASES
202110.5121/csit.2021.111011Probabilistic Models, Machine Learning and Computer Vision
Lung X-ray images, if processed using statistical and computational methods, can distinguish pneumonia from COVID-19. The present work shows that it is possible to extract lung X-ray characteristics to improve the methods of examining and diagnosing patients with suspected COVID-19, distinguishing them from malaria, dengue, H1N1, tuberculosis, and Streptococcus pneumonia. More precisely, an intelligent computational model was developed to process lung X-ray images and classify whether the image is of a patient with COVID-19. The images were processed and extracted their characteristics. These characteristics were the input data for an unsupervised statistical learning method, PCA, and clustering, which identified specific attributes of X-ray images with Covid-19. The introduction of statistical models allowed a fast algorithm, which used the X-means clustering method associated with the Bayesian Information Criterion (CIB). The developed algorithm efficiently distinguished each pulmonary pathology from X-ray images. The method exhibited excellent sensitivity. The average recognition accuracy of COVID-19 was 0.93 ± 0.051.
INTRODUCTION
In November 2019, the first cases of COVID-19 were detecting by China's health authorities. After a few weeks, the virus-infected Wuhan's city and, in March, spread globally.
In Brazil, the Ministry of Health divided the pandemic into two phases for better management: containment and mitigation. The first phase cases were attributing international travel or contact with sick people who traveled abroad. In the mitigation phase, the Ministry of Health recognized the occurrence of community transmission, from person to person, in the country -a late recognition, since there were already deaths unrelated to the transmission chains involving travelers [1].
In this context, one of the first steps towards adopting measures of social isolation and hospitalizations is to know who has been contaminated by COVID-19. The most appropriate tests for COVID-19 are molecular tests, but they can take 24 to 48 hours to be performing. In pandemic conditions, this period can last between 5 and 10 days due to many requests, lack of equipment, and health professionals' help. Therefore, rapid tests for mapping and screening patients are necessary. As X-Ray and lung tomography tests indicate patients with respiratory problems, these tests may answer this demand.
Despite several studies on the diagnosis of pneumonia regarding the criteria for confirmation and classification of pneumonia cases, many questions remain open. Thus, to avoid misinterpretation, this work was based on three underlying assumptions [2,3]:
Pneumonia must be defined as an acute infection of the lung parenchyma by various pathogens, excluding the condition of bronchiolitis.
1. Defining pneumonia as a group of specific co-infections with different characteristics is not a line to be followed since the etiological agents' identification is not always possible. 2. Like other criteria, different types of pneumonia can be classified into more homogeneous groups, producing faster diagnosis advances.
Image recognition and analysis were revolutionizing with the introduction of deep learning, which allowed for unprecedented leaps in performance. The rapid advancement of these technologies expands the possibilities of automated, accurate, accessible, and economical medical diagnostics. Still, smart models are faster than humans and can be implemented on a large scale due to clouds' power or even at the edge. Thus, artificial intelligence techniques can help to compare and group similar types of pneumonia.
The present work proposes a criterion to classify pneumonia cases based on pulmonary radiographs. The images analyzed were of patients with COVID-19 and with common bacterial or viral pneumonia. The extraction method used was Haralick, Wavelets, and we used the Bayesian Information Criterion (CIB) as a probabilistic model, and thus we used a decision tree for the classification of images.
MATERIAL AND METHODS
Material Collection
We searched for articles and repositories that could indicate the signs on chest radiographs (Table 1) before comparing some characteristics present in COVID-19 with tuberculosis, H1N1, dengue, and malaria. The main characteristics highlighted were: pleural effusion, ground-glass opacity, pulmonary edema, rounded morphology of opacities, and bronchitis. Subsequently, we grouped the images into three categories: pneumonia type 1 (tuberculosis and Streptococcus pneumonia), pneumonia type 2 (malaria and dengue), and pneumonia type 3 (COVID-19). The separation was based on similar descriptions characteristic of these pathologies (Table 2) . For this study, we used a total of 3800 chest X-ray images posteroanterior and anteroposterior positions of COVID-19 (1800), dengue (100), tuberculosis (730), Streptococcus pneumonia (200), malaria (270), normal (700). The set of images was acquired from repositories (Figure 1). Also, we use only images that a doctor has already diagnosed. That is, we use images from defined case studies. After separating the set of images, we perform manual segmentation of the pulmonary images and then look for five features that are often detectable on chest radiography was doing considering the descriptions made in case studies and articles published in the medical field to describe the intensity in the respective conditions, as shown in Table 1. Bronchitis COVID-19 X*** (11) X* (10) X** (13) X* (10) X*** (11) Dengue X* (12,17) X** (12) X** (12,17) X*** (12) X*** (12) Malaria X** (8 ) X** (1 ) X** (13 ) X** (1 ) X** (8 ) Streptococcus pneumoniae X** (9) X** (9) X*** (6) X** (6,9) X* (6,9) Tuberculosis X* (15,18) X** (2) X** (14) X* (15,2,18) X* (15,18) * ordinary; ** 40-60% of cases; *** uncommon; mon * are significantly different; ** are not significantly different.
INTELLIGENT ARTIFICIAL MODEL
The model was inspired in ChestNet [26] a Neural Network for support in diagnosis for problems pulmonaries. The workflow for analysis was developed for identifying the characteristics of regions of an image, using algorithms of Wavelets and Haralick extraction attributes of the image. These texture attributes are essential because they determine partners in the lung and rub through clusterization of pixels in different Chest X-Ray regions. Subsequently, we extract the characteristics to determine the hyperparameters of backpropagation from clustering and of data mining descriptive statistics.
The probabilistic model is essential to determine each parameter in a neural network because the descriptors auxiliary the segmentation of regions and classification from the region of interest from the pattern of variation of shades of gray or color of a given region of interest. These existing partners in physicals superficially noticeable to the human eye, bringing a significant amount of information about the superficial nature, such as smoothness and roughness.
Pre-Processing
This step's main objective was to improve the visualization of the bones through the implementation of the image enhancement algorithm of the High-Frequency Emphasis (HEF) filter [27]. HEF helps to sharpen an image by emphasizing the edges; since the edges usually consist of an abrupt change in the pixels' color intensity, representing the high-frequency spectrum of the image.
Lung Segmentation Using the Mask-Regional Convolution Neural Network (Mask-RCNN).
Mask-RCNN is a deep neural network designed to solve instance segmentation problems in machine learning or computer vision [28]. For this model's training, we used 5000 lung images, segmenting the lung on the right and left sides ( Figure 3).
Characteristics of X-Ray Images and Haralick Extractor
Studies show that chest radiographs are initially based on the visualization of the following three characteristics [2][3]. They are (1) Anatomical structures, such as ribs and other bones, must be visible.
(2) The darker (black in the image) the color of the lungs, the more suitable is the functionality.
(3) The heart and peripheral blood vessels must be visible. Using these characteristics, we applied the Haralick method ( Figure 3) [29] to extract texture characteristics through their attributes, using a gray level co-occurrence matrix. The co-occurrence matrix is a square matrix whose size is the number of gray levels in the image to be analyzed. The developed algorithm calculates the distances in all possible 360 degrees and normalizes between 0 and 100. Therefore, the co-occurrence matrix contains 100 rows per 100 columns and generates by combining the distances between the current angle and their respective combinations. 10, 45, 90, and 135 degrees. After calculating this matrix, a matrix of the probability of the combinations between the gray levels was calculating. The following texture characteristics' values were calculated from this matrix: energy, entropy, variance, homogeneity, dissimilarity, and correlation measures.
Characteristics of X-Ray images and Wavelets
The wavelet transform can be time associated with these frequencies, making it very suitable for various fields. As an example, we can mention processing accelerometer signals for motion analysis and fault detection. Success in image compression, using the wavelet transform, is mainly attributed to innovative strategies for organizing and representing data from a transformed image. Such strategies explore implicitly or explicitly the static properties of the coefficients transformed into a wavelet pyramid. Most of the published coders recently used the pyramidal (dyadic) decomposition algorithm in the literature [33] For this project, we use the clustering of significant coefficients in a sub-band.
Selection of probabilistic model
The model selection problem refers to choosing the best model among a set of candidates built from combinations of parameters. Consider a sequence of models M1, M2, and Mn, with the corresponding parameters. There are many techniques for selecting the best model based on the probability ratio, and others add different types of penalty functions to the likelihood ratio. This is the case of the Akaike Information Criterion (CIA) and the Bayesian Information Criterion (CIB), both of which test two models at a time, and the two can be chosen in ascending order of the number of parameters. After that, there is a sequence of CIB and CIA values, which are optimized. This results in the number of parameters to determine which model is the best. Therefore, in the present study, we used X-means [30], an algorithm that efficiently searches the space of the clusters' locations and the number of groups to optimize the measurement of the CIB. To verify the training, testing, and validation of the model, a decision tree was used to find the hyperparameter and the inference tests [31].
Statistical Analysis
We used an Analysis of Variance (ANOVA) followed, when appropriate, by the Tukey-Kramer test of multiple comparisons for different sample sizes.
RESULTS AND DISCUSSION
Before analyzing the 3800 chest X-ray images, we separated the left lung from the right using Mask-RCNN, as described in the methodology, and each segmented lung pair was labeled as type 1 pneumonia, type 2 pneumonia, or type 3 pneumonia (Table 2), based on the data set information and on the literature review on signs and symptoms. The Shapiro normality test resulted in a p-value less than alpha (p-value = 4.899e-33 <0.05). Thus, the null hypothesis was rejected, and we concluded that the data were not extracted from a normal distribution. However, the information obtained from ANOVA and the Tukey-Kramer test was essential to determine the type of data distribution and the type of model. From these results, non-parametric models were used to determine the number of classes and the most appropriate classification model for this type of data (Table 3). Figure 4 shows that the characteristics were grouped according to the pathologies in pneumonia type 1, pneumonia type 2, and pneumonia type 3.
The wavelet transform can be time associated with these frequencies, making it very suitable for various fields. As an example, we can mention processing accelerometer signals for motion analysis and fault detection ( Figure 5).
In this context, model selection is a problem of choosing the set of candidate models with the best performance for training data sets or estimating the model's performance using a resampling technique, such as cross-validation of k-folds. One way to use model selection involves using probabilistic statistical measures to quantify the model's performance in the training data set and the model's complexity, one of which is the Bayesian Information Criterion. The benefit of this information criterion is that it does not require a standby test, although a limitation is that they do not accept the models' uncertainty under consideration and may end up selecting straightforward models.
. Algorithms of (A) Haralick and (B) Wavelets
In Figure 6, we show that the group grouping was efficient. The X-means method allows a variety of cluster K (K-means) to occur, which deals with the allocations of the clusters, repeatedly, trying to partition and maintain the resulting ideal divisions. In this segmentation, we obtained three clusters, validating the data grouping to use a decision tree model. Figure 6. Scatter plot of the partition of the three clusters in the first two main components, for X-means with CIB.
The training was done with 70% of the characteristics and the test with the remaining 30%. The stratified K-Fold approach was using three scores: a minimum score of 0.95, a maximum score of 0.98, and an average score of 0.96. These results show that the average score of 0.96 presents a high assertiveness and indicates the model's high quality with real data. The optimization of the decision tree's hyperparameters was to determine the best criteria, the precision, and the standard deviation of the model ( Table 4). The resulted in three scores: a minimum score of 0.95, a maximum score of 0.98, and an average score of 0.96. These results show that the average score of 0.96 is highly accurate and indicates the model's high quality with real data. The optimization of the decision tree's hyperparameters was to determine the best criteria, the precision, and the standard deviation of the model. With the completion of this first stage, we carried out the test in the municipality of Itapeva. In this first stage, we tested 25 patients with suspected viral or bacterial pneumonia. We tested patients who had at least one complaint about this validation stage, such as headache, changes in taste, fever, and other complaints related to acute pneumonia. However, we tested five patients who did not meet the inclusion criteria for COVID-19 or Tuberculosis, but when the model evaluated the X-Ray, a normal patient presented COVID-19 PCR-RT test was negative, and another patient was also asymptomatic. The model was evaluated as COVID-19 and was confirmed with the PCR-RT test. In the case of the patient with Tuberculosis, he already had Tuberculosis previously, but it was positive for COVID-19 by the PCR-RT. All patients underwent the PCR-RT or Bacilloscopy test to confirm the diagnosis. Currently, some studies have focused on the diagnosis by computed tomography (CT), a technique that has better sensitivity for the detection of soft tissues. However, radiography devices' availability in countries like Brazil is 1: 25,000 inhabitants, while that of CT devices is 1: 100,000 inhabitants. Some countries in Africa also offer more X-ray equipment than CT. Thus, the study of pattern recognition algorithms on chest radiographs can contribute to doctors and radiologists in diagnosing COVID-19 infection in remote regions or regions without CT devices' availability. Besides, chest X-ray diagnosis can be a screening route for isolation and/or hospitalization measures since there is a limited number of molecular test kits (the main one being RT-PCR) and, depending on the manufacturer, a high index of false-negative results may occur.
In this context, several epidemiological relevance diseases have oscillatory and periodic time patterns related to their transmission in the community. These diseases can be associated with intrinsic factors such as immunity, contact pattern, renewal, virulence rates and extrinsic factors, such as temperature, humidity, and precipitation. Among these diseases, the most common are tuberculosis, malaria, Streptococcus pneumonia, and dengue [2][3][4][5][6][7]. As these different pathologies can generate conflicting signals in diagnostic imaging, we investigated metrics that can indicate biomarkers capable of avoiding the false-positive diagnosis of COVID-19. The literature indicates that ground-glass pulmonary opacity patterns, usually with bilateral and peripheral pulmonary distribution, are emerging as a hallmark of COVID-19 infection. This disease pattern, somewhat similar to that described in previous coronavirus outbreaks, such as SARS and MERS, also fits the model that radiologists recognize as the archetypal response to acute lung injury, usually initiated by an infectious or inflammatory condition [4,7]. Inflammation can cause ground-glass opacities in lung images, indicating consolidated dense lesions that can progressively evolve to a linear structure [8][9].
Our research efforts have shown that models using artificial intelligence can determine parameters for different groups with similar symptoms and signs. Our data mostly agree with the work of Pan et al. (2020), showing a preponderance of abnormalities in ground glass in the course of the disease. The recognition of image patterns in this group of images with similar signs and symptoms is an auxiliary tool for understanding the disease's pathophysiology since the definitive diagnosis of COVID-19 requires a positive RT-PCR test. However, current best practices recommend it as an additional test, but not for the final diagnosis of COVID-19. However, the intelligent computational model can help identify complications in the screening systems and the monitoring of pulmonary problems since there is still no effective drug for the disease, and the vaccines are still in the process of validation. The data obtained with our model suggest that the Haralick method can determine the patterns of pulmonary imaging characteristics showing pleural effusion, ground-glass opacity, pulmonary edema, rounded morphological opacities, and bronchitis. These metrics allowed the model to distinguish and significantly classify the three different pneumonia types with high accuracy. Currently, the intelligent computational model is used via Telegram by health professionals in municipalities in Minas Gerais, in Brazil ( Figure 7).
CONCLUSIONS
Haralick's texture descriptors were useful for efficiently representing patterns of interest for image analysis and interpretation, as they showed changes in pixel intensity patterns, which were correlating with pathological changes in COVID-19. However, the new approach to extract texture characteristics by the Haralick method provided more results for the predictive analysis of pixel intensity and, when associated with unsupervised methods (X-means) and supervised methods (Mask-RCNN and Decision Tree), showed results with high accuracy. Thus, characteristics such as homogeneity, energy, dissimilarity, and correlation significantly differentiated some pathologies from the pathology of COVID-19. These results suggest that these characteristics can be used as biomarkers. These biomarkers could also be used to understand the course and stage of the disease. Our results and the preliminary test showed that chest X-rays could help healthcare professionals identify and diagnose COVID-19.
Figure 1 .
1Workflow for analysis and production of the intelligent artificial model in Telegram.
Figure 2 .
2(A) Poor quality image; (B) After using HEF
Figure 3 .
3Mask-RCNN architecture model. (A) Convolutional Backbone of ResNet101 for extracting a map of X-ray scanned image characteristics; (B) Region Proposal Network (RPN): a neural network of small weights that provides for the bounding boxes of the lung under analysis on the characteristics map; (C) Feature map: a result of activating the output of filters applied to the image; (D) RoIAlign layer: a bilinear interpolation of nearby points on the feature map to avoid quantizing the region of interest (RoI); (E) Fixed size feature map: Reduced version extracted from the feature map; (F) Mask Branch: Mask of a fully convolutional network (RTC), which provides a segmentation lung mask for each RoI; (G) Fully connected layers use high-level RoI features by remodeling to a forecast vector; (H) Box regression: predicts the values of the pulmonary coordinates; (I) Classification for the prediction of the lung class.
Figure 3 .
3Regions marked for training. (A) Image of a patient with COVID-19; (B) Pulmonary marking; (C) Heart marking
Figure 7 .
7(A) Implementation of the intelligent computational model in the telegram for access to doctors, nurses, and radiologists.
Table 1 .
1Manifestations in the images: Diseases typical of countries like Brazil that can trigger viral and bacterial pneumonia.Pleural
effusion
Ground
glass
opacity
Pulmonary
edema
Rounded
morphological
opacities
Table 2 .
2Image groups for character extraction and model trainingPneumonia type 1
Pneumonia type 2
Pneumonia type 3
Tuberculosis
Malaria
COVID-19
Streptococcus pneumoniae
Dengue
-
Table 3 .
3Multiple comparison of means -Tukey HSD, FWER = 0.05 to assess Haralick resources.Comparisons
Mean difference
Significance
Contrast
Contrast 1 x Contrast 2
-170,784
Significant
Contrast 1 x Contrast 3
-195,032
Significant
Contrast 2 x Contrast 3
-2.4248
Significant
Energy
Energy 1 x Energy 2
0.031
Significant
Energy 1 x Energy 3
42.3753
Not significant
Energy 2 x Energy 3
423,443
Not significant
Homogeneity
Homogeneity 1 x Homogeneity 2
0.031
Significant
Homogeneity 1 x Homogeneity 3
42.3753
Not significant
Homogeneity 2 x Homogeneity 3
423,443
Not significant
Correlação
Correlation 1 x Correlation 2
0.007
Significant
Correlation 1 x Correlation 3
101.9509
Not significant
Correlation 2 x Correlation 3
1,019,439
Not significant
Dissimilarity
Dissimilarity 1 x Dissimilarity 2
0.8779
Significant
Dissimilarity 1 x Dissimilarity 3
532,676
Not significant
Dissimilarity 2 x Dissimilarity 3
52.3897
Not significant
Table 4 .
4Hyperparameters of the decision tree modelBest criterion
Best maximum tree
depth
Best number of
components
Cross validation for
model evaluation
Entropy
12
3
0.93 ± 0.051
Table 5 .
5Validation in the health unit with patients with suspected COVID-19number of suspect diagnoses
number of diagnoses
Normal
5
4
COVID-19
18
20
Tuberculosis
2
1
Computer Science & Information Technology (CS & IT)
© 2021 By AIRCC Publishing Corporation. This article is published under the Creative Commons Attribution (CC BY) license.
ACKNOWLEDGMENTSThe authors would like to thank the Ministry of Health of Brazil for making the MAIDA system's images. And the health professional Valdirene Bento and the Health Secretary of Itapeva-MG Luciano.AUTHORI obtained three bachelor's degrees in biomedical informatics and speech therapy from the University of Sao Paulo and mathematics from Anhembi Morumbi, a PhD in Sciences with a focus on Bioinformatics data in chronic pain and tinnitus models and a Post-doctorate in Psychology for the development of intelligent models for evaluation of social behaviors by the U. I also acted as Coordinator in projects of Bioinformatics and Artificial Intelligence in Medical Images in the evaluation of good or bad prognosis of childhood cancer by the Department of Pediatrics of the Hospital das Clínicas de Ribeirão Preto.
Folha informativa -COVID-19 (doença causada pelo novo coronavírus). PublishedFolha informativa -COVID-19 (doença causada pelo novo coronavírus). PAHO. https://www.paho.org/bra/index.php?option=com_content&view=article&id=6101:covid19&Itemid= 875. Published June 19, 2020. Updated Jun 19, 2020.
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"Machine Learning Algorithms In User Authentication Schemes",
"Machine Learning Algorithms In User Authentication Schemes"
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[
"Laura Pryor \nDepartment of Computer Science\nDepartment of Computer Science\nUniversity of Wisconsin -Eau Claire Eau Claire\nWIUSA\n",
"DrRushit Dave [email protected] \nDepartment of Computer Science\nUniversity of Wisconsin\n-Eau Claire Eau ClaireWIUSA\n",
"DrJim Seliya [email protected] \nDepartment of Computer Science North Carolina\nUniversity of Wisconsin\n-Eau Claire Eau ClaireWIUSA\n",
"DrEvelyn Sowells Boone \nA&T State University Greensboro\nNCUSA\n"
] |
[
"Department of Computer Science\nDepartment of Computer Science\nUniversity of Wisconsin -Eau Claire Eau Claire\nWIUSA",
"Department of Computer Science\nUniversity of Wisconsin\n-Eau Claire Eau ClaireWIUSA",
"Department of Computer Science North Carolina\nUniversity of Wisconsin\n-Eau Claire Eau ClaireWIUSA",
"A&T State University Greensboro\nNCUSA"
] |
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In the past two decades, the number of mobile products being created by companies has grown exponentially. However, although these devices are constantly being upgraded with the newest features, the security measures used to protect these devices has stayed relatively the same over the past two decades. The vast difference in growth patterns between devices and their security is opening up the risk for more and more devices to easily become infiltrated by nefarious users. Working off of previous work in the field, this study looks at the different Machine Learning algorithms used in user authentication schemes involving touch dynamics and device movement. This study aims to give a comprehensive overview of the current uses of different machine learning algorithms that are frequently used in user authentication schemas involving touch dynamics and device movement. The benefits, limitations, and suggestions for future work will be thoroughly discussed throughout this paper.
|
10.1109/icecet52533.2021.9698440
|
[
"https://arxiv.org/pdf/2110.07826v1.pdf"
] | 239,009,709 |
2110.07826
|
f3742a0e3590a62972fe008efad84f56dae822d9
|
Machine Learning Algorithms In User Authentication Schemes
Laura Pryor
Department of Computer Science
Department of Computer Science
University of Wisconsin -Eau Claire Eau Claire
WIUSA
DrRushit Dave [email protected]
Department of Computer Science
University of Wisconsin
-Eau Claire Eau ClaireWIUSA
DrJim Seliya [email protected]
Department of Computer Science North Carolina
University of Wisconsin
-Eau Claire Eau ClaireWIUSA
DrEvelyn Sowells Boone
A&T State University Greensboro
NCUSA
Machine Learning Algorithms In User Authentication Schemes
behavioral biometricsuser authenticationmachine learning
In the past two decades, the number of mobile products being created by companies has grown exponentially. However, although these devices are constantly being upgraded with the newest features, the security measures used to protect these devices has stayed relatively the same over the past two decades. The vast difference in growth patterns between devices and their security is opening up the risk for more and more devices to easily become infiltrated by nefarious users. Working off of previous work in the field, this study looks at the different Machine Learning algorithms used in user authentication schemes involving touch dynamics and device movement. This study aims to give a comprehensive overview of the current uses of different machine learning algorithms that are frequently used in user authentication schemas involving touch dynamics and device movement. The benefits, limitations, and suggestions for future work will be thoroughly discussed throughout this paper.
I. INTRODUCTION
The technological advancements of the past two decades regarding mobile devices has led society to develop a heightened dependency on the devices in which such personal information is now being stored. However, the security trusted to keep this information safe has stayed stagnant since the beginning of this technological boom. Most devices today still rely on static security methods to authenticate users. Static methods include entering a password, PIN number, or physiological biometrics such as a palm print [1]. While these methods are easy and reliable for the genuine user, their ease has opened the door for more and more attacks by nefarious users leaving users personal and private information vulnerable. This imminent threat on devices has led researchers to investigate possible improvements to security. One possible way of improvement is by using behavioral biometrics as an added layer of protection for devices. Instead of focusing on passwords correctly being inputted, behavioral biometrics uses the behaviors of the user while they are interacting with their device. Behavioral biometrics has been implemented in multiple different schemas and in multiple different devices or systems. For example, [2] looks at behavioral biometrics in healthcare, and papers [3][4][5] look at behavioral biometrics in mobile devices. In order to use behavioral biometrics, the user authentication schema needs to include a machine learning algorithm for classification. Machine learning algorithms have been deemed very beneficial for use in cybersecurity as seen in [6] and can be very useful in behavioral biometric based models. The machine learning algorithms used in this schema often are crucial for the model to be able to perform well. That being said, while there is plenty of research available comparing the different behavioral biometrics, there is little literature done specifically on the machine learning algorithms these models depend on. Along with that, the literature that is being done on the algorithms themselves often look at multiple models using multiple different biometrics. However, some algorithms work differently with different biometrics, thus this study will focus on a common pairing of biometrics: touch dynamics and phone movement. This study aims to create a comprehensive review on the commonly used machine learning algorithms in touch dynamic based user authentication methods.
II. BACKGROUND
This literature review for machine learning algorithms used in touch dynamic and phone movement-based authentication models stems from the findings of our past work on a separate survey paper. In this paper [7], different behavioral biometric based user authentication models were investigated, and it was concluded that the biometric that yielded the most consistent and best results was touch dynamics. Also found in this study was that using multiple biometrics in a model also improved the model's accuracy. While that paper also included an analysis on the machine learning algorithms being used, many of the papers were outdated for the topic and the analysis included algorithms from models that did not include touch dynamics, so it felt necessary to complete a second literature review this time looking specifically at the machine learning algorithms for touch dynamics and phone movement-based models. Four machine learning algorithms are highlighted in this paper, those of which being Support Vector Machine (SVM), Random Forest (RF), K-Nearest Neighbor (K-NN), and Naïve Bayes (NB). SVM is a supervised machine learning algorithm that is widely used in touch dynamic and phone movement-based authentication schemas, for example in papers [8][9][10]. How this algorithm works is that based on the number of features used in the model, an n-dimensional space is created, and, in that space, multiple hyperplanes are found. In this hyperplane all of the data points are classified, in this case the datapoints would be classified as genuine or imposter. The goal of the algorithm is to choose the hyperplane that has the maximum distance between the two classified datapoints groups, from there all future data points are then classified based to how close the fall to the genuine or imposter "sides" of the hyperplane. RF is again another widely used machine learning algorithm in all types of authentication schemes. RF can be seen being tested in papers [11][12][13][14][15][16]. The basic premise of RF is that multiple decision trees are built and each time a new datapoint is run through the algorithm it goes through each decision tree and the results of each tree are then averaged and is then outputted as the final result. Each of the decision trees in the RF are slightly different, which adds an extra level of analysis which is missing from using just one decision tree. KNN is an algorithm that is seeing performing consistently throughout different schema [17]. Similar to SVM, KNN deals with the distance between classified points and the newly added datapoints. The K in KNN stands for a predetermined set of neighbors that are analyzed when a new datapoint is added. So, for example, if the K was 5, when a new datapoint is entered the five closest datapoints to the new point are chosen and then whatever classification is found in the majority of those five points is then "voted" as the classification for the new point. The final algorithm surveyed in this paper is NB is regarded as a very simple and fast algorithm, which can be seen in [18]. NB, unlike the other three algorithms, is a probabilistic approach. This algorithm uses the Bayes Theorem, which is used for calculating conditional probabilities. In this algorithm each feature is deemed to be both independent and equal. Therefore, in the terms of our model, the algorithm is looking for the probability that the user is genuine based on other probabilities that are known to the model based on the previous data that it has been trained on.
III. LITERATURE REVIEW
A. Support Vector Machine
In [19], researchers test SVM, along with other Machine Learning algorithms in a model that looks to authenticate users based on their touch dynamics when using social media and NB algorithms were also tested in this schema. SVM was found to outperform RF when looking at results for testing involving long-term and continuous versus single-use authentication. The researchers used two different scenarios to test both questions, the first scenario being that data was collected during the subject's entire time on their device, and the second scenario collected data only when the user was on social media applications. When looking at short-term authentication, SVM had easily the best average error rate (AER) of 6.02% for scenario 1 and 3.07% for scenario 2. The next best algorithm was the BPNN whose results can be seen on Fig. 1. Overall, SVM had approximately a 4% lower error rate for each scenario. This paper also showed promising results for using SVM to achieve long term accuracies in user authentication schemes. After testing a group of participants on scenario 2 over a two-week span, the researchers found that the AER only increased slightly to 3.68%. The authors also noted that they noticed trending in their data that would suggest that users would be able to achieve more stable behavior changes after two weeks, which would lower that average error rate.
Researchers in [20] look to analyze the effects that behaviorbased profiling systems such as WiFi and application usage have on authentication schema when combined with touch dynamics. In this paper only SVM was tested, however the results proved too strong to be ignored. The authors looked at the accuracy (ACC) for two scenarios: one time authentication and active authentication. The findings, as shown in Fig. 2, concluded that when adding behavior-based profiling systems to the behavioral biometric based schema the ACC of the models increased by an average of 17% for both one time authentication and active authentication. For one time authentication the model achieved an ACC of 82.2% when using the combined method of behavior-based profiling systems and for active authentication that same ACC [20] from touch dynamics with 72.0%. These results show that SVM is an effective algorithm to use for authentication, both for single use and implicit use. The researchers also suggested that the low ACCs that came from the individual biometrics can be explained by limited numbers of data samples during some sessions. This could indicate that SVM may not be a good algorithm to use for smaller samples. Another thing to note is that this data was collected over a two-month period, providing further evidence to support that SVM performs well for longterm authentication.
In [21], SVM and RF were two algorithms used in the researcher's schema that used swipe data from touchscreens to authenticate users. In this study, the authors were experimenting with the use of the capacitive frames that are created when a user swipes on their device, instead of using touch data from the device's sensors. While SVM did not always perform the best in all scenarios, the researchers found it to be the most consistent of the two algorithms and only performed slightly less effectively than the RF algorithm in some tests. As seen in Table 1, while using SVM the model was able to produce maximum authentication accuracy of 79.88% which indicates that SVM can perform well when dealing with models that use capacitive data and not sensor data. However, using SVM also produced a false rejection rate (FRR) of 50%, and while this is much lower than the 90.62% FRR associated with RF, the high FRR still gives cause for concern for the effectiveness of the model. Researchers hypothesized that the high FRRs may be happening due to their dataset only consisting of 160 samples from 8 users, and also the training and testing data including more imposter samples than there were genuine samples. All of these reasons can lead to the model falsely rejecting the user, and it also once again indicates that, as seen in [20], SVM may not perform well with models that will be using small amounts of data to make their decision.
B. Random Forest
Researchers Zhang et. al [22] used RF to experiment with the efficacy of their proposed continuous authentication schema. Along with RF the other algorithms being tested are SVM, NB, and decision tree (DT). In the proposed schema, four implicit swipe gestures are being used to classify the user: a vertical scroll up and down and a horizontal scroll left and right. The researchers are looking to see if using just vertical or horizontal swipe gestures omit the best results or if a combination of the two can achieve the highest accuracies. Three different models were tested, their results can be seen in Fig. 3, and RF was the best performing algorithm for each of the three models. RF achieved the highest accuracies in each of the models getting 96.7%, 96.36%, and 95.03% for the vertical swipe model, horizontal swipe model, and the combined model respectively. The results of this study indicated that the RF algorithm can be used in implicit user authentication to obtain high accuracies. These results also showed that either horizontal or vertical swipe gestures can be used in touch dynamic based schema, however the two gestures should not be combined as it decreases the model's accuracy. It was also noted by the authors that the horizontal or vertical swipe gestures could be combined with other touch dynamics to create a possibly better performing model, as the swipe gestures would not negatively affect the accuracy of the model.
In [23], the authors based their proposed model on the pattern lock that is often seen as an authentication measure for many mobile devices. In this model, a combination of the pattern lock and a password is used to create a simple game that is used for authentication purposes. In order for the user to be granted access to their device they not only need to put in the right password, but they need to put in the password in the same pattern as when they originally created it. This simple game combines the two most common forms of authentication, passwords and patterns, and puts them together, putting multiple letters and symbols included and not included in the password onto the device's screen and the user must connect the letters and symbols of the password using the correct swiping pattern. RF along with SVM and XGBoost were used in this model for testing, and the researchers found that RF was able to achieve the best results getting a False Acceptance Rate of only 1.40% and a False Rejection Rate of 2.08%. The [23] researchers also tested the security of the model and found that even when the imposter users were given the simplest password tested, 50% of the users were not able to gain access to the device. The rest of the security results can be seen in Table 2. This implies that the extra biometric layer was able add additional protection without lengthening the process it takes to authenticate the user. Also noted was that as the password strength increased, the number of imposters able to infiltrate the device, even with previous knowledge of the password, decreased. One final item examined by the researchers was the minimum number of strokes needed for their model to authenticate the user still effectively. It was found that this model only needed four strokes, or five letters or symbols, to be able to accurately authenticate the user. Coupling that information with the low FAR and FRR of the RF classifier, it can be implied that the RF algorithm can be used in models that use a small sample size.
In [24], researchers Wang et. al look at multiple aspects of mobile user authentication schemes and examine how they affect the model's accuracy. One of the aspects looked at is the machine learning algorithm being used as the classifier for the schema. Six algorithms were tested in this study, those being logistic regression (LR), NB, KNN, DT, RF, and SVM. The six different models were tested 21 times with the number of training samples increasing each time, as the authors also wanted to see what the minimum number of training samples is needed to create an accurate model. RF once again performed the best of the six algorithms achieving an accuracy of 97%. It should be noted however, that all algorithms performed well, with the worst-performing algorithm, SVM, still achieving an accuracy of 81%. In this experiment it was also found that after the training sample amount reached 4, the accuracies of the six models started to level off, thus proving that RF does not need a large training sample to still be able to perform well. The researchers then continued their experiment using the RF classifier to look at feature importance. The results concluded that spatial features such as the start and stop positions of a touch gesture hold the strongest effect on the schemas efficiency, and temporal and movement direction features were found to have the weakest effect on the schema. While the authors noted that these findings are not reasoning to completely avoid using temporal and movement direction features, however it does suggest that more researchers should focus on using spatial features in their model, which ultimately increases usability as less features are needed to be collected to create an accurate decision.
C. Other Algorithms
In [25] the machine learning algorithm KNN was highlighted. In this study the authors use KNN to test their schema that combines both the touch and movement data of the device. The results of this study are compared to four previous studies that worked with a similar schema but used different machine learning algorithms for the classifier. The researchers first testing their model by calculating the EERs of 30 unlocking gestures, 20 of which were common gestures and the other 10 were user generated. As seen in Table 3, for the first 20 common gestures, the average EER of the model was 5.24% and the average EER of the final 10 generated gestures was 4.23%. This ends up being an overall average EER of 4.90%. The familiarity of gestures was then tested as researchers were looking to see if the accuracy of the model improves when the user becomes familiar with the gestures. The EER of a user who created their own gesture was compared to the average EER of the other 40 users who were unfamiliar to the gesture and the results indicated that familiarity with the gesture increases the performance of the model. For example, after training the gesture for a week, one user achieved and EER of 2.52%, while the average of the other 40 users was 4.08%. These results could imply that K-NN could be used in long-term authentication schemas, as the EER continued to stay low for the authentic user after a week of use. The researchers also tested the minimum training samples needed for an effective authentication scheme, concluding that a sample size of 15 was the smallest amount that produced similar results to the results from models with more training samples. These results indicate the models that use K-NN do not need to use large training sets, which helps with data collection. Finally, the long-term accuracy of the model was tested, and it was found that as time increases, the EER of the model also increases slightly. While the accuracy of the model does decrease over time, it is only slightly, indicating that models that use K-NN could potentially be used for long-term authentication use, as long as the researchers create a schema that can be implicitly adapted and updated over time. The overall EER of this model was compared to the EERs of four other studies using algorithms such as SVM, RF, dynamic time warping, and MHD. This study achieved the significantly lower results than the other models. Some examples of the EERs of the other models include the model using SVM producing an EER of 9.67% and the model using RF getting an EER of 13.09%. Comparing results of the previous works to the 4.90% EER of the current model indicates that this model and the K-NN classifier can achieve significantly lower results than similar models using other major machine learning classifiers. Authors of [26] proposed a schema similar to others involving collecting touch dynamics while users unlocked a pattern lock, however one big difference with this schema is that instead of testing users with multiple passwords, each user had the same exact pattern. Thus, the accuracy of the proposed model comes solely from the behavioral-biometric layer rather than the biometric layer and knowledge layer combined. Six machine learning algorithms were tested in this model, those being DT, SVM, KNN, gaussian naïve bayes (GNB), RF, and LR. In the researchers first test to find the best performing classifier, the six algorithms were put against each other with the model's accuracy while the user was sitting, walking, and sitting and walking combined being collected. GNB was deemed the best performing algorithm achieving 97.19%, 95.25%, and 95.97% accuracies for sitting, standing, and combined respectively, which can be seeing in Fig 4. The next best algorithm was RF, which also produced similar results, however GNB was chosen due to the optimal feature length for GNB being shorter than RF. Thus, GNB was not only a well performing algorithm, but a cheap algorithm for the researchers to implement. The next experiment conducted by the researchers was to find the optimal number of imposter samples needed for each of the three postures (sitting, walking, and combined), since the amount of imposter users could lead to an imbalance of the FRR and FAR. After testing multiple combinations of imposter user sample sizes, the results concluded that the best imposter counts for each posture were 13 for sitting, 23 for walking, and 8 for the combined postures. The researchers also noted that this data was all collected in one day, so the model would most likely become less accurate as time increased. The results of this study indicate that the GNB algorithm can be a well-performing classifier for touch gesturebased schema.
IV. DISCUSSION & ANALYSIS
In this survey, there were many different machine learning algorithms used in touch dynamic based user authentication schemes introduced and discussed. These algorithms were compared to each other in table [table number]. While many of the algorithms performed well in their respective models, two algorithms excelled among the rest. As seen in the sample of studies discussed, random forest and support vector machine are two heavily used and strong performing algorithms. While many of the other algorithms, such as KNN and NB, were also used quite frequently, they were often outperformed by SVM and RF which is seen throughout the study.
A. Support Vector Machine
SVM's consistent success in mobile user authentication models could be due to the algorithm's ability to capture more complex relationships in the data than many other machine learning algorithms can. However, one downside to SVM found during our research was that SVM often requires larger training samples to be able to perform well, which in turn would increase the number of samples needed from users thus making the model very computationally costly.
B. Random Forest
Unlike SVM, RF was found to perform well with smaller datasets, which can lead to a low-cost model. Also, RF is often considered a strong algorithm for researchers due to the algorithm's usability and forgiveness. RF is an easy algorithm for researchers to understand and implement making it very usable and the algorithm is very forgiving since an error in one tree will not negatively affect the entire model, unlike some other algorithms.
V. LIMITATIONS & FUTURE WORK
While all of the schemes studied in this survey produced very promising results, there are still some limitations that can be affecting these accuracies. Many studies noted using small amounts of participants, which can cause issues with the consistency and validity of their models. Having invalid or inconsistent models can produce false accuracies, both negatively and positively, therefore the results shared by most of these papers could be inaccurate. Along with that, most of the models did not distinguish single touch and multi touch gestures to be different from one another, which can lead to more false positives. Finally, an issue plaguing this study and not the individual papers is that not every model had the same success metrics being used, while it is still pretty easy to evaluate the effectiveness of each of the models, it is hard to compare models that have different metrics for success. Future work should look into increasing participant sizes and creating models that can distinguish different gestures as solving these issues could increase the accuracy of the results being published and would make these model's that much closer to being ready for real-world application.
VI. CONCLUSION
Throughout the review of these papers, it was concluded that the machine learning algorithm that performs the best in touch dynamic and phone movement-based user authentication schemes are random forest and support vector machine. RF was only the best performing algorithm, but it was also found to be one of the easiest and cheapest algorithms to implement. SVM was also found to be a better choice for a classifier than other algorithms since model's using SVM can often achieve high accuracies due to SVM being able to identify more complex relationships that other algorithms cannot see. While there are many other algorithms that can perform well, RF and SVM's usability and reliability puts these algorithms ahead of the rest.
Fig. 3 .
3Performance of the classifiers for each of the vertical, horizontal, & combined models[22]
Fig. 4 .
4Average accuracies of the classifiers[26]
TABLE I .
IPERFORMANCE OF ALGORITHMS FOR AUTHENCATION [21]
Fig. 1. Average Error Rates of the different Classifiers [19]
applications. Along with SVM, the J48 Decision Tree (J48), Radial Basis
Function Network (RBFN), Back Propagation Neural Network (BPNN),
jumped to 97.1%. For comparison, when testing the single behavioral biometrics for one time authentication, the highest ACC cameFig. 2. Performance of model using individual biometric and the biometric paired with the profiling signals both for single use and active authenticationSuccess
Metrics
Left Swipe
Right Swipe
SVM
RF (Number of Trees Below)
SVM
RF (Number of Trees Below)
N/A
10
20
30
40
N/A
10
20
30
40
Accuracy
79.88%
87.69% 87.50% 87.69%
87.89%
57.81% 86.32% 85.93% 87.69% 87.50%
FAR
15.84%
1.33%
1.56%
0.89%
0.89%
17.63%
1.56%
2.00%
0.66%
0.44%
FRR
50.00%
89.06% 89.06% 92.18%
90.62%
62.50% 98.43% 98.43% 93.75% 96.87%
0.00%
5.00%
10.00%
15.00%
20.00%
25.00%
J48
NB
RBFN BPNN SVM
Average
Error
Rate
Classifiers
S1
S2
0
10
20
30
40
50
60
70
80
90
100
Accuracy
(%)
Individual
w/ Profiling Signal
Active Authentication
0
20
40
60
80
100
DT
NB
RF
SVM
Accuracy
(%)
Classifiers
Vertical
Horizontal
Combined
TABLE II .
IISECURITY PERFORMANCE OF THE MODEL
TABLE III .
IIIAVERAGE EERS OF THE 30 TESTED GESTURES [25]
Password
Level
Security Results
Without Instructions
With Instructions
Guessed
Not
Guessed
Guessed
Not Guessed
Level 1
30%
705
48%
52%
Level 2
6%
94%
15%
85%
Level 3
9%
91%
18%
82%
Level 4
18%
82%
21%
79%
Gesture Type
(Average)
Equal
Error Rate
Common (1-20)
5.24%
User-generated (21-30)
4.23%
Overall
4.90%
TABLE IV .
IVCOMPARISON OF THE RESULTS OF THE PAPERS SURVEYEDPapers ML Algorithm SummariesClassifiers Tested
Best Classifier
Best Results
[8]
SVM
SVM
97.40% ACC
[9]
SVM, KNN, NB
SVM
1-2% EER
[10]
J48, NB, SVM,
BPNN
SVM
4.1% AER
[11]
SVM, RF, BN
RF
74.97% AAR
[12]
LR, NB, RF
RF
0.81% EER
[13]
NB, KNN, RF
RF
97.90% ACC
5.1% EER
[14]
BN, NB, SMOP,
KNN, J48, RF
RF
99.35% ACC
[15]
J48, SVM, RF,
BN, NB
RF
76% ACC
[16]
MLP, RF
RF
0.01 EER
[17]
DTW-KNN
DTW-KNN
5.5% Average FAR &
FRR
[18]
SVM, RF, GB,
XGB, NBB, NBG
RF, NBG
98.35% ACC
1.88% EER
[19]
J48, NB, RBFN,
BPNN, SVM
SVM
3.07% AER
[20]
SVM
SVM
97.1% ACC
[21]
SVM, RF
SVM
79.88% ACC
[22]
DT, NB, RF,
SVM
RF
96.7% ACC
[23]
RF, XGB
RF
1.40% FAR
2.08% FRR
[24]
LR, NB, KNN,
DT, RF, SVM
RF
97% ACC
[25]
KNN
KNN
4.90% EER
[26]
DT, SVM, KNN,
NBG, RF, LR
NB
97.19% ACC
ACKNOWLEDGMENT Funding for this project has been provided by the University of Wisconsin-Eau Claire's Office of Research and Special Programs Summer Research Experience Grant. This project was done using Blugold supercomputing Cluster at uwec.
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|
[] |
[
"Convective Organization and Eastward Propagating Equatorial Disturbances in a Simple Excitable System",
"Convective Organization and Eastward Propagating Equatorial Disturbances in a Simple Excitable System"
] |
[
"Geoffrey K Vallis \nUniversity of Exeter\n\n",
"James Penn \nUniversity of Exeter\n\n"
] |
[
"University of Exeter\n",
"University of Exeter\n"
] |
[] |
We describe and illustrate a mechanism whereby convective aggregation and eastward propagating equatorial disturbances, similar in some respects to the Madden-Julian oscillation, arise. We construct a simple, explicit system consisting only of the shallow water equations plus a humidity variable; moisture enters via evaporation from a wet surface, is transported by the flow and removed by condensation, so providing a mass source to the height field. For a broad range of parameters the system is excitable and self-sustaining, even if linearly stable, with condensation producing convergence and gravity waves that, acting together, trigger more condensation. On the equatorial beta-plane the convection first aggregates near the equator, generating patterns related to those in the Matsuno-Gill problem. However, the pattern is unsteady and more convection is triggered on its eastern edge, leading to a precipitating disturbance that progresses eastward. The effect is enhanced by westward prevailing winds that increase the evaporation east of the disturbance. The pattern is confined to a region within a few deformation radii of equator because here the convection can best create the convergence needed to organize into a self-sustaining pattern. Formation of the disturbance preferentially occurs where the surface is warmer and sufficient time (a few tens of days) must pass before conditions arise that enable the disturbance to reform, as is characteristic both of excitable systems and the MJO itself. The speed of the disturbance depends on the efficiency of evaporation and the heat released by condensation, and is typically a few meters per second, much less than the Kelvin wave speed.
|
10.1002/qj.3792
|
[
"https://arxiv.org/pdf/1908.10386v1.pdf"
] | 201,653,422 |
1908.10386
|
9870e8badeb194d091b8494b434dc3b1ba5d25de
|
Convective Organization and Eastward Propagating Equatorial Disturbances in a Simple Excitable System
23 August, 2019
Geoffrey K Vallis
University of Exeter
James Penn
University of Exeter
Convective Organization and Eastward Propagating Equatorial Disturbances in a Simple Excitable System
23 August, 2019
We describe and illustrate a mechanism whereby convective aggregation and eastward propagating equatorial disturbances, similar in some respects to the Madden-Julian oscillation, arise. We construct a simple, explicit system consisting only of the shallow water equations plus a humidity variable; moisture enters via evaporation from a wet surface, is transported by the flow and removed by condensation, so providing a mass source to the height field. For a broad range of parameters the system is excitable and self-sustaining, even if linearly stable, with condensation producing convergence and gravity waves that, acting together, trigger more condensation. On the equatorial beta-plane the convection first aggregates near the equator, generating patterns related to those in the Matsuno-Gill problem. However, the pattern is unsteady and more convection is triggered on its eastern edge, leading to a precipitating disturbance that progresses eastward. The effect is enhanced by westward prevailing winds that increase the evaporation east of the disturbance. The pattern is confined to a region within a few deformation radii of equator because here the convection can best create the convergence needed to organize into a self-sustaining pattern. Formation of the disturbance preferentially occurs where the surface is warmer and sufficient time (a few tens of days) must pass before conditions arise that enable the disturbance to reform, as is characteristic both of excitable systems and the MJO itself. The speed of the disturbance depends on the efficiency of evaporation and the heat released by condensation, and is typically a few meters per second, much less than the Kelvin wave speed.
Introduction
Two, related, problems in tropical dynamics concern convective organization and the Madden-Julian oscillation (the MJO), both of which have been the subject of considerable investigation. However, no consensus has been reached in either case as to the key mechanisms involved and indeed, as recently as 2017, Fuchs & Raymond (2017) wrote that 'the MJO [has] been and still remains a "holy grail" of todayâĂŹs atmospheric science research'.
The MJO is a large-scale precipitating disturbance that propagates eastward at a few meters per second, centered near the equator and extending meridionally about 20°North and South (Madden & Julian, 1971;Zhang, 2005;Lau & Waliser, 2012). Its influence in the geopotential field extends zonally many thousand kilometers, and although the region of intense precipitation is confined to the tropics its influence can be felt in the mid-latitudes and stratosphere. Typically, the disturbance forms over the warm waters of the Indian Ocean and progresses east over the maritime continent and across the Pacific, decaying over the cooler waters of the eastern Pacific. The MJO reforms with a timescale of order a few tens of days -it is sometimes called the 30-60 day oscillation -and its spectral signature is manifest at a lower frequency than a Kelvin wave of the same scale (Wheeler & Kiladis, 1999;Kiladis et al., 2009). The MJO may also be thought of as a translating disturbance that recurs on the above timescale, and that is the perspective we will take here.
The eastward propagation of the MJO suggests the influence of Kelvin waves, yet its propagation speed is much less than that of dry Kelvin waves (about 20 m s −1 ) for any reasonable equivalent depth of the atmosphere. The condensation of water vapor is likely to be a significant influence on the propagation of any precipitating disturbance and this led to theories centered around the notion of a 'moisture mode' (e.g., Raymond, 2001;Raymond & Fuchs, 2009;Sobel & Maloney, 2013). The framework of Majda & Stechmann (2009) also involves moisture in an essential way. Although the various models differ from each other in their assumptions and dynamics, a commonality is to assume some low-mode structure and calculate a dispersion relation or mode of instability that, depending on the form of the model, gives rise to propagation. Rather different types of model were presented by Biello & Majda (2005) and Yang & Ingersoll (2013), who suggested that the MJO is a multi-scale phenomenon with the convective structures resulting from a smaller scale activity.
In contrast to these relatively simple and/or semi-analytic models, comprehensive threedimensional models are becoming able to simulate many aspects of the MJO (e.g., Liu et al., 2009;Khairoutdinov & Emanuel, 2018). In these simulations the MJO seems to be associated with some form of instability associated with a very large scale convective aggregation. However, the complexity of the models can make it difficult to determine the dominant mechanism and to make connections with or distinguish between the multi-scale theories and large-scale moisture-mode models.
Even without reference to the MJO, convective aggregation (e.g., Bretherton et al., 2005;Muller & Bony, 2015;Wing et al., 2017) is a fraught subject with a diversity of results across models and no single accepted dominant mechanism. 'Self'-aggregation, meaning the aggregation of convection over a surface of uniform temperature with no external large-scale influences, may be particularly sensitive to the parameters of the situation. However, as we will discuss, organization in a differentially rotating frame (essentially the beta plane) may be much more robust and is likely to be the building block of the MJO.
In this paper we explore these issues and identify a robust mechanism of equatorial aggregation and eastward propagation. We do so through the use of a minimal but explicit model, specifically the moist shallow water equations with relatively simple physics, a system that retains many of the properties of the three-dimensional equations. We try to make no assumptions that are not transparently connected to the properties of the equations of motions themselves. We find that the resulting system is an excitable one that, even when linearly stable, can produce self-sustained, irregular motion that, when solved on a beta-plane, gives rise to convective organization and eastward propagating equatorial disturbances with many similarities to the MJO. More generally, by considering tropical convection as an excitable system, properties that may have appeared surprising then seem less so, even where not fully understood.
We first describe the equations themselves. We then show the system admits of exact solution of no motion, and examine the stability and excitability properties of the system. After that we describe a number of numerical simulations and discuss the mechanisms of organization and eastward propagation. In the final section we place our results in a broader context and provide some conclusions.
The Moist Shallow Water Equations
The shallow water equations may be derived by way of an expansion of the hydrostatic primitive equations in terms of vertical modes (Matsuno, 1966;Gill, 1980;Vallis, 2017). The first baroclinic mode then obeys an equation set similar to the reduced-gravity shallow water equations although with slight differences in interpretation -in particular the height field is related to the temperature field, and a heating is represented as a source of sink of mass. The usual derivation is valid only in the linear case, since the nonlinear terms bring in additional vertical modes, but nonetheless equations of this general form are often regarded as good analogs of the primitive equations in cases when the vertical structure is dominated by the first baroclinic mode (e.g., Yano et al., 1995;Sobel et al., 2001;Majda & Stechmann, 2009). A deep baroclinic mode is also an observed signature of the MJO (Kiladis et al., 2005;Adames & Wallace, 2014).
We include moisture by introducing humidity variable, q, that is carried by the fluid and that is conserved in the absence of evaporation and condensation, and when condensation occurs it provides a heat source that affects the height field. Since moisture has a much smaller scale height than the troposphere (about 2 km rather than 8 km) we take the velocity field to be that of the lower atmosphere. The equations of motion then become, written using standard shallow water notation and omitting frictional and diffusional terms,
Du Dt + f × u = −g∇h, (1a) ∂ h ∂t + ∇ · (hu) = −γC + R, (1b) ∂ q ∂t + ∇ · (qu) = E − C,(1c)
where E represents evaporation, C represents condensation, R represents thermal forcing (e.g., a radiative relaxation) and γ is a parameter proportional to the latent heat of condensation. With no evaporative or thermal relaxation terms, but even in the presence of condensation, (1b) and (1c) combine to give the conservation equation
∂ M ∂t + ∇ · (Mu) = 0,(2)
where M = h − γq is an analog of a moist static energy or moist enthalpy for the system. The above moist shallow water (MSW) equations are similar to those used by Bouchut et al. (2009) and Rostami & Zeitlin (2018), although the implementation and the experiments, described below, significantly differ. Evaporation from a wet surface is parameterized by a bulk-aerodynamic-type formula,
E = λ|u/U 0 |(q g − q)H (q g − q),(3)
where λ is a constant (similar to a drag coefficient) and q g is the surface humidity. The Heaviside function, H , ensures that evaporation only occurs when the surface humidity is larger than that of the atmosphere, and dew formation is forbidden. The dependence on velocity (|u|/U 0 , where U 0 is a scaling constant) enables wind-induced evaporative effects to occur, and may be omitted.
Condensation is allowed to occur on saturation and we take it to be 'fast', meaning that it occurs on a timescale faster than any other in the system and does not allow the fluid to become significantly supersaturated. We represent this as
C = H (q − q * ) (q − q * ) τ ,(4)
where q * is the saturation specific humidity and τ is the timescale of condensation. (Fast condensation is a simplification of 'fast autoconversion', commonly used in cloud resolving models.) Precipitation schemes of this form are a common feature of various idealized GCMs (e.g., Frierson et al., 2006) and simple (e.g., Betts-Miller Betts (1986) and convective-adjustment style) convection schemes. More elaborate convection parameterization schemes do exist for the shallow water equations (Würsch & Craig, 2014) but here we try to minimize their use. An expression for q * in terms of h is derived in the next section. Equations (1), (3) and (4) form a complete set. We now discuss the form of equations that we numerically integrate and the values of the parameters in relation to the true atmosphere.
Semi-Linear Equations of Motion
We will use the linear form of the momentum and height equations, but keep the nonlinearity in the moisture and Clausius-Clapeyron equations. The reason for this is that our main focus is on equatorial dynamics and the linear shallow water equations are remarkably fecund source of knowledge in that arena, beginning with Matsuno (1966) and Gill (1980). Tropical cyclones are also (deliberately) eliminated by such a linearization. However, a comparable linearization of the moisture equation, about either an unsaturated or saturated state, would be a poor approximation because of the large range of moisture values and the spatial variation of q * , and convective triggering by a cold gravity wave be misrepresented. Since q and q * do vary, it is not the case that moisture convergence alone leads to a convective instability.
The equations of motion become
∂u ∂t + f × u = −g∇h + ν u ∇ 2 u,(5a)∂ h ∂t + H∇ · u = −γC + (h 0 − h) τ r + ν h ∇ 2 h, (5b) ∂ q ∂t + ∇ · (qu) = E − C + ν q ∇ 2 q,(5c)
where H is the equivalent depth of the basic state and h is now the deviation from this. (Only the product of g and H matter but we retain familiar shallow water notation.) The term (h − h 0 )/τ r represents a radiative relaxation back to a prescribed value h 0 (usually taken to be zero) on a timescale τ r , and we now explicitly include diffusive terms with obvious notation.
Parameters and Relation to Atmosphere
Temperature variations may be roughly related to height variations by h/H ∼ −∆T/T 0 where T 0 is a constant (e.g., 300 K), and we choose H = 30 m and g = 10 m s −2 (e.g., Kiladis et al., 2009). An appropriate radiative relaxation timescale τ r is of order a few days, appropriate for a lower atmosphere (e.g., Gill, 1980). We do not include drag on velocity in the simulations shown here, but simulations with a moderate drag are similar. The coefficient of evaporation λ is such that for a typical velocity (which is of order 1 m s −1 ), or with no velocity dependence, the system becomes saturated in a few days. Now consider the other moist parameters. In an ideal gas the latent heat of condensation, L, is such that L∆q a = c p ∆T where q a is the specific humidity in the gas and c p is the heat capacity at constant pressure. Here the analogous relation is γ∆q = −h and so we estimate γ by
γ ∼ LHQ a q 0 c p T 0 ∼ 8.5 m,(6)
where Q a is a typical saturated value of specific humidity near the surface in the tropical atmosphere, q 0 is the corresponding quantity in the MSW equations, which (without loss of generality) we take to be unity. Using L = 2.4 × 10 6 J kg −1 , c p = 1004 J kg −1 K −1 , Q a = 0.035, T 0 = 300 and H = 30 m gives the value above.
To obtain an appropriate expression for saturation humidity for use in (4) we begin with the approximate solution of the Clausius-Clapeyron equation, namely
e s = e 0 exp L R v 1 T 0 − 1 T ≈ e 0 exp L∆T R v T 2 0 ,(7)
where e s is the saturation vapor pressure, e 0 is a constant, R v is the gas constant for water vapor, and ∆T = T − T 0 . Now, since the saturation specific humidity varies approximately in the same way as the saturation vapor pressure, we can write an expression for the saturation humidity in our system as
q * = q 0 exp(−αh/H),(8)where α ∼ L/(R v T 0 ). With R v = 462 J kg −1 K −1 and T 0 = 250 K we have α ∼ 20.
Rather larger values are arguably more realistic since the shallow water equations mainly represent horizontal variations of temperature. The diffusive parameters (ν u etc.) are chosen on a numerical basis. All these parameter estimates are manifestly approximate, and variations of up to an order of magnitude may be reasonably explored. The above MSW equations thus include the effects of evaporation, transport and precipitation of moisture, the heat release associated with precipitation, and a temperature dependent saturation mixing ratio -some of the main features of a tropical atmosphere. Tropical convection is often described as having quasi-equilibrium nature on sufficiently long time scales (e.g., Betts, 1973;Arakawa & Schubert, 1974;Emanuel et al., 1987), associated with the convection occurring on a much faster timescale than that of the larger scale flow. The production of convective available potential energy ( ) by large-scale flow is then nearly balanced by the relaxation of by convection and the temperature profile is constrained to be close to neutrally stable. These effects are represented in our model by the use of a prescribed vertical structure and by condensational effects acting on a fast timescale. Convective instability will result if the fluid is near to saturation and the large-scale flow is convergent at the same location. However, convergence alone does not necessarily lead to convection, nor is convergence the only way to excite it; gravity waves can also trigger convection (as noted by Yang & Ingersoll (2013)), given the height dependence of the Clausius-Clapeyron equation.
Exact Steady Solutions
The equations of motion admit of steady solutions with no motion. In this state the height and moisture equations are then,
0 = −γC − λ r h,(9a)0 = λ(q g − q)H (q g − q) − C,(9b)
where λ r = 1/τ r , C is given by (4) and the saturation humidity is given by (8). (We omit the velocity dependence on the evaporation, but restore this in later sections.) Suppose first that q 0 ≥ q g , which we might interpret as being the case with an unsaturated surface. In this case there can only be evaporation if the fluid is sufficiently cold and h > 0. However, the radiative forcing (and the effects of any condensation) will both warm the fluid and (9a) cannot be satisfied since both terms are negative. The solution in this case is then h = 0, q = q q , C = 0 and, because q = q g , the evaporation is zero. The fluid is thus not saturated unless q g = q 0 . Now suppose q g ≥ q 0 . A balance can now occur when the evaporation equals the condensation and the radiative cooling equals the latent heat release. Eliminating the condensation term in (9a) and (9b) gives
0 = −λ r h − γλ(q g − q)H (q g − q).(10)
Now, the assumed rapidity of the condensation means that the fluid is, to a very good approximation (asymptotically to O(τ)) saturated and (10) becomes
0 = −λ r h − γλ q g − q 0 exp(− αh) ,(11)
where α = α/H and we omit the Heaviside term since q g ≥ q, but retain q 0 for clarity. If q g = q 0 the solution is simply h = 0 and q = q g . If q g > q 0 the exact solution of this equation may be written in terms of the Lambert W function, namely the function that satisfies the equation W(z) exp(W(z)) = z for any z (Corless et al., 1996), with h then given by
h = −R + 1 α W α exp(αR) A ,(12)
where A = λ r /(γλq 0 ) and R = q g γλ/λ r . This solution is the shallow water analog of the drizzle solution in the vertically continuous problem found by Vallis et al. (2019), and the precipitation in both cases is non-convective, involving no fluid motion. A more easily interpreted expression results if we suppose that the saturation humidity varies linearly with the height field and q * = q 0 (1 − αh). Equation (11) becomes
0 = −λ r h − γλ q g − q 0 (1 − αh) (13) with solution h = −γλ(q g − q 0 ) λ r + γλq 0 α ,(14)
and the humidity is the saturated value occurring at this value. If q g = q 0 then h = 0, as expected, giving a solution with no evaporation or precipitation. If q g > q 0 then evaporation occurs, warming the fluid and reducing the value of h. The humidity, q, is then less than that at the surface, q g , and evaporation is continuous. In the steady state the resulting value of h is such that the evaporation exactly replenishes the condensation, and the condensational heating is exactly balanced by the radiative cooling. For q g = 1.1q 0 and the other parameters taking the values previously derived,
gives values of h of order 0.1 m or less, considerably smaller than the variations we will find when the model is in a convective regime. As much their particular form, the expression (12) and its approximation (14) are important because they demonstrate that steady solutions of no motion do exist to the problem. We now examine whether these solutions are stable to small perturbations and whether the system is excitable.
Stability and Excitability
Unless the diffusive and damping terms in the equations are very strong, the solutions found above can be expected to be conditionally unstable, as can be seen by the following argument. In the absence of motion the atmospheric humidity will be relax to the state q = q g , and if q 0 ≤ q g then the atmosphere is saturated, satisfying the steady solution calculated above. However, a localized perturbation, even an infinitesimal one, will lead to further condensation and thence a perturbation in the height equation and the generation of gravity waves, as well as low-level convergence at the source of condensation. These gravity waves will propagate and will induce more condensation nearby, which will in turn generate more gravity waves and so on. Evidently, the steady saturated solution is unstable to an infinitesimal perturbation. If, on the other hand, q 0 > q g then the steady solution Figure 1: a) Boundary of the excitable state for varying values of the radiative damping of the height field and the efficiency of evaporation. The stationary state is saturated with no precipitation of evaporation. In the excitable regime the system maintains a chaotic, convectively active state. In the stable state the system eventually returns to its original state of no motion, even though that state is linearly unstable. (b) Evolution in an excitable and a damped states, both with a damping timescale of 2 days and with evaporative parameters on either side of the critical line. (c) The domain and time-averaged kinetic energy as a function of the latent heat parameter, γ.
will not be saturated and an infinitesimal perturbation will have little effect, but a perturbation large enough to trigger condensation will generate further motion. Thus, unless the damping is unrealistically large any perturbation that causes saturation will lead to the generation of motion and a spreading field of precipitation as the gravity waves propagate.
Following a perturbation, and depending on (a) the rate of evaporation from the surface, (b) the magnitude of the latent heat of condensation, and (c) the size of the damping terms, the system will either evolve into a sustained convective state, or the perturbations will eventually die and the system will return to a state of no motion. The return can occur, even if the initial state is linearly unstable, because the precipitation occurs at a lower temperature than that of the drizzle solution, leaving the atmosphere unsaturated and non-precipitating as it slowly relaxes back to the initial state. The moisture is slowly replenished by surface evaporation, but if that occurs on a timescale that is long compared that on which the gravity waves are damped then the atmosphere only reaches saturation when all the perturbations have died. An external perturbation could nevertheless excite the system again.
On the other hand, with sufficiently small damping and sufficient latent heat release the system will evolve to state of self-sustained motion; that is to say, the system will be 'excitable'. Various forms of excitable system exist (e.g., networks of neurons, auto-catalytic chemical reactions, certain cellular automata) and there is no universal definition, but typically they are extended non-equilibrium systems that may have a linearly stable fixed point but that nevertheless are susceptible to finite perturbations (Meron, 1992;Izhikevich, 2007). In our system, the return or otherwise to the initial state does not sensitively depend on the linear stability properties of the initial state: numerical experiments show that self-sustained states of convection may exist even if q 0 > q g , in which case the steady solution is not saturated and thus linearly stable, and the sustained convection is then subcritical. Conversely, if the damping is moderately large, the system may return to its initial state, and stay there, even if that state is saturated and linearly unstable.
Marginal excitability
The boundary between the stable and excitable regimes, as determined numerically, is illustrated in Fig. 1. To determine the boundary the equations are numerically solved in a domain of size 10 4 km in both xand y-directions, periodic in the x-direction and with a sponge near the meridional walls, and no rotation. (In later simulations with a nonzero beta the equator is in the center at y = 0.) In each case we choose q 0 = q g , meaning that the state with no motion is just saturated and marginally unstable, by analogy to the tropical atmosphere (Xu & Emanuel, 1989). From that state, and for each set of parameters, a small perturbation is added and the system allowed to evolve freely. The system will either evolve into a self-sustained state or, after some time, return to its initial (unperturbed) state. Depending on the result of a particular experiment the damping and/or rate of evaporation is increased or decreased until a marginally critical state is found, and each black dot in Fig. 1 represents the convergence of such a process. There is an almost inverse relation between the critical values of the radiative damping and the evaporation rate; that is, higher values of radiative damping require a larger rate of evaporation to maintain excitability. The other important moisture parameter is the latent heat of vaporization, L v , and the dependence of kinetic energy, once the system reaches statistical equilibrium, is shown in Fig. 1c. The system is largely driven by the release of latent heat and consequently the kinetic energy increases approximately linearly with L v . Similar behavior occurs for a range of values of α from 10 to 100, and here we use α = 60, and there is no velocity dependence in the evaporation.
The evolution from the small perturbation is characterized by an initial spike in the energy (Fig. 1b), very common in excitable systems (e.g., figure 7.1 of Izhikevich (2007)), followed either by a return to the initial state or by self-sustained motion. A typical evolution in physical space from Figure 2: Snapshots of the height and precipitation fields at the times indicated, following an initial small perturbation at y = 0 and x = −0.3 (Units of x and y are 10 7 m.) The disturbance generates a front that propagates away from the disturbance before breaking up, but continuing in excitable, self-sustained motion. the initial state for an excitable state is shown in Fig. 2. A small localized perturbation is applied that triggers convection, and sends out a gravity wave, leading to more convection and precipitation and a propagating, precipitating front. Initially the front is nearly circular but later breaks up into smaller fronts that in turn propagate with no preferred direction (in the absence of differential rotation) and decay, each one triggering convection nearby. In the excitable regime the convection continues indefinitely, but in the stable regime the system slowly decays (typically on a timescale of tens of days), as in Fig. 1b.
Effects of Rotation
We now explore the effects of rotation on the system when it is in an excitable parameter regime. We first briefly discuss the case with constant rotation (i.e., flow on the f -plane) and then, in more detail, the case on a beta-plane. The absolute humidity (specifically q/q 0 − 1, white contours) and relative humidity q/q * (color filled contours), with darkest blue indicating saturation.
Flow on the f -plane
We apply a constant rotation of f = 1 × 10 −4 s −1 and the perform experiments otherwise identical to those shown in Fig. 2. The initial evolution is very similar, but after a few days it is quite different, for the planetary rotation mitigates against the organization of the convection. The reason is that the velocity induced by the condensational heat source tends to give rise to rotational motion around the heat source because of the tendency toward geostrophic balance, rather than convergence toward it. (The effect is well illustrated by moving the heat source away from the equator in the Matsuno-Gill problem, as illustrated in figure 8.14 of Vallis (2017).) This result should not be taken to mean that convective organization is impossible in the presence of rotation, since the omission of the nonlinear terms in the momentum equation means that shallow water cyclones do not form.
Flow on a beta plane
To understand flow on a beta plane we first consider the case with humidity as a passive tracer, with no latent heat release, before considering the case where humidity feeds back on the flow.
Humidity as a passive tracer
We set γ = 0 in (5) but otherwise keep the equations unaltered, so that moisture evaporates from a saturated surface, is advected by the flow and condenses upon saturation. We integrate the equations on an equatorial beta-plane with f = f 0 + βy, with f 0 = 0 and β = 2 × 10 −11 m −1 s −1 . The flow is forced by a static heating in the center of the domain of the form
Q = h 0 exp − x 2 + y 2 L 2 d /2 .(15)
where h 0 is a constant. The flow itself organizes into a familiar, steady, Matsuno-Gill-like pattern, illustrated in Fig. 3, with convergence very near to the center of the heating, depending slightly on the drag and relaxation parameters. Whereas moisture accumulates near the center of the heating, it is the relative humidity that determines whether condensation occurs and this is a function of the height field as well as moisture. The large-scale warm anomalies of the Rossby lobes west and slightly poleward of the heating, and the Kelvin waves east of the heating, inhibit precipitation except in regions close to the heating. Although there is a relative drying out of the equatorial region far to the east of the forcing, in the direct vicinity of the forcing the peak of relative humidity maximum is actually slightly to the east of the forcing center because of the asymmetric temperature distribution in the Matsuno-Gill pattern. This effect provides a mechanism for the eastward propagation of the pattern: the moisture convergence gives rise to condensation slightly east of the original heat source, so providing a new heat source. This in turn creates a pattern slightly east of the original one, and so on, thus providing a mechanism for eastward propagation. We now explore this further in a model with an active moisture tracer.
Humidity as an active tracer
On a beta-plane the initial evolution from a localized perturbation is similar to that in the non-rotating case shown in Fig. 2. At subsequent times rotational effects inhibit the aggregation of convection in mid-latitudes, but close to the equator the convection organizes itself. Condensation near the equator provides a heat source, which tries to generate a pattern similar to that of the left panel of Fig. 3, with moisture convergence maintaining the heat source. Unlike the case with no rotation, there is an east-west asymmetry, for two related reasons. First, the Matsuno-Gill pattern itself is asymmetric, bringing warm, moist air from the east. The associated convergence leads to condensation on the leading (i.e., eastern) edge of the existing condensational heat source, leading to the formation of a precipitating front, related to those described by Frierson et al. (2004) and Lambaerts et al. (2011). Second, close to the equator Kelvin waves are excited and these propagate east, triggering convection and generating more convection in the moist converging fluid just east of the initial disturbance, and so on. If the system is in an excitable regime the mechanism is self-sustaining and the precipitation front propagates eastward. A typical progression is shown in Fig. 4; this simulation has α = 60, γ = 15, τ r = 2 days, λ = 0.08 days −1 and β = 2 × 10 −11 m −1 s −1 . Simulations with resolutions from 100 km to 25 km show very similar behavior. The associated height pattern is qualitatively similar to a Matsuno-Gill pattern with a warm, leading Kelvin lobe eastward of the precipitating confined to the equator and Rossby lobes flanking and slightly westward of the precipitation, broadly consistent with the composite patterns seen in observations (Kiladis et al., 2005;Adames & Wallace, 2014). At Figure 4: Snapshots of the height and precipitation fields at the times indicated, in simulations on a beta plane. The main disturbance forms about 18 days after initialization and propagates eastward at about 6 m s −1 . Units of x and y are 10 7 m and the equatorial deformation radius is about 10 6 m. some times (e.g. at 30 days) the Kelvin lobe itself is flanked by geopotential disturbance of opposite sign, giving a quadrupole nature to the height field. The latitudinal width of the disturbance is of order the equatorial deformation radius, c/β, where c = √ gH, and which in the simulations shown is about 1000 km. For latitudes beyond this distance from the equator the convergence is too weak to generate an organized pattern, as in the f -plane simulations. The speed of the disturbance is not directly associated with the dry gravity wave speed, or even a moisture-modified gravity wave. Rather, it is associated with the time taken for the circulation to respond to a heat source: the disturbance cannot move so quickly that the Matsuno-Gill-like pattern cannot keep up with it and maintain the supply of water vapor to it. The strength of the pattern, and thus its timescale, is determined by the release of latent heat, and as Fig. 5 shows the speed increases as either the latent heat of condensation increases or as the efficiency of evaporation increases. The speed does not increase without bound, for as the effects of moisture increase the pattern formation becomes more irregular, and two or more precipitating patterns may form along the equator, each stealing moisture from the other and slowing the propagation.
Numerical experiments demonstrate that the robustness of the formation of a coherent equatorial structure and its eastward propagation is enhanced by the presence of a wind-evaporation feedback (Neelin et al., 1987;Emanuel et al., 1987). Specifically, suppose evaporation, E, is parameterized via a term of the form
E = λ |u + U | U 0 (q 0 − q),(16)
where u is the model produced wind and U is a constant, background wind due to, say, the trades. If U is directed westward, then evaporation is enhanced eastward of the disturbance and, if |U | ≈ 1 m s −1 (similar to wind speeds produced internally by the disturbance itself) the formation of an eastward propagating disturbance is enhanced. If U is eastward with a similar magnitude then MJO-like disturbances are eliminated. If simulations are performed with a nonzero value of U with no background rotation ( f 0 = β = 0) then the fronts that form preferentially move eastward or westward, depending on whether U is directed westward or eastward, respectively.
A well-known feature of the MJO is that it forms over the warm waters of the Indian Ocean, and that the next event formation will occur some 30-60 days later. It is, in fact, a common feature of an excitable system that it cannot support the passage of a second disturbance over a given location until sufficient time -the 'refractory period' -has passed (Izhikevich, 2007), and this is also a property of our model. To demonstrate this we impose a spatially varying distribution of surface humidity, as one would find with a varying sea-surface temperature anomaly, as in the left panel of Fig. 6. The right panel shows the position of the main disturbance, as determined by the location of the maximum of the precipitation divergence in the velocity field. Disturbances typically form only in the vicinity of the maximum surface humidity and propagate east, decaying where the surface humidity is too low. A second disturbance cannot form until the first disturbance is sufficiently distant from the genesis location, because the first disturbance leaves a wake of dry air that needs sufficient time to reform into a converging pattern and initiate the moisture feedback. In the simulation shown the reformulation time is roughly 20-40 days, the timescale associated with the decay of the disturbance as it moves east. Even without a previous MJO disturbance the atmosphere typically takes many days to organize itself into a quasi-steady state in response to a localized heating (Heckley & Gill, 1984).
Discussion and Conclusions
We have presented a simple, explicit model that reproduces many of the main features of intraseasonal variability and the Madden-Julian oscillation in the tropical atmosphere, and described the mechanism that causes these features. By simple and explicit we mean that we treat the atmosphere as a single baroclinic mode in the vertical and we do not use any convective parameterization (except a rather basic one), or make additional approximations about the scale and nature of the system; rather, we directly solve the resulting equations of motion, which consist simply of the shallow water equations plus a humidity variable. Condensational heat release affects the height field (a proxy for temperature), which induces a velocity field, leading to more convection and so on. If the condensational effects are sufficiently strong, as determined by the latent heat of evaporation and the efficiency of evaporation from the surface, and the radiative damping appropriately weak, the system is excitable and self-sustained motion ensues. Excitable behavior occurs over a wide range of physically reasonable parameters and in this regime the motion -which can develop quite large scales -is driven by the condensational heating at small scales, without the need for any large-scale instabilities.
When integrated on the beta plane the motion in the tropics becomes organized. In a background state that is marginally stable convection aggregates around the equator creating a Matsuno-Gill-like pattern with a scale determined by the equatorial deformation radius. However, the pattern is unstable in the sense that the combination of moist convergence and gravity waves triggers convection nearby and the disturbance propagates, preferentially in an eastward direction. The directionality arises because the pattern draws warm moist air in from the east along the equator, and this air becomes conditionally unstable. Convection is triggered at the eastern edge of the existing disturbance and the whole system then propagates east, as sketched in Fig. 7. Note that the moisture is not advected eastward in the disturbance; rather, it has an evaporative source and is drawn in from the east. The large-scale disturbance may be surrounded by a collection of smaller-scale convective events, and these have no preferred direction because the pressure field they induce is more nearly isotropic. Although simple in construct, the model displays some of the key observed features of the Madden-Julian oscillation, notably: 1. A predominantly dipolar (although at times quadrupolar) Matsuno-Gill-like pattern in the height field, with a Kelvin lobe extending east, flanked by two off-equatorial Rossby lobes. The size of the structure depends on the equatorial radius of deformation.
2. The whole pattern moves eastward at a speed of a few meters per second. The speed is largely determined by the levels of moisture in the system and by the magnitude of the latent heat of condensation, which determine the strength of the pattern and the time it can take to form and move itself.
3. The pattern preferentially forms where there is greater availability of moisture, corresponding to a higher sea surface temperature. If a pattern forms and moves east from that location then an interval of order a few tens of days must pass before a second disturbance can form in the dry wake of the first.
None of these features are built-in to the model; rather, they are all emergent properties. The presence of a wind-induced evaporation, or WISHE, also has an noticeable effect on its eastward propagation, as in Khairoutdinov & Emanuel (2018). Since the energy source is, ultimately, evaporation from the Warm moist air is then drawn in from the east but this is convectively unstable, amenable to triggering by the eastward propagating gravity waves from initial disturbance, and new convection forms on the eastern edge of the original location. The whole pattern then moves unsteadily eastward.
surface, a higher SST (i.e., a higher level of surface humidity) produces a more energetic simulation, as in , and is responsible for the localization of the genesis region. A rather less realistic feature of the model is that the moisture convergence and feedback onto the height field lead to the formation of fronts that are too narrow in the along-propagation direction, compared to observations of composite MJOs. However, both observations from TRMM and high resolution three-dimensional simulations do show that the region of intense precipitation tends to become meridionally extended and zonally confined as the MJO enters the western Pacific (Liu et al., 2009). It is not surprising that the model cannot capture the details of the precipitation distribution but the larger scale features appear broadly consistent with observations. Tight frontal formation is in fact commonly found in idealized moist systems (Yano et al., 1995;Frierson et al., 2004;Lambaerts et al., 2011), and the gridpoint generation of storms and fronts are a common feature of moist GCMs without a convective parameterization. The model also suggests why many GCMs are unable to produce MJO-like variability, even as some cloud permitting models with no convective parameterization but at similar resolution are able to do so (Khairoutdinov & Emanuel, 2018). The mechanism itself is not especially sensitive to resolution, but it does require that the system be excitable. Now, cloud-resolving models in a statistical radiative-convective equilibrium over a uniform SST are almost certainly excitable systems (even if not normally described as such), but a convective parameterization that distributes convective effects more smoothly will perforce diminish the gravity wave generation and that excitability.
Certainly, the nature of the convective parameterization does affect the production of an MJO in a GCM (Benedict et al., 2014), and the explicit introduction of high frequency variability may be beneficial, as in Deng et al. (2015). Relatedly, many GCMs are unable to produce the quasi-biennial oscillation (QBO) without a gravity wave drag parameterization, in part because the upwardly propagating Kelvin and Rossby waves are not properly simulated. The difference is that in the QBO the gravity waves propagate up toward the maximum zonal wind, drawing its maximum down, whereas in the MJO the disturbance itself is the source of the gravity waves and the flow convergence, moving the system east.
The interaction of the moisture field with the pressure and wind fields is essential to the mechanism we have presented here, as it is in the moisture-mode theories discussed in the introduction. The MJO we envision is not, however, that of a single large-scale coupled mode; rather, the large-scale structure and propagation arise because the tropical atmosphere is an unstable system living on the beta-plane. Travelling disturbances and pattern formation are common in excitable systems (Meron, 1992) and moist radiative-convective systems appear to be no exception. The beta-plane dynamics add an extra twist and lead to aggregation at the equator, with warm, moist air drawn from the east toward the disturbance. New convection is then triggered on its eastern flank and the whole pattern bootstraps its way east.
The mechanisms described above seem consistent with the evolution occurring in some more comprehensive 3-D models (Nasuno et al., 2009;Khairoutdinov & Emanuel, 2018), albeit with differences in structure and detail. Nevertheless, the respective importance of the processes at work in the various models is unclear and achieving a better understanding of the connection between them, and the real atmosphere, remains a topic for future work. The ENSO system may be another example of an excitable geophysical system, with a refractory period of a few years and excited by westerly wind bursts, but that too remains to be investigated.
Figure 3 :
3Response to a steady heating centered at the origin on an equatorial beta-plane, where moisture is a passive tracer. (a) The height field (color filled contours) and wind vectors (arrows). (b)
Figure 5 :
5Speed of the main, MJO-like, eastward moving disturbance as function of the evaporation coefficient λ and the latent heat of condensation, γ. Where γ varies the value of λ is fixed at 0.08 and when λ varies the value of γ is 15.
Figure 6 :
6(a) Sea-surface humidity, representing a warm pool in the western equatorial region with a fraction humidity about 25% above a base level (b) Position of the maximum precipitation at the equator, with thicker red lines indicating where the precipitation is associated with MJO-like activity. MJO events form only in the warm western region and decay before reaching the eastern edge, and here recur every 20-40 days.
Figure 7 :
7Schematic of an eastward propagating equatorial disturbance. Convection at the equator gives rise to a feedback producing convective aggregation and a modified Matsuno-Gill-like pattern.
AcknowledgementsThis work was funded by the Leverhulme Trust, NERC and the Newton Fund. I thank Professor Pete Ashwin for a very useful conversation about excitable systems and Professors Mat Collins and John Thuburn for comments about the MJO and possible ENSO relevance.
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[] |
[
"ROUGH CONVERGENCE OF SEQUENCES IN A CONE METRIC SPACE",
"ROUGH CONVERGENCE OF SEQUENCES IN A CONE METRIC SPACE"
] |
[
"Amar Kumar Banerjee ",
"Rahul Mondal "
] |
[] |
[] |
Here we have introduced the idea of rough convergence of sequences in a cone metric space. Also it has been investigated how far several basic properties of rough convergence as valid in a normed linear space are affected in a cone metric space.
|
10.1007/s41478-019-00168-2
|
[
"https://arxiv.org/pdf/1805.10257v1.pdf"
] | 54,805,453 |
1805.10257
|
4cac3aa1085167613d5929497e72583fd1920c1c
|
ROUGH CONVERGENCE OF SEQUENCES IN A CONE METRIC SPACE
25 May 2018
Amar Kumar Banerjee
Rahul Mondal
ROUGH CONVERGENCE OF SEQUENCES IN A CONE METRIC SPACE
25 May 2018
Here we have introduced the idea of rough convergence of sequences in a cone metric space. Also it has been investigated how far several basic properties of rough convergence as valid in a normed linear space are affected in a cone metric space.
Introduction
The idea of cone metric spaces was introduced by Huang and Zhang [5] as a generalization of metric spaces. The distance d(x, y) between two elements x and y in a cone metric space X is defined to be a vector in a ordered Banach space E, where the order relation in the Banach space E is given by using the idea of cone.
In 2001 the notion of rough convergence of sequences was introduced in a normed linear space by Phu [8]. Phu discussed about the implication of rough convergence and the relation between ordinary convergence and rough convergence. There he also introduced the notion of rough Cauchy sequences. In 2003 Phu [9] discussed rough convergence of sequences in an infinite dimensional normed linear space as an extension of [8]. The idea of rough statistical convergence was given by Ayter [1] in 2008. Several works has been done in different direction [2,6,7] by many authors using the idea given by Phu [8].
In our work our motivation is to discuss the idea rough convergence of sequences in a cone metric space. But the structure of ordering associated of a cone metric space is the main problem to discuss this particular idea. However we have found out some basic properties of sequences regarding rough convergence in a cone metric space.
Preliminaries
Definition 2.1. [8] Let {x n } be a sequence in a normed linear space (X, . ), and r be a nonnegative real number. Then {x n } is said to be r-convergent to x if for any ǫ > 0, there exists a natural number k such that x n − x < r + ǫ for all n ≥ k.
Definition 2.2. [5] Let E be a real Banach space and P be a subset of E. Then P is called a cone if and only if (i) P is closed nonempty, and P = {0} 0 2010 AMS Subject Classification: 40A05, 40A99. Key words and phrases: Rough convergence, rough limit point, rough limit set, cone, cone metric space.
(ii) a, b ∈ R, a, b ≥ 0, x, y ∈ P implies ax + by ∈ P .
(iii) x ∈ P and −x ∈ P implies x = 0.
Let E be a real Banach space and P be a cone in E. Let us use the partial ordering [5] with respect to P by x ≤ y if and only if y − x ∈ P . We shall write x < y to indicate that x ≤ y but
x = y.
Also by x << y, we mean y − x ∈ intP , the interior of P. The cone P is called normal if there is a number K > 0 such that for all x, y ∈ E, 0 ≤ x ≤ y implies ||x|| ≤ K||y||.
Definition 2.3. [5]
Let X be a non empty set. If the mapping d : X × X −→ E satisfies the following three conditions (d1) 0 ≤ d(x, y) for all x, y ∈ X and d(x, y) = 0 if and only if x = y;
(d2) d(x, y) = d(y, x) for all x, y ∈ X; (d3) d(x, y) ≤ d(x, z) + d(z, y)
for all x, y, z ∈ X; then d is called a cone metric on X, and (X, d) is called a cone metric space.
It is clear that a cone metric space is a generalization of metric spaces. Throughout (X, d) or simply X stands for a cone metric space which is associated with a real Banach space E with a cone P , R for the set of all real numbers, N for the set of all natural numbers, Sets are always subsets of X unless otherwise stated.
Definition 2.4. [5] Let (X, d) be a cone metric space. A sequence {x n } in X is said to be convergent to x ∈ X if for every c ∈ E with 0 << c there is k ∈ N such that d(x n , x) << c, whenever for all n > k.
We now prove the following results which will be needed in the sequel.
Theorem 2.1. Let E be a real Banach space with cone P . If x 0 ∈ intP and c(> 0) ∈ R then cx 0 ∈ intP .
Proof. Let x 0 ∈ intP . So there exists an open set U such that x 0 ∈ U ⊂ P . Since by definition
x, y ∈ P and a, b ≥ 0 implies ax + by ∈ P , it follows that cU ⊂ P for any real c ≥ 0. Therefore cx 0 ∈ cU ⊂ P . Now cU is open and hence cx 0 ∈ intP .
Theorem 2.2. Let E be a real Banach space and P be a cone in E. If x 0 ∈ P and y 0 ∈ intP
then x 0 + y 0 ∈ intP .
Proof. Let us consider the mapping f : E −→ E be defined by f (x) = x + x 0 for all x ∈ E.
Clearly f being a translation operator is a homeomorphism. Now since y 0 ∈ intP , there exists an open set U such that y 0 ∈ U ⊂ P . Also f (U) is open and hence x 0 + U is open. Also by
definition x 0 + U ⊂ P . Therefore x 0 + y 0 ∈ x 0 + U ⊂ P . So x 0 + y 0 ∈ intP . Corollary 2.1. If x 0 , y 0 ∈ intP then x 0 + y 0 ∈ intP .
The following theorem is widely known. Theorem 2.3. A real normed linear space is always connected.
With the help of above result we can deduce the following theorem.
Theorem 2.4. Let E be a real Banach space with cone P . Then 0 / ∈ intP .
Proof. If possible let 0 ∈ intP . Then for any x ∈ P we have 0 + x ∈ intP and hence x ∈ intP .
Hence every element of P is an interior point P and consequently P becomes an open set. Now P is a non empty set which is both open and closed, This contradicts to the fact that E is connected.
3. Rough convergence in a cone metric space Definition 3.1. [8] Let (X, d) be a cone metric space. A sequence {x n } in X is said to be r-convergent to x for some r ∈ E with 0 << r or r = 0 if for every ǫ with (0 <<)ǫ there exists a k ∈ N such that d(x n , x) << r + ǫ for all n ≥ k.
Usually we denote it by x n r → x, r is said to be the roughness degree of rough convergence of {x n }. It should be noted that when r = 0 the rough convergence becomes the classical convergence of sequences in a cone metric space. If {x n } is r-convergent to x, then x is said to be a r-limit point of {x n }. From the next example we can observe that r-limit point of a sequence {x n } may not be unique. For some r as defined above, the set of all r-limit points of a sequence {x n } is said to be the r-limit set of the sequence {x n } and we will denote it by
LIM r x n . Therefore we can say LIM r x n = x 0 ∈ X : x n r → x 0 . Example 3.1. Let X = R 2 and E = R 2 with P = {(x, y) ∈ E : x, y ≥ 0}. Now let us define d : X × X −→ E by d(x, y) = ( x − y , x − y ) where x − y = (x 1 − y 1 ) 2 + (x 2 − y 2 ) 2 for x = (x 1 , y 1 ) ∈ R 2 and y = (x 2 , y 2 ) ∈ R 2 . Clearly (X, d)
is a cone metric space with P as cone and which is not a metric space. Now let us consider a sequence {x n } in X where x n = (1, 1) if n is odd and x n = (2, 2)
if n is even. We will show that {x n } is not convergent in the usual sense of cone metric spaces. Clearly {x n } can not converge to (1, 1) because choosing ǫ = ( 1 2 , 1 2 ) we have infinitely many natural numbers that is the even natural numbers for which d(x n , x) << ǫ does not
hold, where x = (1, 1). Similarly we can show that {x n } can not converge to (2, 2). Now if we suppose that {x n } converges to p = (x 0 , y 0 ) in X, where p = (1, 1) and p = (2, 2). Let (x 0 − 1) 2 + (y 0 − 1) 2 = c 1 and (x 0 − 2) 2 + (y 0 − 2) 2 = c 2 .
We denote the minimum of c 1 and c 2 by c and let ǫ = (c, c). Then clearly 0 << ǫ, but there exists infinitely many terms of the sequence for which the relation d(x n , p) << ǫ does not hold.
If we consider r = ( √ 2, √ 2) then {x n } is r-convergent to s = (1, 1). Because d(x n , s) equals to ( √ 2, √ 2)
if n is even and d(x n , s) equals to (0, 0) if n is odd. Hence d(x n , s) << r + ǫ for all n ∈ N and for every (0 <<)ǫ ∈ E.
We now recall the definition of boundedness [3] of a sequence in a metric space as follows.
A sequence {x n } in a metric space (X, d) is said to be bounded if for any fixed a ∈ X, there
exists r > 0 such that d(x n , a) < r for all n ∈ N. So d(x n , x m ) ≤ d(x n , a) + d(x m , a) < 2r for all n, m ∈ N.
Using this idea we define boundedness of a sequence in a cone metric space as follows.
Definition 3.2.(cf.[3]) A sequence {x n } in (X, d) is said to be bounded if there exists a (0 << )k ∈ E such that d(x m , x n ) << k for all m, n ∈ N.
As in the case of a normed linear space it is also true in a cone metric space that if a sequence is bounded then the r-limit set of this sequence is non empty for some r as defined above. The following theorem has been given in evidence of that.
Theorem 3.1. If a sequence {x n } in (X, d) is bounded then LIM r x n = φ for some (0 <<)r ∈ E.
Proof. Since {x n } is bounded there exists a (0 <<)p such that d(x n , x m ) << p for all m, n ∈ N.
Hence p − d(x n , x m ) ∈ intP for all m, n ∈ N. Therefore by theorem 2.2 for every (0 <<)ǫ ∈ E we have [p − d(x n , x m )] + ǫ ∈ intP for all m, n ∈ N. Hence d(x n , x m ) << p + ǫ for all m, n ∈ N.
So for any k ∈ N, d(x n , x k ) << p + ǫ for all n ∈ N. Therefore {x n } is p-convergent to x k for all
k ∈ N. So LIM p x n = φ.
It should be noted that in the above theorem we can use any (0 <<)r ∈ E with p << r to
show that LIM r x n = φ.
Remark 3.1. In the example 3.1 let us consider s = (1, 1) and s ′ = (2, 2). Then for
r = ( 1 2 √ 2 , 1 2 √ 2 ), LIM r x n = φ. Because if we suppose {x n } is r-convergent to a then for every (0 <<)ǫ ∈ E there exists a k ∈ N such that d(x n , a) << r + ǫ for all n ≥ k. Let ǫ = ( 1 8 √ 2 , 1 8 √ 2 ), so r + ǫ = ( 5 8 √ 2 , 5 8 √ 2 )
. Since x n = s if n is odd and x n = s ′ if n is even and d(x n , a) << r + ǫ for all n ≥ k, clearly d(s, a) << r + ǫ and d(s ′ , a) << r + ǫ. Now
d(s, s ′ ) ≤ d(s, a) + d(s ′ , a) and hence d(s, s ′ ) << 2(r + ǫ) that is d(s, s ′ ) << ( 5 4 √ 2 , 5 4 √ 2 ), which is a contradiction as d(s, s ′ ) = ( √ 2,√ 2)
. Therefore LIM r x n = φ.
In our previous theorem we have discussed about the choice of r for which a bounded sequence in a cone metric space is r-convergent. Proof. Let {x n } be a r-convergent sequence in (X, d) and r-convergent to x. So for every x). Clearly 0 ≤ M and using theorem 2.2 we have 0 << M + ǫ. For n < k,
(0 <<)ǫ ∈ E, there exists a k ∈ N such that d(x n , x) << r + ǫ for all n ≥ k −→ ( * ). Let M = k n=1 d(x n ,we have M − d(x n , x) ∈ P . Again since 0 << r + ǫ, we have [M − d(x n , x)] + (r + ǫ) ∈ intP , that is, (M + r + ǫ) − d(x n , x) ∈ intP . Hence d(x n , x) << (M + r + ǫ) for all n < k −→ (i).
We have by ( * ), (r + ǫ) − d(x n , x) ∈ intP for all n ≥ k and also M ∈ P . Hence M + [(r + ǫ) − d(x n , x)] ∈ intP for all n ≥ k. Therefore, d(x n , x) << M + (r + ǫ) for all n ≥ k −→ (ii). Hence from (i) and (ii) we can write d(x n , x) << M + (r + ǫ) for all n ∈ N. Proof. Let (0 <<)ǫ be pre-assigned. Since {y n } converges to y there exists a k ∈ N such that d(y n , y) << ǫ for all n ≥ k and hence ǫ − d(y n , y) ∈ intP for all n ≥ k. Also for all n ≥ k, we have d(x n , y n ) ≤ r, so r − d(x n , y n ) ∈ P for all n ≥ k. Therefore by theorem 2.2 (r + ǫ) − [d(x n , y n ) + d(y n , y)] ∈ intP for all n ≥ k. Again since d(x n , y) ≤ d(x n , y n ) + d(y n , y) for all n ∈ N, [d(x n , y n ) + d(y n , y)] − d(x n , y) ∈ P for all n ∈ N. So for all n ≥ k we have [(r + ǫ) − (d(x n , y n ) + d(y n , y))] + [(d(x n , y n ) + d(y n , y)) − d(x n , y)] = (r + ǫ) − d(x n , y) ∈ intP for all n ≥ k. Therefore d(x n , y) << r + ǫ for all n ≥ k and hence {x n } r-converges to y.
Now for i, j ∈ N, we have d(x i , x j ) ≤ d(x i , x)+d(x j , x), that is, [d(x i , x)+d(x j , x)]−d(x i , x j ) ∈ P −→ (iii). Also (M + r + ǫ) − d(x i , x) ∈ intP and (M + r + ǫ) − d(x j , x) ∈ intP and hence, 2(M + r + ǫ) − [d(x i , x) + d(x j , x)] ∈ intP −→ (iv). From (iii) and (iv) we have 2(M + r + ǫ) − d(x i , x j ) ∈ intP and so d(x i , x j ) << 2(M + r + ǫ). Hence {x n } is bounded.
It has been seen in [8] that the diameter of a r-limit set of a sequence in a normed linear space is not greater then 2r. We find out similar kind of property of a sequence in a cone metric space regarding rough convergence as follows. Proof. If possible let there exists elements y, z ∈ LIM r x n such that mr < d(y, z) and m > 2.
Let (0 <<)ǫ be arbitrary. Since y, z ∈ LIM r x n , there exists a i ∈ N such that d(x i , y) << r + ǫ 2 , that is (r+ ǫ 2 )−d(x i , y) ∈ intP −→ (i) and d(x i , z) << r+ ǫ 2 , that is (r+ ǫ 2 )−d(x i , z) ∈ intP −→ (ii). Hence from (i) and (ii) we can write (2r + ǫ) − [d(x i , y) + d(x i , z)] ∈ intP −→ (iii). Now d(y, z) ≤ d(x i , y) + d(x i , z), hence [d(x i , y) + d(x i , z)] − d(y, z) ∈ intP −→ (iv)
. So from (iii) and (iv), (2r + ǫ) − d(y, z) ∈ intP −→ (v). Again d(y, z) − mr ∈ P −→ (vi). Hence from (v) and (vi) we can write (2r + ǫ) − mr ∈ intP that is ǫ − r(m − 2) ∈ intP . This is true for any 0 << ǫ. So choosing ǫ = r(m − 2), we have 0 ∈ intP . This is a contradiction and the result follows.
Theorem 3.5. Let {x n } be r 1 -convergent to x in (X, d). Then {x n } is also r 2 -convergent to x in (X, d) for r 1 < r 2 .
The proof is obvious and so is omitted.
Corollary 3.1. Let {x n } be r 1 -convergent to x in (X, d) and r 1 < r 2 for some 0 << r 2 . Then For 0 << r and a fixed y ∈ X, we define the sets B r (y) and B r (y), the closed and open spheres respectively centred at y with radius r as follows:
LIM r 1 x n ⊂ LIM r 2 x n .
B r (y) = {x ∈ X : d(x, y) ≤ r} and B r (y) = {x ∈ X : d(x, y) << r}.
Theorem 3.6. Let (X, d) be a cone metric space, c ∈ X and 0 << r be such that for any
x ∈ X, either d(x, c) ≤ r or r << d(x, c). If c is a cluster point of a sequence {x n } then LIM r x n ⊂ B r (c).
Proof. Let y ∈ LIM r x n but y / ∈ B r (c). So r << d(y, c). Let ǫ ′ = d(y, c) − r(∈ intP ) and so d(y, c) = r + ǫ ′ , where 0 << ǫ ′ . Let ǫ = ǫ ′ 2 and so we can write d(y, c) = r + 2ǫ. Then B r+ǫ (y)∩B ǫ (c) = φ. Otherwise if p ∈ B r+ǫ (y)∩B ǫ (c) then d(p, y) << r +ǫ and d(p, c) << ǫ and
hence (r + ǫ) − d(p, y) ∈ intP and ǫ − d(p, c) ∈ intP . So we have (r + 2ǫ) − [d(p, y) + d(p, c)] ∈ intP −→ (i). Again d(y, c) ≤ d(y, p) + d(p, c) that is [d(y, p) + d(p, c)] − d(y, c) ∈ P −→ (ii).
Hence from (i) and (ii) we can write (r + 2ǫ) − d(y, c) = 0 ∈ intP , which is a contradiction as 0 / ∈ intP . Therefore B r+ǫ (y) ∩ B ǫ (c) = φ. But since y ∈ LIM r x n , for 0 << ǫ there exists a k 0 ∈ N such that d(x n , y) << r + ǫ for all n ≥ k 0 . Again since c is a cluster point of {x n }, for 0 << ǫ and for k 0 ∈ N, there exists a k 1 ∈ N, with k 1 > k 0 such that d(x k 1 , c) << ǫ. So
x k 1 ∈ B ǫ (c)
. Also d(x k 1 , y) << r + ǫ. So x k 1 ∈ B r+ǫ (y). Thus x k 1 ∈ B r+ǫ (y) ∩ B ǫ (c). Which is a contradiction . Hence y ∈ B r (c).
Theorem 3.7. Let {x n } be a sequences in (X, d) and {y n } be a convergent sequence in LIM r x n converging to y 0 . Then y 0 must belongs to LIM r x n .
Proof. Let (0 <<)ǫ be preassigned. Since {y n } converges to y 0 , for (0 <<)ǫ there exists a k 1 ∈ N such that d(y n , y 0 ) << ǫ 2 for all n ≥ k 1 −→ (i). Now let us choose y m ∈ LIM r x n with m > k 1 . Then there exists a k 2 ∈ N such that d(x n , y m ) << r+ ǫ 2 for all n ≥ k 2 −→ (ii). Also for all n ∈ N we have, d(x n , y 0 ) ≤ d(x n , y m ) + d(y m , y 0 ). Hence we have [d(x n , y m ) + d(y m , y 0 )] − d(x n , y 0 ) ∈ P for all n ∈ N −→ (iii). Since m > k 1 , by (i) we can write ǫ 2 − d(y m , y 0 ) ∈ intP −→ (iv) and for all n ≥ k 2 from (ii) we have (r + ǫ 2 ) − d(x n , y m ) ∈ intP −→ (v). Therefore for all n ≥ k 2 , by (iv) and (v) we can write (r + ǫ) − [d(x n , y m + d(y m , y 0 )] ∈ intP −→ (vi). Hence for all n ≥ k 2 , from (iii) and (vi) we have (r + ǫ) − d(x n , y 0 ) ∈ intP , that is d(x n , y 0 ) << (r + ǫ) for all n ≥ k 2 . Hence y 0 ∈ LIM r x n . Proof. Let {x n } be r-convergent to x and (0 <<)ǫ be preassigned. Then for 0 << ǫ there exist k 1 , k 2 ∈ N such that d(x n , x) << r + ǫ 2 for all n ≥ k 1 and d(x n , y n ) ≤ ǫ 2 for all n ≥ k 2 . Let k > max {k 1 , k 2 }. Then for all n ≥ k we have d(x n , x) << r + ǫ 2 and d(x n , y n ) ≤ ǫ 2 , that is we have
(r + ǫ 2 ) − d(x n , x) ∈ intP for all n ≥ k −→ (i) and [ ǫ 2 − d(x n , y n )] ∈ P for all n ≥ k −→ (ii)
Again for all n ∈ N we can write d(y n , x) ≤ d(x n , y n ) + d(x n , x). Hence we have [d(x n , y n ) + d(x n , x)] − d(y n , x) ∈ P for all n ∈ N −→ (iii). Therefore from (i), (ii) and (iii) it follows that (r + ǫ) − d(y n , x) ∈ intP for all n ≥ k and hence d(y n , x) << r + ǫ for all n ≥ k. Therefore Proof. Let y ∈ LIM r x n and (0 <<)ǫ be arbitrary. Then there exists a m ∈ N such that d(x n , y) << r + ǫ for all n ≥ m. Let n p > m for some p ∈ N. Then n k > m for all k ≥ p.
Therefore d(x n k , y) << r + ǫ for all k > p. Hence y ∈ LIM r x n k .
Theorem 3.10. Let C be the set of all cluster points of a sequence {x n } in (X, d). Also let 0 << r be such that for any x ∈ X either d(x, c) ≤ r or r << d(x, c) for each c ∈ C. Then
LIM r x n ⊂ c∈C B r (c) ⊂ x 0 ∈ X : C ⊂ B r (x 0 ) .
Proof. By theorem 3.6 we can say that LIM r x n ⊂ c∈C B r (c). Now let y ∈ c∈C B r (c). So y ∈ B r (c) for each c ∈ C and hence d(y, c) ≤ r for each c ∈ C. This implies that c ∈ B r (y) for each c ∈ C. Therefore C ⊂ B r (y). So c∈C B r (c) ⊂ x 0 ∈ X : C ⊂ B r (x 0 ) . Therefore we have
LIM r x n ⊂ c∈C B r (c) ⊂ x 0 ∈ X : C ⊂ B r (x 0 ) .
Lemma 3.1. Let (X, d) be a cone metric space, where P is a normal cone with normal constant k. Then for every ǫ > 0, we can choose c ∈ E with c ∈ intP and k c < ǫ.
Proof. Let ǫ > 0 be given and let x ∈ intP . We consider c =
x. ǫ 2
x .k . Clearly c ∈ intP and also c = ǫ 2.k < ǫ k . Hence k c < ǫ.
Lemma 3.2. Let (X, d) be a cone metric space, P be a normal cone with normal constant k.
Then for each c ∈ E with 0 << c, there is a δ > 0, such that x < δ implies c − x ∈ intP .
Proof. Since 0 << c, c ∈ intP and so there exists a δ > 0, such that B δ (c) = {x ∈ E : Proof. First suppose that {x n } is r-convergent to x. Let ǫ > 0 be preassigned. Then we have an element c ∈ E with 0 << c and k c < ǫ. Now for 0 << c there exists a k 1 ∈ N such that d(x n , x) << r + c for all n ≥ k 1 . Hence d(x n , x) − r << c for all n ≥ k 1 . Now d(x n , x) − r ≤ k c < ǫ for all n ≥ k 1 . Therefore {d(x n , x) − r} converges to 0 in E.
x − c < δ} ⊂ P . Now x < δ implies c − (c − x) < δ. Hence c − x ∈ B δ (c) and therefore c − x ∈ intP .
Conversely let {d(x n , x) − r} converges to 0 in E. For any c ∈ E with 0 << c, there is a δ > 0,
such that x < δ implies c−x ∈ intP . For this δ there is a k 1 ∈ N, such that d(x n , x) − r < δ for all n ≥ k 1 . So c − [d(x n , x) − r] ∈ intP for all n ≥ k 1 . Hence d(x n , x) << r + c for all n ≥ k 1 . Therefore {x n } in (X, d) is r-convergent to x.
Theorem 3.12. Let (X, d) be a cone metric space with normal cone P and normal constant k. If {x n } and {y n } be two sequences 1 4k+2 r-convergent to x and y respectively in X, then the sequence {z n } is r -convergent to d(x, y) where 0 << r and z n = d(x n , y n ) for all n ∈ N.
Proof. Let ǫ > 0 be preassigned and let x ∈ intP . Then c = x (4k+2) ∈ intP and c < ǫ 4k+2 . Since {x n } and {y n } are 1 4k+2 r-convergent to x and y respectively, for 0 << c, there exists a k ∈ N such that d(x n , x) << c + r 4k+2 and d(y n , y) << c + r 4k+2 for all n ≥ k. Therefore we have (c + r 4k+2 ) − d(x n , x) ∈ intP and (c + r 4k+2 ) − d(y n , y) ∈ intP for all n ≥ k. Hence 2(c + r 4k+2 ) − [d(x n , x) + d(y n , y)] ∈ intP for all n ≥ k. Now denoting r 4k+2 by r ′ , we have 2(c + r ′ ) − [d(x n , x) + d(y n , y)] ∈ intP for all n ≥ k −→ (i).
Now d(x, y) ≤ d(x n , x) + d(x n , y) for all n ∈ N, that is [d(x n , x) + d(x n , y)] − d(x, y) ∈ P for all n ∈ N −→ (A). Also d(x n , y) ≤ d(x n , y n ) + d(y n , y) for all n ∈ N. Hence Hence by normality of P we can write 2(c + r ′ ) + [d(x, y) − d(x n , y n )] ≤ k 4(c + r ′ ) for all n ≥ k −→ (v). Now d(x, y) − d(x n , y n ) = d(x, y) − d(x n , y n ) + 2(c + r ′ ) − 2(c + r ′ ) ≤ d(x, y) − d(x n , y n ) + 2(c + r ′ ) + 2(c + r ′ ) for all n ∈ N −→ (vi). From (v) and (vi) we have d(x, y) − d(x n , y n ) ≤ 4k (c + r ′ ) + 2 (c + r ′ ) for all n ≥ k. Also 4k (c + r ′ ) + 2 (c + r ′ ) = (4k+2) (c + r ′ ) ≤ (4k+2) c +(4k+2) r ′ < ǫ+ r . Hence d(x n , y n ) − d(x, y) < ǫ + r for all n ≥ k.
[9] H. X. Phu
Theorem 3. 2 .
2Every r-convergent sequence in a cone metric space (X, d) is bounded.
Theorem 3. 3 .
3Let {x n } and {y n } be two sequences in a cone metric space (X, d) and let {y n } converges to y ∈ X. If there exists a (0 <<)r ∈ E such that d(x i , y i ) ≤ r for all i ∈ N. Then {x n } r-converges to y.
Theorem 3 . 4 .
34Let {x n } be a sequence in (X, d). Then there does not exists elements y, z in LIM r x n such that mr < d(x, z), where m is a real number greater then 2.
Definition 3 . 3 .
33Let {x n } be a sequence in a cone metric space (X, d), a point c ∈ X is said to be a cluster point of {x n } if for every (0 <<)ǫ ∈ E and for every k ∈ N, there exists a k 1 ∈ N such that k 1 > k with d(x k 1 , c) << ǫ.
Theorem 3 . 8 .
38Let {x n } and {y n } be two sequences in (X, d). If for every (0 <<)ǫ there exists a k ∈ N such that d(x n , y n ) ≤ ǫ for all n ≥ k. Then {x n } is r-convergent to x if and only if {y n } is r-convergent to x.
{y n } is r-convergent to x. Interchanging the roll of {x n } and {y n } it can be shown that if {y n } is r-convergent to x then {x n } is r-convergent to x. Theorem 3.9. Let {x n k } be a sub sequence of {x n } then LIM r x n ⊂ LIM r x n k .
Theorem 3 . 11 .
311Let (X, d) be a cone metric space, P be a normal cone with normal constant k. Then a sequence {x n } in (X, d) is r-convergent to x if and only if {d(x n , x) − r} converges to 0 in E.
[d(x n , y n ) + d(y n , y)] − d(x n , y) ∈ P for all n ∈ N −→ (B). From (A) and (B) we have[d(x n , x) + d(y n , y) + d(x n , y n )] − d(x, y) ∈ P for all n ∈ N −→ (ii). Again d(x n , y n ) ≤ d(x n , x) + d(y n , y) + d(x, y) for all n ∈ N. So [d(x n , x) + d(y n , y) + d(x, y)] − d(x n , y n ) ∈ P for all n ∈ N −→ (iii). Now from (i) and (iii) we can write 2(c + r ′ ) + [d(x, y) − d(x n , y n )] ∈ intP for all n ≥ k −→ (iv). Now from (i) and (ii) we have 2(c + r ′ ) + [d(x n , y n ) − d(x, y)] ∈ intP for all n ≥ k. Hence 4(c + r ′ ) − [2(c + r ′ ) + d(x, y) − d(x n , y n )] ∈ intP for all n ≥ k, that is [2(c + r ′ ) + d(x, y) − d(x n , y n )] << 4(c + r ′ ) for all n ≥ k. Now by (iv) we can write 0 << 2(c + r ′ ) + [d(x, y) − d(x n , y n )]for all n ≥ k. Also we have 0 << 4(c + r ′ ).
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The University of Burdwan,
Golapbag, Burdwan-713104,
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Rough convergence in normed linear spaces. H X Phu, Numer. Funct. Anal. and Optimiz. 22H. X. Phu, Rough convergence in normed linear spaces, Numer. Funct. Anal. and Optimiz, 22: (2001), 199-222.
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[] |
[
"Could radical pairs play a role in xenon-induced general anesthesia?",
"Could radical pairs play a role in xenon-induced general anesthesia?"
] |
[
"Jordan Smith *[email protected] \nDepartment of Physics and Astronomy\nInstitute for Quantum Science and Technology and Hotchkiss Brain Institute\nUniversity of Calgary\nT2N 1N4CalgaryCanada\n",
"Hadi Zadeh Haghighi \nDepartment of Physics and Astronomy\nInstitute for Quantum Science and Technology and Hotchkiss Brain Institute\nUniversity of Calgary\nT2N 1N4CalgaryCanada\n",
"Christoph Simon \nDepartment of Physics and Astronomy\nInstitute for Quantum Science and Technology and Hotchkiss Brain Institute\nUniversity of Calgary\nT2N 1N4CalgaryCanada\n"
] |
[
"Department of Physics and Astronomy\nInstitute for Quantum Science and Technology and Hotchkiss Brain Institute\nUniversity of Calgary\nT2N 1N4CalgaryCanada",
"Department of Physics and Astronomy\nInstitute for Quantum Science and Technology and Hotchkiss Brain Institute\nUniversity of Calgary\nT2N 1N4CalgaryCanada",
"Department of Physics and Astronomy\nInstitute for Quantum Science and Technology and Hotchkiss Brain Institute\nUniversity of Calgary\nT2N 1N4CalgaryCanada"
] |
[] |
Understanding the mechanisms underlying anesthesia would be a key step towards understanding consciousness. The process of xenon-induced general anesthesia has been shown to involve electron transfer, and the potency of xenon as a general anesthetic exhibits isotopic dependence. We propose that these observations can be explained by a mechanism in which the xenon nuclear spin influences the recombination dynamics of a naturally occurring radical pair of electrons. We develop a simple model inspired by the body of work on the radical-pair mechanism in cryptochrome in the context of avian magnetoreception, and we show that our model can reproduce the observed isotopic dependence of the general anesthetic potency of xenon in mice. Our results are consistent with the idea that radical pairs of electrons with entangled spins could be important for consciousness.
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10.1101/2020.08.24.265082
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[
"https://arxiv.org/pdf/2009.01661v1.pdf"
] | 221,355,590 |
2009.01661
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85900d145f1b89aba3c1316f6283e44f36d0496b
|
Could radical pairs play a role in xenon-induced general anesthesia?
Jordan Smith *[email protected]
Department of Physics and Astronomy
Institute for Quantum Science and Technology and Hotchkiss Brain Institute
University of Calgary
T2N 1N4CalgaryCanada
Hadi Zadeh Haghighi
Department of Physics and Astronomy
Institute for Quantum Science and Technology and Hotchkiss Brain Institute
University of Calgary
T2N 1N4CalgaryCanada
Christoph Simon
Department of Physics and Astronomy
Institute for Quantum Science and Technology and Hotchkiss Brain Institute
University of Calgary
T2N 1N4CalgaryCanada
Could radical pairs play a role in xenon-induced general anesthesia?
Understanding the mechanisms underlying anesthesia would be a key step towards understanding consciousness. The process of xenon-induced general anesthesia has been shown to involve electron transfer, and the potency of xenon as a general anesthetic exhibits isotopic dependence. We propose that these observations can be explained by a mechanism in which the xenon nuclear spin influences the recombination dynamics of a naturally occurring radical pair of electrons. We develop a simple model inspired by the body of work on the radical-pair mechanism in cryptochrome in the context of avian magnetoreception, and we show that our model can reproduce the observed isotopic dependence of the general anesthetic potency of xenon in mice. Our results are consistent with the idea that radical pairs of electrons with entangled spins could be important for consciousness.
Introduction
Understanding consciousness remains one of the big open questions in neuroscience 1 , and in science in general. The study of anesthesia is one of the key approaches to elucidating the processes underlying consciousness 2, 3 , but there are still significant open questions regarding the physical mechanisms of anesthesia itself 4 .
One anesthetic agent that has been studied extensively is xenon. Xenon has been shown experimentally to produce a state of general anesthesia in several species, including Drosophila 5 , mice 6 , and humans 7 . While the anesthetic properties of xenon were discovered in 1939 8 , the exact underlying mechanism by which it produces anesthetic effects remains unclear even after decades of research 9 . Our focus here is on this underlying physical mechanism, and there are important hints about the mechanism provided by two recent publications.
First, Turin et al. showed that when xenon acts anesthetically on Drosophila unpaired electron spin resonance takes place 5 , providing evidence of some form of electron transfer. Turin et al. proposed that the anesthetic action of xenon may be caused by the xenon atom(s) acting as an "electron bridge" 5 , facilitating the electron transfer between a nearby electron donor and electron acceptor. They supported their proposal by density-functional theory calculations showing the effect of xenon on nearby molecular orbitals. Second, Li et al. showed experimentally that isotopes of xenon with non-zero nuclear spin had reduced anesthetic potency in mice compared with isotopes with no nuclear spin 6 .
If the process by which xenon produces anesthetic effects includes free-electron transfer as well as nuclear-spin dependence, a mechanistic framework proposed to explain xenon-induced anesthesia should possess these characteristics. Here we show that a model involving a radical pair of electrons (RP) and the subsequent modulation of the RP spin dynamics by hyperfine interactions is consistent with these assumptions.
The radical pair mechanism (RPM) was first proposed more than 50 years ago 10 . The rupture of a chemical bond can create a pair of electrons that are localized on two different molecular fragments, but whose spins are entangled in a singlet state 11 . The magnetic dipole moment associated with electron spin can interact and couple with other magnetic dipoles, including hyperfine interactions and Zeeman interactions 12 . As a consequence of such interactions, the initial singlet state can evolve into a more complex state that has both singlet and triplet components. Eventually, the coherent oscillation of the RP between singlet and triplet states ceases and the electrons may recombine (for the singlet component of the state) or diffuse apart to form various triplet products 13 . In order for significant interconversion between singlet and triplet states to occur, the RP lifetime and the RP spin-coherence lifetime should be comparable to the electron Larmor precession period, which in the case of the geomagnetic field of the Earth (B ≈ 50µT) is approximately 700 ns [13][14][15] . The coherent spin dynamics and spin-dependent reactivity of radical pairs allow magnetic interactions which are 6 orders of magnitude smaller than the thermal energy, k B T , to have predictable and reproducible effects on chemical reaction yields 14,16 .
The RPM has become a prominent concept in quantum biology 17 . In particular, it has been studied in detail for the cryptochrome protein as a potential explanation for avian magnetoreception 16,[18][19][20][21][22][23] . In the present work we apply the principles and methods used to investigate cryptochrome 14 to xenon-induced anesthesia.
General anesthetics produce widespread neurodepression in the central nervous system by enhancing inhibitory neurotransmission and reducing excitatory neurotransmission across synapses 24,25 . Three ligand-gated ion-channels in particular have emerged as likely molecular targets for a range of anesthetic agents 26 : the inhibitory glycine receptor, the inhibitory γ-aminobutyric acid type-A (GABA A ) receptor, and the excitatory N-methyl-D-aspartate (NMDA) receptor 27 . NMDA receptors require both glycine and glutamate for excitatory activation 28 , and it has been suggested that xenon's anesthetic action is related to xenon atoms participating in competitive inhibition of the NMDA receptor in central-nervous-system neurons by binding at the glycine binding site in the NMDA receptor 26,27,29,30 . Armstrong et al. propose that the glycine binding site of the NMDA receptor contains aromatic rings 26,29 of similar chemical nature to those found in cryptochrome, which is consistent with the possibility that the anesthetic action of xenon uses a spin-dependent process similar to the radical-pair mechanism (RPM) thought to occur in cryptochrome. Let us note that other targets have also been proposed for many anesthetics, e.g. tubulin 31 . In the following we focus on the NMDA receptor to be specific, but our model is very general and could well apply to other targets.
We suggest that the electron transfer evidenced by Turin et al. 5 could affect the recombination dynamics of a naturally occurring radical pair (based on the results of Armstrong et al. this RP may involve aromatic phenylalanine residues 29 located in the binding site), and that an isotope of xenon with a non-zero nuclear spin could couple with the electron spins of such a radical pair, reducing the "electron bridge" effect that is proposed to correlate with anesthesia. Such a mechanism is consistent with the experimental results of Li et al. 6 that xenon isotopes with non-zero nuclear spin have reduced anesthetic potency compared to isotopes with zero nuclear spin.
We hypothesize that in the context of xenon-induced general anesthesia, xenon itself may not be involved in the creation of radical pairs, where the energy for radical-pair creation likely comes from another source such as local ROS. It has been suggested that water (a source of oxygen) may be present in the NMDA receptor 26,28 , and Aizenman et al. 32 as well as Girouard et al. 33 suggest that reactive oxygen species (ROS) may be located in the NMDA receptor. Further, Turin and Skoulakis found that when a sample of xenon gas was administered to Drosophila without oxygen gas present in the sample no spin changes were observed in the flies 34 . Here we propose that ROS could be involved in the formation and existence of a naturally occurring RP in the NMDA receptor.
Given that cryptochrome is one case in which the RPM has been studied extensively, we propose that by recognizing commonalities in the biological and chemical environments in which magnetoreception and xenon-induced general anesthesia are thought to take place, analytical and numerical techniques that have been used to study cryptochrome may be adapted and applied to the case of xenon-induced anesthesia, potentially providing insight into general anesthetic mechanisms. We explore the feasibility of such a mechanism by determining and analyzing the necessary parameters and conditions under which the spin-dependent RP product yields can explain the experimental isotope-dependent anesthetic effects reported by Li et al 6 .
Results
Predicting Experimental Xenon Anesthesia Results using the RPM Model
Quantifying anesthetic potency
In the work of Li et al. 6 , a metric referred to as the "loss of righting reflex ED50" (LRR-ED50) was defined using the concentration of xenon administered to mice, in which the mice were no longer able to right themselves within 10 s of being flipped onto their backs. The LRR-ED50 metric was reported to be correlated with consciousness in mice, and was measured experimentally for 132 Xe, 134 Xe, 131 Xe, and 129 Xe to be 70(4)%, 72(5)%, 99(5)%, and 105(7)%, respectively 6 .
Here we defined the anesthetic potency as the inverse of the LRR-ED50 metric. In order to quantify the anesthetic potency of the various xenon isotopes, the potency of 132 Xe (with I = 0) was normalized to 1. The inverse of the LRR-ED50 value of isotopes 131 Xe and 129 Xe were then divided by that of 132 Xe. The relative isotopic anesthetic potencies were quantified as Pot 0 = 1, Pot 3/2 = 0.71 (8), and Pot 1/2 = 0.67(8) for 132 Xe, 131 Xe, and 129 Xe, respectively, as shown in Table 1, where Pot 0 is the relative potency of xenon with nuclear spin I = 0, and likewise for Pot 3/2 and Pot 1/2 .
RPM model
We have developed an RPM model to predict anesthetic potency by making a connection with the relative singlet yield for different isotopes, as described in more detail below. The model that we have used here was developed using a hypothetical xenon-NMDA receptor RP system based on the information about xenon action sites mentioned previously, involving xenon atoms surrounded by phenylalanine and tryptophan residues located in the glycine-binding site of the NMDA receptor, and also modelled after the cryptochrome case as related to magnetoreception. A spin-correlated radical pair of electrons (termed Here we show that the simplest case of a single xenon atom occupying the active site already allows us to explain the anesthetic potency ratios derived from the experimental results of Li et al. 6 . The additional degrees of freedom implicit in more complex Hilbert spaces, such as the cases of two and three-xenon occupation states, only aid the model in explaining the experimental results of Li et al.. Further, when our model is optimized in order to reproduce the experimental results, the two-xenon occupation state essentially reduces to the single-xenon occupation state, in which one hyperfine interaction between a xenon nucleus and one of the radical electrons is dominant. We therefore focus our analysis on the single atom case, but the two and three-xenon occupation states are discussed in the Supplementary Information.
The Hamiltonian of the RPM in the case of xenon-induced anesthesia depends not only upon the number of xenon atoms present in the glycine binding site of the NMDA receptor, but also upon the assumed hyperfine interactions. In the simulation involving a single xenon atom occupying the active site it was assumed that the xenon nucleus may couple to both radical electrons, and the Hamiltonian is given aŝ
H = ω Ŝ Az +Ŝ Bz + a 1ŜA ·Î 1 + a 2ŜB ·Î 1 ,(1)
whereŜ A andŜ B are the spin operators of radical electrons A and B, respectively,Î i is the nuclear spin operator of xenon nucleus i, a j is a hyperfine coupling constant where a j = γ e a j , and ω is the Larmor precession frequency of the electrons about an external magnetic field 14 . The Larmor precession frequency is defined as ω = γ e B, where γ e is the gyromagnetic ratio of an electron and B is the external magnetic field strength. It should be pointed out that we focus here on the hyperfine interactions between the radical electrons and xenon atoms, and that interactions between the two electron spins, as well as potential interactions between the electron spins and other nuclei are neglected. This is justifiable for our purposes, since we are primarily interested in the differential effect of the xenon nuclear spin.
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Determination of Singlet Yield Ratios
The eigenvalues and eigenvectors of the Hamiltonian can be used to determine the ultimate singlet yield (Φ S ) for all times much greater than the radical-pair lifetime (t τ): 14
Φ S = 1 4 − k 4(k + r) + 1 M 4M ∑ m=1 4M ∑ n=1 m|P S |n 2 k(k + r) (k + r) 2 + (ω m − ω n ) 2 ,(2)
where, following the methodology used by Hore 14 in the context of cryptochrome, M is the total number of nuclear spin configurations,P S is the singlet projection operator, |m and |n are eigenstates ofĤ with corresponding energies of ω m = m|Ĥ |m and ω n = n|Ĥ |n , respectively, k = τ −1 is inverse of the RP lifetime, and r = τ −1 c is the inverse of the RP spin-coherence lifetime.
The spin-dependent RP singlet yield was calculated for each xenon isotope under consideration. The singlet yield ratio (SR) for each nuclear spin value was then calculated by dividing the singlet yield obtained using the given spin value by the singlet yield obtained using spin I = 0, resulting in the ratio of spin-1/2 singlet yield to spin-0 singlet yield being expressed as SR 1/2 , and the ratio of spin-3/2 singlet yield to spin-0 singlet yield expressed as SR 3/2 , with the singlet yield ratio of spin-0 being normalized to SR 0 = 1. The calculated singlet yield ratios were compared with the xenon potency ratios (Pot) derived using the data reported by Li et al. 6 as described above.
We investigated the sensitivity of the singlet yield ratios to changes in the hyperfine interactions (a 1 and a 2 ), RP reaction rate (k), external magnetic field strength (B), and RP spin-coherence relaxation rate (r). The dependence of the quantity |Pot − SR| on a 1 and a 2 is shown in Figs. 2/3, while the dependence of |Pot − SR| on the relationship between r and k is shown in Figs. 4/5. The relationship between SR and B can be seen in Fig. 6. The same dependencies for the two-xenon occupation state can be seen in Supplementary Figs. S1/S2, S3/S4, and S5, respectively. Our goal was to find regions in parameter space such that the spin-dependent singlet yield ratios match the anesthetic isotopic potency ratios, i.e., the quantities |Pot 1/2 − SR 1/2 | and |Pot 3/2 − SR 3/2 | should be smaller than the experimental uncertainties on the anesthetic potency.
In the case of single-xenon occupation the optimized parameter values were found to be a 1 = 69 µT, a 2 = 483 µT, B = 50 µT, k = r = 1.0 × 10 6 s −1 , resulting in SR 1/2 = 0.72 and SR 3/2 = 0.63. It should be noted here that while |Pot 3/2 − SR 1/2 | is greater than |Pot 3/2 − SR 1/2 | for these optimized parameter values, both SR 3/2 and SR 1/2 are consistent with the experimental results of Li et al. 6 , within their uncertainties. Further experiments with smaller uncertainty about the relative anesthetic potencies of spin-3/2 and spin-1/2 xenon isotopes would be of interest.
Discussion
The optimized parameter values for a single xenon atom suggest strong coupling only between the xenon nucleus and one of the two RP electrons, while having weak interactions with the other, see Figs. 2 and 3. To gain a deeper understanding of this coupling, it would be of interest to perform molecular modelling of the electron transfer between xenon and tryptophan/phenylalanine residues using Marcus Theory. Such modelling could be expanded on by exploring the molecular dynamics of the binding site using quantum mechanics/molecular mechanics (QM/MM) simulation techniques, similar to those used in the case of cryptochrome 35 , to determine which aromatic rings are most likely to contain a radical pair, and therefore which molecules the xenon nuclei are most likely to couple with. Further, rather than determining the hyperfine coupling constants by fitting parameters to a model, it could be useful to perform density functional theory (DFT) modelling to determine theoretical hyperfine coupling constants, accounting for the quantity and relative positioning of the xenon atoms within the NMDA receptor glycine binding site.
The sensitivity analysis of the relationship between the RP spin-coherence relaxation rate (r) and the RP first-order reaction rate (k) shown in Figs. 4 and 5 is promising, in that the RP lifetime requirements are within an order of magnitude of the results reported by Hore in the cryptochrome case 14 . The calculated range for the RP lifetime, τ, is comparable to the spin-coherence lifetime, τ c , and also to the electron Larmor precession period. It is interesting to note that the r and k parameter spaces for which SR and Pot agree indicate that for spin I = 1/2 the RP reaction rate (k) should be at least as fast as the RP spin-coherence relaxation rate (r), while in the spin I = 3/2 case the inverse relation appears to be true. For both RP spin-coherence relaxation rates and RP reaction rates much above or below the value outlined by Hore 14 of r = k = 1.0 × 10 6 s −1 , SR and Pot diverge. Relaxation rates much greater than these values result in the RP not having sufficient time to coherently oscillate between singlet and triplet states such that the singlet yield ratios and the derived anesthetic potency ratios are in agreement. As with the cryptochrome case, as the RP lifetime becomes much greater than the RP spin-coherence lifetime, the RP exponentially decays toward the equilibrium state of Φ S = 0.25 for all nuclear spin values. These results emphasizes the importance of the RP spin-coherence lifetime (τ c ) being comparable to the electron Larmor precession period (τ L = 2π/ω) and also to the requisite RP lifetime (τ).
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When considering the sensitivity of the single-xenon model to changes in external magnetic field as shown in Fig. 6, the range of B values that produce agreement between SR and Pot ratios is given as B ∈ [0, 51] µT, and approximates the geomagnetic field at different geographic locations (25 to 65 µT) 36 . This result indicates that for a single-xenon occupation state external field values vastly stronger than the geomagnetic field may result in the anesthetic potency of xenon being reduced significantly. Further, at geographic locations on the Earth where the geomagnetic field is larger than 51 µT, anesthetic potency may be reduced compared to locations where B earth ≤ 51 µT. It is also seen in Fig. 6 that the isotope of xenon with spin I = 0 has no external magnetic field dependence, while isotopes with non-zero spin do show a dependence on B. It would be of interest to investigate the experimental effects of the external magnetic field strength on xenon-induced general anesthesia in vivo. For example, such an experiment could involve the measurement of anesthetic potency of xenon isotopes with various nuclear spin values, including both zero spin and non-zero spin, in an environment with controllable external magnetic field.
In this study the RPM model in the context of cryptochrome 14 was adapted to the case of xenon in the glycine binding site of the N-methyl-D-aspartate receptor, and the experimental isotope-dependent anesthetic potency ratios derived from the results of Li et al. 6 were reproduced theoretically using the RPM model in which a single xenon atom occupies the active site of the NMDA receptor. The RP lifetime and hyperfine coupling parameters were optimized to reproduce the experimental results of Li et al., and the sensitivity of the model to changes in hyperfine coupling constants, RP reaction rate, external magnetic field strength, and RP spin-coherence relaxation rate was explored. The optimized model parameter values and parameter spaces found here seem physiologically feasible, and indicate that the RPM may provide a reasonable explanation for the general anesthetic action of xenon in vivo.
Our results thus suggest that xenon-induced general anesthesia may fall within the realm of quantum biology, and be similar in nature to the proposed mechanism of magnetoreception involving the cryptochrome protein 18 .
This also raises the question whether other anesthetic agents use the same mechanism as xenon to induce general anesthesia. It could prove interesting to explore isotopic nuclear-spin effects as well as magnetic field effects in experiments with other general anesthetic agents that are thought to function similarly to xenon, such as nitrous oxide and ketamine 37,38 .
General anesthesia is clearly related to consciousness, and it has been proposed that consciousness (and other aspects of cognition) could be related to large-scale entanglement [39][40][41][42] . Radical pairs are entangled and could be a key element in the creation of such large-scale entanglement, especially when combined with the suggested ability of axons to serve as waveguides for photons 43 . Viewed in this -admittedly highly speculative -context, the results of the present study are consistent with the idea that general anesthetic agents, such as xenon, could interfere with this large-scale entanglement process, and thus with consciousness.
Data Availability
The datasets generated and analysed during the current study are available from the corresponding author on reasonable request. lifetimes than in the single-xenon case, where both cases are within an order of magnitude of the value suggested by Hore 14 in the cryptochrome context.
The results of the B-sensitivity analysis of the two-xenon occupation model show that for external fields with either very small (< 24 µT) or extremely large (> 1102 µT) magnitudes, SR and Pot values may diverge, see Supplementary Fig. S5.
Considering the three-xenon occupation state in which three xenon atoms occupy the glycine binding site in the NMDA receptor, based on the geometric modelling done by Armstrong et al. 29 it was assumed that radical electron A couples with two xenon nuclei (I 1 and I 2 ) and radical electron B couples only to the third xenon nucleus (I 3 ). In this case the Hamiltonian can be modelled aŝ
H = ω Ŝ Az +Ŝ Bz + a 1ŜA ·Î 1 + a 2ŜA ·Î 2 + a 3ŜB ·Î 3 ,(4)
The optimized parameter values using the three-xenon occupation state were found to be a 1 = 444 µT, a 2 = 444 µT, a 3 = 777 µT, B = 50 µT, k = 2.0 × 10 5 s −1 , and r = 1.0 × 10 6 s −1 , resulting in SR 1/2 = 0.72 and SR 3/2 = 0.71. Note that our three-xenon model includes the two-xenon model as a special case, so it is not surprising that the three-xenon model can also explain the experimental anesthetic potency of xenon reported by Li et al 6 .
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Supplementary Figure S1. The dependence of the two-xenon RPM model on changes in the hyperfine coupling constants a 1 and a 2 for a 1 , a 2 ∈ [0, 1000] µT, using r = 1.0 × 10 6 s −1 , B = 50 µT, and τ = 4.9 × 10 −6 s. The model can explain the experimentally derived 6 relative anesthetic potency of xenon for values of r and k where |Pot 1/2 − SR 1/2 |, |Pot 3/2 − SR 3/2 | ≤ 0.08. (a) The absolute difference between Pot 1/2 and SR 1/2 . (b) The absolute difference between Pot 3/2 and SR 3/2 .
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Supplementary Figure S2. The dependence of the two-xenon RPM model on changes in the hyperfine coupling constants a 1 and a 2 for a 1 , a 2 ∈ [0, 1000] µT, using r = 1.0 × 10 6 s −1 , B = 50 µT, and τ = 4.9 × 10 −6 s. The model can explain the experimentally derived 6 relative anesthetic potency of xenon where |Pot 1/2 − SR 1/2 | ≤ 0.080 and |Pot 3/2 − SR 3/2 | ≤ 0.08 intersect.
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Supplementary Figure S3. The dependence of the two-xenon RPM model on the relationship between r and k for r, k ∈ [1.0 × 10 5 , 1.0 × 10 7 ] s −1 , using a 1 = 0 µT, a 2 = 1000 µT, and B = 50 µT. The model can explain the experimentally derived 6 relative anesthetic potency of xenon for values of r and k where |Pot 1/2 − SR 1/2 |, |Pot 3/2 − SR 3/2 | ≤ 0.08. (a) The absolute difference between Pot 1/2 and SR 1/2 . (b) The absolute difference between Pot 3/2 and SR 3/2 .
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Supplementary Figure S4. The dependence of the two-xenon RPM model on the relationship between r and k for r, k ∈ [1.0 × 10 5 , 1.0 × 10 7 ] s −1 , using a 1 = 0 µT, a 2 = 1000 µT, and B = 50 µT. The model can explain the experimentally derived 6 relative anesthetic potency of xenon where |Pot 1/2 − SR 1/2 | ≤ 0.080 and |Pot 3/2 − SR 3/2 | ≤ 0.08 intersect.
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Supplementary Figure S5. The external field-dependence of the two-xenon RPM model with B ∈ [0, 1200] µT, τ = 4.9 × 10 −6 s, r = 1.0 × 10 6 s −1 , and hyperfine constants of a 1 = 0 µT, a 2 = 1000 µT. (a) The absolute singlet yield using xenon isotopic nuclear spin values of I ∈ {0, 1/2, 3/2}. (b) The relative singlet yield ratios SR 1/2 and SR 3/2 with the singlet yield of spin I = 0 being normalized to SR 0 = 1. Values of SR and Pot agree for B ∈ [24,1102] µT.
Figure 1 .
1Aromatic residues are important for the binding of xenon and glycine at the glycine binding site of the NMDA receptor. The image shows the predicted position of xenon atoms (red spheres) in the glycine site together with the aromatic residues phenylalanine 758, phenylalanine 484, and tryptophan 731.29 electrons A and B), most likely found in chemically excited phenylalanine residues 26 , couple to xenon nuclei with non-zero nuclear spin, as shown inFig. 1.The number of xenon atoms located in the active site when anesthetic action takes place is not yet completely clear, and the work of Dickinson et al. suggests that the number of xenon atoms simultaneously present in the active site ranges probabilistically between zero and three26 .
Figure 2 .Figure 3 .Figure 4 .Figure 5 .
2345The dependence of the single-xenon RPM model on changes in the hyperfine coupling constants a 1 and a 2 for a 1 , a 2 ∈ [0, 1000] µT, using r = 1.0 × 10 6 s −1 , B = 50 µT, and τ = 1.0 × 10 −6 s. The model can explain the experimentally derived 6 relative anesthetic potency of xenon for values of r and k where |Pot 1/2 − SR 1/2 |, |Pot 3/2 − SR 3/2 | ≤ 0.08. (a) The absolute difference between Pot 1/2 and SR 1/2 . (b) The absolute difference between Pot 3/2 and SR 3/2 . The dependence of the single-xenon RPM model on changes in the hyperfine coupling constants a 1 and a 2 for a 1 , a 2 ∈ [0, 1000] µT, using r = 1.0 × 10 6 s −1 , B = 50 µT, and τ = 1.0 × 10 −6 s. The model can explain the experimentally derived 6 relative anesthetic potency of xenon where |Pot 1/2 − SR 1/2 | ≤ 0.080 and |Pot 3/2 − SR 3/2 | ≤ 0.08 intersect. The dependence of the single-xenon RPM model on the relationship between r and k for r, k ∈ [1.0 × 10 5 , 1.0 × 10 7 ] s −1 , using a 1 = 69 µT, a 2 = 483 µT, and B = 50 µT. The model can explain the experimentally derived 6 relative anesthetic potency of xenon for values of r and k where |Pot 1/2 − SR 1/2 |, |Pot 3/2 − SR 3/2 | ≤ 0.08. (a) The absolute difference between Pot 1/2 and SR 1/2 . (b) The absolute difference between Pot 3/2 and SR 3/2 . The dependence of the single-xenon RPM model on the relationship between r and k for r, k ∈ [1.0 × 10 5 , 1.0 × 10 7 ] s −1 , using a 1 = 69 µT, a 2 = 483 µT, and B = 50 µT. The model can explain the experimentally derived 6 relative anesthetic potency of xenon where |Pot 1/2 − SR 1/2 | ≤ 0.080 and |Pot 3/2 − SR 3/2 | ≤ 0.08 intersect.
Figure 6 .
6The external field-dependence of the single-xenon RPM model with B ∈ [0, 100] µT, τ = 1.0 × 10 −6 s, r = 1.0 × 10 6 s −1 , and hyperfine constants of a 1 = 69 µT, a 2 = 483 µT. (a) The absolute singlet yield using xenon isotopic nuclear spin values of I ∈ {0, 1/2, 3/2}. (b) The relative singlet yield ratios SR 1/2 and SR 3/2 with the singlet yield of spin I = 0 being normalized to SR 0 = 1. Values of SR and Pot agree for B ∈ [0, 51] µT.
Table 1 .
1Xenon isotopic nuclear spin, LRR-ED50, and Pot values for xenon isotopes 132 Xe, 134 Xe, 131 Xe, and 129 Xe. LRR-ED50 values as reported in the work of Li et al. 6 Isotope Nuclear Spin, I LRR-ED50 (%) Pot132 Xe
0
70(4)
1
134 Xe
0
72(5)
-
131 Xe
3/2
99(5)
0.71(8)
129 Xe
1/2
105(7)
0.67(8)
AcknowledgementsThe authors would like to thank Peter Hore, Robert Dickinson, Dennis Salahub, Sourabh Kumar, Parisa Zarkeshian, Sumit Goswami, Faezeh Kimiaee Asadi, Stephen Wein, Jiawei Ji, Yufeng Wu, and Kenneth Sharman for their input, comments, and insights on this topic. C.S. particularly thanks Stuart Hameroff for bringing the results of Li et al.6to his attention. This work was supported by the Natural Sciences and Engineering Research Council of Canada.Competing InterestsThe authors declare no competing interests.Supplementary InformationResults with two and three xenon atoms in the active siteIn contrast to the case of the cryptochrome model, in which the nuclear spins are permitted to take on different isotopic spin values leading to various permutations of nuclear spins, in the experimental case of xenon each isotope was supplied separately by Li et al.6, so here all xenon nuclei in the RPM model were constrained to have the same simultaneous isotopic nuclear spinIn the case of two xenon nuclei occupying the active site the assumption was made that each xenon nucleus couples exclusively to a unique electron in the RP, motivated by the proximity of the xenon nuclei to each phenylalenine residue, and by extension to each radical electron, in the work of Dickinson et al. The Hamiltonian describing this case, where each of the two nuclei couples solely to a unique radical electron, can be expressed aŝThe optimized parameter values using the two-xenon occupation state were found to be a 1 = 0 µT, a 2 = 1000 µT, B = 50 µT, k = 0.2 × 10 6 s −1 , and r = 1.0 × 10 6 s −1 , resulting in SR 1/2 = 0.73 and SR 3/2 = 0.71. Similar to the single-xenon case, the two-xenon model produced results indicating that the RPM can explain experimental anesthetic potency results most closely in the case that one xenon nucleus couples with its respective radical electron very strongly, while the other xenon nucleus interacts with its corresponding electron relatively weakly, which essentially renders the two cases equivalent; one radical electron couples strongly with one xenon atom, while the other electron only couples weakly. In the two-xenon case, however, the hyperfine parameter space for which SR is within uncertainty of Pot is relatively large compared with the single-xenon case, as seen inSupplementary Figures S1 and S2. Similar optimized RP lifetime values are seen in this case as in the case of single-xenon occupation, see Supplementary Figs. S3 and S4, but the range of agreeable values extends to somewhat longer RP
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|
[] |
[
"Topological number and Fermion Green's function of Strongly Interacting Topological Superconductors",
"Topological number and Fermion Green's function of Strongly Interacting Topological Superconductors"
] |
[
"Yi-Zhuang You \nDepartment of physics\nUniversity of California\n93106Santa BarbaraCAUSA\n",
"Zhong Wang \nInstitute for Advanced Study\nTsinghua University\n100084BeijingChina\n\nCollaborative Innovation Center of Quantum Matter\n100871BeijingChina\n",
"Jeremy Oon \nDepartment of physics\nUniversity of California\n93106Santa BarbaraCAUSA\n",
"Cenke Xu \nDepartment of physics\nUniversity of California\n93106Santa BarbaraCAUSA\n"
] |
[
"Department of physics\nUniversity of California\n93106Santa BarbaraCAUSA",
"Institute for Advanced Study\nTsinghua University\n100084BeijingChina",
"Collaborative Innovation Center of Quantum Matter\n100871BeijingChina",
"Department of physics\nUniversity of California\n93106Santa BarbaraCAUSA",
"Department of physics\nUniversity of California\n93106Santa BarbaraCAUSA"
] |
[] |
It has been understood that short range interactions can reduce the classification of topological superconductors in all dimensions. In this paper we demonstrate by explicit calculations that when the topological phase transition between two distinct phases in the noninteracting limit is gapped out by interaction, the bulk fermion Green's function G(iω) at the "transition" approaches zero as G(iω) ∼ ω at certain momentum k in the Brillouin zone.
|
10.1103/physrevb.90.060502
|
[
"https://arxiv.org/pdf/1403.4938v2.pdf"
] | 118,613,631 |
1403.4938
|
b1ecb166df90027ead44daa329df84ac515645d0
|
Topological number and Fermion Green's function of Strongly Interacting Topological Superconductors
Yi-Zhuang You
Department of physics
University of California
93106Santa BarbaraCAUSA
Zhong Wang
Institute for Advanced Study
Tsinghua University
100084BeijingChina
Collaborative Innovation Center of Quantum Matter
100871BeijingChina
Jeremy Oon
Department of physics
University of California
93106Santa BarbaraCAUSA
Cenke Xu
Department of physics
University of California
93106Santa BarbaraCAUSA
Topological number and Fermion Green's function of Strongly Interacting Topological Superconductors
It has been understood that short range interactions can reduce the classification of topological superconductors in all dimensions. In this paper we demonstrate by explicit calculations that when the topological phase transition between two distinct phases in the noninteracting limit is gapped out by interaction, the bulk fermion Green's function G(iω) at the "transition" approaches zero as G(iω) ∼ ω at certain momentum k in the Brillouin zone.
Introduction -Unlike bosonic systems, fermionic systems can have nontrivial topological insulator (TI) and topological superconductor (TSC) phases without interaction. Noninteracting fermionic TIs and TSCs in all dimensions with various symmetries have been fully classified in Ref. 1-3. However, it remains an open and challenging question that what role does interaction play in the classification of TIs and TSCs. According to a pioneering work Ref. 4, 5, a 1d TSC with time-reversal symmetry, which in the noninteracting limit has a Z classification [1][2][3], has only a Z 8 classification in the presence of local interactions. This implies that the boundary state of 8 copies of this 1d TSC (which is 8 flavors of 0d Majorana fermion zero modes) can be gapped out by interaction without ground state degeneracy. The work in Ref. 4,5 was soon generalized to 2d TSCs with a 1d boundary [6][7][8][9], and 3d TSCs with 2d boundary [10,11]. All these TSCs have Z classification without interaction, namely in all these cases, for arbitrary flavors of the system, the boundary remains gapless without interaction. But with 8 copies of such TSCs in 2d, and 16 flavors of He 3 B in 3d, a strong enough local interaction that preserves all the symmetries can render the boundary states trivial, namely the boundary states can be gapped out by interaction without degeneracy.
If interaction reduces the classification of a TSC, it not only implies that the boundary becomes trivial under interaction, it also implies that the "TSC" and the trivial state can be adiabatically connected to each other without any phase transition, namely the bulk quantum critical point in the noninteracting limit is gapped out by interaction. A free fermion TI or TSC with Z classification is usually characterized by a quantized topological number, which can be constructed by fermion Green's function [12][13][14][15][16][17]. For example, the so-called TKNN number [12,13] of integer quantum Hall state can be represented as [18,19]
N = 1 24π 2 d 3 k µνρ Tr[G∂ µ G −1 G∂ ν G −1 G∂ ρ G −1 ],(1)
where G(k) is the fermion Green's function in the fre-quency and momentum space k = (iω, k), and iω is the Matsubara frequency. The formula Eq. 1 can be formally applied to interacting systems too, as long as we use the full interacting fermion Green's function [12][13][14][15][16][17].
Recently it was also proved that in the presence of interaction, Eq. 1 is fully determined by zero-frequency Green's function, and the topological invariant takes a simpler form [16] N = 1 2π d 2 kF xy , where F xy is the Berry curvature calculated from the eigenvectors of −G −1 (ω = 0, k), as if it is a "noninteracting" Hamiltonian. This TKNN topological number must be quantized, so it can only change through a sharp "transition" in the phase diagram. In the noninteracting limit, this sharp transition is a physical phase transition which corresponds to closing the bulk gap of the fermion, i.e. the fermion Green's function develops a gapless pole at the transition G(iω) ∼ 1/ω. With strong interaction, the results in Ref. [4][5][6][7][8][9][10][11] imply that this quantum critical point can be gapped out by interactions, while this "transition" of topological number must still occur in the phase diagram. Ref. 20,21 proposed that the quantized topological number such as Eq. 1 not necessarily corresponds to gapless edge states, it can also correspond to zeros of the fermion Green's function at the edge. In our paper we will study the fermion Green's function in the bulk, and we will demonstrate with explicit calculations that the fermion Green's function develops zero at the "transition" of topological numbers, even though there is no transition in bulk spectrum. And the fermion Green's function vanishes analytically as G(iω) ∼ ω.
A simple observation can support this conclusion above: in Eq. (1), G and G −1 are almost equivalent. Thus the topological number should either change through a pole of G (the noninteracting case), or change through a zero of G, which corresponds to a pole of G −1 . This implies that when the "transition" of topological number is gapped out by interactions, the Green's function should approach zero as G(iω) ∼ ω α with positive α. Since at the "transition" the fermion spectrum is fully gapped, the fermion correlation function in real space-time must be short ranged, thus α must be an integer, and the simplest scenario is α = 1. arXiv:1403.4938v1 [cond-mat.str-el] 19 Mar 2014 1d Example -Before providing a general argument of the existence of zeros in the fermion Green's function for arbitrary dimension, we will first demonstrate this behavior explicitly in a 1d example using 8 copies of Kitaev's chain [22], also known as the 1d Fidkowski-Kitaev model [4]. Consider a 1d lattice, on each site i, we introduce 8 Majorana fermions denoted by χ iα with α = 1, · · · , 8 labeling the fermion flavors (species), which satisfy the Majorana fermion anti-commutation relations {χ iα , χ i α } = 2δ ii δ αα . The model Hamiltonian reads
H = H u + H w , H u = 1 2 i,j,α iu ij χ iα χ jα , H w = −w i,{α k } X α1α2α3α4 χ iα1 χ iα2 χ iα3 χ iα4 .(2)
H u is the inter-site coupling term with u ij ∈ R and u ij = −u ji . H w is the on-site Fidkowski-Kitaev interaction, in which the coefficient X α1α2α3α4 ≡ e 1 |χ α1 χ α2 χ α3 χ α4 |e 1 is given as the expectation value of the four-fermion operator on a chosen on-site ground state |e 1 (to be specified later). Here we have omitted the site index i, and the summation of flavor α in H w is to sum over all the possible quartets
α 1 < α 2 < α 3 < α 4 for α k = 1, · · · , 8.
To specify the state |e 1 , we may choose a representation of the Majorana operators as χ 2n−1 = σ 1 n m<n σ 3 m and χ 2n = σ 2 n m<n σ 3 m for n = 1, · · · , 4, with σ µ n being the µ-th Pauli matrix acting on the n-th tensor factor. Then the chosen ground state [4] can be written as |e 1 = (|0000 − |1111 )/ √ 2 (where |0 and |1 are the eigen states of σ 3 ). The choice of |e 1 is not unique, but we will stick to this convention. It turns out that there are 14 non-zero X α1α2α3α4 's, corresponding to the 14 four-fermion terms in H w on each site. The explicit form of H w was given in Ref. 4, and its physical meaning has been discussed in Ref. 7, 21. We would like to introduce an explicit set of eigen basis for H w . Let us focus on a single site, and omit the site index i. In the 16-dimensional Hilbert space of the 8 Majorana fermions on each site, the Hamiltonian H w can be diagonalized. All its on-site many-body eigenstates can be constructed from |e 1 by applying Majorana fermion operators. We can divide these states into even and odd sectors according to their fermion parity. Given the ground state |e 1 is an even parity state, the odd parity states |o α = χ α |e 1 are obtained by acting a single fermion operator; and the even parity states |e α = χ 1 χ α |e 1 can be constructed by acting the two-fermion operator with the first operator fixed as χ 1 . These states |p α (p = e, o and α = 1, · · · , 8) form a set of orthonormal basis, i.e. p α |p α = δ pp δ αα . These states can be defined on each site as |p α i . In this set of basis, the interaction Hamiltonian H w is diagonalized,
H w = i − 14w|e 1 ii e 1 | + 2w 8 α=2 |e α ii e α | . (3)
Odd parity states are annihilated by H w , i.e. of zero eigenvalue, so they do not appear in Eq. (3). The manybody ground state of H w is simply the |e 1 product state, denoted as |0 = i |e 1 i , with the ground state energy E 0 scaling with the system size.
Now we turn on the bilinear fermion coupling term H u perturbatively, assuming u ij w. In this section we will calculate the Green's function up to the first order perturbation of u ij /w. In the next section we will prove that our qualitatively result is valid after summing over the entire infinite perturbation series. H u will take the unperturbed ground state |0 to a group of two-fermion excited states χ iα χ jα |0 , all of which have approximately the same energy E 0 + 28w. So according to the perturbation theory, the perturbed ground state (to the first order in u ij ) reads
|g = 1 − H u 28w |0 .(4)
The Majorana fermion Green's function in the real space can be calculated from [16,17]
G(iω) iα,i α = m g|χ iα |m m|χ i α |g iω − (E m − E 0 ) + m|χ iα |g g|χ i α |m iω + (E m − E 0 ) .(5)
Since
H u H w , we have (E m − E 0 ) 14w, therefore G(iω) iα,i α = g|χ iα χ i α |g iω − 14w + g|χ i α χ iα |g iω + 14w = iω (iω) 2 − (14w) 2 g|{χ iα , χ i α }|g + 14w (iω) 2 − (14w) 2 g|[χ iα , χ i α ]|g .(6)
In the first term, g|{χ iα , χ i α }|g = 2δ ii δ αα according to the Majorana fermion anti-commutation relations. The second term is not vanishing only for i = i , and under such condition, g|[χ iα , χ i α ]|g = 2 g|χ iα χ i α |g = 4 0|χ iα χ i α Hu −28w |0 = 2iu ii δ αα /(14w). Hence
G(iω) iα,i α = 2δ αα (iω) 2 − (14w) 2 (iωδ ii + iu ii ). (7)
Fourier transform to the momentum space, assuming the coupling u ij is alternating between u and v along the 1d chain, s.t. u k = u − ve ik , then on the sublattice basis,
G(iω, k) α,α = 2δ αα (iω) 2 − (14w) 2 iω −iu * k iu k iω .(8)
At the topological "transition" point u = v, u k ∼ −ivk vanishes as k → 0, so the Green's function approaches zero analytically at the zero frequency and momentum iω, k → 0 as expected. In contrast, in the free fermion case, such a topological transition happens through a gapless critical point characterized by the pole in the Green's function instead. Zero in Green's Function in general dimension -The above perturbative calculation is not limited to the 1d model, but can be immediately generalized to higher spatial dimensions. It is found that for a series of d-dimensional models with arbitrary d, the topological number defined with full fermion Green's function can change via the zero in the Green's function at some momentum in the Brillouin zone. This statement actually holds to all orders of perturbation.
We will consider lattice models for a d-dimensional TSC (d ≥ 2) like the follows:
H = 8 α=1 χ α,−k d µ=1 sin k µ Γ µ χ α,k + mχ α,−k d µ=1 cos k µ − d + 1 Γ d+1 χ α,k ;(9)
where Γ 1···d are symmetric matrices, while Γ d+1 is an antisymmetric matrix. When d = 2, Γ 1 = σ x , Γ 2 = σ z , Γ 3 = −σ y ; m = 0 is the quantum critical point between 8 copies of p + ip TSC (m > 0) and p − ip TSC (m < 0), the topological number defined in Eq.
N = 1 48π 2 d 3 k abc Tr[Γ 5 G∂ a G −1 G∂ b G −1 G∂ c G −1 ],(10)
where G = G(0, k) is the zero frequency Green's function. The momentum integral is carried out in the Brillouin zone.
Besides the Majorana fermion hopping terms, we will also turn on the on-site Fidkowski-Kitaev interaction H w to gap out the fermions,
H w = −w j σ,{α k } X {α k } χ σα1,j χ σα2,j χ σα3,j χ σα4,j ,(11)
where σ labels the orbital degrees of freedom, j labels the lattice sites. The on-site interaction in Eq. (11) has a nondegenerate ground state on every site, thus with strong interaction Eq. (11), the quantum critical point m = 0 in Eq. (9) is indeed gapped, and the system is in a trivial direct product state. Now we will demonstrate that in the two dimensional phase diagram of w and m, m = 0 is always the transition line of topological number defined with the Green's function (Fig. 1). With weak interaction w, because the topological number defined in Eq. 1 and Eq. 10 are always quantized, a weak interaction will not change the topological numbers. With strong interaction, the fermion Green's function can be computed by perturbation theory as the previous section. With first order perturbation, the Green's function in any dimension would take the same form as Eq. 8. It is straightforward to verify that the strongly interacting Green's function from first order perturbation theory always leads to the same 2d TKNN number (Eq. 1) as 8 copies of noninteracting p ± ip superconductors with the same m. Also, for strongly interacting He 3 B, according to Eq. 8, the zero frequency Greens function reads G(0, k) ∼ 1/G(0, k) 0 , where G(iω, k) 0 is the free fermion Green's function without interaction at all. In the definition of topological number Eq. 10, G(0, k) and G(0, k) −1 are totally interchangeable since G(0, k) anticommutes with Γ 5 , thus the topological numbers in the noninteracting limit and strong coupling limit are identical. This means that m = 0 is always the "transition" line of topological number, even though the qualitative spectrum of the system might not change across this line. The argument above was based on the explicit form of the fermion Green's function from the first order perturbation theory. But since the topological numbers are always quantized, we expect the higher order perturbations will not change the topological numbers, as long as w is strong enough. Now we analyze the symmetry of the two-fermion Green's function G(k) α,α = − χ kα χ −kα where χ kα denotes the α-flavor Majorana fermion of the frequencymomentum k = (iω, k). In the flavor space (indexed by α, α ), H u has the full SO(8) symmetry and H w has a smaller SO(7) symmetry [4], so the total Hamiltonian H = H u + H w is SO(7) symmetric, and χ α carries a 8-dimensional spinor representation of the SO(7) symmetry. Since with strong interaction Eq. (11) the system is gapped and nondegenerate, the ground state must be SO(7) invariant. Thus the Green's function must be diagonal in the flavor space: G(k) α,α = g(k)δ αα , where g(k) is the Green's function in the orbital space (see Appendix A for a mathematical proof). Physically this can be understood since H w only breaks the SO(8) symmetry by driving a four-fermion ordering without generating any fermion-bilinear order, so the broken symmetry will not be revealed at the two-fermion level. Then we turn to the orbital space Green's function g(k). Because the Fidkowski-Kitaev interaction H w has the full orbital space symmetry, in the strong interaction limit, the orbital degrees of freedom must be degenerate at each en- ergy level. So the Green's function can be written as a polynomial of H u /(iω − H w ) where H w will just behave like a constant in the orbital space. Given that at the "transition" point H u is constructed using symmetric matrices Γ µ only, so the Green's function g(k) must also be a symmetric matrix (in the momentum space), i.e. g(k) = g(k) . However, by definition g(k) = − χ k χ −k , and for the off-diagonal part of g(k), we have g(
−k) = − χ −k χ k = χ k χ −k = −g(k) = −g(k)
, meaning that the off-diagonal part of g(k) must be odd in k. Since in the large w limit the whole system is gapped, the real space fermion Green's function must decay exponentially, thus the off-diagonal components of g(k) must be analytic when k → 0. Thus we conclude that the off-diagonal components of g(k) must scale as g(k) ∼ k α , where α must be a positive odd integer. So now we are left with the diagonal part of g(k). Using the spectral representation [16] of the Green's function, we can show that the diagonal part of g(k) at zeromomentum must take the following form
g(iω, k = 0) σ,σ = m 2iω | m|χ 0σ |g | 2 (iω) 2 − (E m − E g ) 2 ,(12)
where |g is the ground state, and |m denotes a generic excited state. We can see g(iω, k = 0) σ,σ must be odd in iω at k = 0. As long as we are in the fully gapped phase (under strong enough interaction), the excitation energy (E m − E g ) will never vanish, so there will be no singularity at zero frequency, then it follows that diagonal term of the Green's function must also vanish with k → 0. So in conclusion, the fermion Green's function at the topological "transition" in the strong interaction phase must develop a zero at zero frequency around the freefermion Dirac points in the Brillouin zone. Summary and Discussion -Let us summarize our results in Fig. 1. In all dimensions we have considered in this paper, the m = 0 line in the phase diagrams are always the transition lines for topological numbers. And in all dimensions a short range interaction can gap out this transition line, but the fermion Green's function on this line develops zero at zero frequency.
In 1d, the system we consider is a 8 copies of Kitaev's chain with interactions. In 1d, the quantum critical point at the noninteracting limit will be immediately gapped out by infinitesimal interactions, but in 2d and 3d a short range interaction is irrelevant for gapless Dirac and Majorana fermions, thus the interaction can only gap out the critical line m = 0 beyond certain value w c . The 2d example of our paper is a transition between eight p + ip to eight p − ip TSCs, and since these two phases have opposite chiral edge states (opposite thermal Hall effects), thus they cannot be adiabatically connected to each other in the phase diagram. Thus although the m = 0 line is gapped out by strong enough interactions, there must be another phase transition line in the phase diagram (the horizontal line in Fig. 1b) that separates the strongly interacting phase (which is a trivial direct product state) from the p + ip and p − ip TSCs. In 1d and 3d cases discussed in our paper, the two sides across the line m = 0 can be adiabatically connected to each other without any physical phase transition, i.e. the entire phase diagram has only one phase.
Our result implies that when the topological number such as Eq. (1) is nonzero, the system can still remain trivial. Ref. 24 proposed a different method to diagnose TIs and TSCs in 1d and 2d: if |Ψ is a nontrivial TI or TSC, and |Ω is a trivial state, then the following quantity, so called strange correlator, must either saturate to a constant or decay as power-law in the limit |r−r | → +∞: C(r, r ) = Ω|c(r) c † (r )|Ψ Ω|Ψ ,
even though |Ω and |Ψ both only have short range correlation between fermion operators. In another work we will demonstrate that the strange correlator proposed in Ref. 24 can still correctly distinguish trivial states from nontrivial TSCs under interaction. In particular, we calculate C(r, r ) for eight copies of 1d Kitaev's chain [22]. In the TSC phase, C(r, r ) is a constant in the noninteracting limit, while C(r, r ) immediately becomes short range with infinitesimal short range interactions [25]. Cenke Xu and Yizhuang You are supported by the David and Lucile Packard Foundation and NSF Grant No. DMR-1151208, Zhong Wang is supported by NSFC under Grant No. 11304175, Jeremy Oon is funded by the NSS Scholarship from the Agency of Science, Technology and Research (A*STAR) Singapore. The authors would like to thank Shou-Cheng Zhang for helpful discussions.
(1) changes by 16 at this transition. When d = 3, Γ 1,2,3 are 4 × 4 symmetric matrices, Γ 4 and Γ 5 = 4 a=1 Γ a are antisymmetric matrices. m = 0 is the quantum critical point between 8 copies of He 3 B TSC with topological number −1 and +1. The time-reversal symmetry acts on the Majorana fermions as Z T 2 : χ α,k → iΓ 5 Kχ α,−k , where K stands for complex conjugation. The topological number for He 3 B phase can be represented with Fermion Green's function:[14,16,23]
FIG. 1 :
1Schematic phase diagrams in (a) 1d, (b) 2d and (c) 3d. Red line/point marks out the physical phase transition line/point, where the fermion becomes gapless. The background color indicates the topological number.
Explicit Verification of the Emergent SO(8) SymmetryIn the fermion flavor space, the Hamiltonian H = H u + H w only has an SO(7) symmetry, but the resulting two-fermion Green's function actually has an emergent SO(8) symmetry, which is larger than the symmetry of the Hamiltonian. The smaller SO(7) symmetry will be manifested in the four-fermion (and higher order) Green's function, while at the two-fermion level, the Green's function still processes the full SO(8) symmetry.To verify this statement, we note that the 8 flavors of the Majorana fermions form the 8-dimensional spinor representation of the SO(7) group. We may write down the 21 generators of the SO(7) group in this spinor representation explicitly. Let T ij be the generator that performs the SO(7) rotation in the ij-plane. Its spinor representation is an 8 × 8 matrix, which can be written as a direct product of three Pauli matrices, denoted as σ µνλ ≡ σ µ ⊗ σ ν ⊗ σ λ ,To respect the SO(7) symmetry, the two-fermion Greens function G must commute with all the 21 generators listed above. We search over the space of all 8 × 8 matrices by a Mathematica program, and found that σ 000 (the 8 × 8 identity matrix) is the only matrix that commute with all the SO(7)generators. So we must have G ∝ σ 000 (or G α,α ∝ δ αα ) in the fermion flavor space, which indeed has the full SO(8) symmetry.
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|
[] |
[
"Reverse k Nearest Neighbor Search over Trajectories",
"Reverse k Nearest Neighbor Search over Trajectories"
] |
[
"Sheng Wang \nRMIT University\n\n",
"Zhifeng Bao \nRMIT University\n\n",
"J Shane Culpepper \nRMIT University\n\n",
"Timos Sellis \nSwinburne University of Technology\n\n",
"Gao Cong \nNanyang Technological University\n\n"
] |
[
"RMIT University\n",
"RMIT University\n",
"RMIT University\n",
"Swinburne University of Technology\n",
"Nanyang Technological University\n"
] |
[] |
GPS enables mobile devices to continuously provide new opportunities to improve our daily lives. For example, the data collected in applications created by Uber or Public Transport Authorities can be used to plan transportation routes, estimate capacities, and proactively identify low coverage areas. In this paper, we study a new kind of query -Reverse k Nearest Neighbor Search over Trajectories (RkNNT), which can be used for route planning and capacity estimation. Given a set of existing routes DR, a set of passenger transitions DT , and a query route Q, an RkNNT query returns all transitions that take Q as one of its k nearest travel routes. To solve the problem, we first develop an index to handle dynamic trajectory updates, so that the most up-to-date transition data are available for answering an RkNNT query. Then we introduce a filter refinement framework for processing RkNNT queries using the proposed indexes. Next, we show how to use RkNNT to solve the optimal route planning problem MaxRkNNT (MinRkNNT), which is to search for the optimal route from a start location to an end location that could attract the maximum (or minimum) number of passengers based on a pre-defined travel distance threshold. Experiments on real datasets demonstrate the efficiency and scalability of our approaches. To the best of our best knowledge, this is the first work to study the RkNNT problem for route planning.
|
10.1109/tkde.2017.2776268
|
[
"https://arxiv.org/pdf/1704.03978v1.pdf"
] | 3,822,093 |
1704.03978
|
157d8b6f693692dd947240cbbf03a2107ff71dae
|
Reverse k Nearest Neighbor Search over Trajectories
Sheng Wang
RMIT University
Zhifeng Bao
RMIT University
J Shane Culpepper
RMIT University
Timos Sellis
Swinburne University of Technology
Gao Cong
Nanyang Technological University
Reverse k Nearest Neighbor Search over Trajectories
GPS enables mobile devices to continuously provide new opportunities to improve our daily lives. For example, the data collected in applications created by Uber or Public Transport Authorities can be used to plan transportation routes, estimate capacities, and proactively identify low coverage areas. In this paper, we study a new kind of query -Reverse k Nearest Neighbor Search over Trajectories (RkNNT), which can be used for route planning and capacity estimation. Given a set of existing routes DR, a set of passenger transitions DT , and a query route Q, an RkNNT query returns all transitions that take Q as one of its k nearest travel routes. To solve the problem, we first develop an index to handle dynamic trajectory updates, so that the most up-to-date transition data are available for answering an RkNNT query. Then we introduce a filter refinement framework for processing RkNNT queries using the proposed indexes. Next, we show how to use RkNNT to solve the optimal route planning problem MaxRkNNT (MinRkNNT), which is to search for the optimal route from a start location to an end location that could attract the maximum (or minimum) number of passengers based on a pre-defined travel distance threshold. Experiments on real datasets demonstrate the efficiency and scalability of our approaches. To the best of our best knowledge, this is the first work to study the RkNNT problem for route planning.
INTRODUCTION
Reverse k Nearest Neighbor (RkNN) queries have attracted considerable attention [15,16,18,6,21,20]. An RkNN query aims to find all the points among a set of points that take a query point as their k nearest neighbors. The RkNN query has many applications in resource allocation, decision support and profile-based marketing, etc. For example, RkNN can be applied to estimate the number of customers of a planned restaurant among all existing restaurants.
The increasing prevalence of GPS-enabled devices provides new opportunities to obtain many real trajectory data in a short period of time, such as the GPS trajectories of taxi or uber drivers, checkin trajectories of social media users in Foursquare, which describe the movements/transitions of people. In this paper, we will explore the RkNN search over multiple-point trajectories (which is referred to as RkNNT). In a nutshell, the RkNNT query can be described as: taking a planned (or existing) route as a query Q, return all the passengers who will take the query route Q as one of the k nearest routes among the route set DR to travel. Here, a passenger's movement is modeled as a combination of an origin and a destination [10] such as home and office, which is called a transition. As shown in Figure 1, two red points represent a single transition T of a passenger. Among the two available routes, T would favour Route 2 as both points in T can find a point closer in Route 2 than in Route 1.
The main difference between our problem and RkNN is that our query is a route, and our data collections contain routes and user transitions respectively. RkNNT can estimate the passengers that will take the query route to travel. Another significant difference of our RkNNT from previous work is that the transition data is very dynamic and new transitions will arrive continuously, such as the request of uber passengers. Therefore, it is important to take this into consideration in the solution to answering the RkNNT query. RkNNT serves as a core yet fundamental operator in many applications in transportation field. The first and foremost one is to estimate the capacity of a route based on passengers' dynamic movements. On top of capacity estimation, RkNNT can be used for the following promising applications. 1) Optimal Route Planning. (i) Among a set of candidate routes, an RkNNT can be used to find the optimal route which has the maximum (minimum) number of passengers among a set of candidate routes. For Uber drivers, finding a route with the maximum number of passengers can increase profitability (the uber fare will be increased with a surge of passenger requests) and the chance of being hired. For ambulance and fire truck drivers, finding a route which has the fewest people can increase response times in emergency situations. We will address problem (i) in this paper as well.
(ii) Furthermore, by taking the temporal factor into consideration, i.e., user transitions at different time periods, it can help further adjust the frequency of planned vehicles on the planned routes, in order to save running cost for either individual vehicle drivers or public transportation authority [5].
The first challenge in answering RkNNT lies in how to prune the transitions which cannot be in the results without accessing every transition. A straightforward method is to conduct a kNN search for every transition, and then check the resulting ranked lists to see whether the query is a kNN. This method is intractable when there are a large number of transitions and new transitions are being added to the database. For MaxRkNNT, a brute force method can be used to find all candidate routes whose travel distances do not exceed the distance threshold, and then an RkNNT search can be executed on each candidate, and finally the one with maximum number of passengers is selected as the answer. This method is shown to be inefficient in our experimental study. Similar to RkNNT, it is crucial to prune out candidates which cannot be an optimal route. Another challenge is how to support dynamic updates as old transitions expire and new transitions arrive.
To addresses this challenge, we first build two R-tree indexes RR-tree and TR-tree combined with two inverted indexes PList and NList for the route and transition sets, respectively. Then, we choose a set of route points from the existing routes to form a filtering set by traversing the route index RR-tree. By drawing bisectors between the route points in the filtering set and the query, an area can be found where any transition point inside can not have the query as a nearest neighbor. After finding the filtering set, the pruning of transitions starts by traversing TR-tree and checking if a node can be filtered by more than k routes in the filtering set. Finally, all of the candidate transitions are verified using the filtered nodes during the traversal of RR-tree.
Next, we explore the optimal route planning problem, where we consider a graph formed by the bus network. Given a starting location and a destination, we find the optimal route R which connects the two locations in a bus network, and maximizes (minimize) the number of passengers that take R as its kNN without exceeding a distance threshold. We call this problem as the MaxRkNNT (MinRkNNT).
2) Bus Advertisement Recommendation. As an RkNNT query for a route can locate the set of passenger transitions who would take the travel route, and we can obtain the profiles of potential passengers using social networks, a deeper analysis of the common interests among passengers who take similar travel routes can be found. Consequently, we can select and broadcast advertisements that will have the largest influence on passengers taking a route.
For MaxRkNNT, a weighted graph is built by the pre-computed RkNNT set for every vertex. Then we start from the first vertex and access the neighbor vertex v to compute a partial route R. Then a reachability check on R is performed to see whether the estimated lower bound travel distance of R is greater than the distance threshold. Next, the dominance table of v is checked to see if R can dominate other partial routes which terminate at v. Further checks on the route are made when R meets all the conditions.
In summary, the main contributions of this paper are:
• We investigate the RkNNT problem for the first time, which serve as a core yet frequently adopted operator in many practical applications. In particular, we explore how to use RkNNT to plan an optimal route which attracts the most (or fewest) passengers (Section 3).
• We propose a filtering-refinement framework which can prune routes using a filtering set (in Section 4) and a voronoi-based optimization to further improve the efficiency (Section 5).
• We also introduce MaxRkNNT (MinRkNNT) queries which can be used to find the optimal route that attracts the maximum (minimum) number of passengers in a bus network (Section 6).
• We validate the practicality of our approaches using real world datasets (Section 7). In this section, we first compare the difference between our work and classic RkNN search over point and moving object data. Further, we will review related work on route planning.
PRELIMINARY & RELATED WORK
RkNN
RkNN on Spatial Points. Most existing RkNN search work focuses on static point data, and often employ a pruning-refinement frameworks to avoid scanning the entire dataset. However, these approaches cannot easily be translated to route search where both queries and collections are of multi-point trajectories. Given a set of candidates as a query, a maximizing RkNN query finds the optimal trajectory which satisfies the maximum number of results [17,24].
How to improve the search performance has attracted much attentions over last decades. The basic intuition behind filtering out a point p is to find another point which is closer to p than the query point q [16,18,6,21]. Here, we review the half space method and use a simple example to show how pruning works. Figure 2 shows a query point q and a data point p. As we can see, a perpendicular bisector divides the whole space into two sub-spaces, and all points inside the lower subspace would prefer p as a nearest neighbor, such as point r. For the reverse nearest neighbor search, r may be filtered from the candidate set of query q. More specifically, r can be filtered out if it can be pruned by at least k such points.
RkNN on Moving Objects. Given a moving object dataset D and a candidate point set O = (o1, o2, ..., om) as query, Shang et al. [14] find the optimal point from O such that the number of moving objects that choose oi as a nearest neighbor is maximized. Specifically, they proposed a Reverse Path Nearest Neighbor (R-PNN) search which finds the nearest point, and not the k nearest points. Shang et al. introduced the concept of influence-factor to determine the optimal point. The influence-factor of o is f if o is the nearest neighbor of f trajectories for all candidate points in O. Cheema et al. [7] proposed a continuous reverse nearest neighbors query to monitor a moving object and find all static points that take the moving object query as a k nearest neighbor. Recently, Emrich et al. [9] solved a problem of RNN search with "uncertain" moving object trajectories using a Markov model approach. A moving object is treated as a result when it always takes the query object as a nearest neighbor for every time stamp within a given time interval. All of these approaches target a single point rather than a transition of multiple-points, which is the focus of our work.
Route Searching
Bus Route Planning. Bus network design is known to be a complex, non-linear, non-convex, multi-objective NP-hard problem [8]. Based on existing bus networks, Pattnaik et al. [13] proposed a heuristic method which uses a genetic algorithm to minimize the cost of passengers and operators. Yang et al. [22] used ant colony algorithms to maximize the number of direct travelers between two nearby bus stops. Population estimation and user surveys [19] around the planned route are traditional ways to estimate the number of passengers that may use the planned travel route. However, the data is usually out-of-date as census data may only be gathered every five years, and may not reflect current travel patterns.
Chen et al. [8] tried to approximate night time bus route planning by first clustering all points in taxi trajectories to determine "hot spots" which could be bus stops, and then created a bus route graph based on the connectivity between two stops. Based on human mobility patterns, Liu et al. [10] proposed a localized transportation choice model, which can predict bus travel demand for different bus routes by taking into account both bus and taxi travel demands. However, the method must scan the static records for all of the customers, which is inefficient in practice, and the model has to be rebuilt whenever the records are updated.
Shortest Route Searching. Given a starting vertex and an ending vertex, the classical route planning problem is to find the shortest path in a graph. Best First Search (BFS) and Depth First Search (DFS) are two commonly used algorithms for this problem. An extension of this problem is k Shortest Path searching (kSP) [23,1], which aims to find the k shortest paths from a start vertex s to a target vertex t in a directed weighted graph G. Yen's algorithm [23] is a derivative algorithm for ranking the k shortest paths between a pair of nodes. The algorithm always searches the shortest paths in a tree containing the k shortest loop free paths. The shortest one is obtained first, and the second shortest path is explored based on the previous paths. The Constraint Shortest Path (CSP) problem [11,3] applies resource constraints on each edge, and solves the shortest path search problem based on these constraints. An example constraint would be time costs.
PROBLEM DEFINITION
In this section, we formally define the RkNNT problem and important notations are recorded in Table 1. Both a route and a transition are composed of discrete points called route point r and transition point t respectively. We use DT and DR to denote the transition set and route set. DEFINITION 3. (Point-Route Distance) Given a transition point t ∈ T and a route R, the distance dist(t, R) from t to R is the minimum Euclidean distance from t to every point of R, and calculated as:
dist(t, R) = min r∈R distance(t, r)(1)
Based on the point-route distance function, the kNN search of a transition point t is defined as: DEFINITION 4. (kNN) Given a set of routes DR, the kNN search of a transition point t ∈ T retrieves a set S ∈ DR of k routes such that for all R ∈ S, and for all R ∈ DR − S: dist(t, R) ≥ dist(t, R ).
In particular, two types of kNN are supported for a transition T , which can also be found in [9].
1. ∃kNN: T takes R as a kNN iff there exists a point t ∈ T taking R as kNN. So, ∃kNN(T ) = kNN(to) ∪ kNN(t d ).
2. ∀kNN: T takes R as a kNN iff both points to and t d take R as their kNN. So, ∀kNN(T ) =kNN(to) ∩ kNN(t d ). The route and transition sets dist(t, R)
Distance from transition point t to R ⊥(q, r) Perpendicular bisector between q and r Hq:r, Hr:q Two half-planes divided by perpendicular bisector ⊥(q, r) Hr:Q, HR:Q Filtering space formed by Q with r and R C(r) Crossover route set of r S filter , S refine Filtering set and filtered node set S cnd , S result Transition candidates and result set rootr (roott) Root of Route R-tree ( Transition R-tree ) VR,Q Voronoi diagram formed by R and Q G Weighted graph τ Travel distance threshold ω(R), ψ(R) RkNNT set and travel distance of R in G
M ψ [i][j]
Lower bound matrix of ψ(R) where R starts from vertex i to j Now, we can formally define the reverse k nearest neighbor query over trajectories. PROOF. Given a query Q, ∀RkNNT(Q) returns a set of transitions where both origin and destination points have the query as a kNN, then such transitions will also belong to the result of ∃RkNNT(Q), so ∀RkNNT(Q) ⊆ ∃RkNNT(Q). Let ∆(Q) = ∃RkNNT(Q)−∀RkNNT(Q), ∀T ∈ ∆: T only has one point that will take the query as a kNN, so ∆(Q) ∩ ∀RkNNT(Q) = ∅.
Using Lemma 1, the set of transition points which take Q as kNN can be searched for first, and then ∃RkNNT(Q) can be found by adding the corresponding routes. For ∀RkNNT, we need to remove Figure 3: Example of routes and transitions.
transitions that have only one point in ∃RkNNT(Q). Hence, a unified framework can be proposed that answers both ∃RkNNT and ∀RkNNT. In the rest of this paper, we use RkNNT to represent ∃RkNNT by default for ease of composition.
CAPACITY ESTIMATION -A PROCESS-ING FRAMEWORK FOR RkNNT
In this section, we first provide a sketch of our framework to answer the RkNNT query for capacity estimation, which includes the pruning idea based on routing points, and the proposed index structures. Then we describe each step in detail.
Main Idea
All impossible transition points are pruned using a PruneTransition algorithm, and the remaining candidates S cnd are further verified using a RefineCandidates algorithm to generate the final result set S result . Note that before pruning, a subset of routes S filter needs to be generated for efficient pruning, the reasoning and approach are described in Section 4.1.1. In summary, the whole procedure is composed of the three steps in Algorithm 1.
Algorithm 1: RkNNT(Q, rootr, roott)
Output: S result : the result set
1 (S filter , S refine ) ← FilterRoute(rootr, Q, k); // Sec 4.2.1 2 S cnd ← PruneTransition(roott, Q, S filter , k); // Sec 4.2.2 3 S result ← RefineCandidates(Q, S cnd , S refine ); // Sec 4.2.3 4 return S result ;
Pruning Characteristics
By Definition 5, a transition takes a route as a kNN if there exists at least one point (in the transition) that will take the route as a kNN. If there are more than k routes which are closer to a point in a transition than the query, then the point in this transition can be pruned. Such a route which helps prune transitions is called a filtering route. If both points of a transition are pruned, then the transition can be pruned safely, so pruning transitions helps to find the filtering routes to prune the points in the transition. LEMMA 2. If a transition point t is closer to a route point r ∈ R than Q, then t is closer to R than Q.
PROOF. We have dist(t, R) ≤ distance(t, r) according to Equation 1. If distance(t, r) < dist(t, Q), then dist(t, R) < dist(t, Q).
By Lemma 2, a transition point can be removed if it takes a set of routing points from more than k different routes as a kNN rather than the query. These points are called filtering points. Next, we introduce how to prune a transition points using the filtering point r from the routes.
Recall the example in Figure 2 where an RkNN can find an area where the points inside the area will not take the query as the nearest neighbor based on the half space. Similarly, given a query Q, we choose a point r from a route R in DR; then based on the straight line rqi formed by a point qi in Q to r, the perpendicular bisector ⊥(qi, r) is used to cut the space into two half-planes: Hr:q i and Hq i :r which contain r and qi, respectively. For every point qi in Q, there is a Hr:q i .
The intersection of all of the half spaces forms the filtering space Hr:Q defined as:
Hr:Q = q i ∈Q Hr:q i (2)
the point r which belongs to R ∈ DR is called as the filtering point.
As shown in Figure 4, we can see that there are five perpendicular bisectors. They form a polyline abcde which divides the whole space into two sub-spaces, and the left part is the filtering space HR 1:2 :Q. As T5 is entirely located in this area, it cannot take the query as its nearest route. The filtering space can also help filter a set of points using spatial indexes (see Sec. 4.1.2). If a maximum bounded box (MBB) such as MBR2 covering points T o 6 and T d 6 is located entirely inside the filtering space, then T o 6 and T d 6 inside this MBB will not take the query as a nearest neighbor and can be filtered out.
Every point in the route set DR can be a filtering point, but we cannot choose all points in the route set DR to do pruning especially when the whole set is large and located in external memory. To pruning a transition point, if we access all route points every time, the process is costly. In Section 4.2.1, we introduce how to generate a subset from the whole route set. Overall, we can observe three key characteristics based on the above analysis: 1) A filtering space exists between the query and a route point; 2) If a transition point is located in more than k filtering space of query Q simultaneously, then the point can be pruned; and 3) It is important to choose a subset of all routes as the filtering set.
Indexes
• RR-tree & TR-tree are two tree indexes for point data fetched from route dataset DR and transition dataset DT respectively, and referred to as a Route R-tree (RR-tree) and a Transition Rtree (TR-tree). The tree indexes are created first, and every point in the leaf node of RR-tree contains the IDs of the routes it belongs to. Every point in TR-tree also contains the IDs of the transition it belongs to. Through the transition ID and route ID in the node of RR-tree and TR-tree, we can get the corresponding route and transition for further refinement if two points of a transition are both pruned, and the two filtering points belong to the same route (See Section 4.2.3).
• NList. As we need to get all the routes that have a point inside a given node for verification in Section 4.2.3. Hence, for each node in RR-tree, we will create a list for every node of RR-tree by traversing the whole tree bottom-up to store all the IDs of routes inside.
• PList. The inverted list of each route point is created to store the IDs of the corresponding routes. As a bus stop can be shared by many routes in a bus network, we call this index a PList.
Our index supports dynamic updating, where new transitions and routes can be added into the index easily. This is in contrast the previous work [8,10] which needs to train whole dataset from scratch once there are new data inserted.
Key Functions
This section describes: 1) how to generate the filtering set S filter ; 2) how to prune and find all the candidate routes S cnd ; and 3) how to verify the candidate routes and further refine them to find the final query result S result .
Filtering Routes
In order to get a small filtering set S filter for a given query, an empty filtering set S filter is initialized, and new route points are added which cannot be pruned using the existing points of a route in S filter . We organize all the filtered points in a point list, sorted by the number of routes which cover each point, and denote the route set and point set as S filter .R and S filter .P respectively, which are materialized using two dynamic sorted hashtables. Specifically, for S filter .R, the key is the route ID, and the values are points of this route that cannot be filtered. For S filter .P , the key is the route point ID, and the value is a list of routes containing the point.
Note that in a real bus network, a route point can be covered by several routes. If a filtering point is contained by more than k routes, and a transition takes this filtering point as the nearest neighbor rather than the query, and then this transition point can be pruned. We will employ this enhancement to achieve the more efficient pruning. DEFINITION 7. (Crossover Route Set) Given a route point r, the set of routes which cover r is the crossover set of r, and denoted as C(r). The crossover route set of each filter point r ∈ S filter .P can be sorted by |C(r)| to give higher priority to the points which are crossed by more routes in the filtering phase.
Starting from the root node of RR-tree, the filtering algorithm iteratively accesses the entries of RR-tree from a heap in ascending order of their minimum distances to the query Q. The accessed points are used for filtering the search space. If an accessed entry e of index can be filtered -e is pruned by more than k routes -it can be skipped (see Algorithm 3). Otherwise, if e is an intermediate or leaf node, its children are inserted into the heap; if e is a route point and cannot be filtered, it is inserted in the filter set S filter and its half-space is used to filter the search space. The filtering algorithm terminates when the heap is empty. The details can be found in Algorithm 2.
The minimum distance from a child c to the query is computed as the minimum distance from every query point to the node c:
MinDist(Q, c) = min q∈Q MinDist(q, c)(3)
In Line 10 of Algorithm 2, a point that cannot be pruned is a filter point and is added into S filter . First the route ID of the point is found, and inserted into S filter .R. Then the point is inserted into the corresponding sorted point list S filter .P , and each point is affiliated with a list of route IDs containing it. Output: S filter : filtering set, S refine : filtered node set for refinement 1 minheap ← ∅, S filter ← ∅, S refine ← ∅; 2 minheap.push(rootr); 3 minheap.push(c, MinDist(Q, c)); 14 return (S filter , S refine ); Algorithm 3 shows how the filtering works. The filtering of a node is conducted in two steps. In step 1 (Line 2-10), the filter points S filter .P are processed to do the filtering. All points in S filter .P are sorted by the size of their crossover route set, and each point is accessed in a descending order. If a filtering point if found that can filter the node, then all affiliated route IDs are added to S. If S contains more than k unique route IDs, termination occurs and the node can be filtered out. After checking all the filtering points, step 2 (Line 11-16) is initiated, and the routes inside S filter .R are used for filtering. Finally, the filtering method based on Voronoi diagrams is employed (Section 5.1).
Transition Pruning
Based on the filter set S filter , entries e from TR-tree are added to a heap which is sorted by the distance to the query in ascending order, and checked to see if they can be pruned by S filter using Algorithm 3. Algorithm 3 uses the candidates in S filter to check whether e is located in a filtering space of Q. The transition points that cannot be pruned are inserted into the candidate set for further refinement.
Algorithm 4 describes the procedure to prune the transition points using the generated filter set S filter from TR-tree. It is similar to the filtering method for generating the filtering set. The main difference with the traversal of Route R-tree is that only the unpruned points need to be stored, and the filtering set S filter is fixed. As a result, a set of transition points S cnd is obtained which takes the query routes as k nearest neighbors.
Verification
The verification mainly uses the filtered node set S refine during the traversal of RR-tree to find S filter in Algorithm 2. It can be divided into two steps. First, the nodes in RR-tree encountered during the filtering phase are kept in S refine in Line 7 of Algorithm 2. The verification algorithm runs in rounds. In each round, one of the nodes in S refine is opened and its children are inserted into S refine . During each round, the nodes and points in S refine are used to identify the candidates that can be verified using S refine , which are the nodes confirmed as RkNNT or guaranteed not to be RkNNT. Such candidates are verified and removed from S cnd . The algorithm terminates when S cnd is empty. The result set is then stored for a second round of verification.
To verify a candidate effectively, if more than k routes are found in S refine which are closer to the query than the candidates, then it can safely be removed from S cnd . Hence, we maintain a set to store the unique IDs of these routes when every candidate point is checked in S cnd . The route IDs are found, and the set is updated using NList when new filtering points or nodes from S refine are found. When the number of IDs in a set is greater than k, it can be removed from S cnd .
After finding the transition points for the routes, as they will take the query as k nearest neighbors, then for the ∃RkNNT, the transition ID for all remaining points can be returned as the final result S result in the second step. While ∀RkNNT, if a transition only has one point in the result set, then it will be pruned, and only the transitions which have both points in the result will be considered as the real result and added to S result .
OPTIMIZATIONS ON FILTERING PRO-CESS
Voronoi-based Filtering
One problem of the filtering method in Algorithm 3 is that the filtering space obtained from a single point and the query is usually very small. For example in Figure 4, MBR1 cannot be pruned, so it needs to load MBR2 and MBR3 to perform further checks, which require additional pruning time. To further enlarge the pruning space, the available filtering points in a single route can be used rather than a single point to perform the pruning, namely, S filter can be used for additional pruning. Given a query and a filtering route R, a larger filter space can be explored. To find this area, Voronoi cells can be used.
To accomplish this, a plane is partitioned with points into several convex polygons, such that each polygon contains exactly one generating point, and every point in a given polygon is closer to its generating point than to any other. The convex polygon of one point is called as the Voronoi cell, and the point is called the kernel of this cell. By plotting the Voronoi diagram VR,Q between the query Q and a filtering route R, as shown in Figure 5. The Voronoi cell VR,Q[p] of the route R can be found, and any point inside these cells is closer to the filtering route than the query. Furthermore, if a node does not intersect with any cell of the query, then any point inside this node will be closer to the filtering route than the query. If a node can find more than k such filtering routes, then the node can be pruned. DEFINITION 8. (Voronoi Filtering Space) Given a filtering route R and query Q, we define the Voronoi filtering space as:
HR:Q = p∈R VR,Q[p](4)
which is a union of the Voronoi cells of all points from R, and VR,Q is the Voronoi diagram of union of points from R and Q.
Any transition point inside HR:Q cannot have Q as the nearest neighbor. As shown in Figure 5, for any point in the Voronoi filtering space, it can find a point in the filtering route which is closer than any point in the query. Hence, two points in a transition can both find a point in the filtering route rather than the query, so the transition point will not the take query as the nearest neighbor.
The filtering route R is used to further prune the transition point if it cannot be pruned by the filter points in R one by one as shown in Line 2 -10 of Algorithm 3. After scanning all the filtering points of a route in S filter .P , we will use the Voronoi filtering space of the route for the query to prune the transition points, where the space has been created after getting the filtering route set, then the pruning space will be larger, and we can find one more route which is closer to the entry than the query if it can prune the entry.
For example, consider the 4 points belonging to a same route R1 to prune the transition points in Figure 5. The filtering space is larger than the area shown in Figure 4, and MBR1 is entirely
Query
Filtering point set of a route Figure 5: Pruning based on the Voronoi diagram of a query (red) and a filtering route composed of 4 points (black).
located within the filtering space, so it can be pruned from consideration. Since the Voronoi diagram can be produced at the same time as when the perpendicular bisectors from query to every filtering point are computed, then there is no additional cost to generate the Voronoi information. This additional pruning rule improves the probability of a node being pruned.
Divide & Conquer Method
Note that the processing of the proposed method will be complex when the query has many points. The main reason is that a node has to be filtered by every query point, and the probability of a point being pruned will be lower when the query length is larger. To alleviate this problem, we introduce a divide-and-conquer method based on our processing framework. LEMMA 3. The RkNNT of a multi-point query is the union of the RkNNT of all points in a query:
RkNNT(Q) = q i ∈Q RkNNT(qi)(5)
PROOF. For a transition, if it takes a query point as a k nearest neighbor, then it must be a RkNNT result for Q, so q i ∈Q RkNNT(qi) ⊆ RkNNT(Q). For each transition in RkNNT(Q), it must take one query point in Q as the kNN, then RkNNT(Q) ⊆ q i ∈Q RkNNT(qi). Based on above two observations, RkNNT(Q) = q i ∈Q RkNNT(qi).
Based on this observation, a divide and conquer framework is proposed that uses multiple RkNNT searches which were introduced in Section 4. The main idea is that RkNNT search is performed for every query point to find a candidate transition point set for every query point first, and then the transitions containing these points are merged to get the final transition result.
Even though an RkNNT query mainly targets a route query, it can process single-point queries as well since every step in the algorithm does not require that the query have more than one point. According to Definition 6, the filtering space will be the largest when there is only one query point, so the pruning efficiency will be the highest when compared with any multi-point query which extends from this single query point.
OPTIMAL ROUTE PLANNING
In this section, we present a solution to the route planning problem with a distance threshold based on RkNNT. We first define a new query called MaxRkNNT. A baseline method is proposed first, and then an efficient search method based on pre-computation and pruning is described.
Maximizing RkNNT in a Bus Network
In bus route planning, the goal is how to attract the maximum number of passengers within a given distance threshold, since a single bus cannot cover all stops in a city. For Uber drivers, such a route also means high possibility to get high profit by attracting bus passengers to take the Uber to travel alternatively. Next, we will introduce the maximizing RkNNT in the bus network.
Here we take the real bus networks in NYC and LA as an example. Figure 6 shows that the ratio between the travel distance and the straight line distance between start and end bus stops, and does not exceed 2 in most bus routes. Hence, such a distance constraint always exists in real-life route planning.
We first cast the existing bus network as a Weighted Graph. R = (v1, v2, . . . , vn) ∈ V × V × · · · × V such that vi is adjacent to vi+1 for 1 ≤ i < n, v1 and vn are the start and end vertex respectively.
Given a route R, ψ(R) is the travel distance starting from start to end through every vertex in the route:
ψ(R) = p i ∈R&i∈[1,n−1] distance(pi, pi+1)(6)
Recall Definition 5, given a route R in G, among the transition set DT , the RkNNT of R can find all transitions that would choose it as a kNN. The passengers who are likely to take R are the RkNNT set of R. Let ω(R) = RkNNT(R) for simplicity. We now formally define the Maximizing RkNNT (MaxRkNNT) problem for route planning. DEFINITION 10. (MaxRkNNT) Given a distance threshold τ , a starting vertex vs and a destination vertex ve, MaxRkNNT(vs, ve, τ ) returns an optimal route R from Sse such that ∀R ∈ Sse − R, |ω(R)| ≥ |ω(R )| and ψ(R) ≤ τ , where Sse is the set of all possible routes in G that share same start and end vertex.
The definition of MinRkNNT can be defined by changing |ω(R)| ≥ |ω(R )| to |ω(R)| ≤ |ω(R )| in Definition. 10. In this paper, we propose a search algorithm which can solve both MaxRkNNT and MinRkNNT. By default, we choose MaxRkNNT for ease of illustration.
Baseline. A brute force method for
MaxRkNNT is to find all the candidate routes which meet the travel distance threshold constraint. This can be done by extending the k shortest path method proposed by [23,12] with a loop to find the sub-optimal route until the distance threshold τ is met. Then a RkNNT query is ran for each candidate and the one with maximum number of results as the optimal route is selected. We call this method BF. Recall the query in Figure 7, where almost all routes such as abej,acej and acehj will be candidates.
However, the performance of RkNNT decreases as the number of points increases, which is discussed in more detail in Section 7.3. For the bus route planning, it may be tolerable to wait for a few seconds to conduct MaxRkNNT query. However, for real time queries, like identifying profitable routes for Uber drivers, this method will not work well. To achieve better performance, an efficient route searching algorithm is proposed based on the precomputation of the RkNNT set for each vertex in G.
Our Solution
According to Lemma 3, the query Q can be decomposed into a set of |Q| queries, which means that we can get the pre-computed RkNNT set for every vertex, and conduct a union operation on all vertices in a route to get the final RkNNT set for that route.
By using the above property, we introduce a pre-computation based method with a fixed k which provides better performance.
Note that even though k should be fixed in the pre-computation, multiple datasets of representative k can be generated in advance to meet different requirements.
Pre-computation
For every vertex in G, an RkNNT query is ran, and the result stored. A pre-computed matrix M ψ [i][j] is created which stores the pre computed all-pair shortest distance for all vertexes in G using the Floyd-Warshall algorithm [4]. The details of precomputation can be found in Algorithm 5.
foreach vertex v ∈ G − v do 7 M ψ [v][v ] ← ShortestDistance(G, v, v ); 8 return G.V ;
With the pre-computed RkNNT set, we can further improve the performance of the baseline method BF. After getting all candidate routes that do not exceed the distance threshold, the RkNNT set of each route can be found by performing a union operation on the sets. Compared with the baseline method, the on-the-fly RkNNT query is replaced with pre-computation, and the running time for the search is reduced to the search time of k shortest path search. However, it is still possible to leverage distance constraints and dominance relationships to prune additional routes in advance. Figure 7, the red points are the start O = a and end D = j respectively. A query formed by these two points and τ = 6 return the route with largest RkNNT set, where the number on each edge is the distance between two vertices, and the label is the vertex ID. The table shows the pre-computed RkNNT set for each vertex. So, ω(acf hj) = {T1, T2, T3, T4, T6} and ψ(acf hj) = 1 + 1.5 + 1.4 + 1.5 = 5.4.
EXAMPLE 2. As shown in
Route Searching by Pruning
After getting the RkNNT set for every vertex in the graph G, Algorithm 6 can be ran to get the optimal route based on the pre-computed Euclidean distance of every edge. Specifically, the neighbor vertices are accessed around the starting point, and two Figure 7: An exemplar graph with a query (a, j, 6) where a and j are the start and end vertexes, τ = 6 is the distance threshold, and the table shows the RkNNT set for each vertex.
b T d 1 c T d 1 , T o 3 , T o 4 d T o 5 e T o 2 f T o 2 , T d 3 , T d 4 g T o 5 h T d 2 i T o 6 j T d 6
levels of checking are performed to see whether the current partial route R * is feasible. If it is, it is inserted into the priority heap Q, and the partial route is increased until it meets the end point ve and has the maximum result set size. Specifically, the two checking functions work as below:
Algorithm 6: MaxRkNNT(o, d, τ ) Output: R: the optimal route 1 if checkReachability(vs, ve, τ ) then 2 return ∅;
3 R ← ∅, R * ← {vs}; 4 ψ(R * ) ← 0; // travel distance 5 ω(R * ) ← G.V.RkNNT(vs); // RkNNT set 6 Q ← ∅ ;
// queue stores the partial routes 7 push(Q, {R * , ψ(R * ), ω(R * )});
8 max ← |ω(R * )| ; 9 while Q = ∅ do 10 {R * , ψ(R * ), ω(R * )} ← pop(Q); 11
vi ← GetEnd(R * ); 12 foreach vj ∈ Neighbor(G, vi) do 13 if checkReachability(vj, d, τ − ψ(R * )) then 14 if checkDominance(o, vj, ψ(R * ), ω(R * )) then 15 S ← Update(DT [d], ψ(R * ), ω(R * ))); 16 foreach candidate ∈ S do 17 Delete(Q, candidate); checkReachability. This pruning function checks whether the current route meets the distance constraint -namely that the distance from the current vertex to the end vertex is less than τ − ψ(R * ). When M ψ [vj][d] > τ − ψ(R * ), it will return false and move to next neighbor of vertex vi in G.
18 R * ← R * ∪ {vj}; 19 ψ(R * ) ← ψ(R * ) + ψ(vi, vj); 20 ω(R * ) ← ω(R * ) G.V.RkNNT(vj);
checkDominance. This pruning function exploits the dominance relationship between two partial routes. If a partial route exists that ends at the same vertex and has a short route and a larger RkNNT set, then it can dominate the current route. Specifically, a dominating lemma is introduced which works for both ∀RkNNT and ∃RkNNT.
LEMMA 4. Given two partial routes R * 1 and R * 2 which have the same start and end, R * 1 dominates R * 2 in MaxRkNNT (R * 2 dominates R * 1 in MinRkNNT) when |ψ(R * 1 )| < |ψ(R * 2 )| and |∀RkNNT(R * 1 )| > |∃RkNNT(R * 2 )|. PROOF. For purposes of proving the lemma, we use ω(R) and ω * (R) to represent ∃RkNNT(R) and ∀RkNNT(R) to distinguish them. Given any partial route R which starts at vj and ends at d, R * 1 and R * 2 can be connected to form two complete routes R1 and R2. 1) For ∃RkNNT, if |ω * (R * 1 )| > |ω(R * 2 )|, then |ω(R1)| ≥ |ω * (R * 1 )|+|ω(R )|, as there is no intersection between ω * (R * 1 ) and ω(R ) because T ∈ ω * (R * 1 ) is the set of transitions that have kNN in R * 1 for both origin and destination points. Given that |ω(R2)| ≤ |ω(R * 2 )| + |ω(R )|, |ω(R1)| > |ω(R2)|, while ψ(R * 1 ) < ψ(R * 2 ), 2) ∀RkNNT, |ω * (R2)| ≤ |ω(R * 2 )| + |ω * (R )|, while |ω * (R1)| ≥ |ω * (R * 2 )|+|ω * (R )|, so |ω * (R1)| > |ω * (R2)|. Without further spreading, we can see the priority relationship between |ω(R * 1 )| and |ω(R * 2 )| holds.
In Algorithm 6, a dynamic table DT is maintained to store the pairs for every vertex accessed, and updates continue when new feasible partial routes are explored during the search. This is used to compare the RkNNT set and the travel distance of partial routes. The entry for a vertex v inserts a partial route R * which ends at v when an existing partial route cannot be found which dominates R * . After insertion, old entries in DT that are dominated by the new route R * are removed. If a new one is found that dominates R * , the loop terminates, and the next partial route is processed. EXAMPLE 3. In Figure 7, {{a}, 0, 20} is added to the queue Q after checking the reachability from a to j by comparing the pre-computed shortest distance with τ . Then, pop the queue Q to get the partial route R. Next, the last point a of R is checked to see if its neighbor b can be reached, and it can since ψ(bej) = 3.8 < (6 − 1.6). So {{a, b}, 1.6, {T1}} is added to Q. Similarly, {a, c} is inserted into G. {a, d} cannot be enqueued as the shortest distance from d to j is ψ(df hj) = 5. For MinRkNNT, Line 8 is changed to max ← ∞, and Line 23 is changed to |ω(R * )| < max. Moreover, one additional check called checkBounds(max, ω(R * )) after Line 14 in Algorithm 6 must be added. Given a partial route R * and the existing optimal route R and max, R * can be discarded when |ω(R * )| > max as R * can not beat the existing optimal route R. We conducted experiments to evaluate our solutions to RkNNT and MaxRkNNT using real bus route data and check-in data from Foursquare 1 in New York and Los Angeles, which are two largest cities in the USA. We have published our dataset 2 to improve the reproducibility of our results. Figure 8 shows the heatmap of the route and check-in datasets. All experiments were performed on a machine using an Intel Xeon E5 CPU with 256 GB RAM running RHEL v6.3 Linux, implemented in C++, and compiled using GCC 4.8.1 with -O2 optimization enabled.
Route Datasets. We use two real bus network datasets, namely NYC-Route and LA-Route. We extracted the data from the GTFS datasets of New York 3 and Los Angeles 4 . Table 2 provides a breakdown of each dataset.
Transition Datasets. Two real transition datasets, NYC-Transit and LA-Transit, were produced by cleaning the Foursquare check-in data [2], and statistics for the cleaned data is shown in Table 3. Specifically, we divided a user's trajectory with multiple points into several transitions with two points. A trajectory with n points can be divided into n − 1 transitions. Since the real dataset is small, we also generated a synthetic dataset which contains 10 million transitions for the NYC dataset, and refer to it as NYC-Synthetic.
Evaluation of RkNNT
Algorithms for evaluation. We compared the following methods when processing RkNNT over the two datasets. (1) Filter-Refine: The basic framework proposed in Section 4. (2) Voronoi: The Voronoi-based method which can create a larger filtering area by drawing a Voronoi diagram based on the query and filtering route after regular filtering by points. (3) Divide-Conquer: As proposed in Section 5.2.
Queries. We prepared two query sets: the first set is a synthetic query set for the purposes of parameter evaluation, and generated as follows: 1) We randomly generated 1, 000 points from DR. 2) We iteratively chose each point as a start point, and append new points one by one with a limited rotation angle to simulate a realistic case. The rotation angle of every time extension does not exceed 90 • , so the query route will not zigzag [8]. All experimental results are averaged by running all 1, 000 queries. The second query set contains all the routes in NYC-Transit and LA-Transit, which are used as queries to test our most efficient method, Divide-Conquer.
Parameters. Table 4 summarizes all key parameters for a query, and the default values are underlined. I = ψ(Q) |Q| is the interval length between two adjacent points in the query, where ψ(Q) is the travel distance of the query route and can be computed by Equation 6.
(a) LA Effect of |Q|. Figure 11 shows the running time of our three methods. As more points are added into the query, Filter-Refine and Voronoi exhibit a sharp increase in running time. Since these methods need more time to check whether a node can be filtered, the filtering space becomes smaller and the probability of being pruned decreases. In contrast, Divide-Conquer shows almost a linear increase. This is probably a result of the whole query being divided into |Q| queries, and a node is not be pruned by checking every query point. Figure 12 shows a breakdown of the running time to the tasks of filtering and verification on the LA data. We can see that the verification occupies more than 80% for most cases.
F R V O D C F R V O D C F R V O D C F R V O D C F R V O D C F R V O D
Effect of k. Figure 9 shows that the time cost for all three methods will increases as k increases. This is because it is unlikely that a point can be filtered by k filtering routes when k is large.
Effect of I. We observe that the intervals I between two adjacent points vary from route to route in real life. Hence, we conducted experiments to see how the running time is affected in this scenario. As mentioned when describing query generation, the size of the query is increased by appending randomly generated points, one at a time. Figure 13(a) and Figure 13(b) show that there is a slight increase on the running time when I is large. The main reason is that when two query points are close, a node can be filtered by a filtering point easily, while when the intervals are large, it is harder to filter a node.
Real Route Queries. After testing the effect of each individual parameter, we took every route in each dataset as a query to evaluate our best method Divide-Conquer. Note that before running each query, we removed the points of this route from the RR-tree index. Figure 16 shows that over 90% of the queries can be processed in less than 5s. The main reason is the relationship to the number of points in the query.
F R V O D C F R V O D C F R V O D C F R V O D C F R V O D C F R V O D C F R V O D C F R V O D
In summary, our main observations are:
1. Divide-Conquer consistently has the best performance, followed by Voronoi, with Filter-Refine being the worst.
2. All three methods are sensitive to k and |Q|. Only Filter-Refine and Voronoi are sensitive to the interval length I of the query.
3. When taking existing routes as real queries, most queries can be answered efficiently using Divide-Conquer.
Evaluation of MaxRkNNT
Algorithms for evaluation. (1) BruteForce: the baseline method which uses the k shortest paths [23] to find all the routes which have a smaller travel distance than the distance threshold τ , after which an RkNNT is performed on every candidate to choose the maximal one.
(2) Pre: the method that extends the BrouteForce by precomputation of the RkNNT set for every vertex without an on-thefly RkNNT query. For MaxRkNNT and MinRkNNT, both can be solved using the same pruning techniques with little difference in bound checking, which has a small impact on performance. We denote them as (3) Pre-Max and (4) Pre-Min.
Queries. To test the effect of key parameters, we first generated a point set by choosing 1, 000 start points randomly from our route datasets. Then, we searched 6 end points for every start point with different ψ(se), which is the distance between the origin and the destination, as shown in the last row of Table 4. Furthermore, we used existing representative routes as queries and employed MaxRkNNT and MinRkNNT search algorithm to find the new "optimal" routes. Finally, we compared the RkNNT sets of the original routes against the new routes.
Parameters. We discovered two key parameters that affect the performance of MaxRkNNT: (1) the coverage degree of a bus routedenoted by ψ(se) and quantified as the Euclidean distance between the start and end points of a query Q.
(2) τ ψ(se) , which is the ratio of the travel distance over the straight-line distance from origin to destination of Q. The choices of these parameters are from the distribution of all real bus routes, as shown in Figure 17.
F R V O D C F R V O D C F R V O D C F R V O D C F R V O D C F R V O D C CPU Cost ( s )
Interval(km)
Verification Filtering Pre-computation. Table 5 shows the time spent on precomputation. The pre-computation consists of of two steps: the RkNNT query for every vertex, and the shortest distance route search, as shown in Algorithm 5. All-pair shortest distance computation costs about 4 minutes for both datasets, and the RkNNT search of all vertices in G costs less than 5 minutes when k = 10. For the synthetic dataset which contains 10M transitions, the time spent on pre-computation is about 12 minutes when k = 10.
Effect of ψ(se). Figure 17(a) shows that the time spent on the search task increases when the distance between the origin and destination ψ(se) increases. This is because more vertices in the graph need to be scanned between the origin and destination. For Bruteforce, the reasons are twofold: (1) It returns more candidate routes for RkNNT; (2) The candidate routes are longer when ψ(se) is long, so more time has to be spent for every RkNNT query. In contrast, for the remaining three methods, since we have pre-computed the RkNNT set for every vertex, the running time comes from the search over G. Pre-Max has the best performance due to the bound checking during the spreading of partial routes.
Effect of τ ψ(se) . To generate the query, we choose a subset of queries with a fixed ψ(se) as the default value shown in Table 4 and alter τ in the experiment. Figure 19 shows that increasing τ ψ(se) leads to an increased running time. The reason can also be ascribed to the increasing number of candidates between the origin and destination.
Real queries. We took each route in DR as a query to perform an MaxRkNNT search to see whether we can find a better route which has a larger RkNNT set while maintaining an acceptable travel distance threshold. Each query is generated using the start and end bus stop, and the travel distance for each route. Figure 20 shows the running time distribution for the real queries. We can see that most queries in the LA data can be answered in less than a second.
In Figure 21, we show four kinds of routes which share the same start and end locations: 1) the original bus route passes through Manhattan, 2) the shortest distance route, 3) the MaxRkNNT route which attracts the most passengers, 4) the MinRkNNT route which attracts the fewest passengers. The right table shows the search time, number of passengers, travel distance, and number of stops for these four routes. We find: (1) the original route and the MaxRkNNT route are almost the same (in particular, MaxRkNNT finds a route which just is 10 meters longer but can attract 129 extra passengers), which means that the existing bus route is almost optimal between the start and end locations. (2) If a driver wants to save time, the least crowded route can be selected as provided by MinRkNNT; if the car should be shared to increase revenue, the route found by MaxRkNNT is a good choice.
CONCLUSION
In this paper, we proposed and studied the RkNNT query, which can be used directly to support capacity estimation in bus networks. First, we proposed a filter-refine processing framework, and an optimization to increase the filtering space that improves pruning efficiency. Then we employed RkNNT to solve the bus route planning problem. In a bus network, given a start and end bus stop, we can find an optimal route which attracts the most passengers for a given travel distance threshold. To the best of our knowledge, this is the first work studying reverse k nearest neighbors in trajectories, and our solution supports dynamically changing transition data while providing up-to-date answers efficiently.
(a) LA
Figure 1 :
1Bus Routes and User Transition.
Figure 2 :
2Pruning by half-space.
DEFINITION 1 .
1(Route) A route R of length n is a sequence of points (r1, r2, ..., rn) , n ≥ 2, where ri is a point represented by (latitude, longitude).
DEFINITION 2 .
2(Transition) A transition T contains an origin point to and a destination point t d .
DEFINITION 5 .
5(RkNNT) Given a set of routes DR, a set of transitions DT , and a query route Q, ∃RkNNT(Q) (∀RkNNT(Q)) retrieves all transitions T ∈ DT , such that for all T : Q ∈ ∃kNN(T ) (∀kNN(T )).
EXAMPLE 1 .
1In Figure 3, R1, R2, R3 and R4 are routes. T1, T2, T3, T4, T5, T6 are transitions, and T o 1 and T d 1 denote the origin point and destination point for transition T1. The query route Q is composed of 5 query points (in red). If we take the ∀RkNNT query, as point T o 4 and T d 4 take Q as the nearest route, T4 will be the result of ∀RkNNT(Q).
LEMMA 1 .
1Given a query Q, ∀RkNNT(Q) ⊆ ∃RkNNT(Q).
Figure 4 :
4Pruning by half-space for a multi-point query Q. DEFINITION 6. (Filtering Space) Given a route point r and a query Q, the intersection of all Hr:q i forms a filtering space:
For example in Figure 3 ,
3R1 and R4 intersect at the second point R1:2, then C(R1:2) = {R1, R4}. Using the PList, we can retrieve the crossover route set of a point r easily, where C(r) = PList[r].
Algorithm 3 :label = true then 10 SS
310IsFiltered(Q, S filter , node, k)Output: whether the node can be filtered 1 S ← ∅; 2 foreach p ∈ S filter .P do // access list points in ← S ∪ C(p); // crossover route set 11 S ← S filter .R − S; 12 foreach route ∈ ← S ∪ {route}; 17 return false;
Algorithm 4 :
4PruneTransition(roott, Q, S filter , k) Output: S cnd : candidate set 1 minheap ← ∅, S cnd ← ∅; 2 minheap.push(roott); 3 while !minheap.isEmpty() child c of e do 12 minheap.push(c, MinDist(c, Q)); 13 return S cnd ;
DEFINITION 9 .Figure 6 :
96(Weighted Graph) G = (E, V ) is a weighted graph, where V is the vertex set and E is a set of edges which connect two vertices among V . A route in G is a sequence of vertices Frequency histogram of the ratio between travel distance and straight-line distance for all routes in LA and NYC.
Algorithm 5 :
5Precomputation(G, DR, DT , k) Output: G.V : the vertexes with RkNNT set 1 roott ← CreateIndex(DT ); // root of RR-tree 2 rootr ← CreateIndex(DR); // root of TR-tree 3 foreach vertex v ∈ G do 4 S result ← RkNNT(v, rootr, roott);
21 push(Q, {R * , ψ(R * ), ω(R * )}); 22 if GetEnd(R * ) = ve then 23 if |ω(R * )| > max then // new optimal route 24 R ← R * ; 25 max ← |ω(R * )|; 26 return R;
2 > (6 − 1). {{a, b, e}, 3.1, {T1, T2}} and {{a, c, e}, 2.6, {T1, T2, T3, T4}} are enqueued and DT [e] = {{a, b, e}, 3.1, {T1, T2}} is updated. Further, {{a, c, f, h}, 3.9, {T1, T2, T3, T4}} is enqueued. {{a, b, e, h}, 4.5, {T1, T2}} has a greater travel distance, and ω(abeh) = {T1, T2}, and ω * (acf h) = {T1, T2, T3, T4}, so |ω * (acf h)| > |ω(abeh)|, acf h dominates acf h. Based on this extension in the graph, when Q is empty, the algorithm terminates.
Figure 8 :
8The heatmap of the bus route dataset (left) and the transition dataset (right) in NYC (up) and LA (down).
Figure 9 :
9Effect on Running Time with the increasing of k.
Figure 10 :
10Breakdown of running time with increasing k in LA.
Figure 11 :
11Effect on running time with the increasing of |Q|.
Figure 12 :Figure 13 :
1213Breakdown of running time w.r.t. |Q| in LA. Effect on running time with the increasing of |Q| and k in synthetic dataset.
Figure 14 :
14Effect on running time with the increasing of I.
Figure 15 :
15Breakdown of running time with increasing I in LA.
Figure 16 :Figure 17 :
1617The distribution of running time when taking all existing bus routes as query by MaxRkNNT when k = 10. Frequency histogram of ψ(se), I and |R| in LA (left) and NYC (right).
Figure 18 :Figure 19 :
1819Effect on running time as ψ(se) increases. Effect on running time with the increase of τ ψ(se) .
Figure 20 :Figure 21 :
2021Distribution of running time of MaxRkNNT on real route query. Comparison among four routes: ST (searching time), NP (number of passengers), TD (travel distance) and the number of stops.
Table 1 :
1Summary of Notation
Notation
Definition
R
The route composed of points {r1, ..., rn}
T
The transition {to, t d }
Q
Query route {q1, ..., qm}
DR, DT
Table 2 :
2Route Datasets.Dataset
|DR|
|G.E|
|G.V |
LA-Route
1, 208 72, 346 14, 119
NYC-Route 2, 022 61, 118 16, 999
Table 3 :
3Transition Datasets.
Dataset
|DT |
Latitude
Longitude
LA-Transit
109, 036
[32 • ,35 • ] [−120 • ,−117 • ]
NYC-Transit
195, 833
[39 • ,42 • ]
[−75 • , −72 • ]
NYC-Synthetic 10, 000, 000 [39 • ,42 • ]
[−75 • , −72 • ]
Table 4 :
4Parameter Settings.Para
Value
|Q|
3,4,5,6,7,8,9,10
k
1,5,10,15,20,25
I
1km, 2km, 3km, 4km, 5km, 6km
ψ(se) 10km, 20km, 30km, 40km, 50km
τ
ψ(se)
1, 1.2, 1.4, 1.6, 1.8, 2.0
Table 5 :
5Running time(s) for pre-computation when k = 1, 5, 10, which is composed of RkNNT search and all-pair shortest distance computations, the bold numbers are the results for synthetic dataset.LA
NYC
k
1
5
10
1
5
10
RkNNT 80.5 153.2 230.8
140.4 202.1 253.5
201.7 545.8 748.1
Shortest
191.3
251.9
https://foursquare.com/ 2 https://sites.google.com/site/shengwangcs/home/rknnt 3 http://web.mta.info/developers/developer-data-terms.html#data 4 http://developer.metro.net/gtfs/google_transit.zip
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|
[] |
[
"Exploiting Facial Landmarks for Emotion Recognition in the Wild",
"Exploiting Facial Landmarks for Emotion Recognition in the Wild"
] |
[
"Matthew Day [email protected] \nDepartment of Electronics\nUniversity of York\nUK\n"
] |
[
"Department of Electronics\nUniversity of York\nUK"
] |
[] |
In this paper, we describe an entry to the third Emotion Recognition in the Wild Challenge, EmotiW2015. We detail the associated experiments and show that, through more accurately locating the facial landmarks, and considering only the distances between them, we can achieve a surprising level of performance. The resulting system is not only more accurate than the challenge baseline, but also much simpler.
| null |
[
"https://arxiv.org/pdf/1603.09129v1.pdf"
] | 15,512,505 |
1603.09129
|
b8df2c2fa02a37d09b73277ca4edde654ac80953
|
Exploiting Facial Landmarks for Emotion Recognition in the Wild
Matthew Day [email protected]
Department of Electronics
University of York
UK
Exploiting Facial Landmarks for Emotion Recognition in the Wild
1 I210 [Artificial Intelligence]: Vision and Scene Understanding - modeling and recovery of physical attributes, shape, texture General Terms Algorithms, Performance, Design, Experimentation Keywords Machine learningEmotion recognitionFacial landmarksBIFSVMGradient boosting
In this paper, we describe an entry to the third Emotion Recognition in the Wild Challenge, EmotiW2015. We detail the associated experiments and show that, through more accurately locating the facial landmarks, and considering only the distances between them, we can achieve a surprising level of performance. The resulting system is not only more accurate than the challenge baseline, but also much simpler.
INTRODUCTION
Accurate machine analysis of human facial expression is important in an increasing number of applications across a growing number of fields. Human computer interaction (HCI) is one obvious example. Others include medical monitoring, psychological condition analysis, or as a means of acquiring commercially valuable feedback. 1 Whilst the problem was traditionally addressed in highly constrained scenarios, an increasing focus on 'in the wild' (i.e. unconstrained) data has emerged in recent years. In this respect, the EmotiW2015 challenge [5] aims to advance the state of the art. The challenge is divided into two sub-challenges: (1) audiovideo based and (2) static image based. Audio, in particular, has been shown to contain discriminative information [4] and motion intuitively provides valuable cues to a human observer. However, effectively exploiting this information is undoubtedly challenging. This fact is demonstrated by the two baseline systems [5], which achieve negligible accuracy difference across the sub-challenges. It is then clearly a useful pursuit to consider only static images, because accuracy improvements here can be built on in more complex systems that analyze video. Consequently, this work focusses on the image based sub-challenge. 1 This paper was originally accepted to the ACM International Conference on Multimodal Interaction (ICMI 2015), Seattle, USA, Nov 2015. It has been made available through arXiv.org because the author was unable to present.
All images in this sub-challenge contain an expressive face with the goal to assign an emotion label from the set {neutral, happy, sad, angry, surprised, fearful, disgusted}. These labels originate from the work of Ekman [7], who noted that facial expressions are primarily generated by the contraction or relaxation of facial muscles. This causes a change in the location of points on the face surface (i.e. facial landmarks). Whilst other cues may exist, such as coloring of the skin or the presence of sweat or tears, shape changes remain the most significant indicator.
The relationship between muscle movements and emotion has been well studied and is defined by the emotional facial action coding system [9] (EMFACS Section 2 of this paper discusses face detection and landmark location. Sections 3 and 4 describe the features and modelling approaches used in our experiments. Section 5 provides the main experimental results and section 6 discusses what we can take from these as well as offering some miscellaneous considerations.
FACE REGISTRATION
The first stage in any system of facial analysis involves locating and aligning the face. Methods which holistically combine locating a face with locating facial landmark points have clear appeal. In particular, deformable parts models (DPM), introduced in [8] have become one of the most popular approaches in the research community.
On the other hand, face detection and landmark location have both been extensively studied separately. Many excellent solutions have been proposed to both problems and our own experience suggests that tackling the tasks separately may have advantages in some scenarios. In particular, a recent method [12] for facial landmark location has excellent performance on unconstrained images and is therefore well suited to the EmotiW2015 challenge. The method uses a sequence of gradient boosted regression models, where each stage refines the position estimate according to the results of many point-intensity comparisons. We use the implementation of [12] provided by the dlib library [13]. The model, provided for use with this library, was trained using the data from the iBUG 300-W dataset and it positions 68 points on frontal faces, similar to the baseline.
In [15], face detection based on rigid templates, similar to the classic method of [18], achieves comparable accuracy to a detector based on DPM [8], but the former has a substantial speed advantage. We choose the rigid template detector included in the dlib library [13] as a result of informal performance comparisons. This method uses histogram of oriented gradients [3] (HoG) features combined with a linear classifier. It has a very low false positive rate, although it fails to find a face in almost 12% of the challenge images. In these cases, we roughly position a bounding square manually to allow subsequent processing. 2
Figure 1: Example landmarks from baseline (left) and
proposed system (right) Figure 1 shows a representative comparison of the landmark points output by our system and those of the baseline system. To try to quantify the advantage here, we inspect the automatically located points for each image in the training set and give each one a subjective score based on how well they fit the face. In Figure 1 for example, we would consider the baseline points as 'close' and our points as 'very close'. The results from this exercise are shown in Table 1. It is clear from this that we have a better starting point for using shape information to estimate emotionexperiments in later sections quantify this further.
FEATURES
Our final system, i.e. challenge entry, uses only one very simple type of shape feature. However, we performed experiments with various features which we describe in the following paragraphs.
Shape
Intuitively, given accurate facial landmark locations, we can infer a lot about facial expression -arguably more than is possible from only texture. We consider two simple types of shape feature, both derived from automatically located facial landmark locations. For both types, we first normalize the size of the face. 2 After our main experiments were complete, we found a combination of open source detectors could reduce the miss rate to 3% and the landmark estimator is unlikely to perform well on the remaining difficult faces, regardless of initialization.
Distances between Points
We consider the distances between all distinct pairs of landmark points. We have 68 landmarks, giving 2278 unique pairs. Many of these pairs will contain no useful shape information and we could add heuristics to reduce this number considerably, although this is not necessary for the models we subsequently learn.
Axis Distances from Average
We speculate that the point-distances may not capture all shape information alone. We therefore test a second type of feature that considers the displacement from the average landmark location, where the average is taken from all faces in the training set. After up-righting the face, for each point, we take the x-and ydistances from the average location as feature values. This results in a vector of length 136.
Texture
By including texture to complement the shape information, we hope to improve classification accuracy in our experiments. We note that the baseline system [5] is based entirely on texture features and many previous successful approaches have also used texture, e.g. [1].
Biologically Inspired Features (BIF)
BIF [10] are perhaps most well-known for the success they have achieved in facial age estimation. However, they have also demonstrated excellent performance in other face processing problems [16] and have been applied to the classification of facial expressions [14]. As a rich texture descriptor, based on a model of the human visual system, BIF would appear to represent a good candidate for this application.
Evaluation of BIF involves applying a bank of Gabor filters with different orientations and scales to each location in the face image. The responses of these filters are pooled over similar locations and scales via non-linear operators, maximum (MAX) or standard-deviation (STDDEV). In practice, the aligned face image is partitioned into overlapping rectangular regions for pooling and the pooling operation introduces some tolerance to misalignment.
Our implementation closely follows the description in [10]. We extract 60x60 face regions, aligned according to the automatically located landmarks. We use both MAX and STDDEV pooling, with 8 orientations and 8 bands. Our implementation then has 8640 feature values.
Point Texture Features
We speculate that we may gain more information from texture features that are more directly tied to the location of landmarks. We therefore also consider a second type, where the feature values are simply Gabor filter responses at different sizes and orientations, centered on each landmark location. We refer to these as 'point-texture features'. We evaluate filters at 8 scales and 12 orientations, giving a total of 6528 feature values for the 68 landmark points.
MODELLING
To construct predictive models using our features, we use two standard approaches from the machine learning literature: support vector machines (SVM) [11] and gradient boosting (GB) [11]. For the SVM classifiers, we use the implementation provided by libsvm [2] with a RBF kernel. We optimize the C and gamma parameters on the validation data via a grid search, as advocated by the authors of [2]. For the GB classifiers, we use our own implementation. We find that trees with two splits and a shrinkage factor of 0.1 generally work well on this problem, so we fix these parameters and optimize only the number of trees on the validation data.
EXPERIMENTS
In all of the following experiments, only the challenge training data are used to construct the model, with parameters optimized on the validation set. At one point, we experimented with combining the training and validation data and learning using this larger set. However, this did not result in an improvement in accuracy on the test data, so we did not pursue this approach or include the result.
Simple Shape-based Classifiers
We start using only the point-distance features described in 3.1.1. We learn SVM and GB classifiers which give the performance figures shown in Table 2. A confusion matrix for the SVM classifier on the test data is shown in Table 3.
Classifiers using Other Features
Taking each of the other three types of feature described in section 3 in turn, we add to the point-distance features. Surprisingly, in each case, we did not observe any improvement on validation data over using the point-distance features alone.
For the features of 3.1.2, the accuracy on validation data actually dropped slightly. This could be a result of using a slightly different procedure for size normalization with these features. However, there is also a concern that the average point locations were not useful, due to the large variations in pose.
For the texture features of 3.2, the result was more surprising. We expected these to add some useful information, but this appeared not to be the case, despite their quantity far exceeding that of the distance features.
As a consequence, we conclude that the simple point-distance features already contain the most information relevant to the task.
We use the SVM model from Table 2 as our challenge entry.
Improvement over Baseline
To quantify the advantage that our more accurate landmark locations bring over the baseline, we learn directly comparable models using both sets of points. For the baseline system, landmark points are not available for all images, because the face detector fails in some cases. For a fair comparison, we therefore use exactly the same subset of images across train/validation/test sets in both trials. Where no points exist for a test image, we assign a 'Neutral' label. Table 4 shows the results of this comparison. From these we can conclude that the landmarks used in the proposed system provide a very clear advantage over those of the baseline system.
DISCUSSION
Considering the results of Table 3, performance on each class of emotion exhibits the same pattern seen in previous EmotiW challenges. Specifically, performance is promising for faces with neutral (SVM:69%,GB:64%), happy (71%,62%), angry (42%,46%), and surprise (57%,43%) expressions. On the other hand, sad (29%,25%) and fearful (2%,17%) expressions are more difficult to distinguish. The subtleties of disgust (0%,0%) might be impossible to detect using such simple features taken from static images. Indeed, this task is not only difficult for machines, but without contextual information it is also difficult for humans to distinguish disgust from other more prevalent emotions. Our overall accuracy is more than three times better than random guessing, representing a small improvement over the accuracy achieved in [1] on more constrained static images. The final system we propose achieves 47% accuracy on the test data, whilst the baseline achieves 39% accuracy.
Comparing our SVM and GB classifiers, the former lead to slightly better results in most cases, whereas the latter are significantly simpler and faster to evaluate. However, model complexity differences become insignificant if texture features must be evaluated as this dominates time required to evaluate either type. The key advantage of our proposed system is that the distance features are trivial to evaluate in comparison to commonly used features such as BIF [10], LBP [17] or HoG [3].
The GB model allows the influence of its features to be examined and Figure 2 is a result of this analysis. The distance from the eyes to the corners of the mouth clearly has the greatest influence. This seems reasonable considering the degree to which a mouth is upturned or downturned is one of the clearest indicators of emotional state. Figure 2 also includes distances indicative of eye and mouth openings, which are also intuitively discriminative.
Figure 2: From left to right, then top to bottom, the most influential distances in our gradient boosted model
Before concluding, we must note an observation that potentially affects the baseline accuracy. Almost all of the challenge images have an incorrect aspect ratio that results in elongated faces. We manually correct this prior to performing our experiments. If we instead use the images as provided, the face detector finds only around 60% of faces. Given that we are particularly interested in modelling shape here, it is important to work with consistent aspect ratios.
As a final comment, although the landmarks found by our system are more accurate than those in the baseline, there is still much scope for improvement. Given 100% accurate landmark locations, an interesting line of further work might be to tailor the modelling approach to the problem in an attempt to see just how far static shape alone can be used in estimating facial expression.
ACKNOWLEDGMENTS
Our thanks to Professor John A. Robinson for his support in this work.
). For example, happiness is represented by the combination of 'cheek raiser' with 'lip corner puller'. Sadness is demonstrated by 'inner brow raiser' plus 'brow lowerer' together with 'lip corner depresser'. However, in the static image sub-challenge, it is not possible to detect movements, so how well can expression be predicted from a single image? In contrast to the EmotiW2015 challenge, most prior-art has reported results on well-lit well-aligned faces.For a 5-class problem, with
fairly accurate registration, [1] demonstrated classification
accuracies of around 60%.
Table 1 .
1Accuracy of baseline points versus proposed systemExcluded
Fail
Poor
Close
Very
Close
Baseline
67
55
173
663
0
Proposed
0
2
37
215
704
Table 2 :
2Main performance figuresModel
Train
Validate
Test
GB
60.1%
40.8%
44.4%
SVM
52.1%
37.4%
46.8%
Baseline
-
36.0%
39.1%
Table 3 :
3Test data confusion matrix for challenge entryEstimate
→
Angry
Disgust
Fear
Happy
Neutral
Sad
Surprise
Truth ↓
Angry
29
1
4
5
10
7
13
Disgust
3
0
0
6
4
4
0
Fear
13
0
1
3
13
6
5
Happy
5
0
0
67
7
16
0
Neutral
3
0
1
3
40
9
2
Sad
9
0
5
5
12
16
8
Surprise
8
0
2
0
6
0
21
Table 4 :
4Overall accuracy using baseline points and points
from proposed system
Landmarks
Model
Train
Validate
Test
Proposed
GB
65.8%
40.4%
38.4%
SVM
50.0%
41.2%
40.6%
Baseline
GB
53.9%
32.3%
31.2%
SVM
47.9%
34.1%
27.7%
Person-independent facial expression detection using constrained local models. S W Chew, FG'11 IEEE. 915-920Automatic Face & Gesture Recognition and Workshops. Santa Barbara, CaliforniaChew, S. W. et al. Person-independent facial expression detection using constrained local models. 2011. In Automatic Face & Gesture Recognition and Workshops (March 2011, Santa Barbara, California). FG'11 IEEE. 915-920.
LIBSVM : a library for support vector machines. Chih-Chung Chang, Chih-Jen Lin, ACM Trans. on Intelligent Systems and Technology. 23Chih-Chung Chang and Chih-Jen Lin. 2011. LIBSVM : a library for support vector machines. ACM Trans. on Intelligent Systems and Technology. 2, 3 (2011). Software at http://www.csie.ntu.edu.tw/~cjlin/libsvm
Histograms of oriented gradients for human detection. N Dalal, B Triggs, Computer Vision and Pattern Recognition. San Diego, CaliforniaCVPR'05, IEEEDalal, N. and Triggs, B. Histograms of oriented gradients for human detection. 2005. In Computer Vision and Pattern Recognition (June 2005, San Diego, California). CVPR'05, IEEE. 886-893.
Emotion recognition with boosted tree classifiers. M Day, International Conference on Multimodal Interaction. Sydney, AustraliaICMI'13. ACMDay, M. Emotion recognition with boosted tree classifiers. In International Conference on Multimodal Interaction. (December 2013, Sydney, Australia). ICMI'13. ACM. 531- 534.
Video and Image based Emotion Recognition Challenges in the Wild: EmotiW. A Dhall, R Murthy, R Goecke, J Joshi, T Gedeon, International Conference on Multimodal Interaction. Seattle, WashingtonACMICMI'15Dhall, A., Murthy, R., Goecke, R., Joshi, J. and Gedeon, T. 2015. Video and Image based Emotion Recognition Challenges in the Wild: EmotiW 2015. In International Conference on Multimodal Interaction. (November 2015, Seattle, Washington). ICMI'15. ACM.
Collecting large, richly annotated facial-expression databases from movies. A Dhall, R Goecke, S Lucey, T Gedeon, MultiMedia. 19Dhall, A., Goecke, R., Lucey, S. and Gedeon, T. 2012. Collecting large, richly annotated facial-expression databases from movies. In MultiMedia. 19 (2012) IEEE. 34-41.
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Object detection with discriminatively trained part-based models. P Felzenszwalb, IEEE Trans on. 32Pattern Analysis and Machine IntelligenceFelzenszwalb, P. et al. Object detection with discriminatively trained part-based models. 2010. Pattern Analysis and Machine Intelligence, IEEE Trans on. 32, 9 (2010): 1627- 1645.
EMFACS-7: Emotional Facial Action Coding System. W Friesen, P Ekman, University of California, CaliforniaUnpublished manualFriesen, W. and Ekman, P. 1983. EMFACS-7: Emotional Facial Action Coding System. (1983) Unpublished manual, University of California, California.
Human age estimation using bio-inspired features. G Guo, Computer Vision and Pattern Recognition. Miami Beach, FloridaCVPR'09, IEEEGuo, G. et al. Human age estimation using bio-inspired features.2009. In Computer Vision and Pattern Recognition. (June 2009, Miami Beach, Florida). CVPR'09, IEEE. 112- 119.
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Multiresolution gray-scale and rotation invariant texture classification with local binary patterns. T Ojala, M Pietikäinen, T Mäenpää, In Trans. Pattern Analysis and Machine Intelligence. 24Ojala, T., Pietikäinen, M. and Mäenpää, T. 2002. Multiresolution gray-scale and rotation invariant texture classification with local binary patterns. In Trans. Pattern Analysis and Machine Intelligence. 24, 7 (2002) IEEE. 971- 987.
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|
[] |
[] |
[
"Johannes Blümlein \nDeutsches Elektronen-Synchrotron\nDESY\nPlatanenallee 6D-15738ZeuthenGermany\n",
"Alexander Hasselhuhn \nDeutsches Elektronen-Synchrotron\nDESY\nPlatanenallee 6D-15738ZeuthenGermany\n",
"Sebastian Klein \nInstitute for Theoretical Physics E\nRWTH Aachen University\nD-52056AachenGermany\n",
"Carsten Schneider \nResearch Institute for Symbolic Computation (RISC)\nJohannes Kepler University\nAltenbergerstraße 69A-4040LinzAustria\n"
] |
[
"Deutsches Elektronen-Synchrotron\nDESY\nPlatanenallee 6D-15738ZeuthenGermany",
"Deutsches Elektronen-Synchrotron\nDESY\nPlatanenallee 6D-15738ZeuthenGermany",
"Institute for Theoretical Physics E\nRWTH Aachen University\nD-52056AachenGermany",
"Research Institute for Symbolic Computation (RISC)\nJohannes Kepler University\nAltenbergerstraße 69A-4040LinzAustria"
] |
[] |
The O(α 3 s n f T 2 F C A,F ) terms to the massive gluonic operator matrix elements are calculated for general values of the Mellin variable N . These twist-2 matrix elements occur as transition functions in the variable flavor number scheme at NNLO. The calculation uses sum-representations in generalized hypergeometric series turning into harmonic sums. The analytic continuation to complex values of N is provided.
|
10.1016/j.nuclphysb.2012.09.001
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[
"https://arxiv.org/pdf/1205.4184v1.pdf"
] | 117,135,026 |
1205.4184
|
e11ff7ae64ef0611a814ac82e2089304b9b08872
|
18 May 2012 May 2012
Johannes Blümlein
Deutsches Elektronen-Synchrotron
DESY
Platanenallee 6D-15738ZeuthenGermany
Alexander Hasselhuhn
Deutsches Elektronen-Synchrotron
DESY
Platanenallee 6D-15738ZeuthenGermany
Sebastian Klein
Institute for Theoretical Physics E
RWTH Aachen University
D-52056AachenGermany
Carsten Schneider
Research Institute for Symbolic Computation (RISC)
Johannes Kepler University
Altenbergerstraße 69A-4040LinzAustria
18 May 2012 May 2012The O(α 3 s n f T 2 F C A,F ) Contributions to the Gluonic Massive Operator Matrix Elements
The O(α 3 s n f T 2 F C A,F ) terms to the massive gluonic operator matrix elements are calculated for general values of the Mellin variable N . These twist-2 matrix elements occur as transition functions in the variable flavor number scheme at NNLO. The calculation uses sum-representations in generalized hypergeometric series turning into harmonic sums. The analytic continuation to complex values of N is provided.
Introduction
Heavy quark contributions to the deep inelastic scattering structure functions play a crucial role in the QCD analyses to determine the parton distribution functions and the strong coupling constant α s (M 2 Z ) in a consistent manner, cf. [1]. The heavy flavor corrections were calculated at NLO in semianalytic form in [2] 1 . To avoid contributions of higher twist the analysis has to be restricted to large enough values of Q 2 . It has been shown in [4] that for Q 2 > ∼ 10 m 2 , with m the heavy quark mass, the heavy flavor contributions to the structure function F 2 (x, Q 2 ) are rather accurately described using the asymptotic representation in which all power corrections ∝ (m 2 /Q 2 ) k , k ∈ N + are neglected. In this case the heavy flavor Wilson coefficients can be calculated analytically. They are given by convolutions of massive operator matrix elements (OMEs) and the massless Wilson coefficients, cf. Ref. [4,5]. The massless Wilson coefficients are known to 3-loop order [6]. At NLO the massive OMEs were calculated in [4,[7][8][9][10][11][12] in the unpolarized and polarized case, including the O(α 2 s ε) contributions, and in [13] for transversity. The heavy flavor corrections for charged current reactions are available at one loop and in the asymptotic case at two-loops [14,15].
At 3-loop order a series of moments has been calculated for all massive OMEs for N = 2... 10(14) contributing in the fixed and variable flavor scheme, [5]. The 3-loop heavy flavor corrections to F L (x, Q 2 ) in the asymptotic case were calculated in [16]. First results for general values of N have been obtained for the OMEs with operator insertions on the quark lines in case for the color factors n f T 2 F C A,F [17] and 3-loop ladder topologies [18]. First T 2 F C A,Fcontributions at general N were calculated in [19] for two heavy quark lines carrying the same mass. Furthermore, the moments N = 2, 4, 6 in case of the OMEs contributing to the structure function F 2 (x, Q 2 ) with two different heavy quark masses were computed in [19,20]. In all the above cases the massive OMEs are calculated for external massless partons which are on-shell. The case of massive on-shell external lines has been treated in [21] recently.
In the present paper the 3-loop corrections of O(n f T 2 F C A,F ) to the massive OMEs with local operator insertions on the gluonic lines, A gq,Q and A gg,Q , at general values of N are calculated. Together with the corresponding terms with the insertions on the quark lines, [17], these contributions complete all terms corresponding to the case of one massless and one massive fermion line at 3-loop order. These matrix elements contribute to the transition functions needed to describe the parton densities in the variable flavor number scheme (VFNS). In this scheme it is possible to define heavy quark distribution functions assuming that there exists only one heavy quark and all other quarks can be dealt with as massless in the sense of an effective field theory approach. These distributions can be used for effective calculations in some processes at hadron colliders. The picture holds to 2-loop orders. Starting with the 3-loop corrections, [19,20], diagrams containing quarks of two different masses contribute even to the universal corrections. Since m 2 c /m 2 b ≈ 1/10 is not a small number, the original VFNS-picture does not necessarily hold in practice. Here we deal with the O(n f T 2 F C A,F ) contributions which are in accordance with the VFNS. In Section 2 the main formalism is lined out. The calculation is performed in D = 4 + ε dimensions and uses representations in terms of generalized hypergeometric functions. They lead to multiple sum representations, which are solved using modern summation technologies encoded in the package Sigma [22]. The results of the calculation are given in Section 3 both in Mellin-N and in x-space, and Section 4 contains the conclusions.
Parton distribution functions in the VFNS
The neutral current Born cross section of unpolarized deep inelastic scattering (DIS) is given by [23]
d 2 σ NC B dxdy = 2πα 2 xyQ 2 2(1 − y) − 2xy M 2 S + 1 + 4x 2 M 2 Q 2 y 2 1 + R(x, Q 2 ) F 2 (x, Q 2 ) +xy(1 − y)F 3 (x, Q 2 ) ,(1)
neglecting lepton mass contributions. Here x and y denote the Bjorken variables and −q 2 = Q 2 = xyS, with q 2 the 4-momentum transfer. The structure functions F i (x, Q 2 ) contain electroweak effects due to Z-boson exchange and differ for lepton and anti-lepton-nucleon scattering, cf. [23], and
R(x, Q 2 ) = 1 + 4x 2 M 2 Q 2 F 2 (x, Q 2 ) 2xF 1 (x, Q 2 ) − 1 .(2)
In the limit M 2 Z ≫ Q 2 the electromagnetic terms in (1) dominate and only the two structure functions F 1,2 (x, Q 2 ) contribute, with
2xF 1 (x, Q 2 ) = F 2 (x, Q 2 ) − F L (x, Q 2 ) ,(3)
where F L is the longitudinal structure function. Both structure functions contain light and heavy quark contributions. The y-dependence of the differential scattering cross section is used to separate the structure functions [24] and allows precise measurements of the structure function F 2 (x, Q 2 ). In the twist-2 approximation, referring to the fixed flavor number scheme, they are given by
F 2 (x, Q 2 , n f ) = F m=0 2 (x, Q 2 , n f ) + F massive 2,Q (x, Q 2 , n f , m) .(4)
Here F m=0 2 (x, Q 2 ) denotes the well-known massless contribution and the massive contribution in the presence of a single massive quark reads [5] Here f k (x, µ 2 , n f ), Σ(x, µ 2 , n f ), G(x, µ 2 , n f ) denote the kth quark, singlet-quark, and gluon densities, respectively with
Σ(x, n f , µ 2 ) = n f k=1 [f k (x, n f , µ 2 ) + fk(x, n f , µ 2 )] .(6)
The Wilson coefficientsL PS 2,q (n f , Q 2 /m 2 , m 2 /µ 2 ) andL S 2,g (n f , Q 2 /m 2 , m 2 /µ 2 ) have been calculated completely for general values of N in [17].
The renormalization group implies the following representation for the set of (n f + 1) (massless) parton densities expressed in terms of n f parton densities [8] :
f k (n f + 1, µ 2 , m 2 , N) + f k (n f + 1, µ 2 , m 2 , N) = A NS qq,Q n f , µ 2 m 2 , N · f k (n f , µ 2 , N) +f k (n f , µ 2 , N) +Ã PS qq,Q n f , µ 2 m 2 , N · Σ(n f , µ 2 , N) +Ã qg,Q n f , µ 2 m 2 , N · G(n f , µ 2 , N),(7)f Q (n f + 1, µ 2 , m 2 , N) + f Q (n f + 1, µ 2 , m 2 , N) = A PS Qq n f , µ 2 m 2 , N · Σ(n f , µ 2 , N) +A Qg n f , µ 2 m 2 , N · G(n f , µ 2 , N) .(8)
Here f Q (fQ) are the heavy quark densities. The flavor singlet, non-singlet and gluon densities for (n f + 1) flavors are given by
Σ(n f + 1, µ 2 , m 2 , N) = A NS qq,Q n f , µ 2 m 2 , N + n fà PS qq,Q n f , µ 2 m 2 , N +A PS Qq n f , µ 2 m 2 , N · Σ(n f , µ 2 , N) + n fÃqg,Q n f , µ 2 m 2 , N + A Qg n f , µ 2 m 2 , N · G(n f , µ 2 , N) (9) ∆(n f + 1, µ 2 , m 2 , N) = f k (n f + 1, µ 2 , N) + f k (n f + 1, µ 2 , m 2 , N) − 1 n f + 1 Σ(n f + 1, µ 2 , m 2 , N)(10)G(n f + 1, µ 2 , m 2 , N) = A gq,Q n f , µ 2 m 2 , N · Σ(n f , µ 2 , N) +A gg,Q n f , µ 2 m 2 , N · G(n f , µ 2 , N) .(11)
Any relation between the (n f + 1)-and n f -parton density can only contain universal, i.e. processindependent, quantities. Note that the new parton densities depend on the renormalized heavy quark mass m 2 . As outlined above, the corresponding relations for the operator matrix elements depend on the mass-renormalization scheme, with m = m(a s (µ 2 )) in the MS scheme, which we will apply below. These equations describe the transition of one heavy quark becoming light at the time referring to the scale µ 2 .
The matching scales µ 2 are often chosen as µ 2 = m 2 . The comparison of the results in complete calculations to those in which flavor thresholds are matched in the VFNS allows in principle to determine the relevant matching scale. In an analysis of the various deep-inelastic structure function sum rules [25] it has been shown that the scale µ 2 turns out to be significantly different of m 2 . This is not unexpected since mass effects do not turn into the behaviour of the massless case close to the production threshold.
The resummation of large logs, as being performed in the VFNS, has to be performed at very high scales. As has been shown in [26] this is not the case in the kinematic range at HERA. A smooth transition from the threshold region to asymptotic scales has been proposed in terms of the BMSN-scheme [8], (12) which is found to be in excellent agreement with the HERA data [27]. There is a series of other proposals to match between the threshold and asymtotic region [28][29][30], partly with a faster transition to the massless case. Here precise data on F cc 2 (x, Q 2 ) are helpful to distinguish between different descriptions. We would like to mention that a correct treatment of the heavy flavor corrections is of instrumental importance in the QCD analysis of the complete structure functions F 2 (x, Q 2 ), which has been measured to a precision of O(1%) [31].
F cc 2 (x, Q 2 , n f = 4) = F cc,FFNS 2 (x, Q 2 , n f = 3) + F cc,asymp 2 (x, Q 2 , n f = 4) − F cc,asymp 2 (x, Q 2 , n f = 3) ,3 The O(α 3 s n f T 2 F ) contributions to A gg,Q and A gq,Q
The OMEs A gq,Q and A gg,Q are expectation values j|O g |j , i, j = q, g of the gluonic operator
O g,µ 1 ,...,µ N = 2i N −2 SSp[F µ 1 α D µ 2 ...D µ N −1 F α µ N ] − trace terms .(13)
between massless on-shell external states. The corresponding massive OMEs A gq,Q , A gg,Q were calculated to O(α 2 s ) in [8] and including also terms linear in ε in [9] correcting the previous result. The renormalized expressions A gq,Q and A gg,Q to O(a 3 s ) were derived in [5]. In the MS scheme with the heavy quark mass m on-shell they are given by :
A (3),MS gq,Q = − γ (0) gq 24 γ (0) gqγ (0) qg + γ (0) qq − γ (0) gg + 10β 0 + 24β 0,Q β 0,Q ln 3 m 2 µ 2 + 1 8 6γ (1) gq β 0,Q +γ (1) gq γ (0) gg − γ (0) qq − 4β 0 − 6β 0,Q + γ (0) gq γ (1),NS qq +γ (1),PS qq −γ (1) gg + 2β 1,Q ln 2 m 2 µ 2 + 1 8 4γ (2) gq + 4a (2) gq,Q γ (0) gg − γ (0) qq − 4β 0 − 6β 0,Q + 4γ (0) gq a (2),NS qq,Q + a (2),PS Qq − a (2) gg,Q +β (1) 1,Q + γ (0) gq ζ 2 γ (0) gqγ (0) qg + γ (0) qq − γ (0) gg + 12β 0,Q + 10β 0 β 0,Q ln m 2 µ 2 +a (2) gq,Q γ (0) qq − γ (0) gg + 4β 0 + 6β 0,Q + γ (0) gq a (2) gg,Q − a (2),PS Qq − a (2),NS qq,Q − γ (0) gq β (2) 1,Q − γ (0) gq ζ 3 24 γ (0) gqγ (0) qg + γ (0) qq − γ (0) gg + 10β 0 β 0,Q − 3γ (1) gq β 0,Q ζ 2 8 + 2δm (−1) 1 a (2) gq,Q +δm (0) 1γ (1) gq + 4δm (1) 1 β 0,Q γ (0) gq + a (3) gq,Q ,(14)A (3),MS gg,Q = 1 48 γ (0) gqγ (0) qg γ (0) qq − γ (0) gg − 6β 0 − 4n f β 0,Q − 10β 0,Q − 4 γ (0) gg 2β 0 + 7β 0,Q +4β 2 0 + 14β 0,Q β 0 + 12β 2 0,Q β 0,Q ln 3 m 2 µ 2 + 1 8 γ (0) qg γ (1) gq + (1 − n f )γ (1) gq +γ (0) gqγ (1) qg + 4γ (1) gg β 0,Q − 4γ (1) gg [β 0 + 2β 0,Q ] + 4[β 1 + β 1,Q ]β 0,Q +2γ (0) gg β 1,Q ln 2 m 2 µ 2 + 1 16 8γ (2) gg − 8n f a (2) gq,Qγ (0) qg − 16a (2) gg,Q (2β 0 + 3β 0,Q ) +8γ (0) gq a (2) Qg + 8γ (0) gg β (1) 1,Q + γ (0) gqγ (0) qg ζ 2 γ (0) gg − γ (0) qq + 6β 0 + 4n f β 0,Q + 6β 0,Q +4β 0,Q ζ 2 γ (0) gg + 2β 0 2β 0 + 3β 0,Q ln m 2 µ 2 + 2(2β 0 + 3β 0,Q )a (2) gg,Q +n fγ (0) qg a (2) gq,Q − γ (0) gq a (2) Qg − β (2) 1,Q γ (0) gg + γ (0) gqγ (0) qg ζ 3 48 γ (0) qq − γ (0) gg − 2[2n f + 1]β 0,Q −6β 0 + β 0,Q ζ 3 12 [β 0,Q − 2β 0 ]γ (0) gg + 2[β 0 + 6β 0,Q ]β 0,Q − 4β 2 0 −γ (0) qg ζ 2 16 γ (1) gq +γ (1) gq + β 0,Q ζ 2 8 γ (1) gg − 2γ (1) gg − 2β 1 − 2β 1,Q + δm (−1) 1 4 8a (2) gg,Q +24δm (0) 1 β 0,Q + 8δm(1)1 β 0,Q + ζ 2 β 0,Q β 0 + 9ζ 2 β 2 0,Q + δm (0) 1 β 0,Q δm (0) 1 +γ (1) gg +δm (1) 1 γ (0) qg γ (0) gq + 2β 0,Q γ (0) gg + 4β 0,Q β 0 + 8β 2 0,Q − 2δm (0) 2 β 0,Q + a (3) gg,Q .(15)
Here δm (k) i are expansion coefficients of the unrenormalized mass, β i , β i,Q are coefficients of the β-functions (including mass effects), ζ k is the Riemann-ζ function with k ∈ N\{0, 1}, a (2) ij , a (2) ij are two loop contributions to order ε 0 and ε 1 respectively, and γ ij ,γ ij are the anomalous dimensions, and quantities with a hat or a tilde are defined bŷ
f = f (n f + 1) − f (n f ),f = 1 n f f,(16)
see Ref. [5]. The unreormalized OMEÂ (3) gg,Q also receives contributions from the vacuum polarization insertions on the external lineŝ
Π ab µν (p 2 ,m 2 , µ 2 ,â 2 s ) = iδ ab −g µν p 2 + p µ p ν ∞ k−1â k sΠ (k) (p 2 ,m 2 , µ 2 )(17)Π (k) ≡Π (k) (0,m 2 , µ 2 )(18)
such thatÂ
(3) gg,Q =Â (3),1PI gg,Q −Π (3) −Â (2),1PI gg,QΠ (1) − 2Â (1) gg,QΠ (2) +Â (1) gg,QΠ (1)Π(1) (19) ≡ a (3,0) gg,Q ε 3 + a (3,1) gg,Q ε 2 + a (3,2) gg,Q ε + a(3)
gg,Q .
All contributions to (14,15) but the constant terms a
ij,Q are known [4, 7-9, 12, 32]. In particular, all the logarithmic contributions have already been obtained for general values of the Mellin variable N, [33].
In the following we calculate the contributions O(a 3 s n f T 2 F C F,A ) to the massive gluonic OMEs. The Feynman diagrams are generated by QGRAF [34] and the extension allowing to include local operators [5]. The color-algebra is performed using [35]. For a large part of the calculation we use FORM [36]. The momentum integrals are performed introducing a Feynman parameterization. The Feynman parameter integrals are then rewritten in terms of hypergeometric functions ( 2 F 1 , 3 F 2 ), which are represented in terms of absolutely convergent series. The resulting sums, which may still contain finite sums due to binomial expansions, are then processed applying the symbolic summation technology, which is encoded in the package Sigma [22] and making use of a large number of algorithms for processing multi sums using the package EvaluateMultiSums [37,38]. Additionally it is very useful to reduce such sums to a smaller number of 'key sums', by synchronization of the summation ranges and algebraic reduction of the summands. This step helped to reduce the size of the terms from 2GByte to 7.6MByte and the number of sums from 2419 to 29. The algorithms for this step are implemented in the package SumProduction [39]. Details of the corresponding technique are described in [37,38]. The corresponding expressions have simplified using mutual relations and methods applicable to the respective classes of sums encoded in the package HarmonicSums [40]. The results for the individual diagrams have been checked comparing to the moments obtained in [5] using the code MATAD [41]. The constant contributions a (3) gj,Q , j = q, g to (14,15) read :
a (3),n f T 2 F gq,Q = C F T 2 F n f − 16 (N 2 + N + 2) 9(N − 1)N(N + 1) 1 3 S 3 1 + S 2 S 1 + 2 3 S 3 + 14ζ 3 + 3S 1 ζ 2 + 16 (8N 3 + 13N 2 + 27N + 16) 27(N − 1)N(N + 1) 2 3ζ 2 + S 2 1 + S 2 − 32 (35N 4 + 97N 3 + 178N 2 + 180N + 70) 27(N − 1)N(N + 1) 3 S 1 + 32 (1138N 5 + 4237N 4 + 8861N 3 + 11668N 2 + 8236N + 2276) 243(N − 1)N(N + 1) 4 . (21) a (3),n f T 2 F gg,Q = n f T 2 F C A 1 (N − 1)(N + 2) 4P 1 27N 2 (N + 1) 2 S 2 1 + 8P 2 729N 3 (N + 1) 3 S 1 + 160 27 (N − 1)(N + 2)ζ 2 S 1 − 448 27 (N − 1)(N + 2)ζ 3 S 1 + P 3 729N 4 (N + 1) 4 − 2P 4 27N 2 (N + 1) 2 ζ 2 + 56 (3N 4 + 6N 3 + 13N 2 + 10N + 16) 27N(N + 1) ζ 3 − 4P 5 27N 2 (N + 1) 2 S 2 +C F 1 (N − 1)(N + 2) 112 (N 2 + N + 2) 2 27N 2 (N + 1) 2 S 3 1 − 16P 6 27N 3 (N + 1) 3 S 2 1 + 32P 7 81N 4 (N + 1) 4 S 1 + 16 (N 2 + N + 2) 2 3N 2 (N + 1) 2 ζ 2 S 1 + 16 (N 2 + N + 2) 2 3N 2 (N + 1) 2 S 2 S 1 − 32P 8 243N 5 (N + 1) 5 − 16P 9 9N 3 (N + 1) 3 ζ 2 + 448 (N 2 + N + 2) 2 9N 2 (N + 1) 2 ζ 3 + 16P 10 9N 3 (N + 1) 3 S 2 − 160 (N 2 + N + 2) 2 27N 2 (N + 1) 2 S 3 ,(22)
where the polynomials P i are given by
P 1 = 16N 5 + 41N 4 + 2N 3 + 47N 2 + 70N + 32 (23) P 2 = 6944N 8 + 26480N 7 + 23321N 6 − 15103N 5 − 39319N 4 − 27001N 3 − 11178N 2 −2016N + 864 (24) P 3 = 4809N 10 + 24045N 9 − 182720N 8 − 854414N 7 − 1522031N 6 − 1472927N 5 −758234N 4 − 126080N 3 − 1152N 2 − 50688N − 24192 (25) P 4 = 3N 6 + 9N 5 + 307N 4 + 599N 3 + 746N 2 + 448N + 96 (26) P 5 = 40N 6 + 112N 5 − 3N 4 − 166N 3 − 301N 2 − 210N − 96 (27) P 6 = 44N 6 + 123N 5 + 386N 4 + 543N 3 + 520N 2 + 248N + 24 (28) P 7 = 205N 8 + 856N 7 + 3169N 6 + 6484N 5 + 7310N 4 + 4722N 3 + 1534N 2 +48N − 72 (29) P 8 = 1976N 10 + 9385N 9 + 24088N 8 + 38989N 7 + 50214N 6 + 53872N 5 + 35219N 4 +6890N 3 − 4233N 2 − 2844N − 756 (30) P 9 = 14N 6 + 33N 5 + 59N 4 + 39N 3 + 55N 2 + 20N − 12 (31) P 10 = 4N 6 + 3N 5 − 50N 4 − 129N 3 − 100N 2 − 56N − 24 .(32)
Here S b, a =≡ S b, a (N) = N n=1 sign(b) n S a (N)/n |b| ; S ∅ = 1 denote the harmonic sums [42] which only occur as single harmonic sums in the present calculation.
It is convenient to express the renormalized OMEs A gj,Q , j = q, g also referring to the heavy quark mass in the MS scheme, cf. [5]. The OMEs A
(3),n f T 2 F gq,Q and A (3),n f T 2 F gg,Q read : A (3),MS gq,Q,C F T 2 F n f = C F n f T 2 F 32 (N 2 + N + 2) 9(N − 1)N(N + 1) ln 3 m 2 µ 2 + − 16 (N 2 + N + 2) 3(N − 1)N(N + 1) S 2 1 + S 2 + 32 (8N 3 + 13N 2 + 27N + 16) 9(N − 1)N(N + 1) 2 S 1 + 32 (19N 4 + 81N 3 + 86N 2 + 80N + 38) 27(N − 1)N(N + 1) 3 ln m 2 µ 2 + 32 (N 2 + N + 2) 27(N − 1)N(N + 1) S 3 1 + 3S 2 S 1 + 2S 3 − 24ζ 3 − 32 (8N 3 + 13N 2 + 27N + 16) 27(N − 1)N(N + 1) 2 S 2 1 + S 2 + 64 (4N 4 + 4N 3 + 23N 2 + 25N + 8) 27(N − 1)N(N + 1) 3 S 1 + 64 (197N 5 + 824N 4 + 1540N 3 + 1961N 2 + 1388N + 394) 243(N − 1)N(N + 1) 4 (33) A (3),n f T 2 F ,MS gg,Q = n f T 2 F C F 64 (N 2 + N + 2) 2 9(N − 1)N 2 (N + 1) 2 (N + 2) +C A 128 (N 2 + N + 1) 27(N − 1)N(N + 1)(N + 2) − 64 27 S 1 ln 3 m 2 µ 2 (34) −C F 16 3 ln 2 m 2 µ 2 + C A 1 (N − 1)(N + 2) − 4P 11 81N 3 (N + 1) 3 (35) − 16P 12 81N 2 (N + 1) 2 S 1 + C F 1 (N − 1)(N + 2) 16 (N 2 + N + 2) 2 N 2 (N + 1) 2 S 2 1 − 5 3 S 2 − 4P 13 9N 4 (N + 1) 4 − 32P 14 3N 3 (N + 1) 3 S 1 ln m 2 µ 2 +C A 1 (N − 1)(N + 2) − 4P 15 27N 2 (N + 1) 2 S 2 1 − 8P 16 729N 3 (N + 1) 3 S 1 + 512 27 (N − 1)(N + 2)ζ 3 S 1 − 2P 17 729N 4 (N + 1) 4 − 1024 (N 2 + N + 1) 27N(N + 1) ζ 3 + 4P 18 27N 2 (N + 1) 2 S 2 +C F 1 (N − 1)(N + 2) 64 (N 2 + N + 2) 2 9N 2 (N + 1) 2 − 1 3 S 3 1 − 8ζ 3 + 4 3 S 3 + 32P 19 27N 3 (N + 1) 3 S 2 1 − 64P 20 81N 4 (N + 1) 4 S 1 − 32P 21 243N 5 (N + 1) 5 − 32P 22 3N 3 (N + 1) 3 S 2 ,(36)
with the polynomials
P 11 = 297N 8 + 1188N 7 + 640N 6 − 2094N 5 − 1193N 4 + 2874N 3 + 5008N 2 +3360N + 864 (37) P 12 = 136N 6 + 390N 5 + 19N 4 − 552N 3 − 947N 2 − 630N − 288 (38) P 13 = 15N 10 + 75N 9 − 48N 8 − 866N 7 − 2985N 6 − 6305N 5 − 8206N 4 − 7656N 3 −4648N 2 − 1600N − 288 (39) P 14 = 5N 5 + 52N 4 + 109N 3 + 90N 2 + 48N + 16 (40) P 15 = 4N 5 + 17N 4 + 14N 3 + 71N 2 + 70N + 32 (41) P 16 = 3008N 8 + 11600N 7 + 9197N 6 − 10255N 5 − 27739N 4 − 24745N 3 − 12474N 2 −2016N + 864 (42) P 17 = 4185N 10 + 20925N 9 + 1892N 8 − 117118N 7 − 222151N 6 − 176863N 5 − 41446N 4 +22304N 3 − 1296N 2 − 18432N − 6912 (43) P 18 = 16N 6 + 52N 5 − 3N 4 − 106N 3 − 277N 2 − 210N − 96(P 21 = 123N 12 + 738N 11 + 691N 10 − 3526N 9 − 14521N 8 − 29458N 7 − 39189N 6 −37672N 5 − 21920N 4 − 3914N 3 + 2856N 2 + 1872N + 432 (47) P 22 = 2N 6 + 4N 5 + N 4 − 10N 3 − 5N 2 − 4N − 4 .(48)
As has been noted before [5], the above results are free of ζ 2 , which is common to all massive OMEs, and hence is a particular feature of representing also the mass in the MS scheme. Furthermore we note, that the ln 2 (m 2 /µ 2 )-contribution to A (3),MS gg,Q,C F T 2 F n f is particularly simple, while the corresponding contribution to A (3),MS gq,Q,C F T 2 F n f vanishes. As a by-product of the calculation we obtain the corresponding contributions to the anomalous dimensions from the single pole term 1/ε resp. the linear logarithmic contribution, cf. (14,15),
γ (2),n f gq = n f T 2 F C F − 64 (N 2 + N + 2) 3(N − 1)N(N + 1) S 2 1 + S 2 + 128 (8N 3 + 13N 2 + 27N + 16) 9(N − 1)N(N + 1) 2 S 1 − 128 (4N 4 + 4N 3 + 23N 2 + 25N + 8) 9(N − 1)N(N + 1) 3 ,(49)γ (2),n f gg = n f T 2 F C A − 32P 23 27(N − 1)N 2 (N + 1) 2 (N + 2) S 1 − 8P 24 27(N − 1)N 3 (N + 1) 3 (N + 2) +n f T 2 F C F 64 (N 2 + N + 2) 2 3(N − 1)N 2 (N + 1) 2 (N + 2) S 2 1 − 3S 2 + 128P 25 9(N − 1)N 3 (N + 1) 3 (N + 2) S 1 − 16P 26 27(N − 1)N 4 (N + 1) 4 (N + 2) ,(50)
where
P 23 = 8N 6 + 24N 5 − 19N 4 − 78N 3 − 253N 2 − 210N − 96 (51) P 24 = 87N 8 + 348N 7 + 848N 6 + 1326N 5 + 2609N 4 + 3414N 3 + 2632N 2 + 1088N +192 (52) P 25 = 4N 6 + 3N 5 − 50N 4 − 129N 3 − 100N 2 − 56N − 24 ,(53)P 26 = 33N 10 + 165N 9 + 256N 8 − 542N 7 − 3287N 6 − 8783N 5 − 11074N 4 − 9624N 3 −5960N 2 − 2112N − 288 .(54)
Eqs. (49,50) confirm previous results in [32] by a first direct diagrammatic calculation, here in the massive case. The leading singlet eigenvalue for the gluonic anomalous dimensions γ
gj , j = q, g in form of
γ (2),n 2 f gg + γ (2),n 2 f gq γ (0) qg γ (0),n f gg n f (55)
has been calculated in [43] for the leading n f contribution, ∝ n 2 f . We also confirm this result by a direct massive calculation.
Usually the calculation in N-space is being performed multiplying the massive OMEs and the parton distributions analytically 2 , cf. e.g. [44]. The corresponding analytic continuations of harmonic sums up to weight w=8 are given in [45]. Only a single numerical contour integral around the singularities has to be performed, allowing for very fast implementations.
The OMEs (33,34) can also be given in x-space directly for codes operating in x-space only. They are given by : [46]. They can be expressed in terms of elementary functions and the Nielsen integrals [47] : H 0 (x) = ln(x), H 1 (x) = − ln(1 − x), H 0,1 (x) = Li 2 (x), H 0,0,1 (x) = Li 3 (x), H 0,1,1 (x) = S 1,2 (x), H 0,0,0,1 (x) = Li 4 (x), H 0,0,1,1 (x) = S 2,2 (x) and H 0,1,1,1 (x) = S 1,3 (x), with S n,p (x) = (−1) (n+p−1) (n − 1)!p! 1 0 dy y ln (n−1) (y) ln p (1 − xy) ,
A (3),n f T 2 F ,MS gq,Q (x) = C F n f T 2 FA (3),n f T 2 F ,MS gg,Q (x) = n f T 2 F C A − 64x 2 27 + 64x 27 − 64 27(x − 1) + − 128 27 + 64 27x + C F − 256x 2 27 − 64x 9 + 128 9 (1 + x)H 0 + 64 9 + 256 27x ln 3 m 2 µ 2 − 16 3 C F δ(1 − x) ln 2 m 2 µ 2 + C A −608x
Li n (x) = S n−1,n (x) .
Here Li n (x) denotes the polylogarithm. All higher functions but S 2,2 (x) can be reduced to polylogarithms by the argument relation x → (1 − x). Numerical implementations of the functions S n,p (x) were given in [48].
At small values of x the functions A
(3),n f T 2 F ,MS gq(g),Q (x) are singular as ∝ 1/x, or in N-space like ∝ 1/(N − 1), unlike the quarkonic contributions given in [17] with a leading pole ∝ 1/N. One notices that the number of functions needed in x-space to express A (3),n f T 2 F ,MS gq(g),Q is larger than in N-space, as has been found also in other analyses, cf. [7,11,49], requesting very careful numeric implementations.
Conclusions
We have calculated the contributions O(α 3 s n f T 2 F C A,F ) to the massive OMEs with local operator insertions on gluonic lines and veritces at general values of the Mellin variable N. These matrix elements are needed to describe the transition functions in the VFNS. In the calculation representations of the Feynman diagrams by generalized hypergeometric functions play an essential role. They allow the ε-expansion into nested sums, which can be solved using modern summation technologies. The number of these sums is very large, although their structures exhibit similarities. One may synchronize these sums, leading to a low number, however, with voluminous intermediate terms. The solution of the latter sums turns out to be more economic. The final results in N space can be expressed by rational functions in N and single harmonic sums up to S 3 (N). We also derived the corresponding x-space results, which have a more involved structure and depend on six Nielsen integrals.
with the harmonic polylogarithms H a ≡ H a (x) over the alphabet A = {0, 1, −1}2
27
−
16
81
(144ζ 2 − 85)x
+
32
3
(1 + x)H 2
0 −
44
3
δ(1 − x) −
16
81
(144ζ 2 + 149) + −
832x 2
27
+
16x
27
−
800
27
H 0 + −
832x 2
27
+
208x +C F ζ 3
2048x 2
27
+
512x
9
−
1024
9
(1 + x)H 0 −
512
9
−
2048
27x
+ C A
128
81
(1 + x)H 3
0
+ −
208x 2
81
+
812x
81
+
320
81
H 2
0 + −
8624x 2
243
−
8
81
(48ζ 2 − 199)x −
16
27
(8ζ 2 + 19)
−
64
27(x − 1)
H 0 + −
416x 2
81
+
56x
27
−
88
27
+
416
81x
H 1 H 0 +
64
27
ζ 2 −
310
27
δ(1 − x)
+
128
27
(1 + x)H 0,1 H 0 +
208x 2
81
−
20x
9
+
44
27
−
208
81x
H 2
1
−
416
729
x 2 (9ζ 2 + 113) −
8
729
(2088ζ 2 − 864ζ 3 + 6055) −
8
729
x(2601ζ 2 − 864ζ 3 − 4883)
+ −
8624x 2
243
+
2600x
81
−
872
81
+
4592
243x
H 1 +
832x 2
81
+
2144x
81
+
2120
81
−
416
81x
H 0,1
−
128
27
(1 + x)(H 0,0,1 + H 0,1,1 ) −
24064
729(x − 1) +
+
32320
729x
+C F
32
27
(1 + x)H 4
0 + −
128x 2
81
+
256x
81
+
A fast and precise numerical implementation in Mellin space has been given in[3].
For Mellin-space representations of a wide class of parton densities see[15].
Acknowledgment. For discussions we would like to thank J. Ablinger, A. De Freitas, and F. Wißbrock. This work has been supported in part by DFG Sonderforschungsbereich Transregio 9, Computergestützte Theoretische Teilchenphysik, Austrian Science Fund (FWF) grant P203477-N18, and EU Network LHCPHENOnet PITN-GA-2010-264564.
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|
[] |
[
"ASYMMETRIC EMBEDDING IN BRANE COSMOLOGY",
"ASYMMETRIC EMBEDDING IN BRANE COSMOLOGY"
] |
[
"Yuri Shtanov *[email protected]†[email protected] \nBogolyubov Institute for Theoretical Physics\nDepartamento de Física\nUniversidad del Valle\n03680, 25360Kiev, CaliA.AUkraine, Colombia\n",
"Alexander Viznyuk \nBogolyubov Institute for Theoretical Physics\nDepartamento de Física\nUniversidad del Valle\n03680, 25360Kiev, CaliA.AUkraine, Colombia\n",
"Luis Norberto Granda [email protected] \nBogolyubov Institute for Theoretical Physics\nDepartamento de Física\nUniversidad del Valle\n03680, 25360Kiev, CaliA.AUkraine, Colombia\n"
] |
[
"Bogolyubov Institute for Theoretical Physics\nDepartamento de Física\nUniversidad del Valle\n03680, 25360Kiev, CaliA.AUkraine, Colombia",
"Bogolyubov Institute for Theoretical Physics\nDepartamento de Física\nUniversidad del Valle\n03680, 25360Kiev, CaliA.AUkraine, Colombia",
"Bogolyubov Institute for Theoretical Physics\nDepartamento de Física\nUniversidad del Valle\n03680, 25360Kiev, CaliA.AUkraine, Colombia"
] |
[] |
We derive a system of cosmological equations for a braneworld with induced curvature which is a junction between several bulk spaces. The permutation symmetry of the bulk spaces is not imposed, and the values of the fundamental constants, and even the signatures of the extra dimension, may be different on different sides of the brane. We then consider the usual partial case of two asymmetric bulk spaces and derive an exact closed system of scalar equations on the brane. We apply this result to the cosmological evolution on such a brane and describe its various partial cases.
|
10.1142/s0217732308026856
|
[
"https://arxiv.org/pdf/0804.4888v1.pdf"
] | 16,572,487 |
0804.4888
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4f920ba676de4e3663a4b6ce1944dbbd2aad1e92
|
ASYMMETRIC EMBEDDING IN BRANE COSMOLOGY
30 Apr 2008 April 30, 2008
Yuri Shtanov *[email protected]†[email protected]
Bogolyubov Institute for Theoretical Physics
Departamento de Física
Universidad del Valle
03680, 25360Kiev, CaliA.AUkraine, Colombia
Alexander Viznyuk
Bogolyubov Institute for Theoretical Physics
Departamento de Física
Universidad del Valle
03680, 25360Kiev, CaliA.AUkraine, Colombia
Luis Norberto Granda [email protected]
Bogolyubov Institute for Theoretical Physics
Departamento de Física
Universidad del Valle
03680, 25360Kiev, CaliA.AUkraine, Colombia
ASYMMETRIC EMBEDDING IN BRANE COSMOLOGY
30 Apr 2008 April 30, 2008arXiv:0804.4888v1 [gr-qc] 19:34 WSPC/INSTRUCTION FILE mpla Modern Physics Letters A c World Scientific Publishing Companybraneworld model PACS Nos: 0450+h, 9880Es
We derive a system of cosmological equations for a braneworld with induced curvature which is a junction between several bulk spaces. The permutation symmetry of the bulk spaces is not imposed, and the values of the fundamental constants, and even the signatures of the extra dimension, may be different on different sides of the brane. We then consider the usual partial case of two asymmetric bulk spaces and derive an exact closed system of scalar equations on the brane. We apply this result to the cosmological evolution on such a brane and describe its various partial cases.
Introduction
The idea that our four-dimensional world can be described as a timelike hypersurface (brane) embedded in or bounding a five-dimensional manifold continues to be in the focus of modern investigations. Especially this concerns cosmological braneworld solutions which exhibit many interesting and unusual properties (see Refs. 1-3 for recent reviews). Theories with the simplest generic action involving scalar-curvature terms both in the bulk and on the brane allow for the possibilities of superacceleration of the universe expansion, 4,5 cosmological loitering even in a spatially flat universe, 6 and "cosmic mimicry," 7 which is characterized by the property that the braneworld model at low redshifts is virtually indistinguishable from the LCDM (Λ + Cold Dark Matter) cosmology but has renormalized value of the cosmological density parameters. The possibility of explaining other dark-matter phenomena in such models is discussed in Ref. 8.
A vast majority of papers on braneworld cosmology are dealing with the situation of the Z 2 symmetry of reflection with respect to the brane, which is equivalent to the brane being the boundary of the bulk. The present letter is devoted to the investigation of solutions where such symmetry is not imposed.
Braneworld cosmologies without Z 2 symmetry of reflection were a subject of investigation, for example, in Refs. [9][10][11][12][13][14][15][16][17][18][19]. In these papers, however, the curvature term in the action on the brane was absent. The braneworld model without Z 2 symmetry was studied in Ref. 20 in the presence of the curvature term on the brane and with equal bulk cosmological and gravitational constants on either side of the brane. In Ref. 21, the Z 2 -asymmetric model was considered in the braneworld theory with curvature term on the brane and with different cosmological constants in the bulk but with equal bulk gravitational constants on either sides of the brane. In a recent paper, 22 an asymmetric braneworld cosmology was under consideration in the model without the induced-curvature term on the brane but with different bulk cosmological and gravitational constants on either side of the brane.
In this letter, we would like to consider the most generic braneworld model in which the values of the bulk fundamental (gravitational and cosmological) constants are different on the two sides of the brane and, in addition to this, the signatures of the extra dimension on the two sides of the brane may be arbitrary (and different). Note that the Z 2 -symmetric braneworld cosmology with timelike extra dimension was previously considered in Refs. 16, 23-25. Before considering the asymmetric case with a brane separating between two bulk spaces, we consider a generalization in which a brane is a multivolume junction, i.e., a boundary of N bulk spaces with natural junction conditions. The theory of such spaces was initiated in Ref. 26. In the next section, we derive a system of cosmological equations for this configuration without assuming any symmetry between the bulk spaces. In particular, the values of the fundamental constants, and even the signatures of the extra dimension, may be different on different sides of the brane. After that, we study in more detail the usual case of two bulk spaces.
A brane as a multivolume junction
Our starting setup is a four-dimensional hypersurface (brane) B which is simultaneously a boundary of N five-dimensional Riemannian manifolds (bulk spaces) M 1 , . . . , M N and has nondegenerate Lorentzian induced metric (see Fig. 1). The primary junction condition which makes the brane the boundary of all bulk spaces is that the induced metric is one and the same in all M I , I = 1, . . . , N . We assume that the bulk gravitational and cosmological constants on different sides of the brane can be different. The action of the theory has the form
S = N I=1 M 3 I MI (R I − 2Λ I ) − 2ǫ I B K I + B m 2 R − 2σ + B L(h ab , φ) . (1)
Here, R I is the scalar curvature of the five-dimensional metric g I ab on M I , and R is the scalar curvature of the induced metric h ab = g ab − n I a n I b on B, where n a I is the vector of the inner unit normal to the brane. a The quantity K I = K I ab h ab is the a Although the vector of inner unit normal n a I is different in each of the bulk parts M I , the induced metric h ab is one and the same. In this letter, we systematically use the notation and conventions of Ref. 27. In particular, we use the one-to-one correspondence between tensors in B and tensors in M which are invariant under projection to the tangent space to B, i.e., tensors T a1···a k b1···b l such that
T a1···a k b1···b l = h a1 c1 · · · h a k c k h b1 d1 · · · h b l d l T c1···c k d1···d l .(2)
Variation of action (1) gives the equation of motion in the five-dimensional bulk parts M I :
G I ab + Λ I g I ab = 0 , I = 1, . . . , N ,(3)
and on the brane B:
m 2 G ab + σh ab = N I=1 ǫ I M 3 I S I ab + τ ab ,(4)
where G I ab and G ab are the Einstein's tensors of the corresponding spaces, S I ab ≡ K I ab − K I h ab is formed from the tensor of extrinsic curvature, and τ ab denotes the four-dimensional stress-energy tensor of matter on the brane. It is the presence of the tensors S I ab in the equation of motion (4) that makes the inner dynamics on the brane B unusual. b
The Codazzi relations on the brane, in view of the bulk equation (3), read
D a S I ab = R I cd n d I h c b = 0 , I = 1, . . . , N ,(5)
where D a is the (unique) derivative on the brane B associated with the induced metric h ab . Equation (4) then implies the relation
D a τ a b = 0 .(6)
Thus, the four-dimensional stress-energy tensor is covariantly conserved on the brane, which is a consequence of the absence of matter in the bulk. Generalizing the procedure of Refs. 29, 30 to the case of N bulk spaces, we consider the Gauss identity:
R abcd = h a f h b g h c k h d j R I f gkj + ǫ I K I ac K I bd − K I bc K I ad , I = 1, . . . , N .(7)
Contracting this relation and taking into account Eq. (3), one obtains the equation
ǫ I (R − 2Λ I )+K I ab K ab I −K 2 I ≡ ǫ I (R − 2Λ I )+S I ab S ab I − 1 3 S 2 I = 0 , I = 1, . . . , N ,(8)
which is valid on all sides of the brane, and which we expressed in terms of S I ab = K I ab − h ab K I and S I = h ab S I ab . This is the well-known constraint equation on the brane from the viewpoint of the gravitational dynamics in the five-dimensional bulk.
Our aim is to derive the resulting cosmological equation on the brane. One could solve this problem by considering embedding of the brane in the bulk spaces under consideration and calculating the corresponding extrinsic curvatures. However, in a cosmological setup, one can integrate the constraint equations (8) directly on the brane. First, we note that, for any tensor on the brane T ab which is covariantly conserved, i.e., D a T a b = 0, in the cosmological setup
T 0 0 = β(t) , T µ ν = δ µ ν q(t) , µ, ν = 1, 2, 3 ,(9)
one can easily verify the following relation:
T ab T ab − 1 3 T 2 = 1 3a 4 H d dt a 4 β 2 .(10)
In view of the conservation equation, the function q(t) in (9) is uniquely expressed through β(t). Then, setting
S 0 I 0 = −3β I (t) , I = 1, . . . , N ,(11)
and using property (10), valid for all these tensors in view of the conservation equation (
mpla Asymmetric Embedding in Brane Cosmology 5
where C I are integration constants. The zero-zero component of Eq. (4) gives
H 2 + κ a 2 = ρ + σ 3m 2 + 1 m 2 N I=1 ǫ I M 3 I β I .(13)
Substituting β I found from Eq. (12) into this equation, we obtain our main result:
H 2 + κ a 2 = ρ + σ 3m 2 + 1 m 2 N I=1 ζ I M 3 I ǫ I H 2 + κ a 2 − Λ I 6 − C I a 4 1/2 ,(14)
where ζ I = ±1 corresponds to the possibility of different signs in the solution for β I from Eq. (12). Physically, these signs correspond to the two possible ways of bounding each of the spaces M I by the brane.
The integration constants C I generalize the so-called "dark radiation" contribution to the dynamics of the brane. In the case where the brane is a boundary of N independent bulk spaces, there are exactly N such independent integration constants. If nonzero, they reflect the existence of black holes in the corresponding bulk spaces.
The case of two bulk spaces
In this section, we consider in detail the case where the brane is just embedded in one bulk space, so that N = 2, i.e., there are only two "sides" of the brane. Equation (4) involves the tensors S I ab which are constructed from the tensors of extrinsic curvature of the brane, so this equation is not closed with respect to the intrinsic evolution on the brane. However, using (8), it is possible to obtain a system of scalar equations which involves only four-dimensional fields on the brane. Following, with slight modifications, the procedure first adopted in Ref. 21 for the particular case M 1 = M 2 and ǫ 1 = ǫ 2 = 1, we introduce the tensors
Σ ab = ǫ 1 M 3 1 S (1) ab + ǫ 2 M 3 2 S (2) ab , ∆ ab = ǫ 1 M 3 1 S (1) ab − ǫ 2 M 3 2 S (2) ab .(15)
Rewriting (8) in terms of Σ ab and ∆ ab , we easily obtain the following closed system of gravitational equations on the brane:
Σ ab Σ ab − 1 3 Σ 2 + ∆ ab ∆ ab − 1 3 ∆ 2 + 2 ǫ 1 M 6 1 + ǫ 2 M 6 2 R − 4 ǫ 1 M 6 1 Λ 1 + ǫ 2 M 6 2 Λ 2 = 0 ,(16)Σ ab ∆ ab − 1 3 Σ∆ + ǫ 1 M 6 1 − ǫ 2 M 6 2 R − 2 ǫ 1 M 6 1 Λ 1 − ǫ 2 M 6 2 Λ 2 = 0 ,(17)D a ∆ a b = 0 ,(18)
where Σ = Σ ab h ab , ∆ = ∆ ab h ab , and Σ ab is given by Σ ab = m 2 G ab + σh ab − τ ab (19) in view of (4).
This system of equations is to be solved for the metric and matter fields and for the symmetric tensor field ∆ ab on the brane. It constitutes the main system of closed scalar equations on the brane, which arises in the absence of any information and/or boundary conditions for the brane-bulk system. For the particular case M 1 = M 2 and ǫ 1 = ǫ 2 = 1, this system of equations was first obtained in Ref. 21.
In what follows, we consider the cosmological implications of system (16)-(19) for the homogeneous and isotropic cosmological model with the cosmological time t, scale factor a(t), energy density ρ(t), and pressure p(t). Under these conditions, we set the tensor ∆ ab to be homogeneous and isotropic as well:
∆ 0 0 = −β(t) , ∆ µ ν = δ µ ν q(t) , µ, ν = 1, 2, 3 .(20)
Using (6) and (18), it is easy to calculate the quantities
Σ ab Σ ab − 1 3 Σ 2 = − 1 3a 4 H d dt a 2 3m 2 χ − ρ − σ 2 ,(21)∆ ab ∆ ab − 1 3 ∆ 2 = − 1 3a 4 H d dt a 4 β 2 ,(22)Σ ab ∆ ab − 1 3 Σ∆ = − 1 3a 4 H d dt a 4 β 3m 2 χ − ρ − σ ,(23)
where H ≡ȧ/a is the Hubble parameter,
χ ≡ H 2 + κ a 2 ,(24)
and κ = 0, ±1 corresponds to the sign of the spatial curvature of the model. Since the expressions a 3ȧ and a 3ȧ R are also total derivatives, equations (16) and (17) can be integrated with the result
m 4 χ − ρ + σ 3m 2 2 + β 2 9 = 2 ǫ 1 M 6 1 + ǫ 2 M 6 2 χ − 1 3 ǫ 1 M 6 1 Λ 1 + ǫ 2 M 6 2 Λ 2 − C a 4 ,(25)β 3m 2 χ − ρ − σ = 9 ǫ 1 M 6 1 − ǫ 2 M 6 2 χ − 3 2 ǫ 1 M 6 1 Λ 1 − ǫ 2 M 6 2 Λ 2 − 9E a 4 ,(26)
where C and E are integration constants. Then, eliminating β from (26) and using (25), we finally obtain
m 4 H 2 + κ a 2 − ρ + σ 3m 2 2 − 2 ǫ 1 M 6 1 + ǫ 2 M 6 2 H 2 + κ a 2 + 1 3 ǫ 1 M 6 1 Λ 1 + ǫ 2 M 6 2 Λ 2 + C a 4 + ǫ 1 M 6 1 − ǫ 2 M 6 2 H 2 + κ/a 2 − ǫ 1 M 6 1 Λ 1 − ǫ 2 M 6 2 Λ 2 /6 − E/a 4 m 2 (H 2 + κ/a 2 ) − (ρ + σ)/3 2 = 0 . (27)
This is our main result as regards cosmology without Z 2 symmetry in the bulk. It could also be obtained directly from (14) with I = 2. This equation in its general form is rather complicated, so it is useful to consider some partial cases. The general case ǫ 1 = ǫ 2 = +1 will be investigated in more detail elsewhere.
3.1. ǫ 1 = ǫ 2 = +1 and Λ 1 = Λ 2 or M 1 = M 2
We begin with the case Λ 1 = Λ 2 = Λ and also set ǫ 1 = ǫ 2 = +1. Then Eq. (27) becomes c
m 4 H 2 − ρ + σ 3m 2 2 − 2 M 6 1 + M 6 2 H 2 − Λ 6 + C a 4 + M 6 1 − M 6 2 H 2 − Λ/6 − E/a 4 m 2 H 2 − (ρ + σ)/3 2 = 0 .(28)
The last term on the right-hand side of this equation represents the difference between our model and standard Z 2 -symmetric cosmology. Setting additionally M 1 = M 2 = M and E = 0, we obtain the well-known result of the Z 2 -symmetric case: 17,31
m 4 H 2 − ρ + σ 3m 2 2 = 4M 6 H 2 − Λ 6 − C 1 a 4 , C 1 = C 4M 6 .(29)
This equation can easily be solved with respect to the Hubble parameter, giving two branches: 4,5
H 2 = ρ + σ 3m 2 + 2M 6 m 4 ± 2M 6 m 4 1 + m 4 M 6 ρ + σ 3m 2 − Λ 6 − C 1 a 4 1/2 .(30)
Setting ǫ 1 = ǫ 2 = ǫ and M 1 = M 2 = M but different Λ 1 and Λ 2 in Eq. (27), we obtain
m 4 H 2 − ρ + σ 3m 2 2 = 4ǫM 6 H 2 − Λ 1 + Λ 2 12 − ǫC 1 a 4 − M 12 36 Λ 1 − Λ 2 + E 1 /a 4 m 2 H 2 − (ρ + σ)/3 2 ,(31)
where C 1 is defined in (29), and
E 1 = 6E M 6 .(32)
Equation (31) Another interesting possibility is to take all the constants of the theory equal on the two sides of the brane (i. e., Λ 1 = Λ 2 = Λ and M 1 = M 2 = M ), but let E = 0. This means that the Z 2 symmetry is broken only by the difference of the masses of black holes on the two sides of the brane. After some redefinitions, in this case, from (27), we get (again, for simplicity, taking ǫ 1 = ǫ 2 = +1)
m 4 X 4 − 4M 6 X 3 − 4M 6 ρ + σ 3m 2 − Λ 6 − C 1 a 4 X 2 + E 2 m 4 a 8 = 0 ,(33)
c Here and below, for notational simplicity, we omit the spatial curvature term κ/a 2 . It can easily be recovered by the substitution H 2 → H 2 + κ/a 2 in all formulas.
where X stands for the expression H 2 − (ρ + σ)/3m 2 . This result is very similar to that obtained in Ref. 20, where this equation was solved in the limit of small but nonzero m. In the limit m → 0, equation (33) reduces to the equation
H 2 = Λ 6 + C 1 a 4 + (ρ + σ) 2 36M 6 + 9E 2 4a 8 (ρ + σ) 2 ,(34)
which was under consideration in Refs. 17, 18, 19.
3.3. m = 0 and ǫ 1 = ǫ 2 = +1
The limit m = 0 was thoroughly investigated in the previous literature. Taking this limit in Eq. (27) and keeping ǫ 1 = ǫ 2 = +1, one obtains .
This equation was studied by Padilla. 32 He noticed that, in a certain range of parameters, this model cosmologically behaves as the Z 2 -symmetric braneworld model with induced gravity. Specifically, with the upper sign in (35) and with the assumptions
|C 2 | a 4 ≪ (Λ 1 − Λ 2 )(M 6 1 − M 6 2 ) ,(37)
in which the right-hand side is assumed to be positive, and
(ρ + σ) ≪ M 3 1 M 3 2 M 6 1 + M 6 2 (Λ 1 − Λ 2 )(M 6 1 − M 6 2 ) ,(38)
equation (35) reduces to
H 2 = ρ + σ 3m 2 eff + Λ eff 6 + E 2 a 4 ,(39)
where
Λ eff = M 6 1 Λ 1 − M 6 2 Λ 2 M 6 1 − M 6 2 ,(40)m 2 eff = 3 (M 6 1 − M 6 2 ) 3 2M 6 1 M 6 2 (Λ 1 − Λ 2 ) ,(41)
which has the form of the Friedmann equation with cosmological constant and "dark radiation" [the last term in (39)].
Fig. 1 .
1Brane B as a junction between the bulk spaces M 1 , . . . , M N . trace of the symmetric tensor of extrinsic curvature K I ab of B in M I . The parameter ǫ I = 1 if the signature of the corresponding bulk part M I is Lorentzian, so that the extra dimension is spacelike, and ǫ I = −1 if its signature is (−, −, +, +, +), so that the extra dimension is timelike. The symbol L(h ab , φ) denotes the Lagrangian density of the four-dimensional matter fields φ the dynamics of which is restricted to the brane B so that they interact only with the induced metric h ab . All integrations over M I and over B are taken with the corresponding natural volume elements. The symbols M I , I = 1, . . . , N , denote the Planck masses of the corresponding spaces, Λ I , I = 1, . . . , N , are the corresponding five-dimensional cosmological constants, and m and σ are the Planck mass and tension of the brane, respectively.
5), we can integrate Eq. (8) with the result ǫ I H 2 + κ a 2 − Λ I 6 = β 2 I + ǫ I C I a 4 , I = 1, . . . , N ,
coincides with the result of Ref. 21.3.2. Λ 1 = Λ 2 and M 1 = M 2
Yuri Shtanov, Alexander Viznyuk, and Luis Norberto Granda
b Those interested in the derivation of (4) may look into the appendices of Refs.26, 28.
Yuri Shtanov, Alexander Viznyuk, and Luis Norberto Granda
3.4. M 1 = M 2 and ǫ 1 = −ǫ 2 = +1The structure of Eq.(27)allows one to consider the case M 1 = M 2 = M with the (somewhat exotic) condition ǫ 1 = −ǫ 2 = +1. In this case, we havewhere E 1 is defined in(32), and C 1 in (29). In the limit m → 0, this equation reduces toWith the upper sign and for Λ 2 ≫ Λ 1 , this again has the form of (39) with different effective constants m eff and Λ eff .ConclusionIn this letter, we derived a system of cosmological equations for a braneworld which is a multivolume junction, i.e., a boundary of N bulk spaces with natural junction conditions, without assuming any symmetry between the bulk spaces. In particular, the values of the fundamental constants, and even the signatures of the extra dimension, could be different on different sides of the brane. After that, we studied in more detail the usual case of two bulk spaces, and obtained the closed system of scalar gravitational equations (16)-(18) describing the braneworld in the absence of the Z 2 symmetry of reflection of the bulk with respect to the brane, with different fundamental constants in the bulk, and even with possibly different time signature of the fifth dimension on the two sides of the brane. We then derived equation(27)describing the cosmological evolution of such a brane and considered several special cases of this general situation.
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|
[] |
[
"Impurity effect in multi-orbital, sign-reversing s-wave superconductors",
"Impurity effect in multi-orbital, sign-reversing s-wave superconductors"
] |
[
"Y Nagai \nCCSE\nJapan Atomic Energy Agency\n6-9-3 Higashi-Ueno, Taito-ku110-0015TokyoJapan\n\nCREST (JST)\n4-1-8 Honcho332-0012KawaguchiSaitamaJapan\n\nTRIP (JST)\n5 Sanbancho Chiyoda-ku102-0075TokyoJapan\n",
"K Kuroki \nDepartment of Applied Physics and Chemistry\nThe University of Electro -Communications\n182-8585ChofuTokyoJapan\n\nTRIP (JST)\n5 Sanbancho Chiyoda-ku102-0075TokyoJapan\n",
"M Machida \nCCSE\nJapan Atomic Energy Agency\n6-9-3 Higashi-Ueno, Taito-ku110-0015TokyoJapan\n\nCREST (JST)\n4-1-8 Honcho332-0012KawaguchiSaitamaJapan\n\nTRIP (JST)\n5 Sanbancho Chiyoda-ku102-0075TokyoJapan\n",
"H Aoki \nDepartment of Physics\nUniversity of Tokyo\n113-0033HongoTokyoJapan\n\nTRIP (JST)\n5 Sanbancho Chiyoda-ku102-0075TokyoJapan\n"
] |
[
"CCSE\nJapan Atomic Energy Agency\n6-9-3 Higashi-Ueno, Taito-ku110-0015TokyoJapan",
"CREST (JST)\n4-1-8 Honcho332-0012KawaguchiSaitamaJapan",
"TRIP (JST)\n5 Sanbancho Chiyoda-ku102-0075TokyoJapan",
"Department of Applied Physics and Chemistry\nThe University of Electro -Communications\n182-8585ChofuTokyoJapan",
"TRIP (JST)\n5 Sanbancho Chiyoda-ku102-0075TokyoJapan",
"CCSE\nJapan Atomic Energy Agency\n6-9-3 Higashi-Ueno, Taito-ku110-0015TokyoJapan",
"CREST (JST)\n4-1-8 Honcho332-0012KawaguchiSaitamaJapan",
"TRIP (JST)\n5 Sanbancho Chiyoda-ku102-0075TokyoJapan",
"Department of Physics\nUniversity of Tokyo\n113-0033HongoTokyoJapan",
"TRIP (JST)\n5 Sanbancho Chiyoda-ku102-0075TokyoJapan"
] |
[] |
We study the impurity effects on the transition temperature T c with use of the T -matrix approximation. We propose a way to visualize the multi-orbital effect by introducing the hybridization function characterizing the multi-orbital effect for the impurity scattering. Characterizing function does not depend on the superconducting pairing symmetry, since this function is defined by the eigenvectors in normal states. The result indicates that an impurity-robust superconductivity does not necessarily imply a signpreserving pairing. Visualizing the hybridization effect in the effective five-band model for LaFeAsO, we show that the impurity effect on T c is relatively weaker than that in single-band models.
| null |
[
"https://arxiv.org/pdf/1012.5565v1.pdf"
] | 118,581,742 |
1012.5565
|
68bf289268ff4013d7ad1bb47a539121e095a414
|
Impurity effect in multi-orbital, sign-reversing s-wave superconductors
27 Dec 2010
Y Nagai
CCSE
Japan Atomic Energy Agency
6-9-3 Higashi-Ueno, Taito-ku110-0015TokyoJapan
CREST (JST)
4-1-8 Honcho332-0012KawaguchiSaitamaJapan
TRIP (JST)
5 Sanbancho Chiyoda-ku102-0075TokyoJapan
K Kuroki
Department of Applied Physics and Chemistry
The University of Electro -Communications
182-8585ChofuTokyoJapan
TRIP (JST)
5 Sanbancho Chiyoda-ku102-0075TokyoJapan
M Machida
CCSE
Japan Atomic Energy Agency
6-9-3 Higashi-Ueno, Taito-ku110-0015TokyoJapan
CREST (JST)
4-1-8 Honcho332-0012KawaguchiSaitamaJapan
TRIP (JST)
5 Sanbancho Chiyoda-ku102-0075TokyoJapan
H Aoki
Department of Physics
University of Tokyo
113-0033HongoTokyoJapan
TRIP (JST)
5 Sanbancho Chiyoda-ku102-0075TokyoJapan
Impurity effect in multi-orbital, sign-reversing s-wave superconductors
27 Dec 2010Impurity effectsIron-based superconductorsmulti-orbital systems 7420Rp7470Xa
We study the impurity effects on the transition temperature T c with use of the T -matrix approximation. We propose a way to visualize the multi-orbital effect by introducing the hybridization function characterizing the multi-orbital effect for the impurity scattering. Characterizing function does not depend on the superconducting pairing symmetry, since this function is defined by the eigenvectors in normal states. The result indicates that an impurity-robust superconductivity does not necessarily imply a signpreserving pairing. Visualizing the hybridization effect in the effective five-band model for LaFeAsO, we show that the impurity effect on T c is relatively weaker than that in single-band models.
Introduction
The discovery of the iron-based superconductor [1], which is a five-orbital system with three bands involved in the gap function [2]., is stimulating renewed interests in multi-orbital superconductivity. Specifically, a sign-reversing s-wave pairing (s±), which exploits the multi-orbital nature of the system, has been theoretically proposed as a candidate [2,3]. While various experimental measurements for determining the pairing symmetry in the iron-based superconductors are accumulating, which include the penetration depth, thermal conductivity, ARPES, STM, and NMR, one important test is the impurity effect on superconducting transition temperature T c . This has in fact been studied intensively both experimentally and theoretically [4][5][6][7]. There are two trends found in the experiments on the non-magnetic impurity effect on T c for LaFe 1−x Zn x AsO. The experiment by Li et al. suggests that the doping of Zn in LaFeAsO 1−y F y does not reduce T c [4]. The result by Guo et al. suggests that T c decreases significantly for a minimal level of Zn doping in LaFeAsO 0.85 [5]. A theoretical work by Onari and Kontani suggests that a fully gapped sign-reversing s-wave state is very fragile against non-magnetic impurities [7]. On the other hand, the compound containing phosphorus, LaFePO, whose gap function has line-nodes, is robust against non-magnetic impurities [8]. With this background, here we theoretically study the problem from a broader context, i.e., we want to identify which factors are crucial in determining T c in dirty, multi-orbital superconductors, especially for the sign-reversing s-wave.
Formulation
The self-energy in the T -matrix approximation in the orbitalrepresentation is expressed asΣ orbital = n impŤ , where n imp is the density of impurities, andŤ = (1 −VǦ loc ) −1V . HereV = Vσ z , and the local Green's functionǦ loc is defined asǦ loc = 1 N qǦ
orbital (q) witȟ G orbital (q, iω n ) ≡ Ĝ orbital (q, iω n ) −F orbital (q, iω n ) −F orbital † (q, iω n ) −Ĝ orbital (q, −iω n ) .(1)
Throughout the paper,â denotes an n × n matrix in the orbital space whileǎ a 2n × 2n matrix composed of the 2 × 2 Nambu space and the n × n orbital space. With use of the unitary matrixP k that diagonalizes the Hamiltonian in the orbital basis, the self-energy in the band representation is expressed asΣ band
k = n imp [1 −V band (k)Ǧ band loc (k)] −1V band (k) witȟ V band (k) ≡P † kVP k .
Considering the orbital-indepent impurity potentialV = V 0σz introduced in Ref. [7], the self-energy is expressed asΣ band
k = n imp [1 − V 0σzǦ
band loc (k)] −1 V 0σz . We then obtain the diagonal elements of the normal part of the self-energy as
(Σ band,N k ) ii = V 0 + V 2 0 (Ĝ band loc ) ii (k) + · · · ,(2)
with
(Ĝ band loc ) ii (k) = 1 N q l |C li (q, k)| 2 G band ll (q).(3)
Here we have introduced a unitary matrixĈ(q, k) whose element C i j (q, k) can be written as
C i j (q, k) = p † qi p k j ,(4)
where p k j denotes the j-th eigenvector of the Hamiltonian with momentum k in the normal state, while k and q denote, respectively, the initial-and final-state quasiparticle momenta in impurity scatterings. Equations (2) and (3) imply that the l-th band hybridization affects the i-th band self-energy (Σ band,N k ) ii through |C li (q, k)| 2 G band ll (q). Hence the factor |C li (q, k)| 2 characterizes the multi-orbital effect for the impurity scattering. From Eq. (4) we find that the more the eigenvectors with initial-state momentum k and final-state momentum q resemble with each other, the stronger the inter-band impurity effect on T c becomes.
Thus a speciality of a multi-band system is that, on top of the band dispersion and Fermi surface, the character of eigenvectors acts as an important factor determining the way in which the impurity effect appears. We can thus call the factor |C li (q, k)| 2 the impurity scattering intensity. We can use this property to construct a model that is robust against the impurity effect. For instance, the impurity effect does not appear on T c in a twoorbital model described by
H ii k = −[cos(k x ) + cos(k y )], H i j k = t ′ ,
since the impurity scattering intensity in this case reduces to |C li (q, k)| 2 = δ li everywhere in momentum space.
Visualized hybridization effect on the impurity scattering
The general scheme above enables us to visualize the hybridization effect on the impurity scattering. In doing so, we can concentrate on the Fermi momentum, since the Green function has amplitudes localized around the Fermi energy in Eq. (3). Then |C li (q, k)| 2 can be parameterized as |C li (θ l , θ i )| 2 , where θ l(i) describes the position of the final-(initial-) state on the l-th (ith) Fermi surface. Let us visualize the hybridization effect in the effective five-band model proposed by Kuroki et al. for LaFeAsO [2]. In this model with the Fermi energy E F = 10.94eV, there are two hole Fermi pockets around (k x , k y ) = (0, 0), and two electron pockets around (0, π) and (π, 0) as displayed in Fig. 1.
First, we visualize the inter-band impurity scattering intensity of the quasiparticles between the hole and electron Fermi surfaces. Here, we introduce the band-index i whose energy ǫ i satisfies the relation ǫ i > ǫ j (i > j). The hole Fermi surfaces on the 2nd and 3rd bands are mainly constructed from d xz and d yz orbitals. The dominant component of the eigenvectors at the electron Fermi surface on the 4th band depends on the Fermi wave-number. In Fig. 3, we show the dominant components in the orbital basis on each Fermi surface. In the case of the impurity scattering between 2nd and 4th bands, the intensity strongly depends on both of the initial-state θ 2 and the final-state θ 4 as shown in Fig. 2(a). For example, the intensity at (θ 2 , θ 4 ) = (3π/2, π/2) is almost zero, so that the interband impurity scatterings between θ 2 = 3π/2 and θ 4 = π/2 do not affect the superconducting transition temperature T c even if these are sing-reversing scatterings. The intensity at (θ 2 , θ 4 ) = (π/2, π) as shown in Figs. 2 (a) and 3has a maximum value |C 24 (θ 2 , θ 4 )| 2 ≃ 0.6. By contrast, the impurity scattering between 3rd and 4th bands does not affect T c regardless of the initial and final momenta, since the intensity is small everywhere as shown in Fig. 2(b).
Second, we turn to the intra-band impurity scattering intensity |C 22 (θ 2 , θ ′ 2 )| 2 and |C 33 (θ 3 , θ ′ 3 )| 2 . As shown in Fig. 4 , the intensities on a same band are bigger than that between different bands on the whole. On the line satisfying the relation
θ i = θ ′ i , the intensity |C ii (θ i , θ i )| 2 become |C ii (θ i , θ i )| 2 = 1
, since the eigenvectors are same. The intensities |C ii (θ i , θ i + π)| 2 differ from |C ii (θ i , θ i )| 2 = 1. This originates from an absence of the inversion symmetry in the eigenvector p k j . Although this may first seem strange, the Hamiltonian for momentum k in the orbital-basis is not equal to that for momentum −k, since the positional relation between the iron atoms and the arsenic atoms differs between the directions for k and −k.
From these figures 2and 4 we find that the impurity scattering intensities |C li (θ l , θ i )| 2 in this effective five-band model are small on the whole. Focusing the fact that the intensities in single band models are always |C 11 (θ 1 , θ 1 )| 2 = 1, these figures suggest that the impurity effect on T c in the effective five-band model is relatively weaker than that in single-band models.
Finally, we show the impurity intensity dependence of T c in Fig. 5with use of the self-consistent T -matrix approximation in the sign-reversing s-wave superconductor for an impurity density n imp = 0.01. The result, which is essentially the same as the result in Ref. [7], indicates that an attractive impurity potential does not significantly reduce T c . This behavior originates, in the present view, from the multi-orbital hybridization effect characterized by |C li (θ l , θ i )| 2 .
Conclusion
We have studied the impurity effects on T c with use of the T -matrix approximation. We found that the hybridization function |C li (q, k)| 2 (Eq. (4)) characterizes the multi-orbital effect for the impurity scatterings. The more the eigenvectors at initialstate and final-state momenta are similar to each other, the stronger the inter-band impurity effect on T c becomes. We thus proposed a way to visualize the multi-orbital effect. The figures visualizing the multi-orbital effect suggest that the impurity effect on T c in the effective five-band model is relatively weaker than that in single-band models. The above results do not depend on the superconducting pairing symmetry, since |C li (q, k)| 2 is defined only by the eigenvectors in normal states. The message of this result is that there can be impurity-robust sign-reversing s-wave pairing symmetry in the iron-based superconductors. Conversely, an impurity-robust superconductivity does not necessarily imply sign-preserving pairing. In addition, with use of the present visualization, one might find the paring symmetry with line-nodes where the impurity effects do not appear on T c in such materials as LaFePO.
Figure captions
Figure captions
Figure 1 :
1Fermi surfaces in the effective five-band model at the Fermi energy E F = 10.94 eV.
Figure 2 :
2Inter-band impurity scattering intensity of the quasiparticles between the 2nd band and the 3rd band (a) , or between 2nd band and 4th band (b).
Figure 3 :
3Dominant character of the eigenvectors is schematically shown on the Fermi surfaces. The different colors denote different hybridization in the orbital basis. Blue dots represent an example of the initial and final momenta.
Figure 4 :
4Intra-band impurity scattering intensity on the 2nd band (a), or on the 3rd band (b).
Figure 5 :
5Impurity-intensity (I) dependence of T c in the signreversing s-wave superconductor.
Figure 5 :
5Figures
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|
[] |
[
"RAMIFICATION OF INSEPARABLE COVERINGS OF SCHEMES AND APPLICATION TO DIAGONALIZABLE GROUP ACTIONS",
"RAMIFICATION OF INSEPARABLE COVERINGS OF SCHEMES AND APPLICATION TO DIAGONALIZABLE GROUP ACTIONS"
] |
[
"Gabriel Zalamansky "
] |
[] |
[] |
We define the notion of inseparable coverings of schemes and we propose a ramification formalism for them, along the lines of the classical one. Using this formalism we prove a formula analogous to the classical Riemann-Hurwitz formula for generic torsors under infinitesimal diagonalizable group schemes.
| null |
[
"https://arxiv.org/pdf/1603.09284v1.pdf"
] | 119,129,813 |
1603.09284
|
427fc35e8a49683b5f6d8ccc3b32890633071d3e
|
RAMIFICATION OF INSEPARABLE COVERINGS OF SCHEMES AND APPLICATION TO DIAGONALIZABLE GROUP ACTIONS
30 Mar 2016
Gabriel Zalamansky
RAMIFICATION OF INSEPARABLE COVERINGS OF SCHEMES AND APPLICATION TO DIAGONALIZABLE GROUP ACTIONS
30 Mar 2016arXiv:1603.09284v1 [math.AG]
We define the notion of inseparable coverings of schemes and we propose a ramification formalism for them, along the lines of the classical one. Using this formalism we prove a formula analogous to the classical Riemann-Hurwitz formula for generic torsors under infinitesimal diagonalizable group schemes.
Introduction
When f : Y −→ X is a ramified cover of a smooth scheme X, ie a finite, surjective, locally free morphism of smooth schemes which is étale over a dense open subscheme of X, the classical ramification theory associates to f a divisor that measures the obstruction for f to be an étale covering. Let us briefly recall it. If f is as above, the sheaf of first-order differential forms Ω 1 f is trivial on a dense open subscheme of Y and hence is a torsion sheaf, to which one can associate a divisor R f , also denoted R Y /X , (by a process we recall in 3.1) which measures the obstruction for f to be étale everywhere on Y . Such a morphism is classically called a ramified covering and the divisor R f is called the ramification divisor of f . A crucial feature of this construction is that it is transitive with respect to dévissage : if f : Z −→ X is a ramified covering that factors into ramified coverings g : Z −→ Y followed by h : Y −→ X then, as divisors on Z, we have (*) R Z/X = R Z/Y +g * R Y /X . The ramification theory of local rings with perfect residue fields allows for the computation of the local multiplicities of the divisors. For a ramified cover f : Y −→ X, one can relate R f to the geometry of the morphism f via the formula (RH) det(Ω 1 f ) = O Y (R f ) from which is derived, in the case of projective curves, the famous Riemann-Hurwitz formula.
Observe in particular that if f is the quotient morphism of Y by the action of a finite étale group scheme G, then f is étale if and only if the action of G is free everywhere on Y , if and only if f : Y −→ X is a G-torsor. In this case the ramification divisor R f also measures the obstruction for f to be a G-torsor and the formula (RH) relates the action on Y to the geometry of the quotient morphism.
We now raise the question : what if G is no longer assumed to be étale ? More precisely, if Y is a scheme of characteristic p > 0 and G is a finite flat group scheme (possibly infinitesimal) acting on Y , freely on a dense open subset, can one measure the obstruction for the quotient morphism to be a G-torsor ? Our goal is thus to develop a theory of ramified coverings in which the unramified objects would no longer be the étale morphisms but the torsors. Note that torsors under infinitesimal group schemes are purely inseparable. In this case the sheaf of differential 1-forms is no longer torsion and one cannot hope to directly carry over the previous definitions of ramification to this setting. We then have to find a substitute for sheaf of differential 1-forms.
There is, however, an issue which lies in the very formulation of these questions that needs to be addressed first, as illustrated by the following example.
1.1. Example. Let k be a field of characteristic p > 0 and A 1 = Spec(k[x]) be the affine line over k. Consider the action of the infinitesimal k-group scheme µ p,k = Spec( k [s] s p −1 ) on A 1 given by s.x = sx. An easy computation of the invariant ring shows that the quotient morphism is the absolute Frobenius :
F : A 1 −→ A 1 x → x p .
This action is free on the dense open subscheme A 1 \ {0} and has a fixed point in 0. The quotient morphism is thus not a µ p,k -torsor.
However, it is easily seen that F is also the quotient morphism for the action on A 1 of the infinitesimal k-group scheme α p,k = Spec( k[t] t p ) given by t.x = x + t, which is free everywhere. This makes F into an α p,k -torsor.
Finally, F can also be seen as the quotient morphism for the non-free action of α p,k on A 1 , this time given by t.x = x 1+tx = 1 − tx + ... + (−1) p−1 t p−1 x p−1 . Note that the group schemes α p,k and µ p,k are not isomorphic.
This example shows that, contrary to classical ramified coverings, neither the group acting nor the eventuality of being a torsor is determined by the sole quotient morphism. In this situation, the question of measuring the obstruction of a finite locally free morphism to be a torsor makes sense only relatively to a given group action.
Overview of the paper
In the first section of this article, we define the notion of "generalized coverings" which includes the data of a specific action along with a finite flat morphism. Our definitions are formulated in terms of groupoid schemes. This allows our formalism to include finite flat morphisms arising from quotients by vector fields (ie. foliations) that do not necessarily stem from group actions. We recall the relevant basic facts about groupoid schemes.
In a second section we proceed to propose a ramification formalism along the lines of the classical one that we outlined. In the case of a generically étale Galois covering, we recover the ramification divisor of the classical theory.
In the last section we then specify the situation to actions of infinitesimal diagonalizable group schemes to obtain a formula relating the action and the geometry of the quotient morphism, much like the classical Riemann-Hurwitz formula.
Let f : Y −→ X be a morphism between S-schemes. If G ⇒ X is a groupoid one can define its pullback to Y as follows :
• Set f * G = (Y × S Y ) × X×SX G.
• Set f * s(y 1 , y 2 , g) = y 1 and f * t(y 1 , y 2 , g) = y 2 . • Composition is given by f * c((y 1 , y 2 , g), (z 1 , z 2 , h)) = (y 1 , z 2 , gh). One can check that (Y, f * G, f * s, f * t, f * c) is an S-groupoid which we call the pullback of G ⇒ X by f . We shall often denote it by G |Y .
Kernels.
Let f be a morphism between two S-groupoids G ⇒ X and G ′ ⇒ X ′ . The kernel of f , denoted by ker(f ) is the groupoid defined as follows :
• The schemes of arrows is defined by the fibre product
ker(f ) / / X ′ e ′ G f / / G ′ where e ′ is the unit section of G ′ .
• Source and target are given by the compositions ker(f ) −→ G ⇒ X. • Composition is given by the composition in G. By definition points of ker(f ) are those of G which are sent to identities by f . Let us note that the unit section is an immersion (since s ′ • e ′ = id X ′ ). Hence ker(f ) is a subgroupoid of G ⇒ X.
Stabilizers.
Let G ⇒ X be an S-groupoid. We define its stabilizer, which we denote St G , by the fibre product of j = (s, t) : G −→ X × S X with the diagonal morphism of X :
St G / / G j X ∆X / / X × S X
The points of St G are the points of G whose source and target are equal. The composition in G induces a morphism St G × X St G −→ St G which makes St G into an X-group scheme. It is the biggest subgroupoid of G that is an X-group scheme. The action of a groupoid is said to be free if it has trivial stabilizer.
2.2. Generalized covers. We now proceed to give a definition of coverings that would include inseparable morphisms invariant under the generically free action of a finite locally free groupoid scheme.
As explained in the introduction, the same inseparable morphism can be seen as the quotient morphism for several actions of non-isomorphic group schemes. Hence to treat these morphisms as coverings we need to specify a groupoid acting on the source.
We propose the following :
Definition 2.2. Fix a base scheme S and an S-scheme X.
A (generalized) covering of X is a couple (Y −→ X, G ⇒ Y ), where
• Y is an S-scheme.
• G ⇒ Y is a finite locally free X-groupoid whose orbits are included into open affines of Y and whose action on Y is generically free. This means that there exists a dense open subscheme V ⊂ Y such that G |V acts freely on V .
• Y −→ X is a finite surjective locally free morphism which is G-invariant. • The order of G is the same as the order of Y −→ X, ie [G : Y ] = [Y : X].
The word "generalized" will be mostly be employed to stress the difference between classical generically étale morphisms and the objects defined above. When no confusion is likely, the latter will just be called coverings. For short, we shall often write (Y, G) instead of (Y −→ X, G ⇒ Y ). In case G = G × S Y is the action groupoid for the action of a group G, we shall even denote the covering (Y, G). We refer to those as G-coverings.
Let us note that, if (Y, G) is a covering of a scheme X, the hypothesis that the orbits of G ⇒ Y are included into open affines of Y imply, together with local freeness, that the quotient Y /G exists in the category of S-schemes. See [Gro11a, Exp V, th.4.1] for a proof. However, since the action of G is not free, the quotient scheme does not represent the fppf quotient sheaf of Y by G. Hence a priori we do not know the points of Y /G.
The conditions that we imposed in the definition of a covering imply that the quotient Y /G identifies with X, according to the following lemma.
Lemma 2.1. Let (Y, G) be a covering of an S-scheme X. The categorical quotient Y /G identifies with X.
Proof :
This follows from the fact that the morphism
j X = (s, t) : G −→ Y × X Y
is an epimorphism of schemes. Assume this for the moment. We will show that X satisfies the universal property of the quotient Y /G. Let f : Y −→ T be a G-invariant morphism of S-schemes. Since Y −→ X is faithfully flat, to show that f factors through X it suffices, by descent, to show that
f • pr 1 = f • pr 2 , where pr 1 , pr 2 : Y × X Y −→ Y are the two projections. Since by assumption j X is an epimorphism, it is equivalent to show that f • pr 1 • j X = f • pr 2 • j X . But this last equality is just the equality f • s = f • t, which is verified since f is G-invariant.
Hence we are left to show that j X is an epimorphism. Note that it is finite, so in particular quasi-compact and quasi-separated. Hence j X is schematically dominant if and only if j ♯
X : O Y ×X Y −→ j X * O G is injective.V −→ V /G is finite flat of degree [Y : X]. Since Y −→ X is G-invariant, we have a commutative diagram V / / W V /G = = ④ ④ ④ ④ ④ ④ ④ ④
from which we conclude, by the fiberwise criterion for flatness, that V /G −→ W is finite flat of degree 1, hence an isomorphism.
Since the action of G |V is free on V , the morphism j V :
G |V −→ V × W V is an isomorphism. By faithfull flatness of Y −→ X, the immersion V × W V −→ Y × X Y is schematically dominant and we have a commutative diagram G |V / / jV G jX V × W V / / Y × X Y from which we conclude that j X is schematically dominant.
We can then define the notion of morphisms of generalized coverings, in an obvious way.
Definition 2.3. If (Y 1 , G 1 ) and (Y 2 , G 2 ) are two coverings of an S-scheme X, a morphism of coverings is a groupoid morphism f : G 1 −→ G 2 such that the following diagram commutes :
G 1 f / / s1 t1 G 2 s2 t2 Y 1 f0 / / ❆ ❆ ❆ ❆ ❆ ❆ ❆ Y 2~⑥ ⑥ ⑥ ⑥ ⑥ ⑥ ⑥ X
We wish to develop a ramification theory for these objects, along the lines of the classical one, in which the unramified objects would be the coverings given by groupoids acting freely.
Definition 2.4. A covering (Y, G) of an S-scheme X is said to be unramified if the groupoid G ⇒ Y acts freely.
A ramification divisor for generalized coverings.
If (Y, G) is a covering of an S-scheme X, by definition the action of G ⇒ Y is free on a dense open subscheme of Y . Thus its stabilizer group scheme σ : St G −→ Y is trivial over a dense open subscheme of Y . Denote by m G its augmentation ideal, ie the ideal defining the unit section Y −→ St G . The latter is zero if and only if St G is trivial. Hence it follows that the sheaf of O Y -modules σ * m G is a torsion sheaf which is trivial if and only if the groupoid G ⇒ Y acts freely.
We thus have a torsion sheaf of O Y -modules which is zero exactly when the covering (Y, G) is unramified. Accordingly, it is a natural candidate to replace the sheaf of differential 1-forms of the classical theory.
We wish to have a geometric incarnation of this sheaf. We use a construction of Mumford, which we recall, that produces an effective Cartier divisor out of a torsion sheaf. Over a smooth scheme (or at least regular in codimension 1) the corresponding Weil divisor is just the sum of the codimension 1 points of its support with appropriate multiplicities.
3.1. Div of a coherent torsion sheaf. In this section we recall the construction of Mumford that associates an effective Cartier divisor to a coherent torsion sheaf. We refer to [MFK94, Chap. V.3] for greater details.
Let X be a noetherian S-scheme and F a coherent sheaf on X such that : (i) The support of F does not contain any associated point (ie depth 0 point) of X. (ii) For all point x ∈ X, the stalk F x is of finite tor-dimension, ie admits a finite projective resolution. If E is a locally free sheaf of rank r on X, we denote by det(E) the invertible sheaf Λ r E. Let us start with the following lemma, proved in [MFK94, Chap.V, §3, Lemma 5.6].
Lemma 3.1. If 0 −→ E n −→ E n−1 −→ ... −→ E 0 −→ 0 is an exact sequence of locally free sheaves on X, there exists a canonical isomorphism n i=0 det(E i ) (−1) i ≃ O X .
By assumptions, every point x ∈ X has an open neighborhood U over which F has a finite resolution by free O U -modules
0 −→ E n −→ ... −→ E 0 −→ F |U −→ 0. Set U ′ = U \ Supp(F). By definition over U ′ the sequence 0 −→ E n |U ′ −→ ... −→ E 0 |U ′ −→ 0 is exact, hence by the above lemma there is a canonical isomorphism O U ′ ≃ n i=0 det(E i ) (−1) i |U ′ .
Also, since the sheaves E i are free on U , we have an isomorphism
n i=1 det(E i ) (−1) i ≃ O U , unique up to a unit. Composing these we get a morphism O U ′ −→ O U ′ , defined by a section f ∈ O X (U ′ ).
Since f is unique up to unit in U and not a zero-divisor by assumption (i), we get a Cartier divisor (f ) in U . We refer to [MFK94, Chap. V.3] for a proof that these constructions glue to give an effective Cartier divisor div(F) on X.
If x ∈ X is a point of depth 1, it follows from the Auslander-Buschbaum formula that, over some neighborhood of x, such a sheaf F has a free resolution of the form 0 −→ E 1 −→ E 0 −→ F −→ 0, where E 1 and E 0 are free sheaves of the same rank. If h denotes the map E 1 −→ E 0 , we then have div(F) x = (det(h)) x . When X is regular in codimension 1, this allows one to give a simple expression of div(F) :
Lemma 3.2. Suppose X = Spec(A) is the spectrum of a discrete valuation ring. Let π ∈ A be a uniformizer. There exists an A-module M of finite length such that F =M and we have div(F) = (π lA(M) ), where l A (M ) is the A-length of M .
Proof :
There exists an isomorphism of A-modules M ≃ r i=1 A/π ni for an r-uple of integers (n 1 , ..., n r ). We then have a resolution
0 −→ A r h −→ A r −→ M −→ 0, where h is the diagonal matrix (π ni δ ij ) 1≤i,j≤n whose determinant is π lA(M) .
This results globalizes immediately to any scheme that is regular in codimension 1, for if x ∈ X has codimension 1 and U = Spec(A) is an affine neighborhood of x ∈ X, the local ring O X,x is a flat A-module and we can tensor the resolutions used to compute div(F) by O X,x to obtain resolutions that compute div(F x ). Hence the multiplicity of div(F) at x is l OX,x (F x ).
3.2.
A ramification divisor for generalized coverings. We are now ready to define a ramification divisor for generalized coverings.
Definition 3.1. Let (Y, G) be a covering of an S-scheme X. Let σ : St G −→ Y be the stabilizer group scheme of the groupoid G and m G be its augmentation ideal. Suppose that the O Y -module σ * m G has finite projective dimension. Define the ramification divisor of (Y, G) to be
R G := div(σ * m G ).
Remark. In order the use the general construction of 3.1 we have to make sure that the O Y -module σ * m G has finite projective dimension. This will always be the case if Y is a regular scheme.
Let us compute the ramification divisors of the examples 1.1 given in the introduction.
Examples 3.1. Let k be a field of caracteristic p > 0.
• Consider the groupoid G ⇒ A 1 k given by the action of the group scheme G = µ p,k = Spec( k[s] s p −1 ) on the affine line A 1 k = Spec(k[y]) by multiplication. It is defined by the coaction
k[y] −→ k[y,s] s p −1 y → sy .
The stabilizer group scheme is given by the fiber product
St G / / G × k Y j Y ∆Y / / Y × k Y.
The morphism j is defined by the ring map
k[y 1 , y 2 ] −→ k[y,s] s p −1 y 1 → y y 2 → sy . Thus we have O St G = k[y,s] s p −1 ⊗ k[y1,y2] k[y] = k[y,s] s p −1,(s−1)y . Its augmenta- tion ideal is m G = (s − 1) k[y,s]
s p −1,(s−1)y , for which we have the following free resolution as a k[y]-module :
0 −→ k[y] ⊕p−1 ×y −→ k[y] ⊕p−1 −→ m G −→ 0,
the first arrow being the multiplication of each coordinates by y. Its determinant is y p−1 . Hence the ramification divisor of this covering is supported in 0 ∈ A 1 k where is has multiplicity p − 1. As Weil divisors we thus have
R G = (p − 1)[0].
• Consider this time the groupoid G ⇒ A 1 k given by the action of G = α p,k on the affine line defined by the algebra map
k[y] −→ k[y,t] t p y → y 1+ty = 1 − ty + ... + (−1) p−1 t p−1 y p−1 . The morphism j is defined by the ring map k[y 1 , y 2 ] −→ k[y,t] t p y 1 → y y 2 → y 1+ty .
Observe that y − y 1+ty = ty 2 1+ty . Hence O St G = k[y,t] t p ,ty 2 and m G = t k[y,t] t p ,ty 2 . We have the following free resolution of m G as a k[y]-module :
0 −→ k[y] ⊕p−1 ×y 2 −→ k[y] ⊕p−1 −→ m G −→ 0,
the first arrow being the multiplication of each coordinates by y 2 . Its determinant is y 2(p−1) . We thus have
R G = 2(p − 1)[0].
• Finally, it clear that the action of α p,k by translation on the affine line leads to a groupoid with trivial stabilizer, hence no ramification divisor.
We see that our definition of covering allows one to differentiate between these group actions, which was impossible with the sole quotient morphism.
In case the covering is given by the action of a finite étale group scheme, the classical theory already produces a ramification divisor, using first-order differential forms, as recalled in the introduction 1. In the next section we show that, in that case, the latter agrees with the one we just defined.
3.3. The case of generically étale Galois coverings.
Theorem 3.1. Let f : Y −→ X be a generically étale morphism of normal schemes. Suppose that f is a Galois cover of group G, in the sense of [Gro63], and that all the residue fields extensions k(y)/k(f (y)) are separable. Denote by G ⇒ Y the action groupoid of G on Y . Let R G be the divisor associated with the stabilizer of G, defined in 3.1, and R Y /X = div(Ω 1 Y /X ) be the ramification divisor of the classical theory. One has the equality
R G = R Y /X .
Proof :
We will show that both divisors have the same multiplicity in each codimension 1 point of Y . By assumption Y is regular in codimension 1, so we may assume that Y = Spec(A) and X = Spec(A 0 ) are discrete valuation rings, whose corresponding extension K/K 0 of fraction fields is Galois of group G.
If B is an A 0 -algebra we shall denote by B G the algebra of functions from G to B, which is a finite B-module, a basis being given by the functions
e g : G −→ B h → δ h,g .
The multiplication in B G is given by e g e g ′ = δ g,g ′ e g . The action of G on Y is given by algebra automorphisms g ♯ : A −→ A, one for each g ∈ G, satisfying the usual conditions. If we abuse notations and denote by g the automorphism (g ♯ ) −1 , the action map ρ : G × X Y −→ Y corresponds to the algebra map
ρ ♯ : A −→ A ⊗ A0 A 0 [G] ≃ A[G] a → g∈G g(a)e g . The morphism j : G × X Y −→ Y × X Y is then given by j ♯ : A ⊗ A0 A −→ A[G] a ⊗ b → g∈G ag(b)e g .
Let us compute the ideal I defining the stabilizer
St G of the groupoid G × X Y ⇒ Y . It is the ideal generated in A[G] by the image of the ideal defining the diagonal immersion Y ֒→ Y × X Y.
The latter is generated in A ⊗ A0 A by the elements of the form (1 ⊗ a − a ⊗ 1), for a ∈ A. Note that we have
j ♯ (1 ⊗ a − a ⊗ 1) = g∈G (g(a) − a)e g since in A[G]
we have 1 = g∈G e g . These expressions generate the ideal I.
The augmentation ideal of St G is generated by the images in A[G]/I of the e g with g = 1.
Observe that, since e g e g ′ = δ g,g ′ , if t = g∈G t g e g ∈ A[G] and u = g∈G (g(a) − a)e g for some a ∈ A then tu = g∈G t g (g(a) − a)e g .
We thus have an isomorphism of A-algebras
O St G ≃ g∈G A/I g ,
where I g is the ideal generated in A by the expressions (g(a) − a), a ∈ A.
It follows that we have the isomorphism of A-modules
m G ≃ g =1 A/I g .
By assumption, the residue field extension k
(A)/k(A 0 ) is separable. Hence by [Ser68, III, §6, prop.12], A is a monogenic A 0 -algebra. Let x be a generator and v be the valuation in A. For all g ∈ G we have v(I g ) = v(g(x) − x) := i G (g).
With these notations we thus have
m G ≃ g∈G A/π iG(g) ,
where π is a uniformizer of A.
We obtain a free resolution of the A-module m G of the following form :
0 −→ A ⊕|G|−1 M −→ A ⊕|G|−1 −→ m G −→ 0,
where M is a diagonal matrix of size |G|− 1 whose diagonal entries are the elements π iG(g) for g = 1. Its determinant has valuation
g =1 i G (g).
On the other hand, we know from [Ser68, IV, §1, prop.4] that
g =1 i G (g) = v(D A/A0 ),
where D A/A0 stands for the different of the ring extension A/A 0 . The latter is the annihilator of the module Ω 1 A/A0 . Thus we see that the multiplicities of the divisors R G and R Y /X are equal at every codimension 1 points. Hence they are equal.
Remark. The term "generalized covering" that we use to refer to our definition 2.2 is abusive because it is not clear to us how to include the generically étale morphisms that do not arise from group actions. More precisely, if Y −→ X is a finite locally free generically étale morphism we do not know what groupoid to attach to it in order to make it a generalized covering in the sense of 2.2. One can always consider the trivial groupoid Y × X Y ⇒ Y given by the two projections but this would not be a wise choice since its stabilizer is always trivial, even if Y −→ X is not étale everywhere.
On the other hand, if Y −→ X as above is given by the quotient of Y by the action of a finite group, then it becomes a generalized covering in the sense of 2.2 when endowed with the action groupoid.
3.4. Devissage of the ramification divisor. We now tackle the problem of performing dévissage of generalized coverings. Let X be an S-scheme and (f : Z −→ X, G ⇒ Z) be a generalized covering of X. Suppose given a subgroupoid H ⇒ Z of G ⇒ Z. Since G acts generically freely on Z, so does H. Let Y = Z/H be the quotient scheme of Z by H. We then have a covering (g :
Z −→ Y, H ⇒ Z) of Y . Since obviously the morphism f : Z −→ X is H-invariant, we have a factorisation Z g / / f ❅ ❅ ❅ ❅ ❅ ❅ ❅ Y h~⑦⑦ ⑦ ⑦ ⑦ ⑦ ⑦ ⑦ X .
We would like to add some structure on the morphism h in order to make it into a covering of X. The scheme Y should be endowed with the action of the quotient groupoid of G by H. By quotient groupoid we mean a groupoid Q ⇒ Y acting on Y such that every groupoid morphism (G ⇒ Z) −→ (T ⇒ T ) that contains H in its kernel factors through Q ⇒ Y . In case the groupoid G ⇒ Z is given by the action of a finite group scheme G on Z and H ⊳ G is a normal subgroup, it is easy to check that the quotient groupoid is just the residual action groupoid
G/H × S Y ⇒ Y of G/H on Y .
In general, since we want to have a groupoid with source and target defined in Y , it is natural to define Q as the quotient scheme of G by the action of H 2 by preand post-composition, ie we define Q as the quotient of the groupoid
(H × Z H) × (s,s),Z×S Z,(t,s) G ⇒ G whose arrows are of the form (ϕ, ψ, g) : g −→ ϕgψ −1 .
For short, we denote it by G ′ ⇒ G. The compositions G ⇒ Z −→ Y are invariant under the action of the above groupoid so we get maps σ, τ : Q ⇒ Y which will be the source and target of the groupoid we wish to define. It is however not obvious to us how to define the composition of arrows in Q, ie how to fill the diagram
G × s,Z,t G c / / G Q × σ,Y,τ Q Q
in a systematic way. This is because, since the actions involved to construct the quotients are a priori not free, we do not know their points and hence cannot just lift points in Q × σ,Y,τ Q, compose their lifts in G × s,Z,t G and send the result back in Q. This separate problem will be the subject of a subsequent paper.
In the sequel of this section we assume that such a quotient groupoid has been constructed and we investigate the behaviour of the ramification divisor under such a dévissage. We fix an S-scheme X, a covering (f : Z −→ X, G ⇒ Z) of X and a subgroupoid H ֒→ G. We denote the quotient of Z by H by g : Z −→ Y . We let Q be the quotient of the groupoid G ′ ⇒ G defined above.
First we must check that we have the following lemma : Lemma 3.3. With the above notations, suppose that a quotient groupoid Q ⇒ Y has been constructed. If g : Z −→ Y is flat then
(i) (g : Z −→ Y, H ⇒ Y ) is a covering of Z. (ii) (h : Y −→ X, Q ⇒ Y ) is a covering of X.
Proof :
Only (ii) needs a proof. If G acts freely on Z, so does H and then Y represents the fppf quotient sheaf T → Z(T )/H(T ). In the same way Q represents the quotient sheaf T → G(T )/H 2 (T ). We then easily see that Q acts freely on Y . Now if U ⊂ Z is a dense open subscheme on which G acts freely, by faithfull flatness of g : Z −→ Y , its image V ⊂ Y is a dense open subscheme of Y , on which Q acts freely by the above discussion. Furthermore the fiberwise criterion for flatness shows that h is flat, since both f and g are.
The quotient morphisms p : G −→ Q and g : Z −→ Y induce a groupoid morphism which we still denote p :
(G ⇒ Z) −→ (Q ⇒ Y ).
The following lemma shows that, in our situation and under flatness assumptions its kernel will be H, as expected.
Proposition 3.1. With the above notations, if G −→ Q and Z −→ Y are flat, we have ker p ≃ H.
Proof :
Denote by I H ⊂ O G the ideal sheaf defining H in G. We want to show that it agrees with the ideal sheaf defining ker(p). This is a local question so we may assume that all schemes involved are affine and work with global sections. Let x ∈ I H ∩ O Q . It defines a morphism G −→ A 1 , H-invariant and vanishing on H. We have the following commutative diagram :
Z g / / e Yē H i / / s > > ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ G x p / / Q x ⑦ ⑦ ⑦ ⑦ ⑦ ⑦ ⑦ ⑦ A 1 S where e (resp.ē) is the unit section of the groupoid G ⇒ Z (resp. Q ⇒ Y ) and x is the function Q −→ A 1 S induced by the H-invariant function x. We havex •ē • g = x • e = 0 and since g is an epimorphism we havex •ē = 0 and thus x ∈ m Q . Hence I H ∩ O Q ⊂ m Q .
Conversely, since x ∈ m Q we havex •ē = 0. Since by the above diagram we have
(x •ē) • (g • s) = x • i, we see that x vanishes on H and x ∈ I H ∩ O Q . We thus have I H ∩ O Q = m Q .
The kernel of p is defined by the following fibre product :
ker p / / Ȳ e G / / Q . Its structure sheaf is thus O G ⊗ O Q O Y = O G /m Q O G . Since I H ∩ O Q = m Q we have m Q O G ⊂ I H . We thus have a surjection O G /m Q O G −→ O G /I H and a closed immersion H ֒→ ker(p).
Let U ⊂ Z be a dense open subscheme on which H acts freely. We may assume that U is saturated, ie U = g −1 (g(U )). By flatness the preimage U ′ of U in G is also dense. Then the action of G ′ |U ′ on U ′ is also free. Indeed, for (ϕ, ψ, g) ∈ G ′ the equality g = ϕgψ −1 implies that gψ = ϕ and thus that s(ψ) = t(ψ) = s(g) and s(ϕ) = t(ϕ) = t(g). Hence ψ and ϕ must be in the stabilizer of H U , which is trivial. Let us denote V = g(U ). By faithful flatness of g, it is dense in Y . Since the formation of quotients commute with flat base change, Q |V (resp. V ) represents the fppf of U ′ (resp. U ) by G ′ |U ′ (resp. G |U ). We can then verify on points that H |U is the kernel of the projection
G |U −→ Q |V . Indeed, if T is an S-scheme and t ∈ G |U (T ) is such that p(t) = 1 g(s(t)) there exists an fppf covering T ′ −→ T and ϕ, ψ ∈ H(T ′ ) such that, restricting to T ′ , we have 1 s(t) = ϕtψ −1 and hence t = ϕ −1 ψ ∈ H(T ′ ).
Thus we see that the immersion H −→ ker(p) induces an isomorphism H |U ≃ ker(p) |U . By assumption G −→ Q is flat so ker(p) −→ Y is flat. Hence the preimage of U in ker(p) is dense. The closed immersion H ֒→ ker(p) is thus dominant, so it is an isomorphism.
We will use the above proposition to relate the different stabilizers involved. We first need the following two general observations about morphisms of groupoids.
Proof :
(i) Since a is groupoid morphism it maps St A to St B so it induces a morphism a ′ as stated. We have
ker(a) = B × B A so ker(a) × A St A = B × B St A . Since B −→ B factors through the immersions B ֒→ St B ֒→ B we see that B × B St A = B × St B St A = ker(a ′ ). (ii) Recall that a * 0 B = B × B×S B A × S A Since a is a groupoid morphism the diagram A / / B A × S A / / B × S B
is commutative. We thus get a morphism
a : A −→ f * B r → (f (r), s(r), t(r))
where s and t are the source and target in A ⇒ A. The unit section of a * 0 B is given by
A −→ f * B a → (id f0(a) , a, a) so ker(ã) = ker(a) × A St A .
We can now state our result, which we view as a substitute for the first fundamental exact sequence of the sheaf of differential 1-forms.
Theorem 3.2. Let X be an S-scheme, (Z −→ X, G ⇒ Z) be a covering of X and H ֒→ G be a subgroupoid of G. Denote by g : Z −→ Y the quotient of Z by H. Suppose constructed the quotient groupoid Q ⇒ Y .
We have the following exact sequence of Z-group schemes : Accordingly, we obtain an exact sequence relating the augmentation ideals of the stabilizers involved.
1 −→ St H −→ St G α −→ g * St Q , in the sense that (i) St H −→ St G is a closed immersion.
Corollary 3.1. With the notations and hypothesis of the previous theorem, we have the following exact sequence of O Z -modules :
0 −→ g * m Q O St G −→ m G −→ m H −→ 0,
where m Q , m G and m H respectively stand for the augmentation ideals of the group schemes St Q , St G and St H .
Proof :
If I H is the ideal sheaf defining St H in St G we have the following exact sequence
0 −→ I H −→ O St G −→ O St H −→ 0.
Since ker(α) = St G × g * St Q Z, the ideal sheaf defining ker(α) is the one generated by
g * m Q in O St G , namely g * m Q O St G . By 3.2 we have St H ≃ ker(α), hence I H = g * m Q O St G . Thus we have the exact sequence (1) 0 −→ g * m Q O St G −→ O St G −→ O St H −→ 0.
But since the unit section splits the structure maps (1), we get the exact sequence announced.
St G −→ Z and St H −→ Z, as O Z -module we have O St G = m G ⊕ O Z and O St H = m H ⊕ O Z . Modding out by O Z in
Since the length of modules is additive with respect to exact sequences, taking the associated divisors of the modules involved in 3.1, we get the following equality between the associated divisors on Z :
R G = R H +div(g * m Q O St G ).
In particular if the quotient groupoid Q acts freely on Y , we get the equality R G = R H , which is a weak form of the transitivity property (*) for the ramification divisor in this context.
It should be noted that the map α is neither flat nor dominant. Hence it is not obvious to relate the sheaf g * m Q O St G to g * m Q . In general, the formula R G = R H +g * R Q , that one might expect in analogy with the classical situation, is not true, as illustrated by the following example.
F γ : GL n −→ GL (γ) n . We have G γ = Spec k[a ij , 1 ≤ i, j ≤ n] a p γ ij i = j , a p γ ii − 1 .
It is a finite flat k-group scheme of order p γn 2 . The above action of GL n induces an action of all the Frobenius kernels, by the same formula.
Let 0 < β < γ be two integers. By [Jan03, I, §9.4-9.5], G β is a normal subgroup of G γ and G γ /G β ≃ G γ−β . Set Z = M n,k , Y = Z/ G β and X = Z/ G γ . For τ ∈ {β, γ, γ − β} we denote by G τ the action groupoid associated with the action of G τ and by St τ its stabilizer.
We thus have defined a covering (Z, G γ ) of X which we expressed as the covering (Z, G β ) of Y followed by the covering (Y, G γ−β ) of X.
The quotient morphism g : Z −→ Y is easily seen to be defined by the ring map
k[y ij ] −→ k[z ij ] y ij → z p β ij .
For τ ∈ {β, γ} one can show that the stabilizer of the corresponding action is given by
O St τ = k[z ij ][a 1 , . . . , a n ] a p τ 1 , . . . , a p τ n , ∆a 1 , . . . , ∆a n Similarly, we have O St γ−β = k[yij ][b1,...,bn] b γ−β 1 ,...,b γ−β n ,∆ ′ b1,...,∆ ′ bn , where ∆ (resp. ∆ ′ ) stands
for the determinant polynomial in the variables z ij (resp. y ij ).
The corresponding ramification divisors are thus :
R γ = (p nγ − 1)[∆], R β = (p nβ − 1)[∆] and R γ−β = (p n(γ−β) − 1)[∆ ′ ]. Since the determinant ∆ ′ in the variables y ij is mapped to ∆ p β we have g * R γ−β = p β (p n(γ−β) − 1)[∆]. Thus we see that R γ = R β +g * R γ−β .
4. Generalized coverings given by diagonalizable group actions.
4.1.
Diagonalizable group schemes and their actions. We briefly recall some definitions and facts concerning diagonalizable group schemes and their actions which will be useful for us. We refer to [Gro11a, Exp. VIII] for details and proofs. For an integer n ≥ 2 we denote by µ n,S the group scheme D(Z/nZ). According to the lemma above, it is étale if and only if n is prime to the characteristics of all residue fields of S.
It turns out that actions of diagonalizable group schemes are easy to describe in terms of graded algebras. Let us recall the following definition : We analyse the structure of covegins given by actions of diagonalizable group schemes. We are of course interested in the non-étale case.
4.2.
Local structure of D(M )-coverings. Let us fix an S-scheme X. Let G = D(M ) be a finite diagonalizable group scheme acting on an S-scheme Y . Suppose given a morphism f : Y −→ X such that (Y, G) is a covering of X. Then Y is affine over X. As recalled above, there exists an M -graded O X -algebra A such that Y = Spec(A).
The action of G on Y is given by the map
A −→ A ⊗ OS O S [M ] a = m∈M a m → m∈M a m ⊗ X m ,
where we used the notation X m to denote the generator of O S [M ] coreesponding to m ∈ M .
The sub-algebra of invariants is A 0 , so X = Y /G = Spec(A 0 ). By definition of a covering, there exists a schematically dense open subscheme V ⊂ Y on which G acts freely. Replacing V by its G-orbit if necessary, we may assume that V is G-stable. The morphism Furthermore, if M = Z/p n Z, the α i,j are determined by the α i,1 More precisely, let i ∈ Z/p n Z. Denote by s(i) the unique integer in {0, . . . , p n − 1} whose class modulo p n is i. Define the map
j X : G × S Y −→ Y × X Y (g, y) → (y, g.y) then induces an isomorphism G × S V ≃ V × X V . Since G −→ S is flat, G × S V is schematically dense in G × S Y , so j X is schematically dominant. The map j ♯ X : O Y ⊗ OX O Y −→ (j X ) * O G ⊗ OS O Y is thus injective. Since f : Y −→ X is locally free, each A mσ : Z/p n Z × Z/p n Z −→ Z (i, j) → 1 p n (s(i) + s(j) − s(i + j)
) and set β 0 = 1, β i+1 = α 0,1 . . . α i,1 for all i ∈ Z/p n Z such that s(i) = p n − 1 and f = α 0,1 . . . α p n −1,1 . We then have, for all i, j ∈ Z/p n Z,
(1) α i,j = β i+1 β −1 i β −1 j f σi,j .
Conversely, for all (p n − 1)-tuple (α i,1 ) i∈{1,...,p n −1} of non zero-divisors in A 0 we get a µ p n ,S -covering in the following way :
• Set A = A ⊕p n 0 , label each copy of A 0 by an index i ∈ Z/p n Z and set e i = (δ ij ) j∈Z/p n Z .
• For all i, j ∈ Z/p n Z, define α 0,1 = 1 and α i,j according to the formula (1).
• Give A the structure of a Z/p n Z-graded A 0 -algebra by setting, for all i, j ∈ Z/p n Z, e i e j = α i,j e i+j .
Then Spec(A) is a µ p n ,S -covering of Spec(A 0 ).
Proof :
By the preceding remarks, each of the A m is free of rank 1 over A 0 , for which we denote e m a generator, with the convention that e 0 = 1. Since for all (m, n) ∈ M 2 we have A m A n ⊂ A m+n , if we denote by α m,n the determinant of the multiplication
A m ⊗ A0 A n −→ A m+n
we have e m e n = α m,n e m+n .
Let us note that commutativity and associativity of the multiplication in A imply the following relations in A 0 :
-For all (m, n) ∈ M 2 , α m,n = α n,m . -For all (l, m, n) ∈ M 3 , α l,m α l+m,n = α m,n α l,m+n .
Since the morphism j ♯ X :
A ⊗ A0 A −→ A ⊗ B B[M ]
is injective between these two free A 0 -modules of rank |M | 2 , its determinant is a non zero-divisor. Let us compute it on the basis (e m ⊗ e n ) in the source and (e k ⊗ X l ) in the target. We index the matrix by M 2 . We have j ♯ X (e m ⊗ e n ) = e m (e n ⊗ X n ) = α m,n e m+n ⊗ X n . The matrix of j ♯ X in these basis is thus monomial : its coefficient of index ((k, l), (m, n)) is zero if (k, l) = (m + n, n) and α m,n otherwise. We thus have
det(j ♯ X ) = ε(τ ) (m,n)∈M 2 α m,n ,
where ε(τ ) is the signature of the associated permutation. We conclude that the α m,n are all non zero-divisors. Let us specify to the case where M = Z/p n Z, ie G = µ p n ,S . Inside the localization of A 0 in the multiplicative subset of non zero-divisors we consider the multiplicative subgroup generated by the α i,j , which we denote by N . We consider it as a trivial Z/p n Z-module.
By the discussion above we have, for all i, j, k in Z/p n Z, α i,j α i+j,k α −1 j,k α −1 j+k,i = 1. Hence the family (α i,j ) defines a 2-cocycle of Z/p n Z with values in N . Every element f ∈ N determines a 2-cocycle (f σi,j ) in such a way that if f = g p n is a p n -th power then (f σi,j ) is the coboundary induced by the cochain (g −s(i) ). By [Ser68, VIII, §4] we have H 2 (Z/p n Z, N ) = N/N p n . Hence there exists f ∈ N and a coboundary β : Z/p n Z −→ N such that, for all i and j in Z/p n Z,
α i,j = β i+j β −1 i β −1 j f σi,j . Note that the pair (β, f ) is not unique : we still obtain the cocycle α i,j if we replace (β, f ) by ({f ′s(i) β i }, f ′p n f ) for any f ′ ∈ N .
In particular, multiplying by β −1 1 if necessary, we may assume that β 1 = 1. Let us fix i ∈ Z/p n Z. The equation (1) with j = 0 shows that β 0 = 1; with j = 1 we see that β i+1 = α i,1 β i f −σi,1 . We then distinguish between two cases :
• If i = p n − 1 then σ i,1 = 0 and β i+1 = α i,1 β i . Thus by induction we get β i+1 = α i,1 ...α 1,1 . Convention. Our proofs in this section rely on computations of multiplicities of Weil divisors at codimension 1 points. We thus need these points to be regular. Hence, from now on and until the end of this article, we make the additional assumption that the schemes involved in a covering are normal. More precisely, if X is an S-scheme and (Y, G) is a covering of X then Y will always be assumed to be normal. Note that, since normality is preserved by taking invariant rings, this implies that X is itself normal.
Let us fix an S-scheme X, a finite abelian p-group M and let G = D(M ) be the corresponding diagonalizable group scheme. Let (Y, G) be a G-covering of X. We assume that Y is normal and we compute the multiplicities of the ramification divisor defined in 3.1 at each codimension 1 point of Y .
Let y ∈ Y be such a point. By assumption, the local ring O Y,y is a discrete valuation ring which we denote A, with valuation v. We have a graduation of type
M A = m∈M A m
induced by the action of G. Since (Y, G) is a covering, by the discussion 4.2 above, each A m is a free A 0 -module of rank 1 for which we denote e m a generator. We choose e 0 = 1 and define Y y = Spec(A). By definition, the formation of the stabilizer group scheme commutes with base change. Hence the stabilizer St G,y of the covering (Y, G) is given at y by the fibre product
St G,y / / σ G × S Y y jy Y y / / Y y × X Y y .
The diagonal immersion of Y y is given by the ring morphism
A ⊗ A −→ A a ⊗ b → ab
and the morphism j y is defined by the map
A ⊗ A −→ A[M ] a ⊗ b → m∈M ba m X m .
We thus have
O St G,y = A[M ] ⊗ A⊗A A = A[M ] m∈M ba m X m − ab, a, b ∈ A .
It is easily seen that the ideal defining St G,y in G × S Y y is also generated by the elements e m (X m − 1) for m ∈ M . Hence we have
O St G,y = A[M ] e m (X m − 1), m ∈ M .
Its augmentation is generated by the images in O StG,y of the elements X m − 1 for m = 0, ie we have
m St G,y = m∈M\{0} (X m − 1)O St G,y .
Let us remark that the elements X m − 1 for m ∈ M also generate the algebra Hence a k α k,m = 0 for k = n − m and since none of the α i,j is a zero divisor by 4.2, we have a k = 0 for k = n − m and thus a = a n−m e n−m ∈ A n−m . But a ∈ A × then implies e n−m ∈ A × . Since N = {0} by assumption we must have n = m.
We then claim that, for all m ∈ M , we have v(e m ) ≤ |M | − 1. To prove this, suppose there exists some n ∈ M such that v(e n ) ≥ |M |. We can then write e n = π |M| b for some b ∈ A, where π is a uniformizer of A. But for all x ∈ A we have x |M| ∈ A 0 . Indeed, if x = m∈M x m maps to m∈M x m X m via the coaction, since |M | is a p-th power, x |M| maps to m∈M x |M| m X |M|m and since |M | annihilates
M we have X |M|m = 1 for all m ∈ M . Hence x |M| is invariant, ie in A 0 . If we write b = k∈M b k e k on the basis (e i ) we have e n = k∈M π |M| b k e k .
By the preceding remark we have π |M| b k ∈ A 0 for all k ∈ M . Thus b k = 0 if k = n and π |M| b n = 1 which is absurd since π is a uniformizer, in particular non-invertible.
Hence the valuations of the e m are all distinct and lower than |M | − 1. The last proposition allows for the computation of the ramification divisor of a totally ramified covering.
Corollary 4.1. If (Y, G) is totally ramified at y, its ramification divisor has multiplicity |G| − 1 at y.
Proof :
Let us keep the notations of 4.2. This proposition shows that we have the exact sequence of A-modules
0 −→ A ⊕|M|−1 ×e d −→ A ⊕|M|−1 ϕ −→ σ * m StG,y −→ 0,
where the first arrow is the multiplication of each coordinates by e d . Its determinant is e |M|−1 d , so has valuation |M | − 1 = |G| − 1.
When N = {0} the above calculations still allow us to compute the multiplicities of the ramification divisor. We just have to perform a dévissage to reduce to the totally ramified case. (iii) The multiplicity of R G at y is |H y | − 1 = |M/N y | − 1.
Proof :
Let us denote O Y,y = A and Y y = Spec(A). The action of G on Y y is given by a graduation
A = m∈M A m of type M on A. As before set N y = {n ∈ M | A n ⊗ A0 A −n ≃ A 0 }.
It is a subgroup of M . For each m ∈ M , let e m be a generator of A m as an A 0 -module and let (α m,n ) be the corresponding cocyle. Applying the functor D to the exact sequence
0 −→ N y −→ M −→ M/N y −→ 0
we get the exact sequence of group schemes
1 −→ H y −→ G −→ G/H y −→ 1.
Thus the quotient by G can be factored into the quotient by H y followed by the quotient by G/H y , according to the diagram By definition for all n ∈ N the multiplication A n ⊗ A0 A −n −→ A 0 is an isomorphism so Y y /H y −→ Y y /G is a G/H y -torsor. Hence its ramification divisor is trivial and by 3.4 on a neighborhood U of y we have (R G ) U = (R Hy ) U We then claim that Y y −→ Y y /H y is totally ramified. To see this, set B Hence Y y −→ Y y /H y is totally ramified. By the previous corollary, the ramification divisor of (Y, G) has multiplicity |M/N y | − 1 = |H y | − 1 at y.
Y y / / ! ! ❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉ Y y /H y { { ✈ ✈ ✈ ✈ ✈ ✈ ✈ ✈ ✈ Y y /G .
4.4.
Relation with the fixed-point scheme. Another natural object to consider for describing the ramification of a G-covering (Y, G) is its fixed point scheme, as defined in [Fog73]. Let us briefly recall its definition. Consider the fixed point functor
Fix G : Sch /S −→ Ens T → {t ∈ Y (T ) | ρ • (id G ×t) = p 2 • (id g ×t)} ,
where ρ : G × S Y −→ Y is the action of G on Y and p 2 : G × S Y −→ Y is the second projection. In our setting, this functor is representable by a closed subscheme of Y , which we denote Y G . Let I G be the ideal sheaf defining Y G . In case G = D(M ) is an infinitesimal diagonalizable group scheme, we can compute I G . Since the variables X m are A-linearly independent we must have
t ♯ (a m ) = 0 if m = 0 a 0 otherwise.
Such is the case if and only if t ♯ factor through the quotient of A by the ideal generated by the elements e m for m ∈ M \ {0}. We thus have I G =< e m , m = 0 >. Now suppose y ∈ Y is a point of codimension 1. If the covering (Y, G) is totally ramified at y, by 4.2 we know that one of the e m is a uniformizer of the local ring at y. Thus I G,y is just the maximal ideal of O Y,y . Since by 4.2 the multiplicity of the ramification divisor at y is |G| − 1 we have
O Y (− R G ) y = (I G,y ) ⊗|G|−1 .
We thus have the following proposition : As divisors on Z we have
R G = R H +g * R G/H .
Proof :
To show this equality of divisors, we need to show that they have the same multiplicity in each codimension 1 point of Z. Since Z and hence Y and X are normal we may assume that Z = Spec(A), Y = Spec(B) and X = Spec(A 0 ) are spectrums of discrete valuation rings. Let v A denote the valuation in A.
In view of theorem 4.2 we may also assume that the coverings involved are totally ramified (in the sense of definition 4.3). Then by proposition 4.2 there exists positive integers m ≤ n such that G = µ p n , H = µ p m and G/H = µ p n−m . By the same proposition the stabilizers of the actions are given by
O St G = A[t] t p n −1,π(t−1) , O St H = A[s] s p m −1,π(s−1) and O St G/H = B[u]
u p n−m −1,π ′ (u−1) , where π (resp. π ′ ) is a uniformizer of A (resp. B). The graduation of type Z/p n Z defining the action of G on Z is just
A = 0≤i≤p n −1 A 0 π i .
By definition of the residual action of G/H on Y , we have the following commutative digram
O Y ρ ♯ G/H / / g ♯ OY [u] u p n−m −1 O Z ρ ♯ G / / OZ [t] t p n −1 .
We are going to determine the valuation in A of g ♯ (π ′ ). Note that we have
ρ ♯ G/H (π ′ ) = π ′ u. Since the quotient map G −→ G/H is defined by u → t p m , this implies that ρ ♯ G (g ♯ (π ′ )) = g ♯ (π ′ )t p m . Hence we see that g ♯ (π ′ ) ∈ A 0 π p m . Since A is of degree p m over B we cannot have v A (g ♯ (π ′ )) > p m so g ♯ (π ′ ) is of valuation p m . Now O g * St G/H = O St G/H ⊗ B A = A[u]
u p n−m −1, g ♯ (π ′ )(u−1) so the length of g * m G/H as an A-module is l A (g * m G/H ) = (p n−m − 1)p m . Since l A (m G ) = p n − 1 and
l A (m H ) = p m − 1 we have l A (m G ) = l A (m H ) + l A (g * m G/H ).
The divisors R G and R H +g * R G/H thus have the same multiplicity in each codimension 1 point of Z. Hence they are equal.
We wish to relate the ramification divisor of a D(M )-covering to the dualizing sheaf of its quotient morphism. We first give a criterion for the latter to be Gorenstein, in the case D(M ) = µ p n . 4.6. The Gorenstein locus of a µ p n -covering. Let us first recall the following fact about dualizing sheaves of finite morphisms. See [Liu02,6.4.25] for a proof.
Proposition 4.5. If f : T −→ T ′ is a finite locally free morphism between locally noetherian schemes then f has a dualizing sheaf given by
ω f = f ! O T ′ , where f ! O T ′ = Hom O T ′ (f * O T ′ , O T ′ ) is viewed as an O T -module via the law t.θ = (x → θ(tx)).
Recall that a morphism of schemes is said to be Gorenstein if it has a dualizing sheaf which is invertible.
Let us fix a base S and an S-scheme X endowed with a µ p n ,S -covering whose quotient morphism we denote by f : Y −→ X. We wish to give a necessary and sufficient condition for the morphism f to be Gorenstein in terms of the structure constants of the covering. We have the following result.
A i ⊗ OX A j −→ A i+j is an isomorphism. For each l ∈ Z/p n Z set U l = ∩ i+j=l U ij .
The open subscheme of X over which the morphism f is Gorenstein is the union of the U l for l ∈ Z/p n Z.
In particular, f is Gorenstein if and only if we have
X = l∈Z/p n Z U l .
Proof :
Let us first note that, by , since f is finite locally free of rank p n , it admits a dualizing sheaf ω f = Hom(f * O Y , O X ). As f is finite and locally free, f is Gorenstein if and only if, for every point y ∈ Y , ω f,y is free of rank 1.
We may thus assume that Y = Spec(A) and X = Spec(A 0 ) are spectrums of local rings. We denote by (α ij ) i,j∈Z/p n Z the cocyle with values in A 0 inducing the action of µ p n on Y . The dualizing sheaf ω Y /X is then given by the A-module A * := Hom A0 (A, A 0 ) with A-module law a.θ = (x → θ(ax)).
The morphism f is Gorenstein at y if and only if there exists a linear form ϕ : A −→ A 0 such that A * = Aϕ. Let us note (e i ) the A 0 -basis of A associated to the cocycle (α ij ), ie such that e i e j = α ij e i+j for all i, j ∈ Z/p n Z and let (e * i ) be the dual basis. Every linear form θ : A −→ A 0 can be written θ = i θ i e * i . Thus we see that for ϕ ∈ A * we have A * = Aϕ if and only if for every j ∈ Z/p n Z there exists an element b j = i b ij e i in A such that e * j = b j .ϕ. Observe that for all triplet i, j, k ∈ Z/p n Z we have e i .e * j (e k ) = e * j (e i e k ) = e * j (α i,k e i+k ) = α i,k δ i+k,j , where δ is the Kronecker symbol, so that (α ij ϕ i+j ) i,j∈Z/p n Z is invertible. With the notations of 4.1 we have, in the localization of A 0 with respect to the multiplicative subset of non zero-divisors,
e i .e * j = α i,j−i e * j−i . Writing ϕ = m ϕ m e * m we then have b j .ϕ = i,m∈Z/p n Z b ij ϕ m e i .e * m = i,m∈Z/p n Z b ij ϕ m α i,m−i e * m−i = m∈Z/p n Z k+l=m b k,j ϕ l+k α k,l e *α i,j ϕ i+j = 1 β i β j β i+j ϕ i+j f σi,j .
Let N (ϕ) be the matrix (β i+j ϕ i+j f σi,j ) i,j∈Z/p n Z . We then have
det(M (ϕ)) = 1 i∈Z/p n Z β 2 i det(N (ϕ))
We are going to compute det(N (ϕ)). Set γ i = β i ϕ i . We can then write
N (ϕ) = γ 0 γ 1 γ 2 ... γ p n −1 γ 1 γ 2 γ 3 ... γ 0 f . . . . . . . .
.
γ 0 f γ p n −3 f γ p n −1 γ 0 f γ 1 f ... γ p n −2 f .
Let us note that, if n ij denotes the coefficient of index (i, j) in N (ϕ) we have
n ij = γ i+j if s(i) + s(j) ≤ p n − 1 γ i+j f otherwise
Switching the i-th line of the matrix N (ϕ) with the (p n − 1 − i)-th for every i ∈ Z/p n Z we obtain the following matrix :
N ′ (ϕ) = γ p n −1 γ 0 f γ 1 f ... γ p n −2 f γ p n −2 γ p n −1 γ 0 f ... γ p n −3 f . . . . . . . γ 1 γ 2 γ 3 γ 0 f γ 0 γ 1 γ 2 ... γ p n −1
whose coefficients are given by
n ′ ij = γ j−i−1 if s(i) ≥ s(j) γ j−i−1 f otherwise
and thus depend only on the difference between the line and column index. Let S p n be the symetric group of order p n , which we view as the group of bijections of Z/p n Z. For k ∈ Z/p n Z, denote by τ k the permutation (i → i + k).
Observe that for all k, l ∈ Z/p n Z we have τ k • τ l = τ k+l , so that the map
Z/p n Z × S p n −→ S p n (k, σ) → τ k • σ • τ −1 k
defines an action of Z/p n Z on S p n .
If Ω stands for the sets of orbits of this action we can write S p n = ω∈Ω ω and regroup by orbits the terms in det(N ′ (ϕ)) so as to obtain the following expression :
det(N ′ (ϕ)) = σ∈S p n ε(σ) i∈Z/p n Z n ′ i,σ(i) = ω∈Ω σ∈ω ε(σ) i∈Z/p n Z n ′ i,σ(i) ,
where ε stands for the signature of a permutation. Observe that it is constant on orbits. Note furthermore that if σ and θ are in the same orbit, the sets {n ′ i,σ(i) , i ∈ Z/p n Z} and {n ′ i,θ(i) , i ∈ Z/p n Z} are equal. Indeed, if there exists k such that
θ = τ k • σ • τ −1 k we have n ′ i,θ(i) = n ′ i,σ(i−k)+k which by definition of N ′ (ϕ) is also equal to n ′ i−k,σ(i−k)
. Thus, if σ and θ are in the same orbit, we have
ε(θ) i∈Z/p n Z n ′ i,θ(i) = ε(σ) i∈Z/p n Z n ′ i,σ(i) .
By the orbit-stabilizer theorem, the number of element in each orbit is a p-th power. Furthermore the orbit of a permutation σ has only one element if and only if σ • τ 1 = τ 1 • σ since τ k = τ k 1 , from which we see that σ(i) = σ(0) + i for all i ∈ Z/p n Z, meaning that σ is one of the τ k . They all have signature 1. Hence we have det(N ′ (ϕ)) = k∈Z/p n Z i∈Z/p n Z n ′ i,τ k (i) .
Note finally that for k ∈ Z/p n Z we have
n ′ i,τ k (i) = γ k−1 if s(i) ≥ s(i + k)
γ k−1 f otherwise so that i∈Z/p n Z n ′ i,τ k (i) = γ p n k−1 f p n −1−k and hence we have det(N (ϕ)) = (−1) p n −1 2 det(N ′ (ϕ)) = (−1) p n −1 2 k∈Z/p n Z γ p n k−1 f p n −1−k .
We now wish to come back to M (ϕ).
For all i ∈ Z/p n Z, set c i = β p n i f p n −1−i j∈Z/p n Z β 2 j and ǫ = (−1) p n −1 2 , so that det(M (ϕ)) = ǫ i∈Z/p n Z c i ϕ p n i .
Now observe that, taking ϕ = e * l for some l ∈ Z/p n Z, the matrix M (e * l ) is monomial : its term of index (i, j) is δ l,i+j α ij . Its determinant is det(M (e * l )) = ǫ i+j=l α ij . We thus see that for all l ∈ Z/p n Z we have
c l = i+j=l α ij .
Finally we can conclude that, for all ϕ ∈ A * , we have
det(M (ϕ)) = ǫ l∈Z/p n Z ( i+j=l α ij )ϕ p n l .
The A 0 -algebra A is Gorenstein if and only if there exists ϕ ∈ A * such that det(M (ϕ)) is invertible. Since A 0 is a local ring, such is the case if and only if there exists l ∈ Z/p n Z such that α ij is invertible whenever i + j = l, in which case we can take e * l as a generator of A * . 4.7. Application to a Riemann-Hurwitz-type formula. We can now turn to the main main result of this section, which will relate the ramification divisor of a D(M )-covering to the dualizing sheaf of its quotient morphism. We will use a formula known in height one (under the extra assumption that the base is an algebraically closed filed of characteristic p > 0) and extend it to arbitrary height via our formalism. Let us fix an algebraically closed field k of characteristic p > 0 and set S = Spec(k).
Let us first quote the following recent result (see also [RŠ76]) :
Theorem 4.4. [Tzi15, Th 8.1] Let Y be an integral S-scheme. Suppose Y has a dualizing sheaf ω Y , satisfies Serre's S 2 condition and has at worst normal crossing singularities in codimension 1. Suppose Y admits a µ p,S -action. Let I fix be the ideal sheaf defining the scheme of fixed points and f : Y −→ X be the quotient. Then X has a dualizing sheaf ω X and ω Y = (f * ω X ⊗ I Now suppose that a morphism f : Y −→ X is the quotient morphism of a µ p n ,Scovering of the scheme X. Under the extra assumption that f is Gorenstein, which by the previous section can be checked on the cocycle giving the action on Y , we will prove an equality between its dualizing sheaf and the structure sheaf of the ramification divisor.
Theorem 4.5. Let X be a noetherian S-scheme and (f : Y −→ X, µ p n ,S ) be a µ p n ,S -covering of X. Denote by R G the ramification divisor of the covering, as defined in 3.1, ie the divisor associated with the action groupoid G := µ p n ,S ×Y ⇒ Y . Suppose that f is Gorenstein and denote by ω f its dualizing sheaf.
We then have ω f = O Y (R G ).
Proof :
We will show the result by induction on n, using Tziolas's result. For n = 1 this is a direct consequence of 4.4. Indeed, since by assumption f is Gorenstein, ω f is invertible so in particular reflexive. Since (I ⊗p−1 fix ) * is the dual of a finite type module, it is also reflexive. Now by the previous result we have (ω f ) * = I Let n > 1 be an integer and suppose the result is proved for all µ p m ,S -coverings with m < n. Let (f : Z −→ X, µ p n ,S ) be a covering of X. Consider µ p n−1 ,S as a subgroup of µ p n ,S via the obvious closed immersion and the induced action on Z. Set Y = Z/µ p n−1 ,S , which is normal since Z is. We have a µ p n−1 ,S -covering (g : Z −→ Y, µ p n−1 ,S ) of Y a commutative diagram
ω f = O Z (R H ) ⊗ OZ g * O Y (R Q ) = O Z (R H +g * R Q ) = O Z (R G )
which shows that the formula holds for µ p n ,S .
The above formula extends immediately to the case of coverings given by actions of an arbitrary finite diagonalisable group scheme. Indeed, if G = D(M ) is a finite infinitesimal group scheme, the decomposition of M into invariant factors yields a decomposition of G into a product G = r i=1 µ p n i ,S × G ét , where G ét is a finite étale S-group scheme. If (f : Y −→ X, G) is a G-covering of X we can decompose the action of G accordingly and factor f into
Y = Y 0 f1 −→ Y 1 f2 −→ · · · fr−1 −→ Y r g −→ Y r+1 = X
where f i is the quotient by µ p n i ,S and G is the quotient by G ét . Successive applications of the above result 4.5, along with the classical formula (RH) yields the following theorem :
Theorem 4.6. Let X be a scheme defined over an algebraically closed field k and G be a finite diagonalizable k-group scheme. Let (f : Y −→ X, G) be a covering of X given by an action of G on Y . Let R G be the ramification divisor of this covering, associated with the action groupoid G. If f is Gorenstein then we have
ω f = O Y (R G ),
where ω f is the dualizing sheaf of the morphism f .
As in the classical case, when Y is a smooth projective curve over k we can take the degrees in the above formula to relate the genuses of Y ,X and the degree of R G , as follows.
Corollary 4.2. With the notations of the above theorem, suppose furthermore that Y (and hence X) is a smooth projective curve over k. Let g(Y ), respectively g(X), denote the genuses of the curve Y , respectively X. We have the formula 2g(Y ) − 2 = |G|(2g(X) − 2) + deg(R G ).
Proof :
By [Liu02, 6.4, Lemma 4.26] we have ω Y /k ≃ f * ω X/k ⊗ OY ω f . Taking degrees we get deg(ω Y /k ) = deg(f ) deg(ω X/k ) + deg(ω f ). By 4.6 we have deg(ω f ) = deg(R G ). Since the degree of the canonical divisor of a smooth projective curve C is 2g(C)−2, we get the announced formula, noting that f is finite flat of degree |G|.
Note that this formula was proved by Emsalem in [Ems13,cor. 7.3] in the special case of torsors.
Mathematisch Instituut, Universiteit Leiden, Niels Bohrweg 1, 2333 CA Leiden, The Netherlands. E-mail address: [email protected]
Lemma 3 . 4 .
34Let a : (A ⇒ A) −→ (B ⇒ B) be a morphism of S-groupoids. (i) a induces a morphism of S-group schemes a ′ : St A −→ St B and we have ker(a ′ ) = ker(a) × A St A . (ii) let a 0 : A −→ B be the morphism of object schemes induced by a. The morphism a induces a morphism of groupoidsã : (A ⇒ A) −→ (a * 0 B ⇒ A) and we have ker(ã) = ker(a) × A St A .
(ii) ker(α) ≃ St H . Proof : The morphism p : G −→ Q induces a morphismp : G −→ g * Q whose kernel is ker(p) × G St G by the lemma 3.4. Let α : St G −→ g * St Q be the induced morphism on stabilizers. (i) St H −→ St G is the base change of the closed immersion H ֒→ G by the diagonal Z ֒→ Z × Y Z. Thus it is a closed immersion. (ii) By the lemma 3.4 we have ker(α) = ker(p). By proposition 3.1 we have ker(p) ≃ H. Hence ker(α) = ker(p) × G St G ≃ H × G St G = St H .
Example 3. 1 .
1Let n be a positive integer and k a field of positive characteristic p > 0. Consider the action of the group scheme GL n,k of invertible n × n invertible matrices on the n × n matrices M n,k = Spec(k[z ij ]) by left multiplication :GL n × k M n −→ M n (P, M ) → P MFor all positive integer γ, let G γ be the kernel of the γ-th iterated Frobenius morphism
Definition 4 . 1 .
41An S-group scheme G is said to be diagonalizable if it is isomorphic to the character group scheme of a constant group, ie if there exists an abstract abelian group M and an isomorphism of S-groupschemes G ≃ Hom Grp/S (M S , G m,S ), where M S is the constant S-group scheme defined by M . This is equivalent to the existence of an isomorphism of O S -Hopf algebras O G ≃ O S [M ].If M is an abstract group we often denote by D(M ) the S-group scheme associated to it. We obtain a contravariant functor M → D(M ) from abstract groups to diagonalizable S-group schemes.We have the following lemma, from [Gro11a, Exp VIII, prop.2.1] :Lemma 4.1. The group scheme D(M ) −→ S is smooth if and only M is of finite type and the order of its torsion subgroup is prime to all the residue characteristics of S. A diagonalizable group scheme is always faithfully flat and affine over S.
Definition 4. 2 .
2Let M be an abstract abelian group. An O S -algebra A is said to be M -graded if it has a decomposition as O S -moduleA = m∈M A m , where • A 0 is a sub-O S -algebra of A • For all (m, n) ∈ M 2 , A m A n ⊂ A m+nWe have the following proposition, from [Gro11a, Exp I, 4.7.3] and [Gro11b, Exp VIII, Prop 4.1-4.6] : Proposition 4.1. Let M be an abstract abelian group. The functor A → Spec(A) induces an anti-equivalence between the category of M -graded quasi-coherent O Salgebras and the category of affine S-schemes with an action of D(M ). Let Y = Spec(A) be an S-scheme with an action of D(M ). Then Y −→ S is a D(M )-torsor if and only if the following two conditions are satisfied : (a) For all m ∈ M , A m is an invertible O S -module. (b) For all (m, n) ∈ M 2 , the morphism A m ⊗ OS A n −→ A m+n induced by multiplication is an isomorphism. Those two conditions are in turn equivalent to the following : (a') The morphism O S −→ A 0 is an isomorphism. (b') For all m ∈ M , A m A −m = A 0 .
is locally free. One can thus cover X = ∪ i U i by open affines such that the restriction of each of the A i are free in restriction to the open affines f −1 (U i ) ⊂ Y . Hence to investigate the structure of the covering (Y, G), we may assume that S, X and Y are affine schemes and that O Y := A is free (necessarily of finite rank) over O X . Let us denote Y = Spec(A), X = Spec(A 0 ) and S = Spec(B). Up to further localization in X if necessary, we may also assume that each of the pieces A m of the M -grading of A is free. To compute the rank of A m we may restrict to the image in X of a dense open subscheme of Y on which the action of G is free. Then condition (b ′ ) of 4.1 shows that the rank of A m as an A 0 -module is 1.We then have the following result, giving the local structure of covering under diagonalizable groups :Theorem 4.1. With the previous conventions and notations, there exists a basis (e m ) m∈M of A as an A 0 -module with e 0 = 1 and non zero-divisors (α m,n ) m,n∈M in A 0 with α 0,n = α m,0 = 1, α m,n = α n,m and ∀l, m, n ∈ M, α l,m α l+m,n = α m,n α l,m+n , such that the following holds :(i) For all m ∈ M , A m = A 0 e m (ii) For all (m, n) ∈ M 2 , e m e n = α m,n e m+n
•4. 3 .
3If i = p n − 1 then σ i,1 = 1 and since β 0 = 1 we get 1 = α p n −1,1 β p n −1 f Ramification of D(M )-coverings. We now compute the ramification divisor of a D(M )-covering for a finite abelian p-group M .
A[M ]. We thus obtain a surjection of A-modules ϕ : A ⊕|M|−1 −→ σ * m St G,y sending a basis to the basis of the X m − 1 for m ∈ M . We aim at describing its kernel. First note that if e m is invertible in A then X m = 1 in O St G,y . Note furthermore that the equality e m e n = α m,n e m+n implies that if e m and e n are invertible then so is e m+n . Thus the set N := {n ∈ M | e n ∈ A × } is a subgroup of M . Note also that if e m is invertible, its inverse must lie in A −m . Hence we find that e m is invertible if and only if α m,−m is (in which case e −1 m = α −1 m,−m e −m ), so we also have N = {n ∈ M | α n,−n ∈ A × 0 }. We then have the following proposition : Proposition 4.2. If N = {0} there exists d ∈ M such that : (i) d generates M as an abelian group, e d is an uniformizer of A and generates A as an A 0 -algebra. (ii) The kernel of the surjection ϕ is equal to the submodule (e d A) ⊕|M|−1 of A ⊕|M|−1 . Proof : Suppose N = {0}. First, note that the valuations of each of the e m are disjoint. Indeed, let (m, n) ∈ M 2 such that v(e m ) = v(e n ). There exists an invertible element a ∈ A × such that e n = ae m . Write a = k∈M a k e k on the basis (e i ) of A as an A 0 -module. We then have ae m = k∈M a k α k,m e m+k = e n .
We conclude that for all i ∈ {0, ..., |M | − 1} there exists some m ∈ M such that v(e m ) = i. In particular there exists d ∈ M such that v(e d ) = 1, ie such that e d is a uniformizer of A. If m ∈ M we can then write e m = ae v(em) d for some a ∈ A × . On the other hand we know that there exists γ m ∈ A 0 such that e v(em)d m = γ m e v(em)d , namely γ m = α m,m α m,2m . . . α m,v(em)d−1 . Let us write a = k∈M a k e k . We then find e m = k∈M γ m a k α k,v(em)d e k+v(em)d . Thus we must have a k = 0 if k = m−v(e m )d and a = a m−v(em)d e m−v(em)d ∈ A × . Since N = {0} we must have m = v(e m )d and a = a m−v(em) ∈ A × 0 . Thus d generates M as an abelian group and e d generates A as an A 0 -algebra. In O St G,y we thus have e d (X m − 1) = 0 for all m ∈ M and hence (e d A) ⊕|M|−1 ⊂ ker(ϕ). Conversely, if x = (x k ) k∈M * ∈ A ⊕|M|−1 is such that ϕ(x) = 0 let us write x k = y k + z k , with y k ∈ e d A and z k ∈ A × . Then ϕ(x) = k∈M z k (X k − 1) and since the X k − 1 form a basis of k A [M ], where k A is the residue field of A, we must have z k = 0 for all k and thus x ∈ (e d A) ⊕|M|−1 . Hence we have ker(ϕ) = (e d A) ⊕|M|−1 . Definition 4.3. With the preceding notations, if N = {0} we say that the covering (Y, G) is totally ramified at y.
Theorem 4. 2 .
2Let X be a scheme and (Y, G) be a covering of X given by the action of diagonalizable group scheme G = D(M ) on the normal scheme Y . We denote by R G its ramification divisor. For every codimension 1 point of Y there exists a maximal subgroup H y = D(M/N y ) of G such that : (i) The covering (Spec(O Y,y ), H y ) of Spec(O Y,y ) induced by the action of G is totally ramified. (ii) The residual covering Spec(O Y,y )/H y −→ Spec(O Y,y )/G is a G/H y -torsor.
This can be written on the type M graduation of A as follows : for eachm ∈ M choose a lift m ∈ M and write A = m∈M/N n∈N A m+n , the quotient morphisms being given by the inclusions A 0 ֒→ n∈N A n ֒→ A.
O
Yy/Hy = n∈N A 0 e n . For every residue classm ∈ M/N y , fix a lift s(m) ∈ M . Then the action of H y on Y y can be described by the graduation A = m∈M/Ny B 0 e s(m) and the cocycle (βm ,m ′ = α s(m),s(m ′ ) ). By definition,m ∈ M/N y we have e s(m) ∈ A × if and only if s(m) ∈ N y , iem = 0.
Suppose Y = Spec(A) is affine and let A = m∈M A m be the graduation of type M associated with the action of G. Let T be an S-scheme and t : T −→ Y be a T -point of Y , corresponding to a ring morphism t ♯ : A −→ O T (T ). We have t ∈ Fix G (T ) if and only if all a = m∈M a m ∈ A satisfies the equality m∈M t ♯ (a m )X m = t ♯ (a).
Proposition 4 . 3 .
43If (Y, G) is a totally ramified covering of a scheme X given by the action of a diagonalizable group scheme G, with ramification divisor R G , for every codimension 1 point y ∈ Y the ideal sheaf I G of the fixed point scheme verifies the relation O Y (− R G ) y = (I G,y ) ⊗|G|−1 .4.5.Dévissage of D(M )-coverings. We now investigate the behaviour of the ramification divisors through dévissage in the special case of coverings given by diagonalizable group actions. Unlike the general case of 3.4, we will see that in this situation the ramification divisor behaves like the classical one with respect to dévissage.Proposition 4.4. Let X be an S-scheme and (Z, G) be a covering of X given by the action of an infinitesimal diagonalizable group scheme G = D(M ). Suppose given a subgroup H = D(M/N ) of G that gives rise to a covering (Z, H) of the quotient g : Z −→ Y = Z/H. Denote by R G , R H and R G/H the ramification divisors respectively associated to the actions of G, H on Z and of G/H on Y .
Theorem 4. 3 .
3Let f : Y −→ X be a µ p n ,S -covering of X. Denote byA = i∈Z/p n Z A ithe structure sheaf of Y , graded by the action of µ p n ,S . For every i, j ∈ Z/p n Z, denote by U ij the open subscheme of X over which the multiplication
l
Thus b j .ϕ = e * j if and only if for all l ∈ Z/p n Z we have k∈Z/p n Z b k,j ϕ l+k α k,l = δ l,j . Hence ϕ generates the A-module A * if and only if the matrix M (ϕ) =
)[1] , where, for every O Y -module F, we denoted by F [n] = (F ⊗n ) * * its n-th reflexive power.
and hence ω f = (I ⊗p−1 fix ) * . Furthermore since Y is noetherian and I ⊗p−1 fix is coherent by[Har77, III, prop. 6.8] for all point y ∈ Y we have(I ⊗p−1 fix,y ) * = (I p−1 fix ) * y . But by 4.4 for all point y ∈ Y of codimension 1 we have O Y (− R G ) y = I ⊗p−1fix,y . Thus dualizing we get O Y (R G ) y = (I ⊗p−1 fix,y ) * = (I p−1 fix ) * y = ω f,y . Hence the sheaves O Y (R G ) and ω f are invertible sheaves equal in all codimension 1 points so they are equal.
which we see that (h : Y −→ X, µ p,S ) is a covering of X. If G, H, Q are the associated action groupoids, by 4.4 we have R G = R H +g * R Q . By induction hypothesis we have O Y (R Q ) = ω h and O Z (R H ) = ω g . But by [Liu02, 6.4, Lemma 4.26] we have ω f = ω g ⊗ OZ g * ω h . Thus
Note also that a finite schematically dominant morphism is surjective by Cohen-Seidenberg's theorem. Hence, by [Gro63, Exp.VIII, Prop 5.1], to show that j X is an epimorphism we only have to show that it is schematically dominant. Let V ⊂ Y be a saturated schematically dense open subscheme of Y on which G acts freely. Since Y −→ X is faithfully flat, its image W ⊂ X is a schematically dense open of X. By [Gro11a, Exp V, th.4.1], the morphism
AcknowledgementsThe author wishes to thank Matthieu Romagny for his constant support during all stages of this work. We would also like to thank João Pedro Dos Santos for his insightful remarks on a preliminary version of this article.
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We develop a theory of BV and Sobolev Spaces via integration by parts formula in abstract metric spaces; the role of vector fields is played by Weaver's metric derivations. The definition hereby given is shown to be equivalent to many others present in literature.
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Sobolev and BV spaces on metric measure spaces via derivations and integration by parts
19 Sep 2014
Simone Di Marino
Sobolev and BV spaces on metric measure spaces via derivations and integration by parts
19 Sep 2014
We develop a theory of BV and Sobolev Spaces via integration by parts formula in abstract metric spaces; the role of vector fields is played by Weaver's metric derivations. The definition hereby given is shown to be equivalent to many others present in literature.
Introduction
In the last few years a great attention has been devoted to the theory of Sobolev spaces W 1,q on metric measure spaces (X, d, m), see for instance [10,9,6] for an overview on this subject and [3] for more recent developments. These definitions of Sobolev spaces usually come with a weak definition of modulus of gradient, in particular the notion of q-upper gradient has been introduced in [13] and used in [14] for a Sobolev space theory. Also, in [14] the notion of minimal q-upper gradient has been proved to be equivalent to the notion of relaxed upper gradient arising in Cheeger's paper [7]. In [3] the definitions of q-relaxed slope and q-weak upper gradient are given and the minimal ones are seen to be equivalent to the ones in [14].
All of those approach give us a notion of modulus of the gradient instead of the gradient itself, and an integration by parts formula is present only in special cases, and moreover it is often only an inequality. In this paper we want to fill this gap, namely giving a definition of Sobolev spaces more similar to the classical one given with an integration by parts formula; in R n this formula can be written as
R n div(v)f dx = − R n v · ∇f dx,
where f and v are smooth functions and v is with compact support. The usual definition of the Sobolev space W 1,p then can be seen like this: f ∈ L p is a Sobolev function if there exists a function g f ∈ L p (R n ; R n ) such that
R n div(v)f dx = − R n v · g f dx ∀v ∈ C ∞ c (R n ; R n ). (0.1)
Another equivalent formulation is that there exists a constant C such that
R n div(v)f dx ≤ C · v L q (R n ;R n ) ∀v ∈ C ∞ c (R n ; R n ), (0.2)
where one can recover the weak gradient g f by a simple duality argument when p > 1. Note that (0.1) and (0.2) give the same space when 1 < p < ∞, while they differ with p = 1: in that case (0.1) is the definition for W 1,1 (R n ) while (0.2) is the usual definition for the space BV (R n ). This definition can be generalized in any metric measure space; the problem is to find the correct generalization of vector fields in an abstract metric space are the derivations.
The derivations were introduced in the seminal papers by Weaver, and then in more recent times widely used in the Lipschitz theory of metric spaces, for example in connection with Rademacher theory for metric spaces, but also as a generalization of sections of the tangent space [11,12,5,8,4]. Here we see that the derivations are also powerful tools in the Sobolev theory, as already point out in [8]. A derivation is simply a linear map b : Lip 0 (X, d) → L 0 (X, m) such that the Liebniz rule holds and it has the locality property |b(f )| ≤ g · lip a (f ) for some g ∈ L 0 (X, m). Now we simply say that f ∈ L p is a function in W 1,p if there is a linear map L f such that integration by part holds:
X L f (b) dm = − X f · div b dm ∀b ∈ Der q,q ,
where Der q,q is the subset of derivation for which |b|, div b ∈ L q (X, m).
We will see that it is well defined a proper "differential" df : Der q,q → L 1 , and so it is possible to provide also a notion of modulus of the gradient |∇f | in such a way that |df (b)| ≤ |∇f | · |b|; in Section 2 we see that this notion coincides with all the other (equivalent) notion of modulus of the gradient given in [3], and in particular there is also identification of the Sobolev spaces.
The easy part is the inclusion of the Sobolev Space obtained via relaxation of the asymptotic Lipschitz constant into the one defined by derivations. The other inclusion uses the fact that q-plans, namely measures on the space of curves with some integrability assumptions, induces derivations thanks to the basic observation that, even in metric spaces, we can always take the derivative of Lipschitz functions along absolutely continuous curves; this observation has already been used in [11,12,5] to find correlations between the differential structure of (X, d) and the structure of measures on the set of curves (a peculiar role is played by Alberti representation).
In Section 3.1 we extend this equivalence to the BV space, using the results in [1].
Sobolev spaces via derivations
Here (X, d, m) will be any complete separable metric measure space, where m is a nonnegative Borel measure, finite on bounded sets; in particular we don't put assume structural assuption, namely doubling measure nor a Poincaré inequality to hold. In the sequel we will denote by Lip 0 (X, d) the set of Lipschitz functions with bounded support (the support of a continuous function f is defined as supp(f ) = {f = 0}), and with L 0 (X, m) the set of m-measurable function on X, without integrability assumption.
Derivations
We state precisely what we mean here by derivations:
Definition 1.1 A derivation b is a linear map b : Lip 0 (X, d) → L 0 (X, m)
such that the following properties hold:
(i) (Liebniz rule) for every f, g ∈ Lip 0 (X, d), we have b(f g) = b(f )g + f b(g);
(ii) (Weak locality) There exists some function g ∈ L 0 (X, m) such that
|b(f )|(x) ≤ g(x) · lip a f (x) for m-a.e. x, ∀f ∈ Lip 0 (X, d).
The smallest function g with this property is denoted by |b|.
From now on, we will refer to the set of derivation as Der(X, d, m) and when we write b ∈ L p we mean |b| ∈ L p . Since the definition of derivation is local on open sets we can extend b to locally Lipschitz functions. We define also the support of a derivation supp(b) as the essential closure of the set {|b| = 0}; it is easy to see that if supp(f ) ∩ supp(b) = ∅ then b(f ) is identically 0.
In order to get to (0.2), we need also the definition of divergence, and this is done simply imposing the integration by parts formula: whenever b ∈ L 1 loc we define div b as the operator that maps Lip 0 (X, d) ∋ f → − X b(f ) dm (whenever this makes sense). We will say div b ∈ L p when this operator has an integral representation via an L p function:
div b = h ∈ L p if − X b(f ) dm = X h · f dm ∀f ∈ Lip 0 (X, d).
It is obvious that if div b ∈ L p , then is unique. Now we set
Der p (X, d, m) = b ∈ Der(X, d, m) : b ∈ L p (X, m) Der p 1 ,p 2 (X, d, m) = b ∈ Der(X, d, m) : b ∈ L p 1 (X, m), div b ∈ L p 2 (X, m)
We will often drop the dependence on (X, d, m) when it is clear. We notice that Der, Der p and Der p 1 ,p 2 are real vector spaces, the last two being also Banach spaces endowed respectively with the norm b p = |b| p and b p 1 ,p 2 = b p 1 + div b p 2 . For brevity we will denote Der ∞ = Der b , and Der ∞,∞ = Der L (b stands for bounded while L stands for Lipschitz). The last space we will consider is D(div), that will be consisting of derivation b such that |b|, div b ∈ L 1 loc (X, m); it is clear that Der p,q ⊆ D(div) for all p, q ∈ [1, +∞]. In the sequel we will need a simple operation on derivations, namely the multiplication by a scalar function: let u ∈ L 0 (X, m), then we can consider the derivation ub that acts simply as ub(f )(x) = u(x) · b(f )(x): it is obvious that this is indeed a derivation. We now prove a simple lemma about multiplications: Lemma 1.2 Let b a derivation and u ∈ L 0 (X, m); then we have |ub| = |b| · |u|. Moreover, if u ∈ Lip b (X, d) and b ∈ Der p 1 ,p 2 we have that ub is a derivation such that
div(ub) = u div b + b(u) and ub ∈ Der p 1 ,p 3 ,
where p 3 = max{p 1 , p 2 }; in particular we have that Der p,p is a Lip b (X, d)-module.
Proof. Let us prove the first assertion: it is clear that |ub|(x) ≤ |b|(x)·|u(x)| by definition; the other inequality is obvious in the set {u = 0}. In order to prove the converse inequality also in {u = 0} we can choose b u = ub and let
g(x) = u −1 (x) if u(x) = 0 0 otherwise.
Then we know that |gb u | ≤ |g| · |b u |. Noting that b(f ) = gb u (f ) in {u = 0} for every f ∈ Lip b (X, d), we get also |b| = |gb u | in the same set and so |b| = |gb u | ≤ |g| · |b u | ≤ |g| · |u| · |b| = |b| in {u = 0}.
In particular we have |b u | = |b| · |u| in {u = 0} and thus the thesis.
For the second equality we can use Liebniz rule:
let f ∈ Lip b (X, d), and using b(f u) = ub(f ) + f b(u) wehave − X ub(f ) dm = − X b(f u) dm + X f b(u) dm = X f u · div b dm + X f b(u) dm = X f · (u div b + b(u)) dm
and so, thanks to the arbitrariness of f we get div(ub) = u div b + b(u). Lemma 1.3 (Stong locality in D(div)) Let b ∈ D(div). Then for every f, g ∈ Lip(X, d) we have
(i) b(f ) = b(g) m-almost everywhere in {f = g}; (ii) b(f ) ≤ |b| · lip a (f | C ) m-almost everywhere in C, for every closed set C;
Proof. In order to prove (i), it is sufficient to consider g = 0 and f with support contained in B = B r (x 0 ), where we can take r > 0 as small as we want; then we can conclude by linearity and weak locality. So we can suppose that both |b| and div b are integrable in
B. Now we can consider φ ε (x) = (x − ε) + − (x + ε) − ; we have φ ε is a 1-Lipschitz function such that |φ ε (x) − x| ≤ ε and φ(x) = 0 whenever |x| ≤ ε. Let f ε = φ ε (f ); we have b(f ε )
is a family of equi-integrable functions and so there is a subsequence converging weakly in L 1 to some function g. Moreover f ε → f uniformly and in particular
X b(f ε ) dm − X b(f ) dm = − X (f ε − f ) · div b dm → 0; (1.1)
since this is true also for ρb whenever ρ ∈ Lip 0 (X, d) (by Lemma 1.2, we have ρb ∈ D(div)),
we obtain ρ, b(f ε ) → ρ, b(f ) for every ρ ∈ Lip b (X, d) and so g = b(f ) and b(f ε ) ⇀ b(f ). In particular, letting ρ = χ {f =0} sgn(b(f )) and noting that lip a (f ε ) = 0 in the set {|f | < ε} we obtain {f =0} |b(f )| dm = X ρ · b(f ) = lim ε→0 X ρ · b(f ε ) dm = 0.
For (ii) we proceed as follows: for every closed ballB r (y) we consider the McShane extension of the function f restricted to C ∩B r (y) and we call it g r y :
g r y (x) = sup{f (x ′ ) − Ld(x ′ , x) : x ′ ∈ C ∩B r (y)}, L = Lip(f,B r (y) ∩ C)
In particular we have f = g r y on C ∩ B r (y) and Lip(g r y , B r (y)) = Lip(f, B r (y) ∩ C) = Lip(f | C , B r (y)). Applying (i) of this lemma we find that b(f ) = b(g r y ) m-a.e. on C ∩B r (y); in particular
|b(f )|(x) ≤ |b| · Lip(f | C ,B r (y))
m-a.e. on C ∩ B r (y).
Since we haveB r (y) ⊂ B 2r (x) whenever x ∈ B r (y), we obtain
|b(f )|(x) ≤ |b| · Lip(f | C , B 2r (x)) m-a.e. on C ∩ B r (y);
now we can drop the dependance on y and then let r → 0 to get the thesis.
Remark 1.4
Notice that our definition of derivation is slightly different from the classical one of Weaver [15], since we don't require any continuity assumption. However it is easy to see that every derivation in D(div) is also a Weaver derivation, thanks to the integration by part formula. In the sequel only these derivations will play a role in the definition of Sobolev Spaces and so this discrepancy in the definition is harmful.
Definition via derivations
In this whole section we treat the Sobolev spaces W 1,p with 1 ≤ p < +∞; the case of the space BV will be treated separately. We state here the main definition of Sobolev space via derivations: we want to follow the definition (0.2) but in place of the scalar product between the vector field and the weak gradient we assume there is simply a continuous linear map.
Definition 1.5 Let f ∈ L p (X, m); then f ∈ W 1,p (X, d, m) if, setting p = q/(q − 1), there exists a linear map L f : Der q,q → L 1 (X, m) satisfying X L f (b) dm = − X f div b dm for all b ∈ Der q,q , (1.2)
continuous with respect to the Der q norm and such that
L f (hb) = hL f (b) for every h ∈ Lip b , b ∈ Der q,q . When p = 1 assume also that L f can be extended to an L ∞ -linear map in Der ∞ L := L ∞ · Der L .
Since from the definition it is not obvious, we prove that L f (b) is uniquely defined whenever f ∈ W 1,p and b ∈ Der q,q : Remark 1.6 (Well posedness in Der q,q ) Let us fix b ∈ Der q,q , f ∈ W 1,p ; let L f andL f be two different linear maps given in the definition on W 1,p . Let h ∈ Lip b (X, d): using Lemma 1.2 we have hb ∈ Der q,q and so we can use (3.1) and the L ∞ -linearity to get
X hL f (b) dm = X L f (hb) = − X f div(hb) dm,
and the same is true forL f . In particular, since the right hand side does not depend on
L f , we have X hL f (b) = X hL f (b), and thanks to the arbitrariness of h ∈ Lip b (X, d) we conclude that L f (b) =L f (b) m-a.
e. We will call this common value b(f ), since it extends b on Lipschitz functions. The same result is true also for p = 1 and b ∈ Der ∞ L .
Now we can give the definition of weak gradient, in some sense dual to the definition of |b|:
Theorem 1.7 Let f ∈ W 1,p (X, d, m); then there exists a function g f ∈ L p (X, m) such that |b(f )| ≤ g f · |b| m-a.e. in X ∀b ∈ Der q,q . (1.3)
The least function g f (in the m-almost everywhere sense) that realizes this inequality is denoted with |∇f | p , the p-weak gradient of f Proof. We reduce to prove the existence of a weak gradient in the integral sense; then thanks to Lip b -linearity we can prove the theorem. In fact if we find a function g ∈ L p (X, m) such that
X b(f ) dm ≤ X g|b| dm ∀b ∈ Der q,q ,(1.4)
then, choosing b h = hb with h ∈ Lip b (X, d), we can localize the inequality thus obtaining b(f ) ≤ g|b|; using this inequality also with the derivation −b we get (1.3). So, we're given a function f ∈ W 1,p and we want to find g ∈ L p satisfying (1.4); let us note that, by definition, there exists a constant C = L f such that for every b ∈ Der q,q
b(f ) dm ≤ L f (b) 1 ≤ C b q (1.5)
Let us consider two functionals in the Banach space Y = L q (X, m):
Ψ 2 (h) = C h L q (m) (1.6) Ψ 1 (h) = sup X b(f ) dm : |b| ≤ h , b ∈ Der q,q (1.7)
where the supremum of the empty set is meant to be −∞. Equation (1.5) guarantees that
Ψ 1 (h) ≤ Ψ 2 (h) ∀h ∈ Y. (1.8)
Moreover Ψ 2 is convex and continuous while we claim that Ψ 1 is concave: it is clearly positive 1-homogeneus and so it is sufficient to show that
Ψ 1 (h 1 + h 2 ) ≥ Ψ 1 (h 1 ) + Ψ 1 (h 2 ).
We can assume that Ψ 1 (h i ) > −∞ for i = 1, 2 because otherwise the inequality is trivial. In this case for every ε > 0 we can pick two derivations b i ∈ Der q,q such that
X b 1 (f ) dm ≥ Ψ 1 (h 1 ) − ε |b 1 | ≤ h 1 X b 2 (f ) dm ≥ Ψ 1 (h 2 ) − ε |b 2 | ≤ h 2
and so we can consider b 1 +b 2 that still belongs to Der q,q and clearly |b 1 +b 2 | ≤ |b 1 |+|b 2 | ≤ (h 1 + h 2 ) and so
Ψ 1 (h 1 + h 2 ) ≥ X (b 1 + b 2 )(f ) dm ≥ Ψ 1 (h 1 ) + Ψ 1 (h 2 ) − 2ε,
and we get the desired inequality letting ε → 0. By Hahn-Banach theorem we can find a continuous linear functional L on L q (X, m) such that
Ψ 1 (h) ≤ L(h) ≤ Ψ 2 (h).
Case p > 1. We know that (L q ) * = L p and so we can find g ∈ L p such that L(h) = X gh dm. This proves the existence and moreover we have that L(h) ≤ Ψ 2 (h) = C h q for every h ∈ Y and so we have also that g p ≤ C.
Case p = 1, X compact. In this case (notice that here we have to put Der ∞ L in place of Der q,q in (1.7)) if we restrict L : C b (X) → R we can see it as a positive linear such that L(h) ≤ C h ∞ and so, thanks to the compactness of X, it can be represented as a finite measure, i.e. there exists µ ∈ M + (X) such that L(h) = X h dµ for every h ∈ C 0 (X) and µ(X) ≤ C. Now let us fix b ∈ Der ∞ L and let
h ε (x) = 1 |b| if |b|(x) ≥ ε ε −1 otherwise
in such a way that |h ε b| ≤ 1 with equality in {|b| ≥ ε}. Now let us consider for every h ∈ C 0 (X) the derivation h · h ε · b; we know that |h · h ε · b| ≤ |h| and so we can use (1.7) and the L ∞ -linearity to infer that
X hh ε b(f ) dm ≤ X |h| dµ ∀h ∈ C 0 (X);
this permits us to localize the inequality to h ε b(f )m ≤ µ. Now we have a family of measures F = {h ε b(f )m : ∀b ∈ Der ∞ L , ∀ε > 0} such that ν ≤ µ whenever ν ∈ F. Now we can consider the supremum of the measures in F, defined as
µ F (A) = sup N i=1 ν i (A i ) : ν i ∈ F, A i ⊆ A, A i disjoint ;
it is readily seen that this is in fact a measure, and it is the least measure ρ such that ν ≤ ρ for every ρ ∈ F. The existence is clear thanks to the fact that ν ≤ µ, and in particular we have that µ F ≤ µ; moreover, since for every ν ∈ F we have that ν << m, also the supremum inherits this property, in particular we have µ F = gm for some g ∈ L 1 (m). In particular, again fixing b ∈ Der ∞ L , we have that
h ε b(f ) ≤ g m-a.e. ∀ε > 0; (1.9)
in particular, we can divide (1.9) by h ε to obtain
b(f ) ≤ g|b| m-a.e. in {|b| ≥ ε} b(f ) ≤ gε m-a.e. in {|b| < ε}.
(1.10)
Since ε is arbitrary we obtain b(f ) ≤ g|b| for m-almost every x ∈ X, that is the thesis; also in this case p = 1 we have g 1 ≤ µ(X) ≤ L f . Case p = 1, X general. In order to remove the compactness assumption, for every compact non negligible set K ⊆ X let us consider the two functionals in the Banach space
Y K = L ∞ (K, d, m): Ψ 2 (h) = C h L ∞ (K,m) (1.11) Ψ 1 (h) = sup K b(f ) dm : |b| ≤ h m-a.e. on K , b ∈ Der ∞ L .
(1.12)
Now we can argue precisely as before to obtain g K ∈ L 1 (K, m) such that
g K 1 ≤ L f b(f ) ≤ g K |b| m-a.e. on K ∀b ∈ Der q . (1.13)
Now for every increasing sequence of compact sets K n , let us consider g(x) = inf Kn∋x g Kn (x). Denoting Y := n K n , it is easy to note that g ∈ L 1 (Y, m), since g L 1 (Y,m) = sup n g L 1 (Kn,m) ≤ sup n g Kn L 1 (Kn,m) ≤ L f , and we have that
b(f ) ≤ g|b| m-a.e. on Y ∀b ∈ Der q ;
so, in order to conlcude, it is sufficient to find a sequence K n such that m(X \ n K n ) = 0, but this can be done thanks to the hypothesis of m finite on bounded sets (so we can find θ > 0 such that θm is finite and then apply Prokhorov theorem to θm).
Equivalence with other definitions
In this section we want to prove, when p > 1, that Definition 3.1 is equivalent to the other ones W 1,p * and W 1,p w , given in [3]. As a byproduct we obtain the equivalence also with other definitions of Sobolev Spaces, for example the one given in [7], similar to W 1,p * but here the relaxation is made with general L p functions, and the asymptotic Lipschitz constant is replaced by upper gradients, or the one given in [14], similar to W 1,p w but with a slightly stronger notion of negligibility of set of curves.
We will prove that W 1,p * ⊆ W 1,p ⊆ W 1,p w and that the following inequality is true for the weak gradients: |∇f | p, * ≤ |∇f | p ≤ |∇f | p,w m-a.e. in X Then for p > 1, using the equivalence W 1,p * = W 1,p w and |∇f | p,w = |∇f | p, * in [3] will let us conclude; also the coincidence with other definitions can be found in [3]. Let us recall briefly the definitions of W 1,p * (in the stronger version given in [?]) and W 1,p w :
Definition 2.1 (Relaxed Sobolev Space) A function f ∈ L p (X, m) belongs to W 1,p (X, d, m)
if and only if there exists a sequence (f n ) ⊂ Lip 0 (X, d) and a function g ∈ L p (X, m) such that lim
n→∞ f n − f p + lip a (f n ) − g p = 0.
The function g with minimal L p norm that has this property will be denoted with |∇f | p, *
In order to define the space W 1,p w we have to introduce the test plans, that will consent to define a concept of negligibility of set of curves that is crucial in the definition of the weak Sobolev space (see also [2] for a detailed analysis of the different concepts of negligibility given in [3] and [14]) .
Definition 2.2 (Test plans and negligible sets of curves)
We say that a probability measure π ∈ P(C([0, 1], X)) is a q-test plan if π is concentrated on AC q ([0, 1], X), we have 1 0 |γ t | q dt dπ < ∞ and there exists a constant C(π) such that
(e t ) ♯ π ≤ C(π)m ∀t ∈ [0, 1]. (2.1) A set A ⊂ C([0, 1], X)
is said to be p-negligible if it is contained in a π-negligible set for any p-test plan π. A property which holds for every γ ∈ C([0, 1], X), except possibly a p-negligible set, is said to hold for p-almost every curve.
Definition 2.3 (Weak Sobolev Space) A function f ∈ L p (X, m) belongs to W 1,p w (X, d, m) if there exists a function g ∈ L p (X, m) that is a p-weak upper gradient, i.e. it is such that ∂γ f ≤ γ g < ∞ for p-a.e. γ. (2.2)
The minimal p-weak upper gradient (in the pointwise sense) will be denoted by |∇f | p,w .
2.1
W 1,p * ⊆ W 1,p Let f ∈ W 1,p * (X, d,b ∈ Der q,q X f n · div b dm = X b(f n ) dm ≤ X |b| · lip a (f n ) dm.
Taking the limit as n → ∞ we have that
X f · div b dm ≤ X |b| · |∇f | p, * dm ∀b ∈ Der q,q . (2.3)
Now we have to construct the linear functional L f : Der q,q → L 1 . So, fix b ∈ Der q,q and let µ b = |b| · |∇f | p, * m. Notice that µ b is a finite measure. Now let R b : Lip b (X, d) → R be the linear functional defined by
R b (h) = − X f · div(hb) dm.
Notice that, since hb ∈ Der q,q , we can take it as a test derivation in (2.3), obtaining
|R b (h)| ≤ C h ∞ , where C = µ b (X).
In particular R b can be extended to a continuous linear functional on C b (X); since |R b (h)| ≤ X |h| dµ b , we have that R b (h) can be represented as an integral with respect to a signed measure m b , whose total variation is less then µ b , but since µ b is absolutely continuous with respect to m, also m b must have this property; if we denote by L f (b) the density of m b relative to m, we have
− X f · div(hb) dm = h · L f (b) dm ∀ h ∈ Lip b (X, d) (2.4) |L f (b)| ≤ |b| · |∇f | p, * m-almost everywhere (2.5)
Now we have to check the Lip b -linearity, but this is easy since for every h,
h 1 ∈ Lip b by definition we have R hb (h 1 ) = R b (h · h 1 ); in particular X h 1 · L f (hb) d mm = X h 1 · hL f (b) dm ∀h 1 ∈ Lip b (X),
and so L f (hb) = hL f (b).
2.2 W 1,p ⊆ W 1,p w
The crucial observation is that every q-plan induce a derivation:
Proposition 2.4 Let π be a q-plan. For every function f ∈ Lip b (X, d) let us consider b π (f ), the function such that:
X g · b π (f ) dm = AC 1 0 g(γ t ) d(f • γ) ds (t) dt dπ(γ) ∀g ∈ L p . (2.6)
Then we have that b π ∈ Der q,q and moreover
X g · |b π | dm ≤ γ g ds dπ(γ) ∀g ∈ L p , g ≥ 0; (2.7) X f · div(b π ) dm = AC (f (γ 1 ) − f (γ 2 )) dπ(γ) ∀f ∈ L p . (2.8) Proof.
We first fix f ∈ Lip b (X, d) and notice that the right hand side in (2.6) is well defined thanks to Rademacher theorem. Then the Liebniz rule is easy to check thanks to its validity in the right hand side of (2.6). In order to find a good candidate for |b π |, we estimate d(f •γ)
ds
≤ lip a (f )(γ t )|γ t | and so, for every nonnegative g ∈ L p we have
1 0 g(γ t ) d(f • γ) ds (t) dt ≤ 1 0 g(γ t )lip a (f )(γ t )|γ t | dt;
integrating with respect to π and using Fubini theorem we get
X g · b π (f ) dm ≤ X g · lip a (f ) dµ π ,(2.9)
where µ π = 1 0 (e t ) ♯ ( γ t |π) dt is the barycenter of π, and it is such that X g dµ π = γ g ds dπ.
(2.10)
In particular we can use Hölder's inequality to estimate the behavior of µ π :
X g dµ π = AC 1 0 g(γ t )|γ t | dt dπ ≤ |g(γ t )| p dtdπ 1/p |γ t | q dtdπ 1/q ≤ C(π) 1/p · g L p (m) · E q (γ) L q (π) ,
and so, by duality argument, we obtain that µ π = hm with h ∈ L q (X, m); using this representation in (2.9) we obtain
X g · b π (f ) dm ≤ X g · lip a (f )h dm ∀ g ∈ L q , g ≥ 0.
So we deduce that |b π | ≤ h and in particular b π ∈ L q and (2.7) is true thanks to (2.10). It remains to prove the last equality: by definition of divergence we have, for f ∈ Lip 0 (X, d) (2.11) thanks to the fact that the fundamental theorem of calculus holds for Lipschitz functions. By definition of q-plan we have also that (e t ) ♯ π = f t m where f t ≤ C(π) for every t ∈ [0, 1]; since π is a probability measure we have f t dm = 1 and so f t ∈ L 1 ∩ L ∞ and in particular f t ∈ L q and so div b π = (f 1 − f 0 ) ∈ L q . This enables us to extend (2.11) to f ∈ L p and so we proved also (2.8).
f · div b π dm = AC 1 0 d(f • γ) ds (t) dt dπ(γ) = (f (γ 1 ) − f (γ 0 )) dπ,
Lemma 2.5 Let f ∈ W 1,p (X, d, m). Then |∇f | w is a p-weak upper gradient for f .
Proof. By Proposition 2.4 we know that for every q-plan π we can associate a derivation b π ∈ Der q,q ; we use this derivation in the definition of W 1,p and, using also Theorem 1.7, we obtain − X f · div b π dm ≤ |∇f | w · |b π | dm; Now, using (2.7) and (2.8), we obtain precisely
AC (f (γ 0 ) − f (γ 1 ))dπ ≤ AC γ |∇f | w ds dπ. ∀π q-plan (2.12)
We can "localize" this inequality using the fact that for every Borel set A ⊆ C([0, 1]; X) such that π(A) = 0, we have that π A = 1 π(A) π| A is still a q-plan and so we can infer that
A (f (γ 0 ) − f (γ 1 ))dπ ≤ A γ |∇f | w ds dπ. ∀A ⊂ C([0, 1]; X),(2.13)
and so f (γ 0 ) − f (γ 1 ) ≤ γ |∇f | w for π-almost every curve. Applying the same conclusion to −f we get that the upper gradient property is true for π-almost every curve. Since π was an arbitrary q-plan, by definition we have
|f (γ 0 ) − f (γ 1 )| ≤ γ
|∇f | w ds for p-almost every curve γ and so |∇f | w is a p-weak upper gradient.
BV space via derivations
From now on, when µ ∈ M(X), we will denote X dµ = µ(X).
Definition 3.1 Let f ∈ L 1 (X, d, m); we say f ∈ BV (X, d, m) if there exists a linear map L f : Der b → M(X) satisfying X dL f (b) = − X f div b dm ∀ b ∈ Der L , (3.1)
continuous with respect to the Der ∞ norm and such that
L f (hb) = hL f (b) for every h ∈ C b (X), b ∈ Der b .
As in the W 1,p case, we can prove that L f (b) is uniquely defined whenever f ∈ BV and b ∈ Der L : Remark 3.2 (Well posedness in Der L ) Let us fix b ∈ Der L , f ∈ BV ; let L f andL f be two different linear maps given in the definition on BV . Let h ∈ Lip b (X, d): using Lemma 1.2 we have hb ∈ Der L and so we can use the C b linearity and (3.1) to get
X h dL f (b) = X dL f (hb) = − X f div(hb) dm,
and the same is true forL f . In particular X h dL f (b) = X h dL f (b), and thanks to the arbitrariness of h ∈ Lip b (X, d) we conclude that L f (b) =L f (b). We will call this common value Df (b). Now we can give the definition of total variation: Theorem 3.3 Let f ∈ BV (X, d, m); then there exists a finite measure ν ∈ M + (X) such that, for every Borel set A ⊆ X,
A dDf (b) ≤ A |b| * dν ∀b ∈ Der L ,(3.
2)
where g * denotes the upper semicontinuous envelope of g. The least measure that realizes this inequality is denoted with |Df |, the weak total variation of f . Moreover
|Df |(X) = sup{|Df (b)(X)| : |b| ≤ 1, b ∈ Der L }. (3.3)
Proof. We argue similarly to Theorem 1.7: by hypothesis we have that f ∈ BV and so there exists a C b -linear map L f :
Der L → M(X) such that L f (b)(X) ≤ C b L ∞ , where we can take C = sup{|Df (b)(X)| : |b| ≤ 1, b ∈ Der L }. Note that if |b| ≤ h where h ∈ C b (X) then we have that K dL f (b) ≤ C sup x∈K h(x)
∀K ⊆ X compact; (3.4) in fact, denoting with ρ n = min{1 − nd(x, K)}, we have that ρ n → χ K pointwise and 0 ≤ ρ n ≤ 1 so, by dominated convergence theorem,
K dL f (b) = lim n→∞ X ρ n dL f (b) ≤ C lim n→∞ ρ n b ∞ ≤ C lim n sup x∈X ρ n (x)h(x) = C sup x∈K h(x),
where the last equality holds thanks to the compactness of K. Now, for every compact set K ⊆ X and consider two functionals in the Banach space Y = C b (K):
Ψ 2 (h) = C h ∞ (3.5) Ψ 1 (h) = sup K dL f (b) : b ∈ Der b , ∃h ∈ C b (X) such that |b| ≤h,h| K ≤ h (3.6)
where the supremum of the empty set is meant to be −∞. Equation (3.4) guarantees that
Ψ 1 (h) ≤ Ψ 2 (h) ∀h ∈ Y. (3.7)
Moreover, as before, Ψ 2 is convex and continuous while Ψ 2 is concave; by Hahn-Banach theorem we can find a continuous linear functional L on C b (K) such that
Ψ 1 (h) ≤ L(h) ≤ Ψ 2 (h).
In particular there exists a measure µ K such that L(h) = K h dµ K and, thanks to (3.5), we have µ K (K) ≤ C. Moreover, thanks to (3.6) we have that if h ∈ C b (X) is such that |b| ≤ h for some b ∈ Der b then
K dL f (b) ≤ K h dµ K ;
since for every k ∈ C b (X), we have |kb| ≤ |k|h we obtain also
K k dL f (b) ≤ K |k|h dµ K .
In particular, optimizing in k we obtain also that |L f (b)|, the total variation of L f (b), restricted to K, is less then or equal to hµ K . This implies that the following set is nonempty:
A K = {ν ∈ M + (K) : |L f (b)|| K ≤ hν whenever b ∈ Der b , h ∈ C b (X) s.t. |b| ≤ h} .
Clearly this set is convex, weakly- * closed and a lattice, in particular there exists the minimum, that we call ν K . We can drop the dependence on K since it is easy to see that if A ⊂ K 1 ∩ K 2 then ν K 1 (A) = ν K 2 (A); suppose on the contrary that ν K 1 (A) > ν K 2 (A). Then we can consider the measureν(B) = ν K 1 (B \ A) + ν K 2 (B ∩ A) that would be a strictly better competitor than µ K 1 in A K 1 . Now we can extend ν to a measure on the whole space
ν(B) = sup K⊆B ν(K) ∀B ⊆ X Borel;
this is easily seen to be a measure, that is also finite since ν(K) ≤ µ K (K) ≤ C for all K compact and in particular we get ν(X) ≤ C. Thanks to the finiteness of |L f (b)| and ν, using that ν| K ∈ A K , we find that
|L f (b)| ≤ hν whenever b ∈ Der b , h ∈ C b (X) s.t. |b| ≤ h, in particular, integrating in A we get A dL f (b) ≤ A h dν,
and taking the infimum in h we obtain (3.2), recalling that if g ∈ L ∞ then g * (x) = inf{h(x) : h ∈ C b (X), h ≥ g m-a.e.}.
For the last assertion we already proved C ≥ ν(X), while the other inequality is trivial taking A = X in (3.2). Proof. Let us consider two open sets A 1 , A 2 and a closed set C such that A 1 ⋐ C ⋐ A 2 . We will consider (C, d, m) as a separable metric measure space, and relate the definitions of bounded variation in X and C. Let us consider a function f ∈ BV (X, d, m); it is clear that f ∈ BV (C, d, m) since Der L (C) ⊂ Der L (X) (it is sufficient to set b X (f ) = b C (f | C )), and consequently |Df | X ≥ |Df | C by (3.2). Moreover |Df | X (A 1 ) = |Df | C (A 1 ). This is true because there exists a Lipschitz function 0 ≤ χ ≤ 1 such that χ = 0 in X \ C and χ = 1 on a neighborhood of A 1 ; then we have that if b ∈ Der L (X) implies that χb ∈ Der L (C) and so in (3.2) we can imagine that b ∈ Der L (C) whenever A ⊆ A 1 ; but then we get that the measure ν defined as
ν(B) = |Df | X (B \ A 1 ) + |Df | C (B ∩ A 1 )
is a good candidate in (3.2) and so, by the minimality of |Df | X we get |Df | C (A 1 ) = |Df | X (A 1 ). Now, denoting by µ(A) the set function defined in the left hand side of (3.8), it is obvious that µ(A 2 ) ≤ |Df |(A 2 ). But it is also obvious that µ(A 2 ) ≥ |Df | C (C) ≥ |Df | C (A 1 ) = |Df | X (A 1 ). Letting A 1 ↑ A 2 we get the desired inequality.
m). Then, by definition, there exists a sequence of Lipschitz functions with bounded support such that f n p → f and Lip a (f n ) p → |∇f |. Then by the weak locality property of derivations and the definition of divergence have that for every
Theorem 3. 4 (
4Representation formula for |Df |) Let f ∈ BV . Then the classical representation formula holds true: for every open set A |Df |(A) = sup A f · div(b) dm : b ∈ Der L , |b| ≤ 1, supp(b) ⋐ A , (3.8) where B ⋐ A if d(X \ A, B) > 0 and B is bounded.
Lemma 3.5 Let f ∈ BV * (X, d, m). Then we have f ∈ BV (X, d, m) and |Df | ≤ |Df | * as measures.Proof. By hypothesis, we know that there is a sequence (f n ) ⊂ Lip 0 (X, d) such that lip a (f n ) ⇀ |Df | * in duality with C b (X); in particular, for every b ∈ Der L we havetaking limits and recalling that whenever ν n ⇀ ν and g ≥ 0, we have lim inf n→∞ X g dµ n ≤ X g * dµ, we have thatNow this inequality would guarantee that |Df | ≤ |Df | * once we construct the linear functional L f : Der L → M(X) In order to find L f (b) we proceed exactly as in Subsection 2.1, and so we omit the construction. Proof. As for the second inclusion it is sufficient to recall Proposition 2.4: we know that for every ∞-plan π we can associate a derivation b π ∈ Der L ; we use this derivation in the definition of BV and, using also Theorem 3.3, we obtainNow, using (2.7) and (2.8), we obtainNow we can use Remark 7.2 in[1]to conclude that f ∈ w − BV and |Df | w (X) ≤ |Df |(X)Using this two lemmas in conjunction with the equivalence result in[1]we can conlcude.
We just sketch the equivalence with the other definitions given in literature: in particular we refer to [1], where the authors consider the spaces BV * and w − BV and show their equivalence. As we did for W 1. p we show BV * ⊆ BV ⊆ w − BVWe just sketch the equivalence with the other definitions given in literature: in particular we refer to [1], where the authors consider the spaces BV * and w − BV and show their equivalence. As we did for W 1,p we show BV * ⊆ BV ⊆ w − BV .
Thanks to the equivalence theorem in [1] we get BV = BV * = w −BV and |Df | w = |Df | * , in particular |Df | w (X) = |Df | * (X) ≥ |Df |(X), and so |Df |(X) = |Df | * (X) = |Df | w (X). This equality, along with |Df | ≤ |Df | *. − Bv * ⊆ Bv ⊆ W, Bv, | Moreover |df | ≤ |df, X) ≥ |Df | w (X). let us conclude that the three definitions of total variation coincideProof. From Lemma 3.5 and 3.6 we know that BV * ⊆ BV ⊆ w − BV and moreover |Df | ≤ |Df | * and |Df |(X) ≥ |Df | w (X). Thanks to the equivalence theorem in [1] we get BV = BV * = w −BV and |Df | w = |Df | * , in particular |Df | w (X) = |Df | * (X) ≥ |Df |(X), and so |Df |(X) = |Df | * (X) = |Df | w (X). This equality, along with |Df | ≤ |Df | * let us conclude that the three definitions of total variation coincide.
Equivalent definitions of BV space and of total variation on metric measure spaces. L Ambrosio, S Di Marino, J. Funct. Anal. 266PreprintL. Ambrosio and S. Di Marino, Equivalent definitions of BV space and of total variation on metric measure spaces, J. Funct. Anal., 266 7 (2014), 4150-4188. Preprint (2012).
On the duality between p-modulus and probability measures, preprint. L Ambrosio, S Di Marino, G Savaré, L. Ambrosio, S. Di Marino, and G. Savaré, On the duality between p-modulus and probability measures, preprint, 2013.
Density of Lipschitz functions and equivalence of weak gradients in metric measure spaces. L Ambrosio, N Gigli, G Savaré, Rev. Mat. Iberoam. 29L. Ambrosio, N. Gigli, and G. Savaré, Density of Lipschitz functions and equiv- alence of weak gradients in metric measure spaces, Rev. Mat. Iberoam., 29 3 (2013), 969-996
Well posedness of Lagrangian flows and continuity equations in metric measure spaces preprint. L Ambrosio, D Trevisan, L. Ambrosio, D. Trevisan, Well posedness of Lagrangian flows and continuity equations in metric measure spaces preprint, 2014.
D Bate, ArXiv preprint 1208.1954Structure of measures in Lipschitz differentiability spaces. D. Bate, Structure of measures in Lipschitz differentiability spaces, ArXiv preprint 1208.1954, (2012).
Nonlinear potential theory on metric spaces. A Björn, J Björn, EMS Tracts in Mathematics. 17A. Björn and J. Björn, Nonlinear potential theory on metric spaces. EMS Tracts in Mathematics, 17, 2011.
Differentiability of Lipschitz functions on metric measure spaces. J Cheeger, Geom. Funct. Anal. 9J. Cheeger, Differentiability of Lipschitz functions on metric measure spaces, Geom. Funct. Anal., 9 (1999), 428-517.
Nonsmooth differential geometry : an approach tailored for spaces with Ricci curvature bounded from below. N Gigli, PreprintN. Gigli, Nonsmooth differential geometry : an approach tailored for spaces with Ricci curvature bounded from below, Preprint, (2014).
Sobolev met Poincaré. P Hajlasz, P Koskela, Mem. Amer. Math. Soc. 154688P. Hajlasz and P. Koskela, Sobolev met Poincaré, Mem. Amer. Math. Soc., 154 (2000) 688,
Nonsmooth calculus. J Heinonen, Bull. Amer. Math. Soc. 44J. Heinonen, Nonsmooth calculus, Bull. Amer. Math. Soc., 44 (2007), 163-232.
Derivations and Alberti representations, preprint. A Schioppa, A. Schioppa, Derivations and Alberti representations, preprint, (2013).
Metric Currents and Alberti representations, preprint. , Metric Currents and Alberti representations, preprint, (2014).
Quasiconformal mappings and Sobolev spaces. P Koskela, P Macmanus, Studia Math. 131P. Koskela and P. MacManus, Quasiconformal mappings and Sobolev spaces, Studia Math., 131 (1998), 1-17.
Newtonian spaces: an extension of Sobolev spaces to metric measure spaces. N Shanmugalingam, Rev. Mat. Iberoam. 16N. Shanmugalingam, Newtonian spaces: an extension of Sobolev spaces to metric measure spaces, Rev. Mat. Iberoam., 16 (2000), 243-279.
Lipschitz algebras and derivations II. Exterior differentiation. N Weaver, J. Funct. Anal. 178N. Weaver, Lipschitz algebras and derivations II. Exterior differentiation, J. Funct. Anal., 178 1 (2000), 64-112.
|
[] |
[
"The minimum width condition for neutrino conversion in matter",
"The minimum width condition for neutrino conversion in matter"
] |
[
"C Lunardini \nItaly and INFN\nSISSA-ISAS\nvia Beirut 2-434100Trieste\n\nsezione di Trieste\nvia Valerio 234127TriesteItaly\n",
"A Yu Smirnov \nThe Abdus Salam ICTP\nStrada Costiera 1134100TriesteItaly\n\nand Institute for Nuclear Research\nRAS\nMoscowRussia\n"
] |
[
"Italy and INFN\nSISSA-ISAS\nvia Beirut 2-434100Trieste",
"sezione di Trieste\nvia Valerio 234127TriesteItaly",
"The Abdus Salam ICTP\nStrada Costiera 1134100TriesteItaly",
"and Institute for Nuclear Research\nRAS\nMoscowRussia"
] |
[] |
We find that for small vacuum mixing angle θ and low energies (s ≪ M 2 Z ) the width of matter, d 1/2 , needed to have conversion probability P ≥ 1/2 should be larger than d min = π/(2 √ 2G F tan 2θ): d 1/2 ≥ d min . Here G F is the Fermi constant, s is the total energy squared in the center of mass and M Z is the mass of the Z boson. The absolute minimum d 1/2 = d min is realized for oscillations in a uniform medium with resonance density. For all the other density distributions (monotonically varying density, castle wall profile, etc.) the required width d 1/2 is larger than d min . The width d min depends on s, and for Z-resonance channels at s ∼ M 2 Z we get that d min (s) is 20 times smaller than the low energy value. We apply the minimum width condition, d ≥ d min , to high energy neutrinos in matter as well as in neutrino background. Using this condition, we conclude that the matter effect is negligible for neutrinos propagating in AGN and GRBs environments. Significant conversion can be expected for neutrinos crossing dark matter halos of clusters of galaxies and for neutrinos produced by cosmologically distant sources and propagating in the universe.SISSA/13/2000/EP
|
10.1016/s0550-3213(00)00341-2
|
[
"https://arxiv.org/pdf/hep-ph/0002152v2.pdf"
] | 13,301,283 |
hep-ph/0002152
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cbb3a31f04bec457cbf6a4108be6fdddca7dccaa
|
The minimum width condition for neutrino conversion in matter
21 Feb 2000 29th May 2018
C Lunardini
Italy and INFN
SISSA-ISAS
via Beirut 2-434100Trieste
sezione di Trieste
via Valerio 234127TriesteItaly
A Yu Smirnov
The Abdus Salam ICTP
Strada Costiera 1134100TriesteItaly
and Institute for Nuclear Research
RAS
MoscowRussia
The minimum width condition for neutrino conversion in matter
21 Feb 2000 29th May 2018
We find that for small vacuum mixing angle θ and low energies (s ≪ M 2 Z ) the width of matter, d 1/2 , needed to have conversion probability P ≥ 1/2 should be larger than d min = π/(2 √ 2G F tan 2θ): d 1/2 ≥ d min . Here G F is the Fermi constant, s is the total energy squared in the center of mass and M Z is the mass of the Z boson. The absolute minimum d 1/2 = d min is realized for oscillations in a uniform medium with resonance density. For all the other density distributions (monotonically varying density, castle wall profile, etc.) the required width d 1/2 is larger than d min . The width d min depends on s, and for Z-resonance channels at s ∼ M 2 Z we get that d min (s) is 20 times smaller than the low energy value. We apply the minimum width condition, d ≥ d min , to high energy neutrinos in matter as well as in neutrino background. Using this condition, we conclude that the matter effect is negligible for neutrinos propagating in AGN and GRBs environments. Significant conversion can be expected for neutrinos crossing dark matter halos of clusters of galaxies and for neutrinos produced by cosmologically distant sources and propagating in the universe.SISSA/13/2000/EP
Introduction
Since the paper by Wolfenstein [1], the neutrino transformations in matter became one of the most important phenomena in neutrino physics. Neutrinos propagating in matter undergo coherent forward scattering (refraction) described at low energies by the potential
V = √ 2G F n ,(1)
where G F is the Fermi constant, and n is a function of the density and chemical composition of the medium. For the case of ν e − ν µ and ν e − ν τ conversion in matter n coincides with the electron number density, n e . Refraction can lead to an enhancement of oscillations in media with constant density, and to resonant conversion in the varying density case (MSW effect) [2,3]. For periodic, or quasi-periodic density profiles, various parametric effects can occur [4][5][6].
The MSW effect has been applied to solar neutrinos [2], and to neutrinos from supernovae [7]. Oscillations of neutrinos of various origins (solar, atmospheric, supernovae neutrinos, etc.) in the matter of the Earth have been extensively studied. Apart from resonance enhancement of oscillations, parametric effects are expected for neutrinos crossing both the mantle and the core of the Earth [8][9][10][11]. The oscillations and conversion of active neutrinos into a sterile species can be important in the Early Universe [12]. Recently, matter effects on high energy neutrino fluxes from Active Galactic Nuclei (AGN) and Gamma Ray Bursters (GRBs) have been estimated [13]. Propagation of ultra-high energy neutrinos in halos of galaxies has been considered [14].
It is intuitively clear that to have a significant matter effect a sufficiently large amount of matter is needed. Let us define the width of the medium as the integrated density along the path travelled by the neutrino in the matter:
d = n e (L)dL .(2)
This quantity is frequently named "column density" in astrophysical context. We will show that there exists a minimum value d min for the width below which it is not possible to have significant neutrino conversion. This lower bound is independent of the density profile and of the neutrino energy and mass. That allows us to make conclusions on the relevance of matter effect in various situations without knowledge of the density distribution. The paper is organized as follows: in section 2 we derive the minimum width condition for the conversion in matter between two active neutrino flavours, and check it for different density profiles. In section 3 we discuss the generalizations of the condition to the active-sterile case and to conversion induced by flavour changing neutrino-matter interactions. We also study the matter effect in the small width limits. Section 4 presents a study of the minimum width condition for high energy neutrinos both in matter and in neutrino background. Section 5 is devoted to applications of our results to neutrino propagation in AGN and GRBs environments, in dark matter halos and in the Early Universe. Conclusions and discussion follow in section 6.
The minimum width condition
In this section we consider various mechanisms of matter enhancement of neutrino flavour conversion. For each of them we work out the minimum width of the medium needed to have significant conversion probability, showing that a lower bound for the width exists and is realized in the case of uniform medium with resonance density.
The absolute minimum width
Let us consider a system of two mixed flavour states 1 , ν e and ν µ (ν τ ), characterized by vacuum mixing angle θ and mass squared difference ∆m 2 . In a uniform medium the states oscillate and the transition probability, as a function of the distance L, is given by:
P νe→νµ (L) = sin 2 2θ m sin 2 π L l m ,(3)
where θ m and l m are the mixing angle and the oscillation length in the medium:
sin 2θ m = sin 2θ [(2EV /∆m 2 − cos 2θ) 2 + sin 2 2θ] 1/2 l m = l [(2EV /∆m 2 − cos 2θ) 2 + sin 2 2θ] 1/2 .
Here l = 4πE/∆m 2 is the vacuum oscillation length and E is the neutrino energy. We assume that the vacuum mixing is small, so that vacuum oscillations effects are negligible (P vac νe→νµ ≪ 1) and a strong transition in medium, i.e. P νe→νµ = O(1), is essentially due to matter effect. For definitness, we choose the condition of significant conversion to be P νe→νµ ≥ 1 2 .
Let us consider a uniform medium with resonance density [2]:
n res e = ∆m 2 2 √ 2EG F cos 2θ .(6)
In this case the oscillation amplitude is sin 2θ m = 1, and the oscillation length equals l res = l sin 2θ = 4πE ∆m 2 sin 2θ .
According to eq. (3) the condition (5) starts to be satisfied for L = l m /4, and the corresponding width is:
d min = 1 4
n res e l res .
Inserting the expressions of n res e and l res given in (6) and (7), we get:
d min = π 2 √ 2G F tan 2θ = d 0 tan 2θ ,(9)
where
d 0 = π 2 √ 2G F ≃ 1.11 G F .(10)
We will call d 0 the refraction width. Numerically, d 0 = 2.45 · 10 32 cm −2 = 4.08 · 10 8 A cm −2 ,
where A = 6 · 10 23 is the Avogadro number 2 . The widths d min and d 0 have a simple physical interpretation. The refraction width d 0 is a universal quantity: it is determined only by the Fermi coupling constant, and does not depend on the neutrino parameters at all. Using the definition of refraction length
l 0 ≡ 2π V = 2π √ 2n e G F(12)
we can write:
d 0 n e = l 0 4 .(13)
It appears that d 0 corresponds to the distance at which the matter-induced phase difference between the flavour states equals π/2. This can be considered as the definition of refraction width, which by eq. (12) can be written in the general form:
d 0 ≡ π 2 n m V m ,(14)
where V m is the neutrino-medium potential and n m is the concentration of the relevant scatterers in the medium. The minimum width, d min , is inversely proportional to tan 2θ, which represents properties (the mixing) of the neutrino system itself. The smaller the mixing θ, the larger is the width d min needed for strong transition.
2 It can be checked that different choices of the condition (5) lead to analogous results. For instance, taking P νe→νµ ≥ 3 4 we find
d 3/4 0 = 4 3 d 0 = 2π 3 √ 2G F = 5.41 · 10 8 A cm −2 ,
and, for P νe→νµ ≥ 1 4 : d
1/4 0 = 2 3 d 0 = π 3 √ 2G F = 2.7 · 10 8 A cm −2 .
The condition (5) can be generalized. It corresponds to the case of initial state coinciding with a pure flavour state. In general one can require that the change of the probability to detect a given flavour α is larger than 1/2:
∆P ≡ P f (ν α ) − P i (ν α ) ≥ 1 2 ,(15)
where P i and P f are the initial and final probabilities. The condition (5) corresponds to P i (ν µ ) = 0, so that P f (ν α ) = P νe→νµ . Taking P i = 1/4 and P f = 3/4, we get in a similar way:
d 1/2 = 2 3 d min = π 3 √ 2G F tan 2θ .(16)
This d 1/2 is the extreme value, however for most practical situations the condition (5) is more relevant, and from here on we will use the the width d min determined in (9).
In what follows we will show that for all the other density profiles the width d 1/2 required by the condition (5) is larger than d min .
Uniform medium with density out of resonance
For n e = n res e the inequality (5) can be satisfied only if sin 2 2θ m ≥ 1 2 , which means that the density is required to be in the resonance interval: n res e (1−tan 2θ) ≤ n e ≤ n res e (1+tan 2θ). At the edges of the interval we get the width
d 1/2 = π 2G F 1 tan 2θ ± 1 ≃ √ 2d min ,(17)
which is larger than d min . For other values of the density in the resonance interval we have d min < d 1/2 < √ 2d min 3 .
Medium with varying density
In general the neutrino propagation has a character of interplay of resonance conversion and oscillations. Two conditions are needed for strong transition:
1) Resonance condition: the neutrinos should cross the layer with resonance density.
2) Adiabaticity condition: the density should vary slowly enough. This condition can be written in terms of the adiabaticity parameter γ at resonance:
γ ≪ 1 (18) γ ≡ 2E cos 2θ ∆m 2 sin 2 2θ 1 n e dn e dL .(19)
Notice that both the conditions 1) and 2) are local, and can be fulfilled for arbitrarily small widths of the medium. Clearly, they are not sufficient to assure a significant conversion, and a third condition of large enough matter width is needed. Let us consider a linear density profile with length 2L and average density equal to the resonance one, so that n max = n res e + ∆n and n min = n res e − ∆n. Denoting θ 1m and θ 2m the mixings in the initial and final points, we find that in the first order of adiabatic perturbation theory the conversion probability is given by:
P νe→νµ (L) = 1 2 − 1 2 cos 2θ 1m cos 2θ 2m − 1 2 sin 2θ 1m sin 2θ 2m cos 1 γ f (x) − 2 sin(2θ 1m − 2θ 2m )α(x) cos 1 2γ f (x) ,(20)
where
x = 2πγ L l res , f (x) = ln(x + √ 1 + x 2 ) + x √ 1 + x 2 , α(x) = x 0 dy 1 + y 2 cos 1 2γ f (y) .(21)
For γ ≪ 1 we get from eqs. (20) and (21):
d 1/2 = d min 1 + 1 − π 8 γ 2 .(22)
This expression shows that for the adiabatic case d 1/2 ≃ d min and for weak violation of adiabaticity the minimum width increases quadratically with γ. We remark that in this case the effect is dominated by oscillations with large (close to maximal) depth. The change of density gives only small corrections. Let us consider now a situation in which the resonance adiabatic conversion is the main mechanism of flavour transition. A pure conversion effect is realized if the initial neutrino state that enters the medium coincides with one of the eigenstates of the Hamiltonian in matter, and the propagation in matter is adiabatic. In this case no phase effect, and therefore no oscillations occur. Let us denote n i and n f the initial and final densities of the medium, and suppose the initial state is ν i = ν 2m = sin θ m ν e + cos θ m ν µ . The probability to find a ν µ in this state is P i (ν µ ) = cos 2 θ m (n i ). The state evolves following the change of density, so that it remains an eigenstate of the Hamiltonian, and the probability to find ν µ in the final state is P f (ν µ ) = cos 2 θ m (n f ). Since the initial state ν i does not coincide with a pure flavour state we will use the condition (15) as criterion of strong matter effect. Inserting P i and P f in (15), we get the condition for d 1/2 :
cos 2θ m (n f ) − cos 2θ m (n i ) = 1 .(23)
Taking the initial and final values of the density as n i = n res e + ∆n and n f = n res e − ∆n (∆n ≥ 0), and using the definition (4) we find that the equality (23) leads to ∆n = n res
e 1 √ 3 tan 2θ .(24)
Clearly, for a given ∆n the size of the layer, and therefore its width, depend on the gradient of the density which can be expressed in terms of the adiabaticity parameter γ, eq. (19). We get:
n e (L)dL = 2E cos 2θ ∆m 2 sin 2 2θ 1 γ dn e ,(25)
and then integrating this equation we obtain:
d = 2E cos 2θ ∆m 2 sin 2 2θ 1 γ ∆n .(26)
Finally, inserting the expressions (24) and (6) in eq. (26) we find:
d 1/2 = 4 π √ 3 1 γ d min .(27)
Let us comment on this result. As far as the adiabaticity condition is satisfied, the change of probability does not depend on the density distribution; it is a function of the initial and final densities only. If ∆n is fixed, the decrease of the width means the decrease of the length L of the layer, and therefore increase of the gradient of the density. This will lead eventually to violation of the adiabaticity condition. Thus, the minimal width corresponds to the maximal γ for which the adiabaticity is not broken substantially. For strong adiabaticity violation an increase of d 1/2 is expected, due to the increase of the minimum ∆n required by the condition (15), and therefore of the corresponding length. This can be seen if we consider the previous argument taking into account the effect of the adiabaticity breaking from the beginning. Using the Landau-Zener level crossing probability P LZ = exp(−π/2γ), which describes the transition between two eigenstates, we get, instead of (23):
(1 − 2P LZ )(cos 2θ m (n f ) − cos 2θ m (n i )) = 1 ,(28)
where we have averaged out the interference terms. Then instead of eq. (24) we get ∆n = n res
e 1 16P 2 LZ − 16P LZ + 3 tan 2θ .(29)
Finally, the condition for d 1/2 can be written as:
d 1/2 = 4 πγ 16P 2 LZ − 16P LZ + 3 d min .(30)
For γ → 0 eq. (30) gives d 1/2 → ∞, according to the fact that the density changes very slowly and therefore the width needed to have significant conversion increases. With the increase of γ the width d 1/2 decreases and has a minimum at γ ≃0.7, for which we find d 1/2 ≃ 1.5 d min . With further increase of γ (γ > ∼ 0.7) the width d 1/2 increases rapidly. According to (30) it diverges for P → 1/4, when γ → π/(4 ln 2) ≃ 1.13. This value corresponds to the case in which the adiabaticity violation is so strong that even an infinite amount of matter is not enough to satisfy the condition (15). Thus, we have found that also in this case d 1/2 > d min .
Step-like profile
As an extreme case of strong adiabaticity violation, let us consider the profile consisting of two layers of matter, having densities n 1 = n res e + ∆n and n 2 = n res e − ∆n (∆n ≥ 0), and equal lengths L 1 = L 2 = L. At the border between the layers the density has a jump of size 2∆n. We fix L = l res /8, so that d = d min . The result for the conversion probability can be computed exactly:
P step νe→νµ = s 2 sin 2 π 4s + s 2 c 2 1 − cos π 4s 2 ,(31)
where we denote the mixing parameters in the two layers as sin 2θ 2m = sin 2θ 1m ≡ s, cos 2θ 2m = − cos 2θ 1m ≡ c. In absence of the step (∆n = 0), P step νe→νµ equals 1/2, recovering the case n e = n res e = const. The probability (31) decreases monotonically as ∆n increases. Expanding in δ = (∆n/n res e tan 2θ) 2 we get 4 :
P step νe→νµ ≃ 1 2 − ( √ 2 − 1 − π 8 )δ ≃ 1 2 − 0.02 δ .(32)
According to (32), for d = d 0 we have P step νe→νµ < 1/2. This implies that, to have P step νe→νµ = 1/2 one needs d 1/2 > d min .
2.5
Castle-wall profile
The profile consists of a periodical sequence of alternate layers of matter, having two different densities n 1 and n 2 . We denote the corresponding mixing angles as θ 1m and θ 2m . In this case, a strong transformation requires certain conditions on the oscillation phases acquired by neutrinos in the layers [11]; therefore the transformation is a consequence of the specific density profile, rather than of an enhancement of the mixing. Suppose n 1 ≪ n res e and n 2 = 0, and take the width of each layer to be equal to half oscillation length, so that the oscillation phase acquired in each layer is π. It can be shown [6] that for small θ this is the condition under which the conversion probability increases most rapidly with the distance. As a function of the number N of periods (a period corresponds to two layers), the probability is given by [5,6]:
P νe→νµ (N) = sin 2 (2N∆θ) ,(33)
where ∆θ ≡ θ 1m − θ 2m = θ 1m − θ. Using the approximation 2θ 1m − 2θ ≃ sin 2θ 1m − sin 2θ, and expanding sin 2θ 1m in n 1 , we get:
d 1/2 = π 2 2 √ 2G F 1 sin 2θ ≃ πd min .(34)
Again, we find that d 1/2 ≥ d min .
Thus, for all the known mechanisms of matter enhancement of flavour transition (resonant oscillations, adiabatic conversion, parametric effects), we have found that the width d 1/2 is larger than d min , which is realized for the case of uniform medium with resonance density. In fact the constant profile with resonance density could be expected from the beginning to represent an extreme case: this profile is singled out, since it is the simplest distribution with the density fixed at the unique value n res e . It is worthwile to introduce also the total nucleon width. Let us consider a medium made of electrons, protons and neutrons with number densities n e , n p and n n . Defining the number of electrons per nucleon as Y e ≡ n e /(n n + n p ), we can write the total nucleon width that corresponds to d 0 as:
d 0N ≡ d 0 Y e .(35)
We can also introduce the total mass width d ρ :
d ρ ≡ m N d 0N = m N d 0 Y e .(36)
For electrically and isotopically neutral medium (n e = n n = n p ), eq. (35) gives:
d 0N = 2d 0 ,(37)
and numerically:
d 0N = 4.9 · 10 32 cm −2 , d ρ = 8.16 · 10 8 g · cm −2 .(38)
Generalizations
In this section we generalize the previous results to active-sterile transition and to the case of flavour-changing induced conversion. We also discuss the small width limits.
Active-sterile conversion
In this case the scattering both on electrons and on nucleons contributes to the conversion, and the effective potential for an electron neutrino in an electrically neutral medium equals
V = √ 2G F n e 1 − n n 2n p .(39)
Thus, the results for ν e − ν s transition can be obtained from those for ν e − ν µ by the replacement n e → n e (1 − n n /2n p ). For the refraction width we get immediately:
d 0 (ν e → ν s ) = d 0 2n p 2n p − n n .(40)
In particular, for an isotopically neutral medium eq. (40) gives
d 0 (ν e → ν s ) = 2d 0 .(41)
Notice that, for highly neutronized media (n n ≫ n p ) the width d 0 (ν e → ν s ) gets significantly smaller than d 0 . In this case, however, the physical situation is more properly described by the total nucleon width d 0N defined in eq. (35), since the effect is mainly due to the scattering on neutrons. We find:
d 0N (ν e → ν s ) = 2d 0 n p + n n 2n p − n n ,(42)
which gives in the limit n n → ∞:
d 0N (ν e → ν s ) = 2d 0 ,(43)
similarly to eq. (41). For the ν µ − ν s case, the potential, and consequently the width, can be obtained by replacing n e → n e (−n n /2n p ), which gives
d 0 (ν µ → ν s ) = d 0 2n p n n .(44)
For isotopically neutral medium, eq. (44) reduces to eq. (41). For highly neutronized media, the argument is analogous to the one for the ν e − ν s case, and we get the same result as in eq. (43).
Oscillations induced by flavour changing (FC) neutrino-matter interactions
In this case the neutrino masses can be zero, or negligible, and the flavour transition is a pure matter effect. The Hamiltonian of the system has the following form [1]:
H = √ 2G F 0 ǫn f ǫn f ǫ ′ n f ,(45)
where n f is the effective number density of the scatterers, and ǫ and ǫ ′ are parameters of the interaction. As follows from eq. (45), in a uniform medium the neutrinos oscillate with transition probability:
P νe→νµ (L) = 4ǫ 2 4ǫ 2 + ǫ ′2 sin 2 π L l ,(46)l = π √ 2 √ 4ǫ 2 + ǫ ′2 1 G F n f .(47)
We assume ǫ ′ < ǫ, which is needed to have a significant oscillation amplitude. Using eq. (8), we get:
d F C 1/2 = π 2 √ 2G F 1 √ 4ǫ 2 + ǫ ′2 n e n f = d 0 √ 4ǫ 2 + ǫ ′2 n e n f .(48)
Notice that the factor (n e /n f ) tan 2θ/ √ 4ǫ 2 + ǫ ′2 implies that d F C 1/2 can be significantly smaller than d min , and the oscillation effect can be observed in media of smaller width. For a FC neutrino interaction with up (or down) quarks and isotopically neutral medium we have n e /n f = 1/3, and therefore:
d F C 1/2 ≃ 1 6ǫ d 0 .(49)
Notice that there are two sources of decrease of the width: the factor 2 is given by the presence of two off diagonal terms in the Hamiltonian (45) and the factor 3 is due to the larger number of scatterers. Taking ǫ = 1, we find the value:
d F C 1/2 ≃ 1.36 · 10 8 g · cm −2 .(50)
For a density n = 4 g · cm −3 (Earth's crust), this corresponds to the distance L = 337 Km, which is comparable to the length of the present long base-line neutrino experiments: K2K (base-line 250 Km) and ICANOE (740 Km).
The small width limits
In a number of situations (see section 5) the width of the medium is smaller, or much smaller, than d min . We consider, then, the matter effect on oscillations in the limit d/d min ≪ 1. Introducing the two variables:
λ ≡ L l res /4 ρ ≡ n e n res e ,(51)
we can write the small width condition as:
d/d min = λρ ≪ 1.(52)
We focus on two important realizations of this inequality:
1) Small size of the layer and density close to resonance. As we have shown in section 2, a strong transition requires n e ≃ n res e . In case of small width this implies small size of the layer. Therefore we have λ ≪ 1 and ρ ∼ 1. In this case d/d min ≃ λ. We expand the oscillation probability (3) in λ at the lowest (nonzero) order, and find that the matter effect vanishes quadratically with d/d min :
P (λ, ρ) ≃ π 4 2 λ 2 ∼ π 4 2 d d min 2 .
(53)
2) Small density of the medium and length close to the minimum value l res /4. Another condition of strong conversion is to have the size of the layer of the order of the oscillation length. According to eq. (52), this means that the density is small. Thus, we have ρ ≪ 1 and λ ∼ 1, and therefore d/d min ≃ ρ. In order to give a phase-independent description of the matter effect, we perform an expansion in ρ of the oscillation amplitude:
sin 2 2θ m − sin 2 2θ = 2ρ sin 2 2θ cos 2θ ∼ 2 d d min sin 2 2θ cos 2θ .(54)
Unlike the previous case, the relative matter effect is linear in d/d min .
Refraction of high energy neutrinos
In this section we examine the refraction of high energy neutrinos (s > ∼ M 2 Z ), both in matter and in neutrino background.
High energy neutrinos in matter
Let us consider the propagation of high energy neutrinos in medium composed of protons, neutrons and electrons. The expressions (1) and (10) refer to the low energy range, s ≪ M 2 W , where s is the center of mass energy squared of the incoming neutrino and the target electron, and M W is the mass of the W boson. The general formulas, valid for high energies too, can be obtained by restoring the effect of the complete propagator of the W boson in the expression of the potential (an analogous argument holds for the Z boson).
Let us consider ν e − ν µ conversion. Since the refraction effects are determined by the real part of the propagator, the potential (1) is generalized as:
V = √ 2G F n e f (q 2 W ) (55) f (q 2 W ) ≡ 1 − q 2 W (1 − q 2 W ) 2 + γ 2 W ,(56)
where q 2 W ≡ q 2 /M 2 W and γ W ≡ Γ W /M W ; q and Γ W are the four momentum and the width of the W boson.
The only contribution to the potential (55) is given by the forward charged current scattering on electrons (u-channel exchange of W ), for which q 2 ≃ −s. Therefore, in-
troducing s W ≡ s/M 2 W , we have q 2 W ≃ −s W .
From the potential (55) we can find the refraction width d 0 using the definition (14). For the nucleon refraction width (35) we find:
d 0N (s W ) = 1 Y e d 0 f (−s W ) = d 0 Y e (1 + s W ) ,(57)
where we have neglected the width γ W . Eq. (57) shows that d 0N (s W ), and therefore d min (s W ), increase linearly with s W above the threshold of the W boson production. For the active-sterile conversion, one has to take into account also the neutral current interaction channel (t-channel exchange of Z), for which q 2 = 0, so that the low energy formulas (1-10) are still valid. For ν µ −ν s only neutral current processes are involved, thus the low energy result, eq. (44), holds at high energies too. In contrast, for the ν e − ν s case both charged and neutral current interactions contribute, and for an electrically neutral medium the high energy potential can be written as:
V = √ 2G F n e 1 1 + s W − n n 2n p .(58)
The second term in eq. (58) does not depend on s W , thus coinciding with the corresponding term in the low energy expression (39). The potential (58) gives the nucleon refraction width:
d 0N (s W ) = 2d 0 1 + s W (3Y e − 1) − s W (1 − Y e ) .(59)
For isotopically neutral medium (Y e = 1/2) we get:
d 0N (s W ) = 4d 0 1 + s W 1 − s W .(60)
The width d 0N (s W ) diverges for s W → 1 (see fig. 1). At high energies inelastic interactions and absorption become important: at s W ∼ 1 the imaginary part of the interaction amplitude is comparable with the real part. In fig. 1 we show the refraction width d ρ = m N d 0N for ν e − ν µ and ν e − ν s conversion and the absorption width d abs [15,16] as functions of the neutrino energy E in the rest frame of the matter. We have considered isotopically neutral medium, Y e = 1/2. The absorption width d abs is dominated by the contribution of neutrino-nucleon scattering; it decreases monotonically with the energy E. In contrast, d ρ starts to increase at s W ∼ 1, which corresponds to E = 10 6 ÷10 7 GeV, according to eq. (57). For E ≃ 10 6 GeV absorption and refraction become comparable; at higher energies, the former effect dominates: d abs < ∼ d 0 . This means that for a ν e with energy E > 10 6 GeV the conversion in matter is damped by inelastic interactions and absorption [17][18][19], therefore a strong conversion effect can not be expected.
Notice that for small mixing angle θ the minimum width d min is significantly larger than the refraction width d 0 , therefore the absorption starts to be important at lower energies. Taking, for instance, sin 2θ = 0.3 we have d min ≃ d 0 / sin 2θ ≃ 3.3d 0 , and find that d abs < ∼ d min already for E > ∼ 5 · 10 5 GeV. Figure 1: The width d ρ = m N d 0N for ν e − ν µ (dashed line) and for ν e − ν s (dotted line) channels, and the absorption width, d abs , for the electron neutrino (solid line), as functions of the neutrino energy. We have considered isotopically neutral medium, Y e = 0.5. The data for d abs are taken from ref. [15].
Let us consider now the matter effect for conversion of antineutrinos. Forν e −ν µ channel the only contributing interaction is theν e − e scattering with W exchanged in the s-channel. In this case q 2 = s, and using eq. (56) we get:
d 0N (s W ) = 1 Y e d 0 |f (s W )| .(61)
This function has a pole at s W = 1, i.e., in the W -boson resonance. The pole appears because the amplitude becomes purely imaginary in the resonance, so that the potential is zero. The width d 0N (s W ) diverges for s W → ∞, due to the 1/s W decrease of the amplitude. The function (61) has two minima:
d 0N (s min W ) = 2γ W d 0 Y e = 2γ W d 0N (s W = 0) at s min W = 1 ± γ W .(62)
Numerically d 0N (s min W ) = 0.05 d 0N (s W = 0), which shows that refraction effects are enhanced close to the W resonance. However, in this region inelastic interactions become already important.
Forν e −ν s channel the contribution of neutrino-nucleon scattering should be included, and for electrically neutral medium we find:
d 0N (s W ) = 2d 0 (1 − s W ) 2 + γ 2 W (3Y e − 1) + 2s W (1 − 2Y e ) − s 2 W (1 − Y e ) − γ 2 W (1 − Y e ) .(63)
In the case of isotopically neutral matter eq. (63) gives:
d 0N (s W ) = 4d 0 (1 − s W ) 2 + γ 2 W 1 − s 2 W − γ 2 W ,(64)
which has the value 4d 0 in the limits s W ≪ 1 and s W ≫ 1, and a pole at s W ≃ 1. Similarly to eq. (62) we find the minima:
d 0N (s min W ) = 4γ W d 0 = γ W d 0N (s W = 0) at s min W = 1 ± γ W .(65)
In fig. 2 we show the refraction width d ρ = m N d 0N forν e −ν µ andν e −ν s channels and the absorption width for the electron antineutrino, d abs [16], as functions of the neutrino energy. We have considered isotopically neutral medium. For energies outside the W boson resonance interval the main contribution to d abs is given by the neutrino-nucleon scattering; at s W ≃ 1 the effect of the resonantν e − e scattering dominates, providing the narrow peak. It appears that absorption prevails over refraction (d abs < d 0 ) for E > ∼ 6 · 10 6 GeV, corresponding to d ρ ≃ 6 · 10 7 g · cm −2 , for bothν e −ν µ andν e −ν s cases.
The effect of absorption on neutrino conversion starts to be important at lower energies: for sin 2θ = 0.3 we find that d abs < ∼ d min at E > ∼ 6 · 10 5 GeV.
High energy neutrinos in neutrino environment
Let us consider a beam of neutrinos which propagates in a background made of neutrinos of very low energies 5 . This could be the case of beams of low energy neutrinos from supernovae, or high energy neutrinos from AGN and GRBs, or neutrinos produced by the annihilation of superheavy relics, etc.. We assume that the background consists of neutrinos and antineutrinos of various flavours, with number densities n i (i = ν e ,ν e , ν µ , etc.). In the case of relativistic neutrino background we assume its isotropy.
The potential for a neutrino ν α (α = e, µ, τ ) due to neutrino-neutrino interaction can be written as: where the propagator function f (s Z ) has been defined in eq. (56). Here s Z ≡ s/M 2 Z and γ Z ≡ Γ Z /M Z ; M Z and Γ Z are the mass and width of the Z-boson. The first term in eq. (66) is due to ν α − ν α scattering with Z-boson exchange in u-channel, and the second term is the contribution from ν α −ν α annihilation.
V να (s Z ) = √ 2G F n να f (−s Z ) − nν α f (s Z ) + i=νe,νµ,ντ (n i − nī) ,(66)
Neutrino conversion in CP-asymmetric neutrino background
As a first case we consider a strongly CP-asymmetric neutrino background, and suppose n i ≫ nī, so that we can neglect the contributions of antineutrinos in (66). For simplicity, we assume equal concentrations for the neutrino species: n νe = n νµ = n ντ . In terms of the total number density of neutrinos, n ν ≡ n νe + n νµ + n ντ , the potential (66) reduces to:
V να (s Z ) = √ 2G F n ν 1 + 1 3 f (−s Z ) .(67)
The potential for the antineutrino is given by Vν α (s Z ) = −V να (−s Z ).
Let us now find the refraction width d 0 and d min for various channels. 1). For the active-sterile conversion, ν α − ν s , the potential (67) coincides with the difference of the potentials for the two species, and therefore, by eq. (14), it gives immediately the refraction width of neutrinos:
d 0 (s Z ) = d 0 1 + 1 3 f (−s Z ) −1 .(68)
The width d 0 (s Z ) is constant for s Z ≪ 1 and s Z ≫ 1: d 0 (s Z ≪ 1) ≃ 3d 0 /4 = 1.84 · 10 32 cm −2 , and d 0 (s Z ≫ 1) ≃ d 0 (see fig.3).
Let us now compare the refraction and absorption effects. The main contribution to the absorption width d abs [20] is given by the ν α − ν α and ν α − ν β (β = α) scatterings. The width d abs decreases monotonically with s Z and at s Z ≫ 1 it takes the value d abs (s Z ≫ 1) ≃ π/(2G 2 F M 2 Z ) ≃ 3.6 · 10 33 cm −2 . Due to its non-resonant behaviour, d abs is larger than d 0 for any energy of the neutrinos: at s Z ≫ 1 we find that
d 0 /d abs ≃ G F M 2 Z / √ 2 = πα W /(2 cos 2 θ W ) ≃ 0.1,
where θ W and α W = g 2 /4π are the weak mixing angle and coupling constant. Therefore, d abs is also larger than the minimum width, d min , for sin 2θ > ∼ d 0 /d abs ≃ 0.1.
2). For the conversion of an active antineutrino into a sterile species,ν α −ν s , we get the width:
d 0 (s Z ) = d 0 1 + 1 3 f (s Z ) −1 .(69)
This function (see fig.3) has a resonant behaviour with minima at s Z ≃ 1±γ Z :
d 0 (1−γ Z ) ≃ (1/6γ Z + 1) −1 d 0 ≃ d 0 /7 and d 0 (1 + γ Z ) ≃ (1/6γ Z − 1) −1 d 0 ≃ d 0 /5. Outside the Z-boson resonance d 0 (s Z ) is constant: d 0 (s Z ≪ 1) ≃ 3d 0 /4 and d 0 (s Z ≫ 1) ≃ d 0 .
In the range s Z ∼ 1 inelastic scattering and absorption become important. We evaluate the absorption width d abs for antineutrino in neutrino background using the plots in ref. [20]. For s Z < 1, the width d abs decreases with the increasing s Z ; at s Z ≃ 1 it shows the characteristic peak due to the resonant ν α −ν α scattering. For s Z > ∼ 1 the absorption width increases with s Z up to the limit d abs (s Z ≫ 1) ∼ 3 · 10 33 cm −2 , due to the contributions of ν β −ν α scatterings (β = α) and ν α −ν α interaction in the t-channel. We find that d abs > ∼ d 0 (s Z ) for s Z < ∼ 0.8 (s < ∼ 7 · 10 3 GeV 2 ) and for s Z > ∼ 2 (s > ∼ 1.7 · 10 4 GeV 2 ). Furthermore, Notice that, in contrast with the conversion in matter (see figs.1 and 2), for neutrinos and antineutrinos in neutrino environment we can have d abs > ∼ d min (s Z ) even in the high energy range, s Z > ∼ 1: in particular, we find that d abs (s Z ≫ 1) > ∼ d min (s Z ≫ 1) for sin 2θ > ∼ 0.1.
3). Let us now consider the active-active conversion, ν α − ν β . Assuming equal concentrations of neutrinos of different flavours, n νe = n νµ = n ντ , we find from eq. (67) that the difference of the potentials of the two species equals:
∆V α,β = V να − V ν β = 1 3 √ 2G F n ν f (−s α Z ) − f (−s β Z ) ,(70)
where s i Z ≡ s i /M 2 Z (i = α, β), and s i is the center of mass energy squared of the incoming and the background neutrino of the same type i. If the species ν α and ν β in the background have different energies we get that s α Z = s β Z , and therefore ∆V α,β = 0, leading to matter induced neutrino conversion even if ν α and ν β have equal concentrations. This situation is realized if the background neutrinos ν α and ν β have different masses, e.g. m να > m ν β , and are non-relativistic. Denoting by E the energy of the neutrino beam, in the rest frame of the background we have s i = 2m i E, and thus s α Z /s β Z = m να /m ν β > 1. The condition s α Z = s β Z is achieved also if one of the neutrino species is relativistic and the other is not: m να ≫ E β ≫ m ν β , where E β is the energy of ν β in the background. Assuming the isotropy of the neutrino gas, we have that s β ≃ 2E β E.
Using (70) and (14), we get the refraction width:
d 0 (s i Z ) = 3d 0 f (−s α Z ) − f (−s β Z ) −1 ≃ 3d 0 (1 + s α Z )(1 + s β Z ) (s α Z − s β Z )
.
The function (71) diverges for s i Z → ∞ and s i Z → 0. In particular, in the low energy limit, s Z ≪ 1, it reduces to d 0 (s i Z ≪ 1) ≃ 3d 0 /(2E∆m), where ∆m ≡ m να − m ν β . For the realistic case s α Z > ∼ 1 and s β Z ≪ 1, eq. (71) can be written as:
d 0 (s α Z ) ≃ 3d 0 1 + s α Z s α Z ,(72)
which approaches the minimum value 3d 0 when s α Z ≫ 1 (see fig.4). Taking the maximal realistic values for the mass and energy of the neutrino, m να = 5 eV and E = 10 22 eV we get s α Z ≃ 12 at most, so that d 0 (s i Z ) ≃ 3.5d 0 . 4). For theν α −ν β channel the effective potential equals:
∆V α,β = 1 3 √ 2G F n ν f (s α Z ) − f (s β Z ) ,(73)
and therefore we get the width:
d 0 (s i Z ) = 3d 0 f (s α Z ) − f (s β Z ) −1 .(74)
Due to the resonant character of the function f (s Z ), the width d 0 (s i Z ) has the following features (see fig.4): (i) It reaches the local minimum d 0 (s i Z ) ≃ 6γ Z d 0 ∼ d 0 /6 when one of the s i Z 's is at resonance and the other is outside the resonance: s α Z ≃ 1 and s β Z = 1 (or vice versa). (ii) The absolute minimum d 0 (s i Z ) ≃ 3γ Z d 0 is achieved when s α Z ≃ 1 + γ Z and s β Z ≃ 1 − γ Z (or vice versa). These conditions can be satisfied for certain relations between the masses of the background neutrinos. For non-relativistic background: m να /m ν β = (1 + γ Z )/(1 − γ Z ).
Notice that for s i Z discussed in (i) and (ii) the effects of inelastic scattering and absorption can be important. (iii) If one of the s i Z 's is far below the resonance and the other is far above (e.g. s α Z ≫ 1 and
s β Z ≪ 1) then d 0 (s i Z ) ∼ 3d 0 . (iv) d 0 (s i Z ) ≫ d 0 if both the s i Z '
s are far below or far above the resonance. Obviously, for strong CP-asymmetric background with nī ≫ n i the results for ν and ν channels should be interchanged.
Neutrino conversion in CP-symmetric neutrino background
Let us now consider the neutrino conversion in a CP-symmetric neutrino background, n i = nī, with n νe = n νµ = n ντ . In this case the potential (66) can be written as: It vanishes in the low energy limit s Z → 0, but it is unsuppressed at high energies, s Z > ∼ 1, leading to significant matter effect. We will consider the conversion of neutrinos; due to the CP-symmetry antineutrinos undergo analogous effects. 1). For ν α − ν s conversion from the potential (75) we find the refraction width for the neutrinos of flavour α:
V να = √ 2G F n να [f (−s Z ) − f (s Z )] .(75)d 0 (s Z ) = d 0 |f (−s Z ) − f (s Z )| −1 = d 0 (1 + s 2 Z + γ 2 Z ) 2 − 4s 2 Z 2s Z [(1 − s 2 Z ) + γ 2 Z ] .(76)
For s Z ≪ 1 it behaves as d 0 (s Z ) ≃ d 0 /(2s Z ) and for s Z ≫ 1 we have d 0 (s Z ) ≃ d 0 s Z /2 (see fig.5). The width (76) has two minima:
d 0 (s min Z ) ≃ 2γ Z d 0 at s min Z = 1 ± γ Z .(77)
Numerically, d 0 (s min Z ) = 0.055 d 0 ≃ 1.35 · 10 31 cm −2 . The absorption width d abs is dominated by ν α −ν α annihilation, with a resonance peak at s Z ∼ 1. Using the results of ref. [20] we find that d abs is larger than d 0 (s Z ) outside the Z-boson resonance, and the two quantities are comparable at s Z ∼ 1 or at s Z ≫ 1.
For sin 2θ = 0.3 we find that the minimum width d min is larger than d abs in the range 0.7 < ∼ s Z < ∼ 1.6, corresponding to 6 · 10 3 GeV 2 < ∼ s < ∼ 1.3 · 10 4 GeV 2 . At the edges of this interval the width d min takes the value d min (s Z = 0.7) ≃ d min (s Z = 1.6) ≃ 3 · 10 32 cm −2 .
2). For the ν α − ν β channel, eq. (75) gives the difference of potentials:
∆V α,β = V να − V ν β = √ 2G F n να {[f (−s α Z ) − f (s α Z )] − [f (−s β Z ) − f (s β Z )]} .(78)
The corresponding refraction width equals:
d 0 (s i Z ) = d 0 [f (−s α Z ) − f (s α Z )] − [f (−s β Z ) − f (s β Z )] −1 .(79)
We find that d 0 (s i Z ) > ∼ d 0 when both s α Z and s β Z are outside the Z-boson resonance and d 0 (s i Z ) takes its minimum values when either s α Z or s β Z is close to the Z-resonance:
If s α Z ∼ 1 + γ Z and s β Z ∼ 1 − γ Z (or vice versa) d 0 (s i Z ) has the absolute minimum d 0 (s i Z ) ≃ γ Z d 0 . For s α Z ∼ 1 and s β Z = 1 (or vice versa) we have the local minimum d 0 (s i Z ) ≃ 2γ Z d 0 .
Notice that in the realistic case s α Z ≫ s β Z ∼ 0 the width (79) reduces essentially to the one in eq. (76). At resonance, where d 0 (s i Z ) has minima, the effects of inelastic collisions and absorption are important.
Applications
The results derived in the previous sections are now applied to some physical situations of interest. After a brief discussion of well known cases, like the Earth, the Sun and supernovae, we present results for neutrinos in some new astrophysical environments. We find that significant matter induced conversion can be expected for neutrinos crossing the dark matter halos of clusters of galaxies and for neutrinos from cosmologically distant sources.
Minimum width condition and bounds on the mixing
As follows from the analysis in section 2, a significant neutrino conversion in matter requires the fulfilment of the minimum width condition 6 :
d ≥ d min = d 0 tan 2θ .(80)
This condition is independent of the density distribution, and therefore of the specific matter effect involved. Thus the knowledge of the width d allows one to conclude about the significance of the matter effect even if the density profile is unknown. This is the case of some astrophysical objects for which estimates or bounds on d can be obtained directly by observational data with no assumption on their internal structure.
In the Table 1 we show the parameters of interest of some objects, toghether with the values of the ratio
r ≡ d d 0 .(81)
For r < 1, and small mixing angle, the condition (80) can not be satisfied, thus no significant neutrino conversion is expected. Conversely, for r > ∼ 1, (80) can be fulfilled and gives the bound on the mixing:
sin 2θ > ∼ 1 r = d 0 d .(82)
Notice that our analysis holds for small mixings: sin 2θ ≪ 1. For applications we assume sin 2θ < ∼ 0.3, for which we find from eq. (82) that the minimum width condition is satisfied for r > ∼ 3.
The inequality (82) can be considered as the sensitivity limit for the mixing angle that can be achieved by studies of neutrino conversion in a layer of given width d. The real sensitivity can be however much lower than the absolute limit given by the condition (82). This is related to the fact that in the case of varying density only part of the total amount of matter effectively contributes to the conversion. Introducing the corresponding width d conv we have the condition:
sin 2θ > ∼ d 0 d conv ,(83)
instead of the (82). Let us find the expression of d conv for a medium with monotonically varying density. As discussed in section 2.3, the transition occurs mainly in the resonance layer. Using the result (24) we get:
d conv = n res e dL dn 2∆n = 2 √ 3 n res e l n tan 2θ ,(84)
where l n ≡ |( dn dL ) −1 | res n res e . Inserting the expression (84) in the condition (83), we find:
sin 2 2θ > ∼ √ 3d 0 2n res e l n .(85)
Clearly, d conv could be much smaller than the total width d of the object, so that the condition (85) on the mixing could be much stronger than (82). Notice that the bound (85) is quadratic in sin 2θ. Using the definition (19) of the adiabaticity parameter, γ, the condition (85) can be written as γ ≤ 4/(π √ 3), which corresponds to the adiabaticity condition close to its limit of validity.
Another important issue is that the maximal sensitivity for the mixing θ can be achieved for particular values of ∆m 2 /E, which depend on the specific density profile. As follows from (85), for constant (or slowly varying with the distance) l n the smallest sin 2 2θ corresponds to the largest n res e , and therefore to the largest values of ∆m 2 /E. This is the case of exponential density profile. For power-law profile, n e ∼ L −k , we get |l n | = L/k, so that sin 2 2θ ∼ L k−1 . Taking k > 1, fulfilled by practically all the realistic profiles, we find that the smallest θ is achieved for the smallest L, and consequently the highest values of n e and ∆m 2 /E.
Notice that d conv is a local property which depends on the derivative in l n . Of course, the description given by d conv is not correct when the density profile is close to the constant one, so that l n → ∞. In this case d conv can be even larger than the total width d. Thus, the correct condition on the mixing can be written as:
sin 2θ > ∼ d 0 min [d, d conv ]
.
(86)
The Sun, the Earth, the Moon and supernovae
For neutrinos crossing the Earth we consider two types of trajectories, corresponding to different values of the zenith angle θ z . For cos θ z =1 neutrinos travel along the diameter of the Earth, crossing the core and the two layers of the mantle. We get r =13.6, and therefore according to (82) we could expect significant matter conversion for sin 2 2θ > ∼ 5 · 10 −3 . However this maximal sensitivity, which would be achieved for uniform density object density (cm −3 ) size (cm) r = d/d 0 Earth: cos θ z = 1 2.6 · 10 24 1.26 · 10 9 13.6 cos θ z = 0.81 1.5 · 10 24 10 9 6.4 Sun ∼ 7 · 10 24 6.96 · 10 10 2600 Moon ∼ 10 24 3.48 · 10 8 1.4 Supernova 3 · 10 33 10 7 10 9 Universe (n ν = nν) 1.5 · 10 4 10 27 3 · 10 −2 Universe (n ν ≫ nν) ∼ 10 5 10 27 0.3 Galactic halo < ∼ 2 · 10 6 3 · 10 23 5 · 10 −2 Cluster halo < ∼ 5 · 10 7 3 · 10 24 10 AGN d ≃ 10 22 ÷ 10 23 cm −2 10 −10 ÷ 10 −9 GRB 10 10 ÷ 10 12 < 5 · 10 15 < 10 −5 Table 1: The density, the size and the matter width in units of refraction width, r = d/d 0 , for various physical objects. The values given for the densities are averaged along the trajectories of the neutrinos. We quote the number density of electrons for objects made of usual matter, and the concentration of the neutrino background for the halos and the universe. For the Earth the results are given for two trajectories with different zenith angle θ z . The results for the universe correspond to redshift z = 5 for the cases of ν α − ν s andν α −ν s in CP-symmetric and strongly CP-asymmetric neutrino background with η ν ≃ 1.
distribution, is not realized for the Earth profile. For small mixing, the difference between the densities in the core and in the mantle is larger than the resonance interval. As a result, the oscillations are resonantly enhanced either in the mantle or in the core, and only one of the two parts effectively contributes to the effect. At the same time, for certain ranges of ∆m 2 /E, different from both the resonance values in the core and in the mantle, parametric enhancement of oscillations occurs. Numerical calculations [3] give sin 2 2θ > ∼ 2 · 10 −2 as best sensitivity. For cos θ z =0.81 the trajectory is tangential to the core, and therefore it represents the path of maximal length in the mantle. In this case we find r ≃6.4 and the sensitivity limit sin 2 2θ > ∼ 2.5 · 10 −2 . Since this case realizes approximatively the optimal condition of uniform medium, we have good agreement with the results of exact calculations.
In the case of the Moon, r =1.4, and therefore a large mixing is required: sin 2 2θ > ∼ 0.5. A numerical integration of the density profile of the Sun [21] gives d ≃ 1.5·10 12 g · cm −2 . Dividing this result by d ρ = m N d 0N , with Y e = 0.7, we find r ≃2600. From the condition (82) we get then sin 2 2θ > ∼ 1.5 · 10 −7 . This bound is remarkably weaker than the one obtained from the condition (85): taking n res e ≃ 50A cm −3 and l n ≃ 0.3R ⊙ , we get sin 2 2θ ≃ 2.4 · 10 −4 , in good agreement with the results of exact computations [2,3].
For supernovae the total width of the matter above the neutrinosphere gives r ≃ 10 9 , for which the condition (82) would lead to sin 2 2θ > ∼ 10 −18 . Using the density profile n e = n 0 e (R 0 /R) 3 [22], with R 0 = 10 7 cm and n 0 e ≃ 10 34 cm −3 , from (85) we find sin 2 θ > ∼ 10 −8 , which agrees well with the results of numerical calculations [3].
As shown in the previous examples, the maximal sensitivity for sin 2 2θ, given by the total width d, can be achieved in the case of uniform medium at ∆m 2 /E corresponding to the resonance density. Such a situation is realized for neutrinos crossing the mantle of the Earth. In the case of substantial deviations from the constant density, like in the Sun or in supernovae, the sensitivity is much lower. The stronger the deviation from constant density, the smaller d conv , and therefore the lower is the sensitivity.
AGN and GRBs
Let us now turn to high energy neutrinos from Active Galactic Nuclei (AGN) and Gamma Ray Bursters (GRBs) [23][24][25][26].
In AGN, neutrinos are considered to be produced by the interaction of accelerated protons with a photon or proton background [27][28][29]. There is a hope that neutrinos from AGN with energies > ∼ 10 6 GeV could be detected by large scale underwater (ice) and EAS detectors [30].
The width of matter crossed by neutrinos in an AGN can be estimated on the basis of the existing data on the X-ray emission of these objects. The variability of the spectra suggests that the X-radiation is emitted very close to the AGN core [24]. The proton acceleration and therefore the neutrino production are supposed to happen in the same region. For this reason the width of matter crossed by neutrinos equals approximatively the one crossed by the X-radiation. For the later the experimental data [31][32][33] give the value d AGN ≃ (10 −2 ÷ 10 −1 ) A cm −2 , therefore significant neutrino conversion in AGN is excluded 7 (for a short discussion, see also ref. [34]).
A rather successful description of the origin of GRB is provided by the fireball model [35], in which neutrino production is predicted to happen in an analogous way as in AGN [13,36]. A fireball can emit protons, detected as high-energy cosmic rays on Earth, accompanied by a flux of neutrinos. The requirement that the fireball should be transparent to protons gives an estimate of the width of the object: d GRB ≤ d abs , where d abs = 10 ÷ 100 A cm −2 is the total absorption width for the protons. It is possible to evaluate the width in a different way. An estimate of the electron number density in the fireball is given in ref. [13]: n GRB ≃ (10 10 ÷ 10 12 ) cm −3 . Using this value, and taking the fireball mass in the range of star-like objects, M = (1 ÷ 10) M ⊙ , we can get the radius of the object, R GRB = 5 · (10 14 ÷ 10 15 ) cm, and then the width: d GRB = 10 ÷ 10 4 A cm −2 . In agreement with the first argument, we see, then, that also in GRBs the matter effect on neutrino conversion is negligible.
Dark matter halos
According to models [37], part of the dark matter in halos of galaxies and clusters of galaxies should consist of neutrinos 8 . Therefore neutrinos of extragalactic origin crossing the halo on the way to the Earth undergo refraction on the neutrino background. It was suggested in ref. [14] that, due to non uniform neutrino density distribution in the galactic halo, ultrahigh energy neutrinos can be resonantly converted into active and sterile species.
Following ref. [14] we consider a galactic halo composed of non-relativistic neutrinos and antineutrinos of the two species ν µ and ν τ . The electron neutrino is assumed to be lighter, and therefore less clustered, than ν µ and ν τ : n νe ≪ n i , i = ν µ , ν τ . We assume CP-symmetry of the background: n i = nī. We take the density profile [14]:
n ν (r) = n 0 ν 1 1 + (r/a) 2 ,(87)
where n ν ≡ n νµ + n ντ and a is the core radius of the halo. For galactic halos this radius is estimated to be a ≃ 10 ÷ 100 kpc. An upper bound for n 0 ν is given by the Tremaine-Gunn condition [38]: for identical fermions of mass m, the maximum number density n max allowed by the Pauli principle equals
n max = 1 6π 2 (mv esc ) 3 ,(88)
where v esc is the escape velocity of the particle:
v esc = 2GM R 1 2 ∼ 540 M 10 12 M ⊙ 1 2
According to (92) the largest values of d are achieved for objects with big mass M and small radius a; so that compact halos represent the most favourable case.
Let us now check the minimum width condition for ν µ − ν s and ν µ − ν e conversion in galactic halos. We use the refraction width d 0 (s min Z ) = 0.055d 0 ≃ 1.35 · 10 31 cm −2 given in eq. (77). This value was obtained for ν µ − ν s conversion in CP-symmetric neutrino background, and is the absolute minimum of d 0 (s Z ) (eq. (76)), realized at the Z-boson resonance, s Z ∼ 1. Notice that, under the assumption n νe ≪ n i , the result d 0 (s min Z ) holds also for ν µ − ν e conversion: due to the absence of electron neutrinos in the background, n νe ≃ 0, the potential (75) for ν e is negligible, and therefore the electron neutrino behaves as a sterile species.
From eq. (92) we see that, for typical values of M and a of a galaxy, like for instance the Milky Way (M ≃ 10 12 M ⊙ and a ≃ 10 kpc), the minimum width condition is not satisfied: d/d 0 (s min Z ) < ∼ 5 · 10 −3 . For the galaxy M87 (M ≃ 10 13 M ⊙ and a ≃ 100 kpc) we find d/d 0 (s min Z ) < ∼ 5 · 10 −2 . Taking an hypotetical very massive and compact object, with M ≃ 10 13 M ⊙ and a ≃ 10 kpc we get d/d 0 (s min Z ) < ∼ 0.2. Thus, a significant neutrino conversion effect in the galactic halo is excluded, in contrast with the result in ref. [14].
Conversely, significant matter-induced conversion can be realized in halos surrounding a cluster of galaxies. Taking the mass of a cluster as M = (10 13 ÷ 10 15 )M ⊙ and the size a ≃ 1 Mpc, we obtain from eq. (92)
d < ∼ 10 29 ÷ 3 · 10 32 cm −2 ≃ (10 −2 ÷ 20)d 0 (s min Z ) .(93)
For the maximal value , d/d 0 (s min Z ) ≃ 20, from the condition (82) we get the sensitivity to the mixing: sin 2 2θ > ∼ 3 · 10 −3 . With this value we find that the adiabaticity condition (18) is fulfilled for ∆m 2 > ∼ 4 · 10 −6 eV 2 .
The maximal sensitivity is achieved in the energy range of the Z-resonance, where also inelastic scattering and absorption are important. Indeed, in section 4.4, taking sin 2θ = 0.3, we have found that d abs ∼ d min at s ≃ 6·10 3 GeV 2 , where d min ≃ 3·10 32 cm −2 . This value coincides with the upper edge of the interval (93). For smaller sin 2θ d min is larger, and the absorption effect on oscillations becomes even more important. Notice that d 0 (s Z ) takes its minimum value at the Z-resonance: for neutrino energies outside the resonance d 0 (s Z ) is larger, so that d/d 0 (s Z ) < 1 and the minimum width condition is not satisfied.
Thus, we have found that in the halos of clusters of galaxies a significant matterinduced conversion can be achieved in the narrow interval of energies of the Z-boson resonance, where, however, the absorption and the effects of inelastic interactions are important.
Early Universe
In this section we consider neutrinos produced by cosmologically distant sources and propagating in the universe. The refraction occurs due to the interaction of the neutrinos with the particle background of the universe made of neutrinos electrons and nucleons.
The number densities of baryons, n b , and electrons, n e , are given by n b = n e = η b n γ , where η b = 10 −10 ÷ 10 −9 is the baryon asymmetry of the universe and n γ is the concentration of photons. At present time n γ = n 0 γ ≃ 400 cm −3 . We will describe the neutrino background by the total number density n ν + nν and the CP-asymmetry η ν ≡ (n ν − nν)/n γ . The value of η ν is unknown. A natural assumption would be η ν ≃ η b : in this −almost CP-symmetric− case the total concentration of neutrinos equals n ν +nν = 4n γ /11 [39]. However strong asymmetry, η ν ∼ 1, is not excluded. The upper bound η ν ∼ 10 for muon and tau neutrinos follows from the Big Bang Nucleosinthesis [40,41].
For η ν ∼ 1 the contribution of the neutrino background to refraction dominates and the interaction of neutrinos with electrons and nucleons can be neglected. It can be checked that the contributions of neutrino-electron and neutrino-nucleon scattering to the refraction width are smaller than d 0 at any time after the neutrino decoupling epoch, t dec ≃ 1 s.
In the framework of the standard Big-Bang cosmology, the number density of neutrinos in the universe decreases with the increasing time as [39]:
n(t) = n 0 t 0 t 2 t ≥ t eq n eq t eq t 3 2 t < t eq .(94)
Here t 0 ≃ 10 18 s is the age of the universe, and t eq ≃ 10 12 s is the time at which the energy densities of radiation and of matter in the universe were approximatively equal. We denote by n 0 and n eq the neutrino concentrations at t = t 0 and t = t eq respectively. The matter width d(t) crossed by the neutrinos from the time t of their production to the present one is given by the integration of the concentration (94):
d(t) = t 0 t n(τ )dτ = d U t 0 t − 1 t ≥ t eq d eq t eq t 1 2 t < t eq ,(95)
where d U ≡ t 0 n 0 is the present width of the universe and d eq = d U [t 0 /t eq − 1] is the width at t = t eq . In what follows we will focus on the case of matter domination epoch, t ≥ t eq , for which the width (95) can be expressed in terms of the redshift, z ≡ (t 0 /t) 2/3 − 1, as:
d(z) = d U (z + 1) 3 2 − 1 = d i 1 − (z + 1) − 3 2 ,(96)
where d i = tn = d U (z + 1) 3 2 is the width of the universe at the time t of production of the neutrino beam; n is the concentration of the neutrino background at t. According to eq. (96), for large enough z the width at the production time, d i , gives the dominant contribution.
Another important feature of the propagation of neutrinos from cosmological sources is the redshift of energy. The refraction width d 0 depends on the center of mass energy squared s of the incoming and the background neutrinos. As a consequence of the redshift, s Z = s Z (z), the width d 0 changes with time (with z) during the neutrino propagation: d 0 = d 0 (s Z (z)). Thus, the width of matter d should be compared with some effective (properly averaged) refraction width d 0 which in fact depends on the channel of transition. For non-relativistic background neutrinos of mass m ν we have that s increases with z as: s ≃ 2m ν E ∝ (1 + z), where E is the energy of the neutrino beam. For relativistic background with energy
E b one gets s ≃ 2E b E ∝ (1 + z) 2 .
Let us consider neutrino propagation in a strongly CP-asymmetric background, n i ≫ nī, with η ν ∼ 1. For simplicity we assume also flavour symmetry: n νe = n νµ = n ντ .
For the ν α −ν s channel the refraction width (68) increases smoothly from its low energy value, d 0 (s Z ≪ 1) = 3d 0 /4, to the high energy one, d 0 (s Z ≫ 1) = d 0 . For neutrinos produced with energy E < ∼ 10 21 eV and mass of the background neutrino m να ≃ 2 eV we get s Z < ∼ 0.5, which undergoes redshift during the neutrino propagation. Therefore we can use the low energy value of d 0 as the effective refraction width. From (96) we get the ratio r ≡ d/d 0 (s Z ) in terms of the redshift z and the asymmetry η ν :
r(z) ≡ 4d(z) 3d 0 ≃ 2.2 · 10 −2 η ν (z + 1) 3 2 .(97)
Taking the maximal allowed asymmetry, η ν ∼ 10, we find that r = 3 is reached at z ≃ 5, which corresponds to rather recent epoch. Possible sources of high energy neutrinos, the quasars, have been observed at such values of the redshift. With smaller asymmetries, η ν < ∼ 1, the minimum width condition requires much earlier epochs of neutrino production, z > ∼ 27. Notice that in general the minimum width condition, d ≥ d 0 / sin 2θ, is not sufficient to ensure a significant transition effect: as discussed in section 2, the width d 1/2 required to have conversion probability P ≥ 1/2 depends on the specific effect involved. For the adiabatic conversion in varying density (section 2.3) we have found the result d 1/2 ≃ 1.5d min (see eq. (30)). Therefore, the condition P ≥ 1/2 requires larger values of r(z), r > ∼ 4.5. From eq. (97), with η ν ∼ 10, we get r ∼ 4.5 for z ≃ 7.5.
Forν α −ν s channel the refraction width d 0 , eq. (69), has a resonance character with absolute minimum d 0 (s min Z ) ≃ d 0 /7 at s min Z ∼ 1. Outside the resonance it takes the values d 0 (s Z ≪ 1) = 3d 0 /4 and d 0 (s Z ≫ 1) = d 0 . For neutrino energy E < ∼ 10 21 eV at production and mass of the background neutrino m να < ∼ 1 eV we get s Z < 1, so that we can use the low energy value of the refraction width and the result for the ratio r coincides with that in eq. (97). For neutrinos produced at time t = t i with extremely high energies, E ∼ 10 21 ÷ 10 22 eV and mass of the background neutrino m να ≃ 1 ÷ 3 eV we get s Z ≥ 1 at the production time, so that, due to redshift, s Z (z) will cross the resonance interval, in which d 0 has minimum. This, however, does not lead to larger values of the ratio r(z), since the time interval ∆t during which s Z remains in the resonance range is short: ∆t/t i ≃ 1.5γ Z ≃ 0.04.
The matter width collected in the interval ∆t is d res ≃ d∆t/t i ≃ 1.5γ Z d, and the ratio r(z) for the resonance epoch equals r(z) = d res /d 0 (s min Z ) ≃ d/4d 0 . That is even smaller than r outside the resonance, eq. (97); therefore the result (97) holds also in this case.
Thus, in the extreme condition of very large ν −ν asymmetry and production epoch at z > ∼ 5 the matter width crossed by neutrinos satisfies the minimum width condition. Let us comment on the character of the matter-induced neutrino conversion in this case. After its production, the neutrino beam experiences a monotonically decreasing density. Moreover, the energy of neutrinos decreases due to redshift, which also influences the mixing. Taking into account the decrease with time of both the neutrino energy and concentration, the adiabaticity condition (18)- (19) can be generalized as:
γ(t) = γ 0 t t 0 = γ 0 (z + 1) − 3 2 ≪ 1 (98) γ 0 ≃ 8 3V 0 t 0 tan 2 2θ ,(99)
where V 0 is the present value of the neutrino-medium potential. For the present epoch we get γ 0 ≃ 10 2 / tan 2 2θ, so that the adiabaticity is strongly broken. For tan 2 2θ < ∼ 0.1 the adiabaticity can be realized at t/t 0 < 10 −3 , or z > 10 2 . Thus, for z < 10 2 the conversion takes a character of oscillations in the production epoch. A detailed study of the dynamics of neutrino conversion in the universe will be given elsewhere [42].
For the flavour channel ν α −ν β the refraction effect appears at high energies, s Z ∼ 1, as a consequence of different energies of the background neutrinos ν α and ν β . The absolute minimum of the refraction width (71), d min 0 (s i Z ) = 3d 0 , gives a value of r(z) which is 4 times smaller than that in eq. (97). Correspondingly, even for the extreme conditions of η ν ≃ 10 and z = 5 we get r ≃ 0.8. That is, significant matter effect is excluded for neutrinos from the oldest observed sources. The value r > ∼ 3 can be reached for z > ∼ 14. Notice that, according to eq. (71), the refraction disappears for low energies, s i Z ≪ 1, and the redshift spoils the conditions of absolute minimum of d 0 , even if it is realized in certain epoch. Therefore, larger z are required to have significant conversion effect.
Similar conclusions can be obtained forν α −ν β channel, where the refraction width (74) has the local minimum d 0 (s i Z ) ≃ 6γ Z d 0 due to Z-resonance. This minimum is realized however during the resonance epoch ∆t. Taking the corresponding matter width d res ≃ 1.5γ Z d, we get r(z) = d(z)/4d 0 , which is even smaller than in the ν α − ν β case.
Let us consider CP-symmetric neutrino background, n i = nī, with the assumption of flavour symmetry: n νe = n νµ = n ντ . As discussed in section 4.4, unsuppressed refraction effect appears for active-sterile neutrino channels at high energies, s Z ∼ 1, where the propagator corrections become important.
For ν α − ν s (and similarly forν α −ν s , due to CP-symmetry) the refraction width (76) has the absolute minimum d 0 (s min Z ) ≃ 2γ Z d 0 , realized at s min Z = 1 ± γ Z , eq. (77). Assuming that the neutrinos are produced just before the resonance epoch, we find that the width collected during the interval ∆t equals d res ≃ d∆t/t i ≃ 1.5γ Z d, where d is given by eq. (96), with d U = 2n 0 γ t 0 /11 ≃ 7 · 10 29 cm −2 . For the ratio r(z) we find:
r(z) = 3d(z) 4d 0 = 2 · 10 −3 (z + 1) 3 2 ,(100)
which is significantly smaller than r(z) for ν α − ν s channel in CP-asymmetric background, eq. (97). With the maximal redshift z ≃ 5 eq. (100) gives r ≃ 0.03, which excludes matter effect for neutrinos from the most distant observable sources. The result (100) holds also for the ν α − ν β (ν α −ν β ) channel, since the refraction width (79) has analogous behaviour to the one for the ν α − ν s case, eq. (76), with the same local minimum d 0 (s i Z ) ≃ 2γ Z d 0 at resonance.
In conclusion, we have found that the matter effect for neutrinos crossing the universe is mainly due to the neutrino background. For neutrinos from observable sources (z ≃ 5), significant conversion effect can be achieved in the ν α − ν s andν α −ν s channels, if the background has strong CP-asymmetry, close to the maximum value, η ν ≃ 10. The matter effect for the other conversion channels and for the CP-symmetric case is suppressed as a consequence the redshift.
6 Conclusions 1). Matter effects can lead to strong flavour transition even for small vacuum mixing angle: θ ≪ 1. This however requires a sufficiently large amount (width) of matter crossed by neutrinos: the minimum width condition, d ≥ d min , should be satisfied, where d min = π/(2 √ 2G F tan 2θ) = d 0 / tan 2θ, for low neutrino energies, s ≪ M 2 Z , and conversion probability P ≥ 1/2. The absolute minimum d min is realized for uniform medium with resonance density n res e .
2). We have shown that for all the other realistic situations the required width, d 1/2 , is larger than d min . In particular, we have found that d 1/2 /d min = 1 + (1 − π/8)γ 2 for oscillations in medium with slowly varying density (γ ≪ 1); d 1/2 /d min ≥ 1.5 for conversion in medium with varying density; d 1/2 /d min = π for castle wall profile.
3). We discussed the minimum width condition for high energy neutrinos. For s > ∼ M 2 W the minimum width d min becomes function of s, due to the propagator effect: d min = d min (s). The function d min (s) depends on the channel of interactions: in the case of W (or Z) exchange in the s-channel d min (s) decreases in the resonance region by a factor ∼ 20 with respect to the low energy value: d min (0)/d min (M 2 W ) ∼ 20. In this region, however, the inelastic interactions become important, damping the flavour conversion. 4). As a case of special interest we have studied the refraction of high energy neutrinos in neutrino background, which can be important for propagation of cosmic neutrinos in galaxies and intergalactic space. Again we find that the ν α −ν α annihilation channel gives enhancement of refraction at s ≃ M 2 Z , so that d min can be ∼ 1/2γ Z ∼ 20 times smaller than that at low energies. In the case of flavour channels the refraction can appear as the result of the difference of masses of the background neutrinos, even if the concentrations of the various flavours are equal. 5). The minimum width condition allows one to conclude on the relevance of the matter effect without knowledge of the density profile, once the width d is known. In some astrophysical situations the total width on the way of neutrinos can be estimated rather precisely (e.g. by spectroscopical methods) although the density distribution is unknown. Significant matter effect in excluded if d < d 0 , or d < d min if the mixing angle is known. 6). From practical point of view, a study of the matter effects should start with the check of the minimum width condition, d ≥ d min . This condition is necessary but not sufficient for strong conversion effect. If it is fulfilled the ratio d/d 0 allows one to estimate the minimal mixing angle for which significant transition is possible: sin 2θ > d 0 /d. This condition gives an absolute lower bound on θ, which can be achieved for the case of uniform medium with resonance density. In other words, given the width d of the medium, the highest sensitivity to the mixing angle is achieved if the matter is distributed uniformly and the density coincides with the resonance value for a given neutrino energy. For media with non-uniform matter distribution the sensitivity to θ is lower. The stronger the deviation from the constant density, the lower the sensitivity. 7). We applied the minimum width condition to neutrinos in AGN and GRBs environment. For AGN the width d can be estimated by the experimental data on the X-ray spectrum, without assumptions on the the density profile. We got d/d 0 < ∼ 10 −10 for radially moving neutrinos, strongly excluding matter effects. In the case of GRBs the width d can be evaluated under the assumption that the object is transparent to protons. We found d/d 0 < ∼ 10 −5 . Therefore, no significant conversion is expected either. 8). For neutrinos crossing the halos of galaxies and clusters of galaxies the matter effect is given by the interaction of neutrinos with the neutrino component of the halo. We have found that for galactic halos the minimum width condition is not satisfied: the result d(halo)/d 0 ≤ 0.1 excludes any significant conversion effect. For halos of clusters of galaxies we got d(halo)/d 0 > ∼ 10, and the minimum width condition can be satisfied for large enough mixing: sin 2θ > ∼ 0.1. 9). We have considered the refraction of neutrinos from cosmologically distant sources, interacting with the neutrino background of the universe. Significant active-sterile conversion can be expected in case of large ν −ν asymmetry. We have found that for η ν = O(1) the condition d(universe)/d 0 > ∼ 1 can be achieved for neutrinos from sources, galaxies or quasars, with redshift z > ∼ 5. The effect on detected neutrino fluxes from these sources could be a distortion of the energy spectrum.
Figure 2 :
2The width d ρ = m N d 0N forν e −ν µ (dashed line) and forν e −ν s (dotted line) channels, and the absorption width, d abs , for the electron antineutrino (solid line), as functions of the neutrino energy. We have considered isotopically neutral medium, Y e = 0.5. The data for d abs are taken from ref.[16].
d 0 (s Z = 0. 8 )
8≃ 10 32 cm −2 and d 0 (s Z = 2) ≃ 4 · 10 32 cm −2 . Taking sin 2θ = 0.3 we get that d abs > ∼ d min for s Z < ∼ 0.7 and s Z > ∼ 2.4; d min (s Z = 0.7) ≃ 3 · 10 32 cm −2 and d min (s Z = 2.4) ≃ 10 33 cm −2 .
Figure 3 :
3The dependence of the refraction width d 0 for ν α − ν s (dashed line) andν α −ν s (solid line) channels on s Z in strongly CP-asymmetric background, n i ≫ nī. Equal concentrations are assumed for the various flavours.
Figure 4 :
4The dependence of the refraction width d 0 for ν α − ν β (dashed line) andν α −ν β (solid line) channels on s Z in strongly CP-asymmetric background, n i ≫ nī. Equal concentrations are assumed for the various flavours. We have considered s α Z ≫ s β Z ∼ 0.
Figure 5 :
5The dependence of the refraction width d 0 for ν α − ν s andν α −ν s channels on s Z in CP-symmetric neutrino background.
The arguments remain the same for three neutrinos.
It can be checked that the width d 1/2 in eq. (17) is larger than d min for small mixing: sin 2θ < ∼ 0.3. We will use this condition as criterion of smallness of the mixing.
This approximation proves to be very good (relative error ≤ 0.5%) for 0 ≤ δ ≤ 1, i.e. for n 1 and n 2 in the resonance interval.
We will not consider the conversion of neutrinos in the background itself, which can significantly affect the flavour content of the background.
This condition refers to the requirement of conversion probability larger than 1/2, eq. (5). In some circumstances, however, even a small effect, with conversion probability P ≪ 1/2 can be important.
In the present discussion we have considered radial propagation of neutrinos from the inner to the external regions of the object. We have not considered neutrinos travelling through the core of the AGN. In this case a significant matter-induced conversion could occur, however neutrinos crossing the core are supposed to be a small fraction of the total neutrino flux produced.
In what follows, we will not consider the heavy particles present in the halos, because their number density is much smaller than the one of neutrinos, although they provide the largest part of the mass of the halo. Furthermore, the amplitude of the forward scattering of neutrinos on neutrinos and on the heavy particles of dark matter are comparable, or the former is even larger.
AcknowledgmentsThe authors wish to thank O.L.G. Peres for useful comments.Here M and R are the total mass and radius of the galaxy. In what follows we will take R ≃ 3a, corresponding to the radius at which the density reduces at one tenth of its core value, n ν (3a) = n 0 ν /10. From (88) and (89) we get:The integration of the profile (87) gives the matter width:Inserting the expression (90) for n max in (91), we find the upper bound for the width:
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[] |
[
"Analytical Bounce Solution in a Dissipative Quantum Tunneling",
"Analytical Bounce Solution in a Dissipative Quantum Tunneling"
] |
[
"D K Park [email protected] \nDepartment of Physics\nKyungnam University\n631-701MasanKorea\n"
] |
[
"Department of Physics\nKyungnam University\n631-701MasanKorea"
] |
[] |
The analytical bounce solution is derived in terms of the polygamma function in the Caldeira-Leggett's dissipative quantum tunneling model. The classical action for the bounce solution lies between the upper and lower bounds in the full range of α, where α is a dissipation coefficient. The bounce peak point increases from 1 to 4/3 with increase of α. In spite of various nice features we have shown that the solution we have derived is not exact one by making use of the zero mode argument in the linearized fluctuation equation.However, our solution can be a starting point for approximate computation of the prefactor in this model. *
|
10.1142/s0217732304015828
|
[
"https://export.arxiv.org/pdf/hep-th/0312132v1.pdf"
] | 8,650,824 |
hep-th/0312132
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3d1e60ea7ec0ee58d0ab7b99c1e3696e7dd84fdc
|
Analytical Bounce Solution in a Dissipative Quantum Tunneling
12 Dec 2003
D K Park [email protected]
Department of Physics
Kyungnam University
631-701MasanKorea
Analytical Bounce Solution in a Dissipative Quantum Tunneling
12 Dec 2003
The analytical bounce solution is derived in terms of the polygamma function in the Caldeira-Leggett's dissipative quantum tunneling model. The classical action for the bounce solution lies between the upper and lower bounds in the full range of α, where α is a dissipation coefficient. The bounce peak point increases from 1 to 4/3 with increase of α. In spite of various nice features we have shown that the solution we have derived is not exact one by making use of the zero mode argument in the linearized fluctuation equation.However, our solution can be a starting point for approximate computation of the prefactor in this model. *
How to describe a dissipation at the level of quantum mechanics is a long-standing puzzle in physics. Upon our knowledge Feynman and Vernon(FV) [1] firstly described it as an interaction between a system of interest and its enviroments. Especially, they developed a formalism to investigate the quantum dissipation systematically by introducing an influence functional. The influence functional is an extremely important quantity in the sense that it contains all quantum effects of the enviroments and, thus makes it possible to describe the quantum dissipation in terms of only system's coordinates.
Based on FV formalism Caldeira and Leggett(CL) considered in Ref. [2] a quantum tunneling model interacting with the harmonic oscillator enviroments. Introducing a potential which does not have a true vacuum, CL examined the effect of dissipation on quantum tunneling within a semi-classical approximation which was developed several decades ago [3][4][5].
The full application of the semi-classical application, however, is very difficult in this setup because the exact bounce solution is still unknown. In fact, it seems to be extremely hard (or may be impossible) to derive a bounce solution in an analytic form due to the non-local term the CL model involves. Without the analytic bounce it is impossible to exploit the power of the semi-classical approximation maximally.
In Ref. [2] CL obtained the upper and lower bounds of the classical action in terms of the dissipation coefficient α without the explicit bounce solution. These two bounds are monotonically increasing function with respect to α, which indicates that the presence of the dissipation causes the decrease of the tunneling probability within an exponential approximation. The similar physical setup with a double-well potential was examined by a canonical method in Ref. [6,7]. The authors in Ref. [6,7], however, claimed that the dissipation may enhance the tunneling probability. In order to reconcile these two apparently discrepant results we think the prefactor should be examined in the semi-classical side.
However, it is very difficult to compute the prefactor without the analytic bounce solution.
Thus we may need a bounce solution in the analytic form for the computation of the prefactor although it is not exact. It is main purpose of this letter to derive an analytic bounce which interpolates between the exact no-damping solution and the strong-damping solution.
We start with a dimensionless action [
2] 1 σ[z] = ∞ −∞ du dz du 2 + (z 2 − z 3 ) + α π ∞ −∞ du ∞ −∞ du ′ z(u) − z(u ′ ) u − u ′ 2(1)
which yields an equation of motion as
z = z − 3 2 z 2 + 2α π ∞ −∞ du ′ z(u) − z(u ′ ) (u − u ′ ) 2 .(2)
We know the exact no-damping (α → 0) solution
z 0 (u) = sech 2 u 2(3)
and strong-damping (α → ∞) solution
z ∞ (u) = 4 3[1 + u 2α 2 ] .(4)
Thus, the real bounce solution should interpolates between (3) and (4) with increasing the dissipation coefficient α.
The bounce solution we obtained in this letter is
z(u) = 2 π A α 4B 2 α ψ ′ C α + B α + iu 2B α + ψ ′ C α + B α − iu 2B α(5)
where ψ ′ is an usual polygamma function. The A α , B α , and C α are α-dependent but uindependent constants which obey the following equations:
µψ ′′ (µ) + 2ψ ′ (µ) = 2 √ 2π 3 B 2 α A α (6) µ 2 ψ ′′ (µ) + 2µψ ′ (µ) = 4 √ 2πα 3 B α A α (3 − 8α 2 )µ 3 ψ ′′ (µ) + 6(1 − 4α 2 )µ 2 ψ ′ (µ) + 16α 2 µ + 4α 2 = 0
where ψ ′′ is polygamma function and µ = C α /B α . From the first and second equations of Eq.(6) one can show easily C α = 2α. One can show also numerically that A α and B α are monotonically increasing functions with respect to α.
Before we explain how Eq.(5) is derived, we would like to show its nice features. Fig. 1 shows the α-dependence of its classical solution which is represented by the red line. The green and blue lines represent the classical actions for the strong-damping and no-damping solutions respectively. Two black lines are upper and lower bounds of the classical action which were derived explicitly in Ref. [2]. Fig. 1 indicates that the classical action for our bounce solution (5) lies between the upper and lower bounds in the full range of α. Fig. 1 also implies that our solution interpolates between the no-damping and the strong-damping
solutions. The small difference between the red and green lines at the large α region indicates that our solution is an approximate analytical solution. This fact will be proven later by making use of the zero mode argument in the fluctuation equation level. Fig. 2 shows the α-dependence of z(0), which means the peak point of the bounce. As CL predicted in Ref.
[2], the peak point increases from 1 to 4/3 with increase of α.
Now, let me explain how the bounce solution (5) is derived. Taking a Fourier transform
from z(u) toz(ω), one can change the equation of motion (2) in terms ofz(ω) as following
(ω 2 + 2α|ω| + 1)z(ω) = 3 2 √ 2π ∞ −∞ dΩz(ω − Ω)z(Ω).(7)
The explicit form ofz 0 (ω) andz ∞ (ω) which are Fourier transform of z 0 (u) and z ∞ (u) becomẽ
z 0 (ω) = 2 √ 2π ω sinh πω (8) z ∞ (u) = 4 √ 2πα 3 e −2α|ω| .
As expectedz 0 (ω) andz ∞ (ω) are solutions of Eq. (7) without second term and first term in the left-handed side of Eq. (7) respectively. These are properties of no-damping and strongdamping in ω-space. It is worthwhile noting the ω-dependence ofz 0 (ω) andz ∞ (ω). Although
z(ω) = A α ω sinh B α ω e −Cα|ω| .(9)
Then the α-dependent constants A α , B α , and C α should satisfy
A 0 = 2 √ 2π, B 0 = π, C 0 = 0, A ∞ /B ∞ = 4 √
2πα/3 and C ∞ = 2α in order forz(ω) to interpolate between the no-damping and the strong-damping solutions.
Now, we will insert the ansatz (9) into Eq. (7) to extract an information on A α , B α , and C α . The most difficult term we need to compute is the following convolution term
I α (ω) = ∞ −∞ dΩz(ω − Ω)z(Ω) = A 2 α 2 Ĩ 1,α (ω) − ω 2Ĩ 2,α (ω)(10)whereĨ 1,α (ω) = ∞ 0 dy y 2 cosh B α y − cosh B α ω e − Cα 2 (|ω+y|+|ω−y|)(11)I 2,α (ω) = ∞ 0 dy 1 cosh B α y − cosh B α ω e − Cα 2 (|ω+y|+|ω−y|) .
Note thatĨ 1,α (ω) andĨ 2,α (ω) are even function with respect to ω. Thus we can assume ω > 0 without loss of generality. With this assumptionĨ 1,α (ω) andĨ 2,α (ω) are expressed as
followingĨ 1,α (ω) = 2 3 π B α 2 − ω 2 2 ω sinh B α ω + 1 B 3 α ∂ 2 µ K − e −µz 0 ∂ 2 µ K µ=0(12)I 2,α (ω) = − ω sinh B α ω e −Cαω + 1 B α K − e −µz 0 K µ=0 where K ≡ ∞ z 0 dz e −µz cosh z − cosh z 0(13)
and z 0 ≡ B α ω and µ ≡ C α /B α . When we compute the first term ofĨ 1,α (ω) we used the property of the Lerch function in Ref. [8].
Now, the remaining problem for the computation ofĨ α (ω) is to compute K which have an infrared-like infinity as a field theory terminology. In order to take into account the infinity carefully we take a change of variable x = e z , which makes K to be
K = 2(x 0 − x −1 0 ) −1 ∞ x 0 +ǫ dx x −µ x − x 0 − x −µ x − x −1 0(14)
where x 0 = e z 0 . In Eq.(14) we introduced an infinitesimal parameter ǫ explicitly for the regularization of the infrared-like infinity. Performing the integration in Eq.(14) one can express K as a difference of two hypergeometric functions. Making use of the relation between the hypergeometric and digamma function [9] the final expression of K becomes
K = e −µz 0 sinh z 0 (z 0 − ln ǫ) + e 2µz 0 ln(e 2z 0 − 1) + (e 2µz 0 − 1)ψ(µ) − ∞ n=1 (µ) n n! ψ(n + 1)(1 − e −2z 0 ) n(15)
where (µ) n = µ(µ + 1) · · · (µ + n − 1) and ψ is a digamma function. Note that K has a logarithmic divergence as expected. Using Eq.(15) it is straightforward to computeĨ 2,α (ω) which reduces tõ
I 2,α (ω) = e −Cαω B α sinh B α ω z 0 + e 2µz 0 − 1 ψ(µ) + ln e 2z 0 − 1 (16) − ∞ n=1 (µ) n n! ψ(n + 1) 1 − e −2z 0 n .
Note that the infinity term in Eq.(15) disappears in Eq.(16) because of the exact cancellation.
This exact cancellation also takes place inĨ 1,α (ω). After tedious calculation the final form ofĨ α (ω) reduces tõ
I α (ω)z −1 (ω) = 1 3 A α ω 2 + π B α 2 (17) + A α 2B 2 α e 2µz 0 − 1 z 0 ψ ′′ (µ) + 2 e 2µz 0 + 1 ψ ′ (µ) − 4 3 z 2 0 + 2π 2 3 +2 ∞ n=1 (µ) n n! [ψ(n + µ) − ψ(µ)] ψ(n + 1) 1 − e −2z 0 n −2 ∞ n=1 ψ(n + 1) n 1 − e −2z 0 n − ∞ n=2 (µ) n n! [ψ ′ (n + µ) − ψ ′ (µ)] + [ψ(n + µ) − ψ(µ)] 2 ψ(n + 1) (1 − e −2z 0 ) n z 0 +2 ∞ n=2 γ + ψ(n) n ψ(n + 1) (1 − e −2z 0 ) n z 0
where γ is an Euler's constant. In order forz(ω) to be an exact solution the right-handed side of Eq.(17) should be equal to 2 √ 2π(ω 2 + 2αω + 1)/3. To extract an information on A α , B α , and C α we assume this equality. Repeating to take a ω → 0 limit and subsequently to differentiate the right-handed side of Eq.(17) three times, we can derive Eq. (6). Taking an inverse-Fourier transform toz(ω), we can derive Eq.(5).
Although our bounce solution has many nice features as discussed before, it is not an exact solution unfortunately except α = 0. For α = 0 case we can show analytically A 0 = 2 √ 2π, B 0 = π and C 0 = 0 by solving Eq.(6) in the µ → 0 limit. This means our solution exactly coincides with no-damping solution at α → 0 limit. However, for the nonzero α we can show that our bounce solution is an approximate one by using the zero mode of the linearized fluctuation equation as following.
Note that the CL model has a time-translational symmetry in spite of the presence of the non-local term. Although one can show this fact simply from the equation of motion (2), we would like to show it at the fluctuation level for a later use. Inserting z(u) = z cl (u) + η(u)
into Eq.(2) one can construct easily the linearized fluctuation equation
−c 1η + c 2 (1 − 3z cl )η + c 3 2α π ∞ −∞ du ′ η(u) − η(u ′ ) (u − u ′ ) 2 = λη(18)η ≡ dz(u) du = 2 π iA α 8B 3 α ψ ′′ C α + B α + iu 2B α − ψ ′′ C α + B α − iu 2B α(19)
should be a zero mode. Inserting Eq.(19) into the non-local term in Eq.(18) one can show
2α π ∞ −∞ du ′ η(u) − η(u ′ ) (u − u ′ ) 2 = 2 π 3iαA α 4B 4 α ζ 4, C α + B α − iu 2B α − ζ 4, C α + B α + iu 2B α(20)
where ζ(p, q) is a Riemann Zeta function defined as ζ(p, q) = ∞ k=0 1/(q + k) p . Using Eq.(20) the left-handed side of the fluctuation can be plotted numerically. When α is small, the left-handed side of the fluctuation equation is plotted in Fig. 3 in terms of u for various α, which indicates that η is not exact zero mode although approximately it is. Fig. 3 also shows how η can be a zero mode in the α → 0 limit. Fig. 4 is a plot of the left-handed side of Eq.(18) when α is large. This figure also shows how our solution goes to the strong-damping solution in the α → ∞ limit. But we should comment that our bounce solution does not seem to exactly coincide with the strong-damping solution in the α → ∞ limit. That is why there is a small difference in Fig.1 between the classical action for our solution and its lower bound at the large α regime.
In this letter we have derived the analytic bounce solution in the CL model. The classical action for our solution lies between the upper and lower bounds in the full range of α.
Although it has many nice features, we have shown that it is not an exact solution except α = 0 case. However, using this approximate solution, one may be able to compute the prefactor approximately. We guess this prefactor may be important factor to reconcile the discrepancy between the semi-classical method and the canonocal method. We hope to visit this issue in the near future. Our analytic bounce solution might be extended to the non-Ohmic dissipation case. The explicit result will be discussed elsewhere in detail.
ω
is a dimensionless quantity, it should have a dimension if we go back to the CL's original model which has a dimension. Thus, from the dimensional consideration in Eq.(7) we can conjecture thatz(ω) should have a same dimension with ω in the original theory except the strong-damping solution, which should be dimensionless. From this point of view we can understand the ω-dependence ofz 0 (ω) andz ∞ (ω). Thus one can take an ansatz
where λ is an eigenvalue of the fluctuation equation and the constants c 1 , c 2 , and c 3 are introduced for convenience. If c 1 = c 2 = 1 and c 3 = 0, Eq.(18) is a fluctuation equation around the no-damping solution. In this case it is trivial to show that dz 0 (u)/du is a zero mode. If c 1 = 0 and c 2 = c 3 = 1, Eq.(18) corresponds to a fluctuation around the strongdamping solution. In this case also one can show directly that dz ∞ (u)/du is a zero mode. This means that the CL model has a time-translational symmetry in spite of the presence of the dissipation. Now, let us consider the full fluctuation equation, i.e. c 1 = c 2 = c 3 = 1. If our bounce solution (5) is an exact one, by same reason
FIGURESFIG. 1 .
1Plot of α-dependence of classical actions for our bounce solution (5) (red line), no-damping solution (3) (blue line) and strong-damping solution (4) (green line). Two black lines are upper and lower bounds of the classical action which were derived by CL in Ref.[2] without an explicit bounce solution. This figure indicates that the classical action for our solution lies between the upper and lower limits in the full range of α. FIG. 2. Plot of α-dependence of the bounce peak point. As CL have argued in Ref.[2], the bounce peak point increases from 1 to 4/3 with increase of α.FIG. 3. The plot of the left-handed side of Eq.(18) for various small α. The difference from zero indicates that our solution is not an exact one. The increase of the peak point and the wide-spreading shape of peak with increase of α denotes that our bounce solution (5) goes away from the exact one when α becomes larger in the small α regime. FIG. 4. Plot of the left-handed side of Eq.(18) for various large α. The decrease of the peak point with increase of α means that our solution approaches to the exact one when α becomes larger in the large α regime.
Fig. 1
In this letter we will follow the same conventions with Ref.[2]
The Theory of a General Quantum System Interacting with a Linear Dissipative System. R P Feynman, F Vernon, Ann. Phys. 24118R. P. Feynman and F. Vernon, The Theory of a General Quantum System Interacting with a Linear Dissipative System, Ann. Phys. 24 (1963) 118.
Quantum Tunnelng in a Dissipative System. A O Caldeira, A J Leggett, Ann. Phys. 149374A. O. Caldeira and A. J. Leggett, Quantum Tunnelng in a Dissipative System, Ann. Phys. 149 (1983) 374.
Fate of the false vacuum: Semiclassical theory. S Coleman, Phys. Rev. 152929S. Coleman, Fate of the false vacuum: Semiclassical theory, Phys. Rev. D15 (1977) 2929.
Fate of the false vacuum. II. First quantum corrections. C G Callan, S Coleman, Phys. Rev. 161762C. G. Callan and S. Coleman, Fate of the false vacuum. II. First quantum corrections, Phys. Rev. D16 (1977) 1762.
R Rajaraman, Solitons and Instantons. North-Holland, New YorkR. Rajaraman, Solitons and Instantons (North-Holland, New York, 1982).
Canonical Formulation of Quantum Tunneling with Dissipation. K Fujikawa, S Iso, M Sasaki, H Suzuki, Phys. Rev. Lett. 681093K. Fujikawa, S. Iso, M. Sasaki and H. Suzuki, Canonical Formulation of Quantum Tun- neling with Dissipation, Phys. Rev. Lett. 68 (1992) 1093.
Quantum tunneling with dissipation: Possible enhancement by dissipative interactions. K Fujikawa, S Iso, M Sasaki, H Suzuki, Phys. Rev. 4610295K. Fujikawa, S. Iso, M. Sasaki and H. Suzuki, Quantum tunneling with dissipation: Pos- sible enhancement by dissipative interactions, Phys. Rev. B46 (1992) 10295.
I S Gradshteyn, I M Ryzhik, Table of Integrals, Series, and Products. San DiegoAcademic Press10296th editionI. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 6th edition (Academic Press, San Diego, 2000) p375 and p1029.
M Abramowitz, I A Stegun, Handbook of Mathematical Functions. New YorkDover Publications559M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover Publica- tions, New York, 1972) p559.
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"DECOMPOSITION OF HOMOGENEOUS POLYNOMIALS WITH LOW RANK",
"DECOMPOSITION OF HOMOGENEOUS POLYNOMIALS WITH LOW RANK"
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[
"Edoardo Ballico ",
"Alessandra Bernardi "
] |
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Let F be a homogeneous polynomial of degree d in m + 1 variables defined over an algebraically closed field of characteristic 0 and suppose that F belongs to the s-th secant variety of the d-uple Veronese embedding of P m into P ( m+d d )−1 but that its minimal decomposition as a sum of d-th powers of linearWe show that if s+r ≤ 2d+1 then such a decomposition of F can be split in two parts: one of them is made by linear forms that can be written using only two variables, the other part is uniquely determined once one has fixed the first part. We also obtain a uniqueness theorem for the minimal decomposition of F if r is at most d and a mild condition is satisfied.∈ Z (see[19], [17], [18], [9] also for further results on this normal form). If the polynomial is not general, very few things are known.1991 Mathematics Subject Classification. 15A21, 15A69, 14N15.
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10.1007/s00209-011-0907-6
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"https://arxiv.org/pdf/1003.5157v3.pdf"
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1003.5157
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0a483021c6aa54260b3fa94bc51738625c2d266e
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DECOMPOSITION OF HOMOGENEOUS POLYNOMIALS WITH LOW RANK
3 Jun 2011
Edoardo Ballico
Alessandra Bernardi
DECOMPOSITION OF HOMOGENEOUS POLYNOMIALS WITH LOW RANK
3 Jun 2011
Let F be a homogeneous polynomial of degree d in m + 1 variables defined over an algebraically closed field of characteristic 0 and suppose that F belongs to the s-th secant variety of the d-uple Veronese embedding of P m into P ( m+d d )−1 but that its minimal decomposition as a sum of d-th powers of linearWe show that if s+r ≤ 2d+1 then such a decomposition of F can be split in two parts: one of them is made by linear forms that can be written using only two variables, the other part is uniquely determined once one has fixed the first part. We also obtain a uniqueness theorem for the minimal decomposition of F if r is at most d and a mild condition is satisfied.∈ Z (see[19], [17], [18], [9] also for further results on this normal form). If the polynomial is not general, very few things are known.1991 Mathematics Subject Classification. 15A21, 15A69, 14N15.
Introduction
The decomposition of a homogeneous polynomial that combines a minimum number of terms and that involves a minimum number of variables is a problem arising from classical Algebraic Geometry ( [1], [14]), Computational Complexity ( [15]) and Signal Processing ( [20]). Any statement on homogeneous polynomials can be translated in an equivalent statement on symmetric tensors. In fact, if we indicate with V a vector space of dimension m+ 1 defined over a field K of characteristic 0, and with V * its dual space, then, for any positive integer d, there is an obvious identification between the vector space of symmetric tensors S d V * ⊂ (V * ) ⊗d and the space of homogeneous polynomials K[x 0 , . . . , x m ] d of degree d defined over K. In this paper we will always work with an algebraically closed field K of characteristic 0. The requirement that a form (or a symmetric tensor) involves a minimum number of terms is a quite recent and very interesting problem coming from applications. Given a form F ∈ K[x 0 , . . . , x m ] d (or a symmetric tensor T ∈ S d V * ), the minimum positive integer r for which there exist linear forms L 1 , . . . , L r ∈ K[x 0 , . . . , x m ] 1 (vectors v 1 , . . . , v r ∈ V * respectively) such that (1) F = L d 1 + · · · + L d r , (T = v ⊗d 1 + · · · + v ⊗d r ) is called the symmetric rank sr(F ) of F (sr(T ) of T respectively). Computations of the symmetric rank for a given form (or a given symmetric tensor) are studied in [11], [3], [4] and [2]. First of all we focus our attention on those particular decompositions of a form F ∈ K[x 0 , . . . , x m ] d (or T ∈ S d V * ) of the type (1) with r = sr(F ) (r = sr(T ) respectively). What about the possible uniqueness of the decomposition of such a form F (T respectively)? A general form, for example, can have a unique decomposition as in (1) only if 1 n+1 n+d n Let X m,d ⊂ P N , with m ≥ 1, d ≥ 2 and N := m+d m − 1, be the classical Veronese variety obtained as the image of the d-uple Veronese embedding ν d : P m → P N . The s-th secant variety σ s (X m,d ) of the Veronese variety X m,d is the Zariski closure in P N of the union of all linear spans P 1 , . . . , P s with P 1 , . . . , P s ∈ X m,d . For any point P ∈ P N , we indicate with sbr(P ) = s the minimum integer s such that P ∈ σ s (X m,d ). This integer is called the symmetric border rank of P . By a famous theorem of J. Alexander and A. Hirschowitz all integers dim(σ s (X m,d )) are known ( [1], [8], [5]). Since P m ≃ P(K[x 0 , . . . , x m ] 1 ) ≃ P(V * ), the generic element belonging to σ s (X m,d ) is the projective class of a form (a symmetric tensor) of type (1). Unfortunately, for a given P ∈ P N , we only have the inequality sbr(P ) ≤ sr(P ). For the forms F for which the decomposition (1) is not unique, it makes sense to study those different decompositions. There is a uniqueness theorem for general points with prescribed non-maximal symmetric border rank s using the notion of (s − 1)weakly non-defectivity introduced by C. Ciliberto and L. Chiantini ([7], [10], Proposition 1.5). In this paper we are interested in those particular decompositions of a given F ∈ K[x 0 , . . . , x m ] d of the type (1) with r = sr(F ) and sbr(F ) < sr(F ) (T ∈ S d V * respectively). In many applications one would like to reduce the number of variables, at least for a part of the data. For such a particular choice of F , is it possible to find linear forms L 1 , L 2 , M 1 , . . . , M t ∈ K[x 0 , . . . , x m ] 1 and a binary form Q ∈ K[L 1 , L 2 ] d , such that a given polynomial F ∈ K[x 0 , . . . , x m ] d can be written as F = Q + M d 1 + · · · + M d t ? (On normal forms of homogeneous polynomials see also [16], [13], [14].) The main result of this paper is the following.
Theorem 1. Let P ∈ P N with N = m+d d − 1. Suppose that:
sbr(P ) < sr(P ) and sbr(P ) + sr(P ) ≤ 2d + 1.
Let S ⊂ X m,d be a 0-dimensional reduced subscheme that realizes the symmetric rank of P , and let Z ⊂ X m,d be a smoothable 0-dimensional non-reduced subscheme such that P ∈ Z and deg Z ≤ sbr(P ). Let also C d ⊂ X m,d be the unique rational normal curve that intersects S ∪ Z in degree at least d + 2. Then, for all points P ∈ P N as above we have that:
S = S 1 ⊔ S 2 , Z = Z 1 ⊔ S 2 , where S 1 = S ∩ C d , Z 1 = Z ∩ C d and S 2 = (S ∩ Z) \ S 1 .
Moreover deg(Z) = sbr(P ) and the scheme S 2 is unique.
The existence of such a scheme Z was known from [3] and [6] (see Remark 1). The assumption " sbr(P ) + sr(P ) ≤ 2d + 1 " is sharp (see Example 1).
In the language of polynomials, Theorem 1 can be rephrased as follows. An analogous corollary can be stated for symmetric tensors.
Corollary 2. Let T ∈ S d V * be such that sbr(T ) + sr(T ) ≤ 2d + 1 and sbr(T ) < sr(T ). Then there are an integer t ≥ 0, vectors v 1 , v 2 , w 1 , . . . , w t ∈ S 1 V * , and a symmetric tensor v ∈ S d ( v 1 , v 2 ) such that T = v + w ⊗d 1 + · · · + w ⊗d t , t ≤ sbr(T ) + sr(T ) − d − 2, and sr(T ) = sr(v) + t. Moreover t, w 1 , . . . , w t and v 1 , v 2 are uniquely determined by T .
Observe that the variables L 1 , L 2 in Corollary 1 and the vectors v 1 , v 2 in Corollary 2 correspond to the line ℓ ⊂ P m such that C d := ν d (ℓ) is the rational normal curve introduced in Theorem 1. Moreover the integer t in Corollaries 1 and 2 is ♯(S 2 ) where S 2 is as in Theorem 1.
The decompositions Q = R d 1 + · · · + R d r ′ with R i ∈ K[L 1 , L 2 ] 1 , are not unique (analogously the decompositions v = u ⊗d 1 + · · · + u ⊗d r ′ with u i ∈ v 1 , v 2 )
, but one of them may be found using Sylvester's algorithm or any of the available algorithms ( [11], [16], [3]). Unfortunately, given F as in Corollary 1 (T as in Corollary 2 respectively) we do not have any explicit algorithm to find
M 1 , . . . , M t ∈ K[x 0 , . . . , x m ] d and hence Q ∈ K[L 1 , L 2 ] d (w 1 , . . . , w t ∈ S 1 V * and v ∈ S d ( v 1 , v 2 ) respectively).
Using Theorem 1 and a related lemma (Lemma 3) it is also possible to address the question on the uniqueness of the decomposition (1).
Theorem 2. Assume d ≥ 5. Fix a finite set B ⊂ P m such that ρ := ♯(B) ≤ d and no subset of it with cardinality ⌊(d + 1)/2⌋ is collinear. Fix P ∈ ν d (B) such that P / ∈ E for any E ν d (B). Then sr(P ) = sbr(P ) = ρ and ν d (B) is the only 0-dimensional scheme Z ⊂ X m,d such that deg(Z) ≤ ρ and P ∈ Z .
Unfortunately, for a given P ∈ P N that satisfies the hypothesis of Theorem 2 we are not able to give explicitly the set B. Knowing the uniqueness of a decomposition is very interesting both from the applications and the pure mathematical point of view, but very few results are known. Theorem 2 is an extension of [6] with an additional assumption. It is worth noting that without some additional assumption [6], Theorem 1.2.6, cannot be extended (e.g., it is sharp when m = 1). We give an example showing that if m = 2, then Theorem 2 is sharp (see Example 2), even taking B in linearly general position.
Preliminaries
In this section we prove two auxiliary lemmas that will be crucial in the proof of the main result of this paper. Theorems 1 and 2 are well-known if m = 1 since Sylvester. Hence we may assume that m ≥ 2.
Definition 1. We say that a smoothable 0-dimensional scheme Z ⊂ X m,d computes the symmetric border rank sbr(P ) of P ∈ P N if deg(Z) = sbr(P ) and P ∈ Z . A reduced 0-dimensional scheme S ⊂ X m,d computes the symmetric rank sr(P ) of P ∈ P N if ♯(S) = sr(P ) and P ∈ S .
By the definition of symmetric rank, if S computes sr(P ), then P / ∈ S ′ for any reduced 0-dimensional scheme S ′ ⊂ X m,d with deg(S ′ ) < deg(S). Hence S is linearly independent. Lemma 1. Fix any P ∈ P r and two 0-dimensional subschemes A, B of P r such that A = B, P ∈ A , P ∈ B , P / ∈ A ′ for any A ′ A and P / ∈ B ′ for any B ′ B. Then h 1 (P r , I A∪B (1)) > 0.
Proof. Since A and B are 0-dimensional, h 1 (P r , I A∪B (1)) ≥ max{h 1 (P r , I A (1)), h 1 (P r , I B (1))}. Thus we may assume h 1 (P r , I A (1)) = h 1 (P r , I B (1)) = 0, i.e. dim( A ) = deg(A) − 1 and dim( B ) = deg(B) − 1. Set D := A ∩ B (scheme-theoretic intersection). Thus deg(A ∪ B) = deg(A) + deg(B) − deg(D). Since D ⊆ A and A is linearly independent, we have dim( D ) = deg(D) − 1. Since h 1 (P r , I A∪B (1)) > 0 if and only if dim( A ∪ B ) ≤ deg(A ∪ B) − 2, we get h 1 (P r , I A∪B (1)) > 0 if and only if D A ∩ B . Since A = B, then D A. Hence P / ∈ D . Since P ∈ A ∩ B , we are done.
The next observation shows the existence of the scheme Z ⊂ X m,d that computes the symmetric border rank of a point P ∈ P N that satisfies the conditions of Theorem 1.
Remark 1.
Fix integers m ≥ 1, d ≥ 2 and P ∈ P N such that sbr(P ) ≤ d + 1. By [6], Lemma 2.1.5, or [3], Proposition 11, there is a smoothable 0-dimensional scheme E ⊂ X m,d such that deg(E) ≤ sbr(P ) and P ∈ E . Moreover, sbr(P ) is the minimal of the degrees of any such smoothable scheme E.
In the statement of Theorem 1 we claimed the existence of a unique rational normal curve
C d ⊂ X m,d such that deg((S ∪ Z) ∩ C d ) ≥ d + 2.
This will be a consequence of the following lemma where the line ℓ ⊂ P m and the scheme W ⊂ P m will be used in the proof of Theorem 1 with ν d (ℓ) = C d , while as ν d (W ) we will take several different schemes associated to S ∪ Z.
Lemma 2. Fix an integer x ≥ 1. Let W ⊂ P m , m ≥ 2, be a 0-dimensional scheme of degree deg(W ) ≤ 2x + 1 and such that h 1 (P m , I W (x)) > 0. Then there is a unique line ℓ ⊂ P m such that deg(ℓ ∩ W ) ≥ x + 2 and deg(W ∩ ℓ) = x + 1 + h 1 (P m , I W (x)).
Proof. For the existence of the line ℓ ⊂ P m see [3], Lemma 34. Since deg(W ) ≤ 2x + 1 and since the scheme-theoretic intersection of two different lines has length at most one and deg(W ) ≤ 2x + 2, there is no line R = ℓ such that deg(R ∩ W ) ≥ x + 2. Thus ℓ is unique. We prove the formula deg(W ∩ ℓ) = x + 1 + h 1 (I W (x)) by induction on m.
First assume m = 2. In this case ℓ is a Cartier divisor of P m . Hence the residual scheme Res ℓ (W ) of W with respect to ℓ has degree deg(Res ℓ (W )) = deg(W ) − deg(W ∩ ℓ). The exact sequence that defines the residual scheme Res ℓ (W ) is:
(2) 0 → I Res ℓ (W ) (x − 1) → I W (x) → I W ∩ℓ,ℓ (x) → 0. Since dim(Res ℓ (W )) ≤ dim(W ) ≤ 0 and x − 1 ≥ −2, we have h 2 (P m , I Res ℓ (W ) (x − 1)) = 0. Since deg(W ∩ ℓ) ≥ x + 1, we have h 0 (ℓ, I W ∩ℓ (x)) = 0. Since deg(Res ℓ (W )) = deg(W ) − deg(W ∩ ℓ) ≤ x,
we obviously have h 1 (P m , I Resℓ(W ) (x − 1)) = 0 (this is also a particular case of [3], Lemma 34). Thus the cohomology exact sequence of (2) gives h 1 (P m ,
I W (x)) = deg(W ∩ ℓ) − x − 1.
This proves the lemma for m = 2. Now assume m ≥ 3 and that the result is true for P m−1 . Take a general hyperplane H ⊂ P m containing ℓ and set W ′ := W ∩ ℓ. The inductive assumption gives h 1 (H, I W ′ (x)) = deg(W ′ ∩ ℓ) − x − 1. Since deg(Res H (W )) ≤ x − 1, we get, as above, h 1 (P m , I ResH(W ) (x − 1)) = 0. Consider now the analogue exact sequence of (2) using H instead of ℓ:
0 → I Res H (W ) (x − 1) → I W (x) → I W ∩H,H (x) → 0.
Since W ∩ ℓ = W ′ ∩ ℓ, we get, as above, that h 1 (P m , I W (x)) = deg(W ∩ ℓ) − x − 1.
The proofs
In this section we prove Theorems 1 and 2.
Proof of Theorem 1. The existence of the smoothable scheme Z ⊂ X m,d that computes sbr(P ) is assured by Remark 1. Any such smoothable scheme has degree sbr(P ) (Remark 1). Let S (resp. Z) be the only subset (resp. subscheme) of P m such that S = ν d (S) (resp. Z = ν d (Z)). By hypothesis ♯(S) = sr(P ) and deg(Z) = sbr(P ). Set W := S ∪ Z and W := ν d (W ). We have deg(W ) = sr(P ) + sbr(P ) ≤ 2d + 1. Let T be a minimal subscheme of Z such that P ∈ T . Since deg(T ) ≤ deg(Z) < deg(S), we have T = S. Lemma 1 applied to r := N , A := T and B := S gives h 1 (I T ∪S (1)) > 0. Thus h 1 (I W (1)) > 0. Thus dim( W ) ≤ deg(W) − 2. Since deg(W) ≤ deg(Z) + deg(S) = sbr(P ) + sr(P ) ≤ 2d + 1 and h 1 (I W (1)) = h 1 (P m , I W (d)), there is a unique line ℓ ⊂ P m whose image C d := ν d (ℓ) in X m,d contains a subscheme of W with length at least d + 2 (Lemma 2). Since C d = C d ∩ X m,d (scheme-theoretic intersection), we have
W ∩ C d = ν d (W ∩ ℓ), Z ∩ C d = ν d (Z ∩ ℓ) and S ∩ C d = ν d (S ∩ ℓ).
(a) Let S 1 , S 2 ⊂ S be as defined in the statement and set S 3 := S \ (S 1 ∪ S 2 ). Let S 3 ⊂ P m be the only subset such that S 3 = ν d (S 3 ). Set W ′ := W \ S 3 and W ′ := ν d (W ′ ) = W \ S 3 . Notice that W ′ is well-defined, because each point of S 3 is a connected component of the scheme W .
In this step we prove S 3 = ∅, i.e. S 3 = ∅. Assume that this is not the case and that ♯(S 3 ) > 0. Lemma 2 gives h 1 (P m , I W ∩ℓ (d)) = h 1 (P N , I W (1)) and h 0 (
I W (1)) = h 0 (I C d ∩W (1)) − deg(W) + deg(W ∩ C d ). Hence we get dim( W ) = dim( W ′ ) + ♯(S 3 ). Now, by definition, we have that S ∩ W ′ = S 1 ∪ S 2 , W = W ′ ⊔ S 3 and Z ∪ S 1 ∪ S 2 = W ′ . Grass- mann's formula gives dim( W ′ ∩ S ) = dim( W ′ )+dim( S )−dim( W ′ ∪S ) = dim( S )−♯(S 3 ). Since S is linearly independent, we have dim( S 1 ∪ S 2 ) = dim( S ) − ♯(S 3 ). Hence dim( S 1 ∪ S 2 ) = dim( W ′ ∩ S ); since S 1 ∪ S 2 ⊆ W ′ ∩ S we get S 1 ∪ S 2 = W ′ ∩ S . Since P ∈ Z ∩ S ⊆ W ′ ∩ S = S 1 ∪ S 2 ,
we get that P ∈ S 1 ∪ S 2 . Since we supposed that S ⊂ X m,d is a set computing the symmetric rank of P , it is absurd that P belongs to the span of a proper subset of S, then necessarily ♯(S 3 ) = 0, that is equivalent to the fact that S 3 = ∅. Thus in this step we have just proved S = S 1 ⊔ S 2 .
In steps (b), (c) and (d) we will prove Z = (Z ∩ C d ) ⊔ S 2 in a very similar way (using Z instead of S). In each of these steps we take a subscheme W 2 ⊂ W such that S ⊂ W 2 , W 2 ∩ ℓ = W ∩ ℓ and W 2 ∪ Z = W . Then we play with Lemma 2. In steps (b) (resp. (c), resp. (d)) we call
W 2 = W ′′ (resp. W 2 = W Q , resp. W 2 = W 1 ). Since deg(ν d (Z)) ≤ d + 1, the scheme ν d (Z) is linearly independent.
(b) Let Z 4 ⊂ P n be the union of the connected components of Z which do not intersect ℓ ∪ S 2 . Here we prove Z 4 = ∅. Set W ′′ := W \ Z 4 . The scheme W ′′ is well-defined, because Z 4 is a union of some of the connected components of W . Lemma 2 gives dim( ν d (W ) ) = dim( ν d (W ′′ ) ) + deg(Z 4 ). Since W = W ′′ ∪ Z, Grassmann's formula gives dim(
ν d (W ′′ ∪ Z) ) = dim( ν d (W ′′ ) ) + dim( ν d (Z) ) − dim( ν d (W ′′ ) ∩ ν d (Z) ). Thus dim( ν d (Z) ) = dim( ν d (W ′′ ) ∩ ν d (Z) ) + deg(Z 4 ). Since ν d (Z) is linearly independent and Z = (Z ∩ W ′′ ) ⊔ Z 4 , we get dim( ν d (Z) ) = dim( ν d (Z∩W ′′ ) )+deg(Z 4 ). Thus dim( ν d (W ′′ ) ∩ ν d (Z) ) = dim( ν d (Z∩W ′′ ) ). Since ν d (W ′′ ∩ Z) ⊆ ν d (W ′′ ) ∩ ν d (Z) , deg( ν d (W ′′ ) ∩ ν d (Z) ) = dim( ν d (W ′′ ∩ Z) ) + 1, and ν d (W ′′ ) is linearly independent, then the linear space ν d (W ′′ ) ∩ ν d (Z) is spanned by ν d (W ′′ ∩Z).
Since S ⊆ W ′′ and P ∈ ν d (Z) ∩ ν d (S) , we have P ∈ ν d (W ′′ ∩ Z) . Since ν d (Z) computes sbr(P ), we get W ′′ ∩ Z = Z, i.e. Z 4 = ∅.
(c) Here we prove that each point of S 2 is a connected component of Z. Fix Q ∈ S 2 and call Z Q the connected component of Z such that ( (d) To conclude that Z = (Z ∩ ℓ) ⊔ S 2 it is sufficient to prove that every connected component of Z whose support is a point of ℓ is contained in ℓ. Set η := deg(Z ∩ ℓ) and call µ the sum of the degrees of the connected components of Z whose support is contained in ℓ.
Z Q ) red = {Q}. Set Z[Q] := (Z \ Z Q ) ∪ {Q} and W Q := (W \ Z Q ) ∪ {Q}. Since Z Q is a connected component of W , the schemes Z[Q] and W Q are well-defined. Assume Z Q = {Q}, i.e. W Q = W , i.e. Z[Q] = Z. Since W Q ∩ ℓ = W ∩ ℓ, Lemma 2 gives dim( ν d (W ) ) − dim( ν d (W Q ) ) = deg(Z Q ) − 1 > 0. Since ν d (Z) is linearly in- dependent, we have dim( ν d (Z) ) = dim( ν d (Z[Q]) ) + deg(Z Q ) − 1. Grassmann's formula gives dim( ν d (Z[Q]) ) = dim( ν d (W Q ) ∩ ν d (Z) ). Since ν d (Z[Q]) ⊆ ν d (W Q ) ∩ ν d (Z) and Z[Q] is linearly independent, we get ν d (Z[Q]) = ν d (W Q ) ∩ ν d (Z) . Since Q ∈ S 2 ⊆ S, we have S ⊂ W Q . Thus P ∈ ν d (W Q ) . Thus P ∈ ν d (Z) ∩ ν d (W Q ) = ν d (Z[Q]) .Set W 1 := (W ∩ ℓ) ∪ S 2 . Notice that deg(W 1 ) = deg(W ) + η − µ. Lemma 2 gives dim( ν d (W 1 ) ) = dim( ν d (W ) ) + η − µ. Since W = W 1 ∪ Z, Grassmann's formula gives dim( ν d (W 1 ∪ Z) ) = dim( ν d (W 1 ) ) + dim( ν d (Z) ) − dim( ν d (W 1 ) ∩ ν d (Z) ). Thus dim( ν d (Z) ) = dim( ν d (W 1 ) ∩ ν d (Z) ) + µ − η. Notice that Z ∩ W 1 = (Z ∩ ℓ) ⊔ S 2 , i.e. deg(Z ∩ W 1 ) = deg(Z) − η + µ. Since ν d (Z) is linearly independent, we get dim( ν d (Z) ) = dim( ν d (Z ∩ W 1 ) ) + µ − η. Thus dim( ν d (W 1 ) ∩ ν d (Z) ) = dim( ν d (Z ∩ W 1 ) ), i.e. ν d (W 1 ) ∩ ν d (Z) is spanned by ν d (W 1 ∩ Z). Since S ⊂ W 1 and P ∈ ν d (Z) ∩ ν d (S) , we have P ∈ ν d (W 1 ∩ Z) . Since ZZ = Z 1 ⊔ S 2 , Z ′ = Z ′ 1 ⊔ S ′ 2 and S = S ′ 1 ⊔ S ′ 2 with Z 1 = Z ∩ ℓ, Z ′ 1 = Z ′ ∩ ℓ ′ , S 1 = S ∩ ℓ and S ′ 1 = S 1 ∩ ℓ ′ . Now sbr(P ) = deg(Z 1 ) + ♯(S 2 ) = deg(Z ′ 1 ) + ♯(S ′ 2 ), sr(P ) = deg(S 1 ) + ♯(S 2 ) = deg(S ′ 1 ) + ♯(S ′ 2 ), deg(S 1 ) > deg(Z 1 ), deg(S 1 ) + deg(Z 1 ) ≥ d + 2 and deg(S ′ 1 ) + deg(Z ′ 1 ) ≥ d + 2.
Since ℓ ′ = ℓ, at most one of the points of S 1 may be contained in ℓ ′ and at most one of the points of S ′ 1 may be contained in ℓ.
Thus deg(S ′ 1 ) − 1 ≤ ♯(S 2 ) and deg(S 1 ) − 1 ≤ ♯(S ′ 2 ). Since deg(S 1 ) + deg(Z 1 ) + 2(♯(S 2 )) = deg(S ′ 1 ) + deg(Z ′ 1 ) + 2(♯(S ′ 2 )) ≤ 2d + 1, deg(S 1 )+deg(Z 1 ) ≥ d+2 and deg(S ′ 1 )+deg(Z ′ 1 ) ≥ d+2, we get 2(♯(S 2 )) ≤ d−1 and 2(♯(S ′ 2 )) ≤ d−1. Since deg(S 1 ) + deg(Z 1 ) ≥ d + 2 and deg(S 1 ) > deg(Z 1 ), we have deg(S 1 ) ≥ (d + 3)/2. Hence deg(S 1 ) − 1 ≥ (d + 1)/2 > (d − 1)/2 ≥ ♯(S ′ 2 )
, contradiction. Thus all pairs (Z ′ , S) give the same line ℓ. Now assume S ′ = S. Call ℓ ′′ the line associated to the pair (Z, S ′ ). The part of Theorem 1 proved in the previous steps gives that ℓ is the only line containing an unreduced connected component of Z. Thus ℓ ′′ = ℓ. Since we proved that the lines associated to (Z ′ , S ′ ) and (Z, S ′ ) are the same, we are done.
(f) Here we prove the uniqueness of S 2 . Take any pair (Z ′ , S ′ ) with ν d (Z ′ ) computing sbr(P ) and ν d (S ′ ) computing sr(P ). By step (e) the same line ℓ is associated to any pair (Z ′′ , S ′′ ) as above. Hence the set S ′ 2 := S ′ \ (S ′ ∩ ℓ) associated to the pair (Z, S ′ ) is the union of the connected components of Z not contained in ℓ. Thus S ′ 2 = S \ S ∩ ℓ = S 2 . We apply the part of Theorem 1 proved in steps (a), (b), (c) and (d) to the pair (Z ′ , S). We get that S \ S ∩ ℓ is the union of the connected components of Z ′ not contained in ℓ. Applying the same part of Theorem 1 to the pair (Z ′ , S ′ ) we get S ′ \ S ′ ∩ ℓ = S \ S ∩ ℓ, concluding the proof of the uniqueness of S 2 .
The following example shows that the assumption " sbr(P ) + sr(P ) ≤ 2d + 1 " in Theorem 1 is sharp. Example 1. Fix integers m ≥ 2 and d ≥ 4. Let C ⊂ P m be a smooth conic. Let Z ⊂ C be any unreduced degree 3 subscheme. Set Z := ν d (Z). Since d ≥ 2, then Z is linearly independent. Since Z is curvilinear, it has only finitely many degree 2 subschemes. Thus the plane Z contains only finitely many lines spanned by a degree 2 subscheme of Z. Fix any P ∈ Z not contained in one of these lines. Remark 1 gives sbr(P ) = 3. The proof of [3], Theorem 4, gives sr(P ) = 2d − 1 and the existence of a set S ⊂ C such that ♯(S) = 2d − 1 , S ∩ Z = ∅ and ν d (S) computes sr(P ). We have sbr(P ) + sr(P ) = 2d + 2.
Lemma 3. Fix P ∈ P N such that ρ := sbr(P ) = sr(P ) ≤ d. Let Ψ be the set of all 0-dimensional schemes A ⊂ P m such that deg(A) = ρ and P ∈ ν d (A) . Assume ♯(Ψ) ≥ 2. Fix any A ∈ Ψ. Then there is a line ℓ ⊂ P m such that deg(ℓ ∩ A) ≥ (d + 2)/2.
Proof. Since sr(P ) = ρ and ♯(Ψ) ≥ 2, there is B ∈ Ψ such that B = A and at least one among the schemes A and B is reduced. Since deg(A ∪ B) ≤ 2d + 1 and h 1 (P m , I A∪B (d)) > 0, there is a line ℓ ⊂ P m such that deg((A ∪ B) ∩ ℓ) ≥ d + 2. We may repeat verbatim the proof of Theorem Proof of Theorem 2. Since sbr(P ) ≤ ρ ≤ d, the border rank is the minimal degree of a smoothable 0-dimensional scheme A ⊂ X m,d such that P ∈ A (Remark 1). Thus it is sufficient to prove the last assertion. Assume the existence of a 0-dimensional scheme Z ⊂ X m,d such that z := deg(Z) ≤ ρ and P ∈ Z . If z = ρ we also assume Z = ν d (B). Taking z minimal, we may also assume z ≤ sbr(P ). Let Z ⊂ P m be the only scheme such that ν d (Z) = Z. If z < ρ we apply a small part of the proof of Theorem 1 to the pair (Z, ν d (B)) (we just use or reprove that deg((Z ∪ B) ∩ ℓ) ≥ d + 2 and that deg(B ∩ ℓ) = deg(Z ∩ ℓ) + ρ − z ≥ deg(Z ∩ ℓ)). We get a contradiction: indeed B ∩ ℓ must have degree ≥ (d + 1)/2, contradiction. If z = ρ, then we use Lemma 3.
Example 2. Assume m = 2 and d ≥ 4. Let C ⊂ P 2 be a smooth conic. Fix sets S, S ′ ⊂ C such that ♯(S) = ♯(S ′ ) = d + 1 and S ∩ S ′ = ∅. Since no 3 points of C are collinear, the sets S, S ′ and S ∪ S ′ are in linearly general position. Since h 0 (C, O C (d)) = 2d + 1 and C is projectively normal, we have h 1 (P 2 , I S (d)) = h 1 (P 2 , I S ′ (d)) = 0 and h 1 (P 2 , I S∪S ′ (d)) = 1. Thus ν d (S) and ν d (S ′ ) are linearly independent and ν d (S) ∩ ν d (S ′ ) is a unique point. Call P this point. Obviously sr(P ) ≤ d + 1. In order to get the example claimed in the Introduction after the statement of
Corollary 1 .
1Let F ∈ K[x 0 , . . . , x m ] d be such that sbr(F ) + sr(F ) ≤ 2d + 1 and sbr(F ) < sr(F ). Then there are an integer t ≥ 0, linear forms L 1 , L 2 , M 1 , . . . , M t ∈ K[x 0 , . . . , x m ] 1 , and a form Q ∈ K[L 1 , L 2 ] d such that F = Q+M d 1 +· · ·+M d t , t ≤ sbr(F )+sr(F )−d−2, and sr(F ) = sr(Q)+t. Moreover t, M 1 , . . . , M t and the linear span of L 1 , L 2 are uniquely determined by F .
Since Z computes sbr(P ), Z[Q] ⊆ Z and P ∈ ν d (Z[Q]) , we get Z[Q] = Z. Thus each point of S 2 is a connected component of Z.
computes the symmetric border rank of P , we get W 1 ∩ Z = Z, i.e. η = µ. Together with steps (b) and (c) we get Z = (Z ∩ ℓ) ⊔ S 2 . Thus from steps (b), (c) and (d)we get Z = (Z ∩ C d ) ⊔ S 2 .(e) Here we prove the uniqueness of the rational normal curve C d . Notice that ℓ and C d = ν d (ℓ) are uniquely determined by the choice of a pair (Z, S) with ν d (Z) computing sbr(P ) and ν d (S) computing sr(P ). Fix another pair (Z ′ , S ′ ) with ν d (Z ′ ) computing sbr(P ) and ν d (S ′ ) computing sr(P ). Let ℓ ′ be the line associated to Z ′ ∪S ′ . Assume ℓ ′ = ℓ. First assume S ′ = S. The part of Theorem 1 proved before gives
1, because it does not use the inequality deg(A) < deg(B), but only that deg(Z) ≤ deg(S) and Z = S (if T = Z, then deg(T ) < deg(Z) ≤ deg(S) and hence T = S). We get A = A 1 ⊔ A 2 and B = B 1 ⊔ A 2 with A 2 reduced, A 2 ∩ ℓ = ∅ and A 1 ∪ B 1 ⊂ ℓ. Since deg(A) = deg(B), we have deg(A 1 ) = deg(B 1 ). Thus deg(A 1 ) ≥ (d + 2)/2.
The authors were partially supported by CIRM of FBK Trento (Italy), Project Galaad of INRIA Sophia Antipolis Méditerranée (France), Institut Mittag-Leffler (Sweden), Marie Curie: Promoting science (FP7-PEOPLE-2009-IEF), MIUR and GNSAGA of INdAM (Italy). Theorem 2, it is sufficient to prove that sbr(P ) ≥ d + 1. Assume sbr(P ) ≤ d and take Z computing sbr(P ). We may apply a small part of the proof of Theorem 1 to P, S, Z (even if a priori S may not compute sr(P )). We get the existence of a line ℓ such that deg(Z ∩ ℓ) < ♯(S ∩ ℓ) and deg(Z ∩ ℓ) + ♯(S ∩ ℓ) ≥ d + 2. Since d ≥ 4, we get ♯(S ∩ ℓ) ≥ 3, that is a contradiction.We do not have experimental evidence to raise the following question (see[3]for the cases with sbr(P ) ≤ 3). Question 1. Is it true that sr(P ) ≤ d(sbr(P ) − 1) for all P ∈ P N and that equality holds if and only if P ∈ T X m,d \ X m,d where T X m,d ⊂ P N is the tangential variety of the Veronese variety X m,d ?
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06902 Sophia Antipolis, France. E-mail address: [email protected], alessandra. Inria Galaad, Méditerranée, [email protected]. 93GALAAD, INRIA Méditerranée, BP 93, 06902 Sophia Antipolis, France. E-mail address: [email protected], [email protected]
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[] |
[
"A new study of low-energy (p,γ) resonances on Magnesium isotopes",
"A new study of low-energy (p,γ) resonances on Magnesium isotopes"
] |
[
"B Limata \nDipartimento di Scienze Fisiche\nUniversità di Napoli \"Federico II\"\nINFN Sezione di Napoli\nNapoliItaly\n",
"F Strieder \nInstitut für Experimentalphysik\nRuhr-Universität Bochum\nBochumGermany\n",
"A Formicola \nLaboratori Nazionali del Gran Sasso (LNGS)\nINFN\nAssergiAQItaly\n",
"G Imbriani \nDipartimento di Scienze Fisiche\nUniversità di Napoli \"Federico II\"\nINFN Sezione di Napoli\nNapoliItaly\n",
"M Junker \nLaboratori Nazionali del Gran Sasso (LNGS)\nINFN\nAssergiAQItaly\n",
"H W Becker \nFakultät für Physik und Astronomie\nRuhr-Universität Bochum\nBochumGermany\n",
"D Bemmerer \nForschungszentrum Dresden-Rossendorf\nBautzner Landstr. 12801328DresdenGermany\n",
"A Best \nInstitut für Experimentalphysik\nRuhr-Universität Bochum\nBochumGermany\n",
"† R Bonetti \nIstituto di Fisica Generale Applicata\nUniversità degli Studi di Milano\nINFN Sezione di Milano\nItaly\n",
"C Broggini \nIstituto Nazionale di Fisica Nucleare (INFN)\nSezione di Padova\nvia Marzolo 835131PadovaItaly\n",
"A Caciolli \nIstituto Nazionale di Fisica Nucleare (INFN)\nSezione di Padova\nvia Marzolo 835131PadovaItaly\n\nDipartimento di Fisica\nUniversità di Padova\nItaly\n",
"P Corvisiero \nUniversità di Genova\nINFN Sezione di Genova\nGenovaItaly\n",
"H Costantini \nUniversità di Genova\nINFN Sezione di Genova\nGenovaItaly\n",
"A Dileva \nDipartimento di Scienze Fisiche\nUniversità di Napoli \"Federico II\"\nINFN Sezione di Napoli\nNapoliItaly\n",
"Z Elekes \nInstitute of Nuclear Research (ATOMKI)\nDebrecenHungary\n",
"Zs Fülöp \nInstitute of Nuclear Research (ATOMKI)\nDebrecenHungary\n",
"G Gervino \nDipartimento di Fisica Sperimentale\nUniversità di Torino\nINFN Sezione di Torino\nTorinoItaly\n",
"A Guglielmetti \nUniversità degli Studi di Milano\nINFN\nSezione di Milano\nItaly\n",
"C Gustavino \nLaboratori Nazionali del Gran Sasso (LNGS)\nINFN\nAssergiAQItaly\n",
"Gy Gyürky \nInstitute of Nuclear Research (ATOMKI)\nDebrecenHungary\n",
"A Lemut \nUniversità di Genova\nINFN Sezione di Genova\nGenovaItaly\n",
"M Marta \nForschungszentrum Dresden-Rossendorf\nBautzner Landstr. 12801328DresdenGermany\n",
"C Mazzocchi \nUniversità degli Studi di Milano\nINFN\nSezione di Milano\nItaly\n",
"R Menegazzo \nIstituto Nazionale di Fisica Nucleare (INFN)\nSezione di Padova\nvia Marzolo 835131PadovaItaly\n",
"P Prati \nUniversità di Genova\nINFN Sezione di Genova\nGenovaItaly\n",
"V Roca \nDipartimento di Scienze Fisiche\nUniversità di Napoli \"Federico II\"\nINFN Sezione di Napoli\nNapoliItaly\n",
"C Rolfs \nInstitut für Experimentalphysik\nRuhr-Universität Bochum\nBochumGermany\n",
"C Rossi Alvarez \nIstituto Nazionale di Fisica Nucleare (INFN)\nSezione di Padova\nvia Marzolo 835131PadovaItaly\n",
"C Salvo \nLaboratori Nazionali del Gran Sasso (LNGS)\nINFN\nAssergiAQItaly\n",
"E Somorjai \nInstitute of Nuclear Research (ATOMKI)\nDebrecenHungary\n",
"O Straniero \nOsservatorio Astronomico di Collurania\nTeramoItaly\n",
"F Terrasi \nSeconda Università di Napoli\nCaserta\n\nINFN Sezione di Napoli\nNapoliItaly\n",
"H.-P Trautvetter \nInstitut für Experimentalphysik\nRuhr-Universität Bochum\nBochumGermany\n"
] |
[
"Dipartimento di Scienze Fisiche\nUniversità di Napoli \"Federico II\"\nINFN Sezione di Napoli\nNapoliItaly",
"Institut für Experimentalphysik\nRuhr-Universität Bochum\nBochumGermany",
"Laboratori Nazionali del Gran Sasso (LNGS)\nINFN\nAssergiAQItaly",
"Dipartimento di Scienze Fisiche\nUniversità di Napoli \"Federico II\"\nINFN Sezione di Napoli\nNapoliItaly",
"Laboratori Nazionali del Gran Sasso (LNGS)\nINFN\nAssergiAQItaly",
"Fakultät für Physik und Astronomie\nRuhr-Universität Bochum\nBochumGermany",
"Forschungszentrum Dresden-Rossendorf\nBautzner Landstr. 12801328DresdenGermany",
"Institut für Experimentalphysik\nRuhr-Universität Bochum\nBochumGermany",
"Istituto di Fisica Generale Applicata\nUniversità degli Studi di Milano\nINFN Sezione di Milano\nItaly",
"Istituto Nazionale di Fisica Nucleare (INFN)\nSezione di Padova\nvia Marzolo 835131PadovaItaly",
"Istituto Nazionale di Fisica Nucleare (INFN)\nSezione di Padova\nvia Marzolo 835131PadovaItaly",
"Dipartimento di Fisica\nUniversità di Padova\nItaly",
"Università di Genova\nINFN Sezione di Genova\nGenovaItaly",
"Università di Genova\nINFN Sezione di Genova\nGenovaItaly",
"Dipartimento di Scienze Fisiche\nUniversità di Napoli \"Federico II\"\nINFN Sezione di Napoli\nNapoliItaly",
"Institute of Nuclear Research (ATOMKI)\nDebrecenHungary",
"Institute of Nuclear Research (ATOMKI)\nDebrecenHungary",
"Dipartimento di Fisica Sperimentale\nUniversità di Torino\nINFN Sezione di Torino\nTorinoItaly",
"Università degli Studi di Milano\nINFN\nSezione di Milano\nItaly",
"Laboratori Nazionali del Gran Sasso (LNGS)\nINFN\nAssergiAQItaly",
"Institute of Nuclear Research (ATOMKI)\nDebrecenHungary",
"Università di Genova\nINFN Sezione di Genova\nGenovaItaly",
"Forschungszentrum Dresden-Rossendorf\nBautzner Landstr. 12801328DresdenGermany",
"Università degli Studi di Milano\nINFN\nSezione di Milano\nItaly",
"Istituto Nazionale di Fisica Nucleare (INFN)\nSezione di Padova\nvia Marzolo 835131PadovaItaly",
"Università di Genova\nINFN Sezione di Genova\nGenovaItaly",
"Dipartimento di Scienze Fisiche\nUniversità di Napoli \"Federico II\"\nINFN Sezione di Napoli\nNapoliItaly",
"Institut für Experimentalphysik\nRuhr-Universität Bochum\nBochumGermany",
"Istituto Nazionale di Fisica Nucleare (INFN)\nSezione di Padova\nvia Marzolo 835131PadovaItaly",
"Laboratori Nazionali del Gran Sasso (LNGS)\nINFN\nAssergiAQItaly",
"Institute of Nuclear Research (ATOMKI)\nDebrecenHungary",
"Osservatorio Astronomico di Collurania\nTeramoItaly",
"Seconda Università di Napoli\nCaserta",
"INFN Sezione di Napoli\nNapoliItaly",
"Institut für Experimentalphysik\nRuhr-Universität Bochum\nBochumGermany"
] |
[] |
Proton captures on Mg isotopes play an important role in the Mg-Al cycle active in stellar H shell burning. In particular, the strengths of low-energy resonances with E < 200 keV in 25 Mg(p,γ) 26 Al determine the production of 26 Al and a precise knowledge of these nuclear data is highly desirable. Absolute measurements at such low-energies are often very difficult and hampered by γ-ray background as well as changing target stoichiometry during the measurements. The latter problem can be partly avoided using higher energy resonances of the same reaction as a normalization reference. Hence the parameters of suitable resonances have to be studied with adequate precision.In the present work we report on new measurements of the resonance strengths ωγ of the E = 214, 304, and 326 keV resonances in the reactions 24 Mg(p,γ) 25 Al, 25 Mg(p,γ) 26 Al, and 26 Mg(p,γ) 27 Al, respectively. These studies were performed at the LUNA facility in the Gran Sasso underground laboratory using multiple experimental techniques and provided results with a higher accuracy than previously achieved.
|
10.1103/physrevc.82.015801
|
[
"https://arxiv.org/pdf/1006.5281v1.pdf"
] | 119,196,802 |
1006.5281
|
44f772bc376bd14193a7de965d7346572e1606b7
|
A new study of low-energy (p,γ) resonances on Magnesium isotopes
28 Jun 2010 (Dated: June 29, 2010)
B Limata
Dipartimento di Scienze Fisiche
Università di Napoli "Federico II"
INFN Sezione di Napoli
NapoliItaly
F Strieder
Institut für Experimentalphysik
Ruhr-Universität Bochum
BochumGermany
A Formicola
Laboratori Nazionali del Gran Sasso (LNGS)
INFN
AssergiAQItaly
G Imbriani
Dipartimento di Scienze Fisiche
Università di Napoli "Federico II"
INFN Sezione di Napoli
NapoliItaly
M Junker
Laboratori Nazionali del Gran Sasso (LNGS)
INFN
AssergiAQItaly
H W Becker
Fakultät für Physik und Astronomie
Ruhr-Universität Bochum
BochumGermany
D Bemmerer
Forschungszentrum Dresden-Rossendorf
Bautzner Landstr. 12801328DresdenGermany
A Best
Institut für Experimentalphysik
Ruhr-Universität Bochum
BochumGermany
† R Bonetti
Istituto di Fisica Generale Applicata
Università degli Studi di Milano
INFN Sezione di Milano
Italy
C Broggini
Istituto Nazionale di Fisica Nucleare (INFN)
Sezione di Padova
via Marzolo 835131PadovaItaly
A Caciolli
Istituto Nazionale di Fisica Nucleare (INFN)
Sezione di Padova
via Marzolo 835131PadovaItaly
Dipartimento di Fisica
Università di Padova
Italy
P Corvisiero
Università di Genova
INFN Sezione di Genova
GenovaItaly
H Costantini
Università di Genova
INFN Sezione di Genova
GenovaItaly
A Dileva
Dipartimento di Scienze Fisiche
Università di Napoli "Federico II"
INFN Sezione di Napoli
NapoliItaly
Z Elekes
Institute of Nuclear Research (ATOMKI)
DebrecenHungary
Zs Fülöp
Institute of Nuclear Research (ATOMKI)
DebrecenHungary
G Gervino
Dipartimento di Fisica Sperimentale
Università di Torino
INFN Sezione di Torino
TorinoItaly
A Guglielmetti
Università degli Studi di Milano
INFN
Sezione di Milano
Italy
C Gustavino
Laboratori Nazionali del Gran Sasso (LNGS)
INFN
AssergiAQItaly
Gy Gyürky
Institute of Nuclear Research (ATOMKI)
DebrecenHungary
A Lemut
Università di Genova
INFN Sezione di Genova
GenovaItaly
M Marta
Forschungszentrum Dresden-Rossendorf
Bautzner Landstr. 12801328DresdenGermany
C Mazzocchi
Università degli Studi di Milano
INFN
Sezione di Milano
Italy
R Menegazzo
Istituto Nazionale di Fisica Nucleare (INFN)
Sezione di Padova
via Marzolo 835131PadovaItaly
P Prati
Università di Genova
INFN Sezione di Genova
GenovaItaly
V Roca
Dipartimento di Scienze Fisiche
Università di Napoli "Federico II"
INFN Sezione di Napoli
NapoliItaly
C Rolfs
Institut für Experimentalphysik
Ruhr-Universität Bochum
BochumGermany
C Rossi Alvarez
Istituto Nazionale di Fisica Nucleare (INFN)
Sezione di Padova
via Marzolo 835131PadovaItaly
C Salvo
Laboratori Nazionali del Gran Sasso (LNGS)
INFN
AssergiAQItaly
E Somorjai
Institute of Nuclear Research (ATOMKI)
DebrecenHungary
O Straniero
Osservatorio Astronomico di Collurania
TeramoItaly
F Terrasi
Seconda Università di Napoli
Caserta
INFN Sezione di Napoli
NapoliItaly
H.-P Trautvetter
Institut für Experimentalphysik
Ruhr-Universität Bochum
BochumGermany
A new study of low-energy (p,γ) resonances on Magnesium isotopes
28 Jun 2010 (Dated: June 29, 2010)(The LUNA Collaboration)numbers: 2540Ep2540Lw2620Cd2630-k
Proton captures on Mg isotopes play an important role in the Mg-Al cycle active in stellar H shell burning. In particular, the strengths of low-energy resonances with E < 200 keV in 25 Mg(p,γ) 26 Al determine the production of 26 Al and a precise knowledge of these nuclear data is highly desirable. Absolute measurements at such low-energies are often very difficult and hampered by γ-ray background as well as changing target stoichiometry during the measurements. The latter problem can be partly avoided using higher energy resonances of the same reaction as a normalization reference. Hence the parameters of suitable resonances have to be studied with adequate precision.In the present work we report on new measurements of the resonance strengths ωγ of the E = 214, 304, and 326 keV resonances in the reactions 24 Mg(p,γ) 25 Al, 25 Mg(p,γ) 26 Al, and 26 Mg(p,γ) 27 Al, respectively. These studies were performed at the LUNA facility in the Gran Sasso underground laboratory using multiple experimental techniques and provided results with a higher accuracy than previously achieved.
Proton captures on Mg isotopes play an important role in the Mg-Al cycle active in stellar H shell burning. In particular, the strengths of low-energy resonances with E < 200 keV in 25 Mg(p,γ) 26 Al determine the production of 26 Al and a precise knowledge of these nuclear data is highly desirable. Absolute measurements at such low-energies are often very difficult and hampered by γ-ray background as well as changing target stoichiometry during the measurements. The latter problem can be partly avoided using higher energy resonances of the same reaction as a normalization reference. Hence the parameters of suitable resonances have to be studied with adequate precision.
In the present work we report on new measurements of the resonance strengths ωγ of the E = 214, 304, and 326 keV resonances in the reactions 24 Mg(p,γ) 25 Al, 25 Mg(p,γ) 26 Al, and 26 Mg(p,γ) 27 Al, respectively. These studies were performed at the LUNA facility in the Gran Sasso underground laboratory using multiple experimental techniques and provided results with a higher accuracy than previously achieved. Observations from satellites [1,2] have mapped the sky in the light of the prominent γ-ray line at E γ = 1809 keV of the β-decay of 26 Al (T 1/2 = 7 × 10 5 yr). The intensity of the line corresponds to about 3 solar masses of 26 Al in our galaxy [3]. Moreover, evidences for an 26 Al excess in the early solar system was found in CAIs (Calcium Aluminum inclusions) showing a significant correlation of 26 Mg (extinct 26 Al) and 27 Al [4,5]. While the observations from COMPTEL and INTEGRAL provided evidence that 26 Al nucleosynthesis is still active on a large scale, the Mg isotopic variations demonstrate that 26 Mg also was produced at the time of the condensation of the solar-system about 4.6 billion years ago. Any astrophysical scenario for 26 Al nucleosynthesis must be concordant with both observations. The 26 Al is produced mainly via the 25 Mg(p,γ) 26 Al capture reaction. The most important site for the activation of this reaction is the hydrogen-burning shell (HBS), which may be active in off-main-sequence stars of any mass [6][7][8]. In particular, the Mg-Al cycle is at work in the hottest region of the HBS, close to the point of the maximum nuclear energy release. In the HBS, the 25 Mg(p,γ) 26 Al reaction starts when the temperature exceeds about T = 30 × 10 6 K and for T = (40 − 60) × 10 6 K -corresponding to a Gamow energy of about E 0 ≈ 100 keV [9] -almost all the 25 Mg is converted into 26 Al. At higher temperatures, the destruction of 26 Al by 26 Al(p,γ) 27 Si and the refurbishment of 25 Mg by the sequence 24 Mg(p,γ) 25 Al(β + ) 25 Mg begins to play a relevant role. The 25 Mg(p,γ) 26 Al also operates in the carbon and neon burning shells of massive stars during late stellar evolution.
Moreover, a global anticorrelation between the abundances of Mg and Al has been observed, e.g. in Globular Cluster stars (see [10] for a recent analysis and references therein). This observation is to the present knowledge coupled to the nucleosynthesis processes involving the Mg-Al cycle occurring in the HBS of primeval generation AGB or massive stars. A detailed knowledge of these processes is a fundamental step toward a general understanding of the formation of the building blocks of our Galaxy.
The uncertainties in the present stellar models are closely related to a precise evaluation of the relevant reaction rates of the Mg-Al cycle. In particular the reactions 24 Mg(p,γ) 25 Al and 25 Mg(p,γ) 26 Al play a key role in those scenarios.
The reaction 25 Mg(p,γ) 26 Al (Q = 6.306 MeV) is dominated by narrow resonances. These resonances decay in complex γ-ray cascades either to the ground state of 26 Al or an isomeric state at E x = 228 keV. Only the ground state transition is of astrophysical relevance since the ground state decays into the first excited state of 26 Mg with the subsequent γ-ray emission observed by the satellite telescopes. The isomeric state of 26 Al decays (T 1/2 = 6.3 s) exclusively to the ground state of 26 Mg and, thus, is not associated with the emission of γ-rays. The strengths of these 25 Mg(p,γ) 26 Al resonances have been experimentally studied down to an energy 1 of E = 190 keV [11][12][13][14][15][16][17][18][19][20][21]. Nevertheless, the present uncertainty is insufficient for precise models. In particular, a disagreement between resonance strengths measured by γ-ray spectroscopy and delayed AMS (Accelerator Mass Spectrometry) detection of the 26 Al nuclei after a proton irradiation of 25 Mg at the relevant energies has been reported recently [22].
The nuclear reaction rate of 24 Mg(p,γ) 25 Al (Q = 2.272 MeV) at astrophysical energies has a contribution by a low-energy resonance at E = 214 keV. Moreover, a strong direct capture component dominates the resonance contribution. The estimate of the latter contribution [23,24] is solely based on the experimental data from Trautvetter and Rolfs [23]. Additionally, the E = 214 keV resonance strength carries a large systematic discrepancy between the existing data (e.g. [23,24]).
In the present work we report on a new measurement of the strengths of the E = 304 keV resonance in 25 Mg(p,γ) 26 Al, as well as the E = 214 keV resonance in 24 Mg(p,γ) 25 Al. The radiative capture reaction on the third stable Mg isotope, i.e. the E = 326 keV resonance in 26 Mg(p,γ) 27 Al (Q = 8.272 MeV), was studied for completeness. These resonances will serve as a normalization for a subsequent determination of astrophysically important low-energy resonances in 25 Mg(p,γ) 26 Al, i.e. resonance below E< 200 keV. 1 all energies are given in the center-of-mass frame if not indicated differently
The precision and reliability of such normalization standards are important since weak low-energy resonance strengths are often impossible to determine directly from absolute measurements. In particular the target stoichiometry is a critical parameter. Small admixtures of contaminant elements or isotopes in the target, e.g. oxygen as a result of an evaporation process, have already a large effect on the resonance strength determination. Moreover, it is well known in experimental nuclear astrophysics that a solid state target under heavy proton bombardment changes its stoichiometry in the course of the measurement and a frequent control of the target quality is absolutely necessary for long-lasting low-energy measurements. A determination of weak resonance strengths relative to well known resonances can avoid the difficulty of an absolute measurement. The larger yield of highenergy resonances facilitates the determination of the experimental parameters of such resonances. However, these parameters, e.g. target stoichiometry, still need to be measured with high precision: the major goal of the present study.
The resonances were studied using Mg targets with the well known isotopic composition of natural Mg as well as enriched 25 Mg target. The experiments have been performed at the 400 kV LUNA (Laboratory for Underground Nuclear Astrophysics) accelerator in the Laboratori Nazionali del Gran Sasso (LNGS) underground laboratory in Italy [25]. The 1400 m rock overburden (corresponding to 3600 meter water equivalent) of the underground laboratory reduces the γ-ray background by more than three orders of magnitude for energies higher than 3.5 MeV, compared with a measurement on earth's surface [26]. In order to reduce the systematic uncertainties arising from the detection technique several independent methods have been used. The absolute value of the resonance strengths were measured with both a high resolution HPGe detector and a high efficiency 4π BGO summing crystal. The combination of both methods allows for a precise determination of these parameters and the related resonant branching ratios.
As an alternative method -only in case of the 25 Mg(p,γ) 26 Al resonance at E = 304 keV -an enriched Mg target was irradiated with a proton beam and after a proper chemical treatment the number of produced 26 Al nuclei were counted by means of the AMS technique.
In the following sections we will describe in detail the experimental equipment, target preparation and characterization (section II). The data analysis of the γ-ray measurements follows in section III including a description of a GEANT4 [27] Monte Carlo code which was used to obtain the efficiency for the 4π BGO detector (section III C 1). The results of these measurements are given in section III D and new values for the weighted average are recommended. A comparison of the γ-ray measurements with a detection of the reaction products by means of AMS is presented in section IV. Finally, the present work concludes with a discussion and summary of the results (section V).
II. EXPERIMENTAL SETUP AND TARGET PREPARATION
A. The LUNA accelerator
The 400 kV LUNA facility has been described elsewhere [28]. Briefly, the accelerator ( Fig. 1 upper panel) provided in this experiment a proton current on target of up to 250 µA at energies between E p = 180 and 380 keV. The absolute energy is known with an accuracy of 0.3 keV and the energy spread and the long-term energy stability were observed to be 100 eV and 5 eV/h, respectively. The protons are extracted from the radio-frequency ion source and guided under 0 • through a vertical steerer and the first 45 • switching magnet into a second, identical 45 • magnet (distance between both magnets = 1.5 m). With the latter magnet (30 cm radius, 3 cm gap, 1.6 MeV amu) the beam is focussed into the 45 • II beam line of the LUNA facility. The proton beam passed through a circular, retractable collimator (diameter 10 mm), two focussing apertures (diameter 5 mm each), and a copper shroud (ℓ = 1 m; diameter 28 mm) extending to within 2 mm from the target, where the target plane was oriented perpendicularly to the beam direction. The distance between the two focussing apertures was 566 mm and the second aperture prevented the proton beam from hitting the copper shroud (for details see Fig. 1 lower panel). The copper shroud was connected to a cold trap cooled to liquid nitrogen temperature. With a turbo pump installed below the cold trap, the arrangement led to a pressure in the target chamber of better than 5 × 10 −7 mbar; whereby no C deposition was observed on the targets. A voltage of minus 300 V was applied to the cold trap to minimize emission of secondary electrons from both the target and the last aperture; the precision in the current integration was estimated to be about 2%. The beam profile on target was controlled by sweeping the beam in the x and y directions within the geometry of the apertures. The targets were directly water cooled in order to prevent any heat damage during the measurements. The BGO detector was mounted on a movable carriage such that the target could be placed in the center of the borehole of the detector maximizing the efficiency of the setup.
B. 4π BGO summing crystal setup
The BGO crystal is a cylinder (length = 28 cm) with a coaxial hole (diameter = 6 cm) and a radial thickness of 7 cm [29]. The crystal is optically divided in six sectors, each covering a 60 • azimuthal angle. In the original configuration two photomultipliers (PMTs) were coupled to the opposite sides of each sector. In order to allow for a closer distance to the last aperture, all PMTs on one side were replaced with reflecting material. Summing the light produced in all six sectors allows to recover the full energy of detected γ-rays and, thus, leads to an increased detection efficiency in the case of γ-ray cascades. Moreover, single spectra can be acquired due to the optical separation of each sector. The energy resolution of each crystal is on the order of 18% for E γ = 661 keV. The signals from the 6 PMTs of the BGO summing crystal were sent to a 16-fold amplifier (CAEN, module N568) which produced, for each incoming pulse, a linear output signal sent to a 12-bit ADC (Silena FAIR, module 9418 V). The amplifier generated also for each incoming signal a fast output signal. This fast signal generated the acquisition trigger via a constant fraction discriminator (EG&G, module CF8000) if the fast signal from each PMT is higher than a chosen threshold value. When at least one sector generated a trigger signal, the signals arriving from all the 6 PMTs are converted by the ADC. The total processing time of an event is 24 µs. The data acquisition is based on a mixed FAIRVME bus [30]. The spectra of the BGO sectors were displayed on-line on a PC screen, while the raw data, i.e. the 6 PMT signals for each trigger, were saved event-by-event on a hard disk for an off-line data analysis.
C. HPGe detector setup
In the high resolution phase of the experiment the target holder was replaced by a tube that allowed for an orientation of the target with its normal at 45 • with respect to the beam direction. The copper shroud was also cut at 45 • such that an evenly distance of 2 mm to the target was ensured. As in the BGO setup the target was directly water cooled. A HPGe detector (115% relative efficiency, resolution = 2.1 keV at E γ = 1.3 MeV) was placed on another moveable carriage oriented at 55 • with respect to the beam axis. Thus, target and front face of the detector were not parallel but γ-ray attenuation effects were reduced compared to the target holder perpendicular to the beam axis and the influence of any angular distributions was minimized. The distance between target and detector could be varied in a range d = 3.5 to 42.3 cm, where the maximum distance was used for the resonance strength and branching ratio determination. The detector was surrounded by 5 cm of lead, which reduced the background in the low-energy range by a factor 10. Standard electronics was used for processing the detector pulses which were finally stored in a 16k ADC. The acquisition unit was placed close to the experiment and the processed digitized data were sent via Ethernet to a PC for analysis.
D. Target preparation and analysis
A natural Mg target has been produced by evaporation of metallic magnesium of natural isotopic composition on a Ta backing at the IKP of the University of Münster, Germany. A small carbon sample was mounted close to the Ta sheet during the evaporation process and later used for an analysis of the target stoichiometry by means of Rutherford Backscattering (RBS). The Mg target on Ta has been cut into two pieces for the HPGe detector and the BGO crystal measurement, respectively.
The RBS analysis of the Mg target on C backing was performed with a 2 MeV He + beam from the 4 MV Dynamitron-Tandem accelerator of the Ruhr-Universität Bochum, Germany. The beam intensity was about 10 nA and the backscattered particles were detected at an angle of 160 • with respect to the beam. The data were analyzed with the computer code RBX [31].
The result of the RBS analysis is shown in Fig. 2. Three regions with different oxygen content can be identified in the target layer. In particular, there is a thin surface layer (thickness = 1.5 µg/cm 2 ) with a O:Mg ratio of 1:1. In the bulk (thickness = 32 µg/cm 2 ) of the Mg target layer a ratio of O:Mg = (0.12 ± 0.03):1 was found. Finally, the interface to the backing showed another thin MgO layer which was approximated in the analysis by a thickness of 1.5 µg/cm 2 with O:Mg = 1:1. The structure of this latter layer is probably more complex, but matches sufficiently well the general layering of the target; as a result a slight overestimate of the RBS yield with respect to the data points is observed at the low-energy tail of the Mg peak (Fig. 2). Note, however, that the two thin layers at the surface and the interface have no influence on the resonance strength determination if the resonance energy is locate well inside the Mg bulk. The stoichiometry ratio is independent of stopping power and the uncertainty is mainly based on the quality of the fit. Moreover, the homogeneity of the Mg bulk is demonstrate by the flat thick-target yield plateau of the resonance scans (Fig. 3). These stoichiometry results were used in the analysis of the γ-ray data in order to obtain the effective stopping power (see section III A). The target stoichiometry was checked frequently in close geometry, i.e. thick-target yield curves of each resonance were measured. Figure 3 shows, as an example, the results of such a scan for the E = 326 keV resonance of 26 Mg(p,γ) 27 Al. The maximum observed yield decrease during the course of the experiment was 12%. The data were corrected for this stoichiometry change if necessary.
E. Accelerator Mass Spectrometry (AMS)
Complementary to the γ-ray spectroscopy with BGO and HPGe detector the strength of the E = 304 keV resonance of 25 Mg(p,γ) 26 Al was studied by means of Accelerator Mass Spectrometry (AMS). The 25 Mg targets for these AMS measurements were prepared at the Laboratori Nazionali di Legnaro, Padova, Italy, by reducing enriched MgO mixed with zirconium powder and evaporation with an electron gun. The target thickness was between 40 and 60 µg/cm 2 corresponding to an energy loss of 20 and 30 keV at the resonance energy. Finally, two targets were analyzed (labelled in the following as A and B, respectively). In parallel to the irradiation of the AMS samples the corresponding γ-ray yield was observed for each target with the BGO detector in standard geometry. The stoichiometry of the targets could be determined from the γ-ray yield normalized to the new value of the resonance strength and the O:Mg ratio for the targets A and B turned out to be 0.29 ± 0.02 and 0.32 ± 0.02, respectively.
III. EXPERIMENTAL PROCEDURES, DATA ANALYSES AND RESULTS OF γ-RAY MEASUREMENTS
A. Thick-target yield and stopping power
The resonance strengths of the resonances at E = 214, 304, and 326 keV for 24 Mg(p,γ) 25 Al, 25 Mg(p,γ) 26 Al, and 26 Mg(p,γ) 27 Al, respectively, have been measured with both the HPGe and the BGO detector while the γ-ray branchings can only be determined with the HPGe detector. In particular a precise determination of the resonance strength requires unusual efforts in the measurement of all quantities entering the analysis of this value. In general the thick-target yield of a narrow resonance is given by the expression [9]:
Y = λ 2 2 b γ ωγ m Mg + m p m Mg 1 ε eff (1)
where ωγ is the resonance strength, b γ the cross-section fraction that is carried by the observed γ-ray (e.g., the branching ratio for a primary transition), λ the de Broglie wavelength, and ε eff the effective stopping power. The latter quantity accounts for the energy loss of the projectiles in the target layer and can be derived with the formula:
ε eff = ε a + i N i N a ε i ≃ 1 Xy Mg (ε Mg + N O N Mg ε O )(2)
with the number of active atoms N a with respect to the inert atoms N i , Xy Mg the relative isotopic abundance (or enrichment) of the observed Mg isotope (e.g. y = 24, 25, or 26), and the stopping power ε O and ε Mg of protons in oxygen and magnesium at the particular resonance energy, respectively. In this determination of the effective stopping power all other contaminations are neglected since they amount to less than 1% in total. Therefore, the effective stopping power and, in turn, the ωγ scales with the inverse of the relative isotopic abundance of the effective target isotope and is strongly influenced by the oxygen concentration.
The relative isotopic abundance in case of the natural magnesium target as used for the γ-ray measurements is well known: 78.99 ± 0.16% ( 24 Mg), 10.00 ± 0.03% ( 25 Mg) and 11.01 ± 0.03% ( 26 Mg) [32]. Thus, the effective stopping power for the 3 Mg isotopes including the observed oxygen to magnesium ratio (section II D) are ε eff (
Efficiency determination
The efficiency of the HPGe detector was studied for different distances d from the target, i.e. 3.5, 8.5, 13.5, and 42.3 cm, with calibrated γ-ray sources placed at the target position as well as with the E = 259 keV resonance of 14 N(p,γ) 15 O [34]. In addition, the E = 214 keV resonance of 24 Mg(p,γ) 25 Al was used for the relative efficiency determination while at the same time in an iterative process the branching ratios for this resonance were improved. In contrast, the reactions 25 Mg(p,γ) 26 Al and 26 Mg(p,γ) 27 Al could not be used for such a procedure since the decay schemes of 26 Al and 27 Al are too complex. The absolute scale of the efficiency determination was fixed by the data from the γ-ray sources, e.g. 137 Cs, 207 Bi, and 226 Ra, while the energy dependence was determined following the approach described in [34]. Briefly, the latter procedure is based on the assumption that the intensity ratio of primary and secondary γ-ray transition for each excited state including the particular detection efficiency must be unity. These constraints were used in a global fit to the data and the full-energy efficiency ε FE (E γ ) as a function of γ-ray energy and distance d was parameterized by the following empirical expression [35]:
ε FE (E γ ) = A(E γ , d) · e a+b ln Eγ +c(ln Eγ ) 2(3)
with The summing effect of real coincidences were taken into account similarly as in [34]. Figure 4 shows the efficiency ε FE (E γ ) as a function of the γ-ray energy for the four distances. In order to illustrate the importance of the summing correction, the open symbols in the figure represent the efficiency values without summing correction while the filled symbols include this correction. Clearly, for distances much larger than 13.5 cm no influence from summing effects is expected. The absolute uncertainty of the efficiency determination is in the order of 3.5% dominated by the calibration of the radioactive sources. The relative efficiency uncertainty for a measurement at the same distance is lower than 2%. In particular for the far geometry, d = 42.3 cm, the efficiency curve is well determined due to the absence of summing effects and the relative uncertainty is below 1.5%.
A(E γ , d) = 1 − e − d+d 0 α+β √ Eγ (d + d 0 ) 2(4)
Gamma-ray spectra and branching ratio determination
The resonances studied in the present experiment give a relatively high yield. As a consequence of the very low γ-ray background at Gran Sasso these resonances could be observed in far geometry, minimizing the summing effect. For each resonance we could identify all the primary γ-ray transitions with a very low detection limit, in particular in the case of 25 Mg(p,γ) 26 Al in the range from 0.02 % at low-energies, E γ < 1 MeV, to 0.004 % for energies above 6.5 MeV. Therefore, the strengths of the resonances and their branching ratios could be determined with high precision. The γ-ray spectra of the E = 304 keV resonance of 25 Mg(p,γ) 26 Al is shown in Fig. 5 as an example. The spectra for the other resonance are available at [37]. All spectra have been obtained with long runs over several hours -5 hours minimum and a collected charge of about 7.5 C each -on top of the thicktarget yield of each resonance. The background of the γ-ray lines has been subtracted with off-resonance runs at energies slightly lower than the corresponding resonance energies. For energies above E γ = 4 MeV the natural background is negligible and the total background is only determined by beam induced background, e.g. from 19 F(p,αγ) 16 O.
The branching ratios for each resonance are given in tables I, II and III for 24 Mg(p,γ) 25 Al, 25 Mg(p,γ) 26 Al, and 26 Mg(p,γ) 27 Al, respectively. Background subtraction and relative efficiency uncertainty were included in the error budget. In all 3 cases the precision of the branching ratios has been improved with respect to the available literature.
Resonance strengths
The results are summarized in table IV. The uncertainty of the ωγ determination with the HPGe detector is dominated by the absolute error of the γ-ray efficiency curve (3.5% for far geometry). A minor contribution arises from the statistical uncertainty which is almost negligible. Common uncertainties of both detection methods, i.e. stopping power and charge integration, are not considered in the error budget of the single measurements, but for the weighted mean of both detection methods (see below). Note in case of 25 Mg(p,γ) 26 Al we identified transitions feeding either the ground state or the 228 keV isomeric state. The probability for forming the ground state of 26 Al results to f 0 = 87.8 ± 1.0%.
C. Measurements with the BGO detector
Monte Carlo simulation and efficiency determination
In the BGO setup the Mg target is directly located inside the detector, almost in a 4π detection geometry. The detector is a high efficiency detection instrument with the disadvantage of the relatively low resolution of the BGO material. Therefore, the BGO detector is the ideal tool to study resonances at lower energies, e.g. E < 200 keV, with small resonance strengths accessible only in a few cases with a HPGe. The advantage of this 4π geometry is that the influence of any angular distribution and angular correlation effects is strongly reduced compared to smaller detectors. Moreover, the counting statistics in the γ-ray spectra were very high even with a reduced proton beam current minimizing the target deterioration, e.g. the change of stoichiometry (see section II D). On the contrary, the disadvantage of this approach is that the identification of beam induced background is by far more difficult and in some cases the background lines may be located below the γ-ray lines of interest.
However, the efficiency determination for the BGO detector is very complex and experimentally almost not accessible. Due to the different multiplicities of each nuclear reaction and the different γ-ray energies of involved transitions, the total summing efficiency is different for each nuclear reaction. Recently an experimen-tal approach was suggested [41] to first determine these multiplicities, which are then used to derive the corresponding efficiency of the sum peak by means of Monte Carlo simulations. The efficiency determination is simplified in case the multiplicity and the decay scheme are largely known. In the present experiment the efficiency was determined with a Monte Carlo simulation based on GEANT4 [27]. The result of the Monte Carlo code is a simulated γ-ray spectrum. This simulated spectrum can be compared and fitted to the experimental spectrum using only a scaling constant for normalization. The resonance strength can be obtained from the scaling factor and the total event number generated in the Monte Carlo simulation.
The geometry of the BGO detector including beamline, target holder and support structure was implemented in the GEANT4 code. During the initialization of the code the branching ratios and γ-ray energies of the selected resonance are loaded. This includes not only the primary transitions -taken from the HPGe phase of the present work -but also all relevant secondary transitions. All [38] as private communication [39] b numerical values from [38] available information have been used to construct the full decay schemes of the resonances. Thus, the complete γ-ray deexcitation of the compound nucleus is followed down to the ground state, in case of 26 Al also to the isomeric state at E x = 228 keV. For each excited state a random number generator selects the subsequent excited state and, hence, the emitted γ-ray energy according to the implemented feeding probability. In some cases up to six different γ-rays are emitted per simulated event: the multiplicity of the event.
The point of origin of the γ-ray emission in the simulation is located on the target and the γ-rays are tracked through the geometry of the setup. In case the γ-ray deposits energy in the active volume of the BGO crystal the particular energy loss is stored in a histogram. In this way a single spectrum from each of the 6 segments as well as the sum spectrum of the full event is constructed. The energy resolution of each single spectrum was adapted separately to the experimental energy resolution of the BGO sectors. Angular distribution or angu- The efficiency estimate of the simulation was tested with γ-ray sources placed at the target position, i.e. 137 Cs and 60 Co. The results of measurement and simulation agreed to better than 2%. Furthermore, the validity of the Monte Carlo code was verified for a different detector setup, i.e the 12 × 12" NaI detector at the Dynamitron-Tandem Laboratory of the Ruhr-Universität Bochum [42]. The present code delivered the same efficiency curve as a totally independent code based on GEANT 3.21. Finally, the results of both codes agreed very well with measurements of various nuclear reactions testing the characteristics of the NaI detector (for details see [43]). In summary, the efficiency determination of the BGO detector is reliable to better than 3%. a the uncertainty takes into account the statistical error and a 3.5% error for the efficiency b the uncertainty takes into account the background correction, decay scheme uncertanties and an error for the simulation c common uncertainties, i.e. for stopping power and charge integration, are added quadratically d original result ωγ = 9.5 ± 2.0 meV corrected for new stopping power data [33] 2. Data analysis
The same natural Mg target was used for all measurements. The running times were t L = 330, 8150, and 400 s, at the E = 214 ( 24 Mg(p,γ) 25 Al), 304 ( 25 Mg(p,γ) 26 Al), and 326 keV ( 26 Mg(p,γ) 27 Al) resonances, respectively. The dead time was always kept below 4 %. In between these runs the thick-target yield curve for each resonance was obtained in order to determine the best energy for the measurement. The total charge collected during the course of the experiment on target was less than 0.5 C with an average proton current of 10 -40 µA. The target deterioration was checked with resonance scans before and after the measurement and found to be negligible.
The data were stored in an event-by-event mode (list mode) where the energy information and the corresponding crystal segment were recorded. The single spectra for all six BGO segments were extracted from the list mode data and separately energy calibrated. Thus, the total summing spectrum could be reconstructed from the data after the energy calibrations have been matched.
Gamma-ray spectra for the case of 25 Mg(p,γ) 26 Al are displayed in Fig. 6 (other spectra available at [37]) both the incoherent sum of the single crystal spectra (a, in the following called single sum) and the total summing spectrum (b, in the following called total sum). A background measurement at a beam energy slightly lower than the corresponding resonance energy and the simulated spectrum is shown for comparison. The γ-ray energy region used to fit experimental data and simulation is indicated. The reduced yield in the experimental total sum below E γ = 1.3 MeV in on-and off-resonance runs is caused by the energy threshold of the data acquisition trigger on the on-line sum signal (see section II B) and has no effect on the single crystal spectra. This trigger threshold had no impact on the analysis of the γ-ray spectra. Furthermore, a coincidence condition was applied to all events requiring the full energy being in the indicated energy region or above. As a consequence the environmental background is discriminated and the structure of the decay scheme appeared as simulated. Note that the total sum spectra are more sensitive to pile-up. This effect can be observed on the high-energy tail of all full-energy sum peaks and is probably caused by accidental coincidences with low-voltage (low-energy) noise from the PMTs of the BGO detector. An energy cut in order to discriminate those events could not be applied during the off-line analysis since the real low-energy events are necessary to construct the full-energy event. This effect leads in all cases to a slight disagreement between simulated and experimental spectra on top of the full-energy sum peak. The disagreement is not larger than 3% of the total number of counts in the region of interest and in agreement with the number of events found in the pile-up region above the full-energy sum peak.
The agreement between the single sum for all 3 reactions and the corresponding simulations is almost perfect and those spectra have been used to obtain the resonance strengths. However, the pile-up effect becomes almost negligible and the analysis of single and total sum agrees to better than 1% when reducing the proton beam current drastically (below 0.5 µA). Unfortunately, the current measurement gets unreliable due to the bad focussing properties of the accelerator system at such low intensities and consequently enlarged secondary electron emission: those runs could not be used for absolute measurements.
The γ-ray background at the E = 214 keV resonance of 24 Mg(p,γ) 25 Al is dominated by the natural room background. A background measurement acquired directly after the on-resonance run was added to the GEANT4 simulation after gain and run time matching in order to account for this background component in the analysis. The combined spectrum was then fitted to the onresonance run.
In contrast to the 24 Mg(p,γ) 25 Al resonance the measurements of the 25 Mg(p,γ) 26 Al and 26 Mg(p,γ) 27 Al resonances were only influenced by beam-induced background.
Gamma-ray lines from the reactions 19 F(p,αγ) 16 both very close to the 19 F(p,αγ) 16 O resonance energy and, therefore, the yield from this background source is very sensitive to the exact proton energy. As a consequence a background subtraction based on an offresonance measurement is impossible and the energy region of this γ-ray line has been excluded from the analysis [37].
The branching ratios had to be included in the GEANT4 code and as a consequence only the resonance strength can be obtained with this detection technique. However, the fit quality is an indication of the branching ratio precision. The effect of different primary branching ratios was tested in case of 25 Mg(p,γ) 26 Al. In Fig. 7 the experimental single sum spectrum is compared to a simulation based on the data of [16] and the present work, respectively. The relative intensities of the γ-ray peaks are reproduced with both data set. Nevertheless, a slightly better fit is achieved with the present data. This agreement demonstrates the internal consistency of all phases of the present experiment and emphasizes the achieved precision. The uncertainty estimated from the influence of the decay scheme is at most 3 %. The yield in the total sum depends only weakly on these parameters. This is important for weak resonances with largely unknown branching ratios and another advantage of this method.
Resonance strengths
The results on the resonance strength are summarized in Table IV. The systematic error of ωγ BGO is given by the uncertainty of the simulation, the decay scheme and the background correction. This includes also the uncertainty of the pile-up effect on the total sum. The measurements of the 25 Mg(p,γ) 26 Al and 26 Mg(p,γ) 27 Al resonances are essentially background free and the uncertainty of the simulation may be as large as 3%. The influence of the background cannot be neglected in case of the 24 Mg(p,γ) 25 Al resonance and an increased uncertainty of the simulation of 5% accounts for this issue. The decay scheme uncertainty was discussed in the previous section and estimated to 3%. The statistical uncertainties of the measurements were in the order of 0.1% and neglected in the present analysis.
These uncertainties are independent from the HPGe measurement and have to be considered for the calculation of the weighted average of the ωγ values. The additional errors for the stopping power (4.5%) and the charge measurement (2%) have to be added quadratically to the error of the weighted mean in order to evaluate a total uncertainty of the present γ-ray experiment.
Finally, the resonance strengths from the BGO and HPGe measurements are in perfect agreement within their errors. The presented analysis demonstrates that with the BGO detector a similar precision as with a HPGe detector can be reached. The challenging measurements of 25 Mg(p,γ) 26 Al resonances below E > 200 keV with ωγ ≪ 1µeV is feasible with such an approach and will lead to an improved knowledge of the reaction rate at astrophysical energies. The strength of this resonance is the most important parameter in the context of the present work and, therefore, will be discussed first. The weighted average of the present experiment, ωγ = 30.7 ± 1.7 meV, is in very good agreement with previous work, i.e. the latest γray spectroscopy experiment [18], but also with older experiments [12,13]. However, the present result strongly disagrees by more than 3σ with the resonance strength, ωγ = 24 ± 2 meV, measured by means of AMS in [22] -if scaled for the total ωγ. In [22] the authors do not prove the stoichiometry of their target and no test for oxygen contamination is presented. From the present work it is evident that such contaminations cannot be neglected and might have an effect of 20% on the resonance strength. As a consequence the AMS value of [22] will not be considered for the further analysis. The NACRE compilation [40] provided a weighted average over all published γ-ray measurements and recommended a value ωγ = 31 ± 2 meV. This recommendation is in perfect agreement with the present result and we combine the NACRE value with the present ωγ. This procedure yields a new recommended resonance strength of ωγ = 30.8 ± 1.3 meV for the E = 304 keV resonance of the reaction 25 Mg(p,γ) 26 Al. Finally, in a recent series of papers [44][45][46][47] a new reaction rate compilation is presented. Our present result is also in good agreement with the value, ωγ = 30.0±3.5 meV [46], used in this compilation to calculate the rates, while the uncertainty reflects the previous knowledge of the resonance strength. The lowest known resonance in 24 Mg(p,γ) 25 Al at E = 214 keV is less constraint than the reaction 25 Mg(p,γ) 26 Al and only a few measurements exist. However, the most recent experiment of the TUNL group [24] resulted in a 25% higher resonance strength than recommended by NACRE. The NACRE value is essentially based on the result of Trautvetter and Rolfs [23]. This experiment [23] has to be corrected for updated stopping power data [33] which leads to a 6% higher ωγ value, but still only slightly consistent with [24] (see Table IV). The present result, ωγ = 10.6±0.6 meV, is in good agreement with [23] and differs from [24] by about 2σ. A weighted average of these 3 experiments leads to a recommended value of ωγ = 11.2 ± 0.9 where the uncertainty was determined by the scale factor method [48] with an inflation factor of χ 2 /χ 2 (P = 0.5) [49] for χ 2 = 4.8 and 2 d.o.f. The reaction 26 Mg(p,γ) 27 Al was measured for completeness although its astrophysical relevance is very small. However, the available experimental data -compiled in NACRE [40] -differ considerably from each other. Two experiments [18,50] result in a value as low as ωγ = 250 meV while all other experiments [51][52][53][54] give a resonance strength around ωγ ≈ 700 meV. Therefore, the recommended value of the NACRE compilation of ωγ = 590 ± 10 meV is questionable and in particular the uncertainty is not justified. The present result of ωγ = 274 ± 15 meV for the E = 326 keV resonance of the reaction 26 Mg(p,γ) 27 Al is in excellent agreement with the most recent result of [21] and in good agreement with the results of [18,50]. This comparison suggests in view of the internal consistency of the present experiment that experiments giving a 3-fold higher ωγ, e.g. [51][52][53][54], should be discarded. The weighted average of the 3 remaining experiments leads to a recommendation of ωγ = 269 ± 10 meV.
IV. AMS MEASUREMENT
A. Requirements and irradiation
An AMS measurement provides the ratio between a rare and an abundant isotope in the same sample. Therefore, the reaction 25 Mg(p,γ) 26 Al represents an ideal case for an AMS study of the resonance strengths. The lowest observable isotopic ratio determines the sensitivity limit and is usually in the order of 10 −15 . Adding a known amount of stable 27 Al to the sample material after proton irradiation allows for a precise determination of the absolute 26 Al content in the sample from the experimental 26 Al/ 27 Al isotopic ratio. Due to the short lifetime of the 26 Al isomeric state, this off-line technique yields directly the astrophysical important ground state contribution to the resonance strength.
The irradiation of the enriched 25 Mg targets (section II E) was carried out in the BGO standard geometry. The capture γ-ray emission was observed concurrently with the BGO crystal and the ratio between AMS and γ-ray spectroscopy could be determined directly. Any systematic differences between the two methods as suggested by the measurement of Arazi et al. [22] would lead to a ratio different from unity.
The difficulty of such an AMS measurement is a reliable monitoring of the target stoichiometry and quality during the irradiation. In γ-ray spectroscopy this can be easily achieved by a resonance scan of a well-known resonance at higher energy of the same reaction. In general, in AMS the use of this technique is very limited, often it cannot be applied at all since the procedure would lead to a large amount of additional reaction products, i.e. 26 Al. Fortunately, the observed E = 304 keV resonance in 25 Mg(p,γ) 26 Al is by itself strong enough that a complete resonance scan could be measure in a relatively short period. The collected charge during the resonance scans was always kept below 0.1% of the total irradiation charge. Therefore this contribution has been neglected in the analysis. The change of the target stoichiometry during irradiation was taken into account.
In order to study the influence of experimental parameters on the AMS results, e.g. beam power, target A was irradiated with high beam intensity and short irradiation time while target B was irradiated with low intensity over a longer period. A non-irradiated target -target Cserved as an AMS blank sample for background reference (Table V).
B. AMS measurement of 26 Al/ 27 Al isotopic ratios
The details of the AMS system are published elsewhere [56] while the chemical preparation of the sample cathodes is described in Appendix A. The 26 Al/ 27 Al isotopic ratio was obtained measuring the 27 Al current in a Faraday cup and the 26 Al ions in a Silicon detector at the final focal plane. A high voltage applied to the chamber of the analyzing magnet allowed a fast switching between 26 Al and 27 Al measurements.
The obtained isotopic ratio ( 26 Al/ 27 Al) exp , however, depends on the experimental conditions of the AMS system and need to be normalized to a reference sample. The isotopic ratios ( 26 Al/ 27 Al) ref of the reference samples V1 and M11 are well known from other experiments with a precision of 2% [55]. The comparison of ( 26 Al/ 27 Al) exp and ( 26 Al/ 27 Al) ref for the reference samples lead to a correction factor which has to be applied to the experimental isotopic ratios of the 25 Mg samples in order to evaluate their absolute isotopic ratios ( 26 Al/ 27 Al) abs . Finally, the known amount of 27 Al added during the sample preparation (see Table V) allows together with the absolute isotopic ratios for calculating the total number of 26 Al nuclei in the sample and, in turn, the resonance strength for the 25 Mg(p,γ) 26 Al resonance (see below).
The best overall efficiency, i.e. the number of 26 Al detected with respect to the 26 Al pressed into the cathode, was about 2 × 10 −4 . Table VI shows the results of the AMS measurements for the various samples performed at the CIRCE laboratory. The experimental ratios of 304-S1, 304-S2, and 304-S3 are compatible within the statistical errors proving the reproducibility of the AMS measurements. In addition, we report the isotopic ratios of blank cathodes filled with a) standard aluminum oxide (sample Al 2 O 3 ) and b) the material resulting from the chemical procedure applied to the non-irradiated 25 Mg target C (sample 304-BLK). The results of these two blank samples are in perfect agreement confirming that no additional 26 Al background is present in the 25 Mg targets. The 26 Al background level was equal to a isotopic ratio of 8 × 10 −15 and, thus, negligible with respect to the counting rate of the irradiated samples.
C. Results and discussions
The ground state resonance strength ωγ gs = f 0 · ωγ of the E = 304 keV resonance in 25 Mg(p,γ) 26 Al can be evaluated from eq. (1). The factor f 0 is the probability for forming the ground state of 26 Al which in the present experiment was determined with the HPGe detector (section III B 3). However, the stoichiometry of the targets was not determined independently, therefore an independent evaluation of ωγ gs of the E = 304 keV resonance could not be done. Nevertheless, the comparison between the two independent, relative results, i.e. AMS and BGO γ-ray spectroscopy, is testing the precision of both methods. The ratio between the two methods can be determined with high precision by the relation:
(ωγ gs ) AMS (ωγ gs ) BGO = Y26 Al f 0 · Y γ = N abs ( 26 Al) f 0 · ǫ BGO N γ(5)
where Y26 Al and Y γ are 26 Al and γ-ray yield per incident projectile. The parameter N γ /ǫ BGO can be evaluated with the GEANT4 simulation from the related BGO γray spectra, analyzed with the same procedure described previously. The different dead time for both yield values need to be taken into account. The AMS measurement has an uncertainty of 2% arising from the normalization to the reference samples [55]. Additional systematic errors take into account the uncertainties of the carrier weight (0.2%) and the efficiency of the chemical treatment of the samples (1%). This leads to an overall systematic uncertainty of 2.5% for the AMS measurement. The uncertainties of the corresponding γ-ray measurement are given by the absolute efficiency determination for the BGO detector (3%), the decay scheme (3%), and the ground state probability f 0 (1%): a total systematic error of 4%. The uncertainty of the effective stopping power and the charge integration represent common uncertainties to both methods and, therefore, were not taken into account in the error estimate for the ωγ ratio.
The results are shown in Table VII. The ratio of (ωγ gs ) AMS /(ωγ gs ) BGO = 1.02 ± 0.05 is clearly consistent with unity which demonstrates that there is no major systematic uncertainties in the efficiency determination of the present γ-ray measurements. Furthermore, a high quality AMS measurement is possible and no systematic difference exists between AMS result and γ-ray data.
V. SUMMARY AND CONSEQUENCES
In the present work we have measured properties (strengths ωγ and branching ratios) of the E = 214, 304, (Table VIII). We underline that the new results were obtained from measurements with partly independent approaches, yielding a remarkable agreement among them. This is a strong evidence for the internal consistency of the present approach and demonstrates the achieved accuracy. Moreover, particular attention was paid to the critical issue of target stoichiometry and its variation under beam bombardment: a severe problem for Mg targets. The reduction of the uncertainties of these Mg-Al cycle reactions is, by itself, an important improvement in the analysis of the nucleosynthesis scenarios relevant for the production and destruction of Mg and Al isotopes. A detailed discussion of the astrophysical implications is beyond the scope of the present work and will be published elsewhere. Nevertheless, the particular impact on each reaction rate and our recommendations are discussed in the following. The 25 Mg(p,γ) 26 Al resonance strength given by previous compilations, NACRE [40], or recent evalunations [44] is basically confirmed by the present experiment. Therefore, the nuclear reaction rate recommended in these compilations needs only a minor adjustment and is not recalculated here. We suggest to use the existing compilations until the results for low-energy 25 Mg(p,γ) 26 Al resonances, i.e. the resonances at E = 93 and 190 keV, will be available. However, the uncertainty of the E = 304 keV resonance could be reduced to 4%. This is important since this resonance will serve as a normalization standard for the further measurements at lowenergies and, thus, these results will benefit strongly from the present work. Furthermore, the primary branching ratios have been measured with high precision yielding a ground state feeding probability, f 0 = 87.8 ± 1.0%.
An additional AMS measurement based on an irradiation performed simultaneously to a γ-ray detection showed no systematic difference between both detection techniques. The different result of Arazi et al. [22] was not observed in the present experiment and most likely caused by an unidentified oxygen contamination in the sample during the irradiation. In general, the normalization of the AMS measurement, e.g. the irradiation, is a challenging experimental task. In the present experiment the standard approach, i.e. resonance scans of the thick-target yield, appeared to be sufficient. Certainly, at lower energies -in particular for the E = 93 and 190 keV resonances -this approach cannot be applied due to the production of additional 26 Al nuclei yielding a sizeable fraction of the total 26 Al amount. However, such a normalization is of utmost importance for a reliable AMS measurement. Alternatively, the 26 Al yield can be monitored with the E = 214 keV resonance of 24 Mg(p,γ) 25 Al. This resonance is located in an energy window where no additional 26 Al production is expected.
The case of 24 Mg(p,γ) 25 Al is rather complex. The present recommended ωγ of the lowest resonance, E = 214, in this reaction is lower by more than 10% compared to the latest published measurement by Powell et al. [24]. However, as mentioned in the introduction a strong direct capture component dominates the resonance contribution and a reanalysis of the reaction at astrophysical energies by using an R-matrix formalism may prove worthwhile. Moreover, as demonstrated in [24] the narrow resonance approximation [9] cannot be applied to evaluate the reaction rate on the low-energy tail of this resonance. Hence, the γ-width Γ γ need to be known to a high precision which is presently not achieved. A detailed study of the influence of all parameters involved is far beyond the scope of the present study and therefore postponed to a future work. As a consequence we do not give an updated nuclear reaction rate for the 24 Mg(p,γ) 25 Al reaction at the present stage.
The reaction 26 Mg(p,γ) 27 Al proceeds very fast at all temperatures compared to the other Mg-Al cycle reactions and, therefore, its astrophysical implications are negligible. Nevertheless, an apparent discrepancy in the literature (see for example [40]) has been solved and the strength of the E = 326 keV 26 Mg(p,γ) 27 Al resonance was measured with a high precision (Table VIII).
PACS numbers: 25.40.Ep, 25.40.Lw, 26.20.Cd, 26.30.-k I. INTRODUCTION
FIG. 1 .
1(Color online) Floor plan of the 400 kV LUNA accelerator with the 2 beam lines (upper panel) and the 45 • II beam line with the BGO detector setup (lower panel). All measures are given in mm.
FIG. 2 .FIG. 3 .
23Spectrum of the Rutherford backscattering (RBS) analysis of the natural Mg target. The solid line is a fit to the data. The chemical composition of the target and the target layer thickness is derived from this fit. Thick-target yield curves obtained for the E = 326 keV resonance of 26 Mg(p,γ)27 Al. These data serve as a quality check of the target. The height of the plateau is proportional to the effective stopping power and, therefore, related to the stoichiometry change. The filled squares represent the scan on the fresh target and the open circles, filled triangles, and stars are scans after a charge of 7.2, 8, and 9.6 C, respectively.
24 Mg) = 20.7 eVcm 2 /10 15 atoms, ε eff ( 25 Mg) = 140.2 eVcm 2 /10 15 atoms, and ε eff ( 26 Mg) = 123.9 eVcm 2 /10 15 atoms. The error of these effective stopping power values is on the order of 4.5 % each based on the stopping power uncertainties [33] of 2.3 and 4.4 % for protons in oxygen and magnesium, respectively, and including the stoichiometry uncertainty. B. Measurements with the HPGe detector
FIG. 4 .
4Detection efficiency of the HPGe detector as a function of the γ-ray energy for various distances target -detector front face. The efficiency has been determined with γ-ray sources (cross) and the reactions 14 N(p,γ) 15 O (stars) and 24 Mg(p,γ) 25 Al (squares). The filled symbols denote the data corrected for summing while the open symbols are uncorrected. The solid lines represent the fit curves from eq. 3.
where E γ is in MeV, d in cm and a = 8.06 × 10 −2 , b = −0.488, c = −0.141, d 0 = 0.941, α = 10.188 and β = −0.276, respectively, are fit parameters in the global fit.
FIG. 5 .
5The γ-ray spectrum taken at the E = 304 keV 25 Mg(p,γ)26 Al resonance showing the most prominent primary transitions and some very important secondary transitions.
O and 18 O(p,γ)19 F can be observed in the background spectra. In particular the well known E γ = 6.13 MeV line of the strong 19 F(p,αγ)16 O resonance at E = 324 keV is present in the 26 Mg(p,γ) 27 Al measurement. Background and on-resonance measurement are online) Gamma-ray spectra taken with the BGO detector at the E = 304 keV resonance of 25 Mg(p,γ) 26 Al: a) single sum (of all 6 crystals); b) total sum spectrum. The shaded area, solid red line, and dotted line represent the measurement, the GEANT4 simulation, and the background run, respectively. The hatched area illustrates the fitted energy region (for details see text).
FIG
. 7. (Color online) Single sum γ-ray spectrum (shaded area) as shown inFig. 6compared to a simulation with the present branching ratio data (red solid line) and from Ref.[16] (dashed line).
D. Discussion of γ-ray measurements 1. The E = 304 keV resonance in 25 Mg(p,γ) 26 Al
2 .
2The E = 214 keV resonance in 24 Mg(p,γ) 25 Al
3 .
3The E = 326 keV resonance in 26 Mg(p,γ) 27 Al
TABLE I .
IPrimary γ-ray branching ratios of the E = 214 keV 24 Mg(p,γ)25 Al resonance from present and previous work.EX
present work [%]
[24]
[36]
1790
< 0.05
< 0.8
< 0.3
1613
< 0.05
< 0.8
< 0.3
945
15.6 ± 0.3
15.6 ± 1.1
15.7 ± 0.6
452
81.7 ± 1.6
81.7 ± 3.4
81.6 ± 1.1
0
2.70 ± 0.07
2.7 ± 0.3
2.69 ± 0.08
TABLE II .
IIPrimary γ-ray branching ratios of the E = 304 keV 25 Mg(p,γ)26 Al resonance.EX
present work [%]
[16] a
[18] b
5916
0.09 ± 0.02
0.11 ± 0.02
5726
0.10 ± 0.01
0.08 ± 0.02
0.12 ± 0.03
5457
0.14 ± 0.02
5396
0.22 ± 0.02
0.23 ± 0.03
0.35 ± 0.05
4940
0.08 ± 0.01
0.09 ± 0.02
4622
0.28 ± 0.07
0.2 ± 0.04
0.38 ± 0.06
4599
0.12 ± 0.01
0.11 ± 0.02
0.13 ± 0.04
4548
1.30 ± 0.07
1.13 ± 0.06
2.0 ± 0.1
4349
0.03 ± 0.01
0.07 ± 0.02
4206
0.25 ± 0.02
0.24 ± 0.03
0.25 ± 0.05
4192
19.1 ± 0.3
18.7 ± 0.6
14.7 ± 0.8
3963
0.17 ± 0.01
0.22 ± 0.03
0.12 ± 0.05
3750
0.92 ± 0.02
0.97 ± 0.06
1.5 ± 0.1
3681
1.09 ± 0.03
0.90 ± 0.08
0.71 ± 0.08
3675
0.86 ± 0.13
0.61 ± 0.10
0.59 ± 0.06
3596
4.29 ± 0.07
4.2 ± 0.2
3.3 ± 0.2
3160
11.4 ± 0.2
11.4 ± 0.4
15.6 ± 0.9
3073
0.11 ± 0.04
0.18 ± 0.05
0.08 ± 0.05
2913
3.04 ± 0.05
3.0 ± 0.1
4.2 ± 0.3
2661
1.00 ± 0.02
0.97 ± 0.06
1.6 ± 0.1
2545
1.46 ± 0.03
1.38 ± 0.08
0.9 ± 0.1
2365
0.47 ± 0.02
0.87 ± 0.19
0.27 ± 0.07
2069
6.0 ± 0.1
5.7 ± 0.2
6.5 ± 0.4
1759
16.1 ± 0.3
15.8 ± 0.5
22.7 ± 1.3
417
31.8 ± 0.5
33 ± 1
24 ± 1.4
0
0.058 ± 0.004
a branchings <1% are given in
TABLE III .
IIIPrimary γ-ray branching ratios of the E = 326 keV 26 Mg(p,γ)27 Al resonance. lar correlation effects have not been taken into account in the simulation. In order to allow for a full analysis of the experimental spectra also simulations for background reactions like 11 B(p,γ) 12 C, 18 O(p,γ) 19 F or 19 F(p,αγ)16 O could be obtained from the code.EX
present work [%]
[18]
7858
0.09 ± 0.02
0.17 ± 0.03
7280
< 0.01
0.03 ± 0.01
7071
0.30 ± 0.02
0.25 ± 0.02
6993
0.17 ± 0.02
0.20 ± 0.02
6813
12.1 ± 0.1
12.6 ± 0.7
6776
0.06 ± 0.01
0.06 ± 0.02
6651
0.45 ± 0.02
0.50 ± 0.04
6605
1.26 ± 0.03
1.41 ± 0.09
6158
0.71 ± 0.03
0.72 ± 0.05
6116
0.44 ± 0.02
0.34 ± 0.04
6081
0.59 ± 0.03
0.55 ± 0.05
5752
0.80 ± 0.03
0.89 ± 0.06
5551
2.07 ± 0.05
0.39 ± 0.03
5438
0.22 ± 0.03
0.52 ± 0.04
5248
0.94 ± 0.03
0.95 ± 0.06
5156
0.71 ± 0.03
0.03 ± 0.02
4812
0.54 ± 0.03
0.59 ± 0.05
4410
2.96 ± 0.07
3.1 ± 0.2
4055
10.9 ± 0.2
10.7 ± 0.6
3957
2.64 ± 0.07
2.6 ± 0.2
3680
14.5 ± 0.2
13.9 ± 0.8
2982
19.7 ± 0.3
20.2 ± 0.1
2735
4.43 ± 0.09
4.3 ± 0.3
1014
2.04 ± 0.07
2.3 ± 0.2
844
19.3 ± 0.3
20.2 ± 0.1
0
2.06 ± 0.05
2.5 ± 0.2
TABLE IV .
IVResonance strengths of proton captures resonances on magnesium isotopes from the present experiment and previous work.E [keV]
ωγ [meV]
present work
previous work
HPGe a
BGO b
weighted mean c
[40]
[18]
[24]
[23]
[21]
24 Mg(p,γ) 25 Al
214
10.4 ± 0.4 11.1 ± 0.6
10.6 ± 0.6
10 ± 2
12.7 ± 0.9 10.1 ± 2.0 d
25 Mg(p,γ) 26 Al
304
30.7 ± 1.1 30.6 ± 1.3
30.7 ± 1.7
31 ± 2
30 ± 4
26 Mg(p,γ) 27 Al
326
276 ± 11
272 ± 12
274 ± 15
590 ± 10 250 ± 30
273 ± 13
TABLE V .
VExperimental parameters of the 25 Mg AMS sample irradiation and the amount of27 Al added to each sample during the chemical process (for details see text).AMS sample target Ep [keV] Charge [C] 27 Al [mg]
304-S1
A
321
1.316 a
1.0
304-S2
A
321
1.316 a
1.0
304-S3
A
321
1.316 a
1.0
304-R1
B
322
0.0187
0.5
304-BLK
C
−
−
0.5
a Total charge collected on target A was 5.264 C, but the
extracted material was divided into 4 samples.
TABLE VI .
VIResults of the 26 Al/ 27 Al ratios determination with AMS. The values have been obtained from two different measurement periods with independent reference checks as listed. sample tAMS [s]Ī27 Al [pnA] Nmeas( 26 Al) ( Al ) abs b [×10 −11 ] N abs ( 26 Al) b,c [×106 ] The uncertainty includes the statistical error and the accuracy of the reference samples. c The uncertainty of the 27 Al carrier weight was added quadratically. d Average value from the two samples 304-S1 and 304-S2.26 Al
27 Al )exp [×10 −11 ] (
26 Al
27 Al ) ref
a [×10 −11 ] (
26 Al
27 V1
3700
77
26668
1.50 ± 0.01
1.62 ± 0.03
M11
3700
86
17394
0.875 ± 0.007
1.00 ± 0.02
Al2O3
3700
101
17
0.0007 ± 0.0002
304-BLK 3700
20
11
0.0008 ± 0.0002
304-S1
3700
80
16769
0.906 ± 0.008
1.01 ± 0.02 d
225 ± 4
304-S2
3700
61
12602
0.893 ± 0.007
M11
6400
79
26077
0.825 ± 0.007
1.00 ± 0.02
304-S3
10400
21.4
11478
0.825 ± 0.005
1.00 ± 0.02
223 ± 5
304-R1
10400
22.6
327
0.0223 ± 0.0002
0.0270 ± 0.0006
3.02 ± 0.07
a from Ref. [55]
b
TABLE VII .
VIIComparison between AMS result and BGO γray measurement. Target N26 Al [×10 6 ] a Nγ · f0/ǫBGO [×10 6 ] b 02 ± 0.13 1.00 ± 0.04 c average 1.02 ± 0.05 d a includes systematic uncertainty of 2.5% (see text). b includes systematic uncertainty of 4% (see text). c statistical uncertainties only d systematic uncertainty not common to both methods were added quadratically and 326 keV resonances in the reactions 24 Mg(p,γ) 25 Al, 25 Mg(p,γ) 26 Al, and 26 Mg(p,γ) 27 Al, respectively. These new results together with selected previous work (see section III D) are used to calculate updated recommended values for the resonance strengthsAMS
BGO prompt γ-ray
(ωγgs ) AMS
(ωγgs ) BGO
304-S
224 ± 7
218 ± 9
1.03 ± 0.03 c
304-R
3.02 ± 0.12
3.
TABLE VIII .
VIIISummary of the new recommended resonance strength values obtained as weighted average from present and previous work as discussed in section III D. reaction and resonance ωγ [meV] 24 Mg(p,γ) 25 Al E = 214 keV 11.2 ± 0.9 25 Mg(p,γ) 26 Al E = 304 keV 30.8 ± 1.3 26 Mg(p,γ) 27 Al E = 326 keV 271 ± 10
Appendix A: Chemical extraction of26Al from the25Mg bulk Aluminum can be extracted from the sputter ion source either as negative Al − ions or oxide molecules AlO − . In spite of the higher extraction efficiency (about a factor 20[22]) for AlO − in the present measurement the Aluminum was injected as Al − , since Mg does not form negative ions, while MgO does. Hence, isobar interferences were avoided in the AMS measurement. As a consequence all the target material removed from the target backing could in principle be used for preparing the sputter cathode. However, the amount of material needed for a single cathode is very small and, thus, a reduction of the Mg bulk material, i.e. a purification process, was necessary. As a first step a stoichiometric amount of 27 Al -serving as a carrier -had to be added to the target material before the chemical extraction. The standard extraction procedure is based on the precipitation of the Al as hydroxide using ammonia followed by ignition of the precipitate to educe aluminum oxide:This procedure is limited because: i) The aluminum hydroxide precipitates as a gel which strongly retains the mother solution and purification as well as manipulation become difficult. ii) The 26 Al yield scales with the reagent concentrations. In order to extract reasonable yields a high 26 Al concentration is needed implying very small total volumes. Small volumes are difficult to handle. iii) After dehydration the final product (Al 2 O 3 ) is a fine powder adhering to the walls and, thus, increase the risk of losses during the cathodes preparation.Therefore, a new improved procedure, based on liquidliquid extraction of an organic aluminum complex, has been developed and optimized in the LNGS chemistry laboratory. The Al reacts with three 8-hydroxyquinoline molecules to form a coordination compound insoluble in water but highly soluble in chlorinated organic solvents so that it can be extracted and separated. The reaction product can be converted to Aluminum oxide by heating to high temperature. The main advantages of the liquidliquid extraction are:• The handling is easier because of larger liquid volumes.• The yields -measured with ICPMS (inductivelycoupled-plasma mass spectrometry) technique -are fully reproducible and found to be > 99% of the initial value.• The remaining magnesium concentration is lower than 1% of the original value with a proper control of the pH value.• The final product, Al 2 O 3 , is obtained in a well crystalized form, not adhering to the vial walls.After the chemical extraction, the mixture of 26 Al and 27 Al material was pressed together with a comparable amount of Cu powder in Cu cathodes.
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|
[] |
[
"On light dilaton extensions of the Standard Model ⋆",
"On light dilaton extensions of the Standard Model ⋆"
] |
[
"Eugenio Megías \nMax-Planck-Institut für Physik (Werner-Heisenberg-Institut)\nFöhringer Ring 6D-80805MunichGermany\n",
"Oriol Pujolàs \nInstitut de Física d'Altes Energies (IFAE)\nThe Barcelona Institute of Science and Technology (BIST)\nCam-pus UABE-08193Bellaterra (Barcelona)Spain\n",
"Mariano Quirós \nInstitut de Física d'Altes Energies (IFAE)\nThe Barcelona Institute of Science and Technology (BIST)\nCam-pus UABE-08193Bellaterra (Barcelona)Spain\n\nInstitució Catalana de Recerca i Estudis Avançats (ICREA)\nCampus UABE-08193\n\nBellaterra (Barcelona)\nSpain\n\nICTP South American Institute for Fundamental Research\nInstituto de Física Teórica\nUniversidade Es-tadual Paulista\nSão PauloBrazil\n"
] |
[
"Max-Planck-Institut für Physik (Werner-Heisenberg-Institut)\nFöhringer Ring 6D-80805MunichGermany",
"Institut de Física d'Altes Energies (IFAE)\nThe Barcelona Institute of Science and Technology (BIST)\nCam-pus UABE-08193Bellaterra (Barcelona)Spain",
"Institut de Física d'Altes Energies (IFAE)\nThe Barcelona Institute of Science and Technology (BIST)\nCam-pus UABE-08193Bellaterra (Barcelona)Spain",
"Institució Catalana de Recerca i Estudis Avançats (ICREA)\nCampus UABE-08193",
"Bellaterra (Barcelona)\nSpain",
"ICTP South American Institute for Fundamental Research\nInstituto de Física Teórica\nUniversidade Es-tadual Paulista\nSão PauloBrazil"
] |
[] |
We discuss the presence of a light dilaton in Conformal Field Theories deformed by a single scalar operator, in the holographic realization consisting of confining Renormalization Group flows. Then, we apply this formalism to study the extension of the Standard Model with a light dilaton in a 5D warped model. We study the spectrum of scalar and vector perturbations, compare the model predictions with Electroweak Precision Tests and find the corresponding bounds for the lightest modes. Finally, we analyze the possibility that the Higgs resonance found at the LHC be a dilaton.
|
10.1051/epjconf/201612605010
|
[
"https://arxiv.org/pdf/1512.06702v1.pdf"
] | 55,256,038 |
1512.06702
|
117867e76d6545fa90144dfd78fd2b746620fed0
|
On light dilaton extensions of the Standard Model ⋆
21 Dec 2015
Eugenio Megías
Max-Planck-Institut für Physik (Werner-Heisenberg-Institut)
Föhringer Ring 6D-80805MunichGermany
Oriol Pujolàs
Institut de Física d'Altes Energies (IFAE)
The Barcelona Institute of Science and Technology (BIST)
Cam-pus UABE-08193Bellaterra (Barcelona)Spain
Mariano Quirós
Institut de Física d'Altes Energies (IFAE)
The Barcelona Institute of Science and Technology (BIST)
Cam-pus UABE-08193Bellaterra (Barcelona)Spain
Institució Catalana de Recerca i Estudis Avançats (ICREA)
Campus UABE-08193
Bellaterra (Barcelona)
Spain
ICTP South American Institute for Fundamental Research
Instituto de Física Teórica
Universidade Es-tadual Paulista
São PauloBrazil
On light dilaton extensions of the Standard Model ⋆
21 Dec 2015
We discuss the presence of a light dilaton in Conformal Field Theories deformed by a single scalar operator, in the holographic realization consisting of confining Renormalization Group flows. Then, we apply this formalism to study the extension of the Standard Model with a light dilaton in a 5D warped model. We study the spectrum of scalar and vector perturbations, compare the model predictions with Electroweak Precision Tests and find the corresponding bounds for the lightest modes. Finally, we analyze the possibility that the Higgs resonance found at the LHC be a dilaton.
Introduction
The discovery of a resonance at the LHC with the properties of a Standard Model-Higgs in 2012 has incited an increasing interest in the study of the nature of the Higgs boson, as this will provide very likely one of the most important indications for the detection of new physics. An explanation of the light Higgs mass is quite difficult nowadays without the use of unnatural fine tuning of parameters, an issue that is known as the hierarchy problem, see e.g. Ref. [1] and references therein. So, it would be very interesting to study in what kind of scenarios the Standard Model (SM) can be embedded such that the hierarchy problem is solved. One possibility is that the SM be part of a nearly-conformal sector. These models can be realized by means of strong dynamics/extra-dimensions and the Electroweak scale would arise from the spontaneous breaking of conformal invariance (SBCI). It has been recently appreciated that the SBCI can be naturally associated with the appearance of a light dilaton in the spectrum [2][3][4][5][6][7]. Within this realization, the dilaton would be the lightest state of the strong/extradimensional sector. Going further, it comes out the intriguing possibility that, in fact, the dilaton can play the role of a Higgs impostor, so that the 125 GeV resonance would then be the first instance of new physics. The exploration of this possibility will be one of the main motivations of this work. We will do this by considering a class of soft-wall scenario that allows to fully exploit the modelling capabilities of these theories.
Light dilatons in extra dimensional models
SBCI leads to the existence of a non-zero value for the vacuum expectation value of the dilaton field, which is the Goldstone boson associated with this symmetry. While Conformal Field Theories (CFT) do not naturally exhibit SBCI, it has been recently proposed in Ref. [2] a mechanism that allows for light dilatons in Quantum Field Theories that are close to conformal invariance. This is based on the consideration of certain deformations of CFTs
L = L CFT − λ O ,(1)
where O is a nearly marginal operator, i.e. dim(O) = 4 − ∆ with ∆ ≪ 1. A holographic realization of this mechanism has been checked to work in Refs. [3][4][5][6] (see also Ref. [7]). In this Section we describe this mechanism in generic 5D warped models, and then particularize the results to a specific benchmark model.
Renormalization Group flows and CFT deformations
Let us consider a coupled scalar-gravitational system in 5D defined by the action
S = M 3 d 4 xdy √ −g R − 1 2 (∂ M φ) 2 − V(φ) − M 3 α d 4 xdy √ −g 2V α (φ)δ(y − y α ) ,(2)
where φ is a scalar field, V α (α = 0, 1) are UV and IR 4D brane potentials located at y 0 = y(φ 0 ) = 0 and y 1 = y(φ 1 ) respectively, and M is the 5D Planck scale. The deformed CFT of Eq. (1) can be realized by a domain wall geometry in which the scalar field develops a profile in the extra dimension y of the form
ds 2 = dy 2 + e −2A(y) η µν dx µ dx ν , φ = φ(y) .(3)
By seeking for solutions of the equations of motion with the near boundary expansion φ(y) = λ e ∆y + O e (4−∆)y + · · · , y → −∞ ,
one can identify the deformation parameter and the condensate as the leading and sub-leading mode respectively. The holographic Renormalization Group flows can be studied better with the β-function, defined holographically as β(φ) := − ∂φ ∂A . The equations of motion lead to the following first order differential equation
β(φ) β ′ (φ) = 1 2 β(φ) 2 − 24 β(φ) 3 + V ′ (φ) V(φ) .(5)
Associated with this equation there is an integration constant which is identified as the condensate O of the operator along the deformation direction, Eq. (1), and the boundary condition that fixes the physical value of the condensate O phys is that the IR end of the flow be the least singular possible, see e.g. [5,7,8]. We are interested in confining flows, i.e. those which lead to a discrete spectrum in all sectors. In the absence of the IR brane, these flows are characterized by the IR limit
β(φ) → β ∞ with √ 6 < −β ∞ < 2 √ 6 ,(6)
where the lower and upper bounds correspond to the confinement and good IR singularity criterion respectively. A typical profile for such flows is displayed in Fig. 1. It is interesting to see that the flow has three different regimes: i) deformation-dominated regime in the UV, ii) condensate-dominated regime, and iii) confinement region in the IR. V(φ) (dotted blue) and −β(φ) (solid red) as a function of φ. The three regimes (deformation, condensation and confinement) are clearly distinguishable in β(φ). We have considered Model A of Ref. [5], which is characterized by O phys 0.
Spectrum of excitations and type of dilatons
To compute the scalar spectrum and in particular its lightest mode, known as the radion/dilaton, we have to consider a scalar perturbation of the background, and this leads to a contribution in the metric of the form ds 2 ∼ e −2(A+F) . Then the equation of motion of the excitation modes F n is written as [
9] 1 F n − 2ȦḞ n − 4ÄF n − 2φ φḞ n + 4Ȧφ φ F n = −e 2A m 2 n F n .(7)
This is supplemented with boundary conditions on the branesḞ n | y α = f (V α , m n ), which are obtained by integrating in a small interval around y = y α . An approximate solution of Eq. (7) assuming light modes, i.e. m 2 n ≪ Λ 2 IR where Λ IR is the mass gap in the spectrum, leads to a mass formula which allows to distinguish between two kind of light dilatons [5,10]:
• Hard dilatons: those dominated by the value of the β function at the IR brane location. The mass of the dilaton scales like m 2 dil ∼ β 2 IR Λ 2 IR . This is the realization that has been discussed in Refs. [3,4]. • Soft dilatons: those dominated by the value of the β function in the condensation scale. In this case a light dilaton can be realized whenever i) the β function is small at the condensation scale (β cond ≪ 1) and ii) the rise towards confinement (β con f ∼ 1) is fast enough. Then
m 2 dil ∼ (β cond ) κ Λ 2 IR ≪ Λ 2 IR ,(8)
where the precise value of the constant κ > 0 depends on the particular model [5,7]. The dilaton turns out to be naturally light for nearly marginal deformations, for which β cond ∼ ∆ ≪ β con f .
The dilaton corresponds to the fluctuation of the condensate, regardless of whether O vanishes or not. In the following we will simplify the model by setting O = 0, as the physics is qualitatively similar to the one from models with O 0. This can be done by just considering an analytic β function, as then the integration constant in Eq. (5) is set to vanish. We choose
β(φ) = −6ac 1 + e −aφ −1 ,(9)
where c and a are parameters which govern the IR value of β and its slope at φ = 0 respectively. We assume that the brane potentials dynamics have fixed (φ 0 , φ 1 ) to solve the hierarchy problem, i.e. A(φ 1 ) ≃ 35, where we have normalized the warp factor to A(φ 0 ) = 0. We show in Fig. 2 the dilaton mass obtained from a numerical solution of Eq. (7). The realization of the dilaton for a 0.6/c and a 0.6/c is hard and soft respectively, and the existence of the peak is a signal of the change of regime between both pictures. We also display in Fig. 2 (B) the spectrum of vector perturbations, which we refer in the following as KK gauge bosons. See Ref. [10] for details. 1 The dots stand for derivatives with respect to y.
Electroweak breaking
We will now introduce the electroweak sector in the theory. Let us consider the SM propagating in the 5D space described in Section 2. In addition to the 5D SU(2) L × U(1) Y gauge bosons W i M (x, y), B M (x, y) with i = 1, 2, 3 and M = (µ, 5), we define the SM Higgs as
H(x, y) = 1 √ 2 e iχ(x,y) 0 h(y) + ξ(x, y) ,(10)
where h(y) is the Higgs background. The action of the model is
S 5 = d 4 xdy √ −g − 1 4 W 2 MN − 1 4 B 2 MN − |D M H| 2 − V(H) ,(11)
where V(H) is the 5D Higgs potential. Electroweak symmetry breaking is triggered on the IR brane. After the breaking, the lightest modes are separated by a gap from the KK spectrum, and the masses for the W and Z bosons are then approximately given by 2
m 2 V ≈ 1 y 1 y 1 0 dyM 2 V (y) , V = W, Z ,(12)
where the y-dependent bulk masses are defined as
M W (y) = g 5 2 h(y)e −A(y) , M Z (y) = 1 c W M W (y) ,(13)
and g 5 is the 5D gauge coupling, which is related to the 4D one g by g 5 = g √ y 1 . After solving the equations of motion for the Higgs field h(y), one can obtain from Eqs. (12)-(13) the masses for the gauge bosons. To compare the model predictions with Electroweak Precision Tests (EWPT) a convenient parameterization is using the (S , T, U) variables in [11,12]. From the fitted values for S and T as [13]: T = 0.05 ± 0.07, S = 0.00 ± 0.08 , (90% correlation) , one gets the result in Fig. 3 (A). We find a region in a such that m KK = O(TeV) and m dil O(100) GeV for c = 1. 3 Other values of c lead to similar conclusions.
Coupling of the radion/dilaton to Standard Model matter fields
A light dilaton has similar interactions with matter as the Higgs, so an obvious question arises: Is the Higgs-like resonance found at LHC actually a dilaton? There are previous studies in the literature on the coupling of the radion to SM matter fields, and they usually consider that the matter is localized on the IR brane [9,14]. In this work and motivated by EWPT we are assuming that the matter and Higgs fields are localized in the bulk. If the scalar fluctuation decomposes as F(x, y) = F(y)R(x), we can compute the coupling of the dilaton to the massive gauge fields (W µ and Z µ ) normalized as
L rad = − r(x) v 2c W m 2 W W µ W µ + c Z m 2 Z Z µ Z µ ,(15)
where r(x) is the canonically normalized radion field with kinetic term 1 2 ∂ µ r 2 . The case c W = c Z = 1 corresponds to the SM Higgs coupling. After expanding Eq. (11) to linear order in the perturbations, and using the massless radion approximation F = e 2A , one gets the following result for c V (V = W, Z)
c V = 2 v m 2 V √ 6e −A 1 M Pl ρ 2 e −2A e 2A−2A 1 1/2 ky 1 ky 1 0 e 4A−4A 1 y 0 h 2 e −2A y 1 0 h 2 e −2A − ky ky 1 2 d(ky) ,(16)
where M Pl = 2.4 × 10 18 GeV is the 4D (reduced) Planck mass. We show in Fig. 3 (B) the numerical result of c Z in the same regime of parameters as in Fig. 3 (A). The values are very small so that the present dilaton extension of the SM leads to a LHC phenomenology which deviates from SM predictions by an O(10 −4 ) effect. Unfortunately these tiny effects would be unobservable at the LHC. For the same reason, the possibility of a Higgs impostor is excluded for the present model.
Conclusions and discussion
We have studied a mechanism in holography that allows for a naturally light dilaton in CFTs deformed by a single scalar operator. Two different realizations of a light dilaton have been identified: i) the hard realization is induced by the existence of an IR brane, and the dilaton is incarnated by the IR brane location, and ii) the soft realization, in which the dilaton is controlled by the condensation threshold of the CFT operator. These results have been confirmed in a 5D warped model which is large enough to exploit the full capabilities of the extra-dimensional models. The extension of the SM with a light dilaton in the 5D warped model leads to dilaton masses that can naturally be of the order of magnitude of the Higgs mass, and KK vector masses of the order of TeV, all of them compatible with Electroweak Precision Tests. However a first study of the coupling to massive gauge fields suggests that the dilaton couplings to SM particles (affecting e.g. the unitarization of the V L V L elastic and inelastic scattering as well as the strength of production of gauge bosons and fermions) are several orders of magnitude smaller than those predicted for the Higgs in the SM in a way that is unobservable at the LHC. For the same reason our results suggest that in the models discussed here the dilaton cannot be a Higgs impostor. In spite of this, some modifications of the warped model that would allow for a sizeable coupling deserve to be studied [10].
Figure 1 .
13 V ′ (φ)
Figure 2 .
2(A) Mass of the dilaton as a function of the parameter a, normalized to ρ = k e −A(y 1 ) . (B) KK gauge boson masses as a function of the parameter a. In these figures we have plotted the results for c = 0.5, 1, 1.5 and 2, and have considered φ 1 = 5.
Figure 3 .
3(A) Bound on KK mass (solid line) as function of a from electroweak observables. The corresponding dilaton mass is in dashed line. We have drawn horizontal dashed lines corresponding to 125 GeV and 2 TeV. (B) Radion coupling to the Z boson as a function of a. The result for W bosons is c W= (m W /m Z ) 2 · c Z ≈ 0.78 c Z .In these figures we have considered c = 1 and φ 1 = 5.
⋆ Presented by E. Megías at the 4th International Conference on New Frontiers in Physics (ICNFP 2015), 23-30 August2015, Kolymbari, Crete, Greece.
a e-mail: [email protected]
EPJ Web of Conferences
See Ref.[11] for a wide description of the formalism of electroweak breaking by the bulk Higgs.3 Extra-dimensional realizations of Higgs impostors require that the KK scale m KK must not exceed a few TeV, because m KK and the electroweak scale v = 246 GeV are linked by m KK 4πv.
Acknowledgements
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. R Contino, A Pomarol, R Rattazzi, unpublished workR. Contino, A. Pomarol, R. Rattazzi, unpublished work (2010)
. B Bellazzini, C Csaki, J Hubisz, J Serra, J Terning, Eur. Phys. J. 742790B. Bellazzini, C. Csaki, J. Hubisz, J. Serra, J. Terning, Eur. Phys. J. C74, 2790 (2014)
. F Coradeschi, P Lodone, D Pappadopulo, R Rattazzi, L Vitale, JHEP. 1157F. Coradeschi, P. Lodone, D. Pappadopulo, R. Rattazzi, L. Vitale, JHEP 11, 057 (2013)
. E Megias, O Pujolas, JHEP. 0881E. Megias, O. Pujolas, JHEP 08, 081 (2014)
. P Cox, T Gherghetta, JHEP. 026P. Cox, T. Gherghetta, JHEP 02, 006 (2015)
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. E Megias, O Pujolas, M Quiros, 1512.06106E. Megias, O. Pujolas, M. Quiros (2015), 1512.06106
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. C Csaki, J Hubisz, S J Lee, Phys. Rev. 76125015C. Csaki, J. Hubisz, S.J. Lee, Phys. Rev. D76, 125015 (2007)
|
[] |
[
"a Kondo lattice based on intercalation of van der Waals layered transition metal dichalcogenide",
"a Kondo lattice based on intercalation of van der Waals layered transition metal dichalcogenide"
] |
[
"Jingjing Niu \nState Key Laboratory for Artificial Microstructure and Mesoscopic Physics\nBeijing\n\nKey Laboratory of Quantum Devices\nPeking University\n100871BeijingChina\n\nCollaborative Innovation Center of Quantum Matter\n100871BeijingChina\n",
"Zhilin Li \nState Key Laboratory for Artificial Microstructure and Mesoscopic Physics\nBeijing\n\nKey Laboratory of Quantum Devices\nPeking University\n100871BeijingChina\n\nCollaborative Innovation Center of Quantum Matter\n100871BeijingChina\n",
"Sixian Yang \nSynergetic Innovation Center of Quantum Information and Quantum Physics\nCAS Key Laboratory of Quantum Information\nUniversity of Science and Technology of China\n230026Hefei, AnhuiChina\n",
"Wenjie Zhang \nState Key Laboratory for Artificial Microstructure and Mesoscopic Physics\nBeijing\n\nKey Laboratory of Quantum Devices\nPeking University\n100871BeijingChina\n\nCollaborative Innovation Center of Quantum Matter\n100871BeijingChina\n",
"Dayu Yan \nInstitute of Physics\nBeijing National Laboratory for Condensed Matter Physics\nChinese Academy of Sciences\n100190BeijingChina\n",
"Shulin Chen \nSchool of Physics\nElectron Microscopy Laboratory\nPeking University\n100871BeijingChina\n",
"Zhepeng Zhang \nCenter for Nanochemistry\nCollege of Chemistry and Molecular Engineering\nNational Laboratory for Molecular Sciences\nPeking University\n100871Beijing, BeijingChina\n",
"Yanfeng Zhang ",
"Xinguo Ren \nSynergetic Innovation Center of Quantum Information and Quantum Physics\nCAS Key Laboratory of Quantum Information\nUniversity of Science and Technology of China\n230026Hefei, AnhuiChina\n\nCenter for Nanochemistry\nCollege of Chemistry and Molecular Engineering\nNational Laboratory for Molecular Sciences\nPeking University\n100871Beijing, BeijingChina\n",
"Peng Gao \nCollaborative Innovation Center of Quantum Matter\n100871BeijingChina\n\nSchool of Physics\nElectron Microscopy Laboratory\nPeking University\n100871BeijingChina\n\nInternational Center for Quantum Materials\nSchool of Physics\nPeking University\n100871BeijingChina\n",
"Youguo Shi \nInstitute of Physics\nBeijing National Laboratory for Condensed Matter Physics\nChinese Academy of Sciences\n100190BeijingChina\n",
"Dapeng Yu \nState Key Laboratory for Artificial Microstructure and Mesoscopic Physics\nBeijing\n\nKey Laboratory of Quantum Devices\nPeking University\n100871BeijingChina\n\nCollaborative Innovation Center of Quantum Matter\n100871BeijingChina\n\nDepartment of Physics\nSouthern University of Science and Technology of China\n518055ShenzhenChina\n",
"Xiaosong Wu \nState Key Laboratory for Artificial Microstructure and Mesoscopic Physics\nBeijing\n\nKey Laboratory of Quantum Devices\nPeking University\n100871BeijingChina\n\nCollaborative Innovation Center of Quantum Matter\n100871BeijingChina\n\nDepartment of Physics\nSouthern University of Science and Technology of China\n518055ShenzhenChina\n"
] |
[
"State Key Laboratory for Artificial Microstructure and Mesoscopic Physics\nBeijing",
"Key Laboratory of Quantum Devices\nPeking University\n100871BeijingChina",
"Collaborative Innovation Center of Quantum Matter\n100871BeijingChina",
"State Key Laboratory for Artificial Microstructure and Mesoscopic Physics\nBeijing",
"Key Laboratory of Quantum Devices\nPeking University\n100871BeijingChina",
"Collaborative Innovation Center of Quantum Matter\n100871BeijingChina",
"Synergetic Innovation Center of Quantum Information and Quantum Physics\nCAS Key Laboratory of Quantum Information\nUniversity of Science and Technology of China\n230026Hefei, AnhuiChina",
"State Key Laboratory for Artificial Microstructure and Mesoscopic Physics\nBeijing",
"Key Laboratory of Quantum Devices\nPeking University\n100871BeijingChina",
"Collaborative Innovation Center of Quantum Matter\n100871BeijingChina",
"Institute of Physics\nBeijing National Laboratory for Condensed Matter Physics\nChinese Academy of Sciences\n100190BeijingChina",
"School of Physics\nElectron Microscopy Laboratory\nPeking University\n100871BeijingChina",
"Center for Nanochemistry\nCollege of Chemistry and Molecular Engineering\nNational Laboratory for Molecular Sciences\nPeking University\n100871Beijing, BeijingChina",
"Synergetic Innovation Center of Quantum Information and Quantum Physics\nCAS Key Laboratory of Quantum Information\nUniversity of Science and Technology of China\n230026Hefei, AnhuiChina",
"Center for Nanochemistry\nCollege of Chemistry and Molecular Engineering\nNational Laboratory for Molecular Sciences\nPeking University\n100871Beijing, BeijingChina",
"Collaborative Innovation Center of Quantum Matter\n100871BeijingChina",
"School of Physics\nElectron Microscopy Laboratory\nPeking University\n100871BeijingChina",
"International Center for Quantum Materials\nSchool of Physics\nPeking University\n100871BeijingChina",
"Institute of Physics\nBeijing National Laboratory for Condensed Matter Physics\nChinese Academy of Sciences\n100190BeijingChina",
"State Key Laboratory for Artificial Microstructure and Mesoscopic Physics\nBeijing",
"Key Laboratory of Quantum Devices\nPeking University\n100871BeijingChina",
"Collaborative Innovation Center of Quantum Matter\n100871BeijingChina",
"Department of Physics\nSouthern University of Science and Technology of China\n518055ShenzhenChina",
"State Key Laboratory for Artificial Microstructure and Mesoscopic Physics\nBeijing",
"Key Laboratory of Quantum Devices\nPeking University\n100871BeijingChina",
"Collaborative Innovation Center of Quantum Matter\n100871BeijingChina",
"Department of Physics\nSouthern University of Science and Technology of China\n518055ShenzhenChina"
] |
[] |
Since the discovery of graphene, a tremendous amount of two dimensional (2D) materials have surfaced. Their electronic properties can usually be well understood without considering correlations between electrons. On the other hand, strong electronic correlations are known to give rise to a variety of exotic properties and new quantum phases, for instance, high temperature superconductivity, heavy fermions and quantum spin liquids. The study of these phenomena has been one of the main focuses of condensed matter physics. There is a strong incentive to introduce electronic correlations into 2D materials. Via intercalating a van der Waals layered compound VS 2 , we show an emergence of a Kondo lattice, an extensively studied strongly correlated system, by magnetic, specific heat, electrical and thermoelectric transport studies. In particular, an exceptionally large Sommerfeld coefficient, 440 mJ·K −2 ·mol −1 , indicates a strong electron correlation.The obtained Kadowaki-Woods ratio, 2.7 × 10 −6 µΩ·cm·mol 2 ·K 2 ·mJ −2 , also supports the strong electron-electron interaction. The temperature dependence of the resistivity and thermopower corroborate the Kondo lattice picture. The intercalated compound is one of a few rare examples of d-electron Kondo lattices. We further show that the Kondo physics persists in ultra-thin films.This work thus demonstrates a route to generate strong correlations in 2D materials.Here, we demonstrate that the scheme is viable by showing that V 5 S 8 , a self-intercalated compound of VS 2 , is a Kondo lattice. VS 2 is a non-magnetic layered TMD[17]. V 5 S 8 , however, becomes an antiferromagnetic (AFM) metal because of the local moment of the intercalated V ion. In this work, we study the magnetic susceptibility, specific heat, electrical and thermoelectric transport of V 5 S 8 . The Sommerfeld coefficient is found to be 440 and 67 mJ·K −2 per mole of local moments above and below the Néel temperature, respectively.The exceptionally large coefficients indicate renormalization of the effective electron mass due to strong electron correlation. Both the electrical resistivity and thermopower show characteristics of a Kondo lattice, suggesting that the strong correlation results from the
| null |
[
"https://arxiv.org/pdf/1809.04213v1.pdf"
] | 119,337,034 |
1809.04213
|
27daaa7c071caa9efa3462550bcd52f1bd73a285
|
a Kondo lattice based on intercalation of van der Waals layered transition metal dichalcogenide
Sep 2018 V 5 S 8
Jingjing Niu
State Key Laboratory for Artificial Microstructure and Mesoscopic Physics
Beijing
Key Laboratory of Quantum Devices
Peking University
100871BeijingChina
Collaborative Innovation Center of Quantum Matter
100871BeijingChina
Zhilin Li
State Key Laboratory for Artificial Microstructure and Mesoscopic Physics
Beijing
Key Laboratory of Quantum Devices
Peking University
100871BeijingChina
Collaborative Innovation Center of Quantum Matter
100871BeijingChina
Sixian Yang
Synergetic Innovation Center of Quantum Information and Quantum Physics
CAS Key Laboratory of Quantum Information
University of Science and Technology of China
230026Hefei, AnhuiChina
Wenjie Zhang
State Key Laboratory for Artificial Microstructure and Mesoscopic Physics
Beijing
Key Laboratory of Quantum Devices
Peking University
100871BeijingChina
Collaborative Innovation Center of Quantum Matter
100871BeijingChina
Dayu Yan
Institute of Physics
Beijing National Laboratory for Condensed Matter Physics
Chinese Academy of Sciences
100190BeijingChina
Shulin Chen
School of Physics
Electron Microscopy Laboratory
Peking University
100871BeijingChina
Zhepeng Zhang
Center for Nanochemistry
College of Chemistry and Molecular Engineering
National Laboratory for Molecular Sciences
Peking University
100871Beijing, BeijingChina
Yanfeng Zhang
Xinguo Ren
Synergetic Innovation Center of Quantum Information and Quantum Physics
CAS Key Laboratory of Quantum Information
University of Science and Technology of China
230026Hefei, AnhuiChina
Center for Nanochemistry
College of Chemistry and Molecular Engineering
National Laboratory for Molecular Sciences
Peking University
100871Beijing, BeijingChina
Peng Gao
Collaborative Innovation Center of Quantum Matter
100871BeijingChina
School of Physics
Electron Microscopy Laboratory
Peking University
100871BeijingChina
International Center for Quantum Materials
School of Physics
Peking University
100871BeijingChina
Youguo Shi
Institute of Physics
Beijing National Laboratory for Condensed Matter Physics
Chinese Academy of Sciences
100190BeijingChina
Dapeng Yu
State Key Laboratory for Artificial Microstructure and Mesoscopic Physics
Beijing
Key Laboratory of Quantum Devices
Peking University
100871BeijingChina
Collaborative Innovation Center of Quantum Matter
100871BeijingChina
Department of Physics
Southern University of Science and Technology of China
518055ShenzhenChina
Xiaosong Wu
State Key Laboratory for Artificial Microstructure and Mesoscopic Physics
Beijing
Key Laboratory of Quantum Devices
Peking University
100871BeijingChina
Collaborative Innovation Center of Quantum Matter
100871BeijingChina
Department of Physics
Southern University of Science and Technology of China
518055ShenzhenChina
a Kondo lattice based on intercalation of van der Waals layered transition metal dichalcogenide
Sep 2018 V 5 S 8
Since the discovery of graphene, a tremendous amount of two dimensional (2D) materials have surfaced. Their electronic properties can usually be well understood without considering correlations between electrons. On the other hand, strong electronic correlations are known to give rise to a variety of exotic properties and new quantum phases, for instance, high temperature superconductivity, heavy fermions and quantum spin liquids. The study of these phenomena has been one of the main focuses of condensed matter physics. There is a strong incentive to introduce electronic correlations into 2D materials. Via intercalating a van der Waals layered compound VS 2 , we show an emergence of a Kondo lattice, an extensively studied strongly correlated system, by magnetic, specific heat, electrical and thermoelectric transport studies. In particular, an exceptionally large Sommerfeld coefficient, 440 mJ·K −2 ·mol −1 , indicates a strong electron correlation.The obtained Kadowaki-Woods ratio, 2.7 × 10 −6 µΩ·cm·mol 2 ·K 2 ·mJ −2 , also supports the strong electron-electron interaction. The temperature dependence of the resistivity and thermopower corroborate the Kondo lattice picture. The intercalated compound is one of a few rare examples of d-electron Kondo lattices. We further show that the Kondo physics persists in ultra-thin films.This work thus demonstrates a route to generate strong correlations in 2D materials.Here, we demonstrate that the scheme is viable by showing that V 5 S 8 , a self-intercalated compound of VS 2 , is a Kondo lattice. VS 2 is a non-magnetic layered TMD[17]. V 5 S 8 , however, becomes an antiferromagnetic (AFM) metal because of the local moment of the intercalated V ion. In this work, we study the magnetic susceptibility, specific heat, electrical and thermoelectric transport of V 5 S 8 . The Sommerfeld coefficient is found to be 440 and 67 mJ·K −2 per mole of local moments above and below the Néel temperature, respectively.The exceptionally large coefficients indicate renormalization of the effective electron mass due to strong electron correlation. Both the electrical resistivity and thermopower show characteristics of a Kondo lattice, suggesting that the strong correlation results from the
The past decade has seen an explosion in research of two dimensional (2D) materials [1,2].
During discovery of a flood of 2D materials, many salient effects in solid, e.g., integer and fractional quantum Hall effect, quantum spin Hall effect, superconductivity and magnetism, have been observed in 2D materials [3][4][5][6]. Most of the electronic properties of 2D materials can be described by a single-electron picture. On the other hand, strong electron correlations have been one of the most important problems in condensed matter physics and are believed to play a pivotal role in systems, such as high temperature superconductors, heavy fermions, and quantum spin liquids, etc [7][8][9]. There has been little interaction between these two fields.
Very recently, emergence of strong correlations in twisted bilayer graphene has stirred much excitement in both communities [10,11]. The excellent tunability via electrical gating and surface engineering of 2D materials may provide to the experimental investigation of strong correlation phenomena a powerful knob that is not available in research of three dimensional bulk.
2D materials are derived from their three dimensional counterpart, van der Waals layered compounds. Many kinds of atoms and molecules, varying significantly in size and property, can be inserted into the interlayer gap due to weak interlayer coupling. In fact, there have been extensive studies on intercalated graphite and transition metal dichalcogenides (TMD's) [12][13][14]. New electrical, optical and magnetic properties have been introduced to the host material this way. Such intercalation can be achieved in bilayers [15,16], hence making a new 2D material. Therefore, intercalation is expected to greatly expand the spectrum of properties of 2D materials. A naive idea is to insert local magnetic moments so that they couple to the conduction carriers in the layer, forming a Kondo lattice. This could be a versatile scheme for designing strongly correlated 2D electronic systems.
Kondo coupling between itinerant electrons in the plane and intercalated local moments.
The low temperature resistivity follows a T 2 dependence, consistent with the Fermi liquid effect. The Kadowaki-Woods (KW) ratio is ∼0.27a 0 , indicating strong electron-electron scattering. Our results reveal that V 5 S 8 is a d-electron Kondo lattice compound and thus show that intercalation of van der Waals layered materials can be an effective and versatile method to bring new effects, such as strong correlation, into many known 2D materials. V 5 S 8 bulk single crystals were grown by a chemical vapor transport method, using vanadium and sulfur powders as precursors and iodine as a transport agent. These species were loaded into a silica ampule under argon. The ampule was then evacuated, sealed and heated gradually in a two-zone tube furnace to a temperature gradient of 1000 • C to 850 • C. After two weeks, single crystals with regular shapes and shiny facets can be obtained. V 5 S 8 thin flakes were grown by a chemical vapour deposition method [18]. X-ray experiments on grown crystals indicate a pure V 5 S 8 phase. The crystallographic structure of the single crystal was further confirmed by high-angle annular dark field scanning transmission electron microscope (HAADF-STEM). Transport properties were measured using a standard lock-in method in an OXFORD variable temperature cryostat from 1.5 to 300 K. Heat capacity was measured in a Quantum Design Physical Properties Measurement System. A Quantum Design SQUID magnetometer was employed to measure the magnetic susceptibility.
Thermopower of bulk single crystals was measured using a standard four-probe steady-state method with a Chromel/AuFe(0.07%) thermocouple, while a micro-heater method was used for thin films [19]. V 5 S 8 can be seen as a van der Waals layered material VS 2 self-intercalated with V, V 1/4 VS 2 . It crystallizes in a monoclinic structure, space group C2/m. V atoms lie on three inequivalent sites. Intercalated V atoms take the V I site, while V atoms in the VS 2 layer take V II and V III sites. Each V I atom is surrounded by six S atoms, forming a distorted octahedron, shown in Fig. 1(b). The resultant crystal field is believed to be intricately related to the local magnetic moment of V atoms [20,21]. Fig. 1(a) shows a HAADF-STEM image of a single crystal, in which both V and S atoms can be clearly seen, as well as the rectangular arrangement of the intercalated V I atoms. Images have been taken at various spots. All of them show high crystallinity and the same lattice orientation, indicating uniform V intercatlation. A zoom-in and color-enhanced image is shown in the lower right of Fig. 1(a), which is in excellent accordance with the in-plane atomic model of V 5 S 8 . The lattice constants, a = 11.65Å and b = 6.76Å, are determined from a Fast Fourier Transformation (FFT) image shown in the upper right of Fig. 1(a).
The magnetic susceptibility of V 5 S 8 displays a paramagnetic behavior which follows the Curie-Weiss (CW) law at high temperatures and undergoes an antiferromagnetic transition at about 32 K, shown in Fig. 1 [22]. The high temperature CW behavior suggests presence of local moments. Fitting of the paramagnetic susceptibility to the CW law, χ = χ 0 + C T −θ CW , yields a positive CW temperature θ CW ≈ 8.8 K despite the AFM order and an effective magnetic moment of µ eff = 2.43µ B per V 5 S 8 formula unit (f.u.). This value is consistent with reported ones, ranging from 2.12 to 2.49 µ B [22][23][24]. It is generally believed that only V I ions carry a local magnetic moment [20,21,25]. So, the measured moment is close to the theoretical value 2.64µ B for V 3+ (3d 2 ) [20,22,26]. Below the Néel temperature, the moments point in the direction of 10.4 • away from the c axis toward the a axis. They align as ↑↑↓↓ along the a axis, while they align ferromagnetically along the b and c axes [20].
Consequently, the in-plane (B ab plane) susceptibility is larger than the out-of-plane one.
The field dependence of the out-of-plane magnetization displays a sudden change of the slope at about 4.5 T, seen in the inset of Fig. 1(c). This is caused by a metamagnetic spin-flop transition [22]. A neutron scattering study found a moment of 0.7 and 1.5µ B in two single crystals, respectively [27], while a nuclear magnetic resonance study found a much smaller moment, 0.22 µ B [25]. Taking the measured paramagnetic moment of 2.43µ B , S ≈ 0.8 is obtained. The moment below T N mostly seems smaller than the expected value 2S = 1.7 [28]. This discrepancy has been an unresolved puzzle. We would also like to point out that the susceptibility deviates from the CW law below 140 K. In the following, we are going to show that this puzzle and the deviation can be explained in terms of hybridization of the local d-electrons on V I with the conduction electrons in the VS 2 plane, a correlation effect known as the Kondo effect.
There have already been indications for interactions between the localized d-electrons and the conduction electrons in this system. Anomalous Hall effect, due to skew scattering of conduction electrons off from local moments, has been observed in V 5 S 8 [18]. Photoemissionspectroscopy study has shown both local-moment-like and bandlike features for V 3d electrons. To understand the discrepancy, it was thus postulated that the 3d electron that provides the local moment becomes partially itinerant at low temperatures [21]. Recently, in a similar compound, VSe 2 with dilute V intercalation, the Kondo effect has been observed [29].
In V 5 S 8 , where local moments of V I atoms arrange in a periodic array, a Kondo lattice is naturally anticipated. As a result, the effective mass of the conduction electron will be substantially enhanced by the electron correlation. We have carried out specific heat measurements for V 5 S 8 . The data of a 3.18 mg bulk single crystal are presented in Fig. 2(a). A sizeable jump at T N = 32 K manifests the AFM transition. In the inset of Fig. 2(a), C(T )/T is plotted as a function of T 2 . A linear dependence is observed above T N . By a linear fit to C/T = γ p +βT 2 , a very large electronic Sommerfeld coefficient, γ p = 440 mJ·K −2 per mole of V I atoms, is obtained. The value is comparable to those in f -electron heavy-fermion systems that exhibit a magnetic order at low temperatures, e.g., 148 mJ·K −2 ·mol −1 in Ce 2 Rh 3 Ge 5 [30] and 504 mJ·K −2 ·mol −1 in U 2 Zn 17 [31]. Coincidentally, γ = 420 mJ·K −2 ·mol −1 in a vanadium oxide, LiV 2 O 4 , which is the first d-electron heavy fermion metal [32]. The observed large Sommerfeld coefficient provides direct evidence for the mass enhancement in V 5 S 8 .
Below T N , taking into account the spin wave contribution, the non-lattice part of the specific heat can be expressed as
∆C = γ m T + A C ∆ 4 T ∆ e −∆/T 1 + 39 20 T ∆ + 51 32 T ∆ 2 ,(1)
where γ m represents the electron contribution in the magnetically ordered state, ∆ is the spin-wave gap [33][34][35][36]. In Fig. 2(b), we plot the low-T ∆C curve obtained by subtracting the lattice contribution (βT 3 ) according to the high-T fit, i.e., ∆C = C − βT 3 . The data below 20 K can be well fitted to Eq. (1), producing γ m = 67 mJ·K −2 mol −1 and ∆ = 12.2 K. Though much smaller than γ p , γ m is still comparable to some strongly correlated materials [30,37].
The suppression of the Sommerfeld coefficient by magnetic ordering is typical in heavy fermion systems that order magnetically at low temperatures [30,31]. This is because of the competition between the Kondo coupling, which results in mass enhancement, and the Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction, which promotes a magnetic order [38].
The competition is also reflected in the observed negative correlation between γ m and T N . Fig. 2(c) compares the non-lattice part of the low-T specific heat ∆C for three types of V 5 S 8 samples. These samples display variation of T N , possibly due to different intercalation level [22]. As T N decreases, the spin-wave gap ∆ gradually is reduced, but γ m increases from 67 to 156 mJ·K −2 mol −1 . Such a negative correlation is illustrated in the Doniach phase diagram of heavy fermion systems, indicating approaching to the quantum critical point [38].
Given the key effect of intercalated V atoms in introducing the Kondo effect, it would be informative to compare the intercalate V 1/4 VS 2 and the host compound VS 2 . However, it is challenging to grow VS 2 because of self intercalation and contradicting properties have been observed so far [39]. Consequently, we have measured the specific heat of VSe 2 instead, an isoelectronic and isostructural compound of VS 2 , shown in the Supporting Materials. The Sommerfeld coefficient is found to be 46 mJ·K −2 per mol of V 4 Se 8 , smaller than that of V 5 S 8 . This difference offers additional evidence for enhanced electronic correlation by the Kondo coupling to local moments.
To get an idea of the strength of the electron correlation, we estimate the effective quasiparticle mass enhanced by the strong correlation by comparing the experimental Sommerfeld coefficient to that calculated from the Kohn-Sham model to density functional theory (DFT) [40]. Within such a non-interacting electron model, the Sommerfeld coefficient can
be estimated as γ = 1 3 π 2 k 2 B N(ǫ F ),
where k B is the Boltzmann constant and N(ǫ F ) is the density of states (DOS) per f.u. at the Fermi level ǫ F . Our DFT calculation for the antiferromagetic phase of V 5 S 8 yields an electronic DOS of 6.6 states/eV/f.u., which translates to a Sommerfeld coefficient of only 16.8 mJ·K −2 ·mol −1 . To understand the difference between the experimental value and the calculated one here, it is necessary to first take into account the electron-phonon coupling which can enhance the Sommerfeld coefficient by a factor of (1+λ ep ), where λ ep is the mass enhancement factor due to the electron-phonon coupling [41].
A reasonable estimate of λ ep is 1.19 [42], which was obtained in V metal. This leads to an enhanced Sommerfeld coefficient of 36.8 mJ·K −2 ·mol −1 . Thus, one can see that, even after accounting for the electron-phonon coupling effect, the theoretical value estimated from a quasiparticle picture is still a factor of 2 smaller than the experimental one. We attribute the remaining discrepancy to the mass enhancement effect due to strong electron correlations.
In the antiferromagnetic phase, the electron correlation gives rise to an enhanced effective mass of 1.82 m e with m e being the bare electron mass. In the paramagnetic phase, this mass enhancement effect is much more pronounced, where the effective mass is estimated to be 11.4 m e . The details of our DFT calculation for the electronic DOS are included in the supplementary materials.
The temperature dependent resistivity of many metallic Kondo lattice materials exhibits characteristic features, such as a maximum, stemming from the Kondo scattering [31,33] . Fig. 3 shows the resistivity for three V 5 S 8 single crystals, #S1, #S2, and #S3. At 32 K, the resistivity exhibits a kink, which results from the antiferromagnetic transition. Above this temperature, there is an apparent hump at about 140 K, in stark contrast to a linear dependence commonly seen in metals. In fact, VSe 2 displays a typical metallic resistivity linear in T . We tentatively subtract the resistivity of VSe 2 from V 5 S 8 to highlight the effect of intercalated atoms. As shown in the inset of Fig. 3a, the broad maximum is evident. Similar features have been observed in Kondo lattices and believed to originate from a combined effect of the Kondo coupling and the crystal field [30,31,33,43]. On the high temperature side of the maximum, the resistivity roughly follows a − ln T dependence, consistent with the effect of incoherent Kondo scattering. The resistivity maximum is at about T * = 140 K, suggesting that the crystal field splitting is qualitatively of the order of 140 K.
We now turn to the low temperature resistivity in the AFM state. In the magnetically ordered state, the strong decrease of the resistivity below T N is caused by the reduction of spin-disorder scattering. In this case, the resistivity consists of both the electronic contribution and the magnon scattering term, and takes the form of
ρ(T ) = ρ 0 + AT 2 + C∆ 2 T ∆ e −∆/T 1 + 2 3 T ∆ + 2 15 T ∆ 2 ,(2)
where ρ 0 is the residual resistivity, the AT 2 term represents the Fermi liquid contribution, the last term is associated with the spin wave, and ∆ is the spin wave gap [33][34][35][36]. The resistivity below 20 K can be well described by Eq. (2) (see Fig. 3b). The fitting parameters A and ∆ are ∼0.012 µΩ·cm·K −2 and ∼ 28 K, respectively. Moreover, A is found to be nearly independent of the magnetic field, which is consistent with the Fermi liquid contribution [44,45]. On the other hand, with increasing B, ∆ decreases gradually from 30 to 15 K [46,47].
In strongly correlated systems, it has been found that the KadowakiCWoods ratio, The thermoelectric properties of heavy fermion compounds share some common features [53]. Fig. 4 shows the temperature-dependent thermopower S for V 5 S 8 . Instead of a linear T dependence as expected for ordinary metals, S shows a sign change at about 140 K, as well as a negative S minimum around 60 K. A change of sign is generally associated with a change of carrier type. However, this explanation is inconsistent with hole conduction inferred from Hall in the whole temperature range (see the Supplementary Material). In a
r KW =A/γ 2 ,
Kondo lattice system, the interplay between the Kondo and crystal field effects gives rise to a broad peak in thermopower S at high temperatures. More prominently, with decreasing temperature, S changes its sign at T = αT K , where roughly α = 2.5-10. After that, S displays an extremum and may change sign again in some compounds [53][54][55][56]. Our data agree with some of these essential features, e.g., a sign change and a negative peak. It is worth mentioning that the temperature of 140 K, at which S changes its sign, is very close to T * obtained from the resistivity maximum and deviation of the magnetic susceptibility from the CW law.
Based on these observations, we conclude that V 5 S 8 is a d-electron Kondo lattice compound. Under this picture, the magnetic susceptibility can now be understood. The deviation from the CW law beginning at 140 K results from the Kondo coupling and the crystal field effect, which has been seen in other heavy fermion compounds [30,33,36,43,57]. The reduction of the magnetic local moment at low temperatures is due to the Kondo screening, which, though strongly suppressed, persists in the AFM state [31,58].
Finally, we investigate the evolution of the correlation effect with reducing thickness.
Since it is difficult to measure the specific heat of thin layers, we focus on the resistivity and thermopower and monitor how those features that are relevant to the electron correlation change, as shown in Fig. 5. As samples become thinner, the broad maximum of resistivity shifts towards low temperatures, implying reduction of the crystal field splitting, see weak ferromagnetic order emerges [18]. One intriguing feature is that the positive temperature coefficient of resistivity below T N suddenly turns negative when the thickness is reduced to 7.6 nm (see the inset of Fig. 5(a) We have performed detailed magnetoresistance measurements of V 5 S 8 bulk single crystal. Above T N , MR follows a B 2 dependence, consistent with the spin fluctuation scattering [S1].
Moreover, MR data collapse onto a single curve when the field is scaled by temperature,
T + T * , as shown in Fig. S2(b). It is known that in a Kondo impurity model, the magnetore- We studied the evolution of the Hall resistivity ρ xy with temperature, as shown in We carried out density functional theory (DFT) calculations using the Perdew-Burke-Ernzerhof (PBE) [S6] generalized gradient approximation as implemented in the all-electron first-principles code package Fritz Haber Institute ab initio molecular simulations (FHI-aims) package [S7]. In accord with the experimental findings, the monoclinic crystal structure with antiferromagnetic ordering of the V I atoms was used in our calculations. We employed an extended unit cell containing 8 formula unit cells (104 atoms in total), and a 6 × 3 × 3 k grid (with the Γ point included) for the Brillouin zone sampling. The FHI-aims "light" setting for the numerical grid integration and numerical atomic basis sets (5s4p2d1f for V and 4s3p1d for S) were used in the calculations. The unit cell geometry and atomic positions were fully relaxed, with resultant lattice parameters of a = 22.62Å, b = 6.62Å, c = 11.37Å, and α = γ = 90 • , β = 91.7 • . The calculated electronic density of states (DOS) are presented in Fig. S4, where one can see that the DOS at the Fermi level is dominated by contributions from V. The DOS value of 53.0 states/eV for the calculated supercell corresponds to N(ǫ F ) = 6.6 states/eV per formula unit cell. In Fig. S5 the projected DOSs of individual V species are presented, where the V I atoms yield a pronounced peak just below the Fermi level, and contribute a major part of the spin polarization. A Mulliken charge analysis indicates that the local mangetic moments from V I , V II , and V III atoms are respectively 1.93, 0.15 and 0.04 µ B .
is significantly enhanced, around a 0 = 1.0×10 −5 µΩ·cm(mol·K/mJ) 2[48]. Using the low-T Sommerfeld coefficient γ = 67 mJ·K −2 mol −1 and A = 0.012 µΩ·cm, we obtain r KW = 0.27a 0 . This value is similar to those in strongly correlated systems such as LiV 2 O 4 , V 2 O 3 , Sr 2 RuO 4 and Na 0.7 CoO 2 , etc[37,[49][50][51][52].
Fig. 5 (
5a). The main features in thermopower, e.g., the sign change and the negative peak, remain, shown in Fig. 5(c), though the sign change temperature is suppressed. The same trend takes place in T N . Below a critical thickness of 5 nm, AFM disappears and a very
FIG. 1 .FIG. 2 .FIG. 3 .FIG. 4 .FIG. 5 .FIG
12345This work was supported by National Key Basic Research Program of China (No. 2013CBA01603, No. 2016YFA0300600, and No. 2016YFA0300903) and NSFC (Project No. 11574005, No. 11774009, No. 11222436 and No. 11574283). * These authors equally contributed to the work. † [email protected] and magnetic properties of V 5 S 8 bulk single crystal. a, HAADF-STEM image. Lower inset, a zoom-in image with the in-plane atomic model. Upper inset, a reduced FFT image. b, the magnetic unit cell with the V I S 6 octahedron. The blue arrows indicate the direction of the magnetic moments on V I sites. c, the molar magnetic susceptibility χ for B ⊥ ( B ⊥ ab plane) and B (B ab plane). The inset shows the low temperature (T = 2 K) isothermal magnetization curves. d, the inverse magnetic susceptibility 1/(χ − χ 0 ) for B ⊥ . The blue dashed line is a linear fit of the Curie-Weiss law, which gives θ = 8.8 K, χ 0 = 0.01 cm 3 mol −1 and µ ef f = 2.43µ B per V I . Specific heat. a, the specific heat C as a function of temperature. The inset shows a C/T versus T 2 plot. The solid line is a linear fit, which yields γ p = 440 mJ K −2 mol −1 . b, the low-T specific heat ∆C after subtracting the lattice contribution βT 3 according to the high-T fit. The dashed line represents a fit to the Eq. (1). c, the non-lattice part of specific heat ∆C for three types of samples with different T N . The dashed lines are the fits using Eq. (1). d, the fitting parameters as a function of T N . Temperature dependence of resistivity. a, the temperature-dependent resistivity of three samples. The inset illustrates the high-T resistivity after subtracting the non-magnetic ρ of VSe 2 from that of V 5 S 8 . The dotted lines are the − ln T fits. b, the normalized low-temperature resistivity ρ/ρ(T = 50 K). The dashed lines indicate the fits using Eq. (2). The inset shows the low-T resistivity ρ/ρ(T = 50 K) (vertically shifted for clarity.) for Sample #S1 at different magnetic fields (B ⊥ ab plane). Temperature-dependent thermopower S. S changes its sign at about 140 K and displays a negative minimum at 60 K. Electrical and thermoelectric transport of V 5 S 8 thin films. a, the temperature dependent resistivity for samples with different thicknesses. The inset shows the low temperature data and − ln T fits for thin samples. b, a typical device for resistivity ρ and thermopower S measurements.c, the temperature dependent thermopower for thin films. . S1. a Comparison of the temperature dependent resistivity ρ(T ) between V 5 S 8 and VSe 2 .ρ versus T . VSe 2 displays a linear-T dependence, except for an anomaly at ∼ 90 K, which is due to an charge density wave transition. The dotted line is the baseline subtracted from the resistivity of V 5 S 8 so as to highlight the contribution of the intercalation, ∆ρ. b ∆ρ as a function of temperature.
Fig.×
S2(a) shows the magnetoresistance MR = ρ(B)100% at different temperatures.
sistance follows the Schlottmann's relation ρ(B) ρ(0) = f [B/(T +T * )][S2]. The best scaling yields a characteristic temperature of T * = −6 K. The negative sign suggests ferromagnetic correlations in the antiferromagnetic ground state[S3-S5]. This corroborates with the positive Currie-Weiss temperature.
FIG. S3. Anomalous Hall effect and the carrier density of V 5 S 8 . a, Hall resistivity ρ xy (vertically shifted for clarity) at different temperatures. b, Linear relation between the Hall coefficient R H and the magnetic susceptibility χ above T N . The blue line is a linear fit.
Fig.
S2(a). Here, ρ xy includes two contributions, i.e., ρ xy (B) = R 0 B + R AHE µ 0 M, where R 0 and R AHE are the ordinary and anomalous Hall coefficients, respectively, and M is the magnetization, µ 0 the vacuum permeability. So, the Hall coefficient can be expressed asR H = ρ xy /B = R 0 + R AHE µ 0 M/B, namely, R H ∝ χ.We plot R H versus χ at different temperatures above T N and find a good linear relation (SeeFig. S3(b)). The intercept of the linear fit gives a positive value of R 0 = 2.5 × 10 −4 cm 3 C −1 , indicating a temperature independent holes carrier concentration of n = 2.5 × 10 22 cm −3 .
. S4. Calculated total (black lines) and species-projected density of states (V: red lines; S: blue lines) of V 5 S 8 bulk. The dash line at the 0 eV represents the Fermi level.
.S5. The atom-projected density of states(PDOS) of different Vanadium atoms for V I (solid red curve), V II (dashed blue curve) and V III (dot-dashed green curve) in V 5 S 8 bulk. * These authors equally contributed to the work. † [email protected]
). With further reducing thickness, a − ln T dependence in resistivity develops at low temperatures. It is not clear whether the disappearance of the metallic resistivity reflects a beakdown of Fermi liquid behavior or a metal-to-insulator transition. Another possibility would be an appearance of the second incoherent Kondo Our experiments strongly suggest that the intercalated material V 5 S 8 is a d-electron Kondo lattice compound. The Kondo physics persists in nano-thick films. The results have not only discovered a 2D strongly correlated material, but open a door to bring tremendousscattering contribution due to the crystal field effect[30, 43, 59]. Further investigation are
needed. Nevertheless, all these data indicate that the Kondo physics still plays a key role in
ultra-thin samples.
possibilities into 2D material research by intercalation. A recent work has demonstrated
that the intercalation can be extended to heterointerfaces, which will further expand the
scope of the method[60].
FIG. S2. Scaling of magnetoresistance for sample #S1. a, Magnetoresistance M R at various temperatures. The curves are vertically shifted for clarity. b, Normalized resistivity versus B/(T + T * ), where T * is a scaling parameter.200 K
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[
"LAGRANGIAN CONSTANT CYCLE SUBVARIETIES IN LAGRANGIAN FIBRATIONS",
"LAGRANGIAN CONSTANT CYCLE SUBVARIETIES IN LAGRANGIAN FIBRATIONS"
] |
[
"Hsueh-Yung Lin "
] |
[] |
[] |
We show that the image of a dominant meromorphic map from an irreducible compact Calabi-Yau manifold in a wider sense X whose general fiber is of dimension strictly between 0 and dim X is rationally connected. Using this result, we construct for any hyper-Kähler manifold X admitting a Lagrangian fibration a Lagrangian constant cycle subvariety Σ H in X for every divisor class H whose restriction to some smooth Lagrangian fiber is ample. We also show that up to a scalar multiple, the class of a zero-cycle supported on Σ H in CH 0 (X) does not depend neither on H nor on the Lagrangian fibration (provided b 2 (X) ≥ 8).
|
10.1093/imrn/rnx334
|
[
"https://arxiv.org/pdf/1510.01437v2.pdf"
] | 119,614,342 |
1510.01437
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5d1aa1b9010ab5d180d9ed85d34048ed4ea28303
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LAGRANGIAN CONSTANT CYCLE SUBVARIETIES IN LAGRANGIAN FIBRATIONS
6 Oct 2015
Hsueh-Yung Lin
LAGRANGIAN CONSTANT CYCLE SUBVARIETIES IN LAGRANGIAN FIBRATIONS
6 Oct 2015
We show that the image of a dominant meromorphic map from an irreducible compact Calabi-Yau manifold in a wider sense X whose general fiber is of dimension strictly between 0 and dim X is rationally connected. Using this result, we construct for any hyper-Kähler manifold X admitting a Lagrangian fibration a Lagrangian constant cycle subvariety Σ H in X for every divisor class H whose restriction to some smooth Lagrangian fiber is ample. We also show that up to a scalar multiple, the class of a zero-cycle supported on Σ H in CH 0 (X) does not depend neither on H nor on the Lagrangian fibration (provided b 2 (X) ≥ 8).
Introduction
This note is devoted to the construction of some subvarieties Y in a projective hyper-Kähler manifold X admitting a Lagrangian fibration such that every point in Y is rationally equivalent to each other in X. These subvarieties, called constant cycle subvarieties in [12], depend on the choices of a Lagrangian fibration and a divisor class H ∈ Pic(X) whose restriction to some smooth Lagrangian fiber is ample, thus the image of the Gysin map CH 0 (Y) → CH 0 (X) depends a priori on these choices as well. The result of this note is motivated by the conjectural picture on the splitting property of the conjectural Bloch-Beilinson filtration of projective hyper-Kähler manifolds due to Beauville [1] and Voisin [26] as we will explain below.
Note that the study of constant cycle subvarieties in hyper-Kähler manifolds was initiated by Huybrechts in the case of K3 surfaces [12]. Our motivation for studying these subvarieties comes rather from the attempt to generalize the Beauville-Voisin canonical zero-cycle of a projective K3 surface [2] to higher dimensional cases. For a K3 surface S, recall that there are at least two ways to characterize the canonical zero-cycle o S : i) o S is the degree-one generator of the image of the intersection product [2] ⌣ : CH 1 (S) ⊗ CH 1 (S) → CH 2 (S); ii) o S is the class of any point supported on a constant cycle curve in S [24]. More generally, for any n-dimensional subvariety of S [n] parameterizing a family of zero-cycles of constant class z ∈ CH 0 (S), z is proportional to o S .
Each characterization gives a priori different generalization of o S . The first one is related to Beauville's conjecture on the weak splitting property of the Chow ring of projective hyper-Kähler manifolds [1] : [1]). -Let X be a projective hyper-Kähler manifold. The restriction of the cycle class map CH • (X) Q := CH • (X) ⊗ Z Q → H • (X, Q) to the Q-sub-algebra generated by divisor classes is injective.
Conjecture 1.1 (Beauville
The reader is referred to [1,23,10,17,26] for recent developments of this conjecture. In particular, since H 4n (X, Q) = Q, Beauville's conjecture contains as a sub-conjecture the fact that the intersection of any 2n divisor classes in CH • (X) Q is proportional to the same degree one zero-cycle o X ∈ CH 2n (X) Q where 2n is the dimension of X, which generalizes property i) of o S .
The generalization of property ii) is formulated in [26, Conjectures 0.4 and 0.8] : for 0 ≤ i ≤ n, let S i CH 0 (X) denote the subgroup of CH 0 (X) generated by the classes of points whose rational orbit is of dimension ≥ i (1) . One hopes that this decreasing rational orbit filtration S • CH 0 (X) would define a splitting of the conjectural Bloch-Beilinson filtration F • BB in the sense that the inclusion S i CH 0 (X) ֒→ CH 0 (X) induces an isomorphism
S i CH 0 (X) ∼ − → CH 0 (X)/F 2n−2i+1 BB CH 0 (X).
Using the axioms of the Bloch-Beilinson conjecture, the surjectivity of the above map is proved in [26] to be a consequence of the following conjecture :
Conjecture 1.2 ([26]
). -Let X be a projective hyper-Kähler manifold of dimension 2n. The dimension of the set of points in X whose rational orbit has dimension ≥ i is 2n − i.
We refer to [26] for more details on Voisin's circle of ideas for studying the splitting property of the Bloch-Beilinson filtration on CH 0 (X). When i = n, Conjecture 1.2 is equivalent to the existence of constant cycle subvarieties of X of dimension n (which are necessarily Lagrangian, by Roitman-Mumford's theorem) and one would expect to recover the conjectural canonical zero-cycle for any projective hyper-Kähler manifolds X by taking the class of a point in any of these constant cycle Lagrangian subvarieties, hence the second generalization of o S .
In general it is difficult to construct Lagrangian constant cycle subvarieties. However, if X admits a Lagrangian fibration π : X → B, we prove in Section 3 the following theorem, which proves in particular Conjecture 1.2 in the case i = n for every Lagrangian fibration. Theorem 1.3. -Let X be a projective hyper-Kähler manifold admitting a Lagrangian fibration π : X → B. For each divisor class H ∈ Pic(X) whose restriction to some smooth Lagrangian fiber is ample, there exists a Lagrangian constant cycle subvariety Σ π,H ⊂ X all of whose points are rationally equivalent to a multiple of H n · [F] in CH 0 (X).
The fact that [F] ∈ CH n (X) is independent of F is a direct consequence of the following general result. Here a Calabi-Yau manifold is an irreducible (in the sense of Riemannian geometry) compact Kähler manifold with finite fundamental group and trivial canonical bundle. The Riemannian holonomy group of a Calabi-Yau manifold associated to its Kähler metric is either SU(n) or Sp(n). Hyper-Kähler manifolds and Calabi-Yau manifolds in the strict sense are examples of Calabi-Yau manifolds. In the case where X is a projective hyper-Kähler manifold, Theorem 1.4 is to compare with Matsushita's result [14,Theorem 2], saying that if X → B is a surjective morphism over a normal base B such that 0 < dim B < dim X, 1. Precisely, let z ∈ X and O z be the set of points in X which are rationally equivalent to z. O z is called the rational orbit of z and is a countable union of Zariski closed subset of X ; we define dim O z to be the supremum of the dimension of all irreducible components of O z . then B is a Q-factorial klt Fano variety of dimension 1 2 dim X with Picard number 1. Note also that in the case where X is projective and f : X → B is a surjective morphism over a normal Q-Gorenstein variety B without the assumption that dim B < dim X, either K B is numerically trivial or B is uniruled [28,Corollary 2]. Theorem 1.4 also improves and gives a new proof of the main result of [13].
This result will allow to rephrase Theorem 1.3 replacing the fiber F by the cycle L n ∈ CH n (X). Furthermore, under the mild assumption that a very general projective deformation of the Lagrangian fibration π : X → B with ρ(X) ≥ 3 satisfies Matsushita's conjecture (2) (for instance when b 2 (X) ≥ 8 [20]), Theorem 1.3 allow us to define for such a variety X a canonical zero-cycle o X ∈ CH 0 (X) by taking the class of a point supported on any Lagrangian constant cycle subvariety Σ π,H defined above, whose class is also proportional to a product of 2n divisors : iii) Under the same hypothesis as in ii), the class of a point in Σ π,H modulo rational equivalence is independent of the Lagrangian fibration.
Theorem 1.5. -i) Let X
Base variety of rationally fibered Calabi-Yau manifolds
We will prove Theorem 1.4 in this section.
Proof of Theorem 1.4. -Up to a bimeromorphic modification, we suppose that B is smooth. If B has a non-trivial holomorphic 2-form α, then 2 ≤ dim B < dim X and f * α 0 is degenerated, contradicting the Calabi-Yau assumption. Thus B is projective. By Graber-Harris-Starr's theorem [11], it suffices to show that if B satisfies the condition in Theorem 1.4, then B is uniruled. Indeed, suppose that B is not rationally connected, and let B B ′ be the MRC-fibration of B, then the composition map X B B ′ is dominant with 0 < dim B ′ < dim X. So B ′ would be uniruled, contradicting [11,Corollary1.4]. Now suppose B is not uniruled. By [4], the canonical class c 1 (K B ) is pseudo-effective ; let T be a closed positive current of bidegree (1, 1) on B representing c 1 (K B ). This means that the class c 1
(K B ) ∈ H 2 (B, R) is a limit of effective divisor classes. Let X p ← −X q − → B be a resolution of f : X B withX smooth. It is standard to show that p * q * c 1 (K B )
is a pseudo-effective class : indeed, this follows from the fact that p * q * maps effective divisor classes to effective divisor classes.
Since
q :X → B is surjective, the induced map q * K B → Ω kX is non-zero where k = dim B. As X is smooth, this map determines a non-zero morphism L → Ω k X where L is a line bundle such that c 1 (L) = p * q * c 1 (K B ).
Let ω be a Kähler form in X. Since the Riemannian holonomy group Hol(X) of X is either SU(n) or Sp(n/2) where n = dim X, there exists a Kähler-Einstein metric on T X whose corresponding Kähler form is cohomologous to ω [27], which further implies that Ω k X is ω-polystable by Donaldson-Uhlenbeck-Yau 2. We say that a Lagrangian fibration π : X → B satisfies Matsushita's conjecture if either π : X → B is isotrivial or the induced moduli map B A n to some suitable moduli space of abelian varieties is generically injective.
theorem [8,19]. Precisely, Ω k X = E 1 ⊕ · · · ⊕ E m where E i is defined as the parallel transport of each summand in the decomposition of the Hol(X)-module Ω k X|x into irreducible components over any x ∈ X. Each E i is ω-stable of slope µ ω (E) = 0.
The following result can be found in [5, §13, n o 1 and 3] or in [6,Chapter VI.3].
Lemma 2.1. - i) If
Hol(X) = SU(n), then the Hol(X)-module Ω k X|x is irreducible. ii) If Hol(X) = Sp(n/2), the decomposition of Ω k X|x into irreducible Sp(n/2)-sub-modules is described as follows :
Ω k X|x = k≥k−2r≥0 η r |x ∧ P k−2r ,
where P k−2r are irreducible Sp(n/2)-sub-modules of Ω k−2r X |x . Moreover,
dim C P k−2r = 2n k − 2r − 2n k − 2r − 2 .
By virtue of the above lemma, if Hol(X) = Sp(n/2) and k is odd or Hol(X) = SU(n), then dim E i > 1 = rank(L) for all i (since 0 < k < n). As c 1 (L) is pseudo-effective (so µ ω (L) ≥ 0 = µ ω (E)) and E i is stable, there is no non trivial morphism from L to E i for all i, contradicting the non-vanishing of L → Ω k X . Finally if Hol(X) = Sp(n/2) and if k is even, then m ≥ 2 and there exists exactly one i such that rank(E i ) = 1. Moreover, E i ≃ O X and E i ֒→ Ω k X is given by the multiplication by η k/2 . We deduce that if U is a Zariski open subset of X restricted to which f is well-defined, then locally the pullback under f |U of a non-zero holomorphic k-form α on f (U) is proportional to η k/2 , which contradicts the fact that η is non-degenerate.
As an immediate consequence,
Corollary 2.2. -The class of a fiber F in a Lagrangian fibration modulo rational equivalence is independent of F.
In particular, there exists µ ∈ Z\{0} such that L n = µF in CH n (X).
Remark 2.3.
-So far we have been interested in Calabi-Yau manifolds. As for complex tori, which are also Ricci-flat varieties, it is known that the image of a complex torus under a holomorphic map is always a product of projective spaces and a complex torus [7].
Construction of constant cycles subvarieties on Lagrangian fibrations
Let X be a variety. Proof. -It suffices to show that if every point supported on Y is torsion in CH 0 (X), then Y is a constant cycle subvariety. Let α : Y ֒→ X → Alb(X) be the composition of the inclusion map Y ֒→ X with the Albanese map X → Alb(X). If the image of i * : CH 0 (Y) → CH 0 (X) consists of torsion classes, then by Roitman's theorem [18] the map α factorizes through CH 0 (X) tors ≃ Alb(X) tors ֒→ Alb(X) via the cycle class map Y → CH 0 (X) tors . As Y is connected, α is constant, hence Y → CH 0 (X) tors ֒→ CH 0 (X) is constant. [9] implies that a general hypersurface in a projective space with large degree has no constant cycle subvarieties.
The property of being constant cycle for a subvariety is birational in the following sense :
Lemma 3.5. -A subvariety Y of X is constant cycle if and only if there exists a Zariski open subset U of Y such that all points in U are rationally equivalent in X.
Proof. -This follows from the well-known fact that every zero-cycle in Y is rationally equivalent to a zero-cycle supported in U. Proof. -For finiteness, let x 0 be any point in A and alb : A → Alb(A) be the Albanese map of A with respect to x 0 . Since A is an abelian variety, its Albanese map is an isomorphism. Recall that alb factorizes through the Deligne cycle class map α : CH 0 (A) hom → Alb(A), where CH 0 (A) hom denotes the subgroup of CH 0 (A) homologous to zero and the morphism A → CH 0 (A) hom is given by
x → [x] − [x 0 ]. If h = D[x] in CH 0 (A), then D · x = D · alb(x) = α(D[x] − D[x 0 ]) = α(h − D[x 0 ]) in A.∈ A such that (t * a h) = D[x ′ ] in CH 0 (A) Q . Thus (t * a h) = D[x ′ ] + t in CH 0 (A) for some torsion element t ∈ CH 0 (A) hom . Let t ′ be a D-division point of t. Since α (D[x + t ′ ] − D[x 0 ]) = D · x + α(t) = α (t ′ + D[x] − D[x 0 ]) , one gets D[x + t ′ ] = t ′ + D[x] = (t * a h) , therefore h = D[x + t ′ + a].
Let U ⊂ B be a Zariski open subset of B parametrizing smooth fibers of π such that H |π −1 (b) is ample for any b ∈ U. Set X U := π −1 (U). By a standard argument (see for example the proof of [22,Theorem 10.19]), there exist countably many relative Hilbert schemes H i of points of length 1 over U, which can be considered as subvarieties of X U , parametrizing the data of a point t in U and a point x ∈ X t such that D[
x t ] = H |X t in CH 0 (X t ).
Let p : H i → U be the natural projection. Since the X t 's are abelian varieties, p is finite and dominant by Lemma 3.7, so there exists an irreducible component M of H ′ i such that p |M is finite and dominant as well. By construction, viewing H i as a subvariety of X U , M is Zariski locally closed in X of dimension n ; we define Σ π,H as the closure of M in X, which is also of dimension n. Finally, for every x ∈ M, let j : X t ֒→ X be the inclusion of the fiber of π containing x, then D[x] = ( j * H) n in CH 0 (X t ) thus
D[x] = H n · [F] = D ′ H n · L n (3.1)
for some D ′ ∈ Z\{0} in CH 0 (X), where the last equality follows from Corollary 2.2. Hence M is a Zariski open subset of Σ π,H whose points are rationally equivalent to a scalar multiple of H n · L n . We conclude by Lemma 3.5 that Σ π,H is a constant cycle subvariety of X.
Before we start proving of Theorem 1.5, let us recall the following result of Matsushita and Voisin which will be useful later. Let π : X → B be a Lagrangian fibration and let L be the pullback of an ample divisor class from the base. Let j : F ֒→ X be the inclusion map of a smooth Lagrangian fiber in X. where q : Sym 2 H 2 (X, Q) → Q is the Beauville-Bogomolov-Fujiki form associated to X. In particular, the image of j * : H 2 (X, Q) → H 2 (F, Q) is of rank one.
Proof. -The first equality is exactly the statement of [21,Lemme 1.5]. For the second equality, by [15, Lemma 2.2], we have the inclusion ker q(L, ·) ⊂ ker j * . It follows that
b 2 (X) − 1 = dim Q ker q(L, ·) ≤ dim Q ker j * ≤ b 2 (X) − 1
where the last inequality results from the non-vanishing of j * (on any ample divisor class). Hence ker q(L, ·) = ker j * for dimensional reason.
Proof of Theorem 1.5. -Let π : X → B be a Lagrangian fibration on a polarized hyper-Kähler manifold (X, H) and let D ∈ Pic(X). If D |F is cohomologous to 0 on F, then D n · L n = 0 in CH 0 (X) by [25,Theorem 0.9]. If D |F 0 in H 2 (F, Q), then since j * : H 2 (X, Q) → H 2 (F, Q) is of rank one by Lemma 3.8, either D |F or −D |F is ample on F. Suppose without loss of generality that D |F is ample on F, then by (3.1), there exists d ∈ Z\{0}
such that D n · F = d [x] in CH 0 (X) for any point x in the Lagrangian constant cycle subvariety Σ π,D . Since D n · L n is non-trivially proportional to D n · F in CH 0 (X) by Corollary 2.2, this proves i).
To prove ii), let H 1 and H 2 be two divisor classes such that H 1 is ample and H 2 satisfies the assumption of Theorem 1.3. By Lemma 3.8, for a smooth fiber F b := π −1 (b), there exist α, β ∈ Z\{0} such that (αH 1 − βH 2 ) |F b is cohomologous to 0 on F b . It follows that (αH 1 − βH 2 ) |F b is cohomologous to 0 on F b for all b ∈ U where U ⊂ B is the smooth locus of π. This implies by Lemma 3.8 that the product [F b ] · (αH 1 − βH 2 ), hence L n · (αH 1 − βH 2 ), is cohomologous to 0 on X. Since a very general deformation of X preserving the Lagrangian fibration and H 1 , H 2 satisfies Matsushita's conjecture, we can apply [25, Theorem 0.9] so that αH 1 · L n = βH 2 · L n in CH • (X) ⊗ Q. It follows that α n H n 1 · L n − β n H n 2 · L n = (αH 1 − βH 2 ) ·
n−1 i=0 α i β n−1−i H i 1 · H n−1−i 2 · L n = 0 in CH 0 (X).
Since the zero-cycles supported on the constant cycle subvarieties Σ π,H 1 and Σ π,H 2 constructed above are proportional to H n 1 · L n and H n 2 · L n in CH 0 (X) respectively, the second statement of Theorem 1.3 follows. Now we prove iii). Let π : X → B and π ′ : X → B ′ be two Lagrangian fibrations and let L := π * c 1 (O B (1)) and L ′ := π * c 1 (O B ′ (1)). We have to show that H n · L n is proportional to H n · L ′n in CH 0 (X). We also assume that L and L ′ are not proportional, otherwise the proof is finished. By the same argument as in the proof of ii), there exist λ 1 , λ 2 ∈ Q such that λ 1 H n · L n = L ′n · L n = λ 2 H n · L ′n in CH 0 (X) and it suffices to show that deg (L ′n · L n ) 0 to conclude. We have λ 1 0 if and only if deg (L ′n · L n ) 0 if and only if q(L, L ′ ) 0 by Lemma 3.8, hence it suffices to show that q(L, L ′ ) 0. This follows from the fact that the restriction of q to NS(X) Q is of signature (1, 1 − ρ(X)), so the restriction of q to the two-dimensional subspace generated by L and L ′ cannot be zero. Since q(L, L) = q(L ′ , L ′ ) = 0, this implies that q(L, L ′ ) 0.
Theorem 1.4. -Let X be a Calabi-Yau manifold and f : X B a dominant meromorphic map over a Kähler base B. If 0 < dim B < dim X, then B is rationally connected.
be a projective hyper-Kähler manifold admitting a Lagrangian fibration of dimension 2n given by a line bundle L with q(L) = 0. Let H be any divisor class on X ; then the zero-cycle L n · H n is proportional to the class of a point x ∈ X which belongs to a Lagrangian constant cycle subvariety.ii) If a very general projective deformation of X preserving the Lagrangian fibration with ρ(X) ≥ 3 satisfiesMatsushita's conjecture (in particular if b 2 (X) ≥ 8[20]), then the class of a point in Σ π,H modulo rational equivalence is independent of the divisor class H.
Definition 3. 1 .
1-A subvariety Y of X is called constant cycle subvariety if every point in Y is rationally equivalent in X to each other.
Lemma 3. 2 .
2-Let Y ⊂ X be a connected subvariety. If the image of the Gysin map i * : CH 0 (Y) Q → CH 0 (X) Q is generated by an element o Y in CH 0 (X) Q , then Y is a constant cycle subvariety. In this case, we say that Y is represented by the zero-cycle o Y .
Remark 3. 3 .
3-A variety X is called CH 0 -trivial if the degree map deg : CH 0 (X) → Z is injective. Obvious examples of constant cycle subvarieties are provided by subvarieties in a CH 0 -trivial variety and any CH 0 -trivial subvariety in a variety. In particular, rationally connected subvarieties are interesting examples of constant cycle subvarieties from a deformation-theoretic point of view : since rational connectedness is an open and closed property, these subvarieties remain constant cycle as long as they survive under deformations of the ambient variety in which they embed. Note that in general, the property of being constant cycle is not stable under deformation of the embedding of a subvariety. As a counter-example, take an irrational constant cycle curve C inside a K3 surface S (such as examples constructed in[12]). The deformation of C ֒→ S covers a Zariski open subset of S, whereas CH 0 (S) is highly non trivial[16].Remark 3.4. -The main result of N. Fakhruddin in
Now we restrict ourselves to constant cycle subvarieties on projective hyper-Kähler manifolds. Let X be a projective hyper-Kähler manifold of dimension 2n and let η be a holomorphic symplectic 2-form on X.The following result is a direct consequence of Mumford-Roitman's theorem [22, Proposition 10.24] : Proposition-Definition 3.6. -If Y is a constant cycle subvariety of X, then Y is isotropic for η. In particular, dim Y ≤ n. If dim Y = n, then Y is called a Lagrangian constant cycle subvariety. The rest of Section 3 is devoted to the proof of Theorem 1.3 and Theorem 1.5. Proof of Theorem 1.3. -First we prove the following Lemma 3.7. -Let A be an abelian variety of dimension and h an ample divisor of A. There exist a finite number of points x ∈ A such that h = D[x] in CH 0 (A) where D is the degree of h , and the set of these points is nonempty.
Lemma 3. 8 (
8Matsushita [15] + Voisin [21]). -If j * : H 2 (X, Q) → H 2 (F, Q) denotes the restriction map and µ [F] : H 2 (X, Q) → H 2n+2 (X, Q) the cup product against [F], then ker µ [F] = ker j * = ker q(L, ·)
, then h = D[x + a]. Since h is ample, there exists a translation map t a such that t *Hence there are at most D 2 points x ∈ A such that D[x] = h in CH 0 (A).
For existence, first of all we remark that if x ′ ∈ A is a point such that (t *
a h) = D[x ′ ] in CH 0 (A) for some
translation map t a a h is
symmetric. So by Poincaré's formula [3, Corollary 16.5.7], there exists x ′
AcknowledgementI would like to thank my thesis advisor Claire Voisin for introducing me to this beautiful subject and for valuable discussions. I also thank Olivier Benoist, Lie Fu, Christian Lehn, Gianluca Pacienza, and Junliang Shen for interesting remarks and questions.
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[] |
[
"Multilevel Graph Partitioning for Three-Dimensional Discrete Fracture Network Flow Simulations",
"Multilevel Graph Partitioning for Three-Dimensional Discrete Fracture Network Flow Simulations"
] |
[
"Hayato Ushijima-Mwesigwa \nSchool of Computing\nClemson University\nClemsonSouth CarolinaUSA\n",
"Jeffrey D Hyman \nComputational Earth Science (EES-16)\nEarth and Environmental Sciences Division\nLos Alamos National Laboratory\nLos Alamos New Mexico\nUSA\n",
"Aric Hagberg \nComputer, Computational\nStatistical Sciences Division\nLos Alamos National Laboratory\nLos Alamos New Mexico\nUSA\n",
"Ilya Safro \nSchool of Computing\nClemson University\nClemsonSouth CarolinaUSA\n",
"Satish Karra \nComputational Earth Science (EES-16)\nEarth and Environmental Sciences Division\nLos Alamos National Laboratory\nLos Alamos New Mexico\nUSA\n",
"Carl W Gable \nComputational Earth Science (EES-16)\nEarth and Environmental Sciences Division\nLos Alamos National Laboratory\nLos Alamos New Mexico\nUSA\n",
"Gowri Srinivasan \nVerification and Analysis (XCP-8)\nX Computational Physics\nLos Alamos National Laboratory\nLos Alamos New Mexico\nUSA\n"
] |
[
"School of Computing\nClemson University\nClemsonSouth CarolinaUSA",
"Computational Earth Science (EES-16)\nEarth and Environmental Sciences Division\nLos Alamos National Laboratory\nLos Alamos New Mexico\nUSA",
"Computer, Computational\nStatistical Sciences Division\nLos Alamos National Laboratory\nLos Alamos New Mexico\nUSA",
"School of Computing\nClemson University\nClemsonSouth CarolinaUSA",
"Computational Earth Science (EES-16)\nEarth and Environmental Sciences Division\nLos Alamos National Laboratory\nLos Alamos New Mexico\nUSA",
"Computational Earth Science (EES-16)\nEarth and Environmental Sciences Division\nLos Alamos National Laboratory\nLos Alamos New Mexico\nUSA",
"Verification and Analysis (XCP-8)\nX Computational Physics\nLos Alamos National Laboratory\nLos Alamos New Mexico\nUSA"
] |
[] |
We present a topology-based method for mesh-partitioning in three-dimensional discrete fracture network (DFN) simulations that takes advantage of the intrinsic multi-level nature of a DFN. DFN models are used to simulate flow and transport through low-permeability fracture media in the subsurface by explicitly representing fractures as discrete entities. The governing equations for flow and transport are numerically integrated on computational meshes generated on the interconnected fracture networks. Modern high-fidelity DFN simulations require high-performance computing on multiple processors where performance and scalability depends partially on obtaining a high-quality partition of the mesh to balance work work-loads and minimize communication across all processors.The discrete structure of a DFN naturally lends itself to various graph representations, which can be thought of as coarse-scale representations of the computational mesh. Using this concept, we develop a variant of the multilevel graph partitioning algorithm to partition the mesh of a DFN. We compare the performance of this DFN-based mesh-partitioning with standard multi-level graph partitioning using graphbased metrics (cut, imbalance, partitioning time), computational-based metrics (FLOPS, iterations, solver time), and total run time. The DFN-based partition and the mesh-based partition are comparable in terms of the graph-based metrics, but the time required to obtain the partition is several orders of magnitude faster using the DFN-based partition. The computation-based metrics show comparable performance between both methods so, in combination, the DFN-based partition is several orders of magnitude faster than the mesh-based partition.
|
10.1007/s11004-021-09944-y
|
[
"https://arxiv.org/pdf/1902.08029v1.pdf"
] | 67,787,652 |
1902.08029
|
00b64c2ccfe071ecb5a487ab48f87555e2ead3f1
|
Multilevel Graph Partitioning for Three-Dimensional Discrete Fracture Network Flow Simulations
Hayato Ushijima-Mwesigwa
School of Computing
Clemson University
ClemsonSouth CarolinaUSA
Jeffrey D Hyman
Computational Earth Science (EES-16)
Earth and Environmental Sciences Division
Los Alamos National Laboratory
Los Alamos New Mexico
USA
Aric Hagberg
Computer, Computational
Statistical Sciences Division
Los Alamos National Laboratory
Los Alamos New Mexico
USA
Ilya Safro
School of Computing
Clemson University
ClemsonSouth CarolinaUSA
Satish Karra
Computational Earth Science (EES-16)
Earth and Environmental Sciences Division
Los Alamos National Laboratory
Los Alamos New Mexico
USA
Carl W Gable
Computational Earth Science (EES-16)
Earth and Environmental Sciences Division
Los Alamos National Laboratory
Los Alamos New Mexico
USA
Gowri Srinivasan
Verification and Analysis (XCP-8)
X Computational Physics
Los Alamos National Laboratory
Los Alamos New Mexico
USA
Multilevel Graph Partitioning for Three-Dimensional Discrete Fracture Network Flow Simulations
We present a topology-based method for mesh-partitioning in three-dimensional discrete fracture network (DFN) simulations that takes advantage of the intrinsic multi-level nature of a DFN. DFN models are used to simulate flow and transport through low-permeability fracture media in the subsurface by explicitly representing fractures as discrete entities. The governing equations for flow and transport are numerically integrated on computational meshes generated on the interconnected fracture networks. Modern high-fidelity DFN simulations require high-performance computing on multiple processors where performance and scalability depends partially on obtaining a high-quality partition of the mesh to balance work work-loads and minimize communication across all processors.The discrete structure of a DFN naturally lends itself to various graph representations, which can be thought of as coarse-scale representations of the computational mesh. Using this concept, we develop a variant of the multilevel graph partitioning algorithm to partition the mesh of a DFN. We compare the performance of this DFN-based mesh-partitioning with standard multi-level graph partitioning using graphbased metrics (cut, imbalance, partitioning time), computational-based metrics (FLOPS, iterations, solver time), and total run time. The DFN-based partition and the mesh-based partition are comparable in terms of the graph-based metrics, but the time required to obtain the partition is several orders of magnitude faster using the DFN-based partition. The computation-based metrics show comparable performance between both methods so, in combination, the DFN-based partition is several orders of magnitude faster than the mesh-based partition.
Introduction
Discrete Fracture Network (DFN) models are a computational tool for modeling flow and transport through low-permeability subsurface fractured rock. DFN models differ from conventional continuum models by explicitly representing fractures and the networks they form. This allows DFNs to represent a wider range of transport phenomena and makes them a preferred choice when linking network attributes to flow properties [24,35]. DFNs are utilized for characterizing fluid flow and solute transport through low permeability fractured media which is critical for a variety of subsurface applications including the environmental restoration of contaminated fractured media [57,59,75], aquifer storage and management [44], hydrocarbon extraction [36,41,53], longterm storage of spent civilian nuclear fuel [20,24,40],and CO 2 sequestration [39].
The choice to explicitly represent fractures results in a significantly higher computational cost than stochastic continuum [59] or dual porosity/permeability [47] models where upscaled effective properties are used to account for fracture properties. Once a network is constructed, the individual fractures are meshed for computation and the governing equations for flow and transport are numerically integrated on the computational mesh. The number of mesh cells required for a DFN depends on the number of fractures, the density of the network, and the range of length scales being resolved. Even modest sized DFNs containing O(10 2 ) fractures can have a mesh that contains several million nodes. Because of limited computational resources, the first DFN models either represented networks as a set of connected pipes [10,16] or were twodimensional representations [14]. Despite the large scale of DFN computations, high performance computing (HPC) enables flow and transport simulations in large three-dimensional DFNs [4,5,17,38,55,61,62].
These high-fidelity DFN simulations require HPC on multiple processors and the performance and scalability of these simulations necessitate a high-quality partitioning such that the computations are well-balanced across all processes with minimal communication between processors. Common partitioning methods [9] are based on either a global method, e.g., spectral partitioning and max-flow, or iterative local improvement heuristic algorithms, e.g., Kernighan-Lin [43] or Fiduccia-Matthesyses [19]. However, multilevel graph partitioning, which introduces a framework to make global decisions in conjunction with local improvements, is one of the most successful heuristics in practice for partitioning large graphs [12,42,66,69,70,77].
The basic idea behind multilevel graph partitioning is that a graph is successively coarsened, creating a hierarchy of smaller graphs until an initial (coarsest) partition can be computed efficiently. The initial partition is projected back to the next finer level, where local improvements are made. Once at a local optima, the improved partition is projected to the next finer level where further local improvement are made. The process continues until a partition is projected and refined back to the original graph. Multilevel graph partitioning methods are popular because they exhibit excellent trade-off between fast computational time and high-quality solutions compared to other techniques. However, some applications (for example those involving dynamic graphs) require graphs to be repartitioned, and thus require much faster techniques. Thus, depending on the application, even multilevel graph partitioning can take a significant amount of time.
To increase the speed of high-resolution DFN simulations, we propose an approach to graph partitioning the DFN mesh that combines the topological structure of the DFN with multilevel graph partitioning. The discrete structure of a DFN naturally lends itself to various graph representations, for example, vertices in graph can correspond to fractures in the DFN and edges in the graph to fracture intersections. These graphrepresentations of a DFN can also be thought of as coarse-scale representations of the computational mesh, which is the conceptual model that we use here to develop a variant of the multilevel graph partitioning algorithm for mesh partitioning. By using this partitioning on the DFN, we seek to accelerate the HPC computations. The proposed methodology assigns the first coarse level in the multilevel graph partitioning to be a weighted graph based on the topology of the DFN that accounts for the number of mesh nodes on each fracture.
We compare the relative cost of the proposed method with partitioning the full mesh and find that the total run time is reduced by several orders of magnitude using the proposed method. Partitioning the graphrepresentation of the DFN and projecting the solution onto the mesh is computationally cheaper than partitioning the DFN mesh itself since there are orders of magnitude fewer nodes and edges to consider in a graph based on the DFN topology. The method is also sensitive to the mesh resolution on each fracture, i.e., it accounts for the number of mesh nodes on each fracture. The performance of the method compared to partitioning the mesh is measured in terms of graph-based metrics (cut, imbalance, partitioning time), computational-based metrics (FLOPS, iterations, solver time), and total run time. In terms of graph-based metrics, the results obtained using the DFN-based partition are comparable to those obtained using the mesh-based partition, yet the DFN-based partition is several orders of magnitude faster. The results presented here indicate that using the proposed method overall reduces the required time for a single DFN realization simulation and thus allows one to perform more realizations for uncertainty quantification, for a fixed computational budget.
Discrete Fracture Networks
In low-permeability fractured media like shale and crystalline rock, fluid flow and the associated transport of solutes is mainly confirmed to the fractures embedded in the medium [57]. In these physical systems, the structure of the fluid velocity field therein is primarily controlled by the geometry of individual fractures, e.g., size and aperture, and the structure of the network as opposed to matrix properties, e.g., matrix porosity or pore-size distributions [13,15,35,32]. There are a number of methods used to model flow and the associated transport of chemical species including stochastic continuum [58,59,73], dual-porosity / dualpermeability [23,78,47], and discrete fracture network model (DFN) [10,48,49,50,60].
In the DFN methodology, individual fractures are represented as planar N − 1 dimensional objects embedded within an N dimensional space, lines in two dimensions and planes in three dimensions. The size of the domains of interest and the cost of sufficiently sampling relevant quantities in the subsurface, both hydraulic and structural, result in limited availability of data [6,57,78] and requires that DFN models are constructed stochastically. Each fracture within the network is assigned a shape, location, and orientation within the domain by sampling distributions whose parameters are determined by a site characterization [40]. The fractures form a network embedded within the porous medium that are meshed for computation and the governing equations for flow and transport are numerically integrated to simulate physical phenomenon of interest.
The stochastic generation of a DFN is a major obstacle in the creation of a high-quality computational mesh representation of each network. In practice, the planes representing each fracture are randomly included into the domain and can create arbitrarily small features, i.e., length scales, that render the automated meshing of the fracture plans infeasible. Figure 1 shows a DFN composed of 424 fractures in a 15 meter cube, which are represented as circular polygons, to demonstrate the range of length scales that exists in a DFN. Colors on the fractures correspond to the distance on the fracture to the nearest line of intersection and highlight the range of length scales that exists on a fracture plane and throughout the network. Mesh edges must be smaller that the smallest length scale in the network if the physics are to be properly resolved. This requirement is computationally infeasible for arbitrarily small length scales within large domains. There have been a number of methodologies to address this issue by modifying the mesh to remove small features [56,55] or coupling flow between non-conforming meshes using discretization schemes [4,17,61,62].
Meshing Strategy FRAM
The Feature Rejection Algorithm for Meshing (FRAM) introduced by Hyman et al. [33] is one method designed to address the aforementioned mesh generation issues. The cornerstone of FRAM is a user-defined minimum length scale (h) that determines what geometric features are represented in the network. FRAM constrains the generation of the network so that the smallest feature is greater than h through the entire network. This constraint provides a firm lower bound on the required resolution of the mesh and ensures that pathological cases, e.g., arbitrarily small intersections and distances between intersections, that degrade mesh quality do not exist. Then all the features in the network can be resolved by generating triangular cell edges with a minimum length slightly less than ≈ h/2. Once these constraints are met, a conforming Delaunay triangulation algorithm [54] is implemented to mesh each fracture in a manner such that all lines of intersection form a set of connected edges in the Delaunay triangulation. The dual of the Delaunay triangulation is a Voronoi tessellation, which in a certain sense is optimal for two-point flux finite volume solvers [18], that are commonly used in subsurface flow and transport simulators such as FEHM [79], TOUGH2 [63], and PFLOTRAN [46].
One key aspect of FRAM is the provided detailed control of the mesh resolution on each fracture because pathological cases that degrade mesh quality do not exist. Depending upon the physical process to be simulated, the mesh can either have variable or uniform resolution. Points of singularity in the pressure solution occur at the ends of intersection lines and high gradients in the pressure solution and flow fields occur close to the intersections. To properly resolve these gradients the mesh needs to be finer in these regions. If fracture properties are homogeneous within a fracture, i.e., uniform fracture apertures, or only a pressure solution is required, or transport will be simulated using particle tracking then the mesh can be coarsened away from the intersections without significant loss of accuracy. However, if non-uniform apertures are considered, as in [15,50], then the mesh needs to be sufficiently fine that length scales in the aperture field, e.g., correlation lengths, are resolved. Furthermore, if transport is simulated using an Eulerian approach, i.e., a numerical discretization of the advective-dispersion equation, where numerical diffusion/dispersion is controlled by the mesh resolution then a uniform mesh is more appropriate because numerical errors will be uniform across the domain.
Due to these considerations, we propose an extension of the FRAM to allow for variable mesh resolution based on distance from the lines of intersection in a fracture. From a topological point of view, every DFN can be represented as a tuple consisting of a set of fractures and a set of intersections. Formally, let F = { f i } for i = 1, . . . , N denote a fracture network composed of N fractures ( f i ). Every f i ∈ F is assigned a shape, location, and orientation within the domain by sampling distributions whose parameters are determined by a site characterization. Every
f i ∈ R 2 but the network F ∈ R 3 . Let I = {( f i , f j )} be a set of pairs associated with intersections between fractures; if f i ∩ f j = / 0 then ( f i , f j ) ∈ I.
The number of intersections M = |I| depends on the particular shape, orientation, and geometry of the set of fractures in the network. We denote the line of intersection between f i and f j as ( f i , f j ). Using these sets, the topology of a DFN can be defined as the tuple (F , I).
Next, we compute the minimum distance from every point on a fracture x ∈ f i to the lines of intersection on that fracture,
d(x) = min y∈ ( f i , f j ) x − y ∀ j s.t. ( f i , f j ) ∈ I .(1)
The maximum edge length in the mesh at a given distance from an intersection, denoted e(x) max , is determined by a two parameter piecewise linear function
e(x) max = ad(x) + h/2 d(x) ≤ rh, (ar + 1/2)h d(x) > rh.(2)
If an edge in the mesh is greater than e(x) max , then a new point is added to the mesh at the midpoint of that edge to split it in two. In practice, the edge spitting is done using Rivara refinement [64,65].
A few remarks about the method: (i) the mesh is refined to ≈ h/2 along the lines of intersection, (ii) the slope parameter a controls the rate that the mesh is coarsened away from the intersection and ensures gradual refinement, (iii) the distance parameter r determines furthest distance from the intersections that the mesh resolution is variable, (iv) to make the mesh uniform, one can either set a = 0 or r = 0.
Once the DFN is meshed, we can define the following functions
M f : F → Z + ,(3)
returns the number of mesh nodes on a fracture f i ∈ F and
M I : I → Z + ,(4)
returns the number of mesh nodes on the line of intersection ( f i , f j ) ∈ I. These functions allow us to consider the effects of different meshing strategies, uniform sized triangles compared to variable resolution, which we will use later in this study. Figure 2 (a) provides a close up view of uniform mesh resolution on the network shown in Fig. 1 and Fig. 2 (b) shows a close view of variable mesh resolution in the same region. In Fig. 2 (b) the mesh is coarsened away from fracture intersections to reduce the overall size of the mesh using the method described above. The mesh shown in Fig. 2
Graph Partitioning
In HPC computations one wants to minimize the communication between processors and insure that the work performed on each processor is balanced. This problem of minimizing communication and load balancing is identical to the problem of partitioning the graph corresponding to the sparsity pattern of matrix A [45], which, in our problem, is equivalent to partitioning the mesh of the DFN. Thus, for a computer with k processors, we seek a partition of the graph based on the DFN mesh into k parts of equal size where the edges between those parts is minimized.
k-way Graph Partitioning
Formally, given a graph G = (V, E) composed of vertices u ∈ V and edges e i, j = e(u i , u j ) ∈ E, with nonnegative vertex weights w i : V → R + and edge weights, w i, j : E → R + let P = (P 1 , . . . , P k ), be a partition of the vertex set V into k parts such that,
∪ i P i = V,(5)
and
P i ∩ P j = / 0 for i = j .(6)
For a given partition we can measure the volume of each piece of the partition
|P j | := ∑ u i ∈P j w i .(7)
The volume of each piece of the partition is used to provide a measure of imbalance. For an imbalance parameter ε > 0, we can determine if P satisfies the balance constraint
max i |P i | ≤ (1 + ε) |W | k ,(8)
where |W | = ∑ v i ∈V w i . Moreover, we can also measure the cut of a partition
C(P) = ∑ w i, j s.t. e i j ∈ E, u i ∈ P k , u j ∈ P l and k = l.(9)
The k-way graph-partitioning problem (GP) is to find a k-partition, P,that satisfies the balance constraint (8) and minimizes the cut (9). In general, these two desires conflict with one another. Indeed, this graph partition problem is an NP-hard problem [22,31].
Multilevel Graph Partitioning
Multilevel graph partitioning is one of the most successful heuristics for partitioning large graphs [8,11,21,25,27,42,51]. The idea behind multilevel graph partitioning originates from the multiscale optimization and multigrid strategies [7]. A graph is gradually coarsened to one where a k-way partition can be computed efficiently and effectively and then this partition is projected back onto the original graph. To be more specific, let us consider a weighted graph G 0 = (V 0 , E 0 ) that has weights on both vertices and edges. Algorithm 1 summarizes the multilevel framework for graph partitioning.
Input:
: G 0 = (V 0 , E 0 )
with vertex weights w i and edge weights w i, j . Output: : P(G)
1. Coarsening phase : The graph G 0 is transformed into a sequence of smaller graphs G 1 ,
G 2 , . . . , G m such that |V 0 | > |V 1 | > |V 2 | > . . . |V m |.
2. Initial (coarsest graph) partitioning phase: a high-quality algorithm is employed to obtain a k-way partition P m of the graph G m = (V m , E m ).
3. Uncoarsening phase: The partition P m of G m is projected back to G 0 via the intermediate partitions
P m−1 , P m−2 . . . , P 1 , P 0 which are refined at each level l ∈ [0, .., m − 1]. Algorithm 1: Multilevel Graph Partitioning
The approach consists of three main phases: (i) coarsening, (ii) initial partitioning and (iii) uncoarsening. In the coarsening phase the original graph (G 0 ) is gradually approximated by creating a hierarchy of coarsened graphs, G 1 , G 2 , . . . , G m , where there is a decreasing number of vertices in each graph |V 0 | > |V 1 | > |V 2 | > . . . |V m |. This can be achieved by collapsing edges and creating coarse level vertices, which are the nodes in the next level of the hierarchy that represent sets of vertices in next-coarser levels. The coarsening phase is stopped when the graph is small enough to be partitioned using an expensive but accurate algorithm. This phase is referred to as the initial partitioning phase. After the initial partitioning is performed, the uncoarsening phase begins, which is made up of two parts. In the first part of this stage, the partition at the coarser level P i is projected onto the graph one level finer in the hierarchy G i−1 , P i → P i−1 . Next, this projected partition is refined using a variant of the aforementioned improvement algorithms to create a better partition at this level in the hierarchy. This is done until P 0 is obtained. There are other (sometimes more sophisticated) multilevel frameworks for partitioning [52,69] and other cut-based problems on graphs such as the minimum linear arrangement [67], wavefront [29], bandwidth [68], and vertex separators [26].
DFN-based Graph Partitioning
In this section we describe one of the most common graph-representations of a DFN and develop methods to use that graph-representation in the partitioning of the mesh.
We adopt a graph representation of a DFN defined as a tuple(F , I), cf. Section 2.1, where vertices in the graph correspond to fractures in a F and edges correspond to elements in the set of intersections I. Hyman et al. [37] recently showed that this particular graph-representation of a DFN is a projection of a more general bi-partite graph. A simple undirected graph F = (V F , E F ) is constructed in the following way. For every
f i ∈ F , there is a unique vertex u i ∈ V F , φ : f i → u i .(10)
The vertex weight w i for vertex u i ∈ V F is the number of mesh nodes on the fracture f i , obtained using M f (3),
w i = M f ( f i ) ,(11)
Edges are defined in the following way. If two fractures, f i and f j intersect, ( f i , f j ) ∈ I, then there is an edge in E connecting the corresponding vertices,
φ : ( f i , f j ) ∈ I → e i j = (u i , u j ) ,(12)
where (u, v) ∈ E F denotes an edge between vertices u and v. The edge weight w i, j for vertex e(u i , u j ) ∈ E F is the number of mesh nodes on the edge ( f i , f j ) ∈ I, obtained using M I (4),
w i, j = M I [( f i , f j )] ,(13)
This particular mapping has been used by a variety of researchers [1,2,28,30,34,35,72,74,76]. Figure 3 shows a DFN composed of four fractures to demonstrate the connection between the graphrepresentation and the mesh. Figure 3(a) shows the DFN where each fracture has a unique color. Figure 3(b) shows the DFN with the mesh overlaid on the DFN, where the mesh colors correspond to the fracture on which they reside. Figure 3(c) shows the adopted graph-representation of the DFN where vertex colors coincide with the fracture colors and vertex size corresponds to the vertex weight. Figure 3(d) is a plot of the adjacency matrix of graph equivalent of the mesh where colors in the matrix correspond to the fractures on which the nodes reside. We perform a multi-index sort of the mesh nodes -first by fracture number, then x coordinate, y coordinate, and finally z coordinate. This sort reduces the bandwidth of the main diagonal of the adjacency matrix. The block structure of the mesh is a direct result of the fracture network topology, which is captured in the graph plot in Fig. 3(c). The mesh nodes on each fracture make up the main diagonal of the adjacency matrix in the plot shown in Fig. 3(d). The off diagonal nodes (black) correspond to mesh nodes along the fracture intersections. Each of these blocks corresponds to the a single vertex in graph shown in Fig. 3(c) and the number of non-zero entires in each block corresponds to the weight of the vertex. Mesh connections are mostly on a single fracture and there are fewer connections across fracture intersections, as indicated by the few off-diagonal terms in the adjacency matrix.
Multilevel DFN-based Graph Partitioning
We now propose a variant of the multilevel graph partitioning algorithm that takes advantage of the topology of a DFN. The basic idea behind the method is to perform the partitioning on a graph based on the topology of the DFN and then projecting the resulting partition onto the DFN mesh.
Hyman et al. [37] showed that the graph representation F defined by equations (10) and (12) is isomorphic to a DFN F . An implication of that is that for every partition of the graph based on the DFN P(F), there is a corresponding unique partition of the DFN P(F ). This follows directly from the properties of the mapping φ being a bijection. Applying φ −1 to P(F) defines a unique P(F ). Therefore, we can partition a DFN using this graph representation. However, we seek to partition the mesh of the DFN, not just the DFN. Let G = (V G , E G ) be the graph defined by the conforming Delaunay triangulation of the DFN. Note that with the exception of nodes along the lines of intersection in the DFN, every vertex v ∈ V G corresponds to a node in the mesh that resides on a single fracture f i ∈ F. Let f (v) = f i be a function that returns the fracture on which the node corresponding to the vertex v resides. For nodes on intersections between multiple fractures f i and f j , let
f (v) = min( f i , f j ).
We define a mapping Π : G → F to the graph F
Π : v i ∈ V F = {v ∈ V G s.t. f (v) = f i }(14)
and
Π : e i, j ∈ E F = {( f i , f j ) if ∃ v ∈ V G s.t. f (v) = ( f i , f j )}(15)
Define vertex weights on w v ∈ V F by (3), the number of nodes in the mesh that reside on each fracture, and the edge weights in E F by (4), the number of nodes along the lines of intersections between fractures. Note that F is the graph defined according to equations (10) and (12), the graph based on the topology of the DFN where each vertex corresponds to a fracture and edges indicate that fractures intersect. We retain information about the number of vertices that each coarse node in V F represents by using (11) and (13). The graph F is a coarse version of the mesh-based graph but |V F | ≪ |V G | by several orders of magnitude. We can apply the standard multilevel GP method to F and obtain P for a k-way partition. Conceptually, the proposed method defines the first level in the coarsening phase Π : G 0 → G 1 ≡ F and then a partition P(F) is obtained using algorithm 1. Once the partition P(F) is obtained, we project the the partition onto G using Π −1 . In other words, if a fracture f i ∈ P j , then all nodes in the mesh on f i , f (v) = f i , are placed into P j of G.
Theorem 4.1. The projection of a partition P F of the graph F = (V F , E F ) defined by equations (10) and (12) onto graph based on the mesh of the DFN G = (V G , E G )
Π −1 : P(F) → P(G)(16)
Proof. By definition every v ∈ V F is in a unique part of the partition P. Also, note that equation (14) is surjective. Therefore all v ∈ V G in the pre-image of v ∈ V F are in a unique part of the partition P. The proposed procedure drastically simplifies the coarsening phase because it reduces the number of steps that need to be taken to reach a graph F m where a k-way partition can be obtained, because the difference in size between G and F is large. Moreover, it reduces the the complexity of the uncoarsening phase, because P only needs to be obtained on F, not G. In practice, the mesh G is never constructed explicitly, only F needs to be passed to the multilevel GP and the solution passed to the mesh.
As an example, the DFN shown in Fig. 1 is made up of 424 fractures, so the graph-representation has 424 nodes, while the mesh has 870685 nodes for the uniform mesh and 360912 nodes for the variable resolution. Figure 4 (left) shows the graph based on that fracture network F colored according to a four-way partition. The DFN is shown on the right side of the image, where colors correspond to the partitions in F , i.e., the mesh is colored by P(G). Note that the projection Π −1 to obtain the partition P(G) is agnostic to the meshing strategy and resolution. But, as we shall see in the next section, the meshing strategy does affect the quality of the cut in the projected partition P(G).
Numerical Examples
We compare the proposed approach, where the partition of the mesh is based on the partition of the graph representation of the DFN, with the standard approach, where the mesh is partitioned directly. We consider a set of 30 independent identically distributed DFN realizations with both variable and uniform mesh resolution. For each network we consider two partitions: 1) The partition obtained on the mesh itself; we refer to these partitions as P (G) and 2) the partition induced from the partition on the graph representation of the DFN P (F). In our experimental results we use the graph partitioning package KaHIP [71] which among other methods implements the Global path algorithm for matching, and flow-based methods for partition refinement. The quality of the partitions are judged by the cut (number of edges that link between partitions) and the imbalance (the difference in sizes of the partitions). We also compare the impact of the partitions on computational performance by solving porous media flow equations, which are Laplace's equation under steady state, and solve for the distribution of pressure within the network. Here, we compare the number of FLOPS, the run time, and number of iterations to obtain the solution using a bi-conjugate gradient scheme with a block Jacobi preconditioner using the PETSC [3] toolkit. The meshes are partitioned into 2, 4, 8, and 16 partitions.
Generation and meshing of the 30 fracture networks is performed using the DFNWORKS computational suite [38]. A conforming Delaunay triangulation on each network is performed using the feature-rejection algorithm for meshing (FRAM) [33]. The parallelized subsurface flow and reactive transport code PFLO-TRAN [46] which uses PETSC is applied to obtain the solution to Laplace's equation. The 30 test DFN have fracture lengths that are drawn from a power-law distribution (a commonly observed property in the natural world [6]). Each DFN is constructed in a cubic domain with sides of length 15 m and are composed of circular fractures with uniformly random orientations and uniformly random centers. Fracture radii r [m] are sampled from a truncated power law distribution with exponent α = 2.6 and upper and lower cutoffs (r u = 5 m; r 0 = 1 m), with probability density function of
p r (r) = α r 0 (r/r 0 ) −1−α 1 − (r u /r 0 ) −α .(17)
The choice of exponent and cut offs are selected such that no single fracture directly connects inflow and outflow boundaries. Variability in hydraulic properties is included into the network by correlating fracture apertures to their radii. We use a positively correlated power-law relationship b = γr β where γ = 5.0 × 10 −5 and β = 0.5 are dimensionless parameters. On average the networks contain around 470 fractures. In the graph representation, there are around 470 nodes and 645 edges. When using a uniform mesh, there are, on average, one million nodes in the mesh (997,221) and nearly two million triangles (1,964,988). The graph based on the uniform mesh is made up of just under one million vertices and close to 3 million edges (2,962,302), on average. Thus, when partitioning the graph based on the uniform mesh there are 2000 times more vertices than when partitioning the graph based solely on the DFN topology. In the case of the variable mesh, there are around half a million nodes (415,206) and three quarter million triangles (836,452), on average. Therefore the graph based on the variable mesh is made up of just under half a million vertices and over one million edges (1,251,751), on average. Thus, when partitioning graph based on the variable mesh there are about 1000 times more vertices than when partitioning the graph based solely on the DFN topology.
Partition Quality
We begin by reporting the quality of the partitions and computation time. Table 1 reports the cut, imbalance, and times for the uniform and variable mesh resolution. Reported values are the average of the thirty realizations. Columns correspond to each partition and row are sorted by the number of partitions k. For the uniform mesh case, the lowest cuts are all obtained for P (G) for all values of k. The cut values obtained for P (F) are about twice as large as those obtained using P (G) but partitioning P (G) take four orders of magnitude longer than partitioning P (F). The observed difference in cut values for P (F) between uniform mesh and variable mesh is due to the different vertex weights in the DFN-based graph, due to different meshes, which results in slightly different partitions. In all cases, the imbalance values are about the same. Similar observations are made in the variable mesh case, but there are a few subtle differences. The difference in the partition quality in terms of the cut between P (G) and P (F) is substantially larger than in the uniform mesh resolution set. In the case of k = 16, the cut for P (F) is three times larger than for P (G). This increase in the cut values is a result of the fact that cuts in P (F) can only occur along intersections in the fracture network mesh, where the mesh is most refined and the highest number of nodes exists. In contrast, P (G) is not constrained in this manner and can therefore partition the mesh in region of the fracture where the mesh is coarse and fewer edges exists.
All imbalance values are approximately the same. Table 2 reports the number of GFlops, iterations required for the Krylov solver to converge, and run time using the partitions on the 30 networks. For all values of k, the selected metrics for the partitions P (G) and P (F) are roughly the same. An interesting observation is that even though the cuts of P (F) are three times larger than those of P (G) in the case of the uniform mesh, the run times are only slightly larger. Due to the fewer degrees of freedom in the variable mesh than the uniform mesh, the number of FLOPS, iterations, and solve time are lower than those reported for the uniform mesh. In general, the FLOPS and number of iterations are comparable between P (G) and P (F). However, the run times for P (F) are slower than for P (G). This slight slow down is likely related to aforementioned issues with the constrained cut location of P (F). Table 3 reports the total time taken for both the uniform and variable mesh partitions. In all cases, the slowest run times are reported for the P (G), primarily due to the time required for the partition. Note this also drastically affects the scaling of the total run time with number of processors. The fastest times are reported for P (F).
Computational Performance
Total Computational Time
Numerical Examples: Remarks
The examples lead to a few points that are worth discussing.
1. The time required to obtain the partitions using the graph based on the DFN topology is negligible compared to the time required to obtain the partition of the mesh (DFN Delaunay triangulation) due to the drastic difference in the size of the corresponding graphs.
2. In terms of cut, the quality of the partition projected down from the DFN onto the mesh depends upon the adopted meshing strategy -uniform resolution or variable resolution. In the case of uniform mesh resolution, the projected cuts are along the intersection lines that are the same resolution as the mesh within the fractures. However, in the case of a variable resolution mesh the projection of the DFN partition onto the mesh requires that the cuts be made along the intersections where mesh resolution is finest. Due to this, the difference between the cut on P (G) and P (F) is larger than the uniform mesh cases.
3. The quality of the partition influences the requires number of FLOPS, iterations of the Krylov solver, and simulation time. There is little difference in the computational performance between the partitions obtained on the mesh and DFN.
4. The total computational time was either dominated by the partitioning, in the case of mesh based partitioning, or the solver, in the case of the DFN based partitioning. The difference between the relative contribution of partitioning in the two methods are in stark contrast. In the case of the partition based on the mesh, the partitioning was ten to one hundred times slower than the linear solve. In the case of the partition based on the DFN, the partitioning phase was between ten to one hundred times faster. Note that the solver times were generally similar to mesh based partitioning, with the DFN based partition cases being slightly slower than the mesh based partition cases for the variable resolution scenario. These observations result in nearly two orders of magnitude speed up for overall computation when using the DFN based partition.
5. In standard multilevel graph partitioning, there is commonly a step in the uncoarsening phase where local refinements are made to the projected solution from one level higher in the hierarchy. We also applied this concept on the partition of the mesh obtained using the partition of the DFN as an initial condition. While this slightly improved the cut of the final partition, more so in the variable mesh resolution than for the uniform mesh resolution, it did not significantly affect the total run time (details not included). These insignificant changes indicate that the partition obtained by using projection of the DFN is of sufficient quality to not influence the solver run times.
Summary and Conclusions
DFN modeling is a powerful tool to improve our understanding of how the multi-scale structure of fractured media influences flow and transport therein. However, the explicit representation of these fracture networks, which contain length scales that range several orders of magnitude, is computationally demanding. As the number of fractures in a DFN increases, so does the size of the mesh and the associated physical systems to model physical phenomena within the DFN. This increase in computational requirements is compounded by the inherent uncertainty in the subsurface that requires numerous realizations of a DFN to bound system behavior. The combination of these facets requires that DFN models utilize efficient HPC methodologies to accelerate system solving time. Load balancing and minimizing communication between processors are key factors in such methodologies. Thus a cornerstone in the use of HPC for DFN simulations is a high-quality partition of the mesh. We presented a topologically-based method for mesh partitioning in DFN simulations that utilizes the intrinsic multilevel nature of the DFN. The method combines multilevel graph partitioning with a coarsescale graph representation of the DFN to drastically improve the speed of obtaining a high-quality partition of the DFN mesh. We partitioned the graph based on the DFN, rather than the mesh itself, and partition of the mesh is obtained by projecting the DFN partition onto the mesh. The large difference in size between the graph-based on the mesh and the graph-based on the DFN topology with the DFN based partition lead to a mesh based partition that required a fraction of the time.
We demonstrated the utility of the method by applying it to 30 three-dimensional discrete fracture networks composed of approximately five hundred fractures apiece. We also consider two different DFN meshing strategies. In the first, the mesh has uniform resolution and in the second the resolution of the mesh depends on the distance of the vertex on the fracture from the nearest intersection, with the mesh being finer close to the intersection. We compare the proposed method to standard mesh-based partitioning in terms of graph-based metrics (cut, imbalance, time to obtain the partition), computational-based metrics (FLOPS, iterations, solver time), and total run time. In terms of the graph-based metrics, the results obtained using the DFN-based partition are comparable to those obtained using the mesh-based partition, with the exception of the time required for the partition, which is several orders of magnitude faster in the case of the DFN-based partition. In terms of the computation-based metrics, e.g., solver time and FLOPS, the results are similar as well, depending slightly on the adopted meshing scheme. When combined, the DFN-based partition is therefore several orders of magnitude faster than the mesh-based partition. The results presented here indicate that using the proposed method would reduce the overall time for a single DFN realization simulation and thus allowing for an increase in the number of realizations that can be performed at a fixed computational cost.
The intrinsic multilevel structure of a DFN provides an elegant methodology for efficient and high-quality partitioning of the mesh for computational physics solutions. Multilevel graph partitioning coarsens the mesh until a graph that can be readily partitioned is found. Due to the strong block structure of the mesh and few off diagonal terms, the adopted graph representation is a good proxy for the coarsened version of the computational mesh. Moreover, the proposed methodology for hijacking multilevel graph partitioning could be applied to any system that exhibits such a structure and is thus amenable to a coarse-scale graph representation.
Figure 1 :
1A Discrete Fracture Network (DFN) composed of 424 fractures in a 15 meter cube. Colors correspond to the distance on the fracture plane to the nearest fracture intersection. The regions colored white are close to fracture intersections and darker colors indicate larger distances. The variability in colors on a single plane highlights the range of length scales that exist on a single fracture and throughout the network. To properly simulate relevant physical phenomenon the mesh representation of the network must be fine enough to resolve all of these length scales.
(a) is composed of 870,685 nodes and 1,712,924 triangles while the mesh in (b) is made up of 360,912 nodes and 725,787 triangles.
Figure 2 :
2(a) Close view of uniform mesh resolution for the DFN shown in Fig. 1. (b) Close view of variable mesh resolution. In (b) the mesh is coarsened away from fracture intersections to reduce the overall size of the mesh. The mesh shown in (a) is composed of 870,685 nodes and 1,712,924 triangles while the mesh in (b) is made up of 360,912 nodes and 725,787 triangles.
Figure 3 :
3(a) A DFN composed of four fractures. (b) The computational mesh on the DFN. Mesh colors coincide with the fracture on which the mesh node resides. (c) A graph representation of the DFN where vertex colors correspond with the fracture colors in (a). (d) The adjacency matrix for the mesh shown in (c). Colors in the matrix correspond to the fractures on which the nodes reside; black entries correspond to mesh nodes along fracture intersections.
: : F = (V F , E F ) Graph based on DFN Output: : P(G) Partition of the mesh of the DFN F 0 = F Initialize Multilevel method with finest level being the DFN based graph Perform Algorithm 1 to F 1. Coarsening phase : The graph F 0 is transformed into a sequence of smaller graphs F 1 , F 2 , . . . , F m such that |V 0 | > |V 1 | > |V 2 | > . . . |V m | 2. Initial partitioning phase: A local refinement algorithm is employed to obtain a k-way partition P m of the graph G m = (V m , E m ) 3. Uncoarsening phase: The partition P m of F m is projected back to F 0 via the intermediate partitions P m−1 , P m−2 . . . , P 1 , P 0 with subsequent refinements Π −1 : P 0 (F 0 ) → P 0 (G) Project the partition of F 0 onto the mesh of the DFN G Algorithm 2: Multilevel Graph Partitioning For DFN
Figure 4 :
4(Top) Graph-representation of the topology of the DFN. (Bottom) DFN colored based on a fourpart DFN-based partition. The partition of the mesh (Bottom) is obtained by projecting the partition on the DFN-based graph (Top) onto the mesh.
Figure 5
5also reports these values. The left subplot reports the times required using the mesh for partitioning as shown in blue and DFN-based graph are shown in green; hatched bars are for the uniform mesh and solid bars are the variable mesh. The right subplot is Log-Log plot of the total times corresponding to the one on left. Notice that using the DFN-based partitioning demonstrates good strong scaling, the red dotted line is ideal scaling, with increasing number of CPUs while the mesh based partitioning shows poor scaling with total run time increasing with the number of CPUs.
Figure 5 :
5Total time required for partitioning and flow solution. (Left) The times required using the mesh for partitioning are shown in blue and DFN-based graph are in green, hatched bars are for the uniform mesh and solid bars are the variable mesh. (Right) Log-Log plot of the total times corresponding to the one shown on left.
Table 1 :
1Partition MetricsUniform Mesh
Variable Mesh
Metric
k
P (G)
P (F)
P (G)
P (F)
Cut
2
404.73
671.67
237.20
676.37
4
941.87 1657.30 569.03
1632.00
8
1871.33 3412.50 1118.30 3403.23
16 3587.00 7503.43 2119.90 7637.17
Imbalance 2
0.03
0.02
0.02
0.02
4
0.04
0.04
0.04
0.04
8
0.05
0.04
0.05
0.04
16
0.05
0.05
0.05
0.05
Time [sec] 2
313.84
0.13
86.68
0.13
4
375.59
0.18
93.41
0.18
8
515.05
0.25
114.40
0.25
16 415.08
0.35
109.10
0.35
Table 2 :
2Computation MetricsUniform Mesh
Variable Mesh
Metric
k
P (G)
P (F)
P (G)
P (F)
GFlops
2
45.8
44.5
12.7
12.5
4
22.6
22.4
6.29
6.37
8
11.3
11.6
3.24
3.21
16
5.94
5.75
1.62
1.64
Iterations
2
1223.10 1180.20 804.03 788.13
4
1188.97 1175.33 784.97 796.27
8
1180.53 1211.60 806.60 799.90
16 1236.47 1195.17 800.87 812.83
Time [sec] 2
62.35
55.31
14.38
16.29
4
34.94
34.98
8.45
10.10
8
21.48
22.34
5.77
6.43
16
15.64
14.73
4.08
4.45
Table 3 :
3Total Time [sec]Uniform Mesh
Variable Mesh
k
P (G) P (F) P (G)/P (F)
P (G) P (F) P (G)/P (F)
2
376.19 55.44
3.74
101.06 16.42
6.15
4
410.53 35.16
6.64
101.86 10.28
9.90
8
536.53 22.59
11.48
120.17
6.68
17.98
16 430.72 15.07
19.97
113.11
4.80
23.56
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[] |
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"A co-located partitions strategy for parallel CFD-DEM couplings",
"A co-located partitions strategy for parallel CFD-DEM couplings"
] |
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"Gabriele Pozzetti \nUniversité du Luxembourg\nCampus Belval, 6, Avenue de la FonteL-4364Esch-sur-AlzetteLuxembourg\n",
"Xavier Besseron \nUniversité du Luxembourg\nCampus Belval, 6, Avenue de la FonteL-4364Esch-sur-AlzetteLuxembourg\n",
"Alban Rousset \nUniversité du Luxembourg\nCampus Belval, 6, Avenue de la FonteL-4364Esch-sur-AlzetteLuxembourg\n",
"Bernhard Peters \nUniversité du Luxembourg\nCampus Belval, 6, Avenue de la FonteL-4364Esch-sur-AlzetteLuxembourg\n"
] |
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"Université du Luxembourg\nCampus Belval, 6, Avenue de la FonteL-4364Esch-sur-AlzetteLuxembourg",
"Université du Luxembourg\nCampus Belval, 6, Avenue de la FonteL-4364Esch-sur-AlzetteLuxembourg",
"Université du Luxembourg\nCampus Belval, 6, Avenue de la FonteL-4364Esch-sur-AlzetteLuxembourg",
"Université du Luxembourg\nCampus Belval, 6, Avenue de la FonteL-4364Esch-sur-AlzetteLuxembourg"
] |
[] |
In this work, a new partition-collocation strategy for the parallel execution of CFD-DEM couplings is investigated. Having a good parallel performance is a key issue for an Eulerian-Lagrangian software that aims to be applied to solve industrially significant problems, as the computational cost of these couplings is one of their main drawback. The approach presented here consists in co-locating the overlapping parts of the simulation domain of each software on the same MPI process, in order to reduce the cost of the data exchanges. It is shown how this strategy allows reducing memory consumption and inter-process communication between CFD and DEM to a minimum and therefore to overcome an important parallelization bottleneck identified in the literature. Three benchmarks are proposed to assess the consistency and scalability of this approach. A coupled execution on 280 cores shows that less than 0.1% of the time is used to perform inter-physics data exchange.IntroductionEulerian-Lagrangian couplings are nowadays widely used to address engineering and technical problems. In particular, CFD-DEM couplings have been successfully applied to study several configurations ranging from mechanical[14,13], to chemical [11] and environmental [8] engineering.CFD-DEM coupled simulations are normally very computationally intensive, and, as already pointed out in [10], the execution time represents a major issue for the applicability of this numerical approach to complex scenarios. Therefore, optimizing the parallel performance of such a coupling is a fundamental step for allowing large-scale numerical solutions of industrial and technical problems.The parallelization of Eulerian-Lagrangian software is, however, rather delicate. This is mainly due to the fact that the optimal partitioning strategies for those wireframes are different, and the memory requirement of a coupled solution can represent a major performance issue.Furthermore, since the coupling normally affects an extended domain region (often the whole computational domain), the amount of information that is required to be exchanged is normally important. For this reason, highly efficient coupling approaches for boundary problems, like the one proposed in [4] may suffer for the extensive communication layer. At the same time, due to the Eulerian-Lagrangian nature of the coupling, mesh-based communication as the one proposed in [9] cannot, by themselves take care of the information exchange.One of the earliest attempt to parallelize a DEM algorithm was proposed in[21], where the authors distributed inter-particle contacts among processors on a machine featuring 512 cores. In their scheme, all particle data was stored in every process resulting in a memory-intensive computation that leads to a speedup of 8.73 for 512 cores with a 1672 particles assembly. A later work[12]showed how by reducing the DEM inter-process communication, a speedup of ∼11 within 16 process-computation of 100k particles was obtainable. This proved how, for the sole Lagrangian software, the memory usage and the inter-process communication can profoundly affect the parallel performance of the code.The problem of memory consumption and inter-process communication becomes even more important when a Lagrangian code is coupled with an Eulerian one as the single pieces of software generally reach the optimum performance with different load and partitioning strategies. This can lead to massive inter-process communication * Principal corresponding author. that can deeply affect the performance of the overall algorithm. In[5], the authors proposed a mirror domain method for ensuring the correct passage of information between Lagrangian bubbles and an underlying fluid flow in a parallel execution. This method consists in distributing global knowledge of all the Eulerian and Lagrangian domains, by letting a process performing only a part of the computation, and then distributing the information via gather-scatter operations. It was shown in[5], how this strategy allows a correct access to information and an almost linear speedup up to 4 processes for a case with 648000 CFD cells and 100000 particles. Nevertheless, the strategy wasn't able to offer significant advantages when operating on more than 32 processes. Therefore, its application is limited to medium-small scale problems.In order to cope with this problem, in [10], the authors proposed a parallelization strategy based on a Jostle graph for the DEM part and an independent parallelization for the CFD part. This allowed reducing memory consumption while keeping an high inter-physics communication load. The resulting coupling was observed to scale better than what previously seen in the literature obtaining a speedup of ∼35 for 64 processes. Nevertheless, the coupled code was shown to perform significantly worse than the sole DEM part, proving how the inter-physics communication between the Eulerian and the Lagrangian part can induce significant performance issues.This issue was underlined by [1], where a communication strategy based on non-distributed memory was implemented in order to reduce the communication costs. The results proved how this strategy can indeed reduce the DEM and the inter-physics communication, yet its advantages are limited to the usage of ∼30 processes though not allowing very-large-scale simulations.An important attempt to study large-scale problems with an Eulerian-Lagrangian coupling was presented in[19]. In this contributions, the authors propose a coupling between a highly efficient code for molecular dynamics and a well-known open-source software for CFD. The proposed decomposition strategy for CFD and molecular dynamics code are completely independent from each other, and all the required data for the inter-physics exchange is provided through a complex communication layer. It is shown how the specific couple of codes can tackle large-scale problems and scale over more than one hundred of processes. Nevertheless, when a large number of processes is used, the inter-physics communication becomes very important, taking up to 30% of the computation time.Nowadays, there is still no reference solution for the parallel execution of CFD-DEM couplings, and even though several strategies have been proposed, they have been focusing on specific software. The aim of the current work is to investigate the performance of a parallelization strategy for a CFD-DEM four-way coupling that aims to minimize the memory consumption and the inter-physics communication.The article is structured as follows. Firstly, the Lagrangian and Eulerian parts of this coupling are presented, and the systems of equations solved by the DEM and CFD part described. Secondly, the partitioning strategy based on co-located partitions is introduced. Finally, three benchmark cases are proposed in order to assess the consistency and performances of the proposed parallelization strategy.In this work, we refer to the coupling between the XDEM platform[16,15], which is used to treat the Lagrangian entities, and the OpenFOAM [22] libraries, which is used to resolve the fluid flow equations in a Eulerian reference. The parallelization strategy is presented in its general way and can be implemented for a generic CFD-DEM coupling independently from the specific software.To the best of the authors' knowledge, the partitioning strategy for CFD-DEM couplings here proposed is the first one that succeeds in keeping the inter-physics communication negligible while running over hundreds of process.MethodologyDiscrete Element Method (DEM)The Discrete Element Method (DEM) is a well-established approach for granular flows[7]. The XDEM platform[16,3,13]aims to extend the application of the DEM by attaching chemical/thermodynamical variables on the particles, and coupling the DEM core with codes for computational fluid dynamics (CFD). More in general, the coupling between CFD and DEM is a common strategy for addressing industrially relevant problems[13,11]. In this work, the DEM module of the XDEM platform is used to evolve a set of particles describing physical entities in the
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10.1016/j.apt.2018.08.025
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[
"https://arxiv.org/pdf/1802.05029v1.pdf"
] | 3,325,631 |
1802.05029
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5f42cdc15077801c9b0d00f2b375e3b73ea25264
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A co-located partitions strategy for parallel CFD-DEM couplings
Gabriele Pozzetti
Université du Luxembourg
Campus Belval, 6, Avenue de la FonteL-4364Esch-sur-AlzetteLuxembourg
Xavier Besseron
Université du Luxembourg
Campus Belval, 6, Avenue de la FonteL-4364Esch-sur-AlzetteLuxembourg
Alban Rousset
Université du Luxembourg
Campus Belval, 6, Avenue de la FonteL-4364Esch-sur-AlzetteLuxembourg
Bernhard Peters
Université du Luxembourg
Campus Belval, 6, Avenue de la FonteL-4364Esch-sur-AlzetteLuxembourg
A co-located partitions strategy for parallel CFD-DEM couplings
In this work, a new partition-collocation strategy for the parallel execution of CFD-DEM couplings is investigated. Having a good parallel performance is a key issue for an Eulerian-Lagrangian software that aims to be applied to solve industrially significant problems, as the computational cost of these couplings is one of their main drawback. The approach presented here consists in co-locating the overlapping parts of the simulation domain of each software on the same MPI process, in order to reduce the cost of the data exchanges. It is shown how this strategy allows reducing memory consumption and inter-process communication between CFD and DEM to a minimum and therefore to overcome an important parallelization bottleneck identified in the literature. Three benchmarks are proposed to assess the consistency and scalability of this approach. A coupled execution on 280 cores shows that less than 0.1% of the time is used to perform inter-physics data exchange.IntroductionEulerian-Lagrangian couplings are nowadays widely used to address engineering and technical problems. In particular, CFD-DEM couplings have been successfully applied to study several configurations ranging from mechanical[14,13], to chemical [11] and environmental [8] engineering.CFD-DEM coupled simulations are normally very computationally intensive, and, as already pointed out in [10], the execution time represents a major issue for the applicability of this numerical approach to complex scenarios. Therefore, optimizing the parallel performance of such a coupling is a fundamental step for allowing large-scale numerical solutions of industrial and technical problems.The parallelization of Eulerian-Lagrangian software is, however, rather delicate. This is mainly due to the fact that the optimal partitioning strategies for those wireframes are different, and the memory requirement of a coupled solution can represent a major performance issue.Furthermore, since the coupling normally affects an extended domain region (often the whole computational domain), the amount of information that is required to be exchanged is normally important. For this reason, highly efficient coupling approaches for boundary problems, like the one proposed in [4] may suffer for the extensive communication layer. At the same time, due to the Eulerian-Lagrangian nature of the coupling, mesh-based communication as the one proposed in [9] cannot, by themselves take care of the information exchange.One of the earliest attempt to parallelize a DEM algorithm was proposed in[21], where the authors distributed inter-particle contacts among processors on a machine featuring 512 cores. In their scheme, all particle data was stored in every process resulting in a memory-intensive computation that leads to a speedup of 8.73 for 512 cores with a 1672 particles assembly. A later work[12]showed how by reducing the DEM inter-process communication, a speedup of ∼11 within 16 process-computation of 100k particles was obtainable. This proved how, for the sole Lagrangian software, the memory usage and the inter-process communication can profoundly affect the parallel performance of the code.The problem of memory consumption and inter-process communication becomes even more important when a Lagrangian code is coupled with an Eulerian one as the single pieces of software generally reach the optimum performance with different load and partitioning strategies. This can lead to massive inter-process communication * Principal corresponding author. that can deeply affect the performance of the overall algorithm. In[5], the authors proposed a mirror domain method for ensuring the correct passage of information between Lagrangian bubbles and an underlying fluid flow in a parallel execution. This method consists in distributing global knowledge of all the Eulerian and Lagrangian domains, by letting a process performing only a part of the computation, and then distributing the information via gather-scatter operations. It was shown in[5], how this strategy allows a correct access to information and an almost linear speedup up to 4 processes for a case with 648000 CFD cells and 100000 particles. Nevertheless, the strategy wasn't able to offer significant advantages when operating on more than 32 processes. Therefore, its application is limited to medium-small scale problems.In order to cope with this problem, in [10], the authors proposed a parallelization strategy based on a Jostle graph for the DEM part and an independent parallelization for the CFD part. This allowed reducing memory consumption while keeping an high inter-physics communication load. The resulting coupling was observed to scale better than what previously seen in the literature obtaining a speedup of ∼35 for 64 processes. Nevertheless, the coupled code was shown to perform significantly worse than the sole DEM part, proving how the inter-physics communication between the Eulerian and the Lagrangian part can induce significant performance issues.This issue was underlined by [1], where a communication strategy based on non-distributed memory was implemented in order to reduce the communication costs. The results proved how this strategy can indeed reduce the DEM and the inter-physics communication, yet its advantages are limited to the usage of ∼30 processes though not allowing very-large-scale simulations.An important attempt to study large-scale problems with an Eulerian-Lagrangian coupling was presented in[19]. In this contributions, the authors propose a coupling between a highly efficient code for molecular dynamics and a well-known open-source software for CFD. The proposed decomposition strategy for CFD and molecular dynamics code are completely independent from each other, and all the required data for the inter-physics exchange is provided through a complex communication layer. It is shown how the specific couple of codes can tackle large-scale problems and scale over more than one hundred of processes. Nevertheless, when a large number of processes is used, the inter-physics communication becomes very important, taking up to 30% of the computation time.Nowadays, there is still no reference solution for the parallel execution of CFD-DEM couplings, and even though several strategies have been proposed, they have been focusing on specific software. The aim of the current work is to investigate the performance of a parallelization strategy for a CFD-DEM four-way coupling that aims to minimize the memory consumption and the inter-physics communication.The article is structured as follows. Firstly, the Lagrangian and Eulerian parts of this coupling are presented, and the systems of equations solved by the DEM and CFD part described. Secondly, the partitioning strategy based on co-located partitions is introduced. Finally, three benchmark cases are proposed in order to assess the consistency and performances of the proposed parallelization strategy.In this work, we refer to the coupling between the XDEM platform[16,15], which is used to treat the Lagrangian entities, and the OpenFOAM [22] libraries, which is used to resolve the fluid flow equations in a Eulerian reference. The parallelization strategy is presented in its general way and can be implemented for a generic CFD-DEM coupling independently from the specific software.To the best of the authors' knowledge, the partitioning strategy for CFD-DEM couplings here proposed is the first one that succeeds in keeping the inter-physics communication negligible while running over hundreds of process.MethodologyDiscrete Element Method (DEM)The Discrete Element Method (DEM) is a well-established approach for granular flows[7]. The XDEM platform[16,3,13]aims to extend the application of the DEM by attaching chemical/thermodynamical variables on the particles, and coupling the DEM core with codes for computational fluid dynamics (CFD). More in general, the coupling between CFD and DEM is a common strategy for addressing industrially relevant problems[13,11]. In this work, the DEM module of the XDEM platform is used to evolve a set of particles describing physical entities in the
body force C surface curvature F fpi fluid-particle interaction force σ σ σ f fluid stress tensor ρ f fluid density ρ p particle density A p particle surface area ε local porosity M coll torque acting on a particle due to collision events F g gravitational force m particle mass M ext external torque acting on a particle I particle moment of inertia u particle velocity F coll contact force acting on a particle F i j contact force acting on the particle i due to the collision with particle j φ φ φ particle orientation F drag force rising from the particle-fluid interaction ω ω ω particle angular velocity presence of a multiphase fluid. Defining with x i the positions, m i the masses, and φ φ φ the orientations, one can write
m i d 2 dt 2 x i = F coll + F drag + F g ,(1)I i d 2 dt 2 φ φ φ i = M coll + M ext ,(2)
In equation 1, the term F c indicates the collision force,
F coll = ∑ i = j F i j (x j , u j , φ φ φ j , ω ω ω j ),(3)
with u j the velocity of particle j, and ω ω ω the angular velocity. The term M coll indicates the torque acting on the particle due to collisions
M coll = ∑ i = j M i j (x j , u j , φ φ φ j , ω ω ω j ),(4)
with M i j the torque acted from particle j to particle i. The term F drag takes into account the force rising from the interaction with the fluid, and F g corresponds to the gravitational force. For what concerns F drag the analytical expression of the term can be obtained from the literature. Many authors have focused on its description [6,2] that is normally provided in the form of a semi empirical law
F drag = β (u f − u p ),(5)β = β (u f − u p , ρ f , ρ p , d p , A p , µ f , ε),(6)
with u f , u p the fluid and particle velocity respectively, ρ f , ρ p the respective densities, d p , A p the particle characteristic length and area, µ f the fluid viscosity, and ε the porosity, defined as the ratio between the volume occupied by the fluid and the total volume of the CFD cell. For the sake of generality, we took β as described in [18]. During the parallel execution of XDEM, the simulation domain is geometrically decomposed in regularly fixed-size cells that are used to distribute the workload between the processes. Every process has global knowledge of the domain structure and decomposition, but only performs the calculation, and holds knowledge, for the particles that belong to its sub-domain. The load partitioning between processes has been shown to be often crucial for the parallel performance of XDEM [17], and different partitioning strategies are available within the XDEM platform. In this contribution, we investigate the parallel performance of XDEM when operated with geometrically uniform partitions for the coupling with OpenFOAM. In figure 8, standalone XDEM executions compare the different partitioning algorithms, including the geometrically uniform approach, in order to assess their influence on the specific cases used in this work.
Computational Fluid Dynamics (CFD)
For the fluid solution, we refer to an unsteady incompressible flow through porous media with forcing terms arising from particle phase, as described in [16,15,14]. This system must fulfill the Navier-Stokes equations in the form,
∇ · εu f = ∂ ε ∂t ,(7)∂ εu f ∂t + ∇ · (εu f u f ) = −ε∇p + T D + F b + F fpi ,(8)T D = ∇ · ε µ f ∇u f + ∇ T u f ,(9)
with u f the fluid velocity, p the fluid pressure, µ f the fluid viscosity, F b a generic body force, F fpi the fluid-particle interaction force, that is the counterpart of F drag , which is here treated with the semi-implicit algorithm proposed in [23]. This set of equations is solved with the OpenFOAM libraries [22], which parallelization is based on domain decomposition. The CFD domain is split into sub-domains assigned to each process available at run time, over each of them a separate copy of the code is run. The exchange of information between processes is performed at boundaries through a dedicated patch class as described in [22]. In this contribution, the CFD domain is partitioned according to the DEM domain, as described in section 2.3. In figure 5, we compare standalone OpenFOAM execution with different partitioning algorithm to assess the influence of the partitioning on our specific cases.
Co-located partitions strategy for domain decomposition
Since CFD-DEM couplings based on volume averaging technique only consider local exchanges of mass, momentum, and energy, the Lagrangian (DEM) entity will be affected only from the fluid characteristic of an area close to its center. At the same time, the fluid flow will be only locally interested by the action of particles present in the local region.
In this contribution, we are taking advantage of theses characteristics to design an optimized partitioning strategy for our CFD-DEM simulation. This partitioning strategy is illustrated on figure 1:
1. In our problem, the CFD (OpenFOAM) and DEM (XDEM) domains are overlapping, totally or partially. As stated earlier, interactions between the fluid and the particles happen for objects closely located in the simulation space.
2. In consequence, our partitioning strategy considers the 2 domains together and its goal is to maintain the locality of the objects in the simulation. This means the domain elements co-located in the simulation space are assigned to the same process. In this way, the partition i of the CFD and the partition i of the DEM domain will be coinciding.
3. For the parallel execution, partitions are distributed to the computing nodes. It is important to make sure that partition i of the CFD domain and partition i of the DEM domain are actually located in the computing node, or even better, in the same MPI process.
4. With this partitioning strategy, the parallel execution benefits from the resulting communication pattern.
Inter-partition intra-physics data exchanges, i.e. the same that occur in a standalone execution of OpenFOAM or XDEM, are performed with the MPI communication layer using native implementation. This ensures good portability of the approach as the intra-physics communication implementation is not changed. Intra-partition inter-physics data exchanges are implemented with simple read/write operation in memory or direct library function calls because the two partitions share the same intra-process memory space. These data exchanges, which represent the majority of the data exchange due to the co-located partitions strategy, are much faster than the ones based on traditional communication layer like MPI.
Inter-partition inter-physics data exchanges are achieved using the MPI communication layer. Thanks to the co-located partitions strategy, they are reduced to a minimum, or non-existent if the partitions are perfectly aligned.
As a result, our approach has many advantages compared to the other work in the literature (cf section 1). Firstly, we rely on the distribution of the simulation domain among the computing nodes. Every partition of the computational domain keeps only its local data, and therefore reduces memory consumption. This is important as it allows simulating large-scale problems that would not fit in the memory of a single computing node. Secondly, by running the CFD and DEM code in the same process (by linking the two libraries into one executable), the local inter-physics data exchanges can be performed by direct memory accesses and avoid the overhead of an additional communication layer. Thanks to our partitioning strategy that imposes the co-location constraint on the CFD and DEM sub-domains, we ensure that the majority of the inter-physics data exchange will occur in that way. These advantages are meant to reduce the overhead of a parallel execution and to simplify the structure of the interface between the different codes, that will be identical for parallel and sequential execution.
In this study, we implement the proposed strategy by using a simple partitioning algorithm that enforces the co-location constraint. This algorithm controls the partitioning of both the CFD and DEM domain. For the sake of simplicity, the global simulation space is here partitioned with a uniform strategy based on a geometric decomposition. This consists in splitting the domain into regions of similar volume making the spatial volume of the domains, that are assigned to each process, balanced. In this way, every process will handle a similar volume of the computational domain holding data and performing the computation for the CFD and DEM part occupying the specific spatial region. While this strategy is not optimal for a generic case, it greatly simplifies the implementation and the analysis of the results.
The result of this partitioning is then enforced into each software bypassing any internal they may use. For OpenFOAM, this is achieved using the manual decomposition method which allows using a predefined partitioning defined in a text file. In this way, our partitioning algorithm considers the global simulation space of two physics software as a whole, not differencing if a point in space belongs to the CFD mesh or the DEM domain. This strategy is significantly different from what proposed in previous works [19], where the partitioning of CFD and DEM domains was aiming to optimize the load-balance of each standalone software, rather than the communication cost of the coupled execution.
Experimental results
Experimental methodology
In order to assess the validity of our approach and evaluate the scalability of the proposed strategy, we have setup and executed three benchmarks.
The first benchmark, One particle traveling across a processes boundary presented in section 3.2, tests the equivalence of the results between sequential run and parallel run with the current parallelization strategy. This is done to assess the validity of our approach and implementation by checking the continuity of the results when the particle travels across process boundary even without inter-partition inter-physics communication.
The second benchmark, One million particles in two million cells proposed in section 3.3, investigates the effects of an heavy inter-physics communication on the performance of a standalone software. This is done to show how the proposed parallelization strategy allows maintaining the performance of the original single solver in the presence of an heavy inter-physics communication even with hundreds of processes.
The third benchmark, Ten Million Particles in one Million cells described in section 3.4, studies the parallel performance of a coupled solution in case of an heavy coupled case. This is done to show how our solution can handle highly costly simulations and at the same time allows to resolve a main issue underlined in the literature linked to the inter-physics communication.
The experiments were carried out using the Iris cluster of the University of Luxembourg [20] which provides 168 computing nodes for a total of 4704 cores. The nodes used in this study feature a total a 128 GB of memory and have two Intel Xeon E5-2680 v4 processors running at 2.4 GHz, that is to say a total of 28 cores per node. The nodes are connected through a fast low-latency EDR InfiniBand (100Gb/s) network organized in a fat tree. We used OpenFOAM-Extend 3.2 and XDEM version b6e12a86, both compiled with Intel Compiler 2016.1.150 and parallel executions were performed using Intel MPI 5.1.2.150 over the InfiniBand network.
To ensure the stability of the measurement, the nodes were reserved for an exclusive access. Additionally, each performance value reported in this section is the average of at least hundred of measurements. The standard deviation showed no significant variation in the results.
3.2. One particle traveling across a process boundary A first basic test is proposed, featuring a particle traveling across a boundary between processes. This is done in order to test the equivalence of the results obtained in sequential and parallel execution. The particle, initially at rest within the domain assigned to process 0, is accelerated by the fluid according to the law of equation 5. The resulting drag force pushes the particle across the boundary with process 1 causing it to be transferred from the sub-domain 0 to the sub-domain 1.
As shown in figure 2, the boundary conditions imposed on the fluid domain are a uniform Dirichlet at the inlet, no-slip at the wall, and reference pressure at the outlet. The CFD domain is a square channel of dimensions of 1m, 0.2m and 0.2m, and is discretized with 240 identical cubic cells.
In figure 3, the normalized particle velocity and acceleration, and the normalized solid volume and L 1 norm of the fluid acceleration are proposed as a function of times. All the quantity are normalized by dividing them by their maximum value, so that they can be displayed on the same plot. The particle crosses the process boundary at 0.32s leaving the domain assigned to process 0 and entering into the domain assigned to process 1.
It can be observed how the particle velocity and drag force are continuous across the processes boundary. This shows how the information on the fluid velocity at the particle position is correctly exchanged between CFD and DEM code in the whole domain, including the regions between boundaries. Similarly, the porosity and acceleration fields projected by the particle into the Eulerian grid do not suffer discontinuities when the particle switches between processes. In particular, it can be noticed how the results obtained with the parallel execution are perfectly matching the one obtained by running the code in sequential for both particle and fluid quantities. This shows how even without keeping global knowledge of the whole computational domain, continuity across processes of both Eulerian and Lagrangian quantities can be achieved.
This allows greatly limiting both the memory usage and the inter-process communication.
One million particles in two million cells
This second test aims to evaluate the influence of an intense intra-physics communication load on the scalability of the code. For that, a coupling between OpenFOAM for the CFD part and a dummy DEM software, named DummyDEM, is used. DummyDEM, is a modified version of XDEM whose purpose is to trigger all the necessary data exchange, but does not perform any actually DEM computation, i.e. the particles are not moving. This case is similar to the one proposed in the previous section, but in order to increase the amount of data exchange it contains a swarm of one million of particles suspended in a channel flow discretized with two millions of identical cubical CFD cells ( figure 4). The boundary conditions imposed on the fluid domain are as in the previous test the standard inlet, wall and outlet conditions on respectively the channel inlet, the surrounding walls and the outlet section. In figure 5, we first present the scalability of the standalone OpenFOAM run obtained by decomposing the domain with the native SCOTCH and METIS partitioners, and the uniform partitioner that is used in the coupling. One can notice how the behavior of OpenFOAM is not highly dependent on the partitioner adopted. For that case, this can be explained considering that the problem is rather uniform and therefore the optimal partitioning would be close to the uniform one.
Standalone OpenFOAM performance
Coupled OpenFOAM-DummyDEM performance
In a first step to understand the performance of the coupled CFD-DEM approach, we first decided to study the behavior of the coupling in case of uniformly distributed load, so that the final cost is not influenced by dynamic load balancing. In order to obtain this, our coupling OpenFOAM-DummyDEM performs all the coupling data exchange but our simplified DEM implementation, DummyDEM, does not execute any DEM calculation. As a result, in this testing coupling OpenFOAM-DummyDEM case, we have at every timestep:
• data exchange from CFD to DEM: the fluid velocity is used to calculate the drag force applying on the particles • data exchange from DEM to CFD: the distribution of the particles is used to calculate the porosity field of the CFD domain However, DummyDEM will not integrate the particle position which will remain stationary in the simulation domain. Furthermore, this allows adopting the same timestep for the CFD and DEM parts, further simplifying the analysis of the results. The resulting case is therefore equivalent to a pure CFD case, on the top of which a inter-physics communication is triggered. This allows studying the effects of the data exchange on the scalability of the software. In figure 6, we evaluate the scalability of our OpenFOAM-DummyDEM coupling approach, in comparison with standalone OpenFOAM and standalone DummyDEM execution. The aim of this is to underline the influence of the presence of an interface between the Lagrangian and Eulerian domain, on the overall algorithm scalability.
One can observe how in this case, the parallel performance of the DummyDEM part is poor. This is explained considering that the decision of not integrating the particle motion, which makes the DEM computation rather inexpensive, leading to the flat scalability profile observed in figure 6. This also offers the possibility of studying the performance of the coupled solver in a case where the performance of the two basic software is markedly different.
In this configuration, for an execution on 252 cores, we measured that 81% of the execution time is spent in OpenFOAM code, 18% in DummyDEM and less than 1% doing inter-physics data exchange. As a consequence, the performance of the OpenFOAM-DummyDEM coupling is similar to the one of the standalone OpenFOAM execution. This shows how the presence of the interface between the two codes, that in this case is coupling one million of particles with two millions of cells, does not affect the performance significantly.
This can be explained considering that, in this case, the generated partitions are perfectly aligned and, as a result, it does not introduce any inter-physics inter-process communication, and the most of the inter-physics data exchanges use direct memory accesses and avoid any communication layer. This also proves how, by using the co-located partitions strategy, the presence of an intensive inter-physics data exchange does not affect the parallel performance of the software. This is an important step for parallel execution of coupled software, as it should be ensured that its performance matches the one of a heavy loaded standalone algorithm without an excessive overhead.
Ten Million Particles in one Million cells
The third test case consists in a layered bed of ten million particles moving in the presence of a carrier gas. This case was inspired by [19] and chosen to show the capability of the coupling to treat a very large amount of DEM particles while keeping the inter-physics communication cost low.
As shown in figure 7, the test case is of the form proposed in [19] with a domain size of 480mm x 40mm x 480 mm, featuring a layered bed of 10 million of particles moving under the action of an incompressible flow. The boundary conditions for the fluid solution are of constant Dirichlet at the inlet (2m/s), non-slip at the wall boundaries and reference atmospheric pressure at the outlet.
In [19] was pointed out that with heavy coupled cases, when partitioning the two domains independently, the inter-physics communication becomes more and more important with the rise of computing processes, reaching up to 30% of the whole cost of the simulation when using more than 200 computing processes. This means that, when using an independent domain partitioning from each software, the coupled simulation will be negatively affected by the coupling interface, leading to a coupled execution that will perform worse than the single-code one when operated on a high number of computing nodes. This behavior was also identified in [10] where the authors clearly showed how the coupled execution was performing worse than both of the standalone software. In figure 8, the scalability performance of the pure DEM execution obtained with different partitioning algorithms is proposed. One can observe how the results are affected by the choice partitioning algorithm, but the general behavior is rather similar. This can be explained considering that the setup for this case is rather homogeneous and uniformly distributed.
Similarly, in figure 9, the performance of the pure CFD execution obtained with a uniform partitioning and a classic OpenFOAM partitioner (SCOTCH) are compared. Once more, the trends are rather similar and only minor differences can be noticed. In figure 10, we evaluate the performance of our coupled OpenFOAM-XDEM approach in comparison with standalone XDEM and standalone CFD, all executed with a uniform partitioning. It is interesting to notice how the parallel execution of the coupled code is not worse than the one of the single code when operating with more than 200 processes. This can be explained by the repartition of the computational cost: more than 92% is spent in the XDEM part, less than 8% in OpenFOAM and the inter-physics data exchanges represented less than 0.1% as shown on figure 11. It appears that, in this case, the cost of the coupling is negligible and the behavior of the coupled solution is driven by the most computation intensive part of the code, namely XDEM.
Coupled OpenFOAM-XDEM performance
This represents a major improvement compared to previous work. In particular, instead of rising as it was happening for [19], the inter-physics data exchange reduces in magnitude when using our co-located partitions strategy. This can be explained considering that no inter-process communication is involved, and the amount of the data exchanged per each process reduces. In this way, instead of becoming one of the limiting factors for the scalability of a coupled execution, this communication benefits from being executed in parallel when a co-located partitions strategy is adopted. This represents an important advantage for large-scale applications that need to run over hundreds of processes in order to produce results within a reasonable computational time.
Discussion
The main novelty of the current approach presented in section 2.3 is to introduce a new partitioning strategy for CFD and DEM domains. This strategy aims to co-locate partitions from the CFD and DEM domain that will need to perform information exchange, in the same MPI process. This is radically different from what previously proposed in the literature, where most of the works focused on optimizing the load-balancing of each software independently from each other while we focus first on reducing the inter-physics communication.
We implemented this strategy by defining a partitioning algorithm that forces CFD and DEM to be co-located. In the cases proposed in this paper, the co-location constraint is completely fulfilled, meaning that all the information required for the inter-physics exchanges are present within the same MPI process. Therefore, the inter-physics interprocess communications are completely absent. As shown in section 3.4, this strategy allows overcoming the interphysics communication bottleneck that was identified in [19].
Nevertheless, adopting a co-location constraint introduces a significant limitation in the partitioning flexibility of the two domains. First, an hybrid partitioning algorithm must be adopted that can optimize the executions of the two codes. In uniform cases, as the one proposed in this work, a uniform partitioning algorithm allows achieving reasonably good parallel performances. On the other hand, in cases in which the DEM and CFD parts have significantly different parallel requirements (as, for instance, when the computational load of the DEM part is clustered in a specific region of the domain, while the CFD load is uniform) a better partitioning algorithm that optimizes the coupled execution must be designed.
We want to point out how, for more complex partitioning algorithms, a perfect fulfillment of the co-location constraint might be difficult or not possible. In this cases, one might consider relaxing the co-location constraint allowing the partitions to be non-perfectly coinciding. This will require the introduction of an inter-physics interpartition communication layer that will cause a communication overhead with respect to the cases here presented. This cost will be null for perfectly coinciding partitions, and maximum for completely non-coinciding partitions. Therefore, an optimal partitioning algorithm for CFD-DEM couplings must consider both the cost associated with intra and inter-physics communication in order not to be excessively penalized by any bottleneck.
Alternatively, in several cases, it might not be always necessary to have such a complex hybrid partitioning algorithm. When one of the software is clearly dominating in term of computational load, it could be sufficient to partition its domain with one of its native algorithms, and decide the partitions of the second software to match the decision of the dominant one, respecting the co-location principle proposed in this work.
Conclusion
A partitioning strategy for CFD-DEM couplings has been investigated. It consists of the usage of co-located partitions for the CFD and the DEM domains. This allows performing the bulk information exchange between the two codes locally, reducing the global communication time and in particular minimizing the inter-physics parallel communication. This strategy was implemented by using a simple uniform partitioning algorithm that forces the partitions to be co-located.
Three benchmark cases have been introduced in order to assess the consistency and performances of the proposed strategy. Experimental results were carried out using the coupling between the XDEM framework and the OpenFOAM libraries. The validity of the results have been assessed by comparing sequential and parallel executions, and scalability performance over hundreds of processes has been reported showing that less than 0.1% of the time is used for inter-physics data exchange.
The main advantage of the proposed strategy consists in the reduction of the cost associated with the inter-physics communication, that is a fundamental step toward the large-scale computation, as it allows maintaining good parallel performances when operating with hundreds of computing processes. This results proved how the inter-physics communication must be taken into account when choosing a partitioning algorithm for CFD-DEM couplings in order to avoid an important communication bottleneck.
In order to improve the applicability our work to more generic scenarios, an enhanced flexibility in the domain partitioning can be provided. For this, dedicated partitioning algorithms must be designed to find the good tradeoff between a good load-balancing within each software, their own communication requirements and the volume of inter-physics data exchange. This will be studied in future works.
Figure 1 :
1Co-located partitions strategy for a coupled CFD-DEM simulation: 1. OpenFOAM and XDEM domains are overlapping. 2. Domain elements co-located in the simulation space are assigned to the same partitions. 3. Partitions are distributed to the computing nodes. 4. The parallel execution benefits from the resulting communication pattern.
Figure 2 :
2One particle traveling across process boundary: Setup.
Figure 3 :
3One particle traveling across process boundary: comparison between the sequential (SEQ) and parallel (PAR) results for CFD and DEM variables. Continuity across the process boundary can be observed.
Figure 4 :
4One million particles in two million cells: Setup.
Figure 5 :
5One million particles in two millions cells: comparison between the scalability of the pure OpenFOAM partitioned using SCOTCH, METIS, and uniform partitioning algorithm. Similar scalability properties can be observed.
Figure 6 :
6One million particles in two millions cells: speedup of the coupled OpenFOAM-DummyDEM in comparison with standalone OpenFOAM and standalone DummyDEM. The scalability of the coupled OpenFOAM-DummyDEM is similar to the standalone OpenFOAM: the overhead of data exchange of the coupling is negligible. The poor performance of DummyDEM is due to the artificially reduced amount of computation.
Figure 7 :
7Ten million particles in one million cells: Setup.
Figure 8 :
8Ten million particles in one million cells: Speedup of standalone XDEM for different partitioning algorithms.
Figure 9 :
9Ten million particles in one million cells: Speedup of standalone OpenFOAM for different partitioning algorithms.
Figure 10 :
10Ten million particles in one million cells: Speedup of the coupled OpenFOAM-XDEM approach in comparison with standalone XDEM and OpenFOAM: The coupled execution performs better than the standalone XDEM even for more than 200 processes.
Figure 11 :
11Ten million particles in one million cells: Percentage of time spent for inter-physics data exchange for an increasing number of processes. Less than 0.1% of the time is spent in the coupling part and the proposed solution scales well.
Table 1 :
1List of variablesSymbol Variable
Symbol Variable
u f
fluid velocity
p
pressure
µ f
fluid viscosity
u c
compression velocity
F b
http://hpc.uni.lu
AcknowledgementThis research is in the framework of the project DigitalTwin, supported by the programme Investissement pour la compétitivité et emploi -European Regional Development Fund (Grant agreement: 2016-01-002-06).The experiments presented in this paper were carried out using the HPC facilities of the University of Luxembourg 1 .
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|
[] |
[
"Coupled-mode theory for binary optical lattices",
"Coupled-mode theory for binary optical lattices"
] |
[
"Lasha Tkeshelashvili \nAndronikashvili Institute of Physics\nTbilisi State University\nTamarashvili 60177TbilisiGeorgia\n"
] |
[
"Andronikashvili Institute of Physics\nTbilisi State University\nTamarashvili 60177TbilisiGeorgia"
] |
[] |
The coupled-mode theory is developed for description of the nonlinear wave dynamics in binary optical lattices. The obtained equations of motion accurately describe nonlinear wave dynamics close to the band edges and in the gap of the linear spectrum of the system. In order to demonstrate the power of the presented approach, bright gap solitary wave solutions of the nonlinear coupledmode equations are derived and examined both analytically and numerically. The presented results are relevant to nonlinear wave phenomena in coupled waveguide arrays, coupled nano-cavities in photonic crystals, metallo-dielectric systems, and the Bose-Einstein condensates in deep optical lattices.
| null |
[
"https://arxiv.org/pdf/1411.5211v1.pdf"
] | 118,460,806 |
1411.5211
|
d783316a9cc744f2a3d50edbd94ca41d26c8c9a2
|
Coupled-mode theory for binary optical lattices
19 Nov 2014
Lasha Tkeshelashvili
Andronikashvili Institute of Physics
Tbilisi State University
Tamarashvili 60177TbilisiGeorgia
Coupled-mode theory for binary optical lattices
19 Nov 2014numbers: 4265Tg7867Pt4270Qs
The coupled-mode theory is developed for description of the nonlinear wave dynamics in binary optical lattices. The obtained equations of motion accurately describe nonlinear wave dynamics close to the band edges and in the gap of the linear spectrum of the system. In order to demonstrate the power of the presented approach, bright gap solitary wave solutions of the nonlinear coupledmode equations are derived and examined both analytically and numerically. The presented results are relevant to nonlinear wave phenomena in coupled waveguide arrays, coupled nano-cavities in photonic crystals, metallo-dielectric systems, and the Bose-Einstein condensates in deep optical lattices.
Periodically modulated nanostructures may exhibit band gaps in their linear spectrum [1]. That gives rise to a number of unique optical properties of those artificial systems. A prominent example is the existence of so-called gap solitons [2]. Solitons are localized nonlinear wave packets that can propagate undistorted in dispersive media [3]. The gap solitons have carrier wave frequency in the stop gap where the propagation of linear modes is not allowed. A powerful framework for description of nonlinear wave phenomena in photonic band gap materials is proven to be offered by the coupled-mode approach [4][5][6]. Indeed, various types of solitary wave solutions of the nonlinear coupled-mode equations, including bright, dark, and anti-dark gap solitons were found recently [7][8][9][10].
Perhaps, coupled optical waveguide arrays represent the most convenient experimentally accessible systems for studies of linear and nonlinear wave dynamics [11]. That includes the realization of optical analogies of various quantum-mechanical phenomena [12], as well as the demonstration of effects related to the discrete optical solitons [13]. A theoretical model that provides quantitative description of the waveguide arrays is the discrete nonlinear Schrödinger (DNLS) equation which, in the dimensionless form, can be written as follows [14]:
i ∂ψ n ∂z + C(ψ n+1 + ψ n−1 ) + ε n ψ n + N |ψ n | 2 ψ n = 0 . (1)
It is worth to mention here that DNLS equation is used for studies of coupled nano-cavities in photonic crystals [15], metallo-dielectric systems [16], and the Bose-Einstein condensates in deep optical lattices [17] too. In the context of the waveguide arrays, in Eq. (1), z represents the propagation coordinate along the waveguides, ψ n is the eigenmode amplitude in the n-th waveguide, and C gives the coupling rate between adjacent sites. In what follows it is taken to be a positive constant. N is the effective nonlinear constant of the system. The n-dependent term ε n is caused by the different effective refractive index of the individual waveguides. Binary waveguide arrays consist of the sites with alternating effective refractive index. That is, ε n = ε a for sites with n = 0, ±1, · · · , and ε n = ε b for sites with n = ±2, · · · . Before proceeding further, it should be noted that only the variation in ε n is relevant for the wave dynamics. Indeed, by means of the gauge transformation ψ n → exp(iεz)ψ n , all ε n can be shifted by arbitraryε. The choiceε = (ε a + ε b )/2 leads to ε n = (−1) n ε with ε = (ε a − ε b )/2. Thus, without loss of generality, for the modulation term we write
ε n = ε exp(±iπn) = ε cos(2k 0 n) ,(2)
where k 0 = π/2, and ε is assumed to be positive. This recovers the fact that, Eqs. (1) and (2) describe periodically modulated systems with the spatial period π/k 0 = 2.
To obtain the linear dispersion relation for the considered system let us write the linearized DNLS equation for the even-and odd-numbered sites separately
i ∂ψ 2n ∂z + C(ψ 2n+1 + ψ 2n−1 ) + εψ 2n = 0 , i ∂ψ 2n+1 ∂z + C(ψ 2n+2 + ψ 2n ) + εψ 2n+1 = 0 .(3)
Then, inserting the following ansatz [18]:
ψ 2n = Ψ e exp(−i[ωz − 2nk]) , ψ 2n+1 = Ψ o exp(−i[ωz − (2n + 1)k]) ,(4)
with some constants Ψ e and Ψ o , it is straightforward to derive the linear dispersion relation
ω = ± ε 2 + 4C 2 cos 2 (k) .(5)
Thus, the linear spectrum of the system consists of two bands separated by a band gap which ranges from −ε to +ε. As expected, for ε = 0 Eq. (5) reduces to the dispersion relation of the uniform system ω = ±2C cos(k). In this expression the sign is related to the choice of the positive direction. In what follows we choose the minus sign whenever the modulation is neglected.
Recently, a number of exciting wave phenomena associated with the peculiar dispersive properties of the binary DNLS equation were demonstrated in both linear [19] and nonlinear [20][21][22][23] regimes. In the present article the coupled-mode theory is developed for description of the nonlinear wave dynamics in binary waveguide arrays.
If ε and N were zero, a solution of Eq. (1) would consist of the forward and backward propagating plane waves (6) with constant amplitudes f n and b n . Here, ω 0 is given by Eq. (5) at k = k 0 . Note that, since ε = 0 is assumed, ω 0 = 0. That is due to the particular choice of the gauge, and therefore, has no relevance to the system dynamics.
ψ n = f n exp(−i[ω 0 z −k 0 n])+b n exp(−i[ω 0 z +k 0 n]) ,
The coupled-mode approach is based on the observation that the periodic modulation given by Eq. (2) causes strong coupling between the forward and backward propagating waves [4]. Indeed, taking into account that k 0 = π/2 and using Eqs. (2) and (6) we can write
ε n ψ n = εf n exp(−i[ω 0 z − k 0 n] − iπn) + εb n exp(−i[ω 0 z + k 0 n] + iπn) = εb n exp(−i[ω 0 z − k 0 n]) + εf n exp(−i[ω 0 z + k 0 n]) .(7)
As a result, in the modulated systems, f n and b n become dependent on n.
Besides that, f n and b n vary due to the nonlinear effects. By inserting Eq. (6) in |ψ n | 2 ψ n , in the calculations appear so-called "non-phase-matched" terms involving exp(±i3k 0 n). In the optical context it is often argued that such effects are unimportant for the system dynamics [4]. However, since exp(±i3k 0 n) = exp(∓ik 0 n) holds in the present case, those terms must be retained along with the phase-matched ones [24]. Now, assuming that the wave envelope is slowly varying with respect to the lattice spacing, n can be treated as a continuous variable. That yields,
f n±1 ≈ f (z, n) ± ∂f (z, n) ∂n , b n±1 ≈ b(z, n) ± ∂b(z, n) ∂n ,(8)
where f n (z) ≡ f (z, n) and b n (z) ≡ b(z, n). Finally, inserting Eq. (6) in Eq. (1) and collecting the terms with exp(+ik 0 n) and exp(−ik 0 n) respectively, it is straightforward to derive the following nonlinear coupledmode equations [24]:
i 1 v g ∂f ∂z + ∂f ∂n + κb + Γ(|f | 2 +2|b| 2 )f + Γb 2 f * = 0 , i 1 v g ∂b ∂z − ∂b ∂n + κf + Γ(|b| 2 +2|f | 2 )b + Γf 2 b * = 0 ,(9)
where κ = ε/v g , Γ = N/v g , and v g ≡ 2C is the group velocity of waves with k = k 0 in the uniform system, i.e. for ε = 0. In order to estimate accuracy of the involved approximations let us insert
f (z, n) = F exp(−i[Ωz − Kn]) , b(z, n) = B exp(−i[Ωz − Kn]) ,(10)
into the linearized Eq. (9). Here F and B are some constants. That results in the linear dispersion relation
Ω = ± ε 2 + 4C 2 K 2 .(11)
By assuming k = k 0 + K in Eq. (5), it is easy to see that Eq. (11) accurately reproduces the system linear spectrum for |K| ≪ π/2. Moreover, as is noted above, in Eq. (9) the nonlinear response of the system is evaluated exactly. Therefore, Eq. (9) is a valid model for studies of nonlinear wave dynamics in the band gap and/or close to the band edges [4,5]. In order to demonstrate the power of the presented approach, let us consider the gap soliton solutions. The nonlinear coupled-mode equations exhibit rich spectrum of various nonlinear excitations including bright, dark, and anti-dark gap solitary waves [7][8][9][10].
Here, as an example, we consider bright gap solitons. In particular, following Ref. [7], it is straightforward to show that so-called "in-gap" traveling localized solutions of Eq. (9) read
f (z, n) = ∆ −1/2 G(ζ) exp(iθ f ) , b(z, n) = ∆ +1/2 G(ζ) exp(iθ b ) ,(12)
where ζ = n − vz. Moreover, ∆ is a real constant defined as follows
∆ = v g − v v g + v ,(13)
and so, |v/v g | ≤ 1. It is clear that v parametrizes the soliton propagation velocity. The pulse amplitude reads
G(ζ) = ±2κβ Γ cosh(ξ)(1 + β cos(2φ)) 1/2 ,(14)
with
φ = −2 arctan([tanh(ξ/2)] ∓1 ) ,(15)
and ξ = 2κγ(ζ − ζ 0 ), where ζ 0 is an arbitrary constant. In Eq. (14) the choice of the sign must guarantee G(ζ) to be a real function. Furthermore,
γ = 1 1 − (v/v g ) 2 ,(16)
is the Lorentz factor, and
β = 1 1 + 2γ 2 .(17)
Finally, the phases of the forward and backward propagating waves are given by It should be noted that the derived expressions represent a limiting case of more general two-parameter solitary wave solution [7].
θ f = 2βγ 2 1 − β 2 v v g arctan 1 − β 1 + β tan(φ) + 1 2 φ , θ b = 2βγ 2 1 − β 2 v v g arctan 1 − β 1 + β tan(φ) − 1 2 φ .(
In addition, to demonstrate stability of the presented gap soliton solutions, we solve numerically the DNLS equation. In particular, the initial value problem for Eq. (1) is defined by Eqs. (6) and (12) at z = 0. The simulation results depicted in Figs. 1 and 2 show that the stationary as well as the mobile gap solitons are stable and propagate undistorted in the system.
The suggested couple-mode theory provides opportunity to establish close similarity between nonlinear wave phenomena in binary waveguide arrays and other photonic band gap materials such as, for example, fiber Bragg gratings. This fact offers a number of obvious advantages. First of all, the presented approach immediately shows that various types of solitary waves including bright, dark, and anti-dark gap solitons [7][8][9][10] exist in binary waveguide arrays. On the other hand, the coupled waveguide arrays represent one of the most convenient systems to observe those nonlinear excitations which are difficult, or even impossible to study experimentally in another optical context [6].
To conclude, in the present article the coupled-mode theory is developed and applied to describe nonlinear wave dynamics in binary waveguide arrays. It is shown that the suggested equations represent an accurate model for studies of nonlinear waves in the band gap and/or close to the band edges. As an example bright soliton solutions of the nonlinear coupled-mode equations are derived. The stability of the found solitary waves is studied numerically. In particular, the numerical simulations of the DNLS equation show that the gap solitons propagate undistorted over long distances. Finally, it should be stressed that the presented results are relevant to nonlinear wave phenomena in coupled nano-cavities in photonic crystals [15], metallo-dielectric systems [16], and the Bose-Einstein condensates in deep optical lattices [17] as well.
FIG. 1 :FIG. 2 :
12(Color online) Sum of the envelope intensities |fn| 2 + |bn| 2 for the stationary, i.e. moving with v = 0, gap solitary wave. The presented results are obtained from the corresponding numerical solution of the DNLS equation. The chosen parameters are C = 1, N = −1, and ε = 0.02. The plotted variables are dimensionless. (Color online) Sum of the envelope intensities |fn| 2 + |bn| 2 for the gap soliton moving with v = vg/2 velocity. The results are obtained from the corresponding numerical solution of the DNLS equation. The chosen parameters are C = 1, N = −1, and ε = 0.02. Note that v = 1 in this case. The plotted variables are dimensionless.
Acknowledgments. This work is supported by Georgian National Science Foundation (Grant No. 30/12).
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|
[] |
[
"Combining Visual and Textual Features for Semantic Segmentation of Historical Newspapers",
"Combining Visual and Textual Features for Semantic Segmentation of Historical Newspapers"
] |
[
"Raphaël Barman [email protected] ",
"Maud Ehrmann ",
"Simon Clematide \nUniversität Zürich\nSwitzerland\n",
"Sofia Ares Oliveira ",
"Frédéric Kaplan ",
"\n1É cole polytechnique fédérale de Lausanne\nSwitzerland\n"
] |
[
"Universität Zürich\nSwitzerland",
"1É cole polytechnique fédérale de Lausanne\nSwitzerland"
] |
[] |
The massive amounts of digitized historical documents acquired over the last decades naturally lend themselves to automatic processing and exploration. Research work seeking to automatically process facsimiles and extract information thereby are multiplying with, as a first essential step, document layout analysis. Although the identification and categorization of segments of interest in document images have seen significant progress over the last years thanks to deep learning techniques, many challenges remain with, among others, the use of more fine-grained segmentation typologies and the consideration of complex, heterogeneous documents such as historical newspapers. Besides, most approaches consider visual features only, ignoring textual signal. We introduce a multimodal neural model for the semantic segmentation of historical newspapers that directly combines visual features at pixel level with text embedding maps derived from, potentially noisy, OCR output. Based on a series of experiments on diachronic Swiss and Luxembourgish newspapers, we investigate the predictive power of visual and textual features and their capacity to generalize across time and sources. Results show consistent improvement of multimodal models in comparison to a strong visual baseline, as well as better robustness to the wide variety of our material.
|
10.46298/jdmdh.6107
|
[
"https://arxiv.org/pdf/2002.06144v4.pdf"
] | 235,699,592 |
2002.06144
|
57d0877698bd32f4a16d690019ed7fe537c9d30f
|
Combining Visual and Textual Features for Semantic Segmentation of Historical Newspapers
Raphaël Barman [email protected]
Maud Ehrmann
Simon Clematide
Universität Zürich
Switzerland
Sofia Ares Oliveira
Frédéric Kaplan
1É cole polytechnique fédérale de Lausanne
Switzerland
Combining Visual and Textual Features for Semantic Segmentation of Historical Newspapers
historical newspapersimage segmentationmultimodal learningdeep learningdigital humanitites
The massive amounts of digitized historical documents acquired over the last decades naturally lend themselves to automatic processing and exploration. Research work seeking to automatically process facsimiles and extract information thereby are multiplying with, as a first essential step, document layout analysis. Although the identification and categorization of segments of interest in document images have seen significant progress over the last years thanks to deep learning techniques, many challenges remain with, among others, the use of more fine-grained segmentation typologies and the consideration of complex, heterogeneous documents such as historical newspapers. Besides, most approaches consider visual features only, ignoring textual signal. We introduce a multimodal neural model for the semantic segmentation of historical newspapers that directly combines visual features at pixel level with text embedding maps derived from, potentially noisy, OCR output. Based on a series of experiments on diachronic Swiss and Luxembourgish newspapers, we investigate the predictive power of visual and textual features and their capacity to generalize across time and sources. Results show consistent improvement of multimodal models in comparison to a strong visual baseline, as well as better robustness to the wide variety of our material.
INTRODUCTION
For several decades now, digitization efforts are slowly but steadily contributing to an increasing amount of facsimiles of cultural heritage documents. As a result, it is nowadays commonplace for many memory institutions to create and maintain digital repositories that offer rapid, timeand location-independent access to documents, allow to virtually bring together disperse collections, and ensure the preservation of fragile documents thanks to on-line consultation [Terras, 2011]. Beyond this great achievement in terms of preservation and accessibility, the next fundamental challenge -and real promise of digitization-is to exploit the contents of these digital assets, and therefore to adapt and develop appropriate document and language processing technologies to search and retrieve information from this 'Big Data of the Past' [Kaplan and di Lenardo, 2017].
Context. Efforts are, in this regard, well under way and the libraries, digital humanities (DH), natural language processing (NLP), and computer vision (CV) communities are pooling forces and expertise to push forward the processing of facsimiles, as well as the extraction and linking of the information contained therein. 1 This momentum is particularly vivid in the domain of digitized newspaper archives for which there has been a notable increase of research initiatives over the last years. Those range from individual works dedicated to the development of tools [Yang et al., 2011, Dinarelli and Rosset, 2012, Moreux, 2016, Wevers, 2019 or the usage of those tools [Kestemont et al., 2014, Lansdall-Welfare et al., 2017, to evaluation campaigns [Rigaud et al., 2019, Clausner et al., 2019, including the emergence of large consortia projects seeking to apply computational methods to historical newspapers at scale, such as ViralTexts 2 , Oceanic Exchanges 3 , impresso 4 , NewsEye 5 , and Living with Machines 6 [Ridge et al., 2019].
Overall, this research contributes a pioneering set of text and image analysis tools, system architectures, and interfaces covering several aspects of historical newspaper processing. They usually focus on all or part of the typical digitized newspaper pipeline which consists, essentially, of three main steps: facsimile processing, in order to derive the structure and the text from the document image (via, respectively, optical layout recognition and optical character recognition processes); content enrichment, in order to extract and link relevant information from both textual and visual part of the contents; and, finally, exploration support, in order to search and visualize the enriched resources via e.g. application programming or graphical user interfaces.
Motivation. While encouraging, these efforts are still at an early stage and many challenges have yet to be addressed, especially with respect to document layout analysis (first processing phase). Document layout analysis aims at segmenting a document image into meaningful segments and at classifying those segments according to their contents [Eskenazi et al., 2017]. Two types of classification are traditionally distinguished: physical layout analysis, with a focus on the nature of the content (is this segment a textual block, a diagram, a picture, a decoration, a graphic, etc.), and logical layout analysis, with a focus on the function of the content (is this textual block a title, a footer, an article, etc.). Those segments are then fed into optical character recognition (OCR) programs that recognize their textual content.
With newspapers, these image segmentation and classification processes are particularly difficult because of the complexity and diversity of the object. A newspaper page consists of multiple, heterogeneous elements which feature different layout characteristics (text, map, table, illustration), different contents (regular articles, serial, advertisements) and which, additionally, evolve through time, differ according to newspapers, and are in different languages. Besides, facsimiles can be of variable quality due to the conservation state of the originals and this can also affect layout analysis performances.
Although difficult, layout analysis is however essential for historical newspaper understanding and exploitation, and their quality has a direct impact on downstream processes [Binmakhashen and Mahmoud, 2019]. From an information retrieval and user viewpoint, being able to query at the level of meaningful segments such as articles -instead of whole pages-, and to facet over different types of segments are undeniable advantages. From an NLP viewpoint, most analysis of semantic nature such as entity linking, topic modelling or text classification requires and/or performs far better on semantically self-sufficient, autonomous content items. For some processes, it can also be useful to filter out unwanted elements, either because of too noisy in terms of OCR or less relevant in terms of contents (e.g. transport schedule, cross-words, weather reports, TV programs, etc.). Finally, from a media history viewpoint, the automatic classification of content items can enable a better understanding of the evolution of newspaper sections through time and across collections.
Finally, the task of newspaper segmentation is also to be seen within the current context of largescale newspaper projects. Facing both the digitized newspaper material reality and user needs, these initiatives help, on the one hand, reveal the defects of legacy layout and text acquisition outputs from libraries and, on the other, emphasize the needs of finer-grained qualification of newspaper sections for scholarship purposes, as well as of efficient large-scale, trans-collection and diachronic processing of newspaper facsimiles. In this regard, the 'impresso -Media Monitoring of Past' 7 project -in the context of which the present work was carried out-is a case in point. Led by an interdisciplinary team, impresso aims at semantically indexing a multilingual corpus of digitized newspapers and integrating the resulting data into historical research workflows by means of a newly developed user interface 8 . By doing so, it appeared desirable to compensate for the deficiencies of old layout analysis.
Proposition.
In this context, this paper presents an innovative approach for the semantic segmentation of historical newspapers. 'Semantic' in that the targeted image segment typology goes beyond physical and/or logical characteristics and considers fine-grained semantic content item types (e.g. a segment is not only an article, but also e.g. a serial or death notice, or not only a table, but also e.g. election results or stock exchange information). 'Innovative' in that the approach makes joint use of visual and textual features, in an attempt to replicate human comprehension which uses both modalities simultaneously when confronted with document images. Already tested in very few recent studies [Yang et al., 2017, Katti et al., 2018, Dang and Nguyen Thanh, 2019, Denk and Reisswig, 2019, we believe it is the first time a multi-modal document image segmentation approach is applied on newspapers, what is more of historical nature.
Objective.
Our objective is twofold. First, we wish to assess whether the combination of visual with textual features can efficiently segment newspapers images. In this regard, the recent advances of deep learning approaches for semantic image segmentation and text processing suggest that positive results can be achieved: visual-based neural architectures trained for natural images have shown good adaptation to document images, and single architectures have demonstrated their capacities to adapt to different tasks [Ares Oliveira et al., 2018]. As for text, language models based on embeddings have shown their capacity to support a variety of tasks, from named entity recognition to question answering [Collobert et al., 2011]. Second, we wish to investigate whether this multi-modal representation can better support generalization across time and newspapers. The same newspaper section can indeed change drastically in terms of layout through time and across titles while enjoying a certain stability in terms of textual contents.
Contributions.
We present a series of experiments for the segmentation of several newspapers covering different time periods according to four semantic classes. These experiments are based on a modified version of dhSegment, a generic deep-learning approach that operates pixel-wise document segmentation [Ares Oliveira et al., 2018]. Architecture's code, groundtruth data sets as well as models are publicly released.
Section I presents prior works and specifies where the present approach sits with respect to them. Section II introduces the approach, and Section III details the experimental setup. Section IV reports and discusses three series of experiments and Section 4.4.2 considers the limits, but also future application scenarios of the approach and concludes.
I RELATED WORK
The survey of Eskenazi et al. [2017] gives an overview of the approaches for the segmentation of textual document images. Approaches are usually divided into three categories: top-down, when starting from the whole page in order to partition it, bottom up, when starting from small components in order to aggregate them, and hybrid. Classical algorithms heavily rely on specific document priors, e.g. having a "Manhattan" layout, and/or require large amounts of handcrafted features. More recent approaches make use of deep neural networks, trading prior, hand-crafted features for the learning capacities of machine learning, especially deep neural networks. Those include the usage of convolutional neural networks [Chen et al., 2017], as well as several variants of the fully convolutional network (FCN) introduced by Long et al. [2015] [ He et al., 2017a, Xu et al., 2017, Wick and Puppe, 2018, Ares Oliveira et al., 2018.
Considering newspapers images, several works have been proposed for their segmentation. Hebert et al. [2014] proposed an approach that performs physical and logical segmentation, and detects reading order on historical French newspapers. It is based on conditional random fields and a set of heuristics, targets high-level types such as titles, line and articles and achieves state of the art results with ca 85% of accuracy. A similar coarse-grained classification (line, image, illustation, text blocks) is done by Gatos et al. [1999] on Greek newspapers using an hybrid approach, and by Hadjar and Ingold [2003] and Bouressace and Csirik [2018] on contemporary Arabic newspapers using Run Length Smoothing Algorithm (RLSA). Lorang et al. [2015] focuses on a more specific type, that is poetic content items, and make use of manually crafted features to classify crops of newspaper images.
On the other side of the spectrum, another line of research performs newspaper content segmentation using text only (usually when images are not available) via the detection of homogeneous passages based on sentence or paragraph textual similarity [Riedl et al., 2019]. Those approaches can detect and classify segments of textual nature exclusively, but cannot identify their image boundaries, nor take into account more visual items.
Only a few recent work attempt to make use of image and/or localized, two-dimension text information. Meier et al. [2017] use a FCN based on image and OCR output information in order to detect articles in newspaper images (no further segment types). In this case text is reduced to a binary feature information (a pixel has text or not) and the lexical and semantic dimensions are not taken into account. Katti et al. [2018] introduced the concept of chargrid, a two-dimension representation of text where characters are localized on the image (thanks to the box coordinates) and encoded as a one-hot vector. This information is passed through an architecture that uses two encoders, one for the image information, the other for the character one and two decoders, one that produces semantic segmentation and the other that produces bounding boxes. Different model variants (image only, text only, both) are applied on images of administrative documents (invoices), and experiments show that the models based on both signals achieve better results. This is however opposed to a high-computing cost, as emphasized by the authors. Dang and Nguyen Thanh [2019] builds on this work and present an approach based on a multi-stage attentional U-Net using a one-hot encoded character feature. Segmentation of template like administrative documents yield state of the art results in the order of 87% mIoU (see Section 3.4). Denk and Reisswig [2019] also extends Katti et al. [2018], considering not only characters, but words and their corresponding embeddings, with BERTgrid for the automatic extraction of key-value information from invoice images (amount, number, date, etc.). With the same architecture as Katti et al. [2018], they obtain best results with document representation based on one-hot character embeddings and word-level BERT embeddings [Devlin et al., 2019], with no image information. Performances differ quite a lot between classes (of key-value types). Finally, Yang et al. [2017] jointly uses visual and textual features in a network, via text embedding maps where the two-dimension text representation is mapped to the pixel information (cf. Section II). Textual features correspond here to sentence embeddings (average of words vectors obtained with word2vec [Mikolov et al., 2013]), and models are trained on several variants of an end-to-end, multi-modal fully convolutional network for the segmentation and coarse classification of image regions (figure, table, section heading, caption, list, paragraph). Models are tested on various datasets and results show significant, although variable across classes, performance improvements with the model using both visual and textual features.
The method we present builds on the work of Yang et al. [2017] in the sense that it also makes use of text embedding maps. It however differs in that we work with historical newspapers -therefore integrating the diachronic dimension-, target a more fine-grained segment typology and experiment with different embeddings.
II METHOD
Our objective is to segment newspaper images and to classify detected segments according to a fine-grained newspaper section typology. To this end, we introduce a method which performs supervised, pixel-wise multiclass classification using both visual and textual features. The method builds on dhSegment's architecture.
Primary Architecture: dhSegment
dhSegment is an open-source, generic image document segmentation framework 9 [Ares Oliveira et al., 2018]. It consists of a CNN-based pixel-wise predictor coupled with task dependent postprocessing blocks. Its network is based on a U-Net architecture [Ronneberger et al., 2015], where the encoder follows a deep residual network ResNet-50 [He et al., 2016] pre-trained on ImageNet [Deng et al., 2009]. dhSegment has demonstrated competitive results on multiple tasks, e.g. page extraction, baseline extraction, and layout analysis, thereby paving the way for efficient and generic document image segmentation. Its architecture is here modified in order to incorporate textual features.
Text embedding map
Considering textual and visual information at the same time supposes to jointly encode their signals. To this end, and as briefly introduced in Section I, it is possible to map the one-dimensional representation of textual information (e.g. a word vector) into a three-dimensional one by 'positioning' the embedding representation into a two-dimensional space (e.g. a word has a certain width and height when written or printed on a page). This new textual embedding (corresponding to a word or a character) is therefore equivalent to the original vector, augmented with the positioning information (width and height). We refer to this three-dimensional representation of textual information, as introduced in Yang et al. [2017], as a 'text embedding map'.
The three-dimensional encoding of textual information is generated by using the results of an OCR process which outputs text tokens along with their coordinates on the image. Considering for example the left image of Figure 1, an OCR engine produces the token "TEMPS" located in the bounding box [(10, 195), (10, 300), (40, 300), (40,195)]. Looking the token up in an embedding space returns its (textual) vector, which can then be associated with the bounding box information, thereby creating a three-dimension map.
This process can be formally defined as follows. Given an image of size H × W and a list of tokens T where each token t is associated with a bounding box b t on the image, a text embedding map G of size H ×W ×N is produced, where N is the dimension of the embeddings. Specifically, all pixels contained in the bounding box of a token t are defined as the set b t ∈ R 2 and each pixel g i,j ∈ G of the text embedding map is computed with
g ij = E(t) if (i, j) ∈ b t 0 N otherwise where E(t)
is a mapping of t → R N corresponding for example to a word embedding, and 0 N is a null vector in case there is no text in the corresponding pixel. Each pixel overlapping with a bounding box of a token is therefore mapped to its corresponding embedding. A pixel spanning two bounding boxes is attached to the one that has the closest center.
The final result is a text embedding map, i.e. a three-dimension matrix where the first two dimensions correspond to the image-localized representation of the text, and the third to the embedding. Given that it has the same shape as the results of 2D convolutional layers, this construction offers the advantage that it can be processed directly by classical image processing neural network. One way of 'visualizing' this embedding map is to project each word vector, using principal component analysis (PCA), to a new one of dimension three where each dimension corresponds to a color (red, green, blue). This produces a colored text embedding map where the third dimension (the textual one) is transformed into a color value. The idea here is to 'see' the textual information, based on the fact that if two words have the same color, they also share the projection of their embeddings, and therefore textual features. Figure 1 shows such a colouring of textual information with segments of a weather forecast item (left) and a stock exchange table (right). Notwithstanding their similar layouts (i.e. a table with same number of columns, with a title on top and some text below) and the drastic dimension reduction (2048 from the original vector to 3), it is possible to observe information about the text, with differences that could not be easily caught by visual features only. For example, numbers are grey, punctuation is green, stop-words have a yellowish tint, and the weather forecast segment contains a column with letters only.
Model
Our model architecture is a modified version of dhSegment, 10 where the only modification is the addition of the text embedding map. It takes as input an image of a newspaper and its corresponding text embedding map, and outputs a pixel probability map. Figure 2 displays the architecture, with the T marker indicating where the text embedding maps are concatenated (on the channel axis) to the visual feature maps. The size relative to the original image size I is indicated at each step of the network, and the depth of the feature maps is indicated below the blocks for each step, considering that an embedding feature map of size 300 is input at T.
Variants of this model were tested during a pilot phase, in particular different input levels of the text embedding map. Two options were experimented in this regard: at the beginning of the network, in which case the text embedding map passes through most of the network and the textual signal is treated like the visual one, and at the end, in which case it adds further contextual information to the image feature map and support the final decision of the pixel class.
Our preliminary experiments showed that inputting the textual features early in the network is the best option, so do all architectures used in SectionIV.
III EXPERIMENTAL SETUP
We apply this semantic segmentation method on historical newspapers of the impresso project collection, considering four semantic classes. This section introduces the corpora and the typology used for classifying image segments (Section 3.1), presents the embeddings used for the experiments (Section 3.2), specify the training setup (Section 3.3) and details the evaluation framework (Section 3.4).
Datasets
Corpora
Since the only freely available historical newspaper image dataset annotated with content item types considers broad categories alone (e.g. article, caption, header, etc.) [Clausner et al., 2015], we created two new datasets.
Swiss newspapers.
The first one originates from the Swiss National Library and the stillexisting journal Le Temps. 11 It is composed of three titles in French language from the Romandy region with long publication history, namely: the Journal de Genève (JDG, 1826(JDG, -1994 the Gazette de Lausanne (GDL, 1804(GDL, -1991, and the Impartial (IMP, 1881-2017). JDG and GDL, issued in neighboring cities, can be considered as siblings and were merged in 1991. In order to have a long term, diachronic ground truth, newspaper issues were sampled across the whole publication spans with three issues every three years for JDG (used for training and evaluation) and every five years for GDL and IMP (used for evaluation only). Because of misalignment problems between facsimiles and token coordinates of the original OCR, all images of the selected issues were re-OCRed with Abbyy FineReader application 12 .
This material was manually annotated according to the four semantic classes of our typology (see Section 3.1.2), using the VGG Image annotator [Dutta and Zisserman, 2019]. Annotation was done at the pixel level and not at the content item instance level, meaning that each pixel of the image has a label indicating that it belongs to a specific class (or not), but not that it belongs to a specific instance of a class. Several reasons motivate this annotation at pixel level: in most cases, there is only one instance per page, and in case of multiple instances, it might be non-contiguous regions of the same instance. Besides, instance separation and merging can also be done in a post-processing step.
This annotation process yielded a total of 1,982 annotated pages for JDG, 1,008 for GDL and 1,634 for IMP. Table 1 shows the class distribution for the three titles. Pages without annotations do not contain any classified content items.
Luxembourgish newspaper. The second dataset consists of a single title, the Luxemburger Wort (LUXWORT), a Luxembourgish newspaper from the Bibliothèque Nationale du Luxembourg 13 published since 1848 with contents in German, French and Luxembourgish. The library, who outsourced OCR and layout recognition for its newspaper collection, performed a manual check of the recognized segments. Having this at hand, we chose the work with the death notices which, in the LUXWORT, amount to ca. 90,000 segments in 17,000 page images. For our corpus, we sampled 34 issues per year between 1848 and 1950. This resulted in 18,953 images with 1,765 death notices.
Classes
As mentioned earlier, newspapers feature a wide variety of contents which change over time and across newspapers. Given our objectives, we selected four classes of content items likely to, on the one hand, be of historical or practical interest (to use them as search facet or to filter them out before processing) and, on the other, present a mix of visual and textual variation: Table, i.e. a table reporting the values of different national stocks. The degree of confusability of document image segments depends on several dimensions. First, the level of refinement of the typology naturally impacts what is confusable with what: distinguishing generic articles from advertisements is less difficult than distinguishing job adverts from purely commercial ones. The typology we consider here is already finer-grained compared to usual newspaper segmentation with e.g. a specific type of table among the tables (Stock Exchange) and a specific type of article among the articles (Weather Forecast). Next, within a given typology, visual and textual facets are the two main dimensions determining the degree of confusability of segments. Naturally, the more distinct on both dimensions the better. Finally, these dimensions are complemented by the time and source factors, since considering segments from different newspapers, and/or in synchrony or in diachrony also greatly impacts their confusability, not only with other types but also with themselves.
(a) Serial (b) Weather forecast (c) Death notice (d) Stock exchange table
Let's examine our classes, shown in Figure 3, in this light. Regarding the visual confusability of Serials with respect to other items and themselves, in both synchrony and diachrony, they can be considered as rather distinct and stable: during most of their publication through time and across different newspapers, they are located at the bottom of the front page, topped with a thick black line. This makes them visually distinct compared to other items and should ensure good recognition performances using visual features only. As per their textual contents, these can vary and are, to some extent, confusable with regular journalistic contents. In contrast, Weather Forecast segments feature a great visual variability over time (only text, then map and text, then only maps), but a clear textual stability. Here, the consideration of textual features should help improve recall, i.e. removing false negatives. Death notices are visually very similar to advertisements (small textual segments surrounded by a thick, black frame) but have different textual contents. For this class, one can therefore expect a high confusability with advertisements, and taking into account textual features should help to remove false positives. A similar situation holds for Stock Exchange Tables: despite their distinctive layout compared to other content items, they are still visually confusable with other tables (e.g. transport, voting). They, however, enjoy a certain visual and textual stability and their recognition should not drastically suffer across time and sources.
Overall, these four classes contain various combinations of visual and textual confusability, which makes them suitable for exploring the benefit of adding textual features for semantic segmentation.
Embeddings
In order to investigate the effectiveness of different types of embeddings used to build the text embedding maps, we experimented with embeddings having different embedding levels (word or character), contextualized word representations or not (contextual or non-contextual), different languages (mono-or multilingual), and different training data (in-and out-domain). To this end, we use fastText word embeddings, which make use of characters n-grams to learn subword embeddings [Bojanowski et al., 2017]; Byte-Pair encoded subword embeddings (BPEmb), which learn subwords rather than using fixed n-grams [Sennrich et al., 2015]; and characterbased Flair embeddings [Akbik et al., 2018], a character-level variant of the contextual string embeddings introduced in [Peters et al., 2018]. In total, six flavors of these embeddings are considered, with three different stacks. Table 2 Next, in order to test the effect of in-domain embeddings, two models were trained on a corpus of 2GB of text of the Luxemburger Wort for the period 1848-1950. The first is a FastText model trained on lowercase space-separated input, with at least 3 occurrences per token, a context windows of 8 tokens, a sub-word max character n-gram length of 6, resulting in embeddings of size 300 (fastText-luxwort of fastText-fr and flair-fr is used: fastText-flair-fr. For experiments related to LUXWORT, two different configurations are used: a) a combination of pre-trained embeddings BPEmb-multi and Flair-multi, referred to as BPEmb-flair-multi, and b) a combination of in-domain embeddings with fastText-luxwort and flair-luxwort, referred to as fastText-flair-luxwort.
All embeddings and stacks were tested during a pilot phase, and the stacks appeared to be the best in our context. They combine contextual and non-contextual information, as well as word and sub-word information, and seem therefore more suitable to cope with old language and OCR output. Experiments presented in SectionIV are based on the stack embeddings exclusively.
Training and Post-processing
Text embedding maps are pre-computed for all images using the embedding stacks described above and the text associated with the images (original OCR for the Luxembourgish newspaper, ABBYY one for the Swiss). Before training, images are resized to fit in 5 · 10 5 pixels and are augmented by random scaling (s ∈ [0.8, 1.2]) and rotation (r ∈ [−0.01, 0.01]rad). All models are trained for 17,000 steps with a batch size of 4 and batch renormalization [Ioffe, 2017]. We use Adam optimizer [Kingma and Ba, 2014] with an exponentially decaying learning rate of .95 starting at 10 −4 , and a weight regularization of 10 −6 . In order to prevent overfitting a development set containing 10% of the training set is used. The final result for each model is reported on the weights where the loss on the development set was the lowest. Models are trained on a NVIDIA Tesla V100 GPU with 32GB of memory using the Tensorflow library 15 1.13.1.
In terms of post-processing, the class mask is computed from the final output of the network, a pixel probability map. A pixel is considered as belonging to the background if it has probabilities smaller than 50% for all classes, otherwise to the class with the highest probability. In order to avoid small masks, connected components with an area smaller than 5% of the size of the image are discarded.
Evaluation setup
Given an image and a class, we wish to create a mask that contains pixels of the class. Figure 4 illustrates this procedure, with Figure 4a being the input and Figure 4b the ground truth, the latter with the mask coloring each pixel according to its class (here yellow for death notices pixels). On this base, several metrics are used to evaluate the models. Mean Intersection over Union. The Intersection over Union (IoU) is the standard metric for semantic image segmentation and measures how well two sets of pixels are aligned. It is computed as follows. Given an image i belonging to the set of images I, a class c belonging to the set of classes C, a set of predicted pixels P ic of image i belonging to class c, and a set of ground-truth pixels G ic of image i belonging to class c, the IoU for image i and class c is: Figure 4 shows a quantified example, where the prediction on images 4c and 4d are compared to the ground truth of image 4b. Image 4c has a too small and misaligned prediction, and therefore a low IoU, while 4d is better. It is important to note that the metric is computed at the image (or pixel) level, and not the content item instance level. Indeed, although there are four distinct death notice instances in Figure 4, the annotation makes no distinction and the model does not need to separate them to obtain a good score.
Metrics
IoU ic = |P ic ∩ G ic | |P ic ∪ G ic |
The mean Intersection over Union ( The usual way to measure those values in segmentation is to consider an example as positive when above a certain threshold τ ∈ [0, 1] of IoU. In this case, the prediction is well enough aligned with the ground truth to be considered as correct. On this base, it is possible to consider a prediction with an IoU ≥ τ as a TP, a prediction with no IoU (i.e. with a union of zero) as a TN, a prediction with an IoU of zero and no predicted pixels (i.e. intersection of zero and non-zero number of pixel in the ground truth) as a FN and, finally, a non-FN prediction with an IoU < τ as a FP. Given a threshold τ , it is therefore possible to compute precision and recall, as follows:
Precision at τ = P @τ = T P T P + F P Recall at τ = R@τ = T P T P + F N Finally, it is also possible to compute the average precision and recall over a range of thresholds. A range of threshold, is defined by a start τ start an end τ end and step size between two threshold τ step using the following notation: τ start :τ step :τ end , for example a threshold between 50 and 95 with a step of 5 would be written as 50:5:95. Given a threshold range the average metric M (which can be precision, recall or anything else) is then computed as follows:
M @τ start :τ step :τ end = 1 |τ start :τ step :τ end | τ ∈τstart:τstep:τ end M @τ Let us emphasize once again that these metrics (IoU, mIoU, P, R) are computed at the page level and not at the instance level. If a page contains several instances of a class and the prediction matches some instances, but not enough to reach an IoU threshold larger than τ , the whole page is counted as negative.
Reported results
For each experiment, results are reported in terms of mIoU (as a percentage), and precision and recall with, respectively, a threshold of 60% and 80% and the average of threshold 50:5:95 of the IoU.
Since models can have a high variance between runs, each model is trained ten times and the average and standard deviation of their performance are reported in the form of tables and boxplots. Even though most of our analyses are based on the mean, we indicate whether the difference of means between two models is significant by using Welch's t-test, following the recommendation of Reimers and Gurevych [2018]. The significance is indicated using stars (*), where their numbers corresponds to a certain p-value: one star (*) indicates that p ≤ 0.05, two (**) that p ≤ 0.01, three (***) that p ≤ 0.001, and four (****) that p ≤ 0.0001.
Material release
Annotated material is released in the VIA format as open data (under different right statements according to the source on Zenodo) under DOI 10.5281/zenodo.3706863.
The model architecture is thought as a plugin of dhSegment, named dhSegment-text. Two implementations are available. The first one, used for the present experiments and based on the TensorFlow implementation of dhSegment, is available on GitHub 16 and can be used for reproducibility purposes. The second one, based on the new pyTorch implementation of dhSegment, 17 is also available on GitHub 18 and can be used for training new models.
Finally, a selection of (best) trained models are released on the dhSegment-text repository, under a CC BY-SA 4.0 license.
IV EXPERIMENTS
In this section, we motivate and present four series of experiments that address important questions with regard to the automatic recognition of fine-grained semantic segments in historic newspapers. Since our material was published over a long period of time, we are specifically interested in the diachronic robustness of our models.
In Section 4.1, we examine the predictive power of visual and textual features on our four classes, representative of different difficulties. In Section 4.2, we test the generalization ability of our multimodal approach with respect to (a) the changes over time in newspaper layout and content, and (b) the transfer of models from one newspaper to a related one, which has not been part of the training material. In Section 4.3, we examine whether textual features allow to reduce the amount of training data given that they add another source of signal to the models. In Section 4.4, we focus on multilingual death notices from a single newspaper and examine (a) how the increase of training material improves the results, and (b) how valuable in-domain text embedding are, meaning character and word embeddings that were specifically trained on the multilingual and noisy OCR source material from the very newspaper.
Combining Visual and Textual Information
The first series of experiments addresses the following questions: (a) How well does finegrained semantic segmentation perform on our four selected classes under the condition that training and test data are sampled representatively from the same newspaper? (b) How strong is the signal contained in the textual embedding maps? (c) What is the expected benefit of combining visual and textual features?
Experiment description
Here, models are trained on long-term diachronic JDG data only in order to reserve GDL and IMP datasets for generalization experiments (Section 4.2). The JDG dataset was randomly split to compose a training (1,387 images) and test (595 images) sets. Table 3 presents the class distribution, where it can be observed that class ratios are similar between the training and test sets. Given the homogeneity and representativity of the material, the results of this first series of experiments serve as an upper bound for our approach for diachronic, fine-grained image semantic segmentation.
In order to measure the effectiveness of using visual features only, textual features only, or a combination of both, we experimented with three modalities: Results are shown in Table 4 and Figure 5. In general, the Image model outperforms the pure Text model by a large margin in terms of mIoU and precision. Except for recall-oriented setups, there is no advantage of restricting the models to textual features only. As expected, Image model is stronger on classes that are more visually distinct (Weather and Stocks) than on classes that are mainly text based (Serial and Death notice).
With respect to the class average 19 results, Image+Text models perform significantly better than Image for every metric, attesting a real gain in using the combination of visual and textual features for the task. The better precision of Image and the good recall of Text play well together, leading also to less variance across models, as the smaller standard deviations indicate. For all modalities, there is a big drop in precision when augmenting the IoU threshold. This indicates that it is hard to be precise about the location of a segment and that the Image+Text model is more robust than the single modality models.
The recall of Weather is better for both models using textual features. This means that these features are essential for the retrieval of the class Weather. As illustrated in Figure 6, weather reports may contain images, maps, and text. While the first two types are visually distinct, vocabulary might be the only semantically distinct feature for purely textual weather reports.
The mIoU and precision of Death notice is significantly higher for the Image+Text model than any single modality model. In particular, the gains versus the Image model are important for the for the mIoU (+5.8%-10.% at 95% confidence), for the P@60 (+4.4%-11.4% at 95% confidence), for the P@80 (+10.5%-16.5% at 95% confidence) and for the P@50:5:95 (+7.3%-11.6% at 95% confidence). This strong increase in precision shows that the Image+Text model is much more robust against false positives than the Image one. As illustrated in Figure 7, advertisements can have similar layout, but very different textual content.
The absence of significant differences in terms of mIoU and precision between the Image and Image+Text approaches for Serial can be explained by the lack of strong characteristics in either of the modalities. Visually, serials look similar to the rest of the newspaper. Textually, their vocabulary does not differ much from the rest of the newspaper either. However, the fact that it reaches a higher recall score for both models using textual features indicates that they are important for retrieval. The lower precision of the Text model shows that these features are also present in other articles.
Finally, the similar results between all approaches for Stocks show that the visual and textual signals are both strong enough to detect this class. The reason for the slightly lower score of the Text model could be crucially missing visual information about purely visual elements such as the lines of a table.
Summary
The first series of experiments assesses a consistent gain in performance by combining visual and textual features. The gain is particularly strong with content items as Death Notices that exhibit easily confusable visual features, but have distinct textual features. We also observe a better recall for Weather Reports, that consists of a mix of visual and textual elements. Even though the Image+Text model does not improve much on purely textual classes such as Serial or visually distinct classes such as Stocks, it still performed at least as well as a model using only the image.
Generalizing Through Time and Across Newspapers
The second series of experiments addresses the following questions: (a) Do models trained on textual and visual features perform better than purely visual models when applied to an unseen time period of the same newspaper? (b) Do models trained on textual and visual features perform better than purely visual models when applied to a related newspaper, where no issue was part of the training material?
Experiment Description
The first experiment on generalization through time uses the JDG dataset, with material from the periods 1826-1968 and 1992-1998 as training data (1,394 pages), and 1969-1991 as test data (588 pages) Note that the test period has a different layout than the other periods [Buntinx et al., 2017]. The second experiment on generalization across newspapers trains on the same training set as the first series of experiments (Section 4.1), but tests on the data from GDL (1,008 pages) and IMP (1,634 pages). Both experiments compare the generalization ability of the Image and Image+Text models by testing them on layouts never seen before. This setting is therefore more challenging than the previous one where the training set was sampled uniformly over time and representative of the test set. Distributions of classes for each dataset are shown in Table 5. In general, the distribution between the original JDG dataset and the other one changes. The closest dataset is GDL, which is not surprising since both newspapers come from neighbouring cities. The largest difference is between the two time periods, with a much lower ratio of content items of the four classes for the test period. The same embeddings as in Section 4.1 are used, that is to say Fasttext-Flair-fr (c.f. Section 3.2 and Table 2). Results are shown in Table 6 and Figure 8. Compared to the results of the first series of experiments (cf. Section 4.1), the performance is substantially lower. In general, poorer results mean that the examples in the training and test sets are too different. However, it should be noted that all the models using visual and textual features are significantly better than the Image models.
Results and Discussion
When focusing on the time constraint, it is clear that the Image+Text models perform significantly better for every class, except for Stocks. The Image results show that death notices and stocks are visually more stable than serials and weather reports. However, the significant differences between the two models for the classes Weather (+23%-39% at 95% confidence) and Death Notice (+16%-35% at 95% confidence) show that textual features are even more stable for these two classes. The poor results with Serial reveal that this class is neither visually nor textually stable over time. Finally, the results of Stocks suggest that the visual features are more stable than the textual ones.
When focusing on the model transferability to other newspapers, the overall performance drop is much less pronounced with GDL than with IMP, confirming that JDG and GDL have more common features in terms of layout. In particular, the low score of both models for the class Weather in IMP dataset shows that this type of segment has great variability in terms of layout. For IMP, the gain is, once again, particularly significant for Death Notice (+23%-31% at 95% confidence), demonstrating that textual features are particularly good at generalizing for this class.
Summary
The overall performance drop between these experiments and the ones in Section 4.1 confirms the variety of newspaper elements, both through time and across newspapers, and stresses the importance of annotated data representatively sampled across time and newspaper. However, it also demonstrates that model generalization and transferability can be improved by the inclusion of textual features.
When considering the scores, most of them are too low to be considered in any practical use case, except maybe for obituaries. This indicates that even though this method shows some promises in terms of raw performance and generalization, it is still too early to use it at large scale without representative annotated data.
Reducing Training Size
The third series of experiments addresses the following question: Do models combining visual and textual features need less training material?
Experiment Description
These experiments assess the effect of reducing the training size by 60%. The new training size is of 792 training samples againts 1,387 in the first experiments of Section 4.1. The distribution of the re-sampled datasets can be found in Table 7. Once again it can be seen that the ratios of each class are similar between training and testing sets, and also w.r.t the experiments that used 100% of the training data.
Class
Train size (ratio) Test size (ratio) Serial 56 (7.07%) 81 (6.81%) Weather forecast 61 (7.70%) 95 (7.98%) Death Notice 59 (7.45%) 94 (7.90%) Stock Exchange
Results and Discussion
Results are presented in Table 8 and in Figure 9. Overall the performance of the models with less data is inferior to the ones using 100% of the training set. Still considering the average, the drop in mIoU is higher for the Image+Text model than for the Image model. However, this is mainly due to the poor performance of the former on Serial. Indeed, the Image model performs significantly better (+10%-31% at 95% confidence) than the Image+Text model. This may be due to the fact that Image+Text has not enough data to learn the textual features of Serial and is thus more confused.
Regarding Death Notice and Stocks, the Image+Text model improves over the Image model. This may indicate that the textual features of these two classes are easier to learn than for Serial, and that the model using both text and image features leverages better the small amount of training data for these classes by combining the two signals.
Finally, the fact that Weather results have no significant difference between the two models may show that textual features are not as complex to learn as they are for Serial, but do not influence much the results.
Summary
This experiment shows that not all classes are equal when the training size is reduced. It indicates that for content items with non focused textual content, such as Serial, Image+Text requires more data to efficiently combine both signals. In contrast, it suggests that for more domain specific content items, such as Stocks and Death Notice, Image+Text model easily leverages the additional signal provided by the textual information.
Assessing the Benefits of In-domain Embeddings
The fourth series of experiments addresses the following three questions: (a) How big is the advantage if we train in-domain textual embeddings instead of using off-the-shelf embeddings trained on contemporary text data (without noisy OCR)? (b) Can adding more training data compensate for the expected benefit of in-domain embeddings? (c) What is the impact of adding more training material on the performance of the models and is it possible to identify the point at which adding more data becomes ineffective, i.e. a plateau is reached?
Experiment Description
These experiments make uses of several training sets with different numbers of pages, while the test set is kept constant. Training sets of different sizes are iteratively built starting from biggest to smallest by sampling, at each iteration, half of the number of issues per year: the first dataset has 26 issues per year, the next one 13 (therefore a subset of the previous one), the next 6, and so on. Training set statistics are shown in Table 9. The embeddings used are the BPEmb-flair-multi and the fastText-flair-luxwort stacks (cf. Section 3.2). Each experiment is thus characterized by its amount of training data and the embeddings used (or lack of it).
This experiment uses the newspaper Luxemburger Wort and focuses on the Death Notice class only since, as seen in previous experiments, it is the one that benefits most from the addition of textual features.
The results are presented in Table 10 and Figure 10. Overall the results show that there is a significant gain in using textual embeddings maps, both in terms of performance and variance.
The Image+Text out-domain and Image+Text in-domain models always significantly beat the Image model. Moreover, the performance of those models increases and the variance decreases with the size of the training set. However, the performance difference between 550 pages and 14,100 pages is only around 5%, even though more than 25 times as many annotated pages are used.
Dataset # issues/year # issues # pages # death notices (ratio)
# pages
Image Image+Text out-domain Image+Text in-domain In-domain text embeddings are beneficial since they provide a consistent gain (+2-3%) in performance over the out-domain text embeddings. Moreover Image+Text in-domain already surpasses with 1100 examples the performance that Image+Text out-domain achieves with 14100 examples. This is certainly due to the fact that the in-domain embeddings have been trained on enough data to capture the particular semantics of newspapers and the OCR errors present in them.
The results of the model using only the image are quite surprising since the performance decrease with the amount of training data which is counter-intuitive. In order to eliminate the hypothesis that a visual model that uses more data needs more steps for converging, the model with 14,100 pages was trained for the double of training steps, however, it only improved the results by 5%, while still being 5% lower than the model with 1,100 pages and having three times its variance.
Summary
In this experiment, the usage of textual features brings a gain in performances, reduces the variance of the model which converges better even with large amount of data. However, when considering the scores, the addition of training data has little impact and this extra annotation does not seem worth the effort. The use of in-domain embeddings shows a decent improvement that cannot be compensated by more training samples over out-domain embeddings while not requiring additional annotations. Indeed, training in-domain embeddings is done completely unsupervised, making it a worthwhile option if the amount of available text data is sufficiently large.
CONCLUSION AND OUTLOOK
We believe these series of experiments led to a better understanding of the interplay between visual and textual features for semantic segmentation of newspapers document images. The first series of experiments using annotation data that was representatively sampled show a consistent improvement for models that combine textual and visual features relative to a strong baseline using visual features only. Textual features also help to mitigate the problem of high variance that purely visual models have with the varying and diverse material in newspapers published over a long period of time.
The second series of experiments on the generalization ability over time and across newspapers showed that a simple transfer of models leads to a stark drop of performance. However, on average, models with textual features show substantially better results than the ones without. We can conclude that text characteristics are indeed more stable than layout characteristics and that they are vital for improving the model's robustness. For practical applications, these experiments make clear that annotation efforts need to be carefully distributed over the diachronic variety and diversity of the material present in historic newspapers.
The third series of experiments on the reduction of training data gave mixed results. On the one hand, the model using both visual and textual still improves on most classes. On the other, there seems to be, for some classes, a lower bound on the number of samples for the proposed method in order to be able to extract the relevant signals from textual features.
The fourth series of experiments on the benefit of using large amounts of training data-by taking advantage of the fact that a lot of material with useful semantic classification already exists in digitized archives-showed that purely visual models have more difficulties to exploit a larger amount of data than combined models. Another important outcome of these experiments is the fact that more training material can not fully compensate for the availability of text embeddings specifically built on in-domain text data, especially with noisy OCR texts. As in-domain text embeddings can be computed without human annotation and are therefore cost-effective, they should always be considered.
Although proof of concept, the present approach can already support two main use cases. First, similarly as dhSegment, the released framework can help scholars and/or non-specialists to easily process document images, provided they can be associated with text (thanks to e.g. an opensource OCR software), and that embeddings are available. Second, even though not perfect, the models can already be used as support for manual annotation (users only need to correct false positives or negatives) and, for some classes, be used to segment real newspaper collections to offer further search facets and/or filter unwanted material.
As future works, we intend to compare this approach with pure text classification in order to bridge the comparison spectrum from pure pixel to pure text, as well as to apply it to other documents than newspapers. It could also be interesting to integrate a region proposal module (such as in Mask R-CNN [He et al., 2017b]) in order to segment at instance level.
AUTHOR CONTRIBUTIONS
RB designed and carried out the experiments, wrote the paper; ME designed and supervised the project and experiments, wrote the paper; SC designed and supervised the project and experiments, wrote the paper; SO supervised the project and experiments, helped with the paper writing; FK supervised main directions.
Figure 1 :
1Visualization of a Flair-based, PCA-reduced text embedding map projected on three dimensions (red, green, blue).
Figure 2 :
2The model architecture used.
Figure 3 :
3Example images for each of the selected classes. All images are from the Journal de Genève.-Serial, i.e. an excerpt of a bigger work published over time in several issues of a newspaper, corresponding to the French roman-feuilleton. Serials often span several columns in a horizontal layout and can span several pages. -Weather Forecast, i.e. a text or illustration with the prediction of weather, or even a report of past weather measurements. -Death Notice, i.e. a small notice published by relatives of a deceased person.-Stock Exchange
summarizes the main characteristics of the used embeddings.First, four pre-trained embeddings of the Flair library 14 are used with their default implementation settings, as follows:-fastText-fr, i.e the French fastText embeddings of size 300 pre-trained on Common Crawl and Wikipedia; -flair-fr, i.e the French Flair embeddings of size 4096 pre-trained on Wikipedia; -flair-multi, i.e the multilingual Flair embeddings of size 4096 pre-trained on the JW300 corpus [Agić and Vulić, 2019] with more than 300 languages; -BPEmb-multi, i.e the multilingual Byte-pair encoding embeddings of size 300 trained on the 275 most common Wikipedia languages[Heinzerling and Strube, 2018].
Figure 4 :
4Examples of the behaviour of the IoU metric.
mIoU) for a class c over the set of Image I corresponds to the average of the IoUs of all images where the union of the predicted and the ground-truth set Journal of Data Mining and Digital Humanities ISSN 2416-5999, an open-access journal has at least a pixel of class c (the true negatives are thus not counted). This can be more formally defined as all images J = {i ∈ I | |P ic ∪ G ic | > 0}. Then the mIoU is: Recall. The IoU does not qualify performances in terms of true positives (TP), true negatives (TN), false positives (FP) and false negatives (FN). However, those values are of interest when considering whether a model can be used in concrete terms, i.e. if most of the segments are correctly recognized.
Figure 5 :
5Box plots of the mIoU of the first series of experiments.
Figure 6 :
6A Weather forecast with both visual and textual features. The Image + Text model finds both the text and the image, whereas the Image model only finds the map and the table.
Figure 7 :
7A death notice (top) and an advertisement (bottom) with similar layouts, but very different textual features. The Text model correctly detects only the top example, whereas the Image model is misled by the advertisement.
Figure 8 :
8Box plots of the mIoU of the generalization experiments.
Figure 9 :
9Box plots of the miou of the JDG with 60% training data.
Figure 10 :
10Box plots of the mIoU of the Luxwort experiment.
) ,
)Class/Newspaper JDG GDL IMP LUXWORTSerial
137 108 103
-
Weather
156
68
41
-
Death notice
153
69 102
1765
Stocks
275 135
79
-
Pages w/o annotations 1393 697 1326
17188
Total 1982 1008 1634
18953
Table 1: Dataset statistics. Note that page numbers do not add up to the total number
of annotated pages because a single page can contain more than one class.
Journal of Data Mining and Digital Humanities ISSN 2416-5999, an open-access journal
Table 2 :
2Overview of embeddings. The '-' sign means both contextual and non-contextual embeddings
(stacks).
16 https://github.com/dhlab-epfl/dhSegment-text 17 https://github.com/dhlab-epfl/dhSegment-torch 18 https://github.com/dhlab-epfl/dhSegment-text-torchClass
Train size (ratio) Test size (ratio)
Serial
101 (7.28%)
36 (6.05%)
Weather forecast
103 (7.43%)
53 (8.91%)
Death notice
107 (7.71%)
46 (7.73%)
Stock exchange table
189 (13.63%)
86 (14.45%)
Pages w/o annotations
982 (70.80%)
411 (69.08%)
Table 3 :
3Distribution of the classes for the training and test sets. Text: the model receives as input a blank image and a text embedding map of a newspaper page and relies solely on textual features. 3. Image+Text: the model receives as input a newspaper image and its corresponding text embedding map and combines visual and textual features.1. Image: the model receives as input a newspaper image (pixels) only and relies solely on
visual features. It is equivalent to the model described in [Ares Oliveira et al., 2018].
2. Each model with textual features uses the architecture described in Section 2.3, where text and
image information are fused early in the network, as well as the fastText-flair-fr stack embed-
dings.
Metric
Modality
Serial
Weather
Death Notice
Stocks
Average
mIoU
Image
74.12±7.59
81.27±2.18
75.37±2.98
83.11±0.88
79.30±2.29
Text
49.05±9.41
73.55±3.55
71.44±3.26
78.87±2.61
69.30±2.25
Image+Text
76.73±5.90
81.38±3.34 * * * * 83.58±2.02
84.43±1.84 * * 82.16±1.72
P@60
Image
82.02±7.24
91.08±4.93
83.37±4.46
89.21±1.42
86.86±2.60
Text
53.97±12.42
82.29±5.13
82.19±5.42
86.85±2.96
77.27±3.03
Image+Text
83.24±7.17
91.81±4.67 * * * 91.27±2.80
90.11±1.77
* 89.43±1.79
P@80
Image
66.45±14.87
66.94±7.41
67.37±2.04
80.51±2.19
72.29±3.74
Text
29.95±23.47
58.13±3.97
62.01±5.34
74.07±3.90
57.93±5.73
Image+Text
71.54±15.05
71.37±7.28 * * * 80.89±3.88 * * 83.49±2.11 * * 78.07±3.76
P@50:5:95
Image
65.37±10.22
69.10±2.84
66.77±2.43
78.36±1.31
71.53±2.82
Text
36.97±14.21
60.02±3.46
59.78±3.70
70.99±3.32
58.46±2.96
Image + Text
68.12±10.47
70.98±3.81 * * * * 76.18±2.10
79.38±2.55
* 74.80±2.49
R@60
Image
97.91±1.95
78.74±1.98
93.02±2.88
93.78±0.73
90.64±1.03
Text
* * 100.00±0.00 * * * * 90.57±3.64
88.07±2.91
91.71±1.23
91.81±1.82
Image+Text
* * 100.00±0.00 * * * * 87.14±2.75
90.37±1.44
93.73±1.08 * * * 92.39±0.95
R@80
Image
97.52±2.26
73.06±2.57
91.49±3.67
93.15±0.81
88.94±1.40
Text
* * 100.00±0.00 * * * * 87.18±4.82
84.72±4.15
90.42±1.30
89.26±2.78
Image+Text
* * 100.00±0.00 * * * * 84.00±3.32
89.27±1.48
93.27±1.14 * * * 91.37±1.00
R@50:5:95
Image
95.38±3.72
67.87±2.38
88.30±4.73
92.53±0.77
87.13±1.40
Text
85.00±10.8 * * * * 79.09±3.72
75.16±4.00
87.42±1.73
84.89±3.15
Image + Text
96.00±5.16 * * * * 77.68±3.64
85.05±2.63
92.42±1.27 * * 89.35±1.24
Table 4 :
4Results of the first series of experiments reported as mean values ± standard deviation computed from 10 runs. Stars indicate statistically significant improvements from Text and Image+Text relative toImage.
Table 5 :
5Distribution of the classes for the different datasets
Image+Text * * * * 25.08±7.37 * * * * 60.65±10.27 * * * * 77.52± 4.41 60.17±7.42 * * 62.84±6.37 Image+Text * * * 56.70±4.23 * * * * 17.53± 4.11 * * * * 67.36± 4.43 * * * 49.46±3.84 * * * * 54.97±3.25Exp. Modalities
Serial
Weather
Death Notice
Stocks
Average
Time Image
8.00±2.66
29.44± 6.28
51.29±12.88 * * 68.30±3.31
54.65±5.47
Time GDL
Image
67.79±6.62
58.60± 3.00
63.06± 3.26
72.38±2.35
67.59±3.12
GDL Image+Text
* 73.81±4.08
59.16± 2.22 * * * * 75.32± 1.69
72.65±1.77 * * 71.54±1.21
IMP
Image
42.45±7.86
7.04± 4.92
40.14± 3.81
42.45±2.08
40.71±2.82
IMP
Table 6 :
6Results for the mIoU metric. Mean metric ± standard deviation of the metric (in %). Stars show statistical difference of mean between modalities.
Table 7 :
7Distribution of the classes for the training and testing sets.
36±14.22 66.71±3.57 * 71.20±4.03 * * 77.30±1.07 68.22±2.84Modalities # pages
Serial
Weather Death Notice
Stocks
Average
Image
1387
74.12± 7.59 81.27±2.18
75.37±2.98
83.11±0.88 79.30±2.29
Image+Text
1387
76.73± 5.90 81.38±3.34
83.58±2.02
84.43±1.84 82.16±1.72
Image
792
* * 70.27± 2.79 69.67±2.76
65.88±6.48
74.73±2.25 * 70.80±2.32
Image+Text
792
49.
Table 8 :
8Results for the mIoU metric. Mean metric ± standard deviation of the metric (in %). Stars indicate statistical difference of mean with Image. It is only reported for models using 60% of the data.Stocks
Average
Feuilleton
Weather
Death notice
Journal of Data Mining and Digital Humanities ISSN 2416-5999, an open-access journal
Table 9 :
9Distribution of the classes for the training and test sets.
Table 10 :
10Results for the mIoU metric. Mean metric ± standard deviation of the metric (in %). The stars of the Image+Text out-domain column indicate the statistical difference of mean w.r.t to Image, and the ones of Image+Text in-domain the difference w.r.t Image+Text out-domain.
http://jdmdh.episciences.org
More details on dhSegment architecture can be found in the original paper. 11 Both are partners of the impresso project:https://www.nb.admin.ch/snl/en/home.html and https://www.letemps.ch/
https://github.com/flairNLP/flair Journal of Data Mining and Digital Humanities ISSN 2416-5999, an open-access journal
https://www.tensorflow.org Journal of Data Mining and Digital Humanities ISSN 2416-5999, an open-access journal
Journal of Data Mining and Digital Humanities ISSN 2416-5999, an open-access journal
In all experiments, average corresponds to micro-average. Journal of Data Mining and Digital Humanities ISSN 2416-5999, an open-access journal
ACKNOWLEDGMENTSWe warmly thank the journal Le Temps and the Swiss and Luxembourgish National Libraries for giving us access to their newspaper archive collections in the context of the impresso project. We also thank Julien Nguyen Dang for his contribution to the annotation of part of the data. Finally, the second and third authors also gratefully acknowledge the financial support of the Swiss National Science Foundation (SNSF) for the project 'impresso -Media Monitoring of the Past' under grant number CR-SII5 173719.
. * * * * 83, 12±1.53* * * * 83.12±1.53
. * * *, 85.99±0.79* * * 85.99±0.79
. * * *, 85.28±0.46* * * 85.28±0.46
. * * * * 87, 24±0.31* * * * 87.24±0.31
. * * * * 87, 55±0.24* * * * 87.55±0.24
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"https://github.com/dhlab-epfl/dhSegment-text",
"https://github.com/dhlab-epfl/dhSegment-torch",
"https://github.com/dhlab-epfl/dhSegment-text-torchClass",
"https://github.com/flairNLP/flair"
] |
[
"On contact numbers of totally separable unit sphere packings *",
"On contact numbers of totally separable unit sphere packings *"
] |
[
"Károly Bezdek ",
"Balázs Szalkai ",
"István Szalkai "
] |
[] |
[] |
Contact numbers are natural extensions of kissing numbers. In this paper we give estimates for the number of contacts in a totally separable packing of n unit balls in Euclidean d-space for all n > 1 and d > 1. * Keywords: unit sphere packing, touching pairs, density, (truncated) Voronoi cell, union of balls, isoperimetric inequality, spherical cap packing.
|
10.1112/mtk.12102
|
[
"https://arxiv.org/pdf/1501.07907v2.pdf"
] | 42,479,190 |
2010.05091
|
b8d17026c4a53da54867d4df4a15b6b80d60c688
|
On contact numbers of totally separable unit sphere packings *
February 2, 2015
Károly Bezdek
Balázs Szalkai
István Szalkai
On contact numbers of totally separable unit sphere packings *
February 2, 2015
Contact numbers are natural extensions of kissing numbers. In this paper we give estimates for the number of contacts in a totally separable packing of n unit balls in Euclidean d-space for all n > 1 and d > 1. * Keywords: unit sphere packing, touching pairs, density, (truncated) Voronoi cell, union of balls, isoperimetric inequality, spherical cap packing.
Introduction
Let E d denote d-dimensional Euclidean space. Then the contact graph of an arbitrary finite packing of unit balls (i.e., of an arbitrary finite family of closed balls having unit radii and pairwise disjoint interiors) in E d is the (simple) graph whose vertices correspond to the packing elements and whose two vertices are connected by an edge if and only if the corresponding two packing elements touch each other. The number of edges of a contact graph is called the contact number of the given unit ball packing. One of the most basic questions on contact graphs is to find the maximum number of edges that a contact graph of a packing of n unit balls can have in E d . Harborth [15] proved the following optimal result in E 2 : the maximum contact number of a packing of n unit disks in E 2 is 3n − √ 12n − 3 , where · denotes the lower integer part of the given real. In dimensions three and higher the following upper bounds are known for the maximum contact numbers. It was proved in [9] that the contact number of an arbitrary packing of n unit balls in E 3 is always less than 6n − 0.926n 2 3 . On the other hand, it is proved in [6] that for d ≥ 4 the contact number of an arbitrary packing of n unit balls in E d is less than 1 2
τ d n − 1 2 d δ − d−1 d d n d−1 d ,
where τ d stands for the kissing number of a unit ball in E d (meaning the maximum number of non-overlapping unit balls of E d that can touch a given unit ball in E d ) and δ d denotes the largest possible density for (infinite) packings of unit balls in E d . For further results on contact numbers, including some optimal configurations of packings of small number of unit balls in E 3 , we refer the interested reader to [2] and [17]. (See also the relevant section in [8].) On the other hand, [16] offers a focused survey on recognition-complexity results of ball contact graphs. For an overview on sphere packings we refer the interested reader to the recent books [8] and [13].
In this paper we investigate the contact numbers of finite unit ball packings that are totally separable. The notion of total separability was introduced in [11] as follows: a packing of unit balls in E d is called totally separable if any two unit balls can be separated by a hyperplane of E d such that it is disjoint from the interior of each unit ball in the packing. Finding the densest totally separable unit ball packings is a difficult problem, which is solved only in dimensions two ( [11], [5]) and three ( [18]). As a close combinatorial relative we want to investigate the maximum contact number c(n, d) of totally separable packings of n > 1 unit balls in E d , d ≥ 2. Before we state our results we make the following observation. Let B d be a unit ball in an arbitrary totally separable packing of unit balls in E d and assume that B d is touched by m unit balls of the given packing say, at the points t 1 , . . . , t m ∈ S d−1 , where the boundary of B d is identified with the (d − 1)-dimensional spherical space S d−1 . The total separability of the given packing implies in a straightforward way that the spherical distance between any two points of {t 1 , . . . , t m } is at least π 2 . Now, recall that according to [10] (see also [19] and [22]) the maximum cardinality of a point set in S d−1 having pairwise spherical distances at least π 2 , is 2d and that maximum is attained only for the vertices of a regular d-dimensional crosspolytope inscribed in B d . Thus, m ≤ 2d and therefore c(n, d) ≤ dn. In the following we state isoperimetric-type improvements on this upper bound.
A straightforward modification of the method of Harborth [15] implies that
c(n, 2) = 2n − 2 √ n(1)
for all n > 1. For the convenience of the reader a proof of (1) is presented in the Appendix of this paper. Now, let us imagine that we generate totally separable packings of unit diameter balls in E d such that every center of the balls chosen, is a lattice point of the integer lattice Z d in E d . Then let c Z (n, d) denote the largest possible contact number of all totally separable packings of n unit diameter balls obtained in this way. It has been known for a long time ( [14]) that c Z (n, 2) = 2n − 2 √ n , which together with (1) implies that c Z (n, 2) = c(n, 2) for all n > 1. While we do not know any explicit formula for c Z (n, 3) in terms of n, we do have the following simple asymptotic formula for c Z (n, 3) as n → +∞, which follows in a rather straightforward way from the structural-type theorem of [1] characterizing a particular extremal configuration of c Z (n, 3) for any given n > 1: c Z (n, 3) = 3n − 3n
Theorem 1. c Z (n, d) ≤ dn − dn d−1 d
for all n > 1 and d ≥ 2.
We note that the upper bound of Theorem 1 is sharp for d = 2 and all n > 1 and for d ≥ 3 and all n = k d with k > 1. On the other hand, it is not a sharp estimate for example, for d = 3 and n = 5.
Theorem 2. c(n, d) ≤ dn − 1 2d d−1 2 n d−1 d for all n > 1 and d ≥ 4.
Although the method of the proof of Theorem 2 can be extended to include the case d = 3 the following statement is a stronger result.
Theorem 3. c(n, 3) < 3n − 1.346n 2 3 for all n > 1.
In the rest of the paper we prove the theorems stated.
Proof of Theorem 1
A union of finitely many axes parallel d-dimensional orthogonal boxes having pairwise disjoint interiors in E d is called a box-polytope. One may call the following statement the isoperimetric inequality for box-polytopes, which together with its proof presented below is an analogue of the isoperimetric inequality for convex bodies derived from the Brunn-Minkowski inequality. (For more details on the latter see for example, [3].) Lemma 1. Among box-polytopes of given volume the cubes have the least surface volume.
Proof. Without loss of generality we may assume that the volume vol d (A) of the given box-polytope
A in E d is equal to 2 d , i.e., vol d (A) = 2 d . Let B be an axes parallel d-dimensional cube of E d with vol d (B) = 2 d . Let the surface volume of B be denoted by svol d−1 (B). Clearly, svol d−1 (B) = d · vol d (B).
On the other hand, if svol d−1 (A) denotes the surface volume of the box-polytope A, then via the Minkowski definiton of surface volume one obtains that
svol d−1 (A) = lim →0 + vol d (A + B) − vol d (A) ,
where "+" in the numerator stands for the Minkowski addition of the given sets. Using the Brunn-Minkowski inequality ( [3]) we get that
vol d (A + B) ≥ vol d (A) 1 d + vol d ( B) 1 d d = vol d (A) 1 d + · vol d (B) 1 d d . Hence, vol d (A + B) ≥ vol d (A) + d · vol d (A) d−1 d · · vol d (B) 1 d = vol d (A) + · d · vol d (B) = vol d (A) + · svol d−1 (B) . So vol d (A + B) − vol d (A) ≥ svol d−1 (B) and therefore svol d−1 (A) ≥ svol d−1 (B)
, finishing the proof of Lemma 1.
Corollary 1. For any box-polytope P of E d the isoperimetric quotient svol d−1 (P) d vol d (P) d−1
of P is at least as large as the isoperimetric quotient of a cube, i.e.,
svol d−1 (P) d vol d (P) d−1 ≥ (2d) d . Now, let P := {c 1 + B d , c 2 + B d , . . . , c n + B d } denote the totally separable packing of n unit diameter balls with centers {c 1 , c 2 , . . . , c n } ⊂ Z d having contact number c Z (n, d) in E d .
(P might not be uniquely determined up to congruence in which case P stands for any of those extremal packings.) Let U d be the axes parallel d-dimensional unit cube centered at the origin o in E d . Then the unit cubes
{c 1 +U d , c 2 +U d , . . . , c n + U d } have pairwise disjoint interiors and P = ∪ n i=1 (c i + U d ) is a box-polytope. Clearly, svol d−1 (P) = 2dn − 2c Z (n, d). Hence, Corollary 1 implies that 2dn − 2c Z (n, d) = svol d−1 (P) ≥ 2dvol d (P) d−1 d = 2dn d−1 d . So, dn − dn d−1 d ≥ c Z (n, d)
, finishing the proof of Theorem 1.
3 Proof of Theorem 2
Definition 1. Let B d = {x ∈ E d | x ≤ 1}
be the closed unit ball centered at the origin o in E d , where · refers to the standard Euclidean norm of E d . Let R ≥ 1. We say that the packing
P sep = {c i + B d | i ∈ I with c j − c k ≥ 2 for all j = k ∈ I}
of (finitely or infinitely many) non-overlapping translates of B d with centers
{c i | i ∈ I} is an R-separable packing in E d if for each i ∈ I the finite packing {c j + B d | c j + B d ⊆ c i + RB d } is a totally separable packing (in c i + RB d ). Finally, let δ sep (R, d) denote the largest density of all R-separable unit ball packings in E d , i.e., let δ sep (R, d) = sup Psep lim sup λ→∞ ci+B d ⊂Q λ vol d (c i + B d ) vol d (Q λ ) ,
where Q λ denotes the d-dimensional cube of edge length 2λ centered at o in E d having edges parallel to the coordinate axes of E d .
Remark 1. For any 1 ≤ R < 3 we have that δ sep (R, d) = δ d ,
where δ d stands for the supremum of the upper densities of all unit ball packings in E d .
The following statement is the core part of our proof of Theorem 2 and it is an analogue of the Lemma in [6] (see also Theorem 3.1 in [4]).
Theorem 4. If {c i + B d | 1 ≤ i ≤ n} is an R-separable packing of n unit balls in E d with R ≥ 1, n ≥ 1, and d ≥ 2, then nvol d (B d ) vol d (∪ n i=1 c i + 2RB d ) ≤ δ sep (R, d) .
Proof. Assume that the claim is not true. Then there is an > 0 such that
vol d ∪ n i=1 c i + 2RB d = nvol d (B d ) δ sep (R, d) − (2) Let C n = {c i | i = 1, .
. . , n} and let Λ be a packing lattice of C n + 2RB d = ∪ n i=1 c i + 2RB d such that C n + 2RB d is contained in the foundamental parallelotope P of Λ. Recall that for each λ > 0, Q λ denotes the d-dimensional cube of edge length 2λ centered at the origin o in E d having edges parallel to the coordinate axes of E d . Clearly, there is a constant µ > 0 depending on P only, such that for each λ > 0 there is a subset
L λ of Λ with Q λ ⊆ L λ + P and L λ + 2P ⊆ Q λ+µ .(3)
Moreover, let P m (B d ) denote the family of all R-separable packings of m > 1 unit balls in E d . The definition of δ sep (R, d) implies that for each λ > 0 there exists a packing in the family P m (B d ) with centers at the points of C m(λ) such that
C m(λ) + B d ⊂ Q λ and lim λ→∞ m(λ)vol d (B d ) vol d (Q λ ) = δ sep (R, d) .
As lim λ→∞
vol d (Q λ+µ )
vol d (Q λ ) = 1 therefore there exist ξ > 0 and a packing in the family P m(ξ) (B d ) with centers at the points of C m(ξ) and with C m(ξ)
+ B d ⊂ Q ξ such that vol d (P)δ sep (R, d) vol d (P) + < m(ξ)vol d (B d ) vol d (Q ξ+µ ) and nvol d (B d ) vol d (P) + < nvol d (B d )card(L ξ ) vol d (Q ξ+µ ) ,(4)
where card(·) refers to the cardinality of the given set. Now, for each x ∈ P we define an R-separable packing of n(x) translates of B d in E d with centers at the points of
C n(x) = {x + L ξ + C n } ∪ {y ∈ C m(ξ) | y / ∈ x + L ξ + C n + int(2RB d )} ,
where int(·) refers to the interior of the given set in E d . Clearly, (3) implies that C n(x) + B d ⊂ Q ξ+µ . Now, in order to evaluate x∈P n(x)dx, we introduce the function χ y for each y ∈ C m(ξ) defined as follows: χ y (x) = 1 if y / ∈ x + L ξ + C n + int(2RB d ) and χ y (x) = 0 for any other x ∈ P. Then it is easy to see that
x∈P n(x)dx = x∈P ncard(L ξ )+ y∈C m(ξ) χ y (x) dx = nvol d (P)card(L ξ )+m(ξ) vol d (P)−vol d (C n +2RB d ) .
Hence, there is a point p ∈ P with
n(p) ≥ m(ξ) 1 − vol d (C n + 2RB d ) vol d (P) + ncard(L ξ ) and so n(p)vol d (B d ) vol d (Q ξ+µ ) ≥ m(ξ)vol d (B d ) vol d (Q ξ+µ ) 1 − vol d (C n + 2RB d ) vol d (P) + nvol d (B d )card(L ξ ) vol d (Q ξ+µ ) .(5)
Now, (2) implies in a straightforward way that
vol d (P)δ sep (R, d) vol d (P) + 1 − vol d (C n + 2RB d ) vol d (P) + nvol d (B d ) vol d (P) + = δ sep (R, d)(6)
Thus, (4), (5), and (6) yield that
n(p)vol d (B d ) vol d (Q ξ+µ ) > δ sep (R, d) .
As C n(p) + B d ⊂ Q ξ+µ this contradicts the definition of δ sep (R, d), finishing the proof of Theorem 4.
Next, let P = {c 1 + B d , c 2 + B d , . . . , c n + B d } be a totally separable packing of n translates of B d with centers at the points of C n = {c 1 , c 2 , . . . , c n } in E d . Recall that any member of P is tangent to at most 2d members of P and if c i + B d is tangent to 2d members, then the tangent points are the vertices of a regular cross-polytope inscribed in c i + B d and therefore
c i + √ dB d ⊂ 1≤j≤n,j =i c j + √ dB d .
Thus, if m denotes the number of members of P that are tangent to 2d members in P, then the (d − 1)dimensional surface volume svol d−1 bd(C n + √ dB d ) of the boundary bd(C n + √ dB d ) of the non-convex set C n + √ dB d must satisfy the inequality
svol d−1 bd(C n + √ dB d ) ≤ (n − m)d d−1 2 svol d−1 bd(B d )(7)
Finally, the isoperimetric inequality ( [21]) applied to C n + √ dB d yields
Iq(B d ) = svol d−1 bd(B d ) d vol d (B d ) d−1 = d d vol d (B d ) ≤ Iq(C n + √ dB d ) = svol d−1 bd(C n + √ dB d ) d vol d (C n + √ dB d ) d−1 ,(8)
where Iq(·) stands for the isoperimetric quotient of the given set. As d ≥ 4, P is a √ d 2 -separable packing (in fact, it is an R-separable packing for all R ≥ 1) and therefore (7), (8), and Theorem 4 imply in a straightforward way that
n − m ≥ svol d−1 bd(C n + √ dB d ) d d−1 2 svol d−1 (bd(B d )) = svol d−1 bd(C n + √ dB d ) d d+1 2 vol d (B d ) ≥ Iq(B d ) 1 d vol d (C n + √ dB d ) d−1 d d d+1 2 vol d (B d ) ≥ Iq(B d ) 1 d d d+1 2 vol d (B d ) nvol d (B d ) δ sep ( √ d 2 , d) d−1 d = 1 d d−1 2 δ sep ( √ d 2 , d) d−1 d n d−1 d .
Thus, the number of contacts in P is at most
1 2 (2dn − (n − m)) ≤ dn − 1 2d d−1 2 δ sep ( √ d 2 , d) d−1 d n d−1 d < dn − 1 2d d−1 2 n d−1 d ,
finishing the proof of Theorem 2.
Proof of Theorem 3
The following proof is an analogue of the proof of Theorem 1.1 in [7] and as such it is based on the proper modifications of the main (resp., technical) lemmas of [7]. Overall the method discussed below turns out to be more efficient for totally separable unit ball packings than for unit ball packings in general. The more exact details are as follows. Let B 3 denote the (closed) unit ball centered at the origin o of E 3 and let P := {c 1 +B 3 , c 2 +B 3 , . . . , c n + B 3 } denote the totally separable packing of n unit balls with centers c 1 , c 2 , . . . , c n in E 3 , which has the largest number namely, c(n, 3) of touching pairs among all totally separable packings of n unit balls in E 3 . (P might not be uniquely determined up to congruence in which case P stands for any of those extremal packings.)
Lemma 2. 4π 3 n vol 3 n i=1 c i + √ 3B 3 < 0.6401,
where vol 3 (·) refers to the 3-dimensional volume of the corresponding set.
Proof. First, partition n i=1 c i + √ 3B 3 into truncated Voronoi cells as follows. Let P i denote the Voronoi cell of the packing P assigned to c i + B 3 , 1 ≤ i ≤ n, that is, let P i stand for the set of points of E 3 that are not farther away from c i than from any other c j with j = i, 1 ≤ j ≤ n. Then, recall the well-known fact (see for example, [12]) that the Voronoi cells P i , 1 ≤ i ≤ n just introduced form a tiling of E 3 . Based on this it is easy to see that the truncated Voronoi cells P i ∩ (c i + √ 3B 3 ), 1 ≤ i ≤ n generate a tiling of the non-convex container n i=1 c i + √ 3B 3 for the packing P. Second, we prove the following metric properties of the Voronoi cells introduced above. 4 . An easy argument implies that in order to prove the latter claim it is sufficient to check it for triangles abc with the property that the two inner tangent lines of the unit disks a + B 2 and b + B 2 are tangent to the unit disk c + B 2 as well. Furthermore, one can assume that
2 < a − b ≤ 2 √ 2 and 2 < a − c = b − c ≤ 2 √ 2. Now, if x = 1 2 a − b
, then an elementary computation yields that the circumradius of the triangle abc is equal to
f (x) = x 3 2 √ x 2 −1 with 1 < x ≤ √ 2. Finally, f (x) = x 2 (2x 2 −3) 2(x 2 −1) √ x 2 −1 implies in a straightforward way that f ( 3 2 ) = 3 √ 3 4 is a global minimum of f (x) over 1 < x ≤ √ 2.
This finishes the proof of Sublemma 1.
Remark 2.
As one can see from the above proof, the lower bound of Sublemma 1 is a sharp one and it should be compared to the lower bound 2 √ 3 = 1.154 . . . valid for any unit ball packing not necessarily totally separable in E 3 . (For more details on the lower bound 2 √ 3 see for example the discussion on page 29 in [8].) Sublemma 2. The distance between an arbitrary vertex of the Voronoi cell P i and the center c i is always at least √ 2 = 1.414 . . . for any 1 ≤ i ≤ n.
Proof. Clearly, the claim follows from the following statement: If
P 4 = {c 1 + B 3 , c 2 + B 3 , c 3 + B 3 , c 4 + B 3 }
is a totally separable packing of four unit balls with centers c 1 , c 2 , c 3 , c 4 in E 3 , then the circumradius of the terahedron c 1 c 2 c 3 c 4 is at least √ 2. We prove the latter claim by looking at the following two cases possible. P 4 is a totally separable packing with plane H separating either c 1 + B 3 , c 2 + B 3 from c 3 + B 3 , c 4 + B 3 (Case 1) or c 1 + B 3 from c 2 + B 3 , c 3 + B 3 , c 4 + B 3 (Case 2). In both cases it is sufficient to show that if
∪ 4 i=1 c i + B 3 ⊂ x + rB 3
for some x ∈ E 3 and r ∈ R, then r ≥ 1 + Remark 3. As one can see from the above proof, the lower bound of Sublemma 2 is a sharp one and it should be compared to the lower bound 3 2 = 1.224 . . . valid for any unit ball packing not necessarily totally separable in E 3 . (For more details on the lower bound 3 2 see for example the discussion on page 29 in [8].)
Now, let U := conv({o, u 1 , u 2 , u 3 }) be the following special tetrahedron, also called the orthoscheme with vertices o, u 1 , u 2 , u 3 in E 3 (where conv(·) refers to the convex hull of the given set): u 1 is orthogonal to u 2 − u 1 as well as u 3 − u 1 , and u 2 is orthogonal to u 3 − u 2 moreover, u 1 = 1, u 2 = 3 √ 3 4 , and u 3 = √ 2 (where · denotes the Euclidean norm of E 3 ). Rogers's well-known method ( [24]) on dissecting each Voronoi cell P i into special simplices called Rogers simplices combined with Sublemmas 1 and 2 imply the following estimate in a standard way (using the so-called Lemma of Comparison of Rogers (for more details see for example, page 33 in [8])).
Sublemma 3. 4π 3 vol 3 (P i ∩ (c i + √ 2B 3 )) ≤ vol 3 (U ∩ B 3 ) vol 3 (U) < 0.6401. As P i ∩ (c i + √ 2B 3 ) ⊂ P i ∩ (c i + √ 3B 3 )
, therefore Sublemma 3 completes the proof of Lemma 2.
The well-known isoperimetric inequality ( [21]) applied to
n i=1 c i + √ 3B 3 yields Lemma 3. 36π vol 3 n i=1 c i + √ 3B 3 2 ≤ svol 2 bd n i=1 c i + √ 3B 3 3 ,
where svol 2 (·) refers to the 2-dimensional surface volume of the corresponding set.
Thus, Lemma 2 and Lemma 3 generate the following inequality.
Corollary 2. 4π (0.6401) 2 3 n 2 3 < svol 2 bd n i=1 c i + √ 3B 3 .
Now, assume that c i + B 3 ∈ P is tangent to c j + B 3 ∈ P for all j ∈ T i , where T i ⊂ {1, 2, . . . , n} stands for the family of indices 1 ≤ j ≤ n for which dist(c i , c j ) = 2. Then letŜ i := bd(c i + √ 3B) and letĉ ij be the intersection of the line segment c i c j withŜ i for all j ∈ T i . Moreover, let CŜ i (ĉ ij , π 4 ) (resp., CŜ i (ĉ ij , α)) denote the open spherical cap ofŜ i centered atĉ ij ∈Ŝ i having angular radius π 4 (resp., α with 0 < α < π 2 and cos α = 1 √ 3 ). As P is totally separable therefore the family {CŜ i (ĉ ij , π 4 ), j ∈ T i } consists of pairwise disjoint open spherical caps ofŜ i ; moreover,
j∈Ti svol 2 CŜ i (ĉ ij , π 4 ) svol 2 ∪ j∈Ti CŜ i (ĉ ij , α) = j∈Ti Sarea C(u ij , π 4 ) Sarea (∪ j∈Ti C(u ij , α)) ,(9)
where u ij := 1 2 (c j −c i ) ∈ S 2 := bd(B 3 ) and C(u ij , π 4 ) ⊂ S 2 (resp., C(u ij , α) ⊂ S 2 ) denotes the open spherical cap of S 2 centered at u ij having angular radius π 4 (resp., α) and where Sarea(·) refers to the spherical area measure on S 2 .
Lemma 4.
j∈Ti Sarea C(u ij , π 4 ) Sarea (∪ j∈Ti C(u ij , α)) ≤ 3 1 − 1 √ 2 = 0.8786 . . . .
Next, let f i denote the number of internal faces of G n having i sides. As P n is totally separable therefore i ≥ 4. Now, Euler's formula implies that n − c(n) + f 4 + f 5 + . . . = 1 (13) If we add up the number of sides of the internal faces of G n , then every edge of P is counted once and all the other edges twice. Thus, 4(f 4 + f 5 + . . . ) ≤ 4f 4 + 5f 5 + . . . = b + 2(c(n) − b).
Clearly, (13) and (14) imply that 4(1 − n + c(n)) ≤ b + 2(c(n) − b) and so,
2c(n) − 3n + 4 ≤ n − b(15)
Now, let us delete from G n the vertices of P together with the edges incident to them. By the definition of c(n − b), one obtains
c(n) − b − (b 3 + 2b 4 ) ≤ c(n − b).(16)
Next, (12) and (16) imply c(n) ≤ c(n − b) + 2b − 4.
As by induction c(n − b) ≤ 2(n − b) − 2 √ n − b, therefore (17) yields
c(n) ≤ (2n − 4) − 2 √ n − b.(18)
Finally, (15) and (18) imply c(n) ≤ (2n − 4) − 2 2c(n) − 3n + 4, from which it follows easily that 0 ≤ c(n) 2 − 4nc(n) + (4n 2 − 4n).
Notice that the roots of the quadratic equation 0 = x 2 − 4nx + (4n 2 − 4n) are 2n ± 2 √ n. As c(n) < 2n, therefore (19) implies in a straightforward way that c(n) ≤ 2n − 2 √ n, finishing the proof of (1).
. Clearly, c Z (n, 3) ≤ c(n, 3) for all n > 1. So, one may wonder whether c Z (n, 3) = c(n, 3) for all n > 1?The above discussion leads to the natural and rather basic question on upper bounding c Z (n, d) (resp., c(n, d)) in the form of dn − Cn d−1 d , where C > 0 is a proper constant depending on d.
Sublemma 1 .√ 3 4
13The distance between the line of an arbitrary edge of the Voronoi cell P i and the center c i is always at least 3 = 1.299 . . . for any 1 ≤ i ≤ n. Proof. It is easy to see that the claim follows from the following 2-dimensional statement: If {a + B 2 , b + B 2 , c + B 2 } is a totally separable packing of three unit disks with centers a, b, c in E 2 , then the circumradius of the triangle abc is at least 3 √ 3
√ 2 .
2Case 1: Let H + and H − denote the two closed halfspaces bounded by H with c 1 + B 3 ∪ c 2 + B 3 ⊂ H + and c 3 + B 3 ∪ c 4 + B 3 ⊂ H − . Without loss of generality we may assume that vol3 (x + rB 3 ) ∩ H + ≤ vol 3 (x + rB 3 ) ∩ H − . Now, if c 1 (resp., c 2 ) denotes the image of c 1 (resp., c 2 ) under the reflection about H, then clearly P = {c 1 + B 3 , c 2 + B 3 , c 1 + B 3 , c 2 + B 3} is a packing of four unit balls in x + rB 3 symmetric about H. Then using the symmetry of P with respect to H it is easy to see that r ≥ 1 + √ 2.Case 2: Let H + and H − denote the two closed halfspaces bounded by H with c 1 + B 3 ⊂ H + and c 2 + B 3 ∪ c 3 + B 3 ∪ c 4 + B 3 ⊂ H − . If one assumes that r − 1 < √ 2, then using c 1 ∈ (x + (r − 1)B 3 ) ∩ H + and {c 2 , c 3 , c 4 } ⊂ (x + (r − 1)B 3 ) ∩ H − it is easy to see that the triangle c 2 c 3 c 4 is contained in a disk of radius less than 2 √ 2 − 1 = 1.287 . . . . On the other hand, as the unit balls c 2 + B 3 , c 3 + B 3 , c 4 + B 3 form a totally separable packing therefore the proof of Sublemma 1 implies that the radius of any disk containing the triangle c 2 c 3 c 4 must be at least 3 √ 3
= 1.299 . . . , a contradiction.
Proof. By assumption P i (S 2 ) = {C(u ij , π 4 ) | j ∈ T i } is a packing of spherical caps of angular radius π 4 in S 2 . Let V ij (S 2 ) denote the Voronoi region of the packing P i (S 2 ) assigned to C(u ij , π 4 ), that is, let V ij (S 2 ) stand for the set of points of S 2 that are not farther away from u ij than from any other u ik with k = j, k ∈ T i . Recall (see for example[12]) that the Voronoi regions V ij (S 2 ), j ∈ T i are spherically convex polygons and form a tiling of S 2 . Moreover, it is easy to see that no vertex of V ij (S 2 ) belongs to the interior of C(u ij , α) in S 2 . Thus, Hajós Lemma (Hilfssatz 1 in[20]stands for the spherical area of a regular spherical quadrilateral inscribed into C(u ij , α) with sides tangent to C(u ij , π 4 ). Hence,As the truncated Voronoi regions V ij (S 2 ) ∩ C(u ij , α), j ∈ T i form a tiling of ∪ j∈Ti C(u ij , α) therefore (10) finishes the proof of Lemma 4.Lemma 4 implies in a straightforward way that3).(11)Hence, Corollary 2 and(11)yield, finishing the proof of Theorem 3.AppendixWe use the method of Harborth[15]with some natural modifications due to the total separability of the packings under investigation. We prove (1) by induction on n. For simplicity let c(n) := c(n, 2). Clearly, c(2) = 1 = 2 · 2 − 2 √ 2 . So in what follows, we assume that n ≥ 3 and in particular, we assume that (1) holds for all positive integers n with 2 ≤ n ≤ n − 1. Let P n be the totally separable packing of n unit disks in E 2 , which has the largest number namely, c(n) of touching pairs among all totally separable packings of n unit disks in E 2 . (P n might not be uniquely determined up to congruence in which case P n stands for any of those extremal packings.) Let G n denote the embedded contact graph of P n with vertices identical to the centers of the unit disks in P n and with edges represented by line segments connecting two vertices if the unit disks centered at them touch each other. Clearly, the number of edges of G n is equal to c(n). As c(n − 1) + 1 = 2(n − 1) − 2 √ n − 1 + 1 ≤ 2n − 2 √ n and c Z (n, 2) = 2n − 2 √ n ([14]) for all n ≥ 2, therefore one can assume that every vertex of G n is adjacent to at least two other vertices (otherwise there is a vertex of G n of degree one and so, the proof is finished by induction). In addition, using c Z (n, 2) = 2n − 2 √ n again one can assume that G n is 2-connected, that is, G n remains connected after the removal of any of its vertices.Thus, the outer face of G n in E 2 is bounded by a simple closed polygon P . Let b denote the number of vertices of P . As P n is a totally separable unit disk packing therefore the degree of any vertex of P (resp., G n ) is either 2 or 3 or 4 in G n . Let b i stand for the number of vertices of P of degree i with 2 ≤ i ≤ 4. Clearly, b = b 2 + b 3 + b 4 . Due to the total separability of P n , the internal angle of P at a vertex of degree i is at least (i−1)π 2 , and the sum of these angles is (b − 2)π. Thus, b 2 + 2b 3 + 3b 4 ≤ 2b − 4(12)
The three-dimensional polyominoes of minimal area. L Alonso, R Cerf, Electron. J. Combin. 31Research Paper 27, approx. 39 ppL. Alonso and R. Cerf, The three-dimensional polyominoes of minimal area, Electron. J. Combin. 3/1 (1996), Research Paper 27, approx. 39 pp.
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An elementary introduction to modern convex geometry. K Ball, Flavors of Geometry. CambridgeCambridge Univ. PressK. Ball, An elementary introduction to modern convex geometry, in Flavors of Geometry (Ed.: S. Levy), Math. Sci. Res. Inst. Publ., 31, Cambridge Univ. Press, Cambridge, 1997, 1-58.
Finite and infinite packings. U Betke, M Henk, J M Wills, J. Reine Angew. Math. 453U. Betke, M. Henk, and J. M. Wills, Finite and infinite packings, J. Reine Angew. Math. 453 (1994), 165-191.
Locally separable circle packings. A Bezdek, Studia Sci. Math. Hungar. 182-4A. Bezdek, Locally separable circle packings, Studia Sci. Math. Hungar. 18/2-4 (1983), 371-375.
On the maximum number of touching pairs in a finite packing of translates of a convex body. K Bezdek, J. Combin. Theory Ser. A. 981K. Bezdek, On the maximum number of touching pairs in a finite packing of translates of a convex body, J. Combin. Theory Ser. A 98/1 (2002), 192-200.
Contact numbers for congruent sphere packings in Euclidean 3-space. K Bezdek, Discrete Comput. Geom. 482K. Bezdek, Contact numbers for congruent sphere packings in Euclidean 3-space, Discrete Comput. Geom. 48/2 (2012), 298-309.
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Packing and Covering. C A Rogers, Cambridge University PressCambridge; Veszprém, Hungary, E-mailKároly Bezdek Department of Mathematics and Statistics, University of Calgary, Canada, Department of Mathematics, University of PannoniaC. A. Rogers, Packing and Covering, Cambridge University Press, Cambridge, 1964. Károly Bezdek Department of Mathematics and Statistics, University of Calgary, Canada, Department of Mathematics, University of Pannonia, Veszprém, Hungary, E-mail: [email protected]
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"Land Use and Land Cover Classification Using Deep Learning Techniques",
"Land Use and Land Cover Classification Using Deep Learning Techniques"
] |
[
"Nagesh Kumar Uba \nARIZONA STATE UNIVERSITY\n\n",
"ChairJohn Femiani \nARIZONA STATE UNIVERSITY\n\n",
"Anshuman Razdan \nARIZONA STATE UNIVERSITY\n\n",
"Ashish Amresh \nARIZONA STATE UNIVERSITY\n\n"
] |
[
"ARIZONA STATE UNIVERSITY\n",
"ARIZONA STATE UNIVERSITY\n",
"ARIZONA STATE UNIVERSITY\n",
"ARIZONA STATE UNIVERSITY\n"
] |
[] |
Large datasets of sub-meter aerial imagery represented as orthophoto mosaics are widely available today, and these data sets may hold a great deal of untapped information. This imagery has a potential to locate several types of features; for example, forests, parking lots, airports, residential areas, or freeways in the imagery. However, the appearances of these things vary based on many things including the time that the image is captured, the sensor settings, processing done to rectify the image, and the geographical and cultural context of the region captured by the image. This thesis explores the use of deep convolutional neural networks to classify land use from very high spatial resolution (VHR), orthorectified, visible band multispectral imagery. Recent technological and commercial applications have driven the collection a massive amount of VHR images in the visible red, green, blue (RGB) spectral bands, this work explores the potential for deep learning algorithms to exploit this imagery for automatic land use/ land cover (LULC) classification.The benefits of automatic visible band VHR LULC classifications may include applications such as automatic change detection or mapping. Recent work has shown the potential of Deep Learning approaches for land use classification; however, this thesis improves on the state-of-the-art by applying additional dataset augmenting approaches that are well suited for geospatial data. Furthermore, the generalizability of the classifiers is tested by extensively evaluating the classifiers on unseen datasets and we present the accuracy levels of the classifier in order to show that the results actually generalize beyond the small benchmarks used in training. Deep networks have many parameters, and therefore they are often built with very large sets of labeled data. Suitably large datasets for LULC are not easy to come by, but techniques such as refinement learning allow networks trained for one task to be retrained to perform another recognition task. Contributions of this thesis include
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[
"https://arxiv.org/pdf/1905.00510v1.pdf"
] | 133,430,361 |
1905.00510
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470e28b027f6cf917ca97fec6196ff31c8501560
|
Land Use and Land Cover Classification Using Deep Learning Techniques
May 2016 i
Nagesh Kumar Uba
ARIZONA STATE UNIVERSITY
ChairJohn Femiani
ARIZONA STATE UNIVERSITY
Anshuman Razdan
ARIZONA STATE UNIVERSITY
Ashish Amresh
ARIZONA STATE UNIVERSITY
Land Use and Land Cover Classification Using Deep Learning Techniques
May 2016 iApproved April 2016 by the Graduate Supervisory Committee:
Large datasets of sub-meter aerial imagery represented as orthophoto mosaics are widely available today, and these data sets may hold a great deal of untapped information. This imagery has a potential to locate several types of features; for example, forests, parking lots, airports, residential areas, or freeways in the imagery. However, the appearances of these things vary based on many things including the time that the image is captured, the sensor settings, processing done to rectify the image, and the geographical and cultural context of the region captured by the image. This thesis explores the use of deep convolutional neural networks to classify land use from very high spatial resolution (VHR), orthorectified, visible band multispectral imagery. Recent technological and commercial applications have driven the collection a massive amount of VHR images in the visible red, green, blue (RGB) spectral bands, this work explores the potential for deep learning algorithms to exploit this imagery for automatic land use/ land cover (LULC) classification.The benefits of automatic visible band VHR LULC classifications may include applications such as automatic change detection or mapping. Recent work has shown the potential of Deep Learning approaches for land use classification; however, this thesis improves on the state-of-the-art by applying additional dataset augmenting approaches that are well suited for geospatial data. Furthermore, the generalizability of the classifiers is tested by extensively evaluating the classifiers on unseen datasets and we present the accuracy levels of the classifier in order to show that the results actually generalize beyond the small benchmarks used in training. Deep networks have many parameters, and therefore they are often built with very large sets of labeled data. Suitably large datasets for LULC are not easy to come by, but techniques such as refinement learning allow networks trained for one task to be retrained to perform another recognition task. Contributions of this thesis include
ii demonstrating that deep networks trained for image recognition in one task (ImageNet) can be efficiently transferred to remote sensing applications and perform as well or better than manually crafted classifiers without requiring massive training data sets. This is demonstrated on the UC Merced dataset, where 96% mean accuracy is achieved using a CNN (Convolutional Neural Network) and 5-fold cross validation. These results are further tested on unrelated VHR images at the same resolution as the training set. Analysis (I3DEA) labs at the ASU Polytechnic campus. I was utterly fascinated by the kind of work they were doing. They helped me to explore an idea to categorize aerial images and extract features from these images. The approach started by trying to extract features from aerial images using traditional techniques, but these methods gave poor or fair results at best.
Machine Learning and Deep Learning techniques showed promising results in the past Large Scale Visual Recognition Challenge (ILSVRC) ("ImageNet Large Scale Visual Recognition Competition (ILSVRC)," n.d.) competition. The Deep Learning library Caffe is a high performing tool that I could quickly get my hands on and start trying the ideas. The approach is to design and come up with a high-performance Deep Learning classifier that does the job and at the same time quick and easy to build. Thus, the ideas of the approach of transfer learning were suggested. I am thankful that by the guidance of my mentors Dr.
Femiani and Dr. Razdan, I have achieved this feat by doing Land Use Land Cover classification with UC Merced ("UC Merced Land Use Dataset," n.d.) (O. A. Penatti, Nogueira, & dos Santos, 2015) dataset and tested the classifier with unrelated random samples.
INTRODUCTION
Recent technological advancements in remote sensing and high-resolution image capturing by satellite drones and airplanes have been a huge help in gathering datasets for research and development. High-resolution orthoimagery datasets are readily available for download by resources such as United States Geological Survey. With the abundance of the data, the question arises of how to make use of the data for technological applications such as civil engineering, environmental monitoring or data generation for simulation and training. There have been significant advances in remote sensing and high-resolution image processing, and a variety of Land Use and Land Cover(LULC) classification algorithms have been developed in the recent past. One concern about LULC classification is that at high resolutions, there is a significant amount of variability in the data. The benchmark datasets used to test and evaluate classification algorithms may not capture enough of the variability to generalize to unseen examples that may have been acquired at different times or locations.
Recent advances in machine learning have been accomplished through an approach called "deep learning". Deep learning refers to artificial neural networks comprised of many layers of artificial neurons, called perceptrons. Historically the number of layers in a neural network has been limited because the algorithms used to train networks became unstable when the depth (number of layers) of the network is increased. Recent advances in hardware as well as the availability of very large datasets and robust training algorithms have made deep neural networks not only tractable, but they outperform other algorithms by an order of magnitude in some problems ("ImageNet Large Scale Visual Recognition Competition (ILSVRC)," n.d.). One area where deep learning is particularly successful is computer vision, which uses a type of network called a "convolutional neural network", often called a ConvNet or CNN. Convolutional neural networks have the interesting property that the first layers of the network tend to learn patterns that mimic those observed in human vision (Gabor Wavelets, (Daugman, 1988)).
Deep learning frameworks usually need large sets of labeled data to train and classify the images. High-resolution orthoimagery is readily available, but for supervised-learning one first has to label the datasets to train the classifier. For example, the IMAGENET Large Scale Visual Recognition Challenge (ILSVRC) uses 10+ million images belonging to 400+ unique scene categories ("ImageNet Large Scale Visual Recognition Competition (ILSVRC)," n.d.). The challenge data is divided into 8.1M images for training, 20K images for validation and 381K images for testing coming from 401 scene categories.
For the ILSVRC-2012 object localization challenge, the validation and test data consisted of 150K photographs, collected from Flickr and other search engines, hand labeled with the presence or absence of 1000 object categories. The training data for object localization was the subset of ImageNet consisting of 1.2M images used as training data with 1000 categories. Generating such a large amount of annotated data is labor-intensive and takes a large amount of time. It is painstakingly time-consuming for humans to label manually each of the images from any new dataset.
Deep networks require large amount of data or they risk becoming overly specialized, however the process of generating labeled data for each new recognition task is expensive. One approach to solve this problem is to start from a pre-trained convolutional neural network, and use a technique called "refinement" or "transfer learning" to adapt the network to a new task. Transfer learning uses the existing pre-trained classifier and learns only on the top, fully connected layers, of the network.
Artificial neural networks are biologically inspired systems inspired by a human brain (Hopfield, 1988). The processing part of the human brain consists of billions of neurons and each neuron receives information from thousands of other neurons. A neuron can be studied as an input/output device. Neurons fundamentally transmit pulse-coded information. In an ANN (Artificial Neural Network) the input and output of this pulse coded system is a non-linear (sigmoid) function as shown in Figure 1. A sigmoid function is a non-linear function that is saturated at both ends. It is a bounded differentiable real function that is defined for all real input values and has a positive derivative at every point, and this is crucial to neural networks' computational properties (Mira & Sandoval, 1995). neuron. An artificial neural network is an interconnected group of nodes that show a similar character to the vast network of neurons in a biological brain. In Figure 2, each node (circle) represents an artificial neuron, and an arrow represents a connection from the output of one neuron to the input of another (Strickland, 2015). (Hinton, Osindero, & Teh, 2006), (Bengio, Lamblin, Popovici, Larochelle, & others, 2007), and (Ranzato, Poultney, Chopra, & Cun, 2006) demonstrated that deep networks are capable of outperforming other approaches by an order of magnitude on some problems. The papers covered these three principles ("Introduction to Deep Learning Algorithms -Notes de cours IFT6266 Hiver 2010," n.d.):
1. Unsupervised learning of representations is used to train each layer.
2. Training is done on one layer at a time.
3. Supervised training is used in order to fine-tune all the layers.
In this thesis, deep convolutional neural networks are used for handling the LULC classification problems. Recently proposed architectures by Caffe such as AlexNet (Krizhevsky, Sutskever, & Hinton, 2012) and CaffeNet (BVLC | Caffe) have shown promising results with simple deep structures. Caffe's GoogleNet , on the other hand, is a very complex deep network and performs better than the other two networks (AlexNet and CaffeNet) by a small margin. In this thesis, the experiments and conclusions of our approach are shown after evaluating the other existing prior art.
The proposed approach compares the training, validation and testing of models that are trained using transfer learning methods. The resulting classifiers are also tested on unrelated data to test the potential generalizability of labeling the data.
To compare with the existing state-of-the-art methods, the proposed approach has used UC Merced dataset that consists of 21 different classes. Each class has 100 images, and each image measures 256x256 pixels. The images were originally extracted from large images from the Unites States Geological Survey (USGS) National Map Urban Area Imagery ("The National Map: Orthoimagery," n.d.) collection for various urban areas around the country.
The pixel resolution of this public domain imagery is 1 foot ("UC Merced Land Use Dataset," 2016). Please refer to Figure 6 for different image class categories. Figure 6 shows all the samples of the UC Merced dataset.
The thesis makes two main contributions:
1. Transfer learning is used to solve LULC problem using ConvNets and shown to be competitive with the state-of-the-art. However, the generalizability of a ConvNet trained using refinement learning was shown to have issues when applied to other sources of imagery of the same type used in training.
2. Dataset augmentation by rotating training images is shown to improve the generalizability of classifier built using ConvNets.
BACKGROUND LITERATURE
Ojala et al., (Ojala, Pietikainen, & Maenpaa, 2002) have proposed a multi-resolution approach to gray-scale and rotation invariant texture classification based on local binary patterns and distributions. Lowe (Lowe, 2004) has proposed a method for extracting distinctive invariant features from images that can be used to perform reliable matching between different views of an object or scene. The features are invariant to image scale and rotation and are shown to provide robust matching across a substantial range of affine distortion, an addition of noise and change in illumination. These methods do not use convolutional neural networks or support vector machines (SVM) for their work. Dalal et al., (Dalal & Triggs, 2005) propose an approach that uses linear SVMs. They show after reviewing edge and gradient based descriptors that grids of histograms of oriented gradient descriptors significantly outperform other feature sets for human detection existing at the time.
A revolutionary work was published by Hinton et al., (Hinton et al., 2006), a fast learning algorithm for deep belief nets, that proposes a fast, greedy algorithm that can learn deep, directed belief networks one layer at a time. Bengio et al., ("Greedy Layer-Wise Training of Deep Networks -LISA -Publications -Aigaion 2.0," n.d.) (Bengio et al., 2007), which in the context of Hinton et al., continue the research that studies the algorithm empirically and explore variants to better understand its success and extend it to cases where the inputs are continuous or where the structure of the input distribution is not revealing enough about the variable to be predicted in a supervised task. Ranzato et al., (Ranzato et al., 2006) propose a novel unsupervised method for learning sparse, over complete features. Yang et al., (Yang & Newsam, 2010), Bag-of-visual-words(BOVW) and spatial extensions for land use classification is yet another exciting technological enhancement that uses the frequencies but not the locations of quantized image features to discriminate between classes analogous to how words are used for text document classification without regard to their order of occurrence. Their methods are evaluated using UC Merced dataset.
They show an accuracy of 76.81.
Pesaresi et al., (Pesaresi & Gerhardinger, 2011) proposed an automatic recognition of human settlements in the arid regions with scattered vegetation. Their methods are based on subtraction of the vegetated areas from the built-up areas detected using the analysis of image measures extracted using anisotropic rotation-invariant gray-level co-occurrence matrix and on the introduction of a morphological filtering step. They shown an accuracy of 88.69% with morphological filtering and 70.37% with the vegetation subtraction method.
Their methods do not involve neural networks, and compared to them, our approach demonstrates higher accuracy levels.
Rizvi et al., (Rizvi & Mohan, 2011) have proposed object-based image analysis of high-resolution remote sensing images using a kernel called cloud basis function, and they have investigated the probabilistic relaxation labeling process and have shown a 4% higher classification accuracy (91.47%) compared to the conventional Artificial Neural Networks existing at the time.
Yang et al., (Yang & Newsam, 2011) have proposed a spatial pyramid co-occurrence representation that characterizes both photometric and geometric aspects of an image. They show an accuracy of 77.38%.
Krizhevsky et al., (Krizhevsky, Sutskever, & Hinton, 2012) have proposed an ImageNet classification with deep convolutional neural networks. They achieved a winning top-5 test error rate of 15.3% compared to 26.2% obtained by the second best entry, with almost twice as much error rate reduction to the next best entry this paper was crucial to the research and development of the deep learning algorithms and frameworks. Chen et al., (S. Chen & Tian, 2015) have proposed a pyramid-of-spatial-relations model to capture absolute and relative spatial relationships of local features. They employ a novel concept of spatial relation to describe a relative spatial relationship of a group of local features. They have shown that their model is robust to translation and rotation variations and demonstrates excellent performance for the application of remotely sensed land use classification. They achieve an accuracy of 89.10% which is higher than at the time state-ofthe-art classification accuracy. Chen et al., (S. Chen & Tian, 2015) also proposed a spectralspatial classification of hyperspectral data based on deep belief network. They verify the eligibility of restricted Boltzmann machine and deep belief networks and offer a novel deep architecture which combines the spectral-spatial feature extraction and the classification together to get high classification accuracy. They also propose image segmentation, which is extraction and categorization of individual features from the images, and they achieve an overall accuracy of 95.45% on image segmentation. The proposed approach in this thesis concentrates on an overall image classification rather than image segmentation.
Ren et al., (Ren, Jiang, & Yuan, 2015) proposed a new approach to tackling highdimensional local binary patterns. Their objective was to select an optimal subset of binarized-pixel-difference features to compose the local binary pattern structure. They take the advantage of the fact that the local features are closely related, and they propose an incremental Maximal Conditional Mutual Information (MCMI) scheme which learns local binary patterns. Their approach shows an accuracy of 88.20%.
Hu et al., (Hu et al., 2015) present an improved unsupervised feature learning algorithm based on spectral clustering that adaptively learns good local feature representations and also discovers intrinsic structures of local image patches. The approach first maps the original image patches into a small dimensional and inherent features space by linear manifold analysis techniques and then learns a dictionary using K-Means clustering on the patch manifold for feature encoding. They experimented with the UC Merced dataset and have shown to achieve 90.26% accuracy rate.
Shao et al., (Shao, Yang, Xia, & Liu, 2013) propose a classification model based on a hierarchical fusion of multiple features. They employ four discriminative image features and an SVM with histogram intersection kernel in different classification stages, they conduct an extensive evaluation of various configurations and show an accuracy rate of 92.38%. Negrel et al., (Negrel, Picard, & Gosselin, 2014) present an investigation that uses visual features based on second-order statistics, as well as new processing techniques to improve the quality of features. They experiment on UC Merced dataset and show an accuracy of 94.30%.
In their survey paper on transfer learning, Pan et al., (Pan & Yang, 2010) point out that conventional classification approaches have assumed that training data and future inputs to a classifier must be statistically similar. However, significant problems can be solved by knowledge transfer from one task (where a large amount of data is available) into another domain with much fewer examples. Deep networks can make transfer learning especially attractive (Bengio, 2012) because feature modeling and higher order knowledge tend to live at different layers. Recently, Castelluccio et al., (Castelluccio, Poggi, Sansone, & Verdoliva, 2015) have shown that excellent classification results can be obtained by using a pre-trained classifier tuned for a larger dataset (ImageNET) and refining it by replacing the last layers.
They show 97.1% accuracy on UC Merced dataset by improving a pre-trained classifier GoogleNet , but the performance of this model was not tested on Caffe created by Yangqing Jia is one of the popular frameworks for the deep learning methods. It is developed by Berkeley Vision and Learning Centre (BVLC) and community contributors. The main reason for us to use Caffe framework is because it enables us to extensively utilize the GPU for hardware acceleration, and it also provides a framework to build, edit and run custom networks for refining. Caffe can also be run on CPU only. The highly customizable features of Caffe through configuration, prototxt and solver files makes it easy to use and concentrate more on the research techniques instead of worrying about implementing the deep architectures. The code is released under BSD 2-Clause license, and it can be forked and customized to satisfy ones' research requirements. It also provides C++, Python and Matlab APIs for easy accessibility. Figure 3. shows a typical architecture of the CaffeNet. It consists of five convolutional layers, each followed by a pooling layer, and three fully-connected layers. Caffe's model zoo as AlexNet. CaffeNet is a minor variation from the AlexNet with few differences in network topology, data preparation, and augmentation and averaged classification. In the proposed work of this thesis, we have changed the last layer to limit the classifications to 21 classes. So instead of 1000, we have 21 categories in the output layer and these 21 classes are the classes from UC Merced dataset. CaffeNet's Conv1 as shown in Figure 3. is the only layer that is exposed to the raw image. The 96 filter weights of Conv1 are visualized in Figure 4. Visualization of the layers helps in verifying the well trained networks, which usually display smooth filters without any noisy patterns (Alizadeh & Fazel, n.d.). Lin et al., (2013) (Lin, Chen, & Yan, 2013). This module is termed as Inception module. A snapshot of the inception module is shown in Figure 5. The architectures discussed above have been developed to process RGB images to find patterns and extract prominent features and correlate these features with the learned weights to update the weights of the system. Learning from images is a standard and an easy task for human beings to perform. But neural nets need large datasets to train and perform better, and training on large datasets is a time taking process (heavily depends on the hardware acceleration). A typical network may take days/ weeks to train from scratch depending on the hardware capabilities used. Krizhevsky et. al, (2012) (Krizhevsky et al., 2012) claim that the dataset takes 5-6 days to train AlexNet on two NVIDIA GTX 580 3GB GPU's.
In remote sensing, one has access to massive amounts of data, but most of this data is unlabeled. To categorize these remote sensing images, one needs a LULC classifier. But, to train this classifier, one runs into a very limited labeled training data. It is quite a challenge to train a classifier from scratch on these small datasets and obtain good classification accuracy levels. As discussed above, a common problem that we observe by training on small datasets is over-fitting. The classifier works with the best accuracy levels on the training data, but it does not generalize well to test data. The proposed approach in this thesis tries to address this problem, and experiments are conducted on unseen/ unrelated San-Diego remote sensing data to verify the same. The proposed method also evaluates Brazilian Coffee Scenes dataset which is a peculiar dataset much different from the ImageNet dataset.
Our motivation is Razavian et al., (2014) (Razavian, Azizpour, Sullivan, & Carlsson, 2014), Jia et al., (2104) , and Yosinski et al., (2014) (Yosinski, Clune, Bengio, & Lipson, 2014), as they have explored this possibility, published state-of-the-art results and suggested that the features obtained from deep learning with convolutional nets are to be considered as the primary candidate in most visual recognition tasks.
The proposed approach in this thesis takes the advantage of the availability of existing deep network models and the ability to quickly train them by transfer learning and fine-tuning to generate state-of-the-art classifier that performs and generalizes better than most of the prior-art classifiers.
Transfer learning can be categorized into two variations, our approach presents the experiments and results based on these variations: In this thesis, three existing frameworks are considered, AlexNet, CaffeNet and GoogleNet and performances of these are compared using Caffe framework. NVIDIA's Deep Learning GPU Training System (DIGITS) is yet another framework that enables us to harness the power of DNN's running in parallel on multi-GPU systems. In this thesis, TESLA K40c 12 GB and QUADRO K4000 3GB GPUs have been used for hardware acceleration.
DATA ANALYSIS AND RESULTS
Experiments were conducted on three datasets: 1. UC Merced land use scenes, 2.
Brazilian Coffee scenes and 3. San Diego data. UC Merced dataset, as discussed in previous chapter, consists of 21 land-use classes selected from aerial orthoimagery. Each set contains 100 images measuring 256x256 pixels for each of the 21 categories as shown in Figure 6.
These classes include a variety of spatial patterns, some homogeneous on texture, some homogeneous on color and others not homogenous at all (Cusano, Napoletano, & Schettini, 2014). These diverse variations of the dataset make it a good experimental dataset. UC Merced dataset represents similar spatial characteristics to that of the ImageNet (Castelluccio et al., 2015). The Brazilian Coffee scenes (O. A. B. Penatti et al., 2015) dataset, on the other hand, include satellite images with an infra-red band, and these are less similar to that of ImageNet dataset. The dataset is categorized into Coffee and Non-Coffee scenes. Each image of Brazilian dataset measures 64x64 pixels. The San Diego dataset is specially downloaded from the United States Geological Society ("EarthExplorer," n.d.), earth explorer website, to verify the generalizability of the classifier. An aerial patch was selected (from earth explorer website) in a way that it consists of as many classes as possible from the UC Merced dataset.
UC Merced
The images from this dataset share many low-level features with that of ImageNet, and this is the prime reason for this dataset to perform consistently and exceptionally well while fine-tuning the pre-trained networks. Table 1. shows the classification accuracy levels of the proposed solutions. Fine-tuning GoogleNet gives an accuracy of 96%. AlexNet and CaffeNet also show accuracy levels of above 95%. Table 2, shown below, compares the prominent approaches that were built to achieve the state-of-the-art precision. Castelluccio et al., (2015) Table 1 are five-fold validations (80 -Train, 20 -Validate), this is done to compare the accuracy levels with the other state-of-the-art classifiers. Note that all the fine-tuning models show an accuracy level of 95% and above.
AlexNet, CaffeNet, and GoogleNet are fine-tuned with 25,000 iterations. To train a Support
Vector Machine (SVM), the output of the fully connected layer 7 is used as the input for the SVM in the case of AlexNet and CaffeNet, and the output of the penultimate layer of GoogleNet is used as the input for the SVM in the case of GoogleNet. Scikit-learn's SVM is used to achieve these accuracy levels. (1) (2) prior-art chapter for information on the approach used by these methods.
Brazilian Coffee Scenes
As discussed above these images are quite different from that of the training data (ImageNet) used to produce CaffeNet, AlexNet and GoogleNet models. The pretrained models usually take 256x256 size images as input, but the Brazilian Coffee Scenes dataset consists of 64x64 size images. Given the said variations, our approach is to test transfer learning methods on this dataset as this is a remote sensing dataset used for similar land classification purposes. Figure 7 shows some samples of Brazilian Coffee Scenes. The results shown below are obtained after five-fold cross validation. Table 3. CaffeNet feature vectors of FC7 layer trained on SVM shows the accuracy of 85.21%. The standings of these results are presented in Table 4 -Table referenced from Castelluccio et al., (2015) and Penatti et al., (2015). As discussed above, the San Diego dataset is obtained by choosing a random patch from the USGS Earth Explorer website. These images were manually segregated for testing the extent of generalizability of the classifier. The fine-tuned classifier models that report high accuracy levels in Table 1. are taken and tested on this San Diego dataset. The pre-trained models provided by Caffe are used to produce SVM results.
CNN
The models are tested on 661 un-seen, and unrelated San Diego images and the average accuracy results of fine-tuning and SVM approaches are reported in Table 5.
UC Merced data augmentation
The proposed approach also experiments on augmenting the dataset by rotating and cropping each of the images of the UC Merced dataset. The dataset is divided as 60% -training, 20% -validation and 20% -testing (testing the classifier after it is thoroughly trained) per class. Each of the UC Merced dataset images is rotated by +-5, 10, 30 and 40 degrees and added to the augmented dataset. A total of 11340 training images were generated, and these images were used as the training data for CaffeNet fine-tuning approach. Confusion matrix of the results is as shown in Figure 7. The accuracy of the resultant classifier is 85.71%. Some of the categories are closely related (from Figure 6 and Figure 10, we can say that class G -dense residential and class M -medium residential categories are closely related). These classes have subtle differences that are difficult to be differentiated even with the human eye. If these areas (class G and M, Figure 10) are combined, then the accuracy of the classifier on unseen UC Merced dataset boosts up to 89.52%. Figure 10. The confusion matrix of the classified unseen UC Merced testing dataset
To further investigate the data augmentation approach, classifiers are built using 80% of the augmented images for training and the remaining 20% for validation. The training dataset (UC Merced, after data augmentation) consists of a total of 15120 images. The resultant classifiers are tested on San Diego data to verify the generalizability feature. The highest accuracy achieved by this process is 88.07%
by GoogleNet as shown in the Table 6. We can observe from Table 6 that classifiers obtained with data augmentation perform consistently even on the unseen/ unrelated San Diego data. However, AlexNet shows a slight indifferent behavior when compared to CaffeNet or GoogleNet, as accuracy on San Diego data with augmentation does improve over the accuracy obtained with out data augmentation.
To verify this behavior, the classifier (obtained with augmentation) is tested on augmented San Diego dataset (augmented as described above), the average accuracy obtained by this process is 79.83%, which is slightly higher than 79.49% (accuracy w/o data augmentation). This experiment shows that the dataset augmentation by image rotation improves the generalizability of classifier built using CNNs. Figure 11. Error samples by the classifier (CaffeNet) built using the data augmentation approach (a) Airplane à Buildings ( proposed approach is a breakthrough as significant unlabeled remote sensing datasets can now be classified and categorized.
Adapting a deep pre-trained network and fine-tuning the network on a new dataset that has a limited number of labeled images to train quickly, learn and adjust the weights and biases of the network on the new dataset in effect delivers promising results. From Table 2, one can deduce that the other non-CNN methods are at a wide 6% below the proposed accuracy. The near perfect accuracy levels from Table 1 shows that this is the best bleeding edge solution to LULC problem. The next big challenge would be pixel level extraction of cars, trees and other prominent features from the remote sensing datasets. Drones have also become widespread resources for the aerial imagery, we intend to work on these images too in the future for LULC and pixel level image segmentation and feature extraction. Now that the land use and land cover classification is evaluated successfully, future plans are to extend the research to the extraction of trees from aerial/ remote sensing images using deep learning methods. Trees are significant and important features in outdoor scenes, and they can have a tremendous impact on simulation, training, and scientific modeling efforts. In particular, the relationships between tree placement and cultural features such as buildings in urban environments is important because they interact with people and architecture. Trees are important for urban planning applications because the shade cast by trees may impact the walkability of an environment, and also affect the amount of water and energy needed in desert cities. Tree placement is also important for ground based games or simulation applications for realism and immersion, and also because of the features occluded by trees and the cover they provide. In military or unmanned aerial system (UAS) sensor operator simulations, trees can provide cover and hide targets.
Accurate data on the placement of trees can be expensive to produce. Often tree cover can be estimated by extracting foliage from high-resolution Light Detection and Ranging (LiDAR) or stereophotogrammetry data, but trees change quickly compared to other elements like buildings, and it is currently difficult to maintain enough complete and current source material. Recent trends in sensors and commercial applications for satellite and aerial imagery have made orthophoto mosaics at resolutions of around one meter per pixel or less more accessible than ever before. Algorithmic solutions can place statistically likely trees in urban environments, but randomly placed trees may not capture the complicated relationships between cultural features like buildings and trees. There is a need for solutions that can algorithmically place trees guided by existing aerial photographs so that a procedurally generated 3D scene is visually consistent with orthophoto and other mapping data available for a view.
The primary challenge for tree extraction is that the range of colors in trees overlaps the range of colors for other vegetation such as grass. Texture attributes can improve the ability to separate, but the texture of foliage can be obscured by image under-sampling or compression artifacts. Shadows are often a good indication of the presence of a tall feature such as a tree, and shadows also appear to give an indication of the height and shape of a tree. However, in near-nadir viewpoints treetops, there may not be a clear separation between tree top and shadow. Trees are often surrounded by features which obscure or hide their shadow, and the foliage of trees is rough, and so branches of an individual tree cast shadows within the treetop itself. Future work would extend to the auto extraction of individual features from remote sensing images thereby segmenting, separating and representing different pixels of these features clearly to the human eye.
using this opportunity to express my sincere gratitude to my mentors Dr. John Femiani and Dr. Anshuman Razdan, who supported me throughout the course of the Masters thesis and without them, none of this could have happened. I am thankful for their aspiring guidance, friendly advice and invaluably constructive criticism during the status review meetings and the project work. I am sincerely grateful to them for sharing their truthful and illuminating views on some issues related to the research. I express my warm thanks for their constant support and guidance in this research. I would also like to thank Mr. Michael Katic a Senior Software Engineer at the I3DEA labs, ASU Polytechnic and all the people who provided me with the facilities being required to conduct the experiments at the I3DEA lab. Most importantly, none of this could have happened without my family. My mother who offered her encouragement through phone calls despite my limited devotion to correspondence. This dissertation stands as a testament to your unconditional love TABLES .................................................................................................................................... vi LIST OF FIGURES ................................................................................................................................ vii PREFACE........... ..................................................................................................................................... viii CHAPTER 1 INTRODUCTION ................. ................................................................................................. 1 2 BACKGROUND LITERATURE ........................................................................................ 6 3 METHODOLOGY .................. ............................................................................................. 12 4 DATA ANALYSIS AND RESULTS .................................................................................. 20 5 DISCUSSION ................... ...................................................................................................... 33 REFERENCES....... ............................................................................................................................... 35 APPENDIX A TREE EXTRACTION ........................................................................................................ 41 BIOGRAPHICAL SKETCH ................................................................................................................ 44 PREFACE As a beginning graduate student I had the opportunity to see Dr. John Femiani and Dr. Anshuman Razdan present the research work at the Image & 3D Data Exploitation and
Figure 1 .
1Left: A mathematical model of a biological neuron. Right: Sigmoid function A perceptron is an artificial neural network element that is analogous to a biological
Figure 2 .
2Left: A Multi layer perceptron with a single hidden layer Right: Model of a Convolutional Neural Network A typical neural network consists of one or two hidden layers of neurons feeding one another. Deep learning neural networks have many hidden layers. Before 2006, attempts at training deep architectures have not shown much success. A revolutionary work published in 2006-2007 by
These above said published works in 2006-07 have changed the course of Artificial Neural Networks history, and they stand as an inspiration for the later developments in
unrelated data. Catelluccio et al., show the results with five-fold validation (80 -train and 20-validate) without separating out any data for testing the resultant classifier (the resultant classifier is the final model after it has been trained). The proposed approach in this thesis extends it by using a testing dataset along with the training and validation dataset(60training, 20 -validate and 20 -testing) and also verifies the performance of the classifier on wholly unrelated data. This helps us in building a classifier that can categorize large amounts of unseen new data.Deep learning techniques have been very recently applied to the LULC problem, Lv et al.,(Lv et al., 2015) used deep belief networks for LULC on radar imagery. Unlike the proposed solution in this thesis, their approach is not applied to RGB imagery and it not convolutional.Panetti et al., (O. A. B. Penatti, Nogueira, & Santos, 2015) experimented with convolutional networks for aerial and remote sensing images, finding that a network trained for visual RGB data outperformed all other methods on the aerial images. However, low level descriptors, such as BIC (Border Interior Pixel Classification), proposed by Stehling et al.,(Stehling, Nascimento, & Falcão, 2002) outperformed the pre-trained convolutional neural networks on remote sensing images. Castillucio et al. (2015) improve on these results by retraining GoogleNet's classifier by fine-tuning on the same data to outperform all other approaches. The approach in this thesis has achieved similar results as Castillucio et al. (2015). Unaware of each other, our approach in this thesis and the work of Castillucio et al. were experimented at the same time. Romero et al., (2016) (Romero, Gatta, & Camps-Valls, 2016) very recently proposed an unsupervised deep feature extraction for remote sensing image classification. They suggest the use of greedy layer-wise unsupervised pre-training coupled with an algorithm for unsupervised learning of sparse features. They show an average accuracy of 74.34%, which is a decent accuracy for an unsupervised classifier. Pre-training approaches for unsupervised deep networks is currently active and an important research area. METHODOLOGY Convolutional Neural Networks (CNN) are made up of perceptron's that are biologically analogous to the neurons of a human brain. Each perceptron has learnable weights and biases. A dot product of the inputs and their corresponding weights is taken, and a non-linear function is applied to produce the corresponding output. The output of one perceptron is fed into the perceptron of another layer, and this process continues and during this process, the weights and biases of each layer are learned incrementally. The input and the output are the outer layers of a neural network. Each layer applies a function to the output of the previous layer. Hidden layers are the in between layers that do the computation. The hidden layers' job is to apply convolution and pooling operations alternatively and adjust the weights and biases of the network according to the input. In a regular neural network popularly called as Multi-Layered Perceptron (MLP), each perceptron is fully connected to all the perceptrons of the next layer as shown in Figure 2(Left). MLPs are manageable for small-scale images, for example, a 28x28 RGB image would require 28x28x3 weights per perceptron to be computed. The problem with MLP arises with a large scale image as a large number of weights have to be learned, for example, a 256x256 RGB image would need 256x256x3 weights to be computed, moreover one would want to have several such neurons, this will result in a humongous allocation of computational time and resources. Adding to that, learning so many parameters would lead to a problem called Over-fitting. Unlike regular MLPs, CNN's are biologically inspired variants of MLPs ("Convolutional Neural Networks (LeNet) -DeepLearning 0.1 documentation," n.d.). Convolutional Neural Networks have sparse connectivity, for example in Figure 2(Right) each perceptron takes input from three perceptrons from the previous layer. These group perceptrons are also called activations. CNN's take advantage of the input images by localizing the reception of features (features in image are non-dynamic,and are spatially close to each other). This process exploits the spatially-local correlated contiguous fields called receptive fields(Hubel & Wiesel, 1968) by enforcing a local connectivity pattern between neurons of adjacent layers ("Convolutional Neural Networks (LeNet) -DeepLearning 0.1 documentation," n.d.). Moreover, all neurons of a layer are identical to one another, except for their receptive fields, sharing the same weights. This reduces the number of weights to be learned. From the Hubel et al.,(Hubel & Wiesel, 1968) work on cat's visual cortex we know that convolutional neural network closely resembles the biological visual cortex, which is organized in layers composed of similar cells, with different receptive fields over the layers.Convolutional neural networks primarily use the following different types of layers:1. Input Layer: Contains the input image or the raw pixel values. It is the entry layer to all the other layers.2. Convolutional Layer: This layer computes the activations of perceptrons that are connected to the receptive fields of the previous layer. As discussed above, each perceptron is connected to a spatially local region of the input volume. The convolutional parameters of a layer include: a. Number of outputs, is the input for the next layer.b. Kernel size, controls the spatially local region of the input volume.c. Stride, the pixel skips of the sliding window.d. Padding, helps in sizing the layer.3. Pooling Layer: This layer is mainly used to resize and accumulate the spatial representations. For example, using a max() operation is called max pooling. It is quite common to sandwich a pooling layer between convolutional layers periodically. 4. Normalization Layer: Normalizes over local input regions which helps in generalization.5. Fully-connected Layer: These are typically the last couple of layers of the network.Perceptrons in a fully connected layer are fully connected to all activations of the previous layer. The difference between a fully connected layer and a convolution layer is that the perceptrons in the convolution layer are connected only to a local region in the input, whereas all the perceptrons in the fully connected layer are connected to all the perceptrons of the input (input to the fully connected layer).
Figure 4 .
4shows each of the 96 filters (edge detections) that are learned in the first layer of a CaffeNet. Dumitru et. al, (Dumitru Erhan, n.d. "Understanding Representations Learned in Deep Architectures -LISA -Publications -Aigaion 2.0," n.d.) propose a paper to find good qualitative interpretations of high-level features represented by CNN models.These representations are patterns that can be displayed and are meaningful to the human eye.
Figure 3 .
3The original CaffeNet architecture used in this work. fc8 layer from this architecture is modified (number of outputs is 21 instead of 1000) to classify 21 classes Alex Krizhecsky et al., (2012) (Krizhevsky, Sutskever, & Hinton, 2012) have trained a large, deep convolutional neural network to classify 1.2M high-resolution images in the ImageNet LSVRC-2010 contest into 1000 different classes and this model file is available at
Figure 4 .
4Typical looking filters of Conv1 layerGoogleNet, on the other hand, is as exactly as described by in Going Deeper with Convolutions paper. GoogleNet is rather a deep and complex network compared to the CaffeNet or the AlexNet. GoogleNet won the ImageNet Large-Scale Visual Recognition Challenge 2014 -ILSVRC14 challenge ("ImageNet Large Scale Visual Recognition Competition (ILSVRC)," n.d.). It is a 22-layer deep network, and according to Szegedy et al., the the primary hallmark of its architecture is the improved utilization of the computing resources inside the network, a network in network module derived from
Figure 5 .
5Inception module with dimension reductions These inception modules are stacked on one another, thus enabling higher layers to capture features of higher abstraction. For complete description of the 22-layer architecture, please refer to Szegedy et. al. (2014) (Szegedy et al., 2014).
1 .
1SVM (Support Vector Machines): Feature sets from the top most layers of Caffe, Alex or Google Net are taken and are trained on an external classifier (Support Vector Machine -SVM, in this case). The features that are borrowed are generally from fully connected layer features and activations of each perceptron of this layer is dependent on all of the perceptrons of the previous layer. This helps in nonlocalization of the classification. 2. Fine-tuning: Fine tune by replacing the last layer and updating the weights of the pretrained network by changing the base learning rate and the learning rate of the other layers to continue back propagation. The learning rate of other layers can be set to zero if the pre-trained network weights are to be untouched. The learning rates are adjusted in such a way that the last layers learn faster compared to that of the other layers.
Figure 6 .
6Example images associated with 21 land use categories in the UC Merced dataset.(1) Agricultural. (2) Airplane. (3) Baseballdiamond. (4) Beach. (5) Buildings. (6) Chaparral. (7) Denseresidential. (8) Forest. (9) Freeway. (10) Golfcourse. (11) Harbor. (12) Intersection. (13) Mediumresidential. (14) Mobilehomepark. (15) Overpass. (16) Parkinglot. (17) River. (18) Runway. (19) Sparseresidential. (20) Storagetanks. (21) Tenniscourt.
Figure 7 .
7Brazilian Coffee Scenes dataset showing coffee and non coffee samples In fine-tuning, each layer convolves into a set of filters as shown in Figure 1. Conv1 has 96 output features, Conv2-layer 2 has 256, Conv3-layer3 has 344 and so on. As the size of the original input (64x64) is comparatively smaller, the filters get nullified gradually as the features travel deep into the layers. Up-sampling helps in scalability and usage of these images for the experiment. Each image of the dataset has been up-sampled from 64x64 to 256x256 pixels using nearest neighbor resampling filter. Fine-tuning by GoogleNet after up-sampling gives the highest average accuracy of 94.1 when compared to the recent state-of-the-art results by Castellucio et. al., (2015) -GoogleNet/from-scratch. The transfer learning on SVM approach show decent results, as shown in
Figure 8 .
8Some of the error samples found on San Diego dataset using fine-tuning approach. (a) Residential à Mobilehomepark (b) Residential à Mobilehomepark (c) Freeway à Overpass (d) Parkinglot à Storagetanks.
Figure 9 .
9Some of the error samples found on San Diego dataset using SVM approach. (a) River à Golfcourse (b) River à Forest (c) Residential à Intersection (d) River àAgricultural.
, propose the highest accuracylevel with 97.1, a positive difference of 1.1 as compared to the highest accuracy (96)
reported by us.
CNN
Approach
Accuracy
AlexNet
Fine-tuning
svm
95.79
94.76
CaffeNet
Fine-tuning
svm
95.02
92.85
GoogleNet
Fine-tuning
svm
96.0
94.28
Table 1. Classification accuracies levels of the proposed solutions
All the validations that are shown in
b) Baseballdiamond à Sparseresidential (c) Freeway à Overpass (d) Parkinglot à HarborTable 6. Classification accuracies on the UC Merced and San Diego datasets with/ without dataset augmentation. Filter Visualizations: Filter visualization is a way of knowing whether a model learned the correct parameters or not by visual confirmation. Recent developments in filter visualizations ("Research Blog: Inceptionism: Going Deeper into Neural Networks,"n.d.) has been a huge help for us in determining the particular categories on which the GoogleNet model has to be further trained or fine-tuned.Figure12, shows two different classes of the UC Merced dataset (Harbor and Medium residential), (a) shows the original image, (b) shows the image as seen by the pre-trained GoolgeNet model, and (c) shows the image as seen by the fine-tuned GoogleNet model (as proposed in this thesis). The Harbor category is well trained, and its accuracy is close to 100%, whereas the Medium-residential class is often confused with other classes, such as Dense-residential, Sparse-residential and Mobile-home-park. If we compare the fine-tuned images ((c)) of both the categories we can see extra artifacts in the Medium-residential visualization (roof tops as human faces, over-fitting) when compared to that of Harbor. We can deduce by looking at these visualizations that the fine-tuned GoogleNet model is well trained on Harbor (as it shows less/ nil artifacts), but the model needs further fine-tuning/ training on the Mediumresidential category.Figure 12. Filter visualizations of Harbor(top) and Medium-residential(bottom)DISCUSSIONDeep learning has had a transformative effect on computer vision. The proposed approach in this thesis has shown that it can be applied to remote sensing applications for automatic LULC classification from VHR images. The approach demonstrated that networks trained on an unrelated image recognition task can actually be used to solve the LULC classification problem. One would anticipate that a large amount of VHR spatial imagery that already exists and that continues to be collected at higher rates will have a significant impact on a variety of remote sensing applications. The proposed approach has shown two transfer learning methods, 1. Fine-tuning and 2. Feature vector on SVM. Both the methods show accuracies that are at par with the state-of-the-art accuracies on the LULC classification problem. The approach also deduced that these methods are consistent on a similar type of the datasets as the original dataset used for training (ImageNet), this is shown by experimenting with Brazilian Coffee Scenes dataset. The San Diego data classification clearly shows us that the proposed classifiers generalize well on completely unseen data. ThisCNN
Dataset
With Augmentation
Without
Augmentation
AlexNet
UC Merced
San Diego
80.71
78.4
95.79
79.49
CaffeNet
UC Merced
San Diego
82.54
81.63
95.02
79.08
GoogleNet
UC Merced
San Diego
88.58
88.07
96.0
85.59
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he will graduate with a Master of Science in Software Engineering degree, with a focus in Deep Learning Techniques. Before ASU, Nagesh worked as a Software/Application Developer at Tata Elxsi and Oracle Corporation companies for more than five years. Nagesh received his Bachelor's degree in Computer Science and Engineering from Visvesvaraya. K Nagesh, Uba, his 2nd year of study in the Software Engineering. Nagpur -IndiaProgram at the Arizona State University ; National Institute of TechnologyPolytechnic Campus. While at the ASU, he has served as a Teaching Assistant in the Department of CIDSE at the ASU. PolytechnicNagesh K Uba is currently in his 2nd year of study in the Software Engineering Program at the Arizona State University, Polytechnic Campus. In May 2016, he will graduate with a Master of Science in Software Engineering degree, with a focus in Deep Learning Techniques. Before ASU, Nagesh worked as a Software/Application Developer at Tata Elxsi and Oracle Corporation companies for more than five years. Nagesh received his Bachelor's degree in Computer Science and Engineering from Visvesvaraya National Institute of Technology, Nagpur -India. While at the ASU, he has served as a Teaching Assistant in the Department of CIDSE at the ASU, Polytechnic.
|
[] |
[
"Generalization of the Secant Method for Nonlinear Equations (extended version)",
"Generalization of the Secant Method for Nonlinear Equations (extended version)"
] |
[
"Avram Sidi [email protected] \nComputer Science Department\nTechnion -Israel Institute of Technology\n32000HaifaIsrael\n"
] |
[
"Computer Science Department\nTechnion -Israel Institute of Technology\n32000HaifaIsrael"
] |
[
"Original version appeared in: Applied Mathematics E-Notes"
] |
The secant method is a very effective numerical procedure used for solving nonlinear equations of the form f (x) = 0. It is derived via a linear interpolation procedure and employs only values of f (x) at the approximations to the root of f (x) = 0, hence it computes f (x) only once per iteration. In this note, we generalize it by replacing the relevant linear interpolant by a suitable (k + 1)-point polynomial of interpolation, where k is an integer at least 2. Just as the secant method, this generalization too enjoys the property that it computes f (x) only once per iteration. We provide its error in closed form and analyze its order of convergence s k . We show that this order of convergence is greater than that of the secant method, and it increases towards 2 as k → ∞. (Indeed, s 7 = 1.9960 · · · , for example.) This is true for the efficiency index of the method too. We also confirm the theory via an illustrative example.
| null |
[
"https://arxiv.org/pdf/2012.04248v1.pdf"
] | 15,158,018 |
2012.04248
|
e02cbee57a0b8b43bee930b6908a0da23e0d7c14
|
Generalization of the Secant Method for Nonlinear Equations (extended version)
2008
Avram Sidi [email protected]
Computer Science Department
Technion -Israel Institute of Technology
32000HaifaIsrael
Generalization of the Secant Method for Nonlinear Equations (extended version)
Original version appeared in: Applied Mathematics E-Notes
82008
The secant method is a very effective numerical procedure used for solving nonlinear equations of the form f (x) = 0. It is derived via a linear interpolation procedure and employs only values of f (x) at the approximations to the root of f (x) = 0, hence it computes f (x) only once per iteration. In this note, we generalize it by replacing the relevant linear interpolant by a suitable (k + 1)-point polynomial of interpolation, where k is an integer at least 2. Just as the secant method, this generalization too enjoys the property that it computes f (x) only once per iteration. We provide its error in closed form and analyze its order of convergence s k . We show that this order of convergence is greater than that of the secant method, and it increases towards 2 as k → ∞. (Indeed, s 7 = 1.9960 · · · , for example.) This is true for the efficiency index of the method too. We also confirm the theory via an illustrative example.
Introduction
Let α be the solution to the equation
f (x) = 0.(1)
An effective iterative method used for solving (1) that makes direct use of f (x) [but no derivatives of f (x)] is the secant method that is discussed in many books on numerical analysis. See, for example, Atkinson [1], Henrici [7], Ralston and Rabinowitz [11], and Stoer and Bulirsch [14]. See also the recent note [12] by the author, in which the treatment of the secant method and those of the Newton-Raphson, regula falsi, and Steffensen methods are presented in a unified manner. This method is derived by a linear interpolation procedure as follows: Starting with two initial approximations x 0 and x 1 to the solution α of (1), we compute a sequence of approximations {x n } ∞ n=0 , such that the approximation x n+1 is determined as the point of intersection (in the x-y plane) of the straight line through the points (x n , f (x n )) and (x n−1 , f (x n−1 )) with the x-axis. Since the equation of this straight line is
y = f (x n ) + f (x n ) − f (x n−1 ) x n − x n−1 (x − x n ),(2)
x n+1 is given as
x n+1 = x n − f (x n ) f (x n ) − f (x n−1 ) x n − x n−1 .(3)
In terms of divided differences, (3) can be written in the form
x n+1 = x n − f (x n ) f [x n , x n−1 ] .(4)
Again, in terms of divided differences, the error in x n+1 is given as in
x n+1 − α = f [x n , x n−1 , α] f [x n , x n−1 ] (x n − α)(x n−1 − α).(5)
Then, provided f (x) is twice continuously differentiable in a closed interval I containing α in its interior, and provided x n−1 , x n ∈ I, (5) becomes
x n+1 − α = f ′′ (ξ n ) 2f ′ (η n ) (x n − α)(x n−1 − α),
ξ n ∈ int(x n , x n−1 , α), η n ∈ int(x n , x n−1 ).
In case, f ′ (α) = 0 and x 0 and x 1 are sufficiently close to α, there holds lim n→∞ x n = α, and hence lim n→∞
x n+1 − α (x n − α)(x n−1 − α) = f ′′ (α) 2f ′ (α) .
From this, one derives the conclusion that the order of convergence of the secant method is at least (1 + √ 5)/2.
Another way of obtaining the secant method, of interest to us in the present work, is via a variation of the Newton-Raphson method. Recall that in the Newton-Raphson method, we start with an initial approximation x 0 and generate a sequence of approximations {x n } ∞ n=0 to α through
x n+1 = x n − f (x n ) f ′ (x n ) , n = 0, 1, . . . .(7)
We also recall that, when f (x) is twice continuously differentiable in a closed interval I that includes α, and f ′ (α) = 0, this method has order 2. As such, the Newton-Raphson method is extremely effective . To avoid computing f ′ (x) [note that f ′ (x) may not always be available or may be costly to compute], and to preserve the excellent convergence properties of the Newton-Raphson method, we replace f ′ (x n ) in (7) by the approximation
f [x n , x n−1 ] = [f (x n ) − f (x n−1 )]/(x n − x n−1 )
. This results in (4), that is, in the secant method. The justification for this approach is as follows: When convergence takes place, that is, when lim n→∞ x n = α, the difference x n − x n−1 tends to zero, and this implies that, as n increases, the accuracy of f [x n , x n−1 ] as an approximation to f ′ (x n ) increases as well.
In Section 2 of this note, we consider in detail a generalization of the second of the two approaches described above using polynomial interpolation of degree k with k > 1. The (k + 1)-point iterative method that results from this generalization turns out to be very effective. It is of order higher than that of the secant method and requires only one function evaluation per iteration. In Section 3, we analyze this method and determine its order as well. In Section 4, we confirm our theory via a numerical example. This paper is a slightly extended version of the paper [13]. The original version concerns the case in which f (k+1) (α) = 0, while the extension concerns the special case in which f (k+1) (α) = 0, which always occurs when f (x) is a polynomial of degree at most k. In addition, we include a brief discussion of the efficiency index for our method as Section 5.
Generalization of secant method
We start by discussing a known generalization of the secant method (see, for example, Traub [15,Chapters 4,6,and 10]). In this generalization, we approximate f (x) by the polynomial of interpolation p n,k (x), where p n,k (x i ) = f (x i ), i = n, n − 1, . . . , n − k, assuming that x 0 , x 1 , . . . , x n have all been computed. Following that, we determine x n+1 as a zero of p n,k (x), provided a real solution to p n,k (x) = 0 exists. Thus, x n+1 is the solution to a polynomial equation of degree k. For k = 1, what we have is nothing but the secant method. For k = 2, x n+1 is one of the solutions to a quadratic equation, and the resulting method is known as the method of Müller. Clearly, for k ≥ 3, the determination of x n+1 is not easy.
This difficulty prompts us to consider the second approach to the secant method we discussed in Section 1, in which we replaced f ′ (x n ) by the slope of the straight line through the points (x n , f (x n )) and (x n−1 , f (x n−1 )), that is, by the derivative (at x n ) of the (linear) interpolant to f (x) at x n and x n−1 . We generalize this approach by replacing f ′ (x n ) by p ′ n,k (x n ), the derivative at x n of the polynomial p n,k (x) interpolating f (x) at the points x n−i , i = 0, 1, . . . , k, mentioned in the preceding paragraph, with k ≥ 2. Because p n,k (x) is a better approximation to f (x) in the neighborhood of x n , p ′ n,k (x n ) is a better approximation to f ′ (x n ) when k ≥ 2 than f [x n , x n−1 ] used in the secant method. In addition, just as the secant method, the new method computes the function f (x) only once per iteration step, the computation being that of f (x n ). Thus, the new method is described by the following (k + 1)-point iteration:
x n+1 = x n − f (x n ) p ′ n,k (x n ) , n = k, k + 1, . . . ,(8)
with x 0 , x 1 , . . . , x k as initial approximations to be provided by the user. Of course, with k fixed, we can start with x 0 and x 1 , compute x 2 via the method we have described (with k = 1, namely via the secant method), compute x 3 via the method we have described (with k = 2), and so on, until we have completed the list x 0 , x 1 , . . . , x k . We now turn to the computational aspects of this method. What we need is a fast method for computing p ′ n,k (x n ). For this, we write p n,k (x) in Newtonian form as follows:
p n,k (x) = f (x n ) + k i=1 f [x n , x n−1 , . . . , x n−i ] i−1 j=0 (x − x n−j ).(9)
Here
f [x i , x i+1 , . . . , x m ] are divided differences of f (x)
, and we recall that they can be defined recursively via
f [x i ] = f (x i ); f [x i , x j ] = f [x i ] − f [x j ] x i − x j , x i = x j ,(10)
and, for m > i + 1, via
f [x i , x i+1 , . . . , x m ] = f [x i , x i+1 , . . . , x m−1 ] − f [x i+1 , x i+2 , . . . , x m ] x i − x m , x i = x m .(11)
We also recall that f [x i , x i+1 , . . . , x m ] is a symmetric function of its arguments, that is, it has the same value under any permutation of {x i , x i+1 , . . . , x m }. Thus, in (9),
f [x n , x n−1 , . . . , x n−i ] = f [x n−i , x n−i+1 , . . . , x n ].
In addition, when f ∈ C m (I), where I is an open interval containing the points z 0 , z 1 , . . . , z m , whether these are distinct or not, there holds
f [z 0 , z 1 , . . . , z m ] = f (m) (ξ) m! for some ξ ∈ (min{z i }, max{z i }).
Going back to (9), we note that p n,k (x) there is computed by ordering the x i as x n , x n−1 , . . . , x n−k . This ordering enables us to compute p ′ n,k (x n ) easily. Indeed, differentiating p n,k (x) in (9), and letting x = x n , we obtain
p ′ n,k (x n ) = f [x n , x n−1 ] + k i=2 f [x n , x n−1 , . . . , x n−i ] i−1 j=1 (x n − x n−j ).(12)
In addition, note that the relevant divided difference table need not be computed anew each iteration; what is needed is adding a new diagonal (from the south-west to north-east) at the bottom of the existing table. To make this point clear, let us look at the following Table 1: Table of (8).
x 0 f 0 f 01 x 1 f 1 f 012 f 12 f 0123 x 2 f 2 f 123 f 23 f 1234 x 3 f 3 f 234 f 34 f 2345 x 4 f 4 f 345 f 45 f 3456 x 5 f 5 f 456 f 56 f 4567 x 6 f 6 f 567 f 67 x 7 f 7divided differences over {x 0 , x 1 , . . . , x 7 } for use to compute x 8 via p 7,3 (x) inNote that f i,i+1,...,m stands for f [x i , x i+1 , . . . , x m ] throughout.
example: Suppose k = 3 and we have computed x i , i = 0, 1, . . . , 7. To compute x 8 , we use the divided difference table in Table 1.
Letting f i,i+1,...,m stand for f [x i , x i+1 , . . . , x m ],
we have
x 8 = x 7 − f (x 7 ) p 7,3 (x 7 ) = x 7 − f 7 f 67 + f 567 (x 7 − x 6 ) + f 4567 (x 7 − x 6 )(x 7 − x 5 )
.
To compute x 9 , we will need the divided differences f 8 , f 78 , f 678 , f 5678 . Computing first f 8 = f (x 8 ) with the newly computed x 8 , the rest of these divided differences can be computed from the bottom diagonal of Table 1 via the recursion relations (8), after x n has been determined, we need to store only the entries f n , f n−1,n , . . . , f n−k,n−k+1,...,n−1,n along with x n , x n−1 , . . . , x n−k .
f 78 = f 7 − f 8 x 7 − x 8 , f 678 = f 67 − f 78 x 6 − x 8 , f 5678 = f 567 − f 678 x 5 − x 8 ,
Convergence analysis
We now turn to the analysis of the sequence {x n } ∞ n=0 that is generated via (8). Since we already know everything concerning the case k = 1, namely, the secant method, we treat the case k ≥ 2. The following theorem gives the main convergence result for the generalized secant method. I is an open interval containing α, and assume also that f ′ (α) = 0, in addition to f (α) = 0. Let x 0 , x 1 , . . . , x k be distinct initial approximations to α, and generate x n , n = k+1, k+2, . . . , via
x n+1 = x n − f (x n ) p ′ n,k (x n ) , n = k, k + 1, . . . ,(13)
where p n,k (x) is the polynomial of interpolation to f (x) at the points x n , x n−1 , . . . , x n−k . Then, provided x 0 , x 1 , . . . , x k are in I and sufficiently close to α, we have the following cases:
1. If f (k+1) (α) = 0, the sequence {x n } converges to α, and
lim n→∞ ǫ n+1 k i=0 ǫ n−i = (−1) k+1 (k + 1)! f (k+1) (α) f ′ (α) ≡ L; ǫ n = x n − α ∀n.(14)
The order of convergence is s k , 1 < s k < 2, where s k is the only positive root of the equation s k+1 = k i=0 s i and satisfies
2 − 2 −k−1 e < s k < 2 − 2 −k−1 for k ≥ 2; s k < s k+1 ; lim k→∞ s k = 2,(15)
where e is the base of natural logarithms, and
lim n→∞ |ǫ n+1 | |ǫ n | s k = |L| (s k −1)/k .(16)
2. If f (x) is a polynomial of degree at most k, the sequence {x n } converges to α, and
lim n→∞ ǫ n+1 ǫ 2 n = f ′′ (α) 2f ′ (α) ; ǫ n = x n − α ∀n.(17)
Thus {x n } converges of order 2 if f ′′ (α) = 0, and of order greater than 2 if f ′′ (α) = 0.
Remark. Note that, in part 1 of Proof. Below, we shall use the short-hand notation int(a 1 , . . . , a m ) = (min{a 1 , . . . , a m }, max{a 1 , . . . , a m }).
We start by deriving a closed-form expression for the error in x n+1 . Subtracting α from both sides of (13), and noting that
f (x n ) = f (x n ) − f (α) = f [x n , α](x n − α),
we have
x n+1 − α = 1 − f [x n , α] p ′ n,k (x n ) (x n − α) = p ′ n,k (x n ) − f [x n , α] p ′ n,k (x n ) (x n − α).(18)
We now note that
p ′ n,k (x n ) − f [x n , α] = p ′ n,k (x n ) − f ′ (x n ) + f ′ (x n ) − f [x n , α] ,
which, by
f ′ (x n ) − f [x n , α] = f [x n , x n ] − f [x n , α] = f [x n , x n , α](x n − α) = f (2) (η n ) 2! (x n − α) for some η n ∈ int(x n , α), and f ′ (x n ) − p ′ n,k (x n ) = f [x n , x n , x n−1 , . . . , x n−k ] k i=1 (x n − x n−i ) = f (k+1) (ξ n ) (k + 1)! k i=1 (x n − x n−i ) for some ξ n ∈ int(x n , x n−1 , . . . , x n−k ), (19) becomes p ′ n,k (x n ) − f [x n , α] = − f (k+1) (ξ n ) (k + 1)! k i=1 (ǫ n − ǫ n−i ) + f (2) (η n ) 2! ǫ n .(20)
Substituting (19) and (20) in (18), and letting
D n = − f (k+1) (ξ n ) (k + 1)! and E n = f (2) (η n ) 2! ,(21)
we finally obtain
ǫ n+1 = C n ǫ n ; C n ≡ p ′ n,k (x n ) − f [x n , α] p ′ n,k (x n ) = D n k i=1 (ǫ n − ǫ n−i ) + E n ǫ n f ′ (x n ) + D n k i=1 (ǫ n − ǫ n−i ) .(22)
We now prove that convergence takes place. Let M s = max x∈I |f (s) (x)|/s!, s = 1, 2, . . . , and choose the interval I = (α − t/2, α + t/2) sufficiently small to ensure that m 1 = min x∈I |f ′ (x)| > 0 and m 1 > 2M k+1 t k + M 2 t/2. This is possible since α ∈ I and f ′ (α) = 0. It can now be shown that, provided x n−i , i = 0, 1, . . . , k, are all in I, there holds
|C n | ≤ M k+1 k i=1 |ǫ n − ǫ n−i | + M 2 |ǫ n | m 1 − M k+1 k i=1 |ǫ n − ǫ n−i | ≤ M k+1 k i=1 (|ǫ n | + |ǫ n−i )| + M 2 |ǫ n | m 1 − M k+1 k i=1 (|ǫ n | + |ǫ n−i |) ≤ C, where C ≡ M k+1 t k + M 2 t/2 m 1 − M k+1 t k < 1.
Consequently, by (22), |ǫ n+1 | < |ǫ n |, which implies that x n+1 ∈ I, just like x n−i , i = 0, 1, . . . , k. Therefore, if x 0 , x 1 , . . . , x k are chosen in I, then |C n | ≤ C < 1 for all n ≥ k, hence {x n } ⊂ I and lim n→∞ x n = α. As for (14) when f (k+1) (α) = 0, we proceed as follows: By the fact that lim n→∞ x n = α, we first note that
lim n→∞ p ′ n,k (x n ) = f ′ (α) = lim n→∞ f [x n , α],
and thus lim n→∞ C n = 0. This means that lim n→∞ (ǫ n+1 /ǫ n ) = 0 and, equivalently, that {x n } converges of order greater than 1. As a result, lim n→∞ (ǫ n /ǫ n−i ) = 0 for all i ≥ 1, and ǫ n /ǫ n−i = o(ǫ n /ǫ n−j ) as n → ∞, for j < i.
Consequently, expanding in (22) the product
k i=1 (ǫ n − ǫ n−i ), we have k i=1 (ǫ n − ǫ n−i ) = k i=1 − ǫ n−i [1 − ǫ n /ǫ n−i ] = (−1) k k i=1 ǫ n−i [1 + O(ǫ n /ǫ n−1 )] as n → ∞.(23)
Substituting (23) in (22), and defining
D n = D n p ′ n,k (x n ) , E n = E n p ′ n,k (x n ) ,(24)
we obtain
ǫ n+1 = (−1) k D n k i=0 ǫ n−i [1 + O(ǫ n /ǫ n−1 )] + E n ǫ 2 n as n → ∞.(25)
Dividing both sides of (25) by k i=0 ǫ n−i , and defining
σ n = ǫ n+1 k i=0 ǫ n−i ,(26)
we have σ n = (−1) k D n [1 + O(ǫ n /ǫ n−1 )] + E n σ n−1 ǫ n−k−1 as n → ∞.
Now,
lim n→∞ D n = − 1 (k + 1)! f (k+1) (α) f ′ (α) , lim n→∞ E n = f (2) (α) 2f ′ (α) .(28)
Because lim n→∞ D n and lim n→∞ E n are finite, lim n→∞ (ǫ n /ǫ n−1 ) = 0, and lim n→∞ ǫ n−k−1 = 0, it follows that there exist a positive integer N and positive constants β < 1 and D, with |E n ǫ n−k−1 | ≤ β when n ≥ N, for which (27) gives
|σ n | ≤ D + β|σ n−1 | for all n ≥ N.(29)
Using (29), it is easy to show that
|σ N +s | ≤ D 1 − β s 1 − β + β s |σ N |, s = 1, 2, . . . ,
which, by the fact that β < 1, implies that {σ n } is a bounded sequence. Making use of this fact, we have lim n→∞ E n σ n−1 ǫ n−k−1 = 0. Substituting this in (27), and invoking (28), we next obtain lim n→∞ σ n = (−1) k lim n→∞ D n = L, which is precisely (14). That the order of the method is s k , as defined in the statement of the theorem, follows from [15,Chapter 3]. A weaker version can be proved by letting σ n = L for all n and showing that |ǫ n+1 | = Q|ǫ n | s k is possible for s k a solution to the equation s k+1 = k i=0 s i and Q = |L| (s k −1)/k . The proof of this is easy and is left to the reader. This completes the proof of part 1 of the theorem.
When f (x) is a polynomial of degree at most k, we first observe that f (k+1) (x) = 0 for all x, which implies that p n,k (x) = f (x) for all x, hence also p ′ n,k (x) = f ′ (x) for all x. Therefore, we have that p ′ n,k (x n ) = f ′ (x n ) in the recursion of (13). Consequently, (13) becomes
x n+1 = x n − f (x n ) f ′ (x n ) , n = k, k + 1, . . . ,
which is the recursion for the Newton-Raphson method. Thus, (17) follows. This completes the proof of part 2 of the theorem.
A numerical example
We apply the method described in Sections 2 and 3 to the solution of the equation f (x) = 0, where f (x) = x 3 − 8, whose solution is α = 2. We take k = 2 in our method. We also chose x 0 = 0 and x 1 = 6, and compute x 2 via one step of the secant method, namely,
x 2 = x 1 − f (x 1 ) f [x 0 , x 1 ] .(30)
Following that, we compute x 3 , x 4 , . . . , via
x n+1 = x n − f (x n ) f [x n , x n−1 ] + f [x n , x n−1 , x n−2 ](x n − x n−1 )
, n = 2, 3, . . . .
Our computations were done in quadruple-precision arithmetic (approximately 35decimal-digit accuracy), and they are given in Table 2. Note that in order to verify the n x n ǫ n ǫ n+1 ǫ n ǫ n−1 ǫ n−2 log |ǫ n+1 /ǫ n | log |ǫ n /ǫ n−1 | 0 5.00000000000000000000000000000000000D + 00 3.000D + 00 1 4.00000000000000000000000000000000000D + 00 2.000D + 00 1.515 2 3.08196721311475409836065573770491792D + 00 (30) and (31), to the equation
x 3 − 8 = 0.
theoretical results concerning iterative methods of order greater that unity, we need to use computer arithmetic of high precision (preferably, of variable precision, if available) because the number of correct significant decimal digits increases dramatically from one iteration to the next as we are approaching the solution. From Theorem 3.1,
lim n→∞ ǫ n+1 ǫ n ǫ n−1 ǫ n−2 = (−1) 3 3! f (3) (2) f ′ (2) = − 1 12 = −0.08333 · · ·
and lim n→∞ log |ǫ n+1 /ǫ n | log |ǫ n /ǫ n−1 | = s 2 = 1.83928 · · · , and these seem to be confirmed in Table 2. Also, x 9 should have a little under 50 correct significant figures, even though we do not see this in Table 2 due to the fact that the arithmetic we have used to generate Table 2 can provide an accuracy of at most 35 digits approximately.
Discussion of efficiency index of the method
We recall that, for methods that converge superlinearly, that is, with order strictly greater than 1, a good measure of their effectiveness is the so-called efficiency index, a concept introduced originally by Ostrowski [9]. (See Traub [15, pp. 11-13, 260-264] for more on this subject.) If an iterative method for solving f (x) = 0 that requires p evaluations of f (x) (and its derivatives, assuming that their cost is about the same), has order s > 1, the efficiency index EI of the method is defined as EI = s 1/p . Figuratively speaking, EI measures the order of the method per function evaluation. Thus, we may conclude that, the larger EI, the more effective the iterative method, irrespective of its order. In comparing methods, we should examine their performance after we have done a fixed number of function evaluations, this number being the same for all methods. In other words, it makes sense to compare methods that have equal costs. The details of this line of thought follow: Consider two iterative methods M1 and M2 applied to the equation f (x) = 0, and let m 1 and m 2 be the number of function evaluations per iteration for M1 and M2, respectively. Starting with x (1) 0 = x (2) 0 , let the sequences of approximations {x (1) n } ∞ n=0 and {x (2) n } ∞ n=0 be generated by M1 and M2, respectively. Then, for each integer q = 1, 2, . . . , we should compare the approximations x
(q−1)m 1 . In a fundamental paper by Kung and Traub [8], it is conjectured that the order of a multipoint iterative method without memory that uses p function evaluations may not exceed 2 p−1 . This paper contains two such families that use p function evaluations and are of order 2 p−1 . Woźniakowski [16] has proved for some classes of multipoint iterative methods without memory that the order 2 p−1 cannot be exceeded without more information. From this, it is clear that the efficiency index of such methods is at most 2 (p−1)/p = 2 1−1/p < 2. In view of this discussion, we make a few comments on the efficiency index of our method next.
The efficiency index of the generalized secant method developed in this paper is EI k = s k for each k = 1, 2, . . . , because p = 1 for every k. In addition, because lim k→∞ s k = 2, we have lim k→∞ IE k = 2 as well. Actually, even with very small k, we are able to come quite close to this limit. For example, s 7 = 1.9960 · · · and s 10 = 1.9995 · · · .
Over the years, many sophisticated iterative methods with and without memory that do not use derivatives of f (x) and that have high orders have been developed. It is not our purpose here to review these methods; we refer the reader to the papers by Džunić [4], Džunić and Petković [5], [6], Chun and Neta [2], [3], and to the book by Petković et al. [10], for example, and to the bibliographies of these publications. We only would like to comment that the many methods that we have studied have efficiency indices that are strictly less than 2 despite their high order. This may suggest that the method of this paper may be as useful a tool for solving nonlinear equations with simple zeros as other methods that have orders much higher than 2.
Theorem 3. 1
1Let α be the solution to the equation f (x) = 0. Assume f ∈ C k+1 (I), where
1 . Note that, the computation of x
2 entails m 1 m 2 function evaluations and so does the computation of x
in this order, and appended to the bottom ofTable 1. Actually, we can do even better: Since we need only the bottom diagonal ofTable 1to compute x 8 , we need to save only this diagonal,namely, only the entries f 7 , f 67 , f 567 , f 4567 . Once we have computed x 8 and f 8 = f (x 8 ),
we can overwrite f 7 , f 67 , f 567 , f 4567 with f 8 , f 78 , f 678 , f 5678 . Thus, in general, to be able to
compute x n+1 via
rounded to four significant figures. (Recall that s 1 is the order of the secant method.)Theorem 3.1,
s 1= 1.618, s 2= 1.839, s 3= 1.928, s 4= 1.966, s 5= 1.984, s 6= 1.992, s 7= 1.996, etc.
Table 2 :
2Results obtained by applying the generalized secant method with k = 2, as shown in
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Generalization of the secant method for nonlinear equations. A Sidi, Appl. Math. E-Notes. 8A. Sidi. Generalization of the secant method for nonlinear equations. Appl. Math. E-Notes, 8:115-123, 2008.
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Iterative Methods for the Solution of Equations. J F Traub, Prentice HallEnglewood Cliffs, N.J.J.F. Traub. Iterative Methods for the Solution of Equations. Prentice Hall, Englewood Cliffs, N.J., 1964.
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|
[] |
[
"Maximum Rectilinear Crossing Number of Uniform Hypergraphs",
"Maximum Rectilinear Crossing Number of Uniform Hypergraphs"
] |
[
"Rahul Gangopadhyay \nSaint Petersburg State University\nRussia\n",
"Saif Ayan Khan [email protected] \nSaint Petersburg State University\nRussia\n"
] |
[
"Saint Petersburg State University\nRussia",
"Saint Petersburg State University\nRussia"
] |
[] |
We improve the lower bound on the d-dimensional rectilinear crossing number of the complete d-uniform hypergraph having 2d vertices to Ω(2 d d) from Ω(2 d √ d). We also establish that the 3-dimensional rectilinear crossing number of a complete 3-uniform hypergraph having n ≥ 9 vertices is at least 43 42 2 d , where n denotes the number of vertices in each part. We then prove that finding the maximum d-dimensional rectilinear crossing number of an arbitrary d-uniform hypergraph is NP-hard and give a randomized scheme to create a d-dimensional rectilinear drawing of a d-uniform hypergraph H producing the number of crossing pairs of hyperedges up to a constant factor of the maximum d-dimensional rectilinear crossing number of H.
| null |
[
"https://arxiv.org/pdf/1908.04654v5.pdf"
] | 199,551,801 |
1908.04654
|
7860baeb0c1d36dd546e6d5520181b8e5bdd7c71
|
Maximum Rectilinear Crossing Number of Uniform Hypergraphs
23 Mar 2022
Rahul Gangopadhyay
Saint Petersburg State University
Russia
Saif Ayan Khan [email protected]
Saint Petersburg State University
Russia
Maximum Rectilinear Crossing Number of Uniform Hypergraphs
23 Mar 2022‡ Corresponding Author § IIIT-Delhi, India. 2 R. Gangopadhyay et al.Rectilinear Crossing Number · Gale Transform · Moment Curve · NP-Hard
We improve the lower bound on the d-dimensional rectilinear crossing number of the complete d-uniform hypergraph having 2d vertices to Ω(2 d d) from Ω(2 d √ d). We also establish that the 3-dimensional rectilinear crossing number of a complete 3-uniform hypergraph having n ≥ 9 vertices is at least 43 42 2 d , where n denotes the number of vertices in each part. We then prove that finding the maximum d-dimensional rectilinear crossing number of an arbitrary d-uniform hypergraph is NP-hard and give a randomized scheme to create a d-dimensional rectilinear drawing of a d-uniform hypergraph H producing the number of crossing pairs of hyperedges up to a constant factor of the maximum d-dimensional rectilinear crossing number of H.
, and proved this conjecture for d = 3. It is trivially true for d = 2, since any convex drawing of the complete graph Kn produces n 4 pairs of crossing edges. We prove that their conjecture is true for d = 4 by proving that in a 4-dimensional rectilinear drawing of a complete 4-uniform hypergraph having n vertices, the maximum number of crossing pairs of hyperedges is 13 n 8 .
We use Gale transform to prove this result. In fact, we prove a stronger statement. We prove that among all 4-dimensional rectilinear drawings of a complete 4-uniform hypergraph having n vertices, the number of crossing pairs of hyperedges is maximized if and only if all its vertices are placed as the vertices of a 4-dimensional neighborly polytope. We also prove that the maximum d-dimensional rectilinear crossing number of a complete d-partite d-uniform balanced hypergraph is (2 d−1 − 1) n
Introduction
A rectilinear drawing of a graph in R 2 represents its vertices as points in general position, i.e., no three points are colinear, and its edges as straight line segments between the corresponding vertices. In a rectilinear drawing of a graph, a pair of edges are said to be crossing if they are vertex disjoint and contain a common point in their relative interiors. The rectilinear crossing number of a graph G, denoted by cr(G), is the minimum number of crossing pairs of edges in any rectilinear drawing of it. A convex drawing of a graph G is a rectilinear drawing of it where vertices are in a convex position in R 2 . There are other variants of graph crossing numbers which are comprehensively discussed in [19]. Most of the crossing number problems deal with the minimization of crossing in a specific drawing of the graph. Ringel [18] introduced the maximum rectilinear crossing number problem for a graph G, being the maximum number of crossing pairs of edges among all rectilinear drawings of G. Verbitsky [20] gave an approximation algorithm, which in expectation provides a 1/3 approximation guarantee on the maximum rectilinear crossing number problem. The same paper showed that the maximum rectilinear crossing number of a planar graph having n vertices is less than 3n 2 . Bald et al. [7] de-randomized Verbitsky's algorithm and showed that it is NPhard to find the maximum crossing number of an arbitrary graph.
A hypergraph, a natural generalization of a graph, is defined as an ordered pair (V, E) where V is the set of vertices and E ⊆ 2 V \ {∅} is the set of hyperedges. A hypergraph is said to be d-uniform if each hyperedge contains exactly d vertices. Let K d n denote the complete d-uniform hypergraph having n vertices and n d hyperedges. We can partition the vertex set of a d-uniform d-partite hypergraph into d disjoint parts such that each of the d vertices in each hyperedge belongs to a different part and it is balanced if each of the parts has the same number of vertices. A balanced d-uniform d-partite hypergraph having n vertices in each part is complete if it has all n d hyperedges and it is denoted by K d d×n . In [10], Dey and Pach extended the idea of a rectilinear drawing of a graph to a rectilinear drawing of a hypergraph. Consider a set of P having n ≥ d + 1 points in R d . The points in P are said to be in general position if no set of d + 1 points of P lie on a (d − 1)-dimensional hyperplane. The points in P are in convex position if there does not exist any point in P such that it can be expressed as the convex combination of the rest of the points in P . In a d-dimensional rectilinear drawing of a d-uniform hypergraph H, the vertices of H are placed in general position in R d and the hyperedges are drawn as the convex hull of d corresponding vertices, i.e. (d − 1)-simplices. In a d-dimensional rectilinear drawing of H, two hyperedges are said to cross each other if they are vertex disjoint and contain a common point in their relative interiors [6,10]. The d-dimensional rectilinear crossing number of H, denoted by cr d (H), is the minimum number of crossing pairs of hyperedges among all d-dimensional rectilinear drawings of H [6]. Dey and Pach [10] proved that H can have at most O(n d−1 ) hyperedges if cr d (H) = 0. The first non-trivial lower bound of Ω(2 d log d/ √ d) on cr d (K d 2d ) was proved by Anshu and Shannigrahi [6]. Anshu et al. [5] proved that cr d (K d 2d ) = Ω(2 d ) with the bound being later improved to Ω(2 d √ d) [12]. In Section 2, we further improve this bound to Ω(2 d d). We then show that
cr 3 (K 3 n ) ≥
43 42 n 6 when n ≥ 9.
A d-dimensional convex drawing of a d-uniform hypergraph H is a d-dimensional rectilinear drawing of it where all its vertices are in convex position as well as in general position in R d .
In this paper, we define the maximum d-dimensional rectilinear crossing number of a d-uniform hypergraph H, denoted by max-cr d (H), as the maximum number of crossing pairs of hyperedges among all d-dimensional rectilinear drawings of H.
The d-dimensional moment curve γ is defined as γ = {(t, t 2 , . . . , t d ) : t ∈ R}. Let p i = (t i , t 2 i , . . . , t d i ) and p j = (t j , t 2 j , . . . , t d j ) be two points on γ. We say that the point p i precedes the point p j (p i ≺ p j ) if t i < t j .
Consider a set of P having n ≥ d + 1 points in convex position in R d . Let us assume that the affine hull of the points in P is the entire space R d . The convex hull of the points in P is a d-dimensional convex polytope, and it is denoted by Conv(P ). Note that the points in P are the vertices of Conv(P ). A d-dimensional convex polytope is k-neighborly if any subset of its vertex set containing at most k vertices forms a face of it. A d-dimensional convex polytope can be at most d/2 -neighborly unless it is a d-simplex. A d-dimensional d/2 -neighborly polytope is called d-dimensional neighborly polytope. The d-dimensional cyclic polytope is an example of d-dimensional neighborly polytope where all of its vertices are placed on γ.
Anshu et al. [5] proved that placing all the vertices of a K d 2d as the vertices of a d-dimensional cyclic polytope gives rise to a particular d-dimensional rectilinear drawing of K d 2d having c m d crossing pairs of hyperedges, where c m d is defined as follows.
c m d = 2d − 1 d − 1 − d 2 i=1 d i d − 1 i − 1 if d is even 2d − 1 d − 1 − 1 − d 2 i=1 d − 1 i d i if d is odd
In [5], it was conjectured that the maximum number of crossing pairs of hyperedges in any d-dimensional convex drawing of K d 2d is c m d for each d ≥ 2. As mentioned in the abstract, this is evident for d = 2. In [5], the authors also proved that a 3-dimensional rectilinear drawing of K 3 6 can have at most 3 crossing pairs of hyperedges, implying that K 3 n can have at most 3 n 6 crossing pairs of hyperedges in any 3-dimensional rectilinear drawing of it. They also showed that any 3-dimensional convex drawing of K 3 6 has 3 crossing pairs of hyperedges. In Section 4, we prove this conjecture for d = 4 by proving that max-cr 4 (K 4 n ) = 13 n 8 .
Note that we need at least 2d vertices to form a crossing pair of hyperedges since they need to be vertex disjoint, and each set of 2d vertices creates distinct crossing pairs of hyperedges. If placing the vertices of K d 2d on γ maximizes the number of crossing pair of hyperedges in a d-dimensional rectilinear drawing of it, then max-cr d (K d n ) = c m d n 2d since every set of 2d vertices on γ spans c m d crossing pairs of hyperedges. It is natural to ask about the the maximum d-dimensional rectilinear crossing number of K d d×n . In Section 5, we prove that max-cr d (K d d×n ) = (2 d−1 − 1). In Section 6, we prove that for d ≥ 3, finding the maximum d-dimensional rectilinear crossing number of an arbitrary d-uniform hypergraph is NP-hard. Since this problem is NP-hard, we propose a randomized approximation algorithm, which in expectation gives a constantc d approximation guarantee on the maximum d-dimensional rectilinear crossing number problem. The constantc d is dependent on d.
2 Improved Lower Bound on cr d (K d 2d )
In this section, we improve the lower bound on cr d (K d 2d ) to Ω 2 d d . For n ≥ 9, we improve the currently best-known lower bound on the 3-dimensional rectilinear crossing number of K 3 n by proving Theorem 2.
In order to improve the lower bound on cr d (K d 2d ), we need the following two lemmas.
Lemma 1. [12, Proof of Theorem 1] Let C be a set containing d + 4 points in general position in R d . There exist at least (d + 4)/2 pairs of disjoint subsets {C i1 , C i2 } of C for each i satisfying 1 ≤ i ≤ (d + 4)/2 such that the following properties hold.
1. |C i1 | = u i , |C i2 | = v i . 2. C i1 ∪ C i2 = C and u i , v i ≥ (d + 2)/2 3. (u i − 1)-simplex formed by the Conv(C i1 ) crosses the (v i − 1)-simplex formed by the Conv(C i2 ) (i.e., C i1 ∩ C i2 = ∅ and Conv(C i1 ) ∩ Conv(C i2 ) = ∅).
Lemma 2. [12]
Consider a set C that contains 2d points in general position in R d . Let C ⊂ C be a subset such that |C | = d + 4. Let C 1 and C 2 be two disjoint subsets of C such that |C 1 | = c 1 , |C 2 | = c 2 , C 1 ∪ C 2 = C and c 1 , c 2 ≥ (d + 2)/2 . If the (c 1 − 1)-simplex formed by C 1 and the (c 2 − 1)-simplex formed by C 2 form a crossing pair, then the (d−1)-simplex formed by a point set B 1 ⊃ C 1 and the (d − 1)-simplex formed by a point set B 2 ⊃ C 2 satisfying B 1 ∩ B 2 = ∅, |B 1 |, |B 2 | = d and B 1 ∪ B 2 = C also form a crossing pair.
Theorem 1. cr d (K d 2d ) = Ω 2 d d .
Proof. Let V = {v 1 , v 2 , . . . , v 2d } denote the set of 2d points corresponding to the vertices of K d 2d in a d-dimensional rectilinear drawing of it. Let E denote the set of (d − 1)-simplices created by the corresponding hyperedges of K d 2d in that particular d-dimensional rectilinear drawing of it. Let V be any subset of V having d + 4 points. Lemma 1 implies that there exist (d + 4)/2 pairs of subsets {V i1 , V i2 } for each i satisfying 1 ≤ i ≤ (d + 4)/2 such that the following properties hold.
1. |V i1 | = u i , |V i2 | = v i . 2. V i1 ∪ V i2 = V and u i , v i ≥ (d + 2)/2 3. (u i −1)-simplex formed by the Conv(V i1 ) crosses the (v i −1)-simplex formed by the Conv(V i2 ) (i.e., V i1 ∩ V i2 = ∅ and Conv(V i1 ) ∩ Conv(V i2 ) = ∅).
It follows from Lemma 2 that each such crossing pair of (u i − 1)-simplex
and (v i − 1)-simplex can be extended to obtain at least d − 4 d − (d + 2)/2 = Ω 2 d / √
d crossing pairs of (d − 1)-simplices formed by the hyperedges in E. Therefore, the total number of crossing pairs of hyperedges, originated from a particular choice of V , in a d-dimensional rectilinear drawing of K d 2d is at least
(d + 4)/2 Ω 2 d / √ d = Ω 2 d √ d . We can choose V in 2d d + 4 = Θ 4 d / √ d ways. For each choice of V , there exist Ω 2 d √ d crossing pairs of hyperedges in a d-dimensional recti- linear drawing of K d 2d .
On the other hand, note that the same crossing pair of hyperedges may originate from the different choices of subsets having d + 4 points from V . Given a crossing pair of hyperedges, we obtain an upper bound on the number of subsets having d + 4 points from V such that this particular crossing pair of hyperedges originated from them. Note that if d is odd, a particular crossing pair of hyperedges can originate from at most 2 In the following, we state two lemmas that are used to improve the currently best-known lower bound on the 3-dimensional rectilinear crossing number of K 3 n when n ≥ 9. Proof. Let V = {v 1 , v 2 , . . . , v 9 } denote the set of 9 points corresponding to the vertices of K 3 9 in a 3-dimensional rectilinear drawing of it. Lemma 3 implies that in such a 3-dimensional rectilinear drawing of K 3 9 there exist 6 points which are in general as well as convex position in R 3 . Let us consider the sub-hypergraph H of K 3 9 induced by the 6 vertices corresponding to these points. Note that H is isomorphic to K 3 6 . Lemma 4 implies that H contains 3 crossing pairs of hyperedges. Also, note that there are 9 6 distinct sub-hypergraphs of K 3 9 which are isomorphic to K 3 6 . Lemma 4 also implies that each of these 9 6 distinct subhypergraphs contains at least 1 crossing pair of hyperedges and one of them, i.e., H contains 3 crossing pairs of hyperedges. Also, note that the crossing pairs of hyperedges spanned by one set of 6 vertices are distinct from the crossing pairs of hyperedges spanned by another set of 6 vertices. The total number of crossing pairs of hyperedges in a 3-dimensional rectilinear drawing of K 3 9 is at least 9 6 − 1 + 3 = 86. This implies that cr 3 (K 3 9 ) ≥ 86. Consider a 3-dimensional rectilinear drawing of K 3 n where n ≥ 3. Note that K 3 n contains n 9 distinct induced sub-hypergraphs, each of which is isomorphic to K 3 9 . Also, note that each crossing pair of hyperedges is contained in n − 6 3 distinct induced sub-hypergraphs which are isomorphic to K 3 9 . Using these two facts, we obtain that cr 3 K 3 n ≥ 86
d (d+2)/2 d (d+2)/2 +2 + d (d+2)/2 d (d+2)/2 +2 = Θ 4 d /d such d+4 sized subsets of V . If d is
Gale transform and Gale Diagram
We use Gale transform [11] and Gale diagram to prove that the maximum 4dimensional rectilinear crossing number of K 4 n is 13
n 8 .
In this section, we describe Gale transform and Gale diagram of a point set and discuss their properties. Let A = a 1 , a 2 , . . . , a n be a sequence of n points in R d such that their affine hull is R d . The Gale transform of A, denoted by D(A), is a sequence of n vectors g 1 , g 2 , . . . , g n in R n−d−1 .
Let the coordinate of a i be (x i 1 , x i 2 , . . . , x i d ). Let us consider the following matrix M (A).
M (A) = x 1 1 x 2 1 · · · x n 1 x 1 2 x 2 2 · · · x n 2 . . . . . . . . . . . . x 1 d x 2 d · · · x n d 1 1 · · · 1
Since at least d + 1 points of A are affinely independent, the dimension of the This lemma implies that the Gale transform D(A) of A is a totally cyclic vector configuration, and there is a positive dependence among the vectors of D(A). This also implies that there does not exist a hyperplane, passing through the origin, such that all the vectors of D(A) lie on one side of the hyperplane [21]. Note that any totally cyclic vector configuration of n vectors in R n−d−1 that span R n−d−1 can serve as a Gale transform of some point set having n points in R d after proper scaling. We consider the points in A to be in general position. Let D(A) be a Gale transform of A. Due to the general position of the points in A, Lemma 6 implies that none of the vectors in D(A) is equal to zero vector, i.e., ∀i g i = 0.
null space of M (A) is n − d − 1. Let {(b 1 1 , b 1 2 , . . . , b 1 n ), (b 2 1 , b 2 2 , . . . , b 2 n ), . . . , (b n−d−1 1 , b n−d−1
We obtain an affine Gale diagram of A, denoted by D(A), from D(A) by considering a hyperplaneh that is not parallel to any vector in D(A) and not passing through the origin.
For each 1 ≤ i ≤ n, we extend the vector g i ∈ D(A) either in the direction of g i or in its opposite direction until it cutsh at the point g i . We color g i as red (denoted as triangle in Figure 1) if the projection is in the direction of g i , and blue (denoted as squares in Figure 1) otherwise. D(A), the affine Gale diagram of A, is the sequence of n points g 1 , g 2 , . . . , g n in R n−d−2 along with their respective colors.
We Definition 1 (Balanced 2m-partition). Let T be a set of n red and n blue points in R 2 such that all the 2n points are in general position. A balanced 2mpartition of T is partition of it into {X, T \ X} such that the following properties hold.
-The size of the set X is 2m. -X can be separated from T \ X that contains rest of the (2n − 2m) points by a line not passing through any point of T . -X is balanced, i.e., it has an equal number of red and blue points.
Since we are considering distinct balanced 2m-partitions,i.e., the complimentary pairs {X, T \ X}, we only consider them for 1 ≤ m ≤ n/2 . We define a balanced 0-partition to be a partition of T into an empty set and T . Note that there is only one balanced 0-set of a set.
Definition 2 (Monochromatic k-partition). Let T be a set of n red and n blue points in R 2 such that all the 2n points are in general position. A monochromatic k-partition of T is partition of it into {Q, T \ Q} such that the following properties hold.
-The size of the set Q ⊆ T is k.
-Q can be separated from T \ Q that contains rest of the (2n − k) points by a line not passing through any point of T . -Q is monochromatic, i.e., all the points in Q are of the same color.
Properties of the Gale diagram of 8 points in R 4
As already mentioned in the introduction, in order to prove that max-cr 4 (K 4 n ) = 13 n 8 , it is enough to show that placing the vertices of a K 4 8 on the 4-dimensional moment curve maximizes the number of crossing pair of hyperedges among all 4-dimensional rectilinear drawing of it.
Gale diagram maps a configuration of 8 points in R 4 to a configuration of 8 points in a plane along with some color associated with them. We then analyse these planar point sets to prove the desired result.
Consider a set A of 8 points in general position in R 4 . Consider a Gale transform of A , denoted by D(A ), which is a collection of 8 vectors in R 3 . Let us denote an affine Gale diagram of A by D(A ). In the following, we discuss a few properties of D(A ). Definition 3 (Order-type). Consider a sequence of points S = s 1 , s 2 , . . . , s n where points are in general position in R 2 . The order-type of s is a mapping which assigns an orientation (clockwise or counter-clockwise) to each ordered triple s i , s j , s k .
Consider two sequences of points S = s 1 , s 2 , . . . , s n and S = s 1 , s 2 , . . . , s n in R 2 , such that the points in both the sequences are in general position. S and S are said to have same order-type if for any indices i < j < k the orientation of s i , s j , s k is same as the orientation of s i , s j , s k .
Suppose that two sequences of points S = s 1 , s 2 , . . . , s 2n and S = s 1 , s 2 , . . . , s 2n in R 2 have same order-type. Consider a coloring C where n points of S are colored red, and rest of the n points are colored blue. The indices of red-colored points are also the same in S and S , implying that the indices of blue-colored points are also same. For each tuple (i 1 , i 2 , . . . , i 2m ), where 1 ≤ i 1 < i 2 < . . . < i 2m ≤ 2n, {s i1 , s i2 , . . . , s i2m } is a balanced 2m-set of S if and only if {s i1 , s i2 , . . . , s i2m } is a balanced 2m-set of S [21]. There are infinitely many point configurations having n points in general position in R 2 . There are only finitely many order-types for such point configurations. We can think of order-types as equivalence classes. The point configurations that have the same order type share many combinatorial and geometric properties. Consider a 4-dimensional neighborly polytope P having n vertices such that all the vertices of P are in general position in R 4 . Consider a 4-dimensional rectilinear drawing of K 4 n such that the vertices of K 4 n are placed as the vertices of P . Consider any subset P of the vertex set of P having size 8. It is easy to see that the 4-dimensional polytope spanned by the vertices of P is also a neighborly polytope. This implies that in such a drawing every copy of
Maximum Rectilinear Crossing Number of complete d-partite d-uniform Hypergraph
In this section, we prove that max-cr d (K d d×n ) = (2 d−1 − 1) are distinct from the crossing pairs of hyperedges spanned by another copy of
K d d×2 , this implies that max-cr d (K d d×n ) = (2 d−1 − 1) n 2 d .
In the follwoing, we state three lemmas which are used in the proof of Lemma 13.
Lemma 10. [8]
Let p 1 ≺ p 2 ≺ . . . ≺ p d 2 +1 and q 1 ≺ q 2 ≺ . . . ≺ q d 2 +1 be two distinct point sequences on the d-dimensional moment curve such that p i = q j for any 1 ≤ i ≤ d 2 + 1 and 1 ≤ j ≤ d 2 + 1. The d 2 -simplex and the d 2simplex, formed respectively by these point sequences, cross if and only if every interval (q j , q j+1 ) contains exactly one p i and every interval (p i , p i+1 ) contains exactly one q j . Lemma 11. [10] Let P and Q be two vertex-disjoint (d − 1)-simplices such that each of the 2d vertices belonging to these simplices lies on the d-dimensional moment curve. If P and Q cross, then there exists a d 2 -simplex U P and another d 2 -simplex V Q such that U and V cross. We prove Lemma 13 in the following. Let us consider a particular d-dimensional rectilinear drawing of K d d×2 that achieves the above mentioned bound. In this particular drawing, the vertices of K d d×2 are placed on the d-dimensional moment curve such that they satisfy the
ordering p c1 ≺ p c1 ≺ p c2 ≺ p c2 . . . ≺ p c d−1 ≺ p c d−1 ≺ p c d ≺ p c d .
Without loss of generality, let us assume that for any unordered pair {A, B}, A contains the first vertex, i.e., p c1 . Given an unordered pair {A, B}, the vertices of A create d partitions of the d-dimensional moment curve. We call each partition a bucket. Note that the points on the d-dimensional moment curve which precede p c1 are not part of any bucket. Let b i denote the i th bucket. Note that the last bucket has only one endpoint created by the last vertex (according to the order mentioned above) of A and contains all the points over the d-dimensional moment curve which succeed the last vertex of A. Since, both A and B contain exactly one vertex from each part of the vertex set, the following properties hold.
-The first bucket contains either one vertex or two vertices of B, but it can never be empty. Note that to avoid such an alternating chain of d + 2 vertices exactly d/2 buckets should be empty since every bucket can contain at most two vertices of B and all the d vertices of B should be partitioned into the d buckets. Also, note that any two non-empty buckets are not consecutive, and the first bucket is not empty.
When d is even, this implies that each of the odd-numbered buckets contains two vertices, and even-numbered buckets are empty. The only unordered pair {A, B} such that the (d − 1)-simplex formed by the vertices of A does not form a crossing with the (d − 1)-simplex formed by the vertices of B is the following.
A = {p c1 , p c2 , p c3 , p c4 , . . . , p c d−1 , p c d }, B = {p c1 , p c2 , p c3 , p c4 , . . . , p c d−1 , p c d }.
When d is odd, the last bucket should contain exactly one vertex of B. Otherwise, we can form a alternating chain of d + 2 vertices since at least d/2 + 1 of the first d − 1 buckets are non-empty. This implies that for odd d, all the even numbered buckets are empty and each of the odd-numbered buckets contains two vertices except the last bucket which contains one vertex. The only unordered pair {A, B} such that the (d − 1)-simplex formed by the vertices of A does not form a crossing with the (d − 1)-simplex formed by the vertices of B is the following. A = {p c1 , p c2 , p c3 , p c4 , . . . , p cd−2 , p cd−1 , p cd }, B = {p c1 , p c2 , p c3 , p c4 , . . . , p cd−2 , p cd−1 , p cd }.
Theorem 4. max-cr d (K d d×n ) = (2 d−1 − 1) n 2 d .
Proof. For each i satisfying 1 ≤ i ≤ d, let C i denote the i th partition of the vertex set of K d d×n . Let {p i 1 , p i 2 , . . . , p i n } denotes the set of n vertices in C i . Consider the following arrangement of the vertices of K d d×n on the d-dimensional moment curve.
-Any vertex of C i precedes any vertex of C j if i < j.
-For each i satisfying 1 ≤ i ≤ d, p i l ≺ p i m if l < m.
Consider any induced sub-hypergraph of K d d×n which is isomorphic to K d d×2 . In this particular d-dimensional rectilinear drawing of K d d×n , the vertices of the sub-hypergraph follow the same ordering mentioned in the proof of Lemma 13, implying that each of them contains 2 d−1 −1 crossing pairs of hyperedges and the
maximum d-dimensional rectilinear crossing number of K d d×n is (2 d−1 − 1) n 2 d .
On the Maximum Rectilinear Crossing Number of General Hypergraphs
In this section, we turn our focus on finding the Maximum d-dimensional rectilinear crossing number of an arbitrary d-uniform hypergraph H. Given H and an integer l, we show that it is NP-hard to find if there exists a d-dimensional rectilinear drawing D of H having at least l crossing pairs of hyperedges. We reduce MAX-E K -set splitting problem, which is known to be NP-Hard to our problem. Given a K-uniform hypergraph H = (V , E ) and an integer c, the decision version of MAX-E K -set splitting asks whether there exists a partition of V into two parts such that at least c hyperedges of E contain at least one vertex from both the parts.
Lovász [17] proved that given a K-uniform hypergraph H = (V , E ), deciding whether H is 2-colorable is N P -hard when K ≥ 3. For K ≥ 3, this problem is a special case of the decision version of MAX-E K -set splitting where c = |E |. This implies that for K ≥ 3, the decision version of MAX-E K -set splitting problem is also N P -hard.
Note that the MAX-E 2 -set splitting problem is the same as the Max-Cut problem. It is extensively studied in the literature and is known to be N P -hard.
Theorem 5. For d ≥ 3, finding the maximum d-dimensional rectilinear crossing number of an arbitrary d-uniform hypergraph is NP-hard.
Proof. We are given a d-uniform hypergraph H = (V, E) and a constant integer c . We create a d-uniform hypergraphH = (Ṽ ,Ẽ), wherẽ We prove thatH has a d-dimensional rectilinear drawing D having at least tc crossing pairs of hyperedges if and only if there exists a partition of V into two parts such that at least c hyperedges of E contains at least one vertex from both the parts.
V = V ∪ {v 0 , v 1 , v 2 , . . . , v t(d−1) } where t = |E| 2 + 1. E = ∪ i {e i } ∪ E where e i = v 0 , v (i−1)(d−1)+1 , v (i−1)(d−1)+2 , . . . , v (i−1)(d−1)+(d−1) for each i satisfying 1 ≤ i ≤ t. v'0 v'5 v'6 v'4 v'2 v'3 v'1
Let us assume that there exists a partition of V into two parts V 1 and V 2 such that (at least) c hyperedges of E contain at least one vertex from both the parts. Let us denote these hyperedges as cut-hyperedges. We produce a drawing D ofH having at least tc crossing pairs of hyperedges.
Let h be a (d−1)-dimensional hyperplane. We place the points corresponding to the vertices in V 1 and the points corresponding to the vertices in V 2 in general position in R d such that they lie on the different open half-spaces created by h. The hyperedges in E are drawn as the (d − 1)-simplices spanned by the d points corresponding to its vertices. Note that each of the cut-hyperedges has a nontrivial intersection with h. We then create the t hyperedges e 1 , e 2 , . . . , e t . Note that these t hyperedges can not form crossing with each other since each of them contains a common vertex v 0 .
We put the d vertices {v 0 , v 1 , v 2 , . . . , v d−1 } of e 1 on h such that they are in general position with the rest of the points in R d and the convex hull of these d points crosses each of the cut-hyperedges. Note that it is always possible to create such a placement of points since there are only a finite number of cuthyperedges. Note that the position of the vertex v 0 is fixed after the placement of the vertices of e 1 . We then add the other d − 1 vertices of e 2 very close to the d − 1 vertices of e 1 such that they, along with the other vertices, maintain the general position and the (d − 1)-simplex corresponding to the hyperedge e 2 crosses each of the cut-hyperedges. In this way, we keep on adding the vertices of each e i in a very close neighborhood of each other such that they do not violate the general position assumption and each (d − 1)-simplex corresponding to each e i crosses the same number of cut-hyperedges.
Note that in this d-dimensional rectilinear drawing D ofH (as depicted in Figure 4) each of the cut-hyperedges forms a crossing with each e i for 1 ≤ i ≤ t. This implies that there exist at least tc crossing pairs of hyperedges in D.
On the other hand, let us assume thatH has a d-dimensional rectilinear drawing D having at least tc crossing pairs of hyperedges. Suppose each e i crosses at most (c −1) hyperedges of E. Then, the maximum number of crossing
pairs of hyperedges in D is (c − 1)t + |E| 2 < (c − 1)t + t = c t.
This implies that one of the e i must cross at least c hyperedges of E. W.l.o.g suppose that e 1 crosses at least c hyperedges of E. Consider the hyperplane h spanned the d vertices of e 1 , i.e., the affine hull of the points {v 0 , v 1 , v 2 , . . . , v d−1 }.
Consider the partition of V created by h . This implies that there exists a partition of V into two parts V 1 and V 2 such that (at least) c hyperedges of E contain at least one vertex from both the parts. Consider any neighborly d-polytope whose vertices are in general position in R d . Since the vertices are in general position, this class of neighborly polytopes are simplicial. This class of neighborly polytopes have the same f -vectors as the cyclic polytopes [21]. We conjecture that among all d-dimensional rectilinear drawings of K d n , the number of crossing pairs of hyperedges gets maximized if all the vertices of K d n are placed in general position in R d as the vertices of a neighborly d-polytope (whose vertices are in general position). Note that a ddimensional cyclic polytope is also a neighborly polytope with vertices in general position.
It is interesting to come up with a traditional proof of Theorem 3 and Lemma 9. Note that we perform an exhaustive search among all realizable ordertypes of eight points in general position in R 2 . Goodman and Pollack [13] proved that the lower bound on the number of the realizable order-types of n points in general position in R d is n d 2 n+O(n/ log n) . This implies that our method is not effective in higher dimension. Further, we want to ask a more general question in this area. Consider a d-dimensional convex drawing of complete d-uniform hypergraph having 2d vertices. Note that the convex hull of the vertices of K d 2d in a d-dimensional convex drawing of it, is a convex d-polytope. As our results indicate, the convex d-polytopes wih maximum number of facets also maximize the number of crossing pairs of hyperedges. It is an interesting problem to find out the relation between the number of crossing pairs of hyperedges in a d-dimensional convex drawing of K d 2d and the number of facets of the corresponding polytope. Guy [15] noted that in a rectilinear drawing of a complete graph, the number of crossing pairs of edges is minimum when the convex hull of its vertices forms a triangle. Aichholzer et al. [3] proved this claim rigorously using continuous motion of the vertices. It is a nice problem to prove that the convex hull of the vertices of K d n in a d-dimensional recctilinear drawing of it is a d-simplex if the number of crossing pairs of hyperedges is minimum.
Theorem 6 shows that there is a randomized approximation algorithm which in expectation provides ac d guarantee on the maximum d-dimensional rectilinear crossing number problem. It is an interesting open problem to derandomize such algorithm. For d = 2, Bald et al. [7] derandomized the algorithm. Note thatc d is a constant for a given d. It is easy to observe thatc d is upper bounded by 1/2. It would be good to give a lower bound onc d . Our guess isc d ≥ 3/10.
Appendix
Here we give our programs that were used to prove Theorem 3. The source code contains multiple files. This has been done to make it more readable and modular. All files are accessible at https://github.com/ayan-iiitd/maximum-rectilinearcrossing-number-of-uniform-hypergraphs.git . Implementation details are provided in the 'README' available in the aforementioned github repository. # # As an output 3315 csv files are generated each in the following formatimport ast import datetime import os import pandas import subprocess from itertools import combinations , islice # # Data is saved in the same format as before , just we only save only the coloring and its respective configuration for which the total number of balanced colors is 12.
n 6 .
6Anshu et al. [Anshu et al., 2017] conjectured that among all d-dimensional convex drawings of a complete d-uniform hypergraph having n vertices, the number of crossing pairs of hyperedges is maximized if all its vertices are placed on the d-dimensional moment curve. They denoted this number by c m d n 2d
Lemma 3 .
3[9] Every set of 9 points in general position in R 3 contains a subset of 6 points that are in convex position.Note that Lemma 3 is the 3-dimensional analogue of the Erdős-Szekeres theorem.
Lemma 4 .
4[5] The number of crossing pairs of hyperedges in a 3-dimensional rectilinear drawing of K 3 6 is at least 1. The number of crossing pairs of hyperedges in a 3-dimensional rectilinear drawing of K 3 6 is 3 if its vertices are in convex as well as general position in R 3 .Theorem 2. For n ≥ 9, cr 3
be a basis of the null space of M (A). The vector g i in the sequence D(A) of n vectors is g i = (b 1 i , b 2 i , . . . , b separation of vectors in D(A) is a partition of the vectors into D + (A) and D − (A) by a hyperplane passing through the origin. The opposite open half-spaces of the partitioning hyperplane contain the sets D + (A) and D − (A). When |D(A)| is even, a linear separation is called proper if |D + (A)| = |D − (A)| = |D(A)|/2. In the following, we state some interesting properties of the Gale transform of A.
Lemma 5 .
5[16] A sequence D(A) = g 1 , g 2 , . . . , g n of n vectors in R n−d−1 is a Gale transform of some n points in R d if and only if the vectors in D(A) span R n−d
Lemma 6 .
6[16] Every set of n − d − 1 vectors of D(A) span R n−d−1 if and only if the points in A are in general position in R d . Lemma 7. [16] Consider a tuple (i 1 , i 2 , . . . , i k ), where 1 ≤ i 1 < i 2 < . . . < i k ≤ n. The convex hull of {a i1 , a i2 , . . . , a i k } crosses the convex hull of A \ {a i1 , a i2 , . . . , a i k } if and only if there exists a linear separation of the vectors in D(A) into {g i1 , g i2 , . . . , g i k } and D(A) \ {g i1 , g i2 , . . . , g i k }.
Lemma 8 .
8[14] Let the points in A be in general as well convex position inR d . A d-dimensional polytope formed by the convex hull of the points in A is tneighborly if and only if each of the linear separations of D(A) contains at least t + 1 points in each of the open half-spaces created by the corresponding linear hyperplane.
define a separation of the points in D(A) to be a partition of the points in D(A) into two disjoint sets of points D + (A) and D − (A) contained in the opposite open half-spaces created by a hyperplane.
Fig. 1 :
1An affine Gale diagram of 8 points in R 4 Let us define a Balanced 2m-partition for a planar point set having an equal number of blue and red points in general position in R 2 .
Observation 1
1There exists an affine Gale diagram D(A ) of A having 4 red points and 4 blue points in R 2 such that all the 8 points are in general position.Proof. Consider a Gale transform D(A ) of A which is a set of 8 vectors in R 3 . It is easy to note that there exists a 2-dimensional hyperplaneh passing through the origin that partition D(A ) into two equal parts D + (A ) and D − (A ), each having 4 vectors. Consider a hyperplane parallel toh and project the vectors in the way, as mentioned above. The Gale diagram D(A ) obtained in this way has 4 blue points and 4 red points in R 2 , as shown inFigure 1. Also, note that the points in D(A ) are in general position since no three of them are collinear. Thus if three points are collinear, it implies that the corresponding three vectors lie on a plane, which is a contradiction to Lemma 6 since the original points are in general position in R 4 .Observation 2 Consider the Gale diagram D(A ) having four red points and four blue points in R 2 such that all the 8 points are in general position. The total number of proper linear separations (i.e., partition of 8 vectors of D(A ) by a linear hyperplane into equal parts) in D(A ) is equal to the total number of balanced 2-partitions of D(A ) plus the total number of balanced 4-partitions of D(A ) plus 1. Proof. Consider any proper linear separation of vectors in D(A ) into D + (A ) and D − (A ). Note that this proper linear separation of vectors in D(A ) corresponds to a partition of points in D(A ) into D + (A ) and D − (A ) by a line (this line is the intersection of the separating hyperplane with the hyperplane on which we projected the vectors to obtain the affine Gale Diagram). Assume that there be r red points and b blue points in D + (A ). This implies there are 4 − r red points and 4 − b blue points in D − (A ). It is easy to note that the total number of vectors in D + (A ) is equal to the number of red points in D + (A ) plus the number of blue points in D − (A ). This implies that r + (4 − b) is equal to 4. This implies that r = b. This shows that each proper linear separation of vectors in D(A ) corresponds to a balanced 2m-set of D(A ) for some m. Similarly, each balanced 2m-set of D(A ) corresponds to a proper linear separation of vectors in D(A ). The above argument shows that the total number of balanced 2-partitions of D(A ) plus the total number of balanced 4-partitions of D(A ) plus the balanced 0-set of D(A ) is equal to the total number of proper linear separations in D(A ). Note that we have not included a balanced 6-set since each balanced 6-set is the same as a balanced 2-partition. Also, note that there is only one balanced 0-set of D(A ). This balanced 0-set of D(A ) corresponds to the proper linear separation of vectors in D(A ) which was used to obtain this Gale diagram. This proves that total number of balanced 2-partitions of D(A ) plus the total number of balanced 4-partitions of D(A ) plus 1 is equal to the total number of proper linear separations in D(A ).Observation 3 Consider the Gale diagram D(A ) having four red points and four blue points in R 2 such that all the 8 points are in general position. D(A ) is a Gale diagram of a 2-neighborly 4-dimensional polytope if and only if the following conditions hold. -Each 4-set in D(A ) is a balanced 4-partition. -Each 2-set in D(A ) is a balanced 2-partition. -There does not exist a monochromatic 3-partition in D(A ). Proof. Lemma 8 implies that D(A ) is a Gale transform of a 2-neighborly 4dimensional polytope if and only if each of the linear separations of D(A ) contains at least 3 vectors in each of the open half-spaces created by the corresponding linear hyperplane. Consider any linear separation of vectors in D(A ) into D + (A ) and D − (A ). Note that this linear separation of vectors in D(A ) corresponds to a partition of points in D(A ) into D + (A ) and D − (A ) by a line (this line is the intersection of the separating hyperplane with the hyperplane on which we projected the vectors to obtain the affine Gale Diagram) and vice versa. It is easy to note that the total number of vectors in D + (A ) is equal to the number of red points in D + (A ) plus the number of blue points in D − (A ). Similarly, total number of vectors in D − (A ) is equal to the number of red points in D − (A ) plus the number of blue points in D + (A ). (⇒) We first prove that if any of these three conditions mentioned above is violated, D(A ) is not a Gale transform of a 2-neighborly 4-dimensional polytope having 8 vertices. Case 1. For the sake of contradiction, let us assume that there exists a 4-set in D(A ) that is either monochromatic or contains three points of one color and one point of another color. Suppose it is monochromatic. Then, this implies that there exists a linear hyperplane such that all the vectors of D(A ) lie in the same open half-space created by it, leading to a contradiction. Without loss of generality, let us assume that D + (A ) contains 3 points of one color and 1 point of the other color. This implies that there exists a linear separation of D(A ) such that 6 vectors lie in the one side of the linear hyperplane and 2 vectors lie in the other side of the linear hyperplane. Lemma 8 implies that D(A ) is not a Gale transform of a 2 neighborly 4-dimensional polytope. Case 2. For the sake of contradiction, let us assume that the second condition is violated, i.e., there exists a monochromatic 2-set in D(A ). Without loss of generality, we assume that there exists a partition of points in D(A ) into D + (A ) and D − (A ) by a line such that D + (A ) contains 2 points and both the points in D + (A ) are of the same color. This implies that there exists a linear separation of D(A ) such that 6 vectors lie in the one side of the linear hyperplane and 2 vectors lie in the other side of the linear hyperplane, leading to a contradiction.
Case 3 .
3For the sake of contradiction, we assume that there exists a monochromatic 3-partition in D(A ). Without loss of generality, let us assume that D + (A ) contains 3 points having the same color. This implies that there exists a linear separation of D(A ) such that 7 vectors lie in the one side of the linear hyperplane and 1 vectors lie in the other side of the linear hyperplane, leading to a contradiction. (⇐)In the following, we prove that if none of these three conditions is violated, any linear separation of D(A ) contains at least 3 vectors in each of the open half-spaces created by the corresponding linear hyperplane. This implies that D(A ) is a Gale transform of a 2-neighborly 4-dimensional polytope having 8 vertices. Note that for each linear separation of vectors in D(A ), there exists a partition of points in D(A ) into D + (A ) and D − (A ). Let us assume that |D + (A )| = |D − (A )| = 4. Since each 4-set in D(A ) is a balanced 4-partition, any such partition corresponds to a proper linear separation of D(A ) as shown in the proof of Observation 2. Note that in any proper linear separation of D(A ) each open half-space contains 4 vectors of D(A ). Let us assume that |D + (A )| = 2 and |D − (A )| = 6. Since each 2-set in D(A ) is a balanced 2-partition, D − (A ) is also balanced. Any such partition corresponds to a proper linear separation of D(A ) as shown in the proof of Observation 2. Let us assume that |D + (A )| = 3 and |D − (A )| = 5. Since none of the 3sets in D(A ) is monochromatic, D + (A ) contains two points having the same color and one point having another color. Any such partition corresponds to a linear separation of D(A ) such that 5 vectors lie in the one side of the linear hyperplane and 3 vectors lie in the other side of the linear hyperplane. Let us assume that |D + (A )| = 1 and |D − (A )| = 7. Any such partition corresponds to a linear separation of D(A ) such that 5 vectors lie in the one side of the linear hyperplane and 3 vectors lie in the other side of the linear hyperplane. There also exists a unique partition of D(A ) into D + (A ) and D − (A ) where |D + (A )| = 0. As shown in the proof of Observation 2, such a partition corresponds to a proper linear separation of D(A ). The above argument shows that any linear separation of D(A ) contains at least 3 vectors in each of the open half-spaces created by the corresponding linear hyperplane. Lemma 8 implies that D(A ) is a Gale transform of a 2-neighborly 4-dimensional polytope.
Fig. 2 :
2Possible orientations of a triplet in R 2
Aichholzer et al. [1,2] created a database which contains all order-types of 8 points in general position in R 2 . We use those point sets in the proof of Theorem 3.
Lemma 9 .
9result proves Anshu et al.'s conjecture affirmatively for d = 4. We also produce a family of 4-dimensional rectilinear drawings of K Let us consider all order-types of the 8 points in general position in R 2 .[1] and[2] listed all possible 3315 order-types with their representative elements. Let us denote the point sequence corresponding to the i th ordertype with o i . We also generate all possible colorings of a sequence of 8 points where 4 of the points are red, and rest of them are blue. There are8 4 = 70 such colorings. Each coloring can be represented as an 8-bit binary string having an equal number of zeroes and ones. Let us represent the j th coloring in lexicographical order by c j . We consider the point sequence of each order-type and color it according to all the seventy possible ways such that there is an equal number of red and blue points in each coloring. Formally, we consider the setO C = {(o i , c j ) : 1 ≤ i ≤ 3315, 1 ≤ j ≤ 70}containing all possible pairs of (o i , c j ) for each i satisfying 1 ≤ i ≤ 3315, and 1 ≤ j ≤ 70.Consider a 4-dimensional rectilinear drawing of K 4 8 where the vertices ofK 4 8 are points in general position in R 4 . Let us denote these vertices by V = {v 1 , v 2 , . . . , v 8 }. Consider a Gale transform D(V ) of V . Lemma 7 implies that the number of proper linear separations of D(V ) is equal to the number of crossing pairs of hyperedges in this particular drawing of K 4 8 since there exists a bijection between crossing pairs of hyperedges and proper linear separations of D(V ). Consider an affine Gale diagram D(V ) having 4 red and 4 blue points such that all the 8 points are in general position in R 2 . Observation 1 ensures such a D(V ) always exists. Observation 2 ensures that the number of proper linear separations of D(V ) is equal to the total number of balanced 2-partitions of D(V ) plus the total number of balanced 4-partitions of D(V ) plus 1. Note that D(V ) is equivalent to one of the elements of O C . Note that all elements of O C need not be a Gale diagram of some 8 points in R 4 . Consider the point sequence o i under the coloring c j . If there exists a monochromatic 4-set of o i under the colouring c j , then (o i , c j ) is a projection of an acyclic vector configuration, and it can not be a Gale diagram of any set of 8 points in R 4 . We find the maximum value of (total number of balanced 2-partitions + the total number of balanced 4-partitions) over all members of O C by analyzing each of its members. We wrote the program for this purpose in Python 3.7.1 and have provided in the Appendix. . We find the maximum to be 12 when all the 8 points are the vertices of a convex octagon, and the vertices are colored red and blue, alternatively. Observation 2 implies that the maximum number of proper linear separations of D(V ) is 12 + 1 = 13. Lemma 7 implies that the maximum number of crossing pairs of hyperedges in any 4-dimensional rectilinear drawing of K 4 8 is 13. Consider a 4-dimensional rectilinear drawing of K 4 n where all the vertices are placed on the 4-dimensional moment curve. Anshu et al. showed that in this drawing, every K 4 8 has 13 crossing pairs of hyperedges. Since the crossing pairs of hyperedges spanned by a set of 8 vertices are distinct from the crossing pairs of hyperedges spanned by another set of 8 vertices, the above argument shows that max-cr 4 Consider a 4-dimensional neighborly polytope P having n vertices such that all the vertices of P are in general position in R 4 . Consider a 4dimensional rectilinear drawing of K 4 n such that the vertices of K 4 n are placed at the vertices of P . The number of crossing pairs of hyperedges in this 4dimensional rectilinear drawing of K 4 n is 13 n 8 . Proof. As mentioned in the proof of Theorem 3, let us consider the set O C = {(o i , c j ) : 1 ≤ i ≤ 3315 & 1 ≤ j ≤ 70} containing all possible pairs of (o i , c j ) for each i satisfying 1 ≤ i ≤ 3315 and satisfying 1 ≤ j ≤ 70. Consider a 4-dimensional rectilinear drawing of K 4 8 where the vertices of K 4 8 are placed as the vertices of a 4-dimensional neighborly polytope whose vertices are in general position in R 4 . Let us denote these vertices by V = {v 1 , v 2 , . . . , v 8 }. Consider a Gale transform D(V ) of V . Consider an affine Gale diagram D(V ) having 4 red and 4 blue points such that all the 8 points are in general position in R 2 . Observation 3 gives us necessary and sufficient conditions for (o i , c j ) to be a Gale transform of a 4-dimensional neighborly polytope whose vertices are in general position in R 4 . Let us consider all pairs (o i , c j ) such that they satisfy the three conditions mentioned in Observation 3. Let us denote this collection by O . O = {(o i , c j ) : (o i , c j ) f ollows the three conditions mentioned in Observation 3} Note that D(V ) is equivalent to one of the elements of O . Also note that each member of O is an affine Gale diagram of a 4-dimensional neighborly polytope having all its 8 vertices in general position in R 4 .We calculate the sum of the total number of balanced 2-partitions and the total number of balanced 4-partitions over all members of O by analyzing each of it's members. We wrote the program for this purpose in Python 3.7.1 and have provided in the Appendix. . We find the value to be 12 for all members of O .Observation 2 implies that the number of proper linear separations of D(V ) is 12 + 1 = 13. This implies that there exists 13 crossing pairs of hyperedges in a 4-dimensional rectilinear drawing of K 4 8 when the vertices of K 4 8 are placed as the vertices of a 4-dimensional neighborly polytope having all its 8 in general position in R 4 .
pairs of hyperedges. Since the crossing pairs of hyperedges spanned by a set of 8 vertices are distinct from the crossing pairs of hyperedges spanned by another set of 8 vertices, the above argument shows that the number of crossing pairs of hyperedges in a 4-dimensional rectilinear drawing of K 4 n is 13 n 8 if the vertices of K 4 n are placed as the vertices of a 4-dimensional neighborly polytope having all its vertices in general position in R 4 .
Lemma 12 .
12[4] Let us consider d pairwise disjoint sets, each having n points in R d , such that all dn points are in general position. Then there exist n pairwise disjoint (d − 1)-simplices such that each simplex has one vertex from each set.
Fig. 3 :
3Non-crossing pair of hyperedges of K 4 4×2
-
For each i satisfying 2 ≤ i ≤ d − 1, each bucket b i can contain no vertex of B, one vertex of B or two vertices of B depending upon the endpoints of the bucket. The last bucket contains either no vertex or one vertex of B. -For any pair of consecutive buckets, both of them can not contain 2 vertices of B. Lemma 10 and 11 together imply that Conv(A) and Conv(B) do not cross if and only if there does not exist an alternating chain of d+2 vertices as mentioned in Lemma 10.
Fig. 4
4Fig. 4: 3-dimensional Rectilinear Drawing of a 3 uniform hypergraph
Theorem 6 .·
6Let H = (V, E) be a d-uniform hypergraph. Let F be the total number of pairs of vertex disjoint hyperedges. There exists a d-dimensional rectilinear drawing D of H such that there are at leastc d · F crossing pairs of hyperedges in D, wherec d is a constant. Proof. Pick a permutation uniformly at random of the vertices of H. Put the vertices on the d-dimensional moment curve in that order. We draw each hyperedge present in E as a (d − 1)-simplex formed by the corresponding vertices. Let this drawing of H be denoted by D . Let X denote the number of crossing pair of hyperedges in D . Let A and B be two vertex disjoint hyperedges. Let X A ,B denote the indicator random variable. X A ,B is 1 if A and B form a crossing pairs of hyperedges, else it is set to 0. Note that the 2d vertices of can be placed on the d-dimensional moment curve in c m d ways such that the (d − 1)-simplex formed by the vertices of A and the (d − 1)-simplex formed by the vertices of B form a crossing. Note that we can permute {A , B } in two ways to obtain distinct ordered pairs, i.e., (A , B ), and (B , A ). Also, note that vertices of A have d! permutations among themselves. Similarly, vertices of B have d! permutations also. This implies that number of crossing pairs of hyperedges in D is E(X) = E( {A ,B } X A ,B F . This implies that there exists a random ordering of the vertices of H over the d-dimensional moment curve which produces at leastc d · F crossing pairs of hyperedges.Note thatc d is a constant. The following table contains the value ofc d for 2 ≤ d ≤ 10.
paper, we have proved the conjecture of Anshu et al. [5] for d = 4 by proving that max -cr 4 (K 4 n ) = 13 n 8 . The conjecture remains open for d > 4.
list of point sets were downloaded from \ protect \ vrule width0pt \ protect \ href { http :// www . ist . tugraz . at / staff / aichholzer / research / rp / triangulations / ordertypes /}{ http :// www . ist . tugraz . at / staff / aichholzer / research / rp / triangulations / ordertypes /}. The points in the document were in hexadecimal digits so we first covert them to decimal # # The program reads the file " point_set_hex . txt " in the same directory with lines in the format -" da30 9 d36 5842 4 c48 3 d5a 0 db1 37 d2 f335 "# # The output is the file " all_point_sets . txt " with lines in the format -" (218 , 48) , (157 , 54) , (88 , 66) , (76 , 72) , (61 , 90) , (13 , 177) , (55 , 210) , (243 , 53) , " def run () : # # Reading the file and saving it as a list of strings with open ( " ./ point_set_hex . txt " , " r " ) as hexfile : hexlines = hexfile . readlines () all _poi nt_co ordi nates = [] for line in hexlines : # # Splitting the line into list of coordinates , still as a string point_set = [] points = line . split (nt_co ordi nates . append ( list ( point_set ) ) # # Saving list of points in integer format to a filewith open ( " all_point_sets . txt " , " w " ) as point_set_file : for point_set in a ll_po int_ coor dinat es : for point in point_set : point_set_file . write ( we take set of points from the document generated by the code above and calculate all feasible sets of sizes 2 , 3 and 4 for each point set .
row_for_max_balanced . append ( max ( nu mb er _of _b al an ced _s et ) ) row_for_max_balanced . extend ([ ' ' ]* abs ( len ( ro w_ fo r_t ot al _b ala nc ed ) -len ( row_for_max_balanced ) ) ) feasible_ptset . loc [ -1] = ro w_ fo r_t ot al _b ala nc ed feasible_ptset . index = feasible_ptset . index + 1 feasible_ptset . loc [ -1] = row_for_max_balanced if not os . path . exists ( ' .know that the maximum number of balanced configurations can be twelve ( i . e . excluding the one where all points lie on side of the hyperlplane ) . So here we check which of the feasible point sets have a total of 12 balanced color configurations .
when reading directly from a directory , Python converts the file names to byte strings , so a conversion to UTF -8 before they can be worked with is neccesary if ( type ( file_name ) == ' bytes ') : feasible_ptset = pandas . read_csv ( file_name . decode ( 'utf -when reading directly from a directory , Python converts the file names to byte strings , so a conversion to UTF -8 before they can be worked with is neccesary if ( type ( file_name ) == ' bytes ') : feasible_ptset = pandas . read_csv ( file_name . decode ( 'utf -
even, a particular crossing pair of hyperedges can originate from at most 2 = Θ 4 d /d such d + 4 sized subsets of V . This implies that there exist at least Ω 2 d √d
(d+2)/2
d
(d+6)/2
+
d
(d+4)/2
2 d Θ 4 d /
√
d /O 4 d /d =
Ω 2 d d distinct crossing pairs of hyperedges in any d-dimensional rectilinear
drawing of K d
2d .
n 2 d
2. In order to prove this result, we first prove that the maximum d-dimensional rectilinear crossing number of K d d×2 is 2 d−1 − 1 in Lemma 13. We then create a d-Since the crossing pairs of hyperedges spanned by a copy of K ddimensional rectilinear drawing of K d
d×n such that each of the
n
2
d
induced
sub-hypergraphs, which are isomorphic to K d
d×2 , spans 2 d−1 − 1 crossing pairs of
hyperedges. d×2
Lemma 13. The maximum d-dimensional rectilinear crossing number of K d Proof. Consider a K d d×2 . For each i satisfying 1 ≤ i ≤ d, let us denote the i th part of the vertex set of K d d×2 by C i . Let {p ci , p ci } denote the set of 2 vertices in C i . Let A be a set of d vertices of K d d×2 such that each vertex of A is from different parts of K d d×2 .Let B be the set of rest of the vertices of K d d×2 . Note that |B| = d and each vertex of B is from different parts of K d d×2 . The number of unordered pairs {A, pair of disjoint simplices such that each simplex has one vertex from each part of K d d×2 . This implies the maximum number of unordered pairs {A, B} such that (d − 1)-simplex formed by the vertices of A forms a crossing with the (d − 1)-simplex formed by the vertices of B is 2 d−1 − 1.d×2
is 2 d−1 − 1.
B} is
1
2
2 d = 2 d−1 . Our goal is to find the maximum
number of unordered pairs, {A, B} such that the (d − 1)-simplex formed by the
vertices of A forms a crossing with the (d − 1)-simplex formed by the vertices
of B. Lemma 12 implies that in any d-dimensional rectilinear drawing of K d
d×2 ,
there exists a
# # Converting point set from string to usable list of tuples pointset_details = pandas . DataFrame () point_set = list ( ast . literal_eval ( line ) ) optimals = [[] , [] , []] if not os . path . exists ( ' ./ feasible_point_sets ' # # Generating . mod file for solving as LP # # After separating the points into two separate sets , a set can lie on either side of the separating line . To check if a configuration or its inverse is feasible , the contraints need to be checked after reversing their inequalitiesfor repeat in [1 , 2]: lp_file = open ( " run . mod " , " w " ) lp_file . write ( " var x1 ;\ nvar x2 ;\ n " ) lp_file . write ( " maximize obj : x1 + x2 ;\ n " ) # # Using glpsol tool from GLPK GNU tool as a python subprocess and checking for feasibility and if feasible save the details of the point set and move to the next solving_LP = subprocess . run ( " glpsol --math run . mod > LP_result " , shell = True ) with open ( " LP_result " , " r " ) # # File name : c o d e 3 _ c h e c k _ f o r _ b a l a n c e d _ s e t . py # # Now we take each of the 3315 files generated and check the coloring of the partitions of the feasible sets of points and then save the coloring details in a file . # # B represents balanced coloring , M represents monochromatic coloring and I represents imbalanced coloring . # # The output generated is the following format -# If the number of points are three then there are two possiblites , they all are of same color i . e . # If the number of points are four then there are four possiblites , they all are of same color i . e . monochromatic or two points are of same color and the other two are same , i . e . they are balanced and lastly threee are of the same color and one is different i . e . imbalanced feasible_ptset [ color ] = color_result if monochrome_flag_4set == 1: ro w_ fo r_t ot al _ba la nc ed . append ( ' NA ') else : nu mb er _of _b al anc ed _s et . append ( balance_counter ) ro w_ fo r_t ot al _ba la nc ed . append ( balance_counter )def run () :
index_combos = []
# # Generating all possible combinations of points of
sizes 2 , 3 and 4
for size in [2 , 3 , 4]:
if size !=4 :
PointSet Feasible Set Size Feasible Set Indices Feasible Set Points
(218, 48) 2
(0, 1)
[(218, 48), (157, 54)]
(157, 54) 2
(0, 7)
[(218, 48), (243, 53)]
(88, 66) 2
(1, 2)
[(157, 54), (88, 66)]
(76, 72) 2
(2, 3)
[(88, 66), (76, 72)]
(61, 90) 2
(3, 4)
[(76, 72), (61, 90)]
(13, 177) 2
(4, 5)
[(61, 90), (13, 177)]
(55, 210) 2
(5, 6)
[(13, 177), (55, 210)]
(243, 53) 2
(6, 7)
[(55, 210), (243, 53)]
3
(0, 1, 2)
[(218, 48), (157, 54), (88, 66)]
3
(0, 1, 7)
[(218, 48), (157, 54), (243, 53)]
3
(0, 6, 7)
[(218, 48), (55, 210), (243, 53)]
3
(1, 2, 3)
[(157, 54), (88, 66), (76, 72)]
3
(2, 3, 4)
[(88, 66), (76, 72), (61, 90)]
3
(3, 4, 5)
[(76, 72), (61, 90), (13, 177)]
3
(4, 5, 6)
[(61, 90), (13, 177), (55, 210)]
3
(5, 6, 7)
[(13, 177), (55, 210), (243, 53)]
4
(0, 1, 2, 3)
[(218, 48), (157, 54), (88, 66), (76, 72)]
4
(0, 1, 2, 7)
[(218, 48), (157, 54), (88, 66), (243, 53)]
4
(0, 1, 6, 7)
[(218, 48), (157, 54), (55, 210), (243, 53)]
4
(0, 5, 6, 7)
[(218, 48), (13, 177), (55, 210), (243, 53)]
index_combos = index_combos + list (
combinations ( range (0 , 8) , size ) )
else :
index_combos = index_combos + list (
islice ( combinations ( range (0 , 8) ,
4) , 35) )
with open ( " all_point_sets . txt " , " r " ) as ptsfile :
allpts_str = ptsfile . readlines ()
line_no = 1
for line in allpts_str :
) :
os . makedirs ( ' ./ feasible_point_sets ')
pointset_filename = " feasible_point_sets /
point_set_ " + str ( line_no )
for indices in index_combos :
remaining_indices = list ( set ( range (0 ,
8) ) . difference ( indices ) )
combo , remaining_points = [] , []
for index in indices :
combo . append ( point_set [ index
])
for index in remaining_indices :
remaining_points . append (
point_set [ index ])
if repeat == 1:
constraint_count = 1
for coordinates in
combo :
to_print = " s
. t . c " +
str (
constraint_count
) + " : "
+ str (
coordinates
[0]) + "
* x1 + "
+ str (
coordinates
[1]) + " *
x2 >=
1;\ n "
lp_file . write
( to_print
)
constraint_count
+= 1
for coordinates in
remaining_points :
to_print = " s
. t . c " +
str (
constraint_count
) + " : "
+ str (
coordinates
[0]) + "
* x1 + "
+ str (
coordinates
[1]) + " *
x2 <=
1;\ n "
lp_file . write
( to_print
)
constraint_count
+= 1
lp_file . write ( " solve
;\ nend ; " )
lp_file . close ()
else :
constraint_count = 1
for coordinates in
combo :
to_print = " s
. t . c " +
str (
constraint_count
) + " : "
+ str (
coordinates
[0]) + "
* x1 + "
+ str (
coordinates
[1]) + " *
x2 <=
1;\ n "
lp_file . write
( to_print
)
constraint_count
+= 1
for coordinates in
remaining_points :
to_print = " s
. t . c " +
str (
constraint_count
) + " : "
+ str (
coordinates
[0]) + "
* x1 + "
+ str (
coordinates
[1]) + " *
x2 >=
1;\ n "
lp_file . write
( to_print
)
constraint_count
+= 1
lp_file . write ( " solve
;\ nend ; " )
lp_file . close ()
as lp_result :
if ' NO PRIMAL
FEASIBLE ' not in
lp_result . read () :
if combo not
in
optimals
[0]:
optimals
[0].
append
(
combo
)
optimals
[1].
append
(
len
(
combo
)
)
optimals
[2].
append
(
indices
)
if len ( point_set ) > len ( optimals [0]) :
optimals [0]. extend ([ ' ' ]* abs ( len (
point_set ) -len ( optimals [0]) ) )
optimals [1]. extend ([ ' ' ]* abs ( len (
point_set ) -len ( optimals [0]) ) )
optimals [2]. extend ([ ' ' ]* abs ( len (
point_set ) -len ( optimals [0]) ) )
else :
point_set . extend ([ ' ' ]* abs ( len (
optimals [0]) -len ( point_set ) ) )
pointset_details [ ' PointSet '] = point_set
pointset_details [ ' Feasible_Set_Size '] =
optimals [1]
pointset_details [ ' Feasible_Set_Indices '] =
optimals [2]
pointset_details [ ' Feasible_Set_Points '] =
optimals [0]
pointset_details . to_csv ( pointset_filename + "
. csv " , sep = ' , ' , index = False )
print ( ' Checking point set ' , line_no , " \ t
finished at \ t " , datetime . datetime . now () )
line_no += 1
if __name__ == " __main__ " :
run ()
import os
import pandas
def run () :
directory = " ./ feasible_point_sets / "
files = os . listdir ( directory )
# The list of 70 possible colors with equal number of
points of both colors
required_colors = [ ' 00001111 ' , ' 00010111 ' , ' 00011011 '
, ' 00011101 ' , ' 00011110 ' , ' 00100111 ' , ' 00101011 ' ,
' 00101101 ' , ' 00101110 ' ,
' 00110011 ' , '
00110101 ' , '
00110110 ' , '
00111001 ' , '
00111010 ' , '
00111100 ' , '
color_result = []
balance_counter = 0
monochrome_flag_4set = 0
for indices in feasible_indices :
setsize = len ( indices )
# If the number of points is
two then the possiblites
are either both are of
same color i . e . it is
monochromatic or two
points are of different
colors , i . e . equal number
of points of each color ,
therefore balanced
if setsize == 2:
one_side_color =
color [ indices [0]]
+ color [ indices
[1]]
if one_side_color .
count ( '1 ') == 1:
color_result .
append (
one_side_color
+ " -B "
)
balance_counter
+= 1
else :
color_result .
append (
one_side_color
+ " -M "
)
monochromatic or two
points are of same color
and the other one is
different , i . e . they are
imbalanced
elif setsize == 3:
one_side_color =
color [ indices [0]]
+ color [ indices
[1]] + color [
indices [2]]
if one_side_color .
count ( '1 ') == 2
or one_side_color
. count ( '0 ') == 2:
color_result .
append (
one_side_color
+ " -I "
)
elif one_side_color .
count ( '1 ') == 3
or one_side_color
. count ( '0 ') == 3:
color_result .
append (
one_side_color
+ " -M "
)
elif setsize == 4:
one_side_color =
color [ indices [0]]
+ color [ indices
[1]] + color [
indices [2]] +
color [ indices [3]]
if one_side_color .
count ( '1 ') == 2:
color_result .
append (
one_side_color
+ " -B "
)
balance_counter
+= 1
elif one_side_color .
count ( '1 ') == 1
or one_side_color
. count ( '0 ') == 1:
color_result .
append (
one_side_color
+ " -I "
)
elif one_side_color .
count ( '1 ') == 0
or one_side_color
. count ( '1 ') == 4:
color_result .
append (
one_side_color
+ " -M "
)
monochrome_flag_4se t
= 1
https://github.com/ayan-iiitd/maximum-rectilinear-crossing-number-of-uniformhypergraphs.git
Acknowledgementsif not os . path . exists ( ' . # # Data is saved in the same format as before .import os import pandas def run () :
Enumerating order types for small point sets with applications. O Aichholzer, F Aurenhammer, H Krasser, Order. 19O. Aichholzer, F. Aurenhammer and H. Krasser. Enumerating order types for small point sets with applications. Order 19, 265-281 (2002).
New lower bounds for the number of (≤ k)-edges and the rectilinear crossing number of Kn. O Aichholzer, J García, D Orden, P Ramos, Discrete and Computational Geometry. 38O. Aichholzer, J. García, D. Orden and P. Ramos. New lower bounds for the number of (≤ k)-edges and the rectilinear crossing number of Kn. Discrete and Computational Geometry 38, 1-14 (2007).
Disjoint simplices and geometric hypergraphs. J Akiyama, N Alon, Annals of the New York Academy of Sciences. 555J. Akiyama and N. Alon. Disjoint simplices and geometric hypergraphs. Annals of the New York Academy of Sciences 555, 1-3 (1989).
On the rectilinear crossing number of complete uniform hypergraphs. A Anshu, R Gangopadhyay, S Shannigrahi, S Vusirikala, Computational Geometry: Theory and Applications. 61A. Anshu, R. Gangopadhyay, S. Shannigrahi and S. Vusirikala. On the rectilinear crossing number of complete uniform hypergraphs. Computational Geometry: Theory and Applications 61, 38-47 (2017).
A lower bound on the crossing number of uniform hypergraphs. A Anshu, S Shannigrahi, Discrete Applied Mathematics. 209A. Anshu and S. Shannigrahi. A lower bound on the crossing number of uniform hypergraphs. Discrete Applied Mathematics 209, 11-15 (2016).
Approximating the maximum rectilinear crossing number. S Bald, M P Johnson, O Liu, Proceedings of International Computing and Combinatorics Conference. International Computing and Combinatorics ConferenceSpringerS. Bald, M. P. Johnson and O. Liu. Approximating the maximum rectilinear crossing number. In Proceedings of International Computing and Combinatorics Conference. Springer, 455-467(2016).
Primitive Radon partitions for cyclic polytopes. M Breen, Israel Journal of Mathematics. 15M. Breen. Primitive Radon partitions for cyclic polytopes. Israel Journal of Math- ematics 15, 156-157 (1973).
Some Erdős-Szekeres type results about points in space. T Bisztriczky, V Soltan, Monatshefte für Mathematik. 118T. Bisztriczky and V. Soltan. Some Erdős-Szekeres type results about points in space, Monatshefte für Mathematik 118, 33-40 (1994).
Extremal problems for geometric hypergraphs. Algorithms and Computation. T K Dey, J Pach, Proc. ISAAC '96. ISAAC '96OsakaT. K. Dey and J. Pach. Extremal problems for geometric hypergraphs. Algorithms and Computation (Proc. ISAAC '96, Osaka;
T Asano, Also in: Discrete and Computational Geometry. Springer-Verlag1178T. Asano et al., eds.), Lecture Notes in Computer Science 1178, Springer-Verlag, 105-114 (1996). Also in: Discrete and Computational Geometry 19, 473-484 (1998).
Neighboring vertices on a convex polyhedron. D Gale, Linear inequalities and related system. 38D. Gale. Neighboring vertices on a convex polyhedron. Linear inequalities and related system 38, 255-263 (1956).
k-Sets and Rectilinear Crossings in Complete Uniform Hypergraphs. R Gangopadhyay, S Shannigrahi, Computational Geometry: Theory and Applications. 86101578R. Gangopadhyay and S. Shannigrahi. k-Sets and Rectilinear Crossings in Com- plete Uniform Hypergraphs. Computational Geometry: Theory and Applications 86, 101578 (2020).
Upper bounds for configurations and polytopes in R d. J E Goodman, R Pollack, Discrete and Computational Geometry. 1J. E. Goodman and R. Pollack. Upper bounds for configurations and polytopes in R d . Discrete and Computational Geometry 1, 219-227 (1986).
Convex Polytopes. B Grünbaum, SpringerB. Grünbaum. Convex Polytopes. Springer, 2003.
Crossing numbers of graphs. R K Guy, Graph Theory and Applications. R. K. Guy. Crossing numbers of graphs. Graph Theory and Applications, 111-124 (1972).
Lectures in Discrete Geometry. J Matoušek, SpringerJ. Matoušek. Lectures in Discrete Geometry. Springer, 2002.
Coverings and colorings of hypergraphs. L Lovász, Proceedings of the 4th Southeastern Conference on Combinatorics, Graph Theory and Computing. the 4th Southeastern Conference on Combinatorics, Graph Theory and ComputingUtilitas Mathematica PublishingL. Lovász. Coverings and colorings of hypergraphs. In Proceedings of the 4th South- eastern Conference on Combinatorics, Graph Theory and Computing. Utilitas Math- ematica Publishing, 3-12 (1973).
Extremal problems in the theory of graphs. G , Proceedings of Theory of Graphs and its Applications. Theory of Graphs and its ApplicationsG. Ringel. Extremal problems in the theory of graphs. In Proceedings of Theory of Graphs and its Applications, 85-90 (1964).
The graph crossing number and its variants: A survey. The electronic journal of combinatorics 1000. M Schaefer, M. Schaefer. The graph crossing number and its variants: A survey. The electronic journal of combinatorics 1000, 21-22 (2013).
On the obfuscation complexity of planar graphs. O Verbitsky, Theoretical Computer Science. 396O. Verbitsky. On the obfuscation complexity of planar graphs. Theoretical Com- puter Science 396, 294-300 (2008).
G M Ziegler, Lectures on Polytopes. Springer01001101 ' , ' 01001110 ' , ' 01010011 ' , ' 01010101 ' , ' 01010110 ' , ' 01011001 ' , ' 01011010 ' , ' 01011100 ' , ' 01100011 ' , ' 01100101 ' , ' 01100110 ' , ' 01101001 ' , ' 01101010 ' , ' 01101100 ' , ' 01110001 ' , ' 01110010 ' , ' 01110100 ' , ' 01111000 ' , ' 10000111 ' , ' 10001011 ' , ' 10001101 ' , ' 10001110 'G. M. Ziegler. Lectures on Polytopes. Springer, 1995. 01000111 ' , ' 01001011 ' , ' 01001101 ' , ' 01001110 ' , ' 01010011 ' , ' 01010101 ' , ' 01010110 ' , ' 01011001 ' , ' 01011010 ' , ' 01011100 ' , ' 01100011 ' , ' 01100101 ' , ' 01100110 ' , ' 01101001 ' , ' 01101010 ' , ' 01101100 ' , ' 01110001 ' , ' 01110010 ' , ' 01110100 ' , ' 01111000 ' , ' 10000111 ' , ' 10001011 ' , ' 10001101 ' , ' 10001110 ' , '
10011100 ' , ' 10100011 ' , ' 10100101 ' , ' 10100110 ' , ' 10101001 ' , ' 10101010 ' , ' 10101100 ' , ' 10110001 ' , ' 10110010 ' , ' 10110100 ' , ' 10111000 ' , ' 11000011 ' , ' 11000101 ' , ' 11000110 ' , ' 11001001 ' , ' 11001010 ' , ' 11001100 ' , ' 11010001 ' , ' 11010010 ' , ' 11010100 ' , ' 11011000 ' , ' 11100001 ' , ' 11100010 ' , ' 11100100 ' , ' 11101000 ' , ' 11110000 '] for file in files : file_name = directory + file feasible_ptset = pandas . read_csv ( file_name ) ro w_ fo r_t ot al _ba la. R Gangopadhyay, nc ed = [ " " , " " , " " , " Total Balanced Sets "R. Gangopadhyay et al. 10010011 ' , ' 10010101 ' , ' 10010110 ' , ' 10011001 ' , ' 10011010 ' , ' 10011100 ' , ' 10100011 ' , ' 10100101 ' , ' 10100110 ' , ' 10101001 ' , ' 10101010 ' , ' 10101100 ' , ' 10110001 ' , ' 10110010 ' , ' 10110100 ' , ' 10111000 ' , ' 11000011 ' , ' 11000101 ' , ' 11000110 ' , ' 11001001 ' , ' 11001010 ' , ' 11001100 ' , ' 11010001 ' , ' 11010010 ' , ' 11010100 ' , ' 11011000 ' , ' 11100001 ' , ' 11100010 ' , ' 11100100 ' , ' 11101000 ' , ' 11110000 '] for file in files : file_name = directory + file feasible_ptset = pandas . read_csv ( file_name ) ro w_ fo r_t ot al _ba la nc ed = [ " " , " " , " " , " Total Balanced Sets " ]
== ' 12 ') : point_set_details [ ' PointSet '] = feasible_ptset. PointSet '] point_set_details [ ' Feasible_Set_Size ' ] = feasible_ptset [ ' Feasible_Set_Size '] point_set_details [ ' Feasible_Set_Indices '] = feasible_ptset [ ' Feasible_Set_Indices '] point_set_details [ ' Feasible_Set_Points '] = feasible_ptset [ ' Feasible_Set_Points '== ' 12 ') : point_set_details [ ' PointSet '] = feasible_ptset [ ' PointSet '] point_set_details [ ' Feasible_Set_Size ' ] = feasible_ptset [ ' Feasible_Set_Size '] point_set_details [ ' Feasible_Set_Indices '] = feasible_ptset [ ' Feasible_Set_Indices '] point_set_details [ ' Feasible_Set_Points '] = feasible_ptset [ ' Feasible_Set_Points ']
|
[
"https://github.com/ayan-iiitd/maximum-rectilinearcrossing-number-of-uniform-hypergraphs.git",
"https://github.com/ayan-iiitd/maximum-rectilinear-crossing-number-of-uniformhypergraphs.git"
] |
[
"NON-INTEGER CHARACTERIZING SLOPES FOR TORUS KNOTS",
"NON-INTEGER CHARACTERIZING SLOPES FOR TORUS KNOTS"
] |
[
"Duncan Mccoy "
] |
[] |
[] |
A slope p/q is a characterizing slope for a knot K in S 3 if the oriented homeomorphism type of p/q-surgery on K determines K uniquely. We show that for each torus knot its set of characterizing slopes contains all but finitely many non-integer slopes. This generalizes work of Ni and Zhang who established such a result for T 5,2 . Along the way we show that if two knots K and K in S 3 have homeomorphic p/qsurgeries, then for q ≥ 3 and p sufficiently large we can conclude that K and K have the same genera and Alexander polynomials. This is achieved by consideration of the absolute grading on Heegaard Floer homology.
|
10.4310/cag.2020.v28.n7.a5
|
[
"https://arxiv.org/pdf/1610.03283v1.pdf"
] | 119,132,553 |
1610.03283
|
15606ef5ba02756b531d8db0fbff35ee42e5c075
|
NON-INTEGER CHARACTERIZING SLOPES FOR TORUS KNOTS
Duncan Mccoy
NON-INTEGER CHARACTERIZING SLOPES FOR TORUS KNOTS
A slope p/q is a characterizing slope for a knot K in S 3 if the oriented homeomorphism type of p/q-surgery on K determines K uniquely. We show that for each torus knot its set of characterizing slopes contains all but finitely many non-integer slopes. This generalizes work of Ni and Zhang who established such a result for T 5,2 . Along the way we show that if two knots K and K in S 3 have homeomorphic p/qsurgeries, then for q ≥ 3 and p sufficiently large we can conclude that K and K have the same genera and Alexander polynomials. This is achieved by consideration of the absolute grading on Heegaard Floer homology.
Introduction
Given a knot K ⊆ S 3 , we say that p/q ∈ Q is a characterizing slope for K, if the oriented homeomorphism type of the manifold obtained by p/q-surgery on K determines K uniquely. 1 In general determining the set of characterizing slopes for a given knot is challenging. It was a long-standing conjecture of Gordon, eventually proven by Kronheimer, Mrowka, Ozsváth and Szabó, that every slope is a characterizing slope for the unknot [KMOS07]. Ozsváth and Szabó have also shown that every slope is a characterizing slope for the trefoil and the figure-eight knot [OS06]. More recently, Ni and Zhang have studied characterizing slopes for torus knots, showing that T 5,2 has only finitely many non-characterizing slopes which are not negative integers [NZ14]. The aim of this paper is to establish a similar result for arbitrary torus knots. We will be primarily interested in non-integer surgery slopes. For each torus knot, the main result of this paper is to classify the non-integer non-characterizing slopes outside of a finite set of slopes.
Theorem 1.1. For s > r > 1 and q ≥ 2 let K be a knot such that S 3 p/q (K) ∼ = S 3 p/q (T r,s ). If p and q satisfy at least one of the following:
(i) p ≤ min{− 43 4 (rs − r − s), −32q}, (ii) p ≥ max{ 43 4 (rs − r − s), 32q + 2q(r − 1)(s − 1)}, or (iii) q ≥ 9, then we have either (a) K = T r,s , or (b) K is a cable of a torus knot, in which case q = s/r , p = r 2 q 4 −1 q 2 −1 , s = rq 3 ±1 q 2 −1 and r > q. When combined with previously known results about integer characterizing slopes for torus knots, this yields the following corollary [McC14,NZ14].
Corollary 1.2. The knot T r,s with r, s > 1 has only finitely many non-characterizing slopes which are not negative integers.
It is well-known that the manifolds obtained by non-integer surgery on torus knots are Seifert fibred spaces [Mos71]. Conjecturally, the only knots in S 3 with non-integer Seifert fibred surgeries are torus knots and cables of torus knots. Conjecture 1.3. If S 3 p/q (K) is a Seifert fibred space and q ≥ 2, then K is either a torus knot or a cable of a torus knot.
We can use this to obtain a precise conjecture on which non-integer slopes are characterizing slopes for torus knots. In particular, it turns out that conjecturally each torus knot has at most one non-integer non-characterizing slope, which is precisely the non-characterizing slope occurring in the conclusion of Theorem 1.1.
Conjecture 1.4. For the torus knot T r,s with s > r > 1, every non-integer slope is characterizing with the possible exception of p/q for p = r 2 q 4 −1 q 2 −1 and q = s/r ≥ 2, which is non-characterizing only if r > q and s = rq 3 ±1 q 2 −1 . Moreover, for this slope there is a unique knot K = T r,s with S 3 p/q (K) ∼ = S 3 p/q (T r,s ) and K is a cable of a torus knot.
One deduces Conjecture 1.4 from Conjecture 1.3 by checking when a torus knot can share a surgery with a torus knot or a cable of a torus knot. The classification of when a cable of a torus knot and a cable of a torus knot have a common non-integer surgery is recorded in the following proposition. Note that it shows the converse to Conjecture 1.4 is true: for each T r,s satisfying the necessary conditions, there is a cable of a torus knot exhibiting that the required slope is non-characterizing.
Proposition 1.5. For s > r > 1 and q ≥ 2, there exists a non-trivial cable of a torus knot K such that S 3 p/q (K) ∼ = S 3 p/q (T r,s ) if and only if s = rq 3 ± 1 q 2 − 1 , p = r 2 q 4 − 1 q 2 − 1 , q = s/r and r > q, in which case K is the (q, q 2 r 2 −1 q 2 −1 )-cable of T r, rq±1 q 2 −1 . Remark 1.6. If one also allows orientation reversing homeomorphisms in the definition of a characterizing slope, then the list given by Conjecture 1.4 would be incomplete. For example, we have The proof of Theorem 1.1 follows a similar outline to Ni and Zhang's work. Given a knot K ⊆ S 3 such that S 3 p/q (K) ∼ = S 3 p/q (T r,s ), we consider the possibilities that K is a hyperbolic knot, a satellite knot or a torus knot in turn. By applying results from hyperbolic geometry and Heegaard Floer homology, we will show that for the slopes in Theorem 1.1 the only possibilities are that K is a cable of a torus knot or K = T r,s . The bound q ≥ 9 arises as a result of Lackenby and Meyerhoff's bound on the distance between exceptional surgery slopes [LM13]. The other bounds are a consequence of combining restrictions on exceptional surgeries coming from the 6-theorem of Agol [Ago00] and Lackenby [Lac03] with genus bounds on K coming from Heegaard Floer homology. These genus bounds are the key technical results developed in this paper. In general, we show that if S 3 p/q (K) ∼ = S 3 p/q (K ), then under certain circumstances K and K must have the same genera and Alexander polynomials. For arbitrary knots in S 3 , we have the following result.
Theorem 1.7. Let K, K ⊆ S 3 be knots such that S 3 p/q (K) ∼ = S 3 p/q (K ). If |p| ≥ 12 + 4q 2 − 2q + 4qg(K) and q ≥ 3, then ∆ K (t) = ∆ K (t), g(K) = g(K ) and K is fibred if and only if K is fibred.
Here ∆ K (t) denotes the Alexander polynomial of K. We obtain stronger results for L-space knots. 2 Theorem 1.8. Suppose that K is an L-space knot. If S 3 p/q (K) ∼ = S 3 p/q (K ) for some K ⊆ S 3 and either (i) p ≥ 12 + 4q 2 − 2q + 4qg(K) or (ii) p ≤ min{2q − 12 − 4q 2 , −2qg(K)} and q ≥ 2 holds, then ∆ K (t) = ∆ K (t), g(K) = g(K ) and K is fibred.
Both Theorem 1.7 and Theorem 1.8 are proven by making use of the absolute grading in Heegaard Floer homology.
Remark 1.9. Baker and Motegi have recently constructed infinite families of knots {K n } n∈Z in S 3 such that S 3 n (K 0 ) ∼ = S 3 n (K n ), for all n ∈ Z and deg ∆ Kn (t) → ∞ as |n| → ∞ [BM16, Section 3]. This shows that Theorem 1.7 cannot be extended unconditionally to integer surgeries.
Acknowledgements. The author would like to thank his supervisor, Brendan Owens, for his helpful guidance. He also wishes to acknowledge the influential role of the work of Yi Ni and Xingru Zhang which provided inspiration for both the overall strategy and several technical steps in the proof of Theorem 1.1 [NZ14].
Heegaard Floer homology
Heegaard Floer homology is a package of 3-manifold invariants introduced by Ozsváth and Szabó [OS04c]. To each closed oriented 3-manifold Y equipped with a spin c -structure s it associates a family of groups denoted by HF (Y, s), HF ± (Y, s) and HF ∞ (Y, s). Throughout this paper all Heegaard Floer groups are taken with F = Z/2Z coefficients.
We will be primarily concerned with HF + (Y, s), where Y is a rational homology sphere. In this case, the group HF + (Y, s) possesses an absolute Q-grading. There is also a U -action on HF + (Y, s), which gives HF + (Y, s) the structure of an F[U ]-module. Multiplication by U interacts with the Q-grading by decreasing it by 2 [OS03a].
In any spin c -structure HF + (Y, s) can be decomposed as a direct sum:
HF + (Y, s) ∼ = T + ⊕ HF red (Y, s), where T + = F[U, U −1 ]/U F[U ]
and U N HF red (Y, s) = 0 for N sufficiently large. The T + summand is sometimes referred to as the tower. The minimal Q-grading over all elements of the tower is an invariant of (Y, s) called the d-invariant and is denoted d(Y, s). We say that Y is an L-space if HF red (Y, s) = 0 for all s ∈ Spin c (Y ). Heegaard Floer homology is invariant under conjugation of spin c -structures, in the sense that HF + (Y, s) and HF + (Y, s) are isomorphic as F[U ]-modules and as Q-graded groups. In particular, the d-invariants satisfy d(Y, s) = d(Y, s).
2.1. Knot Floer homology. Knot Floer homology was defined independently by Ozsváth and Szabó [OS04b] and Rasmussen [Ras03]. Given a knot in K ⊆ S 3 , it takes the form of a finitely-generated group
HF K(K) = s∈Z HF K(K, s),
where s is known as the Alexander grading. The knot Floer homology also possesses a second grading, known as the Maslov grading such that
HF K(K, s) = d∈Z HF K d (K, s). If K has Alexander polynomial ∆ K (t) = s∈Z a s t s ,
normalized so that a s = a −s and ∆ K (1) = 1, then ∆ K (t) can be recovered by taking the Euler characteristic in each Alexander grading:
a s = χ( HF K(K, s)) = d∈Z (−1) d rk HF K d (K, s).
With this normalization in place, we will take t k (K) to denote the torsion coefficient
t k (K) = i≥0 ia k+i .
Remark 2.1. The coefficients of ∆ K (t) satisfy
a k = t k+1 (K) − 2t k (K) + t k−1 (K)
for all k. Since the Alexander polynomial is normalized so that ∆ K (1) = 1, this means the Alexander polynomial can be computed from the t k (K) for k ≥ 0.
One key geometric property detected by knot Floer homology is the genus [OS04a]:
g(K) = max{s | HF K(K, s) = 0}.
The other geometric property of knot Floer homology that we will use is its ability to detect whether a knot is fibred [Ghi08,Ni07].
Theorem 2.2 (Ni). A knot K of genus g is fibred if and only if rk HF K(K, g) = 1.
2.2. The knot Floer chain complex. The knot Floer homology group HF K(K) can be generalized to the knot Floer chain complex CF K ∞ (K), which takes the form of a bifiltered chain complex
CF K ∞ (K) = i,j∈Z C{(i, j)}, where H * (C{(i, j)}) ∼ = HF K * −2i (K, j − i).
There is also a natural chain complex isomorphism
U : CF K ∞ (K) −→ CF K ∞ (K),
which maps C{(i, j)} isomorphically to C{(i − 1, j − 1)} and lowers the Maslov grading by 2. This gives CF K ∞ (K) the structure of a finitely-generated F[U, U −1 ]-module. The chain homotopy type of CF K ∞ (K) as a bifiltered complex is an invariant of K. In fact, after a suitable chain homotopy, one can assume that
C{(i, j)} ∼ = HF K * −2i (K, j − i).
The knot Floer complex has several important quotient complexes: for each k ∈ Z, the "hook" complexes A + k = C{i ≥ 0 or j ≥ k}, and the complex B + = C{i ≥ 0}.
These complexes admit chain maps
v k , h k : A + k −→ B + ,
where v k is the obvious vertical projection, and h k consists of the composition of a horizontal projection onto C{j ≥ k}, multiplication by U k and a chain homotopy equivalence. We will use A + k = H * (A + k ) and B + = H * (B + ) to denote the homology groups and v k and h k to denote the maps induced on homology by v k and h k respectively. As we are working with a knot in S 3 , we have B + ∼ = HF + (S 3 ) ∼ = T + . The group A + k stabilizes under multiplication by large powers of U , allowing us to define A T k as A T k = U N A + k for N sufficiently large. This always satisfies A T k ∼ = T + . We also define A red,k to be the quotient A red,k = A + k /A T k . When restricted to A T k the map v k is modeled on multiplication by U V k for some nonnegative integer V k [NW15]. Similarly, h k is modeled on multiplication by U H k for some non-negative integer H k when restricted to A T k . These integers V k and H k are known to satisfy
V k = H −k and V k − 1 ≤ V k+1 ≤ V k ,
for all k. For any n ≥ 0, we will use T (n) to denote the F[U ]-submodule of T + generated by U 1−n . For n = 0, we take T (0) = 0.
The following proposition shows how the Alexander polynomial, genus and fiberedness of K are encoded in the V k and A red,k . Proposition 2.3 (Cf. Lemma 3.3 of [NZ14]). For K ⊆ S 3 the following hold:
(i) t k (K) = V k + χ(A red,k ) for all k;
(ii) g(K) = 1 + max{k | V k + rkA red,k > 0}; and (iii) K is fibred if and only if V g−1 + rkA red,g−1 = 1.
Proof. It follows from the definition of v k and A red,k that the kernel of v k admits a splitting as
ker v k ∼ = T (V k ) ⊕ A red,k .
This shows that
χ(ker v k ) = V k + χ(A red,k ).
On the other hand we have the long exact sequence of chain complexes
0 −→ C{i < 0, j ≥ k} −→ A + k v k −→ B + −→ 0.
As v k is surjective on homology, the exact triangle induced by this sequence shows that
ker v k ∼ = H * (C{i < 0, j ≥ k}).
Taking the Euler characteristic this shows that
χ(ker v k ) = i≤−1, j≥k χ( HF K(K, j − i)) = i≥1 iχ( HF K(K, k + i)) = t k (K).
This proves (i).
Since C{i < 0, j ≥ g(K)} = 0, we have ker v k = 0 for k ≥ g. Furthermore, as
C{i < 0, j = g(K)} = C{−1, g − 1} ∼ = HF K(K, g),
we have ker v g−1 ∼ = HF K(K, g). This shows
g(K) − 1 = max{k | V k + rkA red,k > 0},
proving (ii). As Theorem 2.2 shows that K is fibred if and only if rk HF K(K, g) = 1, this also proves (iii).
Let ν + (K) to be the quantity
ν + (K) = min{k | V k = 0}.
It follows from Proposition 2.3 that ν + (K) exists and is at most g(K). In fact, it can be shown that ν + (K) ≤ g 4 (K), where g 4 (K) is the smooth slice genus of K [Ras04, Theorem 2.3]. Recall that K is said to be an L-space knot if S 3 p/q (K) is an L-space for some p/q > 0. Equivalently, K is an L-space knot if and only if A red,k = 0 for all k. The following proposition summarizes the properties of L-space knots that we require.
Proposition 2.4. If K is an L-space knot, then (i) ν + (K) = g(K), (ii) K is fibred, (iii) ν + (K) = 0 and (iv) A red,k (K) ∼ = T (V |k| (K)) for all k.
Proof. As an L-space knot satisfies A red,k (K) = 0 for all k, it follows from Proposition 2.3, that V g(K)−1 > 0 and V g(K) = 0. This shows that ν + (K) = g(K). Since V g(K)−1 ≤ V g(K) +1, it follows that V g(K)−1 = 1 and that K is fibred by Proposition 2.3. The facts about K follow from Lemma 16 and Proposition 17 in [Gai14].
2.3. The mapping cone formula. Given a knot in K ⊆ S 3 , one can determine the Heegaard Floer homology of all manifolds obtained by surgery on it in terms of the knot Floer homology of K via the homology of a mapping cone [OS11]. In this section we summarize the results arising from the mapping cone formula that we will need. More detailed accounts of the mapping cone formula and its consequences can be found in [NW15] or [Gai14], for example.
In order to describe the Heegaard Floer homology of S 3 p/q (K), we need a way to label its spin c -structures. This labeling takes the form of an affine bijection defined in terms of relative spin c -structures on S 3 \ νK, [OS11]:
(2.1) φ K,p/q : Z/pZ −→ Spin c (S 3 p/q (K))
. The exact details of this map are not important, however we note that for any knot K, conjugation of spin c -structures is given by [LN15, Lemma 2.2]:
(2.2) φ K,p/q (q − 1 − i mod p) = φ K,p/q (i) ∈ Spin c (S 3 p/q (K)). If S 3 p/q (K) ∼ = S 3 p/q (K ) ∼ = Y ,
then φ K,p/q and φ K ,p/q will, in general, give rise to different labelings on Spin c (Y ). However, as long it will not cause confusion, we will suppress the map φ K,p/q from the notation.
. When K is the unknot this gives a labeling on the spin c -structures of a lens space. We will use d(p, q, i) to denote the d-invariant
d(p, q, i) = d(S 3 p/q (U ), i) for i ∈ Z/pZ. . Now we describe how CF K ∞ (K) determines HF + (S 3 p/q (K))
. Consider the groups
A + i = s∈Z (s, A + ps+i q ) and B + i = s∈Z (s, B + ), and the maps v ps+i q : (s, A + ps+i q ) → (s, B + ) and h ps+i q : (s, A + ps+i q ) → (s + 1, B + ),
where v k and h k are the maps on homology induced by v k and h k as in the previous section. These maps can be added together to obtain a chain map
D + i,p/q : A + i → B + i , where D + i,p/q (s, x) = (s, v ps+i q (x)) + (s + 1, h ps+i q (x)).
The group HF
+ (S 3 p/q (K), i) is computed in terms of the mapping cone on D + i,p/q . Theorem 2.5 (Ozsváth-Szabó, [OS11]). For any knot K in S 3 . Let X + i,p/q be the mapping cone of D + i,p/q , then there is a graded isomorphism of groups H * (X + i,p/q ) ∼ = HF + (S 3 p/q (K), i).
Remark 2.6. The statement of Theorem 2.5 given here is not quite the one given in [OS11].
Ozsváth-Szabó establish an isomorphism between Heegaard Floer homology and the mapping cone of a map D + i,p/q , whose induced map on homology is D + i,p/q . For surgeries on S 3 , both mapping cones compute the same Heegaard Floer homology groups.
Remark 2.7. The isomorphism in Theorem 2.5 is U -equivariant, so it also provides an isomorphism of F[U ]-modules.
When p/q > 0, the map D + i,p/q is surjective, so Theorem 2.5 gives a graded isomorphism HF + (S 3 p/q (K), i) ∼ = ker D + i,p/q . The grading on ker D + i,p/q is determined by putting a Qgrading on X + i,p/q in such a way that D + i,p/q decreases the grading by one and the grading on B + i , which is independent of K, is fixed to give the correct d-invariants for surgery on the unknot (cf. [OS11, Section 7.2]). In practice, this means that for p/q > 0 and 0 ≤ i ≤ p − 1, the grading on B + i satisfies [NW15] (2.3) gr(0, 1) = d(p, q, i) − 1 and, as
H −k (U ) = V k (U ) = 0 if k ≥ 0 |k| if k ≤ 0,
the gradings of (s, 1) and (s + 1, 1) in B + i are related by [NZ14, Section 3.3]
(2.4) gr(s + 1, 1) = gr(s, 1) + 2 i + ps q , for any s ∈ Z.
With these gradings one finds that for any p/q > 0 and any 0 ≤ i ≤ p−1, the d-invariants S 3 p/q (K) can be calculated by [NW15, Proposition 1.6] (2.5)
d(S 3 p/q (K), i) = d(p, q, i) − 2 max{V i q , V p−i q }.
One can also compute the reduced Heegaard Floer homology groups. We require only the special case when p/q ≥ 2ν + (K) − 1. The following proposition can easily be derived from [Gai14,Corollary 12] or [NZ14, Proposition 3.6].
Proposition 2.8. If p/q ≥ 2ν + (K) − 1, then HF red (S 3 p/q (K), i) ∼ = s∈Z A red, i+ps q as a
Q-graded groups, where the absolute grading on A red, i+ps q is determined by the absolute grading on the summand (s, A + ) ⊂ X + i,p/q . Remark 2.9. When q > 1, the same group A red,k can appear as a summand in HF red (S 3 p/q (K), i) for more than one value of i. Since the grading on X + i,p/q depends on i, these summands will, in general, possess different gradings.
The following lemma shows that under certain circumstances we can recover information about the knot Floer homology of two knots with the same surgery. It is a key technical ingredient in the proofs of Theorem 1.7 and Theorem 1.8.
Lemma 2.10. Suppose that S 3 p/q (K) ∼ = S 3 p/q (K ), for some p/q > 2g(K) − 1, and that φ K,p/q = φ K ,p/q or φ K,p/q = φ K ,p/q . If either (i) S 3 p/q (K) is an L-space; (ii) q ≥ 2 and, for all k, there is N k ≥ 0 such that A red,k (K) ∼ = T (N k ); or (iii) q ≥ 3, then V k (K) = V k (K ) and A red,k (K) ∼ = A red,k (K ) for all k ≥ 0.
Proof. Since conjugation induces a grading preserving isomorphism on Heegaard Floer homology, the assumptions on φ K,p/q and φ K ,p/q , imply that we have an isomorphism HF + (S 3 p/q (K), i) ∼ = HF + (S 3 p/q (K ), i) as Q-graded groups for all 0 ≤ i ≤ p − 1. By comparing the d-invariants of these groups and applying (2.5), this shows that
V k (K) = V k (K ) for all 0 ≤ k ≤ p+q−1 2q . Since p/q > 2ν + (K) − 1, it follows that V p+q−1 2q (K) = V p+q−1 2q (K ) = 0. This shows that V k (K) = V k (K )
for all k ≥ 0 and also that ν + (K) = ν + (K ). If S 3 p/q (K) is an L-space, then we necessarily have A red,k (K) = A red,k (K ) = 0 for all k. Thus we can only need to establish the proposition under conditions (ii) and (iii). This is done by examining the absolute grading on the reduced part of HF + (S 3 p/q (K)). Since p/q > 2g(K) − 1, Proposition 2.8 shows that for any 0 ≤ i ≤ p+q−1 2 the reduced homology group takes the form
HF red (S 3 p/q (K), i) ∼ = A red, i q (K). In particular, for 0 ≤ k < g(K) ≤ p+q−1 2q we have HF red (S 3 p/q (K), kq) ∼ = · · · ∼ = HF red (S 3 p/q (K), kq + q − 1)
. Moreover, by (2.3), these isomorphisms preserve the absolute Q-gradings up to a constant shift. That is, if there is an element of HF red (S 3 p/q (K), kq) with grading x + d(p, q, kq), then for any 0 ≤ j ≤ q − 1, there is an element of HF red (S 3 p/q (K), kq + j) with grading
x + d(p, q, kq + j). It is this constant grading shift property which we will use to prove the proposition. Proposition 2.8 shows that for any 0
≤ i ≤ p+q−1 2 , HF red (S 3 p/q (K), i) ∼ = A red, i q (K) ∼ = A red, i q (K ) ⊕ s =0 A red, i+ps q (K ). Suppose that we do not have A red,k (K) ∼ = A red,k (K ) for all k ≥ 0. Let m ≤ g(K) − 1 be maximal such that A red,m (K) ∼ = A red,m (K ). This means that there is s = 0 such that A red, mq+q−1+ps q (K ) = 0.
The maximality of m implies A red,m+1 (K) ∼ = A red,m+1 (K ). It follows that we must have
mq + q − 1 + ps q < mq + q + ps q ,
and hence that q divides ps . As gcd(p, q) = 1, this shows that s takes the form s = tq for some t ∈ Z.
For any 0 ≤ i ≤ p+q−1 2 , (2.3) and (2.4) show that (s, 1) ∈ (s, A T ps+i q ) ⊆ X + i,p/q has grading given by (2.6) gr(s, 1) = d(p, q, i) − 2V i+sp q + 2 s−1 k=1 i+pk q if s ≥ 1, d(p, q, i) − 2V i+sp q − 2 0 k=−s i+pk q if s ≤ 0. If A red,m (K) ∼ = T (N m ), then it cannot be decomposed as a non-trivial direct sum of F[U ]-modules. So if (ii) holds, then we must have HF red (S 3 p/q (K), mq) ∼ = HF red (S 3 p/q (K), mq + 1) ∼ = A red,m+pt (K )
, for some t = 0. However (2.6) shows that if A red,m+pt (K ) is to be endowed with the correct grading in both X + mq,p/q and X + mq+1,p/q , then
tq−1 k=1 mq + pk q = tq−1 k=1 mq + 1 + pk q if t > 0 and 0 k=−tq mq + pk q = 0 k=−tq mq + 1 + pk q if t < 0.
(2.7)
However, for any r ∈ Z such that rp ≡ −1 mod q, we have mq + pr q < mq + 1 + pr q .
Since we can find such an r in the range 1 ≤ r ≤ q − 1, we see that the equalities in (2.7) cannot hold if t = 0. This completes the proof when condition (ii) holds. If q ≥ 3, then consider any t = 0 for which A red,m+pt (K ) = 0. By comparing the sums in (2.6) for different values of i, we see that if t > 0 and A red,m+pt (K ) contributes a term with grading x + d(p, q, mq + 1) to HF red (S 3 p/q (K), mq + 1), then the corresponding term it contributes to HF red (S 3 p/q (K), mq + 2) has grading strictly greater than x + d(p, q, mq + 2). Similarly, if t < 0 and A red,m+pt (K ) contributes a term with grading x + d(p, q, mq + 1) to HF red (S 3 p/q (K), mq + 1), then the term it contributes to HF red (S 3 p/q (K), mq) has grading strictly greater than x + d(p, q, mq). In particular, such an A red,m+pt (K ) always produces a grading on HF red (S 3 p/q (K), mq + 1) which is too small when compared to the gradings on HF red (S 3 p/q (K), mq) and HF red (S 3 p/q (K), mq+2). This completes the proof when q ≥ 3.
2.4. The d-invariants of lens spaces. In this section, we prove the congruence properties of d-invariants that we will require. Ozsváth and Szabó have shown that the d-invariants of lens spaces can be calculated recursively for 0 ≤ i ≤ p − 1 using
(2.8) d(p, q, i) = − 1 4 + (p + q − 1 − 2i) 2 4pq − d(q, r, i ),
where q ≡ r mod p and i ≡ i mod q, and
d(1, 0, 0) = d(S 3 ) = 0.
It will be temporarily convenient to work with a rescaled version of the d-invariants. Let d(p, q, i) = 2pd(p, q, i). By (2.8), these satisfy
(2.9)d(p, q, i) = (p + q − 1 − 2i) 2 − pq − 2pd(q, r, i ) 2q .
Lemma 2.11. For all i, j in the range 0 ≤ i, j ≤ p − 1, the quantityd(p, q, i) −d(p, q, j) is an integer satisfying
(2.10)d(p, q, i) −d(p, q, j) ≡ 2(i − j)(p + 1) mod 4 and (2.11) q(d(p, q, i) −d(p, q, j)) ≡ 2(pq + q − 1 − i − j)(j − i) mod 4p
Proof. We prove both (2.10) and (2.11) by induction on p. Asd(1, 0, 0) = 0, the required identities are clearly true for p = 1. The inductive step is carried out by performing some elementary but slightly masochistic modular arithmetic. From (2.9), we have (2.12) q(d(p, q, i) −d(p, q, j)) = 2(p + q − 1 − (i + j))(j − i) + p(d(q, r, j ) −d(q, r, i )),
where 0 ≤ i , j , r ≤ q − 1 are congruent modulo q to i, j and p respectively. By the inductive hypothesis we know that (2.13)d(q, r, j ) −d(q, r, i ) ≡ 2(j − i )(q + 1) mod 4 and (2.14)
r(d(q, r, j ) −d(q, r, i )) ≡ 2(qr + r − 1 − i − j )(i − j ) mod 4q.
. We first prove (2.11) by reducing (2.12) modulo 4p. If q is odd, then (2.13) shows that d(q, r, j ) −d(q, r, i ) ≡ 0 mod 4. Therefore, (2.12) gives
q(d(p, q, i) −d(p, q, j)) ≡ 2(p + q − 1 − i − j)(j − i) mod 4p,
as required. If q is even, then i − j ≡ i − j mod 2. So (2.13) shows that
p(d(q, r, j ) −d(q, r, i )) ≡ 2(j − i)p mod 4p.
So when q is even, (2.12) gives
q(d(p, q, i) −d(p, q, j)) ≡ 2(q − 1 − i − j)(j − i) mod 4p,as required.
. To prove (2.10), we consider the result of reducing (2.12) modulo 4q. If we write i = i +αq and j = j + βq, then one can check that
(2.15) 2(p + q − 1 − i − j)(j − i) ≡ 2(p + q − 1 − i − j )(j − i ) + 2q(α + β)(p + 1) mod 4q.
On the other hand, by using (2.13) and (2.14), we find that
p(d(q, r, j ) −d(q, r, i )) ≡ 2(p − r)(i − j )(q + 1) + 2(qr + r − 1 − i − j )(i − j ) mod 4q ≡ 2(i − j )(pq + p − 1 − i − j ) mod 4q.
(2.16)
By summing (2.15) and (2.16), we obtain
q(d(p, q, i) −d(p, q, j)) ≡ 2q(p + 1)(i − j ) + 2q(α + β)(p + 1) mod 4q ≡ 2q(p + 1)(i + j + α + β) mod 4q. (2.17)
Since the right hand side of (2.17) is divisible by q, it follows thatd(p, q, i) −d(p, q, j) is an integer. If p is odd, then p + 1 is even, so (2.17) shows that
q(d(p, q, i) −d(p, q, j)) ≡ 0 mod 4q.
Thus if p is odd, thend(p, q, i) −d(p, q, j) ≡ 0 mod 4, as required. If p is even, then q is necessarily odd. In this case we have i + j ≡ i + j + α + β mod 2. So (2.17) also implies (2.10) when q is odd. This completes the proof.
This allows us to prove the congruence result for d-invariants we require.
Corollary 2.12. We have
d(p, q, i) − d(p, q, j) ∈ 2Z ⇔ (q − 1 − i − j)(j − i) ≡ 0 mod p if p is odd (q − 1 − i − j)(j − i) ≡ 0 mod 2p if p is even.
Proof. We prove this by showing that
d(p, q, i) −d(p, q, j) ∈ 2pZ ⇔ (q − 1 − i − j)(j − i) ≡ 0 mod p if p is odd (q − 1 − i − j)(j − i) ≡ 0 mod 2p if p is even.
. If p is odd, then Lemma 2.11 shows d(p, q, i) −d(p, q, j) ≡ 0 mod 4.
Consequently we see thatd(p, q, i)−d(p, q, j) ≡ 0 mod 4p if and only ifd(p, q, i)−d(p, q, j) ≡ 0 mod p. From (2.11), we see that
q(d(p, q, i) −d(p, q, j)) ≡ 2(q − 1 − i − j)(j − i) mod p.
Since q is coprime to p, this is congruent to 0 if and only if
(q − 1 − i − j)(j − i) ≡ 0 mod p,
as required.
. If p is even, then q is necessarily odd. As an odd q is invertible modulo 4p, it follows from (2.11) thatd(p, q, i) −d(p, q, j) ≡ 0 mod 4p if and only if 2(q − 1 − i − j)(j − i) ≡ 0 mod 4p. Equivalently, if and only if
(q − 1 − i − j)(j − i) ≡ 0 mod 2p,
as required.
One other result we will require is an a bound on the absolute value of the d-invariants of lens spaces.
Lemma 2.13. For any 1 ≤ q ≤ p − 1 and any 0 ≤ i ≤ p − 1, we have
|d(p, q, i)| ≤ p − 1 4 .
Proof. Since the d-invariants of a rational homology sphere satisfy d(−Y, s) = −d(Y, s) for any s ∈ Spin c (Y ), we see that for any 1 ≤ i ≤ p − 1, there is 0 ≤ j ≤ p − 1 such that d(p, q, i) = −d(p, p − q, j). Therefore, to prove the lemma, it is sufficient to show that d(p, q, i) ≥ 1−p 4 . Since d(1, 0, 0) = 0, we can assume p > 1. Suppose that p/q > 1 has a continued fraction expansion
p/q = a 1 − 1 a 2 − 1 . . . − 1 a l ,
where a i ≥ 2 for all i. This expansion has length l ≤ p − 1. If M is the matrix
M = a 1 −1 −1 a 2 −1 −1 . . . −1 −1 a l
, then Ozsváth and Szabó have shown that for any i, there is v ∈ Z l such that [OS03b,OS05] 4d(p, q, i) = v T M −1 v − l.
Since M and, hence M −1 , is positive definite, this shows that 4d(p, q, i) ≥ l ≥ 1 − p.
This gives the desired lower bound.
3. Proving Theorems 1.7 and 1.8
Let Y be a 3-manifold such that Y ∼ = S 3 p/q (K ) ∼ = S 3 p/q (K) for knots K and K in S 3 and some p/q > 0. By (2.5), these two surgery descriptions of Y give labelings φ K,p/q , φ K ,p/q : Z/pZ −→ Spin c (Y ), such that the following diagram commutes
Spin c (Y ) d % % Z/pZ φ K,p/q o o D Z/pZ φ K ,p/q O O D / / Q, where D(i) = d(p, q, i) − 2 max{V i q (K), V p−i q (K)} and D (i) = d(p, q, i) − 2 max{V i q (K ), V p−i q (K )} for 0 ≤ i ≤ p − 1. Thus if we let φ denote map φ := φ −1 K,p/q • φ K ,p/q : Z/pZ → Z/pZ and f (i) = min{ i q , p−i q }, then for 0 ≤ i ≤ p − 1, we have (3.1) 2V f (i) (K) − 2V f (φ(i)) (K ) = d(p, q, i) − d(p, q, φ(i)).
Remark 3.1. There are two important consequences of (3.1). Firstly, it shows
d(p, q, i) − d(p, q, φ(i)) ∈ 2Z.
Secondly, d(p, q, i) − d(p, q, φ(i)) > 0 implies that V f (i) (K) > 0 and hence that ν + (K) ≥ f (i) + 1.
There are three possible forms for φ.
Proposition 3.2. The map φ : Z/pZ → Z/pZ takes one of the following forms:
I: φ(i) = a(i − s) + s mod p, where p is odd, a 2 ≡ 1 mod p and s ∈ { q−1 2 , p+q−1 2 } ∩ Z. II: φ(i) = a(i − s) + s mod p,
where p is even, a 2 ≡ 1 mod 2p and s = q−1 2 ; or III: φ(i) = a(i − s) + s + p 2 mod p, where p ≡ 0 mod 8, a 2 ≡ p + 1 mod 2p and s = q−1 2 . Proof. Let J : Z/pZ → Z/pZ be the map J(i) = q − 1 − i mod p. Since the d-invariants are invariant under conjugation, (2.2) shows that J • φ = φ • J. Since φ is an affine bijection, we may assume that it can be written in the form φ(i) = a(i − s 0 ) + s 1 mod p, for some s 0 ∈ Fix(J) = { q−1 2 , p+q−1 2 } ∩ Z and some a ∈ (Z/pZ) × . Using the invariance of d-invariants under conjugation, we obtain J(s 1 ) = J(φ(s 0 )) = φ(J(s 0 )) = s 1 .
This shows that we also have s 1 ∈ Fix(J). . First assume that p is odd. Since |Fix(J)| = 1 in this case, we have s 0 = s 1 . Since d(p, q, 1 + s 0 ) − d(p, q, φ(1 + s 0 )) ∈ 2Z, Corollary 2.12 shows that
0 ≡ (φ(1 + s 0 ) + 1 + s 0 − q + 1)(φ(1 + s 0 ) − (1 + s 0 )) mod p ≡ (a + s 0 + 1 + s 0 − (q − 1))(a + s 0 − 1 − s 0 )) mod p ≡ a 2 − 1 mod p.
This shows φ takes the form given by type I. . Now assume that p is even and s 0 = s 1 . Since q is necessarily odd, we may assume that s 0 = q−1 2 . Since d(p, q, 1 + s 0 ) − d(p, q, φ(1 + s 0 )) ∈ 2Z, Corollary 2.12 shows that 0 ≡ (φ(1 + s 0 ) + 1 + s 0 − q + 1)(φ(1 + s 0 ) − (1 + s 0 )) mod 2p
≡ (a + s 0 + 1 + s 0 − (q − 1))(a + s 0 − 1 − s 0 )) mod 2p
≡ a 2 − 1 mod 2p.
This shows φ takes the form given by type II.
. Finally, assume that p is even and s 0 = s 1 . We may assume that s 1 = p+q−1 2 = s 0 + p 2 . Since d(p, q, s 0 ) − d(p, q, φ(s 0 )) ∈ 2Z Corollary 2.12 shows that
0 ≡ (φ(s 0 ) + s 0 − q + 1)(φ(s 0 ) − s 0 ) mod 2p ≡ (s 0 + s 1 − (q − 1))(s 1 − s 0 ) mod 2p ≡ p 2 4 mod 2p,
which implies that p ≡ 0 mod 8. Similarly, from d(p, q, s 0 + 1) − d(p, q, φ(s 0 + 1)) ∈ 2Z, we obtain
0 ≡ (φ(1 + s 0 ) + 1 + s 0 − q + 1)(φ(1 + s 0 ) − 1 − s 0 ) mod 2p ≡ (a + s 1 + 1 + s 0 − (q − 1))(a + s 1 − s 0 − 1) mod 2p ≡ (a + p 2 ) 2 − 1 mod 2p ≡ a 2 + p − 1 mod 2p.
This shows φ takes the form given by type III.
This allows us to put bounds on ν + (K) when φ is not the identity or the map corresponding to conjugation. Lemma 3.3. If φ K,p/q = φ K ,p/q and φ K,p/q = φ K ,p/q , then
ν + (K) > p 4q + 1 2 − 3 q − q.
Proof. Consider the map
φ = φ −1 K,p/q • φ K ,p/q : Z/pZ → Z/pZ
Proposition 3.2 shows that if φ K,p/q = φ K ,p/q and φ K,p/q = φ K ,p/q , then φ takes the form φ(x) = a(x − s 0 ) + s 1 for some a ≡ ±1 mod p satisfying a 2 ≡ 1 mod p. Since d-invariants are invariant under conjugation, we can assume that a lies in the range √ p < a < p/2.
. Since φ satisfies φ(x + n) ≡ φ(x) + na mod p for all x and n, we see that for any p+q−1
2 ≥ N ≥ p a , we can find x in the range N − p a ≤ x ≤ N such that p + q − 1 − a 2 ≤ φ(x) ≤ p + q − 1 + a 2 .
For such an x we have
d(p, q, x) − d(p, q, φ(x)) = (p + q − 1 − x − φ(x))(φ(x) − x) pq − d(q, r, x) + d(q, r, φ(x)) > (p + q − 1 + a − 2x)(p + q − 1 − a − 2x) 4pq − q 2 = (p + q − 1 − 2x) 2 − a 2 4pq − q 2 ≥ (p + q − 1 − 2N ) 2 − a 2 4pq − q 2 ,
where we used the bound |d(q, r, φ(x)) − d(q, r, x)| < q 2 arising from Lemma 2.13 to obtain the second line, and the final line was obtained by observing that the quadratic in the preceding line is minimized for x in the range N − p a ≤ x ≤ N by taking x = N .
. Thus if we take
N = p + q − 1 − 2pq 2 + a 2 2 ,
then such an x satisfies d(p, q, x) − d(p, q, φ(x)) > 0. And hence we see that there is k ∈ Z such that V k > 0 and
(3.2) k ≥ p + q − 1 − 2pq 2 + a 2 2q − p aq .
We complete the proof by finding a lower bound for k which is independent of a.
Consider √ 2pq 2 +a 2 2q + p aq as a function of a. For a > 0, this has a single critical value which is a minimum. Thus we see that for a in the range √ p ≤ a ≤ p 2 the minimal value of the right hand side of (3.2) is attained by a = √ p or a = p 2 . Therefore, using the bound 2pq 2 + a 2 ≤ a + pq 2 a when a = p 2 , we obtain
k ≥ p + q − 1 2q − max{ p + 4q 2 + 8 4q , √ p(2 + 2q 2 + 1) 2q } .
However, one can show that 3
max{ p + 4q 2 + 8 4q , √ p(2 + 2q 2 + 1) 2q } < p + 4q 2 + 10 4q .
This gives the bound
k ≥ p 4q + 1 2 − 3 q − q ,
showing that
ν + (K) > p 4q + 1 2 − 3 q − q,
as required.
We now have all the pieces to prove or main technical results.
Theorem 1.7. Let K, K ⊆ S 3 be knots such that S 3 p/q (K) ∼ = S 3 p/q (K ). If |p| ≥ 12 + 4q 2 − 2q + 4qg(K) and q ≥ 3, then ∆ K (t) = ∆ K (t), g(K) = g(K ) and K is fibred if and only if K is fibred.
Proof. Suppose that S 3 p/q (K) ∼ = S 3 p/q (K ). Since S 3 −p/q (K) ∼ = −S 3 −p/q (K) for any K ⊆ S 3 , we may assume that p/q > 0. By Lemma 3.3, the bound p ≥ 12 + 4q 2 − 2q + 4qg(K) implies that either φ K,p/q = φ K ,p/q or φ K,p/q = φ K ,p/q . In either case, the assumption q ≥ 3 allows us to apply Lemma 2.10, which shows that V k (K) = V k (K ) and A red,k (K) ∼ = 3 When considered as a quadratic in √ p, the discriminant of p − 2(2 + 2q 2 + 1) √ p + 4q 2 + 10 is ∆ = 4 2q 2 + 1 − 2q 2 − 4. As this satisfies ∆ < 0 for all q, we see that √ p(2 + 2q 2 + 1) 2q < p + 4q 2 + 10 4q for all p and q.
A red,k (K ) for all k ≥ 0. By Proposition 2.3, this is sufficient information to guarantee that K and K have the same Alexander polynomial, genera and fibredness.
Theorem 1.8. Suppose that K is an L-space knot. If S 3 p/q (K) ∼ = S 3 p/q (K ) for some K ⊆ S 3 and either (i) p ≥ 12 + 4q 2 − 2q + 4qg(K) or (ii) p ≤ min{2q − 12 − 4q 2 , −2qg(K)} and q ≥ 2 holds, then ∆ K (t) = ∆ K (t), g(K) = g(K ) and K is fibred.
Proof. Suppose that S 3 p/q (K) ∼ = S 3 p/q (K ), where K is an L-space knot. First suppose that p ≥ 4q 2 + 12 − 2q + 4qg(K).
In this case, S 3 p/q (K) is an L-space. Lemma 3.3 implies that either φ K,p/q = φ K ,p/q or φ K,p/q = φ K ,p/q . Thus we can apply Lemma 2.10 which shows that V k (K) = V k (K ) and A red,k (K) ∼ = A red,k (K ) = 0 for all k. By Proposition 2.3, this is sufficient information to guarantee that K and K have the same Alexander polynomial and genus. As K is fibred it also shows that K is fibred.
If p ≤ min{2q − 12 − 4q 2 , −2qg(K)} and q ≥ 2, then we use the fact that S 3 −p/q (K) ∼ = −S 3 p/q (K ). As ν + (K) = 0, the condition −p ≥ 4q 2 + 12 − 2q shows that φ K,−p/q = φ K ,−p/q or φ K,−p/q = φ K ,−p/q . As −p/q > 2qg(K) − 1 and A red,k (K) ∼ = T (V |k| (K)) for all k, Lemma 2.10 shows that V k (K) = V k (K ) = 0 and A red,k (K) ∼ = A red,k (K ) for all k ≥ 0. This shows that K and K have the same Alexander polynomial and genus. As K is fibred it also shows that K is fibred.
Surgeries on torus knots
In this section, we prove Theorem 1.1. In order to do this we will need to understand the manifolds obtained by surgery on torus knots. It is well-known that, with the exception of a single reducible surgery for each torus knot, the manifolds obtained by surgery on a torus knot are Seifert fibred spaces [Mos71]. We will use S 2 (e; b1 a1 , b2 a2 , b3 a3 ) to denote the 3-manifold obtained by surgery on the link given in Figure 1. 4 This is a Seifert fibred space when a i = 0 for i = 1, 2, 3 and is a lens space only if |a i | = 1 for some i. Recall that if |a i | > 1 for i = 1, 2, 3, then
S 2 (e; b 1 a 1 , b 2 a 2 , b 3 a 3 ) ∼ = S 2 (e ; d 1 c 1 , d 2 c 2 , d 3 c 3 )
if and only if
e + b 1 a 1 + b 2 a 2 + b 3 a 3 = e + d 1 c 1 + d 2 c 2 + d 3 c 3 and there is a permutation π of {1, 2, 3} such that b i a i ≡ c π(i) d π(i) mod 1 for i = 1, 2, 3 [NR78].
We can describe surgeries on torus knots as follows.
4 In this notation we have S 3 1 (T 3,2 ) ∼ = S 2 (−2;
1 2 , 2 3 , 4 5 ) ∼ = P,
where P is the Poincaré sphere oriented so that it bounds the positive-definite E 8 -plumbing. Proposition 4.1 (cf. Moser [Mos71]). For r, s > 1 and any p/q,
S 3 p/q (T r,s ) ∼ = S 2 (e; s s , r r , q p − rsq ),
where r , s and e are any integers satisfying rs + sr + ers = 1.
Proof. Consider the Seifert fibration of S 3 with exceptional fibres of order r and s for which the regular fibres are isotopic to the torus knot T r,s . We can obtain S 3 p/q (T r,s ) by surgering a regular fibre K ⊆ S 3 . Let µ be a meridian for K and λ a null-homologous longitude. As the linking number between K and a nearby regular fibre is rs, the surgery slope can be written as
pµ + qλ = (p − qrs)µ + qκ,
where κ is a longitude for K given by a regular fibre. This shows that
S 3 p/q (T r,s ) ∼ = S 2 (e; s s , r r , q p − rsq ),
for some s , r , e ∈ Z which are independent of p/q. Considering the order of the homology group H 1 (S 3 p/q (T r,s )) shows that |p| = |rs(p − rsq)(e + s s + r r + q p − rsq )| = |p(rs + sr + rse) − qrs(rs + sr + ers − 1)|.
As this holds for any q it follows that we have rs + sr + ers = 1, as required.
We will also use the Casson-Walker invariant [Wal92]. For any rational homology sphere Y , this is a rational-valued invariant λ(Y ) ∈ Q. For our purposes its most useful property is that it is easily computed for manifolds obtained by surgery. For any knot K ⊆ S 3 , the Casson-Walker invariant satisfies [BL90]:
(4.1) λ(S 3 p/q (K )) = λ(S 3 p/q (K)) + q 2p ∆ K (1).
In particular, this means that if S 3 p/q (K) ∼ = S 3 p/q (T r,s ), then the Alexander polynomial of K satisfies (4.2) ∆ K (1) = ∆ Tr,s (1) = (r 2 − 1)(s 2 − 1) 12 .
Recall also that if K is a satellite knot with pattern P and companion K , then its Alexander polynomial satisfies
∆ K (t) = ∆ K (t w )∆ P (t),
where w is the winding number of P . In particular, this means that (4.3) ∆ K (1) = ∆ P (1) + w 2 ∆ K (1).
Next we consider the possibility that a torus knot and a cable of a torus knot share a non-integer surgery.
Proposition 1.5. For s > r > 1 and q ≥ 2, there exists a non-trivial cable of a torus knot K such that S 3 p/q (K) ∼ = S 3 p/q (T r,s ) if and only if s = rq 3 ± 1 q 2 − 1 , p = r 2 q 4 − 1 q 2 − 1 , q = s/r and r > q, in which case K is the (q, q 2 r 2 −1 q 2 −1 )-cable of T r, rq±1
q 2 −1 .
Proof. Suppose that K is the (w, c)-cable of T a,b , where w > 1 is the winding number of the pattern and that this satisfies
Y ∼ = S 3 p/q (K) ∼ = S 3 p/q (T r,s )
. If Y is reducible, then p/q = wc = rs is an integer. So we can assume from now on that Y is irreducible. We will temporarily drop the assumption that s > r and assume for now only that r, s > 1.
Since Y does not contain an incompressible torus, p/q must take the form p q = wc ± 1 q and Y ∼ = S 3 p/(qw 2 ) (T a,b ) [Gor83]. As q > 1 and gcd(w, c) = 1, we have |qwc ± 1 − abqw 2 | = |qw(c − abw) ± 1| ≥ |qw| − 1.
This shows that Y is not a lens space if q > 1. Thus we can assume Y is a Seifert fibred space with base orbifold S 2 (r, s, |p − rsq|) = S 2 (|a|, |b|, |p − abqw 2 |).
By considering the order of the exceptional fibers, we see we may assume that a = r. Using the Casson-Walker invariant as in (4.2) and (4.3), we see that (a 2 − 1)(s 2 − 1) = w 2 (a 2 − 1)(b 2 − 1) + (c 2 − 1)(w 2 − 1).
This implies that |b| < s. Thus we must have ε 1 s = p − abqw 2 and ε 2 b = p − asq for some ε 1 , ε 2 ∈ {±1}. Solving these simultaneous equations shows that
(4.4) s = p(aqw 2 − ε 2 ) (aqw) 2 − ε 1 ε 2 and b = p(aq − ε 1 ) (aqw) 2 − ε 1 ε 2 .
Since a, s > 1 it follows that p > 0 and b > 0. Using Proposition 4.1 and our two surgery descriptions, we see that Y can be written in the form
Y ∼ = S 2 (−1; a a , s s , ε 2 q b ) ∼ = S 2 (−1; a a , ε 1 qw 2 s , b b )
for some a , b , s ∈ Z, where 1 ≤ b < b and 1 ≤ s < s. Comparing these two descriptions of Y as a Seifert fibred space we see that b satisfies (4.5) b ≡ ε 2 q mod b and b a ≡ 1 mod b; s satisfies (4.6) s ≡ ε 1 qw 2 mod s and s a ≡ 1 mod s;
and we have
(4.7) − 1 + a a + ε 2 q b + s s = −1 + a a + ε 1 qw 2 s + b b .
Claim. We have ε 1 = ε 2 .
Proof of Claim. By (4.7), we see that
s − ε 1 qw 2 s = b − ε 2 q b .
Equations (4.5) and (4.6) show that both sides of this equation are integers. Moreover the assumptions that 1 ≤ s < s and 1 ≤ b < b imply that the right hand side (respectively left hand side) is strictly greater than 0 if and only if ε 1 = −1 (respectively ε 2 = −1). It follows that ε 1 = ε 2 , as required.
In light this claim, we will take ε = ε 1 = ε 2 ∈ {±1} from now on.
Observe that (4.5) and (4.6) imply that b divides aq−ε and s divides aqw 2 −ε. Combining this with (4.4) shows that there is a positive integer k such that kb = aq − ε, ks = aqw 2 − ε and kp = (aqw) 2 − 1.
Since we can write aqw 2 − ε = w 2 (aq − ε) + ε(w 2 − 1), we also see that k also divides w 2 − 1. Now p takes the form p = qwc + δ for some δ ∈ {±1}. Therefore we have kp = qwck + δk = (qwa) 2 − 1.
In particular, we have
(4.8) δk = qwN − 1, where N = qwa 2 − ck.
Claim. We have qN ∈ {0, 1, w}.
Proof of Claim. Assume that qN is non-zero. Since k divides w 2 − 1, (4.8) shows that qwN − 1 divides w 2 − 1. Let α ∈ Z be such that (qwN − 1)α = w 2 − 1. By considering this equation mod w, we see that α takes the form α = βw + 1 for some β ∈ Z. Substituting for α and rearranging shows that β satisfies
(βqN − 1)w = β − qN.
If β ≤ −1 or β ≥ 2, then there are no possible integer choices of qN for which w = β−qN βqN −1 ≥ 2 holds. So it remains only to consider are β = 0 and β = 1. The former implies that qN = w and the latter can only hold if qN = 1, completing the proof of the claim. This gives us three 3 possibilities for qN to consider. If qN = 0, then N = 0 and (4.8) shows that δk = −1. This implies that k = 1 and hence c = qwa 2 . This contradict the fact that gcd(w, c) = 1.
As we are assuming q ≥ 2, qN = 1 can't possibly hold. Thus it remains only to consider the possibility qN = w. In this case (4.8) shows that δ = 1, k = w 2 − 1 and, c = w(q 2 a 2 − 1) q(w 2 − 1) .
As gcd(w, c) = 1, gcd(q 2 a 2 − 1, q) = 1 and gcd(w 2 − 1, w) = 1, this shows that q = w. It follows that s = rq 3 −ε q 2 −1 , p = r 2 q 4 −1 q 2 −1 , c = q 2 r 2 −1 q 2 −1 and b = rq−ε q 2 −1 . This shows that K must take the required form and that s and p satisfy the required conditions.
As K is a non-trivial cable we have b = rq−ε q 2 −1 > 1. This will allow us to derive the condition that r > q. If ε = 1, then b > 1 clearly implies r > q. If ε = −1, then b ∈ Z implies that r ≡ −q mod q 2 − 1. Thus we have r ≥ q 2 − q − 1, which implies that b ≥ q − 1 with equality only if r = q 2 − q − 1. Thus b > 1 implies that either q > 2 or r ≥ 2q 2 − q − 2. In either case this is sufficient to guarantee that r > q when ε = −1.
Finally, observe that
rq < s = rq + rq − ε q 2 − 1 ≤ rq + 2r + 1 3 < r(q + 1),
which implies that q = s/r . This completes the proof of one direction. Conversely, if s = rq 3 ±1 q 2 −1 ∈ Z, then r ≡ ∓q mod q 2 − 1, so we have p = r 2 q 4 −1 q 2 −1 ∈ Z, b = rq±1 q 2 −1 ∈ Z and c = q 2 r 2 −1 q 2 −1 ∈ Z. This allows us to take K to be the (q, c)-cable of T r,b . Note that r > q implies b > 1, so K is cable of a non-trivial torus knot. It is then a straightforward calculation, using Proposition 4.1, that for such a K we have
S 3 p/q (T r,s ) ∼ = S 3 p/q (K) ∼ = S 3 p/q 3 (T r,b ) ∼ = S 2 (0; 1 − q 2 r , ∓q 3 s , ∓q b ),
as required.
Now we consider the possibility that two torus knots share a surgery. The following generalizes [NZ14, Proposition 2.4].
Lemma 4.2. If S 3 p/q (T r,s ) ∼ = S 3 p/q (T a,b ), for some p/q ∈ Q, then T r,s = T a,b or p/q ∈ {rs ± 1}.
Proof. Suppose that Y ∼ = S 3 p/q (T r,s ) ∼ = S 3 p/q (T a,b ). Without loss of generality we may assume that r, s > 1.
If p/q = rs, then Y is reducible. In this case we must have ab = rs and Y ∼ = S 3 r/s (U )#S 3 s/r (U ) ∼ = S 3 a/b (U )#S 3 b/a (U ). Therefore T r,s = T a,b , in this case.
If |p − rsq| > 1, then Y is a Seifert fibred space with base orbifold S 2 (r, s, |p − rsq|) ∼ = S 2 (|a|, |b|, |p − abq|).
It follows that we can assume r = a. By applying the Casson-Walker invariant as in (4.2), we have (a 2 − 1)(b 2 − 1) = (r 2 − 1)(s 2 − 1), which implies that s = |b|. Thus, if T a,b = T r,s , then T a,b = T r,−s . However, T a,b = T r,−s implies that |p − rsq| = |p + rsq|, and hence that p = 0. It is easy to check, using Proposition 4.1, that S 3 0 (T r,s ) ∼ = S 3 0 (T −r,s ). Thus it only remains to consider the case that |p − rsq| = 1 and q > 1. In this case, we also have |p − abq| = 1 and Y is the lens space [Mos71] Y ∼ = L(p, qr 2 ) ∼ = L(p, qa 2 ).
We can assume that 1 < a < b and 1 < r < s. For L(p, qr 2 ) to admit an orientation preserving homeomorphism to L(p, qa 2 ) we must have either qr 2 ≡ qa 2 mod p or q 2 r 2 a 2 ≡ 1 mod p. As qr 2 < p and qa 2 < p, we see that qr 2 ≡ qa 2 mod p implies that r = a and hence that T r,s = T a,b . Thus we can assume that q 2 r 2 a 2 ≡ 1 mod p. Consider the continued fraction expansion s/r = [a 0 , . . . , a k ] + , where a k ≥ 2 and a i ≥ 1 for 0 ≤ i ≤ k − 1. Using some standard identities for continued fractions one can show that 5 a 1 , . . . , a k , q − 1, 1, a k − 1, a k−1 , . . . , a 1 a 1 , . . . , a k−1 , a k − 1, 1, q − 1, a k , . . . , a 1
p qr 2 = [a 0 ,] + if p = qrs + (−1) k [a 0 ,] + if p = qrs − (−1) k .
Since both of these expansions have odd length, we see that if q 2 r 2 a 2 ≡ 1 mod p, then p
(4.10) p qa 2 = [b 0 , b 1 , . . . , b l , q − 1, 1, b l − 1, b l−1 , . . . , b 1 ] + if p = qab + (−1) l [b 0 , b 1 , . . . , b l−1 , b l − 1, 1, q − 1, b l , . . . , b 1 ] + if p = qab − (−1) l .
Since p qa 2 admits a unique continued fraction expansion of odd length with every coefficient strictly positive, we see that the two continued fraction expansions in (4.9) and (4.10) must be the same. Comparing the lengths of these two expansions for p qa 2 shows that l = k. Comparing the coefficients individually and using the assumption that a k , b k > 1 soon shows that s r = b a = [q, . . . , q k+1 ] + . Altogether, this shows that T r,s = T a,b unless p = rs ± 1.
Remark 4.3. When combined with the cyclic surgery theorem of Culler, Gordon, Luecke and Shalen [CGLS87], Lemma 4.2 implies that for any q > 1, any slope of the form p/q = rs ± 1 q is a characterizing slope for T r,s .
Using results of Agol [Ago00], Lackenby [Lac03], and Cao and Meyerhoff [CM01], Ni and Zhang give a restriction on exceptional slopes of a hyperbolic knot.
Proposition 4.4 (Lemma 2.2, [NZ14]). Let K ⊆ S 3 be a hyperbolic knot. If |p| ≥ 43 4 (2g(K) − 1) then S 3 p/q (K) is a hyperbolic manifold. This is the final ingredient we require for the proof of the main theorem.
Theorem 1.1. For s > r > 1 and q ≥ 2 let K be a knot such that S 3 p/q (K) ∼ = S 3 p/q (T r,s ). If p and q satisfy at least one of the following:
(i) p ≤ min{− 43 4 (rs − r − s), −32q}, (ii) p ≥ max{ 43 4 (rs − r − s), 32q + 2q(r − 1)(s − 1)}, or (iii) q ≥ 9, then we have either (a) K = T r,s , or (b) K is a cable of a torus knot, in which case q = s/r , p = r 2 q 4 −1 q 2 −1 , s = rq 3 ±1 q 2 −1 and r > q.
Proof. Let K be a knot such that S 3 p/q (K) ∼ = S 3 p/q (T r,s ) for some p/q with q ≥ 2 such that at least one of conditions (i), (ii) or (iii) are satisfied. Since 4q 2 + 12 − 2q < 32q for q ≤ 8 and g(T r,s ) = (r−1)(s−1) 2 , Theorem 1.8 shows that either (a) q ≥ 9 or (b) K is fibred with g(K) = g(T r,s ) and |p| ≥ 43 4 (2g(K) − 1). According to Thurston, every knot in S 3 is either a hyperbolic knot, a satellite knot or a torus knot [Thu82]. We consider each of these possibilities in turn.
Given Proposition 4.4 and the fact that any exceptional surgery on a hyperbolic knot in S 3 must satisfy q ≤ 8 [LM13], we see that K cannot be a hyperbolic knot.
If K is a satellite knot, then there is an incompressible torus R ⊆ S 3 \ K. This bounds a solid torus V ⊆ S 3 which contains K. Let K be the core of the solid torus V . By choosing R to be innermost, we may assume that K is not a satellite. This means that K is either a torus knot or a hyperbolic knot. Since S 3 p/q (T r,s ) is irreducible and does not contain an incompressible tori, it follows from the work of Gabai that V p/q (K) is again a solid torus and that K is either a 1-bridge knot or a torus knot in V and S 3 p/q (K) ∼ = S 3 p/(qw 2 ) (K ), where w > 1 is the winding number of K in V [Gab89]. Moreover, as q > 1, it follows that K is a torus knot in V .
If q ≥ 9, then the distance bound on exceptional surgeries shows that K is not hyperbolic. If q ≤ 8, then K is fibred, implying that K is also fibred [HMS08]. It follows, for example by considering the degrees of ∆ K (t) and ∆ K (t), that g(K ) < g(K). Therefore, if q ≤ 8, then we have |p| > 43 4 (2g(K ) − 1) and Proposition 4.4 shows that K is not hyperbolic. Thus we can assume that K is a torus knot. Thus, we have shown that if K is a satellite, then it is a cable of a torus knot. In this case Proposition 1.5 applies to give the desired conclusions on p, q, r and s.
If K is a torus knot, then Lemma 4.2 shows that K = T r,s , as required.
Corollary 1.2. The knot T r,s with r, s > 1 has only finitely many non-characterizing slopes which are not negative integers.
Proof. This follows from Theorem 1.1 and the results of [McC14] which show that any slope
p q ≥ 43 4 (rs − r − s)
is a characterizing slope for the torus knot T r,s with r, s > 1.
Figure 1 .
1A surgery presentation for the Seifert fibred space S 2 (e; b1 a1 , b2 a2 , b3 a3 )
1 , . . . , a k−1 , a k − 1, 1, q − 1, a k , . . . , a 0 ] + if p = qrs + (−1) k [a 1 , . . . , a k , q − 1, 1, a k − 1, a k−1 , . . . , a 0 ] + if p = qrs − (−1) k .However, if we consider the continued fraction expansion b/a = [b 0 , . . . , b l ] + , where b l ≥ 2 and b i ≥ 1 for 0 ≤ i ≤ l − 1, then
5
The stated equalities follow from applications of the following identities, all of which admit straightcn, . . . , c 1 ] + , (2) pnq n−1 − qnp n−1 = (−1) n+1 , and (3) for any x ∈ Q, [c 0 , . . . , cn, x] qn = [c 0 , . . . , cn] + denotes the nth convergent of a continued fraction with c i ≥ 1 for all i. 6 Here we are using the fact that if p q = [c 0 , . . . , cn] + , then p q = [cn, . . . , c 0 ] + , where qq ≡ (−1) n mod p.
Throughout the paper, we use Y ∼ = Y to denote the existence of a orientation-preserving homeomorphism between Y and Y .
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|
[] |
[
"A comparison study of medium-modified QCD shower evolution scenarios",
"A comparison study of medium-modified QCD shower evolution scenarios"
] |
[
"Thorsten Renk \nDepartment of Physics\nUniversity of Jyväskylä\nP.O. Box 35FI-40014Finland\n\nHelsinki Institute of Physics\nUniversity of Helsinki\nP.O. Box 64FI-00014Finland\n"
] |
[
"Department of Physics\nUniversity of Jyväskylä\nP.O. Box 35FI-40014Finland",
"Helsinki Institute of Physics\nUniversity of Helsinki\nP.O. Box 64FI-00014Finland"
] |
[] |
The computation of hard processes in hadronic collisions is a major success of perturbative Quantum Chromodynamics (pQCD). In such processes, pQCD not only predicts the hard reaction itself, but also the subsequent evolution in terms of parton branching and radiation, leading to a parton shower and ultimately to an observable jet of hadrons. If the hard process occurs in a heavy-ion collision, a large part of this evolution takes place in the soft medium created along with the hard reaction. An observation of jets in heavy-ion collision thus allows a study of medium-modified QCD shower evolution. In vacuum, Monte-Carlo (MC) simulations are well established tools to describe such showers. For jet studies in heavy-ion collisions, MC models for in-medium showers are currently being developed. However, the shower-medium interaction depends on the nature of the microscopic degrees of freedom of the medium created in a heavy-ion collision which is the very object one would like to investigate. This paper presents a study in comparison between three different possible implementations for the shower-medium interaction, two of them based on medium-induced pQCD radiation, one of them a medium-induced drag force, and shows for which observables differences between the three scenarios become visible. We find that while single hadron observables such as RAA are incapable of differentiating between the scenarios, jet observables such as the longiudinal momentum spectrum of hadrons in the jet show the potential to do so. PACS numbers: 25.75.-q,25.75.Gz
|
10.1103/physrevc.79.054906
|
[
"https://arxiv.org/pdf/0901.2818v2.pdf"
] | 119,267,075 |
0901.2818
|
f0f58d315f55311aaa4b743a0fffbb8add1f9aa9
|
A comparison study of medium-modified QCD shower evolution scenarios
28 Jan 2009
Thorsten Renk
Department of Physics
University of Jyväskylä
P.O. Box 35FI-40014Finland
Helsinki Institute of Physics
University of Helsinki
P.O. Box 64FI-00014Finland
A comparison study of medium-modified QCD shower evolution scenarios
28 Jan 2009
The computation of hard processes in hadronic collisions is a major success of perturbative Quantum Chromodynamics (pQCD). In such processes, pQCD not only predicts the hard reaction itself, but also the subsequent evolution in terms of parton branching and radiation, leading to a parton shower and ultimately to an observable jet of hadrons. If the hard process occurs in a heavy-ion collision, a large part of this evolution takes place in the soft medium created along with the hard reaction. An observation of jets in heavy-ion collision thus allows a study of medium-modified QCD shower evolution. In vacuum, Monte-Carlo (MC) simulations are well established tools to describe such showers. For jet studies in heavy-ion collisions, MC models for in-medium showers are currently being developed. However, the shower-medium interaction depends on the nature of the microscopic degrees of freedom of the medium created in a heavy-ion collision which is the very object one would like to investigate. This paper presents a study in comparison between three different possible implementations for the shower-medium interaction, two of them based on medium-induced pQCD radiation, one of them a medium-induced drag force, and shows for which observables differences between the three scenarios become visible. We find that while single hadron observables such as RAA are incapable of differentiating between the scenarios, jet observables such as the longiudinal momentum spectrum of hadrons in the jet show the potential to do so. PACS numbers: 25.75.-q,25.75.Gz
I. INTRODUCTION
Jet quenching, i.e. the energy loss of hard partons created in the first moments of a heavy ion collision due to interactions with the surrounding soft medium has long been regarded a promising tool to study properties of the soft medium [1,2,3,4,5,6]. The basic idea is to study the changes induced by the medium to a hard process which is well-known from p-p collisions. A number of observables is available for this purpose, among them suppression in single inclusive hard hadron spectra R AA [7], the suppression of back-to-back correlations [8,9], single hadron suppression as a function of the emission angle with the reaction plane [10] and most recently also preliminary measurements of jets have become available [11].
Single hadron observables and back-to-back correlations are well described in detailed model calculations using the concept of energy loss [12,13,14], i.e. under the assumption that the process can be described by a medium-induced shift of the leading parton energy by an amount ∆E, followed by a fragmentation process using vacuum fragmentation of a parton with the reduced energy. However, there are also calculations for these observables in which the evolution of the whole in-medium parton shower is followed in an analytic way [15,16,17]. Recently, also Monte Carlo (MC) codes for in-medium shower evolution have become available [18,19,20,21] which are based on MC shower simulations developed for hadronic collisions, such as PYTHIA [22] or HERWIG * Electronic address: [email protected] [23]. In medium-modified shower computations, energy is not simply lost but redistributed in a characteristic way.
All current in-medium shower MC codes model the interaction between partons and the medium in a different way. JEWEL (Jet Evolution With Energy Loss) [18] assumes either elastic collisions with thermal quasiparticles or, to implement radiative energy loss, an enhancement of the singular part of the parton branching kernels. Ya-JEM (Yet another Jet Energy-loss Model) [19,20] makes the assumption that the virtuality of partons traversing the medium grows according to the medium transport coefficientq which measures the virtuality gain per unit pathlength, and this medium-induced virtuality leads to increased radiation. Finally, Q-PYTHIA, the code presented in [21] is a direct extension of the leading parton energy loss computations done in [4,24] and uses the differential radiation probabilities originally computed from a single hard parton now for each parton propagating in the shower simulation. At this stage, it is hardly surprising that different models employ different implementations of the parton-medium interaction, as the nature of this interaction crucially depends on the microscopic properties of the medium, i.e. the very thing one wishes to determine from the experiments.
A suitable strategy to determine these properties is thus to study the effects of various different implementations of the parton-medium interaction for different observables. In this paper, we begin such a program by investigating the effects on a number of different observables resulting from three different scenarios: Medium-induced radiation by an increase of parton virtuality dependent on the mediumq as used in [19,20], an enhancement of the singular parts of the branching kernel leading to additional radiation as used in [18,25] and a drag force. Momentum-dependent drag forces appear in computations modelling QCD-like N = 4 super Yang-Mills theories via the AdS/CFT conjecture [26], in the present paper we use a simplified ansatz in which a parton in a constant medium undergoes a momentum independent energy loss per unit pathlength. Such a drag term has not been tested in an in-medium shower evolution MC code previously. The paper is organized as follows: First, we briefly review the computation of medium-modified hadron jet as done in [19,20]. In addition, we describe the three different implementations of the parton-medium interaction and its relation to the spacetime structure of the shower in detail. In a first comparison, we make the connection to previous leading parton energy loss calculations by considering a constant medium with fixed length. In this medium, we study the energy loss of the leading parton and present the result in terms of energy loss probability distributions and mean energy loss as a function of the parameters characterizing the medium. In a second comparison, we turn to a medium model which is closer to the experimental situation in so far as it expands and hence dilutes as a function of time. We compute various jet observables in this medium, such as the longitudinal momentum distribuion inside the jet or the angular broadening. Finally, in a last comparison we compute (as done in [19]) the suppression of the inclusive single hard hadron spectrum in terms of the nuclear suppression factor R AA and compare all scenarios with experimental results [7]. From this comparison, we tentatively deduce the relevant medium parameters. We conclude with a discussion of the implications of the results.
II. MEDIUM-MODIFIED SHOWER EVOLUTION
In this section, we describe how the medium-modified fragmentation function (MMFF) is obtained from a computation of an in-medium shower followed by hadronization. Key ingredient for this computation is a pQCD MC shower algorithm. In this work, we employ a modification of the PYTHIA shower algorithm PYSHOW [27]. In the absence of any medium effects, our algorithm therefore corresponds directly to the PYTHIA shower. Furthermore, the subsequent hadronization of the shower is assumed to take place outside of the medium, even if the shower itself was medium-modified. It is computed using the Lund string fragmentation scheme [28] which is also part of PYTHIA.
A. Shower evolution in vacuum
We model the evolution from some initial, highly virtual parton to a final state parton shower as a series of branching processes a → b+c where a is called the parent parton and b and c are referred to as daughters. In QCD, the allowed branching processes are q → qg, g → gg and g → qq. The kinematics of a branching is described in terms of the virtuality scale Q 2 and of the energy fraction z, where the energy of daughter b is given by E b = zE a and of the daughter c by E c = (1 − z)E a . It is convenient to introduce t = ln Q 2 /Λ QCD where Λ QCD is the scale parameter of QCD. t takes a role similar to a time in the evolution equations, as it describes the evolution from some high initial virtuality Q 0 (t 0 ) to a lower virtuality Q m (t m ) at which the next branching occurs. In terms of the two variables, the differential probability dP a for a parton a to branch is [29,30]
dP a = b,c α s 2π P a→bc (z)dtdz (1)
where α s is the strong coupling and the splitting kernels P a→bc (z) read
P q→qg (z) = 4/3 1 + z 2 1 − z (2) P g→gg (z) = 3 (1 − z(1 − z)) 2 z(1 − z) (3) P g→qq (z) = N F /2(z 2 + (1 − z) 2 )(4)
where we do not consider electromagnetic branchings. N F counts the number of active quark flavours for given virtuality. At a given value of the scale t, the differential probability for a branching to occur is given by the integral over all allowed values of z in the branching kernel as
I a→bc (t) = z+(t) z−(t) dz α s 2π P a→bc (z).(5)
The kinematically allowed range of z is given by
z ± = 1 2 1 + M 2 b − M 2 c M 2 a ± |p a | E a (M 2 a − M 2 b − M 2 c ) 2 − 4M 2 b M 2 c M 2 a(6)
where M 2 i = Q 2 i +m 2 i with m i the bare quark mass or zero in the case of a gluon. Given the initial parent virtuality Q 2 a or equivalently t a , the virtuality at which the next branching occurs can be determined with the help of the Sudakov form factor S a (t), i.e. the probability that no branching occurs between t 0 and t m , where
S a (t) = exp − tm t0 dt ′ b,c I a→bc (t ′ ) .(7)
Thus, the probability density that a branching of a occurs at t m is given by
dP a dt = b,c I a→bc (t) S a (t).(8)
These equations are solved for each branching by the PYSHOW algorithm [27] iteratively to generate a shower. For each branching first Eq. (8) is solved to determine the scale of the next branching, then Eqs. (2)-(4) are evaluated to determine the type of branching and the value of z, if the value of z is outside the kinematic bound given by Eq. (6) then the event is rejected. Given t 0 , t m and z, energy-momentum conservation determines the rest of the kinematics except for a radial angle by which the plane spanned by the vectors of the daughter parents can be rotated. In order to account in a schematic way for higher order interference terms, angular ordering is enforced onto the shower, i.e. opening angles spanned between daughter pairs b, c from a parent a are enforced to decrease according to the condition
z b (1 − z)b) M 2 b > 1 − z a z a M 2 a(9)
After a branching process has been computed, the same algorithm is applied to the two daughter partons treating them as new mothers. The branching is continued down to a scale Q min which is set to 1 GeV in the MC simulation, after which the partons are set on-shell, adjusting transverse momentum to ensure energy-momentum conservation. After all possible branchings have been performed, i.e. after for all partons the condition Q ≤ Q min has been reached, the resulting parton shower is connected with a string following the Lund scheme [28] which is subsequently allowed to decay into hadrons. These hadrons form the observable jet, and analyzing the distribution of hadrons, we may for example determine the fragmentation function D f →h (z), i.e. the distribution of hadron species h with an energy E h = zE f originating from a shower initiating parton f where E f is the whole energy of the jet.
B. Spacetime structure of the shower While the vacuum shower evolution equations above are solved in momentum space only, the interaction with the medium requires modelling of the shower evolution in position space as well, because the medium properties in a general medium change as a function of the position space variables. Usually, these are given in the c.m. frame of the collision in terms of the spacetime rapidity η s , the radius r, the proper time τ and the angle φ, and knowledge of the medium evolution implies knowledge of medium properties such as the local medium temperature T in the form T (η s , r, φ, τ ). In order to make the link from momentum space to momentum space, we assume that the average formation time of a shower parton with virtuality Q is developed on the timescale 1/Q, i.e. the average lifetime of a virtual parton with virtuality Q b coming from a parent parton with virtuality Q a is in the rest frame of the original hard collision (the local rest frame of the medium may be different by a flow boost as the medium may not be static) given by
τ b = E b Q 2 b − E b Q 2 a .(10)
Going beyond the ansatz of [19,20] where we used this average formation time for all partons, in the present work we assume that the actual formation time can be obtained from a probability distribution
P (τ b ) = exp − τ b τ b(11)
which we sample to determine the actual formation time of the fluctuation in each branching. This establishes the temporal structure of the shower. With regard to the spatial structure, we make the simplifying assumption that all partons probe the medium along the eikonal trajectory of the shower initiating parton, i.e. we neglect the small difference of the velocity of massive partons to the speed of light and possible (equally small) changes of medium properties within the spread of the shower partons transverse to its axis.
C. The parton-medium interaction
In the following, we assume that any effect of the medium will affect the partonic stage of the evolution, but not the hadronization. This is equivalent to the idea that hadronization takes place outside the medium, an assumption commonly made also for leading parton energy loss calculations. The validity of this assumption will be dicussed below. We use three different scenarios to model the interaction of partons with the medium. The first one, in the following referred to as RAD, has been used previously in [19,20]. The relevant property of the medium probed is the transport coefficientq(η s , r, φ, τ ) which represents the virtuality gain ∆Q 2 per unit pathlength of a parton traversing the medium. Note that this represents an average transfer, i.e. a picture which would be realized in a medium which is characterized by multiple soft scatterings with the hard parton. However, unlike in [19,20] the virtuality transfer to a shower parton is randomized in the present work since the formation time is distributed randomly around its average. Thus, effectively the present scenario includes the possibility to have both a small formation time and hence a small virtuality gain and a large formation time corresponding to a more substantial increase in virtuality.
In practice, we increase the virtuality of a shower parton a propagating through a medium with specified q(η s , r, φ, τ ) by
∆Q 2 a = τ 0 a +τa τ 0 a dζq(ζ)(12)
where the time τ a is given by Eq. (11), the time τ 0 a is known in the simulation as the endpoint of the previous branching process and the integration dζ is along the eikonal trajectory of the shower-initiating parton. If the parton is a gluon, the virtuality transfer from the medium is increased by the ratio of their Casimir color factors, 3/ 4 3 = 2.25. If ∆Q 2 a ≪ Q 2 a , holds, i.e. the virtuality picked up from the medium is a correction to the initial parton virtuality, we may add ∆Q 2 a to the virtuality of parton a before using Eq. (8) to determine the kinematics of the next branching. If the condition is not fulfilled, the lifetime is determined by Q 2 a + ∆Q 2 a and may be significantly shortened by virtuality picked up from the medium. In this case we iterate Eqs. (10), (12) to determine a selfconsistent pair of ( τ a , ∆Q 2 a ). This ensures that on the level of averages, the lifetime is treated consistently with the virtuality picked up from the medium. The actual lifetime is still determined by Eq. (11). In a second scenario, in the following called DRAG, we assume that the medium exerts a drag force on each propagating parton. The medium is thus characterized by a drag coefficient D(η s , r, φ, τ ) which describes the energy loss per unit pathlength. In the simulation, the energy (and momentum) are reduced by
∆E a = τ 0 a +τa τ 0 a dζD(ζ)(13)
Again, for a gluon the energy loss is increased by the color factor ratio 2.25. As in the previous case, the energy loss induced by the drag force is randomized even given the branching kinematics due to the randomized formation time of a branching.
The third scenario has been suggested in [18,25]. In the following, it is referred to as FMED. Here, the modification does not concern the parton kinematics, but rather the evolution kernel, Eqs. (2)(3)(4). In this scenario, the singular part of the branching kernel in the medium is enhanced by a factor 1 + f med , e.g. Eq.
(2) becomes in the medium
P q→qg (z) = 4 3 1 + z 2 1 − z ⇒ 4 3 2(1 + f med ) 1 − z − (1 + z)(14)
The effect of the medium is thus summarized in the value of f med . Note that in the FMED scenario, no explicit reference to the spacetime structure of the shower is made, in this sense, the scenario is rather different from the other two. Note that in the RAD scenario the shower gains energy from the medium by means of the virtuality increase, in the DRAG scenario the shower loses energy to the medium whereas the shower energy is conserved in the FMED scenario. While this appears surprising at first, it is actually rather a matter of book-keeping. For a shower in the medium, there is no conceptual way to separate soft partons from the shower and from the medium. However, the model framework outlined above does not treat the medium as consisting of partons, but rather as an effective influence on the shower. Thus, in a more realistic model one would define a criterion (say a momentum scale) based on which partons are removed from the shower and become part of the medium. In such a model, all three scenarios would lead to a loss of energy from the shower to the medium through the appearance of soft partons in the evolution, in addition to possible other mechanisms of energy transfer to the medium.
III. COMPARISON FOR A CONSTANT MEDIUM
In this section, we perform several computations for the simple case of a constant medium with fixed pathlength. This is chiefly done in order to establish the relation of the models outlined above to older computations based on leading parton energy loss.
A. Presence and absence of scaling
A constant medium corresponds to a choice of a single value ofq, D or f med . However, in the case of both the RAD and the DRAG scenario, also the medium length L has to be specified, thus in principle the medium is characterized by two parameters. In [19] however we found an approximate scaling law for the RAD scenario according to which the modification chiefly depends on the virtuality picked up along the eikonal path of the shower initiating parton ∆Q 2 tot = dζq(ζ) or in the case of a constant medium simplyqL. A similar scaling law can also be established for the DRAG scenario, albeit only in the case of an expanding medium (see below). Whenever such a scaling law holds, a comparison between the different scenarios can be made based on the single parameter ∆Q 2 tot or ∆E tot only. It is clear that the scaling cannot work for all the possible functional formsq(ζ). In two different limits this can be made plausible: If, in the RAD scenario, ∆Q 2 tot is added at once initially, ∆Q 2 /Q 2 is for reasonable values of hard process kinematics and medium properties very small. For example, for typical RHIC kinematics the initial Q 2 from which the evolution starts may be 400 GeV 2 whereas the total virtuality acquired for a parton traversing the whole medium is about 15 GeV 2 according to the results of [19]. However, such a small correction will not influence the shower evolution significantly. On the other hand, if the virtuality is added later when the typical Q 2 is of order of ∆Q 2 , a much stronger modification is expected. Thus, one expects the scaling law to be violated into the direction of less medium effect ifq(ζ) is strongly peaked towards τ = 0. A similar argument, can be made for the DRAG scenario. The drag force acts on every parton in the shower. This means that if D(ζ) is strongly peaked towards τ = 0, then the drag force acts only on one parton, the shower initiator, whereas if it is applied later, its effect is felt by several partons. On the other hand, note that the shower evolution is terminated for every parton which reaches Q 2 ≤ Q 2 min = 1 GeV 2 . This implies that the typical lifetime of the shower for an initial parton with energy E is given by τ max ∼ E/Q 2 min , thus a shower with E = 20 GeV probes the medium on average for a distance of 4 fm (Eq. (11) leads to fluctuations around this average though). Thus, if L is chosen much beyond τ max ,qL or DL are not good parameters any more, as the shower does not effectively probe the whole medium.
The latter effect is clearly not related to an actual physics effect but rather an artefact of the need to switch to a non-perturbative description of hadronization at some point in the simulation. It is unreasonable that a parton (or proto-hadron) would feel no effect from the medium just because its virtuality is small, however it is unclear just how the effect should be implemented properly in the present framework. The behaviour of the simulation thus depends on the actual choice of Q min , and this needs to be optimized eventually in comparison with data. A study of the effect of changing Q min will be presented below.
The resulting MMFF for a light quark into charged hadrons for constantqL or DL and a variation of pathlength is shown in Fig. 1 for both the RAD and the DRAG scenario. In a constant medium, the RAD scenario shows approximate scaling for pathlength between 0.5 and 5 fm. The DRAG scenario does not exhibit a strong scaling in the region of large z, but in the region z ∼ 0.5 which is predominantly probed when computing the single hadron spectra, the variations are not too large for pathlengths between 0.5 and 3 fm.
Note that neither the short pathlength nor the long pathlength limit is actually problematic for a realistic medium evolution taken from a hydrodynamical model. The first limit is avoided by virtue of the thermalization time of order O(0.6) fm for RHIC kinematics. This is a large time compared with the timescale in which the first branchings in a shower occur, thus by the time the medium is present, the shower is already well developed. The second limit is avoided because in an expanding medium q(ζ) or D(ζ) drop rapidly as a function of time, thus the late time contribution to dζq(ζ) or dζD(ζ) is small in any case. Thus, the scaling works much better for a realistic evolution as the constant medium results would suggest.
B. Energy loss and quenching weights
We now proceed to compare the three scenarios on the basis of leading parton energy loss. This is relevant to make the connection to previous calculations in the BDMPS or ASW formalism [2,24] which are formulated using this concept. For this purpose, we select the shower initator to be a c-quark and extract the energy distribution of the leading c-quark dN/dE c after the shower. From the comparison of the distribution dN/dE vac c in vacuum and in the medium dN/dE med c , we can deduce the energy loss probability distribution P (∆E). The idea is to make an ansatz
dN dE med c (E) = d(∆E) dN dE vac c (E ′ )P (∆E)δ(E ′ − E − ∆E)(15)
and solve it for P (∆E). Note that this ansatz contains the rather drastic assumption that there is no parametric dependence on the initial energy E. If we require P (∆E) to be a probability distribution, the assumption may imply that for some partons in the distribution dN/dE vac c the energy loss ∆E is larger than their energy E in which case they have to be considered lost to the medium. A similar situation also occurs in the application of the ASW formalism to finite energy kinematics. The problem of the validity of assuming energy independence however only concern the comparison with the ASW results in which energy loss is formulated in terms of a probability density P (∆E). In all other results presented in this manuscript, the full information of the shower including finite energy kinematics is used and no assumption about energy independence of energy loss needs to be made. The choice of a c-quark as shower initiator has a twofold motivation. First, it allows to define energy loss in the same way as done in the ASW formalism. Note that the ASW formalism assumes infinite parent parton energy and calculates energy loss via the radiation spectrum off the parent. In applying the formalism to finite energy, a process may occur in which a radiated gluon takes 90% of the energy of an initial quark q 1 . This energy is then considered to be lost from the q 1 . However, in the shower language, the radiated gluon would in this case become the new leading parton, and even tagging the leading quark out of a shower would not prevent processes where this gluon splits into a qq pair where the new quark q 2 might still be harder as the original parent q 1 of the gluon. The choice of a c quark as shower initiator effectively suppresses such processes and allows to treat energy loss as closely as possible to ASW [31]. The second advantage of extracting P (∆E) from a c quark is that the c-fragmentation is rather hard, i.e. the probability distribution to find the leading c-quark after a vacuum shower peaks close to z = 1. This effectively means that if one considers an additional, mediuminduced shift in energy, most of the energy range is still available for the dominant part of the distribution. This is very different for a light quark shower where the leading quark distribution typically peaks at z ∼ 0.5 and any energy loss of ∆E > E/2 shifts the bulk of the distribution into the unphysical region of negative energies.
In Fig. 2 we show the leading charm distributions both in vacuum and in medium for a medium pathlength of L = 2 fm. In order to make a meaningful comparison between the different scenarios, the average relative energy loss ∆E /E is fixed to 10% or 20% respectively.
In order to deduce the energy loss probability distribution from these results, we have to solve Eq. (15). By discretizing the integral over ∆E in Eq. (15) we can cast it into the form of a matrix equation
N i (E i ) = n j=1 K ij (E i , ∆E j )P j (∆E j )(16)
where dN/dE c is provided at m discrete values of E labelled N i and P (∆E) is probed at n discrete values of ∆E labelled P j . (16) can in principle be solved for the vector P j by inversion of K ij for m = n. However, in general this does not guarantee that the result is a probability distribution. Especially in the face of statistical errors and finite numerical accuracy the direct matrix inversion may permit negative P j which have no probabilistic interpretation.
Thus, a more promising solution which avoids the above problems is to let m > n and find the vector P which minimizes ||N − KP || 2 subject to the constraints 0 ≤ P i ≤ 1 and n i=1 P i = 1. This guarantees that the outcome can be interpreted as a probability distribution and since the system of equations is overdetermined for m > n errors on individual points R i do not have a critical influence on the outcome any more. This is the approach we have chosen.
The results are shown for L = 2 fm in Fig. 3. Qualitatively, both the radiative energyloss scenarios RAD and FMED produce energy loss probability distributions which are similar to the ASW quenching weights [24] in the sense that they are flat across a wide range in ∆E. In contrast, the DRAG scenario produces a localized peak in the energy loss distribution which reminds of the quenching weights found for elastic energy loss scenarios [32,33]. Especially for larger energy loss, the RAD and the FMED scenario lead to almost identical results.
However, there is an important difference to the ASW quenching weights: While the ASW results typically show a large discrete probability for no energy loss, the results obtained here show no substantial strength in the first bin (the inversion procedure outlined above cannot separate zero energy loss from small energy loss).
C. Parametric dependence of mean energy loss
In order to gain more insight into the different scenarios, we investigate in Fig. 4 for a constant medium with L = 2 fm how the mean energy loss, defined as ∆E = d∆E∆EP (∆E) with P (∆E) obtained as in the previous section behaves as a function of the relevant medium parameters. We include a scenario in which the strong coupling constant is not allowed to run with the virtuality scale in the shower (as is the default option in PYSHOW) but is kept fixed at α s = 0.3. There is no unique way to present and compare the results, as the three relevant parametersq, D and f med are rather different. However, as apparent from Fig. 4, it is possible to find a simple proportionality relation between q and f med such that the rise of the mean energy loss appears very similar. This, in addition to the similarity of P (∆E) for both the RAD and the FMED scenario points towards some generic properties of radiative energy loss scenarios independent of the details of the implementation.
In particular, the RAD and the FMED scenario exhibit saturation of the mean induced energy loss at about 25% of the total energy as the medium effect is increased. This saturation is even more pronounced for a constent α s . In striking contrast, the DRAG scenario in which energy is directly transferred to the medium shows an almost linear rise up to mean energy losses of 50%. Note that the extraction of the energy loss probability based on discretization and matrix inversion as outlined above becomes increasingly problematic at ∆E /E > 0.4 due to the problem of partons being shifted to negative energies mentioned above.
IV. COMPARISON FOR A SINGLE PATH IN AN EXPANDING MEDIUM
We now turn to a more realistic scenario in which the parton propagates in a medium as created in a heavyion collision. Relativistic fluid-dynamical models such as [34] give a good description of many bulk properties of the medium, hence in the following we will assume that hydrodynamics is a valid description of the medium. Both the finite size and the finite lifetime of such a medium are felt by the parton. In particular, the local density may drop a) because of a spatial variation, i.e. the parton reaches the medium edge and b) a temporal variation, i.e. the global expansion of the medium reduces the overall density as a function of time. In addition, there are arguments that the hydrodynamical flow of the medium expansion should also have a direct influence on the medium properties as seen by the medium due to Lorentz transformation between the moving local medium rest frame and the frame of the hard collision [37,38].
A. Characterization of the medium
In [19] we have established that ifq is linked with the medium properties by the relation
q(ζ) = K · 2 · [ǫ(ζ)] 3/4 (cosh ρ(ζ) − sinh ρ(ζ) cos ψ) (17)
with K a parameter determining the interaction strength which is a priori unknown (in an ideal QGP, K = 1 is expected [35] but a comparison study of different energy loss models has shown to be inconclusive in extracting values for K [36]), the medium energy density ǫ, the local flow rapidity ρ with angle ψ between flow and par-ton trajectory [37,38], we find that the vast majority of paths found in the 3-dimensional hydrodynamical model of Bass and Nonaka [34] leads to aq(ζ) which can be described by the rather simple expression . As in III A in the present paper for a constant medium, we found that an approximate scaling in which the medium effects did not depend on details of the trajectories (A), (B), or (C) but only on ∆Q 2 tot = dζq(ζ). The virtue of this scaling law is twofold: First, it allows to present the medium modifications for the relevant class of functionsq(ζ) as a function of a single parameter ∆Q 2 tot only. Second, it considerably speeds up the computation for a comparison with data where a weighted average over all possible paths through the medium has to be computed. In [19] we have made the rather drastic assumption that the medium does not exert any effect before the thermalization of the medium at the time τ in where τ in = 0.6 fm/c in the model studied for RHIC [34]. In the following, we adopt a more realistic approach in which we increase the medium effect linearly from zero at τ = 0 to its value reached at τ in . The idea behind this is that initially no medium can be present, as the timescale for hard processes precedes any other timescale in the system. However, even a medium which is not yet equilibrated may interact with hard partons and lead to scattering processes. A linear interpolation between the initial time and the equilibration time seems a reasonable prescription to capture part of these effects. In practice, qualitative aspects of the results of [19], in particularly the presence of the scaling, are not substantially altered by this modification. There is however an effect on the numerical value of extracted medium parameters. Let us now consider the other scenarios DRAG and FMED. Eq. (17) which linksq with the hydrodynamical properties of the medium is based on counting the potential scattering centers along the parton trajectory. ǫ 3/4 for an ideal gas corresponds to the entropy density, which in turn is proportional to the medium density. The additional factor (cosh ρ(ζ) − sinh ρ(ζ) cos ψ) is nothing but the appropriate transformation to determine how the density seen by the parton is changed under a boost of the restframe of the medium [37]. It is reasonable to assume a similar measure of potential scattering centers to be relevant for the other scenarios. This ansatz leads to D(ζ) = K D · [ǫ(ζ)] 3/4 (cosh ρ(ζ) − sinh ρ(ζ) cos ψ) (19) for the drag coefficient D with an a priori unknown parameter K D specifying the overall strength of the drag force. As discussed above, the FMED scenario has no explicit dependence on the spacetime evolution of the shower, but it seems reasonable the the parameter f med should depend on the total effect of the medium measured in the number of potential scatterers which have been encountered. This leads to the ansatz
q(ζ) = a (b + τ /(1f m/c)) c .(18)f med = K f dζ[ǫ(ζ)] 3/4 (cosh ρ(ζ) − sinh ρ(ζ) cos ψ).(20)
Here, as in the previous scenarios, we also introduce an a priori unknown parameter K f which determines the strength of the parton-medium interaction. The scaling within the RAD scenario of the results with ∆Q 2 tot has been established in [19] and in the present paper also for a constant medium in III A. The DRAG scenario shows no strong scaling for a constant medium, but as anticipated the result is more promising for an expanding medium. The validity of the scaling under these conditions is apprent from Fig. 5 where we compute for fixed ∆E for the three different paths (A), (B) and (C). Note that scaling of the FMED scenario is realized by definition using the ansatz Eq. (20).
B. Longitudinal momentum distribution of the shower
In Fig. 6, we show the longitudinal momentum distribution of charged hadrons inside the shower in therms of the MMFF D(z) for three different scenarios in comparison. There is no a priori criterion at which values of the three medium parameters ∆Q 2 tot , ∆E tot and f med the three different scenarios should be compared. For the comparison in terms of energy loss probability distributions done above we required a fixed value ∆E /E, but this is not a meaningful variable when one wants to compare on the basis of the whole parton shower instead of the leading parton kinematics only. Here, we chose the criterion that the MMFF approximately agree in an interval of 0.4 < z < 0.7. This is the region of the fragmentation function which is predominantly probed when the fragmentation function is folded with a pQCD parton spectrum to compute single inclusive hadron production. The implication is that a computation with MMFFs agreeing in the above interval would yield approximately the same observable hadron spectra. This choice leads to the interesting and amusing numerical coincidence that if the parameters are given in powers of GeV, the relation ∆Q 2 tot /GeV 2 ≈ ∆E/GeV ≈ 10f med holds. We show the MMFF of a 20 GeV d-quark into charged hadrons for ∆Q 2 = 10 GeV 2 (the parameters of the other scenarios adjusted correspondingly) in Fig. 6, right panel.
In order to focus more on the hadron production at low momenta, we introduce the variable ξ = ln(1/x) where x = p/E jet is the fraction of the jet momentum carried by a particular hadron and E jet is the total energy of the jet. The inclusive distribution dN/dξ, the so-called Hump-backed plateau, is an important feature of QCD radiation [39,40] and is in vacuum dominated by color coherence physics. In Fig. 6 (left panel) we show dN/dξ for the three different scenarios in comparison with the unmodified result. It is apparent from the figure that while the three scenarios agree in the high z and consequently low ξ region, they exhibit sizeable differences in the high ξ region where induced radiation is expected to contribute to soft hadron production. Here, both the radiative scenarios RAD and FMED show the expected enhancement of the distribution, but the DRAG scenario is strikingly different -it falls below the vacuum result. However, this is hardly surprising, as in this scenario energy is taken away from the evolving shower and is hence not available for hadron production. While a measurement of dN/dξ would appear to be a promising means to distinguish between induced radiation and a drag force as the microscopic realization of energy loss, it has to be pointed out that there are two things which urge some caution. First, the Lund scheme used to model hadronization in the present framework assumes that hadronization takes place far outside the medium. If the energy of a hadron h of mass m h is E h , the spatial scale at which hadronization occurs can be estimated as l h ≈ E h /m 2 h . For pions, this is not a problem throughout the kinematic range, but for kaons and protons the hadronization length is considerably shortened. Even a 10 GeV proton has only l h ≈ 2 fm, thus heavy hadron production in the high ξ region is not addressed adequately in the model, as one cannot safely assume hadronization takes place outside the medium where the Lund model is applicable. Nevertheless, since pions constitute the bulk of charged hadron production, the essential features of the model are expected to be robust. The second issue concerns the effect of trigger bias. A series of experimental cuts has to be imposed on events in heavy-ion collisions to discriminate hadrons belonging to jets from the background of soft medium hadrons. However, strongly modified jets (for example those emerging from the medium center) are less likely to fall within the cuts than unmodified jets (such as those from the medium edge). As a result there is a trigger bias which suppresses events in which a modification of dN/dξ is visible. A calculation in the RAD scenario taking into account a realistic series of experimental cuts has been performed in [20] and found that there should be no visible enhancement if jets are identified directly via a standard set of cuts.
C. Angular distribution
Another possibility to identify the mechanism of the parton-medium interaction is to study the structure of the jet transverse to the jet axis. This is reflected e.g. in the angular distribution of hadrons around the jet axis. The distribution dN/dφ where φ is the angle between hadron and jet axis for ∆Q 2 tot = 10 GeV 2 (the parameters in the other scenarios adjusted accordingly) for the vacuum and the three different scenarios is shown in Fig. 7 where a cut in momentum of 1 GeV has been applied to focus on hadrons which would appear above the soft background of a heavy-ion collision. It is apparent from the figure that the radiative energy loss scenarios again roughly agree with each other and lead to angular broadening of the jet as compared to the vacuum result, whereas the DRAG scenario shows no indication for broadening.
D. The sensitivity to Qmin
For a constant medium, we noted earlier that there is a sensitivity to the choice of the minimum virtuality scale Q min at which partons in the shower are evolved further. Before comparing the results of this section to data, it is reasonable to ask to what extent a choice of Q min different from its default value Q min = 1 GeV has an influence on the results. In Fig. 8 we show results for the MMFF in both the RAD and the DRAG scenario with a lower Q min = 0.7 GeV where the shower evolves on average a factor two longer. Superimposed are results with the default choice Q min = 1 GeV for which the medium parameters ∆Q 2 tot and ∆E tot respectively have been increased for the best possible agreement of the results. It is evident from the figure that a lower Q min does not substantially influence the shape of the resulting MMFF, but that at least for RHIC kinematics, a lower choice of Q min can be compensated by assuming a different choice of the medium parameters. It is thus not possible to extract definite values forq or D from a mode fit to measured single hadron spectra, rather only pairs (q, Q min ) can be determined.
V. COMPARISON WITH NUCLEAR SUPPRESSION DATA
In this section, we aim at comparing with experimental data. This implies that neither initial position nor initial momentum nor type of the shower initiating parton are known. The probabilities to find a given parton type with given momentum have to be computed in pQCD whereas the probability to produce a parton at a given vertex position can be found from overlap calculations. Note that the need to average over position and initial momentum corresponds to a substantial increase in MC computing time which could not be done without using the scaling laws.
A. The averaging procedure
We begin the analysis by showing how to compute the nuclear suppression factor R AA using the medium-modified fragmentation function in the hydrodynamically evolving medium. For this, we first have to obtain the single inclusive hard hadron spectrum. We treat the partonic subprocesses of the hard reaction in leading order pQCD. The straightforward calculation involves the convolution of the initial nucleon [41,42] (or nuclear [43,44,45]) parton distribution functions with the relevant pQCD subprocesses and yields the single inclusive distribution dσ AB→f +X dp 2 T dy f of hard partons f in transverse momentum p T and rapidity y f where the rest of the reaction X is unobserved (more detailed expressions can be found in [19]). The single inclusive hadron distribution in hadronic momentum P T and rapidity y follows from the parton spectrum through the convolution with the fragmentation function D f →h (z, µ 2 f ) where z is the momentum fraction taken by the hadron and µ f is the hadronic momentum scale as
dσ AB→h+X dP 2 T dy = f dp 2 T dy f dσ AB→f +X dp 2 T dy f 1 zmin dzD f →h (z, µ 2 f )δ m 2 T − M 2 T (p T , y f , z) δ (y − Y (p T , y f , z))(21)
with
M 2 T (p T , y f , z) = (zp T ) 2 + M 2 tanh 2 y f ,(22)z min = 2m T √ s cosh y(23)
and
Y (p T , y f , z) = arsinh P T m T sinh y f .(24)
The nuclear suppression factor is defined as
R AA (P T , y) = dN h AA /dP T dy T AA (0)dσ pp /dP T dy(25)
where T AA (b) is the standard nuclear overlap function. We can compute it by forming the ratio
R AA (P T , y) = dσ AA→h+X medium dP 2 T dy / dσ pp→h+X dP 2 T dy(26)
where dσ pp→h+X /dP 2 T dy follows from Eq. (21) when D f →h (z, µ 2 f ) is set to be the vacuum fragmentation function whereas dσ AA→h+X medium /dP 2 T dy is computed from the same equation with D f →h (z, µ 2 f ) replace by the suitably averaged MMFF D MM (z, µ 2 f ) TAA . This averaging has to be done over all possible paths of partons through the medium. The probability density P (x 0 , y 0 ) for finding a hard vertex at the transverse position r 0 = (x 0 , y 0 ) and impact parameter b is, again in leading order, given by the prod-uct of the nuclear profile functions as
P (x 0 , y 0 ) = T A (r 0 + b/2)T A (r 0 − b/2) T AA (b) ,(27)
where the thickness function is given in terms of Woods-Saxon the nuclear density ρ A (r, z) as T A (r) = dzρ A (r, z). The MMFF must then be averaged over this quantity and all possible directions φ partons could travel from a vertex as
D MM (z, µ 2 ) TAA = 1 2π 2π 0 dφ ∞ −∞ dx 0 ∞ −∞ dy 0 P (x 0 , y 0 )D MM (z, µ 2 , ζ).(28)
Using the approximate scaling relation described in section III A, the medium modified fragmentation function D MM (z, µ 2 , ζ) for a path ζ can be found by computing the line integrals dζq(ζ or dζD(ζ) over Eqs. (17), (19) or by evaluating Eq. (20) respectively. The MC shower code is then used to compute D MM (z, µ 2 , ζ) for each value of ∆Q 2 tot , ∆E tot or f med obtained. As discussed in more detail in [19], there is a conceptual problem with using a MMFF computed for a fixed partonic scale in Eq. (21) where D(z, µ 2 f ) is an object defined at a given hadronic scale. This is a generic problem in obtaining fragmentations from a MC code which starts with given parton properties, however in practice the scale evolution in the RHIC kinematic range is small as compared to other uncertainties in the computation and the resulting uncertainty can be tolerated. In the following, we use a MMFF determined at the partonic scale µ = 20 GeV.
B. Comparison with data
With the medium given by the hydrodynamical evolution model described in [34] and the expressions for hadron production in vacuum and medium Eq. (21), the remaining unknown quantities for a comparison with data are the parameters K, K D and K f which link the medium properties in terms of the energy density ǫ with the parton-medium interaction parametersq, D and f med . Note that according to the results of IV D the value of these parameters cannot be uniquely determined for single inclusive hadron spectra, but depends on the choice of the scale Q min in the shower simulation. In the following, we show the best fit of K, K D and K f to the data given the choice Q min = 1 GeV. In Fig. 9 we show the calculated nuclear suppression factor as a function of hadron momentum P T for all three scenarios in comparison with the data for π 0 production [7]. In all cases, the relevant constant relating medium energy density and parton-medium interaction parameter has been fit to data.
in 200 AGeV central Au-Au collisions. The most striking observation is that all three scenarios are surprisingly similar and could not possibly be distinguished by current data for R AA . Most notably, all scenarios exhibit a falling trend as P T where scenarios based on leading parton energy loss typically exhibit a rising trend (see e.g. [36,46]). In [19], this property was tentatively attributed to the fact that a description of the whole shower keeps track of multiple soft hadron production. It was also suggested that the same physics underlies the enhancement of dN/dξ in the large ξ region and the falling of R AA with P T . However, the present investigation shows that both ideas must be discarded, as the DRAG scenario in which no enhanced soft hadron production occurs shows also no enhancement of dN/dξ, but the same falling trend of R AA with P T as the other scenarios. Thus, the falling trend is not a phenomenon characteristic of radiative energy loss but substantially more general. It also has been observed in other models where a modification of the whole shower by the medium was considered, cf. e.g. [25,49]. In order to gain greater insight into the differences between scenarios which compute energy loss for the leading parton and between those where the whole shower evolution is modified by the medium, we present a schematic comparison between the MMFFs in the RAD, the DRAG and the ASW scenario in Fig. 10 (the FMED scenario, being in essence indistinguishable from the RAD scenario is not shown here). It has been pointed out repeatedly (see e.g. [47,48]) that the ASW scenario applied to RHIC kinematics in essence leads to complete absorption of about 75% of all partons, ∼ 15% emerge without any energy loss and only a small fraction is found after finite energy loss. To good approximation, the MMFF in the ASW scenario is thus just a downward shift of the vacuum baseline. This has been done in Fig. 10 where all parameters have been adjusted such that the curves agree at z = 0.6. It is obvious that the shape of the schematic ASW result is quite different fron the other scenarios. In particular, the difference between RAD and DRAG is much less pronounced than between either of those and ASW. It is in essence given by the presence or absence of soft hadron production and confined to the region z < 0.2. Thus, it appears that the different curvature of D(z) at z > 0.5 is responsible for the rising vs. falling trend in R AA , and thus the way the high P T end of the shower evolves rather than low P T hadron production are seen in the data. At present, the falling trend seems not to be supported by the data. Should this be confirmed by more precise measurements, presumably the possibility of complete absorptions of partons by the medium needs to be intro-duced into the simulation of in-medium shower evolution.
C. Extracting medium parameters
We can use the above results to tentatively extract medium properties. In the RAD scenario, K is a dimensionless parameter and from the fit shown in Fig. 9 the value K = 3 is found. This differs from the result in [19] where K = 1.5 was obtained, note however that in the present work a randomziation of the formation time (see EQ. (11)) has been performed and that the effect of the medium prior to thermalization has been included in a schematic way. These two differences account for the changed value of K. With this value of K,q 0 , i.e. the highest transport coefficient reached in the evolution in the medium center at thermalization time of 0.6 fm/c is found to be 15.6 GeV 2 /fm when Q min = 1 GeV is assumed. For Q min = 0.7 GeV, the extracted value ofK changes to 1.4 andq 0 = 7.2 GeV 2 /fm. K D is a dimensionful parameter which can be expressed in units GeV −1 . The same fit yields (due to the numerical coincidence mentioned before) D 0 = 15.6 GeV/fm for Q min = 1 GeV and D 0 = 7.2 GeV/fm for Q min = 0.7 GeV. While these numbers appear large, it has to be remembered that they reflect a snapshot of the medium at its peak density, from which the energy density drops rapidly as a function of time due to the expansion. Since f med is not in connected to any microscopical properties of the medium, we refrain from analyzing its value here.
VI. DISCUSSION
We have presented a comparison study of three different mechanisms for the parton-medium interaction in the framework of an in-medium shower evolution. In this study, we have considered a variety of assumptions about the evolution of the medium, among them a constant medium with different length L, an evolving medium for parths from the medium center, paths from the medium surface and an average over all possible paths in the medium. We have considered three different types of shower initiators -heavy quarks, light quarks and gluons. We have studied single parton observables such as the distribution of the leading parton momentum or the energy loss probability density P (∆E), single hadron observables like R AA as well as multihadron observables such as the hump-backed plateau dN/dξ. In addition, we have also studied merely technical aspects of modelling such as the role of the cutoff parameter Q min or the effect of randomizing the formation time of partons in branching. From the results in all these different situations, some generic properties can be identified.
• The nuclear suppression factor R AA is not a good observable to distinguish different scenarios of the micro-scopical interaction of partons with the medium. This statement has been made previously from different angles (see e.g. [36,50,51]) and the present results merely confirm previous findings in yet another framework. More differential observables are needed to determine the nature of parton-medium interaction.
• The falling trend of R AA as a function of P T is apparently unrelated to low P T multi hadron production and rather a generic feature observed in models which do not consider energy loss from a leading parton but rather a modification of the whole shower. For example, the results of the Higher Twist approach applied to the leading parton show a rising trend of R AA with P T [52], however when resummed in the shower evolution equations and applied to the whole shower, the Higher Twist approach leads to a falling trend [36], i.e. the same observation is made in quite a different framework. If future data confirm a rising trend, non-trivial modifications to the shower evolution codes, such as the possibility of complete parton absorption by the medium, need to be considered.
• The properties of medium-induced radiation as a mechanism for the parton-medium interaction appear rather generic. There is no observable in this study in which the RAD and the FMED scenarios lead to substantially different results. The useful implication would be that in many observables it is really the underlying physics mechanism one is probing, not technical details of how this mechanism is implemented in a particular model. • In contrast, a different physics mechanism as exemplified here by the DRAG scenario appears distinct in several quantities. Not only is its excitation function in terms of mean energy loss as a function of medium density different than for radiative scenarios (which could be tested by variations in collision centrality), but also the absence of soft hadron production induces pronounced effects in jet observables such as the angular distribution of hadrons around the jet axis or the hump-backed plateau. However, the need to identify a jet in a heavy-ion collision above the soft background introduces additional complications. In essence, a medium-modified jet has properties different from a jet in vacuum and is hence less likely to be identified as jet. A measurement of jets must be carefully designed to avoid this trigger bias which tends to hide the very effect one would like to study [20].
• Technical aspects of the modelling, such as the choice of Q min or the randomization of the formation time as investigated here, do not appear to change the results qualitatively. However, there is a substantial ambiguity once one tries to extract quantitative medium parameters from the computation, especially when this extaction is based on a single observable. As is the case for vacuum shower codes, the relevant technical model parameters should eventually be determined by the best fit to a large body of data. There are several more properties of the parton-medium interaction which could be exploited to distinguish dif-ferent scenarios. A very promising candidate is the pathlength dependence of the medium effect. Experimentally, this can be varied moderately by considering the nuclear suppression as a function of the angle of hard hadron with the reaction plane [10] or more strongly by considering back-to-back correlations (which however require a careful modelling, as the relevant geometry arises as a complicated function of the geometrical bias of the energy loss on the trigger hadron itself [13,14]. For example, in [33] is was argued based on the different pathlength dependence that elastic energy loss cannot be responsible for the suppression of hard back-to-back hadron correlations. In the present paper, we have refrained from making any comparison based on pathlength dependence. Such a study (which is quite substantial on its own) along with a comparison with the experimental results on back-toback correlations and the variation of the suppression as a function of the reaction plane angle will be the topic of a future publication.
VII. OUTLOOK
The study presented here shows that jet observables are more powerful in order to distinguish different microscopical physics process of the parton-medium interaction than observables which are only sensitive to the leading hadron. However, the need to identify a jet above the background medium may quickly eliminate this advantage, at which point one has to resort to what has been termed the 'golden channel' -γ-jet correlations in which the presence of a photon not only allows to get an unbiased jet sample but also reveals the kinematic of the jet. Unfortunately, due to the smallness of the electromagnetic coupling, the statistics in this channel is poor and the measurement is difficult. Future jet measurements at RHIC may overcome this problem by high luminosity, whereas future measurements at LHC where the scale separation between a hard process and the soft medium is considerably larger than at RHIC may not suffer significantly from trigger bias at all. There is now good reason to assume that jet observables will reveal important information about the microscopical properties of the medium, and Monte Carlo simulations of in-medium showers such as YaJEM or JEWEL will most likely be the appropriate tools to extract this information.
FIG. 1 :
1The MMFF of a d-quark into charged hadrons for constant value of ∆Q 2 tot =qL = 5 GeV 2 in the RAD scenario (left panel) and ∆Etot = DL = 5 GeV in the DRAG scenario (right panel) for different pathlengths in a constant medium.
FIG. 2 :
2Energy distribution of the leading c quark for a 20 GeV c quark as shower initiator in the three different scenarios for the parton-medium interaction (see text). Left panel: 10% average energy loss, right panel: 20% average energy loss.
FIG. 3 :
3Energy loss probability distribution P (∆E) for the leading c-quark for a 20 GeV c-quark as shower initiator in the three different scenarios for the parton-medium interaction (see text). Left panel: 10% average energy loss, right panel: 20% average energy loss.
FIG. 4 :
4Mean energy loss as a function of the medium properties in different scenarios for the parton-medium interaction in a constant medium with L = 2 fm.
Based on this expression, we investigated three different scenarios (approximately representing a parton travelling into +x direction originating from x = 4 fm (A), x = 0 (B) and x = −4 fm (C), y = 0 in all cases in the transverse (x, y) plane at midrapidity. These trajectories are characterized by the parameters (b = 1.5, c = 3.3, τ E = 5.8 fm/c) (A), (b = 1.5, c = 2.2, τ E = 10 fm/c) (B) and (b = 1.5, c = 2.2, τ E = 15 fm/c) (C) and are quite typical for partons close to the surface (A), emerging from the central region (B) or traversing the whole medium (C)
FIG. 5 :
5The MMFF of a 20 GeV d-quark into charged hadrons for three different paths (A), (B) and (C) (see text) with ∆Etot = 5 GeV in the DRAG scenario.
FIG. 6 :
6Longitudinal momentum distribution of charged hadrons inside a jet originating from a 20 GeV d-quark shown as fragmentation function D(z) (left panel) and dN/dξ (right panel) for the vacuum and the three different scenarios for the parton-medium interaction (see text). The medium parameters have been chosen to let the modified D(z) approximately agree for 0.4 < z < 0.7 for all in-medium scenarios.
FIG. 7 :
7Angular distribution of charged hadrons above 1 GeV coming from the fragmentation of a 20 GeV d-quark for vacuum and three different scenarios of parton-medium interaction (see text).
for a 20 GeV d-quark as shower initiator for different values of the minimum shower virtuality Qmin in the RAD and the DRAG scenario (see text) where medium parameters have been adjusted to compensate for the choice of Qmin.
FIG. 9 :
9Calculated nuclear suppression factor for three different scenarios for the parton-medium interaction (see text) as a function of hadron momentum PT in comparison with data by the PHENIX collaboration
10: Comparison of the MMFF of a 20 GeV d-quark into charged hadrons for a schematic ASW leading parton energy loss scenario and two different scenarios in which the whole shower is evolves (see text). All parameters are chosen such that the curves agree at z = 0.6.
The kernel K ij is then the calculated dN/dE c for all pairs (E i , ∆E j ) where the energy loss acts as a shift of the distribution, i.e. dN/dE medc
(E) =
dN/dE vac
c (E + ∆E).
Eq.
AcknowledgmentsI'd like to thank Kari J. Eskola for valuable discussions on the problem. This work was financially supported by the Academy of Finland, Project 115262.
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|
[] |
[
"DETECTABILITY OF LOCAL GROUP DWARF GALAXY ANALOGUES AT HIGH REDSHIFTS",
"DETECTABILITY OF LOCAL GROUP DWARF GALAXY ANALOGUES AT HIGH REDSHIFTS"
] |
[
"Anna Patej ",
"Abraham Loeb "
] |
[] |
[] |
The dwarf galaxies of the Local Group are believed to be similar to the most abundant galaxies during the epoch of reionization (z 6). As a result of their proximity, there is a wealth of information that can be obtained about these galaxies; however, due to their low surface brightnesses, detecting their progenitors at high redshifts is challenging. We compare the physical properties of these dwarf galaxies to those of galaxies detected at high redshifts using Hubble Space Telescope and Spitzer observations and consider the promise of the upcoming James Webb Space Telescope on the prospects for detecting high redshift analogues of these galaxies.
|
10.1088/2041-8205/815/2/l28
|
[
"https://arxiv.org/pdf/1510.02101v1.pdf"
] | 119,120,363 |
1510.02101
|
f93973f62d87c77241c04426031c3fd177738c82
|
DETECTABILITY OF LOCAL GROUP DWARF GALAXY ANALOGUES AT HIGH REDSHIFTS
7 Oct 2015 Draft version October 9, 2015 Draft version October 9, 2015
Anna Patej
Abraham Loeb
DETECTABILITY OF LOCAL GROUP DWARF GALAXY ANALOGUES AT HIGH REDSHIFTS
7 Oct 2015 Draft version October 9, 2015 Draft version October 9, 2015Preprint typeset using L A T E X style emulateapj v. 5/2/11
The dwarf galaxies of the Local Group are believed to be similar to the most abundant galaxies during the epoch of reionization (z 6). As a result of their proximity, there is a wealth of information that can be obtained about these galaxies; however, due to their low surface brightnesses, detecting their progenitors at high redshifts is challenging. We compare the physical properties of these dwarf galaxies to those of galaxies detected at high redshifts using Hubble Space Telescope and Spitzer observations and consider the promise of the upcoming James Webb Space Telescope on the prospects for detecting high redshift analogues of these galaxies.
INTRODUCTION
Local Group dwarf galaxies allow investigation into a wide range of astrophysical and cosmological processes; in addition to being representative of the most populous type of galaxy in the universe, their nearness enables detailed investigations of their stellar populations (for a recent overview, see McConnachie 2012). Due to their low masses and typically old stellar populations, many of these dwarfs are believed to have had the majority of their stars produced at early cosmic times and then had further star formation suppressed by reionization at redshifts z ∼ 6 − 10 (e.g, Bullock et al. 2000;Ricotti & Gnedin 2005;Loeb & Furlanetto 2013). Observational arguments in favor of this interpretation for some of the dwarfs have been based on their statistical properties (Bovill & Ricotti 2009) and star formation histories Weisz, Johnson, & Conroy 2014). In this scenario, some of the present-day dwarfs should be similar to their progenitors at higher redshifts.
The advent of optical and infrared space-based telescopes -the Hubble Space Telescope (HST ) and the Spitzer Space Telescope -has allowed for the identification of numerous high-redshift (z 6) galaxies, whose properties, including their sizes, star formation rates, and masses, have now been examined in detail (e.g., Stark et al. 2009;Labbé et al. 2010;Oesch et al. 2010;Bouwens et al. 2011;Ellis et al. 2013;Ono et al. 2013).
However, current observations are missing a population of fainter galaxies that are needed to reionize the universe at these high redshifts (e.g., Alvarez et al. 2012;Finkelstein et al. 2012;Bouwens et al. 2015;Robertson et al. 2015). Discovering some of these fainter galaxies will be within the purview of future observatories like the James Webb Space Telescope (JWST ).
Part of this population of fainter galaxies is likely to consist of the progenitors of galaxies like the Local Group dwarfs. Boylan-Kolchin et al. (2015) used an analysis of the UV luminosities of the dwarfs to determine that JWST should be able to detect progenitors of galax-ies like the Large Magellanic Cloud. Here, we compare the physical properties of the local dwarfs and the highredshift galaxies that have already been detected, and place them in the context of the predicted detection limits for JWST to examine the fraction of dwarf progenitors -and thus the fraction of missing light -that may be observable in the near future. Throughout our discussion, we use the standard cosmological parameters Ω m = 0.3, Ω Λ = 0.7, and H 0 = 70 km/s/Mpc.
DATA
We obtain data for over 100 local galaxies from McConnachie (2012), including their V band Vega magnitude m V , half-light radius r, ellipticity ǫ, and average metallicity [Fe/H] . To provide direct comparison with the high-redshift data, we convert r h to the circularized half-light radius, r h = r √ 1 − ǫ, that is commonly employed. We select only the 87 galaxies that have all these quantities measured in McConnachie (2012), and note in particular that this excludes the Large and Small Magellanic Clouds (LMC and SMC).
We use metallicity as an input to the Flexible Stellar Population Synthesis (FSPS) code (Conroy et al. 2009;Conroy & Gunn 2010) to scale the galaxies back to z = 6, 7. Following the results of Weisz et al. (2014), who analyzed the star formation histories (SFHs) of a subsample of 38 of these dwarfs, we remove those galaxies whose SFHs indicate that the majority of their stellar populations were formed later than these redshifts. We keep those whose SFHs are consistent with at least 50% (within errors) of the stars having been formed prior to z = 6, 7, as well as all the remaining galaxies from McConnachie (2012) whose SFHs have not yet been measured, for a total of 73 galaxies. We use a delayed taumodel with τ = 0.2 Gyr, which assumes an early starburst such that nearly all the stars we see today already existed at the redshifts of interest. The code also calculates the evolution to z = 0, from which we take the predicted V band magnitude and compare it to m V from McConnachie (2012) to obtain a correction for the stellar mass of each galaxy; we then use these values to correct the z = 6, 7 magnitudes since we assume that all the stars we see today were already present at those high redshifts.
The corresponding parameters for observed z = 6 and z = 7 galaxies are obtained from several sources. The Spitzer IRAC 3.6 µm fluxes of a sample of z ∼ 6 galaxies are based on Gonzalez et al. (2012), and the 4.5 µm fluxes of z ∼ 7 galaxies are taken from Labbé et al. (2010). We adopt the 2σ lower and upper limits on the fluxes, corrected to S/N = 5, which gives F 3.6 = [38.9, 667.0] nJy and F 4.5 = [13.3, 637.5] nJy. Oesch et al. (2010) provides a value of r h = 0.7±0.3 kpc for detected galaxies at these redshifts, measured from near-infrared observations using HST. We again adopt the 2σ bounds on this quantity; accordingly, the range of observed sizes that we use is 0.1 − 1.3 kpc.
The future JWST mission's NIRCam imager will have two filters, F356W and F444W, which will cover the bandwidth of the Spitzer filters above. For the purposes of this comparison, we will assume that the Spitzer 3.6 µm filter is identical to F356W and the 4.5 µm filter is identical to F444W.
COMPARISON OF PHYSICAL PARAMETERS
We use the FSPS-derived fluxes and the sizes r h to calculate the surface brightness in Jy/arcsec 2 for the dwarfs at z = 6 and z = 7. We selected those galaxies with cumulative SFHs 0.5 (within the error profile) at early times from Weisz et al. (2014) and used τ = 0.2 Gyr to scale them, as discussed in Section 2, as well as all the remaining galaxies that do not have measured SFHs. We additionally plot the region bounded by the 2σ limits on the surface brightness and half-light radius for galaxies detected at z ∼ 6 and z ∼ 7, using Spitzer 3.6 µm and 4.5 µm fluxes, respectively. Figure 1 shows these regions alongside the scaled galaxies selected from Weisz et al. (2014) and the remaining galaxies from McConnachie (2012). From this, we see that virtually none of the local dwarf analogues have been detected yet.
Nevertheless, the sizes of the dwarfs and the high redshift galaxies agree extremely well, excluding the extreme smallest and faintest objects. However, we do not include in our comparison a scaling of r h with redshift. Luminous galaxies at higher redshifts have been observed to have their sizes scaled by a factor of (1 + z) −s , where s is in the range of 1.0 − 1.5 (e.g., Oesch et al. 2010;Mosleh et al. 2012). There is some indication, however, that at the lowest luminosities yet studied, the half-light radius remains approximately constant with redshift (see Figure 12 of Ono et al. 2013). Our results are consistent with the notion that the Local Group galaxies had roughly the same size at high redshifts as they have at present.
PREDICTIONS FOR JWST
JWST will rely on the NIRCam imager (Rieke et al. 2005;Beichman et al. 2012) to obtain photometry of high-redshift galaxies. We use the prototype exposure time calculator (ETC) for NIRCam 3 to compute the signal-to-noise ratio for the dwarf galaxies scaled back to high redshifts. We assume a total of 100 hours of exposure time; with such a set-up, the ETC predicts that a point source flux of 1.0 nJy can be detected in F356W and 2.0 nJy in F444W.
We can then calculate the minimum surface brightness for the local galaxy analogues in each band. We assume that, as is currently done with HST /Spitzer observations 3 http://jwstetc.stsci.edu/etc/input/nircam/imaging/ of high redshift galaxies, the size of the galaxies is measured from bands at shorter wavelengths. Accordingly, in Figure 1, we also plot a shaded region bounded by the diffraction-limited radius in the 2.0 µm filter at these redshifts and by the surface brightness corresponding to the S/N=5 fluxes calculated by the ETC. We caution, however, that when comparing the sizes care must be taken, as the data for the galaxies uses the half-light radius, whereas the minimum size prediction for JWST is given by the radius of a high redshift object in the diffraction limit.
From Figure 1, we can see that JWST can be expected to discover some of the local dwarfs if their stars already formed at early cosmic times. In particular, we predict that roughly 60-65% of the combined light of the dwarfs will be accessible to JWST. This corresponds to a detection of 9/73 dwarfs at 4.5 µm and 13/73 dwarfs at 3.6 µm, respectively. This differs from the result of Boylan-Kolchin et al. (2015) primarily due to our uniform assumption that these galaxies formed most of their stars at very early times. If it is the case that significant star formation occurs late in the galaxy's evolution, then there will not be enough light emitted to render the galaxy detectable even by JWST. Accordingly, these predicted fluxes will need to be scaled by the fraction of stars formed by z ∼ 6, 7 once more of the galaxies have their star formation histories analyzed.
Our calculated fluxes are predicated upon the assumption that the stellar populations of the z = 0 dwarfs are the modern analogues of the stars at z ∼ 6 − 7. However, these stars may be supplemented by PopIII-like stars, which are predicted to have masses in the range 10 − 1000M ⊙ and short lifetimes (Abel, Bryan, & Norman 2002;Prescott et al. 2009;Cassata, et al. 2013;Loeb & Furlanetto 2013;Sobral et al. 2015). Accordingly, in Figure 2 we consider the scenario in which the dwarf galaxies are 10 times more luminous at high redshifts due to an ancient population of stars that no longer exists.
In this case, we find that some of the dwarf progenitors could be among the galaxies already found with HST and Spitzer. JWST would observe ∼ 67% of the light of the Local Group dwarfs. This scenario can be distinguished from the fiducial one based on spectroscopy; several theoretical works have indicated that such a massive stars would have strong helium line emission, which would distinguish these stellar populations (Bromm et al. 2001;Tumlinson at al. 2001;Schaerer 2003).
An alternative source of an increase in surface brightness at early times may be provided by tidal stripping.
The simulations of Peñarrubia, Navarro, & McConnachie (2008) illustrate the effect of tidal stripping on dwarf galaxies by massive halos. This effect decreases both the sizes and surface brightnesses of the dwarf galaxies. If tidal stripping has played a major role in the history of the observed Local Group, then it is likely that at higher redshifts they were both brighter and larger, which would make the analogues of these galaxies more likely to be detectable.
In addition to the enhanced capabilities of JWST relative to HST and Spitzer, we note that further gains in sensitivity can be made using gravitational lensing (Mashian & Loeb 2013;Atek et al. 2015). Lensing has The surface brightness-size relation for z ∼ 6 (left) and z ∼ 7 (right) galaxies using Spitzer 3.6 µm and 4.5 µm fluxes, respectively, as well as sizes measured from HST. The region bounded by solid lines indicates the portion of the parameter space that has been observed by current programs. The shaded region indicates the part of the parameter space that should be accessible to JWST, with the size limit coming the diffraction limited radius in the given filter (see Section 4). Figure 1, but assuming the ancient stars were 10 times more luminous than those in the dwarfs today.
already permitted the discovery of high redshift galaxies even smaller than the sizes considered here (e.g., Kawamata et al. 2015); this technique using JWST is likely to be able to probe a larger sample of dwarf progenitors.
CONCLUSIONS
We have compared the physical properties of Local Group dwarf galaxies to high-redshift galaxies. We find that the sizes of the two populations agree very well, but when translated to higher redshifts, these dwarfs are too faint to be detected at present. However, in a deep field, the upcoming JWST mission should be able to detect analogues of the brightest of these objects, corresponding to about 60% of the total light of the dwarf population (omitting the LMC and SMC, and galaxies without measurements), assuming that their stars formed early.
This fraction increases if we assume a population of ancient, massive stars in these galaxies at high redshifts; spectroscopy and number counts will enable us to distinguish these two scenarios. Additionally, if these dwarfs have been significantly affected by tidal stripping, then this effect can also amplify the potential for analogues of these galaxies to be detected at high redshifts.
size) + Spitzer 4.5 µm (F ν ) JWST 2.0 µm (size) + 4.5 µm (F ν ) Fig. 1.-
size) + Spitzer 4.5 µm (F ν ) JWST 2.0 µm (size) + 4.5 µm (F ν ) Fig. 2.-Same as
We would like to thank Charlie Conroy and Matt Walker for helpful comments on a draft of this paper. We are also grateful to Charlie Conroy for useful discussions regarding the FSPS code.
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|
[] |
[
"Solution of the Conformable Angular Equation of the Schrodinger Equation",
"Solution of the Conformable Angular Equation of the Schrodinger Equation"
] |
[
"Eqab M Rabei [email protected] \nPhysics Department\nFaculty of Science\nAl al-Bayt University\nP.O. Box 13004025113MafraqJordan\n",
"Mohamed Al-Masaeed [email protected] \nPhysics Department\nFaculty of Science\nAl al-Bayt University\nP.O. Box 13004025113MafraqJordan\n",
"Ahmed Al-Jamel [email protected] \nPhysics Department\nFaculty of Science\nAl al-Bayt University\nP.O. Box 13004025113MafraqJordan\n"
] |
[
"Physics Department\nFaculty of Science\nAl al-Bayt University\nP.O. Box 13004025113MafraqJordan",
"Physics Department\nFaculty of Science\nAl al-Bayt University\nP.O. Box 13004025113MafraqJordan",
"Physics Department\nFaculty of Science\nAl al-Bayt University\nP.O. Box 13004025113MafraqJordan"
] |
[] |
In this work, the conformable Schrodinger equation in spherical coordinates is separated into two parts; radial and angular part, the angular part of the Schrodinger equation is solved. The normalized Spherical harmonics function is obtained as a solution of the angular part
| null |
[
"https://arxiv.org/pdf/2203.11615v1.pdf"
] | 247,597,265 |
2203.11615
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598c1106dcc65764195cc791054aabef1145f808
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Solution of the Conformable Angular Equation of the Schrodinger Equation
22 Mar 2022 March 23, 2022
Eqab M Rabei [email protected]
Physics Department
Faculty of Science
Al al-Bayt University
P.O. Box 13004025113MafraqJordan
Mohamed Al-Masaeed [email protected]
Physics Department
Faculty of Science
Al al-Bayt University
P.O. Box 13004025113MafraqJordan
Ahmed Al-Jamel [email protected]
Physics Department
Faculty of Science
Al al-Bayt University
P.O. Box 13004025113MafraqJordan
Solution of the Conformable Angular Equation of the Schrodinger Equation
22 Mar 2022 March 23, 2022conformable derivativespherical harmonicsSchrodinger equa- tionconformable partial derivative
In this work, the conformable Schrodinger equation in spherical coordinates is separated into two parts; radial and angular part, the angular part of the Schrodinger equation is solved. The normalized Spherical harmonics function is obtained as a solution of the angular part
Introduction
In quantum mechanics, the Schrodinger equation represents a key result to obtain the wave function, and it is the quantum counterpart of Newton's second law in classical mechanics, and to solve it with three-dimensional spherical coordinates, the method of separating the variables was used. It resulted in two equations, the first is a radial equation and the second is an angular equation so that the solution to the radial equation depends on knowing the potential and the solution to the angular equation is using the special functions, specifically the associated Legendre equation [1], where the associated Legendre equation is a generalization of the Legendre differential equation and the solutions P m l (x) to this equation are called the associated Legendre polynomials [2]. The fractional derivative which is a derivative of arbitrary order is as old as calculus. L'Hospital asked the Leibniz about the possibility that the order of the derivative to be 1 2 in 1695. Since then, many researchers tried to put a definition of fractional derivative. Leibniz was the first to offer the idea of a symbolic approach, employing the symbol d n y dx n = D n y for the nth derivative, where n is a non-negative integer [3]. In addition, one of the well-known fractional derivatives is the R.L. fractional derivative [4], and the second one is the Caputo derivative [5] In physics, mathematics, and engineering sciences, the fractional derivative has played an essential role [6][7][8][9][10][11][12][13][14][15]. In 2014, Khalil et.al [16], was introduced a new definition of derivative of α order called the conformable derivative, where 0 < α ≤ 1. This definition is a natural extension of the usual derivative and satisfies the standard properties of the traditional derivative i.e the derivative of the product and the derivative of the quotient of two functions and satisfies the chain rule. The conformable calculus has many applications in several fields, for example in physics , it was used in quantum mechanics to study The effect of fractional calculus on the formation of quantum-mechanical operators [17], and an extension of the approximate methods used in quantum mechanics was made [18][19][20], and the of conformable harmonic oscillator is quantized using the annihilation and creation operators [21], besides, the effect of deformation of special relativity studied by conformable derivative [22], and the conformable Laguerre and associated Laguerre differential equations using conformable Laplace transform are solved [23]. In this work, the conformable Schrodinger equation is separated into two parts radial which depends on the knowing the potential and angular part which we solved and we obtained the conformable spherical harmonic. Besides, as an explanation we plotted |Y 1α 2α | 2 in two and three dimensions.
Conformable derivative
We start by presenting some definitions related to our work. Definition 2.1. The conformable derivative of f with order 0 < α ≤ 1 is defined by [16] T
α (f )(t) = lim ǫ→0 f (t + ǫt 1−α ) − f (t) ǫ ,(1)
where f ∈ [0, ∞) → R. Definition 2.2. The conformable partial derivative of f with order 0 < α ≤ 1 is defined by [24]
∂ α ∂x α i f (x 1 , . . . , x m ) = lim ǫ→0 f (x 1 , . . . , x i−1 , x i + ǫx 1−α i , . . . , x m ) − f (x 1 , . . . , x m ) ǫ (2)
Conformable spherical harmonics
In terms the conformable derivative, we consider the Schrodinger equation as [25]
p 2 α 2m α ψ α (x, t) = (E α − V α (x α ))ψ α (x, t).(3)
and
α α = h (2π) 1 α
. The coordinate and the momentum operators are defined aŝ
x α = x,p α = −i α α ∇ α .(4)
To read more about the conformable quantum mechanics see ref [17,25]. In terms the conformable derivative, the Schrodinger equation in spherical coordinates can be written as
∇ 2α − 2m α 2α α (V α (r α ) − E α ) ψ α (r α , θ α , ϕ α ) = 0.(5)
where ∇ 2α in spherical coordinates is given by
∇ 2α = 1 r 2α D α r [r 2α D α r ] + 1 r 2α sin (θ α ) D α θ [sin (θ α )D α θ ] + 1 r 2α sin 2 (θ α ) D 2α ϕ . (6)
After substituting in eq.(5), we get
1 R α D α r [r 2α D α r R α ] + 1 Y α sin (θ α ) D α θ [sin (θ α )D α θ Y α ] + 1 Y α sin 2 (θ α ) D 2α ϕ Y α − 2m α r 2α 2α α (V α (r α ) − E α ) = 0.(7)
The first part of this equation that depends on r α and equal to a constant is given as
1 R α D α r [r 2α D α r R α ] − 2m α r 2α 2α α (V α (r α ) − E α ) = α 2 ℓ(ℓ + 1).(8)
This equation is called conformable radial equation and the solution of this equation depends on the potential V α (r α ). The second part of equation (7) reads as
1 Y α sin (θ α ) D α θ [sin (θ α )D α θ Y α ] + 1 Y α sin 2 (θ α ) D 2α ϕ Y α = −α 2 ℓ(ℓ + 1).(9)
Using separation of variable Y α (θ α , ϕ α ) = Θ α (θ α )Φ α (ϕ α ) to solve this equation, we get
1 Θ α sin (θ α ) D α θ [sin (θ α )D α θ Θ α ] + 1 Φ α sin 2 (θ α ) D 2α ϕ Φ α = −α 2 ℓ(ℓ + 1),(10)
after multiplied this equation by sin 2 (θ α ), we get
sin (θ α ) Θ α D α θ [sin (θ α )D α θ Θ α ] + α 2 ℓ(ℓ + 1) sin 2 (θ α ) + 1 Φ α D 2α ϕ Φ α = 0.(11)
The part of this equation that depends on ϕ α and equal to a constant is given as
1 Φ α D 2α ϕ Φ α = −α 2 m 2 ,(12)
thus, the solution of this equation is given by
Φ α (ϕ α ) = Ae imϕ α + Be −imϕ α .(13)
In this solution we will adopt the part Ae imϕ α because Φ α is a single valued function where m is integer, so,we get
Φ α (ϕ α ) = Ae imϕ α .(14)
The part of eq.(11) that depends on θ α and equal to a constant is given as
sin (θ α ) Θ α D α θ [sin (θ α )D α θ Θ α ] + α 2 ℓ(ℓ + 1) sin 2 (θ α ) = α 2 m 2 .(15)
Multiplying this equation by Θ α , we get
sin (θ α )D α θ [sin (θ α )D α θ Θ α ] + α 2 ℓ(ℓ + 1) sin 2 (θ α ) − m 2 Θ α = 0.(16)let Θ α (θ α ) = X α (x α ), x α = cos (θ α ) → αx α−1 dx = −αθ α−1 sin (θ α ) → D α θ = − sin (θ α )D α
x , After substituting in this eqution, we get
− (1 − x 2α )D α x [−(1 − x 2α )D α x X α ] + α 2 ℓ(ℓ + 1)(1 − x 2α ) − m 2 X α = 0. (17)
after multiplied this equation by
1 (1−x 2α ) , we get (1 − x 2α )D α x D α x X α − 2αx α D α x X α + α 2 ℓ(ℓ + 1) − m 2 (1 − x 2α ) X α = 0. (18)
This equation is called conformable associated Legendre differential equation and its solution is given by [26] X α = P mα ℓα =
(−1) m (1 − x 2α ) m 2 α ℓ 2 ℓ ℓ! D (ℓ+m)α (x 2α − 1) ℓ .(19)
So, the solution for eq.(9) is given as Y mα ℓα (θ α , ϕ α ) = N mα ℓα e imϕ α P mα ℓα (cos (θ α )), (20) where N mα ℓα is normalization constant, can be calculated using normalization condition |Y mα ℓα | 2 d α Ω = |N mα ℓα | 2 P mα ℓ ′ α (cos (θ α ))P mα ℓα (cos (θ α ))d α Ω
where d α Ω = sin (θ α )d α θd α ϕ.
Using the orthogonality of conformable associated Legendre functions [26], we get (2ℓ + 1)(ℓ − m)! α 2m−2 2(ℓ + m)!(2π) α e imϕ α P mα ℓα (cos (θ α )).
|Y mα ℓα | 2 d α Ω = |N mα ℓα | 2 (2π) α α α 2m−1 2(ℓ + m)! (2ℓ + 1)(ℓ − m)! = 1,
The relation between Y mα ℓα and Y −mα ℓα
The relation between Y mα ℓα and Y −mα ℓα is given by
Y −mα ℓα = (−1) m Y m * α ℓα (23)
Proof. in the first step we need to prove the relation between P mα ℓα and P −mα ℓα , let us define P −mα ℓα using eq.(19) as,
P −mα ℓα = (−1) m (1 − x 2α ) − m 2 α ℓ 2 ℓ ℓ! D (ℓ−m)α (x 2α − 1) ℓ (24) But, D (ℓ+m)α (x 2α −1) ℓ = D (ℓ+m)α (x α −1) ℓ (x α +1) ℓ , now let f = x α −1, g = x α +1 D (ℓ+m)α (f g) ℓ = D (ℓ+m)α (f ) ℓ (g) ℓ = D (ℓ+m)α (f 1 α ) αℓ (g 1 α ) αℓ Let w = f 1 α , z = g 1 α → D (ℓ+m)α [(w) αℓ (z) αℓ ]
. Using Leibniz rule [23], we get
D (ℓ+m)α [(w) αℓ (z) αℓ ] = ℓ+m k=0 ℓ + m k D (ℓ+m−k)α (w) αℓ D kα (z) αℓ = ℓ k=m ℓ + m k D (ℓ+m−k)α (w) αℓ D kα (z) αℓ where D kα (z) αℓ = α k ℓ! (ℓ−k)! (z) (ℓ−k)α , D (ℓ+m−k)α (w) αℓ = α ℓ+m−k ℓ! (k−m)! (w) (k−m)α D (ℓ+m)α [(w) αℓ (z) αℓ ] = ℓ k=m ℓ + m k α k ℓ! (ℓ − k)! (z) (ℓ−k)α α ℓ+m−k ℓ! (k − m)! (w) (k−m)α = ℓ k=m ℓ + m k α ℓ+m (ℓ!) 2 (ℓ − k)!(k − m)! (w) (k−m)α (z) (ℓ−k)α
Thus, we have
D (ℓ+m)α [(w) αℓ (z) αℓ ] = ℓ k=m α ℓ+m (ℓ!) 2 (ℓ + m)! k!(ℓ + m − k)!(ℓ − k)!(k − m)! (w) (k−m)α (z) (ℓ−k)α(25)
In the same way
D (ℓ−m)α [(w) αℓ (z) αℓ ] = ℓ−m r=0 ℓ − m r D (ℓ−m−r)α (w) αℓ D rα (z) αℓ = ℓ−m r=0 ℓ − m r α ℓ−m−r ℓ! (r + m)! (w) (r+m)α α r ℓ! (ℓ − r)! (z) (ℓ−r)α
Thus, we have
D (ℓ−m)α [(w) αℓ (z) αℓ ] = ℓ−m r=0 (ℓ − m)!α ℓ−m (ℓ!) 2 r!(ℓ − m − r)!(r + m)!(ℓ − r)! (w) (r+m)α (z) (ℓ−r)α(26)
Since the omitted terms in the sum vanish D kα (f ) r = 0 if k > r, and change the summation variable to k = r + m and substituting in eq.(26), we get
D (ℓ−m)α [(w) αℓ (z) αℓ ] = ℓ k=m (ℓ − m)!α ℓ−m (ℓ!) 2 (w) (k)α (z) (ℓ+m−k)α (k − m)!(ℓ − k)!(k)!(ℓ + m − k)!(27)
Multiply eq.(27) by α 2m (ℓ+m)!(w) mα (z) mα α 2m (ℓ+m)!(w) mα (z) mα , we have
D (ℓ−m)α [(w) αℓ (z) αℓ ] = (ℓ − m)!(w) mα (z) mα (ℓ + m)!α 2m ℓ k=m (ℓ + m)!α ℓ+m (ℓ!) 2 (w) (k−m)α (z) (ℓ−k)α (k − m)!(ℓ − k)!(k)!(ℓ + m − k)! (28) From eq.(25) , we get D (ℓ−m)α [(w) αℓ (z) αℓ ] = (ℓ − m)!(w) mα (z) mα (ℓ + m)!α 2m D (ℓ+m)α [(w) αℓ(29)
After substitutions, we have
D (ℓ−m)α [(x 2α − 1) ℓ ] = (ℓ − m)!(x 2α − 1) m (ℓ + m)!α 2m D (ℓ+m)α [(x 2α − 1) ℓ ](30)
Now substituting in eq.(24) , we have
P −mα ℓα = (−1) m (ℓ − m)! (ℓ + m)!α 2m (−1) m (1 − x 2α ) m 2 α ℓ 2 ℓ ℓ! D (ℓ+m)α [(x 2α − 1) ℓ ](31)
Using eq.(19), we get
P −mα ℓα = (−1) m (ℓ − m)! α 2m (ℓ + m)! P mα ℓα .(32)
In the second step We define Y −mα ℓα using eq.(22)
Y −mα ℓα = (2ℓ + 1)(ℓ + m)! α −2m−2 2(ℓ − m)!(2π) α e −imϕ α P −mα ℓα (cos (θ α )).(33)
After substituting eq.(32), we get
Y −mα ℓα = (2ℓ + 1)(ℓ + m)! α −2m−2 2(ℓ − m)!(2π) α e −imϕ α (−1) m (ℓ − m)! α 2m (ℓ + m)! P mα ℓα (cos (θ α )) = (−1) m (2ℓ + 1)(ℓ − m)! α 2m−2 2(ℓ + m)!(2π) α e −imϕ α P mα ℓα (cos (θ α )) = (−1) m Y m * α ℓα .(34)
Some of the low-lying conformable spherical harmonic functions are enumerated in the table below, as derived from the above formula.
α 2 2(2π) α 1 -1 α 3 4(2π) α e −iϕ α sin (θ α ) 0 3α 2 2(2π) α cos (θ α ) 1 −α 3 4(2π) α e iϕ α sin (θ α ) 2 -2 15α 2 16(2π) α sin 2 (θ α )e −i2ϕ α -1 α 15 4(2π) α e −iϕ α cos (θ α ) sin (θ α ) 0 5α 2 8(2π) α (3 cos 2 (θ α ) − 1) 1 −α 15 4(2π) α e iϕ α cos (θ α ) sin (θ α ) 2 15α 2 16(2π) α sin 2 (θ α )e i2ϕ α
The conformable spherical harmonic density for Y 1α 2α and for different values of α are plotted in 3D and 2D using Mathematica as follows,
Out[608]=
Conclusions
We have solved the angular part of the conformable Schrodinger equation, and we obtained the conformable spherical harmonic function as solution of this part. We observed that the conformable spherical harmonics goes to spherical harmonic function when α goes to 1. To illustrative our calculation we have drown the conformable spherical harmonic function for ℓ = 2 and m = 1 in 3D and 2D, with different values of α. We observed that in figures 1 the density function gradually convert to the traditional density function given in figures 2. Also the same thing have been seen for density function in polar plot.
Figure 1 :Figure 2 :
12Plot |Y 1α 2α | 2 with different value of α from 0.|Y
Figure 3 :
3Plot |Y 1α 2α | 2 with different value of α from 0.
Figure 4 :
4|Y 1α 2α | 2 when α = 1 in polar plot
Table 1 :
1the first nine conformable spherical harmonics Y mα
ℓα
ℓ m
Y mα
ℓα
0 0
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|
[] |
[
"Study of the generalized quantum isotonic nonlinear oscillator potential",
"Study of the generalized quantum isotonic nonlinear oscillator potential"
] |
[
"Nasser Saad \nDepartment of Mathematics and Statistics\nUniversity of Prince Edward Island\n550 University AvenueC1A 4P3CharlottetownPEICanada\n",
"Richard L Hall \nDepartment of Mathematics and Statistics\nConcordia University\n1455 de Maisonneuve Boulevard WestH3G 1M8MontréalQuébecCanada\n",
"† Hakan \nDepartment of Physics\nFaculty of Arts and Sciences\nGazi University\n06500AnkaraTurkey\n",
"Andözlem Yeşiltaş \nDepartment of Physics\nFaculty of Arts and Sciences\nGazi University\n06500AnkaraTurkey\n"
] |
[
"Department of Mathematics and Statistics\nUniversity of Prince Edward Island\n550 University AvenueC1A 4P3CharlottetownPEICanada",
"Department of Mathematics and Statistics\nConcordia University\n1455 de Maisonneuve Boulevard WestH3G 1M8MontréalQuébecCanada",
"Department of Physics\nFaculty of Arts and Sciences\nGazi University\n06500AnkaraTurkey",
"Department of Physics\nFaculty of Arts and Sciences\nGazi University\n06500AnkaraTurkey"
] |
[] |
We study the generalized quantum isotonic oscillator Hamiltonian given by H = −d 2 /dr 2 + l(l + 1)/r 2 + w 2 r 2 + 2g(r 2 − a 2 )/(r 2 + a 2 ) 2 , g > 0. Two approaches are explored. A method for finding the quasi-polynomial solutions is presented, and explicit expressions for these polynomials are given, along with the conditions on the potential parameters. By using the asymptotic iteration method we show how the eigenvalues of this Hamiltonian for arbitrary values of the parameters g, w and a may be found to high accuracy.
|
10.1155/2011/750168
|
[
"https://arxiv.org/pdf/1104.2591v1.pdf"
] | 119,717,757 |
1104.2591
|
1a98a30bedce8af2ac6663526be3c07d8faeda9d
|
Study of the generalized quantum isotonic nonlinear oscillator potential
13 Apr 2011
Nasser Saad
Department of Mathematics and Statistics
University of Prince Edward Island
550 University AvenueC1A 4P3CharlottetownPEICanada
Richard L Hall
Department of Mathematics and Statistics
Concordia University
1455 de Maisonneuve Boulevard WestH3G 1M8MontréalQuébecCanada
† Hakan
Department of Physics
Faculty of Arts and Sciences
Gazi University
06500AnkaraTurkey
Andözlem Yeşiltaş
Department of Physics
Faculty of Arts and Sciences
Gazi University
06500AnkaraTurkey
Study of the generalized quantum isotonic nonlinear oscillator potential
13 Apr 2011arXiv:1104.2591v1 [math-ph]keyword: Non-linear oscillatorsNon-polynomial potentialsGol'dman and Krivchenkov potentialAsymptotic Iteration MethodQuantum integrable systemsLaguerre polynomials PACS: 0365w, 0365Fd, 0365Ge
We study the generalized quantum isotonic oscillator Hamiltonian given by H = −d 2 /dr 2 + l(l + 1)/r 2 + w 2 r 2 + 2g(r 2 − a 2 )/(r 2 + a 2 ) 2 , g > 0. Two approaches are explored. A method for finding the quasi-polynomial solutions is presented, and explicit expressions for these polynomials are given, along with the conditions on the potential parameters. By using the asymptotic iteration method we show how the eigenvalues of this Hamiltonian for arbitrary values of the parameters g, w and a may be found to high accuracy.
I. INTRODUCTION
Recently, Cariñena et al. [1] studied a quantum nonlinear oscillator potential whose Schrödinger equation reads
− d 2 dx 2 + x 2 + 8 2x 2 − 1 (2x 2 + 1) 2 ψ n (x) = E n ψ(x)(1)
The interest in this problem came from the fact that it is exactly solvable, in a sense that the exact eigenenergies and eigenfunctions can be obtained explicitly. Indeed, Cariñena et al. [1] were able to show that ψ n (x) = Pn(x) (2x 2 +1) e −x 2 /2 , E n = −3 + 2n, n = 0, 3, 4, 5, . . .
where the polynomials factors P n (x) are related to the Hermite polynomials by means of P n (x) = 1 if n = 0 H n (x) + 4nH n−2 (x) + 4n(n − 3)H n−4 (x) if n = 3, 4, 5, . . .
In a more recent work, Fellows and Smith [6] showed that the potential V (x) = x 2 + 8(2x 2 − 1)/(2x 2 + 1) 2 as well as, for certain values of the parameters w, g and a, the potential V (x) = w 2 x 2 +2g(x 2 − a 2 )/(x 2 + a 2 ) 2 of the Schrödinger equation
− d 2 dx 2 + w 2 x 2 + 2g x 2 − a 2 (x 2 + a 2 ) 2 ψ n (x) = 2E n ψ(x),(4)
are indeed supersymmetric partners of the harmonic oscillator potential. Using the supersymmetric approach, the authors were able to construct an infinite set of exact soluble potentials, along with their eigenfunctions and eigenvalues. Very recently, Sesma [9], using a Möbius transformation, was able to transform Eq.(4) into a confluent Heun equation [8] and thereby obtain an efficient algorithm to solve the Schrödinger equation (4) numerically.
The purpose of the present work is to provide a detailed solution, by means of the quasi-polynomial solutions and the application of the asymptotic iteration method [2][3][4][5], for the Schrödinger equation − d 2 dr 2 + l(l + 1) r 2 + w 2 r 2 + 2g r 2 − a 2 (r 2 + a 2 ) 2 ψ(r) = 2Eψ(r), (5) where l is the angular momentum number l = −1, 0, 1, . . . . Our results show that the quasi-exact solutions of Sesma [9] as well the results of Cariñena et al. [1] follow as special cases of our general approach. The present article is organized as follows. In the next section, some preliminary analysis of the Schrödinger equation (5) is presented. A general approach for finding polynomial solutions of Eq.(5), for certain values of parameters w and g, is presented, and is based on a recent work of Ciftci et al. [3] for solving the second-order linear differential equation
3 i=0 a 3,i x i y ′′ + 2 i=0 a 2,i x i y ′ − 1 i=0 τ 1,i x i y = 0.(6)
More general quasi-exact solutions, including the results of Sesma [9], are discussed in section III. Unrestricted solutions of Eq.(5) based on the asymptotic iteration method are discussed in Section IV.
II. GENERALIZED QUANTUM ISOTONIC OSCILLATOR -PRELIMINARY RESULTS
A simple scaling argument, using r = a 2 x, allows us to write the equation (5) as
− d 2 dx 2 + l(l + 1) x 2 + (wa 2 ) 2 x 2 + 2g x 2 − 1 (x 2 + 1) 2 ψ(x) = 2Ea 2 ψ(x).(7)
A further substitution z = x 2 + 1 yields a differential equation with two regular singular points at z = 0, 1 and one irregular singular point of rank 2 at z = ∞. The roots µ's of the indicial equation for the regular singular point z = 0 reads µ ± = 1 2 (1 ± √ 1 + 4g), while the roots of the indicial equation at z = 1 are µ + = (l + 1)/2 and µ − = −l/2. Since the singularity for z → ∞ corresponds to that for x → ∞, it is necessary that the solution for z → ∞ behave as ψ(x) ∼ exp(−wa 2 x 2 /2). Consequently, we may assume the general solution of equation (7) which vanishes at the origin and at infinity takes the form
ψ n (x) = x l+1 (x 2 + 1) µ e − wa 2 2 x 2 f n (x).(8)
A straightforward calculation shows that f n (x) are the solutions of the second-order homogeneous linear differential equation
f ′′ (x) + 2(l + 1) x + 4µx x 2 + 1 − 2wa 2 x f ′ (x) + 2Ea 2 − wa 2 (2l + 3 + 4µ) + 2µ(2l + 3 + 2wa 2 ) + 4µ(µ − 1) − 2g x 2 + 1 + 4(g − µ(µ − 1)) (x 2 + 1) 2 f (x) = 0.(9)
In the next sections, we attempt to give a general solution of this equation. For now, we assume that µ takes the value of the indicial root
µ ≡ µ − = 1 2 (1 − 1 + 4g)(10)
which allows us to write Eq.(9) as
f ′′ n (x) + 2(l + 1) x + 4µx x 2 + 1 − 2wa 2 x f ′ n (x) + 2Ea 2 − wa 2 (2l + 3 + 4µ) + 2µ(2l + 3 + 2wa 2 ) + 2µ(µ − 1) x 2 + 1 f n (x) = 0.(11)
We now consider the cases where the following two equations are satisfied 2µ(2l + 3 + 2wa 2 ) + 2µ(µ − 1) = 0,
g = µ(µ − 1).
The solutions of this system, for g and µ, are given explicitly by
g = 0, µ = 0, or g = 2(1 + l + a 2 w)(3 + 2l + 2a 2 w), µ = −2(1 + l + a 2 w).(12)
In the next, we consider each case of these two sets of solutions.
A. Case 1
The first set of solutions (g, µ) = (0, 0) reduces the differential equation (9) to
xf ′′ n (x) + [−2wa 2 x 2 + 2(l + 1)]f ′ n (x) + (2Ea 2 − wa 2 (2l + 3)) x f n (x) = 0 (13)
which is a special case of the general differential equation
(a 3,0 x 3 + a 3,1 x 2 + a 3,2 x + a 3,3 ) y ′′ + (a 2,0 x 2 + a 2,1 x + a 2,2 ) y ′ − (τ 1,0 x + τ 1,1 ) y = 0,(14)
with a 3,0 = a 3,1 = a 3,3 = a 2,1 = τ 1,1 = 0, a 3,2 = 1, a 2,0 = −2wa 2 , a 2,2 = 2(l + 1), and τ 1,0 = −2Ea 2 + wa 2 (2l + 3). The necessary and sufficient conditions for polynomial solutions of Eq.(14) are given by the following theorem [3].
Theorem 1. The second-order linear differential equation (14) has a polynomial solution of degree n if τ 1,0 = n(n − 1)a 3,0 + na 2,0 , n = 0, 1, 2, . . . ,
along with the vanishing of (n + 1) × (n + 1)-determinant ∆ n+1 given by
∆ n+1 = β 0 α 1 η 1 γ 1 β 1 α 2 η 2 γ 2 β 2 α 3 η 3 . . . . . . . . . . . .
γ n−2 β n−2 α n−1 η n−1 γ n−1 β n−1 α n γ n β n = 0 where β n = τ 1,1 − n((n − 1)a 3,1 + a 2,1 ) α n = −n((n − 1)a 3,2 + a 2,2 ) γ n = τ 1,0 − (n − 1)((n − 2)a 3,0 + a 2,0 ) η n = −n(n + 1)a 3,3
and τ 1,0 is fixed for a given n in the determinant ∆ n+1 = 0.
Thus, the necessary condition for the differential equation (13) to have polynomial solutions
f n (x) = n i=0 c i x i is 2E n a 2 = wa 2 (2n ′ + 2l + 3), n ′ = 0, 1, 2, . . .(17)
while the sufficient condition, Eq(16), is where β n = 0, α n = −n(n + 2l + 1) and γ n = 2wa 2 (n − n ′ − 1).
∆ n+1 = 0 α 1 0 0 γ 1 0 α 2 0 γ 2 0 α 3 0 . . . . . . . . . . . .
If l = −1, the determinant ∆ n+1 is identically zero for all n, which is equivalent to the exact solutions of the one-dimensional harmonic oscillator problem.
For l = −1, we have for n = 0, 2, 4, . . . , ∆ n+1 ≡ 0 and we obtain the exact solutions of the Gol'dman and Krivchenkov (or Isotonic) Hamiltonian H 0 where
H 0 ψ nl (x) ≡ − d 2 dx 2 + l(l + 1) x 2 + w 2 a 4 x 2 ψ nl (x) = 2E g=0 nl a 2 ψ nl (x), 0 ≤ x < ∞.(18)
These exact solutions are given by [7] 2a 2 E g=0 nl = wa 2 (4n + 2l + 3), n = 0, 1, 2, . . .
ψ nl (x) = x l+1 e −wa 2 x 2 /2 1 F 1 (−n; l + 3 2 ; wa 2 x 2 ), n = 0, 1, 2, . . . .(19)
where the confluent hypergeometric function 1 F 1 (−n; a; z) defined, in terms of the Pochhammer symbol (or Gamma function Γ(a))
(a) k = Γ(a + k) Γ(a) = 1 if (k = 0, a ∈ C\{0}) a(a + 1)(a + 2) . . . (a + k − 1) if (k = N, a ∈ C) as 1 F 1 (−n; a; z) = n k=0 (−n) k z k (a) k k! .(20)
The polynomial solutions f n (x) = 1 F 1 (−n; l + 3 2 ; wa 2 x 2 ) are easily obtained by using the asymptotic iteration method (AIM), which is summarized by means of the following theorem.
Theorem 2: (H. Ciftci et al. [4], equations (2.13)-(2.14)) Given λ 0 ≡ λ 0 (x) and s 0 ≡ s 0 (x) in C ∞ , the differential equation
f ′′ (x) = λ 0 (x)f ′ (x) + s 0 (x)f (x)
has the general solution
f (x) = exp − x α(t)dt C 2 + C 1 x exp t (λ 0 (τ ) + 2α(τ )) dτ dt (21) if for some n ∈ N + = {1, 2, . . . } s n λ n = s n−1 λ n−1 = α(x), or δ n (x) = λ n s n−1 − λ n−1 s n = 0,(22)
where
λ n = λ ′ n−1 + s n−1 + λ 0 λ n , s n = s ′ n−1 + s 0 λ n .(23)
For the differential equation (13) with
λ 0 (x) = − (−2wa 2 x 2 +2(l+1)) x , s 0 (x) = −(2Ea 2 − wa 2 (2l + 3),(24)
the first few iterations with δ n = λ n s n−1 − λ n−1 s n = 0, using (21), implies
f 0 (x) = 1 f 1 (x) = 2wa 2 x 2 − (2l + 3) f 2 (x) = 4w 2 a 4 x 4 − 4wa 2 (2l + 5)x 2 + (2l + 3)(2l + 5) . . .(25)
which we may easily generalized using the definition of the confluent hypergeometric function, Eq(20), as
f n (x) = 1 F 1 (−n; l + 3 2 ; wa 2 x 2 )(26)
up to a constant.
B. Case 2
The second set of solutions
(g, µ) = (2(1 + l + a 2 w)(3 + 2l + 2a 2 w), −2(1 + l + a 2 w))
allow us to write the differential equation (9) as
f ′′ n (x) + 2(l + 1) x − 8(l + 1 + a 2 w)x x 2 + 1 − 2wa 2 x f ′ n (x) + 2Ea 2 + wa 2 (6l + 5 + 8wa 2 ) f n (x) = 0.(27)
A further change of variable z = x 2 + 1 allows us to write the differential equation (27) as
4z(z − 1)f ′′ (z) − 4a 2 wz 2 + 2(6l + 5 + 6wa 2 )z − 16(l + 1 + wa 2 ) f ′ (z) + (2Ea 2 + wa 2 (6l + 5 + 8wa 2 ))z f (z) = 0,(28)
Again, Eq.(28) is a special case of the differential equation (14) with a 3,0 = a 3,3 = τ 1,1 = 0, a 3,1 = 4, a 3,2 = −4, a 2,0 = −4wa 2 , a 2,1 = −2(6l + 5 + 6wa 2 ), a 2,2 = 16(l + 1 + wa 2 ) and τ 1,0 = −2Ea 2 − wa 2 (6l + 5 + 8wa 2 ). Consequently, the polynomial solutions f n (x) of (28) are subject to the following two conditions: the necessary condition (15) reads
2E n a 2 = wa 2 (4n ′ − 6l − 5 − 8wa 2 ), n ′ = 0, 1, 2, . . .(29)
and the sufficient condition; namely, the vanishing of the tridiagonal determinant Eq(16), reads
∆ n+1 = β 0 α 1 γ 1 β 1 α 2 γ 2 β 2 α 3 . . . . . . . . . γ n−2 β n−2 α n−1 γ n−1 β n−1 α n γ n β n = 0 where β n = −2n(2n − 6l − 7 − 6wa 2 ) α n = 4n(n − 4l − 5 − 4a 2 w) γ n = 4wa 2 (n − n ′ − 1)(30)
and n ′ = n is fixed for the given dimension of the determinant ∆ n+1 . From the sufficient condition (30) we obtain the following conditions on the parameters
∆ 2 = 0 ⇒ a 2 w(l + 1 + a 2 w) = 0 ∆ 3 = 0 ⇒ a 2 w(l + 1 + a 2 w)(1 + 2l + 2a 2 w) = 0 ∆ 4 = 0 ⇒ a 2 w(l + 1 + a 2 w)(1 + 2l + 2a 2 w)(3(1 + 6l) + 14a 2 w) = 0 ∆ 5 = 0 ⇒ a 2 w(l + 1 + a 2 w)(1 + 2l + 2a 2 w)(3(6l − 1)(6l + 1) + 4(38l + 1)a 2 w + 44a 4 w 2 ) = 0 ∆ 6 = 0 ⇒ a 2 w(l + 1 + a 2 w)(1 + 2l + 2a 2 w)(3(2l − 1)(6l − 1)(6l + 1) + 2(208l 2 − 54l − 5)a 2 w + 200la 4 w 2 ) = 0 . . . = . . .
For a physically meaningful solution we must have a 2 w > 0. This is possible for a very restricted value of the angular momentum number l. Since β 0 = 0, we may observe that
∆ n+1 =(l + 1 + a 2 w)(1 + 2l + 2a 2 w)× β 2 α 3 γ 3 β 3 α 4 γ 4 β 4 α 5 . . . . . . . . . γ n−2 β n−2 α n−1 γ n−1 β n−1 α n γ n β n =(l + 1 + a 2 w)(1 + 2l + 2a 2 w) × Q l n−1 (a 2 w)
where Q l n−1 (a 2 w) are polynomials in the parameter product a 2 w. For physically acceptable solutions, we must have l = −1 and the factor (l + 1 + a 2 w) yields a 2 w = 0, which is not physically acceptable; so we ignore it. The second factor (1 + 2l + 2a 2 w) implies a special value of a 2 w = 1/2, for all n, which we will study shortly in full detail. Meanwhile, the polynomials Q l n (a 2 w)
Q l=−1 n−1 (a 2 w) = 1 if n = 2 14a 2 w − 15 if n = 3 44a 4 w 2 − 148a 2 w + 105 if n = 4 200a 4 w 2 − 514a 2 w + 315 if n = 5 . . .(31)
give new values, not reported before, of a 2 w that yield quasi-exact solutions of the Schrödinger equation (with one eigenstate)
−ψ ′′ n (x) + (wa 2 ) 2 x 2 + 4a 2 w(1 + 2a 2 w) (x 2 − 1) (x 2 + 1) 2 ψ n (x) = wa 2 (4n + 1 − 8a 2 w)ψ n (x)(32)
where
ψ n (x) = (x 2 + 1) −2a 2 w e −wa 2 x 2 /2 f n (x),
and f n (x) are the solutions of
4z(z − 1)f ′′ (z) − 4a 2 wz 2 + 2(−1 + 6wa 2 )z − 16wa 2 f ′ (z) + 4nwa 2 z f (z) = 0, z = x 2 + 1.(33)
For example, ∆ 4 = 0 implies, using (31), that a 2 w = 15 14 , and thus we have for
−ψ ′′ 3 (x) + 225 196 x 2 + 660 49 (x 2 − 1) (x 2 + 1) 2 ψ 3 (x) = 465 98 ψ 3 (x),(34)
the exact solution
ψ 3 (x) = (x 2 + 1) − 15 7 e − 15 28 x 2 (45x 6 + 225x 4 + 315x 2 − 49)
with a plot of the wave function and potential given in Figure 1. Similar results can be obtained for ∆ n+1 = 0, for n ≥ 5.
C. Exactly solvable quantum isotonic nonlinear oscillator
As mentioned above, for l = −1 and a 2 w = 1/2, it clear that ∆ n+1 = 0 for all n and the one-dimensional Schrödinger equation
− d 2 dx 2 + x 2 4 + 4(x 2 − 1) (x 2 + 1) 2 ψ n (x) = (2n − 3 2 )ψ n (x), n = 0, 1, 2, . . .(36)
has the exact solutions
ψ n (x) = (x 2 + 1) −1 e −x 2 /4 f n (x),(37)
where f n (x) are the polynomial solutions of the following second-order linear differential equation (z = x 2 + 1)
4z(z − 1)f ′′ n (z) − 2z 2 + 4z − 8 f ′ n (z) + 2nz f n (z) = 0,(38)
By using AIM (Theorem 2, Eq.(21)), we find that the polynomial solutions f n (x) of Eq.(38) are given explicitly as
f 0 (x) = 1 f 1 (x) = x 2 − 2 f 2 (x) = x 3 − 6x 2 + 8 f 3 (x) = x 4 − 16x 3 + 52x 2 − 52 f 4 (x) = x 5 − 30x 4 + 250x 3 − 580x 2 + 464 . . .(39)
a set of polynomial solutions that can be generated using f 0 (x) = 1, f n (x) = −3x(2n + 1) 1 F 1 (−n; 3 2 ; 1 2 (x − 1)) + 6((n + 1)x − 1) 1 F 1 (−n + 1;
3 2 ; 1 2 (x − 1)),(40)
up to a constant factor, where, again, 1 F 1 refers to the confluent hypergeometric function defined by (20). Note that the polynomials f n (x) in equation (40) can be expressed in terms of the associated Laguerre polynomials [10] as
f 0 (x) = 1, f n (x) = 3(−1) n √ π Γ(n) 2Γ(n + 3 2 ) ((1 + n)(x − 1) 2 + n)L 1 2 n x − 1 2 − (x − 1)((1 + n)x − 1)L 3 2 n x − 1 2 . (41)
III. QUASI-POLYNOMIAL SOLUTIONS OF THE GENERALIZED QUANTUM ISOTONIC OSCILLATOR
In this section we study the quasi-polynomial solutions of the differential equation (9). We note first, using the change of variable z = x 2 , Eq.(9) can be written as
f ′′ n (z) + 2l + 3 2z + 2µ z + 1 − wa 2 f ′ n (z) + 2Ea 2 − wa 2 (2l + 3 + 4µ) 4z + µ(2l + 3 + 2wa 2 ) 2z(z + 1) − g 2 (z − 1) z(z + 1) 2 + µ(µ − 1) (z + 1) 2 f n (z) = 0(42)
By means of the Möbius transformation z = t/(1 − t) that maps the singular points {−1, 0, ∞} into {0, 1, ∞}, we obtain
f ′′ n (t) + 2l + 3 2t(1 − t) + 2(µ − 1) 1 − t − wa 2 (1 − t) 2 f ′ n (t) + µ(2l + 3 + 2wa 2 ) 2t(1 − t) 2 − g 2 (2t − 1) t(1 − t) 2 + µ(µ − 1) (1 − t) 2 f n (t) = 0, (43)
where we assume
2Ea 2 − (2l + 3 + 4µ)wa 2 = 0.(44)
The differential equation (43) can be written as
(t 3 − 2t 2 + t)f ′′ n (t) + −2(µ − 1)t 2 + (2µ − wa 2 − l − 7 2 )t + (l + 3 2 ) f ′ n (t) + (µ(µ − 1) − g)t + g 2 + µ(l + 3 2 + wa 2 ) f n (t) = 0(45)
which we may now compare with equation (14) in Theorem 1 with a 3,0 = 1, a 3,1 = −2, a 3,2 = 1, a 3,3 = 0, a 2,0 = −2(µ − 1), a 2,1 = (2µ − wa 2 − l − 7/2), a 2,2 = (l + 3/2), τ 1,0 = −(µ(µ − 1) − g), τ 1,1 = − g 2 − µ(l + 3 2 + wa 2 ). We, thus, conclude that the quasi-polynomial solutions f n (t) of Eq.(45) are subject to the following conditions:
g = (µ − k)(µ − k − 1), k = 0, 1, 2, . . .(46)
along with the vanishing of the tridiagonal determinant ∆ n+1 = 0
β 0 α 1 γ 1 β 1 α 2 γ 2 β 2 α 3 . . . . . . . . . γ n−1 β n−1 α n γ n β n = 0 where β n = − 1 2 (g + (µ − n)(3 + 2l + 4n + 2a 2 w)), α n = −n(n + l + 1 2 ), γ n = g − (µ − n + 1)(µ − n),(47)
Here, again, g = (µ − k)(µ − k − 1) is fixed for given k = n, the fixed size of the determinant ∆ n+1 .
A. Particular Case: n = 0
For k (f ixed) ≡ n = 0, the differential equation (45) has the exact solution f 0 (t) = 1 if g and µ satisfies, simultaneously, the following system of equations
g + µ(3 + 2l + 2a 2 w) = 0, g = µ(µ − 1).
Solving this system of equations for g and µ, we obtain the following values of g = 2(1 + l + a 2 w)(3 + 2l + 2a 2 w), and µ = −2(l + 1 + wa 2 ),
and the ground-state energy, in this case, is given by Eq.(44), namely,
Ea 2 = − 1 2 a 2 w(5 + 6l + 8a 2 w)(49)
which in complete agreement with the results of Section II.B.
B. Particular Case: n = 1
For k (f ixed) ≡ n = 1, the determinant ∆ 2 = 0 of (47) yields g 2 + g(−1 + 10µ + 2l(2µ + 1) + 2a 2 w(2µ − 1)) + µ(µ − 1)(15 + 4l 2 + 8l(2 + a 2 w) + 4a 2 w(5 + a 2 w)) = 0,
g − (µ − 1)(µ − 2) = 0 (50)
where the energy is given by use of Eq.(44), for the computed values of µ and g, by
E = (l + 3 2 + 2µ)w.(51)
Further, Eq.(50) yields the solutions for l as functions of µ and a 2 w
l = 2 − (5 + 4a 2 w)µ − 2µ 2 ± 4 − 4(3 + 8a 2 w)µ + 9µ 2 4µ ≥ −1,(52)
where the energy states are now given by (51) along with l given by Eq.(52). We may also note that for
a 2 w = 1 2 (k + 1), k = 0, 1, 2, . . .(53)
and
a 2 E ± = − 1 8µ (k + 1) −2 + (2k + 1)µ − 6µ 2 ± 4 − 4(4k + 7)µ + 9µ 2 .(54)
Further, for g = (µ − 1)(µ − 2), we obtain the un-normalized wave function (see Eq. (8))
ψ 1,l (x) = x l+1 (1 + x 2 ) µ−1 e −wa 2 x 2 /2 (1 + 1 + 2l + µ + 2a 2 w 5 + 2l + µ + 2a 2 w x 2 )(55)
Thus, we may summarize these results as follows. The exact solutions of the Schrödinger equation (7) are given by Eqs. (54) and (55) only if g and µ are the solutions of the system given by Eq.(50). In Tables I and II, we report few quasi-exact solutions that can be obtained using this approach.
C. Particular Case n = 2
For k (f ixed) ≡ n = 2, the determinant ∆ 3 = 0 along with the necessary condition (47) yields
g 3 + 3g 2 (7µ − 1 + 2l(1 + µ) + 2a 2 w(µ − 1)) − g[18 + 56l + 8l 2 + 18(7 + 2l)µ − 3(5 + 2l)(7 + 2l)µ 2 −12a 2 w(µ − 1)((7 + 2l)µ − 4) − 4a 4 w 2 (2 + 3(µ − 2)µ)] + µ(µ − 2)(µ − 1)(105 + 142l + 60l 2 + 8l 3 + 6a 2 w(5 + 2l)(7 + 2l) +12a 4 w 2 (7 + 2l) + 8a 6 w 3 ) = 0, g − (µ − 2)(µ − 3) = 0 (56)
where, again, the energy is given, for the computed values of µ and g using Eqs.(44) and (56), by E = (l + 3 2 + 2µ)w.
In Table III, we report the numerical results for some of the exact solutions of µ and g using Eq. (56) and the values of (l, wa 2 ) = (−1, 1 2 ), (l, wa 2 ) = (−1, 1), (l, wa 2 ) = (−1, 3 2 ), (l, wa 2 ) = (−1, 2) (l, wa 2 ) = (0, 1 2 ), and (l, wa 2 ) = (0, 2), respectively. We have also computed the corresponding eigenvalues E wa 2 2,l ≡ E wa 2 2,l (µ, g). n l wa 2 Conditions E wa 2 n,l ≡ E wa 2 n,l (µ, g)
1 −1 1 2 µ = 1 3 −3 − 15A −1/3 − A 1/3 , A = 3(36 − √ 961) g = 1 9 A −2/3 (15 + 6A 1/3 + A 2/3 )(15 + 9A 1/3 + A 2/3 ) E 1 2 1,−1 = −w( 3 2 + 2 3 A 1/3 + 10A −1/3 ) 1 µ = 1 3 −5 − 19A −1/3 − A 1/3 , A = 161 − 3 √ 2118 g = 1 9 A −2/3 (19 + 8A 1/3 + A 2/3 )(19 + 11A 1/3 + A 2/3 ) E 1 1,−1 = −w( 17 6 + 2 3 A 1/3 + 38 3 A −1/3 ) 3 2 µ = 1 3 −7 − 25A −1/3 − A 1/3 , A = 199 − 18 √ 74 g = 1 9 A −2/3 (25 + 10A 1/3 + A 2/3 )(25 + 13A 1/3 + A 2/3 ) E 3 2 1,−1 = −w( 25 6 + 2 3 A 1/3 + 50 3 A −1/3 ) 2 µ = 1 3 −9 − 33A −1/3 − A 1/3 , A = 3(72 − √ 1191) g = 1 9 A −2/3 (33 + 12A 1/3 + A 2/3 )(33 + 15A 1/3 + A 2/3 ) E 2 1,−1 = −w( 11 2 + 2 3 A 1/3 + 22A −1/3 ) 0 1 2 µ = 0 g = 2 E 1 2 1,0 = 3 2 w µ = − 1 2 (7 + √ 17) g = 29 + 5 √ 17 E 1 2 1,0 = − 1 2 (11 + 2 √ 17)w µ = − 1 2 (7 − √ 17) g = 29 − 5 √ 17 E 1 2 1,0 = − 1 2 (11 − 2 √ 17)w 1 µ = −3 + B g = (−4 + B) (−5 + B) B = 1 3 ℜ A 1/3 + 33A −1/3 , A = −108 + 3i √ 2697 E 1 1,0 = − 9 2 − 2B w µ = −3 − B, g = (5 + B) (4 + B)) B = ℜ 11(1+i √ 3)A −1/3 2 + (1−i √ 3)A 1/3 6 , A = −108 + 3i √ 2697 E 1 1,0 = −( 9 2 + 2B)w µ = −3 − B, g = (5 + B) (4 + B)) B = ℜ 11(1−i √ 3)A −1/3 2 + (1+i √ 3)A 1/3 6 , A = −108 + 3i √ 2697 E 1 1,0 = −( 9 2 + 2B)w
IV. NUMERICAL COMPUTATION BY USE OF THE ASYMPTOTIC ITERATION METHOD
For the potential parameters w, a 2 and g, not necessarily obeying the conditions for quasi-polynomial solutions discussed in the previous sections, the asymptotic iteration method can be employed to compute the eigenvalues of Schrödinger equation (7) for arbitrary values w, a 2 and g. The functions λ 0 and s 0 , using Eqs.(43) and (44), are given by n l wa 2 Conditions E wa 2 n,l ≡ E wa 2 n,l (µ, g)
λ 0 (t) = − 2l+3 2t(1−t) + 2( Ea 2 2wa 2 − 2l+3 4 −1) (1−t) − wa 2 (1−t) 2 , s 0 (t) = − ( Ea 2 2wa 2 − 2l+3 4 )(2l+3+2wa 2 ) 2t(1−t) 2 − g 2 (2t−1) t(1−t) 2 + ( Ea 2 2wa 2 − 2l+3 4 )( Ea 2 2wa 2 − 2l+3 4 −1) (1−t) 2 ,(57)0 3 2 µ = − 11 3 + B g = − 14 3 + B − 17 3 + B B = 1 3 ℜ A 1/3 + 43A −1/3 , A = −98 + 9i √ 863 E 3 2 1,0 = − 1 6 (35 − 12B) w µ = − 11 3 − B, g = 1 9 (17 + 3B) (14 + 3B)) B = 1 6 ℜ 43(1 + i √ 3)A −1/3 + (1 − i √ 3)A 1/3 , A = −98 + 9i √ 863 E 1 1,0 = − 1 6 (35 + 12B)w µ = − 11 3 − B, g = 1 9 (17 + 3B) (14 + 3B)) B = 1 6 ℜ 43(1 − i √ 3)A −1/3 + (1 + i √ 3)A 1/3 , A = −98 + 9i √ 863 E 1 1,0 = − 1 6 (35 + 12B)w 2 µ = − 13 3 + B g = 1 9 (−16 + 3B) (−19 + 3B) B = 1 3 ℜ A 1/3 + 55A −1/3 , A = −55 + 165i √ 6 E 3 2 1,0 = − 1 6 (43 − 12B) w µ = − 13 3 − B, g = 1 9 (16 + 3B) (19 + 3B)) B = 1 6 ℜ 55(1 + i √ 3)A −1/3 + (1 − i √ 3)A 1/3 , A = −55 + 165i √ 6 E 1 1,0 = − 1 6 (43 + 12B)w µ = − 13 3 − B, g = 1 9 (16 + 3B) (19 + 3B)) B = 1 6 ℜ 55(1 − i √ 3)A −1/3 + (1 + i √ 3)A 1/3 , A = −55 + 165i √ 6 E 1 1,0 = − 1 6 (43 + 12B)w
where t ∈ (0, 1). The AIM sequence λ n (x) and s n (x) can be calculated iteratively using the iterative sequences (23). The energy eigenvalues of the quantum nonlinear isotonic potential (7) are obtained from the roots of the termination condition (22). According to the asymptotic iteration method, in particular the study of Brodie et al [2], unless the differential equation is exactly solvable, the termination condition (22) produces for each iteration an expression that depends on both t and E (for given values of the parameters wa 2 , g and l). In such a case, one faces the problem of finding the best possible starting value t = t 0 that stabilizes the AIM process [2]. Fortunately, since t ∈ (0, 1), the starting value t 0 doesn't represent a serious issue in our eigenvalue calculation using (57) and the termination condition (22) in contrast to the case of computing the eigenvalues using λ 0 (x) and s 0 (x) as given by, for example, equation (9), where x ∈ (0, ∞). In Table IV, we report our numerical results for energies of the four lowest states of the generalized isotonic oscillator of parameters w and a such that wa 2 = 2 and for different values of g. In this table, we set l = −1 for computing the energies E 0 a 2 and E 2 a 2 , while we put l = 0 for computing the energies E 1 a 2 and E 3 a 2 , respectively. For most of these values, the starting value of t is t 0 = 0.5 and is shifted towards zero as g gets larger in value. For the values of g that admit a quasi-polynomial solution, the number of iteration doesn't exceed three. For most of the other values of g, the total number of iteration didn't exceed 65. We found that for wa 2 = 2 and the values of g reported in Table IV, the number of iteration is relatively small compared to the case of wa 2 = 1/2 and a large value of the parameter g. The numerical computations in the present work were done using Maple version 13 running on an IBM architecture personal computer in a high-precision environment. In order to accelerate our computation we have written our own code for a root-finding algorithm instead of using the default procedure Solve of Maple 13. These numerical results are accurate to the number of decimals reported.
V. CONCLUSION
We have provided a detailed solution of the eigenproblem posed by Schrödiger's equation with a generalized nonlinear isotonic oscillator potential. We have presented a method for computing the quasi-polynomial solutions in cases where the potential parameters satisfy certain conditions. In other more general cases we have used the asymptotic iteration method to find accurate numerical solutions for arbitrary values of the potential parameters g, w and a. IV: Energies of the four lowest states of the generalized isotonic oscillator of parameters w and a given for l = −1 as wa 2 = 2 and for different values of the parameter g. The subscript numbers represents the number of iterations used by AIM.
(− 1 )
12j+1 α 2j+1 γ 2j+1 = 0 if n = 1, 3, 5, . . .
FIG. 1 :
1Plot of the unnormalized wave function ψ3(x) and the potential V3 = 225 196 x 2 + 660 49 (x 2 −1) (x 2 +1) 2 Further, ∆ 5 = 0, Eq.(31) implies a 2 w = 37 22 ± √ 214 22
I: Conditions on the value of the parameters g and µ for the quasi-polynomial solutions in the case of ∆2 = 0 with different values of wa 2 and l.
II: Conditions on the value of the parameters g and µ for the quasi-polynomial solutions in the case of ∆2 = 0 with different values of wa 2 and l.
TABLE
TABLE
TABLE III :
IIIExact eigenvalues for different values of l and wa 2 in the case ∆3 = 0. µ1 = −8.469623341124414 En l wa 2 Conditions
E n,l ≡ E wa 2
n,l (µ, g)
2 −1 1
2
µ1 = −6.301870878994198 E
1
2
2,−1 = −6.051870878994198
g1 = 77.22293097048609
µ2 = −2.4855365082108594 E
1
2
2,−1 = −2.2355365082108594
g2 = 24.605574274703333
1
µ1 = −7.398182984326876 E 1
2,−1 = −7.148182984326876
g1 = 97.7240263912181
µ2 = −3.3550579014968194 E 1
2,−1 = −3.1050579014968194
g2 = 34.03170302988033
µ3 = 0.9498105417574756 E 2
2,−1 = 1.1998105417574756
g3 = 2.1530873564462514
3
2
TABLE
AcknowledgmentsPartial financial support of this work under Grant Nos. GP249507 and GP3438 from the Natural Sciences and Engineering Research Council of Canada is gratefully acknowledged by two of us (respectively NS and RLH).
A quantum exactly solvable nonlinear oscillator related to the isotonic oscillator. J F Cariñena, A M Perelomov, M F Rańada, M Santander, J. Phys. A: Math. Theor. 4185301J. F. Cariñena, A. M. Perelomov, M. F. Rańada and M. Santander, A quantum exactly solvable nonlinear oscillator related to the isotonic oscillator, J. Phys. A: Math. Theor. 41 (2008) 085301.
Asymptotic Iteration Method for singular potentials. B Champion, R L Hall, N Saad, Int. J. Mod. Phys. A. 23B. Champion, R. L. Hall, and N. Saad, Asymptotic Iteration Method for singular potentials, Int. J. Mod. Phys. A 23 (2008) 1405 -1415.
Physical applications of second-order linear differential equations that admit polynomial solutions. H Ciftci, R L Hall, N Saad, E Dogu, J. Phys. A: Math. Theor. 43415206H. Ciftci, R. L. Hall, N. Saad, and E. Dogu, Physical applications of second-order linear differential equations that admit polynomial solutions, J. Phys. A: Math. Theor. 43 (2010) 415206.
Asymptotic iteration method for eigenvalue problems. H Ciftci, R L Hall, N Saad, J. Phys. A: Math. Gen. 3611807H. Ciftci, R. L. Hall, and N. Saad, Asymptotic iteration method for eigenvalue problems, J. Phys. A: Math. Gen. 36 (2003) 11807.
Construction of exact solutions to eigenvalue problems by the asymptotic iteration method. H Ciftci, R L Hall, N Saad, J. Phys. A: Math. Gen. 381147H. Ciftci, R. L. Hall, and N. Saad, Construction of exact solutions to eigenvalue problems by the asymptotic iteration method, J. Phys. A: Math. Gen. 38 (2005) 1147.
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Special functions: an introduction to the classical functions of mathematical physics. N M Temme, WileyNew YorkLaguerre polynomials are discussed in chapter 6N. M. Temme, Special functions: an introduction to the classical functions of mathematical physics, Wiley, New York, (1996). Laguerre polynomials are discussed in chapter 6.
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[] |
[
"Large-scale signatures of unconsciousness are consistent with a departure from critical dynamics",
"Large-scale signatures of unconsciousness are consistent with a departure from critical dynamics"
] |
[
"Enzo Tagliazucchi \nInstitute for Medical Psychology\nChristian Albrechts University Kiel\n24105KielGermany\n\nDepartment of Neurology and Brain Imaging Center\nGoethe University\n\n",
"Dante R Chialvo \nComision Nacional de Investigaciones Cientificas y Tecnologicas (CONICET)\nArgentina\n",
"Michael Siniatchkin \nInstitute for Medical Psychology\nChristian Albrechts University Kiel\n24105KielGermany\n",
"Enrico Amico \nComa Science Group, GIGA Research and Cyclotron Research Center, University and University Hospital of Liège\nLiègeBelgium\n",
"Jean-Francois Brichant \nComa Science Group, GIGA Research and Cyclotron Research Center, University and University Hospital of Liège\nLiègeBelgium\n",
"Vincent Bonhomme \nComa Science Group, GIGA Research and Cyclotron Research Center, University and University Hospital of Liège\nLiègeBelgium\n",
"Quentin Noirhomme \nComa Science Group, GIGA Research and Cyclotron Research Center, University and University Hospital of Liège\nLiègeBelgium\n",
"Helmut Laufs \nDepartment of Neurology and Brain Imaging Center\nGoethe University\n\n\nDepartment of Neurology\nInstitute for Medical Psychology\nChristian Albrechts University Kiel\nUKSH Arnold-Heller-Straße 324104, 24105Kiel, KielGermany., Germany\n",
"Steven Laureys \nComa Science Group, GIGA Research and Cyclotron Research Center, University and University Hospital of Liège\nLiègeBelgium\n",
"Enzo Tagliazucchi "
] |
[
"Institute for Medical Psychology\nChristian Albrechts University Kiel\n24105KielGermany",
"Department of Neurology and Brain Imaging Center\nGoethe University\n",
"Comision Nacional de Investigaciones Cientificas y Tecnologicas (CONICET)\nArgentina",
"Institute for Medical Psychology\nChristian Albrechts University Kiel\n24105KielGermany",
"Coma Science Group, GIGA Research and Cyclotron Research Center, University and University Hospital of Liège\nLiègeBelgium",
"Coma Science Group, GIGA Research and Cyclotron Research Center, University and University Hospital of Liège\nLiègeBelgium",
"Coma Science Group, GIGA Research and Cyclotron Research Center, University and University Hospital of Liège\nLiègeBelgium",
"Coma Science Group, GIGA Research and Cyclotron Research Center, University and University Hospital of Liège\nLiègeBelgium",
"Department of Neurology and Brain Imaging Center\nGoethe University\n",
"Department of Neurology\nInstitute for Medical Psychology\nChristian Albrechts University Kiel\nUKSH Arnold-Heller-Straße 324104, 24105Kiel, KielGermany., Germany",
"Coma Science Group, GIGA Research and Cyclotron Research Center, University and University Hospital of Liège\nLiègeBelgium"
] |
[] |
Loss of cortical integration and changes in the dynamics of electrophysiological brain signals characterize the transition from wakefulness towards unconsciousness. In this study we arrive at a basic model explaining these observations based on the theory of phase transitions in complex systems. We studied the link between spatial and temporal correlations of largescale brain activity recorded with fMRI during wakefulness, propofol-induced sedation and loss of consciousness, and during the subsequent recovery. We observed that during unconsciousness activity in fronto-thalamic regions exhibited a reduction of long-range temporal correlations and a departure of functional connectivity from anatomical constraints.A model of a system exhibiting a phase transition reproduced our findings, as well as the diminished sensitivity of the cortex to external perturbations during unconsciousness. This framework unifies different observations about brain activity during unconsciousness and predicts that the principles we identified are universal and independent from its causes. seems to be partially explained by the underlying anatomy (Hagmann et al. 2008; Hermundstad et al. 2013; Greicius et al. 2009; Wang et al. 2013). This suggests that spontaneous brain activity can be understood as an ever-transient (or metastable) exploration of the wide repertoire of paths offered by the underlying structural connectivity, the extent of such exploration potentially depending on the brain state, with variable repertoires corresponding to different degrees of awareness.Here we put forward an interpretation of propofol-induced loss of consciousness in analogy to the dynamics and connectivity of fluctuations seen on a diversity of complex systems exhibiting different phases. As a system explores the space of possible configurations, its spatio-temporal correlations behave in characteristic ways. In particular, the dynamical changes underlying different degrees of awareness could be analogous to the qualitative 2010; Casali et al. 2013; Pigorini et al. 2014). The response to endogenous fluctuations during deep sleep is also rapidly vanishing, resulting in the loss of temporal long-range correlations (Tagliazucchi et al. 2013a). Finally, spontaneous electrophysiological activity recorded during unconsciousness presents increased stability (Solovey et al. 2015). A mechanistic account of the action of propofol on large-scale brain activity should provide a unified explanation for these seemingly different experimental results.To propose such an explanation, we studied fMRI data acquired during wakefulness, propofol-induced sedation and loss of consciousness, as well as during the subsequent recovery of awareness. We evaluated the presence of two large-scale signatures of a departure from the critical point of a phase transition: loss of long-range temporal correlations and the uncoupling of functional and anatomical connectivity (Stam et al., 2014; Deco et al., 2014), measured using diffusion tensor imaging (DTI) and diffusion spectrum imaging (DSI).5"Finally, we developed a conceptual model presenting a phase transition to assist in the mechanistic interpretation of the experimental results.METHODSExperimental design and participantsParticipants were scanned with fMRI during wakefulness (W), propofol sedation (S), propofolinduced loss of consciousness (LOC) and finally during the recovery of wakefulness (R). Sedation corresponded to Ramsay level 3 (Ramsay et al. 1974). Loss of consciousness corresponded to Ramsay levels 5-6 (subjects did not exhibit responses to verbal instructions). Recovery corresponded to Ramsay level 2. Twenty healthy right-handed volunteers aged between 18 and 31 years (22.4± 2.4 years) were initially included in the study. Following Monti et al (2013), subjects with head displacements exceeding 3 mm during any of the four conditions were discarded from the analysis, resulting in a final set of 12 participants. For all conditions, the resulting average head movement amplitudes did not exceed 1 mm (wakefulness: 0.38 mm, sedation: 0.25 mm, loss of consciousness: 0.17, recovery: 0.36 mm). No significant effect of condition on head displacement was found (! !,!! = 2.63, p=0.062). As noted by Monti et al., this is a conservative approach to limit the impact of head movement. Other methods, such as scan nulling (Power et al. 2014), could affect the estimation of blood-oxygen level dependent (BOLD) signal spectral power and long-range temporal correlations and therefore were not applied. As an additional control, the presence of significant residual correlations between absolute and relative head movement time series and voxel-wise BOLD time series was evaluated after data pre-processing, with no significant residual correlations being detected. Details on fMRI, DTI and DSI data acquisition and preprocessing are provided in the Supplementary Methods. Estimation of long-range temporal dependencies DFA (Kantelhardt et al. 2001) was applied to study the temporal correlations of BOLD fluctuations. This method was developed to obtain estimates of long-range temporal dependence in time series, while accounting for the possibility of non-stationarities. In the Supplementary Methods we provide a formal definition of the procedure followed in the DFA algorithm. Briefly, time series were first de-trended by subtracting the mean and the cumulative sum was then computed. Afterwards, the signal was divided into non-overlapping windows and the intensity of the fluctuations was computed by averaging the standard deviation of the signal across all windows (de-trended within each window). This procedure 6" was repeated for different window sizes and the slope of the standard deviation of the fluctuations vs. the window size ("fluctuation function", in logarithmic scale) was identified with the Hurst exponent (H). Based on the value of H, three qualitatively different scenarios can be distinguished: long-range temporal correlations (slow decay of the autocorrelation function) with 0.5 < H < 1, uncorrelated temporal activity (exponential decay of the autocorrelation function) with H = 0.5 and long-range anti-correlations (switching between high and low values in consecutive time steps) with 0 < H < 0.5. We applied DFA to the first 150 volumes of the BOLD time series of every voxel for each subject and condition, obtaining spatial maps of H values. To compute H, windows of length 10, 15, 25 and 30 volumes were used, as the logarithmic plot of the fluctuation function showed linear behavior within this range. We also estimated H in the frequency domain following a wavelet-based method. The steps followed for the wavelet estimation of H are extensively presented and discussed in the Supplementary Methods.Functional network constructionWe constructed functional networks by extracting average BOLD signals from all regions of interest and computing the linear correlation between all pairs of signals, resulting in the correlation matrix C !" .For comparison with the underlying anatomical connectivity networks, the correlation matrices C !" were thresholded to yield binary adjacency matrices A !" such that A !" = 1 if C !" ≥ ρ and A !" = 0 otherwise. The parameter ρ was chosen to fix the ratio of the connections in the network ( A !" !!! ) to the total possible number of connections (termed link density). It is important to fix the link density when comparing networks since otherwise differences could arise because the means of the respective C !" are different (and therefore the number of nonzeros entries in A !" ) and not because connections are topologically re-organized across conditions.We performed all analyses for a range of link densities between 0.01 and 0.3 in steps of 0.01.When comparing functional networks with their anatomical counterparts, the chosen link density ranges always included the link density of the DTI and DSI anatomical networks.Similarity between functional and anatomical connectivity neighborhoodsWe defined the connectivity neighborhood of node i as n ! = A !" (i.e. the i-th column of the adjacency matrix for a fixed local link density). According to this definition, the j-th entry of n ! is 1 if nodes i and j share a direct connection in the network, and it is zero otherwise. We
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10.1098/rsif.2015.1027
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"https://arxiv.org/pdf/1509.04304v2.pdf"
] | 18,813,495 |
1509.04304
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815e8e39c99cbefc2d2da6eafe0db26fc64d16ec
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Large-scale signatures of unconsciousness are consistent with a departure from critical dynamics
Enzo Tagliazucchi
Institute for Medical Psychology
Christian Albrechts University Kiel
24105KielGermany
Department of Neurology and Brain Imaging Center
Goethe University
Dante R Chialvo
Comision Nacional de Investigaciones Cientificas y Tecnologicas (CONICET)
Argentina
Michael Siniatchkin
Institute for Medical Psychology
Christian Albrechts University Kiel
24105KielGermany
Enrico Amico
Coma Science Group, GIGA Research and Cyclotron Research Center, University and University Hospital of Liège
LiègeBelgium
Jean-Francois Brichant
Coma Science Group, GIGA Research and Cyclotron Research Center, University and University Hospital of Liège
LiègeBelgium
Vincent Bonhomme
Coma Science Group, GIGA Research and Cyclotron Research Center, University and University Hospital of Liège
LiègeBelgium
Quentin Noirhomme
Coma Science Group, GIGA Research and Cyclotron Research Center, University and University Hospital of Liège
LiègeBelgium
Helmut Laufs
Department of Neurology and Brain Imaging Center
Goethe University
Department of Neurology
Institute for Medical Psychology
Christian Albrechts University Kiel
UKSH Arnold-Heller-Straße 324104, 24105Kiel, KielGermany., Germany
Steven Laureys
Coma Science Group, GIGA Research and Cyclotron Research Center, University and University Hospital of Liège
LiègeBelgium
Enzo Tagliazucchi
Large-scale signatures of unconsciousness are consistent with a departure from critical dynamics
Frankfurt am Main. Frankfurt am Main, 60528 Frankfurt am Main, Germany.1" * These authors contributed equally to this work Contact information:Consciousnessanesthesiacomplex systemsphase transitionsfMRI 3"
Loss of cortical integration and changes in the dynamics of electrophysiological brain signals characterize the transition from wakefulness towards unconsciousness. In this study we arrive at a basic model explaining these observations based on the theory of phase transitions in complex systems. We studied the link between spatial and temporal correlations of largescale brain activity recorded with fMRI during wakefulness, propofol-induced sedation and loss of consciousness, and during the subsequent recovery. We observed that during unconsciousness activity in fronto-thalamic regions exhibited a reduction of long-range temporal correlations and a departure of functional connectivity from anatomical constraints.A model of a system exhibiting a phase transition reproduced our findings, as well as the diminished sensitivity of the cortex to external perturbations during unconsciousness. This framework unifies different observations about brain activity during unconsciousness and predicts that the principles we identified are universal and independent from its causes. seems to be partially explained by the underlying anatomy (Hagmann et al. 2008; Hermundstad et al. 2013; Greicius et al. 2009; Wang et al. 2013). This suggests that spontaneous brain activity can be understood as an ever-transient (or metastable) exploration of the wide repertoire of paths offered by the underlying structural connectivity, the extent of such exploration potentially depending on the brain state, with variable repertoires corresponding to different degrees of awareness.Here we put forward an interpretation of propofol-induced loss of consciousness in analogy to the dynamics and connectivity of fluctuations seen on a diversity of complex systems exhibiting different phases. As a system explores the space of possible configurations, its spatio-temporal correlations behave in characteristic ways. In particular, the dynamical changes underlying different degrees of awareness could be analogous to the qualitative 2010; Casali et al. 2013; Pigorini et al. 2014). The response to endogenous fluctuations during deep sleep is also rapidly vanishing, resulting in the loss of temporal long-range correlations (Tagliazucchi et al. 2013a). Finally, spontaneous electrophysiological activity recorded during unconsciousness presents increased stability (Solovey et al. 2015). A mechanistic account of the action of propofol on large-scale brain activity should provide a unified explanation for these seemingly different experimental results.To propose such an explanation, we studied fMRI data acquired during wakefulness, propofol-induced sedation and loss of consciousness, as well as during the subsequent recovery of awareness. We evaluated the presence of two large-scale signatures of a departure from the critical point of a phase transition: loss of long-range temporal correlations and the uncoupling of functional and anatomical connectivity (Stam et al., 2014; Deco et al., 2014), measured using diffusion tensor imaging (DTI) and diffusion spectrum imaging (DSI).5"Finally, we developed a conceptual model presenting a phase transition to assist in the mechanistic interpretation of the experimental results.METHODSExperimental design and participantsParticipants were scanned with fMRI during wakefulness (W), propofol sedation (S), propofolinduced loss of consciousness (LOC) and finally during the recovery of wakefulness (R). Sedation corresponded to Ramsay level 3 (Ramsay et al. 1974). Loss of consciousness corresponded to Ramsay levels 5-6 (subjects did not exhibit responses to verbal instructions). Recovery corresponded to Ramsay level 2. Twenty healthy right-handed volunteers aged between 18 and 31 years (22.4± 2.4 years) were initially included in the study. Following Monti et al (2013), subjects with head displacements exceeding 3 mm during any of the four conditions were discarded from the analysis, resulting in a final set of 12 participants. For all conditions, the resulting average head movement amplitudes did not exceed 1 mm (wakefulness: 0.38 mm, sedation: 0.25 mm, loss of consciousness: 0.17, recovery: 0.36 mm). No significant effect of condition on head displacement was found (! !,!! = 2.63, p=0.062). As noted by Monti et al., this is a conservative approach to limit the impact of head movement. Other methods, such as scan nulling (Power et al. 2014), could affect the estimation of blood-oxygen level dependent (BOLD) signal spectral power and long-range temporal correlations and therefore were not applied. As an additional control, the presence of significant residual correlations between absolute and relative head movement time series and voxel-wise BOLD time series was evaluated after data pre-processing, with no significant residual correlations being detected. Details on fMRI, DTI and DSI data acquisition and preprocessing are provided in the Supplementary Methods. Estimation of long-range temporal dependencies DFA (Kantelhardt et al. 2001) was applied to study the temporal correlations of BOLD fluctuations. This method was developed to obtain estimates of long-range temporal dependence in time series, while accounting for the possibility of non-stationarities. In the Supplementary Methods we provide a formal definition of the procedure followed in the DFA algorithm. Briefly, time series were first de-trended by subtracting the mean and the cumulative sum was then computed. Afterwards, the signal was divided into non-overlapping windows and the intensity of the fluctuations was computed by averaging the standard deviation of the signal across all windows (de-trended within each window). This procedure 6" was repeated for different window sizes and the slope of the standard deviation of the fluctuations vs. the window size ("fluctuation function", in logarithmic scale) was identified with the Hurst exponent (H). Based on the value of H, three qualitatively different scenarios can be distinguished: long-range temporal correlations (slow decay of the autocorrelation function) with 0.5 < H < 1, uncorrelated temporal activity (exponential decay of the autocorrelation function) with H = 0.5 and long-range anti-correlations (switching between high and low values in consecutive time steps) with 0 < H < 0.5. We applied DFA to the first 150 volumes of the BOLD time series of every voxel for each subject and condition, obtaining spatial maps of H values. To compute H, windows of length 10, 15, 25 and 30 volumes were used, as the logarithmic plot of the fluctuation function showed linear behavior within this range. We also estimated H in the frequency domain following a wavelet-based method. The steps followed for the wavelet estimation of H are extensively presented and discussed in the Supplementary Methods.Functional network constructionWe constructed functional networks by extracting average BOLD signals from all regions of interest and computing the linear correlation between all pairs of signals, resulting in the correlation matrix C !" .For comparison with the underlying anatomical connectivity networks, the correlation matrices C !" were thresholded to yield binary adjacency matrices A !" such that A !" = 1 if C !" ≥ ρ and A !" = 0 otherwise. The parameter ρ was chosen to fix the ratio of the connections in the network ( A !" !!! ) to the total possible number of connections (termed link density). It is important to fix the link density when comparing networks since otherwise differences could arise because the means of the respective C !" are different (and therefore the number of nonzeros entries in A !" ) and not because connections are topologically re-organized across conditions.We performed all analyses for a range of link densities between 0.01 and 0.3 in steps of 0.01.When comparing functional networks with their anatomical counterparts, the chosen link density ranges always included the link density of the DTI and DSI anatomical networks.Similarity between functional and anatomical connectivity neighborhoodsWe defined the connectivity neighborhood of node i as n ! = A !" (i.e. the i-th column of the adjacency matrix for a fixed local link density). According to this definition, the j-th entry of n ! is 1 if nodes i and j share a direct connection in the network, and it is zero otherwise. We
INTRODUCTION
Anesthetic drugs transiently impair awareness and thus offer a unique opportunity to investigate the neural correlates of conscious wakefulness. In contrast to other reversible unconscious states (such as sleep), anesthetics simultaneously reduce arousal and awareness and -except in the rare event of intra-operative awareness -result in a brain state incompatible with conscious content (Alkire et al. 2008). Studies of the transition from wakefulness to loss of consciousness induced by propofol (a presumed GABA agonist anesthetic agent) consistently report decreased cortical integration (Alkire et al. 2008;Lee et al. 2009;Boveroux et al. 2010;Schrouff et al. 2011;Monti et al. 2013;Amico et al., 2014) and changes in the dynamics of electrophysiological brain signals, such as delta (1 -4 Hz) and gamma oscillations (30 -70 Hz) (Murphy et al. 2011;Boly et al. 2012). Despite many experimental reports at different temporal and spatial scales, the precise mechanisms underlying propofol-induced unconsciousness remain poorly understood. The need for a mechanistic understanding of this phenomenon is non-trivial, since it could contribute to unraveling how consciousness is constructed and preserved by the brain.
During conscious wakefulness the cortex spontaneously generates a flurry of ever changing activity (Chialvo 2010;Raichle 2011;Sporns 2011). In the temporal domain, this activity is characterized by long-range temporal correlations, meaning that signal fluctuations at the present time influence dynamics up to several minutes in the future (Maxim et al. 2005;He 2011). Lacking any distinctive scale (scale-free), these temporal correlations can be characterized by the computation of scaling exponents, such as the Hurst exponent (Tagliazucchi et al. 2013a). In the spatial domain, these fluctuations are coordinated across networks of regions commonly co-activated during stimulation and cognitive performance, termed resting state networks (RSN) (Beckmann et al. 2005;Smith et al. 2009). While functional connectivity can transiently dissociate from inter-areal anatomical connections (Liegeois et al. 2015), brain activity correlations computed over extended periods of time
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changes observed in the dynamics of complex systems when they move away from a phase transition (Werner 2013). Experimental evidence gathered from functional magnetic resonance imaging (fMRI) data supports the view that during conscious wakefulness the human brain operates near the critical point of such a transition (Chialvo 2010;Tagliazucchi et al., 2012). A robust feature of the critical state is the phenomenon of critical slowing down, which is manifest as increased temporal autocorrelation (i.e. long-range temporal correlations) of fluctuations throughout the system (Werner 2007;Chialvo 2010;Kelso 2012).
Far from the critical transition, the variables describing the system are very stable. As a consequence, any perturbation from equilibrium is dissipated quickly (i.e. dynamics rapidly returns to equilibrium). Conversely, near the transition the effects of any disturbance last longer, thus it is that the dynamics slow down. Since far from the transition the system is stuck into a stable state, its dynamics cannot explore the wider repertoire allowed by structural constraints. The opposite occurs near the phase transition at which the system can switch between a large number of locally stable or metastable states (Werner 2007), and fully explore its structural connectivity (Stam et al., 2014;Deco et al., 2014). Thus, if unconsciousness results in a departure from critical dynamics, we expect to see these two inter-related signatures: 1) Loss of temporal correlations in brain activity time series and 2) a less complete exploration of the activity patterns allowed by the underlying structural connectivity.
Previous experimental results are consistent with a loss of critical slowing down in large-scale brain activity during unconsciousness. For instance, magnetic and electric perturbation of the cortex during different states of consciousness elicits equally different responses: conscious wakefulness is characterized by prolonged and spatiotemporally correlated responses (disturbances last longer), whereas unconsciousness is characterized by a smaller repertoire of rapidly vanishing and spatially localized responses (Massimini et al. 2005;Ferrarelli et al.
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obtained the connectivity neighborhood of all nodes in the anatomical and functional networks across all conditions and participants, as well as for a range of local link densities. To estimate the similarity between the anatomical and functional connectivity neighborhoods of each node we computed the Hamming distance between the anatomical and functional versions of vectors n ! (normalized by their total length). The Hamming distance is defined as the number of symbol substitutions (in this case 0 or 1) needed to transform one sequence into another and vice-versa, and in this case it is equal to twice the number of connections that must be re-wired to turn the functional connectivity neighbor into the anatomical connectivity neighbor. inactive, active and refractory. The rules for the transitions at the i-th node are as follows, 1) Inactive to active: either spontaneously with a probability of 10 -3 or if
W !" > T !!!"!!"#$%&.
2) Active to refractory always occurs.
3) Refractory to inactive with a probability of 10 -1 .
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These rules were used to simulate time series that were subsequently binarized by setting the active state to 1 and the other two to 0, and convolved with the standard hemodynamic response function mimicking the brain neurometabolic coupling. As shown in Haimovici et al.
2013, a second order phase transition exists at T C ≈ 0.05. At this point, activity becomes selfsustained, spatial and temporal correlations are maximized and an optimal agreement with the empirical fMRI data is obtained (including an approximate reproduction of the major RSN reported in the work of Beckmann et al. 2005).
RESULTS
We first obtained Hurst exponent and the low-frequency (0.01 -0.1 Hz) power for each participant and condition (W, S, LOC and R). Also, we investigated the same metrics in a phantom made of water (see Figure values peaked at around 0.5 (corresponding to temporally uncorrelated dynamics) for the water phantom and at H > 0.5 for grey matter brain voxels, i.e. as opposed to brain dynamics, those of the water phantom were temporally uncorrelated.
We conducted voxel-wise statistical tests to assess the effect of the condition on H and lowfrequency power (Fig. 2). We observed a significant effect of the condition ( Results are shown in Fig. 4A (note that this correlation is against structural-functional network distance, not similarity). For both anatomical connectivity networks and almost all link densities, a significant negative correlation between H and the mean Hamming distance was 10"
found. Correlations involving low frequency power were also negative but in most cases slightly above the threshold of statistical significance. Negative correlations imply that the stronger the de-correlation in temporal dynamics, the stronger the uncoupling between anatomical and functional connectivity. We found that the similarity between functional and structural connectivity was maximal near the critical point. This was evident from computing the correlation between functional and anatomical adjacency matrices at each threshold value (Fig. 6A, left) or by computing the Hamming distance between the binarized functional and anatomical connectivities of each node (Fig. 6A, right). The frequency at which activations occurred correlated negatively with 11" the threshold. As shown in Fig. 6B (left), low values facilitated the propagation of activity and induced higher activation rates while higher thresholds caused the opposite effect by hindering the propagation of activity. In the super-critical (T > T C ) regime, the frequency of activations also correlated negatively with the anatomical-functional distance (Fig. 6B, right).
The same result was observed for the Hurst exponent of the average activity generated by the model. Both are consistent with the changes observed under propofol: the higher the uncoupling between anatomical and functional connectivity, the faster and less temporally correlated the dynamics of the system.
The critical slowing down observed when dynamics are close to the phase transition maximizes the response of the model to external perturbations. We studied the average response of the system to a sudden excitation of 60% of the nodes. This computation is of interest to evaluate which phases of the model better correspond to the diminished response to magnetic perturbations of cortical activity observed during loss of consciousness. In Fig. 7 (left) we show the average time course after a perturbation (computed over 100 simulations)
both for T C ≈ 0.05 and for T C < T = 0.01. The response in the critical case was characterized by a sustained oscillation, with temporally persistent activity observed after the perturbation.
On the other hand, the perturbation in the super-critical case induced a transient response rapidly giving way to a baseline of uncorrelated oscillations. We measured the decay of the variance in the activity over short temporal windows of 20 time steps. The activity decay after the perturbations is shown in Fig. 7 (center) for all thresholds. By measuring the time elapsed until a level of low variance (10 -5 ) was crossed, we estimated the time necessary for the activity to decay to its baseline. The decay time peaked near the critical point and quickly decreased both in the super-and sub-critical cases (Fig. 7, right).
DISCUSSION
We studied how propofol-induced loss of consciousness affected the temporal dynamics of BOLD signals and how the changes in large-scale dynamics were related to the exploration of the underlying anatomical connectivity. Loss of consciousness was paralleled by a shift towards faster and temporally uncorrelated BOLD signals in the frontal lobe, the salience network and the thalamus. Within the same regions, functional connectivity departed from the underlying anatomical constraints; this departure covaried with loss of long-range temporal
correlations.
An interpretation for our results is given in Fig. 8. We show a schematic depiction of an elementary system composed of interacting units, the state of the system being symbolized by the position of a particle within a potential landscape with several local equilibria (potential wells). In reality, this potential would span a high-dimensional space, with the state vector describing a multitude of independent variables characterizing the system at each time point.
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However, we adopt this simplified schematic for illustration purposes. Far from the critical point of a phase transition (left panel), the system is more stable and the local minima are deeper; in consequence any external perturbation or internal fluctuation rapidly vanishes and the particle returns quickly to the same local equilibrium. For the same reason, the dynamics Contemporary theories postulate that consciousness is an emergent phenomenon of physical processes in the brain. The explanation of subjective experiences from the objective observation of these processes has remained elusive to neuroscience. However, it is possible to ask what features of brain activity are compatible with the rich subjective phenomenology of consciousness. An aspect common to different theories is that consciousness can be associated to a state of high neural complexity (Tononi et al., 1994;Tononi and Edelman, 13" 1998). This can be understood as a state between the extremes of very high differentiation without information integration (the dynamics of each unit in the system become independent, like in a "disordered" or "random" system at the super-critical state) and very low differentiation (the system presents few possible states, as in an "ordered" or "regular" subcritical system). At the -between ordered and disordered -critical state, dynamics are both integrated (the units of the system present long-range correlations both in time and space) and segregated (the system allows the exploration of a large number of possible metastable states), suggesting this is the state that could maximize the standard definition of neural complexity. Future work will need to formally address a possible equivalence between metrics of neural complexity and metrics of criticality (i.e. order parameters).
Our research provides evidence that the "baseline" state of wakeful rest presents critical dynamics and that unconscious brain states depart from this kind of dynamics. Thus, we identify critical dynamics with the state of consciousness. Since the possibility of having conscious, reportable content ("I see X, hear Y, feel Z") is in general conditional to being in a conscious state; critical dynamics could also be a necessary requirement for conscious content to emerge. This is supported by the observation of comparable neural complexity at rest and during conscious information access (Burguess et al., 2003). Furthermore, the role critical dynamics play in conscious information access could be related to the observation that at criticality the response of the system to external stimuli is maximized (see Fig. 7). Since conscious perception requires the engagement of a distributed set of neurons (dynamical core; Tononi and Edelman, 1998, as Here we introduced the coupling between anatomical and functional connectivity as a signature of the critical state, which is particularly fit for fMRI recordings since both can be measured at the same spatial resolution using this technique. Propofol-induced loss of consciousness resulted in diminished anatomical-functional coupling, interpreted here as a departure from the critical regime characteristic of conscious wakeful rest (Tagliazucchi et al 2012, Expert et al. 2010. Changes in function-anatomy uncoupling can be understood in terms of the emergence of long-range correlations at criticality. The term "long-range" must be treated with caution when discussing the human brain, since regions far away in Euclidean space may be close together in a topological sense (i.e. directly connected anatomically).
Thus, we did not expect to see a breakdown of long-range functional connectivity as a
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function of Euclidean distance following the departure from criticality, but a separation from the structural connectivity backbone instead.
Recent work on anesthetized primates (Barttfeld et al. 2015A) found that transient patterns of functional connectivity strongly resembling the anatomical constraints were more frequent during loss of consciousness relative to wakefulness, which appears to contradict our results.
However, two important differences must be taken into account. First, we found diminished anatomy-function coupling over extended periods of time, which is related to the average of the functional connectivity states visited over time (as an analogy, mapping the average route traced by cars in a city throughout a entire day, as opposed to taking instantaneous snapshots). Second, the effect we report was regionally localized to a set of frontal regions and the thalamus, as opposed to the global effect reported in Barttfeld et al. 2015A. This last distinction is very important since the richness of anatomical connectivity varies throughout the brain, from the regular structure of the cerebellum and primary cortices to the highly complex, variable and phylogenetically advanced frontal and parietal associative cortices (Kaas et al. 2013;Rilling et al. 2014). Indeed, we observed that the functional exploration of fronto-thalamic anatomical connectivity was hindered under propofol, highlighting its importance for the maintenance of conscious awareness. We note that decreased similarity between anatomical and functional connectivity could also result from the selective enhancement of functional connections that are not associated with structural links (as an analogy, cars taking "shortcuts" across regions not directly connected by roads). This possibility is ruled out by the breakdown of within-and between-network functional connectivity observed during propofol-induced unconsciousness (Boveroux et al. 2010).
We also studied the dynamics of BOLD signals, which have received comparatively less attention in the context of anesthesia than electrophysiological recordings. We observed a departure from slow and temporally correlated dynamics in frontal regions and in the thalamus. These areas strongly overlap with those where decreased metabolism under anesthesia was reported (Alkire et al. 1997;Kaisti et al. 2002;Laitio et al. 2007;Bonhomme et al. 2008). Breakdown of long-range temporal correlations was also reported in other unconscious brain states such as deep non-rapid eye movement (NREM) sleep (Tagliazucchi et al. 2013a). This led us to hypothesize that long-range temporal correlations of spontaneous activity fluctuations are a primary characteristic of brain activity during conscious wakeful rest.
Phenomenologically, the subjective feeling of continuity during conscious wakefulness ("stream of consciousness", as famously phrased by William James [James, 1980]) cannot be supported by short-range temporal correlations as exhibited, for example, in Markovian dynamics (when the state of the system depends only on the immediately previous state).
The short-term persistence of conscious information is impossible under these dynamics unless structural changes occur, which likely belong to a completely different temporal scale (Bullmore and Sporns 2009).
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The main limitation of our manuscript arises from the indirect nature of fMRI recordings and the possibility of propofol influencing other physiological variables that are not directly related to the level of consciousness. Experimental evidence shows that the effects of propofol on arterial blood pressure and cerebral blood flow are small (Fiset et al., 2005;Liu et al., 2013;Veselis et al., 2005;Johnston et al., 2003), ruling out confounds related to pressuredependent changes in BOLD signals. As discussed by Hudetz and colleagues (Hudetz et al., 2015), confounds due to alterations in neurovascular coupling are also unlikely given the preservation of functional responses during propofol-induced loss of consciousness (Franceschini et al., 2010). Experiments measuring cardiac and respiration rates simultaneously with fMRI during propofol-induced loss of consciousness did not find a significant difference vs. conscious wakefulness (e.g. Schröter et al., 2012). Another possibility is that our results reflect the concentration of propofol in blood but not the responsiveness of the participants (Barttfeld et al., 2015B). One argument against our results reflecting the increasing concentration of propofol in blood is the fact that we did not observe any significant effects under propofol-induced sedation (a state characterized by responsiveness in spite of non-zero propofol plasma concentration). Our results were specific to unconsciousness, as determined by the onset of state of unresponsiveness (Ramsay et al., 1974).
In summary, we achieved an empirical characterization of large-scale brain activity during propofol-induced unconsciousness in terms of inter-related changes in spatial and temporal correlations. In analogy to other complex systems undergoing phase transitions, the dynamics became temporally uncorrelated during unconsciousness and failed to efficiently explore the underlying structural connections. Since the proposed interpretation is based on general principles of complex systems, further research should reveal the universality of our findings across other brain states of diminished awareness, as well as investigate their applicability for the objective assessment of levels of consciousness.
ACKNOWLEDGMENTS
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Fluctuations in functional connectivity and repertoire of functional networksWe investigated if the fluctuations in the transient connectivity within the frontal executive control RSN were more widespread during wakefulness vs. propofol-induced unconsciousness by computing the average functional connectivity of all nodes in the RSN over short non-overlapping windows of different durations. Afterwards, we computed the variance of the time series of dynamical functional connectivity fluctuations. We investigated the repertoire of functional networks explored over time by means of a new methodology (see Fig. S7 of the Supplemental Information for a schematic). We first computed the connectivity matrices of all nodes within the executive control RSN (Fig. S7A) over non-overlapping segments of 20 volumes. After thresholding at a given link density (ranging from 0.01 to 0.4) this defined a series of binary networks explored over time (Fig. S7B). Afterwards, we computed the average correlation between the adjacency matrices of all these binary networks (Fig. S7C). If the repertoire of explored networks is very constrained this average correlation is high (i.e. all transient networks are very similar). On the other hand, if the system explores a wide range of different transient networks, this average correlation is lower. We termed this index transient network similarity (TNS) index. Computational model The computational model is based on the previous work of Haimovici et al. 2013 (see also a posteriori similar formulation by Stam et al. 2014). It consists of an underlying anatomical network of connections (DSI network) and rules for the transition between three states:
S1 of the Supplementary Figures). The anatomical distribution of Hurst values and low frequency power reflected the division of cortical anatomy into grey and white matter and cerebrospinal fluid. BOLD signals from grey matter voxels were characterized by long-range temporal correlations (H > 0.65) whereas white matter and cerebrospinal fluid voxels generally presented relatively weaker temporal correlations and 0.01-0.1 Hz frequency power(Fig. 1A). A shift towards reduced H and lowfrequency power can be observed in the LOC condition (third row). We computed the global Hurst exponent and low-frequency power values (averaged across all grey matter voxels) andobserved reduced values for the LOC condition relative to W(Fig. 1B). We also observed reduced values of the metrics in the frequency domain for R relative to W, suggesting that the recovery from propofol-induced loss of consciousness might not have been complete.Histograms for H and low-frequency power are shown inFig. S1 (Supplementary Figures). H
W, S, LOC and R) on H (both DFA and wavelet-estimated) and 0.01-0.1 Hz power. This was observed in in a set of regions comprising the thalamus, the ventromedial and orbitofrontal cortices, the frontal and rolandic operculi, the superior and medial frontal gyri and the anterior cingulate and bilateral insular cortices. Post-hoc t-tests between W and all other conditions revealed significant decreases only for the comparison vs. LOC. Similar results can also be observed in the first-order autoregressive coefficient of BOLD signals (Fig. S2, Supplementary Figures). Statistical parametric maps are presented in Fig. 2A (bottom panel). Fig. 2B shows a ranking of the top ten automated anatomical labeling (AAL) atlas (Tzourio-Mazoyer et al. 2002) 9" regions based on the statistical significance of the contrast W vs. LOC. The extent of the overlap between the three different metrics is shown in Fig. 2C as a joint rendering of differences in H (both DFA and wavelet-estimated) and 0.01-0.1 Hz power. No significant differences were observed in terms of the goodness of fit (! ! ) of the DFA fluctuation function. The covariance between the statistical significance maps derived from all three metrics is shown in Fig. S3 (Supplementary Figures). We then studied the coupling between anatomical and functional connectivity. At first, we restricted both functional and anatomical connectivity networks to a sub-network encompassing the executive control network reported in Beckmann et al. 2005, since this RSN overlapped with the regions where we found a breakdown of long-range temporal correlations during LOC (see Fig. S4 of the Supplementary Figures). For both DTI and DSI anatomical connectivity networks and almost all link densities we observed decreased similarity between anatomical and functional connectivity networks during LOC relative to W (Fig. 3A).Afterwards, we studied the local similarity between the anatomical and functional first neighbors of all individual nodes in whole-brain networks. The network nodes associated with decreased anatomical-functional coupling during LOC relative to W are shown inFig. 3B.Differences encompassed the thalamus, as well as the medial prefrontal cortex, anterior cingulate cortex, frontal and rolandic operculi and the bilateral insular cortex. A ranking of AAL regions by their percentage of nodes with significant differences is presented inFig. 3C. The robustness of the results with respect to the two anatomical connectivity networks is manifest in the joint rendering of the nodes presenting significant differences(Fig. 3D). Similar results were obtained using partial correlations instead of linear correlations (seeFig. S6of theSupplementary Figures).As discussed in the introduction, we hypothesized that during LOC the de-correlation of temporal dynamics should be seen together with a less thorough exploration of the repertoire of possible states allowed by anatomical constraints. To address this possibility, we investigated whether changes in H and low-frequency power during LOC were correlated with the degree of anatomy-function coupling. We computed the average anatomy-function Hamming distance within the significant regions inFig. 3B(bottom panel) as a function of the link density, as well as the average H (DFA and wavelet-estimated) and 0.01-0.1 Hz frequency power in the same regions. This was performed for each participant in the LOC condition. We then computed the correlation coefficients and associated p-values between H, low frequency power and the mean Hamming distance as a function of the link density.
Fig. 4Bshows example scatterplots obtained at the reference link density of 0.15.We then investigated the variability of functional connectivity over time to determine if unconsciousness was characterized by diminished fluctuations in dynamic connectivity, as predicted by a departure from criticality (seeHaimovici et al., 2013). Results presented inFig. 5A reveal that the variance of functional connectivity fluctuations (over a wide range of window sizes) were diminished during propofol-induced loss of consciousness. Furthermore, a wider range (repertoire) of functional networks was explored during conscious wakefulness compared to unconsciousness (Fig. 5B), as quantified by the TNS index computed using windows of 20 volumes. To further gauge the significance of our observations we introduced a simple dynamical model to evaluate which qualitative aspects of the propagation of information in anatomical networks were more relevant to replicate our empirical observations. The model allows three possible states for each node in the DSI network. The possible node states and transitions between them are illustrated in Fig. S5 of the Supplementary Figures. The threshold in the model controls the propensity of excitations to propagate throughout the anatomical network. Values higher than the critical threshold of T C ≈ 0.05 difficult the propagation of activity, which eventually dies out. On the other hand, lower thresholds result in self-sustained activity. Very low values result in the extreme of many nodes becoming rapidly activated and then transitioning towards the refractory ("hyperpolarized") state. A critical point exists at T C ≈ 0.05, marked by self-sustained activity allowing the reproduction of many features of large-scale brain activity, such as long-range temporal correlations in space and time and the emergence of coordinated structures strongly resembling RSN (Haimovici et al. 2014). The critical point corresponds to a second order phase transition, characterized by maximal variability in the intrinsic dynamics of the system, critical slowing down and an optimal exploration of the repertoire of metastable state (i.e. states in which the system transiently resides). Examples of the temporal dynamics during the sub-, super-, and critical regimes are shown in Fig. S5 of the Supplementary Figures.
of the system do not allow the exploration of all possibilities offered by the structural connectivity and thus functional correlations reflect only a portion of the anatomical connections. Near the phase transition (right panel) the landscape becomes shallower, the stability decreases and perturbations can induce a more widespread exploration of the potential landscape (see also Fig. 5), resulting in more sustained changes. As the system explores the neighborhood of different local equilibria (or metastable states) spatial correlations better reproduce its structural connectivity. Our observations of large-scale fMRI dynamics and connectivity during loss of consciousness can be interpreted as a departure from a critical state (near the transition) towards more stable fluctuations (far from the transition). Our model also allowed us to simulate the effect of perturbations near and far from its phase transition and thus to connect two robust but seemingly unrelated findings characterizing states of reduced awareness: loss of temporal complexity (i.e. long-range temporal correlations [Tagliazucchi et al. 2013a]) and rapidly vanishing responses to direct magnetic and electric stimulation of the cortex (Massimini et al. 2005; Ferrarelli et al. 2010; Casali et al. 2013; Pigorini et al. 2015). Within our framework, both arise as a result of increased stability, with endogenous as well as exogenous fluctuations failing to displace the system between different metastable states. The mechanisms by which propofol could result in dynamics compatible with a departure from a phase transition deserve further investigation. Most likely, these consist of alterations in the properties of individual units (neurons or groups of neurons) translating into dramatically different collective behaviors. For instance, our model suggests that facilitated spreading of activity results in a state of global hyperpolarization (see Fig. S5) impairing the propagation of external perturbations throughout the system. A possible correlate of this facilitated spreading is the increased power in the gamma frequency band observed during propofol-induced unconsciousness (Murphy et al. 2011; Boly et al. 2012), which is also a main driver of BOLD activity fluctuations (Nir et al. 2007).
well as in the concept of the global workspace; Dehaene and Naccache, 2001), a prerequisite is a high sensitivity to incoming stimuli (high susceptibility). Conversely, at the sub-or super-critical states, sensory stimulation results in a local and transient perturbation failing to propagate to more widespread networks related to conscious perception. At other spatial and temporal scales, evidence for an association between consciousness and critical dynamics has been obtained in the context of deep sleep (Priesemann et al. 2013), anesthesia (Alonso et al. 2014; Scott et al. 2014) and epileptic seizures (Meisel et al. 2012).
FIGURE CAPTIONS Figure 1 :
CAPTIONS1Anatomical specificity of long-range temporal correlations and low frequency (0.01-0.1 Hz) fluctuations. (A) Anatomical overlays of the mean Hurst exponent (DFA and wavelet estimation) and low frequency power for all experimental conditions; long-range temporal correlations and low frequency fluctuations were predominantly observed in cortical and sub-cortical grey matter. (B) Differences in global Hurst exponents and low frequency power relative to the values measured during wakefulness (*p<0.05, Bonferroni corrected for multiple comparisons).
Figure 2 :
2Breakdown of long-range temporal correlations and reduced low frequency power fluctuations during propofol-induced loss of consciousness. (A) Top: Main effect of experimental condition (wakefulness, sedation, loss of consciousness and recovery) on the Hurst exponent (DFA) and low frequency power. Bottom: Reduced Hurst exponent and low frequency power during loss of consciousness compared to wakefulness. Both statistical significance maps were thresholded at p<0.05, FDR-controlled for multiple comparisons. (B) Regions in the AAL atlas ranked according to their differences between wakefulness and loss of consciousness. (C) Combined anatomical overlay of the three metrics presented in Panel A.
Figure 3 :
3Regional dissociation of anatomical and functional connectivity during loss of consciousness. Results in the left column were obtained using the DTI network with 401 nodes, those in the right column using the DSI network with 998 nodes. (A) Similarity (correlation coefficient) between anatomical and functional connectivity networks within the executive control RSN as a function of link density, obtained during wakefulness (blue) and loss of consciousness (red). (B) Anatomical overlay of regions with significant increases in anatomical-functional distance during loss of consciousness vs. wakefulness. (C) Ranking of AAL regions according to the percentage of nodes they contained with significant differences in anatomical-functional distance. (D) Joint anatomical rendering of results obtained using the DTI and DSI anatomical connectivity networks.
Figure 4 :
4Changes in long-range temporal correlations and anatomical-functional coupling are correlated during loss of consciousness. (A) Left: Correlation coefficient between the Hurst exponent (DFA and wavelet estimation) and low frequency power averaged over the regions in Fig. 3B, and the average distance between anatomical (DTI) and functional connectivity; results are shown as a function of the link density. Right: same computation for the DSI network. (B) Scatter plots of anatomy-function distance vs. H (DFA and wavelet estimation) for a reference link density of 0.15 (light blue dashed line in panel A).
Figure 5 :
5Unconsciousness increases network stability and decreases the repertoire of transient network states. (A) The variance of transient average connectivity within the executive control RSN as a function of non-overlapping window size, for wakefulness and loss of consciousness. (B) TNS index (computed using non-overlapping windows of 20 volumes) for wakefulness and loss of consciousness.
Figure 6 :
6Dynamics and connectivity of the model. (A) Left: Similarity between anatomical and functional (i.e. simulated) connectivity as a function of the threshold T. Right: Hamming distance between anatomical and functional node connectivity neighborhoods (averaged across all nodes) as a function of T. In both cases, the highest anatomical-functional coupling is observed close to the critical point (T = T C ). (B) Left: Frequency of node activations as a function of T. Right: Mean anatomical-functional distance as a function of frequency of node activations and Hurst exponent of average activity of the model. As in the experimental data, higher frequency of activations and diminished long-range temporal correlations paralleled the dissociation between anatomical and functional connectivity patterns.
Figure 7 :
7The sensitivity to external perturbations is maximal near the phase transition of the model. (A) Average time course after a perturbation (activation of 60% of the nodes) during the critical and supercritical regimes. (B) Decay of activity after a perturbation for a range of thresholds. (C) Decay time as a function of T. The longest decay times are obtained when T = T C .
Figure 8 :
8Schematic representation of the dynamics of a system far and near the critical point of a phase transition. The state of the system at a given time is represented by the position of the particle on the potential landscape U(x). The system at equilibrium (green) is perturbed at t 0 subsequently relaxing (red, t 1 → t 2 ) at different speeds depending on whether it is far (left) or near (right) a phase transition. Far from the transition (left) the system is stable and the local minima (equilibrium points) are deep, consequently dynamics are rapidly restored and the effects of perturbation are short-lasting. Near the phase transition(right) the landscape of the potential is shallow and consequently the stability of the local minima decreases, which is reflected in the time domain (middle panels) as a slowing down of the system response to fluctuations. The change in stability can be also observed spatially because it affects the exploration of the different metastable states of the system. The graphs denoted by SC (structural connectivity) represent a portion of the underlying structural network. The bottom diagram (FC; functional connectivity) denotes the structural paths traversed during an interval of time. For shallow local minima the structural paths leading from node A to the rest of the nodes (B, C, D, E) are equally likely, resulting in a complete exploration of the underlying structural connectivity. On the contrary, far from the transition
This work was funded by the Bundesministerium für Bildung und Forschung (grant 01 EV 0703) and the LOEWE Neuronale Koordination Forschungsschwerpunkt Frankfurt (NeFF).We thank Ben Palanca and two anonymous reviewers for valuable comments on this manuscript, Ed Bullmore and Nicolas Crossley for sharing the DTI data and Patric Hagmann and Olaf Sporns for sharing the DSI data.
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