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1
+ arXiv:1001.0001v1 [cs.IT] 30 Dec 2009On the structure of non-full-rank perfect codes
2
+ Olof Heden and Denis S. Krotov∗
3
+ Abstract
4
+ The Krotov combining construction of perfect 1-error-corr ecting binary codes
5
+ from 2000 and a theorem of Heden saying that every non-full-r ank perfect 1-error-
6
+ correcting binary code can be constructed by this combining construction is gener-
7
+ alized to the q-ary case. Simply, every non-full-rank perfect code Cis the union of
8
+ a well-defined family of ¯ µ-components K¯µ, where ¯µbelongs to an “outer” perfect
9
+ codeC⋆, and these components are at distance three from each other. Compo-
10
+ nents from distinct codes can thus freely be combined to obta in new perfect codes.
11
+ The Phelps general product construction of perfect binary c ode from 1984 is gen-
12
+ eralized to obtain ¯ µ-components, and new lower bounds on the number of perfect
13
+ 1-error-correcting q-ary codes are presented.
14
+ 1. Introduction
15
+ LetFqdenote the finite field with qelements. A perfect1-error-correcting q-ary code of
16
+ lengthn, for short here a perfect code , is a subset Cof the direct product Fn
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+ q, ofncopies of
18
+ Fq, having the property that any element of Fn
19
+ qdiffers in at most one coordinate position
20
+ from a unique element of C.
21
+ The family of all perfect codes is far from classified or enumerated. We will in this
22
+ short note say something about the structure of these codes. W e need the concept of
23
+ rank.
24
+ We consider Fn
25
+ qas a vector space of dimension nover the finite field Fq. Therank
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+ of aq-ary codeC, here denoted rank( C), is the dimension of the linear span < C >of
27
+ the elements of C. Trivial, and well known, counting arguments give that if there exist s
28
+ a perfect code in Fn
29
+ qthenn= (qm−1)/(q−1), for some integer m, and|C|=qn−m. So,
30
+ for every perfect code C,
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+ n−m≤rank(C)≤n.
32
+ If rank(C) =nwe will say that Chasfull rank.
33
+ ∗This research collaboration was partially supported by a grant from Swedish Institute; the work of
34
+ the second author was partially supported by the Federal Target Program “Scientific and Educational
35
+ PersonnelofInnovation Russia”for 2009-2013(governmentco ntract No. 02.740.11.0429)and the Russian
36
+ Foundation for Basic Research (grant 08-01-00673).
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+ 1We will show thatevery non-full-rankperfect code isa unionofso ca lled ¯µ-components
38
+ K¯µ, and that these components may be enumerated by some other pe rfect codeC⋆, i.e,
39
+ ¯µ∈C⋆. Further, the distance between any two such components will be a t least three.
40
+ This implies that we will be completely free to combine ¯ µ-components from different
41
+ perfect codesofsamelength, toobtainotherperfect codes. Ge neralizing aconstruction by
42
+ Phelps of perfect 1-error correcting binary codes [8], we will obtain further ¯µ-components.
43
+ As an application of our results we will be able to slightly improve the lowe r bound on
44
+ the number of perfect codes given in [6].
45
+ Our results generalize corresponding results for the binary case. In [3] it was shown
46
+ that a binary perfect code can be constructed as the union of diffe rent subcodes (¯ µ-
47
+ components) satisfying some generalized parity-check property , each of them being con-
48
+ structed independently or taken from another perfect code. In [2] it was shown that every
49
+ non-full-rank perfect binary code can be obtained by this combining construction.
50
+ 2. Every non-full-rank perfect code is the union of ¯µ-
51
+ components
52
+ We start with some notation. Assume we have positive integers n,t,n1, ...,ntsuch that
53
+ n1+...+nt≤n. Anyq-aryword ¯xwill berepresented intheblockform ¯ x= (¯x1|¯x2|...|
54
+ ¯xt|¯x0) = (¯x∗|¯x0), where ¯xi= (xi1,xi2,...,x ini),i= 0,1,...,t,n0=n−n1−...−nt,
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+ ¯x∗= (¯x1|¯x2|...|¯xt). For every block ¯ xi,i= 1,2,...,t, we define σi(¯xi) by
56
+ σi(¯xi) =ni/summationdisplay
57
+ j=1xij,
58
+ and, for ¯x,
59
+ ¯σ(¯x) = ¯σ(¯x∗) = (σ1(¯x1),σ2(¯x2),...,σ t(¯xt))
60
+ Recall that the Hamming distance d(¯x,¯y) between two words ¯ x, ¯yof the same length
61
+ means the number of positions in which they differ.
62
+ Amonomial transformation is a map of the space Fn
63
+ qthat can be composed by a
64
+ permutation of the set of coordinate positions and the multiplication in each coordinate
65
+ position with some non-zero element of the finite field Fq.
66
+ Aq-ary codeCislinearifCis a subspace of Fn
67
+ q. A linear perfect code is called a
68
+ Hamming code .
69
+ Theorem 1. LetCbe any non-full-rank perfect code Cof lengthn= (qm−1)/(q−1).
70
+ To any integer r<m, satisfying
71
+ 1≤r≤n−rank(C),
72
+ there is aq-ary Hamming code C⋆of lengtht= (qr−1)/(q−1), such that for some
73
+ monomial transformation ψ
74
+ ψ(C) =/uniondisplay
75
+ ¯µ∈C⋆K¯µ,
76
+ 2where
77
+ K¯µ={(¯x1|¯x2|...|¯xt|¯x0) : ¯σ(¯x) = ¯µ,¯x1,¯x2,...,¯xt∈Fqs
78
+ q,¯x0∈C¯µ(¯x∗)}(1)
79
+ for some family of perfect codes C¯µ(¯x), of length 1+q+q2+...+qs−1, wheres=m−r,
80
+ and satisfying, for each ¯µ∈C⋆,
81
+ d(¯x∗,¯x′
82
+ ∗)≤2 =⇒C¯µ(¯x∗)∩C¯µ(¯x′
83
+ ∗) =∅. (2)
84
+ The codeC⋆will be called an outercode toψ(C). The subcodes K¯µwill be called
85
+ ¯µ-components ofψ(C). As the minimum distance of Cis three, the distance between any
86
+ two distinct ¯ µ-components will be at least three.
87
+ Proof. LetDbe any subspace of Fn
88
+ qcontaining<C >, and of dimension n−r. By
89
+ using a monomial transformation ψof space we may achieve that the dual space of ψ(D)
90
+ is the nullspace of a r×n-matrix
91
+ H=
92
+ | | | | | | | |
93
+ ¯α11···¯α1n1¯α21···¯α2n2···¯αt1···¯αtnt¯0···¯0
94
+ | | | | | | | |
95
+ 
96
+ where ¯αij= ¯αi, fori= 1,2,...,t, the first non-zero coordinate in each vector ¯ αiequals
97
+ 1, ¯αi/ne}ationslash= ¯αi′, fori/ne}ationslash=i′, and where the columns of Hare in lexicographic order, according
98
+ to some given ordering of Fq.
99
+ To avoid too much notation we assume that Cwas such that ψ= id.
100
+ LetC⋆be the null space of the matrix
101
+ H⋆=
102
+ | | |
103
+ ¯α1¯α2···¯αt
104
+ | | |
105
+ 
106
+ Define, for ¯ µ∈C⋆,
107
+ K¯µ={(¯x1|¯x2|...|¯xt|¯x0)∈C: (σ1(¯x1),σ2(¯x2),...,σ(¯xt)) = ¯µ}.
108
+ Then,
109
+ C=/uniondisplay
110
+ ¯µ∈C⋆K¯µ.
111
+ Further, since any two columns of H⋆are linearly independent, for any two distinct words
112
+ ¯µand ¯µ′ofC⋆
113
+ d(K¯µ,K¯µ′)≥3. (3)
114
+ We will show that K¯µhas the properties given in Equation (1).
115
+ Any word ¯x= (¯x1|¯x2|...|¯xt|¯x0) must be at distance at most one from a word of
116
+ C, and hence, the word ( σ1(¯x1),σ2(¯x2),...,σ t(¯xt)) is at distance at most one from some
117
+ word ofC⋆. It follows that C⋆is a perfect code, and as a consequence, as C⋆is linear, it
118
+ is a Hamming code with parity-check matrix H⋆. As the number of rows of H⋆isr, we
119
+ then get that the number tof columns of H⋆is equal to
120
+ t=qr−1
121
+ q−1= 1+q+q2+...+qr−1.
122
+ 3For any word ¯ x∗ofFn1+n2+...+ntq with ¯σ(¯x∗) = ¯µ∈C⋆, we now define the code C¯µ(¯x∗)
123
+ of lengthn0by
124
+ C¯µ(¯x∗) ={¯c∈Fn0
125
+ q: (¯x∗|¯c)∈C}.
126
+ Again, using the fact that Cis a perfect code, we may deduce that for any ¯ x∗such
127
+ that the set C¯µ(¯x∗) is non empty, the set C¯µ(¯x∗) must be a perfect code of length n0=
128
+ (qs−1)/(q−1), for some integer s.
129
+ From the fact that the minimum distance of Cequals three, we get the property in
130
+ Equation (2).
131
+ Let ¯eidenote a word of weight one with the entry 1 in the coordinate positio ni. It
132
+ then follows that the two perfect codes C¯µ(¯x∗) andC¯µ(¯x∗+ ¯e1−¯ei), fori= 2,3,...,n 1,
133
+ must be mutually disjoint. Hence, n1is at most equal to the number of perfect codes in
134
+ a partition of Fn0qinto perfect codes, i.e.,
135
+ n1≤(q−1)n0+1 =qs.
136
+ Similarly,ni≤qs, fori= 2,3,...,t.
137
+ Reversing these arguments, using Equation (3) and the fact that Cis a perfect code,
138
+ we find that ni, for eachi= 1,2,...,t, is at least equal to the number of words in an
139
+ 1-ball ofFn0q.
140
+ We conclude that ni=qs, fori= 1,2,...,t, and finally
141
+ n=qs(1+q+q2+...+qr−1)+1+q+q2+...+qs−1= 1+q+q2+...+qr+s−1.
142
+ Givenr, we can then find sfrom the equality
143
+ n= 1+q+q2+...+qm−1.
144
+
145
+ 3. Combining construction of perfect codes
146
+ In the previous section, it was shown that a perfect code, depend ing on its rank, can
147
+ be divided onto small or large number of so-called ¯ µ-components, which satisfy some
148
+ equation with ¯ σ. The construction described in the following theorem realizes the ide a
149
+ of combining independent ¯ µ-components, differently constructed or taken from different
150
+ perfect codes, in one perfect code.
151
+ A functionf: Σn→Σ, where Σ is some set, is called an n-ary(ormultary)quasigroup
152
+ of order |Σ|if in the equality z0=f(z1,...,z n) knowledge of any nelements of z0,z1,
153
+ ...,znuniquely specifies the remaining one.
154
+ Theorem 2. Letmandrbe integers, m>r,qbe a prime power, n= (qm−1)/(q−1)
155
+ andt= (qr−1)/(q−1). Assume that C∗is a perfect code in Ft
156
+ qand for every ¯µ∈C∗
157
+ we have a distance- 3codeK¯µ⊂Fn
158
+ qof cardinality qn−m−(t−r)that satisfies the following
159
+ generalized parity-check law:
160
+ ¯σ(¯x) = (σ1(x1,...,x l),...,σ t(xlt−l+1,...,x lt)) = ¯µ
161
+ 4for every ¯x= (x1,...,x n)∈K¯µ, wherel=qm−rand¯σ= (σ1,...,σ t)is a collections of
162
+ l-ary quasigroups of order q. Then the union
163
+ C=/uniondisplay
164
+ ¯µ∈C∗K¯µ
165
+ is a perfect code in Fn
166
+ q.
167
+ Proof. It is easy to check that Chas the cardinality of a perfect code. The distance
168
+ at least 3 between different words ¯ x, ¯yfromCfollows from the code distances of K¯µ(if
169
+ ¯x, ¯ybelong to the same K¯µ) andC∗(if ¯x, ¯ybelong to different K¯µ′,K¯µ′′, ¯µ′,¯µ′′∈C∗).△
170
+ The ¯µ-components K¯µcanbeconstructedindependentlyortakenfromdifferentperfec t
171
+ codes. In the important case when all σiare linear quasigroups (e.g., σi(y1,...,y l) =
172
+ y1+...+yl) the components can be taken from any perfect code of rank at m ostn−r, as
173
+ followsfromtheprevioussection(itshouldbenotedthatif ¯ σislinear, thena ¯ µ-component
174
+ can be obtained from any ¯ µ′-component by adding a vector ¯ zsuch that ¯σ(¯z) = ¯µ−¯µ′).
175
+ In general, the existence of ¯ µ-components that satisfy the generalized parity-check law
176
+ for arbitrary ¯ σis questionable. But for some class of ¯ σsuch components exist, as we will
177
+ see from the following two subsections.
178
+ Remark. It is worth mentioning that ¯ µ-components can exist for arbitrary length tof
179
+ ¯µ(for example, in the next two subsections there are no restriction s ont), if we do not
180
+ require the possibility to combine them into a perfect code. This is esp ecially important
181
+ for the study of perfect codes of small ranks (close to the rank o f a linear perfect code):
182
+ once we realize that the code is the union of ¯ µ-components of some special form, we may
183
+ forget about the code length and consider ¯ µ-components for arbitrary length of ¯ µ, which
184
+ allows to use recursive approaches.
185
+ 3.1. Mollard-Phelps construction
186
+ Here we describe the way to construct ¯ µ-components derived from the product construc-
187
+ tion discovered independently in [7] and [9]. In terms of ¯ µ-components, the construction
188
+ in [9] is more general; it allows substitution of arbitrary multary quasig roups, and we will
189
+ use this possibility in Section 4.
190
+ Lemma 1. Let¯µ∈Ft
191
+ qand letC#be a perfect code in Fk
192
+ q. Letvandhbe(q−1)-ary
193
+ quasigroups of order qsuch that the code {(¯y|v(¯y)|h(¯y)) : ¯y∈Fq−1
194
+ q}is perfect. Let
195
+ V1, ...,VtandH1, ...,Hkbe respectively (k+1)-ary and (t+1)-ary quasigroups of order
196
+ q. Then the set
197
+ K¯µ=/braceleftBig
198
+ (¯x11|...|¯x1k|y1|¯x21|...|¯x2k|y2|...|¯xt1|...|¯xtk|yt|z1|z2|...|zk) :
199
+ ¯xij∈Fq−1
200
+ q,
201
+ (V1(v(¯x11),...,v(¯x1k),y1),...,V t(v(¯xt1),...,v(¯xtk),yt)) = ¯µ,
202
+ (H1(h(¯x11),...,h(¯xt1),z1),...,H k(h(¯x1k),...,h(¯xtk),zk))∈C#/bracerightBig
203
+ is a¯µ-component that satisfies the generalized parity-check law with
204
+ σi(·,...,·,·) =Vi(v(·),...,v(·),·).
205
+ 5(The elements of F(q−1)kt+k+t
206
+ q in this construction may be thought of as three-dimensional
207
+ arrays where the elements of ¯xijare z-lined, every underlined block is y-lined, and the
208
+ tuple of blocks is x-lined. Naturally, the multary quasigroups Vimay be named “vertical”
209
+ andHi, “horizontal”.)
210
+ The proof of the code distance is similar to that in [9], and the other pr operties of a
211
+ ¯µ-component are straightforward. The existence of admissible ( q−1)-ary quasigroups v
212
+ andhis the only restriction on the q(this concerns the next subsection as well). If Fqis
213
+ a finite field, there are linear examples: v(y1,...,y q−1) =y1+...+yq−1,v(y1,...,y q−1) =
214
+ α1y1+...+αq−1yq−1whereα1, ...,αq−1are all the non-zero elements of Fq. Ifqis not
215
+ a prime power, the existence of a q-ary perfect code of length q+1 is an open problem
216
+ (with the only exception q= 6, when the nonexistence follows from the nonexistence of
217
+ two orthogonal 6 ×6 Latin squares [1, Th.6]).
218
+ 3.2. Generalized Phelps construction
219
+ Here we describe another way to construct ¯ µ-components, which generalizes the construc-
220
+ tion of binary perfect codes from [8].
221
+ Lemma 2. Let¯µ∈Ft
222
+ q. Let for every ifrom1tot+1the codesCi,j,j= 0,1,...,qk−k
223
+ form a partition of Fk
224
+ qinto perfect codes and γi:Fk
225
+ q→ {0,1,...,qk−k}be the corre-
226
+ sponding partition function:
227
+ γi(¯y) =j⇐⇒¯y∈Ci,j.
228
+ Letvandhbe(q−1)-ary quasigroups of order qsuch that the code {(¯y|v(¯y)|h(¯y)) :
229
+ ¯y∈Fq−1
230
+ q}is perfect. Let V1, ...,Vtbe(k+ 1)-ary quasigroups of order qandQbe a
231
+ t-ary quasigroup of order qk−k+1.
232
+ K¯µ=/braceleftBig
233
+ (¯x11|...|¯x1k|y1|¯x21|...|¯x2k|y2|...|¯xt1|...|¯xtk|yt|z1|z2|...|zk) :
234
+ ¯xij∈Fq−1
235
+ q,
236
+ (V1(v(¯x11),...,v(¯x1k),y1),...,V t(v(¯xt1),...,v(¯xtk),yt)) = ¯µ,
237
+ Q(γ1(h(¯x11),...,h(¯x1k)),...,γ t(h(¯xt1),...,h(¯xtk))) =γt+1(z1,...,zk)/bracerightBig
238
+ is a¯µ-component that satisfies the generalized parity-check law with
239
+ σi(·,...,·,·) =Vi(v(·),...,v(·),·).
240
+ The proof consists of trivial verifications.
241
+ 4. On the number of perfect codes
242
+ In this section we discuss some observations, which result in the bes t known lower bound
243
+ on the number of q-ary perfect codes, q≥3. The basic facts are already contained in
244
+ other known results: lower bounds on the number of multary quasig roups of order q, the
245
+ 6construction [9] of perfect codes from multary quasigroups of or derq, and the possibility
246
+ to choose the quasigroup independently for every vector of the o uter code (this possibility
247
+ was not explicitly mentioned in [9], but used in the previous paper [8]).
248
+ A general lower bound, in terms of the number of multary quasigrou ps, is given by
249
+ Lemma 3. In combination with Lemma 4, it gives explicit numbers.
250
+ Lemma 3. The number of q-ary perfect codes of length nis not less than
251
+ Q/parenleftBiggn−1
252
+ q,q/parenrightBiggRn−1
253
+ q
254
+ whereQ(m,q)is the number of m-ary quasigroups of order qand whereRn′=qn′/(n′q−
255
+ q+1)is the cardinality of a perfect code of length n′.
256
+ Proof. Constructing a perfect code like in Theorem 2 with t=n−1
257
+ q, we combine
258
+ Rn−1
259
+ qdifferent ¯µ-components.
260
+ Constructing every such a component as in Lemma 2, k= 1,t=n−1
261
+ q, we are free
262
+ to choose the t-ary quasigroup Qof orderqinQ(t,q) ways. Clearly, different t-ary
263
+ quasigroups give different components. (Equivalently, we can use L emma 1 and choose
264
+ the (t+1)-ary quasigroup H1, but should note that the value of H1in the construction is
265
+ always fixed when k= 1, because C#consists of only one vertex; so we again have Q(t,q)
266
+ different choices, not Q(t+1,q)). △
267
+ Lemma 4. The number Q(m,q)ofm-ary quasigroups of order qsatisfies:
268
+ (a) [5]Q(m,3) = 3·2m;
269
+ (b) [11]Q(m,4) = 3m+1·22m+1(1+o(1));
270
+ (c) [4]Q(m,5)≥23n/3−0.072;
271
+ (d) [10]Q(m,q)≥2((q2−4q+3)/4)n/2for oddq(the previous bound [4]wasQ(m,q)≥
272
+ 2⌊q/3⌋n);
273
+ (e) [4]Q(m,q1q2)≥Q(m,q1)·Q(m,q2)qm
274
+ 1.
275
+ For oddq≥5, the number of codes given by Lemmas 3 and 4(c,d) improves the
276
+ constantcin the lower estimation of form eecn(1+o(1))for the number of perfect codes, in
277
+ comparison with the last known lower bound [6]. Informally, this can be explained in the
278
+ following way: the construction in [6] can be described in terms of mu tually independent
279
+ small modifications of the linear multary quasigroup of order q, while the lower bounds
280
+ in Lemma 4(c,d) are based on a specially-constructed nonlinear multa ry quasigroup that
281
+ allows a lager number of independent modifications. For q= 3 andq= 2s, the number
282
+ of codes given by Lemmas 3 and 4(a,b,e) also slightly improves the boun d in [6], but do
283
+ not affect on the constant c.
284
+ 7References
285
+ 1. S. W. Golomb and E. C. Posner. Rook domains, latin squares, and e rror-distributing
286
+ codes.IEEE Trans. Inf. Theory , 10(3):196–208, 1964.
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+ 2. O. Heden. On the classification of perfect binary 1-error corre cting codes. Preprint
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+ TRITA-MAT-2002-01, KTH, Stockholm, 2002.
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+ 3. D. S. Krotov. Combining construction of perfect binary codes. Probl. Inf. Transm. ,
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+ 36(4):349–353, 2000. translated from Probl. Peredachi Inf. 36 (4) (2000), 74-79.
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+ 4. D. S. Krotov, V. N. Potapov, and P. V. Sokolova. On reconstru cting reducible n-ary
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+ quasigroups and switching subquasigroups. Quasigroups Relat. Syst. , 16(1):55–67,
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+ 2008. ArXiv:math/0608269
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+ 5. C. F. Laywine and G. L. Mullen. Discrete Mathematics Using Latin Squares . Wiley,
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+ New York, 1998.
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+ 6. A. V. Los’. Construction of perfect q-ary codes by switchings o f simple components.
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+ Probl. Inf. Transm. , 42(1):30–37, 2006. DOI: 10.1134/S0032946006010030 transla ted
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+ from Probl. Peredachi Inf. 42(1) (2006), 34-42.
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+ 7. M. Mollard. A generalized parity function and its use in the constru ction of perfect
300
+ codes.SIAM J. Algebraic Discrete Methods , 7(1):113–115, 1986.
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+ 8. K. T. Phelps. A general product construction for error corre cting codes. SIAM J.
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+ Algebraic Discrete Methods , 5(2):224–228, 1984.
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+ 9. K. T. Phelps. A product construction for perfect codes over a rbitrary alphabets.
304
+ IEEE Trans. Inf. Theory , 30(5):769–771, 1984.
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+ 10. V. N. Potapov and D. S. Krotov. On the number of n-ary quasigroups of finite order.
306
+ Submitted. ArXiv:0912.5453
307
+ 11. V.N.PotapovandD.S.Krotov. Asymptoticsforthenumbero fn-quasigroupsoforder
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+ 4.Sib. Math. J. , 47(4):720–731, 2006. DOI: 10.1007/s11202-006-0083-9 tran slated
309
+ from Sib. Mat. Zh. 47(4) (2006), 873-887. ArXiv:math/0605104
310
+ O. Heden
311
+ Department of Mathematics, KTH
312
+ S-100 44 Stockholm, Sweden
313
314
+ D. Krotov
315
+ Sobolev Institute of Mathematics
316
+ and
317
+ Mechanics and Mathematics Department, Novosibirsk State Univer sity
318
+ Novosibirsk, Russia
319
320
+ 8
1001.0002.txt ADDED
@@ -0,0 +1,771 @@
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
1
+ arXiv:1001.0002v2 [hep-th] 9 Mar 2010Gravity duals for logarithmic conformal field theories
2
+ Daniel Grumiller and Niklas Johansson
3
+ Institute for Theoretical Physics, Vienna University of Te chnology
4
+ Wiedner Hauptstrasse 8-10/136, A-1040 Vienna, Austria
5
6
+ Abstract. Logarithmic conformal fieldtheories with vanishingcentra l charge describe systems
7
+ withquencheddisorder, percolation ordiluteself-avoidi ngpolymers. Inthesetheories theenergy
8
+ momentum tensor acquires a logarithmic partner. In this tal k we address the construction of
9
+ possible gravity duals for these logarithmic conformal fiel d theories and present two viable
10
+ candidates for such duals, namely theories of massive gravi ty in three dimensions at a chiral
11
+ point.
12
+ Outline
13
+ Thistalk isorganized asfollows. Insection 1werecall sali ent featuresof2-dimensionalconformal
14
+ field theories. In section 2 we review a specific class of logar ithmic conformal field theories where
15
+ the energy momentum tensor acquires a logarithmic partner. In section 3 we present a wish-list
16
+ for gravity duals to logarithmic conformal field theories. I n section 4 we discuss two examples
17
+ of massive gravity theories that comply with all the items on that list. In section 5 we address
18
+ possible applications of an Anti-deSitter/logarithmic co nformal field theory correspondence in
19
+ condensed matter physics.
20
+ 1. Conformal field theory distillate
21
+ Conformal field theories (CFTs) are quantum field theories th at exhibit invariance under angle
22
+ preserving transformations: translations, rotations, bo osts, dilatations and special conformal
23
+ transformations. In two dimensions the conformal algebra i s infinite dimensional, and thus
24
+ two-dimensional CFTs exhibit a particularly rich structur e. They arise in various contexts in
25
+ physics, including string theory, statistical mechanics a nd condensed matter physics, see e.g. [1].
26
+ The main observables in any field theory are correlation func tions between gauge invariant
27
+ operators. There exist powerful tools to calculate these co rrelators in a CFT. The operator
28
+ content of various CFTs may differ, but all CFTs contain at leas t an energy momentum tensor
29
+ Tµν. Conformal invariance requires the energy momentum tensor to be traceless, Tµ
30
+ µ= 0,
31
+ in addition to its conservation, ∂µTµν= 0. In lightcone gauge for the Minkowski metric,
32
+ ds2= 2dzd¯z, these equations take a particularly simple form: Tz¯z= 0,Tzz=Tzz(z) :=OL(z)
33
+ andT¯z¯z=T¯z¯z(¯z) :=OR(¯z). Conformal Ward identities determine essentially unique ly the form
34
+ of 2- and3-point correlators between thefluxcomponents OL/Rof theenergy momentum tensor:∝an}b∇acketle{tOR(¯z)OR(0)∝an}b∇acket∇i}ht=cR
35
+ 2¯z4(1a)
36
+ ∝an}b∇acketle{tOL(z)OL(0)∝an}b∇acket∇i}ht=cL
37
+ 2z4(1b)
38
+ ∝an}b∇acketle{tOL(z)OR(0)∝an}b∇acket∇i}ht= 0 (1c)
39
+ ∝an}b∇acketle{tOR(¯z)OR(¯z′)OR(0)∝an}b∇acket∇i}ht=cR
40
+ ¯z2¯z′2(¯z−¯z′)2(1d)
41
+ ∝an}b∇acketle{tOL(z)OL(z′)OL(0)∝an}b∇acket∇i}ht=cL
42
+ z2z′2(z−z′)2(1e)
43
+ ∝an}b∇acketle{tOL(z)OR(¯z′)OR(0)∝an}b∇acket∇i}ht= 0 (1f)
44
+ ∝an}b∇acketle{tOL(z)OL(z′)OR(0)∝an}b∇acket∇i}ht= 0 (1g)
45
+ The real numbers cL,cRare the left and right central charges, which determine key p roperties of
46
+ the CFT. We have omitted terms that are less divergent in the n ear coincidence limit z,¯z→0 as
47
+ well as contact terms, i.e., contributions that are localiz ed (δ-functions and derivatives thereof).
48
+ If someone provides us with a traceless energy momentum tens or and gives us a prescription
49
+ how to calculate correlators,1but does not reveal whether the underlying field theory is a CF T,
50
+ thenwecanperformthefollowing check. Wecalculate all 2- a nd3-point correlators of theenergy
51
+ momentum tensor with itself, and if at least one of the correl ators does not match precisely with
52
+ the corresponding correlator in (1) then we know that the fiel d theory in question cannot be a
53
+ CFT. On the other hand, if all the correlators match with corr esponding ones in (1) we have
54
+ non-trivial evidence that the field theory in question might be a CFT. Let us keep this stringent
55
+ check in mind for later purposes, but switch gears now and con sider a specific class of CFTs,
56
+ namely logarithmic CFTs (LCFTs).
57
+ 2. Logarithmic CFTs with an energetic partner
58
+ LCFTs were introduced in physics by Gurarie [2]. We focus now on some properties of LCFTs
59
+ and postpone a physics discussion until the end of the talk, s ee [3,4] for reviews. There are two
60
+ conceptually different, but mathematically equivalent, way s to define LCFTs. In both versions
61
+ there exists at least one operator that acquires a logarithm ic partner, which we denote by Olog.
62
+ We focus in this talk exclusively on theories where one (or bo th) of the energy momentum
63
+ tensor flux components is the operator acquiring such a partn er, for instance OL. We discuss
64
+ now briefly both ways of defining LCFTs.
65
+ According to the first definition “acquiring a logarithmic pa rtner” means that the
66
+ Hamiltonian Hcannot be diagonalized. For example
67
+ H/parenleftbigg
68
+ Olog
69
+ OL/parenrightbigg
70
+ =/parenleftbigg
71
+ 2 1
72
+ 0 2/parenrightbigg/parenleftbigg
73
+ Olog
74
+ OL/parenrightbigg
75
+ (2)
76
+ Theangularmomentum operator Jmay ormay not bediagonalizable. Weconsider onlytheories
77
+ whereJis diagonalizable:
78
+ J/parenleftbigg
79
+ Olog
80
+ OL/parenrightbigg
81
+ =/parenleftbigg
82
+ 2 0
83
+ 0 2/parenrightbigg/parenleftbigg
84
+ Olog
85
+ OL/parenrightbigg
86
+ (3)
87
+ The eigenvalues 2 arise because the energy momentum tensor a nd its logarithmic partner both
88
+ correspond to spin-2 excitations.
89
+ 1This is exactly what the AdS/CFT correspondence does: given a gravity dual we can calculate the energy
90
+ momentum tensor and correlators.The second definition makes it more transparent why these CFT s are called “logarithmic”
91
+ in the first place. Suppose that in addition to OL/Rwe have an operator OMwith conformal
92
+ weightsh= 2+ε,¯h=ε, meaning that its 2-point correlator with itself is given by
93
+ ∝an}b∇acketle{tOM(z,¯z)OM(0,0)∝an}b∇acket∇i}ht=ˆB
94
+ z4+2ε¯z2ε(4)
95
+ The correlator of OMwithOLvanishes since the latter has conformal weights h= 2,¯h= 0, and
96
+ operators whose conformal weights do not match lead to vanis hing correlators. Suppose now
97
+ that we send the central charge cLand the parameter εto zero, and simultaneously send ˆBto
98
+ infinity, such that the following limits exist:
99
+ bL:= lim
100
+ cL→0−cL
101
+ ε∝ne}ationslash= 0B:= lim
102
+ cL→0/parenleftbigˆB+2
103
+ cL/parenrightbig
104
+ (5)
105
+ Then we can define a new operator Ologthat linearly combines OL/M.
106
+ Olog=bLOL
107
+ cL+bL
108
+ 2OM(6)
109
+ Taking the limit cL→0 leads to the following 2-point correlators:
110
+ ∝an}b∇acketle{tOL(z)OL(0,0)∝an}b∇acket∇i}ht= 0 (7a)
111
+ ∝an}b∇acketle{tOL(z)Olog(0,0)∝an}b∇acket∇i}ht=bL
112
+ 2z4(7b)
113
+ ∝an}b∇acketle{tOlog(z,¯z)Olog(0,0)∝an}b∇acket∇i}ht=−bLln(m2
114
+ L|z|2)
115
+ z4(7c)
116
+ These 2-point correlators exhibit several remarkable feat ures. The flux component OLof the
117
+ energy momentum tensor becomes a zero norm state (7a). Never theless, the theory does not
118
+ become chiral, because the left-moving sector is not trivia l:OLhas a non-vanishing correlator
119
+ (7b) with its logarithmic partner Olog. The 2-point correlator (7c) between two logarithmic
120
+ operators Ologmakes it clear why such CFTs have the attribute “logarithmic ”. The constant
121
+ bL, sometimes called “new anomaly”, defines crucial propertie s of the LCFT, much like the
122
+ central charges do in ordinary CFTs. The mass scale mLappearing in the last correlator above
123
+ has no significance, and is determined by the value of Bin (5). It can be changed to any finite
124
+ value by the redefinition Olog→ Olog+γOLwith some finite γ. We setmL= 1 for convenience.
125
+ Conformal Ward identities determine again essentially uni quely the form of 2- and 3-point
126
+ correlators in a LCFT. For the specific case where the energy m omentum tensor acquires a
127
+ logarithmic partner the 3-point correlators were calculat ed in [5]. The non-vanishing ones are
128
+ given by
129
+ ∝an}b∇acketle{tOL(z,¯z)OL(z′,¯z′)Olog(0,0)∝an}b∇acket∇i}ht=bL
130
+ z2z′2(z−z′)2(8a)
131
+ ∝an}b∇acketle{tOL(z,¯z)Olog(z′,¯z′)Olog(0,0)∝an}b∇acket∇i}ht=−2bLln|z′|2+bL
132
+ 2
133
+ z2z′2(z−z′)2(8b)
134
+ ∝an}b∇acketle{tOlog(z,¯z)Olog(z′,¯z′)Olog(0,0)∝an}b∇acket∇i}ht=lengthy
135
+ z2z′2(z−z′)2(8c)
136
+ If alsoORacquires a logarithmic partner O/tildewiderlogthen the construction above can be repeated,
137
+ changing everywhere L→R,z→¯zetc. In that case we have a LCFT with cL=cR= 0 andbL,bR∝ne}ationslash= 0. Alternatively, it may happen that only OLhas a logarithmic partner Olog. In that
138
+ case we have a LCFT with cL=bR= 0 andbL,cR∝ne}ationslash= 0. This concludes our brief excursion into
139
+ the realm of LCFTs.
140
+ Given that LCFTs are interesting in physics (see section 5) a nd that a powerful way to
141
+ describe strongly coupled CFTs is to exploit the AdS/CFT cor respondence [6] it is natural to
142
+ inquire whether there are any gravity duals to LCFTs.
143
+ 3. Wish-list for gravity duals to LCFTs
144
+ In this section we establish necessary properties required for gravity duals to LCFTs. We
145
+ formulate them as a wish-list and explain afterwards each it em on this list.
146
+ (i) We wishfora 3-dimensional action Sthat dependsonthemetric gµνandpossiblyonfurther
147
+ fields that we summarily denote by φ.
148
+ (ii) We wish for the existence of AdS 3vacua with finite AdS radius ℓ.
149
+ (iii) We wish for a finite, conserved and traceless Brown–Yor k stress tensor, given by the first
150
+ variation of the full on-shell action (including boundary t erms) with respect to the metric.
151
+ (iv) We wish that the 2- and 3-point correlators of the Brown– York stress tensor with itself are
152
+ given by (1).
153
+ (v) We wish for central charges (a la Brown–Henneaux [7]) tha t can be tuned to zero, without
154
+ requiring a singular limit of the AdS radius or of Newton’s co nstant. For concreteness we
155
+ assumecL= 0 (in addition cRmay also vanish, but it need not).
156
+ (vi) We wish for a logarithmic partner to the Brown–York stre ss tensor, so that we obtain a
157
+ Jordan-block structure like in (2) and (3).
158
+ (vii) We wish that the 2- and non-vanishing 3-point correlat ors of the Brown–York stress tensor
159
+ with its logarithmic partner are given by (7) and (8) (and the right-handed analog thereof).
160
+ We explain now why each of these items is necessary. (i) is req uired since the AdS/CFT
161
+ correspondence relates a gravity theory in d+1 dimensions to a CFT in ddimensions, and we
162
+ chosed= 2 on the CFT side. (ii) is required since we are not merely loo king for a gauge/gravity
163
+ duality, butreallyforanAdS/CFTcorrespondence,whichre quirestheexistenceofAdSsolutions
164
+ on the gravity side. (iii) is required since we desire consis tency with the AdS dictionary, which
165
+ relates the vacuum expectation value of the renormalized en ergy momentum tensor in the CFT
166
+ ∝an}b∇acketle{tTij∝an}b∇acket∇i}htto the Brown–York stress tensor TBY
167
+ ij:
168
+ ∝an}b∇acketle{tTij∝an}b∇acket∇i}ht=TBY
169
+ ij=2√−gδS
170
+ δgij/vextendsingle/vextendsingle/vextendsingle
171
+ EOM(9)
172
+ The right hand side of this equation contains the first variat ion of the full on-shell action with
173
+ respect to the metric, which by definition yields the Brown–Y ork stress tensor. (iv) is required
174
+ since the 2- and 3-point correlators of a CFT are fixed by confo rmal Ward identities to take
175
+ the form (1). (v) is required because of the construction pre sented in section 2, where a LCFT
176
+ emerges from taking an appropriate limit of vanishing centr al charge, so we need to be able
177
+ to tune the central charge without generating parametric si ngularities. Actually, there are
178
+ two cases: either left and right central charge vanish and bo th energy momentum tensor flux
179
+ components acquire a logarithmic partner, or only one of the m acquires a logarithmic partner,
180
+ which for sake of specificity we always choose to be left. (vi) is required, since we consider
181
+ exclusively LCFTs where the energy momentum tensor acquire s a logarithmic partner. (vii) is
182
+ required since the 2- and 3-point correlators of a LCFT are fix ed by conformal Ward identities to
183
+ taketheform(7), (8). Ifanyoftheitemsonthewish-listabo veisnotfulfilleditisimpossiblethat
184
+ the gravitational theory under consideration is a gravity d ual to a LCFT of the type discussedin section 2.2On the other hand, if all the wishes are granted by a given grav itational theory
185
+ there are excellent chances that this theory is dual to a LCFT . Until recently no good gravity
186
+ duals for LCFTs were known [8–12].
187
+ Before addressing candidate theories that may comply with a ll wishes we review briefly how
188
+ to calculate correlators on the gravity side [6], since we sh all need such calculations for checking
189
+ several items on the wish-list. The basic identity of the AdS /CFT dictionary is
190
+ ∝an}b∇acketle{tO1(z1)O2(z2)...On(zn)∝an}b∇acket∇i}ht=δ(n)S
191
+ δj1(z1)δj2(z2)...δjn(zn)/vextendsingle/vextendsingle/vextendsingle
192
+ ji=0(10)
193
+ The left hand side is the CFT correlator between noperators Oi, whereOiin our case comprise
194
+ theleft-andright-moving fluxcomponentsoftheenergymome ntumtensor andtheirlogarithmic
195
+ partners. The right hand side contains the gravitational ac tionSdifferentiated with respect to
196
+ appropriate sources jifor the corresponding operators. According to the AdS/CFT d ictionary
197
+ “appropriate sources” refers to non-normalizable solutio ns of the linearized equations of motion.
198
+ We shall be more concrete about the operators, actions, sour ces and non-normalizable solutions
199
+ to the linearized equations of motion in the next section. Fo r now we address possible candidate
200
+ theories of gravity duals to LCFTs.
201
+ The simplest candidate, pure 3-dimensional Einstein gravi ty with a cosmological constant
202
+ described by the action
203
+ SEH=−1
204
+ 8πGN/integraldisplay
205
+ Md3x√−g/bracketleftig
206
+ R+2
207
+ ℓ2/bracketrightig
208
+ −1
209
+ 4πGN/integraldisplay
210
+ ∂Md2x√−γ/bracketleftig
211
+ K−1
212
+ ℓ/bracketrightig
213
+ (11)
214
+ does not comply with the whole wish list. Only the first four wi shes are granted: The 3-
215
+ dimensional action (12) depends on the metric. The equation s of motion are solved by AdS 3.
216
+ ds2
217
+ AdS3=gAdS3µνdxµdxν=ℓ2/parenleftbig
218
+ dρ2−1
219
+ 4cosh2ρ(du+dv)2+1
220
+ 4sinh2ρ(du−dv)2/parenrightbig
221
+ (12)
222
+ The Brown–York stress tensor (9) is finite, conserved and tra celess. The 2- and 3-point
223
+ correlators on the gravity side match precisely with (1). Ho wever, the central charges are given
224
+ by [7]
225
+ cL=cR=3ℓ
226
+ 2GN(13)
227
+ and therefore allow no tuning to cL= 0 without taking a singular limit. Moreover, there is no
228
+ candidate for a logarithmic partner to the Brown–York stres s tensor. Thus, pure 3-dimensional
229
+ Einstein gravity cannot be dual to a LCFT.
230
+ Adding matter fields to Einstein gravity does not help neithe r. While this may lead to other
231
+ kinds of LCFTs, it cannot produce a logarithmic partner for t he energy momentum tensor. This
232
+ is so, because the energy momentum tensor corresponds to gra viton (spin-2) excitations in the
233
+ bulk, and the only field producing such excitations is the met ric.
234
+ Therefore, what we need is a way to provide additional degree s of freedom in the gravity
235
+ sector. The most natural way to do this is by considering high er derivative interactions of the
236
+ metric. Thefirstgravity modelofthistypewas constructedb yDeser, Jackiw andTempleton [13]
237
+ who introduced a Chern–Simons term for the Christoffel connec tion.
238
+ SCS=−1
239
+ 16πGNµ/integraldisplay
240
+ d3xǫλµνΓρσλ/bracketleftig
241
+ ∂µΓσρν+2
242
+ 3ΓσκµΓκσν/bracketrightig
243
+ (14)
244
+ 2Other types of LCFTs exist, e.g. with non-vanishing central charge or with logarithmic partners to operators
245
+ other than the energy momentum tensor. The gravity duals for such LCFTs need not comply with all the items
246
+ on our wish list.Hereµis a real coupling constant. Adding this action to the Einste in–Hilbert action (11)
247
+ generates massive graviton excitations in the bulk, which i s encouraging for our wish list since
248
+ we need these extra degrees of freedom. The model that arises when summing the actions (11)
249
+ and (14),
250
+ SCTMG=SEH+SCS (15)
251
+ is known as “cosmological topologically massive gravity” ( CTMG) [14]. It was demonstrated by
252
+ KrausandLarsen[15]that thecentral charges inCTMG areshi ftedfromtheir Brown–Henneaux
253
+ values:
254
+ cL=3ℓ
255
+ 2GN/parenleftbig
256
+ 1−1
257
+ µℓ/parenrightbig
258
+ cR=3ℓ
259
+ 2GN/parenleftbig
260
+ 1+1
261
+ µℓ/parenrightbig
262
+ (16)
263
+ This is again good news concerning our wish list, since cLcan be made vanishing by a (non-
264
+ singular) tuning of parameters in the action.
265
+ µℓ= 1 (17)
266
+ CTMG (15) with the tuning above (17) is known as “cosmologica l topologically massive gravity
267
+ at the chiral point” (CCTMG). It complies with the first five it ems on our wish list, but we still
268
+ have to prove that also the last two wishes are granted. To thi s end we need to find a suitable
269
+ partner for the graviton.
270
+ 4. Keeping logs in massive gravity
271
+ 4.1. Login
272
+ In this section we discuss the evidence for the existence of s pecific gravity duals to LCFTs that
273
+ has accumulated over the past two years. We start with the the ory introduced above, CCTMG,
274
+ and we end with a relatively new theory, new massive gravity [ 16].
275
+ 4.2. Seeds of logs
276
+ Given that we want a partner for the graviton we consider now g raviton excitations ψaround
277
+ the AdS background (12) in CCTMG.
278
+ gµν=gAdS3µν+ψµν (18)
279
+ Li,SongandStrominger[17]foundanicewaytoconstructthe m,andwefollowtheirconstruction
280
+ here. Imposing transverse gauge ∇µψµν= 0 and defining the mutually commuting first order
281
+ operators
282
+ /parenleftbig
283
+ DM/parenrightbigβ
284
+ µ=δβ
285
+ µ+1
286
+ µεµαβ∇α/parenleftbig
287
+ DL/R/parenrightbigβ
288
+ µ=δβ
289
+ µ±ℓεµαβ∇α (19)
290
+ allows to write the linearized equations of motion around th e AdS background (12) as follows.
291
+ (DMDLDRψ)µν= 0 (20)
292
+ A mode annihilated by DM(DL) [DR]{(DL)2but not by DL}is called massive (left-moving)
293
+ [right-moving] {logarithmic }and is denoted by ψM(ψL) [ψR]{ψlog}. Away from the chiral
294
+ point,µℓ∝ne}ationslash= 1, the general solution to the linearized equations of moti on (20) is obtained from
295
+ linearly combining left, right and massive modes [17]. At th e chiral point DMdegenerates with
296
+ DLand the general solution to the linearized equations of moti on (20) is obtained from linearly
297
+ combining left, right and logarithmic modes [18]. Interest ingly, we discovered in [18] that the
298
+ modesψlogandψLbehave as follows:
299
+ (L0+¯L0)/parenleftbigg
300
+ ψlog
301
+ ψL/parenrightbigg
302
+ =/parenleftbigg
303
+ 2 1
304
+ 0 2/parenrightbigg/parenleftbigg
305
+ ψlog
306
+ ψL/parenrightbigg
307
+ (21)whereL0=i∂u,¯L0=i∂vand
308
+ (L0−¯L0)/parenleftbigg
309
+ ψlog
310
+ ψL/parenrightbigg
311
+ =/parenleftbigg
312
+ 2 0
313
+ 0 2/parenrightbigg/parenleftbigg
314
+ ψlog
315
+ ψL/parenrightbigg
316
+ (22)
317
+ If we define naturally the Hamiltonian by H=L0+¯L0and the angular momentum by
318
+ J=L0−¯L0we recover exactly (2) and (3), which suggests that the CFT du al to CCTMG
319
+ (if it exists) is logarithmic, as conjectured in [18]. It was further shown with Jackiw that the
320
+ existence of the logarithmic excitations ψlogis not an artifact of the linearized approach, but
321
+ persists in the full theory [19].
322
+ Thus, also the sixth wish is granted in CCTMG. The rest of this section discusses the last
323
+ wish.
324
+ 4.3. Growing logs
325
+ We assume now that there is a standard AdS/CFT dictionary [6] available for LCFTs and check
326
+ if CCTMG indeed leads to the correct 2- and 3-point correlato rs. To this end we have to identify
327
+ the sources jithat appear on the right hand side of the correlator equation (10). Following the
328
+ standard AdS/CFT prescription the sources for the operator sOL(OR) [Olog] are given by left
329
+ (right) [logarithmic] non-normalizablesolutions tothel inearized equations of motion (20). Thus,
330
+ our first task is to find all solutions of the linearized equati ons of motion and to classify them
331
+ into normalizable and non-normalizable ones, where “norma lizable” refers to asymptotic (large
332
+ ρ) behavior that is exponentially suppressed as compared to t he AdS background (12).
333
+ A construction of all normalizable left and right solutions was provided in [17], and the
334
+ normalizable logarithmic solutions were constructed in [1 8].3The non-normalizable solutions
335
+ were constructed in [25]. It turned out to be convenient to wo rk in momentum space
336
+ ψL/R/log
337
+ µν(h,¯h) =e−ih(t+φ)−i¯h(t−φ)FL/R/log
338
+ µν(ρ) (23)
339
+ The momenta h,¯hare called “weights”. All components of the tensor Fµνare determined
340
+ algebraically, except for one that is determined from a seco nd order (hypergeometric) differential
341
+ equation. Ingeneral oneofthelinearcombinations of theso lutionsis singularattheorigin ρ= 0,
342
+ whiletheother isregular there. We keep onlyregular soluti ons. For each given set ofweights h,¯h
343
+ the regular solution is either normalizable or non-normali zable. It turns out that normalizable
344
+ solutions exist for integer weights h≥2,¯h≥0 (orh≤ −2,¯h≤0). All other solutions are
345
+ non-normalizable.
346
+ An example for a normalizable left mode is given by the primar y with weights h= 2,¯h= 0
347
+ ψL
348
+ µν(2,0) =e−2iu
349
+ cosh4ρ
350
+ 1
351
+ 4sinh2(2ρ) 0i
352
+ 2sinh(2ρ)
353
+ 0 0 0
354
+ i
355
+ 2sinh(2ρ) 0 −1
356
+
357
+ µν(24)
358
+ Note that all components of this mode behave asymptotically (ρ→ ∞) at most like a constant.
359
+ The corresponding logarithmic mode is given by
360
+ ψlog
361
+ µν(2,0) =−1
362
+ 2(i(u+v)+lncosh2ρ)ψL
363
+ µν(2,0) (25)
364
+ Evidently, it behaves asymptotically like its left partner (24), except for overall linear growth in
365
+ ρ. It is also worthwhile emphasizing that the logarithmic mod e (25) depends linearly on time
366
+ 3All these modes are compatible with asymptotic AdS behavior [20,21], and they appear in vacuum expectation
367
+ values of 1-point functions. Indeed, the 1-point function /angbracketleftTij/angbracketrightinvolves both ψlogandψR[21–24].t= (u+v)/2. Both features are inherent to all logarithmic modes. All o ther normalizable
368
+ modes can be constructed from the primaries (24), (25) algeb raically.
369
+ An example for a non-normalizable left mode is given by the mo de with weights h= 1,
370
+ ¯h=−1
371
+ ψL
372
+ µν(1,−1) =1
373
+ 4e−iu+iv
374
+ 0 0 0
375
+ 0 cosh(2 ρ)−1−2i/radicalig
376
+ cosh(2ρ)−1
377
+ cosh(2ρ)+1
378
+ 0−2i/radicalig
379
+ cosh(2ρ)−1
380
+ cosh(2ρ)+1−4
381
+ cosh(2ρ)+1
382
+ 
383
+ µν(26)
384
+ Note that all components of this mode behave asymptotically (ρ→ ∞) at most like a constant,
385
+ except for the vv-component, which grows like e2ρ. The corresponding logarithmic mode grows
386
+ again faster than its left partner (26) by a factor of ρand depends again linearly on time.
387
+ Given a non-normalizable solution ψLobviously also αψLis a non-normalizable solution,
388
+ with some constant α. To fix this normalization ambiguity we demand standard coup ling of the
389
+ metric to the stress tensor:
390
+ S(ψuL
391
+ v,Tv
392
+ u) =1
393
+ 2/integraldisplay
394
+ dtdφ/radicalig
395
+ −g(0)ψuu
396
+ LTuu=/integraldisplay
397
+ dtdφe−ihu−i¯hvTuu (27)
398
+ HereSis either someCFT action withbackgroundmetric g(0)or adualgravitational action with
399
+ boundary metric g(0). The non-normalizable mode ψLis the source for the energy-momentum
400
+ flux component Tuu. The requirement (27) fixes the normalization. The discussi on above
401
+ focussed on left modes. For the right modes essentially the s ame discussion applies, but with
402
+ the substitutions L↔R,h↔¯handu↔v.
403
+ 4.4. Logging correlators
404
+ Generically the 2-point correlators on the gravity side bet ween two modes ψ1(h,¯h) andψ2(h′,¯h′)
405
+ in momentum space are determined by
406
+ ∝an}b∇acketle{tψ1(h,¯h)ψ2(h′,¯h′)∝an}b∇acket∇i}ht=1
407
+ 2/parenleftbig
408
+ δ(2)SCCTMG(ψ1,ψ2)+δ(2)SCCTMG(ψ2,ψ1)/parenrightbig
409
+ (28)
410
+ where∝an}b∇acketle{tψ1ψ2∝an}b∇acket∇i}htstands for the correlation function of the CFT operators dua l to the graviton
411
+ modesψ1andψ2. On the right hand side one has to plug the non-normalizable m odesψ1
412
+ andψ2into the second variation of the on-shell action and symmetr ize with respect to the two
413
+ modes. The second variation of the on-shell action of CCTMG
414
+ δ(2)SCCTMG=−1
415
+ 16πGN/integraldisplay
416
+ d3x√−g/parenleftbig
417
+ DLψ1∗/parenrightbigµνδGµν(ψ2)+boundary terms (29)
418
+ turns out to be very similar to the second variation of the on- shell Einstein–Hilbert action
419
+ δ(2)SEH=−1
420
+ 16πGN/integraldisplay
421
+ d3x√−gψ1µν∗δGµν(ψ2)+boundary terms (30)
422
+ Thissimilarity allows ustoexploitresultsfromEinsteing ravity forCCTMG,aswenowexplain.4
423
+ The bulk term in CCTMG (29) has the same form as in Einstein the ory (30) with ψ1replaced
424
+ byDLψ1. Now, consider boundary terms. Possible obstructions to a w ell-defined Dirichlet
425
+ boundary value problem can come only from the variation δGµν(ψ2), sinceDLis a first order
426
+ operator. Thus any boundary terms appearing in (29) contain ing normal derivatives must be
427
+ 4Alternatively, one can follow the program of holographic re normalization, as it was done by Skenderis, Taylor
428
+ and van Rees [23]. Their results for 2-point correlators agr ee with the results presented here.identical with those in Einstein gravity upon substituting ψ1→ DLψ1. In addition there can be
429
+ boundary terms which do not contain normal derivatives of th e metric. However, it turns out
430
+ that such terms can at most lead to contact terms in the hologr aphic computation of 2-point
431
+ functions. The upshot of this discussion is that we can reduc e the calculation of all possible 2-
432
+ point functions in CCTMG to the equivalent calculation in Ei nstein gravity with suitable source
433
+ terms. To continue we go on-shell.5
434
+ DLψL= 0 DLψR= 2ψRDLψlog=−2ψL(31)
435
+ These relations together with the comparison between CCTMG (29) and Einstein gravity (30)
436
+ then establish
437
+ ∝an}b∇acketle{tψR(h,¯h)ψR(h′,¯h′)∝an}b∇acket∇i}htCCTMG∼2∝an}b∇acketle{tψR(h,¯h)ψR(h′,¯h′)∝an}b∇acket∇i}htEH (32a)
438
+ ∝an}b∇acketle{tψL(h,¯h)ψL(h′,¯h′)∝an}b∇acket∇i}htCCTMG∼0 (32b)
439
+ ∝an}b∇acketle{tψL(h,¯h)ψR(h′,¯h′)∝an}b∇acket∇i}htCCTMG∼0 (32c)
440
+ ∝an}b∇acketle{tψR(h,¯h)ψlog(h′,¯h′)∝an}b∇acket∇i}htCCTMG∼0 (32d)
441
+ ∝an}b∇acketle{tψL(h,¯h)ψlog(h′,¯h′)∝an}b∇acket∇i}htCCTMG∼ −2∝an}b∇acketle{tψL(h,¯h)ψL(h′,¯h′)∝an}b∇acket∇i}htEH (32e)
442
+ Here the sign ∼means equality up to contact terms. Evaluating the right han d sides in Einstein
443
+ gravity yields
444
+ ∝an}b∇acketle{tψL(h,¯h)ψL(h′,¯h′)∝an}b∇acket∇i}htEH=δh,h′δ¯h,¯h′cBH
445
+ 24h
446
+ ¯h(h2−1)t1/integraldisplay
447
+ t0dt (33)
448
+ and similarly for the right modes, with h↔¯h. The quantity cBHis the Brown–Henneaux
449
+ central charge (13). The calculation of the 2-point correla tor between two logarithmic modes
450
+ cannot be reduced to a correlator known from Einstein gravit y. The result is given by [25]
451
+ ∝an}b∇acketle{tψlog(h,¯h)ψlog(h′,¯h′)∝an}b∇acket∇i}htCCTMG∼ −δh,h′δ¯h,¯h′ℓ
452
+ 4GNh
453
+ ¯h(h2−1)/parenleftbig
454
+ ψ(h−1)+ψ(−¯h)/parenrightbigt1/integraldisplay
455
+ t0dt(34)
456
+ whereψis the digamma function. An ambiguity in defining ψlog, viz.,ψlog→ψlog+γψL, was
457
+ fixed conveniently in the result (34). This ambiguity corres ponds precisely to the ambiguity of
458
+ the LCFT mass scale mLin (7c) (see also the discussion below that equation).
459
+ To compare the results (32)-(34) with the Euclidean 2-point correlators in the short-
460
+ distance limit (1), (7) we take the limit of large weights h,−¯h→ ∞(e.g. lim h→∞ψ(h) =
461
+ lnh+O(1/h)) and Fourier-transform back to coordinate space (e.g. h3/¯his Fourier-transformed
462
+ into∂4
463
+ z/(∂z∂¯z)δ(2)(z,¯z)∝∂4
464
+ zln|z| ∝1/z4). Straightforward calculation establishes perfect
465
+ agreement with the LCFT correlators (1), (7), provided we us e the values
466
+ cL= 0 cR=3ℓ
467
+ GNbL=−3ℓ
468
+ GN(35)
469
+ These are exactly the values for central charges cL,cR[15] and new anomaly bL[23,25] found
470
+ before. Thus, at the level of 2-point correlators CCTMG is in deed a gravity dual for a LCFT.
471
+ 5Above by “on-shell” we meant that the background metric is Ad S3(12) and therefore a solution of the classical
472
+ equations of motion. Here by “on-shell” we mean additionall y that the linearized equations of motion (20) hold.Ψ1
473
+ Ψ3Ψ2
474
+ Figure 1. Witten diagram for three graviton correlator
475
+ We evaluate now the Witten diagram in Fig. 1, which yields the 3-point correlator on the
476
+ gravity side between three modes ψ1(h,¯h),ψ2(h′,¯h′) andψ3(h′′,¯h′′) in momentum space.
477
+ ∝an}b∇acketle{tψ1(h,¯h)ψ2(h′,¯h′)ψ3(h′′,¯h′′)∝an}b∇acket∇i}ht=1
478
+ 6/parenleftbig
479
+ δ(3)SCCTMG(ψ1,ψ2,ψ3)+5 permutations/parenrightbig
480
+ (36)
481
+ On the right hand side one has to plug the non-normalizable mo desψ1,ψ2andψ3into the third
482
+ variation of the on-shell action and symmetrize with respec t to all three modes.
483
+ δ(3)SCCTMG∼ −1
484
+ 16πGN/integraldisplay
485
+ d3x√−g/bracketleftig/parenleftbig
486
+ DLψ1/parenrightbigµνδ(2)Rµν(ψ2,ψ3)+ψ1µν∆µν(ψ2,ψ3)/bracketrightig
487
+ (37)
488
+ The quantity δ(2)Rµν(ψ2,ψ3) denotes the second variation of the Ricci-tensor and the te nsor
489
+ ∆µν(ψ2,ψ3) vanishes if evaluated on left- and/or right-moving soluti ons. All boundary terms
490
+ turn out to be contact terms, which is why only bulk terms are p resent in the result (37) for the
491
+ third variation of the on-shell action. We compare again wit h Einstein gravity.
492
+ δ(3)SEH∼ −1
493
+ 16πGN/integraldisplay
494
+ d3x√−gψ1µνδ(2)Rµν(ψ2,ψ3) (38)
495
+ Once more we can exploit some results from Einstein gravity f or CCTMG, and we find the
496
+ following results [25] for 3-point correlators without log -insertions:
497
+ ∝an}b∇acketle{tψR(h,¯h)ψR(h′,¯h′)ψR(h′′,¯h′′)∝an}b∇acket∇i}htCCTMG∼2∝an}b∇acketle{tψR(h,¯h)ψR(h′,¯h′)ψR(h′′,¯h′′)∝an}b∇acket∇i}htEH (39a)
498
+ ∝an}b∇acketle{tψL(h,¯h)ψR(h′,¯h′)ψR(h′′,¯h′′)∝an}b∇acket∇i}htCCTMG∼0 (39b)
499
+ ∝an}b∇acketle{tψL(h,¯h)ψL(h′,¯h′)ψR(h′′,¯h′′)∝an}b∇acket∇i}htCCTMG∼0 (39c)
500
+ ∝an}b∇acketle{tψL(h,¯h)ψL(h′,¯h′)ψL(h′′,¯h′′)∝an}b∇acket∇i}htCCTMG∼0 (39d)
501
+ with one log-insertion:
502
+ ∝an}b∇acketle{tψR(h,¯h)ψR(h′,¯h′)ψlog(h′′,¯h′′)∝an}b∇acket∇i}htCCTMG∼0 (40a)
503
+ ∝an}b∇acketle{tψL(h,¯h)ψR(h′,¯h′)ψlog(h′′,¯h′′)∝an}b∇acket∇i}htCCTMG∼0 (40b)
504
+ ∝an}b∇acketle{tψL(h,¯h)ψL(h′,¯h′)ψlog(h′′,¯h′′)∝an}b∇acket∇i}htCCTMG∼ −2∝an}b∇acketle{tψL(h,¯h)ψL(h′,¯h′)ψL(h′′,¯h′′)∝an}b∇acket∇i}htEH (40c)and with two or more log-insertions:
505
+ lim
506
+ |weights|→∞∝an}b∇acketle{tψR(h,¯h)ψlog(h′,¯h′)ψlog(h′′,¯h′′)∝an}b∇acket∇i}htCCTMG∼0 (41a)
507
+ lim
508
+ |weights|→∞∝an}b∇acketle{tψL(h,¯h)ψlog(h′,¯h′)ψlog(h′′,¯h′′)∝an}b∇acket∇i}htCCTMG∼δh′′,−h−h′δ¯h′′,−¯h−¯h′Plog(h,h′,¯h,¯h′)
509
+ ¯h¯h′(¯h+¯h′)(41b)
510
+ lim
511
+ |weights|→∞∝an}b∇acketle{tψlog(h,¯h)ψlog(h′,¯h′)ψlog(h′′,¯h′′)∝an}b∇acket∇i}htCCTMG∼δh′′,−h−h′δ¯h′′,−¯h−¯h′lengthy
512
+ ¯h¯h′(¯h+¯h′)(41c)
513
+ Thelast two correlators so far could becalculated qualitat ively only (Plogis a known polynomial
514
+ in the weights and also contains logarithms in the weights, a s expected on general grounds),
515
+ and it would be interesting to calculate them exactly. They a re in qualitative agreement with
516
+ corresponding LCFT correlators. All other correlators hav e been calculated exactly [25], and
517
+ they are in precise agreement with the LCFT correlators (1), (8), provided we use again the
518
+ values (35) for central charges and new anomaly.
519
+ Inconclusion, also theseventh wishisgranted forCCTMG.6Thus, thereareexcellent chances
520
+ that CCTMG is dual to a LCFT with values for central charges an d new anomaly given by (35).
521
+ 4.5. Logs don’t grow on trees
522
+ From the discussion above it is clear that possible gravity d uals for LCFTs are sparse in theory
523
+ space: Einstein gravity (11) does not provide a gravity dual for any tuning of parameters and
524
+ CTMG (15) does potentially provide a gravity dual only for a s pecific tuning of parameters (17).
525
+ Any candidate for a novel gravity dual to a LCFT is therefore w elcomed as a rare entity.
526
+ Very recently another plausible candidate for such a gravit ational theory was found [26].
527
+ That theory is known as “new massive gravity” [16].
528
+ SNMG=1
529
+ 16πGN/integraldisplay
530
+ d3x√−g/bracketleftig
531
+ σR+1
532
+ m2/parenleftbig
533
+ RµνRµν−3
534
+ 8R2/parenrightbig
535
+ −2λm2/bracketrightig
536
+ (42)
537
+ Heremis a mass parameter, λa dimensionless cosmological parameter and σ=±1 the sign of
538
+ the Einstein-Hilbert term. If they are tuned as follows
539
+ λ= 3 ⇒m2=−σ
540
+ 2ℓ2(43)
541
+ then essentially the same story unfolds as for CTMG at the chi ral point. The main difference
542
+ to CCTMG is that both central charges vanish in new massive gr avity at the chiral point
543
+ (CNMG) [27,28].
544
+ cL=cR=3ℓ
545
+ 2GN/parenleftbigg
546
+ σ+1
547
+ 2ℓ2m2/parenrightbigg
548
+ = 0 (44)
549
+ Therefore, both left and right flux component of the energy mo mentum tensor acquire a
550
+ logarithmic partner. It is easy to check that CNMG grants us t he first six wishes from section
551
+ 3. The seventh wish requires again the calculation of correl ators. The 3-point correlators have
552
+ not been calculated so far, but at the level of 2-point correl ators again perfect agreement with
553
+ a LCFT was found, provided we use the values [26]
554
+ cL=cR= 0bL=bR=−σ12ℓ
555
+ GN(45)
556
+ 6The sole caveat is that two of the ten 3-point correlators wer e calculated only qualitatively. It would be
557
+ particularly interesting to calculate the correlator betw een three logarithmic modes (41c), since it contains an
558
+ additional parameter independent from the central charges and new anomaly that determines LCFT properties.Itislikely thatasimilarstorycanberepeatedforgeneralm assivegravity [16], whichcombines
559
+ new massive gravity (42) with a gravitational Chern–Simons term (14). Thus, even though they
560
+ are sparse in theory space we have found a few good candidates for gravity duals to LCFTs:
561
+ cosmological topologically massive gravity, new massive g ravity and general massive gravity. In
562
+ all cases we have to tune parameters in such a way that a “chira l point” emerges where at least
563
+ one of the central charges vanishes.
564
+ 4.6. Chopping logs?
565
+ Sofarwe were exclusively concerned with findinggravitatio nal theories wherelogarithmic modes
566
+ can arise. In this subsection we try to get rid of them. The rat ionale behind the desire to
567
+ eliminate the logarithmic modes is unitarity of quantum gra vity. Gravity in 2+1 dimensions is
568
+ simple and yet relevant, as it contains black holes [29], pos sibly gravity waves [13] and solutions
569
+ that are asymptotically AdS. Thus, it could provide an excel lent arena to study quantum gravity
570
+ in depth provided one is able to come up with a consistent (uni tary) theory of quantum gravity,
571
+ for instance by constructing its dual (unitary) CFT. Indeed , two years ago Witten suggested a
572
+ specific CFT dual to 3-dimensional quantum gravity in AdS [30 ]. This proposal engendered a
573
+ lot of further research (see [31–37] for some early referenc es), including the suggestion by Li,
574
+ Song and Strominger [17] to construct a quantum theory of gra vity that is purely right-moving,
575
+ dubbed“chiral gravity”. To make a long story [18,19,24,38– 81] short, “chiral gravity” is nothing
576
+ but CCTMG with the logarithmic modes truncated in some consi stent way.
577
+ We discuss now two conceptually different possibilities of im plementing such a truncation.
578
+ The first option was proposed in [18]. If one imposes periodic ity in time for all modes, t→t+β,
579
+ then only the left- and right-moving modes are allowed, whil e the logarithmic modes are
580
+ eliminated since they grow linearly in time, see e.g. (25). T he other possibility was pursued
581
+ in [22]. It is based upon the observation that logarithmic mo des grow logarithmically faster in
582
+ e2ρthan their left partners, see e.g. (25). Thus, imposing boun dary conditions that prohibit this
583
+ logarithmic growth eliminates all logarithmic modes.
584
+ Currently it is not known whether chiral gravity has its own d ual CFT or if it exists merely
585
+ as a zero-charge superselection sector of the logarithmic C FT. In the latter case it is unclear
586
+ whether or not the zero-charge superselection sector is a fu lly-fledged CFT. Another alternative
587
+ is that neither the LCFT nor its chiral truncation dual to chi ral gravity exists. In that case
588
+ CTMG is unlikely to exist as a consistent quantum theory on it s own. Rather, it would require
589
+ a UV completion, such as string theory.
590
+ 4.7. Logout
591
+ We summarize now the key results reviewed in this section as w ell as some open issues.
592
+ Cosmological topologically massive gravity (15) at the chi ral point (17) is likely to be dual
593
+ to a LCFT with a logarithmic partner for one flux component of t he energy momentum tensor
594
+ since 2- [23] and 3-point correlators [25] match. The values of central charges and new anomaly
595
+ are given by (35). The detailed calculation of the correlato r with three log-insertions (41c)
596
+ still needs to be performed and will determine another param eter of the LCFT. New massive
597
+ gravity (42) at the chiral point (43) is likely to be dual to a L CFT with a logarithmic partner
598
+ for both flux components of the energy momentum tensor since 2 -point correlators match [26].
599
+ The central charges vanish and the new anomalies are given by (45). The calculation of 3-
600
+ point correlators still needs to be performed and will provi de a more stringent test of the
601
+ conjectured duality to a LCFT. A similar story is likely to re peat for general massive gravity
602
+ (the combination of topologically and new massive gravity) at a chiral point, and it could be
603
+ rewardingtoinvestigate thisissue. Finallyweaddressedp ossibilitiestoeliminatethelogarithmic
604
+ modes and their partners, since such an elimination might le ad to a chiral theory of quantum
605
+ gravity [17], called “chiral gravity”. The issue of whether chiral gravity exists still remains open.5. Towards condensed matter applications
606
+ In this final section we review briefly some condensed matter s ystems where LCFTs do arise,
607
+ see [3,4] for more comprehensive reviews. We focus on LCFTs w here the energy-momentum
608
+ tensor acquires a logarithmic partner, i.e., the class of LC FTs for which we have found possible
609
+ gravity duals.7Condensed matter systems described by such LCFTs are for ins tance systems
610
+ at (or near) a critical point with quenched disorder, like sp in glasses [83]/quenched random
611
+ magnets [84,85], dilute self-avoiding polymers or percola tion [86]. “Quenched disorder” arises
612
+ in a condensed matter system with random variables that do no t evolve with time. If the
613
+ amount of disorder is sufficiently large one cannot study the e ffects of disorder by perturbing
614
+ around a critical point without disorder — standard mean fiel d methods break down. The
615
+ system is then driven towards a random critical point, and it is a challenge to understand its
616
+ precise nature. Mathematically, the essence of the problem lies in the infamous denominator
617
+ arising in correlation functions of some operator Oaveraged over disordered configurations (see
618
+ e.g. chapter VI.7 in [87])
619
+ ∝an}b∇acketle{tO(z)O(0)∝an}b∇acket∇i}ht=/integraldisplay
620
+ DVP[V]/integraltext
621
+ Dφexp/parenleftbig
622
+ −S[φ]−/integraltext
623
+ d2z′V(z′)O(z′)/parenrightbig
624
+ O(z)O(0)/integraltext
625
+ Dφexp/parenleftbig
626
+ −S[φ]−/integraltext
627
+ d2z′V(z′)O(z′)/parenrightbig (46)
628
+ HereS[φ] is some 2-dimensional8quantum field theory action for some field(s) φandV(z) is a
629
+ random potential with some probability distribution. For w hite noise one takes the Gaussian
630
+ probability distribution P[V]∝exp/parenleftbig
631
+ −/integraltext
632
+ d2zV2(z)/(2g2)/parenrightbig
633
+ , wheregis a coupling constant that
634
+ measuresthestrengthoftheimpurities. Ifit werenot forth edenominatorappearingontheright
635
+ hand side of the averaged correlator (46) we could simply per form the Gaussian integral over
636
+ the impurities encoded in the random potential V(z). This denominator is therefore the source
637
+ of all complications and to deal with it requires suitable me thods, see e.g. [88]. One possibility is
638
+ to eliminate the denominator by introducing ghosts. This so -called “supersymmetric method”
639
+ works well if the original quantum field theory described by t he actionS[φ] is very simple, like a
640
+ free field theory. Another option is the so-called replica tr ick, where one introduces ncopies of
641
+ the original quantum field theory, calculates correlators i n this setup and takes the limit n→0
642
+ in the end, which formally reproduces the denominator in (46 ). Recently, Fujita, Hikida, Ryu
643
+ and Takayanagi combined the replica method with the AdS/CFT correspondence to describe
644
+ disordered systems [89] (see [90,91] for related work), ess entially by taking ncopies of the CFT,
645
+ exploiting AdS/CFT to calculate correlators and taking for mally the limit n→0 in the end.
646
+ Like other replica tricks their approach relies on the exist ence of the limit n→0.
647
+ One of the results obtained by the supersymmetric method or r eplica trick is that correlators
648
+ like the one in (46) develop a logarithmic behavior, exactly as in a LCFT [84]. In fact, in
649
+ then→0 limit prescribed by the replica trick, the conformal dimen sions of certain operators
650
+ degenerate. This produces a Jordan block structure for the H amiltonian in precise parallel to
651
+ theµℓ→1 limit of CTMG. More concretely, LCFTs can be used to compute correlators of
652
+ quenched random systems!
653
+ This suggests yet-another route to describe systems with qu enched disorder, and our present
654
+ results add to this toolbox. Namely, instead of taking ncopies of an ordinary CFT we may
655
+ start directly with a LCFT. If this LCFT is weakly coupled we c an work on the LCFT side
656
+ perturbatively, using the results mentioned above [3,4,84 –86]. On the other hand, if the LCFT
657
+ becomes strongly coupled, perturbative methods fail. To ge t a handle on these situations we
658
+ can exploit the AdS/LCFT correspondence and work on the grav ity side. Of course, to this end
659
+ 7A well-studied alternative case is a LCFT with c=−2 [2,82]. There is no obvious way to construct a gravity
660
+ dual for such LCFTs, even when considering CTMG or new massiv e gravity away from the chiral point. We thank
661
+ Ivo Sachs for discussions on this issue.
662
+ 8Analog constructions work in higher dimensions, but we focu s here on two dimensions.one needs to construct gravity duals for LCFTs. The models re viewed in this talk are simple
663
+ and natural examples of such constructions.
664
+ Acknowledgments
665
+ We thank Matthias Gaberdiel, Gaston Giribet, Olaf Hohm, Rom an Jackiw, David Lowe, Hong
666
+ Liu, Alex Maloney, John McGreevy, Ivo Sachs, Kostas Skender is, Wei Song, Andy Strominger
667
+ and Marika Taylor for discussions. DG thanks the organizers of the “First Mediterranean
668
+ Conference on Classical and Quantum Gravity” for the kind in vitation and for all their efforts to
669
+ make the meeting very enjoyable. DG and NJ are supported by th e START project Y435-N16
670
+ of the Austrian Science Foundation (FWF). During the final st age NJ has been supported by
671
+ project P21927-N16 of FWF. NJ acknowledges financial suppor t from the Erwin-Schr¨ odinger-
672
+ Institute (ESI) during the workshop “Gravity in three dimen sions”.
673
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1001.0003.txt ADDED
@@ -0,0 +1,843 @@
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
1
+ arXiv:1001.0003v3 [hep-th] 10 May 2010Preprint typeset in JHEP style - HYPER VERSION KUL-TF-09/28
2
+ HD-THEP-09-31
3
+ A landscape of non-supersymmetric AdS vacua on
4
+ coset manifolds
5
+ Paul Koerber∗
6
+ Instituut voor Theoretische Fysica, Katholieke Universit eit Leuven, Celestijnenlaan
7
+ 200D, B-3001 Leuven, Belgium
8
+ Email:koerber atitf.fys.kuleuven.be
9
+ Simon K¨ ors
10
+ Institut f¨ ur Theoretische Physik, Universit¨ at Heidelbe rg, Philosophenweg 16-19, D-69120
11
+ Heidelberg, Germany
12
+ Email:s.koers atthphys.uni-heidelberg.de
13
+ Abstract: We construct new families of non-supersymmetric sourceles s type IIA AdS 4
14
+ vacua on those coset manifolds that also admit supersymmetr ic solutions. We investigate
15
+ the spectrum of left-invariant modes and find that most, but n ot all, of the vacua are stable
16
+ under these fluctuations. Generically, there are also no mas sless moduli.
17
+ ∗Postdoctoral Fellow FWO – Vlaanderen.Contents
18
+ 1. Introduction 1
19
+ 2. Ansatz 3
20
+ 3. Solutions 6
21
+ 4. Stability analysis 11
22
+ 5. Conclusions 15
23
+ A. SU(3)-structure 15
24
+ B. Type II supergravity 16
25
+ 1. Introduction
26
+ The reasons for studying AdS 4vacua of type IIA supergravity are twofold: first they are
27
+ examples of flux compactifications away from the Calabi-Yau r egime, where all the moduli
28
+ can be stabilized at the classical level. Secondly, they can serve as a gravity dual in the
29
+ AdS4/CFT3-correspondence, which became the focus of attention due to recent progress
30
+ in the understanding of the CFT-side as a Chern-Simons-matt er theory describing the
31
+ world-volume of coinciding M2-branes [1].
32
+ Itismucheasiertofindsupersymmetricsolutionsofsupergr avityasthesupersymmetry
33
+ conditions are simpler than the full equations of motion, wh ile at the same time there
34
+ are general theorems stating that the former – supplemented with the Bianchi identities
35
+ of the form fields – imply the latter [2, 3, 4, 5]. Although spec ial type IIA solutions
36
+ that came from the reduction of supersymmetric M-theory vac ua were already known (see
37
+ e.g. [6, 7, 8]), it was only in [3] that the supersymmetry cond itions for type IIA vacua with
38
+ SU(3)-structure were first worked out in general. It was disc overed that there are natural
39
+ solutions to these equations on the four coset manifolds G/Hthat have a nearly-K¨ ahler
40
+ limit [9, 10, 11, 12, 13, 14] (solutions on other manifolds ca n be found in e.g. [3, 15, 16]).1
41
+ To be precise these are the manifolds SU(2) ×SU(2),G2
42
+ SU(3),Sp(2)
43
+ S(U(2)×U(1))andSU(3)
44
+ U(1)×U(1).2
45
+ These solutions are particularly simple in the sense that bo th the SU(3)-structure, which
46
+ determines the metric, as well as all the form fluxes can be exp anded in terms of forms
47
+ which are left-invariant under the action of the group G. The supersymmetry equations
48
+ 1For an early appearance of these coset manifolds in the strin g literature see e.g. [17].
49
+ 2See [18] for a review and a proof that these are the only homoge neous manifolds admitting a nearly-
50
+ K¨ ahler geometry.
51
+ – 1 –of [3] then reduce to purely algebraic equations and can be ex plicitly solved. Nevertheless,
52
+ these solutions still have non-trivial geometric fluxes as o pposed to the Calabi-Yau or torus
53
+ orientifolds of [15, 16]. Similarly to those papers it is pos sible to classically stabilize all
54
+ left-invariant moduli [14]. Inspired by the AdS 4/CFT3correspondence more complicated
55
+ type IIA solutions have in the meantime been proposed. The so lutions have a more generic
56
+ form for the supersymmetry generators, called SU(3) ×SU(3)-structure [19], and are not
57
+ left-invariant anymore [20, 21, 22, 23] (see also [24]). Sup ersymmetric AdS 4vacua in type
58
+ IIB with SU(2)-structure have also been studied in [25, 26, 2 7, 28] and in particular it has
59
+ been shown in [28] that also in this setup classical moduli st abilization is possible.
60
+ At some point, however, supersymmetry has to be broken and we have to leave
61
+ the safe haven of the supersymmetry conditions. In this pape r we construct new non-
62
+ supersymmetric AdS 4vacua without source terms. This means that the more complic ated
63
+ equations of motion of supergravity should be tackled direc tly3. In order to simplify the
64
+ equations we use a specific ansatz: we start from a supersymme tric AdS 4solution and scan
65
+ for non-supersymmetric solutions with the samegeometry (and thus SU(3)-structure), but
66
+ withdifferent NSNS- and RR-fluxes. Moreover, we expand these form fields in t erms of the
67
+ SU(3)-structure and its torsion classes. This may seem rest rictive at first, but it works for
68
+ 11D supergravity, where solutions like this have been found and are known as Englert-type
69
+ solutions [31, 32, 33] (see [34] for a review). To be specific, for each supersymmetric M-
70
+ theory solution of Freund-Rubin type (which means the M-the ory four-form flux has only
71
+ legs along the external AdS 4space, i.e.F4=fvol4wherefis called the Freund-Rubin
72
+ parameter) it is possible to construct a non-supersymmetri c solution with the same inter-
73
+ nal geometry but with a different four-form flux. The modified fo ur-form of the Englert
74
+ solution has then a non-zero internal part: ˆF4∝η†γm1m2m3m4ηdxm1m2m3m4, whereηis
75
+ the 7D supersymmetry generator, and a different Freund-Rubin parameterfE=−(2/3)f.
76
+ Also the Ricci scalar of the AdS 4space, and thus the effective 4D cosmological constant,
77
+ differs:R4D,E= (5/6)R4D. In type IIA with non-zero Romans mass (so that there is no lif t
78
+ to M-theory) non-supersymmetric solutions of this form hav e been found as well: for the
79
+ nearly-K¨ ahler geometry in [35, 29, 36] and for the K¨ ahler- Einstein geometry in [35, 20, 37].
80
+ In this paper we show that this type of solutions is not restri cted to these limits and sys-
81
+ tematically scan for them. Applying our ansatz to the coset m anifolds with nearly-K¨ ahler
82
+ limit, mentioned above, we find that the most interesting man ifolds areSp(2)
83
+ S(U(2)×U(1))and
84
+ SU(3)
85
+ U(1)×U(1), on which we find several families of non-supersymmetric AdS 4solutions. We
86
+ also find some non-supersymmetric solutions in regimes of th e geometry that do not allow
87
+ for a supersymmetric solution.
88
+ These non-supersymmetric solutions are not necessarily st able. For instance, it is
89
+ known that if there is more than one Killing spinor on the inte rnal manifold (which holds
90
+ in particular for S7, the M-theory lift of CP3=Sp(2)
91
+ S(U(2)×U(1))), the Englert-type solution is
92
+ unstable [38]. We investigate stability of our solutions ag ainst left-invariant fluctuations.
93
+ This means we calculate the spectrum of left-invariant mode s, and check for each mode
94
+ 3Anotherroute would be tofindsome alternative first-ordereq uations, which extendthe supersymmetry
95
+ conditions in that they still automatically imply the full e quations of motion in certain non-supersymmetric
96
+ cases, see e.g. [29, 30].
97
+ – 2 –whether the mass-squared is above the Breitenlohner-Freed man bound [39, 40]. This is not
98
+ a complete stability analysis in that there could still be no n-left-invariant modes that are
99
+ unstable. We do believe it provides a good first indication. I n particular, we find for the
100
+ type IIA reduction of the Englert solution on S7that the unstable mode of [38] is among
101
+ our left-invariant fluctuations and we find the exact same mas s-squared.
102
+ These non-supersymmetric AdS 4vacua are interesting, because, provided they are
103
+ stable, they should have a CFT-dual. For instance in [20] the CFT-dual for a non-
104
+ supersymmetric K¨ ahler-Einstein solution on CP3was proposed. Furthermore, for phe-
105
+ nomenologically more realistic vacua, supersymmetry-bre aking is essential. Really, one
106
+ would like to construct classical solutions with a dS 4-factor, which are necessarily non-
107
+ supersymmetric. Because of a series of no-go theorems – from very general to more specific:
108
+ [41, 42, 43, 44, 45] – this is a very non-trivial task. For pape rs nevertheless addressing this
109
+ problemsee[46,47,45,48,49,28]. Inthiscontext thelands capeofthenon-supersymmetric
110
+ AdS4vacua of this paper can be considered as a playground to gain e xperience before try-
111
+ ing to construct dS 4-vacua. In fact, in [48] an ansatz very similar to the one used in this
112
+ paper was proposed in order to construct dS 4-vacua. Applied to the coset manifolds above,
113
+ it did however not yield any solutions, in agreement with the no-go theorem of [45].
114
+ In section 2 we explain our ansatz in full detail, while in sec tion 3 we present the
115
+ explicit solutions we found on the coset manifolds. In secti on 4 we analyse the stability
116
+ against left-invariant fluctuations before ending with som e short conclusions. We provide
117
+ an appendix with some useful formulae involving SU(3)-stru ctures and an appendix on our
118
+ supergravity conventions.
119
+ Thenon-supersymmetricsolutions of this paperappearedbe forein thesecond author’s
120
+ PhD thesis [50].
121
+ 2. Ansatz
122
+ In this section we explain the ansatz for our non-supersymme tric solutions. The reader
123
+ interested in the details might want to check out our SU(3)-s tructure conventions in ap-
124
+ pendix A, while towards the end of the section we need the type II supergravity equations
125
+ of motion outlined in appendix B.
126
+ We start with a supersymmetric SU(3)-structure solution of type IIA supergravity.
127
+ The SU(3)-structure is defined by a real two-form Jand a complex decomposable three-
128
+ form Ω satisfying (A.1). Moreover, Jand Ω together determine the metric as in (A.2). In
129
+ order for the solution to preserve at least one supersymmetr y (N= 1) [3] one finds that
130
+ the warp factor Aand the dilaton Φ should be constant, the torsion classes W1,W2purely
131
+ imaginary and all other torsion classes zero (for the definit ion of the torsion classes see
132
+ (A.3)). This implies
133
+ dJ=3
134
+ 2W1ReΩ, (2.1a)
135
+ dReΩ = 0, (2.1b)
136
+ dImΩ =W1J∧J+W2∧J, (2.1c)
137
+ – 3 –where we defined W1≡ −iW1andW2≡ −iW2. The fluxes can then be expressed in terms
138
+ of Ω,Jand the torsion classes and are given by
139
+ eΦˆF0=f1, (2.2a)
140
+ eΦˆF2=f2J+f3ˆW2, (2.2b)
141
+ eΦˆF4=f4J∧J+f5ˆW2∧J, (2.2c)
142
+ eΦˆF6=f6vol6, (2.2d)
143
+ H=f7ReΩ, (2.2e)
144
+ where for the supersymmetric solution
145
+ f1=eΦm, f 2=−W1
146
+ 4, f3=−w2, f4=3eΦm
147
+ 10,
148
+ f5= 0, f 6=9W1
149
+ 4, f7=2eΦm
150
+ 5.(2.3)
151
+ Using the duality relation f=˜F0=−⋆6ˆF6=−e−Φf6(see (B.6)) we find that f6is
152
+ proportional to the Freund-Rubin parameter f, whilef1is proportional to the Romans
153
+ massm. Furthermore, we introduced here a normalized version of W2, enabling us later
154
+ on to use (2.2) as an ansatz for the fluxes also in the limit W2→0:
155
+ ˆW2=W2
156
+ w2,withw2=±/radicalbig
157
+ (W2)2, (2.4)
158
+ where one can choose a convenient sign in the last expression .
159
+ The Bianchi identity for ˆF2imposes dW2∝ReΩ. Working out the proportionality
160
+ constant [3] we find
161
+ dW2=−1
162
+ 4(W2)2ReΩ. (2.5)
163
+ Furthermore, using the values for the fluxes (2.3) it fixes the Romans mass:
164
+ e2Φm2=5
165
+ 16/parenleftbig
166
+ 3(W1)2−2(W2)2/parenrightbig
167
+ . (2.6)
168
+ We now want to construct non-supersymmetric AdS solutions o n the manifolds men-
169
+ tioned in the introduction with the samegeometry as in the supersymmetric solution, and
170
+ thus the same SU(3)-structure ( J,Ω), but with different fluxes. We make the ansatz that
171
+ the fluxes can still be expanded in terms of J,Ω and the torsion class ˆW2as in (2.2), but
172
+ with different values for the coefficients fi. To this end we plug the ansatz for the geometry
173
+ (J,Ω) — eqs. (2.1) — and the ansatz for the fluxes — eqs. (2.2) — into the equations of
174
+ motion (B.7) and solve for the fi. We will make one more assumption, namely that
175
+ ˆW2∧ˆW2=cJ∧J+pˆW2∧J, (2.7)
176
+ withc,psome parameters. This is an extra constraint only for theSU(3)
177
+ U(1)×U(1)coset and
178
+ we will discuss its relaxation later.4Wedging with Jwe find then immediately c=−1/6.
179
+ 4With the ansatz (2.2) the constraint is forced upon us. Indee d, suppose that instead ˆW2∧ˆW2=
180
+ −1/6J∧J+pˆW2∧J+P∧J, wherePis a non-zero simple (1,1)-form independent of ˆW2. We find then
181
+ from the equation of motion for Hand the internal part of the Einstein equation respectively f5f3= 0 and
182
+ (f3)2−(f5)2−(w2)2= 0. So the only possibility is then f5= 0 and f3=±w2, which leads in the end to
183
+ the supersymmetric solution. They way out is to also include Pas an expansion form in (2.2).
184
+ – 4 –Furthermore we need expressions for the Ricci scalar and ten sor, which for a manifold with
185
+ SU(3)-structure can be expressed in terms of the torsion cla sses [51]. Taking into account
186
+ that onlyW1,2are non-zero we find:
187
+ R6D=15(W1)2
188
+ 2−(W2)2
189
+ 2, (2.8a)
190
+ Rmn=1
191
+ 6gmnR6D+W1
192
+ 4W2(m·Jn)+1
193
+ 2[W2m·W2n]0+1
194
+ 2Re/bracketleftbig
195
+ dW2|(2,1)m·¯Ωn/bracketrightbig
196
+ ,(2.8b)
197
+ where (P)2andPm·Pnfor a form Pare defined in (B.2) and |0indicates taking the
198
+ traceless part. From eq. (2.5) follows that for our purposes dW2|2,1= 0 so that the last
199
+ term in (2.8b) vanishes. Moreover, using (2.7) [ W2m·W2n]0can be expressed in terms of
200
+ W2(m·Jn).
201
+ Plugging the ansatz for the fluxes (2.2) into the equations of motion (B.7) and using
202
+ eqs. (2.1), (2.5), (A.5), (2.7), (B.5), (B.6) and (2.8b) we fi nd:
203
+ BianchiF2: 0 =3
204
+ 2W1f2−1
205
+ 4w2f3+f1f7,
206
+ eomF4: 0 = 3W1f4+1
207
+ 4w2f5−f6f7,
208
+ eomH: 0 = 6W1f7−3f1f2−12f4f2−6f4f6−f3f5,
209
+ 0 =w2f7+f1f3+f2f5−2f3f4−f5f6+pf3f5, (2.9)
210
+ dilaton eom : 0 = R4D+R6D−2f2
211
+ 7,
212
+ Einstein ext. : 0 = R4D+(f1)2+3(f2)2+12(f4)2+(f6)2+(f3)2+(f5)2,
213
+ Einstein int. : 0 = R6D−6(f7)2+1
214
+ 2/bracketleftbig
215
+ 3(f1)2+3(f2)2−12(f4)2−3(f6)2+(f3)2−(f5)2/bracketrightbig
216
+ ,
217
+ 0 = 4(f2f3+2f4f5)−w2W1−p/bracketleftbig
218
+ (f3)2−(f5)2−(w2)2/bracketrightbig
219
+ .
220
+ In the equation of motion for Hwe get separate conditions from the coefficients of J∧J
221
+ andˆW2∧Jrespectively. In the internal Einstein equation we find like wise a separate
222
+ condition from the trace and the coefficient of W2(m·Jn). In the next section we find
223
+ explicit solutions to these equations for the coset manifol ds with nearly-K¨ ahler limit, the
224
+ stability of which we investigate in section 4.
225
+ Flipping signs
226
+ The Einstein and dilaton equation are quadratic in the form fl uxes and thus insensitive to
227
+ flipping the signs of these fluxes. Taking into account also th e flux equations of motion
228
+ and Bianchi identities, we find that for each solution to the s upergravity equations, we
229
+ automatically obtain new ones by making the following sign fl ips:
230
+ H→ −H,ˆF0→ −ˆF0,ˆF2→ˆF2,ˆF4→ −ˆF4,ˆF6→ˆF6,
231
+ H→ −H,ˆF0→ˆF0,ˆF2→ −ˆF2,ˆF4→ˆF4,ˆF6→ −ˆF6,
232
+ H→H,ˆF0→ −ˆF0,ˆF2→ −ˆF2,ˆF4→ −ˆF4,ˆF6→ −ˆF6.(2.10)
233
+ In particular, these sign flips will transform a supersymmet ric solution into another super-
234
+ symmetric solution (as can be verified using the conditions ( 2.1),(2.3) allowing for suitable
235
+ – 5 –sign flips of J, ReΩ and ImΩ compatible with the metric). If some fluxes are ze ro, more
236
+ sign flips are possible. For instance for ˆF0=ˆF4= 0 we find the following extra sign-flip,
237
+ known as skew-whiffing in the M-theory compactification literature [52] (see also t he review
238
+ [34])
239
+ H→ ±H,ˆF2→ˆF2,ˆF6→ −ˆF6, (2.11)
240
+ which transforms a supersymmetric solution into a non-supersymmetric one. When dis-
241
+ cussing different solutions, we will from now on implicitly co nsider each solution together
242
+ with its signed-flipped counterparts.
243
+ 3. Solutions
244
+ Let us now solve the equations obtained in the previous secti on for the coset manifolds that
245
+ admit sourceless supersymmetric solutions, namelyG2
246
+ SU(3), SU(2)×SU(2),Sp(2)
247
+ S(U(2)×U(1))and
248
+ SU(3)
249
+ U(1)×U(1). For the supersymmetricsolutions on these manifolds we wil l use the conventions
250
+ and presentation of [13, 14]. For moredetails, includingin particular ourchoice of structure
251
+ constants for the relevant algebras, we refer to these paper s.
252
+ On a coset manifold G/Hone can define a coframe emthrough the decomposition of
253
+ the Lie-valued one-form L−1dL=emKm+ωaHain terms of the algebras of GandH. Here
254
+ Lis a coset representative, the Haspan the algebra of Hand theKmspan the complement
255
+ of this algebra within the algebra of G. The exterior derivative on the emis then given
256
+ in terms of the structure constants through the Maurer-Cart an relation. Furthermore,
257
+ the forms that are left-invariant under the action of Gare precisely those forms that are
258
+ constant in the basis spanned by emand for which the exterior derivative is also constant
259
+ in this basis. For these forms the exterior derivative can th en be expressed solely in terms
260
+ of the structure constants only involving the Km. We refer to [53, 54] for a review on coset
261
+ technology or to the above papers for a quick explanation.
262
+ G2
263
+ SU(3)and SU(2) ×SU(2)
264
+ We start from the supersymmetric nearly-K¨ ahler solution o nG2
265
+ SU(3). The SU(3)-structure
266
+ is given by
267
+ J=a(e12−e34+e56),
268
+ Ω =a3/2/bracketleftbig
269
+ (e245−e236−e146−e135)+i(e246+e235+e145−e136)/bracketrightbig
270
+ ,(3.1)
271
+ whereais the overall scale.
272
+ Since this SU(3)-structure corresponds to a nearly-K¨ ahle r geometry the torsion class
273
+ W2is zero. Furthermore we find
274
+ W1=−2√
275
+ 3a−1/2, w2=p= 0. (3.2)
276
+ – 6 –Plugging this into the equations (2.9) we find exactly three s olutions for ( f1,...,f7) (up
277
+ to the sign flips (2.10)):
278
+ a−1/2(√
279
+ 5
280
+ 2,1
281
+ 2√
282
+ 3,0,3
283
+ 4√
284
+ 5,0,−9
285
+ 2√
286
+ 3,1√
287
+ 5),
288
+ a−1/2(/radicalbigg
289
+ 5
290
+ 3,0,0,0,0,5√
291
+ 3,0),
292
+ a−1/2(1,1√
293
+ 3,0,−1
294
+ 2,0,√
295
+ 3,1).(3.3)
296
+ The first is the supersymmetric solution, while the last two a re non-supersymmetric solu-
297
+ tions, which were already found in [35, 29, 36]. Truncating t o the 4D effective theory it
298
+ was shown in [30] that a generalization of this family of solu tions is quite universal as it
299
+ appears in a large class of N= 2 gauged supergravities.
300
+ On the SU(2) ×SU(2) manifold, requiring the same geometry as the supersym metric
301
+ solution and not allowing for source terms will restrict us t o the nearly-K¨ ahler point. The
302
+ analysis is then basically the same as forG2
303
+ SU(3)above.
304
+ Sp(2)
305
+ S(U(2)×U(1))
306
+ The family of supersymmetric solutions on this manifold has , next to the overall scale,
307
+ an extra parameter determining the shape of the solutions. I t is then possible to turn on
308
+ the torsion class W2and venture away from the nearly-K¨ ahler geometry. This mak es this
309
+ class much richer and enables us this time to find new non-supe rsymmetric solutions. The
310
+ SU(3)-structure is given by [12, 13, 14]
311
+ J=a(e12+e34−σe56),
312
+ Ω =a3/2σ1/2/bracketleftbig
313
+ (e245−e236−e146−e135)+i(e246+e235+e145−e136)/bracketrightbig
314
+ ,(3.4)
315
+ whereais the overall scale and σis the shape parameter. We find for the torsion classes
316
+ and the parameter p:
317
+ W1= (aσ)−1/22+σ
318
+ 3,
319
+ (W2)2= (aσ)−18(1−σ)2
320
+ 3⇒w2= (aσ)−1/22√
321
+ 2(1−σ)√
322
+ 3,
323
+ ˆW2=−1√
324
+ 3/parenleftbig
325
+ e12+e34+2σe56/parenrightbig
326
+ ,
327
+ p=−/radicalbig
328
+ 2/3.(3.5)
329
+ We easily read off that σ= 1 corresponds to the nearly-K¨ ahler geometry. Note that ev en
330
+ thoughW2→0 forσ→1,ˆW2is well-defined and non-zero in this limit so that we can
331
+ still use it as an expansion form for the fluxes. The points σ= 2 andσ= 2/5 are also
332
+ special, since eq. (2.6) then implies that the supersymmetr ic solution has zero Romans
333
+ mass and, in particular, can be lifted to M-theory. Moreover , these are the endpoints of
334
+ the interval where supersymmetric solutions exist (since o utside this interval we would find
335
+ from eq. (2.6) that m2<0). They are indicated as vertical dashed lines in the plots.
336
+ – 7 –Figure 1:Sp(2)
337
+ S(U(2)×U(1))-model: plot of aR4Dfor the supersymmetric solutions (light green) and
338
+ the new non-supersymmetric solutions (other colors) in terms of t he shape parameter σ. Unstable
339
+ solutions are indicated in red.
340
+ Pluggingeqs.(3.5) intothesupergravityequationsofmoti on(2.9)wefindnumericallya
341
+ rich spectrumofsolutions, whicharedisplayed infigures1a nd2. Note that thedependence
342
+ on the overall scale can be easily extracted from all plotted quantities by multiplying by
343
+ ato a suitable power. We plotted the value of the 4D Ricci scala rR4Dof the AdS-space
344
+ against the shape parameter σin figure 1. Note that R4Dis inversely proportional to the
345
+ AdS-radius squared and related to the effective 4D cosmologic al constant and the vev of
346
+ the 4D scalar potential Vas follows
347
+ Λ =∝angb∇acketleftV∝angb∇acket∇ight=R4D/4. (3.6)
348
+ The supersymmetric solutions are plotted in light green, wh ile red is used for the non-
349
+ supersymmetric solutions found to be unstable in section 4. For completeness of the pre-
350
+ sentation of our numeric results, we provide the values of ea ch of the coefficients fiof the
351
+ ansatz (2.2) in figure 2.
352
+ The first point to note is that where the supersymmetric solut ions are restricted to
353
+ the interval σ∈[2/5,2], there exist non-supersymmetric solutions in the somewh at larger
354
+ intervalσ∈[0.39958,2.13327]. Furthermore, there are up to five non-supersymmetri c
355
+ solutions for each supersymmetric solution.
356
+ We remark that the parameters σand the overall scale are not continuous moduli since
357
+ they are determined by the vevs of the fluxes, which in a proper string theory treatment
358
+ shouldbequantized. Indeed, inthenextsection wewill show that generically all moduliare
359
+ stabilized. We leave the analysis of flux quantization, whic h is complicated by the fact that
360
+ – 8 –(a) Plot of a1/2f1(Romans mass)
361
+ (b) Plot of a1/2f2(J-part of ˆF2)
362
+ (c) Plot of a1/2f3(ˆW2-part of ˆF2)
363
+ (d) Plot of a1/2f4(J∧J-part of ˆF4)
364
+ (e) Plot of a1/2f5(J∧ˆW2-part of ˆF4)
365
+ (f) Plot of a1/2f6(Freund-Rubin parameter)
366
+ (g) Plot of a1/2f7(ReΩ part of H)
367
+ Figure 2: Plots of the solutions on the cosetSp(2)
368
+ S(U(2)×U(1)). Different colors indicate different
369
+ solutions. Unstable solutions are indicated in red (see section 4) and the supersymmetric solutions
370
+ in light green. By a suitable rescaling of the coefficients the dependen ce on the overall scale ais
371
+ taken out.
372
+ – 9 –there is non-trivial H-flux (twisting the RR-charges), to further work. The expect ation is
373
+ that the continuous line of supergravity solutions is repla ced by discrete solutions.
374
+ Let us now take a look at some special values of σ. Forσ= 1 we find five solutions
375
+ of which three (including the supersymmetric one) are up to s caling equivalent to the
376
+ solutions (3.3) onG2
377
+ SU(3)of the previous section [35, 29, 36, 30]. They have f3=f5= 0 and
378
+ so the fluxes are completely expressed in terms of J. However, there are also two new non-
379
+ supersymmetric solutions (the dark green and the purple one ) which have f3∝negationslash= 0,f5∝negationslash= 0.
380
+ Next we turn to the case σ= 2. This point is special in that the metric becomes
381
+ the Fubini-Study metric on CP3and the bosonic symmetry of the geometry enhances
382
+ from Sp(2) to SU(4). In fact, since the RR-forms of the supers ymmetric solution can be
383
+ expanded in terms of the closed K¨ ahler form ˜J= (1/3)J+(2a)1/2W2of the Fubini-Study
384
+ metric, the symmetry group of the whole supersymmetric solu tion is SU(4). One can also
385
+ show that the supersymmetry enhances from the generic N= 1 toN= 6 [6]. In [37] it
386
+ was found that there is an infinite continuous family of non-s upersymmetric solutions and
387
+ two discrete separate solutions (see also [35] for an incomp lete early discussion), which all
388
+ have SU(4)-symmetry. They are notdisplayed in the plot since they can not be found by
389
+ taking a continuous limit σ→2. For these solutions H= 0 (f7= 0) and ˆF2andˆF4are
390
+ expanded in terms of ˜J(for more details see [37]).
391
+ Instead, in the plot we find apart from the supersymmetric sol ution (which merges
392
+ with the dark green solution at σ= 2) two more discrete non-supersymmetric solutions,
393
+ which have only Sp(2)-symmetry (since the fluxes cannot be ex pressed in terms of ˜Jonly).
394
+ The blue one is new, while the red one turns out to be the reduct ion of the Englert-type
395
+ solution. Indeed for the Englert-type solution we expect
396
+ f1= 0, no Romans mass ,(3.7a)
397
+ f2=f2,susy, f3=f3,susy, same geometry in M ⇒sameˆF2as susy,(3.7b)
398
+ f7=−2f4=−(1/3)f6,susy, f5= 0, fromˆF4in M-theory ,(3.7c)
399
+ f6= (−2/3)f6,susy, Freund-Rubin parameter changes ,(3.7d)
400
+ R4D= (5/6)R4D,susy, 4D Λ changes ,(3.7e)
401
+ which agrees with the values displayed in the figures for the r ed curve at σ= 2.
402
+ Also forσ= 2/5 we find apart from the supersymmetric solution, the Englert solution
403
+ (the purplecurve) andoneextra non-supersymmetricsoluti on (the darkgreen curve). Note
404
+ that while the supersymmetric curve joins the olive green cu rve atσ= 2/5, the purple
405
+ curve only joins the dark green curve at σ= 0.39958.
406
+ SU(3)
407
+ U(1)×U(1)
408
+ For this manifold the SU(3)-structure is given by [13, 14]:
409
+ J=a(−e12+ρe34−σe56),
410
+ Ω =a3/2(ρσ)1/2/bracketleftbig
411
+ (e245+e135+e146−e236)+i(e235+e136+e246−e145)/bracketrightbig
412
+ ,(3.8)
413
+ – 10 –whereρandσare the shape parameters of the model. Furthermore we find for the torsion
414
+ classes:
415
+ W1=−(aρσ)−1/21+ρ+σ
416
+ 3,
417
+ W2=−(2/3)a1/2(ρσ)−1/2/bracketleftbig
418
+ (2−ρ−σ)e12+ρ(1−2ρ+σ)e34−σ(1+ρ−2σ)e56/bracketrightbig
419
+ .(3.9)
420
+ It turns out that the ansatz (2.7) is only satisfied for
421
+ ρ= 1, σ= 1 orρ=σ. (3.10)
422
+ In all three of these cases the equations (2.9) forSU(3)
423
+ U(1)×U(1)reduce to exactly the same
424
+ equations as forSp(2)
425
+ S(U(2)×U(1))so that we obtain the same solution space. However, as we
426
+ will see in the next section, the stability analysis will be d ifferent since the model on
427
+ SU(3)
428
+ U(1)×U(1)has two extra left-invariant modes.
429
+ In order to find further non-supersymmetric solutions, we sh ould go beyond the ansatz
430
+ (2.7). Let us put
431
+ ˆW2∧ˆW2= (−1/6)J∧J+p1ˆW2∧J+p2ˆP∧J, (3.11)
432
+ whereˆPis a primitive normalized (1,1)-form (so that it is orthogon al toJandˆP2= 1).
433
+ Furthermore, we also choose it orthogonal to ˆW2i.e.
434
+ ˆW2·ˆP= 0 or equivalently J∧ˆW2∧ˆP= 0. (3.12)
435
+ From the last equation one finds, using (2.1c), that d ˆP∧ImΩ = 0, which implies on
436
+ SU(3)
437
+ U(1)×U(1)that
438
+ dˆP= 0. (3.13)
439
+ One can now allow the RR-fluxes ˆF2andˆF4to have pieces proportional to ˆPandˆP∧
440
+ Jrespectively and adapt the equations (2.9) accordingly to a ccommodate for the new
441
+ contributions. Now it is possible to numerically find non-su persymmetric solutions for ρ
442
+ andσnot satisfying (3.10). In particular, there are Englert-ty pe solutions on the ellipse of
443
+ values for (ρ,σ) where the supersymmetric solution has zero Romans mass. Fr om eq. (2.6)
444
+ we find that this ellipse is described by
445
+ m2=5
446
+ 16ρσ/bracketleftbig
447
+ −5(ρ2+σ2)+6(ρ+σ+ρσ)−5/bracketrightbig
448
+ = 0. (3.14)
449
+ We will not go into more detail on these solutions in this pape r.
450
+ 4. Stability analysis
451
+ Inthissectionweinvestigate whetherthenewnon-supersym metricsolutionsonSp(2)
452
+ S(U(2)×U(1))
453
+ andSU(3)
454
+ U(1)×U(1)are stable5. To this end we calculate the spectrum of scalar fluctuations . We
455
+ 5In [36] it was found that the non-supersymmetric solutions o nG2
456
+ SU(3)and the similar solutions on the
457
+ nearly-K¨ ahler limits of the other two coset manifolds unde r study are stable. We find exactly the same
458
+ spectrum as the authors of that paper, which provides a consi stency check on our approach. We thank
459
+ Davide Cassani for providing us with these numbers, which ar e not explicitly given in their paper. We did
460
+ not investigate the spectrum of the similar solution on SU(2 )×SU(2), which is more complicated as there
461
+ are more modes.
462
+ – 11 –use the well-known result of [39, 40] that in an AdS 4vacuum a tachyonic mode does not yet
463
+ signal an instability. Only a mode with a mass-squared below the Breitenlohner-Freedman
464
+ bound,
465
+ M2<−3|Λ|
466
+ 4, (4.1)
467
+ where Λ<0 is the 4D effective cosmological constant, leads to an instab ility. We restrict
468
+ ourselves to left-invariant fluctuations, which implies th at even if we do not find any modes
469
+ below the Breitenlohner-Freedman bound, the vacuum might s till be unstable, since there
470
+ might be fluctuations with sufficiently negative mass-square d that are not left-invariant.
471
+ This analysis can however pinpoint many unstable vacua and w e do believe it gives a
472
+ valuable first indication for the stability of the others.
473
+ Truncatingtotheleft-invariant modesonthecoset manifol dsunderstudyleads toa4D
474
+ N= 2 gauged supergravity6. It has been shown in [36] that this truncation is consistent .
475
+ The spectrum of the scalar fields can then be obtained from the 4D scalar potential. In
476
+ fact, this computation is analogous to the one performed in [ 14] for the supersymmetric
477
+ N= 1 vacua on the coset spaces. As opposed to the models here, th e models in that
478
+ paper included orientifolds, which broke the supersymmetr y of the 4D effective theory
479
+ fromN= 2 toN= 1. However, also in the present case the N= 1 approach is applicable
480
+ and effectively we have used exactly the same procedure, i.e. u sing theN= 1 scalar
481
+ fluctuations and obtaining the scalar potential from the N= 1 superpotential and K¨ ahler
482
+ potential (see [55, 56, 57, 58]).7The reason is the following. The N= 2 scalar fluctuations
483
+ in the vector multiplets are
484
+ Jc=J−iB= (ki−ibi)ωi=tiωi, (4.2)
485
+ whereωispan the left-invariant two-forms of the coset manifold. Th e orientifold projection
486
+ of theN= 1 theory would then project out the scalar fluctuations comi ng from expanding
487
+ oneventwo-forms, which are absent for the N= 1 theory on the coset manifolds under
488
+ study. The scalar fluctuations in the N= 2 vector multiplets are thus exactly the same as
489
+ the scalars in the chiral multiplets of the K¨ ahler moduli se ctor of the N= 1 theory. The
490
+ 6It is important to make the distinction between the number of supersymmetries of respectively the
491
+ 4D effective theory, the 10D compactifications, and their 4D t runcation (which are the solutions of the
492
+ 4D effective theory [36]). In the presence of one left-invari ant internal spinor, the effective theory will be
493
+ N= 2 since this same spinor can be used in the 4+ 6 decomposition of both ten-dimensional Majorana-
494
+ Weyl supersymmetry generators, but multiplied with indepe ndent four-dimensional spinors. On the other
495
+ hand, for a certain compactification to preserve the supersy mmetry, certain differential conditions, which
496
+ follow from putting the variations of the fermions to zero mu st be satisfied. In the presence of RR-fluxes,
497
+ these conditions mix both ten-dimensional Majorana-Weyl s pinors, putting the four-dimensional spinors in
498
+ both decompositions equal. A generic supersymmetric compa ctification therefore only preserves N= 1.
499
+ Theσ= 2 supersymmetric K¨ ahler-Einstein solution on CP3on the other hand is non-generic in that it
500
+ preserves N= 6, of which only one internal spinor is left-invariant unde r the action of Sp(2) and remains
501
+ after truncation to 4D.
502
+ 7It is interesting to note that (in N= 1 language) all the D-terms vanish, so that the supersymmet ry
503
+ breaking is purely due to F-terms. Indeed, in [58] it is shown thatD= 0 is equivalent to d H(e2A−ΦReΨ1) =
504
+ 0 in the generalized geometry formalism. For SU(3)-structu re this translates to d( e2A−ΦReΩ) = 0 and
505
+ H∧ReΩ = 0, which is satisfied for our ansatz, eq. (2.1) and (2.2).
506
+ – 12 –(a) Spectrum ofSp(2)
507
+ S(U(2)×U(1))
508
+ (b) Two extra modes of theSU(3)
509
+ U(1)×U(1)-model
510
+ Figure 3: Spectrum of left-invariant modes of the solutions onSp(2)
511
+ S(U(2)×U(1))andSU(3)
512
+ U(1)×U(1).
513
+ expansion forms can then be chosen to bethe same as the Y(2−)
514
+ iof [14]. Furthermore, there
515
+ is one tensor multiplet, which contains the dilaton Φ, the tw o-formBµνand two axions ξ
516
+ and˜ξcoming from the expansion of the RR-potential C3:
517
+ C3=ξα+˜ξβ, (4.3)
518
+ where a choice for αandβspanning the left-invariant three-forms would be Y(3−)and
519
+ Y(3+)of [14] respectively. In the presence of Romans mass or ˆF2-flux the two-form Bµν
520
+ becomes massive and cannot be dualized to a scalar. The dilat on and˜ξappear in a chiral
521
+ multiplet of the complex moduli sector of the N= 1 theory, while Bµνandξare projected
522
+ out by the orientifold. By using the N= 1 approach we thus loose the information on just
523
+ one scalarξ. A proper N= 2 analysis would however learn that ξdoes not appear in the
524
+ scalarpotential (seee.g.[36]), implyingthatitismassle ssandthusabovetheBreitenlohner-
525
+ Freedman bound. Moreover, the scalar potential should be th e same whether it is obtained
526
+ directly from reducing the 10D supergravity action (as in [5 9]) or whether it is obtained
527
+ usingN= 2 orN= 1 technology8. Furthermore we note that the massless scalar field ξ
528
+ not appearing in the potential is not a modulus, since it is ch arged [60, 61], and therefore
529
+ eaten by a vector field becoming massive.
530
+ Thespectraof left-invariant modesforSp(2)
531
+ S(U(2)×U(1))andSU(3)
532
+ U(1)×U(1)aredisplayed infigure
533
+ 3. The Breitenlohner-Freedman bound is indicated as a horiz ontal dashed line. The Sp(2)-
534
+ model has six scalar fluctuations entering the potential: ki,biwithi= 1,2 from the two
535
+ vector multiplets, and Φ ,˜ξfrom the universal hypermultiplet, while the SU(3)-model h as
536
+ two more fluctuations from the extra vector multiplet. These two extra modes make a big
537
+ difference for the stability analysis since one of them tends t o be below the Breitenlohner-
538
+ Freedman bound for the purple and dark green solution. As a re sult, even though the
539
+ solutions for the Sp(2)- and SU(3)-model take the same form, the SU(3)-model has more
540
+ unstable solutions: compare figure 1 and 4.
541
+ 8The only potential difference between the latter two would be the contribution from the orientifold.
542
+ We have checked that this contribution vanishes in the scala r potentials of [14] in the limit of the orientifold
543
+ chargeµ→0.
544
+ – 13 –Figure 4:SU(3)
545
+ U(1)×U(1)-model: plot of aR4Din terms of the shape parameter σ. Unstable solutions
546
+ are indicated in red.
547
+ Inparticular, wenotethatthereductionoftheEnglert-typ esolutionisunstablefor σ=
548
+ 2 in the Sp(2)-model, in agreement with [38], since the M-the ory lift of the corresponding
549
+ supersymmetric solution has eight Killing spinors. We inde ed find the same negative mass-
550
+ squaredM2=−(4/5)|Λ|for the unstable mode as in that paper. On the other hand,
551
+ forσ= 2/5 the Englert-type solution is stable against left-invaria nt fluctuations. This is
552
+ still in agreement with [38] which relied on the existence of at least two Killing spinors,
553
+ while the M-theory lift of the N= 1 supersymmetric solution at σ= 2/5 has only one
554
+ Killing-spinor. For the SU(3)-model, all Englert-type sol utions turn out to be unstable
555
+ (including the ones outside the condition (3.10)).
556
+ We also investigated the stability of the additional soluti ons at the special point σ= 2
557
+ found in [37]. We found that for the Sp(2)-model all these sol utions are stable against left-
558
+ invariant fluctuations. For the SU(3)-model on theother han dit turnsout that thediscrete
559
+ solutions ineqs.(3.16) and(3.17) ofthatreferenceareuns table, whilethecontinuous family
560
+ of eq. (3.18) becomes unstable for
561
+ γ2
562
+ β2>5(75∓16√
563
+ 21)
564
+ 8217, (4.4)
565
+ for the±sign choice in front of the square root in eq. (3.18) of that pa per respectively
566
+ (note that the supersymmetric solution corresponds to the p ointγ2/β2= 0 in this family).
567
+ Finally, we note that generically (i.e. unless an eigenvalu e is crossing zero at a special
568
+ value forσ) all the plotted modes are massive. For a range of values for σone of the
569
+ eigenvalues for the dark green and purple solution takes a sm all, but still non-zero value.
570
+ – 14 –5. Conclusions
571
+ In this paper we presented new families of non-supersymmetr ic AdS 4vacua. In fact,
572
+ extrapolating from our analysis on these specific coset mani folds and under the assumption
573
+ that a proper treatment of flux quantization does not kill muc h more vacua than in the
574
+ supersymmetric case, it would seem that there are more of the se non-supersymmetric
575
+ vacua than supersymmetric ones. This would imply that such v acua cannot be ignored
576
+ in landscape studies. We have moreover shown that many of the m are stable against a
577
+ specific set of fluctuations, namely the ones that can be expan ded in terms of left-invariant
578
+ forms. If these vacua turn out to be stable against all fluctua tions they should also have
579
+ a CFT-dual, which could be studied along the lines of [20], wh ere the three-dimensional
580
+ Chern-Simons-matter theory dual to a particular highly sym metric non-supersymmetric
581
+ vacuum was proposed. Furthermore, the nice property of some IIA vacua that all moduli
582
+ enter the superpotential and thus can be stabilized at a clas sical level [15] also extends to
583
+ our non-supersymmetric vacua.
584
+ A next step would be to relax the constraint that the solution s should have the same
585
+ geometry as the supersymmetric solution. It is also interes ting to investigate whether a
586
+ similar ansatz and techniques can be used to look for tree-le vel dS-vacua [62].
587
+ Acknowledgments
588
+ We thank Davide Cassani for useful email correspondence and proofreading, and further-
589
+ more Claudio Caviezel for active discussions and initial co llaboration. We would further
590
+ like to thank the Max-Planck-Institut f¨ ur Physik in Munich , where both of the authors
591
+ were affiliated during the bulk of the work on this paper. P.K. i s a Postdoctoral Fellow
592
+ of the FWO – Vlaanderen. The work of P.K. is further supported in part by the FWO –
593
+ Vlaanderen project G.0235.05 and in part by the Federal Office for Scientific, Technical and
594
+ Cultural Affairs through the ’Interuniversity Attraction Po les Programme Belgian Science
595
+ Policy’ P6/11-P. S.K. is supported by the SFB – Transregio 33 “The Dark Universe” by
596
+ the DFG.
597
+ A. SU(3)-structure
598
+ A real non-degenerate two-form Jand a complex decomposable three-form Ω define an
599
+ SU(3)-structure on the 6D manifold M6iff:
600
+ Ω∧J= 0, (A.1a)
601
+ Ω∧¯Ω =8i
602
+ 3!J∧J∧J∝negationslash= 0, (A.1b)
603
+ and the associated metric is positive-definite. This metric is determined by Jand Ω as
604
+ follows:
605
+ gmn=−JmpIpn, (A.2)
606
+ withIthe complex structure associated (in the way of [63]) to Ω. Th e volume-form is
607
+ given by vol 6=1
608
+ 3!J3=−(i/8)Ω∧¯Ω.
609
+ – 15 –Theintrinsictorsionofthemanifold M6decomposesintofivetorsionclasses W1,...,W5.
610
+ Alternatively they correspond to the SU(3)-decomposition of the exterior derivatives of J
611
+ and Ω [64]. Intuitively, they parameterize the failure of th e manifold to be of special
612
+ holonomy, which can also be thought of as the deviation from c losure ofJand Ω. More
613
+ specifically we have:
614
+ dJ=3
615
+ 2Im(W1¯Ω)+W4∧J+W3,
616
+ dΩ =W1J∧J+W2∧J+¯W5∧Ω,(A.3)
617
+ whereW1is a scalar, W2is a primitive (1,1)-form, W3is a real primitive (1 ,2)+(2,1)-form,
618
+ W4is a real one-form and W5a complex (1,0)-form. In this paper only the torsion classes
619
+ W1,W2are non-vanishing and they are purely imaginary, so it will b e convenient to define
620
+ W1,2so thatW1,2=iW1,2. A primitive (1,1)-form P(such asW2) transforms under the 8
621
+ of SU(3) and satisfies
622
+ P∧J∧J= 0. (A.4)
623
+ The Hodge dual is given by
624
+ ⋆6P=−P∧J. (A.5)
625
+ A primitive (1 ,2)(or (2,1))-formQon the other hand transforms as a 6(or¯6) under SU(3)
626
+ and satisfies
627
+ Q∧J= 0. (A.6)
628
+ B. Type II supergravity
629
+ The bosonic content of type II supergravity consists of a met ricG, a dilaton Φ, an NSNS
630
+ three-form Hand RR-fields Fn. We use the democratic formalism of [65], in which the
631
+ number of RR-fields is doubled, so that nruns over 0 ,2,4,6,8,10 in type IIA and over
632
+ 1,3,5,7,9 in IIB. We will often collectively denote the RR-fields with the polyform F=/summationtext
633
+ nFn. We have also doubled the RR-potentials, collectively deno ted byC=/summationtext
634
+ nC(n−1).
635
+ These potentials satisfy F= dHC+me−B= (d +H∧)C+me−B. In type IIB there is
636
+ of course no Romans mass m, so that the second term vanishes. In type IIA we find in
637
+ particularF0=m.
638
+ The bosonic part of the pseudo-action of the democratic form alism then simply reads
639
+ S=1
640
+ 2κ2
641
+ 10/integraldisplay
642
+ d10X√
643
+ −G/braceleftbigg
644
+ e−2Φ/bracketleftbigg
645
+ R+4(dΦ)2−1
646
+ 2H2/bracketrightbigg
647
+ −1
648
+ 4F2/bracerightbigg
649
+ , (B.1)
650
+ where we defined F2=/summationtext
651
+ nF2
652
+ nand the square of an l-formPas follows
653
+ P2=P·P=1
654
+ l!Pm1...mlPm1...ml, (B.2a)
655
+ where the indices are raised with the inverse of the metric Gmnor the internal metric gmn
656
+ (defined later on), depending on the context. In the followin g it will also be convenient to
657
+ define:
658
+ Pm·Pn=ιmP·ιnP=1
659
+ (l−1)!Pmm2...mlPnm2...ml. (B.2b)
660
+ – 16 –The extra degrees of freedom for the RR-fields in the democrat ic formalism have to be
661
+ removed by hand by imposing the following duality condition at the level of the equations
662
+ of motion after deriving them from the action (B.1):
663
+ Fn= (−1)(n−1)(n−2)
664
+ 2⋆10F10−n. (B.3)
665
+ That is why (B.1) is only a pseudo-action.
666
+ The fermionic content consists of a doublet of gravitinos ψMand a doublet of dilatinos
667
+ λ. The components of the doublets are of different chirality in t ype IIA and of the same
668
+ chirality in type IIB.
669
+ In this paper we look for vacuum solutions that take the form A dS4×M6. In principle
670
+ there could also be a warp factor A, but it will always be constant for the solutions in this
671
+ paper. We can choose it to be zero. The compactification ansat z for the metric then reads
672
+ ds2
673
+ 10=GmndXmdXn= ds2
674
+ 4+gmndxmdxn, (B.4)
675
+ where ds2
676
+ 4is the line-element for AdS 4andgmnis the metric on the internal space M6. For
677
+ the RR-fluxes the ansatz becomes
678
+ F=ˆF+vol4∧˜F, (B.5)
679
+ whereˆFand˜Fonly have internal indices. The duality constraint (B.3) im plies that ˜Fis
680
+ not independent of ˆF, and given by
681
+ ˜Fn= (−1)(n−1)(n−2)
682
+ 2⋆6ˆF6−n. (B.6)
683
+ What we need in this paper are the type II equations of motion, which can be found
684
+ from the pseudo-action (B.1). We use them as they are written down in [5] (originally they
685
+ were obtained for massive type IIA in [35]), but take some lin ear combinations in order
686
+ to further simplify then. Without source terms (i.e. we put jtotal= 0 in the equations of
687
+ motion of [5]), they then read:
688
+ dHF= 0 (Bianchi RR fields) , (B.7a)
689
+ d−H⋆10F= 0 (eom RR fields) , (B.7b)
690
+ dH= 0 (BianchiH), (B.7c)
691
+ d/parenleftbig
692
+ e−2Φ⋆10H/parenrightbig
693
+ −1
694
+ 2/summationdisplay
695
+ n⋆10Fn∧Fn−2= 0 (eom H), (B.7d)
696
+ 2R−H2+8/parenleftbig
697
+ ∇2Φ−(∂Φ)2/parenrightbig
698
+ = 0 (dilaton eom) , (B.7e)
699
+ 2(∂Φ)2−∇2Φ−1
700
+ 2H2−e2Φ
701
+ 8/summationdisplay
702
+ nnF2
703
+ n= 0 (trace Einstein/dilaton eom) ,(B.7f)
704
+ RMN+2∇M∂NΦ−1
705
+ 2HM·HN−e2Φ
706
+ 4/summationdisplay
707
+ nFnM·FnN= 0 (B.7g)
708
+ (Einstein eq./dilaton/trace) .
709
+ – 17 –References
710
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+ – 21 –
1001.0004.txt ADDED
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1001.0005.txt ADDED
@@ -0,0 +1,441 @@
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
1
+ arXiv:1001.0005v1 [astro-ph.CO] 30 Dec 2009Astronomy& Astrophysics manuscriptno.akari˙RXJ1716˙v5 c∝circlecopyrtESO 2018
2
+ October30,2018
3
+ Environmentaldependenceof 8 µmluminosityfunctionsof
4
+ galaxiesatz ∼0.8
5
+ Comparison between RXJ1716.4 +6708 andthe AKARI NEP deep field.⋆,⋆⋆
6
+ Tomotsugu Goto1,2,⋆⋆⋆, Yusei Koyama3,T.Wada4,C.Pearson5,6,7,H.Matsuhara4,T.Takagi4, H.Shim8, M.Im8,
7
+ M.G.Lee8, H.Inami4,9,10,M.Malkan11, S.Okamura3,T.T.Takeuchi12, S.Serjeant7, T.Kodama2, T.Nakagawa4,
8
+ S.Oyabu4,Y.Ohyama13, H.M.Lee8, N.Hwang2, H.Hanami14, K.Imai15,and T.Ishigaki16
9
+ 1Institute for Astronomy, University of Hawaii,2680 Woodla wnDrive, Honolulu, HI,96822, USA
10
11
+ 2National Astronomical Observatory, 2-21-1 Osawa,Mitaka, Tokyo, 181-8588,Japan
12
+ 3Department of Astronomy, School of Science,The University of Tokyo, Tokyo113-0033, Japan
13
+ 4Institute of Space and Astronautical Science, JapanAerosp ace Exploration Agency, Sagamihara,Kanagawa 229-8510
14
+ 5Rutherford Appleton Laboratory, Chilton, Didcot,Oxfords hire OX110QX, UK
15
+ 6Department of Physics,Universityof Lethbridge, 4401 Univ ersity Drive,Lethbridge, AlbertaT1J 1B1, Canada
16
+ 7Astrophysics Group, Department of Physics, The OpenUniver sity, MiltonKeynes, MK76AA, UK
17
+ 8Department of Physics& Astronomy, FPRD,Seoul National Uni versity, Shillim-Dong,Kwanak-Gu, Seoul 151-742, Korea
18
+ 9Spitzer Science Center,California Institute ofTechnolog y, Pasadena, CA91125
19
+ 10Department of Astronomical Science,The Graduate Universi tyfor Advanced Studies
20
+ 11Department of Physicsand Astronomy, UCLA,Los Angeles, CA, 90095-1547 USA
21
+ 12Institute for Advanced Research, Nagoya University, Furo- cho, Chikusa-ku, Nagoya 464-8601
22
+ 13Academia Sinica,Institute of Astronomyand Astrophysics, Taiwan
23
+ 14Physics Section,Facultyof Humanities and SocialSciences , Iwate University, Morioka, 020-8550
24
+ 15TOMER&D Inc. Kawasaki, Kanagawa 2130012, Japan
25
+ 16Asahikawa National College of Technology, 2-1-6 2-joShunk ohdai, Asahikawa-shi, Hokkaido 071-8142
26
+ Received September 15, 2009; accepted December 16, 2009
27
+ ABSTRACT
28
+ Aims.Weaim to reveal environmental dependence of infraredlumin osity functions (IR LFs)of galaxies at z ∼0.8 using the AKARI
29
+ satellite. AKARI’s wide field of view and unique mid-IR filter s help us to construct restframe 8 µm LFs directly without relying on
30
+ SEDmodels.
31
+ Methods. We construct restframe 8 µm IR LFs in the cluster region RXJ1716.4 +6708 at z=0.81, and compare them with a blank
32
+ field using the AKARI North Ecliptic Pole deep field data at the same redshift. AKARI’s wide field of view (10’ ×10’) is suitable to
33
+ investigate wide range of galaxy environments. AKARI’s 15 µm filter is advantageous here since it directly probes restfr ame 8µm at
34
+ z∼0.8, without relyingona large extrapolation based ona SEDfi t,which was the largestuncertainty inprevious work.
35
+ Results. We have found that cluster IR LFsat restframe 8 µm have a factor of 2.4smaller L∗and a steeper faint-end slope than that
36
+ of the field. Confirming this trend, we also found that faint-e nd slopes of the cluster LFs becomes flatter and flatter with de creasing
37
+ local galaxy density. These changes in LFs cannot be explain ed by a simple infall of field galaxy population into a cluster . Physics
38
+ that canpreferentiallysuppress IR luminous galaxies inhi gh density regions is requiredtoexplain the observed resul ts.
39
+ Keywords. galaxies: evolution, galaxies:interactions, galaxies:s tarburst, galaxies:peculiar, galaxies:formation
40
+ 1. Introduction
41
+ It hasbeenobservedthat galaxypropertieschangeas a funct ion
42
+ of galaxyenvironment;the morphology-densityrelation re ports
43
+ that fractionof elliptical galaxiesis largerat highergal axyden-
44
+ sity(Gotoetal.,2003);thestarformationrate(SFR)ishig herin
45
+ lower galaxy density (G´ omezet al., 2003; Tanakaet al., 200 4)
46
+ . However, despite accumulating observational evidence, w e
47
+ ⋆This research is based on the observations with AKARI, a JAXA
48
+ project withthe participationof ESA.
49
+ ⋆⋆Based on data collected at Subaru Telescope, which is operat ed by
50
+ the National Astronomical Observatory ofJapan.
51
+ ⋆⋆⋆JSPSSPDfellowstill do not fully understand the underlying physics govern ing
52
+ environmental-dependentevolutionofgalaxies.
53
+ Infrared (IR) emission of galaxies is an important
54
+ probe of galaxy activity since at higher redshift, a sig-
55
+ nificant fraction of star formation is obscured by dust
56
+ (Takeuchi,Buat,&Burgarella, 2005; Gotoetal., 2010).
57
+ Although there exist low-z cluster studies (Baiet al., 2006 ;
58
+ Shimet al., 2010; Tranetal., 2010), not much attention has
59
+ been paid to the infrared properties of high redshift cluste r
60
+ galaxies, mainly due to the lack of sensitivity in previous I R
61
+ satellites such as ISO and IRAS. Superb sensitivity of recen tly
62
+ launched Spitzer and AKARI satellites can revolutionize th e
63
+ infraredviewofenvironmentaldependenceofgalaxyevolut ion.2 Gotoet al.:Environemental dependence of 8 µm luminosity functions ofgalaxies atz ∼0.8
64
+ Fig.1.Restframe 8 µm LFs of cluster RXJ1716.4 +6708 at
65
+ z=0.81 in the squares, and those of the AKARI NEP deep
66
+ field in the triangles. For RXJ1716.4 +6708, only photometric
67
+ and spectroscopic cluster member galaxies are used. For the
68
+ NEP deep field, galaxies with photo-z/specz in the range of
69
+ 0.65< z <0.9are used. The dot-dashed lines are 8 µm LFs
70
+ of RXJ1716.4 +6708, but scaled down for easier comparison.
71
+ Thethindottedlinesarethebest-fitdoublepowerlaws.Vert ical
72
+ arrows show the 5 σflux limits of deep/shallow regions of the
73
+ cluster (red) and the NEP deep field (blue) in terms of L8µmat
74
+ z=0.81.
75
+ In this work, we comparerestframe8 µm LFs between clus-
76
+ ter and field regions at z=0.8 using data from the AKARI.
77
+ Monochromaticrestframe 8 µm luminosity ( L8µm) is important
78
+ since it is known to correlate well with the total IR luminosi ty
79
+ (Babbedgeet al., 2006; Huanget al., 2007), andhence,with t he
80
+ SFR of galaxies (Kennicutt, 1998). This is especially true f or
81
+ star-forminggalaxiesbecausethe rest-frame8 µm fluxare dom-
82
+ inatedbyprominentPAHfeaturessuchasat6.2,7.7and8.6 µm
83
+ (Desert,Boulanger,&Puget, 1990).
84
+ ImportantadvantagesbroughtbytheAKARIareasfollows:
85
+ (i) At z=0.8, AKARI’s 15 µm filter (L15) covers the redshifted
86
+ restframe 8 µm, thus we can estimate 8 µm LFs without using
87
+ a large extrapolation based on SED models, which were the
88
+ largest uncertainty in previous work. (ii) Large field of vie w of
89
+ the AKARI’smid-IRcamera(IRC, 10’ ×10’)allowsustostudy
90
+ wider area including cluster outskirts, where important ev olu-
91
+ tionary mechanisms are suggested to be at work (Gotoet al.,
92
+ 2004; Kodamaet al., 2004). For example, passive spiral gala x-
93
+ ies have been observed in such an environment (Gotoet al.,
94
+ 2003). Unless otherwise stated, we adopt a cosmology with
95
+ (h,Ωm,ΩΛ) = (0.7,0.3,0.7)(Komatsuet al., 2009).
96
+ 2. Data & Analysis
97
+ 2.1. LFs ofclusterRXJ1716.4 +6708
98
+ The AKARI is a Japanese infrared satellite (Murakamiet al.,
99
+ 2007), which has continuous filter coverage in the mid
100
+ IR wavelengths ( N2,N3,N4,S7,S9W,S11,L15,L18Wand
101
+ L24). The AKARI has observed a massive galaxy cluster,Fig.2.Restframe 8 µm LFs of cluster RXJ1716.4 +6708 at
102
+ z=0.81, divided according to the local galaxy density ( Σ5th).
103
+ Thestars,circlesandsquaresareforgalaxieswith logΣ5th≥2,
104
+ 1.6≤logΣ5th<2,andlogΣ5th<1.6,respectively.
105
+ RXJ1716.4 +6708, in N3,S7andL15(Koyamaetal., 2008).
106
+ RXJ1716.4 +6708 is at z=0.81 and has σ= 1522+215
107
+ −150km s−1,
108
+ LXbol= 13.86±1.04×1044ergs−1,kT= 6.8+1.0
109
+ −0.6keV.Mass
110
+ estimate from weak lensing and X-ray are 3.7 ±1.3×1014M⊙
111
+ and 4.35 ±0.83×1014M⊙, respectively (see Koyamaet al.,
112
+ 2007, forreferences).
113
+ An important advantage of the AKARI observation is L15
114
+ filter, which corresponds to the restframe 8 µm at z=0.81. With
115
+ 15 (3) pointings, L15reaches 66.5 (96.5) µJy in deep (shal-
116
+ low) regions at 5 σ. Here flux is measured in 11” aperture,
117
+ and coverted to total flux using AKARI’s IRC correction table
118
+ (2009.5.1)1.ClusterstudieswiththeSpitzerareoftenperformed
119
+ in 24µm and thus needed a large extrapolation to estimate ei-
120
+ therL8µmor total infrared luminosity ( LTIR,8−1000µm).
121
+ Note that we do not claim the L8µmis a better indicator of
122
+ thetotalIRluminositythanotherindicators(Brandlet al. ,2006;
123
+ Calzetti et al., 2007; Riekeet al., 2009), but it is importan t that
124
+ theAKARIcanmeausureredshifted 8µmfluxdirectlyinoneof
125
+ thefilters.
126
+ Thanks to the AKARI’s wide field of view (10’ ×10’), the
127
+ total area coverage around the cluster is 200 arcmin2, which
128
+ cover larger area than previous cluster studies with the Spi tzer,
129
+ allowingustostudyIRsourcesintheoutskirts,whereimpor tant
130
+ galaxyevolutiontakesplace(e.g.,Gotoet al.,2003).Prev iously,
131
+ Koyamaet al. (2008) reporteda high fractionof L15sourcesin
132
+ the intermediatedensity regionin the cluster,suggesting a pres-
133
+ enceofenvironmentaleffectintheintermediatedensityen viron-
134
+ ment.
135
+ Thissameregionwasimagedwith Suprime-Camin VRi′z′
136
+ and has a good photometric redshift estimate (Koyamaet al.,
137
+ 2007).Usedinthisworkare54 L15-detectedgalaxieswhichare
138
+ well identifiedwithopticalsourceswith 0.76≤zphoto≤0.83.
139
+ With the L15filter covering the restframe 8 µm, we simply
140
+ convert the observed flux to 8 µm monochromatic luminosity
141
+ 1http://www.ir.isas.jaxa.jp/ASTRO-
142
+ F/Observation/DataReduction/IRC/ApertureCorrection 090501.htmlGotoet al.:Environemental dependence of 8 µm luminosity functions ofgalaxies atz ∼0.8 3
143
+ Table 1.Best doublepower-lawfit parametersforLFs
144
+ Sample L∗
145
+ 8µm(L⊙)φ∗(Mpc−3dex−1)α β
146
+ NEPDeepfield 6.1 ±0.5×10100.0010±0.0003 1.1 ±0.3 5.7 ±1.2
147
+ RXJ1716.4 +67082.5±0.1×10100.74±0.04 2.6 ±0.1 5.5 ±0.4
148
+ (L8µm) using a standard cosmology. Completeness was mea-
149
+ sured by distributing artificial point sources with varying flux
150
+ withinthe field andby examiningwhat fractionofthem wasre-
151
+ coveredasafunctionofinputflux.Sincewehavedeepercover -
152
+ age at the center of the cluster, the completeness was measur ed
153
+ separately in the central deep region and the outer regions o f
154
+ the field. More detail of the method is described in Wada et al.
155
+ (2008).
156
+ Oncethefluxisconvertedtoluminosityandcompletenessis
157
+ takenintoaccount,it is straightforwardto construct L8µmLFs,
158
+ which we show in the squares in Fig.1. Errors of the LFs are
159
+ assumedtofollowPoissondistribution.Here,wetakeanang ular
160
+ distance of the most distant source from the cluster center a s
161
+ a cluster radius ( Rmax= 6.2Mpc). We assumed4
162
+ 3πR3
163
+ maxas
164
+ the volume of the cluster to obtain galaxy density ( φ). This is
165
+ only one of many ways to define a cluster volume, and thus, a
166
+ cautionmustbetakentocompare absolute valuesofourLFsto
167
+ other work such as Shimet al. (2010). This cluster is elongat ed
168
+ inangulardirection(Koyamaet al.,2007),andthus,thevol ume
169
+ mightnotbespherical.Yet,comparisonofthe shapeoftheLFs
170
+ isvalid.
171
+ 2.2. LFs inthe AKARI NEP Deepfield
172
+ Our field LFs are based on the AKARI NEP Deep field
173
+ data. The AKARI performed deep imaging in the North
174
+ Ecliptic Pole Region (NEP) from 2-24 µm, with 4 pointings
175
+ in each field over 0.4 deg2(Matsuharaet al., 2006, 2007;
176
+ Wada etal., 2008). The 5 σsensitivity in the AKARI IR filters
177
+ (N2,N3,N4,S7,S9W,S11,L15,L18WandL24) are 14.2,
178
+ 11.0, 8.0, 48, 58, 71, 117, 121 and 275 µJy (Wada etal., 2008).
179
+ Flux is measured in 3 pix radius aperture (=7”), then correct ed
180
+ tototal flux.
181
+ AsubregionoftheNEP-Deepfield(0.25deg2)hasancillary
182
+ datafromSubaru BVRi′z′(Imaiet al.,2007;Wada etal.,2008),
183
+ CFHTu′(Serjeant et al. in prep.), KPNO2m/FLAMINGOs J
184
+ andKs(Imaietal., 2007), GALEX FUVandNUV(Malkan
185
+ et al. in prep.). For the optical identification of MIR source s,
186
+ we adopt the likelihood ratio method (Sutherland&Saunders ,
187
+ 1992).Usingthesedata,weestimatephotometricredshifto fL15
188
+ detectedsourcesintheregionwiththe LePhare (Ilbertet al.,
189
+ 2006; Arnoutset al., 2007).Themeasurederrorsonthephoto -z
190
+ against 293 spec-z galaxies from Keck/DEIMOS (Takagi et al.
191
+ in prep.) are∆z
192
+ 1+z=0.036 at z≤0.8. We have excluded those
193
+ sourcesbetterfit with QSO templatesfromtheLFs.
194
+ To construct field LFs, we have selected L15sources at
195
+ 0.65< zphotoz<0.9. There remained 289 IR galaxies with
196
+ a median redshift of 0.76. L15flux is converted to L8µmus-
197
+ ing the photometric redshift of each galaxy. LFs are com-
198
+ puted using the 1/ Vmaxmethod. We used the SED templates
199
+ (Lagache,Dole,&Puget, 2003) for k-corrections to obtain the
200
+ maximumobservableredshiftfromthefluxlimit.Completene ss
201
+ of theL15detection is corrected using Pearsonet al. (2009b).
202
+ Thiscorrectionis25%atmaximum,sincewe onlyusethesam-
203
+ plewherethecompletenessisgreaterthan80%.
204
+ The resulting field LFs are shown in the dotted line and tri-
205
+ angles in Fig.1. Errors of the LFs are computed using a 1000Monte Caro simulation with varying zand flux within their er-
206
+ rors. These estimated errors are added to the Poisson errors in
207
+ eachLFbinin quadrature.
208
+ We performed a detaild comparison of restframe 8 µm
209
+ LFs to those in the literature in Gotoetal. (2010). Briefly,
210
+ there is an oder of difference between Caputiet al. (2007) an d
211
+ Babbedgeetal. (2006), reflecting difficulty in estimating L8µm
212
+ dominatedbyPAHemissionsusingSpitzer24 µmflux.Ourfield
213
+ 8µm LF lies between Caputi etal. (2007) and Babbedgeet al.
214
+ (2006). Compared with these work, we have directly observed
215
+ restframe 8 µm using the AKARI L15filter, eliminating the un-
216
+ certaintlyinfluxconversionbasedonSEDmodels.Moredetai ls
217
+ andevolutionoffieldIRLFsaredescribedinGotoet al.(2010 ).
218
+ 3. Results& Discussion
219
+ 3.1. 8µmIRLFs
220
+ In Fig.1, we show restframe 8 µm LFs of cluster
221
+ RXJ1716.4 +6708 in the squares, and LFs of the field re-
222
+ gion in the triangles. First of all, cluster LFs have by a fact or
223
+ of∼700 higher density than the field LFs, reflecting the fact
224
+ the galaxy clusters is indeed high density regions in terms o f
225
+ infraredsources.
226
+ Next, to compare the shape of the LFs, we normalized the
227
+ cluster LF to match the field LFs at the faintest end, and show
228
+ in the dash-dottedline. In contrast to the field LFs, which sh ow
229
+ flattening of the slope at log L8µm<10.8L⊙, the cluster LF
230
+ maintainsthesteepslopeintherangeof 10.0L⊙<logL8µm<
231
+ 10.6L⊙.Thedifferenceissignificant,consideringthesizeofer-
232
+ rorsoneachLF.
233
+ Wefitadouble-powerlawtobothclusterandfieldLFsusing
234
+ thefollowingformulae.
235
+ Φ(L)dL/L∗= Φ∗/parenleftbiggL
236
+ L∗/parenrightbigg1−α
237
+ dL/L∗,(L < L∗) (1)
238
+ Φ(L)dL/L∗= Φ∗/parenleftbiggL
239
+ L∗/parenrightbigg1−β
240
+ dL/L∗,(L > L∗) (2)
241
+ Free parameters are: L∗(characteristic luminosity, L⊙),φ∗
242
+ (normalization, Mpc−3),αandβ(faint and bright end slopes),
243
+ respectively.ThebestfitvaluesforfieldandclusterLFsare sum-
244
+ marisedinTable1andshowninthedottedlinesinFig.1.
245
+ The bright-end slopes are not very different, but L∗of the
246
+ cluster LF is smaller than the field by a factor of 2.4, and the
247
+ faint-endtailofclusterLF issteeperthanthatoffieldLF.
248
+ To further examine the difference at the faint end of the
249
+ LFs, we divide the cluster LF using the local galaxy density
250
+ (Σ5th) measuredbyKoyamaet al. (2008). Thisdensityis based
251
+ on the distance to the 5th nearest neighbor in the transverse di-
252
+ rection using all the optical photo-z members, and thus, is a
253
+ surface galaxy density. We separate LFs using similar crite ria,
254
+ logΣ5th≥2(dense),1.6≤logΣ5th<2(intermediate), and
255
+ logΣ5th<1.6(sparse), then plot LFs of each region in the4 Gotoet al.:Environemental dependence of 8 µm luminosity functions ofgalaxies atz ∼0.8
256
+ stars, circles, and squares in Fig.2. A fraction of the total vol-
257
+ umeofthe clusteris assignedto eachdensitygroupin invers ely
258
+ proportionaltothe sumof Σ3/2
259
+ 5thofeachgroup.
260
+ Interestingly, the faint-end slope becomes flatter and flatt er
261
+ with decreasing local galaxy density. This result is consis tent
262
+ with our comparison with the field in Fig.1. In fact, the lowes t
263
+ densityLF(squares)hasaflatfaint-endtailsimilartothat ofthe
264
+ fieldLF.SincetheseLFsarebasedonthesamedata,changesin
265
+ the faint-end slope are not likely due to the errors in comple te-
266
+ ness correction nor calibration problems. The completenes s of
267
+ the deep and shallow regions of the cluster are measured sep-
268
+ arately. The changes in the slope is much larger than the maxi -
269
+ mumcompletenesscorrectionof25%.Wealsocheckedtheclus -
270
+ ter LFs as a function of cluster centric radius, to find no sign ifi-
271
+ cantdifference,perhapsduetotheelongatedmorphologyof this
272
+ cluster. At the same time, assuming the same cluster volume,
273
+ Fig.2 shows that a possible contamination from the field gala x-
274
+ ies to cluster LFs is only ∼0.1% in the dense region and ∼1%
275
+ eveninthe sparseregion.
276
+ It is interesting that not just the change in the scale of the
277
+ LFs, but there is a change in the L∗and the faint-end slope ( α)
278
+ of the LFs, resulting in the deficit in the 10.2L⊙<logL8µm<
279
+ 10.8L⊙for cluster LFs. One might imaginea change just in L∗
280
+ might explain the difference in Fig.1. However, in Fig.2, th ere
281
+ clearlyisachangein theslopeasafunctionof Σ5th.
282
+ However,interpretationis rathercomplicated;a shapeofL F
283
+ would not change if field galaxies infall into cluster unifor mly
284
+ withoutchangingtheirstar-formationactivity.Although inclus-
285
+ ter environment,a fractionof MIR luminousgalaxiesis smal ler
286
+ than field (Koyamaet al., 2008), uniformand instant quenchi ng
287
+ of star-formation activity of field galaxies can only shift a LF,
288
+ butcannotaccountforachangein L∗andαoftheLFs.
289
+ Two important findings in this work are; (i) L∗is smaller
290
+ in the cluster. (ii) the faint-end slopes become steeper tow ard
291
+ higher-density regions. To explain these changes in LFs, IR -
292
+ luminousgalaxiesneedtobepreferentiallyreduced,witha rela-
293
+ tive increase of IR-faint galaxies. However, an environmen tal-
294
+ driven physical process such as the ram-pressure stripping or
295
+ galaxy-merging would quench star-formation not only in mas -
296
+ sivegalaxiesbutinlessmassivegalaxiesaswell,andthusi snot
297
+ abletoexplaintheobservedchangesinLFs.
298
+ Ontheotherhand,ithasbeenfrequentlyobservedthatmore
299
+ massive galaxies formed earlier in the Universe. This downs iz-
300
+ ing scenario also depends on the environment,in the sense th at
301
+ galaxieswith same mass are moreevolvedin higherdensityen -
302
+ vironmentsthangalaxisin less denseenvironments(Gotoet al.,
303
+ 2005; Tanakaet al., 2005, 2008). Statistically, a good corr ela-
304
+ tionhasbeenfoundbetween LTIRandstellarmass(Elbazet al.,
305
+ 2007). Our finding of the relative lack of IR-luminous galaxi es
306
+ in the cluster environmentmay be consistent with the downsi z-
307
+ ing scenario, where higher density regions have more evolve d
308
+ galaxies and lacks massive star-forming galaxies. In contr ast,
309
+ in lower density regions more massive galaxies are still sta r-
310
+ forming. However, since the data we have shown is in IR lumi-
311
+ nosity, to conclude on this, we need good stellar mass estima te
312
+ basedondeepernear-IRdata.
313
+ Although a specific mechanism is unclear, the steep faint-
314
+ end could also result from the enhanced star-formation in le ss
315
+ massive galaxies. In the above scenario, massive galaxies h ave
316
+ already ceased their star-formation in the cluster, but les s mas-
317
+ sive galaxiesare still formingstars. These less massive ga laxies
318
+ may stop star-formation soon to join the faint-end of the red -
319
+ sequence(Koyamaet al., 2007).Fig.3.TotalinfraredLFsofclusterRXJ1716.4 +6708atz=0.81
320
+ in the solid line, and those of the AKARI NEP deep field in the
321
+ dashed line. Overplottedare the LFs of MS1054 from Bai et al.
322
+ (2007).
323
+ 3.2. Total IRLFs
324
+ To compare the L8µmLF in Fig.1 to those in the literature, we
325
+ needtoconvert L8µmtoLTIR.Weusethethefollowingrelation
326
+ byCaputiet al.(2007);
327
+ LTIR= 1.91×(νLν8µm)1.06(±55%) (3)
328
+ Thisis better tunedfor a similar luminosityrange used here
329
+ than the originalrelationby Bavouzetetal. (2008). The con ver-
330
+ sion, however, has been the largest source of errors in estim at-
331
+ ingLTIRfromL8µm.Caputi etal.(2007)report55%ofdisper-
332
+ sion around the relation. It should be kept in mind that the re st-
333
+ frame8µm is sensitive to the star-formation activity, but at the
334
+ same time, it is where the SED models have strongest discrep-
335
+ anciesduetothecomplicatedPAHemissionlines(seeFig.12 of
336
+ Caputiet al.,2007; Gotoetal., 2010).
337
+ Theestimated LTIRcanbe,then,convertedtoSFRusingthe
338
+ followingrelationfor a Salpeter IMF, φ(m)∝m−2.35between
339
+ 0.1−100M⊙(Kennicutt, 1998).
340
+ SFR(M⊙yr−1) = 1.72×10−10LTIR(L⊙) (4)
341
+ In Fig.3, we show the LTIRLFs. Symbols are the same as
342
+ Fig.1. Inthe topaxis,we showcorrespondingSFR. Overplott ed
343
+ asterisks are cluster LF of MS1054 at z=0.83 with ×2 larger
344
+ mass by Bai et al. (2007), which show good agreement with
345
+ ourLFsofRXJ1716.4 +6708in thesquares.Notethat Bai et al.
346
+ (2007) covered only the central region of MS1054 due to the
347
+ smaller field of view of the Spitzer. The shape of their LF look s
348
+ more similar to our LFs in the highest density bin in Fig.2. A
349
+ shift in scale is perhaps due to difference in esimating clus ter
350
+ volumes.
351
+ Amajordifferenceofourworktothat ofBai etal. (2007)is
352
+ that they were not able to compare in detail on the shape of the
353
+ LFs between field and cluster regions, due mainly to a smaller
354
+ fieldcoverageandlargererrorsonLFs.Theyhadtofixthefain t-
355
+ end slope with a local value. The largest source of errors is i nGotoet al.:Environemental dependence of 8 µm luminosity functions ofgalaxies atz ∼0.8 5
356
+ converting Spitzer 24 µm flux into 8 µm. Both cluster and field
357
+ LFs of this work use L15filter, which measures restframe 8 µm
358
+ fluxdirectly,eliminatingthelargestsourceoferrors.Ina ddition,
359
+ bothclusterandfiledLFsaremeasuredwithanessentiallysa me
360
+ methodology,allowingusa faircomparisonofLFs.
361
+ 4. Summary
362
+ We constructed restframe 8 µm LFs of a massive galaxy cluster
363
+ (RXJ1716.4 +6708) and a rarefied field region (the NEP deep
364
+ field)at z ∼0.8usingessentially thesame methodanddata from
365
+ the AKARI telescope. AKARI’s 15 µm filter nicely covers rest-
366
+ frame8µm at z∼0.8,and thuswe do not needa large interpola-
367
+ tion based on SED models. AKARI’s wide field of view allows
368
+ ustoinvestigatevarietyofclusterenvironmentswith2ord ersof
369
+ differenceinlocal galaxydensity.
370
+ We found that L∗of the cluster 8 µm LF is smaller than the
371
+ field by a factor of 2.4, and the faint-end tail of cluster IR LF s
372
+ becomesteeperandsteeperwithincreasinglocalgalaxyden sity.
373
+ This difference cannot be explained by a simple infall of fiel d
374
+ galaxies into a cluster. Physics that preferentially supre sses IR
375
+ luminous galaxes in higer density regions is needed to expla in
376
+ theobservedresults.
377
+ Acknowledgments
378
+ Wethanktheanonymousrefereeformanyinsightfulcomments ,
379
+ which significantly improved the paper. We are greateful to
380
+ MasayukiTanakaforusefuldiscussion.WethankL.Baiforpr o-
381
+ vidingdataforcomparison.
382
+ T.G.,Y.K. and H.I. acknowledgesfinancial support from the
383
+ JapanSocietyforthePromotionofScience(JSPS)throughJS PS
384
+ ResearchFellowshipsforYoungScientists.MIwassupporte dby
385
+ the Korea Science and Engineering Foundation(KOSEF) grant
386
+ No. 2009-0063616, funded by the Korea government(MEST)”
387
+ HML acknowledgesthe supportfrom KASI throughits cooper-
388
+ ativefundin2008.
389
+ This research is based on the observations with AKARI, a
390
+ JAXA projectwiththe participationofESA.
391
+ Theauthorswishtorecognizeandacknowledgetheverysig-
392
+ nificant cultural role and reverence that the summit of Mauna
393
+ Kea has always had within the indigenous Hawaiian commu-
394
+ nity. We are most fortunate to have the opportunity to conduc t
395
+ observationsfromthissacredmountain.
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1001.0006.txt ADDED
@@ -0,0 +1,447 @@
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
1
+ arXiv:1001.0006v2 [astro-ph.CO] 4 May 2010Draft version November 2, 2018
2
+ Preprint typeset using L ATEX style emulateapj v. 11/10/09
3
+ COMPARISON OF HECTOSPEC VIRIAL MASSES WITH SZE MEASUREMENT S
4
+ Kenneth Rines1,2, Margaret J. Geller2, and Antonaldo Diaferio3,4
5
+ Draft version November 2, 2018
6
+ ABSTRACT
7
+ We present the first comparison of virial masses of galaxy clusters with their Sunyaev-Zel’dovich
8
+ Effect (SZE) signals. We study 15 clusters from the Hectospec Clus ter Survey (HeCS) with
9
+ MMT/Hectospec spectroscopy and published SZE signals. We measu re virial masses of these clusters
10
+ from an average of 90 member redshifts inside the radius r100. The virial masses of the clusters are
11
+ strongly correlated with their SZE signals (at the 99% confidence lev el using a Spearman rank-sum
12
+ test). This correlation suggests that YSZcan be used as a measure of virial mass. Simulations predict
13
+ a powerlaw scaling of YSZ∝Mα
14
+ 200withα≈1.6. Observationally, we find α=1.11±0.16, significantly
15
+ shallower (given the formal uncertainty) than the theoretical pr ediction. However, the selection func-
16
+ tion of our sample is unknown and a bias against less massive clusters c annot be excluded (such a
17
+ selection bias could artificially flatten the slope). Moreover, our sam ple indicates that the relation
18
+ between velocity dispersion (or virial mass estimate) and SZE signal has significant intrinsic scatter,
19
+ comparable to the range of our current sample. More detailed stud ies of scaling relations are therefore
20
+ needed to derive a robust determination of the relation between clu ster mass and SZE.
21
+ Subject headings: galaxies: clusters: individual — galaxies: kinematics and dynamics — co smology:
22
+ observations
23
+ 1.INTRODUCTION
24
+ Clusters of galaxies are the most massive virialized
25
+ systems in the universe. The normalization and evo-
26
+ lution of the cluster mass function is therefore a sen-
27
+ sitive probe of the growth of structure and thus cos-
28
+ mology (e.g., Rines et al. 2007, 2008; Vikhlinin et al.
29
+ 2009; Henry et al. 2009; Mantz et al. 2008; Rozo et al.
30
+ 2008, and references therein). Many methods exist
31
+ to estimate cluster masses, including dynamical masses
32
+ from either galaxies (Zwicky 1937) or intracluster gas
33
+ (e.g., Fabricant et al. 1980), gravitational lensing (e.g.,
34
+ Smith et al.2005;Richard et al.2010), andthe Sunyaev-
35
+ Zel’dovich effect (SZE Sunyaev & Zeldovich 1972). In
36
+ practice, these estimates are often made using simple
37
+ observables, such as velocity dispersion for galaxy dy-
38
+ namics or X-ray temperature for the intracluster gas.
39
+ If one of these observable properties of clusters has a
40
+ well-defined relation to the cluster mass, a large survey
41
+ can yield tight constraints on cosmological parameters
42
+ (e.g., Majumdar & Mohr 2004). There is thus much
43
+ interest in identifying cluster observables that exhibit
44
+ tight scaling relations with mass (Kravtsov et al. 2006;
45
+ Rozo et al. 2008). Numerical simulations indicate that
46
+ X-ray gas observables (Nagai et al. 2007) and SZE sig-
47
+ nals (Motl et al. 2005) are both candidates for tight scal-
48
+ ing relations. Both methods are beginning to gain ob-
49
+ servational support (e.g., Henry et al. 2009; Lopes et al.
50
+ 2009; Mantz et al. 2009; Locutus Huang et al. 2009).
51
+ Dynamical masses from galaxy velocities are unbiased
52
53
+ 1Department of Physics & Astronomy, Western Washington
54
+ University, Bellingham, WA 98225; [email protected]
55
+ 2Smithsonian Astrophysical Observatory, 60 Garden St, Cam-
56
+ bridge, MA 02138
57
+ 3Universit` a degli Studi di Torino, Dipartimento di Fisica G en-
58
+ erale “Amedeo Avogadro”, Torino, Italy
59
+ 4Istituto Nazionale di Fisica Nucleare (INFN), Sezione di
60
+ Torino, Torino, Italyin numerical simulations (Diaferio 1999; Evrard et al.
61
+ 2008), and recent results from hydrodynamical simula-
62
+ tions indicate that virial masses may have scatter as
63
+ small as ∼5% (Lau et al. 2010).
64
+ Previous studies have compared SZE signals to hydro-
65
+ staticX-raymasses(Bonamente et al.2008;Plagge et al.
66
+ 2010) and gravitational lensing masses (Marrone et al.
67
+ 2009, hereafter M09). Here, we make the first compar-
68
+ ison between virial masses of galaxy clusters and their
69
+ SZE signals. We use SZE measurements from the lit-
70
+ erature and newly-measured virial masses of 15 clus-
71
+ ters from extensive MMT/Hectospec spectroscopy. This
72
+ comparison tests the robustness of the SZE as a proxy
73
+ for cluster mass and the physical relationship between
74
+ the SZE signal and cluster mass. Large SZ cluster sur-
75
+ veys are underway and are beginning to yield cosmologi-
76
+ calconstraints(Carlstrom et al.2010;Hincks et al.2010;
77
+ Staniszewski et al. 2009).
78
+ We assume a cosmology of Ω m=0.3, Ω Λ=0.7, and
79
+ H0=70 km s−1Mpc−1for all calculations.
80
+ 2.OBSERVATIONS
81
+ 2.1.Optical Photometry and Spectroscopy
82
+ We are completing the Hectospec Cluster Survey
83
+ (HeCS), a study of an X-ray flux-limited sample of 53
84
+ galaxy clusters at moderate redshift with extensive spec-
85
+ troscopy from MMT/Hectospec. HeCS includes all clus-
86
+ ters with ROSAT X-ray fluxes of fX>5×10−12erg
87
+ s−1at [0.5-2.0]keVfrom the Bright Cluster Survey (BCS
88
+ Ebeling et al.1998)orREFLEXsurvey(B¨ ohringer et al.
89
+ 2004) with optical imaging in the Sixth Data Release
90
+ (DR6) of SDSS (Adelman-McCarthy et al. 2008). We
91
+ use DR6 photometry to select Hectospec targets. The
92
+ HeCS targets are all brighter than r=20.8 (SDSS cata-
93
+ logs are 95% complete for point sources to r≈22.2). Out
94
+ of the HeCS sample, 15 clusters have published SZ mea-
95
+ surements.2 Rines, Geller, & Diaferio
96
+ 2.1.1.Spectroscopy: MMT/Hectospec and SDSS
97
+ HeCS is a spectroscopic survey of clusters in the red-
98
+ shift range 0.10 ≤z≤0.30. We measure spectra with
99
+ the Hectospec instrument (Fabricant et al. 2005) on the
100
+ MMT 6.5m telescope. Hectospec provides simultaneous
101
+ spectroscopy of up to 300 objects across a diameter of
102
+ 1◦. This telescope and instrument combination is ideal
103
+ for studying the virial regions and outskirts of clusters
104
+ at these redshifts. We use the red sequence to preselect
105
+ likely cluster members as primary targets, and we fill
106
+ fibers with bluer targets (Rines et al. in prep. describes
107
+ the details of target selection). We eliminate all targets
108
+ withexistingSDSSspectroscopyfromourtargetlistsbut
109
+ include these in our final redshift catalogs.
110
+ Ofthe15clustersstudiedhere,onewasobservedwitha
111
+ single Hectospec pointing and the remaining 14 were ob-
112
+ served with two pointings. Using multiple pointings and
113
+ incorporatingSDSS redshifts of brighterobjectsmitigate
114
+ fiber collision issues. Because the galaxy targets are rel-
115
+ atively bright ( r≤20.8), the spectra were obtained with
116
+ relativelyshortexposuretimes of3x600sto 4x900sunder
117
+ a variety of observing conditions.
118
+ Figure 1 shows the redshifts of galaxies versus their
119
+ projected clustrocentric radii for the 15 clusters stud-
120
+ ied here. The infall patterns are clearly present in all
121
+ clusters. We use the caustic technique (Diaferio 1999)
122
+ to determine cluster membership. Briefly, the caustic
123
+ technique uses a redshift-radius diagram to isolate clus-
124
+ ter members in phase space by using an adaptive ker-
125
+ nel estimator to smooth out the galaxies in phase space,
126
+ and then determining the edges of this distribution (see
127
+ Diaferio 2009, for a recent review). This technique has
128
+ been successfully applied to optical studies of X-ray clus-
129
+ ters, and yields cluster mass estimates in agreement
130
+ with estimatesfromX-rayobservationsandgravitational
131
+ lensing (e.g., Rines et al. 2003; Biviano & Girardi 2003;
132
+ Diaferio et al. 2005; Rines & Diaferio 2006; Rines et al.
133
+ 2007, and references therein).
134
+ We apply the prescription of Danese et al. (1980) to
135
+ determine the mean redshift cz⊙and projected velocity
136
+ dispersion σpof each cluster from all galaxies within the
137
+ caustics. We calculate σpusing only the cluster members
138
+ projected within r100estimated from the caustic mass
139
+ profile.
140
+ 2.2.SZE Measurements
141
+ The SZE detections are primarily from
142
+ Bonamente et al. (2008, hereafter B08), supplemented
143
+ by three measurements from Marrone et al. (2009,
144
+ hereafter M09). Most of the SZ data were obtained with
145
+ the OVRO/BIMA arrays; the additional clusters from
146
+ M09 were observed with the Sunyaev-Zel’dovich Array
147
+ (SZA; e.g., Muchovej et al. 2007).
148
+ Numerical simulations indicate that the integrated
149
+ Compton y-parameter YSZhas smaller scatter than the
150
+ peak y-decrement ypeak(Motl et al. 2005), so B08 and
151
+ M09 report only YSZ. Although ypeakshould be nearly
152
+ independent of redshift, YSZdepends on the angular size
153
+ of the cluster. The quantity YSZD2
154
+ Aremoves this depen-
155
+ dence. Thus, we compare our dynamical mass estimates
156
+ to this quantity rather than ypeakorYSZ. Table 1 sum-
157
+ marizes the SZ data and optical spectroscopy.
158
+ It is also critical to determine the radius within whichYSZis determined. B08 use r2500, the radius that en-
159
+ closes an average density of 2500 times the critical den-
160
+ sity at the cluster’s redshift; r2500has physical values of
161
+ 300-700kpc forthe massiveclustersstudied by B08(470-
162
+ 670kpcforthesubsamplestudiedhere). M09useaphys-
163
+ ical radius of 350 kpc because this radius best matches
164
+ their lensing data.
165
+ To use both sets of data, we must estimate the con-
166
+ version between YSZ(r2500) measured within r2500and
167
+ YSZ(r= 350 kpc) measured within the smaller radius
168
+ r=350 kpc. There are 8 clusters analyzed in both B08
169
+ and M09 (5 of which are in HeCS). We perform a least-
170
+ squaresfit to YSZ(r2500)−YSZ(r= 350kpc) to determine
171
+ an approximate aperture correction for the M09 clusters.
172
+ We list both quantities in Table 1.
173
+ 3.RESULTS
174
+ We examine two issues: (1) the strength of the corre-
175
+ lation between SZE signal and the dynamical mass and
176
+ (2) the slope of the relationship between them. Figure 2
177
+ shows the YSZ−σprelation. Here, we compute σpfor all
178
+ galaxies inside both the caustics and the radius r100,cde-
179
+ fined by the caustic mass profile [ rδis the radius within
180
+ which the enclosed density is δtimes the critical density
181
+ ρc(z)].
182
+ Because we make the first comparison of dynami-
183
+ cal properties and SZE signals, we first confirm that
184
+ these two variables are well correlated. A nonparametric
185
+ Spearman rank-sum test (one-tailed) rejects the hypoth-
186
+ esis of uncorrelated data at the 98.4% confidence level.
187
+ The strong correlation in the data suggests that both σp
188
+ andYSZD2
189
+ Aincrease with increasing cluster mass.
190
+ Hydrodynamic numerical simulations indicate that
191
+ YSZ(integrated to r500) scales with cluster mass as
192
+ YSZ∝Mα
193
+ 500, whereα=1.60 with radiative cooling and
194
+ star formation, and 1.61 for simulations with radiative
195
+ cooling, star formation, and AGN feedback ( α=1.70 for
196
+ non-radiative simulations, Motl et al. 2005). Combin-
197
+ ing this result with the virial scaling relation of dark
198
+ matter particles, σp∝M0.336±0.003
199
+ 200 (Evrard et al. 2008),
200
+ the expected scaling is YSZ∝σ4.76(we assume that
201
+ M100∝M500). The right panels of Figure 2 shows this
202
+ predicted slope (dashed lines).
203
+ The bisector of the least-squares fits to the data has
204
+ a slope of 2 .94±0.74, significantly shallower than the
205
+ predicted slope of 4.8.
206
+ We recompute the velocity dispersions σp,Afor all
207
+ galaxies within one Abell radius (2.14 Mpc) and in-
208
+ side the caustics. Surprisingly, the correlation is slightly
209
+ stronger (99.4% confidence level). This result supports
210
+ the idea that velocity dispersions computed within a
211
+ fixedphysicalradiusretainstrongcorrelationswith other
212
+ cluster observables, even though we measure the velocity
213
+ dispersion inside different fractions of the virial radius
214
+ for clusters of different masses. Because cluster veloc-
215
+ ity dispersions decline with radius (e.g. Rines et al. 2003;
216
+ Rines & Diaferio 2006), σp,Amay be smaller than σp,100
217
+ (measured within r100,c) for low-mass clusters, perhaps
218
+ exaggerating the difference in measured velocity disper-
219
+ sionsrelativeto the differences in virialmass(i.e., σp,Aof
220
+ a low-mass cluster may be measured within 2 r100while
221
+ σp,Aof a high-mass cluster may be measured within r100;
222
+ the ratio σp,Aof these clusters would be exaggerated rel-Hectospec Virial Masses and SZE 3
223
+ Fig. 1.— Redshift versus projected clustrocentric radius for the 15 HeCS clusters studied here. Clusters are ordered left-to-r ight and
224
+ top-to-bottom by decreasing values of YSZD2
225
+ A(r2500). The solid lines show the locations of the caustics, which w e use to identify cluster
226
+ members. The Hectospec data extend out to ∼8 Mpc; the figure shows only the inner 4 Mpc to focus on the viria l regions.
227
+ ative to the ratio σp,100). Future cluster surveys with
228
+ enough redshifts to estimate velocity dispersions but too
229
+ few to perform a caustic analysis should still be sufficient
230
+ for analyzing scaling relations.
231
+ Because of random errors in the mass estimation, the
232
+ virial mass and the caustic mass within a given radius
233
+ do not necessarily coincide. Therefore, the radius r100
234
+ depends on the mass estimator used. Figure 2 shows
235
+ the scaling relationsfor two estimated masses M100,cand
236
+ M100,v;M100,cis the mass estimated within r100,c(where
237
+ bothquantitiesaredefinedfromthecausticmassprofile),
238
+ andM100,vis the mass estimated within r100,v(both
239
+ quantities are estimated with the virial theorem, e.g.,
240
+ Rines & Diaferio 2006). including galaxies projected in-
241
+ sider100,v. Similar to σp, there is a clear correlation
242
+ between M100,vandYSZD2
243
+ A(99.0% confidence with a
244
+ Spearman test). The strong correlation of dynamical
245
+ mass with SZE also holds for M100,cestimated directlyfrom the caustic technique (99.8% confidence).
246
+ The bisector of the least-squares fits has a slope of
247
+ 1.11±0.16, again significantly shallower than the pre-
248
+ dicted slope of 1.6. This discrepancy has two distinct
249
+ origins. By looking at the distribution of the SZE sig-
250
+ nals in Figure 2, we see that, at a given velocity disper-
251
+ sion or mass, the SZE signals have a scatter which is a
252
+ factor of ∼2. Alternatively, at fixed SZE signal, there
253
+ is a scatter of a factor of ∼2 in estimated virial mass.
254
+ Unless the observational uncertainties are significantly
255
+ underestimated, the data show substantial intrinsic scat-
256
+ ter. Moreover, this scatter is comparable to the range of
257
+ our sample and, therefore, the error on the slope derived
258
+ from our least-squares fit to the data is likely to be un-
259
+ derestimated (see Andreon & Hurn 2010, for a detailed
260
+ discussionofaBayesianapproachtofittingrelationswith
261
+ measurement uncertainties and intrinsic scatter in both
262
+ quantities).4 Rines, Geller, & Diaferio
263
+ TABLE 1
264
+ HeCS Dynamical Masses and SZE Signals
265
+ Cluster z σ p M100,vM100,c YSZD2
266
+ AYSZD2
267
+ ASZE
268
+ (350 kpc) ( r2500)
269
+ km s−11014M⊙1014M⊙10−5Mpc−210−4Mpc2Ref.
270
+ A267 0.2288 743+81
271
+ −616.86±0.82 4.26 ±0.14 3.08 ±0.34 0.42 ±0.06 1
272
+ A697 0.2812 784+77
273
+ −596.11±0.69 5.96 ±3.51 – 1.29 ±0.15 1
274
+ A773 0.2174 1066+77
275
+ −6318.4±1.7 16.3 ±0.7 5.40 ±0.57 0.90 ±0.10 1
276
+ Zw2701 0.2160 564+63
277
+ −473.47±0.42 2.69 ±0.30 1.46 ±0.016 0.17 ±0.02a2
278
+ Zw3146 0.2895 752+92
279
+ −676.87±0.89 4.96 ±0.91 – 0.71 ±0.09 1
280
+ A1413 0.1419 674+81
281
+ −606.60±0.85 3.49 ±0.15 3.47 ±0.24 0.81 ±0.12 1
282
+ A1689 0.1844 886+63
283
+ −5215.3±1.4 9.44 ±5.66 7.51 ±0.60 1.50 ±0.14 1
284
+ A1763 0.2315 1042+79
285
+ −6416.9±1.6 12.6 ±1.5 3.10 ±0.32 0.46 ±0.05a2
286
+ A1835 0.2507 1046+66
287
+ −5519.6±1.6 20.6 ±0.3 6.82 ±0.48 1.37 ±0.11 1
288
+ A1914 0.1659 698+46
289
+ −386.70±0.57 6.21 ±0.21 – 1.08 ±0.09 1
290
+ A2111 0.2290 661+57
291
+ −454.01±0.41 4.77 ±1.23 – 0.55 ±0.12 1
292
+ A2219 0.2256 915+53
293
+ −4512.8±1.0 12.0 ±4.7 6.27 ±0.26 1.19 ±0.05a2
294
+ A2259 0.1606 735+67
295
+ −535.59±0.60 4.90 ±1.69 – 0.27 ±0.10 1
296
+ A2261 0.2249 725+75
297
+ −577.13±0.83 5.10 ±2.07 – 0.71 ±0.09 1
298
+ RXJ2129 0.2338 684+88
299
+ −644.31±0.57 2.94 ±0.13 – 0.40 ±0.07 1
300
+ Note. —aExtrapolated to r2500using the best-fit relation between YSZD2
301
+ A(350kpc) and YSZD2
302
+ A(r2500) for eight clusters in common
303
+ between B08 and M09.
304
+ Note. — Redshift zand velocity dispersion σpare computed for galaxies defined as members using the causti cs. Masses M100,vand
305
+ M100,care evaluated using the virial mass profile and caustic mass p rofile respectively.
306
+ Note. — REFERENCES: SZE data are from (1) Bonamente et al. 2008 and (2) Marrone et al. 2009.
307
+ Our shallow slopes may also arise in part from the fact
308
+ that our sample, which has been assembled from the lit-
309
+ erature and whose selection function is difficult to deter-
310
+ mine, is likely to be biased against clusters with small
311
+ mass and low SZE signal. Larger samples should deter-
312
+ mine whether unknown observational biases or issues in
313
+ the physical understanding of the relation account for
314
+ this discrepancy.
315
+ 4.DISCUSSION
316
+ Thestrongcorrelationbetweenmassesfromgalaxydy-
317
+ namics and SZE signals indicates that the SZE is a rea-
318
+ sonableproxyforcluster mass. B08compareSZEsignals
319
+ toX-rayobservables,inparticularthetemperature TXof
320
+ the intracluster medium and YX=MgasTX, whereMgas
321
+ isthemassoftheICM(seealsoPlagge et al.2010). Both
322
+ of these quantities are measured within r500, a signifi-
323
+ cantly smaller radius than r100where we measure virial
324
+ mass. M09 compare SZE signals to masses estimated
325
+ from gravitational lensing measurements. The lensing
326
+ masses are measured within a radius of 350 kpc. For the
327
+ clusters studied here, this radius is smaller than r2500
328
+ and much smaller than r100. Numerical simulations indi-
329
+ cate that the scatter in masses measured within an over-
330
+ densityδdecreases as δdecreases (White 2002), largely
331
+ because variations in cluster cores are averaged out at
332
+ larger radii. Thus, the dynamical measurement reaching
333
+ to larger radius may provide a more robust indication
334
+ of the relationship between the SZE measurements and
335
+ cluster mass.
336
+ TheYSZD2
337
+ A−Mlensdata presented in M09 show a
338
+ weakercorrelationthanouropticaldynamicalproperties.
339
+ A Spearman test rejects the hypothesis of uncorrelated
340
+ data for the M09 data at only the 94.8% confidence level,
341
+ compared to the 98.4-99.8% confidence levels for our op-
342
+ tical dynamical properties. One possibility is that Mlensis more strongly affected by substructure in cluster cores
343
+ and by line-of-sight structures than are the virial masses
344
+ and velocity dispersions we derive.
345
+ Few measurements of SZE at large radii ( > r500) are
346
+ currently available. Hopefully, future SZ data will allow
347
+ a comparisonbetween virialmass and YSZwithin similar
348
+ apertures.
349
+ 5.CONCLUSIONS
350
+ Our first direct comparison of virial masses, velocity
351
+ dispersions, and SZ measurements for a sizable clus-
352
+ ter sample demonstrates a strong correlation between
353
+ these observables (98.4-99.8% confidence). The SZE sig-
354
+ nal increases with cluster mass. However, the slopes of
355
+ both the YSZ−σrelation ( YSZ∝σ2.94±0.74
356
+ p) and the
357
+ YSZ−M100relation ( YSZ∝M1.11±0.16
358
+ 100) are significantly
359
+ shallower(giventheformaluncertainties)thantheslopes
360
+ predictedbynumericalsimulations(4.76and1.60respec-
361
+ tively).
362
+ This result may be partly explained by a bias against
363
+ less massive clusters that could artificially flatten our
364
+ measured slopes. Unfortunately, the selection function
365
+ of our sample is unknown and we are unable to quan-
366
+ tify the size of this effect. More importantly, our sample
367
+ indicates that the relation between SZE and virial mass
368
+ estimates (or velocity dispersion) has a non-negligible in-
369
+ trinsicscatter. Acomplete, representativeclustersample
370
+ is required to robustly determine the size of this scatter,
371
+ its origin, and its possible effect on the SZE as a mass
372
+ proxy.
373
+ Curiously, YSZis more strongly correlated with both
374
+ σpandM100than with Mlens(M09). Comparison of
375
+ lensingmassesandclustervelocitydispersions(andvirial
376
+ masses)forlarger,complete, objectivelyselected samples
377
+ of clusters may resolve these differences.
378
+ Thefull HeCS sampleof53clusterswill providealargeHectospec Virial Masses and SZE 5
379
+ Fig. 2.— Integrated S-Z Compton parameter YSZD2
380
+ Aversus dynamical properties for 15 clusters from HeCS. Left panels: SZE data
381
+ versus virial mass M100estimated from the virial mass profile (top) and the caustic m ass profile (bottom). Solid and open points indicate
382
+ SZ measurements from B08 and M09 respectively. The dashed li ne shows the slope of the scaling predicted from numerical si mulations:
383
+ YSZ∝M1.6(Motl et al. 2005), while the solid line shows the ordinary le ast-squares bisector. Arrows show the aperture correction s to
384
+ the SZE measurements (see text). Right panels: SZE data versus projected velocity dispersions measured fo r galaxies inside the caustics
385
+ and (top) inside r100,cestimated from the caustic mass profile and (bottom) inside t he Abell radius 2.14 Mpc. The dashed line shows the
386
+ scaling predicted from simulations: YSZ∝M1.6(Motl et al. 2005) and σ∝M0.33(Evrard et al. 2008). The solid line shows the ordinary
387
+ least-squares bisector. Data points and arrows are defined a s in the left panels.
388
+ sample of clusters with robustly measured velocity dis-
389
+ persions and virial masses as a partial foundation for
390
+ these comparisons.
391
+ We thank Stefano Andreon for fruitful discussions
392
+ about fitting scaling relations with measurement errorsand intrinsic scatter in both quantities. AD gratefully
393
+ acknowledges partial support from INFN grant PD51.
394
+ We thank Susan Tokarz for reducing the spectroscopic
395
+ data and Perry Berlind and Mike Calkins for assisting
396
+ with the observations.
397
+ Facilities: MMT (Hectospec)
398
+ REFERENCES
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+ Adelman-McCarthy, J. K. et al. 2008, ApJS, 175, 297
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+ Andreon, S. & Hurn, M. A. 2010, MNRAS in press,
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+ arXiv:1001.4639
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+ B¨ ohringer, H. et al. 2004, A&A, 425, 367
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+ Biviano, A. & Girardi, M. 2003, ApJ, 585, 205
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+ Bonamente, M., Joy, M., LaRoque, S. J., Carlstrom, J. E., Nag ai,
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+ D., & Marrone, D. P. 2008, ApJ, 675, 106Carlstrom, J. et al. 2010, ArXiv e-prints
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+ Danese, L., de Zotti, G., & di Tullio, G. 1980, A&A, 82, 322
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+ Diaferio, A. 1999, MNRAS, 309, 610
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+ —. 2009, ArXiv e-prints
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+ Diaferio, A., Geller, M. J., & Rines, K. J. 2005, ApJ, 628, L97
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+ Ebeling, H., Edge, A. C., Allen, S. W., Crawford, C. S., Fabia n,
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+ A. C., & Huchra, J. P. 1998, MNRAS, 301, 8816 Rines, Geller, & Diaferio
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+ Evrard, A. E. et al. 2008, ApJ, 672, 122
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+ Fabricant, D., Lecar, M., & Gorenstein, P. 1980, ApJ, 241, 55 2
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+ Fabricant, D. et al. 2005, PASP, 117, 1411
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1001.0007.txt ADDED
@@ -0,0 +1,124 @@
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
1
+ arXiv:1001.0007v1 [astro-ph.CO] 30 Dec 2009Cosmicstarformation history
2
+ revealedby the AKARI
3
+ & Spatially-resolvedspectroscopyofan E+A(Post-starbur st)system
4
+ Tomotsugu GOTO∗, the AKARINEPDteam†,M.Yagi∗∗andC.Yamauchi†
5
+ ∗InstituteforAstronomy,Universityof Hawaii,2680Woodla wnDrive, Honolulu,HI,96822,USA
6
+ †JapanAerospaceExplorationAgency,Sagamihara,Kanagawa 229-8510,Japan
7
+ ∗∗NationalAstronomicalObservatory,2-21-1Osawa,Mitaka, Tokyo,181-8588,Japan
8
+ Abstract. We reveal cosmic star-formation history obscured by dust us ing deep infrared observa-
9
+ tionwiththeAKARI.Acontinuousfiltercoverageinthemid-I Rwavelength(2.4,3.2,4.1,7,9,11,
10
+ 15, 18, and 24 µm) by the AKARI satellite allows us to estimate restframe 8 µm and 12 µm lumi-
11
+ nositieswithoutusingalargeextrapolationbasedonaSEDfi t,whichwasthelargestuncertaintyin
12
+ previouswork. We found that restframe 8 µm (0.38<z<2.2), 12µm (0.15<z<1.16), and total
13
+ infrared (TIR) luminosity functions (LFs) (0 .2<z<1.6) constructed from the AKARI NEP deep
14
+ data, show a continuous and strong evolution toward higher r edshift. In terms of cosmic infrared
15
+ luminosity density ( ΩIR), which was obtained by integrating analytic fits to the LFs, we found a
16
+ goodagreementwithpreviousworkat z<1.2,withΩIR∝(1+z)4.4±1.0.Whenweseparatecontri-
17
+ butionsto ΩIRby LIRGs and ULIRGs, we foundmore IR luminoussourcesare inc reasinglymore
18
+ importantathigherredshift.WefoundthattheULIRG(LIRG) contributionincreasesbyafactorof
19
+ 10(1.8)from z=0.35toz=1.4.
20
+ Keywords: galaxies:evolution,galaxies:starburst
21
+ PACS:98.70.Lt
22
+ Introduction .Revealingthecosmicstarformationhistoryisoneofthemaj orgoals
23
+ of the observational astronomy. However, UV/optical estim ation only provides us with
24
+ alowerlimitofthestarformationrate(SFR) duetotheobscu rationbydust.Astraight-
25
+ forward way to overcome this problem is to observe in infrare d, which can capture the
26
+ starformation activityinvisiblein the UV. The superb sens itivitiesofrecently launched
27
+ SpitzerandAKARI satellitescan revolutionizethefield.
28
+ However,most of theSpitzer work relied on a large extrapola tionfrom 24 µm flux to
29
+ estimate the 8, 12 µm or total infrared (TIR) luminosity, due to the limited numb er of
30
+ mid-IR filters. AKARI has continuous filter coverage across t he mid-IR wavelengths,
31
+ thus, allows us to estimate mid-IR luminosity without using a largek-correction based
32
+ on the SED models, eliminating the largest uncertainty in pr evious work. By taking
33
+ advantage of this, we present the restframe 8, 12 µm TIR LFs, and thereby the cosmic
34
+ starformationhistoryderivedfrom theseusingtheAKARINE P-Deep data.
35
+ Data&Analysis .TheAKARIhasobservedtheNEPdeepfield(0.4deg2)in9filters
36
+ (N2,N3,N4,S7,S9W,S11,L15,L18WandL24) to the depths of 14.2, 11.0, 8.0, 48, 58,
37
+ 71, 117, 121 and 275 µJy (5σ)[14]. This region is also observed in BVRi′z′(Subaru),
38
+ u′(CFHT), FUV,NUV(GALEX), and J,Ks(KPNO2m), with which we computed
39
+ photo-zwithΔz
40
+ 1+z=0.043. Objects which are better fit with a QSO template are re movedFIGURE 1. (left) Restframe 8 µm LFs. The blue diamonds, purple triangles, red squares, and orange
41
+ crosses show the 8 µm LFs at 0 .38<z<0.58,0.65<z<0.90,1.1<z<1.4, and 1.8<z<2.2,
42
+ respectively. The dotted lines show analytical fits with a do uble-power law. Vertical arrows show the
43
+ 8µm luminosity corresponding to the flux limit at the central re dshift in each redshift bin. Overplotted
44
+ are Babbedge et al. [1] in the pink dash-dotted lines, Caputi et al. [2] in the cyan dash-dotted lines,
45
+ and Huang et al. [6] in the green dash-dotted lines. AGNs are e xcluded from the sample. (middle)
46
+ Restframe 12 µm LFs. The blue diamonds, purple triangles, and red squares s how the 12 µm LFs at
47
+ 0.15<z<0.35,0.38<z<0.62, and 0 .84<z<1.16, respectively. Overplotted are Pérez-González
48
+ et al. [11] at z=0.3,0.5and 0.9 in the cyan dash-dottedlines, and Rush, Mal kan, & Spinoglio [12] at z=0
49
+ inthegreendash-dottedlines. (right)TIRLFs.
50
+ from the analysis. We compute LFs using the 1/ Vmaxmethod. Data are used to 5 σwith
51
+ completeness correction. Errors of the LFs are from 1000 rea lization of Monte Carlo
52
+ simulation.
53
+ 8µm LF.Monochromatic 8 µm luminosity ( L8µm) is known to correlate well with
54
+ the TIR luminosity [1, 6], especially for star-forming gala xies because the rest-frame
55
+ 8µmfluxaredominatedbyprominentPAHfeaturessuchasat6.2,7 .7and8.6 µm.The
56
+ leftpanelofFig.1showsastrongevoltuionof8 µmLFs.Overplottedpreviousworkhad
57
+ torelyonSEDmodelstoestimate L8µmfromtheSpitzer S24µmintheMIRwavelengths
58
+ whereSEDmodelingisdifficultduetothecomplicatedPAHemi ssions.Here,AKARI’s
59
+ mid-IR bands are advantageous in directly observing redshi fted restframe 8 µm flux in
60
+ one of the AKARI’s filters, leading to more reliable measurem ent of 8µm LFs without
61
+ uncertaintyfromtheSED modeling.
62
+ 12µm LF.12µm luminosity ( L12µm) represents mid-IR continuum, and known to
63
+ correlate closely with TIR luminosity [11]. The middle pane l of Fig.1 shows a strong
64
+ evoltuion of 12 µm LFs. Here the agreement with previous work is better becaus e (i)
65
+ 12µm continuum is easier to be modeled, and (ii) the Spitzer also captures restframe
66
+ 12µm inS24µmat z=1.
67
+ TIRLF.Lastly,weshowtheTIRLFsintherightpanelofFig.1.Weused Lagache,
68
+ Dole, & Puget [8]’s SED templates to fit the photometry using t he AKARI bands
69
+ at>6µm (S7,S9W,S11,L15,L18WandL24). The TIR LFs show a strong evolution
70
+ comparedto localLFs. At 0 .25<z<1.3,L∗
71
+ TIRevolvesas ∝(1+z)4.1±0.4.FIGURE2. Evolutionof TIRluminositydensitybasedon TIRLFs (redcir cles),8µmLFs (stars), and
72
+ 12µm LFs (filled triangles). The blue open squares and orange fill ed squares are for LIRG and ULIRGs
73
+ only,alsobasedonour LTIRLFs.Overplotteddot-dashedlinesareestimatesfromtheli terature:LeFloc’h
74
+ et al. [9], Magnelli et al. [10] , Pérez-González et al. [11], Caputi et al. [2], and Babbedge et al. [1] are
75
+ in cyan, yellow, green, navy, and pink, respectively. The pu rple dash-dotted line shows UV estimate by
76
+ Schiminovichet al.[13].Thepinkdashedlineshowsthe tota lestimateofIR(TIRLF)andUV [13].
77
+ Cosmic star formation history .We fit LFs in Fig.1 with a double-power law, then
78
+ integrate to estimate total infrared luminosity density at various z. The restframe 8
79
+ and 12µm LFs are converted to LTIRusing [11, 2] before integration. The resulting
80
+ evolution of the TIR density is shown in Fig.2. The right axis shows the star formation
81
+ densityassumingKennicutt[7].We obtain ΩIR(z)∝(1+z)4.4±1.0. Comparisonto ΩUV
82
+ [13] suggests that ΩTIRexplains 70% of Ωtotalatz=0.25, and that by z=1.3, 90% of
83
+ the cosmic SFD is explained by the infrared. This implies tha tΩTIRprovides good
84
+ approximationofthe Ωtotalatz>1.
85
+ In Fig.2, we also show the contributions to ΩTIRfrom LIRGs and ULIRGs. From
86
+ z=0.35 to z=1.4,ΩIRby LIRGs increases by a factor of ∼1.6, andΩIRby ULIRGs
87
+ increases byafactorof ∼10. Moredetailsarein Gotoet al. [3].
88
+ Spatially-Resolved Spectroscopy of an E+A (post-starburs t) System .We per-
89
+ formed a spatially-resolved medium resolution long-slit s pectroscopy of a nearby E+A
90
+ (post-starburst) galaxy system with FOCAS/Subaru [4]. Thi s E+A galaxy has an obvi-
91
+ ous companion galaxy 14kpc in front (Fig.3, left) with the ve locity difference of 61.8
92
+ km/s.
93
+ WefoundthatH δequivalentwidth(EW)oftheE+Agalaxyisgreaterthan7Å gal axy
94
+ wide (8.5 kpc) with no significant spatial variation. We dete cted a rotational velocity in
95
+ the companion galaxy of >175km/s. The progenitor of the companion may have beenFIGURE 3. (left) The SDSS g,r,i-composite image of the J1613+5103. The long-slit position s are
96
+ overlayed.The E+A galaxy is to the right (west), with bluer c olour. The companion galaxy is to the left
97
+ (east). (right) H δEW is plotted against D4000. The diamonds and triangles are f or the E+A core/north
98
+ spectra, respectively. The squares and crosses are for the c ompanion galaxy’s core/north spectra. Gray
99
+ lines are population synthesis models with 5-100% delta bur st population added to the 10G-year-old
100
+ exponentially-decaying( τ=1Gyr)underlyingstellarpopulation.SalpeterIMFandmet allicityof Z=0.008
101
+ areassumed.Onthe models,burstagesof0.1,0.25,0.5and2 G yraremarkedwiththefilled circles.
102
+ a rotationally-supported, but yet passive S0 galaxy. The ag e of the E+A galaxy after
103
+ quenching the star formation is estimated to be 100-500Myr, with its centre having
104
+ slightly younger stellar population. The companion galaxy is estimated to have older
105
+ stellarpopulationof >2 Gyrs ofagewithnosignificantspatialvariation(Fig.3, ri ght).
106
+ Thesefindingsareinconsistentwithasimplepicturewheret hedynamicalinteraction
107
+ createsinfallofthegasreservoirthatcausesthecentrals tarburst/post-starburst.Instead,
108
+ ourresultspresentanimportantexamplewherethegalaxy-g alaxyinteractioncantrigger
109
+ agalaxy-widepost-starburstphenomena.
110
+ REFERENCES
111
+ 1. BabbedgeT.S.R., et al.,2006,MNRAS, 370,1159
112
+ 2. CaputiK.I.,et al.,2007,ApJ,660,97
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+ 3. GotoT.,et al. 2010,A&AAKARI specialissue
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+ 4. GotoT.,YagiM.,YamauchiC., 2008,MNRAS, 391,700
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+ 5. HopkinsA.M.,ConnollyA. J.,HaarsmaD. B.,CramL. E.,200 1,AJ, 122,288
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+ 6. HuangJ.-S.,et al.,2007,ApJ, 664,840
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+ 7. KennicuttR. C.,Jr., 1998,ARA&A,36,189
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+ 8. LagacheG., DoleH.,PugetJ.-L.,2003,MNRAS, 338,555
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+ 9. LeFloc’hE.,etal., 2005,ApJ,632,169
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+ 10. MagnelliB., et al.2009,A&A,496,57
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+ 11. Pérez-GonzálezP. G.,etal., 2005,ApJ,630,82
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+ 12. RushB., MalkanM. A.,SpinoglioL.,1993,ApJS,89,1
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+ 13. SchiminovichD.,et al.,2005,ApJ, 619,L47
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+ 14. Wada T.,et al.,2008,PASJ, 60,517
1001.0008.txt ADDED
@@ -0,0 +1,322 @@
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
1
+ arXiv:1001.0008v2 [hep-th] 6 Jan 2010Multi-Stream Inflation: Bifurcations and Recombinations i n the Multiverse
2
+ Yi Wang∗
3
+ Physics Department, McGill University, Montreal, H3A2T8, Canada
4
+ In this Letter, we briefly review the multi-stream inflation s cenario, and discuss its implications in
5
+ the string theory landscape and the inflationary multiverse . In multi-stream inflation, the inflation
6
+ trajectory encounters bifurcations. If these bifurcation s are in the observable stage of inflation, then
7
+ interesting observational effects can take place, such as do main fences, non-Gaussianities, features
8
+ and asymmetries in the CMB. On the other hand, if the bifurcat ion takes place in the eternal stage
9
+ of inflation, it provides an alternative creation mechanism of bubbles universes in eternal inflation,
10
+ as well as a mechanism to locally terminate eternal inflation , which reduces the measure of eternal
11
+ inflation.
12
+ I. INTRODUCTION
13
+ Inflation [1] has become the leading paradigm for the
14
+ very early universe. However, the detailed mechanism
15
+ for inflation still remains unknown. Inspired by the pic-
16
+ ture of string theory landscape [2], one could expect that
17
+ the inflationary potential has very complicated structure
18
+ [3]. Inflation in the string theory landscape has impor-
19
+ tantimplicationsinbothobservablestageofinflationand
20
+ eternal inflation.
21
+ The complicated inflationary potentials in the string
22
+ theory landscape open up a great number of interest-
23
+ ing observational effects during observable inflation. Re-
24
+ searchesinvestigatingthecomplicatedstructureofthein-
25
+ flationary potential include multi-stream inflation [4, 5],
26
+ quasi-single field inflation [6], meandering inflation [7],
27
+ old curvaton [8], etc.
28
+ Thestringtheorylandscapealsoprovidesaplayground
29
+ for eternal inflation. Eternal inflation is an very early
30
+ stage of inflation, during which the universe reproduces
31
+ itself, so that inflation becomes eternal to the future.
32
+ Eternal inflation, if indeed happened (for counter ar-
33
+ guments see, for example [9]), can populate the string
34
+ theory landscape, providing an explanation for the cos-
35
+ mological constant problem in our bubble universe by
36
+ anthropic arguments.
37
+ In this Letter, we shall focus on the multi-stream infla-
38
+ tion scenario. Multi-stream inflation is proposed in [4].
39
+ And in [5], it is pointed out that the bifurcations can
40
+ lead to multiverse. Multi-stream inflation assumes that
41
+ during inflation there exist bifurcation(s) in the inflation
42
+ trajectory. For example, the bifurcations take place nat-
43
+ urally in a random potential, as illustrated in Fig. 1. We
44
+ briefly review multi-stream inflation in Section II. The
45
+ details of some contents in Section II can be found in
46
+ [4]. We discuss some new implications of multi-stream
47
+ inflation for the inflationary multiverse in Section III.
48
49
+ FIG. 1. In this figure, we use a tilted random potential to
50
+ mimic a inflationary potential in the string theory landscap e.
51
+ One can expect that in such a random potential, bifurcation
52
+ effects happens generically, as illustrated in the trajecto ries
53
+ in the figure.
54
+ FIG. 2. One sample bifurcation in multi-stream inflation.
55
+ The inflation trajectory bifurcates into AandBwhen the
56
+ comoving scale k1exits the horizon, and recombines when
57
+ the comoving scale k2exits the horizon.
58
+ II. OBSERVABLE BIFURCATIONS
59
+ In this section, we discuss the possibility that the bi-
60
+ furcation of multi-stream inflation happens during the
61
+ observable stage of inflation. We review the production
62
+ of non-Gaussianities, features and asymmetries [4] in the2
63
+ FIG. 3. In multi-stream inflation, the universe breaks up
64
+ into patches with comoving scale k1. Each patch experienced
65
+ inflation either along trajectories AorB. These different
66
+ patches can be responsible for the asymmetries in the CMB.
67
+ CMB, and investigate some other possible observational
68
+ effects.
69
+ To be explicit, we focus on one single bifurcation, as
70
+ illustrated in Fig. 2. We denote the initial (before bifur-
71
+ cation) inflationary direction by ϕ, and the initial isocur-
72
+ vature direction by χ. For simplicity, we let χ= 0 before
73
+ bifurcation. When comoving wave number k1exits the
74
+ horizon, the inflation trajectory bifurcates into Aand
75
+ B. When comoving wave number k2exits the horizon,
76
+ the trajectories recombines into a single trajectory. The
77
+ universe breaks into of order k1/k0patches (where k0de-
78
+ notes the comoving scale of the current observable uni-
79
+ verse), each patch experienced inflation either along tra-
80
+ jectories AorB. The choice of the trajectories is made
81
+ by the isocurvature perturbation δχat scale k1. This
82
+ picture is illustrated in Fig. 3.
83
+ We shall classify the bifurcation into three cases:
84
+ Symmetric bifurcation . If the bifurcation is symmetric,
85
+ in other words, V(ϕ,χ) =V(ϕ,−χ), then there are two
86
+ potentially observable effects, namely, quasi-single field
87
+ inflation, and a effect from a domain-wall-like objects,
88
+ which we call domain fences.
89
+ As discussed in [4], the discussion of the bifurcation
90
+ effect becomes simpler when the isocurvature direction
91
+ has mass of order the Hubble parameter. In this case,
92
+ except for the bifurcation and recombination points, tra-
93
+ jectoryAand trajectory Bexperience quasi-single field
94
+ inflation respectively. As there are turnings of these tra-
95
+ jectories, the analysis in [6] can be applied here. The
96
+ perturbations, especially non-Gaussianities in the isocur-
97
+ vature directions are projected onto the curvature direc-
98
+ tion, resultingin a correctionto the powerspectrum, and
99
+ potentially large non-Gaussianities. As shown in [6], the
100
+ amount of non-Gaussianity is of order
101
+ fNL∼P−1/2
102
+ ζ/parenleftbigg1
103
+ H∂3V
104
+ ∂χ3/parenrightbigg/parenleftBigg˙θ
105
+ H/parenrightBigg3
106
+ , (1)
107
+ whereθdenotes the angle between the true inflation di-
108
+ rection and the ϕdirection.
109
+ As shown in Fig. 3, the universe is broken into patches
110
+ during multi-stream inflation. There arewall-likebound-
111
+ aries between these patches. During inflation, theseboundaries are initially domain walls. However, after
112
+ the recombination of the trajectories, the tensions of
113
+ these domain walls vanish. We call these objects domain
114
+ fences. As is well known, domain wall causes disasters
115
+ in cosmology because of its tension. However, without
116
+ tension, domain fence does not necessarily cause such
117
+ disasters. It is interesting to investigate whether there
118
+ are observational sequences of these domain fences.
119
+ Nearly symmetric bifurcation If the bifurcation is
120
+ nearly symmetric, in other words, V(ϕ,χ)≃V(ϕ,−χ),
121
+ but not equal exactly, which can be achieved by a spon-
122
+ taneous breaking and restoring of an approximate sym-
123
+ metry, then besides the quasi-single field effect and the
124
+ domain fence effect, there will be four more potentially
125
+ observable effects in multi-stream inflation, namely, the
126
+ features and asymmetries in CMB, non-Gaussianity at
127
+ scalek1and squeezed non-Gaussianity correlating scale
128
+ k1and scale kwithk1< k < k 2.
129
+ The CMB power asymmetries are produced because,
130
+ as in Fig. 3, patches coming from trajectory AorBcan
131
+ have different power spectra PA
132
+ ζandPB
133
+ ζ, which are de-
134
+ termined by their local potentials. If the scale k1is near
135
+ to the scale of the observational universe k0, then multi-
136
+ stream inflation provides an explanation of the hemi-
137
+ spherical asymmetry problem [10].
138
+ The features in the CMB (here feature denotes extra
139
+ large perturbation at a single scale k1) are produced as
140
+ a result of the e-folding number difference δNbetween
141
+ two trajectories. From the δNformalism, the curvature
142
+ perturbation in the uniform density slice at scale k1has
143
+ an additional contribution
144
+ δζk1∼δN≡ |NA−NB|. (2)
145
+ These features in the CMB are potentially observable
146
+ in the future precise CMB measurements. As the addi-
147
+ tional fluctuation δζk1does not obey Gaussian distribu-
148
+ tion, there will be non-Gaussianity at scale k1.
149
+ Finally, there are also correlations between scale k1
150
+ and scale kwithk1< k < k 2. This is because the ad-
151
+ ditional fluctuation δζk1and the asymmetry at scale k
152
+ are both controlled by the isocurvature perturbation at
153
+ scalek1. Thus the fluctuations at these two scales are
154
+ correlated. As estimated in [4], this correlation results in
155
+ a non-Gaussianity of order
156
+ fNL∼δζk1
157
+ ζk1PA
158
+ ζ−PB
159
+ ζ
160
+ PA
161
+ ζP−1/2
162
+ ζ. (3)
163
+ Non-symmetric bifurcation If the bifurcation is not
164
+ symmetric at all, especially with large e-folding number
165
+ differences (of order O(1) or greater) along different tra-
166
+ jectories, the anisotropy in the CMB and the large scale
167
+ structure becomes too large at scale k1. However, in
168
+ this case, regions with smaller e-folding number will have
169
+ exponentially small volume compared with regions with
170
+ larger e-folding number. Thus the anisotropy can behave
171
+ in the form of great voids. We shall address this issue in
172
+ more detail in [11]. Trajectories with e-folding number3
173
+ difference from O(10−5) toO(1) in the observable stage
174
+ of inflation are ruled out by the large scale isotropy of
175
+ the observable universe.
176
+ At the remainderof this section, we would like to make
177
+ several additional comments for multi-stream inflation:
178
+ The possibility that the bifurcated trajectories never re-
179
+ combine. In this case, one needs to worry about the do-
180
+ main walls, which do not become domain fence during
181
+ inflation. These domain walls may eventually become
182
+ domain fence after reheating anyway. Another prob-
183
+ lem is that the e-folding numbers along different tra-
184
+ jectories may differ too much, which produce too much
185
+ anisotropies in the CMB and the large scale structure.
186
+ However, similar to the discussion in the case of non-
187
+ symmetric bifurcation, in this case, the observable effect
188
+ could become great voids due to a large e-folding number
189
+ difference. The case without recombination of trajectory
190
+ also has applications in eternal inflation, as we shall dis-
191
+ cuss in the next section.
192
+ Probabilities for different trajectories . In [4], we con-
193
+ sidered the simple example that during the bifurcation,
194
+ the inflaton will run into trajectories AandBwith equal
195
+ probabilities. Actually, this assumption does not need to
196
+ be satisfied for more general cases. The probability to
197
+ run into different trajectories can be of the same order
198
+ of magnitude, or different exponentially. In the latter
199
+ case, there is a potential barrier in front of one trajec-
200
+ tory, which can be leaped over by a large fluctuation of
201
+ theisocurvaturefield. Alargefluctuationoftheisocurva-
202
+ ture field is exponentially rare, resulting in exponentially
203
+ different probabilities for different trajectories. The bi-
204
+ furcation of this kind is typically non-symmetric.
205
+ Bifurcation point itself does not result in eternal infla-
206
+ tion. As is well known, in single field inflation, if the
207
+ inflaton releases at a local maxima on a “top of the hill”,
208
+ a stage of eternal inflation is usually obtained. However,
209
+ at the bifurcation point, it is not the case. Because al-
210
+ though the χdirection releases at a local maxima, the ϕ
211
+ direction keeps on rolling at the same time. The infla-
212
+ tiondirectionisacombinationofthesetwodirections. So
213
+ multi-stream inflation can coexist with eternal inflation,
214
+ but itself is not necessarily eternal.
215
+ III. ETERNAL BIFURCATIONS
216
+ In multi-stream inflation, the bifurcation effect may ei-
217
+ ther take place at an eternal stage of inflation. In this
218
+ case, it provides interesting ingredients to eternal infla-
219
+ tion. These ingredients include alternative mechanism to
220
+ producedifferentbubble universesandlocalterminations
221
+ for eternal inflation, as we shall discuss separately.
222
+ Multi-stream bubble universes . The most discussed
223
+ mechanisms to produce bubble universes are tunneling
224
+ processes, such as Coleman de Luccia instantons [12] and
225
+ Hawking Moss instantons [13]. In these processes, the
226
+ tunneling events, which are usually exponentially sup-
227
+ pressed, create new bubble universes, while most parts
228
+ FIG. 4. Cascade creation of bubble universes. In this figure,
229
+ we assume trajectory Ais the eternal inflation trajectory, and
230
+ trajectory Bis the non-eternal inflation trajectory.
231
+ of the spatial volume remain in the old bubble universe
232
+ at the instant of tunneling.
233
+ If bifurcations of multi-stream inflation happen dur-
234
+ ing eternal inflation, two kinds of new bubble universes
235
+ can be created with similar probabilities. In this case,
236
+ at the instant of bifurcation, both kinds of bubble uni-
237
+ verseshavenearlyequalspatialvolume. Withachangeof
238
+ probabilities, the measures for eternal inflation should be
239
+ reconsideredformulti-streamtype bubble creationmech-
240
+ anism.
241
+ If the inflation trajectories recombine after a period of
242
+ inflation, the different bubble universes will eventually
243
+ have the same physical laws and constants of nature. On
244
+ the other hand, if the different inflation trajectories do
245
+ not recombine, then the different bubble universes cre-
246
+ ated by the bifurcation will have different vacuum ex-
247
+ pectation values of the scalar fields, resulting to different
248
+ physical laws or constants of nature. It is interesting
249
+ to investigate whether the bifurcation effect is more ef-
250
+ fective than the tunneling effect to populate the string
251
+ theory landscape.
252
+ Note that in multi-stream inflation, it is still possi-
253
+ ble that different trajectorieshaveexponentiallydifferent
254
+ probabilities, as discussed in the previous section. In this
255
+ case, multi-stream inflation behaves similar to Hawking
256
+ Moss instantons during eternal inflation.
257
+ Local terminations for eternal inflation . It is possible
258
+ that during multi-stream inflation, a inflation trajectory
259
+ bifurcates in to one eternal inflation trajectory and one
260
+ non-eternal inflation trajectory with similar probability.
261
+ Inthiscase,theinflatonintheeternalinflationtrajectory
262
+ frequently jumps back to the bifurcation point, resulting
263
+ in a cascade creation of bubble universes, as illustrated
264
+ in Fig. 4. This cascade creation of bubble universes, if4
265
+ realized, is more efficient in producing reheating bubbles
266
+ than tunneling effects. Thus it reduces the measure for
267
+ eternal inflation.
268
+ There are some other interesting issues for bifurcation
269
+ in the multiverse. For example, the bubble walls may
270
+ be observable in the present observable universe, and the
271
+ bifurcations can lead to multiverse without eternal infla-
272
+ tion. These possibilities are discussed in [5].
273
+ IV. CONCLUSION AND DISCUSSION
274
+ To conclude, webriefly reviewedmulti-stream inflation
275
+ during observable inflation. Some new issues such as do-main fences and connection with quasi-single field infla-
276
+ tion are discussed. We also discussed multi-stream infla-
277
+ tion in the context of eternal inflation. The bifurcation
278
+ effect in multi-stream inflation provides an alternative
279
+ mechanism for creating bubble universes and populating
280
+ the string theory landscape. The bifurcation effect also
281
+ provides a very efficient mechanism to locally terminate
282
+ eternal inflation.
283
+ ACKNOWLEDGMENT
284
+ We thank Yifu Cai for discussion. This work was sup-
285
+ ported by NSERC and an IPP postdoctoral fellowship.
286
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+ [8] J. O. Gong and M. Sasaki, JCAP 0901, 001 (2009)
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+ [arXiv:0804.4488 [astro-ph]]. J. O. Gong, C. Lin andY. Wang, arXiv:0912.2796 [astro-ph.CO].
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+ [9] V. F. Mukhanov, L. R. W. Abramo and R. H. Bran-
306
+ denberger, Phys. Rev. Lett. 78, 1624 (1997)
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+ [arXiv:gr-qc/9609026]. Y. F. Cai and Y. Wang,
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+ JCAP0706, 022 (2007) [arXiv:0706.0572 [hep-th]].
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+ Q. G. Huang, M. Li and Y. Wang, JCAP 0709,
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+ 013 (2007) [arXiv:0707.3471 [hep-th]]. Y. Wang,
311
+ arXiv:0805.4520 [hep-th].
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+ [10] H. K. Eriksen, F. K. Hansen, A. J. Banday, K. M. Gorski
313
+ and P. B. Lilje, Astrophys. J. 605, 14 (2004) [Erratum-
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+ ibid.609, 1198 (2004)] [arXiv:astro-ph/0307507].
315
+ A. L. Erickcek, M. Kamionkowski and S. M. Carroll,
316
+ Phys. Rev. D 78, 123520 (2008) [arXiv:0806.0377
317
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318
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319
+ [12] S. R. Coleman and F. De Luccia, Phys. Rev. D 21, 3305
320
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+ [13] S. W. Hawking and I. G. Moss, Phys. Lett. B 110, 35
322
+ (1982).
1001.0009.txt ADDED
@@ -0,0 +1,389 @@
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
1
+ arXiv:1001.0009v1 [q-bio.BM] 30 Dec 2009Jamming proteins with slipknots and their free energy lands cape
2
+ Joanna I. Su/suppress lkowska1, Piotr Su/suppress lkowski2,3,4and Jos´ e N. Onuchic1
3
+ 1Center for Theoretical Biological Physics,
4
+ University of California San Diego,
5
+ Gilman Drive 9500, La Jolla 92037,
6
+ 2Physikalisches Institute and Bethe Center for Theoretical Physics,
7
+ Universit¨ at Bonn, Nussallee 12, 53115 Bonn, Germany
8
+ 3California Institute of Technology, Pasadena, CA 92215,
9
+ 4Institute for Nuclear Studies,
10
+ Ho˙ za 69, 00-681 Warsaw, Poland
11
+ Theoretical studies of stretching proteins with slipknots reveal a surprising growth of their un-
12
+ folding times when the stretching force crosses an intermed iate threshold. This behavior arises as
13
+ a consequence of the existence of alternative unfolding rou tes that are dominant at different force
14
+ ranges. Responsible for longer unfolding times at higher fo rces is the existence of an intermediate,
15
+ metastable configuration where the slipknot is jammed. Simu lations are performed with a coarsed
16
+ grained model with further quantification using a refined des cription of the geometry of the slip-
17
+ knots. The simulation data is used to determine the free ener gy landscape (FEL) of the protein,
18
+ which supports recent analytical predictions.
19
+ PACS numbers: 87.15.ap, 87.14.E-, 87.15.La, 82.37.Gk, 87. 10.+e
20
+ The large increase in determining new protein struc-
21
+ tures has led to the discovery of several proteins with
22
+ complicated topology. This new fact has arised the ques-
23
+ tion if their energy landscape and the folding mechanism
24
+ is similar to typical proteins. One class of such proteins
25
+ includes knotted proteins which comprise around 1% of
26
+ all structures deposited in the PDB database [1, 2]. A
27
+ related class of proteins contains more subtle geometric
28
+ configurations called slipknots [3, 4]. Recent theoretical
29
+ studies using structure-based models (where native con-
30
+ tacts are dominant) suggest that slipknot-like conforma-
31
+ tions act like intermediates during the folding of knotted
32
+ proteins [5]. This entire new mechanism is consistent
33
+ with energy landscape theory (FEL) and the funnel con-
34
+ cept [7, 8]. It was shown that the slipknot formation
35
+ reduces the topological barrier. Complementing regular
36
+ folding studies, additional information about the land-
37
+ scape was obtained by mechanical manipulation of the
38
+ knotted protein with atomic force microscopy [9] both
39
+ experimentally in [10, 11] and theoretically in [12, 13, 14].
40
+ For example, [12] it has been showen that unfolding pro-
41
+ ceeds via a series of jumps between various metastable
42
+ conformations, a mechanism opposite to the smooth un-
43
+ folding in knotted homopolymers.
44
+ Motivated by these early results, we now propose a uni-
45
+ fied picture for the mechanical unfolding of proteins with
46
+ slipknots. In this Letter this question is addressed by
47
+ explaining the role of topological barriers along their me-
48
+ chanical unfolding pathways. Supported by our previous
49
+ results that knotted proteins can still have a minimally
50
+ frustrated funnel-like energy landscape, structure-based
51
+ theoretical coarse-grained models are used [15] to ana-
52
+ lyze the behavior of a slipknot protein under stretching.
53
+ Studies are performed for the α/β class protein thymi-dine kinase (PDB code: 1e2i [17]).
54
+ 2 3 4 5F/LBracket1Ε/Slash1/Angstrom/RBracket17.27.57.88.1logΤ
55
+ FIG. 1: Dependence of the unfolding times τon the stretch-
56
+ ing force Ffor 1e2i (solid line, in red). In this Letter we
57
+ describe this mechanism as a superposition of two unfolding
58
+ pathways: I for small forces (dashed (lower) line, in blue),
59
+ and II for intermidiate and large forces (dashed-dotted (up -
60
+ per) line, green).
61
+ Most of our analysis is based on stretching simulations
62
+ under constant force [16]. The crucial signature for this
63
+ process is the overall unfolding time from the beginning
64
+ of the stretching until the protein fully unfolds. Normally
65
+ one expects that the transition between the native and
66
+ the unfolded basins to be limited by overcoming the free
67
+ energy barrier, which gets effectively reduced upon an
68
+ application of a stretching force. The rate by which this
69
+ barrier is reduced depends on the distance between the
70
+ unfolded basin and the top of the barrier measured along
71
+ the stretching coordinate x. This idea was first devel-
72
+ oped in the phenomenological model of Bell [18], which
73
+ states that the unfolding time τdecreases exponentially
74
+ with applied stretching force Fasτ(F) =τ0e−Fx
75
+ kBT. A2
76
+ refined analysis performed in ref. [19] revealed that this
77
+ dependence is more complicated but still monotonically
78
+ decreasing.
79
+ The unfolding times for 1e2i measured in our simula-
80
+ tions are shown as the red curve in Fig. 1. In contrast to
81
+ the above expectations, increasing the force in the range
82
+ 3-3.5ǫ/˚A surprisingly results in a larger stability of the
83
+ protein. ǫis the typical effective energy of tertiary na-
84
+ tive contacts that is consistent with the value ǫ/˚A≃71
85
+ pN derived in [15]. A solution for this paradox is accom-
86
+ plished by realizing that unfolding is dominated by two
87
+ distinct, alternative routes that are dominant at different
88
+ force regimes. A routing switch occurs when threshold is
89
+ crossed between weak and intermediate forces. At higher
90
+ forces, mechanical unfolding is dominated by a route that
91
+ involves a jammed slipknot. This jamming gives rise to
92
+ the unexpected dependence of unfolding time on applied
93
+ force. Characterizing this mechanism is the central goal
94
+ of this Letter.
95
+ FIG. 2: A slipknot (left) consists of a threaded loop (k1−k2,
96
+ in red) which is partialy threaded through a knotting loop
97
+ (k2−k3, in blue). An example of a protein configuration with
98
+ a tightened slipknot is shown in the right panel.
99
+ To describe the evolution of a slipknot quantitatively
100
+ requires a refined description. A slipknot is character-
101
+ ized by the three points shown in Fig. 2. The first
102
+ pointk1is determined by eliminating amino acids con-
103
+ secutively from one terminus until the knot configura-
104
+ tion is reached (which can be detected e.g. by applying
105
+ the KMT algorithm [20]). The two additional points,
106
+ k2andk3, correspond to the ends of this knot. In the
107
+ native state the protein 1e2i contains a slipknot with
108
+ k1= 10,k2= 128,k3= 298. These three points divide
109
+ the slipknot into two loops, which are called the knotting
110
+ loop and thethreaded loop . The former one is the loop of
111
+ the trefoil knot and the latter one is threaded through the
112
+ knotting loop. Unfolding of the slipknot upon stretch-
113
+ ing depends on the relative shrinking velocity of these
114
+ two loops (see Fig. 3). When the threaded loop shrinks
115
+ faster than the knotting loop, the slipknot unties. In the
116
+ opposite case the slipknot gets (temporarily) tightened
117
+ or jammed, resulting in a metastable state associated
118
+ to a local minimum in the protein’s FEL. Upon further
119
+ stretching, this configuration eventually also unties. The
120
+ evolution of both loops of the slipknot is encoded in thetime dependence of the points k1,k2,k3, see Fig. 3.
121
+ pathway I pathway II
122
+ catch−bonds slip−bondspathway II
123
+ catch−bonds slip−bondspathway I
124
+ FIG. 3: The behavior of the slipknot during stretching (top)
125
+ is determined by the relative behavior of its two loops, en-
126
+ coded in the time dependence of k1,k2andk3(bottom). If the
127
+ threaded loop shrinks faster than the knotting loop, k1merges
128
+ withk2(bottom left) and the slipknot untightens (pathway I,
129
+ top left). If the knotting loop shrinks faster, k2approaches k3
130
+ (bottom right, ≃14000τ) and the slipknot gets temporarily
131
+ tightened (pathway II, top right). This is a metastable stat e
132
+ which can eventually untie further stretching , with k1finally
133
+ merging with k2(bottom right, ≃19000τ). Kinetic stud-
134
+ ies were performed slightly above folding temperature usin g
135
+ overdamped Langevin dynamics with typical folding times of
136
+ 10000τ.
137
+ Before discussing the stretching of 1e2i, we explain why
138
+ a slipknot formed by a uniformly elastic polymer should
139
+ smoothly unfold under stretching. To simplify the discus-
140
+ sion we approximate the threaded and knotting loops by
141
+ circles of size RtandRk. These two loops shrink during
142
+ stretching and, when the threaded one eventually van-
143
+ ishes, the slipknot gets untied. If both loops have similar
144
+ sizes, the slipknot is very unstable and unties immedi-
145
+ ately. When the threaded loop is much larger than the
146
+ knotting one, Rt>> R k, untightening can be explained
147
+ as follows. The elastic energy associated to local bend-
148
+ ing is proportional to the square of the curvature. If the
149
+ loop is approximated by a circle of radius R, then its local
150
+ curvature is constant and equals R−1. The total elastic
151
+ energy is/contintegraltext
152
+ dsR−2∼R−1[21]. From the assumption
153
+ Rt>> R kwe conclude that upon stretching it is ener-
154
+ getically favorable to decrease Rtrather than Rk. This
155
+ happens until both radii become equal and then, just as
156
+ above, the slipknot gets very unstable and untightens. In
157
+ this discussion we have not yet taken into account that
158
+ when a slipknot is stretched some parts of a chain slide
159
+ along each other. This effect could be incorporated by in-
160
+ cluding the friction generated by the sliding [22]. But in
161
+ the slipknot the sliding region associated with the knot-
162
+ ting loop is much longer than the region associated to3
163
+ the threaded loop. Thus this effect results in a faster
164
+ tightening of the threaded rather than the knotting loop,
165
+ facilitating even more the untightening of the slipknot.
166
+ The above argument should apply to slipknots in
167
+ biomolecules because they are characterized by a per-
168
+ sistence length that in principle is simply related to their
169
+ elasticity [23]. For DNA this effect is described by worm-
170
+ like-chain models (WLC) [24] and it has been confirmed
171
+ experimentally. Although WLC models are too simple
172
+ to describe the protein general behavior, they are use-
173
+ ful in some limited applications. Thus at first sight one
174
+ might expect that slipknots in proteins should smoothly
175
+ untie upon stretching. Proteins, however, are much more
176
+ complicated than DNA or uniformly elastic polymers.
177
+ The presence of stabilizing native tertiary contacts leads
178
+ to a jumping character during stretching [12]. In addi-
179
+ tion their bending energy is not uniform along the chain
180
+ due to the heterogeneity of the amino-acid sequence. As
181
+ a consequence it turns out that the intuition obtained
182
+ through the above analysis of polymers or WLC models
183
+ is misleading.
184
+ 2 3 4 5F/LBracket1Ε/Slash1/Angstrom/RBracket10.51Prob/LParen1pathway I/RParen1
185
+ FIG. 4: Dependence on the applied stretching force of the
186
+ probability of choosing pathway I rather than II (see Fig. 3) .
187
+ This varying probability leads to the complicated dependen ce
188
+ of the total unfolding time on the stretching force observed in
189
+ Fig. 1.
190
+ Our analysis of the evolution of the endpoints k1,k2,k3
191
+ (Fig. 3, bottom) reveals that for various stretching forces
192
+ unfolding proceeds along two distinct pathways (Fig. 3,
193
+ top). In pathway I the slipknot smoothly unties, which
194
+ is observed for relatively weak forces. At intermediate
195
+ forces pathway II starts to dominate and the knotting
196
+ loop can shrink tightly before the threaded one vanishes.
197
+ In this regime the protein gets temporarily jammed (Fig.
198
+ 3, right), leading to much longer unfolding times (catch
199
+ pathway). The probability of choosing pathway I at dif-
200
+ ferent forces is shown in Fig. 4. This pathway competi-
201
+ tion explains the nontrivial total unfolding time depen-
202
+ dence observed in Fig. 1.
203
+ The two different pathways I and II arise from com-
204
+ pletely different unfolding mechanisms. Pathway I starts
205
+ and continues mostly from the C-terminal side, along
206
+ 16α, 15β, 14α, 13β, 12(helices bundle), 11 α(here the
207
+ number denotes a consecutive secondary structure ascounted from N-terminal, and αorβspecifies whether
208
+ this is a helix or a β-sheet; for more details about the
209
+ structure of 1e2i see the PDB). This is followed by unfold-
210
+ ing of helices 11 α, 10αthat allows breaking of the con-
211
+ tacts inside the β-sheet created by the N-terminal, with
212
+ unfolding proceeding also from the N-terminal. Pathway
213
+ II also starts from the C-terminal but rapidly (as soon
214
+ as helix 15 is unfolded) switches to the N-terminal. In
215
+ this case, differently from pathway I, the β-sheet from
216
+ the N-terminal unfolds even before 13 β. These scenarios
217
+ indicate that the pathway I should be dominant at weak
218
+ forces since they are not sufficient to break the β-sheet
219
+ during first steps of unfolding. The jammed pathway is
220
+ typical only if stretching forces are sufficiently strong for
221
+ unfolding to proceed from the two terminals of the pro-
222
+ tein.
223
+ A similar phenomenon was firstly proposed in ref. [25]
224
+ and referred to as catch-bonds. Experimental evidence
225
+ suggesting this mechanism was first observed for adhe-
226
+ sion complexes [26, 27]. Using AFM, at large forces the
227
+ ligand-receptor pair becomes entangled and therefore ex-
228
+ pands the unfolding time. A theoretical description of
229
+ this mechanism was given in ref. [28, 29, 30].
230
+ The kinetic data can also be used to determine the as-
231
+ sociated free energy landscape (FEL) [7]. In an initial
232
+ simplification we associate the barriers along the stretch-
233
+ ing coordinate as the the kinetic bottlenecks during the
234
+ mechanical unfolding event. Generalizing Bell’s model,
235
+ a recent description of two-state mechanical unfolding in
236
+ the presence of a single transition barrier has been devel-
237
+ oped in [19], with the rate equation
238
+ τ(F) =τ0/parenleftBig
239
+ 1−νFx†
240
+ ∆G/parenrightBig1−1/ν
241
+ e−∆G
242
+ kBT/parenleftbig
243
+ 1−(1−νFx†/∆G)1/ν/parenrightbig
244
+ ,
245
+ (1)
246
+ whereνencodes the shape of the barrier. Here x†denotes
247
+ the distance between the barrier and the unfolded basin
248
+ (in a first approximation it can be regarded as Findepen-
249
+ dent) and lies on the reaction coordinate along the AFM
250
+ pulling direction. It can be experimentally determined
251
+ by measuring how the stretching force modulates the un-
252
+ folding times τ. The height of the barrier is denoted by
253
+ ∆G. Fig. 1 (unfolding times are given by solid red line)
254
+ shows that this single barrier theory is not sufficient for
255
+ the full range of forces. As described before, in the higher
256
+ force regime, additional basins have to be included in the
257
+ energy landscape. Models with several metastable basins
258
+ have been called multi-state FEL models [31]. Evidence
259
+ supporting the need of multi-states FEL was confirmed
260
+ by AFM experiments in different systems [32, 33].
261
+ To construct a multi-state FEL that incorporates two
262
+ unfolding pathways I and II we use a linear combina-
263
+ tion of eq. (1)-like expressions with different shapes and
264
+ barrier heights. Each one of them essentially accounts
265
+ for the distinct barrier along a relevant unfolding route.
266
+ Fitting the stretching data to eq. (1) with a cusp-like4
267
+ 2.5 33.5 4F/LBracket1Ε/Slash1/Angstrom/RBracket16.67.6logΤ
268
+ N U I
269
+ FIG. 5: Pathway II with two barriers. Left: dependence of the
270
+ unfolding time on the applied force with the data and the fit
271
+ to the formula (1) for the first maximum (lower, in green) and
272
+ for the second maximum (upper, in blue). Right: schematic
273
+ free energy landscape for this pathway, with jammed slipkno t
274
+ in a minimum between two barriers.
275
+ ν= 1/2 approximation (another possibility ν= 2/3 for
276
+ the cubic potential in general leads to similar results [19])
277
+ determines accurately the location and the height of the
278
+ potential barriers. Pathway II involves two barriers: first
279
+ until the moment of creation of the intermediate which
280
+ is followed the untieing event. They are characterized by
281
+ (x1,∆G1) and (x2,∆G2) arising respectively from the
282
+ lower and upper fits in Fig. 5 (left). The superposition
283
+ of these two fits gives the overall mean unfolding time for
284
+ pathway II (dotted-dashed curve in green in Fig. 1). For
285
+ the ordinary slipknot unfolding (pathway I), the results
286
+ xIandGIarise from the dashed blue curve in Fig. 1.
287
+ This analysis leads to the results
288
+ x1= 2.3kBT˚A
289
+ ǫ, x2= 0.7kBT˚A
290
+ ǫ, xI= 1.4kBT˚A
291
+ ǫ,
292
+ ∆G1= 8.0kBT, ∆G2= 4.2kBT, ∆GI= 4.7kBT.
293
+ We conclude that the free energy landscape consists of
294
+ two “valleys”. The force-dependent probability of choos-
295
+ ing one of the valleys during stretching depends on the
296
+ details of the protein structure. It is determined from our
297
+ simulations as shown in Fig. 4. Using these probability
298
+ values and the parameters above for xand ∆G, we can
299
+ accurately represent the simulation data using a linear
300
+ combination of equations of the form (1). This agreement
301
+ supports our analytical analysis and generalizes eq. (1)
302
+ for the full of range forces. In addition it demonstrates
303
+ that structure-based models sufficiently capture the ma-
304
+ jor geometrical properties of a slipknotted protein. A
305
+ schematic representation of the free energy landscape for
306
+ pathway II is shown in Fig. 5 (right).
307
+ Summarizing, we have analyzed the process of tighten-
308
+ ing of the slipknot in protein 1e2i and determined the cor-
309
+ responding free energy landscape. Its main feature is the
310
+ presence of a metastable configuration with a tightened
311
+ slipknot, which is observed for sufficiently large pulling
312
+ forces. This phenomenon does not exist for uniformly
313
+ elastic polymers. In this Letter we concentrated on pro-
314
+ tein 1e2i but similar behavior has also been observed forother proteins with slipknots, e.g. 1p6x. Our results
315
+ provide testable predictions that can now be verified by
316
+ AFM stretching experiments.
317
+ We appreciate useful comments of O. Dudko. The
318
+ work of J.S. was supported by the Center for Theo-
319
+ retical Biological Physics sponsored by the NSF (Grant
320
+ PHY-0822283) with additional support from NSF-MCB-
321
+ 0543906. P.S. acknowledges the support of Hum-
322
+ boldt Fellowship, DOE grant DE-FG03-92ER40701FG-
323
+ 02, Marie-Curie IOF Fellowship, and Foundation for Pol-
324
+ ish Science.
325
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+ [24] C. Bouchiat, M.D. Wang, J.-F. Allemand, T. Strick, S.M.5
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+ Blockand and V. Croquette, Biophys. J. 76409 (1999).
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+ [25] M. Dembo, D.C. Torney, K. Saxman and D. Hammer.
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+ Proc. R. Soc. Lond. B. Biol. Sci. 234 (1988).
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+ McEver and Ch. Zhu, Science319630 (2003).
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+ Vicini, E. Sokurenko and V. Vogel, Biophys. J. 90753
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+ 051910 (2002).
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1
+ arXiv:1001.0011v2 [cond-mat.mes-hall] 16 Apr 2010Guided plasmons in graphene p-njunctions
2
+ E. G. Mishchenko,1A. V. Shytov∗,1and P. G. Silvestrov2
3
+ 1Department of Physics and Astronomy, University of Utah, Sa lt Lake City, Utah 84112, USA
4
+ 2Theoretische Physik III, Ruhr-Universit¨ at Bochum, 44780 Bochum, Germany
5
+ Spatial separation of electrons and holes in graphene gives rise to existence of plasmon waves
6
+ confined to the boundary region. Theory of such guided plasmo n modes within hydrodynamics of
7
+ electron-hole liquid is developed. For plasmon wavelength s smaller than the size of charged domains
8
+ plasmon dispersion is found to be ω∝q1/4. Frequency, velocity and direction of propagation of
9
+ guided plasmon modes can be easily controlled by external el ectric field. In the presence of magnetic
10
+ field spectrum of additional gapless magnetoplasmon excita tions is obtained. Our findings indicate
11
+ that graphene is a promising material for nanoplasmonics.
12
+ PACS numbers: 73.23.-b, 72.30.+q
13
+ Introduction . Breakthrough progress in synthesis and
14
+ characterization has made graphene [2] a promising ob-
15
+ ject for nanoelectronics. Operation of graphene-based
16
+ transistors [3] and other components would rely on the
17
+ propertiesofits single-particle excitations–electronsand
18
+ holes. However, one can also envisage a completely dif-
19
+ ferent set of applications which employ collective excita-
20
+ tions, such as plasmons. Currently, plasmon excitations
21
+ in metallic structures are a subject of nanoplasmonics, a
22
+ new field which has emerged at the confluence of optics
23
+ and condensed matter physics with one of the aims be-
24
+ ing the developing of plasmon-enhanced high resolution
25
+ near-field imaging methods [4, 5]. Another objective is
26
+ possible utilization of plasmons in integrated optical cir-
27
+ cuits. However, perspectives of graphene for nanoplas-
28
+ monics are largely unexplored since plasmon modes of
29
+ graphene flakes have not been addressed so far. As our
30
+ results indicate a great amount of control over graphene
31
+ plasmon properties makes it a very promising material
32
+ for applications.
33
+ Fundamentally, the spectrum of collective chargeoscil-
34
+ lations reflects the long-rangenature of Coulomb interac-
35
+ tion. In conventional two dimensional systems, such as
36
+ those created in semiconducting heterostructures, plas-
37
+ mons are gapless, ω2(q) = 2πe2nq/m∗, withnandm∗
38
+ being electron density and effective mass, respectively
39
+ [6]. Such oscillations can be treated hydrodynamically.
40
+ In clean graphene at zero temperature the plasmon fre-
41
+ quency,ω2∝ |EF|, vanishes with decreasing the doping
42
+ levelEF. It has been argued [7] that the interaction be-
43
+ tweenelectronsandholesinthefinalstatecanmodifythe
44
+ response functions of Dirac fermions and open up a pos-
45
+ sibility for the propagation of charge oscillations at low
46
+ frequencies ω < qv, wherevis electron velocity. Still, hy-
47
+ drodynamic( ω > qv)analogofconventionalplasmonsre-
48
+ mains absent unless either temperature is non-zero [8] or
49
+ graphene is driven away from the charge neutrality point
50
+ by doping or gating [9]. Expectedly, in both cases plas-
51
+ mon spectrum has the conventional form, ω(q)∝q1/2.
52
+ In the present paper we investigate spectra of hydro-
53
+ dynamic plasmons in spatially inhomogeneous grapheneflakes. Realistic graphene samples are typically subject
54
+ to disorder potential and mechanical strain [10] that lead
55
+ totheformationofchargedelectronandholepuddles[11]
56
+ with boundaries between nandpregions being the lines
57
+ ofzerochemicalpotential. Moreover,controlled p-njunc-
58
+ tions can be made with the help of metallic gates [12].
59
+ Alsop-njunctions can be created by applying electric
60
+ field within the plane of a graphene flake, see Fig. 1a.
61
+ The field separates electrons and holes spatially in a way
62
+ that allows control of both the amount of induced charge
63
+ (and thus plasmon frequency) and spatial orientation of
64
+ the junction (the direction of plasmon propagation).
65
+ b)2d 2d
66
+ Ea)
67
+ 0n n
68
+ p p
69
+ FIG.1: Twotypesofgraphene p-njunctions: a)field-induced,
70
+ b) gate-induced. Dot-dashed line indicates boundary betwe en
71
+ electron and hole regions and, correspondingly, the direct ion
72
+ of plasmon propagation. In case of field-induced junction it
73
+ is controlled by the direction of external electric field E0.
74
+ Below, we demonstrate that such p-njunctions can
75
+ guide plasmons. We show the existence of charge oscil-
76
+ lations which are localized at the junction and have the
77
+ amplitude decaying with the distance to the junction.
78
+ For wavelengths shorter than the width of the charged
79
+ domains, we find the plasmon spectrum of the form,
80
+ ω2
81
+ n(q) =αne2v
82
+ ¯h/radicalbigg
83
+ q|ρ′
84
+ 0|
85
+ e, (1)
86
+ whereρ′
87
+ 0is the gradient of equilibrium charge density
88
+ at the junction, vis electron velocity, and n= 0,1,2,...2
89
+ enumerates the solutions. The lowest mode has α0=
90
+ 4√
91
+ 2πΓ(3/4)/Γ(1/4)≈3.39.
92
+ Below we derive this result and discuss plasmon prop-
93
+ erties for the two types of p-njunctions: electric field
94
+ controlled and gate controlled, as shown in Fig. 1.
95
+ Hydrodynamics of charge density oscillations. We uti-
96
+ lize the hydrodynamic approach to describe the motion
97
+ of charged Dirac fermions. The rate of change of electric
98
+ current density Jdue to dynamic electric field Efollows
99
+ from the usual intra-band Drude conductivity with the
100
+ corresponding density of states [13],
101
+ ˙J(r,t) =e2
102
+ π¯h2|µ(r)|E(r,t), (2)
103
+ determined by the local value of chemical potential µ(r)
104
+ as measured from the Dirac point (positive for electrons
105
+ and negative for holes). Electric current is related to the
106
+ variation of charge density δρby means of the continuity
107
+ equation,
108
+ δ˙ρ(r,t)+∇·J(r,t) = 0. (3)
109
+ Finally, the variation of charge density produces electric
110
+ field according to the Coulomb law [14],
111
+ E(r,t) =−∇/integraldisplay
112
+ d2r′δρ(r′,t)
113
+ |r−r′|. (4)
114
+ Equations (2)-(4) give a closed system for plasmon exci-
115
+ tations in graphene flakes. We apply it to a p-njunction
116
+ created in a strip infinite along the y-axis (direction of
117
+ plasmon propagation). Using the Fourier representation,
118
+ δρ(r,t) =δρ(x)exp(iqy−iωt), and eliminating Eand
119
+ Jwe arrive at the equation for the oscillating part of
120
+ electron density,
121
+ ω2δρ(x)+2e2v√π¯h/braceleftBigg
122
+ d
123
+ dx/radicalbigg
124
+ |ρ0(x)|
125
+ ed
126
+ dx−q2/radicalbigg
127
+ |ρ0(x)|
128
+ e/bracerightBigg
129
+ ×/integraldisplayd
130
+ −ddx′δρ(x′)K0(|q||x−x′|) = 0,(5)
131
+ HereK0is the modified Bessel function and 2 dis
132
+ the width of graphene flake. Within the Thomas-
133
+ Fermi approximation equilibrium charge density ρ0(x)
134
+ is related to the chemical potential via ρ0(x) =
135
+ −sgn(µ)eµ2(x)/π¯h2v2(electron charge is taken to be
136
+ −e). This follows from the condition that the electro-
137
+ chemical potential µ(x)−eφ(x) is constant throughout
138
+ the system. The solutions of Eq. (5) will now be consid-
139
+ ered for large and small plasmon momenta separately.
140
+ Short wavelength, q≫1/d. In this case the decay
141
+ of plasmon density δρ(x) occurs over a distance much
142
+ smaller than the width of the system and the limits
143
+ of integration in Eq. (5) can be extended to infinity.
144
+ Assuming (cf. Eq. (11) below) the linear dependence,
145
+ ρ0(x) =ρ′
146
+ 0x, we observe that the integro-differentialequation (5) acquires obvious scaling property. Intro-
147
+ ducing the variable ξ=qxwe arrive at the plasmon
148
+ spectrum in the form (1), with dimensionless constants
149
+ αndetermined from the eigenvalue problem:
150
+ −2√π/parenleftbiggd
151
+ dξ/radicalbig
152
+ |ξ|d
153
+ dξ−/radicalbig
154
+ |ξ|/parenrightbigg
155
+ ×/integraldisplay∞
156
+ −∞dξ′δρ(n)(ξ′)K0(|ξ−ξ′|) =αnδρ(n)(ξ).(6)
157
+ Interestingly, this integro-differential equation allows a
158
+ complete analytic solution, though the detailed analysis
159
+ is beyond the scope of this paper. Our main findings
160
+ are as follows. Solutions are enumerated by n= 0,1,2...
161
+ with even/odd numbers corresponding to even/odd den-
162
+ sity profile, δρ(n)(−ξ) = (−1)nδρ(n)(ξ). Surprisingly,
163
+ eigenvalues are doubly-degenerate and given by
164
+ α2n=α02n+1
165
+ 4n+1·3·7··(4n−1)
166
+ 1·5··(4n−3), α2n+1=α2n.(7)
167
+ At large distances all modes have exponential depen-
168
+ dence,δρ(n)(ξ)∼e−|ξ|, while at |ξ| ≪1 even and
169
+ odd solutions exhibit different behavior, δρ(even)∼1−
170
+ const/radicalbig
171
+ |ξ|andδρ(odd)∼sign(ξ)//radicalbig
172
+ |ξ|. The first pair
173
+ of solutions (belonging to the lowest eigenvalue α0) in
174
+ the Fourier representation δρ(n)(k) =/integraltext
175
+ dξδρ(n)(ξ)eikξ
176
+ acquires a simple form:
177
+ δρ(0)(k)∝1
178
+ (1+k2)3/4, δρ(1)(k)∝k
179
+ (1+k2)3/4.(8)
180
+ Long wavelength, q≪1/d. In contrast to the above
181
+ result (1) plasmon spectrum at small qis sensitive to a
182
+ specific realization of the p-njunction. We address the
183
+ long-wavelength behavior of plasmons in field controlled
184
+ junctions. We expect this case to be of more interest,
185
+ in addition it allows a more complete description. Be-
186
+ fore analyzing plasmons in this structure, we discuss the
187
+ equilibrium density profile. As shown in Fig. 1a the flake
188
+ of width 2 dis placed in external electric field E0applied
189
+ along the x-direction. The equilibrium density distribu-
190
+ tionρ(x) is found from,
191
+ E0x+sgn(x)¯hv
192
+ e/radicalbiggπ
193
+ e|ρ0(x)|+2/integraldisplayd
194
+ 0dx′ρ0(x′)lnx+x′
195
+ |x−x′|= 0,
196
+ (9)
197
+ where it is used that ρ0(x) =−ρ0(−x). Prior to solv-
198
+ ing Eq. (9) it is instructive to analyze validity of the
199
+ semiclassical approach. The first condition implies that
200
+ the change of the electron wavelength is smooth on the
201
+ scale of itself, d/dx(¯hv/µ)≪1. Estimating µ(x)∼eE0x
202
+ we obtain that the distance to the p-njunction line
203
+ (x= 0) should exceed the characteristic electric field
204
+ lengthlE=/radicalbig
205
+ e/E0≪x. The second condition requires
206
+ that the electron wavelength is small compared with the
207
+ width of the system, d≫¯hv/µ. Noting that in graphene3
208
+ ¯hv∼e2we can rewrite this second condition simply as
209
+ lE≪d. Thus, the Thomas-Fermi equation (9) for the
210
+ equilibrium charge density and the hydrodynamic equa-
211
+ tion (5) for its variation are applicable as long as
212
+ lE≪d, q≪1/lE. (10)
213
+ However, the ratio of qand 1/dcan be arbitrary. For a
214
+ moderate external electric field ∼104V/m the value of
215
+ electric length lE∼0.4µm and the first of the conditions
216
+ (10) is satisfied easily for micron-sized samples.
217
+ AnalyticsolutionofEq.(9)ispossiblewhenthe second
218
+ term is small, in which case the charge density is [15]
219
+ ρ0(x) =E0x√
220
+ d2−x2. (11)
221
+ Substituting this expression back into Eq. (9) we ob-
222
+ serve that the second term is indeed negligible as long
223
+ asx≫l2
224
+ E/d. This is assured whenever the condi-
225
+ tions (10) are satisfied. It is also worth pointing out
226
+ that Eq. (11) justifies the linear approximation for the
227
+ charge density used in deriving Eq. (1) for q≫1/d, with
228
+ ρ′
229
+ 0/e= 1/(l2
230
+ Ed).
231
+ We now turn to the analysis of plasma oscillations
232
+ propagating on top of the density distribution, Eq. (11).
233
+ For small plasmon momenta, q≪1/d, electric field ex-
234
+ tends beyond the width of the flake and the equation (5)
235
+ needs to be supplemented with the boundary condition,
236
+ which ensures that electric field (and thus the current)
237
+ vanishes at the edges, x=±d:
238
+ P/integraldisplayd
239
+ −ddxδρ(x)
240
+ x±d= 0. (12)
241
+ The spectrum of the lowest symmetric mode can be most
242
+ easily found by integrating Eq. (5) across the width of
243
+ the flake. The first term in the brackets will then van-
244
+ ish exactly due to the boundary condition (12). The
245
+ remaining integral can now be calculated to the log-
246
+ arithmic accuracy with the help of the approximation
247
+ K0(q|x−x′|) =−lnq|x−x′|:
248
+ /integraldisplayd
249
+ −ddx/radicalbigg
250
+ |ρ0(x)|
251
+ eln(q|x−x′|)≈2dΓ2(3/4)
252
+ lE√πln(qd).
253
+ (13)
254
+ Eqs. (5) and (13) combine to give the equation, [ ω2−
255
+ ω2
256
+ 0(q)]/integraltextd
257
+ −ddxδρ(x) = 0, that yields the dispersion of the
258
+ gapless symmetric plasmon,
259
+ ω2
260
+ 0(q) = Γ2(3/4)4e2vd
261
+ π¯hlEq2ln(1/qd),(14)
262
+ reminiscent of the plasmon spectrum in quasi-one-
263
+ dimensional wires, The remaining modes, n≥1, are
264
+ gapped. For these modes/integraltextd
265
+ −ddxδρ(x) = 0 and simple
266
+ procedure of integrating Eq. (5) over the width of theflake is not useful. Instead, the equation for the n-th fre-
267
+ quency gap can be obtained by setting q= 0 in Eq. (5).
268
+ We observe that
269
+ ω2
270
+ n(0) =βne2v
271
+ ¯hlEd, (15)
272
+ whereβnare the eigenvalues of the equation,
273
+ 2√πd
274
+ dξ/radicalbig
275
+ |ξ|
276
+ (1−ξ2)1/4/integraldisplay1
277
+ −1dξ′δρ(n)(ξ′)
278
+ ξ−ξ′=βnδρ(n)(ξ).(16)
279
+ The zeroth mode β0= 0, see Eq. (14), is found ana-
280
+ lytically: δρ(0)∝1//radicalbig
281
+ 1−ξ2. It describes charge dis-
282
+ tribution in the strip in response to a (uniform along
283
+ xdirection and smooth along y-direction) change of its
284
+ chemical potential [16]. Other solutions of Eq. (16) are
285
+ found numerically:
286
+ β1= 1.41, β2= 6.49, β3= 6.75,... (17)
287
+ With increasing nthe eigenmodes of integro-differential
288
+ equation (16) oscillate faster, but in generaldo not follow
289
+ the oscillation theorem familiar from quantum mechan-
290
+ ics. In particular, the solutions with n= 0 andn= 3 are
291
+ even while n= 1,n= 2 are odd [17].
292
+ Finally, we mention the case of a gate-controlled p-n
293
+ junction, Fig.1b. Theequilibriumdensityprofileislinear
294
+ nearx= 0 and saturates for large |x|[18]. Eq. (1) is still
295
+ applicable for q >1/d. In the limit q <1/done should
296
+ take into account the screening of long-range Coulomb
297
+ interaction by metallic gates. In this case the logarithm
298
+ in the spectrum of the gapless plasmon disappears, and
299
+ the lowest mode Eq. (14) becomes sound-like.
300
+ Magnetoplasmons. If external magnetic field Bis ap-
301
+ plied perpendicularly to the plane of graphene the plas-
302
+ mon spectra acquire new modes. The equation of motion
303
+ (2) should now be modified to include the Lorentz force,
304
+ ˙J(r,t) =e2
305
+ π¯h2|µ(x)|E(r,t)−ev2
306
+ cµ(x)J×B.(18)
307
+ The relative coefficient between electric and magnetic
308
+ terms in this equation follows from the expression for
309
+ the Lorentz force acting on a single particle. The last
310
+ term has opposite sign for electrons and holes. Note that
311
+ the frequency of cyclotron motion ωB(x) =ev2B/cµ(x)
312
+ in graphene p-njunctions is position-dependent. The
313
+ remaining equations (3)-(4) are intact in the presence of
314
+ magnetic field. The boundary condition requires now the
315
+ vanishing of the normal component of electric current at
316
+ the boundary, rather than simply vanishing of the elec-
317
+ tric field, as in Eq. (12). Eliminating JandEwe arrive
318
+ at the generalization of equation (5),
319
+ δρ(x)+2e2
320
+ π/braceleftbigg
321
+ q2Z −q
322
+ ω(ωBZ)′−d
323
+ dxZd
324
+ dx/bracerightbigg
325
+ ×/integraldisplayd
326
+ −ddx′δρ(x′)K0(|q||x−x′|) = 0,(19)4
327
+ whereZ(x) =|µ(x)|/(ω2
328
+ B(x)−ω2).
329
+ The most interesting effect described by Eq. (19) is
330
+ the appearance of a set of new modes, chiral magne-
331
+ toplasmons, similar to those considered in Ref. [19] for
332
+ conventional 2D electron systems with smooth bound-
333
+ aries. To find their dispersion in strong magnetic fields,
334
+ whenω≪ωB(x) (the exact condition is given below),
335
+ one should retain only the second term in Eq. (19).
336
+ Noticing that ( ωBZ)′=πl2
337
+ Bρ′
338
+ 0(x)/e=πl2
339
+ B/(l2
340
+ Ed), where
341
+ lB=/radicalbig
342
+ ¯hc/eBis the magnetic length, we arrive at the
343
+ integral equation
344
+ −2c
345
+ Bq
346
+ ωdρ0(x)
347
+ dx/integraldisplayd
348
+ −ddx′δρ(x′)K0(|q||x−x′|) =δρ(x).(20)
349
+ SinceK0is positive, propagation of magnetoplasmons
350
+ withq >0is quenched, indicative oftheir chiral property
351
+ [20]. As seen from Eq. (20), the plasmon density δρ(x) is
352
+ concentratedwhere ρ′
353
+ 0(x) isthestrongest. Thederivative
354
+ of the charge density in field-induced junctions (11) fea-
355
+ tures strong singularitynearthe edges of the flake. Thus,
356
+ low-frequency magnetoplasmon spectrum is strongly de-
357
+ pendent on the microscopic regularization of this singu-
358
+ lar behavior and is, therefore, beyond the scope of the
359
+ Thomas-Fermi approximation used throughout this pa-
360
+ per.
361
+ Thegate-induced junctions, however, allow a rather
362
+ simple analytical description of these modes if we ap-
363
+ proximate that ρ′
364
+ 0(x) =e/l2
365
+ Edfor|x| ≤dandρ′
366
+ 0(x) = 0
367
+ for|x|> d. The oscillating density δρ(x) then vanishes
368
+ for|x|> d. The solution inside the strip, |x| ≤d, can
369
+ be easily found for q≫1/d, where one can assume the
370
+ range of integration in Eq. (20) to be infinite. The eigen-
371
+ functions of Eq. (20) are simply given by sin[ q⊥(x+d)],
372
+ with the values of q⊥=πn/2ddetermined from the con-
373
+ dition,δρq(±d) = 0. The spectrum of magnetoplasmons
374
+ is then found to be,
375
+ ωn(q) =−2πe2l2
376
+ B
377
+ ¯hl2
378
+ Edq/radicalbig
379
+ q2+π2n2/4d2, n= 1,2...(21)
380
+ The magnetoplasmon spectrum (21) is derived under
381
+ the assumption that magnetic field is strong, ωB(d)≫ω,
382
+ which implies that lB≪lE. In order to neglect the first
383
+ and third terms in the brackets in Eq. (19) one has to
384
+ ensure that q≪(lE/lB)4/d. This condition might turn
385
+ out to be more orless restrictivethan the hydrodynamics
386
+ condition q≪1/lE, depending on the particular value of
387
+ the ratio lB/lE. Note that the smallness of this ratio is
388
+ not in contradiction to the non-quantized description of
389
+ electron motion in magnetic filed. The latter is valid as
390
+ long as the filling factor is large, eEd≫ωB(d), which
391
+ means that lB≫l2
392
+ E/d. For magnetic field ∼1T, and
393
+ lB∼25nm, using the estimate below Eq. (10) that lE∼
394
+ 400nm we conclude that the width of the flake should
395
+ exceedd >10µm. The magnetoplasmon modes (21) are
396
+ ∼(lB/lE)2slowerthan electrons. Note that these modesare undamped since single-particle excitations cannot be
397
+ induced at frequencies below cyclotron frequency ωB.
398
+ Conclusions . Graphene p-njunctions are among the
399
+ most simple and promising applications of this material.
400
+ Single-electron properties of p-njunctions have been ex-
401
+ tensively studied. In the present paper we investigated
402
+ their collective excitations both with and without mag-
403
+ netic field. We anticipate that plasmon modes will be
404
+ crucial for the optical response of graphene nanostruc-
405
+ tures and realistic samples containing electron-hole pud-
406
+ dles. High degree of experimental control should make
407
+ them of special interest to nanoplasmonics and electron-
408
+ ics. Among the most promising applications of plasmons
409
+ inp-njunctions we envisage a possibility of a “plasmon
410
+ transistor” [4]. In particular, by simply switching the
411
+ direction of electric field from across the flake to along
412
+ it (and back) the propagation of plasmons can be facil-
413
+ itated (or prevented). In addition, as follows from the
414
+ above Eqs. (1), (11), the plasmon velocity can be con-
415
+ trolled with simple change in the magnitude of electric
416
+ field. This is in a sharp contrast to plasmons in metal-
417
+ lic nanostructures, whose spectra are typically fixed once
418
+ devices are fabricated.
419
+ Acknowledgments. Useful discussions with M. Raikh
420
+ and O. Starykh are gratefully acknowledged. This
421
+ work was supported by DOE, Grant No. DE-FG02-
422
+ 06ER46313. P.G.S. was supported by the SFB TR 12.
423
+ [*] Present address: School of Physics, University of Exete r,
424
+ EX4 4QL, U.K.
425
+ [2] M. Wilson, Phys. Today 59, No. 1, 21 (2006).
426
+ [3] A.K. Geim and K.S. Novoselov, Nature Mater. 6, 183
427
+ (2007).
428
+ [4] H.A. Atwater, Sci. Am. 296, 56 (2007).
429
+ [5] S.A. Maier, Plasmonics: Fundamentals and Applications
430
+ (Springer, New York, 2007).
431
+ [6] F. Stern, Phys. Rev. Lett. 18, 546 (1967).
432
+ [7] S. Gangadharaiah, A.M. Farid, and E.G. Mishchenko,
433
+ Phys. Rev. Lett. 100, 166802 (2008).
434
+ [8] O. Vafek, Phys. Rev. Lett. 97, 266406 (2006).
435
+ [9] E.H. Hwang and S. Das Sarma, Phys. Rev. B 75, 205418
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+ (2007).
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+ [10] A.H. Castro Neto et al.,Rev. Mod. Phys. 81, 109 (2009).
438
+ [11] J. Martin et al.,Nature Physics 4, 144 (2008).
439
+ [12] J.R. Williams, L. DiCarlo, and C.M. Marcus, Science
440
+ 317, 638 (2007).
441
+ [13] Rigorous derivation of Eq. (2) is based on the “rela-
442
+ tivistic” stress energy-momentum tensor, see L.D. Lan-
443
+ dau and E. M. Lifshitz, Fluid Mechanics , Butterworth-
444
+ Heinemann, Oxford (1987), Ch. 15; M. Mueller, L. Fritz,
445
+ S. Sachdev, and J. Schmalian, arXiv:0810.3657.
446
+ [14] In the case of gate controlled junctions the image charg es
447
+ induced at the gates should be included into Eq. (4).
448
+ [15] T.A. Sedrakyan, E.G. Mishchenko, and M.E. Raikh,
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+ Phys. Rev. B 74, 235423 (2006).
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+ [16] P.G. SilvestrovandK.B. Efetov, Phys.Rev.B 77, 155436
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+ (2008).5
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+ [17] In addition even and odd solutions with n >0 have dif-
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+ ferent singular behavior at |ξ| ≪1:δρ(even)∼/radicalbig
454
+ |ξ|,
455
+ δρ(odd)∼sign(ξ)//radicalbig
456
+ |ξ|. Atξ→ ±1 all solutions diverge
457
+ asδρ∼1//radicalbig
458
+ 1−ξ2.
459
+ [18] L.M. Zhang and M.M. Fogler, Phys. Rev. Lett. 100,116804 (2008).
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+ [19] I.L. Aleiner and L.I. Glazman, Phys. Rev. Lett. 72, 2935
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+ (1994).
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+ (1985).
1001.0012.txt ADDED
@@ -0,0 +1,1189 @@
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
1
+ arXiv:1001.0012v2 [astro-ph.EP] 20 Dec 2010Draft version May 20, 2018
2
+ Preprint typeset using L ATEX style emulateapj v. 8/13/10
3
+ THE STATISTICS OF ALBEDO AND HEAT RECIRCULATION ON HOT EXOPL ANETS
4
+ Nicolas B. Cowan1,2, Eric Agol2,
5
+ Draft version May 20, 2018
6
+ ABSTRACT
7
+ If both the day-side and night-side effective temperatures of a pla net can be measured, it is possible
8
+ to estimate its Bond albedo, 0 < AB<1, as well as its day–night heat redistribution efficiency,
9
+ 0< ε <1. We attempt a statistical analysis of the albedo and redistribution efficiency for 24
10
+ transiting exoplanets that have at least one published secondary e clipse. For each planet, we show
11
+ how to calculate a sub-stellar equilibrium temperature, T0, and associated uncertainty. We then use
12
+ a simple model-independent technique to estimate a planet’s effective temperature from planet/star
13
+ flux ratios. We use thermal secondary eclipse measurements —tho se obtained at λ >0.8 micron—
14
+ to estimate day-side effective temperatures, Td, and thermal phase variations —when available— to
15
+ estimatenight-sideeffectivetemperature. Westronglyruleoutth e“nullhypothesis”ofasingle ABand
16
+ εforall 24planets. If wealloweachplanet to havedifferent paramete rs,we find that lowBond albedos
17
+ are favored ( AB<0.35 at 1σconfidence), which is an independent confirmation of the low albedos
18
+ inferred from non-detection of reflected light. Our sample exhibits a wide variety of redistribution
19
+ efficiencies. When normalized by T0, the day-side effective temperatures of the 24 planets describe
20
+ a uni-modal distribution. The two biggest outliers are GJ 436b (abno rmally hot) and HD 80606b
21
+ (abnormally cool), and these are the only eccentric planets in our sa mple. The dimensionless quantity
22
+ Td/T0exhibits no trend with the presence or absence of stratospheric in versions. There is also no
23
+ clear trend between Td/T0andT0. That said, the 6 planets with the greatest sub-stellar equilibrium
24
+ temperatures ( T >2400 K) have low ε, as opposed to the 18 cooler planets, which show a variety
25
+ of recirculation efficiencies. This hints that the very hottest trans iting giant planets are qualitatively
26
+ different from the merely hot Jupiters. We propose an explanation o f this trend based on how a
27
+ planet’s radiative and advective times scale with temperature: both timescales are expected to be
28
+ shorter for hotter planets, but the temperature-dependance of the radiative timescale is stronger,
29
+ leading to decreased heat recirculation efficiency.
30
+ Subject headings: methods: data analysis — (stars:) planetary systems —
31
+ 1.INTRODUCTION
32
+ Short-period exoplanets are expected to have atmo-
33
+ spheric compositions and dynamics that differ signifi-
34
+ cantly from Solar System giant planets3. These planets
35
+ orbit∼100×closer to their host stars than Jupiter does
36
+ from the Sun. As a result, they receive ∼104×more flux
37
+ andexperiencetidalforces ∼106×strongerthanJupiter.
38
+ In contrast to Jupiter, which releases roughly as much
39
+ power in its interior as it receives from the Sun, short-
40
+ period exoplanets have power budgets dictated by the
41
+ flux they receive from their host stars. Roughly speak-
42
+ ing, the stellar flux incident on a planet does one of two
43
+ things: it is reflected back into space, or advected else-
44
+ where on the planet and re-radiated at different wave-
45
+ lengths. The physical parameters that describe these
46
+ processes are the planet’s Bond albedo and redistribu-
47
+ tion efficiency.
48
+ 1.1.Albedo
49
+ 1CIERA Fellow, Northwestern University, 2131 Tech Dr,
50
+ Evanston, IL 60208
51
52
+ 2Astronomy Department, University of Washington, Box
53
+ 351580, Seattle, WA 98195
54
+ 3For our purposes a “short period” exoplanet is one where the
55
+ periastron distance is less than 0 .1 AU, regardless of its actual
56
+ period, and regardless of mass, which may range from Neptune -
57
+ sized to Brown Dwarf. They are all Class IV and V extrasolar
58
+ giant planets in the scheme of Sudarsky et al. (2003).Giant planets in the Solar System have albedos greater
59
+ than 50%because ofthe presenceofcondensedmolecules
60
+ (H2O, CH 4, NH3, etc.) in their atmospheres. Planets
61
+ with effective temperatures exceeding ∼400 K should be
62
+ cloud free, leading to albedos of 0.05–0.4 (Marley et al.
63
+ 1999). If pressure-broadenedNa and K opacity is impor-
64
+ tant at optical wavelengths (as it is for brown dwarfs,
65
+ Burrows et al. 2000), then the Bond albedos of hot
66
+ Jupiters may be less than 10% (Sudarsky et al. 2000).
67
+ But the very hottest planets, the so-called class V extra-
68
+ solar giant planets ( Teff>1500 K), might have very high
69
+ albedosdue to a high silicate cloud layer(Sudarsky et al.
70
+ 2000). For a planet whose albedo is dominated by
71
+ clouds (as opposed to Rayleigh scattering) the albedo
72
+ depends on the composition and size of cloud particles
73
+ (Seager et al. 2000).
74
+ Earlyattempts to observe reflected light from exoplan-
75
+ ets (Charbonneau et al. 1999; Collier Cameron et al.
76
+ 2002a; Leigh et al. 2003a,b; Rodler et al. 2008, 2010) in-
77
+ dicated that they might not be as reflective as Solar Sys-
78
+ tem gas giants (for a review, see Langford et al. 2010).
79
+ Measurements of HD 209458b taken with the Cana-
80
+ dian MOST satellite revealed a very low albedo ( <8%,
81
+ Rowe et al.2008), andit hassincebeentakenforgranted
82
+ that all short-period planets have albedos on par with
83
+ that of charcoal.
84
+ From the standpoint of the planet’s climate, the im-
85
+ portant factor is not the albedo at any one wavelength,2 Cowan & Agol
86
+ Aλ, but rather the integrated albedo, weighted by the in-
87
+ cident stellar spectrum, known as the Bond albedo and
88
+ denoted in this paper as AB. The relation between Aλ
89
+ and the planet’s Bond albedo is not trivial. If the albedo
90
+ is dominated by gray clouds, then the albedo at a sin-
91
+ gle wavelength can indeed be extrapolated to obtain AB.
92
+ For non-grayreflectance spectra, however, it is critical to
93
+ measureAλat the peak emitting wavelength of the host
94
+ startoobtainagoodestimateofthe planet’senergybud-
95
+ get. For example, as pointed out in Marley et al. (1999),
96
+ planets with identical albedo spectra, Aλ, mayhaveradi-
97
+ cally different ABdepending on the spectraltype oftheir
98
+ host stars.
99
+ 1.2.Redistribution Efficiency
100
+ The first few measurements of hot Jupiter phase vari-
101
+ ations showed signs that these planets are not all cut
102
+ from the same cloth. Harrington et al. (2006) and
103
+ Knutson et al. (2007a) quoted very different phase func-
104
+ tion amplitudes for the υAndromeda and HD 189733
105
+ systems. It was not clear whether the differences were
106
+ intrinsic to the planets, however, because the data
107
+ were taken with different instruments, at different wave-
108
+ lengths, and with very different observation schemes (in
109
+ any case, subsequent re-analysis of the original data and
110
+ newly aquired Spitzerobservations of υAndromeda b
111
+ paint a completely different picture of that system:
112
+ Crossfield et al. 2010).
113
+ The uniform study presented in Cowan et al. (2007),
114
+ on the other hand, showed that HD 179949b and
115
+ HD209458bexhibit significantlydifferentdegreesofheat
116
+ recirculation, confirming suspicions. But it was not clear
117
+ whether hot exoplanets were uni-modal or bi-modal in
118
+ redistribution: are HD 179949b and HD 209458b end-
119
+ members of a single distribution, or prototypes for two
120
+ fundamentally different sorts of exoplanets?
121
+ The presence or lack of a stratospheric tempera-
122
+ ture inversion (Hubeny et al. 2003; Fortney et al. 2006;
123
+ Burrows et al. 2007, 2008; Zahnle et al. 2009) on the
124
+ day-sides of exoplanets has been invoked to explain
125
+ a purported bi-modality in recirculation efficiency on
126
+ hot Jupiters (Fortney et al. 2008). The argument, sim-
127
+ ply put, is that optical absorbers high in the atmo-
128
+ sphere of extremely hot Jupiters (equilibrium temper-
129
+ atures greater than ∼1700 K) would absorb incident
130
+ photons where the radiative timescales are short, mak-
131
+ ingit difficult forthese planets torecirculateenergy. The
132
+ most robust detection of this temperature inversionis for
133
+ HD 209458b (Knutson et al. 2008), but this planet does
134
+ not exhibit a large day-night brightness contrast at 8 µm
135
+ (Cowan et al. 2007). So while temperature inversions
136
+ seem to exist in the majority of hot Jupiter atmospheres
137
+ (Knutson et al. 2010), their connection to circulation ef-
138
+ ficiency —if any— is not clear.
139
+ 1.3.Outline of Paper
140
+ It has been suggested (e.g., Harrington et al. 2006;
141
+ Cowan et al. 2007) that observations of secondary
142
+ eclipses and phase variations each constrain a combina-
143
+ tion of a planet’s Bond albedo and circulation efficiency.
144
+ But observations —even phase variations— at a single
145
+ waveband do little to constrain a planet’s energy bud-
146
+ get. In this work we show how observations in differentwavebands and for different planets can be meaningfully
147
+ combined to estimate these planetary parameters.
148
+ In§2 we introduce a simple model to quantify the
149
+ day-side and night-side energy budget of a short-period
150
+ planet, and show how a planet’s Bond albedo, AB, and
151
+ redistribution efficiency, ε, can be constrained by ob-
152
+ servations. In §3 we use published observations of
153
+ 24 transiting planets to estimate day-side and —where
154
+ appropriate—night-sideeffective temperatures. We con-
155
+ struct a two-dimensionaldistribution function in ABand
156
+ εin§4. We state our conclusions in §5.
157
+ 2.PARAMETERIZING THE ENERGY BUDGET
158
+ 2.1.Incident Flux
159
+ Short-period planets have a power budget entirely dic-
160
+ tated by the flux they receive from their host star,
161
+ which dwarfs tidal heating or remnant heat of forma-
162
+ tion. Following Hansen (2008), we define the equi-
163
+ librium temperature at the planet’s sub-stellar point:
164
+ T0(t) =Teff(R∗/r(t))1/2, whereTeffandR∗are the star’s
165
+ effective temperature and radius, and r(t) is the planet–
166
+ star distance (for a circular orbit ris simply equal to the
167
+ semi-major axis, a). For shorthand, we define the geo-
168
+ metrical factor a∗=a/R∗, which is directly constrained
169
+ by transit lightcurves (Seager & Mall´ en-Ornelas 2003).
170
+ The incident flux on the planet is given by Finc=
171
+ 1
172
+ 2σBT4
173
+ 0, and it is significant that this quantity has some
174
+ associated uncertainty. For a planet on a circular orbit,
175
+ the uncertainty in T0=Teff/√a∗is related —to first
176
+ order— to the uncertainties in the host star’s effective
177
+ temperature, and the geometrical factor:
178
+ σ2
179
+ T0
180
+ T2
181
+ 0=σ2
182
+ Teff
183
+ T2
184
+ eff+σ2
185
+ a∗
186
+ 4a2∗. (1)
187
+ For a planet with non-zero eccentricity, T0varies with
188
+ time, but we are only concerned with its value at su-
189
+ perior conjunction: secondary eclipse occurs at superior
190
+ conjunction, when we are seeing the planet’s day-side.
191
+ At that point in the orbit, the planet–star distance is
192
+ rsc=a(1−e2)/(1−esinω), whereeandωare the
193
+ planet’s orbital eccentricity and argument of periastron,
194
+ respectively.
195
+ For planets with non-zero eccentricity, the uncertainty
196
+ inT0is given by
197
+ σ2
198
+ T0
199
+ T2
200
+ 0=σ2
201
+ Teff
202
+ T2
203
+ eff+σ2
204
+ a∗
205
+ 4a2∗+/parenleftBig
206
+ e2cos2ω
207
+ 1−e2/parenrightBig
208
+ σ2
209
+ ecosω
210
+ +/parenleftBig
211
+ esinω
212
+ 1−e2−1
213
+ 2(1−esinω)/parenrightBig
214
+ σ2
215
+ esinω,(2)
216
+ whereσecosωandσesinωarethe observationaluncertain-
217
+ ties in the two components of the planet’s eccentricity4.
218
+ 2.2.Emergent Flux
219
+ At secondary eclipse, and in the absence of albedo or
220
+ energy circulation, the equilibrium temperature of a re-
221
+ gion on the planet depends on the normalized projected
222
+ 4This formulation is preferable to an error estimate based on σe
223
+ andσω, because the eccentricity and argument of periastron are
224
+ highlycorrelated inorbitalfits. Thatsaid, the uncertaint iesσecosω
225
+ andσesinωare often not included in the literature, in which case
226
+ we use a slightly different —and more conservative— formulat ion
227
+ of the error budget using σeandσω.Albedo and Heat Recirculation on Hot Exoplanets 3
228
+ distance,γ, from the center of the planetary disc as
229
+ T(γ) =T0(1−γ2)1/8. The thermal secondary eclipse
230
+ depth in this limit is given by:
231
+ Fday
232
+ F∗=/parenleftbiggRp
233
+ R∗/parenrightbigg2/parenleftbigghc
234
+ λkT0/parenrightbigg8/parenleftBig
235
+ ehc/λkT∗
236
+ b−1/parenrightBig
237
+ ×/integraldisplay(λkT0/hc)8
238
+ 0dx
239
+ exp(x−1/8)−1, (3)
240
+ whereT∗
241
+ bis the brightness temperature of the star at
242
+ wavelength λ.
243
+ In the no-circulation limit, then, the day-side emer-
244
+ gent spectrum is not exactly that of a blackbody, even
245
+ if each annulus has a blackbody spectrum. But these
246
+ differences are not important for the present study, since
247
+ we are concerned with bolometric flux. By integrating
248
+ Equation 3 over λ, one obtains the effective tempera-
249
+ tureoftheday-sideintheno-albedo,no-circulationlimit:
250
+ Tε=0= (2/3)1/4T0(see also Burrows et al. 2008; Hansen
251
+ 2008). Indeed, treatingtheplanet’sday-sideasauniform
252
+ hemisphere emitting at this temperature gives nearly the
253
+ same wavelength dependence as the more complex Equa-
254
+ tion 3. The Tε=0temperatures for our sample of 24 tran-
255
+ siting planets are shown in Table 1. These set the max-
256
+ imum possible day-side effective temperature we should
257
+ expect to measure.
258
+ The integrated day-side flux in the general —non-zero
259
+ circulation— case is more subtle: heat may be trans-
260
+ ported to the planet’s night-side, and/or to its poles. In
261
+ this paper we neglect the E-W asymetry in the planet’s
262
+ temperature map due to zonal flows and hence phase
263
+ offsets in the thermal phase variations. Under this as-
264
+ sumption, the day-night temperature contrast can more
265
+ directly be extracted from the observed thermal phase
266
+ variations.
267
+ In practice, manystudies haveadopted asingle param-
268
+ eter to represent bothzonal and meridional transport. It
269
+ is instructive to consider the apparent day-side effective
270
+ temperatures in variouslimits: uniform day-sidetemper-
271
+ ature andT= 0 on the night-side (this is often referred
272
+ to as the planet’s “equilibrium temperature”): Tequ=
273
+ (1/2)1/4T0; in the case of perfect longitudinal transport
274
+ but no latitudinal transport: Tlong= (8/(3π2))1/4T0;
275
+ and in the limit of a uniform temperature everywhere
276
+ on the planet: Tuni= (1/4)1/4T0.
277
+ Comparing the apparent day-side temperatures in the
278
+ three limits of circulation above leads to the following
279
+ simple parametrization of the day-side effective temper-
280
+ ature in terms of the planetary albedo, AB, and circula-
281
+ tion efficiency, ε:
282
+ Td=T0(1−AB)1/4/parenleftbigg2
283
+ 3−5
284
+ 12ε/parenrightbigg1/4
285
+ ,(4)
286
+ where 0< ε <1. Note that εis related to —but dif-
287
+ ferent from— the ǫused in (Cowan & Agol 2010). The
288
+ former is merely a parametrization of the observed disk-
289
+ integrated effective temperature, while the latter, which
290
+ can take values from 0 to ∞, is a precisely defined ratio
291
+ of radiative and advective timescales. The ǫ= 0 case is
292
+ precisely equal to the ε= 0 case, while the ǫ→ ∞limit
293
+ is equivalent to ǫ≈0.95.
294
+ Our definition of εis similar to the Burrows et al.(2006) definition of Pnandyieldsthe sameno-circulation
295
+ limit. But our ε= 1 limit produces a lower day-side
296
+ brightness than the Pn= 0.5 limit, because we as-
297
+ sume that the planet’s day-side has a uniform tempera-
298
+ ture distribution in that limit (for a discussion of differ-
299
+ ent redistribution parameterizations, see the appendix of
300
+ Spiegel & Burrows 2010).
301
+ In reality, efficient longitudinal transport (read: fast
302
+ zonalwinds) mayleadtomorebandingandthereforeless
303
+ efficient latitudinal transport. So one could argue that
304
+ in the limit of perfect day-night temperature homoge-
305
+ nization, both the day and night apparent temperatures
306
+ should beTd= (8/(3π2))1/4T0, in between the Burrows
307
+ et al. value of Td= (1/3)1/4T0and that suggested by
308
+ our parameterization, Td= (1/4)1/4T0. At moderate
309
+ day-night recirculation efficiencies, however, there is a
310
+ good deal of latitudinal transport (I. Dobbs-Dixon, priv.
311
+ comm.), so implicitly assuming a constant T∝cos1/4
312
+ latitudinal dependence —as done by Burrows et al.— is
313
+ not founded, either. The bottom line is that any single-
314
+ parameter implementation of advection is incapable of
315
+ capturing the real complexities involved, but longitudi-
316
+ nal transport is the dominant factor in determining day
317
+ and night effective temperatures.
318
+ Not withstanding the subtleties discussed above and
319
+ noting that cooling tends to latitudinaly homogenize
320
+ night-side temperatures (Cowan & Agol 2010), we get a
321
+ night-side temperature of:
322
+ Tn=T0(1−AB)1/4/parenleftBigε
323
+ 4/parenrightBig1/4
324
+ . (5)
325
+ Note thatTdandTnare the equator-weighted tempera-
326
+ tures of their respective hemispheres (ie, as seen by an
327
+ edge-on viewer). As such, they will tend to be slightly
328
+ higher than the hemisphere-averaged temperature, ex-
329
+ cept in the ε= 1 limit. This is also why the quantity
330
+ T4
331
+ d+T4
332
+ nis still a weak function of ε.
333
+ Fig. 1.— Different kinds of idealized observations constrain the
334
+ Bond albedo, ABand circulation efficiency, ε, differently. A mea-
335
+ surement of the secondary eclipse depth at optical waveleng ths is
336
+ a measure of albedo (solid line). A secondary eclipse depth a t
337
+ thermal wavelengths gives a joint constraint on albedo and r ecir-
338
+ culation (dotted line). A measurement of the night-side effe ctive
339
+ temperature from thermal phase variations yields a constra int (the
340
+ dashed line) nearly orthogonal to the day-side measurement .
341
+ In Figure 1 we show how different kinds of observa-4 Cowan & Agol
342
+ tions constrain ABandε. For this example, we chose
343
+ constraints consistent with AB= 0.2 andε= 0.3. The
344
+ solid line is a locus of constant AB; the dotted line is
345
+ the locus of constant Td/T0; the dashed line is a lo-
346
+ cus of constant Tn/T0. From this figure it is clear that
347
+ the measurements complement each other: measuring
348
+ two of the three quantities (Bond albedo, effective day-
349
+ side or night-side temperatures) uniquely determines the
350
+ planet’s albedo and circulation efficiency. When obser-
351
+ vations have some associated uncertainty, they define a
352
+ swath through the AB–εplane.
353
+ 3.ANALYSIS
354
+ 3.1.Planetary & Stellar Data
355
+ We begin by considering all the photometric obser-
356
+ vations of short-period exoplanets published through
357
+ November 2010, summarized in Table 1. We have dis-
358
+ carded photometric observations of non-transiting plan-
359
+ ets because of their unknown radius and orbital inclina-
360
+ tion5. This leaves us with 24 transiting exoplanets for
361
+ which there are observations in at least one waveband
362
+ at superior conjunction, and in some cases in multiple
363
+ wavebands and at multiple planetary phases.
364
+ Stellar and planetary data are taken from the Ex-
365
+ oplanet Encyclopedia (exoplanet.eu), and references
366
+ therein. We repeated parts of the analysis with the
367
+ Exoplanet Data Explorer database (exoplanets.org) and
368
+ found identical results, within the uncertainties. When
369
+ the stellar data are not available, we have assumed typi-
370
+ cal parameters for the appropriate spectral class, and so-
371
+ lar metallicity. Insofar as we are only concerned with the
372
+ broadband brightnesses of the stars, our results should
373
+ not depend sensitively on the input stellar parameters.
374
+ Knowing the stars’ Teff, loggand [Fe/H], we
375
+ use the PHOENIX/NextGen stellar spectrum grids
376
+ (Hauschildt et al. 1999) to determine their brightness
377
+ temperatures at the observed bandpasses. At each wave-
378
+ band for which eclipse or phase observations have been
379
+ obtained, we determine the ratio of the stellar flux to the
380
+ blackbodyfluxatthatgridstar’s Teff. Wethenapplythis
381
+ factor to the Teffof the observed star.
382
+ It is worth noting that the choice of stellar model leads
383
+ to systematic uncertainties in the planetary brightness
384
+ that are of order the photometric uncertainties. For
385
+ example, Christiansen et al. (2010) use stellar models
386
+ for HAT-P-7 from Kurucz (2005), while we use those
387
+ of Hauschildt et al. (1999). The resulting 8 µm bright-
388
+ ness temperatures for HAT-P-7b differ by as much as
389
+ 600 K, or slightly more than 1 σ. Our uniform use
390
+ of Hauschildt et al. (1999) models should alleviate this
391
+ problem, however.
392
+ 3.2.From Flux Ratios to Effective Temperature
393
+ The planet’s albedo and recirculation efficiency gov-
394
+ ern its effective day-side and night-side temperatures, Td
395
+ andTn, respectively. Observationally, we can only mea-
396
+ sure the brightness temperature, ideally at a number of
397
+ different wavelengths: Tb(λ). If one knew, a priori, the
398
+ 5For completeness, these are: τ-Bootis b, υ-Andromeda b,
399
+ 51 Peg b, Gl 876d, HD 75289b, HD 179949b and HD 46375b
400
+ (Charbonneau et al. 1999; Collier Cameron et al. 2002b;
401
+ Leigh et al. 2003a,b; Harrington et al. 2006; Cowan et al. 200 7;
402
+ Seager & Deming 2009; Crossfield et al. 2010; Gaulme et al. 201 0)emergent spectrum of a planet, one could trivially con-
403
+ vert a single brightness temperature to an effective tem-
404
+ perature. Alternatively, if observations were obtained at
405
+ a number of wavelengths bracketing the planet’s black-
406
+ body peak, it would be possible to estimate the planet’s
407
+ bolometric flux and hence its effective temperature in a
408
+ model-independent way (e.g., Barman 2008).
409
+ We adopt the latter empirical approach of converting
410
+ observed flux ratios into brightness temperatures, then
411
+ using these to estimate the planet’s effective tempera-
412
+ ture. The secondary eclipse depth in some waveband di-
413
+ vided by the transit depth is a direct measureofthe ratio
414
+ of the planet’s day-side intensity to the star’s intensity
415
+ at that wavelength, ψ(λ). Knowing the star’s brightness
416
+ temperature at a given wavelength, it is possible to com-
417
+ pute the apparent brightness temperature of the planet’s
418
+ day side:
419
+ Tb(λ) =hc
420
+ λk/bracketleftbigg
421
+ log/parenleftbigg
422
+ 1+ehc/λkT∗
423
+ b(λ)−1
424
+ ψ(λ)/parenrightbigg/bracketrightbigg−1
425
+ .(6)
426
+ On the Rayleigh-Jeans tail, the fractional uncertainty
427
+ in the brightness temperature is roughly equal to the
428
+ fractional uncertainty in the eclipse depth; on the Wien
429
+ tail, the fractional error on brightness temperature can
430
+ be smaller because the flux is very sensitive to tempera-
431
+ ture.
432
+ By the same token, a secondary eclipse depth and
433
+ phase variation amplitude at a given wavelength can be
434
+ combined to obtain a measure of the planet’s night-side
435
+ brightness temperature at that waveband.
436
+ Since the albedo and recirculation efficiency of the
437
+ planet are not known ahead of time, it is not immedi-
438
+ atelyobviouswhich wavelengthsaresensitiveto reflected
439
+ light and which are dominated by thermal emission. For
440
+ each planet, we compute the expected blackbody peak if
441
+ the planet has no albedo and no recirculation of energy,
442
+ λε=0= 2898/Tε=0µm. Insofar as real planets will have
443
+ non-zero albedo and non-zero recirculation, the day side
444
+ should never reach Tε=0, and the actual spectral energy
445
+ distributionwillpeakatslightlylongerwavelengths. The
446
+ coolest planet in our sample, Gl 436b, would exhibit a
447
+ blackbody peak at λε=0= 3.1µm, while the hottest
448
+ planet we consider, WASP-12b, has λε=0= 0.9µm.
449
+ In practice this means that ground-based near-IR and
450
+ space-based mid-IR (e.g., Spitzer) observations are as-
451
+ sumed to measure thermal emission, while space-based
452
+ optical observations (MOST, CoRoT, Kepler) may be
453
+ contaminated by reflected starlight.
454
+ In Figure2, wedemonstratetwo alternativetechniques
455
+ to convert an array of brightness temperatures, Tb(λ),
456
+ into an estimate of a planet’s effective temperature, Teff.
457
+ The solid black line shows a model spectrum of ther-
458
+ mal emission from Fortney et al. (2008), with an ef-
459
+ fective temperature of Teff= 1941 K shown with the
460
+ black dashed line. The expected blackbody peak of
461
+ the planet is marked with a vertical dotted line. The
462
+ red points are the expected brightness temperatures in
463
+ the J, H, and K sbands (crosses), as well as the IRAC
464
+ (asterisks) and MIPS (diamond) instruments on Spitzer
465
+ (Fazio et al. 2004; Rieke et al. 2004; Werner et al. 2004).
466
+ Since the majority of the observations of exoplanets have
467
+ been obtained with SpitzerIRAC, we focus on estimat-
468
+ ingTeffbasedonlyon brightness temperatures in thoseAlbedo and Heat Recirculation on Hot Exoplanets 5
469
+ Fig. 2.— The solid black line shows a model spectrum from
470
+ Fortney et al. (2008) including only thermal emission (ie: n o re-
471
+ flected light). The planet’s effective temperature is shown w ith the
472
+ black dashed line, while the expected wavelength of the blac kbody
473
+ peak of the planet is marked with a black dotted line. The red
474
+ points show the expected brightness temperatures in the J, H , and
475
+ Ksbands (crosses), as well as the IRAC (asterisks) and MIPS (di a-
476
+ mond) instruments on Spitzer. The linear interpolation technique
477
+ described in the text is shown with the red line.
478
+ four bandpasses.
479
+ Wien Displacement: The first approach is to simply
480
+ adopt the brightness temperature of the bandpass clos-
481
+ est to the planet’s blackbody peak (the black dotted
482
+ line). If only the four IRAC channels are available, the
483
+ best one can do is the 3.6 µm measurement, yielding
484
+ Teff= 1925 K. There is —however— some subtlety in
485
+ estimating the peak wavelength, as this is dependent on
486
+ knowing the planet’s temperature (and hence ABandε)
487
+ a priori.
488
+ Linear Interpolation: The linear interpolation tech-
489
+ nique, shown with the red line in Figure 2, obviates the
490
+ need for an estimate of the planet’s temperature. The
491
+ brightness temperature is assumed to be constant short-
492
+ ward of the shortest- λobservation, and longward of the
493
+ longest-λobservation. Between bandpasses, the bright-
494
+ ness temperature changes linearly with λ. As long as
495
+ the various brightness temperatures do not differ grossly
496
+ from one another, this technique implicitly gives more
497
+ weight to observations near the hypothetical blackbody
498
+ peak. The bolometric flux of this “model” spectrum is
499
+ then computed, and admits a single effective tempera-
500
+ ture, which is Teff= 1927 K for the current example.
501
+ Since we hope to apply our routine to planets with well
502
+ sampled blackbody peaks, we adopt the linear interpola-
503
+ tion technique, as it can make use of multiple brightness
504
+ temperature estimates near the peak.
505
+ Thetwotechniquesdescribedaboveproducesimilaref-
506
+ fective temperatures, though —unsurprisingly— neither
507
+ gives precisely the correct answer. But these system-
508
+ atic errors are comparable or smaller than the photo-
509
+ metric uncertainty in observations of individual bright-
510
+ ness temperatures (see Table 1). The best IR observa-
511
+ tions for the nearest, brightest planetary systems (e.g.,
512
+ HD 189733b and HD 209458b) lead to observational un-
513
+ certainties of approximately 50 K in brightness temper-
514
+ ature. For many planets, the uncertainty is 100–200 K.
515
+ By that metric, either the Wien displacement or the lin-
516
+ ear interpolation routines give adequate estimates of the
517
+ effective temperature, with errors of 16 K and 14 K, re-spectively.
518
+ Wemakeamorequantitativeanalysisofthesystematic
519
+ uncertainties involved in the Linear Interpolation tem-
520
+ perature estimates as follows. We produce 8800 mock
521
+ data sets: 100 realizations for 11 models and data in
522
+ up to 8 wavebands (J, H, K, IRAC, MIPS; Since this nu-
523
+ mericalexperiment choosesrandom bands from the eight
524
+ available, the results should not be very different if ad-
525
+ ditional wavebands are considered). We run our Linear
526
+ Interpolation technique on each of these and plot in Fig-
527
+ ure 3 the estimated day-side temperature normalized by
528
+ the actual model effective temperature versus the num-
529
+ ber of wavebands used in the estimate. The temperature
530
+ estimates cluster near Test/Teff= 1, indicating that the
531
+ technique is not significantly biased. The scatter in es-
532
+ timates decreases as more wavebands are used, from a
533
+ standard deviation of 7.6% if only a single brightness
534
+ temperature is used, down to 2.4% if photometry is ac-
535
+ quired in eight bands. We incorporate this systematic
536
+ error into our analysis by adding it in quadrature to
537
+ the observational uncertainties described in the follow-
538
+ ing paragraph. This has the desirable effect that planets
539
+ with fewer observations have a larger systematic uncer-
540
+ tainty on their effective temperature.
541
+ Fig. 3.— The Linear Interpolation technique for estimating day-
542
+ side effective as tested on a suite of eleven hot Jupiter spect ral
543
+ models provided by J.J. Fortney. The y-axis shows the estima ted
544
+ day-side effective temperature normalized by the actual mod el ef-
545
+ fective temperature. The x-axis represents the number of br ight-
546
+ ness temperatures used in the estimate. Each color correspo nds to
547
+ one of the eleven models used in the comparison. The black err or
548
+ bars represent the standard deviation in the normalized tem pera-
549
+ ture estimates.
550
+ Inpractice,wewouldliketopropagatethephotometric
551
+ uncertainties to the estimate of Teff. For the Wien Dis-
552
+ placement technique, this uncertainty propagates triv-
553
+ ially to the effective temperature. For the linear inter-
554
+ polation technique, a Monte Carlo can be used to esti-
555
+ mate the uncertainty in Teff: the input eclipse depths
556
+ are randomly shifted 1000 times in a manner consistent
557
+ with their photometric uncertainties —assuming Gaus-
558
+ sianerrors—andtheeffectivetemperatureisrecomputed
559
+ repeatedly. Thescatterintheresultingvaluesof Teffpro-
560
+ vides an estimate of the observational uncertainty in the
561
+ parameter, to which we add in quadrature the estimate
562
+ ofsystematicerrordescribedabove. The resultinguncer-
563
+ tainties are listed in Table 1. These uncertainties should6 Cowan & Agol
564
+ be compared to the uncertainties in Tε=0(also listed in
565
+ Table 1), which are computed using the uncertainty in
566
+ the star’s properties and the planet’s orbit.
567
+ There are two practical issues with the linear interpo-
568
+ lation temperature estimation technique. In some cases,
569
+ onlyupperlimitshavebeenobtained, thereforeonecould
570
+ setψ= 0, with the appropriate1-sigmauncertainty. But
571
+ this approach leads to huge uncertainties in Tefffor plan-
572
+ ets with a secondary eclipse upper-limit near their black-
573
+ body peak. Instead of “punishing” these planets, we opt
574
+ to not use upper-limits (though for completeness we in-
575
+ clude them in Table 1). Secondly, when multiple mea-
576
+ surements of an eclipse depth have been published for
577
+ a given waveband, we use the most recent observation,
578
+ indicated with a superscript “ e” in Table 1. In all cases
579
+ these observations either explicitly agree with their older
580
+ counterpart, or agree with the re-analyzed older data.
581
+ 4.RESULTS
582
+ 4.1.Looking for Reflected Light
583
+ For each planet, we use thermal observations (essen-
584
+ tially those in the J, H, K s, andSpitzerbands) to es-
585
+ timate the planet’s effective day-side temperature, Td,
586
+ and —when phase variations are available— Tn. These
587
+ values are listed in Table 1. In five cases (CoRoT-
588
+ 1b, CoRoT-2b, HAT-P-7b, HD 209458b, TrES-2b), sec-
589
+ ondary eclipses and/or phase variations have been ob-
590
+ tained at optical wavelengths. Such observations have
591
+ the potential to directly constrain the albedo of these
592
+ planets. One approach is to adopt the Tdfrom thermal
593
+ observations and calculate the expected contrast ratio at
594
+ optical wavelengths, under the assumption of blackbody
595
+ emission (see also Kipping & Bakos 2010). Insofar as
596
+ the observed eclipse depths are deeper than this calcu-
597
+ lated depth, one can invoke the contribution of reflected
598
+ light and compute a geometric albedo, Ag. If one treats
599
+ the planet as a uniform Lambert sphere, the geometric
600
+ albedo is related to the spherical albedo at that wave-
601
+ length byAλ=3
602
+ 2Ag. These values are listed in Table 1.
603
+ But reflected light is not the only explanation for an
604
+ unexpectedly deep optical eclipse. Alternatively, the
605
+ emissivity of the planets may simply be greater at op-
606
+ tical wavelengths than at mid-IR wavelengths, in agree-
607
+ mentwith realisticspectralmodelsofhotJupiters, which
608
+ predict brightness temperatures greater than Teffon the
609
+ Wien tail (see, for example, the Fortney et al. model
610
+ showninFigure2, whichdoesnotincludereflectedlight).
611
+ Note that this increasein emissivityshould occurregard-
612
+ less of whether or not the planet has a stratosphere: by
613
+ definition, the depth at which the optical thermal emis-
614
+ sion is emitted is the depth at which incident starlight
615
+ is absorbed, which will necessarily be a hot layer —
616
+ assuming the incident stellar spectrum peaks in the op-
617
+ tical.
618
+ Determining the albedo directly (ie: by observing re-
619
+ flected light) can be difficult for short period planets,
620
+ because there is no way to distinguish between reflected
621
+ and re-radiated photons. The blackbody peaks of the
622
+ star and planet often differ by less than a micron. There-
623
+ fore, unlike Solar System planets, these worlds do not
624
+ exhibit a minimum in their spectral energy distribution
625
+ between the reflected and thermal peaks. The hottest
626
+ —and therefore most ambiguous case— of the five tran-siting planets with optical constraints is HAT-P-7b. If
627
+ one takes the mid-IR eclipse depths at face value, the
628
+ planet has a day-side effective temperature of ∼2000 K.
629
+ When combined with the Kepler observations, one com-
630
+ putesanalbedoofgreaterthan50%. Thelargeday-night
631
+ amplitude seen in the Kepler bandpass is then simply
632
+ due to the fact that the planet’s night-side reflects no
633
+ starlight, and the cool day-side can be attributed to high
634
+ ABand/orε. If, on the other hand, one takes the op-
635
+ tical flux to be entirely thermal in origin ( Aλ= 0), the
636
+ day-side effective temperature is ∼2800 K. This is very
637
+ close to that planet’s Tε=0, leaving very little power left
638
+ for the night-side, again explaining the large day-night
639
+ contrast observed by Kepler. The truth probably lies
640
+ somewhere between these two extremes, but in any case
641
+ this degeneracy will be neatly broken with Warm Spitzer
642
+ observations: the two scenarios outlined above will lead
643
+ to small and large thermal phase variations, respectively.
644
+ It is telling that the only optical measurement in Table 1
645
+ that is unanimously considered to constrain albedo —
646
+ and not thermal emission— is the MOST observations
647
+ of HD 209458b (Rowe et al. 2008), the coolest of the five
648
+ transiting planets with optical photometric constraints.
649
+ The bottom line is that extracting a constraint on re-
650
+ flected light from optical measurements of hot Jupiters is
651
+ best done with a detailed spectral model. But even when
652
+ reflectedlightcanbedirectlyconstrained,convertingthis
653
+ constraint on Aλinto a constraint on ABalso requires
654
+ detailedknowledgeofboththestarandtheplanet’sspec-
655
+ tral energy distributions, making for a model-dependent
656
+ exercise.
657
+ 4.2.Populating the AB-εPlane
658
+ Setting aside optical eclipses and direct measurements
659
+ of albedo, we may use the rich near- and mid-IR data to
660
+ constrain the Bond albedo and redistribution efficiency
661
+ of short-period giant planets. We define a 20 ×20 grid in
662
+ ABandεand use Equations 4 & 5 to calculate the nor-
663
+ malized day-side and night-side effective temperatures,
664
+ Td/T0andTn/T0, at each grid point, ( i,j). For each
665
+ planet, we have an observational estimate of the day-side
666
+ effective temperature, and in three cases we also have an
667
+ estimate of the night-side effective temperature (as well
668
+ as associated uncertainties).
669
+ We first verifywhether ornot the observationsarecon-
670
+ sistent with a single ABandε. To evaluate this “null
671
+ hypothesis”, we compute the usual χ2=/summationtext24
672
+ i=1(model−
673
+ data)2/error2at each grid point. We use only the esti-
674
+ mates of day-side and (when available) night-side effec-
675
+ tive temperatures to calculate the χ2, giving us 27-2=25
676
+ degreesoffreedom. The“best-fit”has χ2= 132(reduced
677
+ χ2= 5.3), so the current observations strongly rule out
678
+ a single Bond albedo and redistribution efficiency for all
679
+ 24 planets.
680
+ For 21 of the 24 planets considered here, we construct
681
+ a two-dimensional distribution function for each planet
682
+ as follows:
683
+ PDF(i,j) =1/radicalbig
684
+ 2πσ2
685
+ de−(Td−Td(i,j))2/(2σd)2.(7)
686
+ This defines a swath through parameter space with the
687
+ same shape as the dotted line in Figure 1.
688
+ For the three remaining planets (HD 149026b,Albedo and Heat Recirculation on Hot Exoplanets 7
689
+ HD 189733b, HD 209458b), phase variation measure-
690
+ ments help break the degeneracy:
691
+ PDF(i,j) =1√
692
+ 2πσ2
693
+ de−(Td−Td(i,j))2/(2σd)2
694
+ ×1√
695
+ 2πσ2ne−(Tn−Tn(i,j))2/(2σn)2.(8)
696
+ Fig. 4.— The global distribution function for short-period exo-
697
+ planets in the AB–εplane. The gray-scale shows the sum of the
698
+ normalized probability distribution function for the 24 pl anets in
699
+ our sample. The data mostly consist of infrared day-side flux es,
700
+ leading to the dominant degeneracy (see first the dotted line in
701
+ Figure 1).
702
+ We create a two-dimensional normalized probability
703
+ distribution function (PDF) for each planet, then add
704
+ these together to create the global PDF shown in Fig-
705
+ ure 4. This is a democratic way of representing the data,
706
+ since each planet’s distribution contributes equally.
707
+ In Figures 5 and 6 we show the distribution functions
708
+ for the albedo and circulation of the 24 planets in our
709
+ sample,obtainedbymarginalizingtheglobalPDFofFig-
710
+ ure 4 over either ABorε.
711
+ Fig. 5.— The solid black line shows the projection of the 2-
712
+ dimensional probability distribution function (the gray- scale of
713
+ Figure 4) projected onto the ε-axis. The dashed line shows the
714
+ ε-distribution if one requires that all planets have Bond alb edos
715
+ less than 0.1; under this assumption, we see hints of a bimoda l
716
+ distribution in heat circulation efficiency.Fig. 6.— The solid black line shows the projection of the 2-
717
+ dimensionalprobabilitydistributionfunction (the gray- scale ofFig-
718
+ ure 4) projected onto the AB-axis. The cumulative distribution
719
+ function (not shown) yields a 1 σupper limit of AB<0.35.
720
+ The solid line in Figure 5 shows no evidence of bi-
721
+ modality in heat redistribution efficiency, although there
722
+ is a wide range of behaviors. The dashed line in Figure 5
723
+ shows theε-distribution if one requires the albedo to be
724
+ low,AB<0.1. There are then many high-recirculation
725
+ planets, since advection is the only way to depress the
726
+ day-side temperature in the absence of albedo. Inter-
727
+ estingly, the dashed line doesshow tentative evidence of
728
+ two separate peaks in ε: if short-period giant planets
729
+ have uniformly low albedos, then there appear to be two
730
+ modes of heat recirculation efficiency. We revisit this
731
+ idea below.
732
+ Figure 6 shows that planets in this sample are consis-
733
+ tent with a low Bond albedo. Note that this constraint
734
+ is based entirely on near- and mid-infrared observations,
735
+ and is thus independent from the claims of low albedo
736
+ based on searches for reflected light (Rowe et al. 2008,
737
+ and references therein). Furthermore, this is a constraint
738
+ on the Bond albedo, rather than the albedo in any lim-
739
+ ited wavelength range.
740
+ In Figure 7 we plot the dimensionless day-side effec-
741
+ tive temperature, Td/T0, against the maximum expected
742
+ day-side temperature, Tε=0. The cyan asterisks in Fig-
743
+ ure 7 show the four hot Jupiters without temperature
744
+ inversions, while most of the remaining planets have in-
745
+ versions (Knutson et al. 2010). The presence or absence
746
+ of an inversion does not appear to affect the efficiency of
747
+ day–night heat recirculation.
748
+ Planets should lie below the solid red line in Figure 7,
749
+ which denotes Tε=0= (2/3)1/4T0. Of the 24 planets in
750
+ our sample, only one (Gl 436b) has a day-side effective
751
+ temperature significantly above the Tε=0limit6. This
752
+ planet is by far the coolest in our sample, it is on an ec-
753
+ centric orbit, and observations indicate that it may have
754
+ a non-equilibrium atmosphere (Stevenson et al. 2010).
755
+ There is no reason, on the other hand, that planets
756
+ shouldn’t lie below the red dotted line in Figure 7:
757
+ all it would take is non-zero Bond albedo. That said,
758
+ only 3 of the 24 planets we consider are in this region,
759
+ 6This is driven by the abnormally high 3.6 micron brightness
760
+ temperature; including the 4.5 micron eclipse upper limit d oes not
761
+ significantly change our estimate of this planet’s effective temper-
762
+ ature.8 Cowan & Agol
763
+ Fig. 7.— The dimensionless day-side effective temperature,
764
+ Td/T0, plotted against the maximum expected day-side temper-
765
+ ature,Tε=0. The red lines correspond to three fiducial limits of
766
+ recirculation, assuming AB= 0: no recirculation (solid), uniform
767
+ day-hemisphere (dashed), and uniform planet (dotted). The gray
768
+ points indicate the default values (using only observation s with
769
+ λ >0.8 micron) for the four planets whose optical eclipse depths
770
+ may be probing thermal emission rather than just reflected li ght
771
+ (from left to right: TrES-2b, CoRoT-2b, CoRoT-1b, HAT-P-7b ).
772
+ For these planets we have here elected to include optical mea sure-
773
+ ments in our estimate of the day-side bolometric flux and effec tive
774
+ temperature, shown in black. The cyan asterisks denote thos e hot
775
+ Jupiters known notto have a stratospheric inversion according
776
+ to (Knutson et al. 2010). They are, from left to right: TrES-1 b,
777
+ HD 189733b, TrES-3b, WASP-4b. The two red x’s denote the ec-
778
+ centric planets in our sample, which are also the two worst ou tliers.
779
+ with the greatest outlier being HD 80606b, a planet on
780
+ an extremely eccentric orbit with superior conjunction
781
+ nearly coinciding with periastron. As such, it is likely
782
+ that much of the energy absorbed by the planet at that
783
+ point in its orbit performs mechanical work (speeding up
784
+ winds, puffingupthe planet, etc. SeealsoCowan & Agol
785
+ 2010) rather than merely warming the gas. Gl 436b and
786
+ HD 80606b are denoted by red x’s in Figure 7.
787
+ The gray points in Figure 7 indicate the default val-
788
+ ues (using only observationswith λ>0.8 micron) for the
789
+ four planets whose optical eclipse depths may be probing
790
+ thermal emission rather than just reflected light (from
791
+ left to right: TrES-2b, CoRoT-2b, CoRoT-1b, HAT-
792
+ P-7b). For these planets we have here elected to use
793
+ all available flux ratios (including optical observations
794
+ potentially contaminated by reflected light) to estimate
795
+ the day-side bolometric flux and effective temperature,
796
+ shown as black points in Figure 7.
797
+ If one takes these day-side effective temperature es-
798
+ timates at face value, it appears that the planets with
799
+ Tε=0<2400 K exhibit a wide-variety of redistribution
800
+ efficiencies and/or Bond albedos, but are consistent with
801
+ AB= 0. It is worth noting that many of the best char-
802
+ acterized planets in this region have Td/T0≈0.75, and
803
+ this accounts for the sharp peak in the dotted line of Fig-
804
+ ure 5 atε= 0.75. The hottest 6 planets, on the other
805
+ hand, have uniformly high Td/T0, indicating that they
806
+ have both low Bond albedo andlow redistribution effi-
807
+ ciency. These planets must not have the high-altitude,
808
+ reflective silicate clouds hypothesized in Sudarsky et al.
809
+ (2000). But this conclusion is dependent on how one
810
+ interprets the Keplerobservations of HAT-P-7b: if the
811
+ large optical flux ratio is due to reflected light, then this
812
+ planet is cooler than we think, and even the hottest tran-siting planets exhibit a variety of behaviors.
813
+ 5.SUMMARY & CONCLUSIONS
814
+ We have described how to estimate a planet’s incident
815
+ power budget ( T0), where the uncertainties are driven by
816
+ the uncertainties in the host star’s effective temperature
817
+ and size, as well as the planet’s orbit. We then described
818
+ a model-independent technique to estimate the effective
819
+ temperature of a planet based on planet/star flux ra-
820
+ tiosobtained at variouswavelengths. When the observed
821
+ day-side and night-side effective temperatures are com-
822
+ pared, one can constrain a combination of the planet’s
823
+ Bond albedo, AB, and its recirculation efficiency, ε. We
824
+ applied this analysis on 24 known transiting planets with
825
+ measured infrared eclipse depths.
826
+ Our principal results are:
827
+ 1. Essentially all of the planets are consistent with low
828
+ Bond albedo.
829
+ 2. We firmly rule out the “null hypothesis”, whereby all
830
+ transiting planets can be fit by a single ABandε. It
831
+ is not immediately clear whether this stems from differ-
832
+ ences in Bond albedo, recirculation efficiency, or both.
833
+ 3. In the few cases where it is possible to unambiguously
834
+ infer an albedo based on optical eclipse depths, they are
835
+ extremely low, implying correspondingly low Bond albe-
836
+ dos (<10%). If one adopts such low albedos for all
837
+ the planets in our sample, the discrepancies in day-side
838
+ effective temperature must be due to differences in recir-
839
+ culation efficiency.
840
+ 4. These differences in recirculation efficiency do not
841
+ appear to be correlated with the presence or absence of
842
+ a stratospheric inversion.
843
+ 5. Planets cooler than Tε=0= 2400 K exhibit a wide va-
844
+ riety of circulation efficiencies that do not appear to be
845
+ correlated with equilibrium temperature. Alternatively,
846
+ theseplanetsmayhavedifferent (but generallylow)albe-
847
+ dos. Planets hotter than Tε=0= 2400 K have uniformly
848
+ low redistribution efficiencies and albedos.
849
+ The apparent decrease in advective efficiency with
850
+ increasing planetary temperature remains unexplained.
851
+ One hypothesis, mentioned earlier, is that TiO and VO
852
+ would provide additional optical opacity in atmospheres
853
+ hotter than T∼1700 K, leading to temperature in-
854
+ versions and reduced heat recirculation on these plan-
855
+ ets (Fortney et al. 2008). But if our sample shows any
856
+ sharp change it behavior it occurs near 2400 K, rather
857
+ than 1700K. One couldinvokeanotheroptical absorber,
858
+ but in any case the lack of correlation —pointed out in
859
+ thisworkandelsewhere—betweenthepresenceofatem-
860
+ perature inversionand the efficiency of heat recirculation
861
+ makes this explanation suspect. Another possible expla-
862
+ nation for the observed trend is that the hottest planets
863
+ have the most ionized atmospheres and may suffer the
864
+ most severe magnetic drag (Perna et al. 2010).
865
+ The simplest explanation for this trend is simply that
866
+ the radiative time is a steeper function of temperature
867
+ than the advective time: advective efficiency is given
868
+ roughly by the ratio of the radiative and advective times
869
+ (eg: Cowan & Agol 2010). In the limit of Newtonian
870
+ cooling, the radiative time scales as τrad∝T−3. If one
871
+ assumes the wind speed to be of order the local sound
872
+ speed, then the advective time scales as τadv∝T−0.5.
873
+ One might therefore naively expect the advective effi-
874
+ ciency to scale as T−2.5. Such an explanation would notAlbedo and Heat Recirculation on Hot Exoplanets 9
875
+ explain the apparent sharp transition seen at 2400 K,
876
+ however.
877
+ The combination of optical observations of secondary
878
+ eclipses and thermal observations of phase variations is
879
+ the best way to constrain planetary albedo and circu-
880
+ lation. The optical observations should be taken near
881
+ the star’s blackbody peak, both to maximize signal-to-
882
+ noise, and to avoidcontaminationfrom the planet’s ther-
883
+ mal emission, but this separationmay not be possible for
884
+ the hottest transiting planets. The thermal observations,
885
+ likewise, should be near the planet’s blackbody peak to
886
+ better constrain its bolometric flux. Note that this wave-
887
+ length is shortwardof the ideal contrastratio, which typ-
888
+ ically falls on the planet’s Rayleigh-Jeans tail. Further-
889
+ more, the thermal phase observations should span a full
890
+ planetaryorbit: thelightcurveminimumisthemostsen-
891
+ sitive measure of ε, and should occur nearly half an orbit
892
+ apart from the light curve maximum, despite skewed di-
893
+ urnal heatingpatterns (Cowan & Agol 2008, 2010). This
894
+ means that observing campaigns that only cover a little
895
+ more than half an orbit (transit →eclipse) are probably
896
+ underestimating the real peak-trough phase amplitude.A possible improvement to this study would be to per-
897
+ form a uniform data reduction for all the Spitzerexo-
898
+ planet observations of hot Jupiters. These data make up
899
+ the majority of the constraints presented in our study
900
+ and most are publicly available. And while the pub-
901
+ lished observations were analyzed in disparate ways, a
902
+ consensus approach to correcting detector systematics is
903
+ beginning to emerge.
904
+ N.B.C. acknowledges useful discussions of aspects of
905
+ this work with T. Robinson, M.S. Marley, J.J. Fort-
906
+ ney, T.S. Barman and D.S. Spiegel. Thanks to our
907
+ referee B.M.S. Hansen for insightful feedback, and to
908
+ E.D. Feigelson for suggestions about statistical methods.
909
+ N.B.C. was supported by the Natural Sciences and Engi-
910
+ neering Research Council of Canada. E.A. is supported
911
+ by a National Science Foundation Career Grant. Sup-
912
+ port for this work was provided by NASA through an
913
+ award issued by JPL/Caltech. This research has made
914
+ use of the Exoplanet Orbit Database and the Exoplanet
915
+ Data Explorer at exoplanets.org.
916
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1054
+ TABLE 1
1055
+ Secondary Eclipses & Phase Variations of Exoplanets
1056
+ Planet Tε=0[K]aλ[µm]bEclipse DepthcTbright[K] Phase AmplitudecDerived Quantitiesd
1057
+ CoRoT-1b12424(84) 0.60(0.42) 1 .6(6)×10−42726(141) Td=2674(144) K
1058
+ 0.71(0.25) 1 .26(33)×10−42409(75) 1 .0(3)×10−4Aλ<0.1
1059
+ 2.10(0.02) 2 .8(5)×10−32741(125) Td(A= 0)=2515(84) K
1060
+ 2.15(0.32) 3 .36(42)×10−32490(157)
1061
+ 3.6(0.75) 4 .15(42)×10−32098(116)
1062
+ 4.5(1.0) 4 .82(42)×10−32084(106)
1063
+ CoRoT-2b21964(42) 0.60(0.42) 6(2) ×10−52315(85) Td=1864(233) K
1064
+ 0.71(0.25) 1 .02(20)×10−42215(49) Aλ= 0.16(7)
1065
+ 1.65(0.25) <1.7×10−3(3σ) Td(A= 0)=2010(144) K
1066
+ 2.15(0.32) 1 .6(9)×10−31914(292)
1067
+ 3.6(0.75) 3 .55(20)×10−31798(40)
1068
+ 4.5(1.0)e4.75(19)×10−31791(33)
1069
+ 4.5(1.0) 5 .10(42)×10−3
1070
+ 8.0(2.9) 4 .1(1.1)×10−3
1071
+ 8.0(2.9)e4.09(80)×10−31318(143)
1072
+ Gl 436b3934(41) 3.6(0.75) 4 .1(3)×10−41145(23) Td=1082(38) K
1073
+ 4.5(1.0) <1.0×10−4(3σ)
1074
+ 5.8(1.4) 3 .3(1.4)×10−4797(106)
1075
+ 8.0(2.9)e4.52(27)×10−4737(17)
1076
+ 8.0(2.9) 5 .7(8)×10−4
1077
+ 8.0(2.9) 5 .4(7)×10−4
1078
+ 16(5) 1 .40(27)×10−3963(126)
1079
+ 24(9) 1 .75(41)×10−31016(182)
1080
+ HAT-P-1b41666(38) 3.6(0.75) 8 .0(8)×10−41420(47) Td=1439(59) K
1081
+ 4.5(1.0) 1 .35(22)×10−31507(100)
1082
+ 5.8(1.4) 2 .03(31)×10−31626(128)
1083
+ 8.0(2.9) 2 .38(40)×10−31564(151)
1084
+ HAT-P-7b52943(95) 0.65(0.4) 1 .30(11)×10−43037(35) 1 .22(16)×10−4Td=2086(156) K
1085
+ 3.6(0.75) 9 .8(1.7)×10−42063(152) Aλ= 0.58(5)
1086
+ 4.5(1.0) 1 .59(22)×10−32378(179) Td(A= 0)=2830(86) K
1087
+ 5.8(1.4) 2 .45(31)×10−32851(235)
1088
+ 8.0(2.9) 2 .25(52)×10−32512(403)
1089
+ HD 80606b61799(50) 8.0(2.9) 1 .36(18)×10−31137(73) Td=1137(113) K
1090
+ HD 149026b71871(17) 8.0(2.9)e3.7(0.8)×10−4976(276) 2 .3(7)×10−4Td=1571(231) K
1091
+ 8.0(2.9) 8 .4(1.1)×10−4Tn=976(286) K
1092
+ HD 189733b81537(16) 2.15(32) <4.0×10−4(1σ) Td=1605(52) K
1093
+ 3.6(0.75) 2 .56(14)×10−31639(34) Tn=1107(132) K
1094
+ 4.5(1.0) 2 .14(20)×10−31318(45)
1095
+ 5.8(1.4) 3 .10(34)×10−31368(69)
1096
+ 8.0(2.9) 3 .381(55)×10−3
1097
+ 8.0(2.9) 3 .91(22)×10−31.2(2)×10−3
1098
+ 8.0(2.9)e3.440(36)×10−31259(7) 1 .2(4)×10−3
1099
+ 16(5) 5 .51(30)×10−31338(52)
1100
+ 24(9) 5 .98(38)×10−3
1101
+ 24(9)e5.36(27)×10−31202(46) 1 .3(3)×10−3
1102
+ HD 209458b91754(15) 0.5(0.3) 7(9) ×10−62368(156) Td=1486(53) K
1103
+ 2.15(0.32) <3×10−4(1σ) Aλ= 0.09(7)
1104
+ 3.6(0.75) 9 .4(9)×10−41446(45) Td(A= 0)=2031(128) K
1105
+ 4.5(1.0) 2 .13(15)×10−31757(57) Tn=1476(304) K
1106
+ 5.8(1.4) 3 .01(43)×10−31890(149)
1107
+ 8.0(2.9) 2 .40(26)×10−31480(94) <1.5×10−3(2σ)
1108
+ 24(9) 2 .60(44)×10−31131(143)
1109
+ OGLE-TR-56b102874(84) 0.90(0.15) 3 .63(91)×10−42696(116) Td=2696(236) K
1110
+ OGLE-TR-113b111716(33) 2.15(0.32) 1 .7(5)×10−31918(164) Td=1918(219) K
1111
+ TrES-1b121464(16) 3.6(0.75) <1.5×10−3(1σ) Td=998(67) K
1112
+ 4.5(1.0) 6 .6(1.3)×10−4972(56)
1113
+ 8.0(2.9) 2 .25(36)×10−31152(94)
1114
+ TrES-2b131917(21) 0.65(0.4) 1 .14(78)×10−52020(132) Td=1623(76) K
1115
+ 2.15(0.32) 6 .2(1.2)×10−41655(80) Aλ= 0.06(3)
1116
+ 3.6(0.75) 1 .27(21)×10−31490(84) Td(A= 0) = 1751(80) K
1117
+ 4.5(1.0) 2 .30(24)×10−31652(74)
1118
+ 5.8(1.4) 1 .99(54)×10−31373(177)
1119
+ 8.0(2.9) 3 .59(60)×10−31659(163)
1120
+ TrES-3b142093(32) 0.7(0.3) <6.2×10−4(1σ) Td=1761(66) K
1121
+ 1.25(0.16) <5.1×10−4(3σ)
1122
+ 2.15(0.32) 2 .41(43)×10−3
1123
+ 2.15(0.32)e1.33(17)×10−31770(58)
1124
+ 3.6(0.75) 3 .46(35)×10−31818(73)12 Cowan & Agol
1125
+ TABLE 1
1126
+ Secondary Eclipses & Phase Variations of Exoplanets
1127
+ 4.5(1.0) 3 .72(54)×10−31649(107)
1128
+ 5.8(1.4) 4 .49(97)×10−31621(173)
1129
+ 8.0(2.9) 4 .75(46)×10−31480(82)
1130
+ TrES-4b152250(37) 3.6(0.75) 1 .37(11)×10−31889(63) Td=1891(81) K
1131
+ 4.5(1.0) 1 .48(16)×10−31727(83)
1132
+ 5.8(1.4) 2 .61(59)×10−32112(283)
1133
+ 8.0(2.9) 3 .18(44)×10−32168(197)
1134
+ WASP-1b162347(35) 3.6(0.75) 1 .17(16)×10−31678(87) Td=1719(89) K
1135
+ 4.5(1.0) 2 .12(21)×10−31923(91)
1136
+ 5.8(1.4) 2 .82(60)×10−32042(253)
1137
+ 8.0(2.9) 4 .70(46)×10−32587(176)
1138
+ WASP-2b171661(69) 3.6(0.75) 8 .3(3.5)×10−41264(164) Td=1280(121) K
1139
+ 4.5(1.0) 1 .69(17)×10−31380(53)
1140
+ 5.8(1.4) 1 .92(77)×10−31299(232)
1141
+ 8.0(2.9) 2 .85(59)×10−31372(154)
1142
+ WASP-4b182163(60) 3.6(0.75) 3 .19(31)×10−32156(97) Td=2146(140) K
1143
+ 4.5(1.0) 3 .43(27)×10−31971(75)
1144
+ WASP-12b193213(119) 0.9(0.15) 8 .2(1.5)×10−43002(104) Td=2939(98) K
1145
+ 1.25(0.16) 1 .31(28)×10−32894(149)
1146
+ 1.65(0.25) 1 .76(18)×10−32823(88)
1147
+ 2.15(0.32) 3 .09(13)×10−33018(51)
1148
+ 3.6(0.75) 3 .79(13)×10−32704(49)
1149
+ 4.5(1.0) 3 .82(19)×10−32486(68)
1150
+ 5.8(1.4) 6 .29(52)×10−33167(179)
1151
+ 8.0(2.9) 6 .36(67)×10−32996(229)
1152
+ WASP-18b203070(50) 3.6(0.75) 3 .1(2)×10−33000(107) Td=2998(138) K
1153
+ 4.5(1.0) 3 .8(3)×10−33128(150)
1154
+ 5.8(1.4) 4 .1(2)×10−33095(103)
1155
+ 8.0(2.9) 4 .3(3)×10−32991(153)
1156
+ WASP-19b212581(49) 1.65(0.25) 2 .59(45)×10−32677(135) Td=2677(244) K
1157
+ XO-1b221526(24) 3.6(0.75) 8 .6(7)×10−41300(32) Td=1306(47) K
1158
+ 4.5(1.0) 1 .22(9)×10−31265(34)
1159
+ 5.8(1.4) 2 .61(31)×10−31546(89)
1160
+ 8.0(2.9) 2 .10(29)×10−31211(87)
1161
+ XO-2231685(33) 3.6(0.75) 8 .1(1.7)×10−41447(102) Td=1431(98) K
1162
+ 4.5(1.0) 9 .8(2.0)×10−41341(105)
1163
+ 5.8(1.4) 1 .67(36)×10−31497(155)
1164
+ 8.0(2.9) 1 .33(49)×10−31179(219)
1165
+ XO-3241982(82) 3.6(0.75) 1 .01(4)×10−31875(30) Td=1871(63) K
1166
+ 4.5(1.0) 1 .43(6)×10−31965(40)
1167
+ 5.8(1.4) 1 .34(49)×10−31716(330)
1168
+ 8.0(2.9) 1 .50(36)×10−31625(236)
1169
+ aThe planet’s expected day-side effective temperature in the absence of reflection or recirculation ( AB= 0,ε= 0). The 1 σuncertainty is shown
1170
+ in parenthese.
1171
+ bThe bandwidth is shown in parenthese.
1172
+ cEclipse depths and phase amplitudes are unitless, since the y are measured relative to stellar flux.
1173
+ dTdandTndenote the day-side and night-side effective temperatures o f the planet, as estimated from thermal secondary eclipse de pths and
1174
+ thermal phase variations, respectively. The estimated 1 σuncertainties are shown in parentheses. The default day-si de temperature is computed
1175
+ using only observations at λ >0.8µm. Eclipse measurements at shorter wavelengths may then be u sed to estimate the planet’s albedo at those
1176
+ wavelengths, Aλ. Note that this is a spherical albedo; the geometric albedo i s given by Ag=2
1177
+ 3Aλ. If —on the other hand— AB= 0 is assumed,
1178
+ then all the day-side flux is thermal, regardless of waveband , yielding the second Tdestimate.
1179
+ eWhen multiple measurements of an eclipse depth have been pub lished in a given waveband, we use the most recent observatio n. In all cases
1180
+ these observations are either explicitly agree with their o lder counterpart, or agree with the re-analyzed older data.
1181
+ 1Snellen et al. (2009); Alonso et al. (2009b); Gillon et al. (2 009); Rogers et al. (2009); Deming et al. (2010),2Alonso et al. (2009a); Snellen et al.
1182
+ (2010); Gillon et al. (2010); Alonso et al. (2010); Deming et al. (2010),3Deming et al. (2007); Demory et al. (2007); Stevenson et al. ( 2010);
1183
+ Knutson et al. in prep.,4Todorov et al. (2010),5Borucki et al. (2009); Christiansen et al. (2010),6Laughlin et al. (2009),7Knutson et al.
1184
+ (2009b),8Deming et al. (2006); Knutson et al. (2007a); Barnes et al. (2 007); Charbonneau et al. (2008); Knutson et al. (2009c); Ago l et al.
1185
+ (2010),9Richardson et al. (2003); Deming et al. (2005); Cowan et al. ( 2007); Rowe et al. (2008); Knutson et al. (2008),10Sing & L´ opez-Morales
1186
+ (2009),11Snellen & Covino (2007),12Charbonneau et al. (2005); Knutson et al. (2007b),13O’Donovan et al. (2010); Croll et al. (2010a);
1187
+ Kipping & Bakos (2010b),14Fressin et al. (2010); Croll et al. (2010b); Christiansen et al. (2010b),15Knutson et al. (2009a),16,17Wheatley et al.
1188
+ (2010),18Beerer et al. (2010),19L´ opez-Morales et al. (2010); Campo et al. (2010); Croll et a l. (2010c),20Nymeyer et al. (2010),21Anderson et al.
1189
+ (2010),22Machalek et al. (2008),23Machalek et al. (2009),24Machalek et al. (2010)
1001.0013.txt ADDED
@@ -0,0 +1,1084 @@
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
1
+ arXiv:1001.0013v2 [astro-ph.CO] 8 Jan 2010Astronomy& Astrophysics manuscriptno.akari˙LF˙aa˙v7 c∝circlecopyrtESO 2018
2
+ October30,2018
3
+ EvolutionofInfraredLuminosityfunctionsofGalaxiesint he
4
+ AKARINEP-Deepfield
5
+ Revealing thecosmic star formationhistory hidden by dust⋆,⋆⋆
6
+ Tomotsugu Goto1,2,⋆⋆⋆,T.Takagi3,H.Matsuhara3,T.T.Takeuchi4,C.Pearson5,6,7, T.Wada3,T.Nakagawa3,O.Ilbert8,
7
+ E.LeFloc’h9,S.Oyabu3, Y.Ohyama10,M.Malkan11, H.M.Lee12, M.G.Lee12,H.Inami3,13,14, N.Hwang2, H.Hanami15,
8
+ M.Im12, K.Imai16,T.Ishigaki17,S.Serjeant7,and H.Shim12
9
+ 1Institute for Astronomy, University of Hawaii,2680 Woodla wnDrive, Honolulu, HI,96822, USA
10
11
+ 2National Astronomical Observatory, 2-21-1 Osawa,Mitaka, Tokyo, 181-8588,Japan
12
+ 3Institute of Space and Astronautical Science, JapanAerosp ace Exploration Agency, Sagamihara,Kanagawa 229-8510
13
+ 4Institute for Advanced Research, Nagoya University, Furo- cho, Chikusa-ku, Nagoya 464-8601
14
+ 5Rutherford Appleton Laboratory, Chilton, Didcot,Oxfords hire OX110QX, UK
15
+ 6Department of Physics,Universityof Lethbridge, 4401 Univ ersity Drive,Lethbridge, AlbertaT1J 1B1, Canada
16
+ 7Astrophysics Group, Department of Physics, The OpenUniver sity, MiltonKeynes, MK76AA, UK
17
+ 8Laboratoire d’Astrophysique de Marseille, BP8,Traverse d u Siphon, 13376 Marseille Cedex 12, France
18
+ 9CEA-Saclay,Service d’Astrophysique, France
19
+ 10Academia Sinica,Institute of Astronomyand Astrophysics, Taiwan
20
+ 11Department of Physicsand Astronomy, UCLA,Los Angeles, CA, 90095-1547 USA
21
+ 12Department of Physics& Astronomy, FPRD,Seoul National Uni versity, Shillim-Dong,Kwanak-Gu, Seoul 151-742, Korea
22
+ 13Spitzer Science Center,California Institute ofTechnolog y, Pasadena, CA91125
23
+ 14Department of Astronomical Science,The Graduate Universi tyfor Advanced Studies
24
+ 15Physics Section,Facultyof Humanities and SocialSciences , Iwate University, Morioka, 020-8550
25
+ 16TOMER&D Inc. Kawasaki, Kanagawa 2130012, Japan
26
+ 17Asahikawa National College of Technology, 2-1-6 2-joShunk ohdai, Asahikawa-shi, Hokkaido 071-8142
27
+ Received September 15, 2009; accepted December 16, 2009
28
+ ABSTRACT
29
+ Aims.Dust-obscured star-formation becomes much more important with increasing intensity, and increasing redshift. We aim to
30
+ reveal cosmic star-formationhistoryobscured bydust usin g deep infraredobservation withthe AKARI.
31
+ Methods. We construct restframe 8 µm, 12µm, and total infrared (TIR) luminosity functions (LFs) at 0.15< z <2.2using 4128
32
+ infraredsources intheAKARINEP-Deepfield.Acontinuous fil tercoverage inthemid-IRwavelength(2.4,3.2,4.1,7,9,11 , 15,18,
33
+ and 24µm) by the AKARI satellite allows us to estimate restframe 8 µm and 12 µm luminosities without using a large extrapolation
34
+ based ona SEDfit,which was the largestuncertainty inprevio us work.
35
+ Results. Wehavefoundthatall8 µm(0.38< z <2.2),12µm(0.15< z <1.16),andTIRLFs( 0.2< z <1.6),showacontinuous
36
+ andstrongevolutiontowardhigher redshift.Intermsofcos micinfraredluminositydensity( ΩIR),whichwasobtainedbyintegrating
37
+ analytic fits to the LFs,we found a good agreement withprevio us work at z <1.2. We found the ΩIRevolves as ∝(1+z)4.4±1.0.
38
+ Whenweseparatecontributionsto ΩIRbyLIRGsandULIRGs,wefoundmoreIRluminoussourcesareinc reasinglymoreimportant
39
+ at higher redshift. Wefound that the ULIRG(LIRG)contribut ionincreases bya factor of 10(1.8) from z=0.35 toz=1.4.
40
+ Keywords. galaxies: evolution, galaxies:interactions, galaxies:s tarburst, galaxies:peculiar, galaxies:formation
41
+ 1. Introduction
42
+ Studies of the extragalactic background suggest at least ha lf
43
+ the luminous energy generated by stars has been reprocessed
44
+ into the infrared(IR) by dust (Lagacheetal., 1999; Pugetet al.,
45
+ 1996; Franceschini,Rodighiero,&Vaccari, 2008), suggest ing
46
+ that dust-obscured star formation was much more important a t
47
+ higherredshiftsthantoday.
48
+ ⋆This research is based on the observations with AKARI, a JAXA
49
+ project withthe participationof ESA.
50
+ ⋆⋆Based on data collected at Subaru Telescope, which is operat ed by
51
+ the National Astronomical Observatory ofJapan.
52
+ ⋆⋆⋆JSPSSPDfellowBell etal. (2005) estimate that IR luminosity density is 7
53
+ times higher than the UV luminosity density at z ∼0.7 than lo-
54
+ cally. Takeuchi,Buat, &Burgarella (2005) reported that UV -to-
55
+ IRluminositydensityratio, ρL(UV)/ρL(dust),evolvesfrom3.75
56
+ (z=0) to 15.1 by z=1.0 with a careful treatment of the sample
57
+ selection effect, and that 70% of star formation activity is ob-
58
+ scured by dust at 0.5 < z <1.2. Both works highlight the im-
59
+ portance of probing cosmic star formation activity at high r ed-
60
+ shift in the infrared bands. Several works found that most ex -
61
+ tremestar-forming(SF) galaxies,whichareincreasinglyi mpor-
62
+ tant at higher redshifts, are also more heavily obscured by d ust
63
+ (Hopkinsetal., 2001; Sullivanet al., 2001; Buatet al.,200 7).2 Gotoet al.:InfraredLuminosityfunctions withthe AKARI
64
+ Despite the value of infrared observations, studies of
65
+ infrared galaxies by the IRAS and the ISO were re-
66
+ stricted to bright sources due to the limited sensitiv-
67
+ ities (Saundersetal., 1990; Rowan-Robinsonet al., 1997;
68
+ Floreset al., 1999; Serjeantet al., 2004; Takeuchiet al., 2 006;
69
+ Takeuchi,Yoshikawa,&Ishii, 2003), until the recent launc h of
70
+ theSpitzer andtheAKARI satellites. Theirenormousimprov ed
71
+ sensitivitieshaverevolutionizedthefield.Forexample:
72
+ Le Floc’het al. (2005) analyzed the evolution of the total
73
+ and 15µm IR luminosity functions (LFs) at 0< z <1based
74
+ on the the Spitzer MIPS 24 µm data (>83µJy andR <24) in
75
+ the CDF-S, and found a positive evolution in both luminosity
76
+ and density, suggesting increasing importance of the LIRG a nd
77
+ ULIRGpopulationsathigherredshifts.
78
+ P´ erez-Gonz´ alezetal. (2005) used MIPS 24 µm observations
79
+ oftheCDF-SandHDF-N( >83µJy)tofindthatthat L∗steadily
80
+ increasesbyanorderofmagnitudeto z∼2,suggestingthatthe
81
+ luminosity evolution is stronger than the density evolutio n. The
82
+ ΩTIRscalesas(1+z)4.0±0.2fromz=0to0.8.
83
+ Babbedgeet al. (2006) constructed LFs at 3.6, 4.5, 5.8, 8
84
+ and 24µm over0< z < 2using the data from the Spitzer
85
+ Wide-areaInfraredExtragalactic(SWIRE)Surveyin a 6.5de g2
86
+ (S24µm>230µJy). They found a clear luminosity evolu-
87
+ tion in all the bands, but the evolution is more pronounced at
88
+ longer wavelength; extrapolatingfrom 24 µm, they inferred that
89
+ ΩTIR∝(1+z)4.5. They constructed separate LFs for three dif-
90
+ ferentgalaxySED (spectral energydistribution)typesand Type
91
+ 1 AGN, finding that starburst and late-type galaxies showed
92
+ strongerevolution.Comparisonof3.6and4.5 µmLFswithsemi-
93
+ analytic and spectrophotometricmodelssuggested that the IMF
94
+ is skewed towards higher mass star formation in more intense
95
+ starbursts.
96
+ Caputi etal.(2007)estimatedrestframe8 µmLFsofgalaxies
97
+ over 0.08deg2in the GOODS fields based on Spitzer 24 µm (>
98
+ 80µJy) atz=1 and 2. They found a continuousand strong posi-
99
+ tiveluminosityevolutionfrom z=0toz=1,andto z=2.However,
100
+ theyalsofoundthatthenumberdensityofstar-forminggala xies
101
+ withνL8µm
102
+ ν>1010.5L⊙(AGNs are excluded.) increases by a
103
+ factor of 20 from z=0 to 1, but decreases by half from z=1 to 2
104
+ mainlyduetothe decreaseofLIRGs.
105
+ Magnelliet al. (2009) investigated restframe 15 µm, 35µm
106
+ and total infrared (TIR) LFs using deep 70 µm observations
107
+ (∼300µJy) in the Spitzer GOODS and FIDEL (Far Infrared
108
+ Deep Extragalactic Legacy Survey) fields (0.22 deg2in total)
109
+ atz <1.3. They stacked 70 µm flux at the positions of 24 µm
110
+ sources when sources are not detected in 70 µm. They found no
111
+ changeintheshapeoftheLFs,butfoundapureluminosityevo -
112
+ lutionproportionalto(1+z)3.6±0.5,andthatLIRGsandULIRGs
113
+ have increased by a factor of 40 and 100 in number density by
114
+ z∼1.
115
+ Also, see Daiet al. (2009) for 3.6-8.0 µm LFs based on the
116
+ IRACphotometryintheNOAODeepWide-FieldSurveyBootes
117
+ field.
118
+ However, most of the Spitzer work relied on a large
119
+ extrapolation from 24 µm flux to estimate the 8, 12 µm or
120
+ TIR luminosity. Consequently, Spitzer results heavily de-
121
+ pended on the assumed IR SED library (Dale&Helou, 2002;
122
+ Lagache,Dole,&Puget, 2003; Chary& Elbaz, 2001). Indeed
123
+ many authors pointed out that the largest uncertainty in the se
124
+ previous IR LFs came from SED models, especially when one
125
+ computesTIRluminositysolelyfromobserved24 µmflux(e.g.,
126
+ see Fig.5ofCaputiet al.,2007).
127
+ AKARI, the first Japanese IR dedicated satellite, has con-
128
+ tinuous filter coverage across the mid-IR wavelengths, thus , al-Fig.1. Photometric redshift estimates with LePhare
129
+ (Ilbertet al., 2006; Arnoutset al., 2007; Ilbertet al., 200 9)
130
+ for spectroscopically observed galaxies with Keck/DEIMOS
131
+ (Takagi et al. in prep.). Red squares show objects where AGN
132
+ templates were better fit. Errors of the photoz is∆z
133
+ 1+z=0.036 for
134
+ z≤0.8, but becomes worse at z >0.8to be∆z
135
+ 1+z=0.10 due
136
+ mainlyto therelativelyshallownear-IRdata.
137
+ lows us to estimate MIR (mid-infrared)-luminositywithout us-
138
+ ing a large k-correction based on the SED models, eliminating
139
+ thelargestuncertaintyinpreviouswork.Bytakingadvanta geof
140
+ this, we present the restframe 8, 12 µm and TIR LFs using the
141
+ AKARI NEP-Deepdatainthiswork.
142
+ Restframe 8 µm luminosity in particular is of primary rele-
143
+ vance for star-forming galaxies, as it includes polycyclic aro-
144
+ matic hydrocarbon (PAH) emission. PAH molecules charac-
145
+ terize star-forming regions (Desert,Boulanger,&Puget, 1 990),
146
+ and the associated emission lines between 3.3 and 17 µm dom-
147
+ inate the SED of star-forming galaxies with a main bump lo-
148
+ cated around 7.7 µm. Restframe 8 µm luminosities have been
149
+ confirmed to be good indicators of knots of star formation
150
+ (Calzetti etal., 2005) and of the overall star formation act ivity
151
+ of star forming galaxies (Wuet al., 2005). At z=0.375, 0.875,
152
+ 1.25 and 2, the restframe 8 µm is covered by the AKARI S11,
153
+ L15,L18WandL24filters. We present the restframe 8 µm LFs
154
+ at theseredshiftsatSection3.1.
155
+ Restframe 12 µm luminosity functions have also been
156
+ studied extensively (Rush,Malkan,& Spinoglio, 1993;
157
+ P´ erez-Gonz´ alezet al., 2005). At z=0.25, 0.5 and 1, the
158
+ restframe12 µmiscoveredbytheAKARI L15,L18WandL24
159
+ filters. We present the restframe 12 µm LFs at these redshifts in
160
+ Section3.3.
161
+ We also estimate TIR LFs through the SED fit using all
162
+ the mid-IR bands of the AKARI. The results are presented in
163
+ Section3.5.
164
+ Unless otherwise stated, we adopt a cosmology with
165
+ (h,Ωm,ΩΛ) = (0.7,0.3,0.7)(Komatsuet al., 2008).
166
+ 2. Data & Analysis
167
+ 2.1. Multi-wavelength data inthe AKARI NEP Deepfield
168
+ AKARI, the Japanese infraredsatellite (Murakamiet al., 20 07),
169
+ performed deep imaging in the North Ecliptic Region (NEP)
170
+ from 2-24 µm, with 14 pointings in each field over 0.4
171
+ deg2(Matsuharaet al., 2006, 2007; Wada et al., 2008). DueGotoet al.:InfraredLuminosityfunctions withthe AKARI 3
172
+ Fig.2.Photometricredshiftdistribution.
173
+ Fig.3.8µmluminositydistributionsofsamplesusedtocompute
174
+ restframe 8 µm LFs. From low redshift, 533, 466, 236 and 59
175
+ galaxiesarein eachredshiftbin.
176
+ to the solar synchronous orbit of the AKARI, the NEP
177
+ is the only AKARI field with very deep imaging at these
178
+ wavelengths. The 5 σsensitivity in the AKARI IR filters
179
+ (N2,N3,N4,S7,S9W,S11,L15,L18WandL24) are 14.2,
180
+ 11.0, 8.0, 48, 58, 71, 117, 121 and 275 µJy (Wada etal., 2008).
181
+ These filters provide us with a unique continuous wavelength
182
+ coverage at 2-24 µm, where there is a gap between the Spitzer
183
+ IRAC and MIPS, and the ISO LW2andLW3. Please consult
184
+ Wada etal. (2007, 2008); Pearsonet al. (2009a,b) for data ve ri-
185
+ ficationandcompletenessestimateatthesefluxes.ThePSFsi zes
186
+ are 4.4, 5.1, and 5.4” in 2−4,7−11,15−24µm bands. The
187
+ depths of near-IR bands are limited by source confusion, but
188
+ thoseofmid-IRbandsarebyskynoise.In analyzingthese observations,we first combinedthe three
189
+ images of the MIR channels, i.e. MIR-S( S7,S9W, andS11)
190
+ and MIR-L( L15,L18WandL24), in order to obtain two high-
191
+ quality images. In the resulting MIR-S and MIR-L images, the
192
+ residual sky has been reduced significantly, which helps to o b-
193
+ tain more reliable source catalogues. For both the MIR-S and
194
+ MIR-Lchannels,we use SExtractorforthecombinedimagesto
195
+ determineinitialsourcepositions.
196
+ We follow Takagietal. (2007) procedures for photometry
197
+ and band-merging of IRC sources. But this time, to maximize
198
+ the number of MIR sources, we made two IRC band-merged
199
+ catalogues based on the combined MIR-S and MIR-L images,
200
+ andthenconcatenatedthese catalogues,eliminatingdupli cates.
201
+ Intheband-mergingprocess,thesourcecentroidineachIRC
202
+ image has beendetermined,starting fromthe sourcepositio n in
203
+ the combined images as the initial guess. If the centroid det er-
204
+ mined in this way is shifted from the original position by >3′′,
205
+ we reject such a source as the counterpart. We note that this
206
+ band-mergingmethodisusedonlyforIRCbands.
207
+ We comparedraw numbercountswith previouswork based
208
+ on the same data but with different source extraction method s
209
+ (Wadaet al., 2008; Pearsonet al., 2009a,b) and found a good
210
+ agreement.
211
+ A subregion of the NEP-Deep field was observed in the
212
+ BVRi′z′-bands with the Subaru telescope (Imaiet al., 2007;
213
+ Wada etal., 2008), reaching limiting magnitudes of zAB=26
214
+ in one field of view of the Suprime-Cam.We restrict our analy-
215
+ sis to the data in this Suprime-Cam field (0.25 deg2), where we
216
+ have enough UV-opical-NIR coverage to estimate good photo-
217
+ metricredshifts.The u′-bandphotometryinthisareaisprovided
218
+ by the CFHT (Serjeant et al. in prep.). The same field was also
219
+ observed with the KPNO2m/FLAMINGOs in JandKsto the
220
+ depth ofKsVega<20(Imaiet al., 2007). GALEX coveredthe
221
+ entirefieldtodepthsof FUV <25andNUV < 25(Malkanet
222
+ al.in prep.).
223
+ In the Suprime-Cam field of the AKARI NEP-Deep field,
224
+ there are a total of 4128 infrared sources down to ∼100µJy in
225
+ theL18Wfilter. All magnitudesare given in AB system in this
226
+ paper.
227
+ For the optical identification of MIR sources, we adopt the
228
+ likelihood ratio (LR) method (Sutherland&Saunders, 1992) .
229
+ For the probability distribution functions of magnitude an d an-
230
+ gular separation based on correct optical counterparts (an d for
231
+ this purpose only), we use a subset of IRC sources, which are
232
+ detected in all IRC bands. For this subset of 1100 all-band–
233
+ detected sources, the optical counterparts are all visuall y in-
234
+ spected and ambiguous cases are excluded. There are multipl e
235
+ opticalcounterpartsfor35%ofMIRsourceswithin <3′′. Ifwe
236
+ adoptedthenearestneighborapproachfortheopticalident ifica-
237
+ tion,theopticalcounterpartsdiffersfromthat oftheLRme thod
238
+ for20%ofthesourceswith multipleopticalcounterparts.T hus,
239
+ in total we estimate that less than 15% of MIR sources suffer
240
+ fromseriousproblemsofopticalidentification.
241
+ 2.2. Photometric redshift estimation
242
+ For these infrared sources, we have computed photomet-
243
+ ric redshift using a publicly available code, LePhare1
244
+ (Ilbertet al., 2006; Arnoutsetal., 2007; Ilbertet al., 200 9).
245
+ The input magnitudes are FUV,NUV (GALEX), u(CFHT),
246
+ B,V,R,i′,z′(Subaru), J,andK(KPNO2m).Wesummarizethe
247
+ filtersusedinTable1.
248
+ 1http://www.cfht.hawaii.edu/∼arnouts/lephare.html4 Gotoet al.:InfraredLuminosityfunctions withthe AKARI
249
+ Table 1.Summaryoffiltersused.
250
+ Estimate Redshift Filter
251
+ Photoz0.15<z<2.2FUV,NUV ,u,B,V,R,i′,z,J, andK
252
+ 8µm LF 0.38 <z<0.58 S11(11 µm)
253
+ 8µm LF 0.65 <z<0.90 L15(15 µm)
254
+ 8µm LF 1.1 <z<1.4 L18W (18 µm)
255
+ 8µm LF 1.8 <z<2.2 L24(24 µm)
256
+ 12µm LF 0.15 <z<0.35 L15(15 µm)
257
+ 12µm LF 0.38 <z<0.62 L18W (18 µm)
258
+ 12µm LF 0.84 <z<1.16 L24(24 µm)
259
+ TIRLF 0.2 <z<0.5S7,S9W,S11,L15,L18WandL24
260
+ TIRLF 0.5 <z<0.8S7,S9W,S11,L15,L18WandL24
261
+ TIRLF 0.8 <z<1.2S7,S9W,S11,L15,L18WandL24
262
+ TIRLF 1.2 <z<1.6S7,S9W,S11,L15,L18WandL24
263
+ Among various templates and fitting parameters we tried,
264
+ we found the best results were obtained with the following: w e
265
+ used modified CWW (Coleman,Wu,& Weedman, 1980) and
266
+ QSO templates.TheseCWW templatesareinterpolatedandad-
267
+ justed to better match VVDS spectra (Arnoutsetal., 2007). W e
268
+ included strong emission lines in computing colors. We used
269
+ the Calzetti extinction law. More details in training LePhare
270
+ isgiveninIlbertet al.(2006).
271
+ The resulting photometric redshift estimates agree reason -
272
+ ably well with 293 galaxies ( R <24) with spectroscopic red-
273
+ shifts taken with Keck/DEIMOS in the NEP field (Takagi et al.
274
+ inprep.).Themeasurederrorsonthephoto- zare∆z
275
+ 1+z=0.036for
276
+ z≤0.8and∆z
277
+ 1+z=0.10 for z >0.8. The∆z
278
+ 1+zbecomes signifi-
279
+ cantly larger at z >0.8, where we suffer from relative shallow-
280
+ ness of our near-IR data. The rate of catastrophic failures i s 4%
281
+ (∆z
282
+ 1+z>0.2)amongthespectroscopicsample.
283
+ In Fig.1, we compare spectroscopic redshifts from
284
+ Keck/DEIMOS (Takagi et al.) and our photometric red-
285
+ shift estimation. Those SEDs which are better fit with a QSO
286
+ template are shown as red triangles. We remove those red
287
+ triangle objects ( ∼2% of the sample) from the LFs presented
288
+ below. We caution that this can only remove extreme type-1
289
+ AGNs, and thus, fainter, type-2 AGN that could be removedby
290
+ X-raysoropticalspectroscopystill remainin thesample.
291
+ Fig.2showsthedistributionofphotometricredshift.Thed is-
292
+ tributionhasseveralpeaks,whichcorrespondstogalaxycl usters
293
+ in the field (Gotoetal., 2008). We have 12% of sources that do
294
+ nothaveagoodSEDfit toobtainareliablephotometricredshi ft
295
+ estimation.Weapplythisphoto- zcompletenesscorrectiontothe
296
+ LFs we obtain.Readers are referredto Negrelloet atal. (200 9),
297
+ who estimated photometricredshifts using only the AKARI fil -
298
+ terstoobtain10%accuracy.
299
+ 2.3. The1/ Vmaxmethod
300
+ WecomputeLFsusingthe1/ Vmaxmethod(Schmidt,1968).The
301
+ advantage of the 1/ Vmaxmethod is that it allows us to compute
302
+ a LF directly from data, with no parameter dependence or an
303
+ assumed model. A drawback is that it assumes a homogeneous
304
+ galaxy distribution, and is thus vulnerable to local over-/ under-
305
+ densities(Takeuchi,Yoshikawa,&Ishii,2000).
306
+ A comoving volume associated with any source of a given
307
+ luminosity is defined as Vmax=Vzmax−Vzmin, wherezmin
308
+ is the lower limit of the redshift bin and zmaxis the maximum
309
+ redshiftat whichthe objectcouldbe seen giventhe fluxlimit of
310
+ the survey, with a maximum value corresponding to the upperredshiftoftheredshiftbin.Moreprecisely,
311
+ zmax= min(z maxof the bin ,zmaxfromthe flux limit) (1)
312
+ We usedtheSED templates(Lagache,Dole,&Puget, 2003) for
313
+ k-corrections to obtain the maximum observable redshift fro m
314
+ thefluxlimit.
315
+ Foreachluminositybinthen,theLFisderivedas
316
+ φ=1
317
+ ∆L/summationdisplay
318
+ i1
319
+ Vmax,iwi, (2)
320
+ whereVmaxis a comoving volume over which the ith galaxy
321
+ couldbeobserved, ∆Listhesizeoftheluminositybin(0.2dex),
322
+ andwiis the completeness correction factor of the ith galaxy.
323
+ WeusecompletenesscorrectionmeasuredbyWadaet al.(2008 )
324
+ for11and24 µmandPearsonet al.(2009a,b)for15and18 µm.
325
+ Thiscorrectionis25%atmaximum,sincewe onlyusethesam-
326
+ plewherethecompletenessisgreaterthan80%.
327
+ 2.4. Monte Carlo simulation
328
+ Uncertainties of the LF values stem from various factors suc h
329
+ as fluctuations in the numberof sources in each luminosity bi n,
330
+ the photometric redshift uncertainties, the k-correction uncer-
331
+ tainties, and the flux errors. To compute these errors we per-
332
+ formedMonteCarlosimulationsbycreating1000simulatedc at-
333
+ alogs,whereeach catalogcontainsthesame numberof source s,
334
+ but we assign each source a new redshift following a Gaussian
335
+ distribution centered at the photometric redshift with the mea-
336
+ sured dispersion of ∆z/(1 +z) =0.036 for z≤0.8and
337
+ ∆z/(1+z) =0.10forz >0.8(Fig.1). The flux of each source
338
+ is also allowed to vary accordingto the measuredflux error fo l-
339
+ lowingaGaussiandistribution.For8 µmand12µmLFs,wecan
340
+ ignore the errors due to the k-correction thanks to the AKARI
341
+ MIR filter coverage. For TIR LFs, we have added 0.05 dex of
342
+ error for uncertaintyin the SED fitting following the discus sion
343
+ in Magnelliet al. (2009). We did not consider the uncertaint y
344
+ on the cosmic variance here since the AKARI NEP field cov-
345
+ ers a large volume and has comparable number counts to other
346
+ generalfields(Imaiet al.,2007,2008).Eachredshiftbinwe use
347
+ covers∼106Mpc3of volume. See Matsuharaetal. (2006) for
348
+ morediscussion on the cosmic variancein the NEP field. These
349
+ estimated errors are added to the Poisson errors in each LF bi n
350
+ inquadrature.
351
+ 3. Results
352
+ 3.1. 8µm LF
353
+ Monochromatic 8 µm luminosity ( L8µm) is known to cor-
354
+ relate well with the TIR luminosity (Babbedgeet al., 2006;
355
+ Huanget al.,2007),especiallyforstar-forminggalaxiesb ecause
356
+ the rest-frame 8 µm flux are dominated by prominent PAH fea-
357
+ turessuchasat 6.2,7.7and8.6 µm.
358
+ Since the AKARI has continuous coverage in the mid-IR
359
+ wavelengthrange,therestframe8 µmluminositycanbeobtained
360
+ without a large uncertainty in k-correction at a corresponding
361
+ redshift and filter. For example, at z=0.375, restframe 8 µm is
362
+ redshiftedinto S11filter. Similarly, L15,L18WandL24cover
363
+ restframe 8 µm atz=0.875, 1.25 and 2. This continuous filter
364
+ coverageisanadvantagetoAKARIdata.OftenSEDmodelsare
365
+ used to extrapolate from Spitzer 24 µm flux in previous work,Gotoet al.:InfraredLuminosityfunctions withthe AKARI 5
366
+ producingasourceofthe largestuncertainty.We summarise fil-
367
+ tersusedinTable1.
368
+ To obtain restframe 8 µm LF, we applied a flux limit
369
+ of F(S11) <70.9, F(L15) <117, F(L18W) <121.4, and
370
+ F(L24)<275.8µJy atz=0.38-0.58, z=0.65-0.90, z=1.1-1.4
371
+ andz=1.8-2.2,respectively.Thesearethe5 σlimitsmeasuredin
372
+ Wada etal. (2008). We exclude those galaxies whose SEDs are
373
+ betterfit withQSO templates( §2).
374
+ We use the completeness curve presented in Wada et al.
375
+ (2008) and Pearsonet al. (2009a,b) to correct for the incom-
376
+ pleteness of the detection. However, this correction is 25% at
377
+ maximumsincethesampleis80%completeatthe5 σlimit.Our
378
+ mainconclusionsarenotaffectedbythisincompletenessco rrec-
379
+ tion. To compensatefor the increasing uncertaintyin incre asing
380
+ z, we use redshift binsize of 0.38 < z <0.58, 0.65 < z <0.90,
381
+ 1.1< z <1.4,and 1.8 < z <2.2.We show the L8µmdistribution
382
+ in each redshift rangein Fig.3. Within each redshift bin, we use
383
+ 1/Vmaxmethodto compensateforthefluxlimit ineachfilter.
384
+ We show the computed restframe 8 µm LF in Fig.4. Arrows
385
+ show the 8 µm luminosity correspondingto the flux limit at the
386
+ central redshift in each redshift bin. Errorbarson each poi nt are
387
+ basedontheMonteCarlosimulation( §2.3).
388
+ For a comparison, as the green dot-dashed line, we also
389
+ show the 8 µm LF of star-forming galaxies at 0< z < 0.3
390
+ by Huanget al. (2007), using the 1/ Vmaxmethod applied to the
391
+ IRAC 8µm GTO data. Compared to the local LF, our 8 µm LFs
392
+ showstrongevolutionin luminosity.Intherangeof 0.48< z <
393
+ 2,L∗
394
+ 8µmevolvesas ∝(1+z)1.6±0.2. Detailedcomparisonwith
395
+ theliteraturewill bepresentedin §4.
396
+ 3.2. Bolometric IR luminosity density basedonthe 8 µm
397
+ LF
398
+ Constraining the star formation history of galaxies as a fun c-
399
+ tion of redshift is a key to understanding galaxy formation i n
400
+ the Universe. One of the primary purposes in computing IR
401
+ LFs is to estimate the IR luminosity density, which in turn is a
402
+ goodestimatorof thedust hiddencosmic star formationdens ity
403
+ (Kennicutt, 1998). Since dust obscurationis more importan t for
404
+ more actively star forming galaxies at higher redshift, and such
405
+ star formationcannotbeobservedinUV light,it is importan tto
406
+ obtainIR-basedestimateinordertofullyunderstandtheco smic
407
+ star formationhistoryoftheUniverse.
408
+ Weestimatethetotalinfraredluminositydensitybyintegr at-
409
+ ingtheLFweightedbytheluminosity.First, weneedtoconve rt
410
+ L8µmto the bolometric infrared luminosity. The bolometric IR
411
+ luminosity of a galaxy is produced by the thermal emission of
412
+ its interstellarmatter. Instar-forminggalaxies,the UV r adiation
413
+ producedbyyoungstarsheatstheinterstellardust,andthe repro-
414
+ cessed lightisemittedin theIR. Forthisreason,in star-fo rming
415
+ galaxies,thebolometricIRluminosityisagoodestimatoro fthe
416
+ current SFR (star formation rate) of the galaxy. Bavouzetet al.
417
+ (2008) showed a strong correlation between L8µmand total in-
418
+ frared luminosity ( LTIR) for 372 local star-forming galaxies.
419
+ TheconversiongivenbyBavouzetet al.(2008)is:
420
+ LTIR= 377.9×(νLν)0.83
421
+ rest8µm(±37%) (3)
422
+ Caputi etal. (2007) further constrained the sample to lumi-
423
+ nous, high S/N galaxies ( νL8µm
424
+ ν>1010L⊙and S/N>3in all
425
+ MIPS bands) in order to better match their sample, and derive d
426
+ thefollowingequation.Fig.4.Restframe 8 µm LFs based on the AKARI NEP-Deep
427
+ field. The blue diamons, purple triangles, red squares, and o r-
428
+ ange crosses show the 8 µm LFs at 0.38< z <0.58,0.65<
429
+ z <0.90,1.1< z <1.4, and1.8< z <2.2, respectively.
430
+ AKARI’s MIR filters can observe restframe 8 µm at these red-
431
+ shifts in a corresponding filter. Errorbars are from the Mont e
432
+ Caro simulations ( §2.4). The dotted lines show analytical fits
433
+ with a double-power law. Vertical arrows show the 8 µm lumi-
434
+ nosity corresponding to the flux limit at the central redshif t in
435
+ each redshift bin. Overplotted are Babbedgeet al. (2006) in the
436
+ pink dash-dotted lines, Caputiet al. (2007) in the cyan dash -
437
+ dotted lines, and Huanget al. (2007) in the green dash-dotte d
438
+ lines.AGNsareexcludedfromthe sample( §2.2).
439
+ LTIR= 1.91×(νLν)1.06
440
+ rest8µm(±55%) (4)
441
+ Since ours is also a sample of bright galaxies, we use this
442
+ equation to convert L8µmtoLTIR. Because the conversion is
443
+ based on local star-forming galaxies, it is a concern if it ho lds
444
+ at higher redshift or not. Bavouzetet al. (2008) checked thi s by
445
+ stacking 24 µm sources at 1.3< z <2.3in the GOODS fields
446
+ to find the stacked sources are consistent with the local rela -
447
+ tion. They concluded that equation (3) is valid to link L8µm
448
+ andLTIRat1.3< z <2.3. Takagiet al. (2010) also show
449
+ that local L7.7µmvsLTIRrelation holds true for IR galaxies
450
+ at z∼1 (see their Fig.10). Popeetal. (2008) showed that z∼2
451
+ sub-millimeter galaxies lie on the relation between LTIRand
452
+ LPAH,7.7that has been established for local starburst galaxies.
453
+ S70/S24ratios of 70 µm sources in Papovichet al. (2007) are
454
+ also consistent with local SED templates. These results sug gest
455
+ it isreasonabletouse equation(4) foroursample.
456
+ The conversion, however, has been the largest source of er-
457
+ rorinestimating LTIRfromL8µm.Bavouzetet al.(2008)them-
458
+ selvesquote37%ofuncertainty,andthatCaputietal.(2007 )re-
459
+ port 55% of dispersion around the relation. It should be kept in
460
+ mind that the restframe 8µm is sensitive to the star-formation
461
+ activity, but at the same time, it is where the SED models have
462
+ strongest discrepancies due to the complicated PAH emissio n
463
+ lines. A detailed comparison of different conversions is pr e-
464
+ sented in Fig.12 of Caputiet al. (2007), who reported factor of
465
+ ∼5ofdifferencesamongvariousmodels.6 Gotoet al.:InfraredLuminosityfunctions withthe AKARI
466
+ Then the 8 µm LF is weighted by the LTIRand integrated
467
+ to obtain TIR density. For integration, we first fit an ana-
468
+ lytical function to the LFs. In the literature, IR LFs were
469
+ fit better by a double-power law (Babbedgeet al., 2006) or
470
+ a double-exponential (Saunderset al., 1990; Pozziet al., 2 004;
471
+ Takeuchiet al., 2006; Le Floc’het al., 2005) than a Schechte r
472
+ function, which declines too suddenlly at the high luminosi ty,
473
+ underestimating the number of bright galaxies. In this work ,
474
+ we fit the 8 µm LFs using a double-powerlaw (Babbedgeet al.,
475
+ 2006)asshownbelow.
476
+ Φ(L)dL/L∗= Φ∗/parenleftbiggL
477
+ L∗/parenrightbigg1−α
478
+ dL/L∗,(L < L∗) (5)
479
+ Φ(L)dL/L∗= Φ∗/parenleftbiggL
480
+ L∗/parenrightbigg1−β
481
+ dL/L∗,(L > L∗) (6)
482
+ First, the double-powerlaw is fitted to the lowest redshift L F at
483
+ 0.38< z <0.58 to determine the normalization( Φ∗) and slopes
484
+ (α,β).Forhigherredshiftswedonothaveenoughstatisticstosi -
485
+ multaneouslyfit 4parameters( Φ∗,L∗,α,andβ).Therefore,we
486
+ fixedtheslopesandnormalizationat the localvaluesandvar ied
487
+ onlyL∗atforthehigher-redshiftLFs.Fixingthefaint-endslope
488
+ isacommonprocedurewiththedepthofcurrentIRsatellites ur-
489
+ veys (Babbedgeet al., 2006; Caputi etal., 2007). The strong er
490
+ evolution in luminosity than in density found by previous wo rk
491
+ (P´ erez-Gonz´ alezet al., 2005; LeFloc’het al., 2005) also justi-
492
+ fies this parametrization. Best fit parameters are presented in
493
+ Table2.Oncethebest-fitparametersarefound,weintegrate the
494
+ doublepowerlawoutsidetheluminosityrangeinwhichwehav e
495
+ data to obtain estimate of the total infrared luminosity den sity,
496
+ ΩTIR.
497
+ The resulting total luminosity density ( ΩIR) is shown in
498
+ Fig.5 as a function of redshift. Errors are estimated by vary ing
499
+ thefit within1 σofuncertaintyin LFs, thenerrorsin conversion
500
+ fromL8µmtoLTIRare added. The latter is by far the larger
501
+ source of uncertainty. Simply switching from equation (3) ( or-
502
+ ange dashed line) to (4) (red solid line) produces a ∼50% dif-
503
+ ference. Cyan dashed lines show results from LeFloc’het al.
504
+ (2005) for a comparision. The lowest redshift point was cor-
505
+ rectedfollowingMagnellietal. (2009).
506
+ We also show the evolution of monochromatic 8 µm lumi-
507
+ nosity (L8µm), which is obtained by integrating the fits, but
508
+ without converting to LTIRin Fig.6. The Ω8µmevolves as
509
+ ∝(1+z)1.9±0.7.
510
+ The SFR and LTIRare related by the following equation
511
+ for a Salpeter IMF, φ(m)∝m−2.35between0.1−100M⊙
512
+ (Kennicutt,1998).
513
+ SFR(M⊙yr−1) = 1.72×10−10LTIR(L⊙) (7)
514
+ The right ticks of Fig.5 shows the star formation density
515
+ scale,convertedfrom ΩIRusingtheaboveequation.
516
+ In Fig.5, ΩIRmonotonically increases toward higher z.
517
+ Comparedwith z=0,ΩIRis∼10timeslargerat z=1.Theevolu-
518
+ tionbetween z=0.5andz=1.2isalittleflatter,butthisisperhaps
519
+ duetoamoreirregularshapeofLFsat0.65 < z <0.90,andthus,
520
+ wedonotconsideritsignificant.Theresultsobtainedherea gree
521
+ with previous work (e.g., Le Floc’het al., 2005) within the e r-
522
+ rors. We compare the results with previous work in more detai l
523
+ in§4.Fig.5.Evolution of TIR luminosity density computed by inte-
524
+ grating the 8 µm LFs in Fig.4.The red solid lines use the con-
525
+ version in equation (4). The orange dashed lines use equatio n
526
+ (3).ResultsfromLeFloc’hetal.(2005)areshownwiththecy an
527
+ dottedlines.
528
+ Fig.6.Evolution of 8 µm IR luminosity density computed by
529
+ integrating the 8 µm LFs in Fig.4. The lowest redshift point is
530
+ fromHuanget al.(2007).
531
+ 3.3. 12µm LF
532
+ In this subsection we estimate restframe 12 µm LFs based
533
+ on the AKARI NEP-Deep data. 12 µm luminosity ( L12µm)
534
+ has been well studied through ISO and IRAS, and known to
535
+ correlate closely with TIR luminosity (Spinoglioetal., 19 95;
536
+ P´ erez-Gonz´ alezet al.,2005).
537
+ As was the case for the 8 µm LF, it is advantageous that
538
+ AKARI’s continuous filters in the mid-IR allow us to estimate
539
+ restframe 12 µm luminosity without much extrapolation based
540
+ onSEDmodels.Gotoet al.:InfraredLuminosityfunctions withthe AKARI 7
541
+ Table 2.Best fit parametersfor8,12 µmandTIRLFs
542
+ Redshift λ L∗(L⊙)Φ∗(Mpc−3dex−1)α β
543
+ 0.38<z<0.58 8 µm (2.2+0.3
544
+ −0.1)×1010(2.1+0.3
545
+ −0.4)×10−31.75+0.01
546
+ −0.013.5+0.2
547
+ −0.4
548
+ 0.65<z<0.90 8 µm (2.8+0.1
549
+ −0.1)×10102.1×10−31.75 3.5
550
+ 1.1<z<1.4 8 µm (3.3+0.2
551
+ −0.2)×10102.1×10−31.75 3.5
552
+ 1.8<z<2.2 8 µm (8.2+1.2
553
+ −1.8)×10102.1×10−31.75 3.5
554
+ 0.15<z<0.35 12 µm (6.8+0.1
555
+ −0.1)×109(4.2+0.7
556
+ −0.6)×10−31.20+0.01
557
+ −0.022.9+0.4
558
+ −0.2
559
+ 0.38<z<0.62 12 µm (11.7+0.3
560
+ −0.5)×1094.2×10−31.20 2.9
561
+ 0.84<z<1.16 12 µm (14+2
562
+ −3)×1094.2×10−31.20 2.9
563
+ 0.2<z<0.5 Total (1.2+0.1
564
+ −0.2)×1011(5.6+1.5
565
+ −0.2)×10−41.8+0.1
566
+ −0.43.0+1.0
567
+ −1.0
568
+ 0.5<z<0.8 Total (2.4+1.8
569
+ −1.6)×10115.6×10−41.8 3.0
570
+ 0.8<z<1.2 Total (3.9+2.3
571
+ −2.2)×10115.6×10−41.8 3.0
572
+ 1.2<z<1.6 Total (14+1
573
+ −2)×10115.6×10−41.8 3.0
574
+ Fig.7.12µm luminosity distributions of samples used to com-
575
+ puterestframe12 µmLFs. Fromlowredshift,335,573,and213
576
+ galaxiesarein eachredshiftbin.
577
+ Targeted redshifts are z=0.25, 0.5 and 1 where L15,L18W
578
+ andL24filterscovertherestframe12 µm,respectively.Wesum-
579
+ marise the filters used in Table 1. Methodology is the same as
580
+ for the 8µm LF; we used the sample to the 5 σlimit, corrected
581
+ for the completeness, then used the 1/ Vmaxmethod to com-
582
+ pute LF in each redshift bin. The histogram of L12µmdistri-
583
+ bution is presented in Fig.7. The resulting 12 µm LF is shown
584
+ in Fig.8. Compared with Rush,Malkan,& Spinoglio (1993)’s
585
+ z=0 LF based on IRAS Faint Source Catalog, the 12 µm LFs
586
+ show steady evolution with increasing redshift. In the rang e of
587
+ 0.25< z <1,L∗
588
+ 12µmevolvesas ∝(1+z)1.5±0.4.
589
+ 3.4. Bolometric IR luminosity density basedonthe 12 µm
590
+ LF
591
+ 12µm is one of the most frequentlyused monochromaticfluxes
592
+ to estimate LTIR. The total infrared luminosity is computed
593
+ from theL12µmusing the conversionin Chary& Elbaz (2001);
594
+ P´ erez-Gonz´ alezet al.(2005).
595
+ logLTIR= log(0.89+0.38
596
+ −0.27)+1.094logL12µm (8)Fig.8.Restframe 12 µm LFs based on the AKARI NEP-Deep
597
+ field.Thebluediamonds,purpletriangles,andredsquaress how
598
+ the 12µm LFs at 0.15< z <0.35,0.38< z <0.62, and
599
+ 0.84< z <1.16, respectively. Vertical arrows show the 12 µm
600
+ luminosity corresponding to the flux limit at the central red -
601
+ shift in each redshift bin. Overplotted are P´ erez-Gonz´ al ezet al.
602
+ (2005) at z=0.3,0.5 and 0.9 in the cyan dash-dotted lines, and
603
+ Rush,Malkan,& Spinoglio (1993) at z=0 in the green dash-
604
+ dottedlines. AGNsareexcludedfromthesample( §2.2).
605
+ Takeuchietal. (2005) independently estimated the relatio n
606
+ tobe
607
+ logLTIR= 1.02+0.972logL12µm, (9)
608
+ which we also use to check our conversion. As both au-
609
+ thors state, these conversions contain an error of factor of 2-3.
610
+ Therefore, we should avoid conclusions that could be affect ed
611
+ bysucherrors.
612
+ Then the 12 µm LF is weighted by the LTIRand integrated
613
+ to obtain TIR density. Errors are estimated by varying the fit
614
+ within 1σof uncertainty in LFs, and errors in converting from
615
+ L12µmtoLTIRareadded.Thelatter isbyfarthe largestsource
616
+ of uncertainty. Best fit parameters are presented in Table 2. In
617
+ Fig.10,we showtotal luminositydensitybasedonthe12 µmLF8 Gotoet al.:InfraredLuminosityfunctions withthe AKARI
618
+ Fig.9.Evolution of 12 µm IR luminosity density computed by
619
+ integratingthe12 µmLFsinFig.8.
620
+ Fig.10. TIR luminosity density computed by integrating the
621
+ 12µmLFsin Fig.8.
622
+ presented in Fig.8. The results show a rapid increase of ΩIR,
623
+ agreeing with previous work (LeFloc’hetal., 2005) within t he
624
+ errors.
625
+ We also integrate monochromatic L12µmover the LFs
626
+ (without converting to LTIR) to derive the evolution of to-
627
+ tal12µmmonochromatic luminosity density, Ω12µm. The re-
628
+ sults are shown in Fig.9, which shows a strong evolution of
629
+ Ω12µm∝(1 +z)1.4±1.4. It is interesting to compare this to
630
+ Ω8µm∝(1 +z)1.9±0.7obtained in §3.2. Although errors are
631
+ significantonbothestimates, Ω12µmandΩ8µmshowa possibly
632
+ differentevolution,suggestingthatthecosmicinfrareds pectrum
633
+ changesits SED shape.Whetherthisisdueto evolutionindus t,
634
+ or dusty AGN contribution is an interesting subject for futu re
635
+ work.Fig.11.An example of the SED fit. The red dashed line shows
636
+ thebest-fitSEDfortheUV-optical-NIRSED,mainlytoestima te
637
+ photometricredshift.Thebluesolidlineshowsthebest-fit model
638
+ fortheIRSEDat λ >6µm,toestimate LTIR.
639
+ 3.5. TIRLF
640
+ AKARI’scontinuousmid-IRcoverageisalsosuperiorforSED -
641
+ fitting to estimate LTIR, since for star-forming galaxies, the
642
+ mid-IR part of the IR SED is dominated by the PAH emissions
643
+ whichreflectthe SFR ofgalaxies,andthus,correlateswell w ith
644
+ LTIR, which is also a good indicator of the galaxy SFR. The
645
+ AKARI’scontinuousMIRcoveragehelpsustoestimate LTIR.
646
+ After photometric redshifts are estimated using the UV-
647
+ optical-NIRphotometry,we fix the redshift at the photo- z,then
648
+ use the same LePhare code to fit the infrared part of the SED
649
+ to estimate TIR luminosity. We used Lagache,Dole,&Puget
650
+ (2003)’s SED templates to fit the photometryusing the AKARI
651
+ bands at >6µm (S7,S9W,S11,L15,L18WandL24). We
652
+ showanexampleoftheSEDfitinFig.11,wherethereddashed
653
+ and blue solid lines show the best-fit SEDs for the UV-optical -
654
+ NIR and IR SED at λ >6µm, respectively. The obtained total
655
+ infraredluminosity( LTIR) is shown as a functionofredshift in
656
+ Fig.12,withspectroscopicgalaxiesinlargetriangles.Th efigure
657
+ shows that the AKARI can detect LIRGs ( LTIR>1011L⊙)
658
+ up toz=1 and ULIRGs ( LTIR>1012L⊙) toz=2. We also
659
+ checkedthatusingdifferentSEDmodels(Chary& Elbaz,2001 ;
660
+ Dale& Helou,2002) doesnotchangeouressentialresults.
661
+ Galaxies in the targeted redshift range are best sampled in
662
+ the 18µm band due to the wide bandpass of the L18Wfilter
663
+ (Matsuharaet al., 2006). In fact, in a single-band detectio n, the
664
+ 18µm image returns the largest number of sources. Therefore,
665
+ we applied the 1/ Vmaxmethod using the detection limit at
666
+ L18W. We also checked that using the L15flux limit does
667
+ not change our main results. The same Lagache,Dole,&Puget
668
+ (2003)’s models are also used for k-corrections necessary to
669
+ compute VmaxandVmin. The redshift bins used are 0.2 <
670
+ z <0.5,0.5< z <0.8,0.8< z <1.2,and 1.2 < z <1.6. A distri-
671
+ butionof LTIRineachredshiftbinis showninFig.13.
672
+ Theobtained LTIRLFsareshowninFig.14.Theuncertain-
673
+ ties are esimated through the Monte Carlo simulations ( §2.4).
674
+ For a local benchmark, we overplot Sanderset al. (2003) who
675
+ derived LFs from the analytical fit to the IRAS Revised Bright
676
+ Galaxy Sample, i.e., φ∝L−0.6forL < L∗andφ∝L−2.2for
677
+ L > L∗withL∗= 1010.5L⊙. The TIR LFs show a strong evo-
678
+ lutioncomparedtolocalLFs.At 0.25< z <1.3,L∗
679
+ TIRevolvesGotoet al.:InfraredLuminosityfunctions withthe AKARI 9
680
+ Fig.12.TIR luminosity is shown as a function of photometric
681
+ redshift. The photo- zis estimated using UV-optical-NIR pho-
682
+ tometry.LTIRisobtainedthroughSED fit in7-24 µm.
683
+ Fig.13.AhistogramofTIRluminosity.Fromlow-redshift,144,
684
+ 192, 394, and 222 galaxies are in 0.2 < z <0.5, 0.5< z <0.8,
685
+ 0.8< z <1.2,and1.2 < z <1.6,respectively.
686
+ as∝(1 +z)4.1±0.4. We further compare LFs to the previous
687
+ workin§4.
688
+ 3.6. Bolometric IR luminosity density basedonthe TIRLF
689
+ Using the same methodology as in previous sections, we inte-
690
+ grateLTIRLFs in Fig.14 through a double-power law fit (eq.
691
+ 5 and 6). The resulting evolution of the TIR density is shown
692
+ with red diamonds in Fig.15, which in in good agreement with
693
+ LeFloc’hetal.(2005)withintheerrors.Errorsareestimat edby
694
+ varying the fit within 1 σof uncertainty in LFs. For uncertainty
695
+ intheSEDfit,weadded0.15dexoferror.Bestfitparametersar e
696
+ presented in Table 2. In Fig.15, we also show the contributio ns
697
+ toΩTIRfromLIRGsandULIRGswiththebluesquaresandor-
698
+ ange triangles, respectively. We further discuss the evolu tion of
699
+ ΩTIRin§4.Fig.14.TIRLFs.Verticallinesshowtheluminositycorrespond-
700
+ ing to the flux limit at the central redshift in each redshift b in.
701
+ AGNsareexcludedfromthesample( §2.2).
702
+ Fig.15. TIR luminosity density (red diamonds) computed by
703
+ integrating the total LF in Fig.14. The blue squares and oran ge
704
+ trianglesareforLIRG andULIRGsonly.
705
+ 4. Discussion
706
+ 4.1. Comparison with previouswork
707
+ In this section, we compare our results to previous work, esp e-
708
+ ciallythosebasedontheSpitzerdata.Comparisonsarebest done
709
+ inthesamewavelengths,sincetheconversionfromeither L8µm
710
+ orL12µmtoLTIRinvolves the largest uncertainty. Hubble pa-
711
+ rametersinthepreviousworkareconvertedto h= 0.7forcom-
712
+ parison.10 Gotoet al.:InfraredLuminosityfunctions withthe AKARI
713
+ 4.1.1. 8µm LFs
714
+ Recently, using the Spitzer space telescope, restframe 8 µm LFs
715
+ ofz∼1 galaxies have been computed in detail by Caputiet al.
716
+ (2007) in the GOODS fields and by Babbedgeetal. (2006) in
717
+ theSWIREfield.Inthissection,wecompareourrestframe8 µm
718
+ LFs(Fig.4)tothese anddiscusspossibledifferences.
719
+ In Fig.4, we overplot Caputi etal. (2007)’s LFs at z=1 and
720
+ z=2inthecyandash-dottedlines.Their z=2LFisingoodagree-
721
+ ment with our LF at 1.8 < z <2.2. However, their z=1 LF is
722
+ larger than ours by a factor of 3-5 at logL >11.2. Note that
723
+ the brightest ends( logL∼11.4)are consistent with each other
724
+ to within 1 σ. They have excluded AGN using optical-to-X-ray
725
+ flux ratio, and we also have excluded AGN through the optical
726
+ SED fit. Therefore, especially at the faint-end, the contami na-
727
+ tionfromAGN isnot likelyto be the maincauseof differences .
728
+ Since Caputiet al. (2007) uses GOODS fields, cosmic variance
729
+ may play a role here. The exact reason for the difference is un -
730
+ known, but we point out that their ΩIRestimate at z=1 is also
731
+ higherthanotherestimatesbyafactorofafew(seetheirFig .15).
732
+ Once converted into LTIR, Magnelliet al. (2009) also reported
733
+ Caputiet al.(2007)’s z=1LF ishigherthantheirestimatebased
734
+ on 70µm by a factor of several (see their Fig.12). They con-
735
+ cluded the difference is from different SED models used, sin ce
736
+ their LF matched with that of Caputi etal. (2007)’s once the
737
+ same SED models were used. We will compare our total LFs
738
+ tothosein theliteraturebelow.
739
+ Babbedgeet al. (2006) also computed restframe 8 µm LFs
740
+ using the Spitzer/SWIRE data. We overplot their results at
741
+ 0.25< z <0.5and0.5< z <1in Fig.4 with the pink dot-
742
+ dashedlines.Inbothredshiftranges,goodagreementisfou ndat
743
+ higherluminositybins( L8µm>1010.5L⊙).However,atallred-
744
+ shift ranges including the ones not shown here, Babbedgeet a l.
745
+ (2006) tends to show a flatter faint-end tail than ours, and a
746
+ smallerφby a factor of ∼3. Although the exact reason is un-
747
+ known, the deviation starts toward the fainter end, where bo th
748
+ works approach the flux limits of the surveys. Therefore,pos si-
749
+ blyincompletesamplingmaybeoneofthereasons.Itisalsor e-
750
+ portedthat thefaint-endof IRLFsdependson theenvironmen t,
751
+ in the sense that higher-density environment has steeper fa int-
752
+ end tail (Gotoet al., 2010). Note that at z=1, Babbedgeet al.
753
+ (2006)’s LF (pink) deviates from that by Caputiet al. (2007)
754
+ (cyan) by almost a magnitude. Our 8 µm LFs are between these
755
+ works.
756
+ These comparisons suggest that even with the current gen-
757
+ eration of satellites and state-of-the-art SED models, fac tor-of-
758
+ several uncertainties still remain in estimating the 8 µm LFs
759
+ at z∼1. More accurate determination has to await a larger
760
+ and deeper survey by the next generation IR satellites such a s
761
+ HerschelandWISE.
762
+ To summarise, our 8 µm LFs are between those by
763
+ Babbedgeetal.(2006)andCaputiet al.(2007),anddiscrepa ncy
764
+ is by a factor of several at most. We note that both of the previ -
765
+ ous works had to rely on SED models to estimate L8µmfrom
766
+ the Spitzer S24µmin the MIR wavelengths where SED model-
767
+ ing is difficult. Here, AKARI’s mid-IR bands are advantageou s
768
+ indirectlyobservingredshiftedrestframe8 µmfluxinoneofthe
769
+ AKARI’s filters, leading to more reliable measurement of 8 µm
770
+ LFswithoutuncertaintyfromtheSED modeling.
771
+ 4.1.2. 12 µm LFs
772
+ P´ erez-Gonz´ alezet al. (2005) investigated the evolution of rest-
773
+ frame12µmLFsusingthe SpitzerCDF-S andHDF-N data.Weoverplot their results in similar redshift ranges as the cya n dot-
774
+ dashed lines in Fig.8. Consideringboth LFs have significant er-
775
+ ror bars, these LFs are in good agreement with our LFs, and
776
+ show significant evolution in the 12 µm LFs compared with the
777
+ z=012µmLFbyRush,Malkan,&Spinoglio(1993).Theagree-
778
+ ment is in a stark contrast to the comparison in 8 µm LFs in
779
+ §4.1.1, wherewe sufferedfromdifferncesof a factor of sever al.
780
+ Apossiblereasonforthisisthat12 µmissufficientlyredderthan
781
+ 8µm, that it is easier to be extrapolated from S24µmin case of
782
+ the Spitzer work. In fact, at z=1, both the Spitzer 24 µm band
783
+ and AKARI L24observe the restframe 12 µm directly. In addi-
784
+ ton, mid-IR SEDs around 12 µm are flatter than at 8 µm, where
785
+ PAH emissions are prominent.Therefore,SED modelscan pre-
786
+ dict the flux more accurately. In fact, this is part of the rea-
787
+ sonwhyP´ erez-Gonz´ alezet al.(2005)chosetoinvestigate 12µm
788
+ LFs. P´ erez-Gonz´ alezetal. (2005) used Chary&Elbaz (2001 )’s
789
+ SEDtoextrapolate S24µm,andyet,theyagreewellwithAKARI
790
+ results, which are derived from filters that cover the restfr ame
791
+ 12µm. However, in other words, the discrepancy in 8 µm LFs
792
+ highlights the fact that the SED models are perhaps still imp er-
793
+ fect in the 8 µm wavelengthrange, and thus, MIR-spectroscopic
794
+ data that covers wider luminosity and redshift ranges will b e
795
+ necessary to refine SED models in the mid-IR. AKARI’s mid-
796
+ IR slitless spectroscopy survey (Wada, 2008) may help in thi s
797
+ regard.
798
+ 4.1.3. TIRLFs
799
+ Lastly,we compareourTIRLFs(Fig.14) withthoseinthelite r-
800
+ ature.AlthoughtheTIRLFs canalso be obtainedbyconvertin g
801
+ 8µmLFsor12 µmLFs,wealreadycomparedourresultsinthese
802
+ wavelengths in the last subsections. Here, we compare our TI R
803
+ LFstoLe Floc’het al.(2005)andMagnellietal. (2009).
804
+ LeFloc’het al. (2005) obtained TIR LFs using the Spitzer
805
+ CDF-S data. They have used the best-fit SED among various
806
+ templatestoestimate LTIR.WeoverplottheirtotalLFsinFig.14
807
+ with the cyan dash-dotted lines. Only LFs that overlapwith o ur
808
+ redshit ranges are shown. The agreement at 0.3< z <0.45
809
+ and0.6< z <0.8is reasonable, considering the error bars on
810
+ bothsides.However,inallthreeredshiftranges,LeFloc’h et al.
811
+ (2005)’sLFsare higherthanours,especiallyfor 1.0< z <1.2.
812
+ We also overplot TIR LFs by Magnellietal. (2009), who
813
+ used Spitzer 70 µm flux and Chary& Elbaz (2001)’s model to
814
+ estimateLTIR.Inthetwobins(centeredon z=0.55and z=0.85;
815
+ pink dash-dotted lines) which closely overlap with our reds hift
816
+ bins, excellent agreement is found. We also plot Huynhet al.
817
+ (2007)’s LF at 0.6< z <0.9in the navy dash-dotted lines,
818
+ whichis computedfromSpitzer 70µmimagingin the GOODS-
819
+ N, and this also shows very good agreement with ours. These
820
+ LFs are on top of each other within the error bars, despite the
821
+ fact that these measurements are from different data sets us ing
822
+ differentanalyses.
823
+ This means that LeFloc’hetal. (2005)’s LFs is also higher
824
+ thanthatofMagnelliet al.(2009),inadditiontoours.Apos sible
825
+ reasonis that both Magnelliet al. (2009) and we removedAGN
826
+ (at least bright ones), whereas Le Floc’het al. (2005) inclu ded
827
+ them. This also is consistent with the fact that the differen ce
828
+ is larger at 1.0< z <1.2where both surveys are only sen-
829
+ sitive to luminous IR galaxies, which are dominated by AGN.
830
+ Another possible source of uncertainty is that Magnelliet a l.
831
+ (2009) and we used a single SED library, while LeFloc’het al.
832
+ (2005)pickedthebestSEDtemplateamongseverallibraries for
833
+ eachgalaxy.Gotoet al.:InfraredLuminosityfunctions withthe AKARI 11
834
+ Fig.16.EvolutionofTIRluminositydensitybasedonTIRLFs(redcir cles),8µmLFs(stars),and12 µmLFs(filledtriangles).The
835
+ blue open squaresand orangefilled squaresare for LIRG and UL IRGs only, also based on our LTIRLFs. Overplotteddot-dashed
836
+ lines are estimates from the literature: LeFloc’het al. (20 05), Magnelliet al. (2009) , P´ erez-Gonz´ alezet al. (2005) , Caputiet al.
837
+ (2007), and Babbedgeet al. (2006) are in cyan, yellow, green , navy,and pink, respectively.The purple dash-dottedline shows UV
838
+ estimatebySchiminovichetal. (2005).Thepinkdashedline showsthetotalestimateofIR(TIRLF)andUV (Schiminoviche t al.,
839
+ 2005).
840
+ 4.2. Evolution of ΩIR
841
+ In this section, we compare the evolution of ΩIRas a function
842
+ ofredshift.InFig.16, weplot ΩIRestimatedfromTIRLFs(red
843
+ circles), 8 µm LFs (brown stars), and 12 µm LFs (pink filled tri-
844
+ angles),as a functionof redshift.Estimatesbased on12 µmLFs
845
+ and TIR LFs agree each other very well, while those from 8 µm
846
+ LFs show a slightly higher value by a factor of a few than oth-
847
+ ers. This perhaps reflects the fact that 8 µm is a more difficult
848
+ part of the SED to be modeled, as we had a poorer agreement
849
+ amongpapersintheliteraturein8 µmLFs.Thebright-endslope
850
+ of the double-power law was 3.5+0.2
851
+ −0.4in Table 2. This is flat-
852
+ ter than a Schechter fit by Babbedgeet al. (2006) and a double-
853
+ exponential fit by Caputiet al. (2007). This is perhaps why we
854
+ obtainedhigher ΩIRin8µm.
855
+ We overplot estimates from various papers in the litera-
856
+ ture(LeFloc’hetal.,2005; Babbedgeet al.,2006;Caputiet al.,
857
+ 2007; P´ erez-Gonz´ alezet al., 2005; Magnelliet al., 2009) in the
858
+ dash-dottedlines. Our ΩIRhasverygoodagreementwith these
859
+ at0< z <1.2,withalmostallthedash-dottedlineslyingwithin
860
+ ourerrorbarsof ΩIRfromLTIRand12µmLFs.Thisisperhaps
861
+ because an estimate of an integrated value such as ΩIRis more
862
+ reliablethanthat ofLFs.
863
+ Atz >1.2, ourΩIRshows a hint of continuous increase,
864
+ while Caputiet al. (2007) and Babbedgeetal. (2006) observe da slight decline at z >1. However,as both authorsalso pointed
865
+ out, at this high-redshift range, both the AKARI and Spitzer
866
+ satellites are sensitive to onlyLIRGs and ULIRGs, and thust he
867
+ extrapolationto fainterluminositiesassumesthefaint-e ndslope
868
+ of the LFs, which couldbe uncertain.In addition,this work h as
869
+ a poorerphoto-zestimate at z >0.8(∆z
870
+ 1+z=0.10)due to the rel-
871
+ atively shallow near-IR data. Several authors tried to over come
872
+ thisproblembystackingundetectedsources.However,ifan un-
873
+ detectedsourceisalsonotdetectedatshorterwavelengths where
874
+ positions for stacking are obtained, it would not be include d in
875
+ the stacking either. Next generation satellite such as Hers chel,
876
+ WISE, and SPICA (Nakagawa, 2008) will determine the faint-
877
+ endslopeat z >1moreprecisely.
878
+ We parameterize the evolution of ΩIRusing a following
879
+ function.
880
+ ΩIR(z)∝(1+z)γ(10)
881
+ By fitting this to the ΩIRfrom TIR LFs, we obtained γ=
882
+ 4.4±1.0. This is consistent with most previous works.
883
+ For example, LeFloc’hetal. (2005) obtained γ= 3.9±
884
+ 0.4, P´ erez-Gonz´ alezet al. (2005) obtained γ= 4.0±0.2,
885
+ Babbedgeetal. (2006) obtained γ= 4.5+0.7
886
+ −0.6, Magnelliet al.
887
+ (2009) obtained γ= 3.6±0.4. The agreement was expected
888
+ fromFig.16,butconfirmsastrongevolutionof ΩIR.12 Gotoet al.:InfraredLuminosityfunctions withthe AKARI
889
+ Fig.17. Contribution of ΩTIRtoΩtotal= ΩUV+ ΩTIRis
890
+ shownasa functionofredshift.
891
+ 4.3. Differential evolution among ULIRG,LIRG,normal
892
+ galaxies
893
+ In Fig. 15, we also plot the contributions to ΩIRfrom LIRGs
894
+ and ULIRGs (measured from TIR LFs) with the blue open
895
+ squares and orange filled squares, respectively. Both LIRGs
896
+ and ULIRGs show strong evolution, as has been seen for to-
897
+ talΩIRin the red filled circles. Normal galaxies ( LTIR<
898
+ 1011L⊙) are still dominant, but decrease their contribution to-
899
+ ward higher redshifts. In contrast, ULIRGs continueto incr ease
900
+ their contribution. From z=0.35 to z=1.4,ΩIRby LIRGs in-
901
+ creases by a factor of ∼1.6, andΩIRby ULIRGs increases by
902
+ a factor of ∼10. The physical origin of ULIRGs in the local
903
+ Universe is often merger/interaction(Sanders& Mirabel, 1 996;
904
+ Taniguchi&Shioya, 1998; Goto, 2005). It would be interesti ng
905
+ to investigate whether the merger rate also increases in pro por-
906
+ tion to the ULIRG fraction, or if different mechanisms can al so
907
+ produceULIRGsathigherredshift.
908
+ 4.4. Comparison tothe UVestimate
909
+ We have been emphasizing the importance of IR probes of the
910
+ total SFRD of the Universe. However, the IR estimates do not
911
+ take into account the contribution of the unabsorbed UV ligh t
912
+ produced by the young stars. Therefore, it is important to es ti-
913
+ matehowsignificantthisUV contributionis.
914
+ Schiminovichet al. (2005) found that the energy density
915
+ measured at 1500 ˚A evolves as ∝(1+z)2.5±0.7at0< z <1
916
+ and∝(1 +z)0.5±0.4atz >1. using the GALEX data sup-
917
+ plemented by the VVDS spectroscopic redshifts. We overplot
918
+ their UV estimate of ρSFRwith the purple dot-dashed line in
919
+ Fig.16. The UV estimate is almost a factor of 10 smaller than
920
+ the IR estimate at most of the redshifts, confirming the impor -
921
+ tanceofIRprobeswheninvestingtheevolutionofthetotalc os-
922
+ mic star formation density. In Fig.16 we also plot total SFD ( or
923
+ Ωtotal)byadding ΩUVandΩTIR,withthemagentadashedline.
924
+ In Fig.17, we show the ratio of the IR contribution to the to-
925
+ tal SFRD of the Universe ( ΩTIR/ΩTIR+ ΩUV) as a function
926
+ of redshift. Although the errors are large, Fig.17 agrees wi thTakeuchi,Buat,& Burgarella (2005), and suggests that ΩTIR
927
+ explains 70% of Ωtotalatz=0.25, and that by z=1.3, 90% of
928
+ the cosmic SFD is explained by the infrared. This implies tha t
929
+ ΩTIRprovidesgoodapproximationofthe Ωtotalatz >1.
930
+ 5. Summary
931
+ We have estimated restframe 8 µm, 12µm, and total infrared lu-
932
+ minosity functions using the AKARI NEP-Deep data. Our ad-
933
+ vantage over previous work is AKARI’s continuous filter cov-
934
+ erage in the mid-IR wavelengths (2.4, 3.2, 4.1, 7, 9, 11, 15, 1 8,
935
+ and24µm),whichallowustoestimate mid-IRluminositywith-
936
+ out a large extrapolationbased on SED models, which were the
937
+ largest uncertainty in previous work. Even for LTIR, the SED
938
+ fitting is much more reliable due to this continuouscoverage of
939
+ mid-IRfilters.
940
+ Ourfindingsareasfollows:
941
+ –8µm LFs show a strong and continuous evolution from
942
+ z=0.35 to z=2.2. Our LFs are larger than Babbedgeet al.
943
+ (2006), but smaller than Caputi etal. (2007). The differenc e
944
+ perhaps stems from the different SED models, highlighting
945
+ a difficulty in SED modeling at wavelengths crowded by
946
+ strong PAH emissions. L∗
947
+ 8µmshows a continuous evolution
948
+ asL∗
949
+ 8µm∝(1+z)1.6±0.2in therangeof 0.48< z <2.
950
+ –12µm LFs show a strong and continuous evolution from
951
+ z=0.15toz=1.16with L∗
952
+ 12µm∝(1+z)1.5±0.4. Thisagrees
953
+ well with P´ erez-Gonz´ alezet al. (2005), including a flatte r
954
+ faint-endslope. A better agreementthan with 8 µm LFs was
955
+ obtained, perhaps because of smaller uncertainty in model-
956
+ ing the 12 µm SED, and less extrapolationneededin Spitzer
957
+ 24µmobservations.
958
+ –The TIR LFs show good agreement with Magnelliet al.
959
+ (2009), but are smaller than Le Floc’het al. (2005). At
960
+ 0.25< z <1.3,L∗
961
+ TIRevolvesas ∝(1+z)4.1±0.4.Possible
962
+ causes of the disagreement include different treatment of
963
+ SEDmodelsinestimating LTIR,andAGNcontamination.
964
+ –TIR densities estimated from 12 µm and TIR LFs show a
965
+ strong evolution as a function of redshift, with ΩIR∝
966
+ (1 +z)4.4±1.0.ΩIR(z)also show an excellent agreement
967
+ withpreviousworkat z <1.2.
968
+ –We investigated the differential contribution to ΩIRby
969
+ ULIRGsandLIRGs.WefoundthattheULIRG(LIRG)con-
970
+ tribution increases by a factor of 10 (1.8) from z=0.35 to
971
+ z=1.4, suggesting IR galaxies are more dominant source of
972
+ ΩIRathigherredshift.
973
+ –We estimated that ΩIRcaptures80% of the cosmic star for-
974
+ mationatredshiftslessthan1,andvirtuallyallofitathig her
975
+ redshift.Thusaddingtheunobscuredstarformationdetect ed
976
+ at UV wavelengths would not change SFRD estimates sig-
977
+ nificantly.
978
+ Acknowledgments
979
+ We are grateful to S.Arnouts for providing the LePhare code,
980
+ and kindly helping us in using the code. We thank the anony-
981
+ mousrefereeformanyinsightfulcomments,whichsignifican tly
982
+ improvedthe paper.
983
+ T.G. and H.I. acknowledgefinancial supportfrom the Japan
984
+ Society for the Promotion of Science (JSPS) through JSPS
985
+ Research Fellowships for Young Scientists. HML acknowl-
986
+ edges the support from KASI through its cooperative fund in
987
+ 2008. TTT has been supported by Program for Improvement
988
+ of Research Environment for Young Researchers from SpecialGotoet al.:InfraredLuminosityfunctions withthe AKARI 13
989
+ CoordinationFundsforPromotingScienceandTechnology,a nd
990
+ the Grant-in-Aid for the Scientific Research Fund (20740105 )
991
+ commissioned by the Ministry of Education, Culture, Sports ,
992
+ Science and Technology (MEXT) of Japan. TTT has been also
993
+ partially supported from the Grand-in-Aid for the Global CO E
994
+ Program “Quest for Fundamental Principles in the Universe:
995
+ from Particles to the Solar System and the Cosmos” from the
996
+ MEXT.
997
+ This research is based on the observations with AKARI, a
998
+ JAXA projectwiththe participationofESA.
999
+ Theauthorswishtorecognizeandacknowledgetheverysig-
1000
+ nificant cultural role and reverence that the summit of Mauna
1001
+ Kea has always had within the indigenous Hawaiian commu-
1002
+ nity. We are most fortunate to have the opportunity to conduc t
1003
+ observationsfromthissacredmountain.
1004
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1
+ arXiv:1001.0018v2 [quant-ph] 28 Jan 2010Nonadaptive quantum query complexity
2
+ Ashley Montanaro∗
3
+ October 1, 2018
4
+ Abstract
5
+ We studythe powerofnonadaptivequantum queryalgorithms,whic h arealgorithms
6
+ whose queries to the input do not depend on the result of previous q ueries. First, we
7
+ show that any bounded-error nonadaptive quantum query algorit hm that computes
8
+ some total boolean function depending on nvariables must make Ω( n) queries to the
9
+ input in total. Second, we show that, if there exists a quantum algor ithm that uses k
10
+ nonadaptive oracle queries to learn which one of a set of mboolean functions it has
11
+ been given, there exists a nonadaptive classical algorithm using O(klogm) queries to
12
+ solve the same problem. Thus, in the nonadaptive setting, quantum algorithms can
13
+ achieve at most a very limited speed-up over classical query algorith ms.
14
+ 1 Introduction
15
+ Many of the best-known results showing that quantum compute rs outperform their classical
16
+ counterparts are proven in the query complexity model. This model studies the number of
17
+ queries to the input xwhich are required to compute some function f(x). In this work, we
18
+ study two broad classes of problem that fit into this model.
19
+ In the first class of problems, computational problems, one wishes to compute some
20
+ boolean function f(x1,...,x n) using a small number of queries to the bits of the input
21
+ x∈ {0,1}n. The query complexity of fis the minimum number of queries required for any
22
+ algorithm to compute f, with some requirement on the success probability. The dete rmin-
23
+ istic query complexity of f,D(f), is the minimum number of queries that a deterministic
24
+ classical algorithm requires to compute fwith certainty. D(f) is also known as the decision
25
+ tree complexity of f. Similarly, the randomised query complexity R2(f) is the minimum
26
+ number of queries required for a randomised classical algor ithm to compute fwith success
27
+ probability at least 2 /3. The choice of 2 /3 is arbitrary; any constant strictly between 1 /2
28
+ and 1 would give the same complexity, up to constant factors.
29
+ There is a natural generalisation of the query complexity mo del to quantum computa-
30
+ tion, which gives rise to the exact and bounded-error quantu m query complexities QE(f),
31
+ Q2(f) (respectively). In this generalisation, the quantum algo rithm is given access to the
32
+ ∗Department of Computer Science, University of Bristol, Woo dland Road, Bristol, BS8 1UB, UK;
33
34
+ 1inputxthrough a unitary oracle operator Ox. Many of the best-known quantum speed-ups
35
+ can be understood in the query complexity model. Indeed, it i s known that, for certain
36
+ partial functions f(i.e. functions where there is a promise on the input), Q2(f) may be ex-
37
+ ponentially smaller than R2(f)[14]. However, if fis atotal function, D(f) =O(Q2(f)6) [4].
38
+ See [6, 10] for good reviews of quantum and classical query co mplexity.
39
+ In the second class of problems, learning problems, one is given as an oracle an unknown
40
+ functionf?(x1,...,x n), which is picked from a known set Cofmboolean functions f:
41
+ {0,1}n→ {0,1}. These functions can be identified with n-bit strings or subsets of [ n], the
42
+ integers between 1 and n. The goal is to determine which of the functions in Cthe oraclef?
43
+ is, with some requirement on the success probability, using the minimum number of queries
44
+ tof?. Note that the success probability required should be stric tly greater than 1 /2 for
45
+ this model to make sense.
46
+ Borrowing terminology fromthe machinelearning literatur e, each function in Cis known
47
+ as aconcept, andCis known as a concept class [13]. We say that an algorithm that can
48
+ identify any f∈ Cwith worst-case success probability plearnsCwith success probability
49
+ p. This problem is known classically as exact learning from me mbership queries [3, 13],
50
+ and also in the literature on quantum computation as the orac le identification problem [2].
51
+ Many interesting results in quantum algorithmics fit into th is framework, a straightforward
52
+ example being Grover’s quantum search algorithm [9]. It has been shown by Servedio and
53
+ Gortler that the speed-up that may be obtained by quantum que ry algorithms in this model
54
+ is at most polynomial [13].
55
+ 1.1 Nonadaptive query algorithms
56
+ This paper considers query algorithms of a highly restricti ve form, where oracle queries are
57
+ not allowed to depend on previous queries. In other words, th e queries must all be made
58
+ at the start of the algorithm. We call such algorithms nonadaptive , but one could also call
59
+ themparallel, in contrast to the usual serial model of query complexity, w here one query
60
+ follows another. It is easy to see that, classically, a deter ministic nonadaptive algorithm
61
+ that computes a function f:{0,1}n→ {0,1}which depends on all ninput variables must
62
+ query allnvariables (x1,...,x n). Indeed, for any 1 ≤i≤n, consider an input xfor which
63
+ f(x) = 0, butf(x⊕ei) = 1, where eiis the bit string which has a 1 at position i, and is 0
64
+ elsewhere. Then, if the i’th variable were not queried, changing the input from xtox��ei
65
+ would change the output of the function, but the algorithm wo uld not notice.
66
+ In the case of learning, the exact number of queries required by a nonadaptive determin-
67
+ istic classical algorithm to learn any concept class Ccan also be calculated. Identify each
68
+ concept in Cwith ann-bit string, and imagine an algorithm Athat queries some subset
69
+ S⊆[n] of the input bits. If there are two or more concepts in Cthat do not differ on any of
70
+ the bits inS, thenAcannot distinguish between these two concepts, and so canno t succeed
71
+ with certainty. On the other hand, if every concept x∈ Cis unique when restricted to S,
72
+ thenxcan be identified exactly by A. Thus the number of queries required is the minimum
73
+ size of a subset S⊆[n] such that every pair of concepts in Cdiffers on at least one bit in S.
74
+ We will be concerned with the speed-up over classical query a lgorithms that can be
75
+ 2achieved by nonadaptive quantum query algorithms. Interes tingly, it is known that speed-
76
+ ups can indeed be found in this model. In the case of computing partial functions, the
77
+ speed-up can be dramatic; Simon’s algorithm for the hidden s ubgroup problem over Zn
78
+ 2, for
79
+ example, is nonadaptive and gives an exponential speed-up o ver the best possible classical
80
+ algorithm [14]. Thereare also known speed-upsfor computin g total functions. For example,
81
+ the parity of nbits can be computed exactly using only ⌈n/2⌉nonadaptive quantum queries
82
+ [8]. More generally, anyfunction of nbits can be computed with bounded error using only
83
+ n/2+O(√n)nonadaptivequeries, byaremarkablealgorithmofvanDam[ 7]. Thisalgorithm
84
+ in fact retrieves allthe bits of the input xsuccessfully with constant probability, so can also
85
+ be seen as an algorithm that learns the concept class consist ing of all boolean functions on
86
+ nbits usingn/2+O(√n) nonadaptive queries.
87
+ Finally, one of the earliest results in quantum computation can be understood as a
88
+ nonadaptive learning algorithm. The quantum algorithm sol ving the Bernstein-Vazirani
89
+ parity problem [5] uses one query to learn a concept class of s ize 2n, for which any classical
90
+ learning algorithm requires nqueries, showing that there can be an asymptotic quantum-
91
+ classical separation for learning problems.
92
+ 1.2 New results
93
+ We show here that these results are essentially the best poss ible. First, any nonadap-
94
+ tive quantum query algorithm that computes a total boolean f unction with a constant
95
+ probability of success greater than 1 /2 can only obtain a constant factor reduction in the
96
+ number of queries used. In particular, if we restrict to nona daptive query algorithms, then
97
+ Q2(f) = Θ(D(f)). In the case of exact nonadaptive algorithms, we show that the factor of
98
+ 2 speed-up obtained for computing parity is tight. More form ally, our result is the following
99
+ theorem.
100
+ Theorem 1. Letf:{0,1}n→ {0,1}be a total function that depends on all nvariables,
101
+ and letAbe a nonadaptive quantum query algorithm that uses kqueries to the input to
102
+ computef, and succeeds with probability at least 1−ǫon every input. Then
103
+ k≥n
104
+ 2/parenleftBig
105
+ 1−2/radicalbig
106
+ ǫ(1−ǫ)/parenrightBig
107
+ .
108
+ In the case of learning, we show that the speed-up obtained by the Bernstein-Vazirani
109
+ algorithm [5] is asymptotically tight. That is, the query co mplexities of quantum and
110
+ classical nonadaptive learning are equivalent, up to a loga rithmic term. This is formalised
111
+ as the following theorem.
112
+ Theorem 2. LetCbe a concept class containing mconcepts, and let Abe a nonadaptive
113
+ quantum query algorithm that uses kqueries to the input to learn C, and succeeds with
114
+ probability at least 1−ǫon every input, for some ǫ <1/2. Then there exists a classical
115
+ nonadaptive query algorithm that learns Cwith certainty using at most
116
+ 4klog2m
117
+ 1−2/radicalbig
118
+ ǫ(1−ǫ)
119
+ queries to the input.
120
+ 31.3 Related work
121
+ We note that the question of putting lower bounds on nonadapt ive quantum query algo-
122
+ rithms has been studied previously. First, Zalka has obtain ed a tight lower bound on the
123
+ nonadaptive quantum query complexity of the unordered sear ch problem, which is a par-
124
+ ticular learning problem [15]. Second, in [12], Nishimura a nd Yamakami give lower bounds
125
+ on the nonadaptive quantum query complexity of a multiple-b lock variant of the ordered
126
+ search problem. Finally, Koiran et al [11] develop the weigh ted adversary argument of Am-
127
+ bainis [1] to obtain lower bounds that are specific to the nona daptive setting. Unlike the
128
+ situation considered here, their bounds also apply to quant um algorithms for computing
129
+ partial functions.
130
+ We now turn to proving the new results: nonadaptive computat ion in Section 2, and
131
+ nonadaptive learning in Section 3.
132
+ 2 Nonadaptive quantum query complexity of computation
133
+ LetAbe a nonadaptive quantum query algorithm. We will use what is essentially the
134
+ standard model of quantum query complexity [10]. Ais given access to the input x=
135
+ x1...xnvia an oracle Oxwhich acts on an n+1 dimensional space indexed by basis states
136
+ |0/an}brack⌉tri}ht,...,|n/an}brack⌉tri}ht, and performs the operation Ox|i/an}brack⌉tri}ht= (−1)xi|i/an}brack⌉tri}ht. We define Ox|0/an}brack⌉tri}ht=|0/an}brack⌉tri}htfor
137
+ technical reasons (otherwise, Acould not distinguish between xand ¯x). Assume that A
138
+ makeskqueries toOx. As the queries are nonadaptive, we may assume they are made i n
139
+ parallel. Therefore, the existence of a nonadaptive quantu m query algorithm that computes
140
+ fand fails with probability ǫis equivalent to the existence of an input state |ψ/an}brack⌉tri}htand a
141
+ measurement specified by positive operators {M0,I−M0}, such that /an}brack⌉tl⌉{tψ|O⊗k
142
+ xM0O⊗k
143
+ x|ψ/an}brack⌉tri}ht ≥
144
+ 1−ǫfor all inputs xwheref(x) = 0, and /an}brack⌉tl⌉{tψ|O⊗k
145
+ xM0O⊗k
146
+ x|ψ/an}brack⌉tri}ht ≤ǫfor all inputs xwhere
147
+ f(x) = 1.
148
+ The intuition behind the proof of Theorem 1 is much the same as that behind “adver-
149
+ sary” arguments lower bounding quantum query complexity [1 0]. As in Section 1.1, let ej
150
+ denote the n-bit string which contains a single 1, at position j. In order to distinguish two
151
+ inputsx,x⊕ejwheref(x)/n⌉}ationslash=f(x⊕ej), the algorithm must invest amplitude of |ψ/an}brack⌉tri}htin
152
+ components where the oracle gives information about j. But, unless kis large, it is not
153
+ possible to invest in many variables simultaneously.
154
+ We will use the following well-known fact from [5].
155
+ Fact 3(Bernstein and Vazirani [5]) .Imagine there exists a positive operator M≤Iand
156
+ states|ψ1/an}brack⌉tri}ht,|ψ2/an}brack⌉tri}htsuch that /an}brack⌉tl⌉{tψ1|M|ψ1/an}brack⌉tri}ht ≤ǫ, but/an}brack⌉tl⌉{tψ2|M|ψ2/an}brack⌉tri}ht ≥1−ǫ. Then |/an}brack⌉tl⌉{tψ1|ψ2/an}brack⌉tri}ht|2≤
157
+ 4ǫ(1−ǫ).
158
+ We now turn to the proof itself. Write the input state |ψ/an}brack⌉tri}htas
159
+ |ψ/an}brack⌉tri}ht=/summationdisplay
160
+ i1,...,ikαi1,...,ik|i1,...,ik/an}brack⌉tri}ht,
161
+ 4where, for each m, 0≤im≤n. It is straightforward to compute that
162
+ O⊗k
163
+ x|i1,...,ik/an}brack⌉tri}ht= (−1)xi1+···+xik|i1,...,ik/an}brack⌉tri}ht.
164
+ Asfdepends on all ninputs, for any j, there exists a bit string xjsuch thatf(xj)/n⌉}ationslash=
165
+ f(xj⊕ej). Then
166
+ (OxjOxj⊕ej)⊗k|i1,...,ik/an}brack⌉tri}ht= (−1)|{m:im=j}||i1,...,ik/an}brack⌉tri}ht;
167
+ in other words ( OxjOxj⊕ej)⊗knegates those basis states that correspond to bit strings
168
+ i1,...,ikwherejoccurs an odd number of times in the string. Therefore, we hav e
169
+ |/an}brack⌉tl⌉{tψ|(OxjOxj⊕ej)⊗k|ψ/an}brack⌉tri}ht|2=
170
+ /summationdisplay
171
+ i1,...,ik|αi1,...,ik|2(−1)|{m:im=j}|
172
+ 2
173
+ =
174
+ 1−2/summationdisplay
175
+ i1,...,ik|αi1,...,ik|2[|{m:im=j}|odd]
176
+ 2
177
+ =: (1−2Wj)2.
178
+ Now, by Fact 3, (1 −2Wj)2≤4ǫ(1−ǫ) for allj, so
179
+ Wj≥1
180
+ 2/parenleftBig
181
+ 1−2/radicalbig
182
+ ǫ(1−ǫ)/parenrightBig
183
+ .
184
+ On the other hand,
185
+ n/summationdisplay
186
+ j=1Wj=n/summationdisplay
187
+ j=1/summationdisplay
188
+ i1,...,ik|αi1,...,ik|2[|{m:im=j}|odd]
189
+ =/summationdisplay
190
+ i1,...,ik|αi1,...,ik|2n/summationdisplay
191
+ j=1[|{m:im=j}|odd]
192
+ ≤/summationdisplay
193
+ i1,...,ik|αi1,...,ik|2k=k.
194
+ Combining these two inequalities, we have
195
+ k≥n
196
+ 2/parenleftBig
197
+ 1−2/radicalbig
198
+ ǫ(1−ǫ)/parenrightBig
199
+ .
200
+ 3 Nonadaptive quantum query complexity of learning
201
+ In the case of learning, we use a very similar model to the prev ious section. Let Abe a
202
+ nonadaptivequantumqueryalgorithm. Aisgiven access toanoracle Ox, whichcorresponds
203
+ toabit-string xpickedfromaconcept class C.Oxactsonann+1dimensionalspaceindexed
204
+ by basis states |0/an}brack⌉tri}ht,...,|n/an}brack⌉tri}ht, and performs the operation Ox|i/an}brack⌉tri}ht= (−1)xi|i/an}brack⌉tri}ht, withOx|0/an}brack⌉tri}ht=|0/an}brack⌉tri}ht.
205
+ 5Assume that Amakeskqueries toOxand outputs xwith probability strictly greater than
206
+ 1/2 for allx∈ C.
207
+ We will prove limitations on nonadaptive quantum algorithm s in this model as follows.
208
+ First, we show that a nonadaptive quantum query algorithm th at useskqueries to learn C
209
+ is equivalent to an algorithm using one query to learn a relat ed concept class C′. We then
210
+ show that existence of a quantum algorithm using one query th at learns C′with constant
211
+ success probability greater than 1 /2 implies existence of a deterministic classical algorithm
212
+ usingO(log|C′|) queries. Combining these two results gives Theorem 2.
213
+ Lemma 4. LetCbe a concept class over n-bit strings, and let C⊗kbe the concept class
214
+ defined by
215
+ C⊗k={x⊗k:x∈ C},
216
+ wherex⊗kdenotes the (n+ 1)k-bit string indexed by 0≤i1,...,ik≤n, withx⊗k
217
+ i1,...,ik=
218
+ xi1⊕ ··· ⊕xik, and we define x0= 0. Then, if there exists a classical nonadaptive query
219
+ algorithm that learns C⊗kwith success probability pand usesqqueries, there exists a classical
220
+ nonadaptive query algorithm that learns Cwith success probability pand uses at most kq
221
+ queries.
222
+ Proof.Given access to x, an algorithm Acan simulate a query of index ( x1,...,x k) ofx⊗k
223
+ by using at most kqueries to compute x1⊕··· ⊕xk. Hence, by simulating the algorithm
224
+ for learning C⊗k,Acan learn C⊗kwith success probability pusing at most kqnonadaptive
225
+ queries. Learning C⊗ksuffices to learn C, because each concept in C⊗kuniquely corresponds
226
+ to a concept in C(to see this, note that the first nbits ofx⊗kare equal to x).
227
+ Lemma 5. LetCbe a concept class containing mconcepts. Assume that Ccan be learned
228
+ using one quantum query by an algorithm that fails with proba bility at most ǫ, for some
229
+ ǫ<1/2. Then there exists a classical algorithm that uses at most (4log2m)/(1−2/radicalbig
230
+ ǫ(1−ǫ))
231
+ queries and learns Cwith certainty.
232
+ Proof.Associate each concept with an n-bit string, for some n, and suppose there exists a
233
+ quantum algorithm that uses one query to learn Cand fails with probability ǫ<1/2. Then
234
+ by Fact 3 there exists an input state |ψ/an}brack⌉tri}ht=/summationtextn
235
+ i=0αi|i/an}brack⌉tri}htsuch that, for all x/n⌉}ationslash=y∈ C,
236
+ |/an}brack⌉tl⌉{tψ|OxOy|ψ/an}brack⌉tri}ht|2≤4ǫ(1−ǫ),
237
+ or in other words/parenleftBiggn/summationdisplay
238
+ i=0|αi|2(−1)xi+yi/parenrightBigg2
239
+ ≤4ǫ(1−ǫ). (1)
240
+ We now show that, if this constraint holds, there must exist a subset of the inputs S⊆[n]
241
+ such that every pair of concepts in Cdiffers on at least one input in S, and|S|=O(logm).
242
+ By the argument of Section 1.1, this implies that there is a no nadaptive classical algorithm
243
+ that learns Mwith certainty using O(logm) queries.
244
+ We will use the probabilistic method to show the existence of S. For anyk, form a
245
+ subsetSof at most kinputs between 1 and nby a process of krandom, independent
246
+ 6choices of input, where at each stage input iis picked to add to Swith probability |αi|2.
247
+ Now consider an arbitrary pair of concepts x/n⌉}ationslash=y, and letS+,S−be the set of inputs on
248
+ which the concepts are equal and differ, respectively. By the c onstraint (1), we have
249
+ 4ǫ(1−ǫ)≥/parenleftBiggn/summationdisplay
250
+ i=0|αi|2(−1)xi+yi/parenrightBigg2
251
+ =
252
+ /summationdisplay
253
+ i∈S+|αi|2−/summationdisplay
254
+ i∈S−|αi|2
255
+ 2
256
+ =
257
+ 1−2/summationdisplay
258
+ i∈S−|αi|2
259
+ 2
260
+ ,
261
+ so/summationdisplay
262
+ i∈S−|αi|2≥1
263
+ 2−/radicalbig
264
+ ǫ(1−ǫ).
265
+ Therefore, at each stage of adding an input to S, the probability that an input in S−is
266
+ added is at least1
267
+ 2−/radicalbig
268
+ ǫ(1−ǫ). So, after kstages of doing so, the probability that none
269
+ of these inputs has been added is at most/parenleftBig
270
+ 1
271
+ 2+/radicalbig
272
+ ǫ(1−ǫ)/parenrightBigk
273
+ . As there are/parenleftbigm
274
+ 2/parenrightbig
275
+ pairs of
276
+ conceptsx/n⌉}ationslash=y, by a union bound the probability that none of the pairs of con cepts differs
277
+ on any of the inputs in Sis upper bounded by
278
+ /parenleftbiggm
279
+ 2/parenrightbigg/parenleftbigg1
280
+ 2+/radicalbig
281
+ ǫ(1−ǫ)/parenrightbiggk
282
+ ≤m2/parenleftbigg1
283
+ 2+/radicalbig
284
+ ǫ(1−ǫ)/parenrightbiggk
285
+ .
286
+ For anykgreater than
287
+ 2log2m
288
+ log22/(1+2/radicalbig
289
+ ǫ(1−ǫ))<4log2m
290
+ 1−2/radicalbig
291
+ ǫ(1−ǫ)
292
+ this probability is strictly less than 1, implying that ther e exists some choice of S⊆[n]
293
+ with|S| ≤ksuch that every pair of concepts differs on at least one of the in puts inS. This
294
+ completes the proof.
295
+ We are finally ready to prove Theorem 2, which we restate for cl arity.
296
+ Theorem. LetCbe a concept class containing mconcepts, and let Abe a nonadaptive
297
+ quantum query algorithm that uses kqueries to the input to learn C, and succeeds with
298
+ probability at least 1−ǫon every input, for some ǫ <1/2. Then there exists a classical
299
+ nonadaptive query algorithm that learns Cwith certainty using at most
300
+ 4klog2m
301
+ 1−2/radicalbig
302
+ ǫ(1−ǫ)
303
+ queries to the input.
304
+ Proof.LetOxbe the oracle operator corresponding to the concept x. Then a nonadaptive
305
+ quantum algorithm Athat learns xusingkqueries to Oxis equivalent to a quantum
306
+ algorithm that uses one query to O⊗k
307
+ xto learnx. It is easy to see that this is equivalent to
308
+ Ain fact using one query to learn the concept class C⊗k. By Lemma 5, this implies that
309
+ there exists a classical algorithm that uses at most (4 klog2m)/(1−2/radicalbig
310
+ ǫ(1−ǫ)) queries
311
+ to learn C⊗kwith certainty. Finally, by Lemma 4, this implies in turn tha t there exists a
312
+ classical algorithm that uses the same number of queries and learnsCwith certainty.
313
+ 7Acknowledgements
314
+ I would like to thank Aram Harrow and Dan Shepherdfor helpful discussions and comments
315
+ on a previous version. This work was supported by the EC-FP6- STREP network QICS and
316
+ an EPSRC Postdoctoral Research Fellowship.
317
+ References
318
+ [1] A. Ambainis. Polynomial degree vs. quantum query comple xity.J. Comput. Syst. Sci. ,
319
+ 72(2):220–238, 2006. quant-ph/0305028 .
320
+ [2] A. Ambainis, K. Iwama, A. Kawachi, H. Masuda, R. Putra, an d S. Yamashita. Quan-
321
+ tum identification of Boolean oracles. In Proc. STACS 2004 , pages 93–104. Springer,
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+ 2004.quant-ph/0403056 .
323
+ [3] D. Angluin. Queries and concept learning. Machine Learning , 2(4):319–342, 1988.
324
+ [4] R. Beals, H. Buhrman, R. Cleve, M. Mosca, and R. de Wolf. Qu antum lower bounds
325
+ by polynomials. J. ACM, 48(4):778–797, 2001. quant-ph/9802049 .
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+ [5] E. Bernstein and U. Vazirani. Quantum complexity theory .SIAM J. Comput. ,
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+ 26(5):1411–1473, 1997.
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+ [6] H. Buhrman and R. de Wolf. Complexity measures and decisi on tree complexity: a
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+ survey.Theoretical Computer Science , 288:21–43, 2002.
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+ [7] W. vanDam. Quantumoracle interrogation: Gettingall in formation foralmost halfthe
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+ price. In Proc. 39thAnnual Symp. Foundations of Computer Science , pages 362–367.
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+ IEEE, 1998. quant-ph/9805006 .
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+ [8] E. Farhi, J. Goldstone, S. Gutmann, and M. Sipser. A limit on the speed of
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+ quantum computation in determining parity. Phys. Rev. Lett. , 81:5442–5444, 1998.
335
+ quant-ph/9802045 .
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+ [9] L. Grover. Quantum mechanics helps in searching for a nee dle in a haystack. Phys.
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+ Rev. Lett. , 79(2):325–328, 1997. quant-ph/9706033 .
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+ [10] P. Høyer and R. ˇSpalek. Lower bounds on quantum query complexity. Bulletin
339
+ of the European Association for Theoretical Computer Science , 87:78–103, 2005.
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+ quant-ph/0509153 .
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+ [11] P. Koiran, J. Landes, N. Portier, and P. Yao. Adversary l ower bounds for nonadaptive
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+ quantum algorithms. In Proc. WoLLIC 2008: 15th Workshop on Logic, Language,
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+ Information and Computation , pages 226–237. Springer, 2008. arXiv:0804.1440 .
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+ [12] H. Nishimura and T. Yamakami. An algorithmic argument f or nonadaptive query
345
+ complexity lower bounds on advised quantum computation. In Proc. 29th Interna-
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+ tional Symposium on Mathematical Foundations of Computer Sc ience, pages 827–838.
347
+ Springer, 2004. quant-ph/0312003 .
348
+ 8[13] R. Servedio and S. Gortler. Quantum versus classical le arnability. In Proc. 16thAnnual
349
+ IEEE Conf. Computational Complexity , pages 138–148, 2001. quant-ph/0007036 .
350
+ [14] D. R. Simon. Onthepower of quantum computation. SIAM J. Comput. , 26:1474–1483,
351
+ 1997.
352
+ [15] C. Zalka. Grover’s quantum searching algorithm is opti mal.Phys. Rev. A. , 60(4):2746–
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+ 2751, 1999. quant-ph/9711070 .
354
+ 9
1001.0019.txt ADDED
@@ -0,0 +1,378 @@
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
1
+ arXiv:1001.0019v1 [gr-qc] 30 Dec 2009On the instability of Reissner-Nordstr¨ om black holes in de Sitter backgrounds
2
+ Vitor Cardoso∗
3
+ CENTRA, Departamento de F´ ısica, Instituto Superior T´ ecn ico,
4
+ Av. Rovisco Pais 1, 1049-001 Lisboa, Portugal &
5
+ Department of Physics and Astronomy, The University of Miss issippi, University, MS 38677-1848, USA
6
+ Madalena Lemos†and Miguel Marques‡
7
+ CENTRA, Departamento de F´ ısica, Instituto Superior T´ ecn ico,
8
+ Av. Rovisco Pais 1, 1049-001 Lisboa, Portugal
9
+ (Dated: November 3, 2018)
10
+ Recent numerical investigations have uncovered a surprisi ng result: Reissner-Nordstr¨ om-de Sitter
11
+ black holes are unstable for spacetime dimensions larger th an 6. Here we prove the existence of
12
+ such instability analytically, and we compute the timescal e in the near-extremal limit. We find very
13
+ good agreement with the previous numerical results. Our res ults may me helpful in shedding some
14
+ light on the nature of the instability.
15
+ PACS numbers: 04.50.Gh,04.70.-s
16
+ I. INTRODUCTION
17
+ In physics, stability of a given configuration (solution
18
+ of some set of equations), is a useful criterium for rele-
19
+ vance of that solution. Unstable configurations are likely
20
+ not tobe realizablein practice, and representaninterme-
21
+ diate stage in the evolution of the system. Nevertheless,
22
+ the instability itself is of great interest, since an under-
23
+ standing of the mechanism behind it may help one to
24
+ better grasp the physics involved. In particular, it is of
25
+ interest to be able to predict which other systems display
26
+ similar instabilities, or even have a deeper understanding
27
+ of the physics behind the instability (why is the system
28
+ unstable? is there some fundamental principle behind
29
+ the instability?).
30
+ In General Relativity, the Kerr family exhausts the
31
+ blackhole solutionsto the electro-vacEinstein equations.
32
+ Kerr black holes are stable, and can therefore describe
33
+ astrophysicalobjects. However,there aremanyinstances
34
+ of instabilities afflicting objects with an event horizon,
35
+ such as the Gregory-Laflamme [1], the ultra-spinning [2]
36
+ or superradiant instabilities [3] and other instabilities of
37
+ higher-dimensional black holes in alternative theories [4,
38
+ 5](for a review see Ref. [6]).
39
+ Konoplya and Zhidenko (hereafter KZ) recently stud-
40
+ ied small perturbations in the vicinity of a charged black
41
+ hole in de Sitter background, a Reissner-Nordstr¨ om de
42
+ Sitter black hole (RNdS) [7]. Their (numerical) results
43
+ show that when the spacetime dimensionality D >6, the
44
+ spacetime is unstable, provided the charge is larger than
45
+ agiventhreshold, determined byKZforeach D. Because
46
+ ∗Electronic address: [email protected]
47
+ †Electronic address: [email protected]
48
+ ‡Electronic address: [email protected] results are so surprising (the mechanism behind it is
49
+ not yet understood), we set out to to investigate this in-
50
+ stability and hopefully understand it better. Our results
51
+ can be summarized as follows: (i) we can prove analyti-
52
+ cally the existence of unstable modes for charge Qhigher
53
+ thanacertainthreshold. (ii)inthenear-extremalregime,
54
+ we are able to find an explicit solution for the unstable
55
+ modes, determining the instability timescale analytically.
56
+ We hope that our incursion in this topic helps to better
57
+ understand the physics at work.
58
+ II. EQUATIONS
59
+ This work focuses on the higher dimensional RNdS ge-
60
+ ometry, described by the line element
61
+ ds2=−f dt2+f−1dr2+r2dΩ2
62
+ n, (1)
63
+ wheredΩ2
64
+ nis the line element of the nsphere and
65
+ f= 1−λr2−2M
66
+ rn−1+Q2
67
+ r2n−2. (2)
68
+ the background electric field is E0=q/rn, withqthe
69
+ electric charge. The quantities MandQare related to
70
+ the physical mass M and charge qof the black hole [8],
71
+ andλto the cosmological constant. The spacetime di-
72
+ mensionality is D=n+2.
73
+ The above geometry possesses three horizons: the
74
+ black-holeCauchyhorizonat r=ra, the black hole event
75
+ horizon is at r=rband the cosmological horizon is at
76
+ r=rc, whererc> rb> ra, the only real, positive zeroes
77
+ off. For convenience, we set rb= 1, i.e., we measure all
78
+ quantities in terms of the event horizon rb. We thus get
79
+ 2M= 1+Q2−λ, (3)2
80
+ Furthermore, we can also write
81
+ λ=r−4−n
82
+ c(rn+2
83
+ c−r3
84
+ c)(rn+2
85
+ c−Q2r3
86
+ c)
87
+ rn+2c−rc.(4)
88
+ For a fixed rcand spacetime dimension D, the existence
89
+ ofaregulareventhorizonimposesthatthecharge Qmust
90
+ be smaller than a certain value Qext. With our units this
91
+ maximum charge is
92
+ Q2
93
+ ext=rn
94
+ c/parenleftbig
95
+ −2rc+(n+1)rn
96
+ c−(n−1)rn+2
97
+ c/parenrightbig
98
+ −rc/parenleftbig
99
+ rc(n+1)−2nrnc+(n−1)r2n+1c/parenrightbig.(5)
100
+ Gravitational perturbations of this spacetime couple to
101
+ the electromagnetic field, and were completely character-
102
+ ized by Kodama and Ishibashi [8]. They can be reduced
103
+ to a set of two second order ordinary differential equa-
104
+ tions of the form,
105
+ d2
106
+ dr2∗Φ±+/parenleftbig
107
+ ω2−VS±/parenrightbig
108
+ Φ±= 0, (6)where the tortoise coordinate r∗and the potentials VS±
109
+ are defined through
110
+ r∗≡/integraldisplay
111
+ f−1dr, V S±=fU±
112
+ 64r2H2
113
+ ±.(7)
114
+ We have
115
+ H+= 1−n(n+1)
116
+ 2δx, (8)
117
+ H−=m+n(n+1)
118
+ 2(1+mδ)x, (9)
119
+ and the quantities U±are given by
120
+ U+=/bracketleftbig
121
+ −4n3(n+2)(n+1)2δ2x2−48n2(n+1)(n−2)δx
122
+ −16(n−2)(n−4)]y−δ3n3(3n−2)(n+1)4(1+mδ)x4
123
+ +4δ2n2(n+1)2/braceleftbig
124
+ (n+1)(3n−2)mδ+4n2+n−2/bracerightbig
125
+ x3
126
+ +4δ(n+1)/braceleftbig
127
+ (n−2)(n−4)(n+1)(m+n2K)δ−7n3+7n2−14n+8/bracerightbig
128
+ x2
129
+ +/braceleftbig
130
+ 16(n+1)/parenleftbig
131
+ −4m+3n2(n−2)K/parenrightbig
132
+ δ−16(3n−2)(n−2)/bracerightbig
133
+ x
134
+ +64m+16n(n+2)K, (10)
135
+ U−=/bracketleftbig
136
+ −4n3(n+2)(n+1)2(1+mδ)2x2+48n2(n+1)(n−2)m(1+mδ)x
137
+ −16(n−2)(n−4)m2/bracketrightbig
138
+ y−n3(3n−2)(n+1)4δ(1+mδ)3x4
139
+ −4n2(n+1)2(1+mδ)2/braceleftbig
140
+ (n+1)(3n−2)mδ−n2/bracerightbig
141
+ x3
142
+ +4(n+1)(1+ mδ)/braceleftbig
143
+ m(n−2)(n−4)(n+1)(m+n2K)δ
144
+ +4n(2n2−3n+4)m+n2(n−2)(n−4)(n+1)K/bracerightbig
145
+ x2
146
+ −16m/braceleftbig
147
+ (n+1)m/parenleftbig
148
+ −4m+3n2(n−2)K/parenrightbig
149
+ δ
150
+ +3n(n−4)m+3n2(n+1)(n−2)K/bracerightbig
151
+ x
152
+ +64m3+16n(n+2)m2K. (11)
153
+ The variables x,yand parameters µ,mare defined
154
+ through
155
+ x≡2M
156
+ rn−1, y≡λr2, (12)
157
+ µ2≡M2+4mQ2
158
+ (n+1)2, m≡k2−nK,(13)
159
+ andthe quantity δis implicitly givenby µ= (1+2mδ)M.
160
+ Note that the following relations holds Q2= (n+
161
+ 1)2M2δ(1+mδ).
162
+ Note also that for the spacetime considered in this pa-
163
+ perK= 1, whichmeansthatthe eigenvalues k2aregivenbyk2=l(l+n−1), where lis the angular quantum
164
+ number, that gives the multipolarity of the field. The
165
+ behavior of the potentials varies considerably over the
166
+ range of parameters. In Fig. 1 we show V−forD= 8,
167
+ rc= 1/0.95,l= 2andthreedifferentvaluesofthecharge,
168
+ Q= 0.2,0.35,0.44.
169
+ III. A CRITERIUM FOR INSTABILITY
170
+ A sufficient (but not necessary) condition for the exis-
171
+ tence of an unstable mode has been proven by Buell and3
172
+ /s48/s44/s48/s48 /s48/s44/s48/s49 /s48/s44/s48/s50 /s48/s44/s48/s51 /s48/s44/s48/s52 /s48/s44/s48/s53/s45/s50/s48/s50/s52/s54
173
+ /s49/s48/s52
174
+ /s32/s86
175
+ /s45/s49/s48/s52
176
+ /s32/s86
177
+ /s45
178
+ /s32/s32/s86
179
+ /s45
180
+ /s114/s45/s49/s32/s81/s61/s48/s46/s50/s48
181
+ /s32/s81/s61/s48/s46/s51/s53
182
+ /s32/s81/s61/s48/s46/s52/s52/s49/s48/s51
183
+ /s32/s86
184
+ /s45
185
+ FIG. 1: Behavior of V−for different parameters, for D= 8.
186
+ Here we fix the event horizon at rb= 1, and the cosmological
187
+ horizon at rc= 1/0.95. We consider l= 2 modes and three
188
+ different charges, Q= 0.2,0.35,0.44.
189
+ Shadwick [9] and is the following,
190
+ /integraldisplayrc
191
+ rbV
192
+ fdr <0. (14)
193
+ The instability region is depicted in figure 2 for several
194
+ /s48/s44/s48 /s48/s44/s50 /s48/s44/s52 /s48/s44/s54 /s48/s44/s56 /s49/s44/s48/s48/s44/s48/s48/s44/s50/s48/s44/s52/s48/s44/s54/s48/s44/s56/s49/s44/s48/s32
195
+ /s32/s81/s47/s81
196
+ /s101/s120/s116
197
+ /s114
198
+ /s98/s47/s114
199
+ /s99
200
+ FIG. 2: The parametric region of instability in Q/Qext−rb/rc
201
+ coordinates, according to criterim (14), for l= 2. Top to
202
+ bottom, D= 7,8,9,10,11.
203
+ spacetime-dimension D, which can be compared with the
204
+ numerical results by KZ, their figure 4. It is apparent
205
+ that condition (14) very accurately describes the numer-
206
+ ical results for rb/rc∼1, a regime we explore below in
207
+ Section IV. As one moves away from extremality cri-
208
+ terium (14) is just too restrictive. An improved analysis
209
+ and refined criterium would be necessary to describe the
210
+ whole rangeofthe numericalresults. Nevertheless, figure2 is very clear: higher-dimensional ( D >6) RNdS black
211
+ holes are unstable for a wide range of parameters.
212
+ IV. AN EXACT SOLUTION IN THE NEAR
213
+ EXTREMAL RNDS BLACK HOLE
214
+ Let us now specialize to the near extremal RNdS black
215
+ hole, which we define as the spacetime for which the cos-
216
+ mological horizon rcis very close (in the rcoordinate)
217
+ to the black hole horizon rb, i.e.rc−rb
218
+ rb≪1. The wave
219
+ equationin this spacetime can be solvedexactly, in terms
220
+ of hypergeometric functions [10]. The key point is that
221
+ the physical region of interest (where the boundary con-
222
+ ditions are imposed), lies between rbandrc. Thus,
223
+ f∼2κb(r−rb)(rc−r)
224
+ rc−rb, (15)
225
+ where we have introduced the surface gravity κbassoci-
226
+ ated with the event horizon at r=rb, as defined by the
227
+ relationκb=1
228
+ 2df/drr=rb. For near-extremal black holes,
229
+ it is approximately
230
+ κb∼(rc−rb)(n−1)
231
+ 2r2
232
+ b/parenleftbig
233
+ 1−nQ2/parenrightbig
234
+ .(16)
235
+ In this limit, one can invert the relation r∗(r) of (7) to
236
+ get
237
+ r=rce2κbr∗+rb
238
+ 1+e2κbr∗. (17)
239
+ Substituting this on the expression (15) for fwe find
240
+ f=(rc−rb)κb
241
+ 2cosh(κbr∗)2. (18)
242
+ As such, and taking into account the functional form of
243
+ the potentials for wave propagation, we see that for the
244
+ near extremal RNdS black hole the wave equation (6) is
245
+ of the form
246
+ d2Φ(ω,r)
247
+ dr2∗+/bracketleftBigg
248
+ ω2−V0
249
+ cosh(κbr∗)2/bracketrightBigg
250
+ Φ(ω,r) = 0,(19)
251
+ with
252
+ V0=(rc−rb)κb
253
+ 2VS±(rb)
254
+ f(20)
255
+ The potential in (19) is the well known P¨ oshl-Teller po-
256
+ tential [11]. The solutions to (19) were studied and they
257
+ are of the hypergeometric type, (for details see Refs.
258
+ [12, 13]). Itshouldbesolvedunderappropriateboundary
259
+ conditions:
260
+ Φ∼e−iωr∗, r∗→ −∞ (21)
261
+ Φ∼eiωr∗, r∗→ ∞. (22)4
262
+ These boundary conditions impose a non-trivial condi-
263
+ tion onω[12, 13], and those that satisfy both simultane-
264
+ ously are called quasinormal frequencies. For the P¨ oshl-
265
+ Teller potential one can show [12, 13] that they are given
266
+ by
267
+ ω=κb/bracketleftBigg
268
+ −/parenleftbigg
269
+ j+1
270
+ 2/parenrightbigg
271
+ i+/radicalBigg
272
+ V0
273
+ κ2
274
+ b−1
275
+ 4/bracketrightBigg
276
+ , j= 0,1,....
277
+ (23)
278
+ We conclude therefore that an instability is present
279
+ TABLE I: The threshold of instability for near-extremal
280
+ RNdS black holes (i.e., black holes for which the cosmologic al
281
+ and event horizon almost coincide) for l= 2 modes. We show
282
+ the prediction from the exact, analytic expression obtaine d
283
+ in the near extremal limit (24), which we label Q/QN
284
+ extand
285
+ the one from criterium (14) which we label as Q/QV
286
+ ext. Both
287
+ these results are compared to the numerical results by KZ.
288
+ D
289
+ 7 8 9 10 11 D→ ∞
290
+ Q/QN
291
+ ext0.913 0.774 0.683 0.617 0.567p
292
+ 2/D
293
+ Q/QV
294
+ ext0.913 0.775 0.684 0.618 0.568p
295
+ 2/D
296
+ Q/QNum
297
+ ext0.94 0.78 0.68 0.61 0.55 —
298
+ whenever V0is negative. The threshold of stability in
299
+ the near-extremal regime is therefore given by
300
+ VS±(rb)
301
+ f= 0, (24)
302
+ The expression for VS±(rb)/fis lengthy, and we won’t
303
+ presentit here. Thevaluesofthe charge Q/Qextthat sat-
304
+ isfy the condition above are given in Table I (for l= 2),
305
+ and compared to the prediction from the analysis in Sec-
306
+ tion III, criterium (14). The agreement is excellent. Fur-thermore, we compare these predictions against the nu-
307
+ merical results by KZ, extrapolated to the extremal limit
308
+ (ρ= 1 in KZ notation). The agreement is remarkable.
309
+ V. CONCLUSIONS
310
+ We have shown analytically that charged black holes
311
+ in de Sitter backgrounds are unstable for a wide range of
312
+ charge and mass of the black hole, confirming previous
313
+ numerical studies [7]. The stability properties of the ex-
314
+ tremalD= 6 black hole remain unknown. Our methods
315
+ and results and inconclusive at this precise point, further
316
+ dedicated investigations would be necessary.
317
+ Ouranalyticalresultinthenear-extremalregimecould
318
+ be used to investigate further the nature of this instabil-
319
+ ity, something we have not attempted to do here. A
320
+ possible refinement concerns the large- Dlimit of the in-
321
+ stability, where it couldbe possible to find an analytical
322
+ expression throughout all range of parameters. We have
323
+ inmind resultsandtechniquessimilartothoseofKoland
324
+ Sorkin [14]. It would also be interesting to investigate
325
+ the stability properties, using this or other techniques, of
326
+ near-extremal Kerr-dS black holes, which have recently
327
+ been conjectured to have an holographic description [15].
328
+ Acknowledgements
329
+ We warmly thank Roman Konoplya and Alexander
330
+ Zhidenko for useful correspondence and for sharing their
331
+ numerical results with us. This work was partially
332
+ funded by Funda¸ c˜ ao para a Ciˆ encia e Tecnologia (FCT)-
333
+ Portugal through projects PTDC/FIS/64175/2006,
334
+ PTDC/ FIS/098025/2008,PTDC/FIS/098032/2008and
335
+ CERN/FP/109290/2009.
336
+ [1] R. Gregory and R. Laflamme, Phys. Rev. Lett. 70, 2837
337
+ (1993); H. Kudoh, Phys. Rev. D 73, 104034 (2006);
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+ V. Cardoso and O. J. C. Dias, Phys. Rev. Lett. 96,
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+ 181601 (2006).
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+ [2] R. Emparan and R. C. Myers, JHEP 0309, 025 (2003);
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+ O. J. C. Dias, P. Figueras, R. Monteiro, J. E. Santos and
342
+ R. Emparan, arXiv:0907.2248 [hep-th].
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+ [3] V. Cardoso, O. J. C. Dias, J. P. S. Lemos and S. Yoshida,
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+ Phys. Rev. D 70, 044039 (2004) [Erratum-ibid. D 70,
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+ 049903 (2004)]; V. Cardoso and O. J. C. Dias, Phys.
346
+ Rev. D70, 084011 (2004); V. Cardoso and S. Yoshida,
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+ JHEP0507, 009 (2005); V. Cardoso, O. J. C. Dias and
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+ S. Yoshida, Phys. Rev. D 74, 044008 (2006); V. Car-
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+ doso and J. P. S. Lemos, Phys. Lett. B 621, 219 (2005);
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+ H. Kodama, Prog. Theor. Phys. Suppl. 172, 11 (2008);
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+ A. N. Aliev and O. Delice, Phys. Rev. D 79, 024013
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+ (2009); H. Kodama, R. A. Konoplya and A. Zhidenko,
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+ Phys.Rev.D 79, 044003(2009); K.Murata, Prog. Theor.
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+ Phys.121, 1099 (2009); N. Uchikata, S. Yoshida and T.
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+ Futamase, Phys. Rev. D 80, 084020 (2009).[4] G. Dotti and R. J. Gleiser, Phys. Rev. D 72, 044018
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+ (2005); R. J. Gleiser and G. Dotti, Phys. Rev. D 72,
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+ 124002 (2005); M. Beroiz, G. Dotti and R. J. Gleiser,
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+ Phys. Rev. D 76, 024012 (2007).
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+ [5] T. Takahashi and J. Soda, Phys. Rev. D 79, 104025
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+ (2009); T. Takahashi and J. Soda, arXiv:0907.0556 [gr-
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+ qc].
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+ [6] T. Harmark, V. Niarchos and N. A. Obers, Class. Quant.
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+ Grav.24, R1 (2007).
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+ [7] R. A. Konoplya and A. Zhidenko, Phys. Rev. Lett. 103,
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+ 161101 (2009).
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+ [8] H. Kodama and A. Ishibashi, Prog. Theor. Phys. 111,
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+ 29 (2004).
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+ [9] W. F. Buell and B. A. Shadwick, Am. J. Phys. 63, 256
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+ (1995).
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+ [10] V. Cardoso and J. P. S. Lemos, Phys. Rev. D 67, 084020
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+ (2003).
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+ [11] G. P¨ oshl and E. Teller, Z. Phys. 83, 143 (1933).
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+ [12] E. Berti, V. Cardoso and A. O. Starinets, Class. Quant.
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+ Grav.26, 163001 (2009).5
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+ [13] V. Ferrari and B. Mashhoon, Phys. Rev. D 30, 295
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+ (1984).
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+ [14] B. Kol and E. Sorkin, Class. Quant. Grav. 21, 4793(2004).
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+ [15] D. Anninos and T. Hartman, arXiv:0910.4587 [hep-th].
1001.0020.txt ADDED
@@ -0,0 +1,756 @@
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
1
+ arXiv:1001.0020v2 [nlin.SI] 3 Mar 2010Classification of integrable hydrodynamic chains
2
+ A.V. Odesskii1,2, V.V. Sokolov1
3
+ 1L.D. Landau Institute for Theoretical Physics (Russia)
4
+ 2Brock University (Canada)
5
+ Abstract Using the method of hydrodynamic reductions, we find all inte-
6
+ grable infinite (1+1)-dimensional hydrodynamic-type chains of shif t one. A
7
+ class of integrable infinite (2+1)-dimensional hydrodynamic-type c hains is
8
+ constructed.
9
+ MSC numbers: 17B80, 17B63, 32L81, 14H70
10
+ Address : L.D. Landau Institute for Theoretical Physics of Russian Academ y of Sciences,
11
+ Kosygina 2, 119334, Moscow, Russia
12
13
+ 1Contents
14
+ 1 Introduction 3
15
+ 2 Integrable chains and hydrodynamic reductions 4
16
+ 3 GT-systems 5
17
+ 4 Canonical forms of GT-systems associated
18
+ with integrable chains 7
19
+ 5 Generic case 12
20
+ 6 Trivial GT-system and 2+1-dimensional integrable hydrodynamic chains 14
21
+ 7 Infinitesimal symmetries of triangular GT-systems 17
22
+ 21 Introduction
23
+ We consider integrable infinite quasilinear chains of the form
24
+ uα,t=φα,1u1,x+···+φα,α+1uα+1,x, α= 1,2,..., φ α,α+1/negationslash= 0, (1.1)
25
+ whereφα,j=φα,j(u1,...,uα+1).Two chains are called equivalent if they are related by a trans-
26
+ formation of the form
27
+ uα→Ψα(u1,...,uα),∂Ψα
28
+ ∂uα/negationslash= 0, α= 1,2,... (1.2)
29
+ By integrability we mean the existence of an infinite set of hydrodyna mic reductions [1, 2,
30
+ 3, 4, 5, 6].
31
+ Example 1. The Benney equations [7, 8, 9]
32
+ u1,t=u2,x, u 2,t=u1u1,x+u3,x,... u αt= (α−1)uα−1u1,x+uα+1,x,... (1.3)
33
+ provide the most known example of integrable chain (1.1). The hydro dynamic reductions for
34
+ the Benney chain were investigated in [10]. /square
35
+ In [4, 5, 6] integrable divergent chains of the form
36
+ u1t=F1(u1,u2)x, u2t=F2(u1,u2,u3)x,···, uit=Fi(u1,u2,...,ui+1)x,··· (1.4)
37
+ were considered. In [6] some necessary integrability conditions we re obtained. Namely, a non-
38
+ linear overdetermined system of PDEs for functions F1,F2was presented. The general solution
39
+ of the system was not found. Another open problem was to prove t hat the conditions are
40
+ sufficient. In other words, for any solution F1,F2of the system one should find functions
41
+ Fi,i>2 such that the resulting chain is integrable.
42
+ Probably any integrable chain (1.1) is equivalent to a divergent chain. However, the diver-
43
+ gent coordinates are not suitable for explicit formulas. Our main obs ervation is that a conve-
44
+ nient coordinates are those, in which the so-called Gibbons-Tsarev type system (GT-system)
45
+ related to integrable chain is in a canonical form.
46
+ Using our version (see [11, 12]) of the hydrodynamic reduction meth od, we describe all
47
+ integrable chains (1.1). We establish an one-to-one corresponden ce between integrable chains
48
+ (1.1) and infinite triangular GT-systems of the form
49
+ ∂ipj=P(pi,pj)
50
+ pi−pj∂iu1, i/negationslash=j, (1.5)
51
+ ∂i∂ju1=Q(pi,pj)
52
+ (pi−pj)2∂iu1∂ju1, i/negationslash=j, (1.6)
53
+ ∂ium= (gm,0+gm,1pi+···+gm,m−1pm−1
54
+ i)∂iu1, g m,j=gm,j(u1,...,um), gm,m−1/negationslash= 0,
55
+ 3wherem= 2,3,...andi,j= 1,2,3.The functions P,Qare polynomials quadratic in each of
56
+ variablespiandpj,with coefficients being functions of u1,u2.The functions p1,p2,p3,u1,u2,...
57
+ in (3.11) depend on r1,r2,r3,and∂i=∂
58
+ ∂ri.
59
+ Example 1-1 (continuation of Example 1.) The system (1.5),(1.6) corresponding t o the
60
+ Benney chain has the following form
61
+ ∂ipj=∂iu1
62
+ pi−pj, ∂ i∂ju1=2∂iu1∂ju1
63
+ (pi−pj)2, (1.7)
64
+ ∂ium= (−(m−2)um−2−···−2u2pm−2
65
+ i−u1pm−3
66
+ i+pm−1
67
+ i)∂iu1. (1.8)
68
+ Equations (1.7) were firstly obtained in [10]. /square
69
+ Given GT-system (1.5), (1.6) the coefficients of (1.1) are uniquely de fined by the following
70
+ relations
71
+ pi∂ium=φm,1∂iu1+···+φm,m+1∂ium+1, m= 2,3,... (1.9)
72
+ Namely, equating the coefficients at different powers of piin (1.9), we get a triangular system
73
+ of linear algebraic equations for φi,j. Thus, the classification problem for chains (1.1) is reduced
74
+ to a description of all GT-systems (1.5), (1.6) . The latter problem is solved in Section 4-6.
75
+ The paper is organized as follows. Following [11, 12], we recall main defin itions in Section
76
+ 2 (see [1, 2, 3, 11] for details). We consider only 3-component hyd rodynamic reductions since
77
+ the existence of reductions with N >3 gives nothing new [1]. In Section 3 we formulate
78
+ our previous results that are needed in the paper. Section 4 is devo ted to a classification of
79
+ admissible polynomials PandQin (1.5), (1.6). In Sections 5,6 we construct integrable chains
80
+ for the generic case and for some degenerations. Section 6 also co ntains examples of (2+1)-
81
+ dimensional infinite hydrodynamic-type chains integrable from the v iewpoint of the method
82
+ of hydrodynamic reductions. Infinitesimal symmetries of GT-syst ems are studied in Section 7.
83
+ These symmetries seem to be important basic objects in the hydrod ynamic reduction approach.
84
+ Acknowledgments. Authors thank M.V. Pavlov for fruitful discussions. V.S. is gratefu l to
85
+ Brock University for hospitality. He was partially supported by the R FBR grants 08-01-464,
86
+ 09-01-22442-KE, and NS 3472.2008.2.
87
+ 2 Integrable chains and hydrodynamic reductions
88
+ According to [1, 2, 3, 4, 5, 6] a chain (1.1) is called integrable if it admits sufficiently many
89
+ so-called hydrodynamic reductions.
90
+ Definition. A hydrodynamic (1+1)-dimensional N-component reduction of a chain (1.1)
91
+ is a semi-Hamiltonian (see formula (3.18) ) system of the form
92
+ ri
93
+ t=pi(r1,...,rN)ri
94
+ x, i= 1,..,N (2.10)
95
+ 4and functions uj(r1,...,rN), j= 1,2,...such that for each solution of (2.10) functions uj=
96
+ uj(r1,...,rN), i= 1,...satisfy (1.1).
97
+ Substituting ui=ui(r1,...,rN), i= 1,...into (1.1), calculating tandx-derivatives by virtue
98
+ of (2.10) and equating coefficients at rs
99
+ xto zero, we obtain
100
+ ∂suαps=φα,1∂su1+···+φα,α+1∂suα+1, α= 1,2,...
101
+ It is clear from this system that
102
+ ∂suk=gk(ps,u1,...,uk)∂su1, k= 2,3,...
103
+ wheregk(p,u1,...,uk) is a polynomial of degree k−1 inpfor eachk= 2,3,...Compatibility
104
+ conditions∂i∂juk=∂j∂iukgive us a system of linear equations for ∂ipj, ∂jpi, ∂i∂ju1, i/negationslash=j.
105
+ This system should have a solution (otherwise we would not have suffic iently many reductions).
106
+ Moreover, expressions for ∂suk, k= 2,3,..., ∂jpi, ∂i∂ju1, i/negationslash=jshould be compatible and form
107
+ a so-called GT-system.
108
+ Remark. In the sequel we assume N= 3 because the case N >3 gives nothing new [1].
109
+ 3 GT-systems
110
+ Definition. A compatible system of PDEs of the form
111
+ ∂ipj=f(pi,pj,u1,...,un), ∂iu1j/negationslash=i,
112
+ ∂i∂ju1=h(pi,pj,u1,...,un)∂iu1∂ju1, j/negationslash=i, (3.11)
113
+ ∂iuk=gk(pi,u1,...,un)∂iu1, k= 1,...,n−1,
114
+ wherei,j= 1,2,3 is called n-fields GT-system . Herep1,p2,p3,u1,...,unare functions of
115
+ r1,r2,r3and∂i=∂
116
+ ∂ri.
117
+ Definition. Two GT-systems are called equivalent if they are related by a transformation
118
+ of the form
119
+ pi→λ(pi,u1,...,un), (3.12)
120
+ uk→µk(u1,...,un), k= 1,...,n. (3.13)
121
+ Example 2 [13]. Leta0,a1,a2be arbitrary constants, R(x) =a2x2+a1x+a0. Then the
122
+ system
123
+ ∂ipj=a2p2
124
+ j+a1pj+a0
125
+ pi−pj∂iu1, ∂ i∂ju1=2a2pipj+a1(pi+pj)+2a0
126
+ (pi−pj)2∂iu1∂ju1(3.14)
127
+ is an one-field GT-system. The original Gibbons-Tsarev system (1.7 ) corresponds to a2=a1=
128
+ 0,a0= 1.The polynomial R(x) can be reduced to one of the following canonical forms: R= 1,
129
+ 5R=x,R=x2, orR=x(x−1) by a linear transformation (3.12). A wide class of integrable
130
+ 3D-systems of hydrodynamic type related to (3.14) is described in [1 3]. An elliptic version of
131
+ this GT-system and the corresponding integrable 3D-systems wer e constructed in [15]. /square
132
+ Definition. An additional system
133
+ ∂iuk=gk(pi,u1,...,un+m)∂iun, k=n+1,...,n+m (3.15)
134
+ suchthat(3.11)and(3.15)arecompatibleiscalled an extension of(3.11)byfields un+1,...,un+m.
135
+ It turns our that
136
+ ∂iun+1=f(pi,un+1,u1,...,un)∂iu1
137
+ is an extension for GT-system (3.11). Stress that here fis the same function as in (3.11). We
138
+ call this extension the regular extension byun+1.
139
+ Example 2-1. The generic case of Example 2 corresponds to R=x(x−1). The regular
140
+ extension by u2is given by
141
+ ∂iu2=u2(u2−1)
142
+ pi−u2∂iu1.
143
+ If we express u1from this formula and substitute it to (3.14), we get the following one -field
144
+ GT-system
145
+ ∂ipj=pj(pj−1)(pi−u1)
146
+ u1(u1−1)(pi−pj)∂iu1,
147
+ ∂i∂ju1=pipj(pi+pj)−p2
148
+ i−p2
149
+ j+(p2
150
+ i+p2
151
+ j−4pipj+pi+pj)u1
152
+ u1(u1−1)(pi−pj)2∂iu1∂ju1./square(3.16)
153
+ The second basic notion of the hydrodynamic reduction method is so -called GT-family of
154
+ (1+1)-dimensional hydrodynamic-type systems.
155
+ Definition. An (1+1)-dimensional 3-component hydrodynamic-type system o f the form
156
+ ri
157
+ t=vi(r1,...,rN)ri
158
+ x, i= 1,2,3, (3.17)
159
+ is called semi-Hamiltonian if the following relation holds
160
+ ∂j∂ivk
161
+ vi−vk=∂i∂jvk
162
+ vj−vk, i/negationslash=j/negationslash=k. (3.18)
163
+ Definition. A Gibbons-Tsarev family associated with the Gibbons-Tsarev type s ystem
164
+ (4.25) is a (1+1)-dimensional hydrodynamic-type system of the fo rm
165
+ ri
166
+ t=F(pi,u1,...,um)ri
167
+ x, i= 1,2,3, (3.19)
168
+ semi-Hamiltonian by virtue of (3.11).
169
+ 6Example 2-2 [13]. Applying the regular extension to the generic GT-system (3.14) two
170
+ times, we get the following GT-system:
171
+ ∂ipj=pj(pj−1)
172
+ pi−pj∂iw, ∂ ijw=2pipj−pi−pj
173
+ (pi−pj)2∂iw∂jw, i/negationslash=j, (3.20)
174
+ ∂iuj=uj(uj−1)∂iw
175
+ pi−uj, j= 1,2. (3.21)
176
+ Consider the generalized hypergeometric [14] linear system of the f orm
177
+ ∂2h
178
+ ∂uj∂uk=sj
179
+ uj−uk·∂h
180
+ ∂uk+sk
181
+ uk−uj·∂h
182
+ ∂uj, j/negationslash=k, (3.22)
183
+ ∂2h
184
+ ∂uj∂uj=−/parenleftBigg
185
+ 1+n+2/summationdisplay
186
+ k=1sk/parenrightBigg
187
+ sj
188
+ uj(uj−1)·h+sj
189
+ uj(uj−1)n/summationdisplay
190
+ k/negationslash=juk(uk−1)
191
+ uk−uj·∂h
192
+ ∂uk+
193
+ /parenleftBiggn/summationdisplay
194
+ k/negationslash=jsk
195
+ uj−uk+sj+sn+1
196
+ uj+sj+sn+2
197
+ uj−1/parenrightBigg
198
+ ·∂h
199
+ ∂uj.(3.23)
200
+ Herei,j= 1,2 ands1,...,s4are arbitrary parameters. It easy to verify that this system is in
201
+ involution and therefore the solution space is 3-dimensional. Let h1,h2,h3be a basis of this
202
+ space. For any hwe put
203
+ S(p,h) =u1(u1−1)(p−u2)hh1,u1−hu1h1
204
+ h1+u2(u2−1)(p��u1)hh1,u2−hu2h1
205
+ h1.
206
+ Then the formula
207
+ F=S(p,h3)
208
+ S(p,h2)(3.24)
209
+ defines the generic linear fractional GT-family for (3.20). /square
210
+ 4 Canonical forms of GT-systems associated
211
+ with integrable chains
212
+ For integrable chains the corresponding GT-systems involve infinite number of fields ui, i=
213
+ 1,2,...(see Example 1-1). In this Section we show that these GT-systems are equivalent to
214
+ infinite triangular extensions of one-field GT-systems from Example s 2,3.
215
+ A compatible system of PDEs of the form
216
+ ∂ipj=f(pi,pj,u1,...,un)∂iu1, i/negationslash=j,
217
+ ∂iuk=gk(pi,u1,...,uk)∂iu1, k= 1,2,...,, (4.25)
218
+ 7∂i∂ju1=h(pi,pj,u1,...,un)∂iu1∂ju1, i/negationslash=j,
219
+ wherei,j= 1,2,3 is called triangular GT-system . Herep1,p2,p3,u1,u2,...are functions of
220
+ r1,r2,r3,and∂i=∂
221
+ ∂ri.
222
+ Definition. A chain (1.1) is called integrable if there exists a Gibbons-Tsarev type system
223
+ of the form (4.25) and a Gibbons-Tsarev family
224
+ ri
225
+ t=F(pi,u1,...,um)ri
226
+ x, i= 1,2,3, (4.26)
227
+ such that (1.1) holds by virtue of (4.25), (4.26).
228
+ Due to the equivalence transformations (3.12) we can assume witho ut loss of generality that
229
+ F(p,u1,...,um) =p. (4.27)
230
+ Under this assumption we have
231
+ uj,t=/summationdisplay
232
+ s∂sujrs
233
+ t=/summationdisplay
234
+ s∂sujpsrs
235
+ x.
236
+ and similar
237
+ uj,x=/summationdisplay
238
+ s∂sujrs
239
+ x.
240
+ Substituting these expressions into (1.1) and equating coefficients atrs
241
+ xto zero, we obtain
242
+ ∂suαps=φα,1∂su1+···+φα,α+1∂suα+1, α= 1,2,...
243
+ Using (4.25) and replacing psbyp, we get
244
+ p=φ1,1+φ1,2g2, pg2=φ2,1+φ2,2g2+φ2,3g3, pg3=φ3,1+φ3,2g2+φ3,3g3+φ3,4g4,...
245
+ Solving this system with respect to g2, g3,..., we obtain
246
+ gi(p) =ψi,0+ψi,1p+...+ψi,i−1pi−1.
247
+ Hereψi,jare functions of u1,...,ui. For example,
248
+ g2=−p
249
+ φ1,2−φ1,1
250
+ φ1,2. (4.28)
251
+ Remark. Since we assume that φi,i−1/negationslash= 0,we haveψi,i−1/negationslash= 0 for all i. Therefore g1=
252
+ 1,g2,...is a basis in the linear space of all polynomials in p. The coefficients φi,jof our chain
253
+ are just entries of the matrix of multiplication by pin this basis. More generally, if we don’t
254
+ normalizeF=p, then the coefficients φi,jcan be found from the equations
255
+ F(p) =φ1,1+φ1,2g2, F(p)g2=φ2,1+φ2,2g2+φ2,3g3,
256
+ F(p)g3=φ3,1+φ3,2g2+φ3,3g3+φ3,4g4,...(4.29)
257
+ 8Compatibility conditions ∂i∂juα=∂j∂iuα, α= 2,3,4 give a system of linear equations for
258
+ ∂ipj, ∂jpi, ∂i∂ju1. Solving this system, we obtain formulas (1.5),(1.6), where in principa lP, Q
259
+ coulddependon u1,u2,u3,u4. However, itfollowsfromcompatibility conditions ∂i∂jpk=∂j∂ipk
260
+ thatP, Qdepend onu1, u2only.
261
+ Written (1.5) in the form
262
+ ∂ipj=/parenleftbiggR(pj)
263
+ pi−pj+(z4p2
264
+ j+z5pj+z6)pi+z4p3
265
+ j+z3p2
266
+ j+z7pj+z8/parenrightbigg
267
+ ∂iu1, (4.30)
268
+ whereR(x) =z4x4+z3x3+z2x2+z1x+z0,one can derive from the compatibility conditions
269
+ ∂i∂jpk=∂j∂ipk,∂i∂ju1=∂j∂iu1that the equation (1.6) has the following form
270
+ ∂i∂ju1=/parenleftbigg2z4p2
271
+ ip2
272
+ j+z3pipj(pi+pj)+z2(p2
273
+ i+p2
274
+ j)+z1(pi+pj)+2z0
275
+ (pi−pj)2+z9/parenrightbigg
276
+ ∂iu1∂ju1.(4.31)
277
+ It is easy to verify that we can normalize z9=z6−z7, g2=pby a transformation (1.2).
278
+ Then the coefficients zi(x,y),i= 0,...,8 satisfy the following pair of compatible dynamical
279
+ systems with respect to yandx:
280
+ z0,y= 2z0z5−z1z6, z 1,y= 4z0z4+z1z5−2z2z6, z 2,y= 3z1z4−3z3z6,
281
+ z3,y= 2z2z4−z3z5−4z4z6, z 4,y=z3z4−2z4z5, z 5,y=z4z7−z4z6−z2
282
+ 5,
283
+ z6,y=z4z8−z5z6, z 7,y= 2z1z4−2z3z6−z5z6+z4z8, z 8,y= 2z0z4−z2
284
+ 6−z6z7+z5z8,
285
+ and
286
+ z0,x=−z0z2−z0z6+3z0z7−z1z8, z 1,x=−z1z2+3z0z3−z1z6+2z1z7−2z2z8,
287
+ z2,x=−z2
288
+ 2+2z1z3+4z0z4−z2z6+z2z7−3z3z8, z 3,x= 3z1z4−z3z6−4z4z8,
289
+ z4,x=z2z4−z4z6−z4z7, z 5,x=z1z4−z5z6−z4z8, z 6,x=z0z4−z2
290
+ 6,
291
+ z7,x=z1z3+3z0z4+z1z5−z2z6−z2z7+z2
292
+ 7−z3z8−2z5z8,
293
+ z8,x=z0z3+z0z5−z2z8−2z6z8+z7z8.
294
+ These is a complete description of the GT-systems related to integr able chains (1.1).
295
+ To solve the dynamical systems we bring the polynomial Rto a canonical form sacrificing
296
+ to the normalization (4.27).
297
+ It is obvious that linear transformations pi→api+b, wherea,bare functions of u1,u2,
298
+ preserve the form of GT-system (4.30),(4.31). Moreover, there exist transformations of the
299
+ form
300
+ pi=a¯pi+b
301
+ ¯pi−ψ, i= 1,2,3 (4.32)
302
+ 9preserving the form of GT-system (4.30),(4.31). Such admissible tr ansformations are described
303
+ by the following conditions:
304
+ au2=z4(b+aψ), b u2=z4bψ+z5b−z6a, ψ u2=z4ψ2+z5ψ+z6.
305
+ Under transformations (4.32) the polynomial Ris transformed by the following simple way:
306
+ R(pi)→(pi−ψ)4R/parenleftBigapi+b
307
+ pi−ψ/parenrightBig
308
+ .
309
+ Suppose that Rhas distinct roots. It is possible to verify that by an admissible trans formation
310
+ (4.32) we can move three of the four roots to 0 ,1 and∞. It follows from compatibility
311
+ conditionsfortheGT-system thatthenthefourthroot λ(u1,u2)doesnotdependon u2. Making
312
+ transformation of the form u1→q(u1) we arrive at the canonical forms λ=u1orλ=const. It
313
+ is straightforwardly verified that in the first case equations (4.30) , (4.31) coincides with (3.16).
314
+ In the second case the GT-system does not exist.
315
+ In the case of multiple roots the polynomial R(x) can be reduced to one of the following
316
+ forms:R= 0,R= 1,R=x,R=x2, orR=x(x−1).In all these cases equations (4.30),
317
+ (4.31) coincides with the corresponding equations from Example 2.
318
+ Thus, the following statement is valid:
319
+ Proposition 1. There are 6 non-equivalent cases of GT-systems (4.30), (4.31). T he canon-
320
+ ical forms are:
321
+ Case 1: (3.16) (generic case);
322
+ Case 2: (3.14) with R(x) =x(x−1);
323
+ Case 3: (3.14) with R(x) =x2;
324
+ Case 4: (3.14) with R(x) =x;
325
+ Case 5: (3.14) with R(x) = 1.
326
+ Case 6: (3.14) with R(x) = 0./square
327
+ Remark. Cases 2-6 can be obtained from Case 1 by appropriate limit procedur es. For
328
+ example, Case 2 corresponds to the limit u1→u1
329
+ ε, ε→0.
330
+ It follows from (4.27), (4.28) that for any canonical form the func tionsFandg2have the
331
+ following structure:
332
+ g2(pi) =k1pi+k2
333
+ k3pi+k4, F(pi) =f1pi+f2
334
+ k3pi+k4, (4.33)
335
+ where the coefficients are functions of u1,u2.
336
+ Lemma 1. For theCase 1 any function g2can bereduced by anappropriatetransformation
337
+ 10¯u2=σ(u1,u2) to one of the following canonical forms:
338
+ a1:g2(p) =u2(u2−1)(p−u1)
339
+ u1(u1−1)(p−u2)(regular extension);
340
+ b1:g2(p) =1
341
+ p−u1;
342
+ c1:g2(p) =u−λ
343
+ 1(u1−1)λ−1
344
+ p−λλ= 1,0;
345
+ d1:g2(p) =u1−u2
346
+ u1(u1−1)p+u2−1
347
+ u1−1./square
348
+ The GT-system from the Case 1 possesses a discrete automorphis m groupS4interchanging
349
+ the points 0 ,1,∞,u1. The group is defined by generators
350
+ σ1:u1→1−u1, pi→1−pi, σ 2:u1→u1
351
+ u1−1, pi→pi
352
+ pi−1,
353
+ and
354
+ σ3:u1→1−u1, pi→(1−u1)pi
355
+ pi−u1.
356
+ Up to this group the cases b1,c1,d1are equivalent and one can take say the case d1for further
357
+ consideration. The case a1is invariant with respect to the group.
358
+ Remark. The casesb1, c1, d1are degenerations of the case a1. Namely, they can be
359
+ obtained as appropriate limit u2→u1,u2→λ, u2→ ∞correspondingly.
360
+ All possible functions g2for Cases 2-5 are described in the following
361
+ Lemma 2. For the GT-system (3.14) (excluding Case 6) any function g2can be reduced
362
+ by an appropriate transformation ¯ u2=σ(u1,u2) to one of the following canonical forms:
363
+ a2:g2(p) =R(u2)
364
+ p−u2(regular extension);
365
+ b2:g2(p) =1
366
+ p−λ,whereR(λ) = 0;
367
+ c2:g2(p) =p−a2u2.
368
+ The discrete automorphism of the GT-system interchanges the ro ots ofRin the case b2./square
369
+ Lemma 3. For the GT-system (3.14) with R(x) = 0 (Case 6) any function g2can be
370
+ reduced to g2(p) =pby an appropriate transformation ¯ u2=σ(u1,u2). Furthermore, the
371
+ corresponding triangular GT-system has the form
372
+ ∂ipj= 0, ∂ i∂ju1= 0, ∂ iuk=pk−1
373
+ iu1, k= 2,3,.../square (4.34)
374
+ 115 Generic case
375
+ The next step in the classification is to find all functions Fof the form (4.28) for each pair
376
+ consisting of a GT-system from Proposition 1 and the correspondin gg2from Lemmas 1-3.
377
+ The semi-Hamiltonian condition (3.18) yields a non-linear system of PDE s for the functions
378
+ f1(u1,u2),f2(u1,u2).For each case this system can be reduced to the linear generalized h yper-
379
+ geometric system (3.22), (3.23) with a special set of parameters s1,s2,s3,s4or to a degeneration
380
+ of this system.
381
+ The general linear fractional GT-family for the generic case 1, a1is given by (3.24). Ac-
382
+ cording to (4.33), the additional restriction is that the root of the denominator has to be equal
383
+ u2.It is easy to verify that this is equivalent to s2= 0,h1,u2=h2,u2= 0. The latter means that
384
+ h1(u1),h2(u1) are linear independent solutions of the standard hypergeometric equation
385
+ u(u−1)h(u)′′+[s1+s3−(s3+s4+2s1)u]h(u)′+s1(s1+s3+s4+1)h(u) = 0.(5.35)
386
+ The function h3(u1,u2) is arbitrary solution of (3.22), (3.23) with s2= 0 linearly independent
387
+ ofh1(u1),h2(u1). Without loss of generality we can choose
388
+ h3(u1,u2) =/integraldisplayu2
389
+ 0(t−u1)s1ts3(t−1)s4dt.
390
+ Formula (3.24) gives
391
+ F(p,u1,u2) =f1(u1,u2)p−f2(u1,u2)
392
+ p−u2, (5.36)
393
+ where
394
+ f1=u2(u2−1)h1h3,u2+u1(u1−1)(h1h3,u1−h3h′
395
+ 1)
396
+ u1(u1−1)(h1h′
397
+ 2−h2h′
398
+ 1),
399
+ f2=u1u2(u2−1)h1h3,u2+u2u1(u1−1)(h1h3,u1−h3h′
400
+ 1)
401
+ u1(u1−1)(h1h′
402
+ 2−h2h′
403
+ 1).
404
+ Notice that h1h′
405
+ 2−h2h′
406
+ 1=const(u1−1)s1+s4us1+s3
407
+ 1.
408
+ For integer values of s1,s3,s4the hypergeometric system can be solved explicitly. For
409
+ example, if s1=s3=s4= 0, the above formulas give rise to F=g2.Ifs4=−2−s1−s3then
410
+ F=(u2−u1)s1+1us3+1
411
+ 2(u2−1)−1−s1−s3
412
+ p−u2;
413
+ ifs4= 0,then
414
+ F=(p−1)(u2−u1)s1+1us3+1
415
+ 2(u1−1)−1−s1
416
+ p−u2.
417
+ Nowwearetofindthefunctions g3,g4,...in(4.25). Thesefunctionsaredefineuptoarbitrary
418
+ transformation (1.2), where α= 3,4,.... In practice, one can look for functions g3,g4,...linear
419
+ inui,i>2 (cf. (1.8)). An extension linear in ui,i>2 is given by
420
+ g3(p) =−(u1−u2)(u2−1)p
421
+ u1(u1−1)(p−u2)2,
422
+ 12gi(p) =(i−3)(u1−u2)(u2−1)pui
423
+ u1(u1−1)(p−u2)2−(u1−u2)i−3(u2−1)2p(p−u1)(p−1)i−4
424
+ u1(u1−1)i−2(p−u2)i−1−
425
+ i−4/summationdisplay
426
+ s=1(i−s−2)(u1−u2)s(u2−1)2p(p−u1)(p−1)s−1ui−s
427
+ u1(u1−1)s+1(p−u2)s+2.
428
+ The coefficients of the chain (1.1) corresponding to Case 1, a1are determined from (4.29),
429
+ whereFis given by (5.36). Relations (4.29) are equivalent to a triangular syst em of linear
430
+ algebraic equations. Solving this system, we find that for i>4 coefficients of the chain read:
431
+ φi,i+1=(u1−1)(f1u2−f2)
432
+ (u2−1)(u1−u2)def=Q1, φ i,i=f2−f1
433
+ u2−1def=Q2,
434
+ φi,4=−uiQ1, φ i,3=−/parenleftBig
435
+ (u4+i−3)ui+(2−i)ui+1/parenrightBig
436
+ Q1def=Ai,
437
+ andφi,j= 0 for all remaining i,j.Fori≤4 we have
438
+ φ1,1=f1u1−f2
439
+ u1−u2, φ 1,2=−u1
440
+ u2Q1,
441
+ φ2,1=(u2−1)(f1u2−f2)
442
+ (u1−1)(u1−u2), φ 2,2=f2u1−f1u2
443
+ 2
444
+ u2(u1−u2), φ 2,3=f1u2−f2,
445
+ φ3,1=φ3,2= 0, φ 3,3=Q2−(u4−1)Q1, φ 3,4=−Q1,
446
+ φ4,1=φ4,2= 0, φ 4,3=A4, φ 4,4=Q2−u4Q1, φ 4,5=Q1.(5.37)
447
+ The explicit formulas for other cases of Proposition 1 can be obtaine d by limits from the
448
+ above formulas. We outline the limit procedures for the case 1, d1. In this case the limit is
449
+ given byu2→u1+εu2, ε→0.It is easy to check that under this limit the extension a1
450
+ turns tod1. The limit of the system (3.22), (3.23) with s2= 0 can be easily found. The general
451
+ solution of the system thus obtained is given by h=c1(u2−u1)1+s1+s3+s4+h1,whereh1is the
452
+ general solution of (5.35). Let h1,h2be solutions of (5.35), and h3= (u2−u1)1+s1+s3+s4. Then
453
+ the limit procedure in (5.36) gives rise to
454
+ F(p,u1,u2) =Q×/parenleftBig
455
+ (1+s1+s3+s4)h1(p−u1)+u1(u1−1)h′
456
+ 1/parenrightBig
457
+ ,
458
+ where
459
+ Q= (u2−u1)1+s1+s3+s4(u1−1)−1−s1−s4u−1−s1−s3
460
+ 1.
461
+ As usual, the most degenerate cases in classification of integrable P DEs could be interesting
462
+ for applications. In our classification they are Case 5, c2and Case 6. The Benney chain
463
+ (see Examples 1 and 1-1) belongs to Case 5, case c2(i.eg2=p). Any GT-family has the form
464
+ F=f1(u1,u2)p+f2(u1,u2). Iff1= 1 thenF=p+k2u2+k1u1.The Benney case corresponds to
465
+ 13k1=k2= 0. For arbitrary kiwe get the Kupershmidt chain [16]. In the case f1=A(u1),A′/negationslash= 0
466
+ we obtain:
467
+ f1=k2exp(λu1)+k1, f 2=k2k3exp(λu1)+λk1(k3u1−u2).
468
+ In the generic case
469
+ F= exp(λu2)(S1(u1)p+S2(u1)),
470
+ where the functions Sican be expressed in terms of the Airy functions.
471
+ 6 Trivial GT-system and 2+1-dimensional integrable hy-
472
+ drodynamic chains
473
+ It was observed in [11] that (2+1)-dimensional systems of hydro dynamic type with the trivial
474
+ GT-system usually admit some integrable multi-dimensional generaliza tions. For the chains
475
+ such GT-system is defined by (4.34). That is why the Case 6 is of a gre at importance in our
476
+ classification. The automorphisms of (4.34) are given by
477
+ pj→pj, j= 1,...,N, u i→νui+γi, i= 1,2,...; (6.38)
478
+ pj→apj+b, j= 1,...,N, u i→ai−1ui+(i−1)ai−2bui−2+...+bi−1u1, i= 1,2,...
479
+ The corresponding GT-families are of the form F(p) =A(u1,u2)p+B(u1,u2), where
480
+ A(x,y),B(x,y) satisfies the following system of PDEs:
481
+ AByy=AyBy, AB xy=AyBx, AB xx=AxBx,
482
+ AAyy=A2
483
+ y, AA xy=AxAy, AA xx=A2
484
+ x+AxBy−AyBx.(6.39)
485
+ This system can be easily solved in elementary functions. For each so lution formula (4.29)
486
+ defines the corresponding integrable chain (1.1).
487
+ It follows from (6.39) that there are two types of u2-dependence:
488
+ 1(generic case). F(p) = exp(λu2)/parenleftBig
489
+ a(u1)p+b(u1)/parenrightBig
490
+ ,
491
+ 2. F(p) =a(u1)p+λu2+b(u1).
492
+ In the first case there are two subcases: b′/negationslash= 0 andb′= 0.The first subcase gives rise to
493
+ a=σ′, b=k1σ σ(x) =c1exp(µ1x)+c2exp(µ2x),wherec1c2(λk1−µ1µ2) = 0.
494
+ The second subcase leads to
495
+ b=c1, a(x) =c2exp(µx)+c3,wherec2(c1λ−c3µ) = 0.
496
+ The same subcases for the case 2 yield
497
+ a=σ′, b=k1σ σ(x) =c1+c2x+c3exp(µx),wherec3(λ−c2µ) = 0,
498
+ 14and
499
+ b=c1, a(x) =c2exp(µx)+c3,wherec2(λ−c3µ) = 0.
500
+ It is easy to verify that in the generic case the function Fcan be reduced by (6.38) to the
501
+ form
502
+ F(p) =eu2+u1(p−1)+eu2−u1(p+1).
503
+ In this case the corresponding chain reads as
504
+ uk,t= (eu2+u1+eu2−u1)uk+1,x+(eu2−u1−eu2+u1)uk,x, k= 1,2,3,... (6.40)
505
+ Asusual, thischainisthefirstmember ofaninfinitehierarchy. These condflowofthishierarchy
506
+ is given by
507
+ uk,τ= (eu2+u1+eu2−u1)uk+2,x+(u3−u1)(eu2+u1+eu2−u1)uk+1,x+
508
+ (eu2+u1(u1−u3−1)+eu2−u1(u3−u1−1))uk,x, k= 1,2,3,...
509
+ In the case 2 with c3=λ= 0,k1= 1 we get the chain
510
+ uk,t=uk+1,x+u1uk,x, k= 1,2,3,... (6.41)
511
+ This chain is equivalent to the chain of the so-called universal hierarc hy [17]. The chain (6.41)
512
+ is a degeneration of the chain
513
+ uk,t=uk+1,x+u2uk,x, k= 1,2,3,... (6.42)
514
+ Following the line of [3, 11] it is not difficult to find (2+1)-dimensional inte grable generaliza-
515
+ tions for all (1+1)-dimensional integrable chains constructed abo ve. Some families of functions
516
+ Fdescribed above linearly depend on two parameters. Denote these parameters by γ1,γ2.The
517
+ corresponding integrable chain
518
+ uk,t=γ1(φk,1u1,x+···+φk,k+1uk+1,x)+γ2(ψk,1u1,x+···+ψk,k+1uk+1,x)
519
+ is also linear in γ1,γ2.We claim that the following (2+1)-dimensional chain
520
+ uk,t= (φk,1u1,x+···+φk,k+1uk+1,x)+(ψk,1u1,y+···+ψk,k+1uk+1,y) (6.43)
521
+ is integrable from the viewpoint of the method of hydrodynamic redu ctions. For each case the
522
+ reductions can be easily described.
523
+ For example, in the generic case
524
+ F(p) =γ1eu2+u1(p−1)+γ2eu2−u1(p+1)
525
+ formula (6.43) yields (2+1)-dimensional chain
526
+ uk,t=eu2+u1(uk+1,x−uk,x)+eu2−u1(uk+1,y+uk,y), k= 1,2,3,... (6.44)
527
+ 15After a change of variables of the form
528
+ x→ −1
529
+ 2x, y→1
530
+ 2y, u 1→1
531
+ 2u0, u2→u1+1
532
+ 2u0, u3→ −2u2+1
533
+ 2u0,...
534
+ (6.44) can be written as
535
+ u0,t=eu1u0,y+eu1(u1,y−eu0u1,x), u i,t=eu0+u1ui,x+eu1(eu0ui+1,x−ui+1,y),(6.45)
536
+ wherei= 1,2,.... Probably (6.45) is a first example of a (2+1)-dimensional chain integ rable
537
+ from the viewpoint of the hydrodynamic reduction approach.
538
+ TriangularGT-systemsrelatedtointegrable(2+1)-dimensionalch ainswithfields u0,u1,u2,...
539
+ have the form
540
+ ∂ipj=f1(pi,qi,pj,qj,u0,...,un)∂iu0, ∂ iqj=f2(pi,qi,pj,qj,u0,...,un)∂iu0,
541
+ ∂i∂ju0=h(pi,qi,pj,qj,u0,...,un)∂iu0∂ju0, (6.46)
542
+ ∂iuk=gk(pi,qi,u0,...,uk+1)∂iu0, k= 0,1,2,...
543
+ Herei/negationslash=j, i,j= 1,...,3,p1,...,p3, q1,...,q3,u0,u1,u2,...,arefunctionsof r1,r2,r3.Inparticular,
544
+ the GT-system associated with (6.45) has the form:
545
+ ∂ipj=∂i∂ju0= 0, ∂ iqj=/parenleftBigpiqi−pjqj
546
+ pi−pj−qiqj/parenrightBig
547
+ ∂iu0, ∂ iuk=−pi
548
+ (pi−1)k∂iu0.
549
+ Thehydrodynamicreductionsof(6.45)isgivenbythepairofsemi-ha miltonian(1+1)-dimensional
550
+ systems
551
+ ri
552
+ y=eu0/parenleftBig
553
+ 1−1
554
+ qi/parenrightBig
555
+ ri
556
+ x, ri
557
+ t=eu0+u1/parenleftBig1
558
+ (pi−1)qi+1/parenrightBig
559
+ ri
560
+ x.
561
+ Chain (6.45) is the first member of an infinite hierarchy of pairwise com muting flows where
562
+ the corresponding ”times” are t1=t, t2, t3,.... These flows and their hydrodynamic reductions
563
+ can be described in terms of the generating function U(z) =u1+u2z+u3z2+...The hierarchy
564
+ is given by
565
+ D(z)u0=eU(z)/parenleftBig
566
+ u0,y+U(z)y−eu0U(z)x/parenrightBig
567
+ ,
568
+ D(z1)U(z2) =eu0+U(z1)U(z2)x+(1+z1)eU(z1)/parenleftBig
569
+ eu0U(z1)x−U(z2)x
570
+ z1−z2−U(z1)y−U(z2)y
571
+ z1−z2/parenrightBig
572
+ ,
573
+ whereD(z) =∂
574
+ ∂t1+z∂
575
+ ∂t2+z2∂
576
+ ∂t3+...The reductions can be written as
577
+ D(z)ri=eu0+U(z)/parenleftBig
578
+ 1+1+z
579
+ (pi−1−z)qi/parenrightBig
580
+ ri
581
+ x.
582
+ Other (2+1)-dimensional integrable chains related to 2-dimensiona l vector spaces of solu-
583
+ tions for system (6.39) are degenarations of (6.45). In particular F=γ1eu1p+γ2(p+u2) leads
584
+ to the following (2+1)-dimensional integrable generalization of (6.44 ):
585
+ uk,t=eu1uk+1,x+uk+1,y+u2uk,y, k= 1,2,3,....
586
+ 16Conjecture. Any chain of the form (6.43) integrable by the hydrodynamic reduct ion
587
+ method is a degeneration of (6.45).
588
+ We are planning to consider the problem of classification of integrable chains (6.43) in a
589
+ separate paper.
590
+ 7 Infinitesimal symmetries of triangular GT-systems
591
+ A scientific way to construct the functions g3,g4,...for different cases from Proposition 1 is
592
+ related to infinitesimal symmetries of the corresponding GT-syste m1. The whole Lie algebra
593
+ of symmetries is one the most important algebraic structures relat ed to any triangular GT-
594
+ system (4.25). In particular, this algebra acts on the hierarchy of the commuting flows for the
595
+ corresponding chain (1.1).
596
+ A vector field
597
+ S=N/summationdisplay
598
+ j=1X(pj,u1,...,us)∂
599
+ ∂pj+∞/summationdisplay
600
+ m=1Ym(u1,...,ukm)∂
601
+ ∂um,∂Ym
602
+ ∂ukm/negationslash= 0 (7.47)
603
+ is called a symmetry of the triangular GT-system (4.25) if it commutes with all ∂i.Notice that
604
+ it follows from the definition that
605
+ S(∂iu1) =∂i(Y1).
606
+ We call (7.47) a symmetry of shift difkm=m+dform>>0.LetMbe the minimal integer
607
+ such thatkm=m+d,m>M. If the functions gi,i= 1,...,M+dfrom (4.25) are known, then
608
+ the functions X,Y1,...YMcan be found from the compatibility conditions
609
+ S(∂ipj) =∂iS(pj), S(∂iuk) =∂iS(uk), k= 1,...,M.
610
+ The functions YM+1,YM+2,...can be chosen arbitrarily. After that gM+d+1,gM+d+2,...are
611
+ uniquely defined by the remaining compatibility conditions.
612
+ The generic case 1, a 1. Looking for symmetries of shift one, we find X=Y1= 0 and
613
+ M= 1. Hence without loss of generality we can take
614
+ S=∞/summationdisplay
615
+ m=2um+1∂
616
+ ∂um
617
+ for the symmetry. This fact gives us a way to construct all functio nsgi,i >3 in the infinite
618
+ triangular extension for the case 1, a1.Indeed, it follows from the commutativity conditions
619
+ S(∂iuk) =∂iS(uk) thatgk+1=S(gk),wherek= 2,3,.... In particular,
620
+ g3=(pj−u1)(2pju2−pj−u2
621
+ 2)u3
622
+ u1(u1−1)(pj−u2)2.
623
+ 1Note that these functions are not unique because of the triangula r group of symmetries (1.2) acting on the
624
+ fieldsu3,u4,...
625
+ 17The functions githus constructed are not linear in u3.The corresponding chain (1.1) is equiv-
626
+ alent to the chain constructed in Section 5 but not so simple.
627
+ It would be interesting to describe the Lie algebra of all symmetries in this case. Here we
628
+ present the essential part for symmetry of shift 2:
629
+ X=pj(pj−1)u2
630
+ 3
631
+ (pj−u2)u2(u2−1), Y 1=u1(u1−1)u2
632
+ 3
633
+ (u1−u2)u2(u2−1),
634
+ Y2=−3
635
+ 2u4+(2u1−1)u2
636
+ 3
637
+ u2(u2−1)+u3./square
638
+ The case 1, d 1. One can add fields u3,...in such a way that the whole triangular GT-
639
+ system admits the following symmetry of shift 1:
640
+ S=u2
641
+ u1(u1−1)N/summationdisplay
642
+ i=1pi(pi−1)∂
643
+ ∂pi+∞/summationdisplay
644
+ i=1ui+1∂
645
+ ∂ui.
646
+ As in the previous example, one can easily recover the whole GT-syst em. For example,
647
+ ∂iu3=/parenleftbiggu3(pi+u1−1)
648
+ u1(u1−1)+2u2
649
+ 2pi(pi−1)
650
+ u2
651
+ 1(u1−1)2/parenrightbigg
652
+ ∂iu1./square
653
+ Below we describe the symmetry algebra for the case 5, c2(in particular, for the Benney
654
+ chain).
655
+ The case 5, c 2. For the triangular GT-system (1.7), (1.8) there exists an infinite L ie
656
+ algebra of symmetries Si,i∈Z,whereSiis a symmetry of shift i. The simplest symmetries
657
+ are the following:
658
+ S−2=∂
659
+ ∂u1+∞/summationdisplay
660
+ i=3/parenleftBig
661
+ −ui−2+/summationdisplay
662
+ k+m=i−3ukum−/summationdisplay
663
+ k+m+l=i−4ukumul+···/parenrightBig∂
664
+ ∂ui,
665
+ S−1=N/summationdisplay
666
+ j=1∂
667
+ ∂pj+∞/summationdisplay
668
+ i=1(i−1)ui−1∂
669
+ ∂ui,
670
+ S0=N/summationdisplay
671
+ j=1pj∂
672
+ ∂pj+∞/summationdisplay
673
+ i=1(i+1)ui∂
674
+ ∂ui,
675
+ S1=N/summationdisplay
676
+ j=1(p2
677
+ j+3u1)∂
678
+ ∂pj+∞/summationdisplay
679
+ i=1(i+3)ui+1∂
680
+ ∂ui+∞/summationdisplay
681
+ i=2/summationdisplay
682
+ k+m=iukum∂
683
+ ∂ui+∞/summationdisplay
684
+ i=23(i−1)u1ui−1∂
685
+ ∂ui,
686
+ S2=N/summationdisplay
687
+ j=1(p3
688
+ j+4u1pj+5u2)∂
689
+ ∂pj+∞/summationdisplay
690
+ i=1(i+5)ui+2∂
691
+ ∂ui+∞/summationdisplay
692
+ i=14iu1ui∂
693
+ ∂ui+∞/summationdisplay
694
+ i=25(i−1)u2ui−1∂
695
+ ∂ui+
696
+ 18∞/summationdisplay
697
+ i=1/summationdisplay
698
+ k+m=i+13ukum∂
699
+ ∂ui+∞/summationdisplay
700
+ i=3/summationdisplay
701
+ k+m+l=iukumul∂
702
+ ∂ui.
703
+ The whole algebra is generated by S1,S2,S−1,S−2.It is isomorphic to the Virasoro algebra with
704
+ zero central charge.
705
+ LetDtibe the vector fields corresponding to commuting flows for the Benn ey chain. Here
706
+ Dt1=Dx, Dt2=Dt. Then the commutator relations
707
+ [S1,Dti] = (i+1)Dti+1
708
+ hold. Thus the vector field S1plays the role of a master-symmetry for the Benney hierarchy.
709
+ /square
710
+ The case 6 . In this case there exist infinitesimal symmetries of form
711
+ Ti=ui+1∂
712
+ ∂u1+ui+2∂
713
+ ∂u2+..., i= 0,1,2,...
714
+ Si=N/summationdisplay
715
+ j=1pi+1
716
+ j∂
717
+ ∂pj+ui+2∂
718
+ ∂u2+2ui+3∂
719
+ ∂u3+3ui+4∂
720
+ ∂u4+..., i=−1,0,1,2,...
721
+ Note that [Si,Sj] = (j−i)Si+j,[Ti,Tj] = 0,[Si,Tj] =jTi+j./square
722
+ References
723
+ [1]E.V. Ferapontov, K.R. Khusnutdinova , On integrability of (2+1)-dimensional quasilinear
724
+ systems, Comm. Math. Phys. 248(2004) 187-206,
725
+ [2]E.V. Ferapontov, K.R. Khusnutdinova , The characterization of 2-component (2+1)-
726
+ dimensional integrablesystemsofhydrodynamic type, J.Phys. A: Math.Gen. 37(8)(2004)
727
+ 2949 - 2963.
728
+ [3]E.V. Ferapontov, K.R. Khusnutdinova , Hydrodynamic reductions of multidimensional
729
+ dispersionless PDEs: the test for integrability, J. Math. Phys. 45(6) (2004) 2365 - 2377.
730
+ [4]M.V. Pavlov , Classification of the Egorov hydrodynamic chains. Theor. Math. P hys.138
731
+ No. 1 (2004) 55-71.
732
+ [5]M.V. Pavlov , Classification of integrable hydrodynamic chains and generating fu nctions
733
+ of conservation laws, J. Phys. A: Math. Gen. 39(34) (2006) 10803–10819.
734
+ [6]E.V. Ferapontov, D.G. Marshal , Differential-geometric approach to the integrability of
735
+ hydrodynamic chains: the Haanties tensor, Math. Ann. 339(1), (2007) 61–99.
736
+ [7]D.J. Benney , Some properties of long nonlinear waves, Stud. Appl. Math. 52(1973) 45-50.
737
+ 19[8]B.A. Kupershmidt, Yu.I. Manin , Long waves equation with free surface. I. Conservation
738
+ laws and solutions. Func. Anal. and Appl., 11(3) (1977) 31-42.
739
+ [9]V.E. Zakharov , On the Benney’s Equations, Physica 3D (1981) 193-200.
740
+ [10]J. Gibbons, S.P. Tsarev , Reductions of Benney’s equations, Phys. Lett. A, 211(1996)
741
+ 19-24.
742
+ [11]A.V. Odesskii, V.V. Sokolov , Systems of Gibbons-Tsarev type and integrable 3-
743
+ dimensional models, arXiv:0906.3509
744
+ [12]A.V. Odesskii, V.V. Sokolov , Integrable (2+1)-dimensional hydrodynamic type systems,
745
+ Theor. and Math. Phys, to be published.
746
+ [13]A.V. Odesskii, V.V. Sokolov , Integrable pseudopotentials related to generalized hyperge-
747
+ ometric functions, arXiv:0803.0086
748
+ [14]I.M. Gelfand, M.I. Graev, V.S. Retakh , General hypergeometric systems of equations and
749
+ series of hypergeometric type, Russian Math. Surveys 47 (1992) , no. 4, 1–88
750
+ [15]A.V. Odesskii, V.V. Sokolov , Integrable pseudopotentials related to elliptic curves, Teoret.
751
+ and Mat. Fiz., 161(1) (2009) 21–36, arXiv:0810.3879
752
+ [16]B.A. Kupershmidt , Deformations of integrable systems, Proc. Roy. Irish Acad. Sec t. A,
753
+ 83(1) (1983) 45-74.
754
+ [17]L. Martinez Alonso, A.B. Shabat , Hydrodynamic reductions and solutions of a universal
755
+ hierarchy , Teoret. and Mat. Fiz., 140(2) (2004) 1073–1085
756
+ 20
1001.0021.txt ADDED
@@ -0,0 +1,884 @@
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
1
+ arXiv:1001.0021v3 [cond-mat.quant-gas] 8 Oct 2010Strong-coupling expansionforthe two-species Bose-Hubba rd model
2
+ M. Iskin
3
+ Department of Physics, Koc ¸ University, Rumelifeneri Yolu , 34450 Sariyer, Istanbul, Turkey
4
+ (Dated: August 28, 2018)
5
+ Toanalyze the ground-state phase diagram ofBose-Bose mixt ures loadedinto d-dimensional hypercubic op-
6
+ tical lattices, we perform a strong-coupling power-series expansion in the kinetic energy term (plus a scaling
7
+ analysis) for the two-species Bose-Hubbard model with onsi te boson-boson interactions. We consider both
8
+ repulsive and attractive interspecies interaction, and ob tain an analytical expression for the phase boundary be-
9
+ tweentheincompressibleMottinsulatorandthecompressib lesuperfluidphaseuptothirdorderinthehoppings.
10
+ In particular, we find a re-entrant quantum phase transition from paired superfluid (superfluidity of composite
11
+ bosons, i.e. Bose-Bose pairs) to Mott insulator and again to a paired superfluid in all one, two and three di-
12
+ mensions, whentheinterspecies interactionissufficientl ylargeandattractive. Wehope thatsome ofourresults
13
+ couldbe testedwithultracoldatomic systems.
14
+ PACS numbers: 03.75.-b, 37.10.Jk,67.85.-d
15
+ I. INTRODUCTION
16
+ Single-species Bose-Hubbard (BH) model is the bosonic
17
+ generalization of the Hubbard model, and was introduced
18
+ originallytodescribe4Heinporousmediaordisorderedgran-
19
+ ular superconductors [1]. For hypercubic lattices in all di -
20
+ mensions d, there are only two phases in this model: an in-
21
+ compressible Mott insulator at commensurate (integer) fill -
22
+ ings and a compressible superfluid phase otherwise. The su-
23
+ perfluid phase is well described by weak-coupling theories,
24
+ buttheinsulatingphaseisastrong-couplingphenomenonth at
25
+ only appearswhen the system is on a lattice. Transition from
26
+ the Mott insulator to the superfluid phase occurs as the hop-
27
+ ping, particle-particleinteraction,or the chemical pote ntial is
28
+ varied[1].
29
+ It is the recent observation of this transition in effective ly
30
+ three- [2], one- [3], and two-dimensional [4, 5] optical lat -
31
+ tices, which has been considered one of the most remarkable
32
+ achievements in the field of ultracold atomic gases, since it
33
+ paved the way for studying other strongly correlated phases
34
+ in similar setups. Such lattices are created by the intersec tion
35
+ of laser fields, and they are nondissipative periodic potent ial
36
+ energy surfaces for the atoms. Motivated by this success in
37
+ experimentally simulating the single-species BH model wit h
38
+ ultracoldatomic Bose gasesloaded into optical lattices, t here
39
+ has been recently an intense theoretical activity in analyz ing
40
+ BH aswell asFermi-Hubbardtypemodels[6].
41
+ For instance, in addition to the Mott insulator and single-
42
+ species superfluid phases, it has been predicted that the two -
43
+ species BH model has at least two additional phases: an in-
44
+ compressible super-counter flow and a compressible paired
45
+ superfluidphase[7–16]. Ourmaininteresthereisinthelatt er
46
+ phase,wherea directtransitionfromtheMott insulatorto t he
47
+ paired superfluid phase (superfluidity of composite bosons,
48
+ i.e. Bose-Bose pairs) has been predicted, when both species
49
+ have integer fillings and the interspecies interaction is su ffi-
50
+ ciently large and attractive. Given that the interspecies i nter-
51
+ actions can be fine tuned in ongoing experiments, e.g. with
52
+ 41K-87Rb with mixtures [17, 18], via using Feshbach reso-
53
+ nances,we hopethat someof ourresults couldbe tested with
54
+ ultracoldatomicsystems.Inthispaper,weexaminetheground-statephasediagramof
55
+ the two-species BH model with on-site boson-boson interac-
56
+ tionsind-dimensionalhypercubiclattices, includingboth the
57
+ repulsive and attractive interspecies interaction, via a s trong-
58
+ coupling perturbation theory in the hopping. We carry the
59
+ expansion out to third-order in the hopping, and perform a
60
+ scaling analysis using the known critical behavior at the ti p
61
+ of the insulating lobes, which allows us to accurately predi ct
62
+ the critical point, and the shape of the insulating lobes in t he
63
+ plane of the chemical potential and the hopping. This tech-
64
+ niquewaspreviouslyusedtodiscussthephasediagramofthe
65
+ single-species BH model [19–23], extended BH model [24],
66
+ and of the hardcore BH model with a superlattice [25], and
67
+ its results showed an excellent agreement with Monte Carlo
68
+ simulations [23, 25]. Motivated by the success of this tech-
69
+ nique with these models, here we apply it to the two-species
70
+ BH model, hoping to develop an analytical approach which
71
+ couldbeasaccurateasthenumericalones.
72
+ The remaining paper is organized as follows. After in-
73
+ troducing the model Hamiltonian in Sec. II, we develop the
74
+ strong-coupling expansion in Sec. III, where we derive an
75
+ analytical expression for the phase boundary between the in -
76
+ compressible Mott insulator and the compressible superflui d
77
+ phase. Then, in Sec. IV, we proposea chemical-potentialex-
78
+ trapolation technique based on scaling theory to extrapola te
79
+ ourthird-orderpower-seriesexpansioninto a functionalf orm
80
+ thatisappropriatefortheMottlobes,anduse ittoobtainty p-
81
+ ical ground-state phase diagrams. A brief summary of our
82
+ conclusionsisgiveninSec.V.
83
+ II. TWO-SPECIESBOSE-HUBBARDMODEL
84
+ TodescribeBose-Bosemixturesloadedintoopticallattice s,
85
+ weconsiderthe followingtwo-speciesBH Hamiltonian,
86
+ H=−/summationdisplay
87
+ i,j,σtij,σb†
88
+ i,σbj,σ+/summationdisplay
89
+ i,σUσσ
90
+ 2/hatwideni,σ(/hatwideni,σ−1)
91
+ +U↑↓/summationdisplay
92
+ i/hatwideni,↑/hatwideni,↓−/summationdisplay
93
+ i,σµσ/hatwideni,σ, (1)2
94
+ where the pseudo-spin σ≡ {↑,↓}labels the trapped hyper-
95
+ fine states of a given species of bosons, or labels different
96
+ types of bosons in a two-species mixture, tij,σis the tun-
97
+ neling (or hopping) matrix between sites iandj,b†
98
+ i,σ(bi,σ)
99
+ is the boson creation (annihilation) and /hatwideni,σ=b†
100
+ i,σbi,σis
101
+ the boson number operator at site i,Uσσ′is the strength of
102
+ the onsite boson-bosoninteraction between σandσ′compo-
103
+ nents, and µσis the chemical potential. In this manuscript,
104
+ we considera d-dimensionalhypercubiclattice with Msites,
105
+ forwhich we assume tij,σis a real symmetricmatrix with el-
106
+ ementstij,σ=tσ≥0foriandjnearest neighbors and 0
107
+ otherwise. Thelattice coordinationnumber(orthe numbero f
108
+ nearestneighbors)forsuchlatticesis z= 2d.
109
+ We take the intraspecies interactions to be repulsive
110
+ ({U↑↑,U↓↓}>0), but discuss both repulsive and attractive
111
+ interspecies interaction U↑↓as long as U↑↑U↓↓> U2
112
+ ↑↓. This
113
+ guarantees the stability of the mixture against collapse wh en
114
+ U↑↓≪0,andagainstphaseseparationwhen U↑↓≫0. How-
115
+ ever,whentheinterspeciesinteractionissufficientlylar geand
116
+ attractive, we note that instead of a direct transition from the
117
+ Mottinsulatortoasingleparticlesuperfluidphase,itispo ssi-
118
+ bletohaveatransitionfromtheMottinsulatortoa pairedsu -
119
+ perfluid phase (superfluidity of composite bosons, i.e. Bose -
120
+ Bose pairs) [7–16]. Therefore, one needs to consider both
121
+ possibilities,asdiscussednext.
122
+ III. STRONG-COUPLINGEXPANSION
123
+ We use the many-body version of Rayleigh-Schr¨ odinger
124
+ perturbation theory in the kinetic energy term to perform th e
125
+ expansion (in powers of t↑andt↓) for the different energies
126
+ needed to carryout our analysis. The strong-couplingexpan -
127
+ sion technique was previously used to discuss the phase di-
128
+ agram of the single-species BH model [19–21, 23], extended
129
+ BHmodel[24],andofthehardcoreBHmodelwithasuperlat-
130
+ tice [25], and its results showed an excellent agreement wit h
131
+ Monte Carlo simulations [23, 25]. Motivated by the success
132
+ of this technique with these models, here we apply it to the
133
+ two-speciesBH model.
134
+ To determine the phase boundary separating the incom-
135
+ pressible Mott phase from the compressible superfluid phase
136
+ within the strong-coupling expansion method, one needs the
137
+ energyoftheMottphaseandofits‘defect’states(thosesta tes
138
+ whichhaveexactlyoneextraelementaryparticleorholeabo ut
139
+ the ground state) as a function of t↑andt↓. At the point
140
+ where the energy of the incompressible state becomes equal
141
+ to its defect state, the system becomes compressible, assum -
142
+ ing that the compressibility approaches zero continuously at
143
+ the phaseboundary. Here,we remarkthat thistechniquecan-
144
+ notbeusedtocalculatethephaseboundarybetweentwocom-
145
+ pressiblephases.A. Ground-StateWave Functions
146
+ The perturbation theory is performed with respect to the
147
+ ground state of the system when t↑=t↓= 0, and therefore
148
+ we first need zeroth order wave functions of the Mott phase
149
+ and of its defect states. To zerothorderin t↑andt↓, the Mott
150
+ insulatorwavefunctioncanbewrittenas,
151
+ |Ψins(0)
152
+ Mott/an}bracketri}ht=1/radicalbig
153
+ n↑!n↓!/productdisplay
154
+ i(b†
155
+ i,↑)n↑(b†
156
+ i,↓)n↓|0/an}bracketri}ht,(2)
157
+ where/an}bracketle{t/hatwideni,σ/an}bracketri}ht=nσis anintegernumbercorrespondingto the
158
+ ground-stateoccupancyofthe pseudo-spin σbosons,/an}bracketle{t···/an}bracketri}htis
159
+ thethermalaverage,and |0/an}bracketri}htisthevacuumstate. Ontheother
160
+ hand, the wave functions of the defect states are determined
161
+ by degenerate perturbation theory. The reason for that lies
162
+ in the fact that when exactly one extra elementary particle o r
163
+ hole is added to the Mott phase, it could go to any of the M
164
+ lattice sites, since all of those states share the same energ y
165
+ whent↑=t↓= 0. Therefore, the initial degeneracy of the
166
+ defectstates isoforder M.
167
+ Whentheelementaryexcitationsinvolveasingle- σ-particle
168
+ (exactly one extra pseudo-spin σboson) or a single- σ-hole
169
+ (exactly one less pseudo-spin σboson), this degeneracy is
170
+ lifted at first order in t↑andt↓. The treatment for this case is
171
+ very similar to the single-species BH model [19, 24], and the
172
+ wave functions(to zerothorderin t↑andt↓) forthe single- σ-
173
+ particleandsingle- σ-holedefectstates turnouttobe
174
+ |Ψsσp(0)
175
+ def/an}bracketri}ht=1√nσ+1/summationdisplay
176
+ ifsσp
177
+ ib†
178
+ i,σ|Ψins(0)
179
+ Mott/an}bracketri}ht,(3)
180
+ |Ψsσh(0)
181
+ def/an}bracketri}ht=1√nσ/summationdisplay
182
+ ifsσh
183
+ ibi,σ|Ψins(0)
184
+ Mott/an}bracketri}ht, (4)
185
+ wherefsσp
186
+ i=fsσh
187
+ iis the eigenvector of the hopping matrix
188
+ tij,σwith the highest eigenvalue (which is ztσwithz= 2d)
189
+ such that/summationtext
190
+ jtij,σfsσp
191
+ j=ztσfsσp
192
+ i.The normalizationcondi-
193
+ tion requires that/summationtext
194
+ i|fsσp
195
+ i|2= 1. Notice that we choose the
196
+ highest eigenvalue of tij,σbecause the hoppingmatrix enters
197
+ theHamiltonianas −tij,σ,andweultimatelywantthelowest-
198
+ energystates.
199
+ However,whentheelementaryexcitationsinvolvetwopar-
200
+ ticles (exactly one extra boson of each species) or two holes
201
+ (exactly one less boson of each species), the degeneracy is
202
+ lifted at second order in t↑andt↓. Such elementary excita-
203
+ tions occur when U↑↓is sufficiently large and attractive [26],
204
+ and the wave functions (to zeroth order in t↑andt↓) for the
205
+ two-particleandtwo-holedefectstatescanbewrittenas
206
+ |Ψtp(0)
207
+ def/an}bracketri}ht=1/radicalbig
208
+ (n↑+1)(n↓+1)/summationdisplay
209
+ iftp
210
+ ib†
211
+ i,↑b†
212
+ i,↓|Ψins(0)
213
+ Mott/an}bracketri}ht,(5)
214
+ |Ψth(0)
215
+ def/an}bracketri}ht=1√n↑n↓/summationdisplay
216
+ ifth
217
+ ibi,↑bi,↓|Ψins(0)
218
+ Mott/an}bracketri}ht, (6)
219
+ whereftp
220
+ i=fth
221
+ iturns out to be the eigenvector of the
222
+ tij,↑tij,↓matrix with the highest eigenvalue (which is zt↑t↓
223
+ withz= 2d)suchthat/summationtext
224
+ jtij,↑tij,↓ftp
225
+ j=zt↑t↓ftp
226
+ i.Sincethe
227
+ elementaryexcitationsinvolvetwo particlesor two holes, the3
228
+ degeneratedefectstatescannotbeconnectedbyonehopping ,
229
+ but rather require two hoppings to be connected. Therefore,
230
+ oneexpectsthedegeneracytobeliftedatleastatsecondord er
231
+ int↑andt↓, asdiscussednext.
232
+ B. Ground-StateEnergies
233
+ Next, we employ the many-body version of Rayleigh-
234
+ Schr¨ odinger perturbation theory in t↑andt↓with respect to
235
+ the ground state of the system when t↑=t↓= 0, and cal-
236
+ culate the energy of the Mott phase and of its defect states.
237
+ The energy of the Mott state is obtained via nondegenerate
238
+ perturbation theory, and to third order in t↑andt↓it is given
239
+ by
240
+ Eins
241
+ Mott
242
+ M=/summationdisplay
243
+ σUσσ
244
+ 2nσ(nσ−1)+U↑↓n↑n↓−/summationdisplay
245
+ σµσnσ
246
+ −/summationdisplay
247
+ σnσ(nσ+1)zt2
248
+ σ
249
+ Uσσ+O(t4). (7)Thisis anextensivequantity,i.e. Eins
250
+ Mottis proportionalto the
251
+ number of lattice sites M. The odd-order terms in t↑andt↓
252
+ vanishforthe d-dimensionalhypercubiclatticesconsideredin
253
+ thismanuscript,whichissimplybecausetheMott state give n
254
+ in Eq. (2) cannot be connected to itself by only one hopping,
255
+ but ratherrequirestwo hoppingsto be connected. Notice tha t
256
+ Eq. (7) recovers the known result for the single-species BH
257
+ modelwhenoneofthepseudo-spincomponentshavevanish-
258
+ ingfilling,e.g. n↓= 0[19,24].
259
+ Thecalculationofthedefect-stateenergiesismoreinvolv ed
260
+ since it requires using degenerate perturbation theory. As
261
+ mentioned above, when the elementary excitations involve a
262
+ single-σ-particleorasingle- σ-hole,thedegeneracyisliftedat
263
+ firstorderin t↑andt↓. Alengthybutstraightforwardcalcula-
264
+ tionleadstotheenergyofthesingle- σ-particledefectstateup
265
+ tothirdorderin t↑andt↓as
266
+ Esσp
267
+ def=Eins
268
+ Mott+U↑↓n−σ+Uσσnσ−µσ−(nσ+1)ztσ
269
+ −nσ/bracketleftbiggnσ+2
270
+ 2+(nσ+1)(z−3)/bracketrightbiggzt2
271
+ σ
272
+ Uσσ−2n−σ(n−σ+1)U2
273
+ ↑↓
274
+ U2
275
+ −σ−σ−U2
276
+ ↑↓zt2
277
+ −σ
278
+ U−σ−σ
279
+ −nσ(nσ+1)/bracketleftbig
280
+ nσ(z−1)2+(nσ+1)(z−1)(z−4)+(nσ+2)(3z/4−1)/bracketrightbigzt3
281
+ σ
282
+ U2σσ
283
+ −4(nσ+1)n−σ(n−σ+1)U2
284
+ ↑↓
285
+ U2
286
+ −σ−σ−U2
287
+ ↑↓/parenleftBigg
288
+ z−1−U2
289
+ −σ−σ
290
+ U2
291
+ −σ−σ−U2
292
+ ↑↓/parenrightBigg
293
+ ztσt2
294
+ −σ
295
+ U2
296
+ −σ−σ+O(t4), (8)
297
+ where(− ↑)≡↓and vice versa. Here, we assume Uσσ≫tσand{U−σ−σ,|U−σ−σ±U↑↓|} ≫t−σ. Equation(8) is valid for
298
+ alld-dimensionalhypercubiclattices,andit recoversthe know nresult forthesinglespeciesBH modelwhen n−σ= 0[19, 24].
299
+ Note that this expression also recovers the known result for the single species BH model when U↑↓= 0, which provides an
300
+ independentcheckofthe algebra. To thirdorderin t↑andt↓, we obtaina similarexpressionfortheenergyofthe single- σ-hole
301
+ defectstate givenby
302
+ Esσh
303
+ def=Eins
304
+ Mott−U↑↓n−σ−Uσσ(nσ−1)+µσ−nσztσ
305
+ −(nσ+1)/bracketleftbiggnσ−1
306
+ 2+nσ(z−3)/bracketrightbiggzt2
307
+ σ
308
+ Uσσ−2n−σ(n−σ+1)U2
309
+ ↑↓
310
+ U2
311
+ −σ−σ−U2
312
+ ↑↓zt2
313
+ −σ
314
+ U−σ−σ
315
+ −nσ(nσ+1)/bracketleftbig
316
+ (nσ+1)(z−1)2+nσ(z−1)(z−4)+(nσ−1)(3z/4−1)/bracketrightbigzt3
317
+ σ
318
+ U2σσ
319
+ −4nσn−σ(n−σ+1)U2
320
+ ↑↓
321
+ U2
322
+ −σ−σ−U2
323
+ ↑↓/parenleftBigg
324
+ z−1−U2
325
+ −σ−σ
326
+ U2
327
+ −σ−σ−U2
328
+ ↑↓/parenrightBigg
329
+ ztσt2
330
+ −σ
331
+ U2
332
+ −σ−σ+O(t4), (9)
333
+ which is also valid for all d-dimensional hypercubic lattices, and it also recovers the known result for the single-species BH
334
+ modelwhen n−σ= 0orU↑↓= 0[19, 24]. Here, we againassume Uσσ≫tσand{U−σ−σ,|U−σ−σ±U↑↓|} ≫t−σ. We also
335
+ checkedtheaccuracyofthesecond-ordertermsinEqs.(8)an d(9)viaexactsmall-cluster(two-site)calculationswith oneσand
336
+ two−σparticles.
337
+ We note that the mean-field phase boundarybetween the Mott ph ase and its single- σ-particle and single- σ-holedefect states
338
+ canbecalculatedas
339
+ µpar,hol
340
+ σ=Uσσ(nσ−1/2)+U↑↓n−σ−ztσ/2±/radicalbig
341
+ U2σσ/4−Uσσ(nσ+1/2)ztσ+z2t2σ/4. (10)4
342
+ This expression is exact for infinite-dimensionalhypercub iclattices, and it recoversthe knownresult for the single s pecies BH
343
+ model when n−σ= 0orU↑↓= 0[1]. In the d→ ∞limit (while keeping dtσconstant), we checked that our strong-coupling
344
+ perturbationresultsgiveninEqs.(8)and(9)agreewiththi sexactsolutionwhenthelatterisexpandedouttothirdorde rint↑and
345
+ t↓,providinganindependentcheckofthealgebra. Equation(1 0)alsoshowsthat,forinfinite-dimensionallattices,theM ottlobes
346
+ are separatedby U↑↓n−σ, but theirshapesandcritical points(thelatter are obtain edbysetting µpar
347
+ σ=µhol
348
+ σ) are independentof
349
+ U↑↓. This is not the case for finite-dimensional lattices as can b e clearly seen from our results. It is also important to menti on
350
+ herethat boththe shapesandcritical pointsare independen tofthe sign of U↑↓in finite dimensions(at the third-orderpresented
351
+ here)ascanbeseen inEqs.(8) and(9).
352
+ However, when the elementary excitations involve two parti cles or two holes (which occurs when U↑↓is sufficiently large
353
+ and attractive [26]), the degeneracyis lifted at second ord erint↑andt↓. A lengthybut straightforwardcalculationleads to the
354
+ energyofthetwo-particledefectstate uptothirdorderin t↑andt↓as
355
+ Etp
356
+ def=Eins
357
+ Mott+U↑↓(n↑+n↓+1)+/summationdisplay
358
+ σ(Uσσnσ−µσ)+2(n↑+1)(n↓+1)
359
+ U↑↓zt↑t↓
360
+ +/summationdisplay
361
+ σ/bracketleftbigg(nσ+1)2
362
+ U↑↓−nσ(nσ+2)
363
+ 2Uσσ+U↑↓+2nσ(nσ+1)
364
+ Uσσ/bracketrightbigg
365
+ zt2
366
+ σ+O(t4). (11)
367
+ Here, we assume {Uσσ,|U↑↓|,2Uσσ+U↑↓} ≫tσ. Equation (11) is valid for all d-dimensional hypercubiclattices, where the
368
+ odd-ordertermsin t↑andt↓vanish[27]. Tothirdorderin t↑andt↓,weobtainasimilarexpressionfortheenergyofthetwo-hol e
369
+ defectstate givenby
370
+ Eth
371
+ def=Eins
372
+ Mott−U↑↓(n↑+n↓−1)−/summationdisplay
373
+ σ[Uσσ(nσ−1)−µσ]+2n↑n↓
374
+ U↑↓zt↑t↓
375
+ +/summationdisplay
376
+ σ/bracketleftbiggn2
377
+ σ
378
+ U↑↓−(n2
379
+ σ−1)
380
+ 2Uσσ+U↑↓+2nσ(nσ+1)
381
+ Uσσ/bracketrightbigg
382
+ zt2
383
+ σ+O(t4), (12)
384
+ which is also valid for all d-dimensional hypercubic lattices,
385
+ where the odd-order terms in t↑andt↓vanish [27]. Here,
386
+ we again assume {Uσσ,|U↑↓|,2Uσσ+U↑↓} ≫tσ. Since
387
+ the single- σ-particleandsingle- σ-holedefectstateshavecor-
388
+ rections to first order in the hopping, while the two-particl e
389
+ and two-hole defect states have corrections to second order
390
+ in the hopping, the slopes of the Mott lobes are finite as
391
+ {t↑,t↓} →0in the former case, but they vanish in the lat-
392
+ tercase. Hence,theshapeoftheinsulatinglobesareexpect ed
393
+ to be very different for two-particle or two-hole excitatio ns.
394
+ In addition, the chemical-potential widths ( µσ) of all Mott
395
+ lobes are Uσσin the former case, but they [ (µ↑+µ↓)/2] are
396
+ U↑↓+(U↑↑+U↓↓)/2inthelatter.
397
+ We note that in the limit when t↑=t↓=t,U↑↑=U↓↓=
398
+ U0,U↑↓=U′,n↑=n↓=n0,µ↑=µ↓=µ, andz= 2
399
+ (ord= 1), Eq. (12) is in complete agreementwith Eq. (3) of
400
+ Ref. [11], providing an independent check of the algebra. In
401
+ addition, in the limit when t↑=t↓=J,U↑↑=U↓↓=U,
402
+ U↑↓=W≈ −U,n↑=n↓=m, andµ↑=µ↓=µ,
403
+ Eqs. (11) and (12) reduce to those given in Ref. [12] (after
404
+ settingUNN= 0there). However, the terms that are propor-
405
+ tional tot↑t↓are not included in their definitions of the two-
406
+ particle and two-hole excitation gaps. We also checked the
407
+ accuracy of Eqs. (11) and (12) via exact small-cluster (two-
408
+ site) calculationswithoneparticleofeachspecies.
409
+ Wewouldalsoliketoremarkinpassingthattheenergydif-
410
+ ferencebetweentheMottphaseanditsdefectstatesdetermi ne
411
+ the phase boundaryof the particle and hole branches. This is
412
+ because at the point where the energy of the incompressiblestate becomes equal to its defect state, the system becomes
413
+ compressible, assuming that the compressibility approach es
414
+ zero continuously at the phase boundary. While Eins
415
+ Mottand
416
+ its defects Esσp
417
+ def,Esσh
418
+ def,Etp
419
+ defandEth
420
+ defdepend on the lattice
421
+ sizeM, their difference do not. Therefore, the chemical po-
422
+ tentialsthatdeterminetheparticleandholebranchesarei nde-
423
+ pendentof Mat thephaseboundaries. Thisindicatesthat the
424
+ numerical Monte Carlo simulations should not have a strong
425
+ dependenceon M.
426
+ It is known that the third-order strong-coupling expansion
427
+ isnotveryaccuratenearthetipoftheMottlobes,as t↑andt↓
428
+ arenotverysmallthere[19,24]. Forthisreason,anextrapo la-
429
+ tion technique is highly desirable to determine more accura te
430
+ phase diagrams. Therefore, having discussed the third-ord er
431
+ strong-coupling expansion for a general two-species Bose-
432
+ Bose mixtures with arbitary hoppings tσ, interactions Uσσ′,
433
+ densities nσ, and chemical potentials µσ, next we show how
434
+ todevelopa scalingtheory.
435
+ IV. EXTRAPOLATIONTECHNIQUE
436
+ In this section, we propose a chemical potential extrapo-
437
+ lation technique based on scaling theory to extrapolate our
438
+ third-orderpower-seriesexpansionintoafunctionalform that
439
+ is appropriate for the entire Mott lobes. It is known that the
440
+ criticalpointatthetipofthelobeshasthescalingbehavio rof
441
+ a(d+1)-dimensional XYmodel,andthereforethelobeshave5
442
+ Kosterlitz-Thouless shapes for d= 1and power-law shapes
443
+ ford >1. For illustration purposes, here we analyze only
444
+ the latter case, but this techniquecan be easily adapted to t he
445
+ d= 1case [19].
446
+ A. ScalingAnsatz
447
+ Fromnowonwe considera two-speciesmixturewith t↑=
448
+ t↓=t,U↑↑=U↓↓=U,U↑↓=V,n↑=n↓=n, and
449
+ µ↑=µ↓=µ. Whend >1, we proposethe followingansatz
450
+ which includes the known power-law critical behavior of the
451
+ tipofthe lobes
452
+ µ±
453
+ U=A(x)±B(x)(xc−x)zν, (13)
454
+ whereA(x) =a+bx+cx2+dx3+···andB(x) =α+βx+
455
+ γx2+δx3+···areregularfunctionsof x= 2dt/U,xcisthe
456
+ critical point which determines the location of the lobes, a nd
457
+ zνis the critical exponent for the ( d+ 1)-dimensional XY
458
+ model which determines the shape of the lobes near xc=
459
+ 2dtc/U. In Eq. (13), the plus sign correspondsto the particle
460
+ branch, and the minus sign corresponds to the hole branch.
461
+ Theformoftheansatzistakentobethesameforbothsingle-
462
+ and two-partice (or single- and two-hole) excitations, but the
463
+ parametersareverydifferent.
464
+ The parameters a,b,candddepend on U,Vandn, and
465
+ they are determined by matching them with the coefficients
466
+ givenbyourthird-orderexpansionsuchthat A(x) = (µpar+
467
+ µhol)/(2U).Here,µparandµholare our strong-couplingex-
468
+ pansion results determined from Eqs. (8) and (9) for the
469
+ single-particle and single-hole excitations, or from Eqs. (11)
470
+ and(12)forthetwo-particleandtwo-holeexcitations,res pec-
471
+ tively. Writing our strong-coupling expansion results for the
472
+ particleandhole branchesin the form µpar=U/summationtext3
473
+ n=0e+
474
+ nxn
475
+ andµhol=U/summationtext3
476
+ n=0e−
477
+ nxn, leads to a= (e+
478
+ 0+e−
479
+ 0)/2,
480
+ b= (e+
481
+ 1+e−
482
+ 1)/2,c= (e+
483
+ 2+e−
484
+ 2)/2, andd= (e+
485
+ 3+e−
486
+ 3)/2.
487
+ To determine the U,Vandndependence of the parameters
488
+ α,β,γ,δ,xcandzν, we first expand the left hand side of
489
+ B(x)(xc−x)zν= (µpar−µhol)/(2U)in powers of x, and
490
+ matchthecoefficientswiththecoefficientsgivenbyourthir d-
491
+ orderexpansion,leadingto
492
+ α=e+
493
+ 0−e−
494
+ 0
495
+ 2xzνc, (14)
496
+ β
497
+ α=zν
498
+ xc+e+
499
+ 1−e−
500
+ 1
501
+ e+
502
+ 0−e−
503
+ 0, (15)
504
+ γ
505
+ α=zν(zν+1)
506
+ 2x2c+zν
507
+ xce+
508
+ 1−e−
509
+ 1
510
+ e+
511
+ 0−e−
512
+ 0+e+
513
+ 2−e−
514
+ 2
515
+ e+
516
+ 0−e−
517
+ 0,(16)
518
+ δ
519
+ α=zν(zν+1)(zν+2)
520
+ 6x3c+zν(zν+1)
521
+ 2x2ce+
522
+ 1−e−
523
+ 1
524
+ e+
525
+ 0−e−
526
+ 0
527
+ +zν
528
+ xce+
529
+ 2−e−
530
+ 2
531
+ e+
532
+ 0−e−
533
+ 0+e+
534
+ 3−e−
535
+ 3
536
+ e+
537
+ 0−e−
538
+ 0. (17)
539
+ We fixzνat its well-known values such that zν≈2/3for
540
+ d= 2andzν= 1/2ford >2. If the exact value of xcis known via other means, e.g. numerical simulations, α,β,
541
+ γandδcan be calculated accordingly, for which the extrap-
542
+ olation technique gives very accurate results [23, 25]. If t he
543
+ exact value of xcis not known, then we set δ= 0, and solve
544
+ Eqs. (14), (15), (16) and the δ= 0equation to determine
545
+ α,β,γandxcself-consistently, which also leads to accurate
546
+ results [19, 24]. Next we present typical ground-state phas e
547
+ diagrams for (d= 2)- and (d= 3)-dimensional hypercubic
548
+ latticesobtainedfromthisextrapolationtechnique.
549
+ B. Numerical Results
550
+ In Figs. 1 and 2, the results of the third-order strong-
551
+ couplingexpansion(dottedlines)arecomparedtothoseoft he
552
+ extrapolationtechnique(hollowpink-squaresandsolidbl ack-
553
+ circles) when V= 0.5UandV=−0.85U, respectively, in
554
+ two (d= 2orz= 4) andthree ( d= 3orz= 6) dimensions.
555
+ We recall here that t↑=t↓=t,U↑↑=U↓↓=U,U↑↓=V,
556
+ n↑=n↓=n, andµ↑=µ↓=µ.
557
+ In Fig. 1, we show the chemical potential µ(in units of U)
558
+ versusx= 2dt/Uphasediagramfor(a)two-dimensionaland
559
+ (b) three-dimensional hypercubic lattices, where we choos e
560
+ the interspecies interaction to be repulsive V= 0.5U. Com-
561
+ paring Eqs. (8) and (9) with Eqs. (11) and (12), we expect
562
+ that the excited state of the system to be the usual superfluid
563
+ for allV >0for allt. The dotted lines correspond to phase
564
+ boundary for the Mott insulator to superfluid state as deter-
565
+ mined from the third-order strong-coupling expansion, and
566
+ the hollow pink-squares correspond to the extrapolation fit s
567
+ forthesingle-particleandsingle-holeexcitationsdiscu ssedin
568
+ the text. We recall here that an incompressible super-count er
569
+ flow phase [7–9, 13] also exists outside of the Mott insulator
570
+ lobes, but our current formalism cannot be used to locate its
571
+ phaseboundary.
572
+ TABLE I. List of the critical points (location of the tips) xc=
573
+ 2dtc/Ufor the first two Mott insulator lobes that are found from
574
+ the chemical potential extrapolation technique described in the text.
575
+ Here,t↑=t↓=t,U↑↑=U↓↓=U,U↑↓=V,n↑=n↓=n, and
576
+ µ↑=µ↓=µ. These critical points for the single-particle or single-
577
+ hole excitations are determined from Eqs. (8) and (9), and th ey tend
578
+ tomove inas Vincreases, andare independent of the signof V.
579
+ d= 2 d= 3
580
+ V/Un= 1n= 2n= 1n= 2
581
+ 0.00.234 0.138 0.196 0.116
582
+ 0.10.234 0.138 0.196 0.115
583
+ 0.20.233 0.137 0.195 0.115
584
+ 0.30.230 0.136 0.194 0.114
585
+ 0.40.227 0.134 0.193 0.113
586
+ 0.50.223 0.131 0.190 0.112
587
+ 0.60.217 0.128 0.187 0.110
588
+ 0.70.208 0.123 0.182 0.107
589
+ 0.80.197 0.116 0.174 0.102
590
+ 0.90.193 0.113 0.163 0.0956
591
+ 0 1.5 3 4.5
592
+ 0 0.09 0.18 0.27µ/U
593
+ x = 2dt/U(a) Two dimensions (V=0.5U)
594
+ n=1n=2n=3sp/sh ext
595
+ third order
596
+ 0 1.5 3 4.5
597
+ 0 0.09 0.18 0.27µ/U
598
+ x = 2dt/U(a) Two dimensions (V=0.5U)
599
+ sp/sh ext
600
+ third order
601
+ 0 1.5 3 4.5
602
+ 0 0.09 0.18 0.27µ/U
603
+ x = 2dt/U(b) Three dimensions (V=0.5U)
604
+ n=1n=2n=3sp/sh ext
605
+ third order
606
+ 0 1.5 3 4.5
607
+ 0 0.09 0.18 0.27µ/U
608
+ x = 2dt/U(b) Three dimensions (V=0.5U)
609
+ sp/sh ext
610
+ third order
611
+ FIG. 1. (Color online) Chemical potential µ(in units of U) versus
612
+ x= 2dt/Uphase diagram for (a) two- and (b) three-dimensional
613
+ hypercubic lattices with t↑=t↓=t,U↑↑=U↓↓=U,U↑↓=
614
+ V= 0.5U,n↑=n↓=n, andµ↑=µ↓=µ. The dotted lines
615
+ correspond to phase boundary for the Mott insulator to super fluid
616
+ state as determined from the third-order strong-coupling e xpansion,
617
+ and the hollow pink-squares to the extrapolation fit for the s ingle-
618
+ particle or single-hole excitations discussed in the text. Recall that
619
+ anincompressiblesuper-counterflowphasealsoexistsouts ideofthe
620
+ Mott insulator lobes.
621
+ Att= 0, the chemical potential width of all Mott lobes
622
+ areU(similar to the single-species BH model), but they are
623
+ separated from each other by Vas a function of µ. Astin-
624
+ creasesfromzero,therangeof µaboutwhichthegroundstate
625
+ is a Mott insulator decreases, and the Mott insulator phasedisappears at a critical value of t, beyond which the system
626
+ becomes a superfluid. In addition, similar to what was found
627
+ forthesingle-speciesBH model[19,24],thestrong-coupli ng
628
+ expansionoverestimatesthe phase boundaries,and it leads to
629
+ unphysical pointed tips for all Mott lobes, which is expecte d
630
+ since a finite-order expansion cannot describe the physics o f
631
+ thecriticalpointcorrectly. Ashortlistof V/Uversusthecrit-
632
+ ical points xc= 2dtc/Uis presented for the first two Mott
633
+ insulator lobes in Table I, where it is shown that the criti-
634
+ cal points tend to move in as Vincreases. This is because
635
+ presence of a second species (say −σones) screens the on-
636
+ site intraspeciesrepulsion Uσσbetweenσ-species, and hence
637
+ increasesthesuperfluidregion.
638
+ In Fig. 2, we show the chemical potential µ(in units of
639
+ U)versusx= 2dt/Uphasediagramfor(a) two-dimensional
640
+ and (b) three-dimensionalhypercubiclattices, where in th ese
641
+ figures we choose the interspecies interaction to be attract ive
642
+ V=−0.85U. Comparing Eqs. (8) and (9) with Eqs. (11)
643
+ and (12), we expect that the excited state of the system to
644
+ be a paired superfluid for all V <0whent→0. This is
645
+ clearlyseen inthefigurewherethedottedlinescorrespondt o
646
+ phaseboundaryfortheMottinsulatortosuperfluidstateasd e-
647
+ termined from the third-orderstrong-couplingexpansion, the
648
+ hollow pink-squares correspond to the extrapolation fits fo r
649
+ thesingle-particleandsingle-holeexcitations(shownon lyfor
650
+ illustration purposes), and the solid black-circles corre spond
651
+ to the extrapolation fits for the two-particle and two-hole e x-
652
+ citations(thisisthe expectedtransition)discussedin th etext.
653
+ Att= 0, the chemical potential width of all Mott lobes
654
+ areV+U= 0.15U, which is in contrast with the single-
655
+ species BH model. As tincreases from zero, the range of µ
656
+ aboutwhichthegroundstateisaMottinsulatordecreaseshe re
657
+ as well, and the Mott insulator phase disappears at a critica l
658
+ value oft, beyondwhich the system becomesa paired super-
659
+ fluid. The strong-couplingexpansionagain overestimatest he
660
+ phaseboundaries,anditagainleadstounphysicalpointedt ips
661
+ for all Mott lobes. In addition, a short list of V/Uversus the
662
+ critical points xc= 2dtc/Uare presented for the first two
663
+ MottinsulatorlobesinTableI. Ourresultsareconsistentw ith
664
+ the expectation that, for small V, the locations of the tips in-
665
+ crease as a function of V, because the presence of a nonzero
666
+ Viswhatallowedthesestatestoforminthefirstplace. How-
667
+ ever, when Vis largerthan some critical value ( ∼0.6U), the
668
+ locationsofthetipsdecrease,andtheyeventuallyvanishw hen
669
+ V=−U. Thismay indicatean instabilitytowardsa collapse
670
+ sinceat thispoint U↑↑U↓↓is exactlyequalto U2
671
+ ↑↓.
672
+ Compared to the V >0case shown in Fig. 1, note that
673
+ shape of the Mott insulator to paired superfluidphase bound-
674
+ ary is very different, showing a re-entrant behavior in all d i-
675
+ mensions from paired superfluid to Mott insulator and again
676
+ to a paired superfluid phase, as a function of t. Our results
677
+ are consistent with an early numerical time-evolving block
678
+ decimation (TEBD) calculation [11], where such a re-entran t
679
+ quantumphasetransitionin onedimensionwaspredicted.
680
+ The re-entrant quantum phase transition occurs when co-
681
+ efficient of the hopping term in Eq. (12) is negative [so7
682
+ -0.45-0.3-0.15 0
683
+ 0 0.1 0.2 0.3 0.4µ/U
684
+ x = 2dt/U(a) Two dimensions (V=-0.85U)
685
+ n=1n=2n=3tp/th ext
686
+ sp/sh ext
687
+ third order
688
+ -0.45-0.3-0.15 0
689
+ 0 0.1 0.2 0.3 0.4µ/U
690
+ x = 2dt/U(a) Two dimensions (V=-0.85U)
691
+ n=1n=2n=3tp/th ext
692
+ sp/sh ext
693
+ third order
694
+ -0.45-0.3-0.15 0
695
+ 0 0.1 0.2 0.3 0.4µ/U
696
+ x = 2dt/U(b) Three dimensions (V=-0.85U)
697
+ n=1n=2n=3tp/th ext
698
+ sp/sh ext
699
+ third order
700
+ -0.45-0.3-0.15 0
701
+ 0 0.1 0.2 0.3 0.4µ/U
702
+ x = 2dt/U(b) Three dimensions (V=-0.85U)
703
+ n=1n=2n=3tp/th ext
704
+ sp/sh ext
705
+ third order
706
+ FIG. 2. (Color online) Chemical potential µ(in units of U) versus
707
+ x= 2dt/Uphase diagram for (a) two- and (b) three-dimensional
708
+ hypercubic lattices with t↑=t↓=t,U↑↑=U↓↓=U,U↑↓=
709
+ V=−0.85U,n↑=n↓=n, andµ↑=µ↓=µ. The dotted lines
710
+ correspond to phase boundary for the Mott insulator to super fluid
711
+ statedeterminedfromthethird-order strong-coupling exp ansion, the
712
+ hollow pink-squares to the extrapolation fit for the single- particle or
713
+ single-hole excitations (shown only for illustration purp oses), and
714
+ the solid black-circles to the extrapolation fit for the two- particle or
715
+ two-hole excitations (the expected transition) discussed inthe text.
716
+ that the two-hole excitation branch has a negative slope in
717
+ (µ↑+µ↓)/2versustσphase diagram when tσ→0], i.e.
718
+ −(2n↑n↓/U↑↓)zt↑t↓−/summationtext
719
+ σ[n2
720
+ σ/U↑↓−(n2
721
+ σ−1)/(2Uσσ+
722
+ U↑↓)+2nσ(nσ+1)/Uσσ]zt2
723
+ σterm,whichoccursforthefirst
724
+ few Mott lobes beyond a critical U↑↓. When this coefficient
725
+ is negative, its value is most negative for the first Mott lobe ,TABLE II. List of the critical points (location of the tips) xc=
726
+ 2dtc/Uthat are found from the chemical potential extrapolation
727
+ techniquedescribedinthetext. Here, t↑=t↓=t,U↑↑=U↓↓=U,
728
+ U↑↓=V,n↑=n↓=n, andµ↑=µ↓=µ. These critical
729
+ points for the two-particle or two-hole excitations are det ermined
730
+ from Eqs. (11) and (12) when V <0. Note that, for small V,xc’s
731
+ tend to increase as a function of V, since the presence of a nonzero
732
+ Vis what allowed these states to form in the first place. Howeve r,
733
+ xc’s decrease beyond a critical V, and they eventually vanish when
734
+ V=−U,which mayindicate an instabilitytowards a collapse.
735
+ d= 2 d= 3
736
+ V/Un= 1n= 2n= 1n= 2
737
+ -0.010.0543 0.0337 0.0611 0.0379
738
+ -0.030.0937 0.0582 0.105 0.0655
739
+ -0.050.121 0.0749 0.136 0.0843
740
+ -0.070.142 0.0883 0.160 0.0994
741
+ -0.10.169 0.105 0.190 0.118
742
+ -0.20.233 0.145 0.262 0.164
743
+ -0.30.277 0.173 0.311 0.195
744
+ -0.40.307 0.193 0.345 0.217
745
+ -0.50.325 0.205 0.366 0.230
746
+ -0.60.331 0.209 0.372 0.235
747
+ -0.70.321 0.203 0.362 0.228
748
+ -0.80.291 0.183 0.327 0.206
749
+ -0.90.225 0.141 0.253 0.159
750
+ -0.930.193 0.121 0.217 0.136
751
+ -0.950.166 0.103 0.187 0.116
752
+ -0.970.1304 0.0812 0.147 0.0913
753
+ -0.990.0764 0.0474 0.0860 0.0534
754
+ and thereforethe effect is strongest there. However,the co ef-
755
+ ficientincreasesandeventuallybecomespositiveasafunct ion
756
+ offilling,andthusthere-entrantbehaviorbecomesweakera s
757
+ fillingincreases,anditeventuallydisappearsbeyondacri tical
758
+ filling. For the parametersused in Fig. 2, this occursonlyfo r
759
+ the first lobe, as can be seen in the figures. We also note that
760
+ the sign of this coefficientis independentof the dimensiona l-
761
+ ity of the lattice, since z= 2dentersinto the coefficient only
762
+ asanoverallfactor.
763
+ What happenswhen t↑/ne}ationslash=t↓and/orU↑↑/ne}ationslash=U↓↓? We donot
764
+ expectany qualitativechangefor attractiveinterspecies inter-
765
+ actions. However, for repulsive interspecies interaction s, this
766
+ lifts the degeneracyof the single-particle or single-hole exci-
767
+ tation energies. While the transition is from a double Mott
768
+ insulator to a double superfluid of both species in the degen-
769
+ erate case, it is from a double-Mott insulator of both specie s
770
+ toaMottinsulatorofonespeciesandasuperfluidoftheother
771
+ inthenondegeneratecase.
772
+ V. CONCLUSIONS
773
+ We analyzed the zero temperature phase diagram of the
774
+ two-species Bose-Hubbard (BH) model with on-site boson-
775
+ boson interactions in d-dimensional hypercubic lattices, in-8
776
+ cluding both the repulsive and attractive interspecies in-
777
+ teraction. We used the many-body version of Rayleigh-
778
+ Schr¨ odinger perturbation theory in the kinetic energy ter m
779
+ with respect to the ground state of the system when the ki-
780
+ netic energy term is absent, and calculate ground state ener -
781
+ gies needed to carry out our analysis. This technique was
782
+ previously used to discuss the phase diagram of the single-
783
+ speciesBH model[19–21, 23], extendedBH model[24],and
784
+ of the hardcore BH model with a superlattice [25], and its
785
+ resultsshowedanexcellentagreementwithMonteCarlosim-
786
+ ulations [23, 25]. Motivated by the success of this techniqu e
787
+ with these models, here we generalized it to the two-species
788
+ BH model, hoping to develop an analytical approach which
789
+ couldbeasaccurateasthe numericalones.
790
+ We derived analytical expressions for the phase boundary
791
+ betweentheincompressibleMottinsulatorandthecompress -
792
+ iblesuperfluidphaseuptothirdorderinthehoppings. Weals o
793
+ proposed a chemical potential extrapolation technique bas ed
794
+ on the scaling theory to extrapolateour third-orderpower s e-
795
+ riesexpansionintoafunctionalformthatisappropriatefo rthe
796
+ Mott lobes. In particular, when the interspecies interacti on is
797
+ sufficiently large and attractive, we found a re-entrant qua n-
798
+ tum phase transition from paired superfluid (superfluidity o f
799
+ compositebosons,i.e. Bose-Bosepairs)toMottinsulatora nd
800
+ again to a paired superfluid in all one, two and three dimen-sions. SincetheavailableMonteCarlocalculations[9,10] do
801
+ not provide the Mott insulator to superfluid transition phas e
802
+ boundary in the experimentally more relevant chemical po-
803
+ tentialversushoppingplane,wecouldnotcompareourresul ts
804
+ with them. This comparison is highly desirable to judge the
805
+ accuracyofourstrong-couplingexpansionresults.
806
+ A possible direction to extend this work is to consider the
807
+ limit where hopping of one-species is much larger than the
808
+ other. In this limit, the two-species BH model reduces to
809
+ theBose-BoseversionoftheFalicov-Kimballmodel[28],th e
810
+ Fermi-Fermi version of which has been widely discussed in
811
+ the condensed-matter literature and the Fermi-Bose versio n
812
+ has just been studied [29]. It is known for such models that
813
+ thereisa tendencytowardsbothphaseseparationanddensit y
814
+ wave order [30], which requires a new calculation partially
815
+ similar to that of Ref. [24]. One can also examine how the
816
+ momentumdistributionchangeswiththehoppingintheinsu-
817
+ latingphases[23, 31], whichhasdirect relevanceto ultrac old
818
+ atomicexperiments.
819
+ VI. ACKNOWLEDGMENTS
820
+ The author thanks Anzi Hu, L. Mathey and J. K. Freer-
821
+ icksfordiscussions,andTheScientificandTechnologicalR e-
822
+ searchCouncilofTurkey(T ¨UB˙ITAK)forfinancialsupport.
823
+ [1] M.P.A.Fisher,P.B.Weichman,G.Grinstein,andD.S.Fis her,
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+ Phys.Rev. B 40, 546(1989).
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+ [2] M. Greiner, O. Mandel, T. Esslinger, T. W. H¨ ansch, and I.
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+ Bloch, Nature (London) 415, 39(2002).
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+ [10] A. Isacsson, Min-Chul Cha, K. Sengupta, and S. M. Girvin ,
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+ [11] A.Arg¨ uelles andL.Santos, Phys.Rev. A 75, 053613 (2007).
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+ [12] C. Trefzger, C. Menotti, and M. Lewenstein, Phys. Rev. L ett.
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+ [13] Anzi Hu, L. Mathey, I. Danshita, E. Tiesinga, C. J. Willi ams,
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+ [14] P.Buonsante, S.M. Giampaolo, F.Illuminati, V. Penna, and A.
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+ Vezzani, Eur.Phys.J.B 68, 427 (2009).
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+ [15] A. Hubener, M. Snoek, and W. Hofstetter, Phys. Rev. B 80,
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+ 245109 (2009).
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+ [16] C.Menotti and S.Stringari,Phys.Rev. A 81, 045604 (2010).[17] J. Catani, L. De Sarlo, G. Barontini, F. Minardi, and M. I ngus-
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+ cio, Phys.Rev. A 77, 011603(R) (2008).
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+ [18] G. Thalhammer, G. Barontini, L. De Sarlo, J. Catani, F. M i-
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+ nardi, and M. Inguscio, Phys. Rev. Lett. 100, 210402 (2008).
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+ [19] J. K.Freericks andH. Monien, Phys.Rev. B 53, 2691 (1996).
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+ [20] T.D.K¨ uhnerandH.Monien, Phys.Rev.B, 58,R14741(1998).
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+ [21] P. Buonsante, V. Penna, and A. Vezzani, Phys. Rev. B 70,
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+ 184520 (2004).
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+ [22] K. Sengupta andN.Dupuis, Phys.Rev. A 71, 033629 (2005).
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+ [23] J. K. Freericks, H. R. Krishnamurthy, Y. Kato, N. Kawash ima,
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+ and N.Trivedi, Phys.Rev. A 79, 053631 (2009).
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+ [24] M. Iskinand J.K. Freericks,Phys.Rev. A 79, 053634 (2009).
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+ [25] Itay Hen, M. Iskin, and M. Rigol, Phys. Rev. B 81, 064503
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+ (2010).
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+ [26] Recallthat U2
867
+ ↑↓cannot be greaterthanorequalto U↑↑U↓↓,oth-
868
+ erwise the mixture would be unstable against collapse. In ad di-
869
+ tion, see e.g. Fig. 7 in [13], where TEBD calculations show in
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+ one dimension that V/lessorsimilar−0.06Uis already sufficient for the
871
+ Mott insulator topaired superfluidtransition.
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+ [27] Note that, unlike those of single-particle and single- hole exci-
873
+ tations where dtσis a constant when d→ ∞, in the case of
874
+ two-particle and two-hole excitations, dt2
875
+ σmust be kept con-
876
+ stant when d→ ∞. In this respect, Eqs. (11) and (12) do not
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+ contain any finite- dcorrectionat the second order inhopping.
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+ [28] L. M. Falicov and J. C. Kimball, Phys. Rev. Lett. 22, 997
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+ (1969).
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+ [29] M. Iskin and J. K. Freericks, Phys. Rev. A 80, 053623 (2009);
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+ and see references therein.
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+ [30] S ¸. G. S¨ oyler, B. Capogrosso-Sansone, N. V. Prokof’ev , and B.
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+ V. Svistunov, New J. Phys. 11, 073036 (2009).
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+ [31] M. Iskinand J.K. Freericks,Phys.Rev. A 80, 063610 (2009).
1001.0022.txt ADDED
@@ -0,0 +1,651 @@
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
1
+ arXiv:1001.0022v2 [hep-ph] 17 Mar 2010Preprint typeset in JHEP style - HYPER VERSION MADPH–09-1552
2
+ µτProduction at Hadron Colliders
3
+ Tao Han∗, Ian Lewis†
4
+ Department of Physics, University of Wisconsin, Madison, W I 53706, U.S.A.
5
+ Marc Sher‡
6
+ Particle Theory Group, College of William and Mary, William sburg, Virginia 23187
7
+ Abstract: Motivated by large νµ−ντflavor mixing, we consider µτproduction at hadron
8
+ colliders via dimension-6 effective operators, which can be a ttributed to new physics in the
9
+ flavor sector at a higher scale Λ. Current bounds on many of the se operators from low energy
10
+ experiments are very weak or nonexistent, and they may lead t o cleanµ+τ−andµ−τ+signals
11
+ at hadron colliders. At the Tevatron with 8 fb−1, one can exceed current bounds for most
12
+ operators, with most 2 σsensitivities being in the 6 −24 TeV range. We find that at the LHC
13
+ with 1 fb−1(100 fb−1) integrated luminosity, one can reach a2 σsensitivity for Λ ∼3−10 TeV
14
+ (Λ∼6−21 TeV), depending on the Lorentz structure of the operator. For some operators,
15
+ an improvement of several orders of magnitude in sensitivit y can be obtained with only a few
16
+ tens of pb−1at the LHC.
17
+ Keywords: Lepton flavor physics; Hadron collider phenomenology..
18
19
20
21
+ 1. Introduction 1
22
+ 2.µτProduction at Hadron Colliders 3
23
+ 3. Signal Identification and Backgrounds 4
24
+ 3.1τDecay to Electrons 5
25
+ 3.1.1 Signal Reconstruction 5
26
+ 3.1.2 Backgrounds and their Suppression 6
27
+ 3.2τDecay to Hadrons 9
28
+ 3.3 Sensitivity Reach at the Tevatron 10
29
+ 3.4 Sensitivity Reach at the LHC 10
30
+ 4. Discussions and Conclusions 13
31
+ A. New Physics Bounds 14
32
+ B. Partial Wave Unitarity Bounds 14
33
+ 1. Introduction
34
+ The most important discovery in particle physics in the past decade has only deepened the
35
+ mystery of “flavor” of quarks and leptons. The fact that the mi xing angles in the leptonic
36
+ sector are large [1, 2] stands in sharp contrast with the obse rved small mixing angles in the
37
+ quarksector. Inparticular, mixingbetweenthesecondandt hirdgeneration neutrinosappears
38
+ to be maximal. Of course, this large mixing could occur from d iagonalizing the neutrino mass
39
+ matrix, the charged lepton mass matrix, or both. At present, the source of this large mixing
40
+ is a mystery.
41
+ In view of this, it is tempting to explore other interactions which change lepton flavor
42
+ between the second and third generations. Several years ago , two of us (TH, MS), along with
43
+ Black and He (BHHS) [3], performed a comprehensive analysis of constraints on these inter-
44
+ actions based on low energy meson physics. BHHS chose an effect ive field theory approach,
45
+ in which all dimension-6 operators of the form
46
+ (¯µΓτ)(¯qαΓqβ), (1.1)
47
+ – 1 –were studied, where Γ contains possible Dirac γ-matrices. With six flavors of quarks, there
48
+ were 12 possible combinations of qaandqb(assuming Hermiticity), six diagonal and six off-
49
+ diagonal, and four choices S,P,V,A of the gamma matrices were considered. All of these
50
+ operators were considered, and most were bounded by conside ringτ,K,Bandtdecays.
51
+ In particular, BHHS considered operators of the form
52
+ ∆L= ∆L(6)
53
+ τµ=/summationdisplay
54
+ j,α,βCj
55
+ αβ
56
+ Λ2(µΓjτ)/parenleftBig
57
+ qαΓjqβ/parenrightBig
58
+ + H.c., (1.2)
59
+ where Γ j∈(1, γ5, γσ, γσγ5) denotes relevant Dirac matrices, specifying scalar, pseu doscalar,
60
+ vector and axial vector couplings, respectively. They did n ot consider tensor operators since
61
+ the hadronic matrix elements were not known and the bounds we re expected to be weak in
62
+ any event. They chose a value of
63
+ Cj
64
+ αβ= 4πO(1) (default) , (1.3)
65
+ which corresponds to an underlying theory with a strong gaug e coupling of αS=O(1).
66
+ Arguments can be made for multiplying or dividing this by 4 π, for naive dimensional analysis
67
+ or for weakly coupled theories, respectively. A discussion is found in BHHS; we simply choose
68
+ the above definition of Λ and other choices can be made by simpl e rescaling.
69
+ Besides the four fermion operators in Eq. (1.2), there may be other induced operators
70
+ involving the SM gauge bosons, such as the electroweak trans ition operator
71
+ ∆L=κv
72
+ Λ2¯µσµντFµν, (1.4)
73
+ wherevis the vacuum expectation value of the Standard Model Higgs fi eld andFµνis the
74
+ electroweak field tensor. However, when these operators are compared to the underlying
75
+ new strong dynamics of the four fermion interaction in Eq. (1 .2), it is found that they are
76
+ suppressed by O(MW/Λ), where MWis the mass of the electroweak gauge boson. For new
77
+ physics scales of order 1 TeV or greater, this is at least an or der of magnitude suppression.
78
+ Thus, we ignore these operators.
79
+ BHHS found that operators involving the three lightest quar ks were strongly bounded,
80
+ with bounds ranging from 3 to 13 TeV on the related value of Λ. T hese bounds can be found
81
+ in Appendix A. Not surprisingly, operators involving the to p quark were either unbounded or
82
+ very weakly bounded, with only the tuoperator for vector and axial vector couplings being
83
+ bounded by Λ <650 GeV (the bound arises through a loop in B→µτdecay). Operators
84
+ involving the b-quark and a light quark also have bounds on Λ which were gener ally in
85
+ the several TeV range. However, there were some surprises. T he scalar and pseudoscalar
86
+ operators involving cuandccwere completely unbounded, and the bboperator was essentially
87
+ unbounded for all S,P,V,A operators. And, as noted above, noneof the tensor operators
88
+ were considered at all, for all quark combinations.
89
+ In this note, we point out that the operators in Eq. (1.1) (wit hout involving top quarks)
90
+ will contribute to µ−τproduction at hadron colliders. Given that many of the possi ble
91
+ – 2 –operators, as noted above, are completely unbounded or weak ly bounded from the current
92
+ low energy data, study of pp→µτat the LHC or pp→µτat the Tevatron will probe
93
+ unexplored territory.
94
+ There have been some previous discussions of µ−τproduction at hadron colliders. Han
95
+ and Marfatia [4] looked at the lepton-violating decay h→µτat hadron colliders, and a very
96
+ detailed analysis of signals and backgrounds was carried ou t by Assamagan et al. [5] after-
97
+ wards. Other work looking at Higgs decays focused on mirror f ermions [6], supersymmetric
98
+ models [7], seesaw neutrino models [8], and Randall-Sundru m models [9]. In addition to
99
+ Higgs decays, others have considered lepton-flavor violati on in the decays of supersymmetric
100
+ particles [10] and in horizontal gauge boson models [11]. Th ese analyses, however, were done
101
+ in the context of very specific models (often relying on the as sumption that the µandτare
102
+ emitted in the decay of a single particle). Here, we will use a much more general effective
103
+ field theory approach.
104
+ This paper is organized as follows. In the next section, we di scuss the cross sections
105
+ forµτproduction via the various operators. A detailed analysis o f the signal identification
106
+ and background subtraction is in Section 3, and Section 4 con tains some discussions and our
107
+ conclusions. Appendix A reiterates the bounds from BHHS for comparison, and Appendix B
108
+ outlines the calculation of partial-wave unitarity bounds .
109
+ 2.µτProduction at Hadron Colliders
110
+ Dueto the absenceof appreciable µτproductionin theSM, their production can beestimated
111
+ via the effective operators in Eq. (1.1). On dimensional groun ds, the cross section for ¯ qiqj→
112
+ µτgrows with center of mass energy, i.e.,
113
+ σ(¯qiqj→µτ)∝s
114
+ Λ4, (2.1)
115
+ where√sis the center of mass energy for the partonic system. This gro wth of cross section
116
+ with energy will eventually violate unitarity bounds. Expa nding the scattering amplitudes in
117
+ partial waves, we find the unitarity bounds to be (see Appendi x B)
118
+ s≤/braceleftBigg
119
+ 2Λ2for scalar ,pseudoscalar ,and tensor;
120
+ 3Λ2vector and axial vector case .(2.2)
121
+ The total cross sections for µτproduction at the hadronic level after convoluting with
122
+ the parton distribution functions (pdfs) are
123
+ σScalar=π
124
+ 3S
125
+ Λ4/integraldisplayτmax
126
+ τ0dτ(q⊗q)(τ)/parenleftbigg
127
+ 1−τ0
128
+ τ/parenrightbigg2
129
+ τ (2.3)
130
+ σVector=4π
131
+ 9S
132
+ Λ4/integraldisplayτmax
133
+ τ0dτ(q⊗q)(τ)/parenleftbigg
134
+ 1−τ0
135
+ τ/parenrightbigg2/parenleftbigg
136
+ 1+τ0
137
+ 2τ/parenrightbigg
138
+ τ (2.4)
139
+ σTensor=8π
140
+ 9S
141
+ Λ4/integraldisplayτmax
142
+ τ0dτ(q⊗q)(τ)/parenleftbigg
143
+ 1−τ0
144
+ τ/parenrightbigg2/parenleftbigg
145
+ 1+2τ0
146
+ τ/parenrightbigg
147
+ τ, (2.5)
148
+ – 3 –whereτ=s/S,τ0=m2
149
+ τ/S,mτis the tau mass, and√
150
+ Sis the center of mass energy in the
151
+ lab frame. The pseudoscalar cross section is of the same form as the scalar cross section, and
152
+ the axial vector cross section is of the same form as the vecto r cross section. Our perturbative
153
+ calculation will become invalid at the unitarity bound, hen ce there is a maximum on the τ
154
+ integration. It is given by τmax= 2Λ2/Sfor the scalar, pseudoscalar, and tensor cases, and
155
+ τmax= 3Λ2/Sfor the vector and axial-vector cases. Also, q(x) is the quark distribution
156
+ function with flavor sum suppressed, and ⊗denotes the convolution defined as
157
+ (g1⊗g2)(y) =/integraldisplay1
158
+ 0dx1/integraldisplay1
159
+ 0dx2g1(x1)g2(x2)δ(x1x2−y). (2.6)
160
+ The CTEQ6L parton distribution function set is used for all o f the results [12].
161
+ Results for the cross sections for the scalar, pseudoscalar , vector, axial vector, and tensor
162
+ structures at the Tevatron, LHC at 10 TeV and 14 TeV are given i n Table 1. Thecross section
163
+ for the pseudoscalar (axial vector) current is the same as fo r the scalar (vector) current. For
164
+ all cases, Λ is set equal to 2 TeV and the unitarity bounds are t aken into consideration. At
165
+ this rather high scale, the production rates are dominated b y the valence quark contributions.
166
+ The cross sections at the LHC are larger than those at the Teva tron by roughly an order of
167
+ magnitude, reaching about 100 pb.
168
+ For some cases the bounds from BHHS are greater than 2 TeV, hen ce the cross section
169
+ needs to be scaled to determine a realistic cross section at h adron colliders. The partonic
170
+ cross sections scale at Λ−4, but at the hadronic level a complication arises since the un itarity
171
+ bounds introduce a dependence on the new physics scale in the integration over pdfs. If
172
+ the unitarity bounds are ignored ( τmax= 1), one finds that with Λ = 2 TeV neglecting the
173
+ unitarity bounds has at most a 10% effect on the cross sections a t the LHC for both 10 TeV
174
+ and 14 TeV and no effect at the Tevatron since the unitarity boun ds are greater than the lab
175
+ frame energy. Hence, if Λ is increased from 2 TeV, at the LHC it is a good approximation to
176
+ assume the cross section scales as Λ−4and at the Tevatron the cross section scales exactly as
177
+ Λ−4. For example, the lower bound on Λ for the vector u¯ucoupling from BHHS is 12 TeV, so
178
+ the maximum cross section at the 14 TeV LHC from this operator would be approximately
179
+ 160×(2/12)4pb = 120 fb. On the other hand, there is no bound whatsoever for the vector
180
+ u¯ccoupling, and thus a cross section limit of 110 pb would yield a new limit of 2 TeV on the
181
+ scale of this operator. This would constitute an improvemen t of many orders of magnitude.
182
+ 3. Signal Identification and Backgrounds
183
+ Upon production at hadron colliders, τ’s will promptly decay and are detected via their decay
184
+ products. About 35% of the time the τdecays to two neutrinos and an electron or muon, the
185
+ other 65% of the time the τdecays to a few hadrons plus a neutrino. We will consider the τ
186
+ decay to an electron as well as hadronic decays in this work. T he decay to a muon will result
187
+ in aµ+µ−final state that has a large Drell-Yan background. We will stu dy the signal reach
188
+ at the Tevatron and at the 14 TeV LHC.
189
+ – 4 –Table 1: Cross sections for all the scalar, pseudoscalar, vector, axial ve ctor, and tensor structures at
190
+ the Tevatron at 2 TeV, the LHC at 10 TeV, and the LHC at 14 TeV. Th e pseudoscalar (axial vector)
191
+ cross section is the same as the scalar (vector) cross section. All cross sections were evaluated with
192
+ the new physics scale Λ = 2 TeV and the unitarity bounds are taken int o consideration.
193
+ Tevatron 2 TeV ( p¯p)LHC 10 TeV ( pp) LHC 14 TeV ( pp)
194
+ σ(pb)1,γ5γµ,γµγ5σµν1,γ5γµ,γµγ5σµν1,γ5γµ,γµγ5σµν
195
+ u¯u8.4 11 22 63 85 170 120 160 310
196
+ d¯d2.5 3.3 6.7 38 51 100 72 98 190
197
+ s¯s0.18 0.24 0.49 5.5 7.4 15 11 15 30
198
+ d¯s1.3 1.7 3.4 34 45 91 66 89 180
199
+ d¯b0.50 0.67 1.3 17 22 45 34 46 90
200
+ s¯b0.13 0.17 0.34 5.0 6.7 13 11 14 28
201
+ u¯c1.5 2.0 3.9 41 55 110 80 110 210
202
+ c¯c0.070 0.094 0.19 2.6 3.5 7.0 5.5 7.3 15
203
+ b¯b0.021 0.028 0.056 1.1 1.5 2.9 2.4 3.2 6.4
204
+ 3.1τDecay to Electrons
205
+ 3.1.1 Signal Reconstruction
206
+ Theτdecays to an electron plus two neutrinos about 18% of the time . We thus search for a
207
+ final state of an electron and muon
208
+ e+µ. (3.1)
209
+ The electromagnetic calorimeter resolution is simulated b y smearing the electron energies
210
+ according to a Gaussian distribution with a resolution para meterized by
211
+ σ(E)
212
+ E=a/radicalbig
213
+ E/GeV⊕b, (3.2)
214
+ where the constants are a= 10% and b= 0% at the Tevatron [13], a= 5% and b= 0.55% at
215
+ the LHC [14], and ⊕indicates addition in quadrature. For simplicity, we have u sed the same
216
+ form of smearing for the muons.
217
+ The decay of the τleaves us with some missing energy and we need to consider how to
218
+ effectively reconstructthe τmomentum. Forourprocessallthemissingtransversemoment um
219
+ is coming from the τ, hence
220
+
221
+ T=pe
222
+ T+pmiss
223
+ T. (3.3)
224
+ At hadron colliders, we have no information on the longitudi nal component of the missing
225
+ momentum on an event-by-event basis. However, the τwill be highly boosted and its decay
226
+ – 5 –products will be collimated. Hence, the missing momentum sh ould be aligned with the
227
+ electron momentum and the ratio pe
228
+ z/pmiss
229
+ zshould be the same as the ratio of the magnitudes
230
+ of the transverse momenta, pe
231
+ T/pmiss
232
+ T. Therefore, the longitudinal component of the τcan be
233
+ reconstructed as [4]
234
+
235
+ z=pe
236
+ z/parenleftbigg
237
+ 1+pmiss
238
+ T
239
+ pe
240
+ T/parenrightbigg
241
+ . (3.4)
242
+ Once the three-momentum is reconstructed, we can solve for t heτenergy,E2
243
+ τ=p2
244
+ τ+m2
245
+ τ.
246
+ Figure 1 illustrates the effectiveness of this method at the Te vatron. Figure 1(a) (Figure 1(b))
247
+ shows the transverse momentum (longitudinal momentum) dis tribution for the theoretically
248
+ generated (solid) and kinematically reconstructed (dashe d)τmomenta. As can be seen, the
249
+ τmomentum is reconstructed effectively.
250
+ We first apply some basic cuts on the transverse momentum and t he pseudo rapidity
251
+ to simulate the detector acceptance and triggering, as well as to isolate the signal from the
252
+ background,
253
+
254
+ T>20 GeV,|ηµ|<2.5,
255
+ pe
256
+ T>20 GeV,|ηe|<2.5. (3.5)
257
+ Since the signal does not contain any jets, we also require a j et veto such that there are no
258
+ jets with pT>50 GeV and |η|<2.5.
259
+ There are several distinctive kinematic features of our sig nal. The decay products of the
260
+ τwill be highly collimated, and the electron transverse mome ntum will be traveling in the
261
+ same direction as the missing transverse momentum. Also, in the transverse plane the muon
262
+ and tau should be back to back. Since the electron will mostly be in the direction of the τ,
263
+ it will also be nearly back to back with the muon. Finally, the τandµhave equal transverse
264
+ momenta; hence, the decay products of the τhave less transverse momentum than the µ. We
265
+ can measure this discrepancy using the momentum imbalance
266
+ ∆pT=pµ
267
+ T−pe
268
+ T. (3.6)
269
+ For the signal, this observable should be positive. Based on the kinematics of our signal, we
270
+ apply the further cuts [5]
271
+ δφ(pµ
272
+ T,pe
273
+ T)>2.75 rad, δφ(pmiss
274
+ T,pe
275
+ T)<0.6 rad, (3.7)
276
+ ∆pT>0.
277
+ 3.1.2 Backgrounds and their Suppression
278
+ Theleadingbackgrounds are W+W−pair production, Z0/γ⋆→τ+τ−, andt¯tpair production
279
+ [5]. The total rates for these backgrounds at the Tevatron an d the LHC are given in Table 2
280
+ with consecutive cuts. We consider both of the final states wi thµ+andµ−.
281
+ – 6 –0 100 200 300
282
+
283
+ T (GeV)10-410-310-210-1dσ/dpT (pb/GeV)Generated
284
+ Reconstructed
285
+ (a)-300 -200 -100 0 100 200 300
286
+
287
+ z (GeV)10-310-210-1dσ/dpz (pb/GeV)Generated
288
+ Reconstructed
289
+ (b)
290
+ Figure 1: Distributions of the theoretically generated (solid line) and kinematic ally reconstructed
291
+ (dashed line) τmomentum at the Tevatron at 2 TeV with a u¯cinitial state, scalar coupling, and new
292
+ physics scale of 1 TeV. Fig. (a) is the τtransverse momentum distribution, and Fig. (b) is the τ
293
+ longitidunal momentum distribution.
294
+ Table 2: Leading backgrounds to the τ’s electronic decay before and after consecutive kinematic and
295
+ invariant mass cuts for (a) the Tevatron at 2 TeV and (b) the LHC a t 14 TeV.
296
+ Backgrounds (pb) No Cuts Cuts Eq. (3.5) + Eq. (3.7) + Eq. (3.8)
297
+ (a) Tevatron 2 TeV
298
+ W+W−→µ±νµτ∓ντ0.032 0.0046 0.0012 2.6×10−4
299
+ W+W−→µ±νµe∓νe0.18 0.13 0.0060 9.8×10−4
300
+ Z0/γ⋆→τ+τ−→µ±νµτ∓610 0.21 0.091 1.4×10−4
301
+ t¯t→µ±νµbτ∓ντ¯b 0.020 6.5×10−47.4×10−54.4×10−5
302
+ t¯t→µ±νµbe∓νe¯b 0.11 0.0099 7.3×10−42.7×10−4
303
+ (b) LHC 14 TeV
304
+ W+W−→µ±νµτ∓ντ0.34 0.030 0.0088 0.0031
305
+ W+W−→µ±νµe∓νe 1.9 0.99 0.051 0.014
306
+ Z0/γ⋆→τ+τ−→µ±νµτ∓2300 1.1 0.49 0.0014
307
+ t¯t→µ±νµbτ∓ντ¯b 1.9 0.070 0.010 0.0077
308
+ t¯t→µ±νµbe∓νe¯b 11 1.5 0.10 0.050
309
+ The partonic cross section of our signal increases with ener gy while the cross sections
310
+ of the backgrounds will decrease with energy. Hence, the inv ariant mass distribution of our
311
+ signal does not fall off as quickly as the backgrounds.
312
+ Figure 2(a) shows the invariant mass distributions of backg rounds and our signal at the
313
+ Tevatron with initial states c¯candu¯cwith various couplings and a new physics scale of
314
+ 1 TeV after applying the cuts in Eqs. (3.5) and (3.7). The cros s section for the pseudoscalar
315
+ – 7 –0 200 400 600800 1000
316
+ Mµτ (GeV)10-510-410-310-2dσ/dMµτ (pb/GeV)eµνν
317
+ τ+τ−
318
+ Tensor
319
+ Vector
320
+ Scalaru c-bar
321
+ c c-barTevatron
322
+ (a)0 200 400 600800 1000
323
+ Mµτ (GeV)10-510-410-310-2dσ/dMµτ (pb/GeV)eµνν
324
+ τ+τ−1 TeV
325
+ 2 TeV
326
+ 3 TeVTevatron
327
+ (b)
328
+ Figure 2: The invariant mass distributions of the reconstructed τ−µsystem at the Tevatron at 2
329
+ TeV. Fig. (a) shows the distributions of the leading backgrounds (d otted and dot-dot-dash) and of
330
+ our signal for the u¯candc¯cinitial states with coupling of various Lorentz structures and a new physics
331
+ scale of 1 TeV. Fig. (b) shows the distributions of the leading backgr ounds (dotted and dashed) and
332
+ of our signal (solid) for the u¯cinitial state with scalar coupling and various new physics scales. The
333
+ cuts in Eqs. (3.5) and (3.7) have been applied.
334
+ (axial-vector) couplings are the same as those for the scala r (vector) couplings. The decline
335
+ in the signal rates is due to a suppression of the pdfs at large x. Although the signal rates
336
+ steeply decline with invariant mass the background falls off faster. The u¯csignal is still clearly
337
+ above background due to a valence quark in the initial state, but thec¯csignal distribution is
338
+ much closer to the background distribution due to the steep f all with invariant mass and a
339
+ lack of an initial state valence quark. Figure 2(b) shows the invariant mass distributions of
340
+ backgroundsandoursignalattheTevatronwithinitial stat eu¯candscalarcouplingforvarious
341
+ new physics scales. The 3 TeV new physics scale invariant mas s distribution is approaching
342
+ the background distribution. A higher cutoff on the invarian t mass will be needed to separate
343
+ the weak signal from the backgrounds. Based on Fig. 2, we prop ose a selection cut on
344
+ Mµτ>250 GeV . (3.8)
345
+ Table 2 shows the effects of the invariant mass cut on the backgr ounds in the last column.
346
+ Similar analyses can be carried out for the LHC. Figure 3(a) s hows the invariant mass
347
+ distribution for our signal with the u¯candc¯cinitial states and various Lorentz structures, as
348
+ well as the backgroundsafter thecuts in Eqs. (3.5) and(3.7) . Thenewphysics scale was set to
349
+ 1 TeV and the unitarity bound is imposed. Figure 3(b) shows th e invariant mass distribution
350
+ of theu¯cinitial state with various new physics scales. The cutoff on t he invariant mass
351
+ corresponds to the unitarity bound, the scale at which the pe rturbative calculation becomes
352
+ untrustworthy. In the lack of the knowledge for the new physi cs to show up at the scale Λ,
353
+ we simply impose a sharp cutoff at the unitarity bound. As comp ared with the Tevatron,
354
+ the LHC signal rates fall off much less quickly with invariant mass since the Tevatron’s lower
355
+ – 8 –0 500 1000 1500
356
+ Mµτ (GeV)10-410-310-210-1dσ/dMµτ (pb/GeV)Tensor
357
+ Vector
358
+ Scalar
359
+ bbeµνντ+τ−u c-bar
360
+ c c-barLHC
361
+ (a)0 1 2 3 4
362
+ Mµτ (TeV)10-510-410-310-210-1dσ/dMµτ (pb/GeV)bbeµνν
363
+ τ+τ−1 TeV
364
+ 2 TeV
365
+ 3 TeV
366
+ 4 TeV
367
+ 5 TeVLHC
368
+ (b)
369
+ Figure 3: The invariant mass distributions of the reconstructed τ−µsystem at the LHC at 14 TeV.
370
+ Fig. (a) shows the distributions of the leading backgrounds (dotte d and dot-dot-dash) and of our
371
+ signal for the u¯candc¯cinitial states with coupling of various Lorentz structures and a new physics
372
+ scale of 1 TeV. Fig. (b) shows the distributions of the leading backgr ounds (dotted and dashed) and
373
+ of our signal (solid) for the u¯cinitial state with scalar coupling and various new physics scales. The
374
+ cut offs in the distributions at high invariant mass are due to the unita rity bounds. The cuts in Eqs.
375
+ (3.5) and (3.7) have been applied.
376
+ energy leads to a suppression from the pdfs at large x. As can be seen, as the new physics
377
+ scale increases the cross section decreases and the backgro und becomes more problematic at
378
+ lower invariant mass. Also, as the new physics scale increas es the unitarity bound becomes
379
+ less strict. Hence, although the backgrounds at the LHC are c onsiderably larger than at the
380
+ Tevatron, for large new physics scales the LHC has an enhance ment in the signal cross section
381
+ from the large invariant mass region.
382
+ 3.2τDecay to Hadrons
383
+ Although with significantly larger backgrounds, the signal fromτhadronic decays can be
384
+ very distinctive as well. We limit the hadronic τdecays to 1-prong decays to pions, i.e.,
385
+ τ±→π±ντ,τ±→π±π0ντ, andτ±→π±2π0ντ. Theτ’s have 1-prong decays to these final
386
+ states about 50% of the time. We thus search for a final state of aτjet and a muon
387
+ jτ+µ. (3.9)
388
+ To simulate detector resolution effects, the energy is smeare d according to Eq. (3.2) with
389
+ a= 80% and b= 0% for the jet at the Tevatron [13] and a= 100% and b= 5% at the LHC
390
+ [14]. As in the electronic decay, the τis highly boosted and its decay products are collimated.
391
+ Hence, all the missing energy in the event should be aligned w ith theτ. The signal is then
392
+ reconstructed as described in Eqs. (3.3) and (3.4) with the e lectron momentum replaced by
393
+ the momentum of the τ-jet.
394
+ – 9 –The hadronic decay of the τalso has the backgrounds W+W−pair production, Z0/γ⋆→
395
+ τ+τ−, andt¯tpair production plus an additional background of W+jet, where the jet is
396
+ misidentified as a τ-jet. At the Tevatron, we assume a τ-jet tagging efficiency of 67% and
397
+ that a light jet is mistagged as a τ-jet 1.1% of the time [15] and at the LHC we assume a τ-jet
398
+ tagging efficiency of 40% and a light jet misidentification rat e of 1% [14]. Even with a low
399
+ rate of misidentification, the W+jet background is large. To suppress this background, we
400
+ note that for hadronic decays most of the τtransverse momentum will be carried by the jet.
401
+ Hence the τ-jet should be traveling in the same direction as the reconst ructedτmomentum.
402
+ Motivated by this observation, we apply the same cuts as Eqs. (3.5), (3.7), and (3.8) with the
403
+ electron momentum replaced by the τ-jet momentum and the additional cuts
404
+ pτ−jet
405
+ T
406
+
407
+ T>0.6 ∆ R(pτ−jet
408
+ T,pτ
409
+ T)<0.2 rad. (3.10)
410
+ 3.3 Sensitivity Reach at the Tevatron
411
+ One can determine the sensitivity of the Tevatron to the new p hysics scale with 8 fb−1of
412
+ data. Table 3 shows the sensitivity of the Tevatron for (a) el ectronic and (b) hadronic τ
413
+ decays. The tables list the maximum new physics scale sensit ivity at 2 σand 5σlevel at the
414
+ Tevatron. The reaches for scalar (vector) and pseudoscalar (axial-vector) are the same at the
415
+ Tevatron, although the previous bounds from BHHS for the sca lar (vector) and pseudoscalar
416
+ (axial-vector) couplings may not be the same. The bounds fro m BHHS can be found in
417
+ Appendix A. If only one of the bounds for scalar (vector) or ps eudoscalar (axial-vector)
418
+ coupling from BHHS is greater than the Tevatron reach one sta r is placed next to the new
419
+ physics scale, if both bounds are greater than the Tevatron r each two stars are placed next
420
+ to the new physics scale. Due to the larger backgrounds from W+jet, the Tevatron is much
421
+ less sensitive to the τhadronic decays than the τelectronic decays.
422
+ There were no bounds from BHHS for the tensor couplings, so th e Tevatron will be
423
+ able to exlude some of the parameter space. Since the tensor c ross sections are generally at
424
+ least twice as large as the scalar cross sections, the Tevatr on is more sensitive to the tensor
425
+ couplings than it is to scalar couplings. Also, in general, t he Tevatron is more sensitive to
426
+ processes with initial state valence quarks than those with out initial state quarks. With 8
427
+ fb−1of data most of the bounds can be increased, some quite string ently.
428
+ Somewhat similar leptonic final states have been searched fo r in a model-independent
429
+ way at the Tevatron [16], although these included substanti al missing energy and possible
430
+ jets. We encourage the Tevatron experimenters to carry out t he analyses as suggested in this
431
+ article.
432
+ 3.4 Sensitivity Reach at the LHC
433
+ The LHC is also sensitive to flavor changing operators. For th e signal and background anal-
434
+ ysis, we used the same kinematical cuts as we used at the Tevat ron, see Eqs. (3.5), (3.7),
435
+ and (3.8). Table 4 shows the sensitivity of the LHC to all poss ible initial states and the
436
+ – 10 –Table 3: Maximum new physics scales the Tevatron is sensitive to with 8 fb−1of data at the 2 σ
437
+ and 5σlevels. The sensitivities are presented for both (a) electronic and ( b) hadronic τdecays with
438
+ various initial states. One star indicates that the Tevatron reach is less than only one of the scalar
439
+ (vector) or pseudoscalar (axial-vector) bounds from BHHS, and two stars indicates that the Tevatron
440
+ reach is less than both bounds from BHHS. BHHS does not contain bo unds on the tensor coupling.
441
+ (a)τ→e
442
+ ΛNP(TeV) 2σsensitivity 5σdiscovery
443
+ Coupling 1,γ5γµ,γµγ5σµν1,γ5γµ,γµγ5σµν
444
+ u¯u20 21 24 14 15 17
445
+ d¯d17 18 21 12 13 15
446
+ s¯s9.9 10 12 7.2* 7.7** 8.7
447
+ d¯s15 16 18 10 11* 13
448
+ d¯b13 14 16 9.8 10* 11
449
+ s¯b9.5 10 11 6.9 7.3 8.3
450
+ u¯c17 18 20 12 13 14
451
+ c¯c7.9 8.3 9.5 5.7 6.0 6.9
452
+ b¯b6.4 6.8 7.7 4.6 4.9 5.6
453
+ (b)τ→h±
454
+ ΛNP(TeV) 2σsensitivity 5σdiscovery
455
+ Coupling 1,γ5γµ,γµγ5σµν1,γ5γµ,γµγ5σµν
456
+ u¯u8.6** 9.2** 10 6.5** 6.9** 8.1
457
+ d¯d5.7** 6.1** 7.1 4.3** 4.6** 5.4
458
+ s¯s1.8* 1.9** 2.3 1.4** 1.4** 1.7
459
+ d¯s3.7 4.0* 4.6 2.8* 3.0** 3.5
460
+ d¯b2.7* 2.9* 3.4 2.0** 2.2 2.5
461
+ s¯b1.5** 1.6** 1.9 1.1** 1.2** 1.4
462
+ u¯c3.9 4.1 4.8 2.9 3.1 3.6
463
+ c¯c1.1 1.2 1.4 0.89 0.95** 1.1
464
+ b¯b0.91 0.97 1.1 0.68 0.73 0.86
465
+ couplings under consideration with 100 fb−1of data. The table contains the maximum new
466
+ physics scales the LHC is sensitive to at the 2 σand 5σlevels. As with the Tevatron, the
467
+ LHC reach for scalar (vector) couplings is the same as that fo r pseudoscalar (axial-vector)
468
+ couplings, although the bounds from BHHS may be different. If o nly one of the bounds for
469
+ scalar (vector) or pseudoscalar (axial-vector) coupling f rom BHHS is greater than the LHC
470
+ reach one star is placed next to the new physics scale, if both bounds are greater than the
471
+ LHC reach two stars are placed next to the new physics scale. D espite the larger backrounds
472
+ for the hadronic τdecays, at the LHC the reaches for the hadronic and electroni cτdecays are
473
+ – 11 –Table 4: Maximum new physics scales the LHC is sensitive to at 14 TeV with 100 fb−1of data at the
474
+ 2σand 5σlevels. The sensitivities are presented for both (a) electronic and ( b) hadronic τdecays with
475
+ various initial states. One star indicates that the LHC reach is less t han only one of the scalar (vector)
476
+ or pseudoscalar (axial-vector) bounds from BHHS, and two stars indicates that the LHC reach is less
477
+ than both bounds from BHHS. BHHS does not contain bounds on the tensor coupling.
478
+ (a)τ→e
479
+ ΛNP(TeV) 2σsensitivity 5σdiscovery
480
+ Coupling 1,γ5γµ,γµγ5σµν1,γ5γµ,γµγ5σµν
481
+ u¯u18 19 21 14 15 17
482
+ d¯d16 17 19 12 13 15
483
+ s¯s9.0* 9.6* 11 7.1* 7.6** 8.6
484
+ d¯s13 14 16 10 11* 13
485
+ d¯b12 13 14 9.7 10 11
486
+ s¯b8.7 9.2 10 6.8 7.3 8.2
487
+ u¯c15 16 18 12 13 14
488
+ c¯c7.2 7.6 8.6 5.7 6.0 6.8
489
+ b¯b5.8 6.2 7.0 4.6 4.9 5.5
490
+ (b)τ→h±
491
+ ΛNP(TeV) 2σsensitivity 5σdiscovery
492
+ u¯u15 16 18 12 13 14
493
+ d¯d13 14 16 10* 11* 13
494
+ s¯s7.9* 8.4** 9.7 6.2* 6.7** 7.7
495
+ d¯s11 12* 14 9.3 9.9* 11
496
+ d¯b10 11 13 8.4* 8.9 10
497
+ s¯b7.6 8.1 9.3 6.0 6.4 7.4
498
+ u¯c13 14 16 10 11 12
499
+ c¯c6.3 6.7 7.8 5.0 5.3 6.2
500
+ b¯b5.1 5.5 6.3 4.1 4.3 5.0
501
+ much more similar than at the Tevatron since the LHC cross sec tion receives an enhancement
502
+ from the large invariant mass region. For electronic (hadro nic)τdecays the LHC with 100
503
+ fb−1of data is less (more) sensitive than the Tevatron with 8 fb−1of data.
504
+ Figure 4 shows the integrated luminosities needed for 2 σand 5σobservation at the LHC
505
+ with various initial states and τdecay to electrons as a function of the new physics scale.
506
+ For some initial states and Lorentz structures BHHS had a bou nd on the new physics scale
507
+ larger than 1 TeV. In those cases the distribution does not be gin until the BHHS bound on
508
+ the new physics scale. The sensitivity for the pseudoscalar (axial-vector) is the same as the
509
+ scalar (vector) state, although the bounds from BHHS are diffe rent. Note that extraordinary
510
+ – 12 –1 2 3 4 5 6 78 9 10
511
+ ΛNP (TeV)10-310-210-1100101102103L (fb-1)Scalar
512
+ Vector
513
+ Tensor
514
+ 2σu c-bar Initial State
515
+ 5σLHC 14 TeV
516
+ (a)1 2 3 4 5 6 78 9 10
517
+ ΛNP (TeV)10-310-210-1100101102103L (fb-1)
518
+ Scalar
519
+ Vector
520
+ Tensor
521
+ 2σc c-bar Initial State
522
+ 5σLHC 14 TeV
523
+ (b)
524
+ 1 2 3 4 5 6 78 9 10
525
+ ΛNP (TeV)10-310-210-1100101102103L (fb-1)Scalar
526
+ Vector
527
+ Tensor
528
+ 2σd b-bar Initial State
529
+ 5σLHC 14 TeV
530
+ (c)1 2 3 4 5 6 78 9 10
531
+ ΛNP (TeV)10-310-210-1100101102103L (fb-1)
532
+ Scalar
533
+ Vector
534
+ Tensor
535
+ 2σs b-bar Initial State
536
+ 5σLHC 14 TeV
537
+ (d)
538
+ Figure 4: The luminosity at the 14 TeV LHC needed for 2 σand 5σobservation as a function of the
539
+ new physics scales with couplings of various Lorentz structures an d electronic τdecay. The sensitivity
540
+ for theu¯cinitial state is shown in (a), for the c¯cinitial state in (b), for the d¯binitial state in (c), and
541
+ for thes¯binitial state in (d). The lower bounds on the new physics scale were ta ken from BHHS.
542
+ improvementintheboundscouldbefound(oradiscoverymade )withrelatively lowintegrated
543
+ luminosity. Consider, for example, the u¯cinitial state. There is currently no bound at all;
544
+ in principle, Λ could be tens of GeV. The figure shows that a tot al integrated luminosity of
545
+ an inverse picobarn would give a 5 σsensitivity for a Λ of 1 TeV. An integrated luminosity of
546
+ an inverse femtobarn would give substantial improvements f or all of the operators shown in
547
+ Fig. 4.
548
+ 4. Discussions and Conclusions
549
+ In a previous article, motivated by discovery of large νµ−ντmixing in charged current
550
+ interactions, bounds on the analogous mixing in neutral cur rent interactions were explored.
551
+ A general formalism for dimension-6 fermionic effective oper ators involving τ−µmixing with
552
+ – 13 –typical Lorentz structure ( µΓτ)(qαΓqβ) was presented, and the low-energy constraints on
553
+ the new physics scale associated with each operator were der ived, mostly from experimental
554
+ bounds on rare decays of τ, hadrons or heavy quarks. Tensor operators were not conside red,
555
+ and some of the operators, such as cuµτ, were completely unbounded.
556
+ Inthis article, weconsider µτproductionat hadroncolliders viatheseoperators. Tables 3
557
+ and4 list thenewphysics scales that are accessible at the Te vatron and theLHC, respectively.
558
+ Duetomuchsmallerbackgrounds, boththeLHCandTevatronar emoresensitivetoelectronic
559
+ τdecays than hadronic τdecays. For hadronic τdecays, the LHC receives an enhancement
560
+ from the large invariant mass region and is more sensitive th an the Tevatron. Since the
561
+ backgrounds to electronic τdecays at the Tevatron are much smaller than those at the LHC,
562
+ the Tevatron is more sensitive than the LHC to electronic τdecays. We found that at the
563
+ Tevatron with 8 fb−1, one can exceed current bounds for most operators, with most 2σ
564
+ sensitivities being in the 6 −24 TeV range. We find that at the LHC with 1 fb−1(100 fb−1)
565
+ integrated luminosity, one can reach a 2 σsensitivity for Λ ∼3−10 TeV (Λ ∼6−21 TeV),
566
+ depending on the Lorentz structure of the operator.
567
+ Acknowledgments
568
+ We would like to thank Vernon Barger and Xerxes Tata for discu ssions. MS would like to
569
+ thank the Wisconsin Phenomenology Institute, in particula r Linda Dolan, for hospitality
570
+ during his visit. The work of TH and IL was supported by the US D OE under contract
571
+ No. DE-FG02-95ER40896, and that of MS was supported in part b y the National Science
572
+ Foundation PHY-0755262.
573
+ A. New Physics Bounds
574
+ The bounds from BHHS in units of TeV are presented in Table 5. T he *s indicate there are
575
+ no bounds on the new physics scale. Also, there are no bounds f rom BHHS for the tensor
576
+ coupling.
577
+ B. Partial Wave Unitarity Bounds
578
+ Since the cross section from our higher-dimensional operat ors increases as s, it is necessary
579
+ to determine the unitarity bound for q¯q→µτ. The partial wave expansion for a+b→1+2
580
+ can be written as
581
+ M(s,t) = 16π∞/summationdisplay
582
+ J=M(2J+1)aJ(s)dJ
583
+ µµ′(cosθ)
584
+ where
585
+ aJ(s) =1
586
+ 32π/integraldisplay1
587
+ −1M(s,t)dJ
588
+ µµ′(cosθ)dcosθ,
589
+ µ=sa−sb,µ′=s1−s2andJ≤max(|µ|,|µ′|). The condition for unitarity is |ℜ(aJ)| ≤1/2.
590
+ – 14 –Coupling type 1 γ5 γµ γµγ5
591
+ u¯u 2.6 12 12 11
592
+ d¯d 2.6 12 12 11
593
+ s¯s 1.5 9.9 14 9.5
594
+ d¯s 2.3 3.7 13 3.6
595
+ d¯b 2.2 9.3 2.2 8.2
596
+ s¯b 2.6 2.8 2.6 2.5
597
+ u¯c * * 0.55 0.55
598
+ c¯c * * 1.1 1.1
599
+ b¯b * * 0.18 *
600
+ Table 5: Bounds on the new physics scales from BHHS in units of TeV for variou s operators and the
601
+ scalar, pseudoscalar, vector, and axial-vector couplings. The *s indicate there were no bounds.
602
+ It is straightforward to calculate the coefficients for the S, V,T operators. For example,
603
+ for the scalar operator
604
+ M=4π
605
+ Λ2¯vλ1(p1)uλ2(p2)¯uλ3(p3)vλ4(p4)
606
+ one can just plug in the explicit expressions:
607
+ uλ(p)≡/parenleftBigg/radicalbig
608
+ E−λ|p|χλ(ˆp)/radicalbig
609
+ E+λ|p|χλ(ˆp)/parenrightBigg
610
+ vλ(p)≡/parenleftBigg
611
+ −/radicalbig
612
+ E+λ|p|χ−λ(ˆp)/radicalbig
613
+ E−λ|p|χ−λ(ˆp)/parenrightBigg
614
+ whereχ+(ˆz) =/parenleftbig1
615
+ 0/parenrightbig
616
+ ,χ−(ˆz) =/parenleftbig0
617
+ 1/parenrightbig
618
+ . In the massless limit, this simply gives a0=s/(4Λ2) and
619
+ so the unitarity bound gives s≤2Λ2. For the vector case, a0= 0 and a1=s/(6Λ2) giving
620
+ the unitarity bound s≤3Λ2. The tensor case gets contributions from both a0anda1, and
621
+ the stronger bound then applies.
622
+ References
623
+ [1] S. Fukuda, et al., [Super-Kamiokande Collaboration], Phys. Rev. Lett. 85, 3999 (2000); 86,
624
+ 5656 (2001); 82, 1810 (1999); 81, 1562 (1998); 81, 1158 (1998); and T. Toshito,
625
+ [Super-Kamiokande Collaboration], hep-ex/0105023 .
626
+ [2] Q. R. Ahmad, et al.,[SNO collaboration], Phys. Rev. Lett. 87, 071301 (2001). Q. R. Ahmad, et
627
+ al.,[SNO Collaboration], Phys. Rev. Lett. (2002), nucl-ex/0204008 andnucl-ex/0204009 .
628
+ [3] D. Black, T. Han, H. J. He and M. Sher, Phys. Rev. D 66, 053002 (2002)
629
+ [arXiv:hep-ph/0206056].
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+ – 15 –[4] T. Han and D. Marfatia, Phys. Rev. Lett. 86, 1442 (2001) [arXiv:hep-ph/0008141].
631
+ [5] K. A. Assamagan, A. Deandrea and P. A. Delsart, Phys. Rev. D 67, 035001 (2003)
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+ [arXiv:hep-ph/0207302].
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+ [6] U. Cotti, J. L. Diaz-Cruz, R. Gaitan, H. Gonzales and A. Hernand ez-Galeana, Phys. Rev. D 66,
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+ 015004 (2002) [arXiv:hep-ph/0205170]; J. L. Diaz-Cruz, D. K. Gho sh and S. Moretti, Phys.
635
+ Lett. B679, 376 (2009) [arXiv:0809.5158 [hep-ph]].
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+ [7] A. Brignole and A. Rossi, Phys. Lett. B 566, 217 (2003) [arXiv:hep-ph/0304081].
637
+ [8] E. Arganda, A. M. Curiel, M. J. Herrero and D. Temes, Phys. Rev . D71, 035011 (2005)
638
+ [arXiv:hep-ph/0407302].
639
+ [9] A. Azatov, M. Toharia and L. Zhu, Phys. Rev. D 80, 035016 (2009) [arXiv:0906.1990 [hep-ph]].
640
+ [10] F. Deppisch, J. Kalinowski, H. Pas, A. Redelbach and R. Ruckl, ar Xiv:hep-ph/0401243.
641
+ [11] H. U. Bengtsson, W. S. Hou, A. Soni and D. H. Stork, Phys. Re v. Lett.55, 2762 (1985).
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+ [12] D. Stump, J. Huston, J. Pumplin, W. K. Tung, H. L. Lai, S. Kuhlma nn and J. F. Owens, JHEP
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+ 0310(2003) 046 [arXiv:hep-ph/0303013].
644
+ [13] M. S. Carena et al.[Higgs Working Group Collaboration], Report of the Tevatron Higgs
645
+ working group, arXiv:hep-ph/0010338.
646
+ [14] G. L. Bayatian et al.[CMS Collaboration], J. Phys. G 34, 995 (2007). G. Aad et al.[The
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+ ATLAS Collaboration], arXiv:0901.0512 [hep-ex].
648
+ [15] P. Svoisky [D0 Collaboration], Nucl. Phys. Proc. Suppl. 189, 338 (2009).
649
+ [16] B. Abbott et al.[D0 Collaboration], Phys. Rev. D 62, 092004 (2000) [arXiv:hep-ex/0006011];
650
+ J. Piper [CDF Collaboration and D0 Collaboration], arXiv:0906.3676 [hep- ex].
651
+ – 16 –
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1001.0024.txt ADDED
@@ -0,0 +1,659 @@
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
1
+ arXiv:1001.0024v1 [q-fin.CP] 30 Dec 2009November 10, 2018 20:49 WSPC/INSTRUCTION FILE svJCSC3
2
+ Journal of Circuits, Systems, and Computers
3
+ c/circlecopyrtWorld Scientific Publishing Company
4
+ BAYESIAN INFERENCE OF STOCHASTIC VOLATILITY MODEL
5
+ BY HYBRID MONTE CARLO
6
+ Tetsuya Takaishi†
7
+ Hiroshima University of Economics,
8
+ Hiroshima 731-0192 JAPAN
9
10
+ Received (Day Month Year)
11
+ Revised (Day Month Year)
12
+ Accepted (Day Month Year)
13
+ The hybrid Monte Carlo (HMC) algorithm is applied for the Bay esian inference of the
14
+ stochastic volatility (SV) model. We use the HMC algorithm f or the Markov chain Monte
15
+ Carloupdates of volatility variables of the SV model. First we compute parameters of the
16
+ SV model by using the artificial financial data and compare the results from the HMC
17
+ algorithm with those from the Metropolis algorithm. We find t hat the HMC algorithm
18
+ decorrelates the volatility variables faster than the Metr opolis algorithm. Second we
19
+ make an empirical study for the time series of the Nikkei 225 s tock index by the HMC
20
+ algorithm. We find the similar correlation behavior for the s ampled data to the results
21
+ from the artificial financial data and obtain a φvalue close to one ( φ≈0.977), which
22
+ means that the time series has the strong persistency of the v olatility shock.
23
+ Keywords : Hybrid Monte Carlo Algorithm, Stochastic Volatility Mode l, Markov Chain
24
+ Monte Carlo, Bayesian Inference, Financial Data Analysis
25
+ 1. Introduction
26
+ Many empirical studies of financial prices such as stock indexes, ex change rates
27
+ have confirmed that financial time series of price returns shows va rious interesting
28
+ properties which can not be derived from a simple assumption that th e price re-
29
+ turns follow the geometric Brownian motion. Those properties are n ow classified
30
+ as stylized facts1,2. Some examples of the stylized facts are (i) fat-tailed distribu-
31
+ tion of return (ii) volatility clustering (iii) slow decay of the autocorre lation time
32
+ of the absolute returns. The true dynamics behind the stylized fac ts is not fully
33
+ understood. In order to imitate the real financial markets and to understand the
34
+ origins of the stylized facts, a variety of models have been propose d and examined.
35
+ Actually many models are able to capture some of the stylized facts3-14.
36
+ In empirical finance the volatilityis an important value to measurethe risk. One
37
+ of the stylized facts of the volatility is that the volatility of price retu rns changes
38
+ in time and shows clustering, so called ”volatility clustering”. Then the histogram
39
+ of the resulting price returns shows a fat-tailed distribution which in dicates that
40
+ 1November 10, 2018 20:49 WSPC/INSTRUCTION FILE svJCSC3
41
+ 2Authors’ Names
42
+ the probability of having a large price change is higher than that of th e Gaussian
43
+ distribution. In order to mimic these empirical properties of the vola tility and to
44
+ forecast the future volatility values, Engle advocated the autore gressive conditional
45
+ hetroskedasticity (ARCH) model15where the volatility variable changes determin-
46
+ istically depending on the past squared value of the return. Later t he ARCH model
47
+ is generalized by adding also the past volatility dependence to the vola tility change.
48
+ This model is known asthe generalizedARCH(GARCH) model16. The parameters
49
+ of the GARCH model applied to financial time series are conventionally determined
50
+ by the maximum likelihood method. There are many extended versions of GARCH
51
+ models, such as EGARCH17, GJR18, QGARCH19,20models etc., which are de-
52
+ signed to increase the ability to forecast the volatility value.
53
+ The stochastic volatility (SV) model21,22is another model which captures the
54
+ propertiesofthevolatility.IncontrasttotheGARCHmodel,thevo latilityoftheSV
55
+ model changes stochastically in time. As a result the likelihood functio n of the SV
56
+ model is given as a multiple integral of the volatility variables. Such an in tegral in
57
+ general is not analytically calculable and thus the determination of th e parameters
58
+ of the SV model by the maximum likelihood method becomes difficult. To o vercome
59
+ this difficulty in the maximum likelihood method the Markov Chain Monte Ca rlo
60
+ (MCMC) method based on the Bayesian approach is proposed and de veloped21. In
61
+ the MCMC of the SV model one has to update not only the parameter variables
62
+ but also the volatility ones from a joint probability distribution of the p arameters
63
+ and the volatility variables. The number of the volatility variables to be updated
64
+ increases with the data size of time series. The first proposed upda te scheme of
65
+ the volatility variables is based on the local update such as the Metro polis-type
66
+ algorithm21. It is however known that when the local update scheme is used for
67
+ the volatility variables having interactions to their neighbor variables in time, the
68
+ autocorrelationtime ofsampledvolatilityvariablesbecomeslargeand thusthe local
69
+ update scheme becomes ineffective23. In order to improve the efficiency of the local
70
+ update method the blocked scheme which updates several variable s at once is also
71
+ proposed23,24. A recent survey on the MCMC studies of the SV model is seen in
72
+ Ref.25.
73
+ In our study we use the HMC algorithm26which had not been considered
74
+ seriously for the MCMC simulation of the SV model. In finance there ex ists an
75
+ application of the HMC algorithm to the GARCH model27where three GARCH
76
+ parameters are updated by the HMC scheme. It is more interesting to apply the
77
+ HMC for updates of the volatility variables because the HMC algorithm is a global
78
+ update scheme which can update all variables at once. This feature of the HMC
79
+ algorithm can be used for the global update of the volatility variables which can not
80
+ be achieved by the standard Metropolis algorithm. A preliminary stud y28shows
81
+ that the HMC algorithmsamplesthe volatilityvariableseffectively.In t his paperwe
82
+ give a detailed description of the HMC algorithm and examine the HMC alg orithm
83
+ with artificial financial data up to the data size of T=5000. We also ma ke an
84
+ empirical analysis of the Nikkei 225 stock index by the HMC algorithm.November 10, 2018 20:49 WSPC/INSTRUCTION FILE svJCSC3
85
+ Instructions for Typesetting Manuscripts (Condensed Titl e for the Paper) 3
86
+ 2. Stochastic Volatility Model
87
+ The standard version of the SV model21,22is given by
88
+ yt=σtǫt= exp(ht/2)ǫt, (1)
89
+ ht=µ+φ(ht−1−µ)+ηt, (2)
90
+ whereyt= (y1,y2,...,yn) represents the time series data, htis defined by ht= lnσ2
91
+ t
92
+ andσtiscalledvolatility.Wealsocall htvolatilityvariable.Theerrorterms ǫtandηt
93
+ are taken from independent normal distributions N(0,1) andN(0,σ2
94
+ η) respectively.
95
+ We assume that |φ|<1. When φis close to one, the model exhibits the strong
96
+ persistency of the volatility shock.
97
+ For this model the parameters to be determined are µ,φandσ2
98
+ η. Let us use θ
99
+ asθ= (µ,φ,σ2
100
+ η). Then the likelihood function L(θ) for the SV model is written as
101
+ L(θ) =/integraldisplayn/productdisplay
102
+ t=1f(ǫt|σ2
103
+ t)f(ht|θ)dh1dh2...dhn, (3)
104
+ where
105
+ f(ǫt|σ2
106
+ t) =/parenleftbig
107
+ 2πσ2
108
+ t/parenrightbig−1
109
+ 2exp/parenleftbigg
110
+ −y2
111
+ t
112
+ 2σ2
113
+ t/parenrightbigg
114
+ , (4)
115
+ f(h1|θ) =/parenleftBigg
116
+ 2πσ2
117
+ η
118
+ 1−φ2/parenrightBigg−1
119
+ 2
120
+ exp/parenleftbigg
121
+ −[h1−µ]2
122
+ 2σ2η/(1−φ2)/parenrightbigg
123
+ , (5)
124
+ f(ht|θ) =/parenleftbig
125
+ 2πσ2
126
+ η/parenrightbig−1
127
+ 2exp/parenleftbigg
128
+ −[ht−µ−φ(ht−1−µ)]2
129
+ 2σ2η/parenrightbigg
130
+ . (6)
131
+ As seen in Eq.(3), L(θ) is constructed as a multiple integral of the volatility vari-
132
+ ables. For such an integral it is difficult to apply the maximum likelihood me thod
133
+ which estimates values of θby maximizing the likelihood function. Instead of using
134
+ the maximum likelihood method we perform the MCMC simulations based o n the
135
+ Bayesian inference as explained in the next section.
136
+ 3. Bayesian inference for the SV model
137
+ From the Bayes’ rule, the probability distribution of the parameter sθis given by
138
+ f(θ|y) =1
139
+ ZL(θ)π(θ), (7)
140
+ whereZis the normalization constant Z=/integraltext
141
+ L(θ)π(θ)dθandπ(θ) is a prior disti-
142
+ bution of θfor which we make a certian assumption. The values of the paramete rs
143
+ are inferred as the expectation values of θgiven by
144
+ /an}bracketle{tθ/an}bracketri}ht=/integraldisplay
145
+ θf(θ|y)dθ. (8)
146
+ In general this integral can not be performed analytically. For tha t case, one can
147
+ use the MCMC method to estimate the expectation values numerically .November 10, 2018 20:49 WSPC/INSTRUCTION FILE svJCSC3
148
+ 4Authors’ Names
149
+ In the MCMC method, we first generate a series of θwith a probability of
150
+ P(θ) =f(θ|y). Letθ(i)= (θ(1),θ(2),...,θ(k)) be values of θgenerated by the MCMC
151
+ sampling. Then using these kvalues the expectation value of θis estimated by an
152
+ average as
153
+ /an}bracketle{tθ/an}bracketri}ht=1
154
+ kk/summationdisplay
155
+ i=1θ(i). (9)
156
+ The statistical error for kindependent samples is proportional to1√
157
+ k. When the
158
+ sampled data are correlated the statistical error will be proportio nal to/radicalbigg
159
+
160
+ kwhere
161
+ τis the autocorrelation time between the sampled data. The value of τdepends
162
+ on the MCMC sampling scheme we take. In order to reduce the statis tical error
163
+ within limited sampled data it is better to choose an MCMC method which is able
164
+ to generate data with a small τ.
165
+ 3.1.MCMC Sampling of θ
166
+ For the SV model, in addition to θ, volatility variables htalso have to be updated
167
+ sincetheyshouldbeintegratedoutasinEq.(3).Let P(θ,ht)be thejointprobability
168
+ distribution of θandht. ThenP(θ,ht) is given by
169
+ P(θ,ht)∼¯L(θ,ht)π(θ), (10)
170
+ where
171
+ ¯L(θ,ht) =n/productdisplay
172
+ t=1f(ǫt|ht)f(ht|θ). (11)
173
+ For the prior π(θ) we assume that π(σ2
174
+ η)∼(σ2
175
+ η)−1and for others π(µ) =π(φ) =
176
+ constant.
177
+ The MCMC sampling methods for θare given in the following21,22. The prob-
178
+ ability distribution for each parameter can be derived from Eq.(10) b y extracting
179
+ the part including the corresponding parameter.
180
+ •σ2
181
+ ηupdate scheme.
182
+ The probability distribution of σ2
183
+ ηis given by
184
+ P(σ2
185
+ η)∼(σ2
186
+ η)−n
187
+ 2−1exp/parenleftbigg
188
+ −A
189
+ σ2η/parenrightbigg
190
+ , (12)
191
+ where
192
+ A=1
193
+ 2{(1−φ2)(h1−µ)2+n/summationdisplay
194
+ t=2[ht−µ−φ(ht−1−µ)]2}.(13)
195
+ Since Eq.(12) is an inverse gamma distribution we can easily draw a value
196
+ ofσ2
197
+ ηby using an appropriate statistical library in the computer.November 10, 2018 20:49 WSPC/INSTRUCTION FILE svJCSC3
198
+ Instructions for Typesetting Manuscripts (Condensed Titl e for the Paper) 5
199
+ •µupdate scheme.
200
+ The probability distribution of µis given by
201
+ P(µ)∼exp/braceleftbigg
202
+ −B
203
+ 2σ2η(µ−C
204
+ B)2/bracerightbigg
205
+ , (14)
206
+ where
207
+ B= (1−φ2)+(n−1)(1−φ)2, (15)
208
+ and
209
+ C= (1−φ2)h1+(1−φ)n/summationdisplay
210
+ t=2(ht−φht−1). (16)
211
+ µis drawn from a Gaussian distribution of Eq.(14).
212
+ •φupdate scheme.
213
+ The probability distribution of φis given by
214
+ P(φ)∼(1−φ2)1/2exp{−D
215
+ 2σ2η(φ−E
216
+ D)2}, (17)
217
+ where
218
+ D=−(h1−µ)2+n/summationdisplay
219
+ t=2(ht−1−µ)2, andE=/summationtextn
220
+ t=1(ht−µ)(ht−1−µ).(18)
221
+ In order to update φwith Eq.(17), we use the Metropolis-Hastings
222
+ algorithm30,31. Let us write Eq.(17) as P(φ)∼P1(φ)P2(φ) where
223
+ P1(φ) = (1−φ2)1/2, (19)
224
+ P2(φ)∼exp{−D
225
+ 2σ2η(φ−E
226
+ D)2}. (20)
227
+ SinceP2(φ) is a Gaussian distribution we can easily draw φfrom Eq.(20).
228
+ Letφnewbe a candidate given from Eq.(20). Then in order to obtain the
229
+ correct distribution, φnewis accepted with the following probability PMH.
230
+ PMH= min/braceleftbiggP(φnew)P2(φ)
231
+ P(φ)P2(φnew),1/bracerightbigg
232
+ = min/braceleftBigg/radicalBigg
233
+ (1−φ2new)
234
+ (1−φ2),1/bracerightBigg
235
+ .(21)
236
+ In addition to the abovestep we restrict φwithin [−1,1]to avoida negative
237
+ value in the calculation of square root.
238
+ 3.2.Probability distribution for ht
239
+ The probability distribution of the volatility variables htis given by
240
+ P(ht)≡P(h1,h2,...,hn)∼ (22)
241
+ exp/parenleftBig
242
+ −/summationtextn
243
+ i=1{ht
244
+ 2+ǫ2
245
+ t
246
+ 2e−ht}−[h1−µ]2
247
+ 2σ2
248
+ η/(1−φ2)−/summationtextn
249
+ i=2[ht−µ−φ(ht−1−µ)]2
250
+ 2σ2
251
+ η/parenrightBig
252
+ .November 10, 2018 20:49 WSPC/INSTRUCTION FILE svJCSC3
253
+ 6Authors’ Names
254
+ Thisprobabilitydistributionisnotasimplefunction todrawvaluesof ht.Aconven-
255
+ tional method is the Metropolis method30,31which updates the variables locally.
256
+ There are several methods21,22,23,24developed to update htfrom Eq.(22). Here
257
+ we use the HMC algorithm to update htglobally. The HMC algorithm is described
258
+ in the next section.
259
+ 4. Hybrid Monte Carlo Algorithm
260
+ Originallythe HMCalgorithmis developedforthe MCMCsimulationsofthe lattice
261
+ QuantumChromoDynamics(QCD) calculations26. Amajordifficultyofthe lattice
262
+ QCDcalculationsistheinclusionofdynamicalfermions.Theeffectoft hedynamical
263
+ fermions is incorporated by the determinant of the fermion matrix. The computa-
264
+ tional work of the determinant calculation requires O(V3) arithmetic operations29,
265
+ whereVis the volume of a 4-dimensional lattice. A typical size of the volume is
266
+ V >104. The standard Metropolis algorithm which locally updates variables do es
267
+ not work since each local update requires O(V3) arithmetic operations for a deter-
268
+ minant calculation,which results in unacceptable computational cos t in total. Since
269
+ the HMC algorithm is a global update method, the computational cos t remains in
270
+ the acceptable region.
271
+ The basic idea of the HMC algorithm is a combination of molecular dynamic s
272
+ (MD) simulation and Metropolis accept/reject step. Let us conside r to evaluate the
273
+ following expectation value /an}bracketle{tO(x)/an}bracketri}htby the HMC algorithm.
274
+ /an}bracketle{tO(x)/an}bracketri}ht=/integraldisplay
275
+ O(x)f(x)dx=/integraldisplay
276
+ O(x)elnf(x)dx, (23)
277
+ wherex= (x1,x2,...,xn),f(x) is a probability density and O(x) stands for an
278
+ function of x. First we introduce momentum variables p= (p1,p2,...,pn) conjugate
279
+ to the variables xand then rewrite Eq.(23) as
280
+ /an}bracketle{tO(x)/an}bracketri}ht=1
281
+ Z/integraldisplay
282
+ O(x)e−1
283
+ 2p2+lnf(x)dxdp=1
284
+ Z/integraldisplay
285
+ O(x)e−H(p,x)dxdp. (24)
286
+ whereZis a normalization constant given by
287
+ Z=/integraldisplay
288
+ exp/parenleftbigg
289
+ −1
290
+ 2p2/parenrightbigg
291
+ dp, (25)
292
+ andp2stands for/summationtextn
293
+ i=1p2
294
+ i.H(p,x) is the Hamiltonian defined by
295
+ H(p,x) =1
296
+ 2p2−lnf(x). (26)
297
+ Note that the introduction of pdoes not change the value of /an}bracketle{tO(x)/an}bracketri}ht.
298
+ In the HMC algorithm, new candidates of the variables ( p,x) are drawn by
299
+ integrating the Hamilton’s equations of motion,
300
+ dxi
301
+ dt=∂H
302
+ ∂pi, (27)
303
+ dpi
304
+ dt=−∂H
305
+ ∂xi. (28)November 10, 2018 20:49 WSPC/INSTRUCTION FILE svJCSC3
306
+ Instructions for Typesetting Manuscripts (Condensed Titl e for the Paper) 7
307
+ In general the Hamilton’s equations of motion arenot solved analytic ally. Therefore
308
+ wesolvethemnumericallybydoingthe MDsimulation.Let TMD(∆t) beanelemen-
309
+ tary MD step with a step size ∆ t, which evolves ( p(t),x(t)) to (p(t+∆t),x(t+∆t)):
310
+ TMD(∆t) : (p(t),x(t))→(p(t+∆t),x(t+∆t)). (29)
311
+ Any integrator can be used for the MD simulation provided that the f ollowing
312
+ conditions are satisfied26
313
+ •area preserving
314
+ dp(t)dx(t)dx=dp(t+∆t)dx(t+∆t). (30)
315
+ •time reversibility
316
+ TMD(−∆t) : (p(t+∆t),x(t+∆t))→(p(t),x(t)). (31)
317
+ The simplest and often used integrator satisfying the above two co nditions is
318
+ the 2nd order leapfrog integrator given by
319
+ xi(t+∆t/2) =xi(t)+∆t
320
+ 2pi(t)
321
+ pi(t+∆t) =p(t)i−∆t∂H
322
+ ∂xi
323
+ xi(t+∆t) =xi(t+∆t/2)+∆t
324
+ 2pi(t+∆t). (32)
325
+ In this study we use this integrator.The numericalintegration is pe rformedNsteps
326
+ repeatedly by Eq.(32) and in this case the total trajectory length λof the MD is
327
+ λ=N×∆t.
328
+ At the end of the trajectory we obtain new candidates ( p′,x′). These candidates
329
+ are accepted with the Metropolis test, i.e. ( p′,x′) are globally accepted with the
330
+ following probability,
331
+ P= min{1,exp(−H(p′,x′))
332
+ exp(−H(p,x))}= min{1,exp(−∆H)}, (33)
333
+ where∆Histhe energydifferencegivenby∆ H=H(p′,x′)−H(p,x). Sinceweinte-
334
+ grate the Hamilton’s equations of motion approximately by an integra tor, the total
335
+ Hamiltonianisnotconserved,i.e.∆ H/ne}ationslash= 0.Theacceptanceorthe magnitudeof∆ H
336
+ is tuned by the step size ∆ tto obtain a reasonable acceptance. Actually there ex-
337
+ ists the optimal acceptance which is about 60 −70%for 2nd order integrators32,33.
338
+ Surprisingly the optimal acceptance is not dependent of the model we consider. For
339
+ the n-th order integrator the optimal acceptance is expected to be32∼exp/parenleftbigg
340
+ −1
341
+ n/parenrightbigg
342
+ .
343
+ We could also use higher order integrators which give us a smaller ener gy dif-
344
+ ference ∆ H. However the higher order integrators are not always effective sin ce
345
+ they need more arithmetic operations than the lower order integra tors32,33. The
346
+ efficiency of the higher order integrators depends on the model we consider. ThereNovember 10, 2018 20:49 WSPC/INSTRUCTION FILE svJCSC3
347
+ 8Authors’ Names
348
+ also exist improved integrators which have less arithmetic operation s than the con-
349
+ ventional integrators34.
350
+ For the volatility variables ht, from Eq.(22), the Hamiltonian can be defined by
351
+ H(pt,ht) =n/summationdisplay
352
+ i=11
353
+ 2p2
354
+ i+n/summationdisplay
355
+ i=1{hi
356
+ 2+ǫ2
357
+ i
358
+ 2e−hi}+[h1−µ]2
359
+ 2σ2η/(1−φ2)+n/summationdisplay
360
+ i=2[hi−µ−φ(hi−1−µ)]2
361
+ 2σ2η,(34)
362
+ wherepiis defined as a conjugate momentum to hi. Using this Hamiltonian we
363
+ perform the HMC algorithm for updates of ht.
364
+ 5. Numerical Studies
365
+ In order to test the HMC algorithm we use artificial financial time ser ies data
366
+ generatedbythe SVmodel with a setofknownparametersand per formthe MCMC
367
+ simulations to the artificial financial data by the HMC algorithm. We als o perform
368
+ the MCMC simulations by the Metropolis algorithm to the same artificial data and
369
+ compare the results with those from the HMC algorithm.
370
+ Using Eq.(1) with φ= 0.97,σ2
371
+ η= 0.05 andµ=−1 we have generated 5000
372
+ time series data. The time series generated by Eq.(1) is shown in Fig.1. From those
373
+ data we prepared 3 data sets: (1)T=1000 data (the first 1000 of the time series),
374
+ (2)T=2000data (the first 2000ofthe time series)and (3) T=5000 (the whole data).
375
+ To these data sets we made the Bayesian inference by the HMC and M etropolis
376
+ algorithms.Preciselyspeakingboth algorithmsareusedonlyfor the MCMC update
377
+ of the volatility variables. For the update of the SV parameters we u sed the update
378
+ schemes in Sec.3.1.
379
+ For the volatility update in the Metropolis algorithm, we draw a new can didate
380
+ of the volatility variables randomly, i.e. a new volatility hnew
381
+ tis given from the
382
+ previous value hold
383
+ tby
384
+ hnew
385
+ t=hold
386
+ t+δ(r−0.5), (35)
387
+ whereris a uniform random number in [0 ,1) andδis a parameter to tune the
388
+ acceptance. The new volatility hnew
389
+ tis accepted with the acceptance Pmetro
390
+ Pmetro= min/braceleftbigg
391
+ 1,P(hnew
392
+ t)
393
+ P(hold
394
+ t)/bracerightbigg
395
+ , (36)
396
+ whereP(ht) is given by Eq.(22).
397
+ The initial parameters for the MCMC simulations are set to φ= 0.5,σ2
398
+ η= 1.0
399
+ andµ= 0. The first 10000 samples are discarded as thermalization or burn -in
400
+ process. Then 200000samples are recorded for analysis. The tot al trajectory length
401
+ λof the HMC algorithm is set to λ= 1 and the step size ∆ tis tuned so that the
402
+ acceptance of the volatility variables becomes more than 50%.
403
+ First we analyze the sampled volatility variables. Fig.2 shows the Mont e Carlo
404
+ (MC) history of the volatility variable h100fromT= 2000 data set. We take h100
405
+ as the representative one of the volatility variables since we have ob served theNovember 10, 2018 20:49 WSPC/INSTRUCTION FILE svJCSC3
406
+ Instructions for Typesetting Manuscripts (Condensed Titl e for the Paper) 9
407
+ 0 1000 2000 3000 4000 5000t-6-4-20246yt
408
+ Fig. 1. The artificial SV time series used for this study.
409
+ 50000 55000 60000
410
+ Monte Carlo history-2-10123h100HMC
411
+ 50000 55000 60000
412
+ Monte Carlo history-2-10123h100Metropolis
413
+ Fig. 2. Monte Carlo histories of h100generated by HMC (left) and Metropolis (right) with
414
+ T= 2000 data set. The Monte Carlo histories in the window from 5 0000 to 60000 are shown.
415
+ similar behavior for other volatility variables. See also Fig.3 for the sim ilarity of the
416
+ autocorrelation functions of the volatility variables.
417
+ AcomparisonofthevolatilityhistoriesinFig.2clearlyindicatesthatth ecorrela-
418
+ tion of the volatility variable sampled from the HMC algorithm is smaller th an that
419
+ from the Metropolis algorithm. To quantify this we calculate the auto correlation
420
+ function (ACF) of the volatility variable. The ACF is defined as
421
+ ACF(t) =1
422
+ N/summationtextN
423
+ j=1(x(j)−/an}bracketle{tx/an}bracketri}ht)(x(j+t)−/an}bracketle{tx/an}bracketri}ht)
424
+ σ2x, (37)
425
+ where/an}bracketle{tx/an}bracketri}htandσ2
426
+ xare the average value and the variance of xrespectively.
427
+ Fig.3 shows the ACF for three volatility variables, h10,h20andh100sampled
428
+ by the HMC. It is seen that those volatility variables have the similar co rrelation
429
+ behavior. Other volatility variables also show the similar behavior. Thu s hereafter
430
+ we only focus on the volatility variable h100as the representative one.
431
+ Fig.4 compares the ACF of h100by the HMC and Metropolis algorithms. It
432
+ is obvious that the ACF by the HMC decreases more rapidly than that by theNovember 10, 2018 20:49 WSPC/INSTRUCTION FILE svJCSC3
433
+ 10Authors’ Names
434
+ 0 20 40 60 80t0.010.11ACFh10
435
+ h20
436
+ h100
437
+ Fig. 3. Autocorrelation functions of three volatility vari ablesh10,h20andh100sampled by the
438
+ HMC algorithm for T= 2000 data set. These autocorrelation functions show the si milar behavior.
439
+ 0 100 200 300 400 500t0.010.11ACFHMC
440
+ Metropolis
441
+ Fig. 4. Autocorrelation function of the volatility variabl eh100by the HMC and Metropolis
442
+ algorithms for T= 2000 data set.
443
+ Metropolis algorithm. We also calculate the autocorrelation time τintdefined by
444
+ τint=1
445
+ 2+∞/summationdisplay
446
+ t=1ACF(t). (38)
447
+ The results of τintof the volatility variables are given in Table 1. The values in
448
+ the parentheses represent the statistical errors estimated by the jackknife method.
449
+ We find that the HMC algorithm gives a smaller autocorrelation time tha n the
450
+ Metropolis algorithm, which means that the HMC algorithm samples the volatility
451
+ variables more effectively than the Metropolis algorithm.
452
+ Next we analyze the sampled SV parameters. Fig.5 shows MC histories of the
453
+ φparameter sampled by the HMC and Metropolis algorithms. It seems t hat both
454
+ algorithms have the similar correlationfor φ. This similarity is also seen in the ACF
455
+ in Fig.6(left), i.e. both autocorrelation functions decrease in the sim ilar rate with
456
+ timet. The autocorrelation times of φare very large as seen in Table 1. We also
457
+ find the similar behavior for σ2
458
+ η, i.e. both autocorrelation times of σ2
459
+ ηare large.
460
+ On the other hand we see small autocorrelations for µas seen in Fig.6(right).
461
+ Furthermore we observe that the HMC algorithm gives a smaller τintforµthanNovember 10, 2018 20:49 WSPC/INSTRUCTION FILE svJCSC3
462
+ Instructions for Typesetting Manuscripts (Condensed Titl e for the Paper) 11
463
+ φ µ σ2
464
+ η h100
465
+ true 0.97 -1 0.05
466
+ T=1000 HMC 0.973 -1.13 0.053
467
+ SD 0.010 0.51 0.017
468
+ SE 0.0004 0.003 0.001
469
+ 2τint 360(80) 3.1(5) 820(200) 12(1)
470
+ Metropolis 0.973 -1.14 0.053
471
+ SD 0.011 0.40 0.017
472
+ SE 0.0005 0.003 0.0013
473
+ 2τint 320(60) 10.1(8) 720(160) 190(20)
474
+ T=2000 HMC 0.978 -0.92 0.053
475
+ SD 0.007 0.26 0.012
476
+ SE 0.0003 0.001 0.0009
477
+ 2τint 540(60) 3(1) 1200(150) 18(1)
478
+ Metropolis 0.978 -0.92 0.052
479
+ SD 0.007 0.26 0.011
480
+ SE 0.0003 0.003 0.0009
481
+ 2τint 400(100) 13(2) 1000(270) 210(50)
482
+ T=5000 HMC 0.969 -1.00 0.056
483
+ SD 0.005 0.11 0.009
484
+ SE 0.0003 0.0004 0.0007
485
+ 2τint 670(100) 4.2(7) 1250(170) 10(1)
486
+ Metropolis 0.970 -1.00 0.054
487
+ SD 0.005 0.12 0.008
488
+ SE 0.00023 0.0011 0.0005
489
+ 2τint 510(90) 30(10) 960(180) 230(28)
490
+ Table 1. Results estimated by the HMC and Metropolis algorit hms.SDstands for Standard
491
+ Deviation and SEstands for Statistical Error. The statistical errors are es timated by the jackknife
492
+ method. We observe no significant differences on the autocorr elation times among three data sets.
493
+ that of the Metropolis algorithm, which means that HMC algorithm sam plesµ
494
+ more effectively than the Metropolis algorithm although the values of τintforµ
495
+ take already very small even for the Metropolis algorithm.
496
+ The values of the SV parameters estimated by the HMC and the Metr opolis
497
+ algorithms are listed in Table 1. The results from both algorithms well r eproduce
498
+ the true values used for the generation of the artificial financial d ata. Furthermore
499
+ for each parameter and each data set, the estimated parameter s by the HMC and
500
+ the Metropolis algorithms agree well. And their standard deviations a lso agree
501
+ well. This is not surprising because the same artificial financial data, thus the same
502
+ likelihood function is usedfor both MCMC simulationsby the HMC and Met ropolis
503
+ algorithms. Therefore they should agree each other.November 10, 2018 20:49 WSPC/INSTRUCTION FILE svJCSC3
504
+ 12Authors’ Names
505
+ 40000 45000 50000
506
+ MC history0.940.950.960.970.980.991φ
507
+ HMC
508
+ 40000 45000 50000
509
+ MC history0.940.950.960.970.980.991φ
510
+ Metropolis
511
+ Fig. 5. Monte Carlo histories of φgenerated by HMC (left) and Metropolis (right) for T= 2000
512
+ data set.
513
+ 0 1000t0.010.11ACFHMC
514
+ Metropolis
515
+ 0 100 200 300t0.0010.010.1 ACFHMC
516
+ Metropolis
517
+ Fig. 6. Autocorrelation functions of φ(left) and µ(right) by the HMC and Metropolis algorithm
518
+ forT= 2000 data set.
519
+ 6. Empirical Analysis
520
+ In this section we make an empirical study of the SV model by the HMC algorithm.
521
+ The empirical study is based on daily data of the Nikkei 225 stock inde x. The
522
+ sampling period is 4 January 1995 to 30 December 2005 and the numbe r of the
523
+ observations is 2706. Fig.7(left) shows the time series of the data. Letpibe the
524
+ Nikkei 225 index at time i. The Nikkei 225 index piare transformed to returns as
525
+ ri= 100ln( pi/pi−1−¯s), (39)
526
+ where ¯sis the average value of ln( pi/pi−1). Fig.7(right) shows the time series of
527
+ returns calculated by Eq.(39). We perform the same MCMC sampling b y the HMC
528
+ algorithm as in the previous section. The first 10000 MC samples are d iscarded and
529
+ then 20000 samples are recorded for the analysis. The ACF of samp ledh100and
530
+ sampled parameters are shown in Fig.8. Qualitatively the results of t he ACF are
531
+ similar to those from the artificial financial data, i.e. the ACF of the v olatility and
532
+ µdecrease quickly although the ACF of φandσ2
533
+ ηdecrease slowly. The estimated
534
+ values of the parameters are summarized in Table 2. The value of φis estimated to
535
+ beφ≈0.977. This value is very close to one, which means the time series has th e
536
+ strong persistency of the volatility shock. The similar values are also seen in theNovember 10, 2018 20:49 WSPC/INSTRUCTION FILE svJCSC3
537
+ Instructions for Typesetting Manuscripts (Condensed Titl e for the Paper) 13
538
+ HMC φ µ σ2
539
+ η h100
540
+ 0.977 0.52 0.020
541
+ SD 0.006 0.13 0.005
542
+ SE 0.001 0.0016 0.001
543
+ 2τint560(190) 4(1) 1120(360) 21(5)
544
+ Table 2. Results estimated by the HMC for the Nikkei 225 index data.
545
+ 050010001500200025003000t10000150002000025000
546
+ Nikkei 225 Index
547
+ 050010001500200025003000t-505rt
548
+ Fig. 7. Nikkei 225 stock index from 4 January 1995 to 30 Decemb er 2005(left) and returns(right).
549
+ 0 20 40 60t0.010.11ACFh100
550
+ 0 200 400 600800 1000t0.010.11ACFφ
551
+ ση2
552
+ µ
553
+ Fig. 8. Autocorrelation functions of the volatility variab leh100(left) and the sampled parameters
554
+ (right).
555
+ previous studies21,22.
556
+ 7. Conclusions
557
+ We applied the HMC algorithm to the Bayesian inference of the SV mode l and
558
+ examined the property of the HMC algorithm in terms of the autocor relation times
559
+ of the sampled data. We observed that the autocorrelation times o f the volatility
560
+ variables and µparameter are small. On the other hand large autocorrelation times
561
+ are observed for the sampled data of φandσ2
562
+ ηparameters. The similar behavior
563
+ for the autocorrelation times are also seen in the literature22.
564
+ From comparison of the HMC and Metropolis algorithms we find that th e HMCNovember 10, 2018 20:49 WSPC/INSTRUCTION FILE svJCSC3
565
+ 14Authors’ Names
566
+ algorithmsamplesthevolatilityvariablesand µmoreeffectivelythantheMetropolis
567
+ algorithm. However there is no significant difference for φandσ2
568
+ ηsampling. Since
569
+ the autocorrelation times of µfor both algorithms are estimated to be rather small
570
+ the improvement of sampling µby the HMC algorithm is limited. Therefore the
571
+ overall efficiency is considered to be similar to that of the Metropolis a lgorithm.
572
+ By using the artificial financial data we confirmed that the HMC algor ithm cor-
573
+ rectly reproduces the true parameter values used to generate t he artificial financial
574
+ data. Thus it is concluded that the HMC algorithm can be used as an alt ernative
575
+ algorithm for the Bayesian inference of the SV model.
576
+ If we are only interested in parameter estimations of the SV model, t he HMC
577
+ algorithm may not be a superior algorithm. However the HMC algorithm samples
578
+ thevolatilityvariableseffectively.ThustheHMC algorithmmayservea sanefficient
579
+ algorithm for calculating a certain quantity including the volatility varia bles.
580
+ Acknowledgments.
581
+ The numerical calculations were carried out on SX8 at the Yukawa In stitute for
582
+ Theoretical Physics in Kyoto University and on Altix at the Institute of Statistical
583
+ Mathematics.
584
+ Note added in proof. After this work was completed the author noticed a sim-
585
+ ilar approach by Liu35. The author is grateful to M.A. Girolami for drawing his
586
+ attention to this.
587
+ References
588
+ 1. R.Mantegna and H.E.Stanley, Introduction to Econophysics (Cambride University
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+ Press, 1999).
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+ 2. R. Cont, Empirical Properties of Asset Returns: Stylized Facts and Statistical Issues,
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+ 35. J.S. Liu, Monte Carlo Strategies in Scientific Computing (Springer, 2001).
1001.0025.txt ADDED
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1
+ arXiv:1001.0025v1 [cs.CR] 30 Dec 2009GNSS-based Positioning: Attacks and Countermeasures
2
+ Panos Papadimitratos and Aleksandar Jovanovic
3
+ EPFL
4
+ Switzerland
5
+ Email: firstname.lastname@epfl.ch
6
+ Abstract
7
+ Increasing numbers of mobile computing devices, user-
8
+ portable, or embedded in vehicles, cargo containers, or the
9
+ physical space, need to be aware of their location in order
10
+ to provide a wide range of commercial services. Most often,
11
+ mobile devices obtain their own location with the help of
12
+ Global Navigation Satellite Systems (GNSS), integrating,
13
+ for example, a Global Positioning System (GPS) receiver.
14
+ Nonetheless, an adversary can compromise location-aware
15
+ applications by attacking the GNSS-based positioning: It
16
+ can forge navigation messages and mislead the receiver into
17
+ calculating a fake location. In this paper, we analyze this
18
+ vulnerability and propose and evaluate the effectiveness of
19
+ countermeasures. First, we consider replay attacks, which
20
+ can be effective even in the presence of future cryptographic
21
+ GNSS protection mechanisms. Then, we propose and an-
22
+ alyze methods that allow GNSS receivers to detect the re-
23
+ ception of signals generated by an adversary, and then re-
24
+ ject fake locations calculated because of the attack. We
25
+ consider three diverse defense mechanisms, all based on
26
+ knowledge, in particular, own location ,time, andDoppler
27
+ shift, receivers can obtain prior to the onset of an attack.
28
+ We find that inertial mechanisms that estimate location
29
+ can be defeated relatively easy. This is equally true for the
30
+ mechanism that relies on clock readings from off-the-shelf
31
+ devices; as a result, highly stable clocks could be needed.
32
+ On the other hand, our Doppler Shift Test can be effective
33
+ without any specialized hardware, and it can be applied to
34
+ existing devices.
35
+ 1 Introduction
36
+ As wireless communications enable an ever-broadening
37
+ spectrum of mobile computing applications, location or
38
+ position information becomes increasingly important for
39
+ those systems. Devices need to determine their own posi-
40
+ tion,1to enable location-based or location-aware function-
41
+ ality and services. Examples of such systems include: sen-
42
+ sors reporting environmental measurements; cellular tele -
43
+ phones or portable digital assistants (PDAs) and comput-
44
+ ers offering users information and services related to their
45
+ 1In this paper, we are not concerned with the related but ortho g-
46
+ onal localization problem of allowing a specific entity to de termine
47
+ and ascertain the location of other devices.surroundings; mobile embedded units, such as those for
48
+ Vehicular Communication (VC) systems seeking to pro-
49
+ vide transportation safety and efficiency; or, merchandize
50
+ (container) and fleet (truck) management systems.
51
+ Global navigation satellite systems (GNSS), such as the
52
+ Global Positioning System (GPS), its Russian counter-
53
+ part (GLONAS), and the upcoming European GALILEO
54
+ system, are the most widely used positioning technology.
55
+ GNSS transmit signals bearing reference information from
56
+ a constellation of satellites; computing platforms nodes),
57
+ equipped with the appropriate receiver, can decode them
58
+ and determine their own location.
59
+ However, commercial instantiations of GNSS systems,
60
+ which are within the scope of this paper, are open to
61
+ abuse: An adversary can influence the location informa-
62
+ tion,loc(V), a node Vcalculates, and compromise the node
63
+ operation. For example, in the case of a fleet management
64
+ system, an adversary can target a specific truck. First, the
65
+ adversary can use a transmitter of forged GNSS signals
66
+ that overwrite the legitimate GNSS signals to be received
67
+ by the victim node (truck) V. This would cause a false
68
+ loc(V) to be calculated and then reported to the fleet cen-
69
+ ter, essentially concealing the actual location of Vfrom the
70
+ fleet management system. Once this is achieved, physical
71
+ compromise of the truck (e.g., breaking into the cargo or
72
+ hijacking the vehicle) is possible, as the fleet management
73
+ system would have limited or no ability to protect its as-
74
+ sets.
75
+ This is an important problem, given the consequences
76
+ such attacks can have. In this paper, we are concerned
77
+ with methods to mitigate such a vulnerability. In partic-
78
+ ular, we propose mechanisms to detect and reject forged
79
+ GNSS messages, and thus avoid manipulation of GNSS-
80
+ based positioning. Our investigation is complementary
81
+ to cryptographic protection, which commercial GNSS sys-
82
+ tems do not currently provide but are expected to do so
83
+ in the future (e.g., authentication services by the upcom-
84
+ ing GALILEO system [5]). Our approach is motivated by
85
+ the fundamental vulnerability of GNSS-based positioning
86
+ toreplay attacks [9], which can be mounted even against
87
+ cryptographically protected GNSS.
88
+ The contribution of this paper consists of three mecha-
89
+ nisms that allow receivers to detect forged GNSS messages
90
+ and fake GNSS signals. Our countermeasures rely on in-
91
+ formation the receiver obtained before the onset of an at-
92
+ 1tack, or more precisely, before the suspected onset of an
93
+ attack. We investigate mechanisms that rely on own (i)
94
+ location information, calculated by GNSS navigation mes-
95
+ sages, (ii) clock readings, without any re-synchronization
96
+ with the help of the GNSS or any other system, and (iii)
97
+ received GNSS signal Doppler shift measurements. Based
98
+ on those different types of information, our mechanisms
99
+ can detect if the received GNSS signals and messages orig-
100
+ inate from adversarial devices. If so, location informatio n
101
+ induced by the attack can be rejected and manipulation
102
+ of the location-aware functionality be avoided. We clarify
103
+ that the reaction to the detection of an attack, and mecha-
104
+ nisms that mitigate unavailability of legitimate GNSS sig-
105
+ nals is out of the scope of this paper.
106
+ We briefly introduce the GNSS operation and related
107
+ work in Sec. 2. We discuss the adversary model and specific
108
+ attack methods in Sec. 3.2. We then present and analyze
109
+ the three defensive mechanisms in Sec. 4. Our findings
110
+ support that highly accurate clocks can be very effective
111
+ at the expense of appropriate clock hardware; but they
112
+ can otherwise be susceptible, when off-the-shelf hardware
113
+ is used. Location-based mechanisms can also be defeated
114
+ relatively easily. On the contrary, our Doppler Shift Test
115
+ (DST) provides accurate detection of attacks, even against
116
+ a sophisticated adversary.
117
+ 2 GNSS Overview
118
+ 2.1 Basic Operation
119
+ Each GNSS-equipped node Vcan receive simultaneously
120
+ a set of navigation messages NAV ifrom each satellite Si
121
+ in the visible constellation . Satellite transmitters utilize a
122
+ spread-spectrum technique and each satellite is assigned a
123
+ unique spreading code Ci. These codes are a priori pub-
124
+ licly known. Navigation messages allow Vto determine its
125
+ position, loc(V) = (XV, YV, ZV), in a Cartesian system, as
126
+ well global time, by obtaining a clock correction or time
127
+ offset,tV, also called the synchronization error . At least
128
+ four satellites should be visible in order for a receiver to
129
+ compute position and exact time, the so-called PVT (Po-
130
+ sition, Velocity and Time) or navigation solution [6]. This
131
+ computation relies on the pseudo-range measurements per-
132
+ formed by V, one pseudo-range per visible satellite, that is,
133
+ estimating the satellite-receiver distance based on the es ti-
134
+ mated signal propagation delay, ρi. For each pseudo-range
135
+ ρiestimated at V, the following equation is formed:
136
+ ρi=|si−loc(V)|+c·tV (1)
137
+ The satellite Siposition is si, the receiver position is
138
+ loc(V),cis the speed of light, and tVis the synchronization
139
+ error for V.2.2 Future Cryptographic GNSS Protec-
140
+ tion
141
+ Cryptographic protection ensures the authenticity and in-
142
+ tegrity of GNSS messages, i.e., ensures that NAV messages
143
+ generated solely by GNSS entities, with no modification,
144
+ are accepted and used by nodes. Currently, cryptography is
145
+ used in military systems, but it is not available for commer-
146
+ cial systems to provide authenticity and integrity. Public
147
+ or asymmetric key cryptography is a flexible and scalable
148
+ approach that does not require tamper-resistant receivers .2
149
+ Independently of the number of receivers present in the sys-
150
+ tem (possibly, millions or eventually hundreds of millions ),
151
+ a pair of private/public keys ki, Kican be assigned to each
152
+ satellite Si, with the public key bound to the satellite iden-
153
+ tity via a certificate provided by a Certification Authority.
154
+ Each receiver obtains the certified public keys of all satel-
155
+ lites in order to be able to validate NAV messages digitally
156
+ signed with the corresponding ki.Navigation Message Au-
157
+ thentication (NMA) [5] will be available as a GALILEO
158
+ service.
159
+ To further enhance protection, a different public-key
160
+ NMA approach was proposed in [7]. Each Sichooses a
161
+ secret spreading code for each NAV message but discloses
162
+ this, along with a hidden timing marker , in a delayed and
163
+ authenticated manner to the receiving nodes. If nodes can
164
+ maintain accurate clocks by means other than the GNSS
165
+ system alone, they can then safely detect messages that are
166
+ forged or replayed between the time of their creation and
167
+ the code disclosure. A similar idea using Secret Spreading
168
+ Codes (SSC) was presented in [11].
169
+ 3 Attacking GNSS
170
+ 3.1 Adversary model
171
+ The location (position) GNSS-equipped nodes obtain can
172
+ be manipulated by an external adversary, without any ad-
173
+ versarial control on the GNSS entities (the system ground
174
+ stations, the satellites, the ground-to-satellite commun ica-
175
+ tion, and the receiver). If any cryptographic protection
176
+ is present, we assume that cryptographic primitives are
177
+ not breakable and that the private keys of satellites can-
178
+ not be compromised. The adversary can receive signals
179
+ from all available satellites (depending on the locations o f
180
+ the adversary-controlled receivers). It is also fully awar e
181
+ of the GNSS implementation specifics and thus can pro-
182
+ duce fully compliant signals, i.e., with the same modula-
183
+ tion, transmission frequency equal to the nominal one, ft,
184
+ or any frequency in the range of received ones, fr; similarly,
185
+ transmitted and received signal powers, as well as message
186
+ preambles and body format (header, content).
187
+ We classify adversaries based on their ability to re-
188
+ produce GNSS messages and signals, considering ones
189
+ equipped with:
190
+ 2To prevent the compromise of a single, system-wide symmetri c
191
+ key, shared among the GNSS and all nodes.
192
+ 21. Single or multiple radios, each transmitting at the
193
+ same constant power, Pc
194
+ t, and frequency fc
195
+ t.
196
+ 2. Single or multiple radios, each being ability to adapt
197
+ its transmission frequency, fj
198
+ t, over time; jis an index
199
+ of adversarial radios.
200
+ 3. Multiple radios with adaptive transmission capabili-
201
+ ties as above, and additionally the ability to estab-
202
+ lish fast communication among any of the adversarial
203
+ nodes equipped with those radios.
204
+ Adversarial radios in all above cases can record GNSS
205
+ signals and navigation messages for long periods. For all
206
+ adversaries above, we consider a nominal range R, within
207
+ which adversarial transmissions can be received, with this
208
+ value varying for different adversarial radios. We denote
209
+ this as the area under attack . Clearly, the more powerful
210
+ and the more numerous radios an adversary has, the higher
211
+ its potential impact can be. In the sense, it can influence a
212
+ larger system area and potentially mislead more receivers.
213
+ We assume that the area under attack does not coin-
214
+ cide with the wireless system area. In other words, the
215
+ adversary has limited physical presence and communica-
216
+ tion capabilities. This implies that nodes can lock on ac-
217
+ tual GNSS signals for a period of time before entering an
218
+ area under attack. We do not dwell on how frequently and
219
+ under what circumstances nodes are under attack. Rather,
220
+ we investigate the strength of different defense mechanisms
221
+ given that a node is under attack. We abstract the phys-
222
+ ical properties of the adversarial equipment and consider
223
+ the periods of time it can cause unavailability and maintain
224
+ the receiver locked on the spoofed signal.
225
+ We emphasize that our attack model is notthe worst
226
+ case; this would be a receiver under attack during its cold
227
+ start, that is, the first time it is turned on and searches for
228
+ GNSS signals to lock on. However, our adversary model
229
+ corresponds to a broad range of realistic cases and it is a
230
+ powerful one. For example, returning to the cargo example
231
+ of the introduction: It will be hard for an adversary to
232
+ control a receiver from its installation, e.g., on a contain er,
233
+ and then throughout a trip. But it would be rather easy to
234
+ select a location and time to mount its attack. Regarding
235
+ the strength of the attacker, it is noteworthy that attacks
236
+ are possible without any physical access to and without
237
+ tampering with the victim node(s) software and hardware.
238
+ 3.2 Mounting Attacks against GNSS Re-
239
+ ceivers
240
+ The adversary can construct a transmitter that emits sig-
241
+ nals identical to those sent by a satellite, and mislead the
242
+ receiver that signals originate from a visible satellite. H ow-
243
+ ever, the attacker has to first force the receiver to lose
244
+ its “lock” on the satellite signals. This can be achieved
245
+ byjamming legitimate GNSS signals, by transmitting a
246
+ sufficiently powerful signal that interferes with and ob-
247
+ scures the GNSS signals [12]. Jammers are simple to con-
248
+ struct with low cost and very effective: for example, withReceived GNSS signal delayed
249
+ Transmit after treplay NAV message buffering
250
+ Preamble
251
+ detection
252
+ Victim receiver
253
+ V
254
+ Total
255
+ delay treplay Adversary
256
+ Figure 1: Illustration of the replay attack: the adversary
257
+ captures and replays the signal after some time treplay =
258
+ tmin
259
+ replay +τ, with the τ≥0 chosen by the adversary, and
260
+ tmin
261
+ replay >0 imposed by the specifics of the attack configu-
262
+ ration and the adversary capabilities.
263
+ 1 Watt of transmission power, the reception of GNSS sig-
264
+ nals is stopped within a radius of approximately 35 km
265
+ radius [6,12].
266
+ Then, the adversary can spoof GNSS signals, i.e., forge
267
+ and transmit signals at the same frequency and with power
268
+ thatexceeds that of the legitimate GNSS signal at the re-
269
+ ceiver’s antenna. Satellite simulators are capable of broa d-
270
+ casting simultaneously signals carrying counterfeit navi ga-
271
+ tion data from ten satellites.3The spoofed signal can also
272
+ be generated by manipulating and rebroadcasting actual
273
+ signals ( meaconing ). As long as the lock of the victim re-
274
+ ceiver Von the spoofed signal persists, loc(V) is under the
275
+ influence or full control of the adversary.
276
+ Apart from jamming, the adversary could take advan-
277
+ tage of gaps in coverage , i.e., areas and periods of time for
278
+ which Vcannot lock on to more than three satellite sig-
279
+ nals. Clearly, this can be often possible in urban areas or
280
+ because of the terrain, such as tunnels or obstructions from
281
+ high-rise buildings. We do not consider further this case,
282
+ as such loss of satellite signals is not under the control of
283
+ the attacker. Nonetheless, the tests we propose here are ef-
284
+ fective independently of what causes receivers to loose loc k
285
+ on GNSS signals.
286
+ 3.3 Replay attack
287
+ Thereplay attack can be viewed as a part of a more general
288
+ class of relay attacks : the attacker receives at one location
289
+ legitimate GNSS signals, relays those to another location
290
+ 3The adversary can deceive the receiver after down-conversi on
291
+ of the satellite signal, with one component in-phase and one in-
292
+ quadrature:
293
+ I(t) =aiCa(t)M(t)cos(ft) (2)
294
+ Q(t) =aqCa(t)M(t)sin(ft) (3)
295
+ Cais the C/A (Course/Aquisition) code, M(t) is the NAV message,
296
+ and coefficients aiandaqrepresent the signal attenuation. The at-
297
+ tacker could pick the amplifying coefficients aiandaqsuch that the
298
+ received signal power exceeds the nominal power od a GPS sign al [13].
299
+ 3where it retransmits them without any modification. This
300
+ way the adversary can avoid detection if cryptography is
301
+ employed, while it can “present” a victim with GNSS sig-
302
+ nals that are not normally visible at the victim’s location.
303
+ In this paper, we abstract away the placement of adversar-
304
+ ial nodes, and we characterize the replay attack by two fea-
305
+ tures: (i) the adversarial node capability to receive, reco rd
306
+ and replay GNSS signals, and (ii) the delay treplay between
307
+ reception and re-transmission of a signal.
308
+ The GNSS signal reception and replay can be done
309
+ at the message or symbol level, or it can be done by
310
+ recording the entire frequency band and replaying it with-
311
+ out de-spreading signals. The latter, more involved and
312
+ thus costly, would enable the attacker to mount an at-
313
+ tack against the delayed-disclosure secret spreading code
314
+ approach, as pointed out in [7], not only for long replay-
315
+ ing delays but also for very short ones. Clearly, such an
316
+ instantiation of the replaying attack implies a more sophis -
317
+ ticated adversary than one replaying symbols or messages.
318
+ For example, the adversary would need to infer, possibly by
319
+ possessing a legitimate receiver, the start of NAV messages
320
+ to replay signals accordingly
321
+ Thetreplay delay between reception and re-transmission
322
+ depends on the attack configuration (e.g., the distance be-
323
+ tween the receiving and re-transmitting adversarial radio s,
324
+ the physics of the signal propagation, and, when applica-
325
+ ble, the delay for the adversary to decode the GNSS signal).
326
+ We capture such factors by considering tmin
327
+ replay >0, a min-
328
+ imum delay that the adversary cannot avoid. Beyond this,
329
+ the attacker can choose some additional delay τ≥0, such
330
+ that it replays the signal after treplay =tmin
331
+ replay +τ. We
332
+ illustrate a replay attack in Fig. 1: The recording of the
333
+ NAV message starts after its beginning is detected, due to
334
+ the preamble 10001011, with length of eight chips, and the
335
+ decoding of the NAV message first bit. This corresponds
336
+ totmin
337
+ replay = 20ms: the transmission rate of 50 bit/s implies
338
+ that 20ms are needed for the first bit to be received by an
339
+ adversarial radio.
340
+ The adversary can choose different treplay values for sig-
341
+ nals from different satellites, even though “blind” replayi ng
342
+ of all NAV signals with the same delay can be effective. The
343
+ selection of which signals (from which satellites) to relay of-
344
+ fer flexibility. But even the “blind” replaying of all NAV
345
+ signals (the entire band) can be effective: treplay controls
346
+ the “shift” in the PVT solution. Essentially, treplay con-
347
+ trols the “shift” in the PVT solution the adversary induces
348
+ to the victim node(s).
349
+ Fig. 2 shows the impact of a replay attack as a function
350
+ of the spoofing stage of the attack: (i) the location offset
351
+ or error, i.e., the distance between the attack-induced and
352
+ the actual victim receiver position, and (ii) the time offset
353
+ or error, that is, the time difference between the attack-
354
+ induced clock value and the actual time. We consider for
355
+ this example trelay= 20ms, as the first bit decoding de-
356
+ lay dwarfs the preamble detection and propagation delays.
357
+ This is indeed a very subtle attack we refer to [9] for a range
358
+ oftreplay values, which shows that the larger the treplay, as0 50 100 150 200 250 300010002000300040005000600070008000900010000
359
+ Attack duration [s]Distance offset [m]
360
+ (a)
361
+ 0 50 100 150 200 250 300050100150200250300350
362
+ Attack duration [s]Time offset [ms]
363
+
364
+ (b)
365
+ Figure 2: Impact of the replay attack, as a function of
366
+ thespoofing attack duration. (a) Location offset or er-
367
+ ror: Distance between the attack-induced and the actual
368
+ victim receiver position. (b) Time offset or error: Time
369
+ difference between the attack-induced clock value and the
370
+ actual time.
371
+ the adversary tunes its τvalue, the higher the location and
372
+ time offsets.
373
+ Even for a very low treplay, while the mobile node re-
374
+ ceiver is still locked on the attacker-transmitted signals , the
375
+ location error increases, with the victim receiver “dragge d”
376
+ away from its actual position. Each millisecond of trelay
377
+ translates approximately into 300m of location offset for
378
+ each pseudorange (as the speed of light, c, is taken into
379
+ account), with the actual “displacement” of the victim de-
380
+ pending on the geometry (e.g., position of the satellite
381
+ whose signals were replayed).
382
+ As for the time offset, which can be viewed as a side-
383
+ effect of the attack: it is in the order of less than one mil-
384
+ lisecond per second, and it can very well go easily unnoticed
385
+ by the user. With a given trelay, every time the victim re-
386
+ ceiver re-synchronizes, typically at the end of a NAV mes-
387
+ sage that lasts 30 sec, treplay will emerge as tVfrom the
388
+ PVT solution and thus will be accumulated as part of the
389
+ time offset shown in Fig. 2.
390
+ 4 Defense mechanisms
391
+ We investigate three defense mechanisms that rely on a
392
+ common underlying three-step idea. First, the receiver col -
393
+ lects data for a given parameter during periods of time it
394
+ deems it is not under attack; we term this the normal mode .
395
+ 4Second, based on the normal mode data, the receiver pre-
396
+ dicts the value of the parameter in the future. When it
397
+ suspects it is under attack, it enters what we term alert
398
+ mode. In this mode, the receiver compares the predicted
399
+ values with the ones it obtains from the GNSS functional-
400
+ ity. If the GNSS-obtained values differ, beyond a protocol-
401
+ selectable threshold, from the predicted ones, the receive r
402
+ deems it is under attack . In that case, all PVT solutions
403
+ obtained in alert mode are discarded. Otherwise, the sus-
404
+ pected PVT solutions are accepted and the receiver reverts
405
+ to the normal mode.
406
+ In this work, we consider three parameters: location ,
407
+ time, andDoppler Shift , and we present the corresponding
408
+ detection mechanisms, Location Inertial Test ,Clock Offset
409
+ Test, andDoppler Shift Test . We emphasize again that all
410
+ three mechanisms rely on the availability of prior informa-
411
+ tion collected in normal mode. But they are irrelevant if
412
+ the receiver starts its operation without any such informa-
413
+ tion (i.e., a cold start ).
414
+ To evaluate the proposed schemes, we use GPS traces
415
+ collected by an ASHTECH Z-XII3T receiver that out-
416
+ puts observation and navigation (.obs and .nav) data into
417
+ RINEX ( Receiver Independent Exchange Format ) [8]. We
418
+ implement the PVT solution functionality in Matlab, ac-
419
+ cording to the receiver interface specification [8]. Our im-
420
+ plementation operates on the RINEX data, which include
421
+ pseudoranges and Doppler frequency shift and phase mea-
422
+ surements. We simulate the movement of receivers over a
423
+ period of T= 300 s, with their position updated at steps of
424
+ Tstep= 1sec.
425
+ 4.1 Location Inertial Test
426
+ At the transition to alert mode, the node utilizes own lo-
427
+ cation information obtained from the PVT solution, to
428
+ predict positions while in attack mode. If those positions
429
+ match the suspected as fraudulent PVT ones, the receiver
430
+ returns to normal mode. We consider two approaches for
431
+ the location prediction: (i) inertial sensors and (ii) Kalm an
432
+ filtering.
433
+ Inertial sensors , i.e., altimeters, speedometers, odome-
434
+ ters, can calculate the node (receiver) location indepen-
435
+ dently of the GNSS functionality.4However, the accuracy
436
+ of such (electro-mechanical) sensors degrades with time.
437
+ One example is the low-cost inertial MEMS Crista IMU-15
438
+ sensor (Inertial Measurement Unit).
439
+ Fig. 3 shows the position error as a function of time [4],
440
+ which is in our context corresponds to the period the re-
441
+ ceiver is in the alert mode. As the inertial sensor inaccurac y
442
+ increases, the node has to accept as normal attack-induced
443
+ locations. Fig. 4 shows a two-dimensional projection of
444
+ two trajectories, the actual one and the estimated and er-
445
+ roneously accepted one. We see that over a short period
446
+ 4They have already been used to provide continuous navigatio n
447
+ between the update periods for GNSS receivers, which essent ially are
448
+ discrete-time position/time sensors with sampling interv al of approx-
449
+ imately one second0102030405060708090100050100150200250300
450
+ GNSS unavailability period [s]Inertial navigation error [m]
451
+
452
+ Figure 3: Location error of Crista IMU-15 inertial sensor,
453
+ as a function of the GNSS unavailability period.
454
+ 3.456 3.458 3.46 3.462 3.464 3.466 3.468
455
+ x 1065.295.35.315.325.335.345.355.365.375.38x 105
456
+
457
+ X coordinate [m] Y coordinate [m]
458
+ Attacker−induced trajectory
459
+ Actual trajectory
460
+ Figure 4: Illustration of location error using inertial sen -
461
+ sors: Actual vs. estimated when under attack trajectory.
462
+ of time, a significant difference is created because of the
463
+ attack.
464
+ A more effective approach is to rely on Kalman filtering
465
+ of location information obtained during normal mode. Pre-
466
+ dicted locations can be obtained by the following system
467
+ model:
468
+ Sk+1= Φ kSk+Wk (4)
469
+ withSkbeing the system state, i.e., location ( Xk, Yk, Zk)
470
+ and velocity ( V xk, V yk, V zk) vectors, Φ kthe transition
471
+ matrix, and Wkthe noise. Fig. 5 illustrates the location
472
+ offset for a set of various trajectories. Unlike the case that
473
+ only inertial sensors are used, with measurements of iner-
474
+ tial sensors (with the error characteristics of Fig. 3 used
475
+ as data when GNSS signals are unavailable, filtering pro-
476
+ vides a linearly increasing error with the period of GNSS
477
+ unavailability.
478
+ Overall, for short unavailability periods, inertial mech-
479
+ anisms can be effective. As long as the error (Y axes of
480
+ Figs. 4, 5) does not grow significantly, the replay attack
481
+ can be detected. But for sufficiently high errors, the re-
482
+ play attack impact can remain undetected. We remind the
483
+ reader that the x-axes in Fig. 2 provide the duration of the
484
+ spoofing attack - the transmission (replay) of GNSS signals
485
+ - and they are not to be confused with the duration of the
486
+ GNSS period of unavailability in the x-axis of Figs. 4, 5.
487
+ 50 50 100 150 200 250 300020040060080010001200
488
+ Time [s]Distance offset [m]
489
+ Figure 5: Distance error of inertial mechanisms with
490
+ Kalman filtering, as a function of the GNSS unavailabil-
491
+ ity period.
492
+ 0 5 10 15 20 25 30−9−8.5−8−7.5−7−6.5−6x 10−3
493
+ Time [30s step]Time offset [s]
494
+
495
+ Figure 6: Clock offset for the ASHTECH Z-XII3T receiver,
496
+ during a 900 sec period with no re-synchronization.
497
+ 4.2 Clock Offset Test
498
+ Each receiver has a clock that is in general imprecise, due
499
+ to the drift errors of the quartz crystal. If the reception
500
+ of GNSS signals is disrupted, the oscillator switches from
501
+ normal to holdover mode. Then, the time accuracy de-
502
+ pends only on the stability of the local oscillator [2,6]. Th e
503
+ quartz crystals of different clocks run at slightly different
504
+ frequencies, causing the clock values to gradually diverge
505
+ from each other (skew error).
506
+ A simulation based study [2] of quartz clocks claims that
507
+ coarse time synchronization can be maintained at microsec-
508
+ ond accuracy without GPS reception for 350 sec in 95%
509
+ cases. This means that quartz oscillators can maintain
510
+ millisecond synchronization for few hours, including ran-
511
+ dom errors and temperature change inaccuracies. Indeed,
512
+ in such a case, the adversary would need to cause GNSS
513
+ availability for long periods of time, for example, tens of
514
+ hours, before being able to mount a relay attack that causes
515
+ a time offset in the order of tens of milliseconds.
516
+ However, without highly stable clocks, mounting attacks
517
+ against the Clock Offset Test can be significantly easier.
518
+ This can be the case for a ASHTECH receiver, for which
519
+ time offset values are shown at successive points in time,
520
+ each 30 seconds apart, in Fig. 6. We clarify this is notto be perceived as criticism for a given receiver or to be
521
+ the basis for the suitability of the Clock Offset Test. As
522
+ explained above, the stability of the receiver clock deter-
523
+ mines the strength of this test. But the data in Fig. 6,
524
+ over a period of 900 seconds, exactly demonstrates that
525
+ for commodity receivers significant instability is observe d;
526
+ time offset values are in the order of ten milliseconds (or
527
+ slightly less). Consequently, the adversary would need to
528
+ jam for roughly a couple of minutes, force the receiver to
529
+ consider as acceptable a time offset of 20 to 32 millisec-
530
+ onds, and thus be mislead by a replay attack as detailed in
531
+ Sec. 3.
532
+ Finally, we note that we do not consider here the case
533
+ of synchronization by means external to the GNSS system.
534
+ For example, if the receiver could connect to the Internet
535
+ and run NTP, it could obtain accurate time. But this would
536
+ be an infrequent operation (in the order of magnitude of
537
+ days), thus useful only if highly stable clock hardware were
538
+ available.
539
+ 4.3 Doppler Shift Test (DST)
540
+ Based on the received GNSS signal Doppler shift, with
541
+ respect to the nominal transmitter frequency ( ft=
542
+ 1.575GHz), the receiver can predict future Doppler Shift
543
+ values. Once lock to GNSS signals is obtained again, pre-
544
+ dicted Doppler shift values are compared to the ones cal-
545
+ culated due to the received GNSS signal. If the latter are
546
+ different than the predicted ones beyond a threshold, the
547
+ GNSS signal is deemed adversarial and rejected. What
548
+ makes this approach attractive is the smooth changes of
549
+ Doppler shift and the ability to predict it with low, es-
550
+ sentially constant errors over long periods of time. This
551
+ in dire in contrast to the inertial test based on location,
552
+ whose error grows exponentially with time.
553
+ The Doppler shift is produced due to the relative motion
554
+ of the satellite with respect to the receiver. The satellite
555
+ velocity is computed using ephemeris information and an
556
+ orbital model available at the receiver. The received fre-
557
+ quency, fr, increases as the satellite approaches and de-
558
+ creases as it recedes from the receiver; it can be approxi-
559
+ mated by the classical Doppler equation:
560
+ fr=ft·(1−vr·a
561
+ c) (5)
562
+ where ftis nominal (transmitted) frequency, frreceived
563
+ frequency, vris the satellite-to-user relative velocity vector
564
+ andcspeed of radio signal propagation. The product vr·
565
+ arepresents the radial component of the relative velocity
566
+ vector along the line-of-sight to the satellite.
567
+ If the frequency shift differs from the predicted shift for
568
+ each visible satellite Siin the area depending on the data
569
+ obtained from the almanac (in the case when the naviga-
570
+ tion history is available), for more than defined thresholds
571
+ (∆fmin,∆fmax) or estimated Doppler shift from naviga-
572
+ tion history differs for more than the estimated shift, know-
573
+ ing the rate ( r), the receiver can deem the received signal
574
+ as product of attack.
575
+ 650 100 150 200 250 3002300235024002450250025502600265027002750
576
+ Time [s]Frequency offset [Hz]
577
+
578
+ Measured Doppler shift [Hz ]
579
+ Linear approximation
580
+ Prediction bounds
581
+ Figure 7: Measured and approximated Doppler frequency
582
+ shift.
583
+ TheAlmanac contains approximate position of the satel-
584
+ lites, ( Xsi, Y si, Zsi), time and the week number ( WN, t ),
585
+ and the corrections, such that the receiver is aware of the
586
+ expected satellites, their position, and the Doppler offset .
587
+ Because of the high carrier frequencies and large satel-
588
+ lite velocities, large Doppler shifts are produced ( ±5kHz),
589
+ and vary rapidly (1 Hz/s). The oscillator of the receiver
590
+ has frequency shift of ±3KHz, thus the resultant frequency
591
+ shift goes therefore up to ±9KHz. Without the knowledge
592
+ of the shift, the receiver has to perform a search in this
593
+ range of frequencies in order to acquire the signal. The
594
+ rate of Doppler shift receiving frequency caused by the rel-
595
+ ative movement between GPS satellite and vehicles approx-
596
+ imately 40 Hz per minute to the maximum. These varia-
597
+ tions are linear for every satellite. If the receiver is mobi le,
598
+ the Doppler shift variation can be estimated knowing the
599
+ velocity of the receiver( [3]).
600
+ In our simulations, Doppler shift is analyzed for each
601
+ available satellite (number of available satellites varie s). To
602
+ be consistent with results shown for other mechanisms, we
603
+ present results for DST for the 300sec period.
604
+ We observe in Fig. 7 the Doppler shift variation based
605
+ on data collected by an ASHTECH receiver: the maximum
606
+ change in rate is within + /−20Hz around a linear curve
607
+ fitted to the data. This clues that with sufficient samples,
608
+ the future Doppler Shift rate, and thus the shift per se,
609
+ values can be predicted. In practice, we observe that 50
610
+ sec of samples, with one sample per second, appear to be
611
+ sufficient.
612
+ More precisely, the rate of change of the frequency shift,
613
+ Di(t), is computed for each satellite, Si, as:
614
+ ri=dDi(t)
615
+ dt(6)
616
+ which can be approximated by numerical methods. Based
617
+ on prior samples for each Di, available for some time win-
618
+ dow the frequency shift can be predicted based those sam-
619
+ ples and the estimate rate of change of the Doppler shift.
620
+ Based on prior measured statistics of the signal at the re-
621
+ ceiver, the variance σ2of a random component, assumed
622
+ to beN(0, σ2), can be estimated. This random component0 50 100 150 200 250 300−10000100020003000
623
+ Time [s]Frequency offset [Hz]SV−1
624
+ 0 50 100 150 200 250 300−10000−50000
625
+ Time [s]Frequency offset [Hz]SV−4
626
+
627
+ 0 50 100 150 200 250 3000200040006000
628
+ Time [s]Frequency offset [Hz]SV−7
629
+ 0 50 100 150 200 250 3000100020003000
630
+ Time [s]Frequency offset [Hz]SV−13
631
+ 0 50 100 150 200 250 300−4000−20000
632
+ Time [s]Frequency offset [Hz]SV−20
633
+ 0 50 100 150 200 250 300−10000100020003000
634
+ Time [s]Frequency offset [Hz] SV−24
635
+ 0 50 100 150 200 250 300−4000−20000
636
+ Time [s]Frequency offset [Hz] SV−25
637
+ Figure 8: Doppler shift attack; unsophisticated adversary .
638
+ The dotted line represents the predicted and the solid line
639
+ the measured frequency offset.
640
+ is due to signal variation (including receiver mobility, RF
641
+ multipath, scattering). Its estimation can serve to deter-
642
+ mine an acceptable interval around the predicted values.
643
+ The adversary is mostly at the ground and static or mov-
644
+ ing with speed that is much smaller than the satellite ve-
645
+ locity, which is in a range around 3km/s. Thus, the adver-
646
+ sary will not be able to produce the same Doppler shift as
647
+ the satellites, unless it changes its transmission frequen cy
648
+ to match the one receivers would obtain from GNSS sig-
649
+ nals due to the Doppler shift. An unsophisticated attacker
650
+ would then be easily detected. This is illustrated in Fig. 8:
651
+ After a “gap” corresponding to jamming, there is a striking
652
+ difference, between 100 and 150 seconds, when comparing
653
+ the Doppler shift due to the attack to the predicted one.
654
+ The case of A sophisticated adversary that controls its
655
+ transmission frequency (the attack starts at 160 s)is shown
656
+ in the Fig. 9. The adversary has multiple adaptive ra-
657
+ dios and it operates according to the following principle: i t
658
+ predicts the Doppler frequency shift at the location of the
659
+ receiver, and it then changes its transmission frequency
660
+ accordingly. If the attacker is not precisely aware of the
661
+ actual location and motion dynamics of the victim node
662
+ (receiver), there is still a significant difference between t he
663
+ predicted and the adversary-caused Doppler shift. This
664
+ is shown, with a magnitude of approximately 300 Hz, in
665
+ Fig. 9; a difference that allows detection of the attack.
666
+ 5 Conclusion
667
+ Existing GNSS receivers are vulnerable to a number of
668
+ attacks that manipulate the location and time the re-
669
+ ceivers compute. We qualitatively and quantitatively ana-
670
+ lyze those in this paper, and identify memory-based mech-
671
+ anisms that can help in securing GNNS signals. In particu-
672
+ lar, we realize that location-based inertial mechanisms an d
673
+ a clock offset test can be relatively easily defeated, with th e
674
+ adversary causing (through jamming) a sufficiently long
675
+ period of unavailability. In the latter case, only special-
676
+ ized highly stable clock hardware could enable detection of
677
+ fraudulent GNSS signals. Our Doppler Shift Test provides
678
+ 70 50 100 150 200 250 300020004000
679
+ Time [s]Frequency offset [Hz]SV−1
680
+
681
+ 0 50 100 150 200 250 300−10000−50000
682
+ Time [s]Frequency offset [Hz]SV−21
683
+
684
+ 0 50 100 150 200 250 3000500010000
685
+ Time [s]Frequency offset [Hz]SV−7
686
+
687
+ 0 50 100 150 200 250 300020004000
688
+ Time [s]Frequency offset [Hz]SV−25
689
+
690
+ 0 50 100 150 200 250 300−4000−20000
691
+ Time [s]Frequency offset [Hz]SV−9
692
+
693
+ 0 50 100 150 200 250 3000100020003000
694
+ Time [s]Frequency offset [Hz]SV−29
695
+
696
+ 0 50 100 150 200 250 300−4000−20000
697
+ Time [s]Frequency offset [Hz]SV−13
698
+
699
+ Figure 9: Doppler shift attack; sophisticated adversary.
700
+ The dotted line represents the predicted and the solid line
701
+ the measured frequency offset.
702
+ resilience to long unavailability periods without special ized
703
+ equipment.
704
+ Our results are the first, to the best of our knowledge,
705
+ to provide tangible demonstration of effective mechanisms
706
+ to secure mobile systems from location information manip-
707
+ ulation via attacks against the GNSS systems.
708
+ As part of on-going and future work, we intent to further
709
+ refine and generalize the simulation framework we utilized
710
+ here, to consider precisely the effect of counter-measures
711
+ that only partially limit the attack impact. Moreover, we
712
+ will consider more closely the cost of mounting attacks of
713
+ differing sophistication levels, especially through proof -of-
714
+ concept implementations.
715
+ References
716
+ [1] N. Bertelsen, K. Borre, The GPS Code Software Re-
717
+ ceiver , Aalborg University, Birkhauser, 2007
718
+ [2] W. Franz and H. Hartenstein, Inter-Vehicle Communi-
719
+ cations, FleetNet project , University Karlruhe, 2005
720
+ [3]http://www.freepatentsonline.com/5036329.html
721
+ [4] S. Godha, Performance Evaluation of Low Cost
722
+ MEMS-Based IMU Integrated with GPS for Land Ve-
723
+ hicle Navigation Appplication , University of Calgary,
724
+ 2006
725
+ [5] G.W. Hein and F. Kneissl, Authenticating GNSS Proofs
726
+ Against Spoofs , InsideGNSS, September/October 2007
727
+ [6] E.D. Kaplan, Understanding GPS - Principles and Ap-
728
+ plications , Artech House, 2006
729
+ [7] M. Kuhn, An asymetric Security Mechanism for Nav-
730
+ igation Signals , Sixth Information Hiding Workshop,
731
+ Toronto, Canada, 2004
732
+ [8] NAVSTAR GPS Joint Program Office, NAVSTAR
733
+ Global Positioning System - Interface Specification IS-
734
+ GPS 200 Space Segment/Navigation User Interfaces ,
735
+ SMC/GP, CA, USA, 2004[9] P. Papadimitratos and A. Jovanovic, Protection and
736
+ Fundamental Vulnerability of GNSS , IWSSC, Toulouse,
737
+ 2008
738
+ [10] A.D. Rabbany, Introduction to GPS , Artech House,
739
+ 2002
740
+ [11] L. Scott, Anti-Spoofing and Authenticated Signal Ar-
741
+ chitectures for Civil Navigation Signals , ION-GNNS,
742
+ Portand, Oregon, 2003
743
+ [12] J.A. Volpe, Vulnearability Assesment of the Trans-
744
+ portation Infrastructure Relying on GPS , NTSC, NAV-
745
+ CEN draft report, 2001
746
+ [13] H. Wen, P. Huang, and J. Fagan, Countermeasures for
747
+ GPS signal spoofing , The University of Oklahoma, 2004
748
+ [14] J. Zogg, GPS Basics - Introduction to the System , U-
749
+ blox AG, 2002
750
+ 8
1001.0026.txt ADDED
@@ -0,0 +1,318 @@
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
1
+ arXiv:1001.0026v1 [astro-ph.SR] 30 Dec 2009Detectionof solar-likeoscillations from Keplerphotometry ofthe open
2
+ cluster NGC 6819
3
+ DennisStello,1Sarbani Basu,2HansBruntt,3Benoˆ ıt Mosser,3Ian R. Stevens,4
4
+ TimothyM.Brown,5Jørgen Christensen-Dalsgaard,6Ronald L. Gilliland,7Hans Kjeldsen,6
5
+ Torben Arentoft,6J´ erˆ omeBallot,8CarolineBarban,3TimothyR. Bedding,1WilliamJ. Chaplin,4
6
+ YvonneP. Elsworth,4Rafael A.Garc´ ıa,9Marie-Jo Goupil,3SaskiaHekker,4Daniel Huber,1
7
+ SavitaMathur,10Søren Meibom,11Reza Samadi,3VinothiniSangaralingam,4
8
+ Charles S. Baldner,2KevinBelkacem,12KatiaBiazzo,13Karsten Brogaard,6
9
+ Juan Carlos Su´ arez,14Francesca D’Antona,15Pierre Demarque,2LisaEsch,2NingGai,2,16
10
+ Frank Grundahl,6YvelineLebreton,17Biwei Jiang,16NadaJevtic,18ChristofferKaroff,4
11
+ AndreaMiglio,12JoannaMolenda- ˙Zakowicz,19JosefinaMontalb´ an,12ArletteNoels,12
12
+ Teodoro RocaCort´ es,20,21Ian W. Roxburgh,22AldoM. Serenelli,23VictorSilvaAguirre,23
13
+ ChristiaanSterken,24Peter Stine,18Robert Szab´ o,25AchimWeiss,23WilliamJ. Borucki,26
14
+ DavidKoch,26JonM. Jenkins27– 2 –
15
+ 1SydneyInstituteforAstronomy(SIfA),SchoolofPhysics,U niversityofSydney,NSW2006,Australia
16
+ 2DepartmentofAstronomy,YaleUniversity,P.O.Box 208101, New Haven,CT 06520-8101
17
+ 3LESIA,CNRS,Universit´ ePierreetMarieCurie,Universit´ eDenisDiderot,ObservatoiredeParis,92195Meudon,
18
+ France
19
+ 4SchoolofPhysicsandAstronomy,UniversityofBirmingham, Edgbaston,BirminghamB152TT,UK
20
+ 5LasCumbresObservatoryGlobalTelescope,Goleta,CA 93117 ,USA
21
+ 6DepartmentofPhysicsandAstronomy,AarhusUniversity,80 00AarhusC,Denmark
22
+ 7SpaceTelescopeScienceInstitute,3700San MartinDrive,B altimore,Maryland21218,USA
23
+ 8Laboratoired’AstrophysiquedeToulouse-Tarbes,Univers it´ edeToulouse,CNRS,14avE.Belin,31400Toulouse,
24
+ France
25
+ 9Laboratoire AIM, CEA/DSM-CNRS, Universit´ e Paris 7 Didero t, IRFU/SAp, Centre de Saclay, 91191, Gif-sur-
26
+ Yvette,France
27
+ 10IndianInstituteofAstrophysics,Koramangala,Bangalore 560034,India
28
+ 11Harvard-SmithsonianCenterforAstrophysics,60GardenSt reet,Cambridge,MA,02138,USA
29
+ 12Institutd’AstrophysiqueetdeG´ eophysiquedel’Universi t´ edeLi` ege,17All´ eedu6Aoˆ ut,B-4000Li` ege,Belgium
30
+ 13ArcetriAstrophysicalObservatory,LargoE.Fermi5,50125 ,Firenze,Italy
31
+ 14InstitutodeAstrof´ ısicadeAndaluc´ ıa(CSIC),Dept. Stel larPhysics,C.P. 3004,Granada,Spain
32
+ 15INAF -Osservatoriodi Roma,via diFrascati 33,I-00040,Mon teporzio,Italy
33
+ 16DepartmentofAstronomy,BeijingNormalUniversity,Beiji ng100875,China
34
+ 17GEPI,ObservatoiredeParis,CNRS, Universit´ eParisDider ot,5Place JulesJanssen,92195Meudon,France
35
+ 18Departmentof Physics& EngineeringTechnology,Bloomsbur gUniversity,400East SecondSt, BloomsburgPA
36
+ 17815,USA
37
+ 19AstronomicalInstitute,UniversityofWrocław,ul.Kopern ika11,51-622Wrocław,Poland
38
+ 20DepartmentodeAstrof´ ıca,Universidadde LaLaguna,38207 LaLaguna,Tenerife,Spain
39
+ 21InstitutodeAstrof´ ıcadeCanarias,38205La Laguna,Tener ife,Spain
40
+ 22QueenMaryUniversityofLondon,Mile EndRoad,LondonE14NS ,UK
41
+ 23MaxPlanckInstituteforAstrophysics,KarlSchwarzschild Str. 1,GarchingbeiM¨ unchen,D-85741,Germany
42
+ 24Vrije UniversiteitBrussel, Pleinlaan2,B-1050Brussels, Belgium
43
+ 25KonkolyObservatory,H-1525Budapest,P.O. Box67,Hungary
44
+ 26NASA AmesResearchCenter,MS 244-30,MoffatField,CA 94035 ,USA
45
+ 27SETIInstitute/NASA AmesResearchCenter,MS244-30,Moffa tField, CA 94035,USA– 3 –
46
+ ABSTRACT
47
+ Asteroseismology of stars in clusters has been a long-sough t goal because the as-
48
+ sumption of a common age, distance and initial chemical comp osition allows strong
49
+ tests of the theory of stellar evolution. We report results f rom the first 34 days of sci-
50
+ encedatafromthe KeplerMission fortheopenclusterNGC6819—oneoffourclus-
51
+ ters in the field of view. We obtain the first clear detections o f solar-like oscillations
52
+ in the cluster red giants and are able to measure the large fre quency separation, ∆ν,
53
+ andthefrequencyofmaximumoscillationpower, νmax. Wefindthattheasteroseismic
54
+ parameters allow us to test cluster-membership of the stars , and even with the limited
55
+ seismicdatainhand,wecan alreadyidentifyfourpossiblen on-membersdespitetheir
56
+ havinga betterthan 80% membershipprobabilityfrom radial velocitymeasurements.
57
+ We are also able to determine the oscillation amplitudes for stars that span about two
58
+ orders of magnitude in luminosity and find good agreement wit h the prediction that
59
+ oscillation amplitudesscale as the luminosityto the power of 0.7. These early results
60
+ demonstrate the unique potential of asteroseismology of th e stellar clusters observed
61
+ byKepler.
62
+ Subjectheadings: stars: fundamentalparameters—stars: oscillations—star s: interi-
63
+ ors—techniques: photometric—openclustersandassociati ons: individual(NGC6819)
64
+ 1. Introduction
65
+ Openclustersprovideuniqueopportunitiesinastrophysic s. Starsinopenclustersarebelieved
66
+ to be formed from the same cloud of gas at roughly the same time . The fewer free parameters
67
+ available to model cluster stars make them interesting targ ets to analyze as a uniform ensemble,
68
+ especiallyforasteroseismicstudies.
69
+ Asteroseismology is an elegant tool based on the simple prin ciple that the frequency of a
70
+ standing acoustic wave inside a star depends on the sound spe ed, which in turn depends on
71
+ the physical properties of the interior. This technique app lied to the Sun (helioseismology) has
72
+ provided extremely detailed knowledge about the physics th at governs the solar interior, (e.g.,
73
+ Christensen-Dalsgaard2002). Allcoolstarsareexpectedt oexhibitsolar-likeoscillationsofstand-
74
+ ing acoustic waves – called p modes – that are stochastically driven by surface convection. Using
75
+ asteroseismology to probe the interiors of cool stars in clu sters, therefore, holds promise of re-
76
+ warding scientific return (Gough& Novotny 1993; Brown& Gill iland 1994). This potential has
77
+ resulted in several attempts to detect solar-like oscillat ions in clusters using time-series photome-
78
+ try. These attempts were often aimed at red giants, since the iroscillation amplitudesare expected– 4 –
79
+ tobelargerthanthoseofmain-sequenceorsubgiantstarsdu etomorevigoroussurfaceconvection.
80
+ Despite these attempts, only marginal detections have been attained so far, limited either by the
81
+ lengthofthetimeseriesusuallyachievablethroughobserv ationswiththe HubbleSpaceTelescope
82
+ (Edmonds& Gilliland 1996; Stello&Gilliland 2009) or by the difficulty in attaining high preci-
83
+ sion from ground-based campaigns (e.g., Gillilandetal. 19 93; Stelloet al. 2007; Frandsen et al.
84
+ 2007).
85
+ InthisLetterwereportcleardetectionsofsolar-likeosci llationsinred-giantstarsintheopen
86
+ cluster NGC 6819 using photometry from NASA’s Kepler Mission (Borucki et al. 2009). This
87
+ cluster,oneoffourinthe Keplerfield, isabout2.5Gyrold. Itisatadistanceof2.3kpc, andha sa
88
+ metallicityof[Fe/H] ∼ −0.05(see Holeet al. 2009, and references herein).
89
+ 2. Observations anddata reduction
90
+ The data were obtained between 2009 May 12 and June 14, i.e., t he first 34 days of con-
91
+ tinuous science observations by Kepler(Q1 phase). The spacecraft’s long-cadence mode ( ∆t≃
92
+ 30minutes) used in this investigation provided a total of 1639 data points in the time series of
93
+ each observed star. For this Letter we selected 47 stars in th e field of the open cluster NGC 6819
94
+ with membership probability PRV>80% from radial velocity measurements (Holeet al. 2009).
95
+ Figure1showsthecolor-magnitudediagram(CMD)oftheclus terwiththeselectedstarsindicated
96
+ by green symbols. The eleven annotated stars form a represen tative subset, which we will use to
97
+ illustrate our analyses in Sections 3 and 4. We selected the s tars in this subset to cover the same
98
+ brightnessrangeasourfullsample,whilegivinghighweigh ttostarsthatappeartobephotometric
99
+ non-members (i.e., stars located far from the isochrone in t he CMD). Data for each target were
100
+ checked carefully to ensure that the time-series photometr y was not contaminated significantly
101
+ by other stars in the field, which could otherwise complicate the interpretation of the oscillation
102
+ signal.
103
+ Fourteen data points affected by the momentum dumping of the spacecraft were removed
104
+ from the time series of each star. In addition, we removed poi nts that showed a point-to-point
105
+ deviation greater than 4σ, whereσis the local rms of the point-to-point scatter within a 24 hou r
106
+ window. This process removed on average one data-point per t ime series. Finally, we removed a
107
+ linear trend from each time series and then calculated the di screte Fourier transform. The Fourier
108
+ spectraathighfrequencyhavemeanlevelsbelow5partsperm illion(ppm)inamplitude,allowing
109
+ usto search forlow-amplitudesolar-likeoscillations.– 5 –
110
+ 3. Extractionofasteroseismicparameters
111
+ Figure 2 shows the Fourier spectra (in power) of 9 stars from o ur subset. These range from
112
+ thelowerred-giant branch to thetip ofthe branch (see Figur e1). The stars are sorted by apparent
113
+ magnitude, which for a cluster is indicative of luminosity, with brightest at the top. Note that the
114
+ redgiantsinNGC6819aresignificantlyfainter( 12/lessorsimilarV/lessorsimilar14)thanthesampleof Keplerfieldred
115
+ giants (8/lessorsimilarV/lessorsimilar12) studied by Beddinget al. (2010). Nevertheless, it is clear from Figure 2 that
116
+ we can detect oscillations for stars that span about two orde rs of magnitude in luminosity along
117
+ theclustersequence.
118
+ Weusedfourdifferentpipelines(Hekkeret al.2009a;Huber et al.2009a;Mathuret al.2009;
119
+ Mosser& Appourchaux 2009) to extract the average frequency separation between modes of the
120
+ same degree (the so-called large frequency separation, ∆ν). We have also obtained the frequency
121
+ of maximum oscillation power, νmax, and the oscillation amplitude. The measured values of ∆ν
122
+ are indicated by vertical dotted lines in Figure 2 centered o n the highest oscillation peaks near
123
+ νmax. While the stars in Figure 2, particularly in the lower panel s, show the regular series of
124
+ peaks expected for solar-likeoscillations,the limitedle ngth of the time-series datadoes not allow
125
+ such structureto be clearly resolved for the mostluminouss tars in our sample— thosewith νmax
126
+ /lessorsimilar20µHz. We do, however, see humps of excess power in the Fourier sp ectra (see Figure 2 star
127
+ no. 2 and 8) with νmaxand amplitude in mutual agreement with oscillations. With l onger time
128
+ series weexpectmorefirm resultsforthesehigh-luminosity giants.
129
+ 4. Cluster membership from asteroseismology
130
+ It isimmediatelyclear fromFigure2thatnotallstars follo wtheexpected trendofincreasing
131
+ νmaxwith decreasing apparent magnitude, suggesting that some o f the stars might be intrinsically
132
+ brighterorfainterthanexpected. Sinceoscillationsinas taronlydependonthephysicalproperties
133
+ of the star, we can use asteroseismology to judge whether or n ot a star is likely to be a cluster
134
+ member independentlyof its distanceand of interstellarab sorption and reddening. For cool stars,
135
+ νmaxscaleswiththeacousticcut-offfrequency,anditiswelles tablishedthatwecanestimate νmax
136
+ by scalingfromthesolarvalue(Brownet al. 1991; Kjeldsen& Bedding 1995):
137
+ νmax
138
+ νmax,⊙=M/M⊙(Teff/Teff,⊙)3.5
139
+ L/L⊙, (1)
140
+ whereνmax,⊙= 3100µHz. The accuracy of such estimates is good to within 5% (Stell oet al.
141
+ 2009)assumingwehavegoodestimatesofthestellarparamet ersM,L, andTeff.
142
+ In thefollowingweassumetheidealisticscenario whereall clustermembersfollowstandard
143
+ stellar evolutiondescribed by the isochrone. Stellar mass along the red giant branch of thecluster– 6 –
144
+ isochrone varies by less than 1%. The variation is less than 5 % even if we also consider the
145
+ asymptoticgiant branch. For simplicity,we therefore adop t a mass of 1.55M⊙for all stars, which
146
+ is representativefortheisochronefrom Marigoet al. (2008 )(Figure 1) and a similarisochroneby
147
+ VandenBerg etal. (2006). Neglectingbinarity (see Table 1) , we derivethe luminosityof each star
148
+ in our subset from its V-band apparent magnitude, adopting reddening and distance modulus of
149
+ E(B−V) = 0.1and(M−m)V= 12.3,respectively(obtainedfromsimpleisochronefitting,see
150
+ Holeetal.2009). WeusedthecalibrationofFlower(1996)to convertthestellar (B−V)0colorto
151
+ Teff. BolometriccorrectionswerealsotakenfromFlower(1996) . Thederivedquantitieswerethen
152
+ used toestimate νmaxfor each star(Eq.1), and compared withtheobservedvalue(s eeFigure3).
153
+ Figure 3 shows four obvious outliers (no. 1, 3, 8 and 11), thre e of which are also outliers in
154
+ theCMD (no. 1, 3, and11). Fortherest ofthestars weseegood a greement between theexpected
155
+ andobservedvalue,indicatingthattheuncertaintyonthe νmaxestimatesarerelativelysmall. Since
156
+ thevariationsinmassandeffectivetemperatureamongthec lustergiantstarsaresmall,deviations
157
+ fromthedottedlinemustbecausedbyanincorrectestimateo ftheluminosity. Thisimpliesthatthe
158
+ luminositiesofstarsfallingsignificantlyaboveorbelowt helinehavebeenover-orunderestimated,
159
+ respectively. The simplest interpretation is that these ou tliers are fore- or background stars, and
160
+ hence not members of the cluster. To explain the differences between the observed and expected
161
+ value ofνmaxwould require the deviant stars to have Verrors of more than 1 magnitude, and in
162
+ some cases B−Verrors of about 0.2 magnitude if they were cluster members. B inarity may
163
+ explain deviations above the dotted line, but only by up to a f actor of two in L(and hence, in the
164
+ ratio of the observed to expected νmax). The deviation of only one star (no.1) could potentially
165
+ be explained this way. However, that would be in disagreemen t with its single-star classification
166
+ from multi-epoch radial velocity measurements, assuming i t is not a binary viewed pole-on (see
167
+ Table 1). Hence, under the assumptionof a standard stellar e volution, the most likely explanation
168
+ forallfouroutliersinFigure3isthereforethatthesestar sarenotclustermembers. Thisconclusion
169
+ is, however,in disagreementwith theirhighmembershippro babilityfrom measurementsofradial
170
+ velocity (Holeet al. 2009) and proper motion (Sanders 1972) (see Table 1). Another interesting
171
+ possibility is that the anomalous pulsation properties mig ht be explained by more exotic stellar
172
+ evolutionscenariosthan isgenerally anticipatedforopen -clusterstars.
173
+ 5. Asteroseismic“color-magnitude diagrams”
174
+ ItisclearfromFigure2thattheamplitudesoftheoscillati onsincreasewithluminosityforthe
175
+ seismicallydeterminedclustermembers. Basedoncalculat ionsbyChristensen-Dalsgaard& Frandsen
176
+ (1983), Kjeldsen& Bedding (1995) have suggested that the ph otometric oscillation amplitude of
177
+ p modes scale as (L/M)sTeff−2, withs= 1(the velocity amplitudes, meanwhile, would scale as– 7 –
178
+ (L/M)s). This was revised by Samadi etal. (2007) to s= 0.7based on models of main sequence
179
+ stars. Takingadvantageofthefewerfreeparameterswithin thisensembleofstars,ourobservations
180
+ allow us to make some progress towards extrapolating this sc aling to red giants and determining
181
+ thevalueof s.
182
+ In Figure4 weintroduceanewtypeofdiagramthatissimilart oaCMD, butwithmagnitude
183
+ replaced by an asteroseismicparameter – in thiscase, theme asured oscillationamplitude. Ampli-
184
+ tudeswereestimatedforallstarsinoursample(exceptfort hefouroutliers)usingmethodssimilar
185
+ tothatofKjeldsenet al.(2008)(seealsoMichelet al.2008) ,whichassumethattherelativepower
186
+ betweenradialandnon-radialmodesisthesameasintheSun. Thisdiagramconfirmstherelation-
187
+ ship between amplitude and luminosity. Despite a large scat ter, which is not surprising from this
188
+ relatively short timeseries, we see that s= 0.7provides a much better match than s= 1.0. Once
189
+ verifiedwithmoredata,thisrelationwillallowtheuseofth emeasuredamplitudeasanadditional
190
+ asteroseismic diagnostic for testing cluster membership a nd for isochrone fitting in general. We
191
+ notethat theother clusters observed by Keplerhave different metallicitiesthan NGC 6819, which
192
+ willallowfutureinvestigationon themetallicitydepende nce oftheoscillationamplitudes.
193
+ We expect to obtain less scatter in the asteroseismic measur ements when longer time series
194
+ become available. That will enable us to expand classical is ochrone fitting techniques to include
195
+ diagramslikethis,whereamplitudecouldalsobereplacedb yνmaxor∆ν. Inparticular,weshould
196
+ beabletodeterminetheabsoluteradiiaidedby ∆νoftheredgiantbranchstars,whichwouldbean
197
+ importantcalibratorfor theoretical isochrones. Additio nally,thedistributionsoftheasteroseismic
198
+ parameters – such as νmax– can potentially be used to test stellar population synthes is models
199
+ (Hekkeret al.2009b;Miglioet al.2009b). Applyingthisapp roachtoclusterscouldleadtofurther
200
+ progress in understanding of physical processes such as mas s loss during the red-giant phase (see
201
+ e.g.,Miglioet al.2009a). Notethatafewclearoutliersare indicativeofnon-membershiporexotic
202
+ stellarevolution,asaresultoffactorssuchasstellarcol lisionsorheavymassloss,whileageneral
203
+ deviationfromthetheoreticalpredictionsbyalargegroup ofstarswouldsuggestthatthestandard
204
+ theorymay need revision.
205
+ Finally, we note that NGC 6819 and another Keplercluster, NGC 6791, contain detached
206
+ eclipsingbinaries(Talamantes& Sandquist2009;Street et al.2005;deMarchi et al.2007;Mochejskaetal.
207
+ 2005). For these stars masses and radii can be determined ind ependently (Grundahl et al. 2008),
208
+ whichwillfurtherstrengthenresultsofasteroseismicana lyses.– 8 –
209
+ 6. Discussion& Conclusions
210
+ PhotometricdataofredgiantsinNGC6819obtainedbyNASA’s KeplerMission haveenabled
211
+ ustomakethefirst cleardetectionofsolar-likeoscillatio nsin clusterstars. Thegeneral properties
212
+ of the oscillations ( ∆ν,νmax, and amplitudes) agree well with results of field red giants m ade by
213
+ Kepler(Bedding etal.2010)andCoRoT(deRidderet al.2009;Hekker et al.2009b). Wefindthat
214
+ the oscillation amplitudes of the observed stars scale as (L/M)0.7Teff−2, suggesting that previous
215
+ attemptstodetect oscillationsinclustersfrom groundwer eat thelimitofdetection.
216
+ We find that the oscillation properties provide additional t ests for cluster membership, al-
217
+ lowing us to identify four stars that are either non-members or exotic stars. All four stars have
218
+ membership probability higher than 80% from radial-veloci ty measurements, but three of them
219
+ appear to be photometric non-members. We further point out t hat deviations from the theoretical
220
+ predictionsoftheasteroseismicparametersamongalarges ampleofclusterstarshavethepotential
221
+ ofbeingusedasadditionalconstraintsintheisochronefitt ingprocess,whichcanleadtoimproved
222
+ stellarmodels.
223
+ Our results, based on limited data of about one month, highli ght the unique potential of as-
224
+ teroseismologyon the brighteststars in thestellarcluste rs observed by Kepler. With longerseries
225
+ sampled at the spacecraft’s short cadence ( ≃1 minute), we expect to detect oscillations in the
226
+ subgiantsand turn-offstars, as wellas inthebluestraggle rsinthiscluster.
227
+ FundingforthisDiscoverymissionisprovidedbyNASA’sSci enceMissionDirectorate. The
228
+ authorswouldliketothanktheentire Keplerteamwithoutwhomthisinvestigationwouldnothave
229
+ been possible. The authors also thank all funding councils a nd agencies that have supported the
230
+ activitiesofWorkingGroup 2ofthe KeplerAsteroseismicScience Consortium(KASC).
231
+ Facilities: Kepler.
232
+ REFERENCES
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+ Bedding,T. R., et al. 2010,ApJL,inpress
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+ Borucki, W.,et al. 2009,inIAU Symposium,Vol.253, IAUSymp osium,289
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+ Brown, T.M.,& Gilliland,R. L. 1994,ARA&A,32, 37
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+ Brown, T.M.,Gilliland,R. L., Noyes,R. W.,& Ramsey,L. W.19 91,ApJ, 368,599
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+ Christensen-Dalsgaard,J.2002,ReviewsofModern Physics ,74, 1073– 9 –
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+ Christensen-Dalsgaard,J.,& Frandsen, S. 1983,Sol. Phys. ,82,469
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+ deMarchi,F., etal. 2007,A&A,471, 515
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+ deRidder, J.,et al. 2009,Nature, 459,398
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+ Edmonds,P. D., &Gilliland,R. L.1996, ApJ,464,L157
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+ Flower, P. J.1996,ApJ, 469,355
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+ Frandsen, S., et al. 2007,A&A,475,991
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+ Gilliland,R. L., et al. 1993,AJ,106,2441
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+ Gough, D. O., & Novotny, E. 1993, in ASP Conf. Ser. 42: GONG 199 2. Seismic Investigationof
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+ theSunand Stars, ed. T.M. Brown,355
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+ Grundahl,F., Clausen, J. V.,Hardis, S., &Frandsen, S. 2008 ,A&A,492,171
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+ Hekker, S., et al. 2009a,MNRAS, in press(astro-ph/0911.26 12)
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+ —.2009b,A&A, 506,465
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+ Hole, K. T., Geller, A. M., Mathieu, R. D., Platais, I., Meibo m, S., & Latham, D. W. 2009, AJ,
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+ 138,159
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+ Huber, D., Stello, D., Bedding, T. R., Chaplin, W. J., Arento ft, T., Quirion, P., & Kjeldsen, H.
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+ 2009a,Commun.Asteroseismol.,160,74
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+ Kjeldsen,H., &Bedding,T. R. 1995,A&A,293, 87
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+ Kjeldsen,H., etal. 2008,ApJ, 682,1370
256
+ Latham,D. W.,Brown, T.M.,Monet,D. G., Everett,M.,Esquer do,G. A.,& Hergenrother, C. W.
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+ 2005,inBulletinoftheAmerican AstronomicalSociety,Vol . 37,1340
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+ Marigo, P., Girardi, L., Bressan, A., Groenewegen, M. A. T., Silva, L., & Granato, G. L. 2008,
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+ A&A,482,883
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+ Mathur,S., et al. 2009,A&A,inpress (arXiv:0912.3367)
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+ Michel,E., etal. 2008,Science, 322,558
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+ Miglio, A., Montalb´ an, J., Eggenberger, P., Hekker, S., & N oels, A. 2009a, in American Institute
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+ ofPhysicsConference Series, Vol.1170,AmericanInstitut eofPhysicsConference Series,
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+ ed. J.A. Guzik& P. A. Bradley,132– 10 –
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+ Miglio,A., et al.2009b,A&A,503, L21
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+ Mochejska,B. J., et al.2005,AJ, 129,2856
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+ Mosser,B., & Appourchaux,T.2009,A&A,508, 877
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+ Samadi, R., Georgobiani, D., Trampedach, R., Goupil, M. J., Stein, R. F., & Nordlund, ˚A. 2007,
269
+ A&A,463,297
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+ Sanders, W. L. 1972,A&A,19,155
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+ Stello,D., Chaplin,W. J.,Basu, S., Elsworth,Y., &Bedding , T.R. 2009, MNRAS, 400,80
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+ Stello,D., &Gilliland,R. L.2009,ApJ, 700,949
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+ Stello,D., et al. 2007,MNRAS, 377,584
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+ Street, R. A.,et al. 2005,MNRAS, 358,795
275
+ Talamantes, A., & Sandquist, E. L. 2009, in Bulletin of the Am erican Astronomical Society,
276
+ Vol.41,320
277
+ VandenBerg, D. A., Bergbusch, P. A., &Dowler,P. D. 2006,ApJ S, 162,375
278
+ ThispreprintwaspreparedwiththeAAS L ATEXmacrosv5.2.– 11 –
279
+ Table1:Cross identificationsandmembership.
280
+ ID ID WOCS ID ID Mem.ship Mem.ship Mem.ship
281
+ Thiswork KICaHoleet al. Sanders Holeet al.bSanderscThiswork
282
+ 1 5024272 003003 SM95% no
283
+ 2 5024750 001004 141 SM93% 83% yes
284
+ 3 5023889 004014 42 SM95% 90% no
285
+ 4 5023732 005014 27 SM94% 90% yes
286
+ 5 5112950 003005 148 SM95% 92% yes
287
+ 6 5112387 003007 73 SM95% 88% yes
288
+ 7 5024512 003001 116 SM93% 90% yes
289
+ 8 4936335 007021 9 SM95% 68% no
290
+ 9 5024405 004001 100 SM93% 91% yes
291
+ 10 5112072 009010 39 SM95% 91% yes
292
+ 11 4937257 009015 144 SM88% 80% no
293
+ aIDfromthe KeplerInputCatalogue (Lathamet al. 2005).
294
+ bClassification (SM:singlemember)andmembershipprobabil ityfromradialvelocity(Holeetal. 2009).
295
+ cMembershipprobabilityfrompropermotion(Sanders1972).– 12 –
296
+ Fig. 1.— Color-magnitude diagram of NGC 6819. Plotted stars have membership probability
297
+ PRV>80% as determined by Holeet al. (2009). Photometric indices ar e from the same source.
298
+ Theisochroneis from Marigoet al. (2008)(Age=2.4 Gyr, Z=0. 019,modified for theadopted red-
299
+ dening of 0.1mag). Color-coded stars have been analyzed, an d the annotated numbers refer to the
300
+ legend in panels of Figure 2 and star numbers in Figure 3 (see a lso Table 1). Insets show light
301
+ curves in parts per thousand of two red giants oscillating on different timescales. The variations
302
+ ofthelightcurves inPanelA and Baredominatedby thestella roscillationswithperiodsofafew
303
+ days andofaboutsix hours,respectively.– 13 –
304
+ Fig. 2.— Fourierspectraofa representativeset ofred giant salongtheclustersequence sortedby
305
+ apparent magnitude. Annotated numbers in each panel refer t o the star identification (see Fig. 1
306
+ and Table 1). ‘AM’ indicates that the star is an asteroseismi c member. Red solid curves show the
307
+ smoothed spectrum for stars with νmax<20µHz. To guide the eye, we have plotted dotted lines
308
+ toindicatethemeasuredaveragelargefrequencyseparatio n. Thecentraldottedlineiscenteredon
309
+ thehighestoscillationpeaksnear νmax. Notethatsince ∆νisgenerallyfrequencydependent,only
310
+ thecentraldottedlineisexpectedtolineupwithapeakinth eoscillationspectrum. Theredarrows
311
+ indicate the position of the expected νmax(see Eq. 1) for stars where the observed value does not
312
+ agree withtheexpectationsforthiscluster(seeSection 4) .– 14 –
313
+ Fig. 3.— Ratioofobservedandexpected νmax. 1-σerrorbarsindicatetheuncertaintyon νmax(obs).
314
+ Stars clearly above or below the dotted line are either not cl uster members or members whose
315
+ evolutionhavenot followedthestandardscenario.– 15 –
316
+ Fig. 4.— Amplitude color diagram of red giant stars in NGC 681 9 with the Marigoet al. (2008)
317
+ isochrone overlaid with three values of sin the amplitude scaling relation: (L/M)sTeff−2. The
318
+ solarvalueusedin thisscalingis 4.7ppm(Kjeldsen &Bedding 1995).
1001.0027.txt ADDED
@@ -0,0 +1,389 @@
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
1
+ arXiv:1001.0027v1 [astro-ph.GA] 30 Dec 2009New candidate Planetary Nebulae in the IPHAS survey: the cas e of
2
+ PNe with ISM interaction.
3
+ Laurence SabinA, Albert A. ZijlstraA, Christopher WareingB, Romano L.M.
4
+ CorradiC, Antonio MampasoC, Kerttu ViironenC, Nicholas J. WrightDand
5
+ Quentin A. ParkerE
6
+ AJodrell Bank Center for Astrophysics, School of Physics and Astronomy, University of Manchester,
7
+ Manchester M13 9PL, UK
8
+ BDepartment of Applied Mathematics, University of Leeds, Le eds, LS2 9JT, UK
9
+ CInstituto de Astrofisica de Canarias, Tenerife, Spain
10
+ DHarvard-Smithsonian Center for Astrophysics, 60 Garden St reet, Cambridge, MA, 02138, USA
11
+ EMacquarie University/Anglo-Australian Observatory, Dep artment of Physics, North Ryde, Sydney
12
+ NSW 2190, AUSTRALIA
13
+ AEmail: [email protected]
14
+ Abstract: We present the results of the search for candidate Planetary Nebulae interacting with
15
+ the interstellar medium (PN-ISM) in the framework of the INT Photometric H αSurvey (IPHAS)
16
+ and located in the right ascension range 18h-20h. The detect ion capability of this new Northern
17
+ survey, in terms of depth and imaging resolution, has allowe d us to overcome the detection problem
18
+ generally associated to the low surface brightness inheren t to PNe-ISM. We discuss the detection of
19
+ 21 IPHAS PN-ISM candidates. Thus, different stages of intera ction were observed, implying various
20
+ morphologies i.e. from the unaffected to totally disrupted s hapes. The majority of the sources belong
21
+ to the so-called WZO2 stage which main characteristic is a br ightening of the nebula’s shell in the
22
+ direction of motion. The new findings are encouraging as they would be a first step into the reduction
23
+ of the scarcity of observational data and they would provide new insights into the physical processes
24
+ occurring in the rather evolved PNe.
25
+ Keywords: Planetary nebulae, ISM interaction, survey.
26
+ 1 Introduction
27
+ Large Hαsurveys have so far allowed the detection of
28
+ ∼3000 planetary nebulae (PNe) in the Galaxy. The
29
+ data can be principally found in the Strasbourg-ESO
30
+ Catalogue (Acker et al.1992)andtherecentMacquarie-
31
+ AAO-StrasbourgH αPlanetaryNebulaCatalogues: MASH
32
+ IandII(Parker et al.(2006)andMiszalski et al(2008)).
33
+ Unfortunatelyalimitation inour understandingofthis
34
+ short and rather complex phase of stellar evolution lies
35
+ either in the deepness of the detections realised or the
36
+ type of PNe investigated. Indeed, although enormous
37
+ progress has been made over the years in terms of ob-
38
+ servations, the well-studied PNe are generally bright
39
+ and often young. This hampers the study of:
40
+ •PNe hidden by the interstellar medium, partic-
41
+ ularly those located at low galactic height.
42
+ •PNe with (very)low surface brightness where we
43
+ find the group of old PNe.
44
+ •Very distant PNe which appear as unresolved
45
+ and not recognisable as nebulae.•PNe located in crowded areas such as the galac-
46
+ tic plane.
47
+ Moreover, excluding these objects from global studies
48
+ (morphology, abundances,luminosityfunction...etc)may
49
+ bias our understanding of planetary nebulae. As an il-
50
+ lustration, few PNe are described in the literature as
51
+ “PNe with ISM interaction”, which is the step before
52
+ the complete dilution of the nebulae in the interstel-
53
+ lar medium (Borkowski et al. (1990), Ali et al. (2000),
54
+ Xilouris et al. (1996) and Tweedy et al. (1996)). The
55
+ study of the interaction process would give new in-
56
+ sights intoseveral aspects of the PNevolution. Indeed,
57
+ the density difference between ISM and PNe will affect
58
+ their shape. This is expected to be observable in old
59
+ objects where the nebular density declines sufficiently
60
+ to be overcome by the ISM density. Other phenom-
61
+ ena like the flux and brightness enhancement following
62
+ the compression of the external shell, the increase of
63
+ the recombination rate in the PN Rauch et al. (2000),
64
+ the occurrence of turbulent Rayleigh-Taylor instabili-
65
+ ties and the implication of magnetic fields Dgani et al.
66
+ (1998) are among the physical processes which need
67
+ to be addressed not only from a theoretical but also
68
+ observational point of view.
69
+ 12 Publications of the Astronomical Society of Australia
70
+ The low surface brightness generally associated to
71
+ PNe-ISM has for a long time prevented any deeper ob-
72
+ servation and good statistical study of these interac-
73
+ tions, where only the interacting rim is well seen. New
74
+ generations of H αsurveys have overcome this prob-
75
+ lem. A perfect example is the discovery of PFP 1 by
76
+ Pierce et al. (2004)intheframeworkoftheAAO/UKST
77
+ SuperCOSMOS H αsurvey (SHS) (Parker et al. 2005).
78
+ This PN, starting to interact with the ISM at the
79
+ rim, is very large (radius = 1.5 ±0.6 pc) and very
80
+ faint (logarithm of the H αsurface brightness equal
81
+ to -6.05 ergcm−2.s−1.sr−1). In order to unveil and
82
+ study this “missing PN population” in the Northern
83
+ hemisphere we need surveys providing the necessary
84
+ observing depth: the Isaac Newton Telescope (INT)
85
+ Photometric H αSurvey (IPHAS) is one of them and
86
+ will complete the work done in the South by the SHS.
87
+ 2 IPHAS contribution
88
+ IPHAS is a new fully photometric CCD survey of the
89
+ Northern Galactic Plane, started in 2003 (Drew et al.
90
+ (2005), Gonzalez-Solares et al (2008)) and which has
91
+ now been completed1. Using the 2.5m Isaac Newton
92
+ Telescope (INT)in LaPalma (Canary Islands, SPAIN)
93
+ and the Wide Field Camera (WFC) offering a field of
94
+ view of 34.2 ×34.2 arcmin2, IPHAS targets the Galac-
95
+ tic plane in the Northern hemisphere, at a latitude
96
+ range of -5◦<b<5◦and covers 1800 deg2. This
97
+ international survey is conducted not only in H αbut
98
+ also makes use of two continuum filters, respectively
99
+ the Sloan r’ and i’. IPHAS is viewed as an enhance-
100
+ ment to former narrow-band surveys, first due to the
101
+ use of CCD and the particularly small pixel scale al-
102
+ lowed bytheWFCwith0.33 arcsec pix−1butalso (and
103
+ mainly) due to the depth reached for point sources de-
104
+ tection. Thus sources with a r’ magnitude between 13
105
+ and 19.5-20 could be detected with a very good pho-
106
+ tometric accuracy. The most interesting characteristic
107
+ for our purpose is the ability to detect resolved ex-
108
+ tended emissions with an H αsurface brightness down
109
+ to 2×10−17erg cm−2s−1arcsec−2.
110
+ In this paper we will focus on extended (candi-
111
+ date) PNe (i.e. objects with a size greater than 5 arc-
112
+ sec). They were searched for via a visual inspection of
113
+ 2 deg2Hα-r(continuum removal) mosaics made from
114
+ the different IPHAS observations. And in order to al-
115
+ low the detection of objects of multiple size and bright-
116
+ ness level, the mosaics were binned at respectively 15
117
+ pixels×15 pixels (5 arcsec) and 5 pixels ×5 pixels (1.7
118
+ arcsec). The first binning level, which is of particu-
119
+ lar interest to us, helps to detect resolved, low surface
120
+ brightness objects (down to the IPHAS limit) and to
121
+ accentuate the contours/shape of the nebulae (this is
122
+ particularly useful to see, for example, the full extent
123
+ of an outflow or a tail). The second set, is used to de-
124
+ tect intermediate size nebulae i.e. smaller than ∼15-20
125
+ arcsec in diameter.
126
+ 1http://www.iphas.orgThe first area that has been fully investigated is the re-
127
+ gion between RA=18h and RA=20h. We detected 233
128
+ candidate PNe among which other nebulosities may be
129
+ found e.g. small HII regions (Sabin, PhD thesis, to be
130
+ published). Around 20% of this sample have been so
131
+ far spectroscopically confirmed as PNe (Sabin et al.,
132
+ in preparation). If we look at the particularities of the
133
+ PNe and candidate PNe uncovered, we observe that
134
+ from thepointofviewofthesize, large objects (greater
135
+ than 20 arcsec) constitute the main new group (Fig.
136
+ 1). As large objects are generally considered as more
137
+ evolved, we are confident in finding in this group new
138
+ old PNe and byextension new cases of PNe interacting
139
+ with the surrounding ISM (PNe/ISM).
140
+ Figure 1: Galactic distribution of the IPHAS neb-
141
+ ulae according to their size.
142
+ 3 Candidate PNewith ISMin-
143
+ teraction
144
+ Fundamental in PN development, the interaction with
145
+ the ISM does not only concern old PNe, as may be
146
+ commonly thought. Indeed, the PN-ISM interaction
147
+ has mainly been detected in a rather small number of
148
+ nebulae, which are generally bright objects (“young”
149
+ and“mid-age”PNe). Rauch et al.(2000)andWareing et al.
150
+ (2007) showed that different stages of interaction are
151
+ exhibited during the PNe life. The low surface bright-
152
+ ness, generally associated with nebulae mixing with
153
+ the ISM and “old” PNe, has for a long time prevented
154
+ any deeper observation and good statistical study of
155
+ these interactions. Although faint objects will still re-
156
+ main difficult to detect, the IPHAS survey provides
157
+ a noticeable improvement. Nevertheless, a caveat is
158
+ the difficulty to visually separate PNe-ISM from other
159
+ faint and extended structures like old HII regions, Su-
160
+ pernovae (SNRs) or diffuse H αstructures. As an ex-
161
+ ample, faint bow shocks generally characteristics ofwww.publish.csiro.au/journals/pasa 3
162
+ PNemixingwiththeISMcanalsobefilamentarystruc-
163
+ tures from old SNRs. A spectroscopic analysis is the
164
+ only way to have a clear identification.
165
+ The work presented here is based on the classification
166
+ from Wareing et al. (2007) (WZO 1-4 called after the
167
+ authors’ names) and will allow us to establish the de-
168
+ gree of interaction for each nebula. Their classifica-
169
+ tion is the result of the first extensive investigation
170
+ of the applicable parameter space, varying stellar pa-
171
+ rameters, relative velocities through the ISM and ISM
172
+ densities.
173
+ The depth reached by the IPHAS survey combined
174
+ with the binning detection method allowed us to iden-
175
+ tify 21 cases of interacting candidate PNe.
176
+ 3.1 WZO1 type
177
+ The first group of PNe/ISM concerns those where the
178
+ main PN is still unaffected and which may display a
179
+ distant bow shock. In our area of study (18h-20h), the
180
+ majority of candidates answer the first condition, but
181
+ none show the outer bow shock. Outside this area, the
182
+ nicknamed“EarNebula”orIPHASXJ205013.7+465518
183
+ with a 6 arcmin size may be coincident with a WZO1
184
+ description as this object is a confirmed bipolar PN
185
+ (Fig. 3) surrounded by a shell which may be an AGB
186
+ remnant shell or would indicate a multiple shell nebula
187
+ (Fig. 2).
188
+ 3.2 WZO2 type
189
+ This category concerns PNe showing a bright rim in
190
+ the direction of motion. This is the most common fea-
191
+ ture found in our sample and 17 objects out of 21 fall
192
+ under this classification. Fig. 4 presents 3 examples
193
+ with different angular sizes, although they all display
194
+ a diameter on the order of a few arcmin (we consid-
195
+ ered the assumed full extent of the round nebulae).
196
+ We point out in Fig. 4-Top the difficulty to determine
197
+ the true direction of motion regarding the CS position
198
+ and off-axis bow shock. Such a geometry could be ex-
199
+ plained by an ISM gradient from high on the left to
200
+ low on the right. We also notice a particularly low
201
+ observed surface brightness (SB) which may explain
202
+ previous non detections.
203
+ 3.3 WZO3 type
204
+ This type is exemplified by PNe whose geometric cen-
205
+ tres are shifted away from the central star (CS): both
206
+ are no longer coincident. An example, is the ancient
207
+ PN Sh 2-188 around which IPHAS has uncovered an
208
+ extended structure (Wareing et al. 2006). We identi-
209
+ fied 3 candidate PNe coincident with this description.
210
+ The most probing WZO3 type in our sample is pre-
211
+ sented in figure 5 and corresponds, according to hy-
212
+ drodynamical models, to a PN with a CS velocity of
213
+ about 100 km/s.3.4 WZO4 type
214
+ The WZO4 corresponds to the most difficult types of
215
+ PN to be detected: the CS has left the vicinity of
216
+ the now totally disrupted PN, leaving an amorphous
217
+ structure. The challenge does not lie in the detection
218
+ ability (it enters in the IPHAS range of detection) but
219
+ more intheselection oftheobjects as possible PNedue
220
+ to the total lack of symmetry or axi-symmetry. This
221
+ type of interaction is also discussed in more detail by
222
+ Wareing et al. in these proceedings.
223
+ We identified 1 candidate PN which could fit the
224
+ given description. Fig. 6 presents the selected can-
225
+ didate in the top panel. We suggest the the nebular
226
+ material has been moved from the front to the rear
227
+ leaving a remnant “wall of material”. We also notice
228
+ that some features may be linked to turbulence effects.
229
+ The comparison with the hydrodynamical model (bot-
230
+ tom panel) seems to support this hypothesis. Nev-
231
+ ertheless a spectroscopic confirmation of the nebula’s
232
+ nature will be needed. The model implies a velocity
233
+ relative to the ISM of 100 Km/s and an evolution in
234
+ the post-AGB phase of 10 000 years.
235
+ 3.5 Distribution of the candidates
236
+ Fig. 7-top shows that the majority of the WZO2 nebu-
237
+ lae typesare locatedinzones ofrelatively lowISMden-
238
+ sity (compared to the Galactic Centre). The low stress
239
+ exerted on the nebulae may explain why they still keep
240
+ their quasi circular shape. The ISM is more dense in
241
+ the Galactic Plane than in the zone towards the anti-
242
+ centreor thezoneaboveaheightof100pc(from obser-
243
+ vation of neutral hydrogen gas, Dickey et al. (1990)).
244
+ We therefore expected a greater influence of the in-
245
+ teraction process in this area. Indeed, we observed
246
+ that the most advanced stages of interaction, namely
247
+ WZO3 and WZO4, are detected in areas of high ISM
248
+ density, where PN are more likely to be affected by
249
+ such densities.
250
+ Thesizedistribution, Fig. 7-bottom, indicatesthat
251
+ althoughmostofthedetectedcandidatePNearelarge2,
252
+ i.e with a size greater than 100 arcsec, or of medium
253
+ size i.e. between 20 and 100 arcsec, small nebulae
254
+ also show signs of interaction. This confirms that the
255
+ ISM interaction process does not “a priori” only im-
256
+ ply “old” nebulae. We also observe that large objects
257
+ mainly lie at higher latitudes than smaller nebulae but
258
+ it is also interesting to notice that we detect large ob-
259
+ jects inzones ofhighextinction; large PNeseemtosur-
260
+ vive at relatively low latitudes. They would undergo
261
+ strong alteration by the ISM and would display more
262
+ advanced stages of interaction. Those disruptions tend
263
+ to affect them more than smaller size nebulae at the
264
+ same latitude range.
265
+ 4 Conclusion and Perspectives
266
+ In the first fully analysed area of the Galactic plane,
267
+ RA=18h to RA=20h, the new H αphotometric survey
268
+ 2The sizes here are defined in terms of angular sizes, so
269
+ the physical correspondence will depend on the distance.4 Publications of the Astronomical Society of Australia
270
+ IPHASappearstobeanexcellenttooltostudyPNein-
271
+ teracting with the ISM. Indeed the survey contributes
272
+ to the detection of nebulae so far hidden mainly due to
273
+ their faintness. Thus, 21 objects have been identified
274
+ aspossible planetarynebulaeinteractingwiththeISM.
275
+ They show diverse sizes (although the majority display
276
+ a diameter greater than 100 arcsec) and morphologies
277
+ corresponding to the four different cases of interaction
278
+ commonly defined going from the unaffected to the to-
279
+ tally disrupted nebula. The most common stage is the
280
+ WZO2correspondingtonebulaeshowingabrightening
281
+ of their rim in the direction of motion. This is coinci-
282
+ dent with the observations made by Wareing and al (in
283
+ these proceedings) crossing different H αsurveys. We
284
+ were also able to reach those targets at low latitudes
285
+ and found that some could survive in those environ-
286
+ ments although they would be strongly affected by the
287
+ ISM. The total lack of PNe/ISM at the highest point
288
+ of ISM density (b= ±0.5 deg and 30 deg <l<50 deg)
289
+ can either be due to the limitation of IPHAS or be-
290
+ cause they have been totally destroyed by the effects
291
+ of ISM interaction.
292
+ The next logical step is the spectroscopic identification
293
+ of these sources, their central star study and physical
294
+ size determination. The low surface brightness implies
295
+ the use of particular means such as integral field spec-
296
+ troscopy to be able to retrieve the maximum infor-
297
+ mation. Therefore a new programme of IPHAS PN
298
+ candidate follow-up spectroscopy led by Q. Parker, A.
299
+ Zijlstra and R. Corradi is now underway.
300
+ References
301
+ Acker, A., Marcout, J., Ochsenbein, F., Stenholm, B.
302
+ , Tylenda, R.,1992, Garching: European Southern
303
+ Observatory,Strasbourg - ESO catalogue of galactic
304
+ planetary nebulae. Part 1; Part 2.
305
+ Ali, A., El-Nawawy, M. S. and Pfleiderer, J., 2000,
306
+ APSS, 271, 245.
307
+ Borkowski, K. J., Sarazin, C. L. and Soker, N., 1990,
308
+ Apj, 360,173.
309
+ Dgani, R. and Soker, N., 1998, 495, 337.
310
+ Dickey, J. M. and Lockman, F. J.,1990,ARAA,28,215
311
+ Drew, J. E., Greimel, R., Irwin, M. J., Aungwero-
312
+ jwit, A., Barlow, M. J., Corradi, R. L. M., Drake,
313
+ J. J.., G¨ ansicke, B. T., Groot, P. and 26 co-
314
+ authors,2005,MNRAS,362,753.
315
+ Gonzalez-Solares, E. A., Walton, N. A., Greimel, R.
316
+ and Drew, J. E., Irwin, M. J., Sale, S. E., Andrews,
317
+ K., Aungwerojwit, A., Barlow, M. J. and 41 co-
318
+ authors, 2008,MNRAS,388,89.
319
+ Miszalski, B., Parker, Q. A., Acker, A.,
320
+ Birkby, J. L., Frew, D. J. and Kovacevic,
321
+ A.,2008,MNRAS,384,525.Parker, Q. A., Phillipps, S., Pierce, M. J., Hartley, M.,
322
+ Hambly, N. C., Read, M. A., MacGillivray, H. T.,
323
+ Tritton, S. B., Cass, C. P., Cannon, R. D., Cohen,
324
+ M., Drew, J. E., Frew, D. J., Hopewell, E., Mader,
325
+ S., Malin, D. F., Masheder, M. R. W., Morgan,
326
+ D. H., Morris, R. A. H., Russeil, D., Russell, K. S.
327
+ and Walker, R. N. F., 2005, MNRAS, 362, 689.
328
+ Parker,Q. A., Acker, A., Frew, D. J., Hartley,
329
+ M., Peyaud, A. E. J., Ochsenbein, F., Phillipps,
330
+ S., Russeil, D., Beaulieu, S. F., Cohen, M.,
331
+ K¨ oppen, J., Miszalski, B., Morgan, D. H., Mor-
332
+ ris, R. A. H., Pierce, M. J. and Vaughan,
333
+ A. E.,2006,MNRAS,373,79.
334
+ Pierce, M. J., Frew, D. J., Parker, Q. A. and K¨ oppen,
335
+ J., 2004, PASP, 21, 334.
336
+ Rauch,T., Furlan, E., Kerber, F. and Roth,
337
+ M.,2000,ASP Conf. Ser:Asymmetrical Planetary
338
+ Nebulae II,199,341.
339
+ Riesgo, H. and L´ opez, J. A.,2006,Revista Mexicana de
340
+ Astronomia y Astrofisica,42,47-51
341
+ Tweedy, R. W. and Kwitter, K. B., 1996, ApJS, 107,
342
+ 255.
343
+ Wareing, C. J., O’Brien, T. J., Zijlstra, A. A., Kwitter,
344
+ K. B., Irwin, J., Wright, N., Greimel, R. and Drew
345
+ , J. E.,2006,MNRAS,366,387.
346
+ Wareing, C. J., Zijlstra, A. A. and O’Brien,
347
+ T. J.,2007,MNRAS,382,1233.
348
+ Xilouris, K. M., Papamastorakis, J., Paleologou, E.
349
+ and Terzian, Y., 1996, AAP, 310, 603.www.publish.csiro.au/journals/pasa 5
350
+ Direction of motionThick outer
351
+ Filamentsshell: rim
352
+ + Bipolar outflowBright edges of the bipolar
353
+ Sharp structures
354
+ Faint opposite edge
355
+ Figure 2: An example of WZO1 type: The “Ear Nebula” IPHAS PN. Nort h on the top and East on the
356
+ left.
357
+ /0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0
358
+ /1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0
359
+ /1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1erg/cm2/s/A erg/cm2/s/A
360
+ Figure 3: WHT spectra of the “EarNebula” using the R300Band R158 R gratings. This nebula, for which
361
+ we show some of the “strongest” emission lines useful for an identifi cation, presents a clear [NII] over-
362
+ intensity and it has been confirmed as true PN using the revised diagn ostic diagram from Riesgo et al.
363
+ (2006) (particularly the log [H α/[SII]]vslog [Hα/[NII]] diagram).6 Publications of the Astronomical Society of Australia
364
+ Interacting rimDirection of motion
365
+ CS candidate
366
+ Direction of motion
367
+ Geometric centerBow shock CS candidates
368
+ Direction of motionCS candidateBright rim
369
+ Figure 4: Examples of WZO2 types. Top: Size=1.2 arcmin and SB=3.4e−17erg cm−2s−1arcsec−2.
370
+ Middle: Size= 8.5 arcmin and SB=1.1e−16erg cm−2s−1arcsec−2, Bottom: 4.3 arcmin and SB=2.7e−16
371
+ erg cm−2s−1arcsec−2. North on the top and East on the left.
372
+ Direction of motion
373
+ Most probable CS
374
+ Bright rim
375
+ Figure 5: WZO3 type of ISM interaction in a IPHAS candidate PNe. Nor th on the top and East on the
376
+ left.www.publish.csiro.au/journals/pasa 7
377
+
378
+ Direction of motion
379
+ Dense "wall" of nebular material gas and dust
380
+ or
381
+ turbulences Traces ofFaint frontal bow shock
382
+ Figure 6: A possible example of WZO4 ISM interaction in one IPHAS PN ca ndidate (top: North on the
383
+ top and East on the left) with the corresponding hydrodynamical m odel (bottom) [reproduction of figure
384
+ 5(d) from Wareing et al. (2007)].8 Publications of the Astronomical Society of Australia
385
+ Figure 7: Galactic distribution of the candidate PNe/ISM according t o their stage of interaction and
386
+ their size.
387
+ Figure 8: Example of candidates with sizes greater than 100 arcsec (respectively 7.7 and 2.9 arcmin) and
388
+ located at b= ±1 deg. These objects present a WZO2 stage of interaction and only their (very) faint
389
+ interacting rim are seen.
1001.0028.txt ADDED
@@ -0,0 +1,1777 @@
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
1
+ arXiv:1001.0028v2 [math.CO] 28 Feb 2012CYCLIC SIEVING FOR GENERALISED NON-CROSSING
2
+ PARTITIONS ASSOCIATED WITH COMPLEX REFLECTION
3
+ GROUPS OF EXCEPTIONAL TYPE
4
+ Christian Krattenthaler†andThomas W. M ¨uller‡
5
+ †Fakult¨ at f¨ ur Mathematik, Universit¨ at Wien,
6
+ Nordbergstraße 15, A-1090 Vienna, Austria.
7
+ WWW:http://www.mat.univie.ac.at/ ~kratt
8
+ ‡School of Mathematical Sciences,
9
+ Queen Mary & Westfield College, University of London,
10
+ Mile End Road, London E1 4NS, United Kingdom.
11
+ WWW:http://www.maths.qmw.ac.uk/ ~twm/
12
+ Dedicated to the memory of Herb Wilf
13
+ Abstract. We prove that the generalised non-crossing partitions associated with
14
+ well-generated complex reflection groups of exceptional type obe y two different cyclic
15
+ sieving phenomena, as conjectured by Armstrong, and by Bessis a nd Reiner. The
16
+ computational details are provided in the manuscript “Cyclic sieving for generalised
17
+ non-crossing partitions associated with complex reflectio n groups of exceptional type
18
+ — the details” [arχiv:1001.0030 ].
19
+ 1.Introduction
20
+ In his memoir [2], Armstrong introduced generalised non-crossing partitions asso-
21
+ ciated with finite (real) reflection groups, thereby embedding Krew eras’ non-crossing
22
+ partitions [22], Edelman’s m-divisible non-crossing partitions [12], thenon-crossing par-
23
+ titions associated with reflection groups due to Bessis [6] and Brady and Watt [10] into
24
+ one uniform framework. Bessis and Reiner [9] observed that Arms trong’s definition can
25
+ be straightforwardly extended to well-generated complex reflection groups (see Section 2
26
+ for the precise definition). These generalised non-crossing partit ions possess a wealth
27
+ of beautiful properties, and they display deep and surprising relat ions to other combi-
28
+ natorial objects defined for reflection groups (such as the gene ralised cluster complex
29
+ 2000Mathematics Subject Classification. Primary 05E15; Secondary 05A10 05A15 05A18 06A07
30
+ 20F55.
31
+ Key words and phrases. complex reflection groups, unitary reflection groups, m-divisible non-
32
+ crossing partitions, generalised non-crossing partitions, Fuß–Ca talan numbers, cyclic sieving.
33
+ †Research partially supported by the Austrian Science Foundation F WF, grants Z130-N13 and
34
+ S9607-N13, the latter in the framework of the National Research Network “Analytic Combinatorics
35
+ and Probabilistic Number Theory.”
36
+ ‡Research supported by the Austrian Science Foundation FWF, Lise Meitner grant M1201-N13.
37
+ 12 C. KRATTENTHALER AND T. W. M ¨ULLER
38
+ of Fomin and Reading [13], or the extended Shi arrangement and the geometric multi-
39
+ chains of filters of Athanasiadis [4, 5]); see Armstrong’s memoir [2] and the references
40
+ given therein.
41
+ Ontheotherhand, cyclic sieving isaphenomenonbroughttolightbyReiner, Stanton
42
+ and White [30]. It extends the so-called “( −1)-phenomenon” of Stembridge [34, 35].
43
+ Cyclic sieving can be defined in three equivalent ways (cf. [30, Prop. 2.1]). The one
44
+ which gives the name can be described as follows: given a set Sof combinatorial
45
+ objects, an action on Sof a cyclic group G=/an}bracketle{tg/an}bracketri}htwith generator gof ordern, and
46
+ a polynomial P(q) inqwith non-negative integer coefficients, we say that the triple
47
+ (S,P,G)exhibits the cyclic sieving phenomenon , if the number of elements of Sfixed
48
+ bygkequalsP(e2πik/n). In [30] it is shown that this phenomenon occurs in surprisingly
49
+ many contexts, and several further instances have been discov ered since then.
50
+ In [2, Conj. 5.4.7] (also appearing in [9, Conj. 6.4]) and [9, Conj. 6.5], Ar mstrong,
51
+ respectively Bessis and Reiner, conjecture that generalised non- crossing partitions for
52
+ irreducible well-generated complex reflection groups exhibit two diffe rent cyclic sieving
53
+ phenomena (see Sections 3 and 7 for the precise statements).
54
+ According to the classification of these groups due to Shephard an d Todd [32], there
55
+ are two infinite families of irreducible well-generated complex reflectio n groups, namely
56
+ the groups G(d,1,n) andG(e,e,n), wheren,d,eare positive integers, and there are 26
57
+ exceptional groups. For the infinite families of types G(d,1,n) andG(e,e,n), the two
58
+ cyclic sieving conjectures follow from the results in [19].
59
+ Thepurposeofthepresent articleistopresent aproofofthecyc licsieving conjectures
60
+ of Armstrong, and of Bessis and Reiner, for the 26 exceptional ty pes, thus completing
61
+ the proof of these conjectures. Since the generalised non-cros sing partitions feature a
62
+ parameterm, from the outset this is nota finite problem. Consequently, we first need
63
+ several auxiliary results to reduce the conjectures for each of t he 26 exceptional types
64
+ to afiniteproblem. Subsequently, we use Stembridge’s Maplepackagecoxeter [36]
65
+ and theGAPpackageCHEVIE[14, 28] to carry out the remaining finitecomputations.
66
+ The details of these computations are provided in [21]. In the presen t paper, we con-
67
+ tent ourselves with exemplifying the necessary computations by go ing through some
68
+ representative cases. It is interesting to observe that, for the verification of the type
69
+ E8case, it is essential to use the decomposition numbers in the sense o f [17, 18, 20] be-
70
+ cause, otherwise, the necessary computations would not be feas ible in reasonable time
71
+ with the currently available computer facilities. We point out that, fo r the special case
72
+ where the aforementioned parameter mis equal to 1, the first cyclic sieving conjecture
73
+ has been proved in a uniform fashion by Bessis and Reiner in [9]. (See [3 ] for a —
74
+ non-uniform — proof of cyclic sieving for non-crossing partitions as sociated with real
75
+ reflection groups under the action of the so-called Kreweras map, a special case of the
76
+ second cyclic sieving phenomenon discussed in the present paper.) T he crucial result on
77
+ which the proof of Bessis and Reiner is based is (5.5) below, and it plays an important
78
+ rolein our reduction of the conjectures forthe 26 exceptional gr oupsto a finite problem.
79
+ Our paper is organised as follows. In the next section, we recall the definition of
80
+ generalised non-crossing partitions for well-generated complex re flection groups and of
81
+ decomposition numbers in the sense of [17, 18, 20], and we review so me basic facts.
82
+ The first cyclic sieving conjecture is subsequently stated in Section 3. In Section 4, we
83
+ outline an elementary proof that the q-Fuß–Catalan number, which is the polynomial
84
+ Pin the cyclic sieving phenomena concerning the generalised non-cros sing partitionsCYCLIC SIEVING FOR GENERALISED NON-CROSSING PARTITIONS 3
85
+ for well-generated complex reflection groups, is always a polynomial with non-negative
86
+ integer coefficients, as required by the definition of cyclic sieving. (F ull details can be
87
+ found in [21, Sec. 4]. The reader is referred to the first paragraph of Section 4 for
88
+ comments on other approaches for establishing polynomiality with no n-negative coeffi-
89
+ cients.) Section 5 contains the announced auxiliary results which, fo r the 26 exceptional
90
+ types, allow a reduction of the conjecture to a finite problem. In Se ction 6, we discuss
91
+ a few cases which, in a representative manner, demonstrate how t o perform the re-
92
+ maining case-by-case verification of the conjecture. For full det ails, we refer the reader
93
+ to [21, Sec. 6]. The second cyclic sieving conjecture is stated in Sect ion 7. Section 8
94
+ contains the auxiliary results which, for the 26 exceptional types, allow a reduction of
95
+ the conjecture to a finite problem, while in Section 9 we discuss some r epresentative
96
+ cases of the remaining case-by-case verification of the conjectu re. Again, for full details
97
+ we refer the reader to [21, Sec. 9].
98
+ 2.Preliminaries
99
+ Acomplex reflection group isa groupgeneratedby(complex) reflections in Cn. (Here,
100
+ a reflection is a non-trivial element of GLn(C) which fixes a hyperplane pointwise and
101
+ which hasfiniteorder.) Wereferto[24]foranin-depthexpositionof thetheorycomplex
102
+ reflection groups.
103
+ Shephard and Todd provided a complete classification of all finitecomplex reflection
104
+ groups in [32] (see also [24, Ch. 8]). According to this classification, a n arbitrary
105
+ complex reflection group Wdecomposes into a direct product of irreducible complex
106
+ reflection groups, acting on mutually orthogonal subspaces of th e complex vector space
107
+ onwhichWisacting. Moreover, thelistofirreduciblecomplexreflectiongroups consists
108
+ of the infinite family of groups G(m,p,n), wherem,p,nare positive integers, and 34
109
+ exceptional groups, denoted G4,G5,...,G 37by Shephard and Todd.
110
+ In this paper, we are only interested in finite complex reflection grou ps which are
111
+ well-generated . A complex reflection group of rank nis called well-generated if it is
112
+ generated by nreflections.1Well-generation can be equivalently characterised by a
113
+ duality property due to Orlik and Solomon [29]. Namely, a complex reflec tion group of
114
+ ranknhastwo sets ofdistinguished integers d1≤d2≤ ··· ≤dnandd∗
115
+ 1≥d∗
116
+ 2≥ ··· ≥d∗
117
+ n,
118
+ called its degreesandcodegrees , respectively (see [24, p. 51 and Def. 10.27]). Orlik and
119
+ Solomon observed, using case-by-case checking, that an irreduc ible complex reflection
120
+ groupWof ranknis well-generated if and only if its degrees and codegrees satisfy
121
+ di+d∗
122
+ i=dn
123
+ for alli= 1,2,...,n. The reader is referred to [24, App. D.2] for a table of the degree s
124
+ and codegrees of all irreducible complex reflection groups. Togeth er with the classi-
125
+ fication of Shephard and Todd [32], this constitutes a classification o f well-generated
126
+ complex reflection groups: the irreducible well-generated complex r eflection groups are
127
+ — the two infinite families G(d,1,n) andG(e,e,n), whered,e,nare positive inte-
128
+ gers,
129
+ — the exceptional groups G4,G5,G6,G8,G9,G10,G14,G16,G17,G18,G20,G21of
130
+ rank 2,
131
+ 1We refer to [24, Def. 1.29] for the precise definition of “rank.” Roug hly speaking, the rank of a
132
+ complex reflection group Wis the minimal nsuch that Wcan be realized as reflection group on Cn.4 C. KRATTENTHALER AND T. W. M ¨ULLER
133
+ — the exceptional groups G23=H3,G24,G25,G26,G27of rank 3,
134
+ — the exceptional groups G28=F4,G29,G30=H4,G32of rank 4,
135
+ — the exceptional group G33of rank 5,
136
+ — the exceptional groups G34,G35=E6of rank 6,
137
+ — the exceptional group G36=E7of rank 7,
138
+ — and the exceptional group G37=E8of rank 8.
139
+ In this list, we have made visible the groups H3,F4,H4,E6,E7,E8which appear as
140
+ exceptional groups in the classification of all irreducible realreflection groups (cf. [16]).
141
+ LetWbe a well-generated complex reflection group of rank n, and letT⊆Wdenote
142
+ theset of all(complex) reflections inthegroup. Let ℓT:W→Zdenotethewordlength
143
+ in terms of the generators T. This word length is called absolute length orreflection
144
+ length. Furthermore, we define a partial order ≤TonWby
145
+ u≤Twif and only if ℓT(w) =ℓT(u)+ℓT(u−1w). (2.1)
146
+ This partial order is called absolute order orreflection order . As is well-known and
147
+ easy to see, the equation in (2.1) is equivalent to the statement tha t every shortest
148
+ representation of uby reflections occurs as an initial segment in some shortest produc t
149
+ representation of wby reflections.
150
+ Now fix a (generalised) Coxeter element2c∈Wand a positive integer m. The
151
+ m-divisible non-crossing partitions NCm(W) are defined as the set
152
+ NCm(W) =/braceleftbig
153
+ (w0;w1,...,w m) :w0w1···wm=cand
154
+ ℓT(w0)+ℓT(w1)+···+ℓT(wm) =ℓT(c)/bracerightbig
155
+ .
156
+ A partial order is defined on this set by
157
+ (w0;w1,...,w m)≤(u0;u1,...,u m) if and only if ui≤Twifor 1≤i≤m.
158
+ We have suppressed the dependence on c, since we understand this definition up to
159
+ isomorphism of posets. To be more precise, it can be shown that any two Coxeter
160
+ elements are related to each other by conjugation and (possibly) a n automorphism on
161
+ the field of complex numbers (see [33, Theorem 4.2] or [24, Cor. 11.2 5]), and hence the
162
+ resulting posets NCm(W) are isomorphic to each other. If m= 1, thenNC1(W) can
163
+ be identified with the set NC(W) of non-crossing partitions for the (complex) reflection
164
+ groupWasdefined byBessis andCorran(cf.[8]and[7, Sec.13]; theirdefinit ionextends
165
+ the earlier definition by Bessis [6] and Brady and Watt [10] for real r eflection groups).
166
+ The following result has been proved by a collaborative effort of seve ral authors (see
167
+ [7, Prop. 13.1]).
168
+ 2An element of an irreducible well-generated complex reflection group Wof ranknis called a
169
+ Coxeter element if it isregularin the sense of Springer [33] (see also [24, Def. 11.21]) and of order dn.
170
+ An element of Wis called regular if it has an eigenvector which lies in no reflecting hyperp lane of a
171
+ reflection of W. It follows from an observation of Lehrer and Springer, proved un iformly by Lehrer
172
+ and Michel [23] (see [24, Theorem 11.28]), that there is always a regu lar element of order dnin an
173
+ irreducible well-generated complex reflection group Wof rankn. More generally, if a well-generated
174
+ complex reflection group Wdecomposes as W∼=W1×W2×···×Wk, where the Wi’s are irreducible,
175
+ then a Coxeter element of Wis an element of the form c=c1c2···ck, whereciis a Coxeter element of
176
+ Wi,i= 1,2,...,k. IfWis arealreflection group, that is, if all generators in Thave order 2, then the
177
+ notion of generalised Coxeter element given above reduces to that of a Coxeter element in the classical
178
+ sense (cf. [16, Sec. 3.16]).CYCLIC SIEVING FOR GENERALISED NON-CROSSING PARTITIONS 5
179
+ Theorem 1. LetWbe an irreducible well-generated complex reflection group, and let
180
+ d1≤d2≤ ··· ≤dnbe its degrees and h:=dnits Coxeter number. Then
181
+ |NCm(W)|=n/productdisplay
182
+ i=1mh+di
183
+ di. (2.2)
184
+ Remark1.(1) The number in (2.2) is called the Fuß–Catalan number for the reflection
185
+ groupW.
186
+ (2) Ifcis a Coxeter element of a well-generated complex reflection group Wof rank
187
+ n, thenℓT(c) =n. (This follows from [7, Sec. 7].)
188
+ We conclude this section by recalling the definition of decomposition nu mbers from
189
+ [17, 18, 20]. Although we need them here only for (very small) real re flection groups,
190
+ and although, strictly speaking, they have been only defined for re al reflection groups in
191
+ [17, 18, 20], this definition can be extended to well-generated comple x reflection groups
192
+ without any extra effort, which we do now.
193
+ Given a well-generated complex reflection group Wof rankn, typesT1,T2,...,T d(in
194
+ the sense of the classification of well-generated complex reflection groups) such that the
195
+ sumoftheranksofthe Ti’sequalsn, andaCoxeter element c, thedecompositionnumber
196
+ NW(T1,T2,...,T d) is defined as the number of “minimal” factorisations c=c1c2···cd,
197
+ “minimal” meaning that ℓT(c1) +ℓT(c2) +···+ℓT(cd) =ℓT(c) =n, such that, for
198
+ i= 1,2,...,d, the type of cias a parabolic Coxeter element is Ti. (Here, the term
199
+ “parabolic Coxeter element” means a Coxeter element in some parab olic subgroup. It
200
+ follows from [31, Prop.6.3] that any element ciis indeed a Coxeter element in a unique
201
+ parabolic subgroup of W.3By definition, the type of ciis the type of this parabolic
202
+ subgroup.) Since any two Coxeter elements are related to each oth er by conjugation
203
+ plus field automorphism, the decomposition numbers are independen t of the choice of
204
+ the Coxeter element c.
205
+ The decomposition numbers for real reflection groups have been c omputed in [17,
206
+ 18, 20]. To compute the decomposition numbers for well-generated complex reflection
207
+ groups is a task that remains to be done.
208
+ 3.Cyclic sieving I
209
+ In this section we present the first cyclic sieving conjecture due to Armstrong [2,
210
+ Conj. 5.4.7], and to Bessis and Reiner [9, Conj. 6.4].
211
+ Letφ:NCm(W)→NCm(W) be the map defined by
212
+ (w0;w1,...,w m)/mapsto→/parenleftbig
213
+ (cwmc−1)w0(cwmc−1)−1;cwmc−1,w1,w2,...,w m−1/parenrightbig
214
+ .(3.1)
215
+ It is indeed not difficult to see that, if the ( m+ 1)-tuple on the left-hand side is an
216
+ element ofNCm(W), then so is the ( m+1)-tuple on the right-hand side. For m= 1,
217
+ this action reduces to conjugation by the Coxeter element c(applied to w1). Cyclic
218
+ sieving arising from conjugation by chas been the subject of [9].
219
+ 3The uniqueness can be argued as follows: suppose that ciwere a Coxeter element in two parabolic
220
+ subgroups of W, sayU1andU2. Then it must also be a Coxeter element in the intersection U1∩U2.
221
+ On the other hand, the absolute length of a Coxeter element of a co mplex reflection group Uis always
222
+ equal to rk( U), the rank of U. (This follows from the fact that, for each element uofU, we have
223
+ ℓT(u) = codim/parenleftbig
224
+ ker(u−id)/parenrightbig
225
+ , with id denoting the identity element in U; see e.g. [31, Prop. 1.3]). We
226
+ conclude that ℓT(ci) = rk(U1) = rk(U2) = rk(U1∩U2), This implies that U1=U2.6 C. KRATTENTHALER AND T. W. M ¨ULLER
227
+ It is easy to see that φmhacts as the identity, where his the Coxeter number of W
228
+ (see (5.1) and Lemma 29 below). By slight abuse of notation, let C1be the cyclic group
229
+ of ordermhgenerated by φ. (The slight abuse consists in the fact that we insist on C1
230
+ to be a cyclic group of order mh, while it may happen that the order of the action of
231
+ φgiven in (3.1) is actually a proper divisor of mh.)
232
+ Given these definitions, we are now in the position to state the first c yclic sieving
233
+ conjecture of Armstrong, respectively of Bessis and Reiner. By t he results of [19] and
234
+ of this paper, it becomes the following theorem.
235
+ Theorem 2. For an irreducible well-generated complex reflection group Wand any
236
+ m≥1, the triple (NCm(W),Catm(W;q),C1), whereCatm(W;q)is theq-analogue of
237
+ the Fuß–Catalan number defined by
238
+ Catm(W;q) :=n/productdisplay
239
+ i=1[mh+di]q
240
+ [di]q, (3.2)
241
+ exhibits the cyclic sieving phenomenon in the sense of Reine r, Stanton and White [30].
242
+ Here,nis the rank of W,d1,d2,...,d nare the degrees of W,his the Coxeter number
243
+ ofW, and[α]q:= (1−qα)/(1−q).
244
+ Remark2.We write Catm(W) for Catm(W;1).
245
+ By definition of the cyclic sieving phenomenon, we have to prove that Catm(W;q) is
246
+ a polynomial in qwith non-negative integer coefficients, and that
247
+ |FixNCm(W)(φp)|= Catm(W;q)/vextendsingle/vextendsingle
248
+ q=e2πip/mh, (3.3)
249
+ for allpin the range 0 ≤p<mh. The first fact is established in the next section, while
250
+ the proof of the second is achieved by making use of several auxiliar y results, given
251
+ in Section 5, to reduce the proof to a finite problem, and a subseque nt case-by-case
252
+ analysis. Alldetails ofthisanalysiscanbefoundin[21, Sec. 6]. Inthe present paper, we
253
+ content ourselves with discussing the cases where W=G24and whereW=G37=E8,
254
+ since these suffice to convey the flavour of the necessary comput ations.
255
+ 4.Theq-Fusz–Catalan numbers Catm(W;q)
256
+ The purpose of this section is to provide an elementary, self-conta ined proof of the
257
+ fact that, for all irreducible complex reflection groups W, theq-Fuß–Catalan number
258
+ Catm(W;q) is a polynomial in qwith non-negative integer coefficients. For most of
259
+ the groups, this is a known property. However, aside from the fac t that, for many of
260
+ the known cases, the proof is very indirect and uses deep algebraic results on rational
261
+ Cherednik algebras, there still remained some cases where this pro perty had not been
262
+ formally established. The reader is referred to the “Theorem” in Se ction 1.6 of [15],
263
+ whichsaysthat, undertheassumptionofacertainrankcondition( [15, Hypothesis2.4]),
264
+ theq-Fuß–Catalan number Catm(W;q) is a Hilbert series of a finite-dimensional quo-
265
+ tient of the ring of invariants of Wand also the graded character of a finite-dimensional
266
+ irreducible representation of a spherical rational Cherednik algeb ra associated with
267
+ W. At present, this rank condition has been proven for all irreducible well-generated
268
+ complex reflection groups apart from G17,G18,G29,G33,G34; see [26, Tables 8 and 9,
269
+ column “rank”], and the recent paper [27], which establishes the res ult in the case of
270
+ G32.CYCLIC SIEVING FOR GENERALISED NON-CROSSING PARTITIONS 7
271
+ In the sequel, aside from the standard notation [ α]q= (1−qα)/(1−q) forq-integers,
272
+ we shall also use the q-binomial coefficient, which is defined by
273
+ /bracketleftbigg
274
+ n
275
+ k/bracketrightbigg
276
+ q:=/braceleftBigg
277
+ 1, ifk= 0,
278
+ [n]q[n−1]q···[n−k+1]q
279
+ [k]q[k−1]q···[1]q,ifk>0.
280
+ We begin with several auxiliary results.
281
+ Proposition 3. For all non-negative integers nandk, theq-binomial coefficient [n
282
+ k]q
283
+ is a polynomial in qwith non-negative integer coefficients.
284
+ Proof.This is a well-known fact, which can be derived either from the recurr ence rela-
285
+ tion(s) satisfied by the q-binomial coefficients (generalising Pascal’s recurrence relation
286
+ for binomial coefficients; cf. [1, eqs. (3.3.3) and (3.3.4)]), or from th e fact that the q-
287
+ binomial coefficient [n
288
+ k]qis the generating function for (integer) partitions with at most
289
+ kparts all of which are at most n−k(cf. [1, Theorem 3.1]). /square
290
+ Proposition 4. For all non-negative integers mandn, theq-Fuß–Catalan number of
291
+ typeAn,
292
+ 1
293
+ [(m+1)n+1]q/bracketleftbigg
294
+ (m+1)n+1
295
+ n/bracketrightbigg
296
+ q,
297
+ is a polynomial in qwith non-negative integer coefficients.
298
+ Proof.In [25, Sec. 3.3], Loehr proves that
299
+ 1
300
+ [(m+1)n+1]q/bracketleftbigg
301
+ (m+1)n+1
302
+ n/bracketrightbigg
303
+ q
304
+ =/summationdisplay
305
+ v∈V(m)
306
+ nqm(n
307
+ 2)+/summationtext
308
+ i≥0(m(vi
309
+ 2)−ivi)/productdisplay
310
+ i≥1qvi/summationtextm
311
+ j=1(m−j)vi−j/bracketleftbigg
312
+ vi+vi−1+···+vi−m−1
313
+ vi/bracketrightbigg
314
+ q,(4.1)
315
+ whereV(m)
316
+ ndenotes the set of all sequences v= (v0,v1,...,v s) (for some s) of non-
317
+ negative integers with v0>0,vs>0, andv0+v1+···+vs=n, and such that there
318
+ is never a string of mor more consecutive zeroes in v. By convention, vi= 0 for all
319
+ negativei. His proof works by showing that the expressions on both sides of ( 4.1)
320
+ satisfy the same recurrence relation and initial conditions, using cla ssicalq-binomial
321
+ identities. We refer the reader to [25] for details. By Proposition 3, the expression on
322
+ the right-hand side of (4.1) is manifestly a polynomial in qwith non-negative integer
323
+ coefficients. /square
324
+ Lemma 5. Ifaandbare coprime positive integers, then
325
+ [ab]q
326
+ [a]q[b]q(4.2)
327
+ is a polynomial in qof degree (a−1)(b−1), all of whose coefficients are in {0,1,−1}.
328
+ Moreover, if one disregards the coefficients which are 0, then+1’s and(−1)’s alternate,
329
+ and the constant coefficient as well as the leading coefficient o f the polynomial equal +1.
330
+ Proof.LetΦn(q)denotethe n-thcyclotomicpolynomialin q. Usingtheclassicalformula
331
+ 1−qn=/productdisplay
332
+ d|nΦd(q),8 C. KRATTENTHALER AND T. W. M ¨ULLER
333
+ we see that
334
+ (1−q)(1−qab)
335
+ (1−qa)(1−qb)=/productdisplay
336
+ d1|a,d1/ne}ationslash=1
337
+ d2|a,d2/ne}ationslash=1Φd1d2(q),
338
+ so that, manifestly, the expression in (4.2) is a polynomial in q. The claim concerning
339
+ the degree of this polynomial is obvious.
340
+ In order to establish the claim on the coefficients, we start with a sub -expression of
341
+ (4.2),
342
+ (1−qab)
343
+ (1−qa)(1−qb)=/parenleftbiggb−1/summationdisplay
344
+ i=0qia/parenrightbigg/parenleftbigg∞/summationdisplay
345
+ j=0qjb/parenrightbigg
346
+ =∞/summationdisplay
347
+ k=0Ckqk, (4.3)
348
+ say. The assumption that aandbare coprime implies that 0 ≤Ck≤1 fork≤
349
+ (a−1)(b−1). Multiplying both sides of (4.3) by 1 −q, we obtain the equation
350
+ [ab]q
351
+ [a]q[b]q= (1−q)(a−1)(b−1)/summationdisplay
352
+ k=0Ckqk+(1−q)∞/summationdisplay
353
+ k=(a−1)(b−1)+1Ckqk. (4.4)
354
+ By our previous observation on the coefficients Ckwithk≤(a−1)(b−1), it is obvious
355
+ that the coefficients of the first expression on the right-hand side of (4.4) are alternately
356
+ +1 and−1, when 0’s are disregarded. Since we already know that the left-ha nd side is
357
+ a polynomial in qof degree (a−1)(b−1), we may ignore the second expression.
358
+ The proof is concluded by observing that the claims on the constant and leading
359
+ coefficients are obvious. /square
360
+ Corollary 6. Letaandbbe coprime positive integers, and let γbe an integer with
361
+ γ≥(a−1)(b−1). Then the expression
362
+ [γ]q[ab]q
363
+ [a]q[b]q
364
+ is a polynomial in qwith non-negative integer coefficients.
365
+ Proof.Let
366
+ [ab]q
367
+ [a]q[b]q=(a−1)(b−1)/summationdisplay
368
+ k=0Dkqk.
369
+ We then have
370
+ [γ]q[ab]q
371
+ [a]q[b]q=(a−1)(b−1)+γ−1/summationdisplay
372
+ N=0qNN/summationdisplay
373
+ k=max{0,N−γ+1}Dk. (4.5)
374
+ IfN≤γ−1, then, by Lemma 5, the sum over kon the right-hand side of (4.5) equals
375
+ 1−1+1−1+···, which is manifestly non-negative. On the other hand, if N >γ−1,
376
+ then we may rewrite the sum over kon the right-hand side of (4.5) as
377
+ N/summationdisplay
378
+ k=max{0,N−γ+1}Dk=(a−1)(b−1)/summationdisplay
379
+ k=N−γ+1Dk=(a−1)(b−1)+γ−1−N/summationdisplay
380
+ k=0D(a−1)(b−1)−k.
381
+ Again, by Lemma 5, this sum equals 1 −1 + 1−1 +···, which is manifestly non-
382
+ negative. /squareCYCLIC SIEVING FOR GENERALISED NON-CROSSING PARTITIONS 9
383
+ The next lemmas all have a very similar flavour, and so do their proofs . In order to
384
+ avoid repetition, proof details are only provided for Lemmas 7 and 16 ; the proofs of
385
+ Lemmas 9–15, 22–24 follow the pattern exhibited in the proof of Lem ma 7, while the
386
+ proofs of Lemmas 17–21 follow that of the proof of Lemma 15. Full d etails are found
387
+ in [21, Sec. 4].
388
+ Lemma 7. Letαandβbe positive integers with α≥6andβ≥8. Then the expression
389
+ [α]q3[β]q4[72]q[3]q[4]q
390
+ [8]q[9]q[12]q
391
+ is a polynomial in qwith non-negative integer coefficients.
392
+ Proof.We have
393
+ [72]q[3]q[4]q
394
+ [8]q[9]q[12]q
395
+ = (1−q3+q9−q15+q18)(1−q4+q8−q12+q16−q20+q24−q28+q32).
396
+ It should be observed that both factors on the right-hand side ha ve the property that
397
+ coefficients are in {0,1,−1}and that (+1)’s and ( −1)’s alternate, if one disregards the
398
+ coefficients which are 0. If we now apply the same idea as in the proof o f Corollary 6,
399
+ then we see that [ α]q3times the first factor is a polynomial in qwith non-negative
400
+ integer coefficients, as is [ β]q4times the second factor. Taken together, this establishes
401
+ the claim. /square
402
+ Lemma 8. Letαandβbe positive integers with α≥26andβ≥8. Then the expression
403
+ [α]q[β]q4[15]q
404
+ [3]q[5]q[72]q[3]q[4]q
405
+ [8]q[9]q[12]q
406
+ is a polynomial in qwith non-negative integer coefficients.
407
+ Lemma 9. Letαandβbe positive integers with α≥18andβ≥3. Then the expression
408
+ [α]q3[β]q4[90]q[3]q[4]q
409
+ [5]q[6]q[9]q
410
+ is a polynomial in qwith non-negative integer coefficients.
411
+ Lemma 10. Letαandβbe positive integers with α≥20andβ≥18. Then the
412
+ expression
413
+ [α]q[β]q3[90]q[3]q
414
+ [5]q[6]q[9]q
415
+ is a polynomial in qwith non-negative integer coefficients.
416
+ Lemma 11. Letαbe a positive integer with α≥26. Then the expression
417
+ [α]q[15]q
418
+ [3]q[5]q[12]q3
419
+ [3]q3[4]q3
420
+ is a polynomial in qwith non-negative integer coefficients.10 C. KRATTENTHALER AND T. W. M ¨ULLER
421
+ Lemma 12. Letαbe a positive integer with α≥14. Then the expression
422
+ [α]q[15]q
423
+ [3]q[5]q[6]q3
424
+ [2]q3[3]q3
425
+ is a polynomial in qwith non-negative integer coefficients.
426
+ Lemma 13. Letαandβbe positive integers with α≥30andβ≥20. Then the
427
+ expression
428
+ [α]q[β]q2[84]q[2]q
429
+ [4]q[6]q[7]q
430
+ is a polynomial in qwith non-negative integer coefficients.
431
+ Lemma 14. Letαandβbe positive integers with α≥24andβ≥68. Then the
432
+ expression
433
+ [α]q[β]q[105]q
434
+ [3]q[5]q[7]q
435
+ is a polynomial in qwith non-negative integer coefficients.
436
+ Lemma 15. Letαandβbe positive integers with α≥24andβ≥34. Then the
437
+ expression
438
+ [α]q[β]q[70]q
439
+ [2]q[5]q[7]q
440
+ is a polynomial in qwith non-negative integer coefficients.
441
+ Lemma 16. Letαandβbe positive integers with α≥4andβ≥2. Then the expression
442
+ [α]q2[β]q5[30]q[2]q[3]q[5]q
443
+ [6]q[10]q[15]q
444
+ is a polynomial in qwith non-negative integer coefficients.
445
+ Proof.We have
446
+ [30]q[2]q[3]q[5]q
447
+ [6]q[10]q[15]q= 1+q−q3−q4−q5+q7+q8.
448
+ If we multiply this expression by [ α]q2, then, forα= 4 we obtain
449
+ 1+q+q2−q5−q9+q12+q13+q14,
450
+ forα= 5 we obtain
451
+ 1+q+q2−q5+q8−q11+q14+q15+q16,
452
+ and, forα≥6, we obtain
453
+ 1+q+q2−q5+q8+q10+p1(q)+q2α−4+q2α−2−q2α+1+q2α+4+q2α+5+q2α+6,
454
+ wherep1(q) is a polynomial in qwith non-negative coefficients of order at least 11 and
455
+ degree at most 2 α−5. In all cases it is obvious that the product of the result and [ β]q5,
456
+ withβ≥2, is a polynomial in qwith non-negative coefficients. /squareCYCLIC SIEVING FOR GENERALISED NON-CROSSING PARTITIONS 11
457
+ Lemma 17. Letαandβbe positive integers with α≥14andβ≥2. Then the
458
+ expression
459
+ [α]q[β]q5[14]q
460
+ [2]q[7]q[30]q[2]q[3]q[5]q
461
+ [6]q[10]q[15]q
462
+ is a polynomial in qwith non-negative integer coefficients.
463
+ Lemma 18. Letαandβbe positive integers with α≥32andβ≥12. Then the
464
+ expression
465
+ [α]q[β]q2[35]q
466
+ [5]q[7]q[30]q[2]q[3]q[5]q
467
+ [6]q[10]q[15]q
468
+ is a polynomial in qwith non-negative integer coefficients.
469
+ Lemma 19. Letαandβbe positive integers with α≥16andβ≥2. Then the
470
+ expression
471
+ [α]q2[β]q5[60]q[2]q[3]q[5]q
472
+ [10]q[12]q[15]q
473
+ is a polynomial in qwith non-negative integer coefficients.
474
+ Lemma 20. Letαandβbe positive integers with α≥56andβ≥4. Then the
475
+ expression
476
+ [α]q[β]q2[35]q
477
+ [5]q[7]q[60]q[2]q[3]q[5]q
478
+ [10]q[12]q[15]q
479
+ is a polynomial in qwith non-negative integer coefficients.
480
+ Lemma 21. Letαandβbe positive integers with α≥38andβ≥2. Then the
481
+ expression
482
+ [α]q[β]q5[14]q
483
+ [2]q[7]q[60]q[2]q[3]q[5]q
484
+ [10]q[12]q[15]q
485
+ is a polynomial in qwith non-negative integer coefficients.
486
+ Lemma 22. Letαandβbe positive integers with α≥30andβ≥26. Then the
487
+ expression
488
+ [α]q[β]q3[126]q[3]q
489
+ [6]q[7]q[9]q
490
+ is a polynomial in qwith non-negative integer coefficients.
491
+ Lemma 23. Letαandβbe positive integers with α≥66andβ≥54. Then the
492
+ expression
493
+ [α]q[β]q3[252]q[3]q
494
+ [7]q[9]q[12]q
495
+ is a polynomial in qwith non-negative integer coefficients.
496
+ Lemma 24. Letαandβbe positive integers with α≥54andβ≥34. Then the
497
+ expression
498
+ [α]q[β]q2[140]q[2]q
499
+ [4]q[7]q[10]q
500
+ is a polynomial in qwith non-negative integer coefficients.12 C. KRATTENTHALER AND T. W. M ¨ULLER
501
+ We are now ready for the proof of the main result of this section.
502
+ Theorem 25. For all irreducible well-generated complex reflection grou ps and posi-
503
+ tive integers m, theq-Fuß–Catalan number Catm(W;q)is a polynomial in qwith non-
504
+ negative integer coefficients.
505
+ Proof.First, letW=An. In this case, the degrees are 2 ,3,...,n+1, and hence
506
+ Catm(An;q) =1
507
+ [(m+1)n+1]q/bracketleftbigg
508
+ (m+1)n+1
509
+ n/bracketrightbigg
510
+ q,
511
+ which, by Proposition 4, is a polynomial in qwith non-negative integer coefficients.
512
+ Next, letW=G(d,1,n). In this case, the degrees are d,2d,...,nd , and hence
513
+ Catm(G(d,1,n);q) =/bracketleftbigg
514
+ (m+1)n
515
+ n/bracketrightbigg
516
+ qd,
517
+ which, by Proposition 3, is a polynomial in qwith non-negative integer coefficients.
518
+ Now, letW=G(e,e,n). In this case, the degrees are e,2e,...,(n−1)e,n, and hence
519
+ Catm(G(e,e,n);q) =[m(n−1)e+n]q
520
+ [n]qn−1/productdisplay
521
+ i=1[m(n−1)e+ie]q
522
+ [ie]q
523
+ =/bracketleftbigg
524
+ (m+1)(n−1)
525
+ n−1/bracketrightbigg
526
+ qe+qn[e]qn/bracketleftbigg
527
+ (m+1)(n−1)
528
+ n/bracketrightbigg
529
+ qe,
530
+ which, by Proposition 3, is a polynomial in qwith non-negative integer coefficients.
531
+ It remains to verify the claim for the exceptional groups.
532
+ For the groups W=G6,G9,G14,G17,G21,and partially for the groups W=G20,G23,
533
+ G28,G30,G33,G35,G36,G37(depending on congruence properties of the parameter m),
534
+ polynomiality and non-negativity of coefficients of the correspondin gq-Fuß–Catalan
535
+ number can be directly read off by a proper rearrangement of the t erms in the defining
536
+ expression; for example, for W=G21(with degrees given by 12 ,60) we have
537
+ Catm(G21;q) =[60m+12]q[60m+60]q
538
+ [12]q[60]q= [5m+1]q12[m+1]q60,
539
+ which is manifestly a polynomial in qwith non-negative integer coefficients.
540
+ For the groups G5,G10,G18,G26,G27,G29,G34, the terms in the defining expres-
541
+ sion of the corresponding q-Fuß–Catalan number can be arranged in a manner so
542
+ that aq-binomial coefficient appears; polynomiality and non-negativity of co efficients
543
+ then follow from Proposition 3. For example, for W=G34(with degrees given by
544
+ 6,12,18,24,30,42) we have
545
+ Catm(G34;q) =[42m+6]q[42m+12]q[42m+18]q[42m+24]q[42m+30]q[42m+42]q
546
+ [6]q[12]q[18]q[24]q[30]q[42]q
547
+ = [m+1]q42/bracketleftbigg
548
+ 7m+5
549
+ 5/bracketrightbigg
550
+ q6,
551
+ which, written in this form, is obviously a polynomial in qwith non-negative integer
552
+ coefficients.CYCLIC SIEVING FOR GENERALISED NON-CROSSING PARTITIONS 13
553
+ On the other hand, for the groups G4,G8,G16,G25,G32, the terms in the defining
554
+ expression of the corresponding q-Fuß–Catalan number can be arranged in a manner so
555
+ that aq-Fuß–Catalannumber of type Aappears andProposition 4 applies; for example,
556
+ forW=G32(with degrees given by 12 ,18,24,30) we have
557
+ Catm(G32;q) =[30m+12]q[30m+18]q[30m+24]q[30m+30]q
558
+ [12]q[18]q[24]q[30]q
559
+ =1
560
+ [5m+6]q6/bracketleftbigg
561
+ 5m+6
562
+ 5/bracketrightbigg
563
+ q6,
564
+ which indeed fits into the framework of Proposition 4 and, hence, is a polynomial in q
565
+ with non-negative integer coefficients.
566
+ In the other cases, the more “specialised” auxiliary results given in C orollary 6 and
567
+ Lemmas7–24havetobeapplied. Forthesakeofillustration, weexhib it oneexample for
568
+ each of them below, with full details being provided in [21, Sec. 4]. In ge neral, the idea
569
+ is that, given a rational expression consisting of cyclotomic factor s, as in the definition
570
+ oftheq-Fuß–Catalannumbers, onetriestoplacedenominator factorsbe lowappropriate
571
+ numerator factors so that one can divide out the denominator fac tor completely. For
572
+ example, if we were to encounter the expression
573
+ [30m+12]q·(other terms)
574
+ [12]q·(other terms)
575
+ and know that mis even, then we would try to simplify this to
576
+ /bracketleftbig5m+2
577
+ 2/bracketrightbig
578
+ q12·(other terms)
579
+ (other terms),
580
+ where [5m+2
581
+ 2]q12is manifestly a polynomial in qwith non-negative integer coefficients.
582
+ On the other hand, in a situation where twodenominator factors “want” to divide a
583
+ singlenumerator factor, we “extract” as much as we can from the nume rator factor and
584
+ compensate by additional “fudge” factors. To be more concrete , if we encounter the
585
+ expression
586
+ [14m+14]q·(other terms)
587
+ [6]q[14]q·(other terms)
588
+ and we know that m≡0 (mod 3), then we would try the rewriting
589
+ /bracketleftbigm+1
590
+ 3/bracketrightbig
591
+ q42[21]q2
592
+ [3]q2[7]q2[2]q·(other terms)
593
+ (other terms),
594
+ with the idea that we might find somewhere else a term [2 α]q, which could be combined
595
+ with the term[2] qin the denominator into [2 α]q/[2]q= [α]q2, andthen apply Corollary6
596
+ to see that
597
+ [α]q2[21]q2
598
+ [3]q2[7]q2
599
+ is a polynomial in qwith non-negative integer coefficients (provided αis at least 12),
600
+ with/bracketleftbigm+1
601
+ 3/bracketrightbig
602
+ q42being such a polynomial in any case.
603
+ In situations where threedenominator factors “want” to divide a singlenumerator
604
+ factor, one has to perform more complicated rearrangements, in order to be able to
605
+ apply one of the Lemmas 7–24.14 C. KRATTENTHALER AND T. W. M ¨ULLER
606
+ For example, for W=G24, the degrees are 4 ,6,14, and hence
607
+ Catm(G24;q) =[14m+4]q[14m+6]q[14m+14]q
608
+ [4]q[6]q[14]q.
609
+ We have
610
+ Catm(G24;q) =
611
+ 
612
+ /bracketleftbig7m
613
+ 2+1/bracketrightbig
614
+ q4/bracketleftbig14m
615
+ 6+1/bracketrightbig
616
+ q6[m+1]q14,ifm≡0 (mod 6),/bracketleftbig7m+2
617
+ 3/bracketrightbig
618
+ q6/bracketleftbig7m+3
619
+ 2/bracketrightbig
620
+ q4[m+1]q14, ifm≡1 (mod 6),
621
+ /bracketleftbig7m
622
+ 2+1/bracketrightbig
623
+ q4[7m+3]q2/bracketleftbigm+1
624
+ 3/bracketrightbig
625
+ q42[21]q2
626
+ [3]q2[7]q2,ifm≡2 (mod 6),
627
+ [7m+2]q2/bracketleftbig7m
628
+ 3+1/bracketrightbig
629
+ q6/bracketleftbigm+1
630
+ 2/bracketrightbig
631
+ q28[14]q2
632
+ [2]q2[7]q2,ifm≡3 (mod 6),
633
+ /bracketleftbig7m+2
634
+ 6/bracketrightbig
635
+ q12[6]q2
636
+ [2]q2[3]q2[7m+3]q2[m+1]q14,ifm≡4 (mod 6),
637
+ [7m+2]q2/bracketleftbig7m+3
638
+ 2/bracketrightbig
639
+ q4/bracketleftbigm+1
640
+ 3/bracketrightbig
641
+ q42[21]q2
642
+ [3]q2[7]q2,ifm≡5 (mod 6),
643
+ which, by Corollary 6, are polynomials in qwith non-negative integer coefficients in all
644
+ cases.
645
+ ForW=G30=H4, the degrees are 2 ,12,20,30, and hence
646
+ Catm(H4;q) =[30m+2]q[30m+12]q[30m+20]q[30m+30]q
647
+ [2]q[12]q[20]q[30]q.
648
+ Ifmis odd, then we may write
649
+ Catm(H4;q) =/bracketleftbig15m+1
650
+ 2/bracketrightbig
651
+ q4[5m+2]q6[3m+2]q10/bracketleftbigm+1
652
+ 2/bracketrightbig
653
+ q60[30]q2[2]q2[3]q2[5]q2
654
+ [6]q6[10]q2[15]q2,
655
+ which, by Lemma 16, is a polynomial in qwith non-negative integer coefficients.
656
+ ForW=G35=E6, the degrees are 2 ,5,6,8,9,12, and hence
657
+ Catm(E6;q) =[12m+2]q[12m+5]q[12m+6]q[12m+8]q[12m+9]q[12m+12]q
658
+ [2]q[5]q[6]q[8]q[9]q[12]q.
659
+ Ifm≡5 (mod 30),then we have
660
+ Catm(E6;q) = [6m+1]q2/bracketleftbig12m+5
661
+ 5/bracketrightbig
662
+ q5[2m+1]q6
663
+ ×[3m+2]q4[4m+3]q3/bracketleftbigm+1
664
+ 6/bracketrightbig
665
+ q72[72]q[3]q[4]q
666
+ [8]q[9]q[12]q,
667
+ which, by Lemma 7, is a polynomial in qwith non-negative integer coefficients.
668
+ Ifm≡7 (mod 30),then we have
669
+ Catm(E6;q) =/bracketleftbig6m+1
670
+ 2/bracketrightbig
671
+ q4[12m+5]q/bracketleftbig2m+1
672
+ 15/bracketrightbig
673
+ q90
674
+ ×[90]q[3]q[4]q
675
+ [5]q[6]q[9]q[3m+2]q4[4m+3]q3/bracketleftbigm+1
676
+ 2/bracketrightbig
677
+ q24[6]q4
678
+ [2]q4[3]q4,
679
+ which, by Corollary 6 and Lemma 9, is a polynomial in qwith non-negative integer
680
+ coefficients.CYCLIC SIEVING FOR GENERALISED NON-CROSSING PARTITIONS 15
681
+ Ifm≡8 (mod 30),then we have
682
+ Catm(E6;q) = [6m+1]q2[12m+5]q[2m+1]q6/bracketleftbig3m+2
683
+ 2/bracketrightbig
684
+ q8
685
+ ×/bracketleftbig4m+3
686
+ 5/bracketrightbig
687
+ q15[15]q
688
+ [3]q[5]q/bracketleftbigm+1
689
+ 3/bracketrightbig
690
+ q36[12]q3
691
+ [3]q3[4]q3,
692
+ which, by Lemma 11, is a polynomial in qwith non-negative integer coefficients.
693
+ Ifm≡13 (mod 30) ,then we have
694
+ Catm(E6;q) = [6m+1]q2[12m+5]q/bracketleftbig2m+1
695
+ 3/bracketrightbig
696
+ q18[6]q3
697
+ [2]q3[3]q3
698
+ ×[3m+2]q4/bracketleftbig4m+3
699
+ 5/bracketrightbig
700
+ q15[15]q
701
+ [3]q[5]q/bracketleftbigm+1
702
+ 2/bracketrightbig
703
+ q24[6]q4
704
+ [2]q4[3]q4,
705
+ which, by Lemma 12, is a polynomial in qwith non-negative integer coefficients.
706
+ Ifm≡22 (mod 30) ,then we have
707
+ Catm(E6;q) = [6m+1]q2[12m+5]q/bracketleftbig2m+1
708
+ 15/bracketrightbig
709
+ q90[90]q[3]q
710
+ [5]q[6]q[9]q
711
+ ×/bracketleftbig3m+2
712
+ 2/bracketrightbig
713
+ q8[4m+3]q3[m+1]q12,
714
+ which, by Lemma 10, is a polynomial in qwith non-negative integer coefficients.
715
+ Ifm≡23 (mod 30) ,then we have
716
+ Catm(E6;q) = [6m+1]q2[12m+5]q[2m+1]q6
717
+ ×[3m+2]q4/bracketleftbig4m+3
718
+ 5/bracketrightbig
719
+ q15[15]q
720
+ [3]q[5]q/bracketleftbigm+1
721
+ 6/bracketrightbig
722
+ q72[72]q[3]q[4]q
723
+ [8]q[9]q[12]q,
724
+ which, by Lemma 8, is a polynomial in qwith non-negative integer coefficients.
725
+ ForW=G36=E7, the degrees are 2 ,6,8,10,12,14,18, and hence
726
+ Catm(E7;q) =[18m+2]q[18m+6]q[18m+8]q[18m+10]q
727
+ [2]q[6]q[8]q[10]q
728
+ ×[18m+12]q[18m+14]q[18m+18]q
729
+ [12]q[14]q[18]q.
730
+ Ifm≡18 (mod 140) ,then we have
731
+ Catm(E7;q) = [9m+1]q2/bracketleftbig3m+1
732
+ 5/bracketrightbig
733
+ q30[15]q2
734
+ [3]q2[5]q2
735
+ ×/bracketleftbig9m+4
736
+ 2/bracketrightbig
737
+ q4[9m+5]q2/bracketleftbig3m+2
738
+ 28/bracketrightbig
739
+ q168[84]q2[2]q2
740
+ [4]q2[6]q2[7]q2[9m+7]q2[m+1]q18,
741
+ which, by Corollary 6 and Lemma 13, is a polynomial in qwith non-negative integer
742
+ coefficients.16 C. KRATTENTHALER AND T. W. M ¨ULLER
743
+ Ifm≡23 (mod 140) ,then we have
744
+ Catm(E7;q) =/bracketleftbig9m+1
745
+ 4/bracketrightbig
746
+ q8/bracketleftbig3m+1
747
+ 35/bracketrightbig
748
+ q210[105]q2
749
+ [3]q2[5]q2[7]q2[9m+4]q2[9m+5]q2
750
+ ×[3m+2]q6[9m+7]q2/bracketleftbigm+1
751
+ 2/bracketrightbig
752
+ q36[6]q6
753
+ [2]q6[3]q6,
754
+ which, by Corollary 6 and Lemma 14, is a polynomial in qwith non-negative integer
755
+ coefficients.
756
+ Ifm≡54 (mod 140) ,then we have
757
+ Catm(E7;q) = [9m+1]q2[3m+1]q6/bracketleftbig9m+4
758
+ 70/bracketrightbig
759
+ q140[70]q2
760
+ [2]q2[5]q2[7]q2[9m+5]q2
761
+ ×/bracketleftbig3m+2
762
+ 4/bracketrightbig
763
+ q24[6]q4
764
+ [2]q4[3]q4[9m+7]q2[m+1]q18.
765
+ Ifonedecomposes[9 m+7]q2as[9m
766
+ 2+4]q4+q2[9m
767
+ 2+3]q4, thenoneseesthat, byCorollary6
768
+ and Lemma 15, this is a polynomial in qwith non-negative integer coefficients.
769
+ ForW=G37=E8, the degrees are 2 ,8,12,14,18,20,24,30, and hence
770
+ Catm(E7;q) =[30m+2]q[30m+8]q[30m+12]q[30m+14]q
771
+ [2]q[8]q[12]q[14]q
772
+ ×[30m+18]q[30m+20]q[30m+24]q[30m+30]q
773
+ [18]q[20]q[24]q[30]q.
774
+ Ifm≡3 (mod 84),then we have
775
+ Catm(E8;q) =/bracketleftbig15m+1
776
+ 2/bracketrightbig
777
+ q4/bracketleftbig15m+4
778
+ 7/bracketrightbig
779
+ q14[5m+2]q6/bracketleftbig15m+7
780
+ 4/bracketrightbig
781
+ q8/bracketleftbig5m+3
782
+ 6/bracketrightbig
783
+ q36[6]q6
784
+ [2]q6[3]q6
785
+ ×[3m+2]q10[5m+4]q6/bracketleftbigm+1
786
+ 4/bracketrightbig
787
+ q120[60]q2[2]q2[3]q2[5]q2
788
+ [10]q2[12]q2[15]q2,
789
+ which, by Corollary 6 and Lemma 19, is a polynomial in qwith non-negative integer
790
+ coefficients.
791
+ Ifm≡8 (mod 84),then we have
792
+ Catm(E8;q) = [15m+1]q2/bracketleftbig15m+4
793
+ 4/bracketrightbig
794
+ q8/bracketleftbig5m+2
795
+ 42/bracketrightbig
796
+ q252[126]q2[3]q2
797
+ [6]q2[7]q2[9]q2[15m+7]q2[5m+3]q6
798
+ ×/bracketleftbig3m+2
799
+ 2/bracketrightbig
800
+ q20/bracketleftbig5m+4
801
+ 4/bracketrightbig
802
+ q24[m+1]q30,
803
+ which, by Lemma 22, is a polynomial in qwith non-negative integer coefficients.
804
+ Ifm≡11 (mod 84) ,then we have
805
+ Catm(E8;q) =/bracketleftbig15m+1
806
+ 2/bracketrightbig
807
+ q4[15m+4]q2/bracketleftbig5m+2
808
+ 3/bracketrightbig
809
+ q18/bracketleftbig15m+7
810
+ 4/bracketrightbig
811
+ q8/bracketleftbig5m+3
812
+ 2/bracketrightbig
813
+ q12
814
+ ×/bracketleftbig3m+2
815
+ 7/bracketrightbig
816
+ q70[35]q2
817
+ [5]q2[7]q2[5m+4]q6/bracketleftbigm+1
818
+ 4/bracketrightbig
819
+ q120[60]q2[2]q2[3]q2[5]q2
820
+ [10]q2[12]q2[15]q2,CYCLIC SIEVING FOR GENERALISED NON-CROSSING PARTITIONS 17
821
+ which, by Corollary 6 and Lemma 20, is a polynomial in qwith non-negative integer
822
+ coefficients.
823
+ Ifm≡16 (mod 84) ,then we have
824
+ Catm(E8;q) = [15m+1]q2/bracketleftbig15m+4
825
+ 4/bracketrightbig
826
+ q8/bracketleftbig5m+2
827
+ 2/bracketrightbig
828
+ q12[15m+7]q2[5m+3]q6
829
+ ×/bracketleftbig3m+2
830
+ 2/bracketrightbig
831
+ q20/bracketleftbig5m+4
832
+ 84/bracketrightbig
833
+ q504[252]q2[3]q2
834
+ [7]q2[9]q2[12]q2[m+1]q30,
835
+ which, by Lemma 23, is a polynomial in qwith non-negative integer coefficients.
836
+ Ifm≡18 (mod 84) ,then we have
837
+ Catm(E8;q) = [15m+1]q2/bracketleftbig15m+4
838
+ 2/bracketrightbig
839
+ q4/bracketleftbig5m+2
840
+ 4/bracketrightbig
841
+ q24[15m+7]q2/bracketleftbig5m+3
842
+ 3/bracketrightbig
843
+ q18
844
+ /bracketleftbig3m+2
845
+ 28/bracketrightbig
846
+ q280[140]q2[2]q2
847
+ [4]q2[7]q2[10]q2/bracketleftbig5m+4
848
+ 2/bracketrightbig
849
+ q12[m+1]q30,
850
+ which, by Lemma 24, is a polynomial in qwith non-negative integer coefficients.
851
+ Ifm≡21 (mod 84) ,then we have
852
+ Catm(E8;q) =/bracketleftbig15m+1
853
+ 4/bracketrightbig
854
+ q8[15m+4]q2[5m+2]q6/bracketleftbig15m+7
855
+ 14/bracketrightbig
856
+ q28[14]q2
857
+ [2]q2[7]q2/bracketleftbig5m+3
858
+ 12/bracketrightbig
859
+ q72[12]q6
860
+ [3]q6[4]q6
861
+ ×[3m+2]q10[5m+4]q6/bracketleftbigm+1
862
+ 2/bracketrightbig
863
+ q60[30]q2[2]q2[3]q2[5]q2
864
+ [6]q2[10]q2[15]q2,
865
+ which, by Corollary 6 and Lemma 17, is a polynomial in qwith non-negative integer
866
+ coefficients.
867
+ Ifm≡25 (mod 84) ,then we have
868
+ Catm(E8;q) =/bracketleftbig15m+1
869
+ 4/bracketrightbig
870
+ q8[15m+4]q2[5m+2]q6/bracketleftbig15m+7
871
+ 2/bracketrightbig
872
+ q4/bracketleftbig5m+3
873
+ 4/bracketrightbig
874
+ q24
875
+ ×/bracketleftbig3m+2
876
+ 7/bracketrightbig
877
+ q70[35]q2
878
+ [5]q2[7]q2/bracketleftbig5m+4
879
+ 3/bracketrightbig
880
+ q18/bracketleftbigm+1
881
+ 2/bracketrightbig
882
+ q60[30]q2[2]q2[3]q2[5]q2
883
+ [6]q2[10]q2[15]q2,
884
+ which, by Lemma 18, is a polynomial in qwith non-negative integer coefficients.
885
+ Ifm≡27 (mod 84) ,then we have
886
+ Catm(E8;q) =/bracketleftbig15m+1
887
+ 14/bracketrightbig
888
+ q28[14]q2
889
+ [2]q2[7]q2[15m+4]q2[5m+2]q6/bracketleftbig15m+7
890
+ 4/bracketrightbig
891
+ q8/bracketleftbig5m+3
892
+ 6/bracketrightbig
893
+ q36[6]q6
894
+ [2]q6[3]q6
895
+ ×[3m+2]q10[5m+4]q6/bracketleftbigm+1
896
+ 4/bracketrightbig
897
+ q120[60]q2[2]q2[3]q2[5]q2
898
+ [10]q2[12]q2[15]q2,
899
+ which, by Corollary 6 and Lemma 21, is a polynomial in qwith non-negative integer
900
+ coefficients.
901
+ All other cases are disposed of in a similar fashion. /square
902
+ 5.Auxiliary results I
903
+ This section collects several auxiliary results which allow us to reduce the problem
904
+ of proving Theorem 2, or the equivalent statement (3.3), for the 2 6 exceptional groups
905
+ listed in Section 2 to a finite problem. While Lemmas 27 and 28 cover spec ial choices
906
+ of the parameters, Lemmas 26 and 30 afford an inductive procedur e. More precisely,18 C. KRATTENTHALER AND T. W. M ¨ULLER
907
+ if we assume that we have already verified Theorem 2 for all groups o f smaller rank,
908
+ then Lemmas 26 and 30, together with Lemmas 27 and 31, reduce th e verification of
909
+ Theorem 2 for the group that we are currently considering to a finit e problem; see
910
+ Remark 3. The final lemma of this section, Lemma 32, disposes of com plex reflection
911
+ groups with a special property satisfied by their degrees.
912
+ Letp=am+b, 0≤b<m. We have
913
+ φp/parenleftbig
914
+ (w0;w1,...,w m)/parenrightbig
915
+ = (∗;ca+1wm−b+1c−a−1,ca+1wm−b+2c−a−1,...,ca+1wmc−a−1,
916
+ caw1c−a,...,cawm−bc−a/parenrightbig
917
+ ,(5.1)
918
+ where∗stands for the element of Wwhich is needed to complete the product of the
919
+ components to c.
920
+ Lemma 26. It suffices to check (3.3)forpa divisor of mh. More precisely, let pbe
921
+ a divisor of mh, and letkbe another positive integer with gcd(k,mh/p) = 1, then we
922
+ have
923
+ Catm(W;q)/vextendsingle/vextendsingle
924
+ q=e2πip/mh= Catm(W;q)/vextendsingle/vextendsingle
925
+ q=e2πikp/mh (5.2)
926
+ and
927
+ |FixNCm(W)(φp)|=|FixNCm(W)(φkp)|. (5.3)
928
+ Proof.For (5.2), this follows immediately from
929
+ lim
930
+ q→ζ[α]q
931
+ [β]q=/braceleftBigg
932
+ α
933
+ βifα≡β≡0 (modd),
934
+ 1 otherwise ,(5.4)
935
+ whereζis ad-th root of unity and α,βare non-negative integers such that α≡β
936
+ (modd).
937
+ In order to establish (5.3), suppose that x∈FixNCm(W)(φp), that is,x∈NCm(W)
938
+ andφp(x) =x. It obviously follows that φkp(x) =x, so thatx∈FixNCm(W)(φkp).
939
+ To establish the converse, note that, if gcd( k,mh/p) = 1, then there exists k′with
940
+ k′k≡1 (modmh
941
+ p). It follows that, if x∈FixNCm(W)(φkp), that is, if x∈NCm(W) and
942
+ φkp(x) =x, thenx=φk′kp(x) =φp(x), whencex∈FixNCm(W)(φp). /square
943
+ Lemma 27. Letpbe a divisor of mh. Ifpis divisible by m, then(3.3)is true.
944
+ Proof.According to (5.1), the action of φponNCm(W) is described by
945
+ φp/parenleftbig
946
+ (w0;w1,...,w m)/parenrightbig
947
+ = (∗;cp/mw1c−p/m,...,cp/mwmc−p/m/parenrightbig
948
+ .
949
+ Hence, if (w0;w1,...,w m) is fixed by φp, then each individual wimust be fixed under
950
+ conjugation by cp/m.
951
+ Using the notation W′= Cent W(cp/m), theprevious observationmeans that wi∈W���,
952
+ i= 1,2,...,m. Springer [33, Theorem 4.2] (see also [24, Theorem 11.24(iii)]) prove d
953
+ thatW′is a well-generated complex reflection group whose degrees coincide with those
954
+ degrees ofWthat are divisible by mh/p. It was furthermore shown in [9, Lemma 3.3]
955
+ that
956
+ NC(W)∩W′=NC(W′). (5.5)CYCLIC SIEVING FOR GENERALISED NON-CROSSING PARTITIONS 19
957
+ Hence, the tuples ( w0;w1,...,w m) fixed byφpare in fact identical with the elements of
958
+ NCm(W′), which implies that
959
+ |FixNCm(W)(φp)|=|NCm(W′)|. (5.6)
960
+ Application of Theorem 1 with Wreplaced by W′and of the “limit rule” (5.4) then
961
+ yields that
962
+ |NCm(W′)|=/productdisplay
963
+ 1≤i≤n
964
+ mh
965
+ p|dimh+di
966
+ di= Catm(W;q)/vextendsingle/vextendsingle
967
+ q=e2πip/mh. (5.7)
968
+ Combining (5.6) and (5.7), we obtain (3.3). This finishes the proof of t he lemma. /square
969
+ Lemma 28. Equation (3.3)holds for all divisors pofm.
970
+ Proof.Using (5.4) and the fact that the degrees of irreducible well-genera ted complex
971
+ reflection groups satisfy di<hfor alli<n, we see that
972
+ Catm(W;q)/vextendsingle/vextendsingle
973
+ q=e2πip/mh=/braceleftBigg
974
+ m+1 ifm=p,
975
+ 1 ifm/ne}ationslash=p.
976
+ On the other hand, if ( w0;w1,...,w m) is fixed by φp, then, because of the action (5.1),
977
+ we must have w1=wp+1=···=wm−p+1andw1=cwm−p+1c−1. In particular,
978
+ w1∈CentW(c). By the theorem of Springer cited in the proof of Lemma 27, the
979
+ subgroup Cent W(c) is itself a complex reflection group whose degrees are those degre es
980
+ ofWthat are divisible by h. The only such degree is hitself, hence Cent W(c) is the
981
+ cyclic group generated by c. Moreover, by (5.5), we obtain that w1=ε, the identity
982
+ element of W, orw1=c. Therefore, for m=pthe set Fix NCm(W)(φp) consists of the
983
+ m+1 elements ( w0;w1,...,w m) obtained by choosing wi=cfor a particular ibetween
984
+ 0 andm, all otherwj’s being equal to ε, while, for m/ne}ationslash=p, we have
985
+ FixNCm(W)(φp) =/braceleftbig
986
+ (c;ε,...,ε)/bracerightbig
987
+ ,
988
+ whence the result. /square
989
+ Lemma 29. LetWbe an irreducible well-generated complex reflection group a ll of
990
+ whose degrees are divisible by d. Then each element of Wis fixed under conjugation by
991
+ ch/d.
992
+ Proof.By the theorem of Springer cited in the proof of Lemma 27, the subg roupW′=
993
+ CentW(ch/d) is itself a complex reflection group whose degrees are those degre es ofW
994
+ that are divisible by d. Thus, by our assumption, the degrees of W′coincide with the
995
+ degrees ofW, and hence W′must be equal to W. Phrased differently, each element of
996
+ Wis fixed under conjugation by ch/d, as claimed. /square
997
+ Lemma 30. LetWbe an irreducible well-generated complex reflection group o f rankn,
998
+ and letp=m1h1be a divisor of mh, wherem=m1m2andh=h1h2. Without loss of
999
+ generality, we assume that gcd(h1,m2) = 1. Suppose that Theorem 2has already been
1000
+ verified for all irreducible well-generated complex reflect ion groups with rank <n. Ifh2
1001
+ does not divide all degrees di, then Equation (3.3)is satisfied.20 C. KRATTENTHALER AND T. W. M ¨ULLER
1002
+ Proof.Let us write h1=am2+b, with 0 ≤b < m 2. The condition gcd( h1,m2) = 1
1003
+ translates into gcd( b,m2) = 1. From (5.1), we infer that
1004
+ φp/parenleftbig
1005
+ (w0;w1,...,w m)/parenrightbig
1006
+ = (∗;ca+1wm−m1b+1c−a−1,ca+1wm−m1b+2c−a−1,...,ca+1wmc−a−1,
1007
+ caw1c−a,...,cawm−m1bc−a/parenrightbig
1008
+ .(5.8)
1009
+ Supposing that ( w0;w1,...,w m) is fixed by φp, we obtain the system of equations
1010
+ wi=ca+1wi+m−m1bc−a−1, i= 1,2,...,m 1b,
1011
+ wi=cawi−m1bc−a, i=m1b+1,m1b+2,...,m,
1012
+ which, after iteration, implies in particular that
1013
+ wi=cb(a+1)+(m2−b)awic−b(a+1)−(m2−b)a=ch1wic−h1, i= 1,2,...,m.
1014
+ It is at this point where we need gcd( b,m2) = 1. The last equation shows that each wi,
1015
+ i= 1,2,...,m, and thus also w0, lies in Cent W(ch1). By the theorem of Springer cited
1016
+ in the proof of Lemma 27, this centraliser subgroup is itself a complex reflection group,
1017
+ W′say, whose degrees are those degrees of Wthat are divisible by h/h1=h2. Since,
1018
+ by assumption, h2does not divide alldegrees,W′has rank strictly less than n. Again
1019
+ by assumption, we know that Theorem 2 is true for W′, so that in particular,
1020
+ |FixNCm(W′)(φp)|= Catm(W′;q)/vextendsingle/vextendsingle
1021
+ q=e2πip/mh.
1022
+ The arguments above together with (5.5) show that Fix NCm(W)(φp) = Fix NCm(W′)(φp).
1023
+ On the other hand, using (5.4) it is straightforward to see that
1024
+ Catm(W;q)/vextendsingle/vextendsingle
1025
+ q=e2πip/mh= Catm(W′;q)/vextendsingle/vextendsingle
1026
+ q=e2πip/mh.
1027
+ This proves (3.3) for our particular p, as required. /square
1028
+ Lemma 31. LetWbe an irreducible well-generated complex reflection group o f rank
1029
+ n, and letp=m1h1be a divisor of mh, wherem=m1m2andh=h1h2. We assume
1030
+ thatgcd(h1,m2) = 1. Ifm2>nthen
1031
+ FixNCm(W)(φp) =/braceleftbig
1032
+ (c;ε,...,ε)/bracerightbig
1033
+ .
1034
+ Proof.Let us suppose that ( w0;w1,...,w m)∈FixNCm(W)(φp) and that there exists a
1035
+ j≥1 such that wj/ne}ationslash=ε. By (5.8), it then follows for such a jthat alsowk/ne}ationslash=εfor
1036
+ allk≡j−lm1b(modm), where, as before, bis defined as the unique integer with
1037
+ h1=am2+band 0≤b < m 2. Since, by assumption, gcd( b,m2) = 1, there are
1038
+ exactlym2suchk’s which are distinct mod m. However, this implies that the sum of
1039
+ the absolute lengths of the wi’s, 0≤i≤m, is at least m2> n, a contradiction to
1040
+ Remark 1.(2). /square
1041
+ Remark 3.(1) If we put ourselves in the situation of the assumptions of Lemma 30,
1042
+ then we may conclude that equation (3.3) only needs to be checked f or pairs (m2,h2)
1043
+ subject to the following restrictions:
1044
+ m2≥2,gcd(h1,m2) = 1,andh2divides all degrees of W. (5.9)
1045
+ Indeed, Lemmas 27 and 30 together imply that equation (3.3) is alway s satisfied in all
1046
+ other cases.CYCLIC SIEVING FOR GENERALISED NON-CROSSING PARTITIONS 21
1047
+ (2) Still putting ourselves in the situation of Lemma 30, if m2>nandm2h2does not
1048
+ divide any of the degrees of W, then equation (3.3) is satisfied. Indeed, Lemma 31 says
1049
+ thatinthiscasetheleft-handsideof (3.3)equals1,whileastraightf orwardcomputation
1050
+ using (5.4) shows that in this case the right-hand side of (3.3) equals 1 as well.
1051
+ (3)It shouldbeobserved that thisleaves afinitenumber of choices form2to consider,
1052
+ whence a finite number of choices for ( m1,m2,h1,h2). Altogether, there remains a finite
1053
+ number of choices for p=h1m1to be checked.
1054
+ Lemma 32. LetWbe an irreducible well-generated complex reflection group o f rankn
1055
+ with the property that di|hfori= 1,2,...,n. Then Theorem 2is true for this group
1056
+ W.
1057
+ Proof.By Lemma 26, we may restrict ourselves to divisors pofmh.
1058
+ Suppose that e2πip/mhis adi-th rootof unity for some i. In other words, mh/pdivides
1059
+ di. Sincediis a divisor of hby assumption, the integer mh/palso divides h. But this
1060
+ is equivalent to saying that mdividesp, and equation (3.3) holds by Lemma 27.
1061
+ Now assume that mh/pdoes not divide any of the di’s. Then, by (5.4), the right-
1062
+ hand side of (3.3) equals 1. On the other hand, ( c;ε,...,ε) is always an element of
1063
+ FixNCm(W)(φp). To see that there are no others, we make appeal to the classific a-
1064
+ tion of all irreducible well-generated complex reflection groups, whic h we recalled in
1065
+ Section 2. Inspection reveals that all groups satisfying the hypot heses of the lemma
1066
+ have rank n≤2. Except for the groups contained in the infinite series G(d,1,n)
1067
+ andG(e,e,n) for which Theorem 2 has been established in [19], these are the grou ps
1068
+ G5,G6,G9,G10,G14,G17,G18,G21. We now discuss these groups case by case, keeping
1069
+ the notation of Lemma 30. In order to simplify the argument, we not e that Lemma 31
1070
+ implies that equation (3.3) holds if m2>2, so that in the following arguments we
1071
+ always may assume that m2= 2.
1072
+ CaseG5. The degrees are 6 ,12, and therefore Remark 3.(1) implies that equa-
1073
+ tion (3.3) is always satisfied.
1074
+ CaseG6. The degrees are 4 ,12, and therefore, according to Remark 3.(1), we need
1075
+ only consider the casewhere h2= 4andm2= 2, that is, p= 3m/2. Then (5.8) becomes
1076
+ φp/parenleftbig
1077
+ (w0;w1,...,w m)/parenrightbig
1078
+ = (∗;c2wm
1079
+ 2+1c−2,c2wm
1080
+ 2+2c−2,...,c2wmc−2,cw1c−1,...,cw m
1081
+ 2c−1/parenrightbig
1082
+ .
1083
+ (5.10)
1084
+ If (w0;w1,...,w m) isfixed by φpandnot equal to ( c;ε,...,ε), there must exist an iwith
1085
+ 1≤i≤m
1086
+ 2such thatℓT(wi) =ℓT(wm
1087
+ 2+i) = 1,wm
1088
+ 2+i=cwic−1,wiwm
1089
+ 2+i=wicwic−1=c,
1090
+ and allwj, withj/ne}ationslash=i,m
1091
+ 2+i, equalε. However, with the help of the GAPpackage
1092
+ CHEVIE[14, 28], one verifies that there is no wiinG6such that
1093
+ ℓT(wi) = 1 and wicwic−1=c
1094
+ are simultaneously satisfied. Hence, the left-hand side of (3.3) is eq ual to 1, as required.
1095
+ CaseG9. The degrees are 8 ,24, and therefore, according to Remark 3.(1), we need
1096
+ only consider the case where h2= 8 andm2= 2, that is, p= 3m/2. This is the same p
1097
+ as forG6. Again, CHEVIEfinds no solution. Hence, the left-hand side of (3.3) is equal
1098
+ to 1, as required.
1099
+ CaseG10. The degrees are 12 ,24, and therefore Remark 3.(1) implies that equa-
1100
+ tion (3.3) is always satisfied.22 C. KRATTENTHALER AND T. W. M ¨ULLER
1101
+ CaseG14. The degrees are 6 ,24, and therefore Remark 3.(1) implies that equa-
1102
+ tion (3.3) is always satisfied.
1103
+ CaseG17. The degrees are 20 ,60, and therefore, according to Remark 3.(1), we need
1104
+ only consider the cases where h2= 20 orh2= 4. In the first case, p= 3m/2, which is
1105
+ the samepas forG6. Again,CHEVIEfinds no solution. In the second case, p= 15m/2.
1106
+ Then (5.8) becomes
1107
+ φp/parenleftbig
1108
+ (w0;w1,...,w m)/parenrightbig
1109
+ = (∗;c8wm
1110
+ 2+1c−8,c8wm
1111
+ 2+2c−8,...,c8wmc−8,c7w1c−7,...,c7wm
1112
+ 2c−7/parenrightbig
1113
+ .(5.11)
1114
+ By Lemma 29, every element of NC(W) is fixed under conjugation by c3, and, thus, on
1115
+ elements fixed by φp, the above action of φpreduces to the one in (5.10). This action
1116
+ was already discussed in the first case. Hence, in both cases, the le ft-hand side of (3.3)
1117
+ is equal to 1, as required.
1118
+ CaseG18. The degrees are 30 ,60, and therefore Remark 3.(1) implies that equa-
1119
+ tion (3.3) is always satisfied.
1120
+ CaseG21. The degrees are 12 ,60, and therefore, according to Remark 3.(1), we need
1121
+ only consider the cases where h2= 12 orh2= 4. In the first case, p= 5m/2, so that
1122
+ (5.8) becomes
1123
+ φp/parenleftbig
1124
+ (w0;w1,...,w m)/parenrightbig
1125
+ = (∗;c3wm
1126
+ 2+1c−3,c3wm
1127
+ 2+2c−3,...,c3wmc−3,c2w1c−2,...,c2wm
1128
+ 2c−2/parenrightbig
1129
+ .(5.12)
1130
+ If (w0;w1,...,w m) is fixed by φpand not equal to ( c;ε,...,ε), there must exist an i
1131
+ with 1≤i≤m
1132
+ 2such thatℓT(wi) = 1 andwic2wic−2=c. However, with the help of
1133
+ theGAPpackageCHEVIE[14, 28], one verifies that there is no such solution to this
1134
+ equation. In the second case, p= 15m/2. Then (5.8) becomes the action in (5.11).
1135
+ By Lemma 29, every element of NC(W) is fixed under conjugation by c5, and, thus,
1136
+ on elements fixed by φp, the action of φpin (5.11) reduces to the one in the first case.
1137
+ Hence, in both cases, the left-hand side of (3.3) is equal to 1, as re quired.
1138
+ This completes the proof of the lemma. /square
1139
+ 6.Exemplification of case-by-case verification of Theorem 2
1140
+ It remains to verify Theorem 2 for the groups G4,G8,G16,G20,G23=H3,G24,G25,
1141
+ G26,G27,G28=F4,G29,G30=H4,G32,G33,G34,G35=E6,G36=E7,G37=E8. All
1142
+ details can be found in [21, Sec. 6]. We content ourselves with illustra ting the type of
1143
+ computation that is needed here by going through the case of the g roupG24, and by
1144
+ discussing some of the arguments needed for the group G37=E8.
1145
+ In the sequel we write ζdfor a primitive d-th root of unity.
1146
+ CaseG24.The degrees are 4 ,6,14, and hence we have
1147
+ Catm(G24;q) =[14m+14]q[14m+6]q[14m+4]q
1148
+ [14]q[6]q[4]q.
1149
+ Letζbe a 14m-th root of unity. In what follows, we abbreviate the assertion tha t “ζis
1150
+ a primitive d-th root of unity” as “ ζ=ζd.” The following cases on the right-hand sideCYCLIC SIEVING FOR GENERALISED NON-CROSSING PARTITIONS 23
1151
+ of (3.3) occur:
1152
+ lim
1153
+ q→ζCatm(G24;q) =m+1,ifζ=ζ14,ζ7, (6.1a)
1154
+ lim
1155
+ q→ζCatm(G24;q) =7m+3
1156
+ 3,ifζ=ζ6,ζ3,3|m, (6.1b)
1157
+ lim
1158
+ q→ζCatm(G24;q) =7m+2
1159
+ 2,ifζ=ζ4,2|m, (6.1c)
1160
+ lim
1161
+ q→ζCatm(G24;q) = Catm(G24),ifζ=−1 orζ= 1, (6.1d)
1162
+ lim
1163
+ q→ζCatm(G24;q) = 1,otherwise. (6.1e)
1164
+ We must now prove that the left-handside of (3.3) in each case agre es with the values
1165
+ exhibited in (6.1). The only cases not covered by Lemma 27 are the on es in (6.1b),
1166
+ (6.1c), and (6.1e). (In both (6.1a) and (6.1d) we have d|h.)
1167
+ We first consider (6.1b). By Lemma 26, we are free to choose p= 7m/3 ifζ=ζ6,
1168
+ respectively p= 14m/3 ifζ=ζ3. In both cases, mmust be divisible by 3.
1169
+ We start with the case that p= 7m/3. From (5.1), we infer
1170
+ φp/parenleftbig
1171
+ (w0;w1,...,w m)/parenrightbig
1172
+ = (∗;c3w2m
1173
+ 3+1c−3,c3w2m
1174
+ 3+2c−3,...,c3wmc−3,c2w1c−2,...,c2w2m
1175
+ 3c−2/parenrightbig
1176
+ .
1177
+ Supposing that ( w0;w1,...,w m) is fixed by φp, we obtain the system of equations
1178
+ wi=c3w2m
1179
+ 3+ic−3, i= 1,2,...,m
1180
+ 3, (6.2a)
1181
+ wi=c2wi−m
1182
+ 3c−2, i=m
1183
+ 3+1,m
1184
+ 3+2,...,m. (6.2b)
1185
+ There are two distinct possibilities for choosing the wi’s, 1≤i≤m: either all the wi’s
1186
+ are equal to ε, or there is an iwith 1≤i≤m
1187
+ 3such that
1188
+ ℓT(wi) =ℓT(wi+m
1189
+ 3) =ℓT(wi+2m
1190
+ 3) = 1.
1191
+ Writingt1,t2,t3forwi,wi+m
1192
+ 3,wi+2m
1193
+ 3, respectively, the equations (6.2) reduce to
1194
+ t1=c3t3c−3, (6.3a)
1195
+ t2=c2t1c−2, (6.3b)
1196
+ t3=c2t2c−2. (6.3c)
1197
+ One of these equations is in fact superfluous: if we substitute (6.3b ) and (6.3c) in
1198
+ (6.3a), then we obtain t1=c7t1c−7which is automatically satisfied due to Lemma 29
1199
+ withd= 2.
1200
+ Since (w0;w1,...,w m)∈NCm(G24), we must have t1t2t3=c. Combining this with
1201
+ (6.3), we infer that
1202
+ t1(c2t1c−2)(c4t1c−4) =c. (6.4)
1203
+ With the help of CHEVIE, one obtains 7 solutions for t1in this equation, each of them
1204
+ giving rise to m/3 elements of Fix NCm(G24)(φp) sincei(inwi) ranges from 1 to m/3.
1205
+ In total, we obtain 1 + 7m
1206
+ 3=7m+3
1207
+ 3elements in Fix NCm(G24)(φp), which agrees with
1208
+ the limit in (6.1b).
1209
+ The case where p= 14m/3 can be treated in a similar fashion. In the end, it
1210
+ turns out that we have to solve the same enumeration problem as fo rp= 7m/3, and,24 C. KRATTENTHALER AND T. W. M ¨ULLER
1211
+ consequently, the number of elements of Fix NCm(G24)(φp) is the same, namely7m+3
1212
+ 3, as
1213
+ required.
1214
+ Our next case is (6.1c). Proceeding in a similar manner as before, we s ee that there is
1215
+ againthe trivial possibility ( c;ε,...,ε), and otherwise we have to find t1withℓT(t1) = 1
1216
+ satisfying the inequality
1217
+ t1(c3t1c−3)≤Tc. (6.5)
1218
+ With the help of CHEVIE, one obtains 7 solutions for t1in this relation, each of them
1219
+ giving rise to m/2 elements of Fix NCm(G24)(φp) sincei(inwi) ranges from 1 to m/2.
1220
+ In total, we obtain 1 + 7m
1221
+ 2=7m+2
1222
+ 2elements in Fix NCm(G24)(φp), which agrees with
1223
+ the limit in (6.1c).
1224
+ Finally, we turn to (6.1e). By Remark 3, the only choices for h2andm2to be consid-
1225
+ ered areh2= 1 andm2= 3,h2=m2= 2, andh2= 2 andm2= 3. These correspond
1226
+ to the choices p= 14m/3,p= 7m/2, respectively p= 7m/3, all of which have already
1227
+ been discussed as they do not belong to (6.1e). Hence, (3.3) must n ecessarily hold, as
1228
+ required.
1229
+ CaseG37=E8.The degrees are 2 ,8,12,14,18,20,24,30, and hence we have
1230
+ Catm(E8;q) =[30m+30]q[30m+24]q[30m+20]q[30m+18]q
1231
+ [30]q[24]q[20]q[18]q
1232
+ ×[30m+14]q[30m+12]q[30m+8]q[30m+2]q
1233
+ [14]q[12]q[8]q[2]q.
1234
+ Letζbe a 30m-th root of unity. The cases occurring on the right-hand side of (3 .3)
1235
+ not covered by Lemma 27 are:
1236
+ lim
1237
+ q→ζCatm(E8;q) =5m+4
1238
+ 4,ifζ=ζ24,4|m, (6.6a)
1239
+ lim
1240
+ q→ζCatm(E8;q) =3m+2
1241
+ 2,ifζ=ζ20,2|m, (6.6b)
1242
+ lim
1243
+ q→ζCatm(E8;q) =5m+3
1244
+ 3,ifζ=ζ18,ζ9,3|m, (6.6c)
1245
+ lim
1246
+ q→ζCatm(E8;q) =15m+7
1247
+ 7,ifζ=ζ14,ζ7,7|m, (6.6d)
1248
+ lim
1249
+ q→ζCatm(E8;q) =(5m+4)(5m+2)
1250
+ 8,ifζ=ζ12,2|m, (6.6e)
1251
+ lim
1252
+ q→ζCatm(E8;q) =(5m+4)(15m+4)
1253
+ 16,ifζ=ζ8,4|m, (6.6f)
1254
+ lim
1255
+ q→ζCatm(E8;q) =(5m+4)(3m+2)(5m+2)(15m+4)
1256
+ 64,ifζ=ζ4,2|m,(6.6g)
1257
+ lim
1258
+ q→ζCatm(E8;q) = Catm(E8),ifζ=−1 orζ= 1, (6.6h)
1259
+ lim
1260
+ q→ζCatm(E8;q) = 1,otherwise. (6.6i)
1261
+ We now have to prove that the left-hand side of (3.3) in each case ag rees with the
1262
+ values exhibited in (6.6). Since the corresponding computations in th e various cases are
1263
+ very similar, we concentrate here only on the cases (6.6f) and (6.6g ), these two being
1264
+ representative of the types of arguments arising. As before, we refer the reader to [21,
1265
+ Sec. 6] for full details.CYCLIC SIEVING FOR GENERALISED NON-CROSSING PARTITIONS 25
1266
+ Letusconsiderthecasein(6.6f)first. ByLemma26, wearefreeto choosep= 15m/4.
1267
+ In particular, mmust be divisible by 4. From (5.1), we infer
1268
+ φp/parenleftbig
1269
+ (w0;w1,...,w m)/parenrightbig
1270
+ = (∗;c4wm
1271
+ 4+1c−4,c4wm
1272
+ 4+2c−4,...,c4wmc−4,c3w1c−3,...,c3wm
1273
+ 4c−3/parenrightbig
1274
+ .
1275
+ Supposing that ( w0;w1,...,w m) is fixed by φp, we obtain the system of equations
1276
+ wi=c4wm
1277
+ 4+ic−4, i= 1,2,...,3m
1278
+ 4, (6.7a)
1279
+ wi=c3wi−3m
1280
+ 4c−3, i=3m
1281
+ 4+1,3m
1282
+ 4+2,...,m. (6.7b)
1283
+ There are several distinct possibilities for choosing the wi’s, 1≤i≤m, which we
1284
+ summarise as follows:
1285
+ (i) all thewi’s are equal to ε(andw0=c),
1286
+ (ii) there is an iwith 1≤i≤m
1287
+ 4such that
1288
+ 1≤ℓT(wi) =ℓT(wi+m
1289
+ 4) =ℓT(wi+2m
1290
+ 4) =ℓT(wi+3m
1291
+ 4)≤2, (6.8a)
1292
+ and the other wj’s, 1≤j≤m, are equal to ε,
1293
+ (iii) there are i1andi2with 1≤i1<i2≤m
1294
+ 4such that
1295
+ ℓT(wi1) =ℓT(wi2) =ℓT(wi1+m
1296
+ 4) =ℓT(wi2+m
1297
+ 4)
1298
+ =ℓT(wi1+2m
1299
+ 4) =ℓT(wi2+2m
1300
+ 4) =ℓT(wi1+3m
1301
+ 4) =ℓT(wi2+3m
1302
+ 4) = 1,(6.8b)
1303
+ and all other wjare equal to ε.
1304
+ Moreover, since ( w0;w1,...,w m)∈NCm(E8), we must have
1305
+ wiwi+m
1306
+ 4wi+2m
1307
+ 4wi+3m
1308
+ 4≤Tc,
1309
+ or
1310
+ wi1wi2wi1+m
1311
+ 4wi2+m
1312
+ 4wi1+2m
1313
+ 4wi2+2m
1314
+ 4wi1+3m
1315
+ 4wi2+3m
1316
+ 4=c.
1317
+ Together with equations (6.7)–(6.8), this implies that
1318
+ wi=c15wic−15andwi(c11wic−11)(c7wic−7)(c3wic−3)≤Tc, (6.9)
1319
+ or that
1320
+ wi1=c15wi1c−15, wi1=c15wi2c−15,
1321
+ andwi1wi2(c11wi1c−11)(c11wi2c−11)(c7wi1c−7)(c7wi2c−7)(c3wi1c−3)(c3wi2c−3) =c.
1322
+ (6.10)
1323
+ Here, the first equation in (6.9) and the first two equations in (6.10) are automatically
1324
+ satisfied due to Lemma 29 with d= 2.
1325
+ With the help of Stembridge’s Maplepackagecoxeter [36], one obtains 30 solutions
1326
+ forwiin (6.9) with ℓT(wi) = 1, 45 solutions for wiwithℓT(wi) = 2 and wiof type
1327
+ A2
1328
+ 1(as a parabolic Coxeter element; see the end of Section 2), and 20 s olutions for
1329
+ wiwithℓT(wi) = 2 and wiof typeA2. Each of them gives rise to m/4 elements of
1330
+ FixNCm(E8)(φp) sinceiranges from 1 to m/4.
1331
+ The number of solutions in Case (iii) can be computed from our knowled ge of the
1332
+ solutions in Case (ii) according to type, using some elementary count ing arguments.
1333
+ Namely, the number of solutions of (6.10) is equal to
1334
+ 45·2+20·3 = 150,26 C. KRATTENTHALER AND T. W. M ¨ULLER
1335
+ since an element of type A2
1336
+ 1can be decomposed in two ways into a product of two
1337
+ elements of absolute length 1, while for an element of type A2this can be done in 3
1338
+ ways.
1339
+ In total, we obtain 1 + (30 + 45 + 20)m
1340
+ 4+ 150/parenleftbigm/4
1341
+ 2/parenrightbig
1342
+ =(5m+4)(15m+4)
1343
+ 16elements in
1344
+ FixNCm(E8)(φp), which agrees with the limit in (6.6f).
1345
+ Next, we discuss the case in (6.6g). By Lemma 26, we are free to cho osep= 15m/2.
1346
+ In particular, mmust be divisible by 2. From (5.1), we infer
1347
+ φp/parenleftbig
1348
+ (w0;w1,...,w m)/parenrightbig
1349
+ = (∗;c8wm
1350
+ 2+1c−8,c8wm
1351
+ 2+2c−8,...,c8wmc−8,c7w1c−7,...,c7wm
1352
+ 2c−7/parenrightbig
1353
+ .
1354
+ Supposing that ( w0;w1,...,w m) is fixed by φp, we obtain the system of equations
1355
+ wi=c8wm
1356
+ 2+ic−8, i= 1,2,...,m
1357
+ 2, (6.11a)
1358
+ wi=c7wi−m
1359
+ 2c−7, i=m
1360
+ 2+1,m
1361
+ 2+2,...,m. (6.11b)
1362
+ There are several distinct possibilities for choosing the wi’s, 1≤i≤m:
1363
+ (i) all thewi’s are equal to ε(andw0=c),
1364
+ (ii) there is an iwith 1≤i≤m
1365
+ 2such that
1366
+ 1≤ℓT(wi) =ℓT(wi+m
1367
+ 2)≤4, (6.12a)
1368
+ and the other wj’s, 1≤j≤m, are equal to ε,
1369
+ (iii) there are i1andi2with 1≤i1<i2≤m
1370
+ 2such that
1371
+ ℓ1:=ℓT(wi1) =ℓT(wi1+m
1372
+ 2)≥1, ℓ2:=ℓT(wi2) =ℓT(wi2+m
1373
+ 2)≥1,andℓ1+ℓ2≤4,
1374
+ (6.12b)
1375
+ and the other wj’s, 1≤j≤m, are equal to ε,
1376
+ (iv) there are i1,i2,i3with 1≤i1<i2<i3≤m
1377
+ 2such that
1378
+ ℓ1:=ℓT(wi1) =ℓT(wi1+m
1379
+ 2)≥1, ℓ2:=ℓT(wi2) =ℓT(wi2+m
1380
+ 2)≥1,
1381
+ ℓ3:=ℓT(wi3) =ℓT(wi3+m
1382
+ 2)≥1,andℓ1+ℓ2+ℓ3≤4,(6.12c)
1383
+ and the other wj’s, 1≤j≤m, are equal to ε,
1384
+ (v) there are i1,i2,i3,i4with 1≤i1<i2<i3<i4≤m
1385
+ 2such that
1386
+ ℓT(wi1) =ℓT(wi2) =ℓT(wi3) =ℓT(wi4)
1387
+ =ℓT(wi1+m
1388
+ 2) =ℓT(wi2+m
1389
+ 2) =ℓT(wi3+m
1390
+ 2) =ℓT(wi4+m
1391
+ 2) = 1,(6.12d)
1392
+ and all other wj’s are equal to ε.
1393
+ Moreover, since ( w0;w1,...,w m)∈NCm(E8), we must have wiwi+m
1394
+ 2≤Tc, respec-
1395
+ tivelywi1wi2wi1+m
1396
+ 2wi2+m
1397
+ 2≤Tc, respectively
1398
+ wi1wi2wi3wi1+m
1399
+ 2wi2+m
1400
+ 2wi3+m
1401
+ 2≤Tc,
1402
+ respectively
1403
+ wi1wi2wi3wi4wi1+m
1404
+ 2wi2+m
1405
+ 2wi3+m
1406
+ 2wi4+m
1407
+ 2=c.
1408
+ Together with equations (6.11)–(6.12), this implies that
1409
+ wi=c15wic−15andwi(c7wic−7)≤Tc, (6.13)
1410
+ respectively that
1411
+ wi1=c15wi1c−15, wi2=c15wi2c−15,andwi1wi2(c7wi1c−7)(c7wi2c−7)≤Tc,(6.14)CYCLIC SIEVING FOR GENERALISED NON-CROSSING PARTITIONS 27
1412
+ respectively that
1413
+ wi1=c15wi1c−15, wi2=c15wi2c−15, wi3=c15wi3c−15,
1414
+ andwi1wi2wi3(c7wi1c−7)(c7wi2c−7)(c7wi3c−7)≤Tc,(6.15)
1415
+ respectively that
1416
+ wi1=c15wi1c−15, wi2=c15wi2c−15, wi3=c15wi3c−15, wi4=c15wi4c−15,
1417
+ andwi1wi2wi3wi4(c7wi1c−7)(c7wi2c−7)(c7wi3c−7)(c7wi4c−7) =c.(6.16)
1418
+ Here, the first equation in (6.13), the first two in (6.14), the first t hree in (6.15), and
1419
+ the first four in (6.16), are all automatically satisfied due to Lemma 2 9 withd= 2.
1420
+ With the help of Stembridge’s Maplepackagecoxeter [36], one obtains
1421
+ — 45 solutions for wiin (6.13) with ℓT(wi) = 1,
1422
+ — 150 solutions for wiin (6.13) with ℓT(wi) = 2 andwiof typeA2
1423
+ 1,
1424
+ — 100 solutions for wiin (6.13) with ℓT(wi) = 2 andwiof typeA2,
1425
+ — 75 solutions for wiin (6.13) with ℓT(wi) = 3 andwiof typeA3
1426
+ 1,
1427
+ — 165 solutions for wiin (6.13) with ℓT(wi) = 3 andwiof typeA1∗A2,
1428
+ — 90 solutions for wiin (6.13) with ℓT(wi) = 3 andwiof typeA3,
1429
+ — 15 solutions for wiin (6.13) with ℓT(wi) = 4 andwiof typeA2
1430
+ 1∗A2,
1431
+ — 45 solutions for wiin (6.13) with ℓT(wi) = 4 andwiof typeA1∗A3;
1432
+ — 5 solutions for wiin (6.13) with ℓT(wi) = 4 andwiof typeA2
1433
+ 2,
1434
+ — 18 solutions for wiin (6.13) with ℓT(wi) = 4 andwiof typeA4,
1435
+ — 5 solutions for wiin (6.13) with ℓT(wi) = 4 andwiof typeD4.
1436
+ Each of them gives rise to m/2 elements of Fix NCm(E8)(φp) sinceiranges from 1 to m/2.
1437
+ There are no solutions for wiin (6.13) with wiof typeA4
1438
+ 1.
1439
+ Letting the computer find all solutions in cases (iii)–(v) would take ye ars. However,
1440
+ the number of these solutions can be computed from our knowledge of the solutions
1441
+ in Case (ii) according to type, if this information is combined with the de composition
1442
+ numbers in the sense of [17, 18, 20] (see the end of Section 2) and some elementary
1443
+ (multiset) permutation counting. The decomposition numbers for A2,A3,A4, andD4
1444
+ of which we make use can be found in the appendix of [18].
1445
+ To begin with, the number of solutions of (6.14) with ℓ1=ℓ2= 1 is equal to
1446
+ n1,1:= 150·2+100·NA2(A1,A1) = 600,
1447
+ since an element of type A2
1448
+ 1can be decomposed in two ways into a product of two
1449
+ elements of absolute length 1, while for an element of type A2this can be done in
1450
+ NA2(A1,A1) = 3 ways. Similarly, the number of solutions of (6.14) with ℓ1= 2 and
1451
+ ℓ2= 1 is equal to
1452
+ n2,1:= 75·3+165·(1+NA2(A1,A1))+90·NA3(A2,A1) = 1425,
1453
+ the number of solutions of (6.14) with ℓ1= 3 andℓ2= 1 is equal to
1454
+ n3,1:= 15·(2+NA2(A1,A1))+45·(1+NA3(A2,A1))+5·(2NA2(A1,A1))
1455
+ +18·(NA4(A3,A1)+NA4(A1∗A2,A1))+5·(ND4(A3,A1)+ND4(A3
1456
+ 1,A1)) = 660,28 C. KRATTENTHALER AND T. W. M ¨ULLER
1457
+ the number of solutions of (6.14) with ℓ1=ℓ2= 2 is equal to
1458
+ n2,2:= 15·(2+2NA2(A1,A1))+45·(2NA3(A2,A1))+5·(2+NA2(A1,A1)2)
1459
+ +18·(NA4(A2,A2)+NA4(A2
1460
+ 1,A2
1461
+ 1)+2NA4(A2,A2
1462
+ 1))
1463
+ +5·(ND4(A2,A2)+2ND4(A2,A2
1464
+ 1)) = 1195,
1465
+ the number of solutions of (6.15) with ℓ1=ℓ2=ℓ3= 1 is equal to
1466
+ n1,1,1:= 75·3!+165·(3NA2(A1,A1))+90NA3(A1,A1,A1) = 3375,
1467
+ the number of solutions of (6.15) with ℓ1= 2 andℓ2=ℓ3= 1 is equal to
1468
+ n2,1,1:= 15·(2+NA2(A1,A1)+2·2·NA2(A1,A1))+45·(2NA3(A2,A1)+NA3(A1,A1,A1))
1469
+ +5·(2NA2(A1,A1)+2NA2(A1,A1)2)+18·(NA4(A2,A1,A1)+NA4(A2
1470
+ 1,A1,A1))
1471
+ +5·(ND4(A2,A1,A1)+ND4(A2
1472
+ 1,A1,A1)) = 2850,
1473
+ and the number of solutions of (6.16) is equal to
1474
+ n1,1,1,1:= 15·(12NA2(A1,A1))+45·(4NA3(A1,A1,A1))+5·(6NA2(A1,A1)2)
1475
+ +18·NA4(A1,A1,A1,A1)+5·ND4(A1,A1,A1,A1) = 6750.
1476
+ In total, we obtain
1477
+ 1+(45+150+100+75+165+90+15+45+5+18+5)m
1478
+ 2+(n1,1+2n2,1+2n3,1+n2,2)/parenleftbiggm/2
1479
+ 2/parenrightbigg
1480
+ +(n1,1,1+3n2,1,1)/parenleftbiggm/2
1481
+ 3/parenrightbigg
1482
+ +n1,1,1,1/parenleftbiggm/2
1483
+ 4/parenrightbigg
1484
+ =(5m+4)(3m+2)(5m+2)(15m+4)
1485
+ 64
1486
+ elements in Fix NCm(E8)(φp), which agrees with the limit in (6.6g).
1487
+ 7.Cyclic sieving II
1488
+ In this section we present the second cyclic sieving conjecture due to Bessis and
1489
+ Reiner [9, Conj. 6.5].
1490
+ Letψ:NCm(W)→NCm(W) be the map defined by
1491
+ (w0;w1,...,w m)/mapsto→/parenleftbig
1492
+ cwmc−1;w0,w1,...,w m−1/parenrightbig
1493
+ . (7.1)
1494
+ Form= 1, we have w0=cw−1
1495
+ 1, so that this action reduces to the inverse of the
1496
+ Kreweras complement Kc
1497
+ idas defined by Armstrong [2, Def. 2.5.3].
1498
+ It is easy to see that ψ(m+1)hacts as the identity, where his the Coxeter number of
1499
+ W(see (8.1) below). By slight abuse of notation as before, let C2be the cyclic group
1500
+ of order (m+1)hgenerated by ψ.
1501
+ Given these definitions, we are now in the position to state the secon d cyclic sieving
1502
+ conjecture of Bessis and Reiner. By the results of [19] and of this p aper, it becomes the
1503
+ following theorem.
1504
+ Theorem 33. For an irreducible well-generated complex reflection group Wand any
1505
+ m≥1, the triple (NCm(W),Catm(W;q),C2), whereCatm(W;q)is theq-analogue of
1506
+ the Fuß–Catalan number defined in (3.2), exhibits the cyclic sieving phenomenon.CYCLIC SIEVING FOR GENERALISED NON-CROSSING PARTITIONS 29
1507
+ By definition of the cyclic sieving phenomenon, we have to prove that
1508
+ |FixNCm(W)(ψp)|= Catm(W;q)/vextendsingle/vextendsingle
1509
+ q=e2πip/(m+1)h, (7.2)
1510
+ for allpin the range 0 ≤p<(m+1)h.
1511
+ 8.Auxiliary results II
1512
+ This section collects several auxiliary results which allow us to reduce the problem of
1513
+ proving Theorem 33, respectively the equivalent statement (7.2), for the 26 exceptional
1514
+ groups listed in Section 2 to a finite problem. The corresponding lemma s, Lemmas 34–
1515
+ 39, are analogues of Lemmas 26–28 and 30–32 in Section 5.
1516
+ Letp=a(m+1)+b, 0≤b<m+1. We have
1517
+ ψp/parenleftbig
1518
+ (w0;w1,...,w m)/parenrightbig
1519
+ = (ca+1wm−b+1c−a−1;ca+1wm−b+2c−a−1,...,ca+1wmc−a−1,
1520
+ caw0c−a,...,cawm−bc−a/parenrightbig
1521
+ .(8.1)
1522
+ Lemma 34. It suffices to check (7.2)forpa divisor of (m+1)h. More precisely, let pbe
1523
+ a divisor of (m+1)h, and letkbe another positive integer with gcd(k,(m+1)h/p) = 1,
1524
+ then we have
1525
+ Catm(W;q)/vextendsingle/vextendsingle
1526
+ q=e2πip/(m+1)h= Catm(W;q)/vextendsingle/vextendsingle
1527
+ q=e2πikp/(m+1)h (8.2)
1528
+ and
1529
+ |FixNCm(W)(ψp)|=|FixNCm(W)(ψkp)|. (8.3)
1530
+ Proof.For (8.3), this follows in the same way as (5.3) in Lemma 26.
1531
+ For (8.2), we must argue differently than in Lemma 26. Let us write ζ=e2πip/(m+1)h.
1532
+ For a given group W, we writeS1(W) for the set of all indices isuch thatζdi−h= 1,
1533
+ and we write S2(W) for the set of all indices isuch thatζdi= 1. By the rule of de
1534
+ l’Hospital, we have
1535
+ Catm(W;q)/vextendsingle/vextendsingle
1536
+ q=e2πip/(m+1)h=
1537
+
1538
+ 0 if |S1(W)|>|S2(W)|,/producttext
1539
+ i∈S1(W)(mh+di)/producttext
1540
+ i∈S2(W)di/producttext
1541
+ i/∈S1(W)(1−ζdi−h)
1542
+ /producttext
1543
+ i/∈S2(W)(1−ζdi),if|S1(W)|=|S2(W)|.
1544
+ (8.4)
1545
+ Since, by Theorem 25, Catm(W;q) is a polynomial in q, the case |S1(W)|<|S2(W)|
1546
+ cannot occur.
1547
+ We claim that, for the case where |S1(W)|=|S2(W)|, the factors in the quotient of
1548
+ products/producttext
1549
+ i/∈S1(W)(1−ζdi−h)/producttext
1550
+ i/∈S2(W)(1−ζdi)
1551
+ cancel pairwise. If we assume the correctness of the claim, it is obv ious that we get
1552
+ the same result if we replace ζbyζk, where gcd( k,(m+1)h/p) = 1, hence establishing
1553
+ (8.2).
1554
+ In order to see that our claim is indeed valid, we proceed in a case-by- case fash-
1555
+ ion, making appeal to the classification of irreducible well-generated complex reflection
1556
+ groups, which werecalled inSection2. Firstofall, since dn=h, thesetS1(W)isalways
1557
+ non-empty as it contains the element n. Hence, if we want to have |S1(W)|=|S2(W)|,30 C. KRATTENTHALER AND T. W. M ¨ULLER
1558
+ the setS2(W) must be non-empty as well. In other words, the integer ( m+ 1)h/p
1559
+ must divide at least one of the degrees d1,d2,...,d n. In particular, this implies that,
1560
+ for each fixed reflection group Wof exceptional type, only a finite number of values of
1561
+ (m+1)h/phas to be checked. Writing Mfor (m+1)h/p, what needs to be checked is
1562
+ whether the multisets (that is, multiplicities of elements must be taken into account)
1563
+ {(di−h) modM:i /∈S1(W)}and{dimodM:i /∈S2(W)}
1564
+ are the same. Since, for a fixed irreducible well-generated complex r eflection group,
1565
+ thereisonlyafinitenumber ofpossibilities for M, thisamountstoaroutineverification.
1566
+ /square
1567
+ Lemma 35. Letpbe a divisor of (m+ 1)h. Ifpis divisible by m+ 1, then(7.2)is
1568
+ true.
1569
+ We leave the proof to the reader as it is completely analogous to the p roof of
1570
+ Lemma 27.
1571
+ Lemma 36. Equation (7.2)holds for all divisors pofm+1.
1572
+ Proof.We have
1573
+ Catm(W;q)/vextendsingle/vextendsingle
1574
+ q=e2πip/(m+1)h=/braceleftBigg
1575
+ 0 ifp<m+1,
1576
+ m+1 ifp=m+1.
1577
+ Here, the first case follows from (8.4) and the fact that we have S1(W)⊇ {n}and
1578
+ S2(W) =∅ifp|(m+1) andp<m+1.
1579
+ Ontheother hand, if ( w0;w1,...,w m) is fixed by ψp, then onecanapply anargument
1580
+ similar to that in Lemma 28 with any witaking the role of w1, 0≤i≤m. It follows
1581
+ that ifp=m+1, the set Fix NCm(W)(ψp) consists of the m+1 elements ( w0;w1,...,w m)
1582
+ obtained by choosing wi=cfor a particular ibetween 0 and m, all otherwj’s being
1583
+ equal toε. Ifp<m+1, then there is no element in Fix NCm(W)(ψp). /square
1584
+ Lemma 37. LetWbe an irreducible well-generated complex reflection group o f rank
1585
+ n, and letp=m1h1be a divisor of (m+1)h, wherem+1 =m1m2andh=h1h2. We
1586
+ assume that gcd(h1,m2) = 1. Suppose that Theorem 33has already been verified for
1587
+ all irreducible well-generated complex reflection groups w ith rank< n. Ifh2does not
1588
+ divide all degrees di, then equation (7.2)is satisfied.
1589
+ We leave the proof to the reader as it is completely analogous to the p roof of
1590
+ Lemma 30.
1591
+ Lemma 38. LetWbe an irreducible well-generated complex reflection group o f rank
1592
+ n, and letp=m1h1be a divisor of (m+1)h, wherem+1 =m1m2andh=h1h2. We
1593
+ assume that gcd(h1,m2) = 1. Ifm2>nthen
1594
+ FixNCm(W)(ψp) =∅.
1595
+ We leave the proof to the reader as it is analogous to the proof of Le mma 31.
1596
+ Remark 4.By applying the same reasoning as in Remark 3 with Lemmas 30 and 31
1597
+ replaced by Lemmas 37 and 38, respectively, it follows that we only ne ed to check (7.2)
1598
+ for pairs (m2,h2) satisfying (5.9) and m2≤n. This reduces the problem to a finite
1599
+ number of choices.CYCLIC SIEVING FOR GENERALISED NON-CROSSING PARTITIONS 31
1600
+ Lemma 39. LetWbe an irreducible well-generated complex reflection group o f rankn
1601
+ with the property that di|hfori= 1,2,...,n. Then Theorem 33is true for this group
1602
+ W.
1603
+ Proof.Proceeding in a fashion analogous to the beginning of the proof of Le mma 32, we
1604
+ mayrestricttothecasewhere p|(m+1)hand(m+1)h/pdoesnotdivideanyofthe di’s.
1605
+ Inthiscase, itfollowsfrom(8.4)andthefactthatwehave S1(W)⊇ {n}andS2(W) =∅
1606
+ that the right-hand side of (7.2) equals 0. Inspection of the classifi cation of all irre-
1607
+ ducible well-generated complex reflection groups, which we recalled in Section 2, reveals
1608
+ that all groups satisfying the hypotheses of the lemma have rank n≤2. Except for the
1609
+ groups contained in the infinite series G(d,1,n) andG(e,e,n) for which Theorem 2 has
1610
+ been established in [19], these are the groups G5,G6,G9,G10,G14,G17,G18,G21. The
1611
+ verification of (7.2) can be done in a similar fashion as in the proof of Le mma 32. We
1612
+ illustrate this by going through the case of the group G6. In analogy with the earlier
1613
+ situation, we note that Lemma 38 implies that equation (7.2) holds if m2>2, so that
1614
+ in the following arguments we may assume that m2= 2.
1615
+ CaseG6. The degrees are 4 ,12, and therefore, according to Remark 4, we need only
1616
+ consider the case where h2= 4 andm2= 2, that is, p= 3(m+1)/2. Then the action
1617
+ ofψpis given by
1618
+ ψp/parenleftbig
1619
+ (w0;w1,...,w m)/parenrightbig
1620
+ = (c2wm+1
1621
+ 2c−2;c2wm+3
1622
+ 2c−2,...,c2wmc−2,cw0c−1,...,cw m−1
1623
+ 2c−1/parenrightbig
1624
+ .
1625
+ (8.5)
1626
+ If (w0;w1,...,w m) is fixed by ψp, there must exist an iwith 0≤i≤m−1
1627
+ 2such that
1628
+ ℓT(wi) = 1,wicwic−1=c, and allwj,j/ne}ationslash=i,m+1
1629
+ 2+i, equalε. However, with the help of
1630
+ CHEVIE, one verifies that there is no such solution to this equation. Hence, the left-hand
1631
+ side of (7.2) is equal to 0, as required.
1632
+ This completes the proof of the lemma. /square
1633
+ 9.Exemplification of case-by-case verification of Theorem 3 3
1634
+ It remains to verify Theorem 33 for the groups G4,G8,G16,G20,G23=H3,G24,G25,
1635
+ G26,G27,G28=F4,G29,G30=H4,G32,G33,G34,G35=E6,G36=E7,G37=E8. All
1636
+ details can be found in [21, Sec. 9]. We content ourselves with discuss ing the case of
1637
+ the groupG24, as this suffices to convey the flavour of the necessary computat ions.
1638
+ In order to simplify our considerations, it should be observed that t he action of ψ
1639
+ (given in(7.1)) is exactly the same as the actionof φ(given in (3.1)) with mreplaced by
1640
+ m+1on the components w1,w2,...,w m+1, that is, if we disregard the 0-th component
1641
+ of the elements of the generalised non-crossing partitions involved . The only difference
1642
+ which arises is that, while the ( m+ 1)-tuples ( w0;w1,...,w m) in (7.1) must satisfy
1643
+ w0w1···wm=c, forw1,w2,...,w m+1in (3.1) we only must have w1w2···wm+1≤Tc.
1644
+ Consequently, we may use the counting results from Section 6, exc ept that we have to
1645
+ restrict our attention to those elements ( w0;w1,...,w m,wm+1)∈NCm+1(W) for which
1646
+ w1w2···wm+1=c, or, equivalently, w0=ε.
1647
+ CaseG24.The degrees are 4 ,6,14, and hence we have
1648
+ Catm(G24;q) =[14m+14]q[14m+6]q[14m+4]q
1649
+ [14]q[6]q[4]q.32 C. KRATTENTHALER AND T. W. M ¨ULLER
1650
+ Letζbe a 14(m+ 1)-th root of unity. The following cases on the right-hand side of
1651
+ (7.2) occur:
1652
+ lim
1653
+ q→ζCatm(G24;q) =m+1,ifζ=ζ14,ζ7, (9.1a)
1654
+ lim
1655
+ q→ζCatm(G24;q) =7m+7
1656
+ 3,ifζ=ζ6,ζ3,3|(m+1), (9.1b)
1657
+ lim
1658
+ q→ζCatm(G24;q) = Catm(G24),ifζ=−1 orζ= 1, (9.1c)
1659
+ lim
1660
+ q→ζCatm(G24;q) = 0,otherwise. (9.1d)
1661
+ We must now prove that the left-handside of (7.2) in each case agre es with the values
1662
+ exhibited in (9.1). The only cases not covered by Lemma 35 are the on es in (9.1b) and
1663
+ (9.1d). On the other hand, the only cases left to consider accordin g to Remark 4 are
1664
+ the cases where h2= 1 andm2= 3,h2= 2 andm2= 3, andh2=m2= 2. These
1665
+ correspond to the choices p= 14(m+1)/3,p= 7(m+1)/3, respectively p= 7(m+1)/2.
1666
+ The first two cases belong to (9.1b), while p= 7(m+1)/2 belongs to (9.1d).
1667
+ In the case that p= 7(m+1)/3, the action of ψpis given by
1668
+ ψp/parenleftbig
1669
+ (w0;w1,...,w m)/parenrightbig
1670
+ = (c3w2m+2
1671
+ 3c−3;c3w2m+5
1672
+ 3c−3,...,c3wmc−3,c2w0c−2,...,c2w2m−1
1673
+ 3c−2/parenrightbig
1674
+ .
1675
+ Hence, for an iwith 0≤i≤m−2
1676
+ 3, we must find an element wi=t1, wheret1satisfies
1677
+ (6.4), so that we can set wi+m+1
1678
+ 3=c2t1c−2,wi+2m+2
1679
+ 3=c4t1c−4, and all other wj’s equal
1680
+ toε. We have found seven solutions to the counting problem (6.4), and e ach of them
1681
+ gives rise to ( m+1)/3 elements in Fix NCm(G24)(ψp) since the index iranges from 0 to
1682
+ (m−2)/3.
1683
+ On the other hand, if p= 14(m+1)/3, then the action of ψpis given by
1684
+ ψp/parenleftbig
1685
+ (w0;w1,...,w m)/parenrightbig
1686
+ = (c5wm+1
1687
+ 3c−5;c5wm+4
1688
+ 3c−5,...,c5wmc−5,c4w0c−4,...,c4wm−2
1689
+ 3c−4/parenrightbig
1690
+ .
1691
+ By Lemma 29, every element of NC(W) is fixed under conjugation by c7, and, thus, the
1692
+ equations for t1in this case are the same as in the previous one where p= 7(m+1)/3.
1693
+ Hence, in either case, we obtain 7m+1
1694
+ 3=7m+7
1695
+ 3elements in Fix NCm(G24)(ψp), which
1696
+ agrees with the limit in (9.1b).
1697
+ Ifp= 7(m+ 1)/2, the relevant counting problem is (6.5). However, no element
1698
+ (w0;w1,...,w m)∈FixNCm(G24)(ψp) can be produced in this way since the counting
1699
+ problem imposes the restriction that ℓT(w0) +ℓT(w1) +···+ℓT(wm) be even, which
1700
+ contradicts the fact that ℓT(c) =n= 3. This is in agreement with the limit in (9.1d).
1701
+ Acknowledgements
1702
+ The authors thank an anonymous referee for a very careful rea ding of the original
1703
+ manuscript, and for numerous pertinent suggestions which have h elped to considerably
1704
+ improve the original manuscript.CYCLIC SIEVING FOR GENERALISED NON-CROSSING PARTITIONS 33
1705
+ References
1706
+ [1] G. E. Andrews, The Theory of Partitions , Encyclopedia of Math. and its Applications, vol. 2,
1707
+ Addison–Wesley, Reading, 1976.
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+ [2] D. Armstrong, Generalized noncrossing partitions and combinatorics of C oxeter groups , Mem.
1709
+ Amer. Math. Soc., vol. 202, no. 949, Amer. Math. Soc., Providence , R.I., 2009.
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+ [3] D.Armstrong, C.Stump andH. Thomas, A uniform bijection between nonnesting and noncrossing
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+ partitions , Trans. Amer. Math. Soc. (to appear).
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+ [4] C. A. Athanasiadis, Generalized Catalan numbers, Weyl groups and arrangements of hyperplanes ,
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+ Bull. London Math. Soc. 36(2004), 294–302.
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+ 51(2004), Article B51b, 16 pp.
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+ [14] M. Geck, G. Hiss, F. L¨ ubeck, G. Malle and G. Pfeiffer, CHEVIE— a system for computing and
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+ Comput. 7(1996), 175–210.
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+ [15] I. Gordon and S. Griffeth, Catalan numbers for complex reflection groups , Amer. J. Math. (to
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+ appear).
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+ [16] J. E. Humphreys, Reflection groups and Coxeter groups , Cambridge University Press, Cambridge,
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+ 1990.
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+ [17] C. Krattenthaler, TheF-triangle of the generalised cluster complex , in: Topics in Discrete Mathe-
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+ matics, dedicated to Jarik Neˇ setˇ ril on the occasion of his 60th bir thday, M. Klazar, J. Kratochvil,
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+ M. Loebl, J. Matouˇ sek, R. Thomas and P. Valtr (eds.), Springer–V erlag, Berlin, New York, 2006,
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+ S´ eminaire Lotharingien Combin. 54(2006), Article B54l, 34 pages.
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+ alised non-crossing partitions , Trans. Amer. Math. Soc. 362(2010), 2723–2787.
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+ [21] C. Krattenthaler and T. W. M¨ uller, Cyclic sieving for generalised non-crossing partitions
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+ associated with complex reflection groups of exceptional ty pe — the details , manuscript;
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+ arχiv:1001.0030 .
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+ [26] G. Malle and J. Michel, Constructing representations of Hecke algebras for comple x reflection
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+ groups, LMS J. Comput. Math. 13(2010), 426–450.
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+ [27] I. Marin, The cubic Hecke algebra on at most 5 strands , preprint, arχiv:1110.6621 .
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+ [28] J. Michel, TheGAP-part of the CHEVIEsystem,GAP3-package available for download from
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+ http://people.math.jussieu.fr/jmichel/chevie/chevie .html.34 C. KRATTENTHALER AND T. W. M ¨ULLER
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+ [29] P. Orlik and L. Solomon, Unitary reflection groups and cohomology , Invent. Math. 59(1980),
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+ 77–94.
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+ [30] V. Reiner, D. Stanton and D. White, The cyclic sieving phenomenon , J. Combin. Theory Ser. A
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+ 108(2004), 17–50.
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+ [31] V. Ripoll, Orbites d’Hurwitz des factorisations primitives d’un ´ el´ ement de Coxeter , J. Algebra
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+ 323(2010), 1432–1453.
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+ [32] G. C. Shephard and J. A. Todd, Finite unitary reflection groups , Canad. J. Math. 6(1954),
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+ 274–304.
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+ [33] T. A. Springer, Regular elements of finite reflection groups , Invent. Math. 25(1974), 159–198.
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+ [34] J. R. Stembridge, Some hidden relations involving the ten symmetry classes of plane partitions ,
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+ J. Combin. Theory Ser. A 68(1994), 372–409.
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+ [35] J.R. Stembridge, Canonical bases and self-evacuating tableaux , DukeMath. J. 82(1996).585–606,
1771
+ [36] J. R. Stembridge, coxeter,Maplepackagefor workingwith root systems and finite Coxetergroups;
1772
+ available at http://www.math.lsa.umich.edu/~jrs .
1773
+ Fakult¨at f¨ur Mathematik, Universit ¨at Wien, Nordbergstraße 15, A-1090 Vienna,
1774
+ Austria. WWW: http://www.mat.univie.ac.at/ ~kratt.
1775
+ School of Mathematical Sciences, Queen Mary & Westfield Col lege, University of
1776
+ London, Mile End Road, London E1 4NS, United Kingdom.
1777
+ http://www.maths.qmw.ac.uk/ ~twm/.
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1
+ arXiv:1001.0032v1 [astro-ph.SR] 30 Dec 2009Draft version November 15, 2018
2
+ Preprint typeset using L ATEX style emulateapj v. 08/22/09
3
+ ASTEROSEISMIC INVESTIGATION OF KNOWN PLANET HOSTS IN THE KEPLER FIELD
4
+ J. Christensen-Dalsgaard1,2, H. Kjeldsen1,2, T. M. Brown3, R. L. Gilliland4, T. Arentoft1,2, S. Frandsen1,2,
5
+ P.-O. Quirion1,2,5, W. J. Borucki6, D. Koch6, and J. M. Jenkins7
6
+ Draft version November 15, 2018
7
+ ABSTRACT
8
+ In addition to its great potential for characterizing extra-solar p lanetary systems the Kepler mis-
9
+ sionis providing unique data on stellar oscillations. A key aspect of Keplerasteroseismology is the
10
+ application to solar-like oscillations of main-sequence stars. As an ex ample we here consider an ini-
11
+ tial analysis of data for three stars in the Keplerfield for which planetary transits were known from
12
+ ground-based observations. For one of these, HAT-P-7, we obt ain a detailed frequency spectrum and
13
+ hence strong constraints on the stellar properties. The remaining two stars show definite evidence for
14
+ solar-like oscillations, yielding a preliminary estimate of their mean dens ities.
15
+ Subject headings: stars: fundamental parameters — stars: oscillations — planetary systems
16
+ 1.INTRODUCTION
17
+ The main goal of the Kepler mission is to character-
18
+ ize extra-solar planetary systems, particularly Earth-like
19
+ planets in the habitable zone (e.g., Borucki et al. 2009).
20
+ The mission detects the presence of planets through the
21
+ minute reduction of the light from a star as a planet
22
+ crosses the line of sight. Several observations of such
23
+ reductions at fixed time intervals for a given star, and
24
+ extensive follow-up observations, are used to verify that
25
+ the effect results from planet transits and to characterize
26
+ the planet. To ensure a reasonable chance of detection
27
+ Keplerobserves more than 100,000 stars simultaneously,
28
+ in a fixed field in the Cygnus-Lyra region. Most stars
29
+ are observed at a cadence of 29.4 min, but a subset of
30
+ up to 512 stars can be observed at a short cadence (SC)
31
+ of 58.85s. Keplerwas launched on 6 March 2009 and
32
+ data from the commissioning period and the first month
33
+ of regular observations are now available.
34
+ The very high photometric accuracy required to detect
35
+ planet transits (Borucki et al. 2010; Koch et al. 2010)
36
+ also makes the Keplerobservations of great interest for
37
+ asteroseismic studies of stellar interiors. In particular,
38
+ the SC data allow investigations of solar-like oscillations
39
+ in main-sequence stars. Apart from the great astrophys-
40
+ ical interest of such investigations they also provide pow-
41
+ erful tools to characterize stars that host planetary sys-
42
+ tems (Kjeldsen et al. 2009).
43
+ In stars with effective temperature Teff<∼7000K we
44
+ expect to see oscillations similar to those observed in the
45
+ Sun (e.g., Christensen-Dalsgaard 2002), excited stochas-
46
+ ticallybythe near-surfaceconvection. Theseareacoustic
47
+ modes of high radial order; in main-sequence stars such
48
+ 1Department of Physics and Astronomy, Aarhus University,
49
+ DK-8000 Aarhus C, Denmark: e-mail [email protected]
50
+ 2Danish AsteroSeismology Centre
51
+ 3Las Cumbres Observatory Global Telescope, Goleta, CA 93117
52
+ 4Space Telescope Science Institute, 3700 San Martin Drive, B al-
53
+ timore, MD 21218
54
+ 5Canadian Space Agency, 6767 Route de l’A´ eroport, Saint-
55
+ Hubert, QC, J3Y 8Y9 Canada (present address)
56
+ 6NASA Ames Research Center, MS 244-30, Moffett Field, CA
57
+ 94035, USA
58
+ 7SETI Institute/NASA Ames Research Center, MS244-30, Mof-
59
+ fett Field, CA 94035, USAmodes approximately satisfy the asymptotic relation
60
+ νnl≃∆ν0(n+l/2+ǫ)−l(l+1)D0 (1)
61
+ (Vandakurov 1967; Tassoul 1980). Here νnlis the cyclic
62
+ frequency, nis the radial order of the mode and lis
63
+ the degree, l= 0 corresponding to radial (i.e., spher-
64
+ ically symmetric) oscillations. Also, ∆ ν0is essentially
65
+ the inverse sound travel time across the stellar diameter;
66
+ this is closely related to the mean stellar density ∝angbracketleftρ∗∝angbracketright:
67
+ ∆ν0∝ ∝angbracketleftρ∗∝angbracketright1/2.D0depends sensitively on conditions
68
+ near the center of the star; for stars during the central
69
+ hydrogenburningphasethisprovidesameasureofstellar
70
+ age. Finally, ǫis determined by conditions near the stel-
71
+ lar surface. This regular form of the frequency spectrum
72
+ simplifies the analysis of the observations, and the close
73
+ relation between the stellar properties and the param-
74
+ eters characterizing the frequencies make them efficient
75
+ diagnostics of the properties of the star. This has been
76
+ demonstrated in the last few years through observations
77
+ of solar-like oscillations from the ground and from space
78
+ (for reviews, see Bedding & Kjeldsen 2008; Aerts et al.
79
+ 2009; Gilliland et al. 2010a).
80
+ Even observations allowing a determination of ∆ ν0
81
+ provide useful constraints on ∝angbracketleftρ∗∝angbracketright. With a reliable de-
82
+ termination of individual frequencies ∝angbracketleftρ∗∝angbracketrightis tightly con-
83
+ strained and an estimate of the stellar age can be ob-
84
+ tained. This can greatly aid the interpretation of obser-
85
+ vations of planetary transits (e.g., Gilliland et al. 2010b;
86
+ Nutzman et al. 2010). We note that photometric obser-
87
+ vations such as those carried out by Keplerare predom-
88
+ inantly sensitive to modes of degree l= 0−2. As indi-
89
+ catedbyEq.(1)thesearesufficienttoobtaininformation
90
+ about the core properties of the star.
91
+ Ground-based transit observations have identified
92
+ three planetary systems in the Keplerfield: TrES-2
93
+ (O’Donovan et al. 2006; Sozzetti et al. 2007), HAT-P-7
94
+ (P´ al et al. 2008), and HAT-P-11 (Dittmann et al. 2009;
95
+ Bakos et al. 2010). These systems have been observed
96
+ byKeplerin SC mode. Their properties (cf. Table 1)
97
+ indicate that they should display solar-like oscillations
98
+ at observable amplitudes, and hence they are obvious
99
+ targets for Keplerasteroseismology. Here we report the
100
+ results of a preliminary asteroseismic characterization of2 Christensen-Dalsgaard et al.
101
+ TABLE 1
102
+ Properties of transiting systems.
103
+ Name KIC No Teff(K) [Fe/H] L/L⊙log(g) (cgs) vsiniSource
104
+ (kms−1)
105
+ HAT-P-7 10666592 6350 ±80 0.26±0.08 4 .9±1.1 4.07±0.06 3.8±0.5 (a)
106
+ 6525±61 0.31±0.07 4 .09±0.08 (b)
107
+ HAT-P-11 10748390 4780 ±50 0.31±0.05 0.26±0.02 4.59±0.03 1.5±1.5 (c)
108
+ TrES-2 11446443 5850 ±50−0.15±0.10 1.17±0.10 4.4±0.1 2 ±1 (d)
109
+ 5795±73 0.06±0.08 4 .30±0.13 (b)
110
+ Note. — Sources: (a): P´ al et al. (2008); (b): Ammler-von Eif et al . (2009); (c): Bakos et al. (2010); (d): Sozzetti et al. (2007 ). In some
111
+ cases asymmetric error bars have been symmetrized.
112
+ the central stars in the systems, based on the early Ke-
113
+ plerdata.
114
+ 2.OBSERVATIONS AND DATA ANALYSIS
115
+ We have analyzed data from Kepler for three
116
+ planet-hosting stars using a pipeline developed for
117
+ fast and robust analysis of all Keplerp-mode data
118
+ (Christensen-Dalsgaard et al. 2008; Huber et al. 2009).
119
+ Each time series contains 63324 data points. SC data
120
+ characteristics and minor post-pipeline processing are
121
+ discussed in Gilliland et al. (2010c). In addition a limb-
122
+ darkened transit light curve model fit has been removed
123
+ and 5-σclipping applied to remove outlying data points
124
+ from each of the time series. The frequency analysis con-
125
+ tains four main steps:
126
+ 1. We calculate an oversampled (factor of four) ver-
127
+ sion ofthe power spectrum by using a least-squares
128
+ fitting. We smoothed the spectrum to 3 µHz reso-
129
+ lution to remove the fine structure caused by the
130
+ finite mode lifetime.
131
+ 2. We correlated the smoothed power spectrum with
132
+ an equally spaced comb of delta functions, sepa-
133
+ ratedby∆ ν0/2,andconfinedtoaGaussian-shaped
134
+ band with a full width at half maximum of 5∆ ν0.
135
+ We adopted the maximum of this convolution over
136
+ lags between 0 and 0.5 ∆ ν0as the filter output for
137
+ each ∆ν0.
138
+ 3. After identifying the peak correlation for the best
139
+ matched model filter and extracting the large sep-
140
+ aration corresponding to this peak we calculate the
141
+ folded spectrum (see Fig. 1b), i.e., the sum of the
142
+ power as a function of frequency modulo the opti-
143
+ mumlargeseparation(theonecorrespondingtothe
144
+ peak correlation). The summed power is used to
145
+ locate the p-mode structure and identify the ridges
146
+ corresponding to the different mode degrees (based
147
+ on the asymptotic relation).
148
+ 4. From the asymptoticrelationandthe identification
149
+ of mode degrees we finally identify the position of
150
+ the individual p-mode frequencies in the smoothed
151
+ version of the power spectrum; when more than
152
+ one mode is seen near the expected frequency we
153
+ use the power-weighted average of the two peaks.
154
+ Those extracted frequencies and the mode identifi-
155
+ cations are used in the modeling.
156
+ For observations with low signal-to-noise ratio it may
157
+ not be possible to identify the individual frequencies. In 0 1 2
158
+ Fig. 1.— (a)PowerspectrumofHAT-P-7forfrequencies between
159
+ 300 and 3000 µHz. The spectrum is smoothed with a gaussian filter
160
+ with a FWHM of 3 µHz. The noise level at high frequencies corre-
161
+ sponds to 1.1 ppm in amplitude. The white curve is a smoothed
162
+ power spectrum with a gaussian filter (150 µHz FWHM). A fit to
163
+ the background (dashed white curve) is also shown. The exces s
164
+ power and the individual p-modes are evident. (b) Folded pow er
165
+ spectrum, between 750 and 1500 µHz, for HAT-P-7 for a large sep-
166
+ aration of 59 .22µHz. Indicated are the positions corresponding to
167
+ radial modes ( l= 0) and non-radial modes with l= 1 and 2. The
168
+ measured positions are used to identify the individual osci llation
169
+ modes in panel (a). (c) ´Echelle diagram (see text) for frequencies
170
+ of degree l= 0, 1, and 2 in HAT-P-7; a frequency separation of
171
+ 59.36µHz and a starting frequency of 10 .8µHz were used. The
172
+ filled symbols, coded for degree as indicated, show the obser ved
173
+ frequencies, while the open symbols are for Model 3 in Table 2 ,
174
+ minimizing χ2
175
+ ν.3
176
+ such cases the analysis is carried through step 2, to de-
177
+ termine the maximum response and hence an estimate of
178
+ the large separation.
179
+ Results on the three individual cases are presented in
180
+ §4.
181
+ 3.MODEL FITTING
182
+ Stellar evolution models and adiabatic oscillation
183
+ frequencies were computed using the Aarhus codes
184
+ (Christensen-Dalsgaard 2008a,b), with the OPAL
185
+ equation of state (Rogers et al. 1996) and opacity
186
+ (Iglesias & Rogers 1996) and the NACRE nuclear reac-
187
+ tion parameters (Angulo et al. 1999). In some cases (see
188
+ below) diffusion and settling of helium were included,
189
+ using the simplified formulation of Michaud & Proffitt
190
+ (1993). Convection was treated with the B¨ ohm-Vitense
191
+ (1958) mixing-length formulation, with a mixing length
192
+ αML= 2.00 in units of the pressure scale height roughly
193
+ corresponding to a solar calibration. In some models
194
+ with convective cores, overshoot was included over a dis-
195
+ tance of αovpressure scale heights. Evolution started
196
+ from chemically homogeneous zero-age models. The ini-
197
+ tial abundances by mass X0andZ0of hydrogen and
198
+ heavy elements were characterized by the assumed value
199
+ of [Fe/H], using as reference a present solar surface com-
200
+ position with Zs/Xs= 0.0245 (Grevesse & Noels 1993)
201
+ and assuming, from galactic chemical evolution, that
202
+ X0= 0.7679−3Z0.
203
+ From the observed ∆ ν0, effective temperature and
204
+ composition an initial estimate of the stellar parame-
205
+ ters was obtained using the grid-based SEEK pipeline
206
+ (Quirion et al., in preparation). Smaller grids were then
207
+ computed in the vicinity of these initial parameters, to
208
+ obtaintighterconstraintsonstellarproperties. ForHAT-
209
+ P-7 the analysis of the observations yielded frequencies
210
+ of individually identified modes; here the analysis was
211
+ based on
212
+ χ2
213
+ ν=1
214
+ N−1/summationdisplay
215
+ nl/parenleftBigg
216
+ ν(obs)
217
+ nl−ν(mod)
218
+ nl
219
+ σν/parenrightBigg2
220
+ ,(2)
221
+ whereν(obs)
222
+ nlandν(mod)
223
+ nlare the observed and model fre-
224
+ quencies, σνis the standard error in the observed fre-
225
+ quencies (assumed to be constant) and Nis the num-
226
+ ber of observed frequencies. In addition, we considered
227
+ χ2=χ2
228
+ ν+χ2
229
+ T, whereχ2
230
+ Tis the corresponding normalized
231
+ square difference between the observed and model effec-
232
+ tive temperature. When χ2
233
+ νwas available we minimized
234
+ it along each evolution track and considered the result-
235
+ ing minimum values, and the corresponding value of χ2,
236
+ as a function of the parameters characterizing the mod-
237
+ els (see Gilliland et al. 2010b, for details). When only
238
+ the large separation ∆ ν0could be determined from the
239
+ observations, we identified the model along each track
240
+ which matched ∆ ν0and considered the resulting χ2
241
+ Tas
242
+ a function of the model parameters.
243
+ 4.RESULTS
244
+ 4.1.HAT-P-7
245
+ The observed power spectrum for HAT-P-7 is shown
246
+ in Fig. 1a. The presence of solar-like p-mode peaks, with
247
+ a maximum power around 1.1mHz, is evident. At high
248
+ frequency the noise level in the amplitude spectrum is1.1 parts per million (ppm), with some increase at lower
249
+ frequency, likely due to the effects of stellar granulation.
250
+ Carrying out the correlation analysis described in §2
251
+ we determined the large separation as ∆ ν0= 59.22µHz.
252
+ Figure 1b shows the resulting folded spectrum. This
253
+ clearly shows two closely spaced peaks, identified as cor-
254
+ responding to modes of degree l= 0 and 2, and single
255
+ peak separated from these two by approximately ∆ ν0/2,
256
+ corresponding to l= 1. On this basis we finally deter-
257
+ mined the individual frequencies, identifying the modes
258
+ from the asymptotic relation; the final set includes 33
259
+ p-mode frequencies, determined with a standard error
260
+ σν= 1.4µHz. These frequencies, corresponding to ra-
261
+ dial orders between 11 and 24, are illustrated in Fig. 1c
262
+ in an ´ echelle diagram (see below).
263
+ A grid of models was computed for masses between
264
+ 1.41 and 1 .61M⊙, [Fe/H] between 0.17 and 0.38, and
265
+ αov= 0,0.1 and 0.2, extending well beyond the end of
266
+ central hydrogen burning. The modeling did not include
267
+ diffusion and settling. At the mass of this star the outer
268
+ convection zone is quite thin, and as a result the set-
269
+ tling timescale is much shorter than the age of the star.
270
+ Including settling, without compensating effects such as
271
+ partial mixing in the radiative region or mass loss, leads
272
+ to a rapid change in the surface composition which is
273
+ inconsistent with the observed [Fe/H]; for simplicity we
274
+ therefore neglected these effects for HAT-P-7.8
275
+ The computed frequencies were corrected according to
276
+ the procedure of Kjeldsen et al. (2008) for errors in the
277
+ modeling of the near-surface layers, by adding a(ν/ν0)b
278
+ wherea= 0.1158µHz,ν0= 1000µHz andb= 4.9. As
279
+ discussed in §3, for each evolution track, characterized
280
+ by a set of model parameters, we minimized the depar-
281
+ tureχ2
282
+ νof the model frequencies from the observations,
283
+ defining the best model for this set.
284
+ We first consider χ2
285
+ νas a function of the effective
286
+ temperature of the models (Fig. 2a). It is evident
287
+ that there is a clear minimum in χ2
288
+ ν; this is consistent
289
+ with the determination of Teffby P´ al et al. (2008) but
290
+ not with the somewhat higher temperature obtained by
291
+ Ammler-von Eif et al. (2009) (see also Table 1). Thus
292
+ in the following we use the observed quantities from
293
+ P´ al et al. (2008).
294
+ Since the frequencies to leading order are determined
295
+ by the mean stellar density ∝angbracketleftρ∗∝angbracketright, Fig. 2b,c show χ2
296
+ νand
297
+ χ2as functions of ∝angbracketleftρ∗∝angbracketright. It is evident that the best-fitting
298
+ modelsoccupyanarrowrangeof ∝angbracketleftρ∗∝angbracketright, withawell-defined
299
+ minimum. Fittingaparabolato χ2inpanel(c)weobtain
300
+ the estimate ∝angbracketleftρ∗∝angbracketright= 0.2712±0.0032gcm−1. In Fig. 2d
301
+ χ2is shown against model age. Here the variation with
302
+ model parameters is substantially stronger, resulting in
303
+ a greater spread in the inferred age; in particular, it is
304
+ evident, not surprisingly, that the results depend on the
305
+ extent of convective overshoot. From the figure we esti-
306
+ matethattheageofHAT-P-7isbetween1.4and2.3Gyr.
307
+ Examples of evolution tracks are shown in Fig. 3; pa-
308
+ rameters for these models are provided in Table 2. They
309
+ were chosen to give the smallest χ2
310
+ νfor each of the three
311
+ values of αovconsidered. Also shown are the locations
312
+ 8Artificially suppressing settling in the outer layers, whil e in-
313
+ cluding diffusion and settling in the core, leads to results t hat are
314
+ very similar to those presented here.4 Christensen-Dalsgaard et al.
315
+ TABLE 2
316
+ Stellar evolution models fitting the observed frequencies for HAT-P-7.
317
+ No M ∗/M⊙Age Z0X0αovR∗/R⊙/angbracketleftρ∗/angbracketrightTeffL∗/L⊙χ2
318
+ νχ2
319
+ (Gyr) (gcm−3) (K)
320
+ 1 1.53 1.758 0.0270 0.6870 0.0 1.994 0.2718 6379 5.91 1.08 1.2 1
321
+ 2 1.52 1.875 0.0290 0.6809 0.1 1.992 0.2708 6355 5.81 1.04 1.0 4
322
+ 3 1.50 2.009 0.0270 0.6870 0.2 1.981 0.2718 6389 5.87 1.00 1.2 4
323
+ Note. — Models minimizing χ2
324
+ ν(cf. Eq. 2) along the evolution tracks, illustrated in Fig. 3 . The models have been selected as providing
325
+ the smallest χ2
326
+ νfor each of the three values of the overshoot parameter αov. The smallest value of χ2
327
+ νis obtained for Model 3.
328
+ Fig. 2.— Results of fitting the observed frequencies to a grid of
329
+ stellar models (see text for details). Plusses, stars and di amonds
330
+ correspond to modelswith αov= 0 (no overshoot), 0.1, and 0.2. (a)
331
+ Minimum mean square deviation χ2
332
+ νof the frequencies (cf. Eq. 2)
333
+ along each evolution track, against the effective temperatu reTeff
334
+ of the corresponding models. The vertical dashed and dotted lines
335
+ indicate the effective temperatures found by P´ al et al. (200 8) and
336
+ Ammler-von Eif et al. (2009). (b) Minimum mean square devia-
337
+ tionχ2
338
+ νagainst the mean density /angbracketleftρ∗/angbracketrightof the corresponding models.
339
+ (c) As (b), but showing the combined χ2. (d)χ2against the age
340
+ for the models that minimize χ2
341
+ ν; the different ridges correspond to
342
+ the different masses in the grid, the more massive models resu lting
343
+ in a lower estimate of the age.Fig. 3.— Theoretical HR diagram with selected evolutionary
344
+ tracks, corresponding to the models defined in Table 2. The ’+ ’ in-
345
+ dicate the models along the full set of evolutionary sequenc es mini-
346
+ mizing the difference between the computed and observed freq uen-
347
+ cies. The box is centered on the LandTeffas given by P´ al et al.
348
+ (2008), with a size matching the errors on these quantities.
349
+ of the models minimizing χ2
350
+ νalong each of the computed
351
+ tracks; these evidently fall close to a line in the HR di-
352
+ agram, corresponding to the small range in ∝angbracketleftρ∗∝angbracketright. The
353
+ range of luminosities, from P´ al et al. (2008), is based on
354
+ modeling and hence has not been used in our fit; even
355
+ so, it is gratifying that the present models are essentially
356
+ consistentwiththesevalues. Also,asindicatedbyFig.2a
357
+ andTable 2, the best-fitting models areclose to the value
358
+ ofTeffobtained by P´ al et al. (2008).
359
+ The match of the best-fitting model (Model 3 of Ta-
360
+ ble 2) to the observed frequencies is illustrated in a so-
361
+ called´ echelle diagram (Grec et al. 1983) in Fig. 1c. In
362
+ accordance with Eq. (1) the frequency spectrum is di-
363
+ vided into slices of length ∆ ν, starting at a frequency of
364
+ 10.8µHz; the figure shows the location of the observed
365
+ (filled symbols) and computed (open symbols) frequen-
366
+ cies within each slice, against the starting frequency of
367
+ the slice; the model results extend to the acoustical cut-
368
+ off frequency, 1930 µHz, of the model. There is clearly a
369
+ very good overall agreement between model and obser-
370
+ vations, including the detailed variation with frequency
371
+ which reflects the frequency dependence of the large sep-
372
+ aration, as a possible diagnostics of the outer layers of
373
+ the star (e.g., Houdek & Gough 2007).
374
+ We have finally made a fit of the inferred ∝angbracketleftρ∗∝angbracketright, as
375
+ well asTeffand [Fe/H] from P´ al et al. (2008), to com-
376
+ puted evolutionary tracks from the Yonsei-Yale compi-
377
+ lation (Yi et al. 2001). This was based on a Markov
378
+ Chain Monte Carlo analysis to obtain the statistical
379
+ properties of the inferred quantities (see Brown 2010,
380
+ for details). This resulted in M= 1.520±0.036M⊙,
381
+ R= 1.991±0.018R⊙and an age of 2 .14±0.26Gyr.
382
+ We note that the age estimate reflects the specific as-5
383
+ sumptions in the Yonsei-Yale evolution calculations; as
384
+ indicated by Fig. 2d the true uncertainty in the age de-
385
+ termination is likely somewhat larger.
386
+ 4.2.HAT-P-11
387
+ For HAT-P-11 the oscillation amplitudes were much
388
+ smaller than in HAT-P-7, as expected from the general
389
+ scaling of amplitudes with stellar mass and luminosity
390
+ (e.g., Kjeldsen & Bedding 1995). Thus with the present
391
+ short run of data it has only been possible to determine
392
+ thelargeseparation∆ ν0= 180.1µHzfromthemaximum
393
+ in the correlation analysis. We have matched this to a
394
+ grid of models, including diffusion and settling of helium,
395
+ with masses between 0.7 and 0 .9M⊙and [Fe/H] between
396
+ 0.21 and 0.41. These models provide a good fit to the
397
+ observed TeffandL/L⊙; note that in the presentcase the
398
+ luminosity is based on a reasonably well-determined par-
399
+ allax. We havedetermined an estimateof ∝angbracketleftρ∗∝angbracketrightbyaverag-
400
+ ing the results of those models which match the observed
401
+ ∆ν0and lie within 2 standard deviations ( ±100K) from
402
+ the value of Teffprovided by Bakos et al. (2010); the re-
403
+ sult is∝angbracketleftρ∗∝angbracketright= 2.5127±0.0009gcm−3. Although the for-
404
+ mal error is extremely small, owing to a tight relation
405
+ between the large separation and the mean density for
406
+ stars in this region in the HR diagram, the true error is
407
+ undoubtedly substantially larger. In particular, we ne-
408
+ glected the error in the determination of ∆ ν0and these
409
+ data have not allowed a correction for the systematic
410
+ errors in the modeling of the near-surface layers of the
411
+ star.
412
+ 4.3.TrES-2
413
+ Here also we were unable to determine individual fre-
414
+ quencies from the present set of data. The expected am-
415
+ plitudes are smaller than for HAT-P-7, and the noise
416
+ level higher due to the fainter magnitude of TrES-2.
417
+ The correlation analysis yielded two possible values of
418
+ ∆ν0: 97.7µHz and 130 .7µHz. For this star ∝angbracketleftρ∗∝angbracketrighthas
419
+ been determined from the analysis of the transit light
420
+ curve. Sozzetti et al. (2007) obtained ∝angbracketleftρ∗∝angbracketright= 1.375±
421
+ 0.065gcm−3, while Southworth (2009) found ∝angbracketleftρ∗∝angbracketright=
422
+ 1.42±0.13gcm−3. From the scaling with ∝angbracketleftρ∗∝angbracketright1/2thesmaller of the two possible values of ∆ ν0is clearly incon-
423
+ sistent with these values of ∝angbracketleftρ∗∝angbracketright, while ∆ ν0= 130.7µHz
424
+ yields models that are consistent with the observed Teff
425
+ and log(g) of Sozzetti et al. (2007) as well as with these
426
+ values of the mean density. Here we considered a grid
427
+ of models with helium diffusion and settling, masses be-
428
+ tween 0.85 and 1 .1M⊙and [Fe/H] between −0.25 and
429
+ −0.05. Determining again the mean value of ∝angbracketleftρ∗∝angbracketrightfor
430
+ those models that matched ∆ ν0and had Teffwithin two
431
+ standard deviations of the value of Sozzetti et al. (2007)
432
+ we obtained ∝angbracketleftρ∗∝angbracketright= 1.3233±0.0027gcm−3. As in the
433
+ case of HAT-P-11 the true error is likely substantially
434
+ higher.
435
+ 5.DISCUSSION AND CONCLUSION
436
+ The present preliminary analysis provides a striking
437
+ demonstration of the potential of Keplerasteroseismol-
438
+ ogyanditssupportingroleintheanalysisofplanethosts.
439
+ Thesestarswill undoubtedly be observedthroughout the
440
+ mission and hence the quality of the data will increase
441
+ substantially. For HAT-P-7 the detected frequencies are
442
+ already close to what will be required for a detailed anal-
443
+ ysis of the stellar interior, beyond the determination of
444
+ the basic parameters of the star. Thus here we can look
445
+ forward to a test of the assumptions of the stellar mod-
446
+ eling; the resulting improvements will further constrain
447
+ the overall properties of the star, in particular its age.
448
+ Also, given the observed vsiniwe expect a rotational
449
+ splitting comparable to that observed in the Sun, and
450
+ hence likely detectable with a few months of observa-
451
+ tions. For the other two stars there is strong evidence
452
+ for the presence of solar-like oscillations; thus continued
453
+ observations will very likely result in the determination
454
+ of individual frequencies and hence further constraints
455
+ on the properties of the stars.
456
+ Funding for this Discovery mission is provided by
457
+ NASA’s Science Mission Directorate. We are very grate-
458
+ ful to the entire Keplerteam, whose efforts have led to
459
+ this exceptional mission. The present work was sup-
460
+ ported by the Danish Natural Science Research Council.
461
+ Facilities: The Kepler Mission
462
+ REFERENCES
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+ Aerts, C., Christensen-Dalsgaard, J., & Kurtz, D. W. 2009,
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1001.0033.txt ADDED
@@ -0,0 +1,1148 @@
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
1
+ arXiv:1001.0033v1 [astro-ph.SR] 30 Dec 2009WIYN OPEN CLUSTER STUDY. XXXVIII. STELLAR RADIAL VELOCITIE S IN THE YOUNG OPEN
2
+ CLUSTER M35 (NGC 2168)
3
+ Aaron M. Geller∗, Robert D. Mathieu∗, Ella K. Braden∗, Søren Meibom∗,†
4
+ Department of Astronomy, University of Wisconsin - Madison , WI 53706, USA
5
+ and
6
+ Imants Platais
7
+ Department of Physics and Astronomy, The Johns Hopkins Univ ersity, Baltimore, MD 21218, USA
8
+ and
9
+ Christopher J. Dolan∗
10
+ Department of Astronomy, University of Wisconsin - Madison , WI 53706, USA
11
+ ABSTRACT
12
+ We present 5201 radial-velocity measurements of 1144 stars, as p art of an ongoing study of the
13
+ young (150 Myr) open cluster M35 (NGC 2168). We have observed M 35 since 1997, using the Hydra
14
+ Multi-Object Spectrograph on the WIYN 3.5m telescope. Our stellar sample covers main-sequence
15
+ stars over a magnitude range of 13.0 ≤V≤16.5 (1.6 - 0.8 M ⊙) and extends spatially to a radius of
16
+ 30 arcminutes (7 pc in projection at a distance of 805 pc or ∼4 core radii). Due to its youth, M35
17
+ provides a sample of late-type stars with a range of rotation period s. Therefore, we analyze the radial-
18
+ velocity measurement precision as a function of the projected rot ational velocity. For narrow-lined
19
+ stars (vsini≤10 km s−1), the radial velocities have a precision of 0.5 km s−1, which degrades to 1.0
20
+ km s−1for stars with vsini= 50 km s−1. The radial-velocitydistribution shows a well-defined cluster
21
+ peak with a central velocity of -8.16 ±0.05 km s−1, permitting a clean separation of the cluster and
22
+ field stars. For stars with ≥3 measurements, we derive radial-velocity membership probabilities a nd
23
+ identify radial-velocity variables, finding 360 cluster members, 55 of which show significant radial-
24
+ velocity variability. Using these cluster members, we construct a co lor-magnitude diagram for our
25
+ stellar sample cleaned of field star contamination. We also compare th e spatial distribution of the
26
+ single and binary cluster members, finding no evidence for mass segr egation in our stellar sample.
27
+ Accounting for measurement precision, we place an upper limit on the radial-velocity dispersion of
28
+ the cluster of 0 .81±0.08 km s−1. After correction for undetected binaries, we derive a true radia l-
29
+ velocity dispersion of 0 .65±0.10 km s−1.
30
+ (galaxy:) open clusters and associations: individual (NGC 2168) - (s tars:) binaries: spectroscopic
31
+ 1.INTRODUCTION
32
+ Young open clusters are laboratories for the direct
33
+ study of the near-primordial characteristics of stellar
34
+ populations. Their properties, and particularly those of
35
+ the binary systems, offer unique insights into how stars
36
+ arebornand provideessentialguidancefor N-bodystud-
37
+ ies of star clusters. Indeed, with sophisticated N-body
38
+ simulations now able to model real open clusters (e.g.,
39
+ Hurley et al. 2005), knowledge of the correct initial con-
40
+ ditions are all the more important. In particular, the ini-
41
+ tial binary population has a vast impact on the dynami-
42
+ cal evolution of the cluster, and the characteristics of the
43
+ initial binary population will affect the overallfrequency,
44
+ formation rate and formation mechanisms of anomalous
45
+ stars, like blue stragglers, as interactions with binaries
46
+ are thought to be catalysts for the formation of these ex-
47
+ otic objects (Hurley et al. 2005; Knigge et al. 2009). As
48
+ a rich open cluster with an age of ∼150 Myr, M35 is a
49
+ prime cluster to define these hitherto poorly known ini-
50
+ tial conditions for the binary population required for any
51
+ ∗Visiting Astronomer, Kitt Peak National Observatory, Nati onal
52
+ Optical Astronomy Observatory, which is operated by the Ass o-
53
+ ciation of Universities for Research in Astronomy (AURA) un der
54
+ cooperative agreement with the National Science Foundatio n.
55
+ †Current address: Harvard-Smithsonian Center for Astrophy sics,
56
+ 60 Garden Street, Cambridge, MA 02138, USAopen cluster simulation.
57
+ M35 is a fundamental cluster in the WIYN Open Clus-
58
+ ter Study (WOCS; Mathieu 2000), and as such has a
59
+ strong base of astrometric and photometric observations
60
+ fromboththeWOCScollaborationandothers. Theclus-
61
+ ter is centered at α= 6h09m07.s5 andδ= +24◦20′28′′
62
+ (J2000),towardstheGalacticanticenter. Numerouspho-
63
+ tometric studies have identified the rich main-sequence
64
+ population (e.g., Kalirai et al. 2003; von Hippel et al.
65
+ 2002; Sung & Bessell 1999). WOCS CCD photometry
66
+ places the cluster at a distance of 805 ±40 pc, with an
67
+ ageof150 ±25Myr, ametallicityof[Fe/H]=-0.18 ±0.05
68
+ and a reddening of E(B−V)=0.20±0.01 (C. Deliyan-
69
+ nis, private communication). The most recent published
70
+ parameters, from Kalirai et al. (2003), place the cluster
71
+ at a distance of 912+70
72
+ −65pc ((m−M)0= 9.80±0.16)
73
+ with an age of 180 Myr, adopting a E(B−V)=0.20 and
74
+ [Fe/H] = -0.21. (See Kalirai et al. (2003) for a thorough
75
+ review of previous photometry references and their de-
76
+ rived cluster parameters). We note that these two recent
77
+ studies used different isochrone families.
78
+ There have been multiple proper-motion studies
79
+ of the cluster (Ebbighausen 1942; Cudworth 1971;
80
+ McNamara & Sekiguchi 1986a), although none deter-
81
+ mine clustermembership for individual starsfainter than2 Geller et al.
82
+ V≈15.0. Using proper motions, Leonard & Merritt
83
+ (1989) derive a cluster mass from 1600-3200 M ⊙within
84
+ 3.75 pc. Detailed observations have also been made
85
+ in M35 to study tidal evolution in binary stars
86
+ (Meibom & Mathieu 2005; Meibom et al. 2006, 2007),
87
+ lithium abundances (Steinhauer & Deliyannis 2004;
88
+ Barrado y Navascu´ es et al. 2001), and white dwarfs
89
+ (Reimers & Koester 1988; Williams et al. 2004, 2006,
90
+ 2009).
91
+ This is the first paper in a series studying the dy-
92
+ namical state of M35 through the use of radial-velocity
93
+ (RV) measurements. The data and results presented
94
+ in this series will form the largest database of spec-
95
+ troscopic cluster membership and variability in M35 to
96
+ date. In this paper, we present results from our ongo-
97
+ ing radial-velocity study of the cluster, which we began
98
+ in September 1997. Our stellar sample includes solar-
99
+ type main-sequence stars within the magnitude range of
100
+ 13.0≤V≤16.5, which corresponds to a mass range1of 1.6
101
+ - 0.8 M ⊙. The main-sequence turnoff is at V∼9.5,∼4
102
+ M⊙. In Section 2, we describe this stellar sample, ob-
103
+ servations and data reduction in detail. We thoroughly
104
+ investigate our RV measurement precision and the effect
105
+ of stellar rotation in Section 3. Then in Section 4 we de-
106
+ rive RV membership probabilities, and use our study of
107
+ the RV precision to identify RV variables, which we as-
108
+ sume to be binaries or higher-order systems. Within this
109
+ mass range, we identify 360 solar-type main-sequence
110
+ members; 305 are single2(non-RV-variable) stars while
111
+ 55 show significant RV variability. We then use these re-
112
+ sults to plot a color-magnitude diagram (CMD) cleaned
113
+ offield starcontamination, tosearchfor evidenceofmass
114
+ segregation and to study the cluster RV dispersion (Sec-
115
+ tion5). Finally, inSection6, weprovideabriefsummary.
116
+ In future papers, we will study the binary population of
117
+ M35in detail, providingobservationsthat will be used to
118
+ directly constrain the initial binary population of open
119
+ cluster simulations.
120
+ 2.OBSERVATIONS AND DATA REDUCTION
121
+ In the following section, we define our stellar sample,
122
+ provide a detailed description of our observations and
123
+ data reduction process, and discuss the completeness of
124
+ our spectroscopic observations.
125
+ 2.1.Photometric Target Selection
126
+ Initially, we created our M35 target list from the stars
127
+ in three wide-field CCD images centered on M35, taken
128
+ by T. von Hippel with the Kitt Peak National Observa-
129
+ tory (KPNO) Burrell Schmidt telescope on November 18
130
+ and 19, 1993. These images have Vexposures of 4 s, 20 s
131
+ and 180 s and Bexposures of 4 s, 25 s and 240s covering
132
+ a 70′×70′field. We obtained BandVphotometry with
133
+ a limiting magnitude of V= 17, denoted as source 1 in
134
+ 1This mass range is derived from a 180 Myr Padova isochrone
135
+ (Marigo et al. 2008) using the distance, reddening and metal licity
136
+ from Kalirai et al. (2003).
137
+ 2In the following, we use the term “single” to identify stars
138
+ with no significant RV variation. Certainly, many of these st ars
139
+ are also binaries, although generally with longer periods a nd/or
140
+ lower mass ratios ( q=m2/m1) than the binaries identified in this
141
+ study. When applicable, we have attempted to reduce this bin ary
142
+ contamination amongst the single star sample by photometri cally
143
+ identifying objects as binaries that lie well above the sing le-star
144
+ main-sequence (see Section 4.2).Fig. 1.— Color-magnitude diagram for stars in the field
145
+ of M35 highlighting the selected region used in this survey.
146
+ We plot all stars in the field with the gray points to show
147
+ the location of our selected sample relative to the full clus -
148
+ ter. Our stellar sample is bounded by the solid black lines.
149
+ Within this region, we plot observed stars in the solid black
150
+ points. Additionally, for reference we plot a 180 Myr Padova
151
+ isochrone (Marigo et al. 2008) using the distance, reddenin g
152
+ and metallicity from Kalirai et al. (2003)
153
+ Table 3. Additionally, we derive astrometry from these
154
+ plates, tied to the Tycho catalogue.
155
+ More recently, we added to our database the BVpho-
156
+ tometry of Deliyannis (private communication), taken
157
+ on the WIYN30.9m telescope with the S2KB 2K by
158
+ 2K CCD. This photometry derives from a mosaic of five
159
+ fields. Each field has a 20′×20′field-of-view, with one
160
+ central field and four tiled around the center, for a total
161
+ field-of-view of of 40′×40′. This photometry is denoted
162
+ as source 2 in Table 3, and covers 74% of the objects
163
+ we have observed in this study. We note that this pho-
164
+ tometry is more precise than that of source 1. The star-
165
+ by-star difference in Vmagnitudes for the two sources is
166
+ roughly Gaussian with σ= 0.06 mag. However there is
167
+ a tail that extends beyond three times this sigma value.
168
+ Therefore we caution the reader when using magnitudes
169
+ from source 1.
170
+ We selected stars for the RV master list based on three
171
+ constraints. The faintest sources that can be observed
172
+ efficiently at echelle resolution using the Hydra Multi-
173
+ Object Spectrograph (MOS) on the WIYN 3.5m have
174
+ V=16.5; this therefore sets our faint limit for observa-
175
+ tions. Stars bluer than ( B−V)∼0.6 ((B−V)0∼0.4)
176
+ do not provide precise RV measurementsdue to rapid ro-
177
+ tationandpaucityofspectrallines; thisthereforesetsour
178
+ blue limit for observations. Finally, we perform a photo-
179
+ metric selection of cluster member candidates, shown as
180
+ the outlined region4in Figure 1. This region includes a
181
+ wide swath above the main sequence so as to not select
182
+ against binary stars (e.g. Dabrowski & Beardsley 1977),
183
+ yet also removes stars that are very likely cluster non-
184
+ members. This photometric selection allows for an effi-
185
+ cient survey of the cluster. Our sample extends radially
186
+ to 30 arcminutes from the cluster center. At a distance
187
+ of 805 pc, this corresponds to the inner ∼7 pc of the
188
+ 3The WIYN Observatory is a joint facility of the University of
189
+ Wisconsin-Madison, Indiana University, Yale University, and the
190
+ National Optical Astronomy Observatories.
191
+ 4Specifically, we select stars with 0 .6<(B−V)<1.5, 13.0<
192
+ V <16.5 and between the lines defined by 5 .7(B−V)+8.6< V <
193
+ 5.7(B−V)+11.0.WOCS. RV Measurements in M35 3
194
+ Fig. 2.— Completeness of our observations as a function of Vmagnitude (left) and projected radius (right). We plot the
195
+ completeness in stars observed ≥3 times with the dashed line, and stars observed ≥1 time with the solid line.
196
+ cluster in projection. Given the core radius derived by
197
+ Mathieu (1983) of1.9 ±0.1pc, oursample isdrawn from
198
+ the inner ∼4 core radii.
199
+ Note that we lack BVphotometry for ∼11% of the
200
+ point sources found within 30 arcminutes from the clus-
201
+ tercenterin2MASS.Formostoftheseobjects, itislikely
202
+ that there is a nearby or overlapping additional object
203
+ which has prevented accurate BVphotometric measure-
204
+ ments from either of our sources. These objects, by de-
205
+ fault, are not included in our stellar sample. In total, our
206
+ stellar sample contains 1344 stars.
207
+ 2.2.Spectroscopic Observations
208
+ Since September 1997, we have collected 5201 spec-
209
+ tra of 1144 stars within this stellar sample as part of
210
+ an ongoing observing program using the WIYN Hydra
211
+ MOS. For the majority of these observations, we use Hy-
212
+ dra’s blue-sensitive 300 µm fibers, which project to a 3.1′
213
+ aperture on the sky. We use the 316 lines mm−1echelle
214
+ grating, isolating the 11th order with the X14 filter. The
215
+ resulting spectra span a wavelength range of ∼25 nm,
216
+ with a dispersion of 0.015 nm pixel−1, centered on 512.5
217
+ nm. We have also occasionally centered our observation
218
+ on 637.5 nm using a very similar setup. In this region,
219
+ we use the same grating, but isolate the 9th order with
220
+ the X18 filter. These observations span a slightly larger
221
+ wavelength range of ∼30 nm, and have a dispersion of
222
+ 0.017 nm pixel−1. Due to a broken filter, observations
223
+ taken after the spring of 2008 use different observing se-
224
+ tups than discussed above; most are centered on 560 nm
225
+ and all use the echelle grating. We have not noticed any
226
+ decrease in performance from the new wavelength range,
227
+ but we caution the reader that we lack sufficient obser-
228
+ vations in these setups to reliably determine our RV pre-
229
+ cision for these measurements. During this same period
230
+ certain upgrades were made to the spectrograph collima-
231
+ tor5. All observed regions are rich in metal lines. The
232
+ typical velocity resolution is 15 km s−1. In a two-hour
233
+ integration, the spectra have signal-to-noise (S/N) ra-
234
+ tios ranging from ∼18 per resolution element for V=16.5
235
+ stars to∼100 per resolution element for V=13 stars.
236
+ We create fiber configurations ( pointings ) for our ob-
237
+ servations using a similar method as Geller et al. (2008).
238
+ Monte Carlo simulations show that we require at least
239
+ 5http://www.astro.wisc.edu/ ∼mab/research/bench upgrade/threeobservationsoverthecourseofayearinordertoen-
240
+ sure 90% confidence that a star is either constant or vari-
241
+ able in RV out to binary periods of 1000 days (Mathieu
242
+ 1983, Geller & Mathieu, in preparation). Given three
243
+ observations with consistent RV measurements over a
244
+ timespan of at least a year and typically longer, we clas-
245
+ sify a given star as single (strictly, non-RV variable) and
246
+ finished, andmoveittothelowestpriority. Ifagivenstar
247
+ has three RV measurements with a standard deviation
248
+ >2.0 km s−1(four times our precision for narrow-lined
249
+ stars; see Section 4.2), we classify the star as RV vari-
250
+ able and give it the highest priority for observation on a
251
+ schedule appropriate to its timescale of variability. This
252
+ prioritization allows us to most efficiently derive orbital
253
+ solutions for our detected binaries.
254
+ We place our shortest-period binaries at the highest
255
+ priority for observations each night, followed by longer-
256
+ period binaries to obtain 1-2 observations per run. Be-
257
+ low the confirmed binaries we place, in the following or-
258
+ der, “candidate binaries” (once-observedstars with a RV
259
+ measurement outside the cluster RV distribution or stars
260
+ with a few measurements that span only 1.5 - 2.5 km
261
+ s−1), once observed and then twice observed non-RV-
262
+ variable likely members, twice observed non-RV-variable
263
+ likely non-members, unobserved stars, and finally, “fin-
264
+ ished” stars. Within each group, we prioritize by dis-
265
+ tance from the cluster center, giving those stars nearest
266
+ to the center the highest priority. A typical pointing
267
+ will contain ∼70 fibers placed on individual stars in our
268
+ sample and ∼10 sky fibers.
269
+ For a given pointing we obtain three consecutive ex-
270
+ posures, each of 40 minutes. In poor transparency or
271
+ with a particularly bright sky, we restrict the targets
272
+ toV <15.0 and shorten the integration time, gener-
273
+ ally to 20 minute exposures. We obtain Thorium-Argon
274
+ (ThAr), oroccasionallyCopper-Argon(CuAr), emission-
275
+ lamp comparison spectra (300 s integrations) before and
276
+ after each set of science integrations for wavelength cal-
277
+ ibration and to check for wavelength shifts during the
278
+ observing sequence. For each set of integrations we also
279
+ obtain one flat-field image (200 s) of a white spot on
280
+ the dome illuminated by incandescent lights. Associat-
281
+ ing the flat-field images with the science integrations is
282
+ particularly critical for calibrating throughput variations
283
+ between the fibers in order to apply sky subtractions. In
284
+ total, we have observed 106 distinct pointings in M354 Geller et al.
285
+ over the roughly 11 years since our survey began.
286
+ 2.3.Data Reduction
287
+ For a thorough description of our data reduction pro-
288
+ cess, seeGeller et al.(2008). Inshort, weperformastan-
289
+ dard bias and flat-field correction to the images using
290
+ the overscan strip and the flat-field images, respectively.
291
+ The flat-field spectra are used to trace each aperture in
292
+ a given pointing and thereby extract the science spec-
293
+ tra. Wavelength solutions derived from the emission-
294
+ lamp spectra are applied, followed by sky-subtraction
295
+ using sky fibers from each pointing. The three sets of
296
+ spectra (one set from each integration in a given con-
297
+ figuration) are then combined via a median filter to re-
298
+ movecosmicraysignalsandimproveS/N.These reduced
299
+ spectra are cross-correlated with a high S/N solar spec-
300
+ trum, obtained using a dusk sky exposure taken on the
301
+ WIYN 3.5m with the same instrument setup as the given
302
+ pointing. A Gaussian fit to the cross-correlation func-
303
+ tion (CCF) yields a RV and a full width at half maxi-
304
+ mum (FWHM, in km s−1) for each stellar observation.
305
+ The mean UT time is used to find and correct each RV
306
+ measurement for the Earth’s heliocentric velocity. Fi-
307
+ nally we apply the unique fiber-to-fiber RV offsets de-
308
+ rived by Geller et al. (2008) for the WIYN-Hydra data
309
+ to these RVs. As in Geller et al. (2008), to ensure a
310
+ sufficient quality of measurement, we incorporate into
311
+ our database only those spectra with a CCF peak height
312
+ higher than 0.4. Additionally, we examine the distri-
313
+ bution of RVs for each individual star and visually in-
314
+ spect any measurements that are outliers in the distri-
315
+ bution. Occasionally we remove a measurement whose
316
+ CCF, though having a peak height above 0.4, clearly
317
+ provides a spurious measurement (e.g., inadequate sky
318
+ subtraction).
319
+ 2.4.Completeness of Spectroscopic Observations
320
+ We have at least one observation for 1144 of the
321
+ 1344 stars in our stellar sample, for a completeness of
322
+ 85% across our entire sample. 60% of the stars in our
323
+ stellar sample have sufficient observations for their RVs
324
+ to be considered final (813/1344). For these stars, we ei-
325
+ ther have ≥3 RV measurements that show no variation,
326
+ or, if we do see RV variability, we have found a binary
327
+ orbital solution. (These 813 stars comprise the SM, SN,
328
+ BM and BN classes; see Section 4.1). Of those stars not
329
+ finalized, 231 have only one or two observations, and an-
330
+ other 100stars arevariable but do not yet havedefinitive
331
+ orbital solutions.
332
+ In Figure 2, we show the completeness of our observa-
333
+ tions as functions of Vmagnitude (left) and projected
334
+ radius (right). We plot the completeness in stars ob-
335
+ served≥3 times with the dashed line and stars observed
336
+ ≥1 time with the solid line. Our prioritization of stars
337
+ by distance from the cluster center is evident by our de-
338
+ creasing completeness with cluster radius. The decreas-
339
+ ing completeness towards fainter stars reflects the need
340
+ for dark skies with minimal sky contamination in order
341
+ to obtain sufficient S/N in our spectra to derive reliable
342
+ RVsforfaint stars. Thereare37starswith V <15in our
343
+ stellar sample that do not have RV measurements, one of
344
+ which is a proper-motion member. 15 were observed but
345
+ did not yield reliable RVs, mostly due to rapid rotation.22 were not observed, 15 of which are farther than 20
346
+ arcminutes in radius from the cluster center.
347
+ Thedifferenceincompletenessbetweenbrightstarsob-
348
+ served≥1 and≥3 times is also a result of an increas-
349
+ ing population of rapidly rotating stars towards bluer
350
+ (B−V) color. For many of these stars, we have multi-
351
+ ple observations of which only a few, and sometimes one,
352
+ exceed this cutoff value of CCF peak height >0.4 and
353
+ therefore are included in our database. For purposes of
354
+ future research we also include seven rapid rotators in
355
+ Table 3 for which we have been unable to derive RVs
356
+ from our spectra.
357
+ 3.EFFECTS OF STELLAR ROTATION ON
358
+ MEASUREMENT PRECISION
359
+ 3.1.Observed Rotation
360
+ Because of its youth, M35 provides a sample of
361
+ late-type stars with a range of rotational periods
362
+ (Meibom et al. 2009); some of these stars have projected
363
+ rotational velocities that exceed our spectral resolution.
364
+ As such, the cluster presents an opportunity to explore
365
+ empiricallythedependence ofourmeasurementprecision
366
+ on increasing vsini, whereiis the inclination angle of
367
+ the stellar rotation axis to our line of sight.
368
+ Fig. 3.— FWHM as a function of vsinifor observations
369
+ in the 512.5 nm region. FWHM values are measured from
370
+ the CCF peaks derived from a series of artificially broadened
371
+ templates, of known vsini, correlated against the original
372
+ narrow-lined spectrum. We also show a polynomial fit to the
373
+ data, which we then use to derive vsinivalues for observed
374
+ stars in M35. Additionally we plot a dashed line at vsini
375
+ = 10 km s−1, below which the curve flattens out due to our
376
+ spectral resolution. We impose a floor in vsiniat this value
377
+ as we are unable to reliably measure slower rotation.
378
+ The measured FWHM of the CCF for a given star is
379
+ directly related to the vsini(Rhode et al. 2001). Thus
380
+ in order to derive a vsinivalue, we first measure the
381
+ FWHM of the CCF peak. To do so, we fit a Gaussian
382
+ function to the peak, forcing the baseline of the Gaus-
383
+ sian to start at the background level of the CCF. Specif-
384
+ ically, we subtract from the CCF a polynomial fit to this
385
+ backgroundlevel, and then fit the Gaussian to the subse-
386
+ quent “continuum subtracted” CCF. We only use spec-
387
+ tra from the 512.5 nm region to measure the FWHM, as
388
+ the FWHM is dependent on the setup (i.e., the disper-
389
+ sion, etc.), and most of our observations were taken in
390
+ the 512.5 nm region. We then use a similar technique as
391
+ Rhode et al. (2001), to convert this FWHM to a vsini.WOCS. RV Measurements in M35 5
392
+ Fig. 4.— Histogram of vsinimeasurements (left) and vsinias a function of ( B−V)0(right) for the cluster members of M35.
393
+ We have removed double-lined binaries and any binaries with known periods less than 10.2 days, the circularization peri od in
394
+ M35 (Meibom & Mathieu 2005). We only show stars with mean vsinivalues derived from ≥3 observations within the 512.5
395
+ nm region. Notice that the stars with the largest rotation ar e generally also the bluest stars in our sample.
396
+ We create a series of artificially broadened templates by
397
+ convolving our standard solar template with a series of
398
+ theoretical rotation profiles of specific vsinivalues. We
399
+ then cross correlate this series of broadened templates
400
+ with the original narrow-lined template and measure the
401
+ FWHM of the CCF peak as described above. In Fig-
402
+ ure 3 we show the results of this analysis along with a
403
+ polynomial fit to the data. We use this curve to derive
404
+ vsinivalues for all observations of stars in M35 in the
405
+ 512.5 nm region. We then take the mean vsinifor each
406
+ star, using only our highest quality (CCF peak height
407
+ >0.4) spectra, and provide these values in Table 3. We
408
+ are unable to reliably measure vsinivalues below 10 km
409
+ s−1, due to the spectral resolution; we therefore impose
410
+ a floor to the vsiniat this value.
411
+ The median FWHM value that we observe is 46.1 km
412
+ s−1which corresponds to vsini= 10.3 km s−1. Exclud-
413
+ ing stars rotating slower than 10 km s−1, we find a preci-
414
+ sion of 1.4 km s−1for individual vsinivalues of ≤25 km
415
+ s−1, which increasesto 1.6km s−1forvsini >25km s−1.
416
+ These precision values were derived in the same manner
417
+ as for our RV precision, with a fit to a χ2function; see
418
+ Section 3.2 and Geller et al. (2008). Where possible, we
419
+ derive a mean vsinifor a given star from multiple, gen-
420
+ erally≥3, observations within the 512.5 nm region. We
421
+ have compared our vsinimeasurements to the rotation
422
+ periods from Meibom et al. (2009) for stars observed in
423
+ both studies, and find the vsiniand rotation periods to
424
+ be consistent.
425
+ In the left panel of Figure 4, we plot a histogram of
426
+ the mean vsinimeasurements for M35 cluster members.
427
+ (See Section 4.1 for our membership criteria.) In this
428
+ and the other panel, we have excluded any binaries with
429
+ periods known to be less than the circularization period
430
+ in M35 of 10.2 days (Meibom & Mathieu 2005), as the
431
+ rotation of the stars in these binaries have likely been af-
432
+ fectedbytidalprocesses. Wehavealsoremovedanystars
433
+ that appearto be indouble-lined binaries, asthe spectral
434
+ lines in many of these observations are broadened due to
435
+ the secondary spectrum at similar, though slightly off-
436
+ set, RV. In the right panel of Figure 4, we plot the mean
437
+ vsinias a function of ( B−V)0for M35 cluster mem-
438
+ bers. We see a clear trend of increasing rotation towards
439
+ bluer stars, as has also been observed in other young
440
+ open clusters and the field (e.g., field, Hyades, Pleiades,Kraft 1967; Pleiades, Soderblom et al. 1993; Blanco 1,
441
+ Mermilliod et al. 2008; IC 2391, Platais et al. 2007).
442
+ 3.2.Radial-Velocity Precision
443
+ We determine the RV measurement precision following
444
+ Geller et al. (2008), where a χ2distribution is fit to the
445
+ distribution of the standard deviations of the first three
446
+ RV measurements for each star in an ensemble of stars.
447
+ Here we do this operation on samples of stars with dif-
448
+ feringvsini. Specifically, we consider stars with vsini
449
+ of≤10 km s−1, 10 - 20 km s−1and 20 - 80 km s−1.
450
+ The bin sizes were chosen arbitrarily in order to pro-
451
+ vide sufficiently large samples. The first bin contains all
452
+ narrow-lined stars for which we have imposed a floor to
453
+ thevsini(see Section 3.1); these stars have line widths
454
+ characteristicof the auto-correlationof our spectralreso-
455
+ lution. The remainingbins containstarswith line widths
456
+ increased by stellar rotation.
457
+ A detailed study of the RV measurement precision of
458
+ our observation and data-reduction pipeline has been
459
+ done by Geller et al. (2008) for late-type stars in the
460
+ old open cluster NGC 188. For the narrow-lined stars
461
+ in NGC 188 they find a single-measurement precision of
462
+ 0.4 km s−1. This precision is also a function of the S/N
463
+ of the spectrum, as shown in Geller et al. (2008) by the
464
+ degrading precision with increasing Vmagnitude as well
465
+ as decreasing CCF peak height. The largest S/N effect
466
+ seen for narrow-lined stars in NGC 188 is to degrade the
467
+ precision by 0.25 km s−1. The effect of rotation is larger
468
+ than this amount. Here, we derive a relationship be-
469
+ tween the measurement precision and vsiniand use this
470
+ relationship in our analysis throughout this paper.
471
+ In Figure 5 we show the RV precision as a function
472
+ ofvsiniin M35 for observations taken in the 512.5 nm
473
+ region. The narrow-lined stars have a RV precision of
474
+ 0.5 km s−1, similar to that found for the narrow-lined
475
+ stars in NGC 188 observed with this same setup. As ex-
476
+ pected, the value of the measurement precision increases
477
+ with increasing line width. For the most rapidly rotating
478
+ stars (vsini >50 km s−1), the measurement precision
479
+ degrades to ∼1.0 km s−1. We fit a linear relationship to
480
+ the points in Figure 5, shown as the dashed line:
481
+ σi= 0.38+0.012(vsini) km s−1,(1)
482
+ whereσiis our precision. We use this equation with
483
+ the mean measured vsinifor a given star to calculate6 Geller et al.
484
+ the single-measurement RV precision for that star. We
485
+ adopt a floor to our precision at 0.5 km s−1, as found for
486
+ our narrow-lined stars, and shown by the break in the
487
+ dashed line in Figure 5.
488
+ Fig. 5.— RV measurement precision as a function of the
489
+ averagevsini(in km s−1) for single lined stars with ≥3 ob-
490
+ servations. The bins are vsiniof≤10 km s−1, 10 - 20 km
491
+ s−1and 20 - 80 km s−1, chosen to provide sufficiently large
492
+ samples. The gray horizontal bars indicate the bin sizes for
493
+ each point. The black vertical error bars show the one sigma
494
+ errors on the precision fit values. The dotted line shows the
495
+ fit to these data, and provided in Equation 1; we impose a
496
+ floor to our precision at 0.5 km s−1.
497
+ We lack sufficient observations to perform this same
498
+ analysis using observations in the 637.5 nm region or
499
+ for observations taken after the spring of 2008 (see Sec-
500
+ tion 2.2). Therefore, for the 129 stars that do not have
501
+ anyobservationsinthe512.5nmregion( ∼11%ofourob-
502
+ served stars), we visually inspect the spectra and CCFs.
503
+ For narrow-lined stars, we set the precision to 0.5 km
504
+ s−1, and for rotating stars we set the precision to 1.0 km
505
+ s−1. We can then use this RV precision value for a given
506
+ star to determine whether our observations for this star
507
+ are constant or variable in velocity (see Section 4.2). We
508
+ note that only 13 of these stars have sufficient observa-
509
+ tions for their RVs to be considered final, and only 2 are
510
+ probable members.
511
+ 4.RESULTS
512
+ The full M35 database is available with the electronic
513
+ version of this paper; here we show a sample of our re-
514
+ sults in Table 3. The first column in Table 3 contains
515
+ the WOCS identification number ( IDW). These num-
516
+ bers are defined in the same manner as in Hole et al.
517
+ (2009), with the cluster center set at α= 6h9m7.s5 and
518
+ δ= +24◦20′28′′(J2000). Nextwe givethe corresponding
519
+ IDs from Meibom et al. (2009), McNamara & Sekiguchi
520
+ (1986a) and Cudworth (1971) ( IDM,IDMcandIDC).
521
+ The next few columns provide the right ascension ( RA),
522
+ declination ( DEC), theBVphotometry and the source
523
+ number ( S) for this photometry (see Section 2.1). Next,
524
+ we show the number of RV measurements ( N) and the
525
+ mean and standard error of the RV measurements. For
526
+ stars with only one RV measurements, we show the
527
+ single-measurementRV precision instead of the standard
528
+ error. Next we provide this single-measurement RV pre-
529
+ cision (σi, derived using equation 1), the mean and stan-TABLE 1
530
+ Gaussian Fit Parameters For Cluster
531
+ and Field RV Distributions
532
+ Cluster Field
533
+ Ampl. (Number) 69.0 ±2.0 2.4 ±0.4
534
+ RV(km s��1) -8.17 ±0.05 13 ±4
535
+ σ(km s−1) 0.92 ±0.08 34 ±4
536
+ dard error of the vsinimeasurements6, thee/ivalue
537
+ (see Section 4.2), the calculated RV membership proba-
538
+ bility(P RV, seeSection4.1), theproper-motionmember-
539
+ ship probability from McNamara & Sekiguchi (1986a)
540
+ (PPM1) and Cudworth (1971) ( PPM2), where available,
541
+ andthen, theclassificationoftheobject(seeSection4.3).
542
+ For RV-variable stars with orbital solutions, we present
543
+ the center-of-mass ( γ) RV with the derived error in place
544
+ of the mean RV and its standard error, and add the com-
545
+ ment SB1 or SB2 for single- and double-lined binaries,
546
+ respectively. Additionally, for binaries without orbital
547
+ solutionsthatappeartobedouble-lined, weaddthecom-
548
+ ment of SB2. Finally, for purposes of future research we
549
+ include seven rapid rotators for which we have been un-
550
+ able to derive RVs from our spectra, and label them with
551
+ the comment RR.
552
+ 4.1.Membership
553
+ The RV distribution of M35 is clearly distinguished
554
+ from that of the field when we plot a histogram of the
555
+ mean RVs for the observedstars in our stellar sample. In
556
+ Figure 6, we show a histogram of the mean RVs for stars
557
+ with≥3 RV measurements whose standard deviations
558
+ are<2 km s−1, as well as the γ-RVs for binary stars
559
+ with orbitalsolutions, thus excludingfrom the fit anyRV
560
+ variables whose γ-RVs are unknown. The cluster shows
561
+ a well-defined peak rising above the broad distribution
562
+ of the field stars. We simultaneously fit one-dimensional
563
+ Gaussian functions, Fc(v) andFf(v), to represent the
564
+ cluster and field RV distributions, respectively, and then
565
+ use these fits to calculate RV membership probabilities
566
+ for each individual star. We compute the membership
567
+ probability PRV(v) with the usual formula:
568
+ PRV(v) =Fc(v)
569
+ Ff(v)+Fc(v)(2)
570
+ (Vasilevskis et al. 1958). We plot these Gaussian fits in
571
+ Figure 6 with the dashed lines, and show the fit param-
572
+ eters in Table 1.
573
+ For a given single star, we use the mean RV to com-
574
+ pute the RV membership probability. For a given binary
575
+ star with an orbital solution, we compute the RV mem-
576
+ bership probability from the γ-RV. For RV-variable stars
577
+ without orbital solutions, the γ-RVs are not known, and
578
+ therefore we cannot calculate RV membership probabil-
579
+ ities. For these stars, we provide a preliminary member-
580
+ ship classification, described in Section 4.3.
581
+ 6For double-lined binaries and stars with no observation in t he
582
+ 512.5 nmregion, wedo notderive a vsinivalue. For starswith only
583
+ one measurement in the 512.5 nm region, we convert the 1-sigm a
584
+ error on the FWHM (derived from the Gaussian fit to the CCF
585
+ peak) to an error on the vsiniusing the fit shown in Figure 3. As
586
+ this relationship is not linear, we provide the mean of the de rived
587
+ upper and lower errors on vsini.WOCS. RV Measurements in M35 7
588
+ Fig. 6.— RV histogram for stars in the field of M35. We
589
+ include the mean RVs for stars observed ≥3 times with RV
590
+ standarddeviations <2kms−1andtheγ-RVsfor binarystars
591
+ with orbital solutions, excluding RV variables whose γ-RVs
592
+ are unknown. The bin sizes are 0.5 km s−1, equal to our
593
+ RV precision for narrow-lined stars, as found in Section 3.
594
+ The dashed lines show the simultaneous Gaussian fits to the
595
+ cluster and field RV distributions.
596
+ Fig. 7.— Histogram of membership probabilities, P RV, for
597
+ stars observed ≥3 times with RV standard deviations <2 km
598
+ s−1and for binaries whose γ-RVs are known. For the single
599
+ stars, we compute P RVusing the mean observed RV; for bi-
600
+ naries with orbital solutions, P RVis based on the γ-RV. We
601
+ show our membership cutoff of P RV=50% with the dashed
602
+ line, above which we classify a star as a cluster member. Note
603
+ that we do not show the full height of the bin at lowest mem-
604
+ bership probability for clarity.
605
+ In Figure 7, we show the distribution of RV member-
606
+ ship probabilities, displaying a clean separation between
607
+ the cluster members and field stars. In the following
608
+ analysis, we use a probability cutoff of P RV≥50 % to
609
+ define our cluster member sample. Using the 344 single
610
+ clustermembersandbinaryclustermemberswithorbital
611
+ solutions, we find a mean cluster RV of -8.16 ±0.05 km
612
+ s−1. From the area under the fit to the cluster and field
613
+ distributions, we estimate a field contamination of 6%
614
+ within our cluster member sample (P RV≥50%). Though
615
+ this estimate is derived excluding the RV variables that
616
+ do not have orbital solutions, the percent contamination
617
+ should be valid for the cluster as a whole.
618
+ Our RV membership probabilities agree well with the
619
+ proper-motion memberships of Cudworth (1971) and
620
+ McNamara & Sekiguchi (1986a). We note that our stel-
621
+ lar sample covers only the faintest portion of either
622
+ proper-motion study. There are 24 Cudworth (1971)proper-motionmembers within our observed stellar sam-
623
+ ple, of which we find 14 (58%) to also have ≥50% RV
624
+ membership probabilities. Cudworth (1971) note that
625
+ forV >13 they begin to find significant errors in
626
+ their photometry and expect many field stars to con-
627
+ taminate their proper-motion member sample; this can
628
+ likely explain the 10 discrepant stars. There are 70
629
+ McNamara & Sekiguchi(1986a)proper-motionmembers
630
+ within our observed stellar sample, of which we find 64
631
+ (91%) to also have ≥50% RV membership probabilities.
632
+ McNamara & Sekiguchi (1986a) expects up to 15 field
633
+ stars contaminating their cluster member sample from
634
+ 13< V <15, which can easily account for the 6 dis-
635
+ crepant stars.
636
+ We also note that NGC 2158 is only ∼28 arcminutes
637
+ away from the center of M35, at α= 6h07m25sandδ=
638
+ +24◦05′48′′(J2000), and thus is within the spatialregion
639
+ that we have surveyed. Scott et al. (1995) find a mean
640
+ RV for NGC 2158 of 28 ±4 km s−1. There are five stars
641
+ within our sample that lie within the cluster radius of 2.5
642
+ arcminutes Carraro et al. (2002) from the center of NGC
643
+ 2158 and have RVs within three times the standard error
644
+ (12 km s−1) of the mean RV : 125044, 39017, 111050,
645
+ 57037, 54048. Two of these stars (125044 and 57037)
646
+ have less than three observations; the remaining three
647
+ have≥3observationsandappeartobenon-RV-variables.
648
+ 4.2.Radial-Velocity Variability
649
+ RV-variable stars are distinguishable by the larger
650
+ standard deviations of their RV measurements. Here,
651
+ we assume that such velocity variability is the result of
652
+ a binary companion, or perhaps multiple companions.
653
+ Specifically, we consider a star to be a RV variable if the
654
+ ratio of the standard deviation of its RV measurements
655
+ to the single-measurement RV precision7(e/i) for that
656
+ star is greater than four (Geller et al. 2008). We provide
657
+ thee/ivalue for each single-lined star in Table 3; we
658
+ label double-lined systems as RV variables directly, and
659
+ include the comment of SB2 in Table 3.
660
+ MonteCarloanalysishasshownthat, forsimilarobser-
661
+ vations ofsolar-typestars in NGC 188, Geller & Mathieu
662
+ (in preparation) can detect the majority of binaries with
663
+ periods less than 104days and a negligible fraction of
664
+ longer-period binaries. Though the slightly poorer preci-
665
+ sionforthe M35datawill effect the specific completeness
666
+ numbers, we can assume a similarly high completeness in
667
+ detected binaries with periods less than 104days and a
668
+ corresponding drop in completeness for longer-period bi-
669
+ naries. Some of the undetected systems are evident from
670
+ their separation from the main sequence (see Figure 8).
671
+ We have currently identified 55 RV-variable members
672
+ of M35, and have derived orbital solutions for 71%
673
+ (39/55) of this sample. In following papers we will pro-
674
+ vide the orbital solutions for these systems, including
675
+ all derived parameters. We will then perform a detailed
676
+ analysis of the distributions of these orbital parameters
677
+ as well as the binary frequency of the cluster.
678
+ 7We use the same nomenclature of “ e/i” as in Geller et al.
679
+ (2008), though in other sections, for clarity, we have label ed the
680
+ precision as σi, so as not to confuse the precision with an inclina-
681
+ tion angle.8 Geller et al.
682
+ TABLE 2
683
+ Number of Stars
684
+ or Star Systems
685
+ Within Each
686
+ Membership
687
+ Class
688
+ Class Number
689
+ SM 305
690
+ SN 452
691
+ BM 39
692
+ BN 17
693
+ BLM 16
694
+ BU 16
695
+ BLN 68
696
+ U 231
697
+ 4.3.Membership Classification of Radial-Velocity
698
+ Variable Stars
699
+ We follow the same classification system as
700
+ Geller et al. (2008) and Hole et al. (2009) in order
701
+ to provide a qualitative guide to a given star’s mem-
702
+ bership and variability, in addition to the calculated
703
+ RV memberships and e/ivalues. We provide these
704
+ classifications for all observed stars, while the member-
705
+ ships and e/ivalues are only provided for a subset of
706
+ appropriate stars.
707
+ For stars with e/i<4, we classify those with P RV≥50%
708
+ as single members (SM), and those with P RV<50% as
709
+ singlenon-members(SN). Ifa starhas e/i≥4and enough
710
+ measurements from which we are able to derive an or-
711
+ bital solution, we use the γ-RV to compute a secure
712
+ RV membership. For these binaries, we classify those
713
+ with P RV≥50% as binary members (BM) and those with
714
+ PRV<50% as binary non-members (BN). For RV vari-
715
+ ables without orbital solutions, we split our classifica-
716
+ tions into three categories. If the mean RV results in
717
+ PRV≥50%,weclassifythesystemasabinarylikelymem-
718
+ ber (BLM). If the mean RV results in P RV<50% but the
719
+ range of measured RVs includes the cluster mean RV, we
720
+ classify the system as a binary with unknown member-
721
+ ship (BU). Finally, if the RV measurements for a given
722
+ star all lie either at a lower or higher RV than the clus-
723
+ ter distribution, we classify the system as a binary likely
724
+ non-member (BLN), since it is unlikely that any orbital
725
+ solution could place the binary within the cluster distri-
726
+ bution. We classify stars with <3 RV measurements as
727
+ unknown (U), as these stars do not meet our minimum
728
+ criterion for deriving RV memberships or e/imeasure-
729
+ ments. In the following analysis, we include the SM,
730
+ BM and BLM stars as cluster members. Including these
731
+ stars, we find 360 total cluster members in our sample.
732
+ We list the number of stars within each class in Table 2.
733
+ 5.DISCUSSION
734
+ In the following section, we present a CMD for M35
735
+ cleaned of field star contamination (Section 5.1), com-
736
+ pare the spatial distribution of the single and binary
737
+ members (Section 5.2), and analyze the RV dispersion
738
+ of the cluster (Section 5.3).
739
+ 5.1.Color-Magnitude Diagram
740
+ In Figure 8, we show the CMD for all RV cluster mem-
741
+ bers in M35 from this study for which we have pho-
742
+ tometry from the WIYN 0.9m, as this set of photom-Fig. 8.— Color-magnitude diagram of M35 including only
743
+ cluster members (P RV≥50%) with photometry from WIYN
744
+ 0.9m (source 2). We plot the RV variables with orbital so-
745
+ lutions with circles and without orbital solutions with dia -
746
+ monds. We show the 180 Myr Padova isochrone as the black
747
+ line. The solid gray line shows where binaries with mass ra-
748
+ tios of 1.0 lie on the CMD, and the dashed gray line shows the
749
+ deviation from the isochrone of twice the photometric error .
750
+ etry is of higher precision than that taken on the Bur-
751
+ rell Schmidt (see Section 2.1). We also plot a 180 Myr
752
+ Padova isochrone using the cluster parameters derived
753
+ by Kalirai et al. (2003) in the black curve. Binaries with
754
+ orbital solutions are circled and RV variables without or-
755
+ bital solutions are marked by diamonds.
756
+ Additionally, we use the Padova isochrone to plot the
757
+ location on the CMD of binaries with mass ratios q= 1,
758
+ shown as the gray line. We note that there are a number
759
+ ofstarsobservedbrighterandtothe redofthis line, some
760
+ that we have not identified as RV variables. In this loca-
761
+ tion on the CMD one would expect to find either higher-
762
+ order systems or field stars. There are 33 RV members
763
+ that lie above the q= 1 line; 22 are single and 11 show
764
+ RV variability. We expect a 6% field star contamination
765
+ within the cluster members sample (Section 4.1). If we
766
+ include only the 309 cluster members that have photom-
767
+ etry from the WIYN 0.9m (and are therefore shown in
768
+ Figure 8), this results in 19 possible field stars; including
769
+ our entire cluster member sample results in 22 possible
770
+ field stars. Therefore field star contamination cannot ac-
771
+ count for all of these sources, suggesting that a subset
772
+ of these stars are indeed higher-order systems. We also
773
+ notethatthereareanadditional12clustermemberswith
774
+ photometry from the Burrell Schmidt that lie above the
775
+ q= 1 line, but recall that this source of photometry is of
776
+ poorer precision.
777
+ Finally, for use in Sections 5.2, we follow a similar pro-
778
+ cedure as Montgomery et al. (1993) to attempt to photo-
779
+ metricallyidentify binariesthatlie farfromtheisochrone
780
+ ontheCMD. Wederivethedistanceofeachstarfromthe
781
+ main-sequenceisochroneand fit a Gaussianfunction rep-
782
+ resenting the photometric error distribution to the dis-
783
+ tribution of these distances. We notice a clear excess in
784
+ the observed distribution from the Gaussian fit at 2 σ,
785
+ shown as the dashed gray line in Figure 8. We attribute
786
+ this excess to photometric binaries. A 1 M ⊙star in M35
787
+ with the additional light from a companion of mass-ratio
788
+ q= 0.78 would lie on this line. Therefore sources ob-
789
+ served above this line are likely binaries with larger mass
790
+ ratios (q >0.78), or very infrequently, field stars. We
791
+ observe 42 cluster members above this line that showWOCS. RV Measurements in M35 9
792
+ no significant RV variation (and therefore fall into the
793
+ SM class). Many of these are likely long-period binaries
794
+ that are outside of our detection limits, as the hard-soft
795
+ boundary for solar-type stars in M35 is ∼105−106days,
796
+ and we only detect binaries with P/lessorsimilar104days (Geller
797
+ & Mathieu, in preparation).
798
+ 5.2.Spatial Distribution and Mass Segregation
799
+ In Figure 9 we compare the cumulative projected ra-
800
+ dial distributions of the single and binary members of
801
+ M35. We have attempted to reduce the contamination
802
+ from undetected binaries within our single-star sample
803
+ by only including stars with no detectable RV variation
804
+ (SM) that arefainter and bluer than the dashed grayline
805
+ in Figure 8. This conservative cut removes large- q(i.e.,
806
+ high total mass) binaries that have periods longer than
807
+ our detection limit. We have not applied any correc-
808
+ tion for the spatial bias found in our observations (Sec-
809
+ tion 2.4), because this bias will be present in both the
810
+ single- and binary-star samples and should therefore not
811
+ effect this analysis. A Kolmogorov-Smirnov test shows
812
+ no significant difference between these two populations
813
+ with a value of 60%. We therefore conclude that the
814
+ solar-type main-sequence binaries in M35 show no evi-
815
+ dence for central concentration as compared to the single
816
+ stars.
817
+ Mathieu (1983) finds a half-mass relaxation time for
818
+ the cluster of 150 Myr, comparable to the cluster age.
819
+ This study of the radial spatial distributions for proper-
820
+ motion-selected member stars in the 8 .0< V < 14.5
821
+ (∼4.4 - 1.2 M ⊙) range revealed mass segregation only
822
+ amongstarsmoremassivethan 2M ⊙. The degreeofseg-
823
+ regationlessenswith decreasingmass, and is largelynon-
824
+ existent among solar-like stars. McNamara & Sekiguchi
825
+ (1986b) found similar results in their proper-motion se-
826
+ lected sample, which covered stars down to V= 14.5 (∼
827
+ 1.2 M⊙). We only include primary stars with masses of
828
+ 1.6 - 0.8 M ⊙, and therefore most of our binaries have
829
+ total masses that are lower than the higher-mass stars
830
+ that have been shown to be mass segregated. Mathieu
831
+ (1983) found that M35 is fit well with a multi-mass King
832
+ model. In such models the reduction in mass segregation
833
+ for lower-mass systems derives from more severe tidal
834
+ truncation of higher-dispersion velocity distributions in
835
+ a cluster potential dominated by the solar-like stars.
836
+ 5.3.Cluster Radial-Velocity Dispersion
837
+ To determine the true RV dispersion of the cluster, we
838
+ followtheprocedureofGeller et al.(2008). Wefirstlimit
839
+ oursampletoonlyincludeSMstarsthathave vsini≤10
840
+ km s−1. We limit the vsinivalue to ensure that we only
841
+ use the highest precision RV measurements for this anal-
842
+ ysis. These narrow-lined stars have a precision σi= 0.5
843
+ km s−1. We will discuss the effect of undetected binaries
844
+ that likely remain within this sample in Section 5.3.1.
845
+ Usingthissampleof67SMstars,wefirstderivetheob-
846
+ serveddispersion σobsbytakingthestandarddeviationof
847
+ themeanRVsforeachstar,andwefind σobs= 0.86±0.07
848
+ km s−1. This observed dispersion is a function of our
849
+ measurement precision and is also inflated by undetected
850
+ binaries. Therefore, in order to derive the true RV dis-
851
+ persion, we must first account for the precision on theseFig. 9.— Cumulative projected radial spatial distributions
852
+ of the M35 single and binary cluster members. We have ex-
853
+ cluded any stars from the single-star sample that are bright er
854
+ and redder than the dashed grey line in Figure 8, as these
855
+ stars are likely long-period binaries that are outside of ou r de-
856
+ tection limits. We plot the single stars with the black point s
857
+ andtheRVvariables with theopen diamonds. We findnosig-
858
+ nificant evidence for central concentration of the RV-varia ble
859
+ population.
860
+ RV measurements. We derive8the “combined RV dis-
861
+ persion” σcbfrom :
862
+ σ2
863
+ cb=σ2
864
+ obs−1
865
+ nn/summationdisplay
866
+ i=1ξ2
867
+ i. (3)
868
+ Here,n= 67 is the number of stars used in this analysis,
869
+ andξiis the mean errorof the RV for the ith star defined
870
+ as,
871
+ ξi=
872
+ m/summationdisplay
873
+ j=1/parenleftbig
874
+ RVj−RVi/parenrightbig2
875
+ m(m−1)
876
+ 1/2
877
+ (4)
878
+ whereRVjis one of the mnumber of RV measurements
879
+ for a given star i, andRViis the mean RV for that star.
880
+ WenotethatthesecondterminEquation3isverynearly
881
+ equal to σ2
882
+ i/3, as we have 3 RVs for most of the stars in
883
+ this sample, and these narrow-lined stars all have the
884
+ same precision of σi= 0.5 km s−1. Following this proce-
885
+ dure, wederiveacombineddispersionof σcb= 0.81±0.08
886
+ km s−1. The error on this combined dispersion is almost
887
+ entirely due to the statistical error on σobs.
888
+ We find no significant difference in the combined RV
889
+ dispersion of the SM or BM stars. For the BM stars,
890
+ we use the γ-RVs in place of the mean RVs, and substi-
891
+ tute the measurement precision for the standard devia-
892
+ tion portion in Equation 4. There is also no significant
893
+ variation in the combined RV dispersion as a function
894
+ of radius, although due to the small sample sizes our
895
+ binned RV dispersion values have large uncertainties (of
896
+ 0.1 - 0.15 km s−1for bins of 10 arcmin).
897
+ This combined RV dispersion is inflated by undetected
898
+ binaries. In the following section, we quantify this effect
899
+ and apply the correctionto derivethe true RV dispersion
900
+ ofM35, an improvementon the procedureofGeller et al.
901
+ (2008).
902
+ 8The use of Equations 3 and 4 is an improvement over the pro-
903
+ cedure of Geller et al. (2008) adapted from McNamara & Sander s
904
+ (1977). The uncertainty on σcbalso follows McNamara & Sanders
905
+ (1977).10 Geller et al.
906
+ 5.3.1.Contribution from Undetected Binaries
907
+ The combined RV dispersion defined in Equation 3 is
908
+ also described by,
909
+ σcb=σc+β (5)
910
+ whereσcis the true RV dispersion of the cluster and
911
+ βrepresents the contribution from undetected binaries
912
+ within our sample. Therefore, in order to derive the true
913
+ RV dispersion of the cluster we have performed a Monte
914
+ Carlo analysis to determine this contribution from unde-
915
+ tected binaries.
916
+ We first create a set of simulated binaries with or-
917
+ bital parameters distributed according to the Galactic
918
+ field solar-type binaries studied by Duquennoy & Mayor
919
+ (1991). Specifically, these binaries have a log-normal pe-
920
+ riod distribution centered on log( P[days] ) = 4.8 with
921
+ σ= 2.3, and a Gaussian eccentricity distribution cen-
922
+ tered on e= 0.3. For binaries with periods below the
923
+ circularization period of 10.2 days (Meibom & Mathieu
924
+ 2005), we set the eccentricity to zero. We use only solar-
925
+ massprimary stars, and a distribution in secondarymass
926
+ between0.08-1M ⊙describedby aGaussiancenteredon
927
+ M2= 0.23 M⊙withσ= 0.42 M⊙(Kroupa et al. 1990).
928
+ Duquennoy & Mayor (1991) found this Gaussian to be
929
+ the best fit to their solar-type field binaries, and this dis-
930
+ tribution is also consistent with that of Goldberg et al.
931
+ (2003) for their field binaries with primary masses >0.67
932
+ M⊙. The orbital inclinations and phases of the binaries
933
+ are chosen randomly. We then generate three RVs for
934
+ these simulated binaries distributed in time according to
935
+ the actual distribution of our first three observations for
936
+ starsin M35. The majorityofthe SM starsin oursample
937
+ haveonlythreeobservations. TotheseRVs, wealsoadda
938
+ randomerrorgeneratedfromaGaussiancenteredonzero
939
+ and with σ= 0.5 km s−1, the RV precision for narrow-
940
+ lined stars in M35. We also add a random velocity offset
941
+ generated from a Gaussian centered on zero with a stan-
942
+ dard deviation equal to an adopted one-dimensional RV
943
+ dispersion.
944
+ To this sample, we add a number of simulated single
945
+ stars to produce a desired binary frequency. We generate
946
+ three RVs for each single star from a Gaussian described
947
+ byourprecision. To themean RVforeverysinglestarwe
948
+ also add a random offset described by the assumed RV
949
+ dispersion in the same manner as for the simulated bi-
950
+ naries. We then keep only those simulated binaries (and
951
+ single stars) whose first three RVs result in an e/i <4,
952
+ and whose mean RVs are within three standard devia-
953
+ tions of the mean RV from a Gaussian fit to the simu-
954
+ lated RV distribution. This cutoff in standard deviation
955
+ reflects our membership criterion of P RV≥50% for the
956
+ M35 observations. These binaries would be undetected
957
+ within the SM sample.
958
+ We then follow the equations given above to derive β
959
+ fora rangeofbinaryfrequenciesand velocitydispersions,
960
+ σc. In Figure 10, we plot the true cluster RV dispersion
961
+ (σc) as a function of the combined RV dispersion ( σcb)
962
+ for a range of total binary frequencies, where each line
963
+ corresponds to a different binary frequency between 0%
964
+ (far left) to 100% (far right) in steps of 10%. We can
965
+ then use the results shown in Figure 10 to derive the
966
+ true RV dispersion for M35. Furthermore, the results
967
+ shown in this figure are also applicable to RV dispersionFig. 10.— The true cluster RV dispersion ( σc) plotted
968
+ against the combined RV dispersion ( σcb) for a range of total
969
+ binaryfrequencies. Eachlinecorrespondstoadifferentbin ary
970
+ frequency in steps of 10%, with 0% at the far left and 100%
971
+ at the far right. With the vertical gray rectangle, we plot th e
972
+ region included in the combined RV dispersion for M35 of
973
+ 0.81±0.08 km s−1. The diagonal gray region covers the pos-
974
+ sible lines within our extrapolated true binary frequency i n
975
+ M35 of 66% ±8% (derived assuming the M35 binaries follow
976
+ a Duquennoy & Mayor (1991) period distribution). Finally,
977
+ we plot the resulting true RV dispersion in M35 of 0.65 ±
978
+ 0.10 km s−1with the black point at the intersection of these
979
+ two shaded regions.
980
+ analyses for other star clusters, provided that the binary
981
+ population is consistent with the Duquennoy & Mayor
982
+ (1991) field binaries.
983
+ 5.3.2.True Radial-Velocity Dispersion
984
+ To date, we have detected 55 binaries in M35 out
985
+ of 360 cluster members. If we assume a similar com-
986
+ pleteness as in Geller & Mathieu (in preparation), for
987
+ NGC 188, then we can assume that we have detected
988
+ 63% of the binaries with periods less than 104days
989
+ (and a negligible fraction of binaries with longer peri-
990
+ ods). This correction results in a binary frequency of
991
+ 24%±3% forP <104days. This binary frequency
992
+ is consistent with that of solar-type stars in the Galac-
993
+ tic field Duquennoy & Mayor (1991) out to the same
994
+ period limit. If we assume the M35 binaries follow
995
+ a Duquennoy & Mayor (1991) period distribution, then
996
+ our binary frequency for P <104days implies a total bi-
997
+ nary frequency of 66% ±8%, with the inclusion of wider
998
+ binaries currently beyond our detection limits. We then
999
+ take this value for the total binary frequency and correct
1000
+ our combined RV dispersion for undetected binaries.
1001
+ In the filled gray areas in Figure 10 we show the re-
1002
+ gions defined by our M35 combined RV dispersion and
1003
+ the total binary frequency. At the intersection, we plot
1004
+ the derivedtrue RV dispersionin M35 of σc= 0.65±0.10
1005
+ km s−1. Using a flat distribution in secondary mass (and
1006
+ mass ratio), as has been suggested by some studies (e.g.,
1007
+ Mazeh et al. 1992, 2003), has a negligible effect on the
1008
+ derived true RV dispersion. This true RV dispersion is
1009
+ consistent with the projected velocity dispersion of 1.0
1010
+ ±0.15 km s−1, derived by Leonard & Merritt (1989) us-
1011
+ ingtheproper-motiondatafromMcNamara & Sekiguchi
1012
+ (1986a).
1013
+ 6.SUMMARY
1014
+ This is the first paper in a series studying the dynam-
1015
+ ical state of the young ( ∼150 Myr) open cluster M35WOCS. RV Measurements in M35 11
1016
+ (NGC 2168). In this first paper, we present our RV ob-
1017
+ servations and provide initial results from this survey.
1018
+ Our stellar sample extends to 30 arcminutes in radius
1019
+ from the cluster center (7 pc in projection at a distance
1020
+ of 805 pc or ∼4 core radii), and we have selected a region
1021
+ from aV, (B−V) CMD (Figure 1) which covers a mass
1022
+ range of 1.6 - 0.8 M ⊙. We have used the WIYN 3.5m
1023
+ telescope with the Hydra MOS to obtain 5201 spectra of
1024
+ 1144 stars within this stellar sample. From these spec-
1025
+ tra, we derive RV measurements with a precision of 0.5
1026
+ km s−1for narrow-lined stars. The vast majority of the
1027
+ observed stars have multiple measurements, allowing de-
1028
+ termination of cluster membership and identification of
1029
+ spectroscopic binary stars. We detect 360 cluster mem-
1030
+ bers, 55 of which show significant variability in their RV
1031
+ measurements. Binary orbital solutions have been ob-
1032
+ tained for 39 of these RV variables, which we will present
1033
+ in detail in the next paper in this series. Observations
1034
+ of the rest of the RV variables and the remainder of our
1035
+ stellar sample are ongoing. Table 3 provides the first RV
1036
+ membership database for M35 and extends ∼1.5 magni-
1037
+ tudes deeper than any previous membership catalogue.
1038
+ Using the RV cluster members, we study the spa-
1039
+ tial distribution and velocity dispersion of the single
1040
+ and binary stars. We find their spatial distributions
1041
+ to be indistinguishable. This lack of central concentra-
1042
+ tion for the binaries is consistent with earlier observa-
1043
+ tional studies of stars in M35 as well as with a fully re-
1044
+ laxed dynamical model for the cluster (Mathieu 1983;
1045
+ McNamara & Sekiguchi 1986b). In these studies, mass
1046
+ segregationisseeninhigher-massstars,butdiminishesto
1047
+ being undetectable for stars in our observed mass range.After correcting for measurement precision, but not for
1048
+ binaries, we place an upper limit on the RV dispersion of
1049
+ the cluster of 0 .81±0.08 km s−1. When we also correct
1050
+ for undetected binaries, we derive a true RV dispersion
1051
+ of 0.65±0.10 km s−1.
1052
+ The WOCS group will continue our survey of M35 in
1053
+ ordertoderiveRVmembershipsforallstarsin ourstellar
1054
+ sample and obtain orbital solutions for all binaries with
1055
+ periods less than a few thousand days, as well as some
1056
+ with longer periods. In future papers, we will study the
1057
+ binary population of M35 in detail, providing all orbital
1058
+ solutions and analyzing the binary frequency and dis-
1059
+ tributions of orbital parameters. These data will form
1060
+ essential constraints on the hitherto poorly known initial
1061
+ binary populations used in sophisticated N-body models
1062
+ of open clusters.
1063
+ The authors would like to express their gratitude to
1064
+ the staff of the WIYN Observatory for their skillful and
1065
+ dedicated work that have allowed us to obtain these ex-
1066
+ cellent spectra. We thank Ata Sarajedini and Ted von
1067
+ Hippel for the acquisition of the Schmidt images, Vera
1068
+ Platais for work on the astrometry and photometry as
1069
+ well as John Bjorkman for early photometry work. We
1070
+ also thank the many undergraduate and graduate stu-
1071
+ dents who have contributed late nights to obtain the
1072
+ spectra for this project. This work was supported by
1073
+ NSF grant AST 0406615and the Wisconsin Space Grant
1074
+ Consortium.
1075
+ Facilities: WIYN 3.5m
1076
+ REFERENCES
1077
+ Barrado y Navascu´ es, D., Deliyannis, C. P., & Stauffer, J. R. 2001,
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+ ApJ, 549, 452
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+ Carraro, G., Girardi, L., & Marigo, P. 2002, MNRAS, 332, 705
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+ Cudworth, K. M. 1971, AJ, 76, 475
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+ Dabrowski, J. P., & Beardsley, W. R. 1977, PASP, 89, 225
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+ Duquennoy, A., & Mayor, M. 1991, A&A, 248, 485
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+ Ebbighausen, E. G. 1942, AJ, 50, 1
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+ Geller, A. M., Mathieu, R. D., Harris, H. C., & McClure, R. D.
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+ Marigo, P., Girardi, L., Bressan, A., Groenewegen, M. A. T., Silva,
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+ Mathieu, R. D. 1983, Ph.D. thesis (University of California ,
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+ Mathieu, R. D. 2000, in ASPC, Vol. 198, ”Stellar Clusters
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+ and Associations: Convection, Rotation, and Dynamos”, ed.
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+ R. Pallavicini, G. Micela, & S. Sciortino, 517
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+ ApJ, 599, 1344McNamara, B., & Sekiguchi, K. 1986a, AJ, 91, 557
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+ McNamara, B. J., & Sanders, W. L. 1977, A&A, 54, 569
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+ Meibom, S., & Mathieu, R. D. 2005, ApJ, 620, 970
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+ Meibom, S., Mathieu, R. D., & Stassun, K. G. 2006, ApJ, 653, 62 1
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+ Montgomery, K. A., Marschall, L. A., & Janes, K. A. 1993, AJ,
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+ Platais, I., Melo, C., Mermilliod, J.-C., Kozhurina-Plata is, V.,
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+ Fulbright, J. P., M´ endez, R. A., Altmann, M., & Sperauskas,
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+ J. 2007, A&A, 461, 509
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+ Reimers, D., & Koester, D. 1988, A&A, 202, 77
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+ Rhode, K. L., Herbst, W., & Mathieu, R. D. 2001, AJ, 122, 3258
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+ Scott, J. E., Friel, E. D., & Janes, K. A. 1995, AJ, 109, 1706
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+ Soderblom, D. R., Stauffer, J. R., Hudon, J. D., & Jones, B. F.
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+ Sung, H., & Bessell, M. S. 1999, MNRAS, 306, 361
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+ Vasilevskis, S., Klemola, A., & Preston, G. 1958, AJ, 63, 387
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+ von Hippel, T., Steinhauer, A., Sarajedini, A., & Deliyanni s, C. P.
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+ Williams, K. A., Bolte, M., & Koester, D. 2004, ApJ, 615, L49
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+ Williams, K. A., Bolte, M., & Koester, D. 2009, ApJ, 693, 355
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+ Williams,K. A., Liebert, J., Bolte, M., & Hanson, R. B. 2006, ApJ,
1135
+ 643, L12712 Geller et al.TABLE 3
1136
+ Radial-Velocity Data Table
1137
+ IDWIDMIDMcIDCRA DEC V (B−V)S N RV RV eσivsini(vsini)ePRVPPM1PPM2e/i Class Comment
1138
+ 96041 410 ··· ··· 6:10:28.65 24:11:52.0 16.416 0.960 1 4 51.89 1.55 0.52 11.8 1 .1 ··· ·· · · ·· 5.93 BLN ···
1139
+ 36042 209 ··· ··· 6:10:34.30 24:14:07.8 14.835 0.859 1 3 -8.74 0.55 0.64 21.6 0 .8 96 ·· · · ·· 1.49 SM ···
1140
+ 36045 209 ··· ··· 6:10:43.69 24:16:08.9 14.497 0.824 1 17 -9.58 0.19 0.50 10.3 0.2 91 ·· · · ·· 77.46 BM SB1
1141
+ 138057 366 ··· ··· 6:10:50.20 24:04:50.7 16.368 1.165 1 1 -25.13 0.50 0.50 ··· ··· ··· ·· · · ·· · ·· U ···
1142
+ 64052 312 ··· ··· 6:10:43.70 24:07:00.8 15.884 1.023 1 4 -8.85 0.64 0.55 14.2 4 .2 96 ·· · · ·· 2.34 SM ···
1143
+ 15036 180 ··· 731 6:10:15.70 24:11:31.7 13.450 0.690 2 1 57.98 0.50 0.50 ··· ··· ··· ·· · 0 · ·· U ···
1144
+ 49051 227 ··· ··· 6:10:51.32 24:11:10.6 15.086 0.890 1 4 88.83 0.34 0.50 10.0 ··· 0 ·· · · ·· 1.35 SN ···
1145
+ 29047 87 ··· ··· 6:10:44.59 24:13:44.3 14.948 0.895 1 4 56.95 0.28 0.50 10.0 ··· 0 ·· · · ·· 1.10 SN ···
1146
+ 40032 ··· ··· ··· 6:10:11.15 24:14:01.5 15.280 0.820 2 3 -7.72 0.32 0.55 14.3 0 .8 96 ·· · · ·· 1.02 SM ···
1147
+ 20037 193 ··· 758 6:10:22.30 24:14:39.3 14.270 0.650 2 1 3.88 0.50 0.50 ··· ··· ··· ·· · 7 · ·· U ···
1148
+ The contents of each column are defined in Section 4.
1001.0034.txt ADDED
@@ -0,0 +1,594 @@
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
1
+ arXiv:1001.0034v1 [math.NT] 4 Jan 2010NEW IDENTITIES INVOLVING q-EULER
2
+ POLYNOMIALS OF HIGHER ORDER
3
+ T. Kim AND Y. H. Kim
4
+ Abstract. In this paper, we present new generating functions which are relat ed to
5
+ q-Euler numbers and polynomials of higher order. From these genera ting functions, we
6
+ give new identities involving q-Euler numbers and polynomials of higher order.
7
+ §1. Introduction/ Preliminaries
8
+ LetCbe the complex number field. We assume that q∈Cwith|q|<1 and
9
+ theq-number is defined by [ x]q=1−qx
10
+ 1−qin this paper. The q-factorial is given by
11
+ [n]q! = [n]q[n−1]q···[2]q[1]qand theq-binomial formulae are known that
12
+ (x:q)n=n/productdisplay
13
+ i=1(1−xqi−1) =n/summationdisplay
14
+ i=0/parenleftbiggn
15
+ i/parenrightbigg
16
+ qq(i
17
+ 2)(−x)i,(see [3, 14, 15]) ,
18
+ and
19
+ 1
20
+ (x:q)n=n/productdisplay
21
+ i=1/parenleftbigg1
22
+ 1−xqi−1/parenrightbigg
23
+ =∞/summationdisplay
24
+ i=0/parenleftbiggn+i−1
25
+ i/parenrightbigg
26
+ qxi,(see [3, 5, 14, 15]) ,
27
+ where/parenleftbign
28
+ i/parenrightbig
29
+ q=[n]q!
30
+ [n−i]q![i]q!=[n]q[n−1]q···[n−i+1]q
31
+ [i]q!.
32
+ The Euler polynomials are defined by2
33
+ et+1ext=/summationtext∞
34
+ n=0En(x)tn
35
+ n!, for|t|< π. In the
36
+ special case x= 0,En(=En(0)) are called the n-th Euler numbers. In this paper, we
37
+ consider the q-extensions of Euler numbers and polynomials of higher orde r. Barnes’
38
+ multiple Bernoulli polynomials are also defined by
39
+ (1)
40
+ tr
41
+ /producttextr
42
+ j=1(eajt−1)ext=∞/summationdisplay
43
+ n=0Bn(x,r|a1,···,ar)tn
44
+ n!,where|t|<max
45
+ 1≤i≤r2π
46
+ |ai|, (see [1, 14]).
47
+ Key words and phrases. : multiple q-zeta function, q-Euler numbers and polynomials, higher
48
+ order q-Euler numbers, Laurent series, Cauchy integral.
49
+ 2000 AMS Subject Classification: 11B68, 11S80
50
+ The present Research has been conducted by the research Grant of Kw angwoon University in 2010
51
+ Typeset by AMS-TEX
52
+ 1In one of an impressive series of papers (see [1, 6, 14]), Barn es developed the so-called
53
+ multiple zeta and multiple gamma function. Let a1,···,aNbe positive parameters.
54
+ Then Barnes’ multiple zeta function is defined by
55
+ ζN(s,w|a1,···,aN) =/summationdisplay
56
+ m1,···,mN=0(w+m1a1+···+mNaN)−s,(see [1]),
57
+ whereℜ(s)> N,ℜ(w)>0. Form∈Z+, we have
58
+ ζN(−m,w|a1,···,aN) =(−1)mm!
59
+ (N+m)!BN+m(w,N|a1,···,aN).
60
+ In this paper, we consider Barnes’ type multiple q-Euler numbers and polynomials.
61
+ The purpose of this paper is to present new generating functi ons which are related
62
+ toq-Euler numbers and polynomials of higher order. From the Mel lin transformation
63
+ of these generating functions, we derive the q-extensions o f Barnes’ type multiple
64
+ zeta functions, which interpolate the q-Euler polynomials of higher order at negative
65
+ integer. Finally, we give new identities involving q-Euler numbers and polynomials of
66
+ higher order.
67
+ §2.q-Euler numbers and polynomials of higher order
68
+ In this section, we assume that q∈Cwith|q|<1. Letx,a1,... ,a rbe complex
69
+ numbers with positive real parts. Barnes’ type multiple Eul er polynomialsare defined
70
+ by
71
+ (2)2r
72
+ /producttextr
73
+ j=1(eajt+1)ext=∞/summationdisplay
74
+ n=0E(r)
75
+ n(x|a1,... ,a r)tn
76
+ n!,for|t|<max
77
+ 1≤i≤rπ
78
+ |wi|,(see [6]),
79
+ andE(r)
80
+ n(a1,... ,a r)(=E(r)
81
+ n(0|a1,... ,a r)) are called the n-th Barnes’ type multiple
82
+ Euler numbers. First, we consider the q-extension of Euler polynomials. The q-Euler
83
+ polynomials are defined by
84
+ (3)Fq(t,x) =∞/summationdisplay
85
+ n=0En,q(x)tn
86
+ n!= [2]q∞/summationdisplay
87
+ m=0(−q)me[m+x]qt,(see [8, 11, 13, 14, 15]) .
88
+ From (3), we have
89
+ En,q(x) =[2]q
90
+ (1−q)nn/summationdisplay
91
+ l=0/parenleftbiggn
92
+ l/parenrightbigg(−1)lqlx
93
+ (1+ql+1).
94
+ In the special case x= 0,En,q(=En,q(0)) are called the n-thq-Euler numbers. From
95
+ (3), we can easily derive the following relation.
96
+ E0,q= 1,andq(qE+1)n+En,q= 0 ifn≥1,(see [8, 16, 17]) ,
97
+ 2where we use the standard convention about replacing EkbyEk,q.It is easy to show
98
+ that
99
+ lim
100
+ q→1Fq(t,x) =2
101
+ et+1ext=∞/summationdisplay
102
+ n=0En(x)tn
103
+ n!,(see [2, 3, 19-23]) ,
104
+ whereEn(x) are the n-th Euler polynomials. For r∈N, the Euler polynomials of
105
+ orderris defined by
106
+ (4)/parenleftbigg2
107
+ et+1/parenrightbiggr
108
+ ext=∞/summationdisplay
109
+ n=0E(r)
110
+ n(x)tn
111
+ n!,for|t|< π.
112
+ Now we consider the q-extension of (4).
113
+ (5)F(r)
114
+ q(t,x) = [2]r
115
+ q∞/summationdisplay
116
+ m1,...,m r=0(−q)m1+···+mre[m1+···+mr+x]qt=∞/summationdisplay
117
+ n=0E(r)
118
+ n,q(x)tn
119
+ n!,
120
+ whereE(r)
121
+ n,q(x) are called the n-thq-Euler polynomials of order r(see [10-15]). From
122
+ (5), we can derive
123
+ (6) E(r)
124
+ n,q(x) =[2]r
125
+ q
126
+ (1−q)nn/summationdisplay
127
+ l=0/parenleftbiggn
128
+ l/parenrightbigg(−1)lqlx
129
+ (1+ql+1)r.
130
+ By (5) and (6), we see that
131
+ (7) F(r)
132
+ q(t,x) = [2]r
133
+ q∞/summationdisplay
134
+ m=0/parenleftbiggm+r−1
135
+ m/parenrightbigg
136
+ (−q)me[m+x]qt.
137
+ Thus, we note that lim q→1F(r)
138
+ q(t,x) =/parenleftBig
139
+ 2
140
+ et+1/parenrightBigr
141
+ ext=/summationtext∞
142
+ n=0E(r)
143
+ n(x)tn
144
+ n!.In the special
145
+ casex= 0,E(r)
146
+ n,q(=E(r)
147
+ n,q(0)) are called the n-thq-Euler numbers of order r. By (5),
148
+ (6) and (7), we obtain the following proposition.
149
+ Proposition 1. Forr∈N, let
150
+ F(r)
151
+ q(t,x) = [2]r
152
+ q/summationdisplay
153
+ m1,...,m r=0(−q)m1+···+mre[m1+···+mr+x]qt=∞/summationdisplay
154
+ n=0E(r)
155
+ n,q(x)tn
156
+ n!.
157
+ Then we have
158
+ E(r)
159
+ n,q(x) =[2]r
160
+ q
161
+ (1−q)nn/summationdisplay
162
+ l=0/parenleftbiggn
163
+ l/parenrightbigg(−1)lqlx
164
+ (1+ql+1)r= [2]r
165
+ q∞/summationdisplay
166
+ m=0/parenleftbiggm+r−1
167
+ m/parenrightbigg
168
+ (−q)m[m+x]n
169
+ q.
170
+ 3From the Mellin transformation of F(r)
171
+ q(t,x), we can derive the following equation.
172
+ 1
173
+ Γ(s)/integraldisplay∞
174
+ 0F(r)
175
+ q(−t,x)ts−1dt= [2]r
176
+ q∞/summationdisplay
177
+ m1,...,m r=0(−q)m1+···+mr
178
+ [m1+···+mr+x]sq
179
+ = [2]r
180
+ q∞/summationdisplay
181
+ m=0/parenleftbiggm+r−1
182
+ m/parenrightbigg
183
+ (−q)m1
184
+ [m+x]sq, (8)
185
+ wheres∈C,x/negationslash= 0,−1,−2,.... By (8), we can define the multiple q-zeta function
186
+ related to q-Euler polynomials.
187
+ Definition 2. Fors∈C,x∈Rwithx/negationslash= 0,−1,−2,..., we define the multiple q-zeta
188
+ function related to q-Euler polynomials as
189
+ ζq,r(s,x) = [2]r
190
+ q∞/summationdisplay
191
+ m1,...,m r=0(−q)m1+···+mr
192
+ [m1+···+mr+x]sq.
193
+ Note that ζq,r(s,x) is a meromorphic function in whole complex s-plane. From (8),
194
+ we also note that
195
+ ζq,r(s,x) = [2]r
196
+ q∞/summationdisplay
197
+ m=0/parenleftbiggm+r−1
198
+ m/parenrightbigg
199
+ (−q)m1
200
+ [m+x]sq.
201
+ By Laurent series and the Cauchy residue theorem in (5) and (8 ), we see that
202
+ ζq(−n,x) =E(n)
203
+ n,q(x),forn∈Z+.
204
+ Therefore, we obtain the following theorem.
205
+ Theorem 3. Forr∈N,n∈Z+, andx∈Rwithx/negationslash= 0,−1,−2,..., we have
206
+ ζq(−n,x) =E(r)
207
+ n,q(x).
208
+ Letχbe the Dirichlet’s character with conductor f∈Nwithf≡1 (mod 2). Then
209
+ the generalized q-Euler polynomial attached to χare considered by
210
+ Fq,χ(x) =∞/summationdisplay
211
+ n=0En,χ,q(x)tn
212
+ n!= [2]q∞/summationdisplay
213
+ m=0(−q)mχ(m)e[m+x]qt.
214
+ From (3) and (9), we have
215
+ En,χ,q(x) =[2]q
216
+ [2]qff−1/summationdisplay
217
+ a=0(−q)aχ(a)En,qf(x+a
218
+ f).
219
+ 4In the special case x= 0,En,χ,q=En,χ,q(0) are called the n-th generated q-Euler
220
+ number attached to χ.
221
+ It is known that the generalized Euler polynomials of order rare defined by
222
+ (10) (2/summationtextf−1
223
+ a=0(−1)aχ(a)eat
224
+ eft+1)rext=∞/summationdisplay
225
+ n=0E(r)
226
+ n,χ(x)tn
227
+ n!,
228
+ for|t|<π
229
+ f.
230
+ We consider the q-extension of (10). The generalized q-Euler polynomials of order
231
+ rattached to χare defined by
232
+ F(r)
233
+ q,χ(t,x) = [2]r
234
+ q∞/summationdisplay
235
+ m1,...,m r=0(−q)m1+···+mr(r/productdisplay
236
+ i=1χ(mi))e[m1+···+mr+x]qt
237
+ =∞/summationdisplay
238
+ n=0E(r)
239
+ n,χ,q(x)tn
240
+ n!,(see [14, 15]) . (11)
241
+ Note that
242
+ lim
243
+ q→1F(r)
244
+ q,χ(t,x) = (2/summationtextf−1
245
+ a=0(−1)aχ(a)eat
246
+ eft+1)r.
247
+ By (11), we easily see that
248
+ E(r)
249
+ n,χ,q(x) =[2]r
250
+ q
251
+ (1−q)nn/summationdisplay
252
+ l=0/parenleftbiggn
253
+ l/parenrightbigg
254
+ (−qx)lf−1/summationdisplay
255
+ a1,...,ar=0(r/productdisplay
256
+ j=1χ(aj))(−ql+1)/summationtextr
257
+ i=1ai
258
+ (1+q(l+1)f)r
259
+ = [2]r
260
+ q∞/summationdisplay
261
+ m1,...,m r=0(−q)m1+···+mr(r/productdisplay
262
+ i=1χ(mi))[m1+···+mr+x]n
263
+ q.
264
+ Fors∈C,x∈Rwithx/negationslash= 0,−1,−2,..., we have
265
+ 1
266
+ Γ(s)/integraldisplay∞
267
+ 0F(r)
268
+ q,χ(−t,x)ts−1dt
269
+ = [2]r
270
+ q∞/summationdisplay
271
+ m1,...,m r=0(−q)m1+···+mr(/producttextr
272
+ i=1χ(mi))
273
+ [m1+···+mr+x]sq,(see [15]) . (12)
274
+ From (12), we can consider the Dirichlet’s type multiple q-l-function as follows :
275
+ Definition 4. Fors∈C,x∈Rwithx/negationslash= 0,−1,−2,..., we define the Dirichlet’s
276
+ type multiple q-l-function as
277
+ lq(s,x|χ) = [2]r
278
+ q∞/summationdisplay
279
+ m1,...,m r=0(−q)m1+···+mr(/producttextr
280
+ i=1χ(mi))
281
+ [m1+···+mr+x]sq,(see [15]) .
282
+ By Laurent series and the Cauchy residue theorem in (11) and ( 12), we obtain the
283
+ following theorem.
284
+ 5Theorem 5. Forn∈Z+, we have
285
+ lq(−n,x|χ) =E(r)
286
+ n,χ,q(x).
287
+ Forh∈Zandr∈N, we consider the extended r-pleq-Euler polynomials.
288
+ F(h,r)
289
+ q(t,x) = [2]r
290
+ q∞/summationdisplay
291
+ m1,...,m r=0q/summationtextr
292
+ j=1(h−j+1)mj(−1)/summationtextr
293
+ j=1mje[m1+···+mr+x]qt
294
+ =∞/summationdisplay
295
+ n=0E(h,r)
296
+ n,q(x)tn
297
+ n!. (13)
298
+ Note that
299
+ lim
300
+ q→1F(h,r)
301
+ q(t,x) = (2
302
+ et+1)rext=∞/summationdisplay
303
+ n=0E(r)
304
+ n(x)tn
305
+ n!.
306
+ From (13), we note that
307
+ E(h,r)
308
+ n,q(x) =[2]r
309
+ q
310
+ (1−q)nn/summationdisplay
311
+ l=0/parenleftbiggn
312
+ l/parenrightbigg(−qx)l
313
+ (−qh−r+l+1:q)r
314
+ = [2]r
315
+ q∞/summationdisplay
316
+ m=0/parenleftbiggm+r−1
317
+ m/parenrightbigg
318
+ q(−qh−r+1)m[m+x]n
319
+ q. (14)
320
+ By (14), we easily see that
321
+ (15)F(h,r)
322
+ q(t,x) = [2]r
323
+ q∞/summationdisplay
324
+ m=0/parenleftbiggm+r−1
325
+ m/parenrightbigg
326
+ q(−qh−r+1)me[m+x]qt,(see [11, 13, 14]) .
327
+ Using the Mellin transform for F(h,r)
328
+ q(t,x), we have
329
+ 1
330
+ Γ(s)/integraldisplay∞
331
+ 0F(r)
332
+ q(−t,x)ts−1dt
333
+ = [2]r
334
+ q∞/summationdisplay
335
+ m1,...,m r=0(−1)m1+···+mrq/summationtextr
336
+ j=1(h−j+1)mj
337
+ [m1+···+mr+x]sq,(see [13, 14, 15]) ,(16)
338
+ fors∈C,x∈Rwithx/negationslash= 0,−1,−2,.... Now we can define the extended q-zeta
339
+ function associated with E(h,r)
340
+ n,q(x).
341
+ 6Definition 6. Fors∈C,x∈Rwithx/negationslash= 0,−1,−2,..., we define the (h, q)-zeta
342
+ function as
343
+ ζ(h)
344
+ q,r(s,x) = [2]r
345
+ q∞/summationdisplay
346
+ m1,...,m r=0(−1)m1+···+mrq/summationtextr
347
+ j=1(h−j+1)mj
348
+ [m1+···+mr+x]sq.
349
+ Notethat ζ(h)
350
+ q,r(s,x)isalsoa meromorphic function inwholecomplex s-plane. From
351
+ (16) and (15), we note that
352
+ (17) ζ(h)
353
+ q,r(s,x) = [2]r
354
+ q∞/summationdisplay
355
+ m=0/parenleftbiggm+r−1
356
+ m/parenrightbigg
357
+ q(−qh−j+1)m1
358
+ [m+x]sq.
359
+ Using the Cauchy residue theorem and Laurent series in (16), we obtain the following
360
+ theorem.
361
+ Theorem 7. Forn∈Z+, we have
362
+ ζ(h)
363
+ q,r(−n,x) =E(h,r)
364
+ n,q(x).
365
+ We consider the extended r-ple generalized q-Euler polynomials as follows :
366
+ F(h,r)
367
+ q,χ(t,x)
368
+ = [2]r
369
+ q∞/summationdisplay
370
+ m1,...,m r=0q/summationtextr
371
+ j=1(h−j+1)mj(−1)/summationtextr
372
+ j=1mj(r/productdisplay
373
+ j=1χ(mj))e[m1+···+mr+x]qt(18)
374
+ =∞/summationdisplay
375
+ n=0E(h,r)
376
+ n,χ,q(x)tn
377
+ n!.
378
+ By (18), we see that
379
+ E(h,r)
380
+ n,χ,q(x) =[2]r
381
+ q
382
+ (1−q)nf−1/summationdisplay
383
+ a1,...,ar=0(−1)/summationtextr
384
+ j=1aj(r/productdisplay
385
+ j=1χ(aj))n/summationdisplay
386
+ l=0/parenleftbiggn
387
+ l/parenrightbigg(−1)lqlxq(h−j+l+1)aj
388
+ (−q(h−r+l+1)f:qf)r
389
+ =[2]r
390
+ q
391
+ [2]r
392
+ qf[f]n
393
+ qf−1/summationdisplay
394
+ a1,...,ar=0(−1)/summationtextr
395
+ j=1aj(r/productdisplay
396
+ j=1χ(aj))q/summationtextr
397
+ j=1(h−j+1)ajζ(h)
398
+ qf,r(−n,x+/summationtextr
399
+ j=1aj
400
+ f).(19)
401
+ Therefore, we obtain the following theorem.
402
+ 7Theorem 8. Forn∈Z+, we have
403
+ E(h,r)
404
+ n,χ,q(x)
405
+ =[2]r
406
+ q
407
+ [2]r
408
+ qf[f]n
409
+ qf−1/summationdisplay
410
+ a1,...,ar=0(−1)/summationtextr
411
+ j=1aj(r/productdisplay
412
+ j=1χ(aj))q/summationtextr
413
+ j=1(h−j+1)ajζ(h)
414
+ qf,r(−n,x+/summationtextr
415
+ j=1aj
416
+ f).
417
+ From (18), we note that
418
+ 1
419
+ Γ(s)/integraldisplay∞
420
+ 0F(h,r)
421
+ q,χ(−t,x)ts−1dt
422
+ = [2]r
423
+ q∞/summationdisplay
424
+ m1,...,m r=0q/summationtextr
425
+ j=1(h−j+1)mj(/producttextr
426
+ j=1χ(mj))(−1)m1+···+mr
427
+ [m1+···+mr+x]sq, (20)
428
+ wheres∈C,x∈Rwithx/negationslash= 0,−1,−2,....
429
+ From (20), we define the Dirichlet’s type multiple ( h,q)-l-function associated with
430
+ the generalized multiple q-Euler polynomials attached to χ.
431
+ Definition 9. Fors∈C,x∈Rwithx/negationslash= 0,−1,−2,..., we define the Dirichlet’s
432
+ type multiple q-l-function as follows :
433
+ l(h)
434
+ q(s,x|χ) = [2]r
435
+ q∞/summationdisplay
436
+ m1,...,m r=0q/summationtextr
437
+ j=1(h−j+1)mj(/producttextr
438
+ i=1χ(mi))(−1)m1+···+mr
439
+ [m1+···+mr+x]sq.
440
+ Note that l(h)
441
+ q(s,x|χ) is a meromorphic function in whole complex plane. It is easy
442
+ to show that
443
+ l(h)
444
+ q(s,x|χ)
445
+ =[2]r
446
+ q
447
+ [2]r
448
+ qf1
449
+ [f]sqf−1/summationdisplay
450
+ a1,...,ar=0(−1)/summationtextr
451
+ j=1aj(r/productdisplay
452
+ j=1χ(aj))q/summationtextr
453
+ j=1(h−j+1)ajζ(h)
454
+ qf,r(s,x+/summationtextr
455
+ j=1aj
456
+ f).
457
+ By (19) and (20), we obtain the following theorem.
458
+ Theorem 10. Forn∈Z+, we have
459
+ l(h)
460
+ q(−n,x|χ) =E(h,r)
461
+ n,χ,q(x).
462
+ Finally, we give the q-extension of Barnes’ type multiple Euler polynomials in (2 ).
463
+ Forx,a1,... ,a r∈Cwith positive real part, let us define the Barnes’ type mutipl e
464
+ 8q-Euler polynomials in Cas follows :
465
+ F(r)
466
+ q(t,x|a1,... ,a r;b1,... ,b r)
467
+ = [2]r
468
+ q∞/summationdisplay
469
+ m1,...,m r=0(−1)m1+···+mrq(b1+1)m1+···+(br+1)mre[a1m1+···+armr+x]t(21)
470
+ =∞/summationdisplay
471
+ n=0E(r)
472
+ n,q(x|a1,... ,a r;b1,... ,b r)tn
473
+ n!,
474
+ whereb1,... ,b r∈Z. By (21), we see that
475
+ E(r)
476
+ n,q(x|a1,... ,a r;b1,... ,b r)
477
+ =[2]r
478
+ q
479
+ (1−q)nn/summationdisplay
480
+ l=0/parenleftbiggn
481
+ l/parenrightbigg(−1)lqlx
482
+ (1+qla1+b1+1)···(1+qlar+br+1)
483
+ = [2]r
484
+ q∞/summationdisplay
485
+ m1,...,m r=0(−1)m1+···+mrq(b1+1)m1+···+(br+1)mr[a1m1+···+armr+x]n
486
+ q.
487
+ From (21), we note that
488
+ 1
489
+ Γ(s)/integraldisplay∞
490
+ 0F(r)
491
+ q(−t,x|a1,... ,a r;b1,... ,b r)ts−1dt
492
+ = [2]r
493
+ q∞/summationdisplay
494
+ m1,...,m r=0(−q)m1+···+mrqb1m1+···+brmr
495
+ [a1m1+···+armr+x]sq. (22)
496
+ By (22), we define the Barnes’ type multiple q-zeta function as follows :
497
+ ζq,r(s,x|a1,... ,a r;b1,... ,b r)
498
+ = [2]r
499
+ q∞/summationdisplay
500
+ m1,...,m r=0(−q)m1+···+mrqb1m1+···+brmr
501
+ [a1m1+···+armr+x]sq,
502
+ wheres∈C,x∈Rwithx/negationslash= 0,−1,−2,.... By (21), (22) and (23), we obtain the
503
+ following theorem.
504
+ Theorem 11. Forn∈Z+, we have
505
+ ζq,r(s,x|a1,... ,a r;b1,... ,b r) =E(r)
506
+ n,q(x|a1,... ,a r;b1,... ,b r).
507
+ Letχbe the Dirichlet’s character with conductor f∈Nwithf≡1 (mod 2). Then
508
+ the generalized Barnes’ type multiple q-Euler polynomials attached to χare defined
509
+ 9by
510
+ F(r)
511
+ q,χ(t,x|a1,... ,a r;b1,... ,b r)
512
+ = [2]r
513
+ q∞/summationdisplay
514
+ m1,...,m r=0(−q)m1+···+mrqb1m1+···+brmr(r/productdisplay
515
+ i=1χ(mi))e[a1m1+···+armr+x]qt(24)
516
+ =∞/summationdisplay
517
+ n=0E(r)
518
+ n,χ,q(x|a1,... ,a r;b1,... ,b r)tn
519
+ n!,
520
+ From (24), we note that
521
+ 1
522
+ Γ(s)/integraldisplay∞
523
+ 0F(r)
524
+ q,χ(−t,x|a1,... ,a r;b1,... ,b r)ts−1dt
525
+ = [2]r
526
+ q∞/summationdisplay
527
+ m1,...,m r=0(−q)m1+···+mrqb1m1+···+brmr(/producttextr
528
+ i=1χ(mi))
529
+ [a1m1+···+armr+x]sq. (25)
530
+ By (25), we can define Barnes’ type multiple q-l-function in C. Fors∈C,x∈Rwith
531
+ x/negationslash= 0,−1,−2,..., let us define the Barnes’ type multiple q-l-function as follows :
532
+ l(r)
533
+ q(s,x|a1,... ,a r;b1,... ,b r)
534
+ = [2]r
535
+ q∞/summationdisplay
536
+ m1,...,m r=0(−q)m1+···+mrqb1m1+···+brmr(/producttextr
537
+ i=1χ(mi))
538
+ [a1m1+···+armr+x]sq. (26)
539
+ Note that l(r)
540
+ q(s,x|a1,... ,a r;b1,... ,b r) is a meromorphic function in whole complex
541
+ s-plane. By (24), (25) and (26), we easily see that
542
+ l(r)
543
+ q(−n,x|a1,... ,a r;b1,... ,b r) =E(r)
544
+ n,χ,q(x|a1,... ,a r;b1,... ,b r)
545
+ forn∈Z+, (see [1-18]).
546
+ References
547
+ [1] E. W. Barnes, On the theory of multiple gamma function , Trans. Camb. Ohilos. Soc. A
548
+ 196(1904), 374-425.
549
+ [2] I. N. Cangul,V. Kurt, H. Ozden, Y. Simsek, On the higher-order w-q-Genocchi numbers ,
550
+ Adv. Stud. Contemp. Math. 19(2009), 39–57.
551
+ [3] N. K.Govil, V. Gupta, Convergence of q-Meyer-Konig-Zeller-Durrmeyer operators , Adv.
552
+ Stud. Contemp. Math. 19(2009), 97–108.
553
+ [4] T. Kim, On aq-analogue of the p-adic log gamma functions and related integrals , J.Number
554
+ Theory76(1999), 320–329.
555
+ [5] T. Kim, q-Volkenborn integration , Russ. J. Math. Phys. 9(2002), 288–299.
556
+ [6] T. Kim, On Euler-Barnes multiple zeta functions , Russ. J. Math. Phys. 10(2003), 261–267.
557
+ 10[7] T. Kim, Analytic continuation of multiple q-zeta functions and their values at negative
558
+ integers, Russ. J. Math. Phys. 11(2004), 71–76.
559
+ [8] T. Kim, The modified q-Euler numbers and polynomials , Adv. Stud. Contemp. Math. 16
560
+ (2008), 161–170.
561
+ [9] T. Kim, Note on the q-Euler numbers of higher order , Adv. Stud. Contemp. Math. 19
562
+ (2009), 25–29.
563
+ [10] T. Kim, Note on Dedekind type DC sums , Adv. Stud. Contemp. Math. 18(2009), 249–260.
564
+ [11] T. Kim, Note on the Euler q-zeta functions , J. Number Theory 129(2009), 1798–1804.
565
+ [12] T. Kim, A note on the generalized q-Euler numbers , Proc. Jangjeon Math. Soc. 12(2009),
566
+ 45–50.
567
+ [13] T. Kim, Some identities on the q-Euler polynomials of higher order a nd q-stirling numbers
568
+ by the fermionic p-adic integral on Zp, Russ. J. Math. Phys. 16(2009), 1061-9208.
569
+ [14] T. Kim, Barnes type multiple q-zeta functions and q-Euler polynomials , arXiv:0912.5119v1.
570
+ [15] T. Kim, Note on multiple q-zeta functions , to be appeared in Russ. J. Math. Phys.,
571
+ arXiv:0912.5477v1.
572
+ [16] T. Kim, On theq-extension of Euler and Genocchi numbers , J. Math. Anal. Appl. 326,
573
+ 1458–1465.
574
+ [17] T. Kim, Onp-adicq-l-functions and sums of powers , J. Math. Anal. Appl. 329, 1472–1481.
575
+ [18] T. Kim, Y. Simsek, Analytic continuation of the multiple Daehee q-l-functions associated
576
+ with Daehee numbers , Russ. J. Math. Phys. 15(2008), 58–65.
577
+ [19] Y. H. Kim, W. Kim, C. S. Ryoo, On the twisted q-Euler zeta function associated with
578
+ twistedq-Euler numbers , Proc. Jangjeon Math. Soc. 12(2009), 93-100.
579
+ [20] H.Ozden, I.N.Cangul, Y.Simsek, Remarks on q-Bernoulli numbers associated with Daehee
580
+ numbers , Adv. Stud. Contemp. Math. 18(2009), 41-48.
581
+ [21] K. Shiratani, S. Yamamoto, On ap-adic interpolation function for the Euler numbers and
582
+ its derivatives , Mem. Fac. Sci., Kyushu University Ser. A 39(1985), 113-125.
583
+ [22] Y. Simsek, Theorems on twisted L-function and twisted Bernoulli numbers , Advan. Stud.
584
+ Contemp. Math. 11(2005), 205–218.
585
+ [23] Z. Zhang, Y. Zhang, Summation formulas of q-series by modified Abel’s lemma , Adv. Stud.
586
+ Contemp. Math. 17(2008), 119–129.
587
+ Taekyun Kim
588
+ Division of General Education-Mathematics, Kwangwoon Uni versity,
589
+ Seoul 139-701, S. Korea e-mail: [email protected]
590
+ Young-Hee Kim
591
+ Division of General Education-Mathematics,
592
+ Kwangwoon University,
593
+ Seoul 139-701, S. Korea e-mail: [email protected]
594
+ 11
1001.0035.txt ADDED
@@ -0,0 +1,934 @@
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
1
+ arXiv:1001.0035v2 [hep-th] 25 Mar 2010Reconstruction of Baxter Q-operator fromSklyanin SOV
2
+ for cyclic representations ofintegrable quantum models
3
+ G. Niccoli
4
+ DESY,Notkestr. 85, 22603 Hamburg, GermanyDESY 09-227
5
+ Abstract
6
+ In [1], the spectrum (eigenvaluesand eigenstates) of a latt ice regularizationsof the Sine-
7
+ Gordon model has been completely characterized in terms of p olynomial solutions with
8
+ certain propertiesof the Baxter equation. This characteri zation for cyclic representations
9
+ hasbeenderivedbytheuseofthe SeparationofVariables(SO V)methodofSklyaninand
10
+ bythedirectconstructionoftheBaxter Q-operatorfamily. Here,wereconstructtheBaxter
11
+ Q-operatorandthesamecharacterizationofthespectrumbyo nlyusingtheSOVmethod.
12
+ This analysis allows us to deduce the main features required for the extension to cyclic
13
+ representationsofotherintegrablequantummodelsofthis kindofspectrumcharacteriza-
14
+ tion.
15
+ Keywords: Integrable Quantum Systems;Separation of Variables;Baxt erQ-operator; PACScode 02.30.IK2
16
+ 1. Introduction
17
+ Theintegrabilityofa quantummodelisbydefinitionrelated to theexistenceofamutuallycommu-
18
+ tativefamily Qofself-adjointoperators Tsuchthat
19
+ (A) [T,T′] = 0,
20
+ (B) [T,U] = 0,
21
+ (C) if [ T,O] = 0,∀T,T′∈ Q,
22
+ ∀T∈ Q,
23
+ ∀T∈ Q,thenO=O(Q),(1.1)
24
+ whereUis the unitary operatordefining the time-evolutionin the mo del; note that the property(C)
25
+ stays for the completeness of the family Q. In the framework of the quantum inverse scattering
26
+ method [2, 3, 4] the Lax operator L(λ)is the mathematical tool which allows to define the transfer
27
+ matrix:
28
+ T(λ) = trC2M(λ),M(λ)≡/parenleftbiggA(λ)B(λ)
29
+ C(λ)D(λ)/parenrightbigg
30
+ ≡LN(λ)...L1(λ),(1.2)
31
+ aoneparameterfamilyofmutualcommutativeself-adjointo perators. Theintegrabilityofthemodel
32
+ follows from T(λ)if the properties (B) and (C) of definition (1.1) can be proven for it. In some
33
+ quantummodeltheintegrabilityisderivedbyprovingtheex istenceofafurtherone-parameterfamily
34
+ ofself-adjointoperatorsthe Q-operatorwhichbydefinitionsatisfiesthefollowingproper ties:
35
+ [Q(λ),Q(µ)] = 0,[T(λ),Q(µ)] = 0,∀λ,µ∈C, (1.3)
36
+ plusthe Baxterequationwith thetransfermatrix:
37
+ T(λ)Q(λ) =a(λ)Q(q−1λ)+d(λ)Q(qλ). (1.4)
38
+ This is in particular the case for those models (like Sine-Go rdon [1]) for which the time-evolution
39
+ operatorUis expressedin terms of Q. A naturalquestionarises: Is the integrablestructure of t hese
40
+ quantummodelscompletelycharacterizedbythetransferma trixT(λ)?
41
+ Note that a standard procedure1to prove the existence of Q(λ)is by a direct construction of an
42
+ operatorsolutionoftheBaxterequation(1.4). Moreover,t hecoefficients a(λ)andd(λ)aswellasthe
43
+ analytic and asymptotics properties of Q(λ)are some model dependent features which are derived
44
+ bythe construction. Let usrecall thatthe generalstrategy [11, 12,13,14, 15] ofthisconstructionis
45
+ to findagaugetransformation2such that the action of each gaugetransformedLax matrixon Q(λ)
46
+ becomesupper-triangular. Thenthe Q-operatorassumesa factorized localformandthe problemof
47
+ its existence in such a form is reduced to the problem of the ex istence of some model dependent
48
+ specialfunction3.
49
+ 1Itis worth recalling that there are also others constructio ns of theQ-operator. An interesting example is presented in the
50
+ series of works [5,6,7] by V.V. Bazhanov, S.L.Lukyanov and A .B.Zamolodchikov on the integrable structure of conformal
51
+ field theories. In [6,7] the Q-operator is obtained asatransfer-matrix byatrace proced ure ofafundamental L-operator with
52
+ q-oscillator representation for the auxiliary space (see al so [8, 9]). This construction can be extended to massive inte grable
53
+ quantum field theories as itwas argued by thesame authors in [ 10].
54
+ 2Itleaves unchanged thetransfer matrix while modifies the mo nodromy matrix M(λ)defined in (1.2) .
55
+ 3Thequantumdilogarithm functions [16,17,18,19,20,21,22,23,24,25]forexample a ppear intheSinh-Gordon model
56
+ [26],in their non-compact form,and in the Sine-Gordon model[1],in their cyclicform.3
57
+ It is worth pointing out that on the one hand the construction of these special functionsfor general
58
+ modelscanrepresenta concretetechnicalproblem4andthat ontheotherhandtheexistenceofsuch
59
+ functionsis onlya sufficientcriterionforthe existence of Q(λ). It is then a relevantquestionif it is
60
+ possibletobypassthiskindofconstructionprovidingadif ferentproofofthe existenceof Q(λ).
61
+ Given an integrable quantum model the first fundamental task to solve is the exact solution of its
62
+ spectral problem , i.e. the determination of the eigenvalues and the simultan eous eigenstates of the
63
+ operator family Q, defined in (1.1). There are several methods to analyze this s pectral problem as
64
+ thecoordinate Bethe ansatz [27, 28, 29], the TQmethod [28], the algebraic Bethe ansatz (ABA)
65
+ [2, 3, 4], the analyticBethe ansatz [30] and the separation of variables (SOV) meth od of Sklyanin
66
+ [31, 32, 33]; this last one seems to be more promising. Indeed , on the one hand it resolves the
67
+ problems related to the reduced applicability of other meth ods (like ABA) and on the other hand
68
+ it directly implies the completeness of the characterizati on of the spectrum which instead for other
69
+ methodshastobeproven.
70
+ For cyclic representations [34] of integrable quantum mode ls the SOV method should lead to the
71
+ characterizationoftheeigenvaluesandthesimultaneouse igenstatesofthetransfermatrix T(λ)bya
72
+ finite5systemofBaxter-likeequations. However,itisworthpoint ingoutthatsuchacharacterization
73
+ of the spectrum is not the most efficient; this is in particula r true in view of the analysis of the
74
+ continuum limit. Here the main question reads: Is it possibl e to define a set of conditions under
75
+ whichtheSOVcharacterizationofthespectrumcanbereform ulatedintermsofafunctionalBaxter
76
+ equation? In fact, this is equivalent to ask if we can reconst ruct theQ-operator from the finite
77
+ system of Baxter-like equations. In this case the solution o f the spectral problem is reduced to the
78
+ classification of the solutions of the Baxter equation which satisfy some analytic and asymptotic
79
+ propertiesfixedbythe operators TandQ.
80
+ The lattice Sine-Gordonmodelis used asa concreteexamplew herethese questionsaboutquantum
81
+ integrability find a complete and affirmative answer. Indeed , in section 3, we show that the SOV
82
+ characterization of the transfer matrix spectrum is exactl y equivalent to a functional equation of
83
+ the form detD(Λ) = 0, whereD(λ)(see (3.21)) is a one-parameter family of quasi-tridiagonal
84
+ matrices. In section 4, we show that this functional equatio n is indeed equivalent to the Baxter
85
+ functional equation and, in section 5, we use these results t o reconstruct the Baxter Q-operator
86
+ with the same level of accuracy obtained by the direct constr uction presented in [1]. It is worth
87
+ pointingoutthat these resultsallowusto provethat thetra nsfermatrix T(λ)(plustheΘ-chargefor
88
+ even chain) describes the family Qof complete commuting self-adjoint charges which implies t he
89
+ quantum integrabilityof the model accordingto definition ( 1.1). So that in the Sine-Gordonmodel
90
+ theBaxter Q-operatorplaysonlytheroleofa usefulauxiliaryobject.
91
+ Let us point out that one of the main advantages of the spectru m characterization derived for the
92
+ Sine-Gordonmodelisthe possibilityto proveanexactrefor mulationin termsof non-linearintegral
93
+ 4TheSine-Gordon model at irrational values of the coupling β2is asimple case where this kind of problem emerges.
94
+ 5Thenumber of equations in thesystem is finite and related to t hedimension of thecyclic representation.4
95
+ equations6(NLIE).Thiswill bethe subjectof a futurepublicationwher ethe NLIEcharacterization
96
+ will lead us by the implementation of the continuum limit to t he description of the Sine-Gordon
97
+ spectrum in all the interesting regimes. These results will be shown to be consistent with those
98
+ obtained previously in the literature7[37, 38, 39, 40, 41, 42] (see [43, 44] for reviews). Note that
99
+ the methodbasedon thereformulationofthe spectralproble min termsofNLIEhasbeenalso used
100
+ recently [49] to derive the Sinh-Gordonspectrum in finite vo lume and to characterize the spectrum
101
+ in theinfraredandultravioletlimits.
102
+ The analysis of the Sine-Gordon model allows us to infer the m ain features required to extend this
103
+ kindofspectrumcharacterizationtocyclicrepresentatio nsofotherintegrablequantummodels. This
104
+ is particularly relevant for those models for which a direct construction of the Baxter Q-operator
105
+ encounterstechnicaldifficulties.
106
+ Acknowledgments. I would like to thank J. Teschner for stimulating discussion s and suggestions on a prelimi-
107
+ nary versionof this workand J.-M.Maillet for the interests hown.
108
+ I gratefullyacknowledge support from the ECbythe Marie Cur ie Excellence GrantMEXT-CT-2006-042695.
109
+ 2. The Sine-Gordon model
110
+ Weusethissectiontorecallthemainresultsderivedin[1]o nthedescriptionintermsofSOVofthe
111
+ lattice Sine-Gordonmodel. This will be used as the starting point to introducea characterizationof
112
+ the spectrumof the transfermatrix T(λ)which will lead to the constructionof the Q-operatorfrom
113
+ SOV.
114
+ 2.1 Definitions
115
+ Thelattice Sine-Gordonmodelcanbecharacterizedbythefo llowingLaxmatrix8:
116
+ LSG
117
+ n(λ) =κn
118
+ i/parenleftigg
119
+ iun(q−1
120
+ 2κnvn+q+1
121
+ 2κ−1
122
+ nv−1
123
+ n)λnvn−λ−1
124
+ nv−1
125
+ n
126
+ λnv−1
127
+ n−λ−1
128
+ nvniu−1
129
+ n(q+1
130
+ 2κ−1
131
+ nvn+q−1
132
+ 2κnv−1
133
+ n)/parenrightigg
134
+ ,(2.5)
135
+ whereλn≡λ/ξnfor anyn∈ {1,...,N}withξnandκnparameters of the model. For any n∈
136
+ {1,...,N}thecoupleofoperators( un,vn)defineaWeyl algebra Wn:
137
+ unvm=qδnmvmun,whereq=e−πiβ2. (2.6)
138
+ We will restrictourattentiontothecase inwhich qisarootofunity,
139
+ β2=p′
140
+ p, p,p′∈Z>0, (2.7)
141
+ 6Thistype of equations werebefore introduced in adifferent framework in [35,36]
142
+ 7See [45, 46] for a related model analyzed in the framework of A BA and [47, 48] for the corresponding finite volume
143
+ continuum limit.
144
+ 8Thelattice regularization of the Sine-Gordon model that we consider here goes back to [4,50] and is related to formula-
145
+ tions which have morerecently been studied in [51,52,53].5
146
+ withp≡2l+ 1odd andp′even so that qp= 1. In this case each Weyl algebra Wnadmits a
147
+ finite-dimensional representation of dimension p. In fact, we can represent the operators un,vnon
148
+ thespace ofcomplex-valuedfunctions ψ:SN
149
+ p→Cas
150
+ un·ψ(z1,...,z N) =unznψ(z1,...,z n,...,z N),
151
+ vn·ψ(z1,...,z N) =vnψ(z1,...,q−1zn,...,z N).(2.8)
152
+ whereSp={q2n;n= 0,...,2l}is a subset of the unit circle; note that Sp={qn;n= 0,...,2l}
153
+ sinceq2l+2=q.
154
+ Themonodromymatrix M(λ)definedin(1.2)intermsoftheLax-matrix(2.5)satisfiesthe quadratic
155
+ relations:
156
+ R(λ/µ)(M(λ)⊗1)(1⊗M(µ)) = (1⊗M(µ))(M(λ)⊗1)R(λ/µ), (2.9)
157
+ wheretheauxiliary R-matrixisgivenby
158
+ R(λ) =
159
+ qλ−q−1λ−1
160
+ λ−λ−1q−q−1
161
+ q−q−1λ−λ−1
162
+ qλ−q−1λ−1
163
+ . (2.10)
164
+ The elements of M(λ)generate a representation RNof the so-called Yang-Baxter algebra char-
165
+ acterized by the 4Nparametersκ= (κ1,...,κ N),ξ= (ξ1,...,ξ N),u= (u1,...,u N)and
166
+ v= (v1,...,v N); in the present paper we will restrict to the case un= 1,vn= 1,n= 1,...,N.
167
+ The commutation relations (2.9) are at the basis of the proof of the mutual commutativity of the
168
+ T-operators.
169
+ Inthecase ofa latticewith Nevenquantumsites, we havealso tointroducetheoperator:
170
+ Θ =N/productdisplay
171
+ n=1v(−1)1+n
172
+ n, (2.11)
173
+ whichplaystheroleofa gradingoperator inthe Yang-Baxteralgebra:
174
+ Proposition 6 of [1] Θcommuteswiththetransfermatrixandsatisfiesthefollowin gcommutation
175
+ relationswith theentriesofthemonodromymatrix:
176
+ ΘC(λ) =qC(λ)Θ,[A(λ),Θ] = 0, (2.12)
177
+ B(λ)Θ =qΘB(λ),[D(λ),Θ] = 0. (2.13)
178
+ Moreover,the Θ-chargeallowstoexpressthe asymptoticsofthetransferma trixas:
179
+ lim
180
+ logλ→∓∞λ±NT(λ) =/parenleftiggN/productdisplay
181
+ a=1κaξ±1
182
+ a
183
+ i/parenrightigg
184
+ /parenleftbig
185
+ Θ+Θ−1/parenrightbig
186
+ . (2.14)6
187
+ Let us denotewith ΣTthe spectrum(the set of the eigenvaluefunctions t(λ)) of the transfer matrix
188
+ T(λ). By the definitions(1.2) and (2.5), then ΣTis contained9inC[λ2,λ−2](N+eN−1)/2, where we
189
+ haveusedthenotatione N= 0forNoddand1forNeven.
190
+ Notethat inthecase of Neven,the Θ-chargenaturallyinducesthegrading ΣT=/uniontextl
191
+ k=0Σk
192
+ T,where:
193
+ Σk
194
+ T≡/braceleftigg
195
+ t(λ)∈ΣT: lim
196
+ logλ→∓∞λ±Nt(λ) =/parenleftiggN/productdisplay
197
+ a=1κaξ±1
198
+ a
199
+ i/parenrightigg
200
+ (qk+q−k)/bracerightigg
201
+ .(2.15)
202
+ This simply follows by the asymptotics of T(λ)and by its commutativity with Θ. In particular,
203
+ anyt(λ)∈Σk
204
+ Tis aT-eigenvalue corresponding to simultaneous eigenstates of T(λ)andΘwith
205
+ Θ-eigenvalues q±k.
206
+ 2.2 CyclicSOVrepresentations
207
+ TheseparationofvariablesmethodofSklyaninisbasedonth eobservationthatthespectralproblem
208
+ forT(λ)simplifies considerablyif one worksin an auxiliaryreprese ntationwherethe commutative
209
+ familyofoperators B(λ)isdiagonal.
210
+ InthecaseoftheSine-Gordonmodelthevectorspace10CpNunderlyingtheSOVrepresentationcan
211
+ beidentifiedwiththespaceoffunctions Ψ(η)definedforηtakenfromthediscreteset
212
+ BN≡/braceleftbig
213
+ (qk1ζ1,...,qkNζN); (k1,...,k N)∈ZN
214
+ p/bracerightbig
215
+ , (2.16)
216
+ onthesefunctions B(λ)actsasa multiplicationoperator,
217
+ BN(λ)Ψ(η) =ηeN
218
+ Nbη(λ)Ψ(η), b η(λ)≡N/productdisplay
219
+ n=1κn
220
+ i[N]/productdisplay
221
+ a=1(λ/ηa−ηa/λ) ; (2.17)
222
+ where[N]≡N−eNandη1,...,η[N]are the zerosof bη(λ). In the case of even Nit turns out that
223
+ we needa supplementaryvariable ηNinordertobeable toparameterizethe spectrumof B(λ).
224
+ In[1]wehaveproventhatforgeneralvaluesoftheparameter sκandξoftheoriginalrepresentation
225
+ it is possible to construct these SOV representationsand mo reoverwe have defined the map which
226
+ fixestheSOVparameter ηintermsoftheparameters κandξ.
227
+ In these SOV representations the spectral problem for T(λ)is reduced to the following discrete
228
+ system ofBaxter-likeequationsin thewave-function Ψt(η) =/a\}bracketle{tη|t/a\}bracketri}htofaT-eigenstate |t/a\}bracketri}ht:
229
+ t(ηr)Ψ(η) =a(ηr)T−
230
+ rΨ(η)+d(ηr)T+
231
+ rΨ(η)∀r∈ {1,...,[N]},(2.18)
232
+ 9Herewith C[x,x−1]Mwearedenoting the linear space ofthe Laurentpolynomials o f degreeMin thevariable x∈C.
233
+ 10It is always possible to provide the structure of Hilbert spa ce to this finite-dimensional linear space. In particular, t he
234
+ scalar product in the SOVspace is naturally introduced by th e requirement that the transfer matrix is self-adjoint in th e SOV
235
+ representation. Appendix B addresses this issue.7
236
+ whereT±
237
+ raretheoperatorsdefinedby
238
+
239
+ rΨ(η1,...,η N) = Ψ(η1,...,q±1ηr,...,η N),
240
+ whilethe coefficients a(λ)andd(λ)are definedby:
241
+ a(λ) =N/productdisplay
242
+ n=1κn
243
+ iλn(1−iq−1/2λnκn)(1−iq−1/2λn
244
+ κn),d(λ) =qNa(−λq).(2.19)
245
+ Inthecase of Nevenwe haveto addto thesystem(2.18) thefollowingequatio ninthevariable ηN:
246
+ T+
247
+ NΨ±k(η) =q±kΨ±k(η), (2.20)
248
+ fort(λ)∈Σk
249
+ Twithk∈ {0,...,l}.NotethatthecyclicityoftheseSOVrepresentationsisexpr essed
250
+ bytheidentificationof (T±
251
+ j)pwith theidentityforany j∈ {1,...,N}.
252
+ 3. SOV characterization of T-eigenvalues
253
+ Let usintroducetheoneparameterfamily D(λ)ofp×pmatrix:
254
+ D(λ)≡
255
+ t(λ)−d(λ) 0 ··· 0 −a(λ)
256
+ −a(qλ)t(qλ)−d(qλ) 0 ··· 0
257
+ 0......
258
+ ... ···...
259
+ ... ···...
260
+ ...... 0
261
+ 0... 0−a(q2l−1λ)t(q2l−1λ)−d(q2l−1λ)
262
+ −d(q2lλ) 0 ... 0−a(q2lλ)t(q2lλ)
263
+ (3.21)
264
+ wherefornow t(λ)isjust anevenLaurentpolynomialofdegree N+eN−1inλ.
265
+ Lemma 1. Thedeterminant detpDisanevenLaurentpolynomialofmaximaldegree N+eN−1in
266
+ Λ≡λp.
267
+ Proof.Let us start observingthat D(λq)is obtainedby D(λ)exchangingthe first and p-th column
268
+ andafterthefirst and p-throw,so that
269
+ det
270
+ pD(λq) = det
271
+ pD(λ)∀λ∈C, (3.22)
272
+ whichimpliesthat detpDisfunctionof Λ. Let usdevelopthedeterminant:
273
+ det
274
+ pD(Λ) =p/productdisplay
275
+ h=1a(λqh)+p/productdisplay
276
+ h=1a(−λqh)−qNa(λ)a(−λ) det
277
+ 2l−1D(1,2l+1),(1,2l+1)(λ)
278
+ −qNa(λq)a(−λq) det
279
+ 2l−1D(1,2),(1,2)(λ)+t(λ)det
280
+ 2lD1,1(λ), (3.23)8
281
+ whereD(h,k),(h,k)(λ)denotes the (2l−1)×(2l−1)sub-matrix of D(λ)obtained removing the
282
+ rowsandcolumns handkwhileDh,k(λ)denotesthe 2l×2lsub-matrixof D(λ)obtainedremoving
283
+ therowhandcolumn k. Theinteresttowardthisdecompositionof detpD(Λ)isduetothefact that
284
+ the matrices D(1,2),(1,2)(λ),D(1,2l+1),(1,2l+1)(λ)andD1,1(λ)aretridiagonal matrices. Following
285
+ thesamereasoningusedinLemma4toprovethat det2lD1,1(λ)isanevenfunctionof λwecanalso
286
+ showthatthisistruefor det2l−1D(1,2),(1,2)(λ)anddet2l−1D(1,2l+1),(1,2l+1)(λ). Fromtheparityof
287
+ these functionsthe parityof detpD(Λ)followsbyusing(3.23).
288
+ Beinga(λ),d(λ)andt(λ)Laurentpolynomialofdegree Ninλ,inthecaseof Neventhestatement
289
+ ofthelemmaisalreadyproven;so wehavejust toshowthat:
290
+ lim
291
+ logΛ→∓∞Λ±Ndet
292
+ pD(Λ) = 0 (3.24)
293
+ forNoddwhichfollowsobservingthat:
294
+ lim
295
+ logΛ→∓∞Λ±Ndet
296
+ pD(Λ) =i±pNN/productdisplay
297
+ n=1κp
298
+ nξ±p
299
+ ndet
300
+ p/vextenddouble/vextenddouble/vextenddoubleq−(1∓1)N/2δh,k+1−q(1∓1)N/2δh,k−1/vextenddouble/vextenddouble/vextenddouble.
301
+ (3.25)
302
+ The interesttowardthe function detpD(Λ)isdueto the fact thatit allowsthefollowingcharacteri-
303
+ zationofthe T-spectrum:
304
+ Lemma 2. ΣTis the set of all the functions t(λ)∈C[λ2,λ−2](N+eN−1)/2which satisfy the system
305
+ of equations:
306
+ det
307
+ pD(ηp
308
+ a) = 0∀a∈ {1,...,[N]}and(η1,...,η[N])∈BN, (3.26)
309
+ plusin thecaseof Neven:
310
+ lim
311
+ logΛ→∓∞Λ±Ndet
312
+ pD(Λ) = 0. (3.27)
313
+ Proof.The requirement that the system of equations (2.18) admits a non-zerosolution leads to the
314
+ equations(3.26),while theequation(3.27) foreven Nsimplyfollowsbyobservingthat:
315
+ lim
316
+ logΛ→∓∞Λ±Ndet
317
+ pD(Λ) = det
318
+ p/vextenddouble/vextenddouble/vextenddoubleq(1∓1)N/2δi,j−1+q−(1∓1)N/2δi,j+1−(qk+q−k)δi,j/vextenddouble/vextenddouble/vextenddouble
319
+ ×(−1)N/productdisplay
320
+ n=1/parenleftbig
321
+ iκnξ±
322
+ n/parenrightbigp= 0. (3.28)
323
+ Note that the above characterization of the T-spectrum ΣTrequires as input the knowledge of BN,
324
+ i.e. the lattice of zeros of the operator B(λ). It is so interesting to notice that this characterization9
325
+ has in fact a reformulation which is independent from the kno wledge of BN. To explain this let us
326
+ notethatLemma1allowsto introducethefollowingmap:
327
+ Dp,N:t(λ)∈C[λ2,λ−2](N+eN−1)/2→ Dp,N(t(λ))≡det
328
+ pD(Λ)∈C[Λ2,Λ−2](N+eN−1)/2.
329
+ (3.29)
330
+ Intermsofthismapwecanintroduceafurthercharacterizat ionofthespectrumofthetransfermatrix
331
+ T(λ).
332
+ Theorem 1. The spectrum ΣTof the transfer matrix T(λ)coincides with the kernel NDp,N⊂
333
+ C[λ2,λ−2](N+eN−1)/2ofthe map Dp,N.
334
+ Proof.The inclusion NDp,N⊂ΣTis trivial thanks to Lemma 2, vice-versa if t(λ)∈ΣTthen
335
+ the function detpD(Λ)is zero in N+eNdifferent values of Λ2which thanks to Lemma 1 implies
336
+ detpD(Λ)≡0,i.e.ΣT⊂ NDp,N.
337
+ That is the set of eigenvalues of the transfer matrix T(λ)is exactly characterized as the subset of
338
+ C[λ2,λ−2](N+eN−1)/2whichcontainsallthesolutionsofthefunctionalequation detpD(Λ) = 0. In
339
+ thenextsectionwewill showthat thisfunctionalequationi s nothingelse thattheBaxterequation.
340
+ Remark 1. Let us note that the same kind of functional equation detD(Λ) = 0 also appears
341
+ in [54, 55, 56]. There it recasts, in a compact form, the funct ional relations which result from the
342
+ truncatedfusionsoftransfermatrixeigenvalues. Itissor elevanttopointoutthatforthe BBS-model11
343
+ in the SOV representation the non-triviality condition of t he solutions of the system of Baxter-like
344
+ equations has been shown [60] to be equivalent to the truncat ion identity in the fusion of transfer
345
+ matrixeigenvalues.
346
+ 4. Baxterfunctional equation
347
+ The main consequence of the previous analysis is that it natu rally leads to the complete character-
348
+ ization of the transfer matrix spectrum in terms of polynomi al solutions of the Baxter functional
349
+ equation.
350
+ Theorem2. Lett(λ)∈ΣTthent(λ)definesuniquelyuptonormalizationapolynomial Qt(λ)that
351
+ satisfiestheBaxterfunctionalequation:
352
+ t(λ)Qt(λ) =a(λ)Qt(λq−1)+d(λ)Qt(λq)∀λ∈C. (4.30)
353
+ Proof.The fact that given a t(λ)∈C[λ2,λ−2](N+eN−1)/2there exists up to normalizationat most
354
+ one polynomial Qt(λ)that satisfies the Baxter functional equationhas been prove nin Lemma 2 of
355
+ [1]. So we have to prove only the existence of Qt(λ)∈C[λ]. An interesting point about the proof
356
+ givenhereisthatit isa constructiveproof.
357
+ 11TheBBS-model [12, 57,58,59] has been analyzed in the SOVapp roach in aseries of works [60,61,62].10
358
+ Let usnoticethatthe condition t(λ)∈ΣT≡ NDp,Nimpliesthatthe p×pmatrixD(λ)hasrank2l
359
+ foranyλ∈C\{0}. Letusdenotewith
360
+ Ci,j(λ) = (−1)i+jdet
361
+ 2lDi,j(λ) (4.31)
362
+ the(i,j)cofactorof the matrix D(λ); then the matrix formedout of these cofactorshasrank 1, i.e.
363
+ all thevectors:
364
+ Vi(λ)≡(Ci,1(λ),Ci,2(λ),...,Ci,2l+1(λ))T∈Cp∀i∈ {1,...,2l+1}(4.32)
365
+ areproportional:
366
+ Vi(λ)/Ci,1(λ) =Vj(λ)/Cj,1(λ)∀i,j∈ {1,...,2l+1},∀λ∈C. (4.33)
367
+ Theproportionality(4.33)oftheeigenvectorsV i(λ)implies:
368
+ C2,2(λ)/C2,1(λ) =C1,2(λ)/C1,1(λ) (4.34)
369
+ which,byusingtheproperty(A.69),canberewrittenas:
370
+ C1,1(λq)/C1,2l+1(λq) =C1,2(λ)/C1,1(λ). (4.35)
371
+ Moreover,thefirst elementinthe vectorialcondition D(λ)V1(λ) =0¯reads:
372
+ t(λ)C1,1(λ) =a(λ)C1,2l+1(λ)+d(λ)C1,2(λ). (4.36)
373
+ Let us note that from the form of a(λ),d(λ)andt(λ)∈ΣTit follows that all the cofactors are
374
+ Laurentpolynomialofmaximaldegree122lNinλ:
375
+ Ci,j(λ) = Ci,jλ−2lN+ai,j4lN−(ai,j+bi,j)/productdisplay
376
+ h=1(λ(i,j)
377
+ h−λ). (4.37)
378
+ In Lemma 5, we show that the equations (4.35) and (4.36) imply that if C 1,1(λ)has a common
379
+ zero with C 1,2(λ)then this is also a zero of C 1,2l+1(λ)and that the same statement holds ex-
380
+ changing C 1,2(λ)with C 1,2l+1(λ). So we can denote with C1,1C1,1(λ),C1,2l+1C1,2l+1(λ)and
381
+ C1,2C1,2(λ)the polynomials of maximal degree 4lNobtained simplifying the common factors in
382
+ C1,1(λ), C1,2l+1(λ)and C1,2(λ). Then,byequation(4.35),theyhavetosatisfythe relation s:
383
+ C1,2l+1(λ) =q¯N1,1C1,1(λq−1),C1,2(λ) =q−¯N1,1C1,1(λq)andC1,2l+1=ϕC1,1,(4.38)
384
+ whereϕ≡C1,1/C1,2and¯N1,1is the degree of the polynomial C1,1(λ). So that equation (4.36)
385
+ assumestheformofa Baxterequationin thepolynomial C1,1(λ):
386
+ t(λ)C1,1(λ) = ¯a(λ)C1,1(λq−1)+¯d(λ)C1,1(λq), (4.39)
387
+ 12Theai,jandbi,jare non-negative integers and λ(i,j)
388
+ h/ne}ationslash= 0for anyh∈ {1,...,4lN−(ai,j+bi,j)}.11
389
+ with coefficients ¯a(λ)≡q¯N1,1ϕa(λ)and¯d(λ)≡q−¯N1,1ϕ−1d(λ). Note that the consistence of the
390
+ aboveequationimpliesthat ϕisap-rootoftheunity. Indeed,denotingwith ¯D(Λ)thematrixdefined
391
+ asin(3.21) butwithcoefficients ¯a(λ)and¯d(λ), equation(4.39) implies:
392
+ 0 = det
393
+ p¯D(Λ)≡(ϕp−1)/parenleftiggp/productdisplay
394
+ h=1a(λqh)−ϕ−pp/productdisplay
395
+ h=1a(−λqh)/parenrightigg
396
+ . (4.40)
397
+ The expansionfor detp¯D(Λ)in (4.40) is derivedby using the expansion(3.23) for detp¯D(Λ), the
398
+ formulae13:
399
+ det
400
+ 2lD1,1(λ) = det
401
+ 2lD1,1(λ), (4.41)
402
+ det
403
+ 2l−1D(1,2),(1,2)(λ) = det
404
+ 2l−1D(1,2),(1,2)(λ), (4.42)
405
+ det
406
+ 2l−1D(1,2l+1),(1,2l+1)(λ) = det
407
+ 2l−1D(1,2l+1),(1,2l+1)(λ), (4.43)
408
+ andthecondition t(λ)∈ΣT. Finally,if wedefine:
409
+ Qt(λ)≡λaC1,1(λ), (4.44)
410
+ whereq−a=q¯N1,1ϕwitha∈ {0,..,2l},we getthestatementofthetheorem.
411
+ Remark2. Theprevioustheoremimpliesthatforany t(λ)∈ΣTthepolynomialsolution Qt(λ)of
412
+ theBaxterequationcanberelatedtothedeterminantofatri diagonalmatrixoffinitesize p−1. Note
413
+ thatthe spectrumoftheSine-Gordonmodelinthecase ofirra tionalcoupling ¯β2shouldbededuced
414
+ fromβ2=p′/prational in the limit β2→¯β2. In particular, this implies that underthis limit ( p→
415
+ +∞)thedimensionoftherepresentationdivergesaswellasthe sizeofthetridiagonalmatrixwhose
416
+ determinant is associated to the solution Qt(λ)of the Baxter equation. It is then relevant to point
417
+ out that in the case of the quantum periodic Toda chain the sol utions of the corresponding Baxter
418
+ equationareexpressedintermsofdeterminantsofsemi-infi nitetridiagonalmatrices[63,13, 64].
419
+ It is worth noticing that the set of polynomials Qt(λ), introducedin the previoustheorem,admitsa
420
+ moreprecisecharacterization:
421
+ Theorem 3. Lett(λ)∈ΣTthent(λ)defines uniquely up to normalization a polynomial solution
422
+ Qt(λ)oftheBaxterfunctionalequation(4.30) ofmaximaldegree 2lN.
423
+ Inthecase Nodd,it results:
424
+ Qt(0)≡Q0/\e}atio\slash= 0,andlim
425
+ λ→∞λ−2lNQt(λ)≡Q2lN/\e}atio\slash= 0. (4.45)
426
+ In the case Neven, the condition (4.45) selects t(λ)∈Σ0
427
+ Twhile fort(λ)∈Σk
428
+ Twithk∈ {1,...,l}
429
+ we havethecharacterization Q0=Q2lN= 0and:
430
+ lim
431
+ λ→0Qt(λq)
432
+ Qt(λ)=q±k,lim
433
+ λ→∞Qt(λq)
434
+ Qt(λ)=q−(N±k). (4.46)
435
+ 13They follow from the tridiagonality of these matrices and by using Lemma3.12
436
+ Proof.Thankstoformula(A.74),thecofactor C 1,1(λ)∈C[λ,λ−1]2lNiseveninλandso it admits
437
+ theexpansions:
438
+ C1,1(λ) = C1,1λ−2lN+2˜a1,12lN−(˜a1,1+˜b1,1)/productdisplay
439
+ i=1(λ(1,1)
440
+ i−λ)(λ(1,1)
441
+ i+λ).(4.47)
442
+ Let us note now that by using the properties(A.69) and (A.74) , the relation (4.34) can be rewritten
443
+ as:
444
+ C1,1(λq)C1,1(λ) =qNC1,2(λ)C1,2(−λ). (4.48)
445
+ Usingthat andthegeneralrepresentation(4.37)forthe cof actor C 1,2(λ), weget:
446
+ a1,2= 2˜a1,1≡2a,b1,2= 2˜b1,1≡2b,C2
447
+ 1,2=C2
448
+ 1,1q−2(N+b)(4.49)
449
+ and:/parenleftig
450
+ λ(1,1)
451
+ i/parenrightig2
452
+ =/parenleftig
453
+ λ(1,2)
454
+ i/parenrightig2
455
+ ≡¯λ2
456
+ i,/parenleftig
457
+ λ(1,2)
458
+ i+2lN−(a+b)/parenrightig2
459
+ =/parenleftbig¯λi/q/parenrightbig2(4.50)
460
+ with¯λi/\e}atio\slash= 0for anyi∈ {1,...,2lN−(a+b)}withaandb∈Z≥0. Note that the equation (4.49)
461
+ andthefactthat ϕ≡C1,1/C1,2isap-rootofthe unityimply ϕ=qb+N. Thenwecanwrite:
462
+ C1,1(λ) = Cλ−2lN+2a2lN−(a+b)/productdisplay
463
+ i=1(¯λi+λ)(¯λi−λ), (4.51)
464
+ C1,2(λ) =qaCλ−2lN+2a2lN−(a+b)/productdisplay
465
+ i=1(¯λi+λ)((−1)H(x−i)¯λi−λq), (4.52)
466
+ whereC≡C1,1andH(n)≡ {0forn <0,1forn≥0}is the Heaviside step function. Here, x
467
+ isanon-negativeintegerwhichisfixedtozerothankstoform ula(4.38). Thenthesolution Qt(λ)of
468
+ theBaxter equation(4.30) belongsto C[λ]2lNandhastheform:
469
+ Qt(λ)≡λa2lN−(a+b)/productdisplay
470
+ i=1(¯λi−λ). (4.53)
471
+ Let usshow nowthe remainingstatementsof thetheoremconce rningthe asymptoticsof Qt(λ). To
472
+ thisaimwe computethe limits:
473
+ lim
474
+ logλ→∓∞λ±2lNC1,1(λ) = det
475
+ 2l/vextenddouble/vextenddouble/vextenddoubleq−(1∓1)N/2δi,j+1+q(1∓1)N/2δi,j−1−(qk+q−k)δeN,1δi,j/vextenddouble/vextenddouble/vextenddouble
476
+ i/ne}ationslash=1,j/ne}ationslash=1
477
+ ×N/productdisplay
478
+ h=1(κhξ±1
479
+ h
480
+ i)2l= (δeN,1(1+(2l+1)δk,0)−1)N/productdisplay
481
+ h=1(κhξ∓1
482
+ h
483
+ i)2l,(4.54)
484
+ whichimply:
485
+ a=b= 0, (4.55)13
486
+ forNoddandNevenwitht(λ)∈Σ0
487
+ T,i.e. thecondition(4.45). Inthe remainingcases, Nevenand
488
+ t(λ)/∈Σ0
489
+ T,the sameformulaimplies:
490
+ a/\e}atio\slash= 0,b/\e}atio\slash= 0, (4.56)
491
+ sothatQ0=Q2lN= 0,whiletheasymptoticsbehaviors(4.46)simplyfollowtaki ngtheasymptotics
492
+ oftheBaxterequationsatisfied by Qt(λ).
493
+ 5.Q-operator: Existence andcharacterization
494
+ Let us denote with Σtthe eigenspace of the transfer matrix T(λ)corresponding to the eigenvalue
495
+ t(λ)∈ΣT,then:
496
+ Definition 1. LetQ(λ)betheoperatorfamily definedby:
497
+ Q(λ)|t/a\}bracketri}ht ≡Qt(λ)|t/a\}bracketri}ht ∀|t/a\}bracketri}ht ∈Σtand∀t(λ)∈ΣT, (5.57)
498
+ withQt(λ)the element of C[λ]2lNcorresponding to t(λ)∈ΣTby the injection defined in the
499
+ previoustheorem.
500
+ Under the assumptions ξandκreal or imaginarynumbers, which assure the self-adjointne ssof the
501
+ transfermatrix T(λ)forλ∈R,thefollowingtheoremholds:
502
+ Theorem4. Theoperatorfamily Q(λ)isaBaxter Q-operator:
503
+ (A)Q(λ)satisfieswith T(λ)thecommutationrelations:
504
+ [Q(λ),T(µ)] = [Q(λ),Q(µ)] = 0∀λ,µ∈C, (5.58)
505
+ plusthe Baxterequation:
506
+ T(λ)Q(λ) =a(λ)Q(λq−1)+d(λ)Q(λq)∀λ∈C. (5.59)
507
+ (B)Q(λ)isa polynomialofdegree 2lNinλ:
508
+ Q(λ)≡2lN/summationdisplay
509
+ n=0Qnλn,
510
+ with coefficients Qnself-adjointoperators.
511
+ (C)Inthecase Nodd,the operator Q2lN=idandQ0isaninvertibleoperator.
512
+ (D)Inthecase Neven,Q(λ)commuteswiththe Θ-chargeandtheoperator Q2lNistheorthogonal
513
+ projectionontothe Θ-eigenspacewith eigenvalue1. Q0hasnon-trivialkernel coincidingwith
514
+ theorthogonalcomplementto the Θ-eigenspacewith eigenvalue1.14
515
+ Proof.Note that the self-adjointness of the transfer matrix T(λ)implies that Q(λ)is well defined,
516
+ indeed its action is defined on a basis. The property (A) is a tr ivial consequence of Definition 1.
517
+ Notethat theinjectivityofthemap t(λ)∈ΣT→Qt(λ)∈C[λ]2lNimplies:
518
+ (Qt(λ))∗=Qt(λ∗)∀λ∈C (5.60)
519
+ being(a(λ))∗=d(λ∗)and(t(λ))∗=t(λ∗). So we get the Hermitian conjugation property
520
+ (Q(λ))†=Q(λ∗), i.e. the self-adjointness of the operators Qn. The properties (C) and (D) of
521
+ the operators Q0andQ2lNdirectly follow from the asymptotics of the eigenfunction Qt(λ)while
522
+ thecommutativityof Q(λ)andΘisa directconsequenceofthecommutativityof T(λ)andΘ.
523
+ 6. Conclusion
524
+ Intheprevioussectionwehaveshownthatbyonlyusingthech aracterizationofthespectrumofthe
525
+ transfer matrix obtained by the SOV method we were able to rec onstruct the Q-operator. It is also
526
+ interestingto pointoutastheresultsderivedin [1]togeth erwiththoseofthepresentarticleyield:
527
+ Theorem5. Thefamily Qwhichcharacterizesthequantumintegrabilityofthelatti ceSine-Gordon
528
+ model(see definition(1.1)) isdescribedby thetransfermat rixT(λ)fora chainwith Noddnumber
529
+ of siteswhile by T(λ)plustheΘ-chargefora chainwith Nevennumberof sites.
530
+ Proof.LetusstartnoticingthatProposition3andTheorem4of[1]a rederivedonlyusingtheSOV
531
+ method (i.e. without any assumption about the existence of t heQ-operator). So only using SOV
532
+ analysis we have derived that for Nodd the transfer matrix T(λ)has simple spectrum while for
533
+ Neven this is true for T(λ)plus theΘ-charge; i.e. they define a complete family of commuting
534
+ observables and so satisfy the properties(A) and (C) of the d efinition (1.1). In this article we have
535
+ moreover shown that the Q-operator is defined as a function of the transfer matrix whic h implies
536
+ the property(B) of (1.1) recalling that in [1] the time-evol utionoperator Uhas been expressed as a
537
+ functionofthe Q-operator.
538
+ Let us shortly point out the main features required in abstra ct to extend to cyclic representationsof
539
+ other integrable quantum models the same kind of spectrum ch aracterization derived here for the
540
+ lattice Sine-Gordonmodel.
541
+ R1.The model admits an SOV description and the spectrum of the tr ansfer matrix can be charac-
542
+ terizedbyasystem ofBaxter-likeequationsin the T-wave-function Ψ(η) =/a\}bracketle{tη|t/a\}bracketri}ht:
543
+ t(ηr)Ψ(η) =a(ηr)Ψ(η1,...,q−1ηr,...,η N)+d(ηr)Ψ(η1,...,qη r,...,η N),(6.61)
544
+ where(η1,...,ηN)∈BNwithBNtheset ofzerosofthe B-operatorintheSOV representation.
545
+ Here,theparameter qisa rootofunitydefinedasin (2.6) and(2.7).15
546
+ Note that for cyclic representationsof an integrable quant um model the set BNis a finite subset of
547
+ CN. So the coefficients a(ηr)andd(ηr)are specified only in a finite number of points where they
548
+ satisfy thefollowingaveragevaluerelations14:
549
+ A(ηp
550
+ r) =p/productdisplay
551
+ k=1a(qkηr),D(ηp
552
+ r) =p/productdisplay
553
+ k=1d(qkηr). (6.62)
554
+ HereA(Λ)andD(Λ)are the average values of the operator entries A(λ)andD(λ)of the mon-
555
+ odromy matrix. Let us recall that the operator entries of the monodromymatrix are expected to be
556
+ polynomials(orLaurentpolynomials)inthespectralparam eterλsothecorrespondingaverageval-
557
+ uesarepolynomials(orLaurentpolynomials)in Λ≡λp. Itisthennaturaltointroducethefunctions
558
+ a(λ)andd(λ)aspolynomial(orLaurentpolynomial)solutionsofthefoll owingaveragerelations:
559
+ A(Λ)+γB(Λ) =p/productdisplay
560
+ k=1a(qkλ),D(Λ)+δB(Λ) =p/productdisplay
561
+ k=1d(qkλ), (6.63)
562
+ whereB(Λ)istheaveragevalueoftheoperator B(λ)andγandδare constanttobe fixed.
563
+ R2.Let usdenotewith Zf(λ)the set ofthezerosofthefunctions f(λ), then:
564
+ ∃λ0∈Za(λ):λ0/∈ ∪2l−1
565
+ h=0Zd(λqh). (6.64)
566
+ R3.Theaveragevaluesofthefunctions aanddarenotcoincidinginallthezerosofthe B-operator:
567
+ A(ηp
568
+ a)/\e}atio\slash=D(ηp
569
+ a)∀a∈ {1,...,[N]}and(η1,...,η[N])∈BN. (6.65)
570
+ The requirement R1yields the introduction of the p×pmatrixD(λ), defined as in (3.21), by
571
+ the functions a(λ)andd(λ)solutions of (6.63). This should allow us to reformulate the spectral
572
+ problem for the transfer matrix as the problem to classify al l the solutions t(λ)to the functional
573
+ equationdetpD(Λ) = 0ina modeldependentclassoffunctions.
574
+ The requirement R2implies that the rank of the matrix D(λ)is almost everywhere 2l. Indeed, the
575
+ condition (6.64) implies C 1,p(λ0)/\e}atio\slash= 0, independently from the function t(λ). Being the cofactor
576
+ C1,p(λ)acontinuousfunctionofthespectralparametertheabovest atementontherankofthematrix
577
+ D(λ)follows. Underthisconditionwecanfollowtheprocedurepr esentedinTheorem2toconstruct
578
+ the solutionsof the Baxter equation. Then the self-adjoint nessof the transfer matrix Tallows us to
579
+ proceedasinsection5to showthe existenceofthe Q-operatorasa functionof T.
580
+ The requirement R3is a sufficient criterion15to show the simplicity of the spectrum of Twhich
581
+ should imply that the full integrable structure of the quant um model should be described by the
582
+ 14Theequations in (6.62) are trivial consequences of the SOVr epresentation and of the cyclicity.
583
+ 15It is worth noticing that in the case of the Sine-Gordon model the criterion R3does not apply to the representations
584
+ withun=vn= 1. Nevertheless, we have shown the simplicity of Tby using some model dependent properties of the
585
+ coefficients a(λ)andd(λ), see section 5of [1].16
586
+ transfermatrixassoonastheproperty(B)indefinition(1.1 )isshownforthemodelunderconsider-
587
+ ation.
588
+ Following the schema here presented, in a future publicatio n we will address the analysis of the
589
+ spectrumfortheso-called α-sectorsoftheSine-Gordonmodel(see[1]). Theuseofthisapproachi s
590
+ in particularrelevantin these sectorsof theSine-Gordonm odelbecausea direct constructionof the
591
+ Q-operatorleadstosometechnicaldifficulty.
592
+ A. Properties ofthecofactors C i,j(λ)
593
+ Let usconsideran M×Mtridiagonal matrix16O:
594
+ O≡
595
+ z1y10··· 0 0
596
+ x1z2y20··· 0
597
+ 0x2z3y3...
598
+ .........
599
+ ...... 0
600
+ 0...0xM−2zM−1yM−1
601
+ 0 0...0xM−1zM
602
+ (A.66)
603
+ i.e. a matrix with non-zero entries only along the principal diagonal and the next upper and lower
604
+ diagonals.
605
+ Lemma 3. The determinantof atridiagonalmatrix is invariantundert he transformation ̺αwhich
606
+ multiplies for αthe entries above the diagonal and for α−1the entries below the diagonal leaving
607
+ theentriesonthediagonalunchanged.
608
+ Proof.Letusnotethatthedeterminantofa tridiagonalmatrixadmi tsthefollowingexpansion:
609
+ det
610
+ MO=z1det
611
+ M−1O1,1+x1y1det
612
+ M−2O(1,2),(1,2), (A.67)
613
+ wherewe haveused thesame notationsintroducedafterformu la(3.23). By usingit, we getthat the
614
+ actionof̺αreads:
615
+ det
616
+ M̺α(O) =z1det
617
+ M−1̺α(O)1,1+x1y1det
618
+ M−2̺α(O)(1,2),(1,2). (A.68)
619
+ Then the statement follows by induction noticing that the tr ansformation ̺αleaves always un-
620
+ changedthedeterminantofa 2×2matrix.
621
+ 16An interesting analysis of the eigenvalue problem for tridi agonal matrices is presented in [65].17
622
+ Lemma 4. Thefollowingpropertieshold:
623
+ Ch+i,k+i(λ) =Ch,k(λqi)∀i,h,k∈ {1,...,2l+1}, (A.69)
624
+ and:
625
+ C1,1(λ) =C1,1(−λ)andC2,1(λ) =qNC1,2(−λ). (A.70)
626
+ Proof.Note that by the definition (4.31) of the cofactors C i,j(λ)the equations (A.69) are simple
627
+ consequencesof qp= 1andareprovenexchangingrowsandcolumnsin thedeterminan ts.
628
+ Let us provenow that the cofactor C 1,1(λ) = det 2lD1,1(λ)is an even function of ��. The tridiago-
629
+ nalityofthematrix D1,1(λ)allowsusto usethepreviouslemma:
630
+ C1,1(λ)≡det
631
+ 2l/vextenddouble/vextenddoublet(λqh)δh,k−a(λqh)δh,k+1−qNa(−λqh+1)δh,k−1/vextenddouble/vextenddouble
632
+ h>1,k>1
633
+ = det
634
+ 2l/vextenddouble/vextenddoublet(λqh)δh,k−qNa(λqh)δh,k+1−a(−λqh+1)δh,k−1/vextenddouble/vextenddouble
635
+ h>1,k>1
636
+ = det
637
+ 2l/vextenddouble/vextenddoublet(λqh)δh,k−d(−λqk)δk,h−1−a(−λqk)δk,h+1/vextenddouble/vextenddouble
638
+ h>1,k>1
639
+ ≡det
640
+ 2l(D1,1(−λ))T=C1,1(−λ). (A.71)
641
+ To provenowthesecondrelationin (A.70) weexpandthe cofac tors:
642
+ C2,1(λ) =2l+1/productdisplay
643
+ h=2a(λqh)+d(λ) det
644
+ 2l−1D(1,2),(1,2)(λ), (A.72)
645
+ C1,2(λ) =2l/productdisplay
646
+ h=1d(λqh)+a(λq) det
647
+ 2l−1D(1,2),(1,2)(λ). (A.73)
648
+ Byusingthesamestepsshownin(A.71),thetridiagonalityo fthematrixD (1,2),(1,2)(λ)impliesthat
649
+ its determinant is an even function of λfrom which the statement C 2,1(λ) =qNC1,2(−λ)follows
650
+ recallingthat d(λ) =qNa(−λq).
651
+ Remark 3. Inthisarticlewe needonlytheproperties(A.70);however, it isworthpointingoutthat
652
+ theyarespecialcasesofthefollowingpropertiesofthe cof actors:
653
+ Ci,j(λ) =qN(i−j)Cj,i(−λ)∀i,j∈ {1,...,2l+1}. (A.74)
654
+ Theproofof(A.74)canbedonesimilarlytothatof(A.70) but we omitit forsimplicity.
655
+ Let ususe onceagainthenotation Zfforthe set ofthezerosofafunction f(λ), then:
656
+ Lemma 5. Theequations(4.35)and(4.36) imply:
657
+ ZC1,1∩ZC1,2≡ZC1,1∩ZC1,2l+1. (A.75)18
658
+ Proof.The inclusions/parenleftbig
659
+ ZC1,1∩ZC1,2/parenrightbig
660
+ \Za⊂ZC1,1∩ZC1,2l+1and/parenleftbig
661
+ ZC1,1∩ZC1,2l+1/parenrightbig
662
+ \Zd⊂
663
+ ZC1,1∩ZC1,2triviallyfollowbyequation(4.36).
664
+ Let us observe now that C 1,2(λq−1)has no common zero with a(λ)and that C 1,2l+1(λq)has no
665
+ common zero with d(λ). These statements simply follow from (A.73), (A.69)and(A. 72) when we
666
+ recall that a(λ)has no common zero with/producttext2l−1
667
+ h=0d(λqh)and thatd(λ)has no common zero with/producttext2l+1
668
+ h=2a(λqh). So,if/parenleftbig
669
+ ZC1,1∩ZC1,2/parenrightbig
670
+ ∩Zaisnotemptyand λ0∈/parenleftbig
671
+ ZC1,1∩ZC1,2/parenrightbig
672
+ ∩Za,theequation
673
+ (4.35) computed in λ=q−1λ0implies C 1,2l+1(λ0) = 0being C 1,2(λ0q−1)/\e}atio\slash= 0, i.e.λ0∈
674
+ ZC1,1∩ZC1,2l+1. Similarly,if/parenleftbig
675
+ ZC1,1∩ZC1,2l+1/parenrightbig
676
+ ∩Zdisnotemptyand λ0∈/parenleftbig
677
+ ZC1,1∩ZC1,2l+1/parenrightbig
678
+ ∩Zd,
679
+ the equation (4.35) computed in λ=λ0implies C 1,2(λ0) = 0being C 1,2l+1(λ0q)/\e}atio\slash= 0, i.e.λ0∈
680
+ ZC1,1∩ZC1,2. So that (4.35) implies the inclusions/parenleftbig
681
+ ZC1,1∩ZC1,2/parenrightbig
682
+ ∩Za⊂ZC1,1∩ZC1,2l+1and/parenleftbig
683
+ ZC1,1∩ZC1,2l+1/parenrightbig
684
+ ∩Zd⊂ZC1,1∩ZC1,2inthiswaycompletingthe proofofthelemma.
685
+ B. Scalarproduct inthe SOV space
686
+ Here is described as a natural structure of Hilbert space can be provided to the linear space of the
687
+ SOV representationbypreservingtheself-adjointnessoft hetransfermatrix.
688
+ B.1 CyclicrepresentationsoftheWeylalgebra
689
+ Here,we considerthecyclicrepresentationsoftheWeyl alg ebraW(n)
690
+ qinthecase:
691
+ up
692
+ n=vp
693
+ n= 1forβ2=p′/pwithp′evenandp= 2l+1odd. (B.76)
694
+ At anysitenofthechain,weintroducethe quantumspace Rnwithvn-eigenbasis:
695
+ vn|k,n/a\}bracketri}ht=qk|k,n/a\}bracketri}ht ∀|k,n/a\}bracketri}ht ∈Bn={|k,n/a\}bracketri}ht,∀k∈ {−l,...,l}}. (B.77)
696
+ Note that the eigenvaluesof vndescribe the unit circle Sp={qk:k∈ {−l,...,l}},indeedql+1=
697
+ q−l. OnRnisdefinedap-dimensionalrepresentationoftheWeyl algebrabysetting :
698
+ un|k,n/a\}bracketri}ht=|k+1,n/a\}bracketri}ht ∀k∈ {−l,...,l} (B.78)
699
+ with thecyclicitycondition:
700
+ |k+p,n/a\}bracketri}ht=|k,n/a\}bracketri}ht. (B.79)
701
+ B.2 Representationin the SOVbasis
702
+ The analysis developed in [1] define recursively the eigenba sis{|¯η1qh1,...,¯ηNqhN/a\}bracketri}ht}of theB-
703
+ operator in the original representation, i.e. as linear com binations of the elements of the basis
704
+ {|h1,...,hN/a\}bracketri}ht ≡/circlemultiplytextN
705
+ n=1|hn,n/a\}bracketri}ht}, where|hn,n/a\}bracketri}htare the elements of the vn-eigenbasis defined in
706
+ (B.77). To writethischangeofbasisin amatrixformlet usin troducethe followingnotations:
707
+ |yj/a\}bracketri}ht ≡ |¯η1qh1,...,¯ηNqhN/a\}bracketri}htand|xj/a\}bracketri}ht ≡ |h1,...,hN/a\}bracketri}ht (B.80)19
708
+ where:
709
+ j:=h1+N/summationdisplay
710
+ a=2(2l+1)(a−1)(ha−1)∈ {1,...,(2l+1)N}, (B.81)
711
+ notethat thisdefinesa oneto onecorrespondencebetween N-tuples(h1,...,hN)∈ {1,...,2l+1}N
712
+ and integers j∈ {1,...,(2l+1)N}, which just amountsto chose an orderingin the elementsof th e
713
+ two basis. Underthisnotation,wehave:
714
+ |yj/a\}bracketri}ht=W|xj/a\}bracketri}ht=(2l+1)N/summationdisplay
715
+ i=1Wi,j|xi/a\}bracketri}ht, (B.82)
716
+ where we are representing |xj/a\}bracketri}htas the vector |j/a\}bracketri}htin the natural basis in C(2l+1)NandW=||Wi,j||
717
+ is a(2l+1)N×(2l+1)Nmatrix. The matrix Wis defined by recursion in terms of the kernel K
718
+ constructedinappendixCof[1], letususethenotation:
719
+ K({h1,...,hN},k1,{k2,...,kN})≡KN(η|χ2;χ1), (B.83)
720
+ whereweareconsideringthecase N−M = 1. Thenthe recursionreads:
721
+ W(N)
722
+ i,j=2l+1/summationdisplay
723
+ k2,...,kN=1K({h1(j),...,hN(j)},h1(i),{k2,...,kN})W(N−1)
724
+ ¯h(i),a(k2,...,kN),(B.84)
725
+ where we have introduced the index (N)and(N−1)in the matrices Wto make clear the step
726
+ of the recursion. Here, (h1(j),...,hN(j))is the unique N-tuples corresponding to the integer j∈
727
+ {1,...,(2l+ 1)N}andh1(i)is the first entry in the unique N-tuples corresponding to the integer
728
+ i∈ {1,...,(2l+1)N}. Moreover,wehavedefined:
729
+ ¯h(i) := 1+i−h1(i)
730
+ 2l+1∈ {1,...,(2l+1)(N−1)}anda(k2,...,kN) =k2+N/summationdisplay
731
+ a=3(2l+1)(a−2)(ka−1),
732
+ (B.85)
733
+ Remarks:
734
+ a)Underthechangeofbasis {|xj/a\}bracketri}ht} → {|yj/a\}bracketri}ht}thegenericoperatorX transformsforsimilarity:
735
+ XSOV≡W−1XW, (B.86)
736
+ so fromtheactionofthezerooperators ηaandtheshift operators T±
737
+ aontheB-eigenbasis |yj/a\}bracketri}ht:
738
+ ηa|yj/a\}bracketri}ht= ¯ηaqha(j)|yj/a\}bracketri}htandT±
739
+ a|yj/a\}bracketri}ht=|yj±(2l+1)(a−1)/a\}bracketri}ht (B.87)
740
+ we havethat:
741
+ (ηa)SOV= ¯ηa||qha(j)δi,j||and/parenleftbig
742
+
743
+ a/parenrightbig
744
+ SOV=||δi,j±(2l+1)(a−1)||. (B.88)20
745
+ Fromtheaboveexpressionwe have17:
746
+ (ηa)†
747
+ SOV= (ηa)∗
748
+ SOVand/parenleftbig
749
+
750
+ a/parenrightbig†
751
+ SOV=/parenleftbig
752
+ T∓
753
+ a/parenrightbig
754
+ SOV. (B.89)
755
+ b) The known transformation properties of the entries of the monodromy matrix in the original
756
+ representationimply:
757
+ /parenleftbiggDSOV(λ)CSOV(λ)
758
+ BSOV(λ)ASOV(λ)/parenrightbigg
759
+ =/parenleftigg
760
+ G−1(ASOV(λ∗))†G−G−1(BSOV(λ∗))†G
761
+ −G−1(CSOV(λ∗))†G G−1(DSOV(λ∗))†G/parenrightigg
762
+ ,(B.90)
763
+ withGisapositiveself-adjointmatrixdefinedby G:=W†W.
764
+ c)Thequantumdeterminantrelationis invariantundersimi laritytransformationsandso we have:
765
+ a(λ)d(λq−1) =ASOV(λ)DSOV(λq−1)−BSOV(λ)CSOV(λq−1), (B.91)
766
+ Lemma 6. Thebasis {|yj/a\}bracketri}ht}isnotanorthogonalbasisw.r.t. thenaturalscalarproduct on{|xj/a\}bracketri}ht}.
767
+ Proof.Note that the condition {|yj/a\}bracketri}ht}is an orthogonal basis is equivalent to the statement Gis
768
+ a diagonal matrix (with positive diagonal entries). Let us r ecall that the Hermitian conjugation
769
+ propertyofB(λ)togetherwiththeYang-Baxtercommutationrelationsimply :
770
+ [B†(λ),B(µ)] = [B(µ),C(λ∗)] =q−q−1
771
+ λ∗/µ−µ/λ∗(A(λ∗)D(µ)−A(µ)D(λ∗))/\e}atio\slash= 0 (B.92)
772
+ that is the operator B(λ)is not a normal operator. Now let us show that the non-normali tyofB(λ)
773
+ impliesthat Gisnotdiagonal. Indeed,wecanwrite:
774
+ [B†(λ),B(µ)] =/parenleftbig
775
+ W†/parenrightbig−1(BSOV(λ))†GBSOV(µ)W−1−WBSOV(µ)G−1(BSOV(λ))†W†
776
+ =W(G−1(BSOV(λ))†GBSOV(µ)−BSOV(µ)G−1(BSOV(λ))†G)W−1.(B.93)
777
+ Notenowthatifweassume Gdiagonal,then Gcommutesbothwith BSOV(λ)andwith(BSOV(λ))†,
778
+ being all diagonal matrices in the SOV representation, whic h implies the absurd [B†(λ),B(µ)] =
779
+ 0.
780
+ B.3 Scalarproductin theSOVspace
781
+ The self-adjointness of the family T(λ)implies that the transfer matrix eigenstates are orthogona l
782
+ undertheoriginalscalar product:
783
+ δi,j= (|ti/a\}bracketri}ht,|tj/a\}bracketri}ht), (B.94)
784
+ we have chosen the orthonormal ones. Note that the above equa tion naturally induces a scalar
785
+ productintheSOV representationobtainedunderchangeofb asis:
786
+ (|b/a\}bracketri}ht,|a/a\}bracketri}ht)SOV≡(G|b/a\}bracketri}ht,|a/a\}bracketri}ht) (B.95)
787
+ 17Here, weare using the standard notation for the adjoint X†≡(X∗)t.21
788
+ thatisascalarproductforwhichtheadjointofavector |a/a\}bracketri}htisthenaturaladjointtimesthematrix G:
789
+ |b/a\}bracketri}ht†SOV≡ /a\}bracketle{tb|Gwith/a\}bracketle{tb|=/parenleftig
790
+ (|b/a\}bracketri}ht)t/parenrightig∗
791
+ , (B.96)
792
+ andsoforthegenericoperator Xwehave:
793
+ X†SOV≡G−1X†G. (B.97)
794
+ It istrivialtonoticethat:
795
+ Lemma 7. The family of operators TSOV(λ)is self-adjoint w.r.t. †SOVand the eigenstates
796
+ |tj/a\}bracketri}htSOV≡W−1|tj/a\}bracketri}htare orthonormal w.r.t. the scalar product defined in (B.95). Moreover, it
797
+ results:
798
+ /parenleftigg
799
+ (ASOV(λ∗))†SOV(BSOV(λ∗))†SOV
800
+ (CSOV(λ∗))†SOV(DSOV(λ∗))†SOV/parenrightigg
801
+ =/parenleftbiggDSOV(λ)−CSOV(λ)
802
+ −BSOV(λ)ASOV(λ)/parenrightbigg
803
+ .(B.98)
804
+ References
805
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806
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807
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808
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809
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810
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815
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816
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817
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819
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1001.0036.txt ADDED
@@ -0,0 +1,1085 @@
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
1
+ The Computational Structure of Spike Trains
2
+ Robert Haslinger,1, 2Kristina Lisa Klinkner,3and Cosma Rohilla Shalizi3, 4
3
+ 1Martinos Center for Biomedical Imaging,Massachusetts General Hospital, Charlestown MA
4
+ 2Department of Brain and Cognitive Sciences, Massachusetts Institute of Technology, Cambridge MA
5
+ 3Department of Statistics, Carnegie Mellon University, Pittsburgh PA
6
+ 4Santa Fe Institute, Santa Fe NM
7
+ (Dated: September 2008; January 2009)
8
+ Neurons perform computations, and convey the results of those computations
9
+ through the statistical structure of their output spike trains. Here we present a
10
+ practical method, grounded in the information-theoretic analysis of prediction, for
11
+ inferring a minimal representation of that structure and for characterizing its com-
12
+ plexity. Starting from spike trains, our approach nds their causal state models
13
+ (CSMs), the minimal hidden Markov models or stochastic automata capable of
14
+ generating statistically-identical time series. We then use these CSMs to objec-
15
+ tively quantify both the generalizable structure and the idiosyncratic randomness
16
+ of the spike train. Speci cally, we show that the expected algorithmic informa-
17
+ tion content (the information needed to describe the spike train exactly) can be
18
+ split into three parts describing (1) the time-invariant structure (complexity) of
19
+ the minimal spike-generating process, which describes the spike train statistically ,
20
+ (2) the randomness (internal entropy rate) of the minimal spike-generating process,
21
+ and (3) a residual pure noise term not described by the minimal spike generating
22
+ process. We use CSMs to approximate each of these quantities. The CSMs are in-
23
+ ferred non-parametrically from the data, making only mild regularity assumptions,
24
+ via the Causal State Splitting Reconstruction (CSSR) algorithm. The methods
25
+ presented here complement more traditional spike train analyses by describing not
26
+ only spiking probability, and spike train entropy, but also the complexity of a spike
27
+ train's structure. We demonstrate our approach using both simulated spike trains
28
+ and experimental data recorded in rat barrel cortex during vibrissa stimulation.
29
+ I. INTRODUCTION
30
+ The recognition that neurons are computational devices is one of the foundations of modern neuroscience (McCulloch
31
+ & Pitts, 1943). However, determining the functional form of such computation is extremely dicult, if only because
32
+ while one often knows the output (the spikes) the input (synaptic activity) is almost always unknown. Often, therefore,
33
+ scientists must draw inferences about the computation from its results, namely the output spike trains and their
34
+ statistics. In this vein, many researchers have used information theory to determine, via calculation of the entropy
35
+ rate, a neuron's channel capacity, i.e., how much information the neuron could conceivably transmit, given the
36
+ distribution of observed spikes (Rieke et al., 1997). However, entropy quanti es randomness, and says little about
37
+ how much structure a spike train has, or the amount and type of computation which must have, at a minimum, taken
38
+ place to produce this structure. Here, and throughout this paper, we mean \computational structure" information-
39
+ theoretically, i.e., the most compact e ective description of a process capable of statistically reproducing the observed
40
+ spike trains. The complexity of this structure is the number of bits needed to describe it. This is di erent from the
41
+ algorithmic information content of a spike train, which is the number of bits needed to reproduce the latter exactly ,
42
+ describing not only its regularities, but also its accidental, noisy details.
43
+ Our goal is to develop rigorous yet practical methods for determining the minimal computational structure necessary
44
+ and sucient to generate neural spike trains. We are able to do this through non-parametric analysis of the directly-
45
+ observable spike trains, without resorting to a priori assumptions about what kind of structure they have. We do this
46
+ by identifying the minimal hidden Markov model (HMM) which can statistically predict the future of the spike train
47
+ without loss of information. This HMM also generates spike trains with the same statistics as the observed train.
48
+ It thus de nes a program which describes the spike train's computational structure, letting us quantify, in bits, the
49
+ structure's complexity.
50
+ From multiple directions, several groups, including our own, have shown that minimal generative models of time
51
+ series can be discovered by clustering histories into \states", based on their conditional distributions over future events
52
+ (Crutch eld & Young, 1989; Grassberger, 1986; Jaeger, 2000; Knight, 1975; Littman et al., 2002; Shalizi & Crutch eld,
53
+ 2001). The observed time series need notbe Markovian (few spike trains are), but the construction always yieldsarXiv:1001.0036v1 [q-bio.NC] 30 Dec 20092
54
+ the minimal HMM capable of generating and predicting the original process. Following Shalizi (2001); Shalizi &
55
+ Crutch eld (2001), we will call such a HMM a \Causal State Model" (CSM). Within this framework, the model
56
+ discovery algorithm called Causal State Splitting Reconstruction , or CSSR (Shalizi & Klinkner, 2004) is an adaptive
57
+ non-parametric method which consistently estimates a system's CSM from time-series data. In this paper we adapt
58
+ CSSR for use in spike train analysis.
59
+ CSSR provides us with non-parametric estimates of the time- and history- dependent spiking probabilities found by
60
+ more familiar parametric analyses. Unlike those analyses, it is also capable, in the limit of in nite data, of capturing all
61
+ the information about the computational structure of the spike-generating process contained in the spikes themselves.
62
+ In particular, the CSM quanti es the complexity of the spike-generating process by showing how much information
63
+ about the history of the spikes is relevant to their future, i.e., how much information is needed to reproduce the
64
+ spike train statistically. This is equivalent to the log of the e ective number of statistically-distinct states of the
65
+ process (Crutch eld & Young, 1989; Grassberger, 1986; Shalizi & Crutch eld, 2001). While this is not the same as
66
+ the algorithmic information content, we show that CSMs can also approximate the average algorithmic information
67
+ content, splitting it into three parts: (1) The generative process's complexity in our sense; (2) the internal entropy
68
+ rateof the generative process, the extra information needed to describe the exact state transitions the undergone while
69
+ generating the spike train; and (3) the residual randomness in the spikes, unconstrained by the generative process.
70
+ The rst of these quanti es the spike train's structure, the last two its randomness.
71
+ Below, we give precise de nitions of these quantities, both their ensemble averages ( xII.C) and their functional
72
+ dependence on time ( xII.D). The time-dependent versions allow us to determine when the neuron is traversing states
73
+ requiring complex descriptions. Our methods put hard numerical lower bounds on the amount of computational
74
+ structure which must be present to generate the observed spikes. They also quantify, in bits, the extent to which the
75
+ neuron is driven by external forces. We demonstrate our approach using both simulated and experimentally recorded
76
+ single-neuron spike trains. We discuss the interpretation of our measures, and how they add to our understanding of
77
+ neuronal computation.
78
+ II. THEORY AND METHODS
79
+ Throughout this paper we treat spike trains as stochastic binary time series, with time divided into discrete, equal-
80
+ duration bins steps (typically at one millisecond resolution); \1" corresponds to a spike and \0" to no spike. Our aim is
81
+ to nd a minimal description of the computational structure present in such a time series. Heuristically, the structure
82
+ present in a spike train can be described by a \program" which can reproduce the spikes statistically. The information
83
+ needed to describe this program (loosely speaking the program length) quanti es the structure's complexity. Our
84
+ approach uses minimal, optimally predictive HMMs, or Causal State Models (CSMs), reconstructed from the data, to
85
+ describe the program. (We clarify our use of \minimal" below.) The CSMs are then used to calculate various measures
86
+ of the computational structure, such as its complexity.
87
+ The states are chosen so that they are optimal predictors of the spike train's future, using only the information
88
+ available from the train's history. (We discuss the limitations of this below.) Speci cally the states Stare de ned
89
+ by grouping the histories of past spiking activity Xt
90
+ 1which occur in the spike train into equivalence classes, where
91
+ all members of a given equivalence class are statistically equivalent in terms of predicting the future spiking X1
92
+ t+1.
93
+ (Xt
94
+ t0denotes the sequence of random observables, i.e., spikes or their absence, between t0andt>t0whileXtdenotes
95
+ the random observable at time t. The notation is similar for the states.) This construction ensures that the causal
96
+ states are Markovian, even if the spike train is not (Shalizi & Crutch eld, 2001, Lemma 6, p. 839). Therefore, at all
97
+ timestthe system and its possible future evolution(s) can be speci ed by the state St. Like all HMMs, a CSM can
98
+ be represented pictorially by a directed graph, with nodes standing for the process's hidden states and directed edges
99
+ the possible transitions between these states. Each edge is labeled with the observable/symbol emitted during the
100
+ corresponding transition (\1" for a spike and \0" for no spike), and the probability of traversing that edge given that
101
+ the system started in that state. The CSM also speci es the time-averaged probability of occupying any state (via
102
+ the ergodic theorem for Markov chains).
103
+ The theory is described in more detail below, but at this point examples may clarify the ideas. Figures 1 A and B
104
+ show two simple CSMs. Both are built from simulated 40 Hz spike trains 200 seconds in length (1 msec time bins,
105
+ p= 0:04 IID at each time when spiking is possible). However, spike trains generated from the CSM in Figure 1 B
106
+ have a 5 msec refractory period after each spike (when p= 0), while the spiking rate in non-refractory periods is still
107
+ 40 Hz (p= 0:04). The refractory period is additional structure, represented by the extra states. State Arepresents
108
+ the status of the neuron during 40 Hz spiking, outside of the refractory periods. While in this state, the neuron either
109
+ emits no spike ( Xt+1= 0), staying in state A, or emits a spike ( Xt+1= 1) with probability p= 0:04 and moves to
110
+ stateB. The equivalence class of past spiking histories de ning state Atherefore includes all past spiking histories
111
+ for which the most recent ve symbols are 0, symbolically f00000g. StateBis the neuron's state during the rst3
112
+ msec of the refractory period. It is de ned by the set of spiking histories f1g. No spike can be emitted during a
113
+ refractory period so the transition to state Cis certain and the symbol emitted is always '0'. In this manner the
114
+ neuron proceeds through states CtoFand back to state Awhereupon it is possible to spike again.
115
+ The rest of this section is divided into four subsections. First, we brie
116
+ (for details, see Shalizi (2001); Shalizi & Crutch eld (2001)) and discuss why they can be considered a good choice for
117
+ understanding the structural content of spike trains. Second, we describe the Causal State Splitting Reconstruction
118
+ (CSSR) algorithm used to reconstruct CSMs from observed spike trains (Shalizi & Klinkner, 2004). We emphasize
119
+ that CSSR requires no a priori knowledge of the structure of the CSM which is discovered from the spike train. Third,
120
+ we discuss two di erent notions of spike train structure, namely statistical complexity and algorithmic information
121
+ content. These two measures can be interpreted as di erent aspects of a spike train's computational structure, and
122
+ each can be related to the reconstructed CSM. Fourth and nally, we show how the reconstructed CSM can be used
123
+ to predict spiking, measure the neural response and detect the in
124
+ A. Causal State Models
125
+ The foundation of the theory of causal states is the concept of a predictively sucient statistics . A statistic, ,
126
+ on one random variable, X, is sucient for predicting another random variable, Y, when(X) andXhave the
127
+ same information1aboutY,I[X;Y] =I[(X);Y]. This holds if and only if XandYare conditionally independent
128
+ given(X):P(YjX;(X)) = P(Yj(X)). This is a close relative of the familiar idea of parametric suciency;
129
+ in Bayesian statistics, where parameters are random variables, parametric suciency is a special case of predictive
130
+ suciency (Bernardo & Smith, 1994). Predictive suciency shares all of parametric suciency's optimality properties
131
+ (Bernardo & Smith, 1994). However, a statistic's predictive suciency depends only on the actual joint distribution of
132
+ XandY, not on any parametric model of that distribution. Again as in the parametric case, a minimal predictively
133
+ sucient statistic is one which is a function of every other sucient statistic , i.e.,(X) =h((X)) for some h.
134
+ Minimal sucient statistics are the most compact summaries of the data which retain all the predictively-relevant
135
+ information. A basic result is that a minimal sucient statistic always exists and is (essentially) unique, up to
136
+ isomorphism (Bernardo & Smith, 1994; Shalizi & Crutch eld, 2001).
137
+ In the context of stochastic processes, such as spike trains, is the minimal sucient statistic of the history Xt
138
+ 1
139
+ for predicting future of the process, X1
140
+ t+1. This statistic is the optimal predictor of the observations. The sequence
141
+ of values of the minimal sucient statistic, St=(Xt
142
+ 1), is another stochastic process. This process is always a
143
+ homogeneous Markov chain, whether or not the Xtprocess is (Knight, 1975; Shalizi & Crutch eld, 2001). Turned
144
+ around, this means that the original Xtprocess is always a random function of a homogeneous Markov chain, whose
145
+ latent states, named the causal states by Crutch eld & Young (1989), are optimal, minimal predictors of the future
146
+ of the time series.
147
+ Acausal state model orcausal state machine is a stochastic automaton or HMM constructed so that its Markov
148
+ states are minimal sucient statistics for predicting the future of the spike train, and consequently can generate spike
149
+ trains statistically identical to those observed.2Causal state reconstruction means inferring the causal states from
150
+ the observed spike train. Following Crutch eld & Young (1989); Shalizi & Crutch eld (2001), the causal states can be
151
+ seen as equivalence classes of spike-train histories Xt
152
+ 1which maximize the mutual information between the state(s)
153
+ and the future of the spike train X1
154
+ t+1. Because they are sucient, they predict the future of the spike train as well
155
+ as it can be predicted from its history alone. Because they are minimal, the number of states or equivalence classes
156
+ is as small as it can be without discarding predictive power.3
157
+ Formally, two histories, xandy, are equivalent when P(X1
158
+ t+1jXt
159
+ 1=x) =P(X1
160
+ t+1jXt
161
+ 1=y). The equiva-
162
+ lence class of xis [x]. De ne the function which maps histories to their equivalence classes:
163
+ (x)[x]
164
+ =
165
+ y:P(X1
166
+ t+1jXt
167
+ 1=y) =P(X1
168
+ t+1jXt
169
+ 1=x)
170
+ 1See Cover & Thomas (1991) for information-theoretic de nitions and notation.
171
+ 2Some authors use \hidden Markov Model" only for models where the current observation is independent of all other
172
+ variables given the current state, and call the broader class which includes CSMs \partially observable Markov
173
+ model".
174
+ 3There may exist more compact representations, but then the states, or their equivalents, can never be empirically
175
+ identi ed | see Shalizi & Crutch eld (2001, thm. 3, p. 846), or L ohr & Ay (2009).4
176
+ The causal states are the possible values of , i.e., the equivalence classes; each corresponds to a distinct distribution
177
+ for the future. The state at time tisSt=(Xt
178
+ 1). Clearly,(x) is a sucient statistic. It is also minimal, since if 
179
+ is sucient, then (x) =(y) implies(x) =(y). One can further show (Shalizi & Crutch eld, 2001, Theorem
180
+ 3) thatis the unique minimal sucient statistic, meaning that any other must be isomorphic to it.
181
+ In addition to being minimal sucient statistics, the causal states have some other important properties which
182
+ make them ideal for quantifying structure (Shalizi & Crutch eld, 2001). (1) As mentioned, fStgis a Markov process,
183
+ and one can write the observed process Xas a random function of the causal state process, i.e., Xhas a natural
184
+ hidden-Markov-model representation. (2) The causal states are recursively calculable; there is a function Tsuch
185
+ thatSt+1=T(St;Xt+1) | see Appendix A. (3) CSMs are closely related to the \predictive state representations" of
186
+ controlled dynamical systems (Littman et al., 2002); see Appendix C.
187
+ B. Causal State Splitting Reconstruction
188
+ Our goal is to nd a minimal sucient statistic for the spike train, which will form a hidden Markov model. As
189
+ stated previously, the states of this model are equivalence classes of spiking histories Xt
190
+ 1. In practice, we need an
191
+ algorithm which can both cluster histories into groups which preserve their conditional distribution of futures, and
192
+ nd the history length  at which the past may be truncated while preserving the computational structure of the
193
+ spike train. The former is accomplished by the CSSR algorithm (Shalizi & Klinkner, 2004) for inferring causal states
194
+ from data by building a recursive next-step-sucient statistic.4We do the latter by minimizing Schwartz's Bayesian
195
+ Information Criterion (BIC) over .
196
+ To save space, we just sketch the CSSR algorithm here.5CSSR starts by treating the process as an independent,
197
+ identically-distributed sequence, with one causal state. It adds states when statistical tests show that the current set
198
+ of states is not sucient. Suppose we have a sequence xN
199
+ 1=x1;x2;:::xNof lengthNfrom a nite alphabet Aof
200
+ sizek. We wish to derive from this an estimate ^ of the minimal sucient statistic . We do this by nding a set  of
201
+ states, each of which will be a set of strings, or nite-length histories. The function ^ will then map a history xto
202
+ whichever state contains a sux of x(taking \sux" in the usual string-manipulation sense). Although each state
203
+ can contain multiple suxes, one can check (Shalizi & Klinkner, 2004) that the mapping ^ will never be ambiguous.
204
+ The null hypothesis is that the process is Markovian on the basis of the states in ,
205
+ P(XtjXt1
206
+ tL=axt1
207
+ tL+1) =P(Xtj^S= ^(xt1
208
+ tL+1)) (1)
209
+ for alla2A. In words, adding an extra piece of history does not change the conditional distribution for the next
210
+ observation. We can check this with standard statistical tests, such as 2or Kolmogorov-Smirnov. In this paper, we
211
+ used a KS test of size = 0:01.6If we reject this hypothesis, we fall back on a restricted alternative hypothesis , that
212
+ we have the right set of conditional distributions, but have matched them with the wrong histories. That is,
213
+ P(XtjXt1
214
+ tL=axt1
215
+ tL+1) =P(Xtj^S=s) (2)
216
+ for somes2, buts6= ^(xt1
217
+ tL+1). If this hypothesis passes a test of size , thensis the state to which we assign
218
+ the history7. Only if the (2) is itself rejected do we create a new state, with the sux axt1
219
+ tL+1.8
220
+ 4A next-step-sucient statistic contains all the information needed for optimal one-step-ahead prediction,
221
+ I[Xt+1;(Xt
222
+ 1)] =I[Xt+1;Xt
223
+ 1], but not necessarily for longer predictions. CSSR relies on the theorem that
224
+ ifis next-step sucient, and it is recursively calculable, then is sucient for the whole of the future (Shalizi &
225
+ Crutch eld, 2001, pp. 842{843). CSSR rst nds a next-step sucient statistic, and then re nes it to be recursive.
226
+ 5In addition to Shalizi & Klinkner (2004), which gives pseudocode, some details of convergence, and applications to
227
+ process classi cation, are treated in Klinkner & Shalizi (2009); Shalizi et al. (2009). An open-source C++ imple-
228
+ mentation is available at http://bactra.org/CSSR/ . The CSMs generated by CSSR can be displayed graphically,
229
+ as we do in this paper, with the open-source program dot(http://www.graphviz.org/
230
+ 6For niteN, decreasing tends to yield simpler CSMs with fewer states. In a sense, it is a sort of regularization
231
+ coecient. The in
232
+ of this paper, varying in the range 0 :001< < 0:1 made little di erence.
233
+ 7If more than one such state sexists, we chose the one for which bP(Xtj^S=s) di ers least, in total variation
234
+ distance, from bP(Xtjt1
235
+ tL=axt1
236
+ tL+1), which is plausible and convenient. However, which state we chose is irrelevant
237
+ in the limit N!1 , so long as the di erence between the distributions is not statistically signi cant.
238
+ 8The conceptually similar algorithm of Kennel & Mees (2002) in e ect always creates a new state, which leads to
239
+ more complex models, sometimes in nitely more complex ones; see Shalizi & Klinkner (2004).5
240
+ The algorithm itself has three phases. Phase I initializes  to a single state, which contains only the null sux ;.
241
+ (That is,;is a sux of any string.) The length of the longest sux in  is L; this starts at 0. Phase II iteratively
242
+ tests the successive versions of the null hypothesis, Eq. 1, and Lincreases by one each iteration, until we reach some
243
+ maximum length . At the end of II, ^ is (approximately) next-step sucient. Phase III makes ^ recursively calculable,
244
+ by splitting the states until they have deterministic transitions. Under mild technical conditions (a nite true number
245
+ of states, etc.), CSSR converges in probability on the correct CSM as N!1 , provided only that  is long enough
246
+ to discriminate all of the states. The error of the predicted distributions of futures P(X1
247
+ t+1jXt
248
+ 1), measured by total
249
+ variation distance, decays as N1=2. Section 4 of Shalizi & Klinkner (2004) details CSSR's convergence properties.
250
+ Comparisons of CSSR's performance with that of more traditional expectation maximization based approaches can
251
+ also be found in Shalizi & Klinkner (2004) as can time complexity bounds for the algorithm. Depending upon the
252
+ machine used, CSSR can process an N= 106time series in under a minute.
253
+ 1. Choosing 
254
+ CSSR requires no a priori knowledge of the CSM's structure, but does need a choice of of ; here pick it by
255
+ minimizing the BIC of the reconstructed models over , i.e.,
256
+ BIC2 logL+dlogN (3)
257
+ whereLis the likelihood, Nis the data length and dis the number of model parameters, in our case the number
258
+ of predictive states9BIC's logarithmic-with- Npenalty term helps keep the number of causal states from growing
259
+ too quickly with increased data size, which is why we use it instead of the Akaike Information Criterion (AIC). Also,
260
+ BIC is known to be consistent for selecting the order of Markov chains and variable-length Markov models (Csisz ar
261
+ & Talata, 2006), both of which are sub-classes of CSMs.
262
+ Writing the observed spike train as xN
263
+ 1, and the state sequence as sN
264
+ 0, the total likelihood of the spike train is
265
+ L=X
266
+ sN
267
+ 02N+1P(XN
268
+ 1=xN
269
+ 1jSN
270
+ 0=sN
271
+ 0)P(SN
272
+ 0=sN
273
+ 0); (4)
274
+ the sum over all possible causal state sequences of the joint probability of the spike train and the state sequence.
275
+ Since the states update recursively, st+1=T(st;xt+1), the starting state s0and the spike train xN
276
+ 1 x the entire state
277
+ sequencesN
278
+ 0. Thus the sum over state sequences can be replaced by a sum over initial states
279
+ L=X
280
+ si2P(XN
281
+ 1=xN
282
+ 1jS0=si)P(S0=si) (5)
283
+ with the state probabilities P(S0=si) coming from the CSM. By the Markov property,
284
+ P(XN
285
+ 1=xN
286
+ 1jS0=si) =NY
287
+ j=1P(Xj=xjjSj1=sj1) (6)
288
+ Selecting  is now straightfoward: for each value of , we build the CSM from the spike train, calculate the
289
+ likelihood using Eq. 5 and 6, and pick the value, and CSM, minimizing Eq. 3. We try all values of  up to a model-
290
+ independent upper bound. For a wide range of stochastic processes, Marton & Shields (1994) showed that the length
291
+ mof subsequences for which probabilities can be consistently and non-parametrically estimated can grow as fast as
292
+ logN=h, wherehis the entropy rate, but no faster. CSSR estimates the distribution of the next symbol given the
293
+ previous  symbols, which is equivalent to estimating joint probabilities of blocks of length m=  + 1. Thus Marton
294
+ and Shield's result limits the usable values of :
295
+ logN
296
+ h1 (7)
297
+ 9The number of independent parameters dinvolved in describing the CSM will be (number of states)*(number of
298
+ symbols - 1) since the sum of the outgoing probabilities for each state is constrained to be 1. Thus, for a binary
299
+ alphabet,d= number of states.6
300
+ Using Eq. 7 requires the entropy rate h. The latter can either be upper bounded as the log of the alphabet size (here,
301
+ log 2 = 1), or by some other, less pessimistic, estimator of the entropy rate (such as the output of CSSR with  = 1).
302
+ Use of an upper bound on hresults in a conservative maximum value for . For example, a 30 minute experiment
303
+ with 1 msec time bins lets us use at least 20 by the most pessimistic estimate of h= 1; the actual maximum
304
+ value of  may be much larger. We use  25 in this paper but see no indication that this can't be extended further,
305
+ if need be.
306
+ 2. Condensing the CSM
307
+ For real neural data, the number of causal states can be very large | hundreds or more. This creates an interpreta-
308
+ tion problem, if only because it is hard to t such a CSM on a single page for inspection. We thus developed a way to
309
+ reduce the full CSM while still accounting for most of the spike train's structure. Our \state culling" technique found
310
+ the least-probable states and selectively removed them, appropriately redirecting state transitions and reassigning
311
+ state occupation probabilities. By keeping the most probable states, we focus on the ones which contribute the most
312
+ to the spike train's structure and complexity. Again, we used BIC as our model selection criterion.
313
+ First, we sorted the states by probability, nding the least probable state (\remove" state) with a single incoming
314
+ edge from a state (its \ancestor") with outgoing transitions to two di erent states, the remove state and a second,
315
+ \keep" state. We redirected both of the ancestor's outgoing edges to the keep state. Second, we reassigned the remove
316
+ state's outgoing transitions to the keep state. If the outgoing transitions from the keep state were still deterministic (at
317
+ most a single 0 emitting edge and a single 1 emitting edge), we stopped. If the transitions were non-deterministic, we
318
+ merged states reached by emitting 0s with each other (likewise those reached by 1s), repeating this until termination.
319
+ Third, we checked that there existed a state sequence of the new model which could generate the observed spikes. If
320
+ there was, we accepted the new CSM. If not, we rejected the new CSM and chose the next lowest probability state
321
+ from the original CSM to remove.
322
+ This culling was iterated until removing any state made it impossible for the CSM to generate the spike train. At
323
+ each iteration, we calculated BIC (as described in the previous section), and ultimate chose the culled CSM with the
324
+ minimum BIC. This gave a culled CSM for each value of ; the nal one we used was chosen after also minimizing
325
+ BIC over . The CSMs shown below in the results section paper result from this minimizing of BIC over  and state
326
+ culling.
327
+ 3. ISI Bootstrapping
328
+ While we do model selection with BIC, we also want to do model checking or adequacy-testing. For the most part,
329
+ we do this by using the CSM to bootstrap point-wise con dence bounds on the interspike interval (ISI) distribution,
330
+ and checking their coverage of the empirical ISI distribution. Because this distribution is not used by CSSR in
331
+ reconstructing the CSM, it provides a check on the latter's ability to accurately describe the spike-train's statistics.
332
+ Speci cally, we generated con dence bounds as follows. To simulate one spike train, we picked a random starting
333
+ state according to the CSM's inferred state-occupation probabilities, and then ran the CSM forward for Ntime-steps,
334
+ Nbeing the length of the original spike train. This gives a binary time-series, where a \1" stands for a spike and
335
+ a \0" for no-spike, and gave us a sample of inter-spike intervals from the CSM. This in turn gave an \empirical"
336
+ ISI distribution. Repeated over 104independent runs of the CSM, and taking the 0 :005 and 0:995 quantiles of the
337
+ distributions at each ISI length, gives 99% pointwise con dence bounds. (Pointwise bounds are necessary because of
338
+ the ISI distribution often modulates rapidly with ISI length.) If the CSM is correct, the empirical ISI will, by chance,
339
+ lie outside the bounds at 1% of the ISI lengths.
340
+ If we split the data into training and validation sets, a CSM reconstructed from the training set can be used to
341
+ bootstrap ISI con dence bounds, which can be compared to the ISI distribution of the test set. We discuss this sort
342
+ of of cross validation, as well as an additional test based on the time rescaling theorem, in Appendix B.7
343
+ C. Complexity and Algorithmic Information Content
344
+ The algorithmic information content K(xn
345
+ 1) of a sequence xn
346
+ 1is the length of the shortest complete (input-free)
347
+ computer program which will output xn
348
+ 1exactly and then halt (Cover & Thomas, 1991)10. In general, K(xn
349
+ 1) is
350
+ strictly uncomputable, but when xn
351
+ 1is the realization of a stochastic process Xn
352
+ 1, the ensemble-averaged algorithmic
353
+ information essentially coincides with the Shannon entropy (\Brudno's theorem"; see Badii & Politi (1997)), re
354
+ the fact that both are maximized for completely random sequences (Cover & Thomas, 1991). Both the algorithmic
355
+ information and the Shannon entropy can be conveniently written in terms of a minimal sucient statistic Q:
356
+ E[K(Xn
357
+ 1)] =H[Xn
358
+ 1] +o(n)
359
+ =H[Q] +H[Xn
360
+ 1jQ] +o(n) (8)
361
+ The equality H[Xn
362
+ 1] =H[Q] +H[Xn
363
+ 1jQ] holds because Qis a function of Xn
364
+ 1, soH[QjXn
365
+ 1] = 0.
366
+ The key to determining a spike train's expected algorithmic information is thus to nd a minimal sucient statistic.
367
+ By construction, causal state models provide exactly this; a minimal sucient statistic for xn
368
+ 1is the state sequence
369
+ sn
370
+ 0=s0;s1;:::sn(Shalizi & Crutch eld, 2001). Thus the ensemble-averaged algorithmic information content, dropping
371
+ termso(n) and smaller, is
372
+ E[K(Xn
373
+ 1)] =H[Sn
374
+ 0] +H[Xn
375
+ 1jSn
376
+ 0]
377
+ =H[S0] +nX
378
+ i=1H[SijSi1] +nX
379
+ i=1H[XijSi;Si1] (9)
380
+ Going from the rst to the second line uses the causal states' Markov property. Assuming stationarity, Eq. 9 becomes
381
+ E[K(Xn
382
+ 1)] =H[St] +n(H[StjSt1] +H[XtjSt;St1])
383
+ =C+n(J+R) (10)
384
+ This separates terms representing structure from those representing randomness.
385
+ The rst term in Eq. 10 is the complexity ,C, of the spike-generating process (Crutch eld & Young, 1989; Grass-
386
+ berger, 1986; Shalizi et al., 2004).
387
+ C=H[St] =E[logP(St)] (11)
388
+ Cis the entropy of the causal states, quantifying the structure present in the observed spikes. This is distinct from
389
+ the entropy of the spikes themselves, which quanti es not their structure but their randomness (and is approximated
390
+ by the other two terms). Intuitively, Cis the (time-averaged) amount of information about the past of the system
391
+ which is relevant to predicting its future. For example, consider again the IID 40 Hz Bernoulli process of Figure
392
+ 1 A. With p= 0:04, this has an entropy of 0 :24 bits/msec, but because it can be described by a single state, the
393
+ complexity is zero. (That state emits either a \0" or a \1", with respective probabilities 0 :96 and 0:04, but either way
394
+ the state transitions back to itself.) In contrast, adding a 5 ms refractory period to the process means six states are
395
+ needed to describe the spike trains (Figure 1 B). The new structure of the refractory period is quanti ed by the higher
396
+ complexity, C= 1:05 bits.
397
+ The second and third terms in Eq. 10 both describe randomness, but of distinct kinds. The second term, the
398
+ internal entropy rate J, quanti es the randomness in the state transitions; it is the entropy of the next state given
399
+ the current state.
400
+ J=H[St+1jSt] =E[logP(St+1jSt)] (12)
401
+ This is the average number of bits per time-step needed to describe the sequence of states the process moved through
402
+ (beyond those given by C). The last term in Eq. 10 accounts for any residual randomness in the spikes which is not
403
+ captured by the state transitions.
404
+ R=H[Xt+1jSt;St+1] =E[logP(Xt+1jSt;St+1)] (13)
405
+ 10The algorithmic information content is also called the Kolmogorov complexity. We do not use this term, to avoid
406
+ confusion with our \complexity" Cthe information needed to reproduce the spike train statistically rather than
407
+ exactly (Eq. 11). See Badii & Politi (1997) for a detailed comparison of complexity measures.8
408
+ For long trains, the entropy of the spikes, H[Xn
409
+ 1], is approximately the sum of these two terms, H[Xn
410
+ 1]n(J+R).
411
+ Computationally, Crepresents the xed generating structure of the process, which needs to be described once, at
412
+ the beginning of the time series, and n(J+R) represents the growing list of details which pick out a particular time
413
+ series from the ensemble which could be generated; this needs, on average, J+Rextra bits per time-step. (Cf. the
414
+ \sophistication" of G acs et al. (2001).)
415
+ Consider again the 40 Hz Bernoulli process. As there is only one state, the process always stays in that state. Thus
416
+ the entropy of the next state J= 0. However, the state sequence yields no information about the emitted symbols
417
+ (the process is IID), so the residual randomness R= 0:24 bits/msec | as it must be, since the total entropy rate is
418
+ 0:24 bits/msec. In contrast, the states of the 5 msec refractory process are informative about the process's future. The
419
+ internal entropy rate J= 0:20 bits/msec and the residual randomness R= 0. All of the randomness is in the state
420
+ transitions, because they uniquely de ne the output spike train. The randomness in the state transition is con ned
421
+ to stateA, where the process \decides" whether it will stay in A, emitting no spike, or emit a spike and go to B. The
422
+ decision needs, or gives, 0 :24 bits of information. The transitions from BthroughFand back to Aare xed and
423
+ contribute 0 bits, reducing the expected J.
424
+ The important point is that the structure present in the refractory period makes the spike train less random,
425
+ lowering its entropy. Averaged over time, the mean ring rate of the process is p= 0:0333. Were the spikes IID, the
426
+ entropy rate would be 0 :21 bits/msec, but in fact J+R= 0:20 bits/msec. This is because a minimal description
427
+ of a long sequence Xt1:::XtN=XtN
428
+ t1, the generating process only needs to be described once (C), while the internal
429
+ entropy rate and randomness need to be updated at each time step ( n(J+R)). Simply put, a complex, structured
430
+ spike train can be exactly described in fewer bits than one which is entirely random. The CSM lets us calculate this
431
+ reduction in algorithmic information, and quantify the structure by means of the complexity.
432
+ D. Time-Varying Complexity and Entropies
433
+ The complexity and entropy are ensemble-averaged quantities. In the previous section the ensemble was the entire
434
+ time series, and the averaged complexity and entropies were analogous to a mean ring rate. The time-varying
435
+ complexity and entropies are also of interest, for example their variation after stimuli. A peri-stimulus time histogram
436
+ (PSTH) shows how the ring probability varies with time; the same idea works for the complexity and entropy.
437
+ Since the states form a Markov chain, and any one spike train stays within a single ergodic component, we can
438
+ invoke the ergodic theorem (Gray, 1988), and (almost surely) assert that
439
+ X
440
+ St;St+1P(St;St+1;Xt+1)f(St;St+1;Xt+1) = lim
441
+ N!11
442
+ NNX
443
+ t=1f(St;St+1;Xt+1)
444
+ = lim
445
+ N!1hf(St;St+1;Xt+1)iN (14)
446
+ for arbitrary integrable functions f(St;St+1;Xt+1).
447
+ In the case of the mean ring rate, the function to time-average is l(t)Xt+1. For the time averaged-complexity,
448
+ internal entropy and residual randomness, the functions (respectively c,jandr) are
449
+ c(t) =logP(St)
450
+ j(t) =logP(St+1jSt)
451
+ r(t) =logP(Xt+1jSt;St+1) (15)
452
+ and time-varying entropy h(t) =j(t) +r(t).
453
+ The PSTH averages over an ensemble of stimulus presentations, rather than time:
454
+ PSTH (t) =1
455
+ MMX
456
+ i=1li(t) =1
457
+ MMX
458
+ i=1Xt+1;i (16)
459
+ withMbeing number of stimulus presentations, and tre-set to zero at each presentation. Analogously, the \PSTH"
460
+ of the complexity is
461
+ CPSTH (t) =1
462
+ MMX
463
+ i=1ci(t) =1
464
+ MMX
465
+ i=1logP(St;i) (17)9
466
+ For the entropies, replace cwithj,rorhas appropriate. Similar calculations can be made with any well-de ned
467
+ ensemble of reference times, not just stimulus presentations; we will also calculate cand the entropies as functions of
468
+ the time since the latest spike.
469
+ We can estimate the error these time-dependent quantities as the standard error of the mean as a function of time,
470
+ SEt=st=p
471
+ Mwherestis the sample standard deviation in each time bin tandMis the number of trials. The
472
+ probabilities appearing in the de nitions of c(t),j(t),r(t) also have some estimation errors, either because of sampling
473
+ noise or, more interestingly, because the ensemble is being distorted by outside in
474
+ between their averages (over time or stimuli) and what the CSM predicts for those averages. In the next section, we
475
+ explain how to use this to measure the in
476
+ E. The In
477
+ If we know that St=s, the CSM predicts that ring probability is (t) =P(Xt+1= 1jSt=s). By means of the
478
+ CSM's recursive ltering property (Appendix A), once a transient regime has passed, the state is always known with
479
+ certainty. Thereafter, the CSM predicts what the ring probability should be at all times, incorporating the e ects of
480
+ the spike train's history. As we show in the next section, these predictions give good matches to the actual response
481
+ function in simulations where the spiking probability depends only on the spike history. But real neurons' spiking
482
+ rates generally also depend on external processes, e.g., stimuli. As currently formulated, the CSM is (or, rather,
483
+ converges on) the optimal predictor of the future of the process given its own past. Such an \output only" model
484
+ does not represent the (possible) e ects of other processes, and so ignores external covariates and stimuli. Presently,
485
+ determining the precise form of spike trains' responses to external forces is best left to parametric models.
486
+ However, we can use output-only CSMs to learn something about the computation: the PSTH-calculated entropy
487
+ rateHPSTH (t) =JPSTH (t) +RPSTH (t) quanti es the extent to which external processes drive the neuron. (The
488
+ PSTH subscript is henceforce supressed.) Suppose we know the true ring probability true(t). At each time step,
489
+ the CSM predicts the ring probability CSM(t). IfCSM(t) =true(t), then the CSM correctly describes the spiking
490
+ and the PSTH entropy rate is
491
+ HCSM(t) =CSM(t) log [CSM(t)](1CSM(t)) log [1CSM(t)] (18)
492
+ However, if CSM(t)6=true(t), then the CSM mis-describes the spiking, because it neglects the in
493
+ processes. Simply put, the CSM has no way of knowing when the stimuli happen. The PSTH entropy rate calculated
494
+ using the CSM becomes
495
+ HCSM(t) =true(t) log [CSM(t)](1true(t)) log [1CSM(t)] (19)
496
+ Solvingtrue(t),
497
+ true(t) =HCSM(t) + log [1CSM(t)]
498
+ log [1CSM(t)]log [CSM(t)](20)
499
+ The discrepancy between CSM(t) andtrue(t) indicates how much of the apparent randomness in the entropy rate
500
+ is actually due to external driving. The true PSTH entropy rate Htrue(t) is
501
+ Htrue(t) =true(t) log [true(t)](1true(t)) log [1true(t)] (21)
502
+ The di erence between HCSM(t) andHtrue(t) quanti es, in bits, the driving by external forces as a function of the
503
+ time since stimulus presentation.
504
+ H=HCSM(t)Htrue(t)
505
+ =true(t) logtrue(t)
506
+ CSM(t)
507
+ + (1true(t)) log1true(t)
508
+ 1CSM(t)
509
+ (22)
510
+ This stimulus-driven entropy His the relative entropy or Kullback-Leibler divergence D(XtruekXCSM) between the
511
+ true distribution of symbol emissions and that predicted by the CSM. Information-theoretically, this relative entropy
512
+ is the error in our prediction of the next state due to assuming the neuron is running autonomously when it's actually
513
+ externally driven. Since every state corresponds to a distinct distribution over future behavior, this is our error in
514
+ predicting the future due to ignorance of the stimulus.11
515
+ 11Cf. the informational coherence introduced by Klinkner et al. (2006) to measure of information-sharing between10
516
+ III. RESULTS
517
+ We now present a few examples. (All of them use a time-step of 1 millisecond.) We begin with idealized model
518
+ neurons to illustrate our technique. We recover CSMs for the model neurons using only the simulated spike trains
519
+ as input to our algorithms. From the CSM we calculate the complexity, entropies, and, when appropriate, stimulus-
520
+ driven entropy (Kullback-Leibler divergence between the true and CSM predicted ring probabilities) of each model
521
+ neuron. We then analyze spikes recorded in vivo from a neuron in layer II/III of rat SI (barrel) cortex. We use spike
522
+ trains recorded both with and without external stimulation of the rat's whiskers. See Andermann & Moore (2006)
523
+ for experimental details.
524
+ A. Model neuron with a \soft" refractory period and bursting
525
+ We begin with a refractory, bursting model neuron, whose spiking rate depends only on the time since the last
526
+ spike. The baseline rate is 40 Hz. Every spike is followed by a 2 msec \hard" refractory period, during which spikes
527
+ never occur. The spiking rate then rebounds to twice its baseline, to which it slowly decays. (See dashed line in the
528
+ rst panel of Figure 3 B.) This history dependence mimics that of a bursting neuron, and is, intuitively, more complex
529
+ than the simple refractory period of the model in Figure 1.
530
+ Figure 2 shows the 17-state CSM reconstructed from a 200 second spike train (at 1 msec resolution) generated by
531
+ this model. It has a complexity of C= 3:16 bits (higher than that of the model in Figure 1, as anticipated), an
532
+ internal entropy rate of J= 0:25 bits/msec and a residual randomness of R= 0 bits/msec. The CSM was obtained
533
+ with  = 17 (selected by BIC). Figure 3 A shows how the 99% ISI bounds bootstrapped from the CSM enclose the
534
+ empirical ISI distribution, with the exception of one short segment.
535
+ The CSM is easily interpreted. State Ais the baseline state. When it emits a spike, the CSM moves to state
536
+ B. There are then two deterministic transitions, to Cand thenD, which never emit spikes; this is the hard 2 msec
537
+ refractory period. Once in Dit is possible to spike again, and if that happens, the transition is back to state B.
538
+ However, if no spike is emitted, the transition is to state E. This is repeated, with varying ring probabilities, as
539
+ statesEthroughQare traversed. Eventually, the process returns to Aand so to baseline.
540
+ Figure 3 B plots the ring rate, complexity, and internal entropies as functions of the time since the last spike
541
+ conditional on no subsequent spike emission . This lets us compare the ring rate predicted by the CSM (solid line
542
+ squares) to the speci cation of the model which generated the spike train (dashed line) and a PSTH calculated by
543
+ triggering on the last spike (solid line). Except at 16 and 17 msec post spike, the CSM-predicted ring rate agrees
544
+ with both the generating model and the PSTH. The discrepancy arises because the CSM only discerns the structure
545
+ in the data, and most of the ISIs are shorter than 16 msec. There is much closer agreement between the CSM and
546
+ the PSTH if ring rates are plotted as a function of time since a spike without conditioning on no subsequent spike
547
+ emission (not shown).
548
+ The second and third panels of Figure 3 plot the time-dependent complexity and entropies. The complexity is
549
+ much higher after the emission of a spike than during baseline, because the states traversed (B-Q) are less probable,
550
+ and represent the additional structures of refractoriness and bursting. The time-dependent entropies (third panel)
551
+ show that just after a spike, the refractory period imposes temporary determinism on the spike train, but burstiness
552
+ increases the randomness before the dynamics return to the baseline state.
553
+ B. Model neuron under periodic stimulation
554
+ Figure 4 shows the CSM for a periodically-stimulated model neuron . This CSM was reconstructed from 200 seconds
555
+ of spikes with a baseline ring rate of 40 Hz ( p= 0:04). Each second, the ring rate rose over the course of 5 msec to
556
+ p= 0:54 spikes/msec, falling slowly back to baseline over the next 50 msec. This mimics the periodic presentation of
557
+ a strong external stimulus. (The exact inhomogeneous ring rate used was (t) = 0:93[et=10et=2] + 0:04 witht
558
+ in msec. See Figure 5 B, rst panel, dashed line.) In this model, the ring rate does not directly depend on the spike
559
+ train's history, but there is a sort of history dependence in the stimulus time-course, and this is what CSSR discovers.
560
+ BIC selected  = 7, giving a 16 state CSM with C= 0:89 bits,J= 0:27 bits/msec and R= 0:0007 bits/msec. The
561
+ baseline is again state Aand if no spike is emitted then the process stays in A. Spikes are either spontaneous and
562
+ neurons, by quantifying the error in predicting the distribution of the future of one neuron due to ignoring its
563
+ coupling with another.11
564
+ random, or stimulus-driven. Because the stimulus is external, it is not immediately clear which of these two causes
565
+ produced a given spike. Thus, if a spike is emitted, the CSM traverses states BthroughF, deciding, so to speak,
566
+ whether or not the spike is due to a stimulus. If two spikes happen within 3 msec of each other, then the CSM decides
567
+ that it is being stimulated and goes to one of states G,HorM. StatesGthroughPrepresent the response to the
568
+ stimulus. The CSM moves between these states until no spike is emitted for 3 msec, when it returns to the baseline,
569
+ A.
570
+ The ISI distribution from the CSM matches that from the model (Figure 5 A). However, because the stimulus
571
+ doesn't depend on spike train's history, the CSM makes inaccurate predictions during stimulation. The rst panel of
572
+ Figure 5 B plots the ring rate as a function of time since stimulus presentation, comparing the model (dashed line)
573
+ and the PSTH (solid line) with the CSM's prediction (line with squares). The discrepancy between these is due to
574
+ the CSM having no way of knowing that an external stimulus has been applied until several spikes in a row have been
575
+ emitted (represented, as we just say, by states B{F)12. Despite this, c(t) shows that something more complex than
576
+ simple random ring is happening (second panel of Figure 5 B), as do j(t) andr(t) (third panel). Further, something
577
+ is clearly wrong with the entropy rate, because it should be upper-bounded by h= 1 bit/msec (when p= 0:5). The
578
+ fact thath(t) exceeds this bound indicates an external force, not fully captured by the CSM, is at work.
579
+ As discussed in Methods ( xII.E), drive from the stimulus can be quanti ed with a relative entropy (Figure 5 C).
580
+ Stimuli are presented at t= 1 msec, where  H(t)>1 bit. It is not until 25 msec post-stimulus that  H(t)0 and
581
+ the CSM once again correctly describes the internal entropy rate. Thus, as expected, the stimulus strongly in
582
+ neuronal dynamics immediately after its presentation. The true internal entropy rate Htrue(t) is slightly less than 1
583
+ bit/msec shortly after stimulation, when the true spiking rate has a maximum of pmax= 0:54. The fact that the CSM
584
+ gives an inaccurate value for Jactually lets us nd the number of bits of information gain supplied by the stimulus,
585
+ e.g., H > 1 bit immediately after the stimulus is presented.
586
+ C. Spontaneously Spiking Barrel Cortex Neuron
587
+ We reconstructed a CSM from 90 seconds of spontaneous (no vibrissa de
588
+ II/III FSU barrel cortex neuron. CSSR, using  = 21, discovered a CSM with 315 states, a complexity of C= 1:78
589
+ bits, and internal entropy rate of J= 0:013 bits/msec. After state culling ( xII.B.2), the reduced CSM, plotted in
590
+ Figure 6, has 14 states, C= 1:02,J= 0:10 bits/msec, and residual randomness of R= 0:005 bits/msec. We focus on
591
+ the reduced CSM from this point onwards.
592
+ This CSM resembles that of the spontaneously- ring model neuron of xIII.A and Fig. 2. The complexity and
593
+ entropies are lower than those of our model neuron because the mean spike rate is much lower, and so simple
594
+ descriptions suce most of the time. (Barrel cortex neurons exhibit notoriously low spike rates, especially during
595
+ anesthesia.) There is a baseline state Awhich emits a spike with probability p= 0:01, i.e., 10 Hz. When a spike is
596
+ emitted, the CSM moves to state Band then on through the chain of states CthroughN, return to Aif no spike is
597
+ subsequently emitted. However, the CSM can emit a second or even third spike after the rst, and indeed this neuron
598
+ displays spike doublets and triplets. In general, emitting a spike moves the CSM to B, with some exceptions that
599
+ show the structure to be more intricate than the model neuron's.
600
+ Figure 7 A shows the CSM's 99% con dence bounds almost completely enclosing the empirical ISI distribution.
601
+ The rst panel of Figure 7 B plots the history-dependent ring probability predicted by the CSM as a function of the
602
+ time since the latest spike, according to both the PSTH and the CSM's prediction. They are highly similar in the
603
+ rst 13 msec post-spike, indicating that the CSM gets the spiking statistics right in this epoch. The CSM and PSTH
604
+ the diverge after this, for two reasons. First, as with the model neuron, there are few ISIs of this length. Most of the
605
+ ISIs are either shorter, due to the nueron's burstiness, or much longer, due to the low baseline ring rate. Secondly,
606
+ 90 seconds is not very much data. We show in Figure 10 that a CSM reconstructed from a longer spike train does
607
+ capture all of the structure. We present the results of this shorter spike train to emphasize that, as a non-parametric
608
+ method, CSSR only uncovers the statistical structure in the data , no more, no less.
609
+ Finally, the second and third panels of Figure 6 B show, respectively, the complexity and entropies as functions of
610
+ the time since the latest spike. As with the model of xIII.A, the structure in the process occurs after spiking, during
611
+ the refractory and bursting periods. This is when the complexity is largest, and also when the entropies vary most.
612
+ 12In e ect, this part of the CSM implements Bayes's rule, balancing the increased likelihood of a spike after a stimulus
613
+ against the low a priori probability or base-rate of stimulation.12
614
+ D. Periodically Stimulated Barrel Cortex Neuron
615
+ We reconstructed CSMs from 335 seconds of spike trains taken from the same neuron used above, but recorded
616
+ while it was being periodically stimulated by vibrissa de
617
+ in Figure 8. (Before state culling, the original CSM had 1916 states, C= 2:55 andJ= 0:11.) The reduced CSM has a
618
+ complexity of C= 1:97 bits, an internal entropy rate of J= 0:10 bits/msec, and a residual randomness of R= 0:005
619
+ bits/msec. Note that Cis higher when the neuron is being stimulated as opposed to when it is spontaneously ring,
620
+ indicating more structure in the spike train.
621
+ While at rst the CSM may seem to only represents history-dependent refractoriness and bursting, ignoring the
622
+ external stimulus, this is not quite true. Once again, there is a baseline state A, and most of the other states ( B{X)
623
+ comprise a refractory/bursting chain, like this neuron has during spontaneous ring. However, the transition upon A
624
+ emitting a spike is not back to Band then down the chain again, but to either state C1, and subsequently C2, or more
625
+ often to state ZZ. These three states represent the structure induced by the external stimulus, as we saw with the
626
+ model stimulated neuron of xIII.B and Figure 4. (The state ZZis comparable to the state Mof the model stimulated
627
+ neuron: both loop back to themselves if they emit a spike.) Three states are enough because, in this experiment,
628
+ barrel cortex neurons spike extremely sparsely, 0 :1{0:2 spikes per stimulus presentation.
629
+ Figure 9 A plots the ISI distribution, nicely enclosed by the bootstrapped con dence bounds. Figure 9 B shows
630
+ the ring rate, complexity and entropies as functions of the time since stimulus presentation (averaged over all
631
+ presentations). These plots look much like those in Figure 7 B. However, there is a clear indication that something
632
+ more complex takes place after stimulation: the CSM's ring-rate predictions are wrong. The stimulus-driven entropy
633
+ Hturns out to be as large as 0 :02 bits within 5{15 msec post-stimulus. This agrees with the known 5{10 msec
634
+ stimulus propagation time between vibrissae and barrel cortex (Andermann & Moore, 2006). The reason that  H
635
+ is so much smaller for the real neuron than the stimulated model neuron of xIII.B is that the former's ring rate is
636
+ much lower. Although the ring rate post-stimulus can be almost twice as large as the CSM's prediction, the actual
637
+ rate is still low, max (t)0:04 spikes/msec. Most of the time the neuron does not spike, even when stimulated, so
638
+ on average, the stimulus provides little information per presentation. For completeness, Figure 10 shows the spike
639
+ probability, complexity and entropies as functions of the time since the latest spike. Averaged over this ensemble, the
640
+ CSM's predictions are highly accurate.
641
+ IV. DISCUSSION
642
+ The goal of this paper was to present methods for determining the structural content of spike trains while making
643
+ minimal a priori assumptions as to the form which that structure takes. We use the CSSR algorithm to build
644
+ minimal, optimally predictive hidden Markov models (CSMs) from spike trains, Schwartz's Bayesian Information
645
+ Criterion to nd the optimal history length  of the CSSR algorithm, and bootstrapped con dence bounds on the
646
+ ISI distribution from the CSM to check goodness-of- t. We demonstrated how CSMs can estimate a spike train's
647
+ complexity, thus quantifying its structure, and its mean algorithmic information content, quantifying the minimal
648
+ computation necessary to generate the spike train. Finally we showed how to quantify, in bits, the in
649
+ external stimuli upon the spike-generating process. We applied these methods both to simulated spike trains, for
650
+ which the resulting CSMs agreed with intuition, and to real spike trains recorded from a layer II/II rat barrel cortex
651
+ neuron, demonstrating increased structure, as measured by the complexity, when the neuron was being stimulated.
652
+ We are unaware of any other practical techniques for quantifying the complexity and computational structure of
653
+ a spike train as we de ne them. Intuitively, neither random (Poisson), nor highly ordered (e.g., strictly periodic, as
654
+ in Olufsen et al. (2003)) spike trains should be thought of as complex since they do not possess structure requiring a
655
+ sophisticated program to generate. Instead, complexity lies between order and disorder (Badii & Politi, 1997), in the
656
+ non-random variation of the spikes. Higher complexity means a greater degree of organization in neural activity than
657
+ would be implied by random spiking. It is the reconstruction of the CSM through CSSR which allows us to calculate
658
+ the complexity.
659
+ Our de nition of complexity stands in stark contrast to other complexity measures which assign high values to
660
+ highly disordered systems. Some of these, such as Lempel Ziv complexity (Amigo et al., 2002, 2004; Jimenez-Montano
661
+ et al., 2002; Szczepanski et al., 2004) and context free grammar complexity (Rapp et al., 1994) have been applied to
662
+ spike trains. However, both of these are measures of the amount of information required to reproduce the spike train
663
+ exactly , and take on very high values for completely random sequences. These \complexity" measures are therefore
664
+ much more similar to total algorithmic information content and even to the entropy rate than to our sort of complexity.
665
+ Our measure of complexity is the entropy of the distribution of causal states. This has the desired property of
666
+ being maximized for structured, rather than ordered or disordered systems, because the causal states are de ned
667
+ statistically, as equivalence classes of histories conditioned on future events. Other researchers have also calculated13
668
+ complexity measures which are entropies of state distributions, but have de ned their states di erently. Amigo et al.
669
+ (2002) uses the observables (symbol strings) present in the spike train to de ne a k-th order Markov process and calls
670
+ each individual length k string which appears in the spike train a state. Gorse and Taylor (1990) similarly use single
671
+ sux symbol strings to de ne the states of a Markov process. In both cases, IID Bernoulli sequences could exhibit
672
+ up to 2kstates (in long enough sequences), and possess an extremely high \complexity". However, all of these states
673
+ make the same prediction for the future of the process. The minimal representation is a single causal state, a CSM
674
+ with a complexity of zero.
675
+ It should be noted that there are also many works which model spike trains using HMMs, but in which the hidden
676
+ states represent macro -states of the system (awake/asleep, Up/Down, etc.), and spiking rates are modeled separately
677
+ in each macro-state (Abeles et al., 1995; Achtman et al., 2007; Chen et al., 2008; Danoczy & Hahnloser, 2005; Jones
678
+ et al., 2007). Although the graphical representation of such HMMs may look like those of CSMs, the two kinds of
679
+ states have very di erent meanings. Finally, there are also state space methods which model the dynamical state of
680
+ the system as a continuous hidden variable, the most well known of which is the linear Gaussian model with Kalman
681
+ ltering. These have been extensively applied to neural encoding and decoding problems (Eden et al., 2007; Smith
682
+ et al., 2004; Srinivasan et al., 2007). Interestingly, for a univariate Gaussian ARMA model in state-space form, the
683
+ Kalman lter's one-step-ahead prediction and mean-squared prediction error are, jointly, minimal-sucient for next-
684
+ step prediction, and since they can be updated recursively they in fact constitute the minimal sucient statistic, and
685
+ hence the causal state in this special case.
686
+ Neurons are driven by their a erent synapses. Although as discussed in Appendix C, there is a parallel \transducer"
687
+ formalism for generating CSMs which take external in
688
+ mented, and our current approach reconstructs CSMs only from the spike train. Since the history of the neuron under
689
+ study is typically connected with the history of the network in which it is located, this CSM will, in general, re
690
+ more than a neuron's internal biophysical properties. Nonetheless, in both our model neurons and in the real barrel
691
+ cortex neuron, states not interpretable as simple refractoriness or bursting appeared when a stimulus was present,
692
+ proving we can detect stimulus-driven complexity. Further, we showed that the CSM can be used to determine the
693
+ extent (in bits) to which a neuron is driven by external stimuli.
694
+ The methods presented here complement more established modes of spike-train analysis, which have di erent goals.
695
+ Parametric methods, such as PSTHs or maximum likelihood estimation (Brown et al., 2004; Truccolo et al., 2005)
696
+ generally focus on determining a neuron's ring rate (mean, instantaneous or history-dependent), and on how known
697
+ external covariates modulate that rate. They have the advantage of requiring less data than non-parametric methods
698
+ such as CSSR, but the disadvantage, for our purposes, of imposing the structure of the model at the outset. When
699
+ the experimenter wants to know how a neuron encodes a particular aspect of a covariate, e.g., how neurons in the
700
+ sensory periphery or primary sensory cortices encode stimuli, parametric methods have proved highly illuminating.
701
+ However, in many cases the identity or even existence of relevant external covariates is uncertain. For example, one
702
+ could envision using CSMs to analyze recordings in pre-frontal cortex during di erent cognitive tasks, or to perhaps
703
+ compare spiking structure during di erent attentional states. In both cases, the relevant external covariates are not
704
+ at all clear, but CSMs could still be used to quantify changes in computational structure, for single neurons or for
705
+ groups of them. For neural populations one can envision generating distributions (over the population) of complexities
706
+ and examining how these distributions change in di erent cortical macro-states. This would be entirely analagous to
707
+ analyzing distributions of ring rates or tuning curves.
708
+ In addition to calculations of the complexity, the whole array of mutual-information analyses can be applied to
709
+ CSMs, but instead of calculating mutual information between the spikes and the covariates (which could include other
710
+ spike trains), one can calculate the mutual information between the covariates and the causal states . The advantage
711
+ is that the causal states represent the behavioral patterns of the spike-generating process, and so are closer to the
712
+ actual state of the system than the spikes (output observables) are themselves. Results on calculating the mutual
713
+ information between the causal states of di erent neurons (informational coherence) in a large simulated network
714
+ show that synchronous neuronal dynamics are more e ectively revealed than when calculated directly from the spikes
715
+ (Klinkner et al., 2006).
716
+ In closing, our methods provide a way to understand structure in spike trains, and should be considered as comple-
717
+ ments to traditional analysis methods. We rigorously de ne structure, and show how to discover it from the data itself.
718
+ Our methods go beyond those which seek to describe the observed variation in the spiking rates by also describing
719
+ the underlying computational process (in the form of a CSM) needed to generate that variation. A CSM can show
720
+ not only that the spike rate has changed, but also show exactly howit has changed.
721
+ Acknowledgments The authors thank Mark Andermann and Christopher Moore for the use of their data. RH thanks
722
+ Emery Brown, Anna Dreyer and Christopher Moore for valuable discussions. CRS thanks Anthony Brockwell, Dave
723
+ Feldman, Chris Genovese, Rob Kass and Alessandro Rinaldo for valuable discussions.14
724
+ APPENDIX A: Filtering with CSMs
725
+ A common diculty with hidden Markov models is that predictions can only be made from a knowledge of the state,
726
+ which must itself be guessed at from the time series, since it is, after all, hidden. This creates the state estimation
727
+ or ltering problem. Under strong assumptions (linear Gaussian stochastic dynamics, linearly observed through IID
728
+ additive Gaussian noise) the Kalman lter is an optimal yet tractable solution. For non-linear processes, however,
729
+ optimal ltering essentially amounts to maintaining a posterior distribution over the states and updating it via Bayes's
730
+ rule (Ahmed, 1998). (This distribution is sometimes called the process's \information state".)
731
+ One convenient and important feature of CSMs is that this whole machinery of ltering is unnecessary, because of
732
+ their recursive-updating property. Given the state at time t,St, and the observation at time t+ 1,Xt+1, the state at
733
+ timet+ 1 is xed, St+1=T(St;Xt+1) for some transition function T. Clearly, if the state is known with certainty
734
+ at any time, it will remain known. However, the same recursive updating property also allows us to show that the
735
+ state does become certain, i.e., that after some nite (but possibly random) time ,P(S=sjX
736
+ 1) is either 0 or 1 for
737
+ all statess. For Markov chains of order k, clearlyk; under more general circumstances P(t) goes to zero
738
+ exponentially or faster.
739
+ Thus, after a transient period, the state is completely unambiguous. This will be useful to us in multiple places,
740
+ including understanding the computational structure of the process and predicting the ring rate of the neuron. It also
741
+ leads to considerable numerical simpli cations, compared to approaches which demand conventional ltering. Further,
742
+ recursive ltering is easily applied to a new spike train, not merely the one from which the CSM was reconstructed.
743
+ This helps in cross-validating CSMs, as discussed in the next appendix.
744
+ APPENDIX B: Cross-Validation
745
+ It is often desirable to cross-validate a statistical model by spliting one's data set in two, using one part (generally
746
+ the larger) as a training set for the model and the other part to validate the model by some statistical test. In the
747
+ case of CSMs it is particularly important to check the validity of the BIC used to regularize the  control-setting.
748
+ One possible test is the ISI bootstrapping of xII.B.3. A second, somewhat stronger, goodness-of- t test is based
749
+ on the time rescaling theorem of Brown et al. (2002). This test rescales the interspike intervals as a function of the
750
+ integrated history-dependent spiking rate over the ISI:
751
+ k= 1eRtk+1
752
+ tk(t)dt(B1)
753
+ where theftkgare the spike times and (t) is the history-dependent spiking rate from the CSM. If the CSM describes
754
+ the data well, then rescaled ISI's fkgshould follow a uniform distribution. This can be tested using either a
755
+ Kolmogorov Smirnov test or by plotting the empirical CDF of the rescaled times against the CDF of the uniform
756
+ distribution (Kolmogorov Smirnov or \KS" plot) (Brown et al., 2002).
757
+ Figure 11 gives cross-validation results for the rat barrel cortex neuron, during both spontaneous ring and periodic
758
+ vibrissae de
759
+ validation set. The 335 seconds of stimulus-evoked ring were split into a 270 second training set and a 65 second
760
+ validation set. Panels A and B show the ISI bootstrapping results for the spontaneous and stimulus evoked ring
761
+ respectively. The dashed lines are 99% con dence bounds from a CSM reconstructed from the training set and the
762
+ solid line is the ISI distribution of the validation set. The ISI distribution largely falls within these bounds for both
763
+ the spontaneous and stimulus evoked data.
764
+ Panels C-F display the time rescaling test. Panels C and D show the time rescaling plots for the spontaneous and
765
+ stimulus evoked training data respectively. The dashed lines are 95% con dence bounds. The spontaneous KS plot
766
+ largely falls within the bounds. The stimulus-evoked does not, but this is expected because, as discussed, the CSM
767
+ does not completely capture the imposition of the external stimulus. (The jagged \steps" in both plots result from
768
+ the 1 msec temporal discretization.) Panels E and F show the time rescaling plots for, respectively, the spontaneous
769
+ and stimulus evoked validation data. The ts here are somewhat worse. In the stimulated case, this is not surprising.
770
+ In the spontaneous case the cause is likely non-stationarity in the data, a problem shared with other spike train
771
+ analysis techniques, such as the Generalized Linear Model approaches described in the next Appendix. It should be
772
+ emphasized that the point of reconstructing CSMs is not to obtain perfect ts to the data, but instead to estimate
773
+ the structure inherent in the spike train, and the cross-validation results should be viewed in this light.15
774
+ APPENDIX C: Causal State Transducers and Predictive State Representations
775
+ Mathematically, CSMs can be expanded to include the in
776
+ state transducers , which are optimal representations of the history-dependent mapping from inputs to outputs (Shalizi,
777
+ 2001, ch. 7). Such causal state transducers are a type of partially-observable Markov decision process, closely related
778
+ to predictive state representations (PSRs) (Littman et al., 2002). In both formalisms, the right notion of \state"
779
+ is a statistic, a measurable function of the observable past of the process. Causal states represent this through an
780
+ equivalence relation on the space of observable histories. For PSRs, the representation is through \tests", i.e., a dis-
781
+ tinguished set input/output sequence pairs; the idea is that states can be uniquely characterized by their probabilities
782
+ of producing the output sequences conditional on the input sequences.
783
+ An algorithm for reconstructing causal state transducers would begin by estimating probability distributions of
784
+ future histories conditioned on both the history of the spikes and the history of an external covariate Y, e.g.
785
+ P(X1
786
+ t+1jXt
787
+ 1;Yt
788
+ 1), and otherwise be entirely parallel to CSSR. This has not yet implemented.
789
+ References
790
+ Abeles, M., Bergman, H., Gat, I., Meilijson, I., Seidemann, E., Tishby, N. & Vaadia, E. (1995). Cortical ativity
791
+ quasi-stationary states. Proc. Natl. Acad. Sci. USA. ,92, 8616{8620.
792
+ Achtman, N., Afshar, A., Santhanam, G., Yu, B. M., Ryu, S. I., & Shenoy, K. V. (2007). Free paced high-performance
793
+ brain-computer interfaces. Journal of Neural Engineering ,4, 336{347.
794
+ Ahmed, N. U. (1998). Linear and nonlinear ltering for scientists and engineers . Singapore: World Scienti c.
795
+ Amigo, J. M., Szczepanski, J.,Wajnryb, E., Sanchez-Vives, M. V. (2002). On the Number of States of the Neuronal Sources.
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+ Biosystems ,68, 57{66.
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+ Amigo, J. M., Szczepanski, J.,Wajnryb, E., Sanchez-Vives, M. V. (2004). Estimating the Entropy Rate of Spike Trains via
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+ Lempel-Ziv Complexity. Neural Computation ,16, 717{736.
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+ Andermann, M. L. & Moore, C. I. (2006). A sub-columnar direction map in rat barrel cortex. Nature Neuroscience ,9, 543{551.
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+ Badii, R. & Politi, A. (1997). Complexity: Hierarchical structures and scaling in physics . Cambridge, England: Cambridge
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+ University Press.
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+ Bernardo, J. M. & Smith, A. F. M. (1994). Bayesian Theory . New York: Wiley.
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+ Brown, E. N., Barbieri, R., Ventura, V., Kass, R. E., & Frank, L. M. (2002). The time rescaling theorem and its application
804
+ to neural spike train data analysis. Neural Computation ,14, 325{346.
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+ Brown, E. N., Kass, R. E., & Mitra, P. P. (2004). Multiple neural spike train data analysis: State-of-the-art and future
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+ challanges. Nature Neuroscience ,7, 456{461.
807
+ Chen, Z., Vijayan, S., Barbieri, R., Wilson, M. A., & Brown, E. N. (2008) Discrete- and continuous- time probablistic models
808
+ and inference algorithms for neuronal decoding of Up and Down states. In review at Neural Computation.
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+ Cover, T. M. & Thomas, J. A. (1991). Elements of Information Theory. New York: Wily.
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+ Crutch eld, J. P. & Young, K. (1989). Inferring statistical complexity. Physical Review Letters ,63, 105{108.
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+ Csisz ar, I. & Talata, Z. (2006). Context tree estimation for not necessarily nite memory processes, via BIC and MDL. IEEE
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+ Transactions on Information Theory ,52, 1007{1016.
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+ Danoczy, M. G., Hahnloser, R. H. R. (2005). Ecient Estimation of Hidden State Dynamics. Advances in Neural Information
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+ Processing Systems (NIPS 2005) Cambridge, Massachusetts. MIT Press.
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+ Eden, U. T., Frank, L. M., Barbieri, R., Solo, V. & Brown, E. N.. (2004). Dynamic analysis of neural encoding by point process
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+ adaptive ltering Neural Computation ,16, 971{998.
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+ G acs, P., Tromp, J. T., & Vitanyi, P. M. B. (2001). Algorithmic statistics. IEEE Transactions on Information Theory ,47,
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+ 2443{2463.
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+ Gorse, D., Taylor, J. G. (1990). A General Model of Stochastic Neural Processing. Biological Cybernetics ,63, 299{306.
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+ Grassberger, P. (1986). Toward a quantitative theory of self-generated complexity. International Journal of Theoretical Physics ,
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+ 25, 907{938.
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+ Gray, R. M. (1988). Probability, random processes, and ergodic properties . New York: Springer-Verlag.
823
+ Jaeger, H. (2000). Observable operator models for discrete stochastic time series. Neural Computation ,12, 1371{1398.
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+ Jimenez-Montano, M. A., Ebeling, W., Pohl, T., Rapp, P. E. (2002). Entropy and Complexity of Finite Sequences and
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+ Fluctuating Quantities. Biosystems ,64, 23{32.
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+ Jones, L. M., Fontanini, A., Sadacca, B. F., & Katz, D. B. (2007). Natural stimuli evoke dynamic sequences of states in sensory
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+ cortical ensembles. Proc. Natl. Acad. Sci. USA. ,104, 18772{18777.
828
+ Kennel, M. B. & Mees, A. I. (2002). Context-tree modeling of observed symbolic dynamics. Physical Review E ,66, 056209.
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+ Klinkner, K. L. & Shalizi, C. R. (2009). CSSR: A nonparametric algorithm for predicting and classifying time series. Manuscript
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+ in preparation.
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+ Klinkner, K. L., Shalizi, C. R., & Camperi, M. F. (2006). Measuring shared information and coordinated activity in neuronal
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+ networks. In Weiss, Y., Sch olkopf, B., & Platt, J. C. (Eds.), Advances in neural information processing systems 18 (NIPS
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+ 2005) , (pp. 667{674), Cambridge, Massachusetts. MIT Press.
834
+ Knight, F. B. (1975). A predictive view of continuous time processes. Annals of Probability ,3, 573{596.16
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+ Littman, M. L., Sutton, R. S., & Singh, S. (2002). Predictive representations of state. In Dietterich, T. G., Becker, S., &
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+ Ghahramani, Z. (Eds.), Advances in neural information processing systems 14 (NIPS 2001) , (pp. 1555{1561)., Cambridge,
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+ Massachusetts. MIT Press.
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+ Lohr, W. & Ay, N. (2009). On the Generative Nature of Prediction. Advances in Complex Systems , forthcoming.
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+ Marton, K. & Shields, P. C. (1994). Entropy and the consistent estimation of joint distributions. Annals of Probability ,22,
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+ 960{977. Correction, Annals of Probability ,24(1996): 541{545.
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+ McCulloch, W. S. & Pitts, W. (1943). A logical calculus of the ideas immanent in nervous activity. Bulletin of Mathematical
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+ Biophysics ,5, 115{133.
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+ Olufsen, M. S., Whittington, M. A., Camperi, M., & Kopell, N. (2003). New Roles for the Gamma Rhythm: Population Tuning
844
+ and Processing for the Beta Rhythm. Journal of Computation Neuroscience ,14, 33{54.
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+ Rapp, P. E., Zimmerman, I. D., Vining, E. P., Cohen, N.,Albano, A. M., Jimenez-Montano, M. A. (1994). The Algorithmic
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+ Complexity of Neural Spike Trains Increases During Focal Seizures. Journal of Neuroscience ,14, 4731{4739.
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+ Rieke, F., Warland, D., de Ruyter van Steveninck, R., & Bialek, W. (1997). Spikes: Exploring the neural code . Cambridge,
848
+ Massachusetts: MIT Press.
849
+ Shalizi, C. R. (2001). Causal architecture, complexity and self-organization in time series and cellular automata . PhD thesis,
850
+ University of Wisconsin-Madison.
851
+ Shalizi, C. R. & Crutch eld, J. P. (2001). Computational mechanics: Pattern and prediction, structure and simplicity. Journal
852
+ of Statistical Physics ,104, 817{879.
853
+ Shalizi, C. R. & Klinkner, K. L. (2004). Blind construction of optimal nonlinear recursive predictors for discrete sequences. In
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+ Chickering, M. & Halpern, J. Y. (Eds.), Uncertainty in arti cial intelligence: Proceedings of the twentieth conference (UAI
855
+ 2004) , (pp. 504{511)., Arlington, Virginia. AUAI Press.
856
+ Shalizi, C. R., Klinkner, K. L., & Haslinger, R. (2004). Quantifying self-organization with optimal predictors. Physical Review
857
+ Letters ,93, 118701.
858
+ Shalizi, C. R., Rinaldo, A., & Klinkner, K. L. (2009). Adaptive nonparametric prediction and bootstrapping of discrete time
859
+ series. Manuscript in preparation.
860
+ Singh, S., Littman, M. L., Jong, N. K., Pardoe, D., & Stone, P. (2003) Learning predictive state representations. In T. Fawcett
861
+ and N. Mishra, editors, Proceedings of the Twentieth International Conference on Machine Learning (ICML-2003) , (pp.
862
+ 712-719). AAAI Press.
863
+ Smith, A. C., Frank, L. M., Wirth, S., Yanike, M., Hu, D., Kubota, Y., Graybiel, A. M., Suzuki, W. A., & Brown, E. N. (2004).
864
+ Dynamic analysis of learning in behavioral experiments. Journal of Neuroscience ,24, 447{461.
865
+ Srinivasan, L., Eden, U. T., Mitter, S. K., & Brown, E. N. (2007). General purpose lter design for neural prosthetic devices.
866
+ Journal of Neurophysiology ,98, 2456{2475.
867
+ Szczepanski, J., Amigo, J. M., Wajnryb, E., Sanchez-Vives, M. V. (2004). Characterizing spike trains with Lempel-Ziv com-
868
+ plexity. Neurocomputing ,58, 79{84.
869
+ Truccolo, W., Eden, U. T., Fellow, M. R., Donoghue, J. P., & Brown, E. N. (2005). A point process framework for relating
870
+ neural spiking activity to spiking history, neural ensemble and covariate e ects. Journal of Neurophysiology ,93, 1074{1089.17
871
+ A 0 | 0.96 1 | 0.04
872
+ A0 | 0.96 B 1 | 0.04 C0 | 1 D0 | 1
873
+ E0 | 1
874
+ F0 | 1
875
+ 0 | 1A
876
+ B
877
+ FIG. 1 Two simple CSMs reconstructed from 200 sec of simulated spikes using CSSR. States are represented as the nodes
878
+ of a directed graph. The transitions between states are labeled with the symbol emitted during the transition (1 = spike, 0
879
+ = no spike) and the probability of the transition given the origin state. (A) The CSM for a 40 Hz Bernoulli spiking process
880
+ consists of a single state Awhich always transitions back to itself, emitting a spike with probability p= 0:04 per msec. (B)
881
+ CSM for 40 Hz Bernoulli spiking process with a 5 msec refractory period imposed after each spike. State Aagain spikes with
882
+ probability p= 0:04. Upon spiking the CSM transitions through a deterministic chain of states B{F(squares) which represent
883
+ the refractory period. The increased structure of the refractory period requires a more complex representation.18
884
+ A0 | 0.957
885
+ B1 | 0.043
886
+ C0 | 1.000
887
+ D0 | 1.000
888
+ H1 | 0.053
889
+ I0 | 0.9471 | 0.069
890
+ J0 | 0.931F1 | 0.024
891
+ G0 | 0.9761 | 0.039
892
+ 0 | 0.9611 | 0.076
893
+ K0 | 0.924E1 | 0.012
894
+ 0 | 0.9881 | 0.003
895
+ 0 | 0.997
896
+ P1 | 0.069
897
+ Q0 | 0.9310 | 0.933
898
+ 1 | 0.067
899
+ O1 | 0.074
900
+ 0 | 0.926N1 | 0.078
901
+ 0 | 0.922M1 | 0.080
902
+ 0 | 0.920L1 | 0.075
903
+ 0 | 0.9251 | 0.079
904
+ 0 | 0.921
905
+ FIG. 2 CSM reconstructed from a 200 sec simulated spike train with a \soft" refractory/bursting structure. C= 3:16,J= 0:25,
906
+ R= 0. State A(circle) is the baseline 40 Hz spiking state. Upon emitting a spike the transition is to state B. States Band
907
+ C(squares) are \hard" refractory states from which no spike may be emitted. States Dthrough Q(hexagons) compromise a
908
+ refractory/bursting chain from which if a spike is emitted the transition is back to state B. Upon exiting the chain the CSM
909
+ returns to the baseline state A.19
910
+ 0 5 10 15 20 25 3000.20.4Entropies0 5 10 15 20 25 300246Complexity (C(t))0 5 10 15 20 25 3000.050.1History Dependent Firing ProbabilityISI Distribution
911
+ 0 5 10 15 20 25 30 35 40 45 5000.020.040.060.08
912
+ time since most recent spike (msec)msecspikes/msec bits bitsA
913
+ B
914
+ λCSM(t)
915
+ λPSTH (t)
916
+ J(t)
917
+ R(t)H(t)λmodel (t)
918
+ FIG. 3 \Soft" refractory and bursting model ISI distribution and time dependent ring probability, complexity and entropies.
919
+ (A) ISI distribution and 99% con dence bounds bootstrapped from the CSM. (B) First panel: Firing probability as a function
920
+ of time since the most recent spike. Line with squares = ring probability predicted by CSM. Solid line = ring probability
921
+ deduced from PSTH. Dashed line = model ring rate used to generate spikes. Second panel: Complexity as a function of time
922
+ since most recent spike. Third panel: Entropies as a function of time since most recent spike. Squares = internal entropy rate,
923
+ circles = residual randomness, solid line = entropy rate. (overlaps squares)20
924
+ A0 | 0.957
925
+ B1 | 0.043
926
+ C0 | 0.944
927
+ H1 | 0.056D1 | 0.055
928
+ F0 | 0.945
929
+ E0 | 0.885
930
+ M1 | 0.1150 | 0.882
931
+ 1 | 0.1180 | 0.948
932
+ G1 | 0.0520 | 0.821
933
+ 1 | 0.179
934
+ I0 | 0.791
935
+ 1 | 0.209 J0 | 0.839
936
+ K1 | 0.1610 | 0.878
937
+ 1 | 0.122
938
+ L0 | 0.677 1 | 0.323
939
+ 1 | 0.205
940
+ N0 | 0.795
941
+ 1 | 0.392 0 | 0.6081 | 0.366
942
+ O0 | 0.6341 | 0.265
943
+ P0 | 0.7350 | 0.816
944
+ 1 | 0.184
945
+ FIG. 4 16-state CSM reconstructed from 200 sec of simulation of periodically-stimulated spiking. C= 0:89,J= 0:27,
946
+ R= 0:0007. State Ais the baseline state. States Bthrough F(octagons) are \decision" states in which the CSM evaluates
947
+ whether a spike indicates a stimulus or was spontaneous. Two spikes within 3 msec cause the CSM to transition to states G
948
+ through P, which represent the structure imposed by the stimulus. If no spikes are emitted within 5 (often fewer) sequential
949
+ msec, the CSM goes back to the baseline state A.21
950
+ 0 5 10 15 20 25 30 35 40 45 5000.020.040.060.080.10.12
951
+ 0 5 10 15 20 25 30 35 40 45 5000.20.40.6Time Dependent Firing Probability
952
+ 0 5 10 15 20 25 30 35 40 45 500510Complexity C(t)
953
+ 0 5 10 15 20 25 30 35 40 45 500123EntropiesISI Distribution
954
+ msec
955
+ Time since stimulus (msec)
956
+ Time since stimulus (msec)bits bitsspikes/msec
957
+ Stimulus Driven Entropy (ΔH(t))A
958
+ B
959
+ CλCSM(t)
960
+ λPSTH (t)
961
+ J(t)
962
+ R(t)
963
+ H(t)λmodel (t)
964
+ 0 5 10 15 20 25 30 35 40 45 5000.511.5 bits
965
+ FIG. 5 Stimulus model ISI distribution and time-dependent complexity and entropies. (A) ISI distribution and 99% con dence
966
+ bounds. (B) First panel: Firing probability as a function of time since stimulus presentation. Second panel: Time dependent
967
+ complexity. Third panel: time-dependent entropies. (C) The stimulus-driven entropy is >1 bit, indicating strong external
968
+ drive. See text for discussion.22
969
+ A0 | 0.990
970
+ B1 | 0.010
971
+ D
972
+ E0 | 0.9281 | 0.072
973
+ 1 | 0.065
974
+ F0 | 0.935C
975
+ 0 | 0.9611 | 0.039 0 | 0.9911 | 0.009
976
+ H1 | 0.046 G0 | 0.954
977
+ M1 | 0.042
978
+ N0 | 0.9580 | 0.959
979
+ 1 | 0.041
980
+ L1 | 0.036
981
+ 0 | 0.964K1 | 0.041
982
+ 0 | 0.959J
983
+ 1 | 0.0320 | 0.968I1 | 0.047
984
+ 0 | 0.9531 | 0.052
985
+ 0 | 0.9481 | 0.053
986
+ 0 | 0.947
987
+ FIG. 6 14-state CSM reconstructed from 90 sec of spiking recorded from a spontaneously spiking (no stimulus) neuron located
988
+ in layer II/III of rat barrel cortex. C= 1:02,J= 0:10,R= 0:005. State A(circle) is baseline 10 Hz spiking. States Bthrough
989
+ Ncomprise a refractory/bursting chain similar to, but with a somewhat more intricate structure than, that of the model neuron
990
+ in Figure 223
991
+ 0 5 10 15 20 25 30 35 40 45 5000.020.040.060.080.1
992
+ 0 5 10 15 20 25 3000.020.040.06History Dependent Firing Probability
993
+ 0 5 10 15 20 25 3002468Complexity C(t)
994
+ 0 5 10 15 20 25 3000.20.4EntropiesISI Distribution
995
+ msecspikes/msecbits bits
996
+ time since most recent spike (msec)BA
997
+ J(t)
998
+ R(t)
999
+ H(t)λCSM(t)
1000
+ λPSTH (t)
1001
+ FIG. 7 Spontaneously spiking barrel cortex neuron. (A) ISI distribution and 99% bootstrapped con dence bounds. (B) First
1002
+ panel: Time dependent ring probability as a function of time since most recent spike. See text for explanation of discrepancy
1003
+ between CSM and PSTH spike probabilities. Second Panel: Complexity as a function of time since most recent spike. Third
1004
+ Panel: Entropy rates as a function of time since most recent spike.24
1005
+ A0 | 0.99
1006
+ B1 | 0.01
1007
+ ZZ
1008
+ 1 | 0.02
1009
+ C2 0 | 0.99C1 | 0.010 | 0.99
1010
+ X0 | 0.98
1011
+ 1 | 0.02L
1012
+ 1 | 0.03M0 | 0.97
1013
+ 1 | 0.04N0 | 0.96D0 | 0.95
1014
+ C11 | 0.05H2
1015
+ 1 | 0.050 | 0.95
1016
+ 1 | 0.06E0 | 0.94
1017
+ H1
1018
+ 1 | 0.01 0 | 0.99
1019
+ 1 | 0.040 | 0.96
1020
+ O1 | 0.030 | 0.97
1021
+ 1 | 0.03P0 | 0.977J
1022
+ 1 | 0.04K0 | 0.96
1023
+ 1 | 0.040 | 0.97H
1024
+ 1 | 0.04
1025
+ I0 | 0.96
1026
+ 1 | 0.040 | 0.96G
1027
+ 1 | 0.050 | 0.95F
1028
+ 1 | 0.050 | 0.95
1029
+ 1 | 0.060 | 0.94
1030
+ 0 | 0.991 | 0.01 W
1031
+ 0 | 0.981 | 0.02V
1032
+ 1 | 0.020 | 0.98S
1033
+ 1 | 0.03T0 | 0.97
1034
+ 1 | 0.03U0 | 0.98Q
1035
+ 1 | 0.03R0 | 0.97
1036
+ 1 | 0.030 | 0.97
1037
+ 1 | 0.030 | 0.97
1038
+ 1 | 0.020 | 0.98
1039
+ FIG. 8 29-state CSM reconstructed from 335 seconds of spikes recorded from a layer II/III barrel cortex neuron undergoing
1040
+ periodic (125 msec inter-stimulus interval) stimulation via vibrissa de
1041
+ states are devoted to refractory/bursting behavior, however states \C1", \C2" and \ZZ" represent the structure imposed by the
1042
+ external stimulus. See text for discussion.25
1043
+ 0 5 10 15 20 25 30 35 40 45 5000.020.040.060.08
1044
+ 0 5 10 15 20 25 30 35 40 45 5000.020.04Time Dependent Firing Probability
1045
+ 0 5 10 15 20 25 30 35 40 45 501.522.53Complexity C(t)
1046
+ 0 5 10 15 20 25 30 35 40 45 5000.20.4EntropiesISI Distribution A
1047
+ msecspikes/msecbits bits
1048
+ time since stimulus presentation (msec)λCSM(t)
1049
+ λPSTH (t)
1050
+ J(t)
1051
+ R(t)
1052
+ H(t)B
1053
+ C
1054
+ time since stimulus presentation (msec)0 5 10 15 20 25 30 35 40 45 5000.0050.010.0150.020.025Stimulus Driven Entropy (ΔH(t))bits
1055
+ FIG. 9 Stimulated barrel cortex neuron ISI distribution and time-dependent complexity and entropies. (A) ISI distribution
1056
+ and 99% con dence bounds. (B) First panel: Firing probability as a function of time since stimulus presentation. Second
1057
+ panel: Time-dependent complexity. Third panel: time-dependent entropies. (C) The stimulus driven entropy (maximum of
1058
+ 0:02 bits/msec) is low because the number of spikes per stimulus ( 0:10:2) is very low and hence the stimulus does not
1059
+ supply much information.26
1060
+ 0 5 10 15 20 25 3000.020.040.06History Dependent Firing Probability
1061
+ 0 5 10 15 20 25 30 35 40 45 500510Complexity C(t)
1062
+ 0 5 10 15 20 25 3000.20.4Entropiesspikes/msecbits bits
1063
+ time since most recent spike (msec)J(t)
1064
+ R(t)
1065
+ H(t)λCSM(t)
1066
+ λPSTH (t)
1067
+ FIG. 10 Firing probability, complexity and entropies of the stimulated barrel cortex neuron as a function of time since the
1068
+ most recent spike.27
1069
+ 0 10 20 30 40 5000.020.040.060.080.10.12
1070
+ 0 10 20 30 40 5000.020.040.060.080.1
1071
+ 0 0.2 0.4 0.6 0.8 100.20.40.60.81
1072
+ 0 0.2 0.4 0.6 0.8 100.20.40.60.81
1073
+ 0 0.2 0.4 0.6 0.8 100.20.40.60.81
1074
+ 0 0.2 0.4 0.6 0.8 100.20.40.60.81ISI Distrib. Spont. Spiking
1075
+ Validation DataISI Distrib. Stimulated Spiking Validation Data
1076
+ KS Plot, Spont. Spiking Training Data
1077
+ Validation DataKS Plot, Stimulated Spiking Training Data
1078
+ Validation Datamsec msecA B
1079
+ C D
1080
+ E F
1081
+ FIG. 11 Cross validation of CSMs reconstructed from spontaneously ring and stimulus evoked rat barrel cortex on an inde-
1082
+ pendent validation training set. (A,B) ISI distribution of spontaneously and stimulus evoked ring validation sets and 99%
1083
+ con dence bounds bootstrapped from CSM. (C-D) Time rescaling plots of training data sets for spontaneously ring and stim-
1084
+ ulus evoked ring respectively. Dashed lines are 95% con dence bounds and the solid line is the rescaled ISIs. The solid line
1085
+ along the digagonal is for visual comparison to an ideal t. (E-F) Similar time rescaling plots for the validation data sets.
1001.0037.txt ADDED
Binary file (31.5 kB). View file
 
1001.0038.txt ADDED
@@ -0,0 +1,1123 @@
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
1
+ arXiv:1001.0038v1 [math.CA] 30 Dec 2009BERGMAN-TYPE SINGULAR INTEGRAL OPERATORS ON METRIC
2
+ SPACES
3
+ ALEXANDER VOLBERG†AND BRETT D. WICK‡
4
+ Abstract. In this paper we study “Bergman-type” singular integral ope rators on Ahlfors regular
5
+ metric spaces. The main result of the paper demonstrates tha t if a singular integral operator on a
6
+ Ahlfors regular metric space satisfies an additional estima te, then knowing the “T(1)” conditions
7
+ for the operator imply that the operator is bounded on L2. The method of proof of the main result
8
+ is an extension and another application of the work originat ed by Nazarov, Treil and the first author
9
+ on non-homogeneous harmonic analysis.
10
+ 1.Introduction and Statements of results
11
+ We are interested in Calder´ on-Zygmund operators living on metric spaces. In particular, these
12
+ kernels will live on a metric space of homogeneous type. We br iefly recall these types of metric
13
+ spaces. A metric space of homogeneous type is a space X, a quasi-metric ρ, and a non-negative
14
+ Borel measure νon the space X. The key property that defines these spaces is that all balls B(x,r)
15
+ defined byρare open, and the measure νsatisfies the doubling condition
16
+ ν(B(x,2r))≤Cdoubν(B(x,r))∀x∈X, r∈R+.
17
+ We also require that ν(B(x,r))<∞for allx∈Xandr∈R+. The main example of the metric
18
+ spaces that the reader should keep in mind is the case of Rnwith the standard metric and Lebesgue
19
+ measure. Instead of the standard doubling condition, we wil l impose a slightly stronger condition.
20
+ Let(X,ν,ρ) beaAhlforsregularmetricmeasurespace. By thiswemeanth at (X,ρ) isacomplete
21
+ metric space, ν≥0 is a Borel measure on X, and there exist constants 0 < c1< c2,n >0, such
22
+ that, for all r≥0 andx∈X:
23
+ c1rn≤ν(B(x,r))≤c2rn. (1.1)
24
+ It is easy to see that condition 1.1implies the doubling condition on νwithCdoub=c2
25
+ c12n.
26
+ We next recall the definition of Calder´ on-Zygmund operator s on metric spaces as introduced by
27
+ Christ, [1]. For anyx,y∈X, we set
28
+ λ(x,y) =ν(B(x,ρ(x,y)))≈ρ(x,y)n.
29
+ A simple calculation shows that λ(x,y)≈λ(y,x) because of the doubling condition on ν. Then
30
+ astandard kernel is a function k:X×X\ {x=y} →Csuch that there exists constants CCZ,
31
+ τ,δ>0
32
+ |k(x,y)| ≤CCZ
33
+ λ(x,y)=CCZ
34
+ ρ(x,y)n∀x/\e}atio\slash=y∈X;
35
+ and
36
+ |k(x,y)−k(x,y′)|+|k(x,y)−k(x′,y)| ≤CCZρ(x,x′)τ
37
+ ρ(x,y)τ1
38
+ λ(x,y)=CCZρ(x,x′)τ
39
+ ρ(x,y)τ+n
40
+ †Research supported in part by a National Science Foundation DMS grant.
41
+ ‡The second author is supported by National Science Foundati on CAREER Award DMS# 0955432 and an
42
+ Alexander von Humboldt Fellowship.
43
+ 12 A. VOLBERG AND B. D. WICK
44
+ provided that ρ(x,x′)≤δρ(x,y). In this situation, we say that the kernel ksatisfies the standard
45
+ estimates. Again, the canonical examples to keep in mind are the usual Calder´ on-Zygmund kernels
46
+ onRn.
47
+ However, we will be interested in kernels that satisfy estim ates as if they lived on a “smaller
48
+ space”. First, suppose that we have another measure µon the metric space X(which need not be
49
+ doubling), but satisfies the following relationship, for so me 0≤m<n
50
+ µ(B(x,r))/lessorsimilarrm∀x∈X,∀r. (H)
51
+ Then, we define a standard kernel of order 0<m≤nas a function k:X×X\{x=y} →C
52
+ such that there exists constants CCZ,τ,δ >0
53
+ |k(x,y)| ≤CCZ
54
+ ρ(x,y)m∀x/\e}atio\slash=y∈X;
55
+ and
56
+ |k(x,y)−k(x,y′)|+|k(x,y)−k(x′,y)| ≤CCZρ(x,x′)τ
57
+ ρ(x,y)τ+m
58
+ provided that ρ(x,x′)≤δρ(x,y). In this situation, we say that the kernel ksatisfies the standard
59
+ estimates. In this case, we then define the Calder´ on-Zygmun d operator associated to µas
60
+ Tµ(f)(x) :=/integraldisplay
61
+ Xk(x,y)f(y)dµ(y).
62
+ For “nice” functions f, this integral is well defined and
63
+ These definitions are motivated by the Calder´ on-Zygmund ke rnels that live in Rn, but satisfy
64
+ estimates as if they lived in Rmwithm≤n. One should think of the measure µas given by the
65
+ m-dimensional Lebesgue measure after restricting to a m-dimensional hyperplane.
66
+ The constants CCZ,τ,δandmwill be referred to as the Calder´ on-Zygmund constants of th e
67
+ kernelk(x,y).
68
+ We will also be interested in the kernels that have the additi onal property that satisfy
69
+ |k(x,y)| ≤1
70
+ max(dm(x),dm(y)),
71
+ whered(x) := dist(x,X\Ω) = inf{ρ(x,y) :y∈X\Ω}and Ω being an open set in X.
72
+ Our main result is the following theorem:
73
+ Theorem 1. Let(X,ρ,ν)be a Ahlfors regular metric space. Let k(x,y)be a Calder´ on-Zygmund
74
+ kernel of order mon(X,ρ,ν), with Calder´ on-Zygmund constants CCZandτ, that satisfies
75
+ |k(x,y)| ≤1
76
+ max(d(x)m,d(y)m),
77
+ whered(x) := dist(x,X\Ω). Letµbe a probability measure with compact support in Xand all
78
+ balls such that µ(B(x,r))>rmlie in an open set Ω. Finally, suppose also that a “ T1Condition”
79
+ holds for the operator Tµ,mwith kernel kand for the operator T∗
80
+ µ,mwith kernel k(y,x):
81
+ /bardblTµ,mχQ/bardbl2
82
+ L2(X;µ)≤Aµ(Q),/bardblT∗
83
+ µ,mχQ/bardbl2
84
+ L2(X;µ)≤Aµ(Q). (1.2)
85
+ Then/bardblTµ,m/bardblL2(X;µ)→L2(X;µ)≤C(A,m,d,τ ).
86
+ The balls for which we have µ(B(x,r))>rmwill be called “non-Ahlfors balls”. The key hypoth-
87
+ esis is that we can capture all the non-Ahlfors balls in some o pen set Ω. To mitigate against this
88
+ difficulty, we will have to suppose that our Calder´ on-Zygmun d kernels have an additional estimate
89
+ in terms of the behavior in terms of the distance to the comple ment of Ω.
90
+ An immediate application of Theorem 1is a new proof of results by the authors in [ 9]. In
91
+ [9] a variant of Theorem 1was obtained in the Euclidean setting, and then is further ex tended
92
+ to Calder´ on-Zygmund kernels in the natural metric associa ted to the Heisenberg group on theBERGMAN-TYPE SINGULAR INTEGRAL OPERATORS ON METRIC SPACES 3
93
+ unit ball. This was then used to characterize the Carleson me asures for the analytic Besov–Sobolev
94
+ spaces on the unit ball in Cn. The connection between Carleson measures and a variant of T heorem
95
+ 1is provided since a measure is Carleson if and only if a certai n naturally occurring Calder´ on-
96
+ Zygmund operator is bounded on L2. The operator to be studied is amenable to the methods of
97
+ non-homogeneous harmonic analysis.
98
+ Themethod of proof of Theorem 1will beto usethe tools of non-homogeneous harmonicanalysi s
99
+ as developed by F. Nazarov, S. Treil, and the first author in th e series of papers [ 3–6] and further
100
+ explained in the book by the first author [ 8]. We essentially adapt the proof given by the authors in
101
+ [9] to the case of metric spaces considered in this paper. Const ants will bedenoted by Cthroughout
102
+ the paper.
103
+ 2.Proof of Theorem 1
104
+ The proof of this theorem will be divided in several parts. We first recall the construction of M.
105
+ Christ of “dyadic cubes” on a metric space of homogeneous typ e, see [1]. The interested reader
106
+ can also consult the paper by E. Sawyer and R. Wheeden, [ 7], where a similar construction is
107
+ performed.
108
+ Theorem 2 (M. Christ, [ 1]).There exists a collection of open sets {Qk
109
+ α⊂X:k∈Z,α∈Ik}and
110
+ constantsκ∈(0,1),a0>0, andη>0andC1,C2<∞such that
111
+ (i)ν(X\/uniontext
112
+ αQk
113
+ α) = 0∀k∈Z;
114
+ (ii)Ifl≥k, then either Ql
115
+ β⊂Qk
116
+ αorQl
117
+ β⊂Qk
118
+ α=∅;
119
+ (iii)For each (k,α)and eachl<kthere is a unique βsuch thatQk
120
+ α⊂Ql
121
+ β;
122
+ (iv)The diameter of Qk
123
+ αis an absolute constant multiple of κk;
124
+ (v)EachQk
125
+ αcontains some ball B(zα,a0κk);
126
+ (vi)ν{x∈Qk
127
+ α: dist(x,X\Qk
128
+ α)≤tκk}/lessorsimilartην(Qk
129
+ α)
130
+ HereIkis a (possibly finite) index set, depending only on k∈Z.
131
+ The construction of these cubes uses only the properties of t he homogeneous space ( X,ρ,ν).
132
+ One can think of the cubes Qk
133
+ αas being cubes or balls of diameter κkand centerzk
134
+ α. We will let D
135
+ denote the collection of dyadic cubes on Xthat exists by the above Theorem.
136
+ We further remark that it is possible to “randomize” this con struction. In a recent paper by
137
+ Hyt¨ onen and Martikainen, [ 2], they studied this construction in and showed that it is pos sible to
138
+ construct several random dyadic grids of the type above. The details of this construction aren’t
139
+ immediately important for the proof of the main results in th is paper, only the existence of these
140
+ randomgrids. We recommendthat thereaderconsultthewell- written paper[ 2] fortheconstruction
141
+ of these grids. In particular, Section 10 of that paper conta ins the necessary modifications of
142
+ Theorem 2to construct the random dyadic lattices in a metric space.
143
+ We also define the dilation of a set E⊂Xby a parameter λ≥1 by
144
+ λE:={x∈X:ρ(x,E)≤(λ−1)diam(E)}.
145
+ 2.1.Terminal and transit cubes. We will call the cube Q∈ Daterminal cube if the parent of
146
+ Q(which exists and is unique by (iii) of Theorem 2) is contained in our open set Ω or µ(Q) = 0.
147
+ All other cubes are called transitcubes. Then, denote by DtermandDtranas the terminal and
148
+ transit cubes from D. We first state two obvious Lemmas.
149
+ Lemma 3. IfQbelongs to Dterm, then
150
+ |k(x,y)| ≤1
151
+ κm.
152
+ This follows since Qbelongs to its parent which is a subset of Ω and so for x,y∈Qwe have
153
+ thatd(x)≥κand similarly for y. Another obvious lemma:4 A. VOLBERG AND B. D. WICK
154
+ Lemma 4. IfQbelongs to Dtran, then
155
+ µ(B(x,r))/lessorsimilarrm.
156
+ We assume that F= suppµlies in a grand child cube of Qwhere, this Qis a certain (fixed) unit
157
+ cube. We then take two “random” lattices as constructed by Hy t¨ onen and Martikainen in [ 2]. Now,
158
+ letD1andD2be two such dyadic lattices, that have the property that the u nit cube contains the
159
+ support ofµdeep inside a unit cube of the corresponding lattice. We will decompose our functions
160
+ fandgwith respect to the lattices D1andD2.
161
+ We would like to denote Qjas a dyadic cube belonging to the dyadic lattice Dj. Unfortunately,
162
+ this makes the notation later very cumbersome. So, we will us e the letter Qto denote a dyadic
163
+ cube belonging to the lattice D1and the letter Rto denote a dyadic cube belonging to the lattice
164
+ D2. We will also let s(Q) denote the “size” or “scale” of the cube, namely, what gener ation of the
165
+ construction from Theorem 2the cube belongs to.
166
+ From now on, we will always denote by Qjthe dyadic subcubes of a cube Qenumerated in some
167
+ “natural order”. Similarly, we will always denote by Rjthe dyadic subcubes of a cube RfromD2.
168
+ Next, notice that there are special unit cubes Q0andR0of the dyadic lattices D1andD2
169
+ respectively. They have the property that they are both tran sit cubes and contain Fdeep inside
170
+ them.
171
+ 2.2.Projections Λand∆Q.LetDbe one of the dyadic lattices above. For a function ψ∈
172
+ L1(X;µ) and for a cube Q⊂X, denote by /a\}bracketle{tψ/a\}bracketri}htQthe average value of ψoverQwith respect to the
173
+ measureµ, i.e.,
174
+ /a\}bracketle{tψ/a\}bracketri}htQ:=1
175
+ µ(Q)/integraldisplay
176
+ Qψdµ
177
+ (of course, /a\}bracketle{tψ/a\}bracketri}htQmakes sense only for cubes Qwithµ(Q)>0). Put
178
+ Λϕ:=/a\}bracketle{tϕ/a\}bracketri}htQ0.
179
+ Clearly, Λϕ∈L2(X;µ) for allϕ∈L2(X;µ), and Λ2= Λ, i.e., Λ is a projection. Note also, that
180
+ actually Λ does not depend on the lattice Dbecause the average is taken over the whole support
181
+ of the measure µregardless of the position of the cube Q0(orR0).
182
+ Below we will start almost every claim by “Assume (for definit eness) that s(Q)≤s(R)...”.
183
+ Below, for ease of notation, we will write that a cube Q∈ X ∩Y to mean that the dyadic cube Q
184
+ has both property XandYsimultaneously.
185
+ For every transit cube Q∈ D1, define ∆Qϕby
186
+ ∆Qϕ/vextendsingle/vextendsingle
187
+ X\Q:= 0, ,∆Qϕ/vextendsingle/vextendsingle
188
+ Qj:=
189
+
190
+ /bracketleftBig
191
+ /a\}bracketle{tϕ/a\}bracketri}htQj−/a\}bracketle{tϕ/a\}bracketri}htQ/bracketrightBig
192
+ ifQjis transit;
193
+ ϕ−/a\}bracketle{tϕ/a\}bracketri}htQifQjis terminal.
194
+ Observe that for every transit cube Q, we haveµ(Q)>0, so our definition makes sense since no
195
+ zero can appear in the denominator. We repeat the same definit ion forR∈ D2.
196
+ We then have have following Lemma that collects several easy properties of ∆Qϕ. To check these
197
+ properties is left to the reader as an exercise.
198
+ Lemma 5. For everyϕ∈L2(X;µ)and every transit cube Q,
199
+ (1) ∆Qϕ∈L2(X;µ);
200
+ (2)/integraltext
201
+ X∆Qϕdµ= 0;
202
+ (3) ∆Qis a projection, i.e., ∆2
203
+ Q= ∆Q;
204
+ (4) ∆QΛ = Λ∆Q= 0;BERGMAN-TYPE SINGULAR INTEGRAL OPERATORS ON METRIC SPACES 5
205
+ (5) IfQ,/tildewideQare transit,/tildewideQ/\e}atio\slash=Q, then ∆Q∆eQ= 0.
206
+ We next note that it is possible to decompose functions ϕinto the corresponding projections Λ
207
+ and ∆Q.
208
+ Lemma 6. LetQ0be a transit cube. For every ϕ∈L2(X;µ)we have
209
+ ϕ= Λϕ+/summationdisplay
210
+ Qtransit∆Qϕ,
211
+ the series converges in L2(X;µ)and, moreover,
212
+ /bardblϕ/bardbl2
213
+ L2(µ)=/bardblΛϕ/bardbl2
214
+ L2(µ)+/summationdisplay
215
+ Qtransit/bardbl∆Qϕ/bardbl2
216
+ L2(µ).
217
+ Proof.Note first of all that if one understands the sum
218
+ /summationdisplay
219
+ Qtransit
220
+ as limk→∞/summationtext
221
+ Qtransit:s(Q)>δk, then forµ-almost every x∈X, one has
222
+ ϕ(x) = Λϕ(x)+/summationdisplay
223
+ Qtransit∆Qϕ(x).
224
+ Indeed, the claim is obvious if the point xlies in some terminal cube. Suppose now that this is not
225
+ the case. Observe that
226
+ Λϕ(x)+/summationdisplay
227
+ Qtransit:s(Q)>κk∆Qϕ(x) =/a\}bracketle{tϕ/a\}bracketri}htQk,
228
+ whereQkis the dyadic cube of size κk, containing x. Therefore, the claim is true if
229
+ /a\}bracketle{tϕ/a\}bracketri}htQk→ϕ(x).
230
+ But, the exceptional set for this condition has µ-measure 0. Now the orthogonality of all ∆ Qϕ
231
+ between themselves, and their orthogonality to Λ ϕproves the lemma. /square
232
+ 3.Good and bad functions
233
+ We consider the functions fandg∈L2(X;µ). We fix two dyadic lattices D1andD2as before
234
+ and define decompositions of fandgvia Lemma 6,
235
+ f= Λf+/summationdisplay
236
+ Q∈Dtran
237
+ 1∆Qf, g= Λg+/summationdisplay
238
+ R∈Dtran
239
+ 2∆Rg.
240
+ For a dyadic cube Rwe denote ∪i∈Ik∂RibyskR, called the skeleton ofR. Here theRiare the
241
+ dyadic children of R.
242
+ Letτ,mbe parameters of the Calder´ on-Zygmundkernel k. We fixα=τ
243
+ 2τ+2m.
244
+ Definition 7. Fix a small number δ >0 andS≥2 to be chosen later. Choose an integer rsuch
245
+ that
246
+ κ−r≤δS<κ−r+1. (3.1)
247
+ A cubeQ∈ D1is called bad(δ-bad) if there exists R∈ D2such that
248
+ (1)s(R)≥κrs(Q);
249
+ (2) dist(Q,skR)<s(Q)αs(R)1−α.6 A. VOLBERG AND B. D. WICK
250
+ LetB1denote the collection of all bad cubes and correspondingly l etG1denote the collection of
251
+ good cubes. The symmetric definition gives the collection of badcubesR∈ D2, denotes as B2.
252
+ We say, that ϕ=/summationtext
253
+ Q∈Dtran
254
+ 1∆Qϕisbadif in the sum only bad Q’s participate in this decompo-
255
+ sition with the same appling to ψ=/summationtext
256
+ Q∈Dtran
257
+ 2∆Qψ. In particular, given two distinct lattices D1
258
+ andD2we fix the decomposition of fandginto good and bad parts:
259
+ f=fgood+fbad,wherefgood= Λf+/summationdisplay
260
+ Q∈Dtran
261
+ 1∩G1∆Qf.
262
+ The same applies to g= Λg+/summationtext
263
+ R∈Dtran
264
+ 2∆Rg=ggood+gbad.
265
+ Theorem 8. One can choose S=S(α)in such a way that for any fixed Q∈ D1,
266
+ P{Qis bad} ≤δ2. (3.2)
267
+ By symmetry P{Ris bad} ≤δ2for any fixed R∈ D2.
268
+ Theproof of this Theorem can befoundin thepaper [ 2]. Theuseof Theorem 8gives usS=S(α)
269
+ in such a way that for any fixed Q∈ D1,
270
+ P{Qis bad} ≤δ2. (3.3)
271
+ We are now ready to prove
272
+ Theorem 9. Consider the decomposition of ffrom Lemma 6. Then one can choose S=S(α)in
273
+ such a way that
274
+ E(/bardblfbad/bardblL2(X;µ))≤δ/bardblf/bardblL2(X;µ). (3.4)
275
+ The proof depends only on the property ( 3.3) and not on a particular definition of what it means
276
+ to be a bad or good function.
277
+ Proof.By Lemma 6(its left inequality),
278
+ E(/bardblfbad/bardblL2(X;µ))≤E/parenleftBig/summationdisplay
279
+ Q∈Dtran
280
+ 1∩B1/bardbl∆Qf/bardbl2
281
+ L2(X;µ)/parenrightBig1/2
282
+ .
283
+ Then
284
+ E(/bardblfbad/bardblL2(X;µ))≤/parenleftBig
285
+ E/summationdisplay
286
+ Q∈Dtran
287
+ 1∩B1/bardbl∆Qf/bardbl2
288
+ L2(X;µ)/parenrightBig1/2
289
+ .
290
+ LetQbe a fixed cube in D1; then, using ( 3.3), we conclude:
291
+ E/bardbl∆Qf/bardbl2
292
+ L2(X;µ)=P{Qis bad}/bardbl∆Qf/bardbl2
293
+ L2(X;µ)≤δ2/bardbl∆Qf/bardbl2
294
+ L2(X;µ).
295
+ Therefore, we can continue as follows:
296
+ E(/bardblfbad/bardblL2(X;µ))≤δ/parenleftBig/summationdisplay
297
+ Q∈Dtran
298
+ 1∩B1/bardbl∆Qf/bardbl2
299
+ L2(X;µ)/parenrightBig1/2
300
+ ≤δ/bardblf/bardblL2(X;µ).
301
+ The last inequality uses Lemma 6again (its right inequality). /square
302
+ This theorem can also be found in the paper [ 2].BERGMAN-TYPE SINGULAR INTEGRAL OPERATORS ON METRIC SPACES 7
303
+ 3.1.Reduction to Estimates on Good Functions. We consider two random dyadic lattices
304
+ D1andD2as constructed in [ 2]. Take now two functions fandg∈L2(X;µ) decomposed according
305
+ to Lemma 6
306
+ f= Λf+/summationdisplay
307
+ Q∈Dtran
308
+ 1∆Qf, g= Λg+/summationdisplay
309
+ R∈Dtran
310
+ 2∆Rg.
311
+ Recall that we can now write f=fgood+fbad,g=ggood+gbad. Then
312
+ (Tf,g) = (Tfgood,ggood)+R(f,g),whereR(f,g) = (Tfbad,g)+(Tfgood,gbad).
313
+ Theorem 10. LetTbe any operator with bounded kernel. Then
314
+ E|R(f,g)| ≤2δ/bardblT/bardblL2(X;µ)→L2(X;µ)/bardblf/bardblL2(X;µ)/bardblg/bardblL2(X;µ).
315
+ Remark 11.Notice that the estimate depends on the norm of Tnot on the bound on its kernel.
316
+ Proof.The procedure of taking the good and bad part of a function are projections in L2(X;µ)
317
+ and so they do not increase the norm. Since we have that the ope ratorTis bounded, then
318
+ |R(f,g)| ≤ /bardblT/bardblL2(X;µ)→L2(X;µ)/parenleftbig
319
+ /bardblg/bardblL2(X;µ)/bardblfbad/bardblL2(X;µ)+/bardblf/bardblL2(X;µ)/bardblgbad/bardblL2(X;µ)/parenrightbig
320
+ Therefore, upon taking expectations we find
321
+ E|R(f,g)| ≤ /bardblT/bardblL2(X;µ)→L2(X;µ)/parenleftbig
322
+ /bardblg/bardblL2(X;µ)E(/bardblfbad/bardblL2(X;µ))+/bardblf/bardblL2(X;µ)E(/bardblgbad/bardblL2(X;µ))/parenrightbig
323
+ .
324
+ Using Theorem 9we finish the proof.
325
+ /square
326
+ We see that we need now only to estimate
327
+ |(Tfgood,ggood)| ≤C(τ,m,d,T 1)/bardblf/bardblL2(X;µ)/bardblg/bardblL2(X;µ). (3.5)
328
+ In fact, considering any operator Twith bounded kernel we conclude
329
+ (Tf,g) =E(Tf,g) =E(Tfgood,ggood)+ER(f,g).
330
+ Using Theorem 10and (3.5) we have
331
+ |(Tf,g)| ≤C/bardblf/bardblL2(X;µ)/bardblg/bardblL2(X;µ)+2δ/bardblT/bardblL2(X;µ)→L2(X;µ)/bardblf/bardblL2(X;µ)/bardblg/bardblL2(X;µ).
332
+ From here, taking the supremum over fandgin the unit ball of L2(X;µ), and choosing δ=1
333
+ 4we
334
+ get
335
+ /bardblT/bardblL2(X;µ)→L2(X;µ)≤2C.
336
+ 3.2.Splitting (Tfgood,ggood)into Three Sums. First let us get rid of the projection Λ. We fix
337
+ two corresponding dyadic lattices D1andD2. Recall that F= suppµis deep inside a unit cube Q
338
+ of the standard dyadic lattice Das well as inside the shifted unit cubes Q0∈ D1andR0∈ D2. If
339
+ f∈L2(X;µ), we have
340
+ /bardblTΛf/bardblL2(X;µ)=/a\}bracketle{tf/a\}bracketri}htQ0/bardblTχQ0/bardblL2(X;µ)
341
+ ≤A1.2/bardblf/bardblL2(X;µ)µ(Q0)1/2
342
+ µ(Q0)µ(Q0)1/2
343
+ =A1.2/bardblf/bardblL2(X;µ).
344
+ So we can replace fbyf−Λfand identically we can repeat this argument with gand from now
345
+ on we may assume further that/integraldisplay
346
+ Xf(x)dµ(x) = 0 and/integraldisplay
347
+ Xg(x)dµ(x) = 0.
348
+ Based on the reductions above, we can now think that fandgare good functions with zero
349
+ averages. We skip mentioning below that Q∈ Dtran
350
+ 1andR∈ Dtran
351
+ 2, since this will always be the
352
+ case by the convention established above.8 A. VOLBERG AND B. D. WICK
353
+ To study the action of the Calder´ on-Zygmund operator Tonfandg, we split the pairing in
354
+ the following manner,
355
+ (Tf,g) =/summationdisplay
356
+ Q∈G1,R∈G2,s(Q)≤s(R)(∆Qf,∆Rg)+/summationdisplay
357
+ Q∈G1,R∈G2,s(Q)>s(R)(∆Qf,∆Rg).
358
+ The question of convergence of the infinite sum can be avoided here, as we can think that the
359
+ functionsfandgare only finite sums. This removes the question of convergenc e and allows us to
360
+ rearrange and group the terms in the sum in any way we want.
361
+ We need to estimate only the first sum, as the second will follo w by symmetry. For the sake of
362
+ notational simplicity we will skip mentioning that the cube sQandRare good and we will skip
363
+ mentioning s(Q)≤s(R). So, for now on,
364
+ /summationdisplay
365
+ Q,R:other conditionsmeans/summationdisplay
366
+ Q,R:s(Q)≤s(R),Q∈G1,R∈G2,other conditions.
367
+ Remark 12.It is convenient sometimes to think that the summation/summationdisplay
368
+ Q,R:other conditions
369
+ goes over good QandallR. Formally, this does not matter, since the functions fandgare good
370
+ functions, and so this merely reduces to adding or omitting s everal zeros to the sum. For the
371
+ symmetric sum over Q,R:s(Q)> s(R) the roles of QandRin this remark must of course be
372
+ interchanged.
373
+ The definition of δ-badness involved a large integer r, see (3.1). Use this notation to write our
374
+ sum overs(Q)≤s(R) as follows
375
+ /summationdisplay
376
+ Q,R(∆Qf,∆Rg) =/summationdisplay
377
+ Q,R:s(Q)≥κ−rs(R)+/summationdisplay
378
+ Q,R:s(Q)<κ−rs(R)=/summationdisplay
379
+ Q,R:s(Q)≥κ−rs(R),dist(Q,R)≤s(R)+
380
+ /bracketleftbigg/summationdisplay
381
+ Q,R:s(Q)≥κ−rs(R),dist(Q,R)>s(R)+/summationdisplay
382
+ Q,R:s(Q)<κ−rs(R),Q∩R=∅/bracketrightbigg
383
+ +/summationdisplay
384
+ Q,R:s(Q)<κ−rs(R),Q∩R/\egatio\slash=∅
385
+ =:σ1+σ2+σ3.
386
+ 3.3.Three Potential Estimates of/integraltext
387
+ X/integraltext
388
+ Xk(x,y)f(x)g(y)dµ(x)dµ(y).Recall that the kernel
389
+ k(x,y) ofTsatisfies the estimate
390
+ |k(x,y)| ≤1
391
+ max(d(x)m,d(y)m), d(x) = dist(x,X\Ω),
392
+ Ω being an open set in X, and
393
+ |k(x,y)| ≤CCZ
394
+ ρ(x,y)m∀x/\e}atio\slash=y∈X;
395
+ and
396
+ |k(x,y)−k(x,y′)|+|k(x,y)−k(x′,y)| ≤CCZρ(x,x′)τ
397
+ ρ(x,y)τ+m
398
+ provided that ρ(x,x′)≤δρ(x,y), with some fixed constants numbers CCZ,τ,m.
399
+ First, we will sometimes write
400
+ /integraldisplay
401
+ X/integraldisplay
402
+ Xk(x,y)f(x)g(y)dµ(x)dµ(y) =/integraldisplay
403
+ X/integraldisplay
404
+ X[k(x,y)−k(x0,y)]f(x)g(y)dµ(x)dµ(y)
405
+ using the fact that our fandgwill actually be ∆ Qfand ∆ Rgand so their integrals are zero.
406
+ Temporarily write K(x,y) for either k(x,y) ork(x,y)−k(x0,y).BERGMAN-TYPE SINGULAR INTEGRAL OPERATORS ON METRIC SPACES 9
407
+ After that we have three logical possibilities to estimate
408
+ /integraldisplay
409
+ X/integraldisplay
410
+ XK(x,y)f(x)g(y)dµ(x)dµ(y).
411
+ (1) Estimate |K|inL∞, andf,ginL1norms;
412
+ (2) Estimate |K|inL∞L1norm, and finL1norm,ginL∞norm (or maybe, do this
413
+ symmetrically);
414
+ (3) Estimate |K|inL1norm, andf,ginL∞norms.
415
+ The third method is widely used for Calder´ on–Zygmund estim ates on homogeneous spaces (say
416
+ with respect to Lebesgue measure), but it is very dangerous t o use in the case of a nonhomogeneous
417
+ measure. Here is the reason. After fandgare estimated in the L∞norm, one needs to continue
418
+ these estimates to have L2norms. There is nothing strange in that as usually fandgare almost
419
+ proportional to characteristic functions. But for fliving onQsuch thatf=cQχQ(cQis a
420
+ constant),
421
+ /bardblf/bardblL∞(X:µ)≤1
422
+ µ(Q)1/2/bardblf/bardblL2(X;µ).
423
+ The same reasoning applies for gonR. Then
424
+ /vextendsingle/vextendsingle/vextendsingle/integraldisplay
425
+ X/integraldisplay
426
+ XK(x,y)f(x)g(y)dµ(x)dµ(y)/vextendsingle/vextendsingle/vextendsingle≤1
427
+ (µ(Q)µ(R))1/2/bardblf/bardblL2(X;µ)/bardblg/bardblL2(X;µ).
428
+ And the nonhomogeneous measure has no estimate from below. H aving two uncontrollable almost
429
+ zeroes in the denominator is a very bad idea. We will never use the estimate of type (3).
430
+ On the other hand, estimates of type (2) are much less dangero us (although requires the care as
431
+ well). This is because, in this case one applies
432
+ /bardblf/bardblL1(X;µ)≤µ(Q)1/2/bardblf/bardblL2(X;µ)and/bardblg/bardblL∞(X;µ)≤1
433
+ µ(R)1/2/bardblg/bardblL2(µ),
434
+ and gets
435
+ /vextendsingle/vextendsingle/vextendsingle/integraldisplay
436
+ X/integraldisplay
437
+ XK(x,y)f(x)g(y)dµ(x)dµ(y)/vextendsingle/vextendsingle/vextendsingle≤/parenleftbiggµ(Q)
438
+ µ(R)/parenrightbigg1/2
439
+ /bardblf/bardblL2(X;µ)/bardblg/bardblL2(X;µ).
440
+ If we choose to use estimate of the type (2) only for pairs Q,Rsuch thatQ⊂Rwe are in good
441
+ shape. This approach is what we will end up going when estimat ingσ3.
442
+ Plan. The first sum is the “diagonal” part of the operator, σ1. The second sum, σ2is the “long
443
+ range interaction”. The final sum, σ3, is the “short range interaction”. The diagonal part will be
444
+ estimated using our T1 assumption of Theorem 1.2, for the long range interaction we will use the
445
+ first type of estimates described above, for the short range i nteraction we will use estimates of types
446
+ (1) and (2) above. But all this will be done carefully!
447
+ 4.The Long Range Interaction: Controlling Term σ2
448
+ We first prove a lemma that demonstrates that for functions wi th supports that are far apart,
449
+ we have some good control on the bilinear form induced by our C alder´ on-Zygmund operator T.
450
+ For two dyadic cubes QandR, we set
451
+ D(Q,R) :=s(Q)+s(R)+dist(Q,R).10 A. VOLBERG AND B. D. WICK
452
+ Lemma 13. Suppose that QandRare two cubes in X, such that s(Q)≤s(R). LetϕQ,ψR∈
453
+ L2(X;µ). Assume that ϕQvanishes outside Q, andψRvanishes outside R;/integraltext
454
+ XϕQdµ= 0and, at
455
+ last,dist(Q,suppψR)≥s(Q)αs(R)1−α. Then
456
+ |(ϕQ,TψR)| ≤ACs(Q)τ
457
+ 2s(R)τ
458
+ 2
459
+ D(Q,R)m+τ/radicalbig
460
+ µ(Q)µ(R)/bardblϕQ/bardblL2(X;µ)/bardblψR/bardblL2(X;µ).
461
+ Remark 14.Note that we require only that the support of the function ψRlies far from the cube
462
+ Q; the cubes QandRthemselves may intersect! Such situations will arise when e stimating the
463
+ termσ2.
464
+ Proof.LetxQbe the center of the cube Q. Note that for all x∈Q,y∈suppψR, we have
465
+ ρ(xQ,y)≥s(Q)
466
+ 2+dist(Q,suppψR)≥s(Q)
467
+ 2+2r(1−α)s(Q)/greaterorsimilars(Q)/greaterorsimilarρ(x,xQ).
468
+ Therefore,
469
+ |(ϕQ,TψR)|=/vextendsingle/vextendsingle/vextendsingle/integraldisplay
470
+ X/integraldisplay
471
+ Xk(x,y)ϕQ(x)ψR(y)dµ(x)dµ(y)/vextendsingle/vextendsingle/vextendsingle
472
+ =/vextendsingle/vextendsingle/vextendsingle/integraldisplay
473
+ X/integraldisplay
474
+ X[k(x,y)−k(xQ,y)]ϕQ(x)ψR(y)dµ(x)dµ(y)/vextendsingle/vextendsingle/vextendsingle
475
+ /lessorsimilars(Q)τ
476
+ dist(Q,suppψR)m+τ/bardblϕQ/bardblL1(X;µ)/bardblψR/bardblL1(X;µ).
477
+ There are two possible cases.
478
+ Case 1: dist(Q,suppψR)≥s(R).Then
479
+ D(Q,R) :=s(Q)+s(R)+dist(Q,R)≤3dist(Q,suppψR)
480
+ and therefore
481
+ s(Q)τ
482
+ dist(Q,suppψR)m+τ/lessorsimilars(Q)τ
483
+ D(Q,R)m+τ/lessorsimilars(Q)τ
484
+ 2s(R)τ
485
+ 2
486
+ D(Q,R)m+τ.
487
+ Case 2:s(Q)αs(R)1−α≤dist(Q,suppψR)≤s(R).ThenD(Q,R)≤3s(R) and we get
488
+ s(Q)τ
489
+ dist(Q,suppψR)m+τ≤s(Q)τ
490
+ [s(Q)αs(R)1−α]m+τ=s(Q)τ
491
+ 2s(R)τ
492
+ 2
493
+ s(R)m+τ/lessorsimilars(Q)τ
494
+ 2s(R)τ
495
+ 2
496
+ D(Q,R)m+τ.
497
+ Here, key to the proof was the choice of α=τ
498
+ 2(τ+m). Now, to finish the proof of the lemma, it
499
+ remains only to note that
500
+ /bardblϕQ/bardblL1(X;µ)≤/radicalbig
501
+ µ(Q)/bardblϕQ/bardblL2(X;µ)and/bardblψR/bardblL1(X;µ)≤/radicalbig
502
+ µ(R)/bardblψR/bardblL2(X;µ).
503
+ /square
504
+ Applying this lemma to ϕQ= ∆QfandψR= ∆Rg, we obtain
505
+ |σ2|/lessorsimilar/summationdisplay
506
+ Q,Rs(Q)τ
507
+ 2s(R)τ
508
+ 2
509
+ D(Q,R)m+τ/radicalbig
510
+ µ(Q)/radicalbig
511
+ µ(R)/bardbl∆Qf/bardblL2(X;µ)/bardbl∆Rg/bardblL2(X;µ). (4.1)
512
+ To control term σ2the computations above suggest that we will define a matrix op erator, de-
513
+ pending on the cubes QandRand show that it is a bounded operator on ℓ2.
514
+ Lemma 15. Define
515
+ TQ,R:=s(Q)τ
516
+ 2s(R)τ
517
+ 2
518
+ D(Q,R)m+τ/radicalbig
519
+ µ(Q)/radicalbig
520
+ µ(R) (Q∈ Dtr
521
+ 1, R∈ Dtr
522
+ 2, s(Q)≤s(R)).BERGMAN-TYPE SINGULAR INTEGRAL OPERATORS ON METRIC SPACES 11
523
+ Then, for any two families {aQ}Q∈Dtr
524
+ 1and{bR}R∈Dtr
525
+ 2of nonnegative numbers, one has
526
+ /summationdisplay
527
+ Q,RTQ,RaQbR≤AC/bracketleftBig/summationdisplay
528
+ Qa2
529
+ Q/bracketrightBig1
530
+ 2/bracketleftBig/summationdisplay
531
+ Rb2
532
+ R/bracketrightBig1
533
+ 2.
534
+ Remark 16.Note thatTQ,Ris defined for all QandRwiths(Q)≤s(R) and that the conditions
535
+ dist(Q,R)≥s(Q)αs(R)1−α(or even the condition Q∩R=∅) no longer appears as a condition in
536
+ the summation!
537
+ Assuming Lemma 15for the moment, the estimate of σ2then proceeds in an obvious fashion.
538
+ |σ2|/lessorsimilar/summationdisplay
539
+ Q,Rs(Q)τ
540
+ 2s(R)τ
541
+ 2
542
+ D(Q,R)m+τ/radicalbig
543
+ µ(Q)/radicalbig
544
+ µ(R)/bardbl∆Qf/bardblL2(X;µ)/bardbl∆Rg/bardblL2(X;µ)
545
+ /lessorsimilar
546
+ /summationdisplay
547
+ Q/bardbl∆Qf/bardbl2
548
+ L2(X;µ)
549
+ 1/2/parenleftBigg/summationdisplay
550
+ R/bardbl∆Rg/bardbl2
551
+ L2(X;µ)/parenrightBigg1/2
552
+ /lessorsimilar/bardblf/bardblL2(X;µ)/bardblg/bardblL2(X;µ).
553
+ Here the first line follows by ( 4.1), the second by Lemma 15, and finally the last by Lemma 6. We
554
+ now turn to the proof of Lemma 15.
555
+ Proof.Let us “slice” the matrix TQ,Raccording to the ratios(Q)
556
+ s(R). Namely, let
557
+ T(k)
558
+ Q,R=/braceleftBigg
559
+ TQ,Rifs(Q) =κ−ks(R);
560
+ 0 otherwise ,
561
+ (k= 0,1,2,...). To prove the lemma, it is enough to show that for every k≥0,
562
+ /summationdisplay
563
+ Q,RT(k)
564
+ Q,RaQbR≤C2−τ
565
+ 2k/bracketleftBig/summationdisplay
566
+ Qa2
567
+ Q/bracketrightBig1
568
+ 2/bracketleftBig/summationdisplay
569
+ Rb2
570
+ R/bracketrightBig1
571
+ 2.
572
+ The matrix {T(k)
573
+ Q,R}has a “block” structure since the variables bRcorresponding to the cubes
574
+ R∈ Dtr
575
+ 2for whichs(R) =κjcan only interact with the variables aQcorresponding to the cubes
576
+ Q∈ Dtr
577
+ 1, for which s(Q) =κj−k. Thus, to get the desired inequality, it is enough to estimat e each
578
+ block separately, i.e., to demonstrate that
579
+ /summationdisplay
580
+ Q,R:s(Q)=κj−k,s(R)=κjT(k)
581
+ Q,RaQbR≤C/bracketleftBig/summationdisplay
582
+ Q:s(Q)=κj−ka2
583
+ Q/bracketrightBig1
584
+ 2/bracketleftBig/summationdisplay
585
+ R:s(R)=κjb2
586
+ R/bracketrightBig1
587
+ 2.
588
+ Let us introduce the functions
589
+ F(x) :=/summationdisplay
590
+ Q:s(Q)=κj−kaQ/radicalbig
591
+ µ(Q)χQ(x) and G(x) :=/summationdisplay
592
+ R:ℓ(R)=κjbR/radicalbig
593
+ µ(R)χR(x).
594
+ Note that the cubes of a given size in one dyadic lattice do not intersect (Property (ii) of Theorem
595
+ 2), and therefore at each point x∈X, at most one term in the sum can be non-zero. Also observe
596
+ that
597
+ /bardblF/bardblL2(X;µ)=/bracketleftBig/summationdisplay
598
+ Q:s(Q)=κj−ka2
599
+ Q/bracketrightBig1
600
+ 2and/bardblG/bardblL2(X;µ)=/bracketleftBig/summationdisplay
601
+ R:s(R)=κjb2
602
+ R/bracketrightBig1
603
+ 2.
604
+ Then the estimate we need can be rewritten as/integraldisplay
605
+ X/integraldisplay
606
+ XKj,k(x,y)F(x)G(y)dµ(x)dµ(y)≤C/bardblF/bardblL2(X;µ)/bardblG/bardblL2(X;µ),12 A. VOLBERG AND B. D. WICK
607
+ where
608
+ Kj,k(x,y) =/summationdisplay
609
+ Q,R:s(Q)=κj−k,s(R)=κjs(Q)τ
610
+ 2s(R)τ
611
+ 2
612
+ D(Q,R)m+τχQ(x)χR(y).
613
+ Again, for every pair of points x,y∈X, only one term in the sum can be nonzero. Since ρ(x,y)+
614
+ s(R)≤3D(Q,R) for anyx∈Qandy∈R, we obtain
615
+ Kj,k(x,y) =Cκ−τ
616
+ 2ks(R)τ
617
+ D(Q,R)m+τ
618
+ /lessorsimilarκ−τ
619
+ 2kκjτ
620
+ [κj+ρ(x,y)]m+τ=:κ−τ
621
+ 2kkj(x,y).
622
+ So, it is enough to check that/integraldisplay
623
+ X/integraldisplay
624
+ Xkj(x,y)F(x)G(y)dµ(x)dµ(y)/lessorsimilar/bardblF/bardblL2(X;µ)/bardblG/bardblL2(X;µ).
625
+ We remind the reader that we called the balls “non-Ahlfors ba lls” if
626
+ µ(B(x,r))>rm.
627
+ According to the Schur test, it would suffice to prove that for e veryy∈X, one has the estimate/integraltext
628
+ Xkj(x,y)dµ(x)/lessorsimilar1 and vice versa (i.e., for every x∈X, one has/integraltext
629
+ Xkj(x,y)dµ(y)/lessorsimilar1). Then
630
+ the norm of the integral operator with kernel kjinL2(X;µ) would be bounded by a constant and
631
+ the proof of Lemma 15would be over. If we assumed a priori that the supremum of radi i of all
632
+ non-Ahlfors balls centered at y∈Rwiths(R) =κj,were less than κj+1, then the needed estimate
633
+ would be immediate. In fact, we can write/integraldisplay
634
+ Xkj(x,y)dµ(x) =/integraldisplay
635
+ B(y,κj+1)kj(x,y)dµ(x)+/integraldisplay
636
+ X\B(y,κj+1)kj(x,y)dµ(x)
637
+ /lessorsimilarκ−jmµ(B(y,κj+1))+/integraldisplay
638
+ X\B(y,κj+1)κjτ
639
+ ρ(x,y)m+τdµ(x)
640
+ /lessorsimilarκ−jmµ(B(y,κj+1))+∞/summationdisplay
641
+ k=0κjτ
642
+ (κkκj+1)m+τµ(B(y;κkκj+1))
643
+ /lessorsimilar/parenleftBig
644
+ 1+∞/summationdisplay
645
+ k=01
646
+ κkτ/parenrightBig
647
+ ≈1.
648
+ The passage from the second to the third line follows by exhau sting the the space X\B(y,κj+1)
649
+ by “annular regions” and making obvious estimates using con dition (H).
650
+ The difficulty with this approach is that we cannot guarantee t he supremum of the radii of all
651
+ non-Ahlfors balls centered at ybe less than κj+1for everyy∈X. Our measure may not have this
652
+ uniform property.
653
+ So, generally speaking, we are unable to show that the integr al operator with kernel kj(x,y) acts
654
+ inL2(X;µ); but we do not need that much! We only need to check that the corresponding bilin ear
655
+ form is bounded on two givenfunctionsFandG. So, we are not interested in the points y∈X
656
+ for whichG(y) = 0 (or in the points x∈X, for which F(x) = 0). But, by definition, Gcan be
657
+ non-zero only on the transit cubes in D2. Here we used our convention that we omit in all sums
658
+ the fact that QandRare transit cubes, however they are!
659
+ Now let us notice that if (and this is the case for all Rin the sum we estimate in our lemma)
660
+ R∈ Dtran
661
+ 2, then the supremum of radii of all non-Ahlfors balls centere d aty∈Ris bounded by
662
+ s(R) for every y∈R. Indeed, this is just Lemma 4. The same reasoning shows that if Q∈ Dtran
663
+ 1,
664
+ then the supremum of radii of all non-Ahlfors balls centered atx∈Qis bounded by κj−k+1≤κj+1
665
+ wheneverF(x)/\e}atio\slash= 0, and we are done with Lemma 15. /squareBERGMAN-TYPE SINGULAR INTEGRAL OPERATORS ON METRIC SPACES 13
666
+ Now, wehope, thereader will agree that thedecision to decla re thecubescontained inΩterminal
667
+ was a good one. As a result, the fact that the measure µis not Ahlfors did not put us in any real
668
+ trouble – we barely had a chance to notice this fact at all. But , it still remains to explain why we
669
+ were so eager to have the extra condition
670
+ |k(x,y)| ≤1
671
+ max(dm(x),dm(y)), d(x) := dist(x,X\Ω)
672
+ on our Calder´ on–Zygmund kernel. The answer is found in the n ext two sections.
673
+ 5.Short Range Interaction and Nonhomogeneous Paraproducts: Controlling
674
+ Termσ3.
675
+ Recall that the sum σ3is taken over the pairs Q,R, for which s(Q)<κ−rs(R) andQ∩R/\e}atio\slash=∅.
676
+ We would like to improve this condition to the demand that Qlie “deep inside” one of the subcubes
677
+ Rj. Recall also that we defined the skeletonskRof the cube Rby
678
+ skR:=/uniondisplay
679
+ j∂Rj.
680
+ We have declared a cube Q∈ D1bad if there exists a cube R∈ D2such thats(R)>κrs(Q) and
681
+ dist(Q,skR)≤s(Q)αs(R)1−α. Now, for every good cube Q∈ D1, the conditions s(Q)<κ−rs(R)
682
+ andQ∩R/\e}atio\slash=∅together imply that Qlies inside one of the children RjofR. We will denote this
683
+ subcube by RQ. The sum σ3can now be split into
684
+ σterm
685
+ 3:=/summationdisplay
686
+ Q,R:Q⊂R,s(Q)<κ−rs(R),RQis terminal(∆Qf,T∆Rg)
687
+ and
688
+ σtran
689
+ 3:=/summationdisplay
690
+ Q,R:Q⊂R,s(Q)<κ−rs(R),RQis transit(∆Qf,T∆Rg).
691
+ 5.1.Estimation of σterm
692
+ 3.First of all, write (recall that Rjdenote the children of R):
693
+ σterm
694
+ 3=/summationdisplay
695
+ j/summationdisplay
696
+ Q,R:s(Q)<κ−rs(R),Q⊂Rj∈Dterm
697
+ 2(∆Qf,T∆Rg).
698
+ Clearly, it is enough to estimate the inner sum for every fixed , and so let us do this for j= 1. We
699
+ have/summationdisplay
700
+ Q,R:s(Q)<κ−rs(R),Q⊂R1∈Dterm
701
+ 2(∆Qf,T∆Rg) =/summationdisplay
702
+ R:R1∈Dterm
703
+ 2/summationdisplay
704
+ Q:s(Q)<κ−rs(R),Q⊂R1(∆Qf,T∆Rg).
705
+ Recall that the kernel kof our operator Tsatisfies the estimate of Lemma 3
706
+ |k(x,y)|/lessorsimilar1
707
+ s(R)mfor allx∈R1,y∈X. (5.1)
708
+ Hence,
709
+ |T∆Rg(x)|/lessorsimilar/bardbl∆Rg/bardblL1(X;µ)
710
+ s(R)mfor allx∈R1, (5.2)
711
+ and therefore
712
+ /bardblχR1·T∆Rg/bardblL2(X;µ)/lessorsimilar/bardbl∆Rg/bardblL1(X;µ)/radicalbig
713
+ µ(R1)
714
+ s(R)m
715
+ /lessorsimilarµ(R)
716
+ s(R)m/bardbl∆Rg/bardblL2(X;µ)≤AB/bardbl∆Rg/bardblL2(X;µ).14 A. VOLBERG AND B. D. WICK
717
+ This follows because /bardbl∆Rg/bardblL1(X:µ)≤/radicalbig
718
+ µ(R)/bardbl∆Rg/bardblL2(X;µ)andµ(R1)≤µ(R) hold trivially. Ad-
719
+ ditionally, by Lemma 4we have
720
+ µ(R)/lessorsimilars(R)m(5.3)
721
+ becauseR(the father of the cube R1) is a transit cube if R1is terminal.
722
+ Now, recalling Lemma 6, and taking into account that ∆Qf≡0 outsideQ, we get
723
+ /summationdisplay
724
+ Q:Q⊂R1|(∆Qf,T∆Rg)|=/summationdisplay
725
+ Q:Q⊂R1|(∆Qf,χR1·T∆Rg)|
726
+ /lessorsimilar/bardblχR1·T∆Rg/bardblL2(X;µ)/bracketleftBig/summationdisplay
727
+ Q:Q⊂R1/bardbl∆Qf/bardbl2
728
+ L2(X;µ)/bracketrightBig1
729
+ 2
730
+ /lessorsimilar/bardbl∆Rg/bardblL2(X;µ)/bracketleftBig/summationdisplay
731
+ Q:Q⊂R1/bardbl∆Qf/bardbl2
732
+ L2(X;µ)/bracketrightBig1
733
+ 2.
734
+ So, we obtain/summationdisplay
735
+ R:R1∈Dterm
736
+ 2/summationdisplay
737
+ Q:Q⊂R1|(∆Qf,T∆Rg)|
738
+ /lessorsimilar/summationdisplay
739
+ R:R1∈Dterm
740
+ 2/bardbl∆Rg/bardblL2(X;µ)/bracketleftBig/summationdisplay
741
+ Q:Q⊂R1/bardbl∆Qf/bardbl2
742
+ L2(X;µ)/bracketrightBig1
743
+ 2
744
+ /lessorsimilar/bracketleftBig/summationdisplay
745
+ R:R1∈Dterm
746
+ 2/bardbl∆Rg/bardbl2
747
+ L2(X;µ)/bracketrightBig1
748
+ 2/bracketleftBig/summationdisplay
749
+ R:R1∈Dterm
750
+ 2/summationdisplay
751
+ Q:Q⊂R1/bardbl∆Qf/bardbl2
752
+ L2(X;µ)/bracketrightBig1
753
+ 2.
754
+ But the terminal cubes in D2do not intersect! Therefore every ∆Qfcan appear at most once in
755
+ the last double sum, and we get the bound
756
+ /summationdisplay
757
+ R:R1∈Dterm
758
+ 2/summationdisplay
759
+ Q:Q⊂R1|(∆Qf,T∗∆Rψ)|
760
+ /lessorsimilar/bracketleftBig/summationdisplay
761
+ R/bardbl∆Rg/bardbl2
762
+ L2(X;µ)/bracketrightBig1
763
+ 2/bracketleftBig/summationdisplay
764
+ Q/bardbl∆Qf/bardbl2
765
+ L2(X;µ)/bracketrightBig1
766
+ 2/lessorsimilar/bardblf/bardblL2(X;µ)/bardblψ/bardblL2(X;µ).
767
+ Lemma6has been used again in the last inequality.
768
+ 5.2.Estimation of σtran
769
+ 3.Recall that
770
+ σtran
771
+ 3=/summationdisplay
772
+ Q,R:Q⊂R,s(Q)<κ−rs(R),RQis transit(∆Qf,T∗∆Rg).
773
+ Split every term in the sum as
774
+ (∆Qf,T∆Rψ) = (∆Qf,T(χRQ∆Rg))+(∆Qf,T∗(χR\RQ∆Rg)).
775
+ Observe that since Qis good,Q⊂R, ands(Q)<κ−rs(R), we have
776
+ dist(Q,suppχR\RQ∆Rg)≥dist(Q,skR)≥s(Q)αs(R)1−α.
777
+ Using Lemma 13and taking into account that the norm /bardblχR\RQ∆Rψ/bardblL2(X;µ)does not exceed
778
+ /bardbl∆Rψ/bardblL2(X;µ), we conclude that the sum
779
+ /summationdisplay
780
+ Q,R:Q⊂R,s(Q)<κ−rs(R),RQis transit|(∆Qf,T∗(χR\RQ∆Rg))|BERGMAN-TYPE SINGULAR INTEGRAL OPERATORS ON METRIC SPACES 15
781
+ can be estimated by the sum ( 4.1). Thus, our task is to find a good bound for the sum
782
+ /summationdisplay
783
+ Q,R:Q⊂R,s(Q)<κ−rs(R),RQis transit(∆Qf,T∗(χRQ∆Rg)).
784
+ Recalling the definition of ∆Rψand recalling that RQis atransitcube, we get
785
+ χRQ∆Rg=cR,QχRQ,
786
+ where
787
+ cRQ=/a\}bracketle{tψ/a\}bracketri}htRQ−/a\}bracketle{tg/a\}bracketri}htR
788
+ is aconstant. So, our sum can be rewritten as
789
+ /summationdisplay
790
+ Q,R:Q⊂R,s(Q)<κ−rs(R),RQis transitcRQ(∆Qf,T∗(χRQ)).
791
+ Our next goal will be to extend the function χRQto the function 1 in every term.
792
+ Let us observe that
793
+ (∆Qf,T∗(χX\RQ)) =/integraldisplay
794
+ X/integraldisplay
795
+ X\RQk(x,y)∆Qf(x)dµ(x)dµ(y)
796
+ =/integraldisplay
797
+ X/integraldisplay
798
+ X\RQ[k(x,y)−k(xQ,y)]∆Qf(x)dµ(x)dµ(y).
799
+ Note again that for every x∈Q,y∈X\RQ, we have
800
+ ρ(xQ,y)≥s(Q)
801
+ 2+dist(Q,X\RQ)/greaterorsimilars(Q)/greaterorsimilarρ(x,xQ).
802
+ Therefore,
803
+ |k(x,y)−k(xQ,y)|/lessorsimilar/parenleftBiggρ(x,xQ)
804
+ ρ(xQ,y)/parenrightBiggτ1
805
+ ρ(x,y)m/lessorsimilars(Q)τ
806
+ ρ(xQ,y)m+τ,
807
+ and
808
+ |(∆Qf,T(χX\RQb))|/lessorsimilars(Q)τ/bardbl∆Qf/bardblL1(X;µ)/integraldisplay
809
+ X\RQdµ(y)
810
+ ρ(xQ,y)m+τ.
811
+ Now let us consider the sequence of cubes R(j)∈ D2, beginning with R(0)=RQand gradually
812
+ ascending ( R(j)⊂R(j+1),s(R(j+1)) =κs(R(j))) to the starting cube R0=R(N)of the lattice D2.
813
+ Clearly, all these cubes R(j)are transit cubes.
814
+ We have
815
+ /integraldisplay
816
+ X\RQdµ(y)
817
+ ρ(xQ,y)m+τ=/integraldisplay
818
+ R0\RQdµ(y)
819
+ ρ(xQ,y)m+τ=N/summationdisplay
820
+ j=1/integraldisplay
821
+ R(j)\R(j−1)dµ(y)
822
+ ρ(xQ,y)m+τ.
823
+ We call the j-th term of this sum Ij. Note now that, since Qis good and s(Q)< κ−rs(R)≤
824
+ κ−rs(R(j)) for allj, we have
825
+ dist(Q,R(j)\R(j−1))≥dist(Q,skR(j))≥s(Q)αs(R(j))1−α.
826
+ Hence
827
+ Ij≤1
828
+ [s(Q)αs(R(j))1−α]m+τ/integraldisplay
829
+ R(j)dµ.16 A. VOLBERG AND B. D. WICK
830
+ Recalling that α=τ
831
+ 2(m+τ), we see that the first factor equals
832
+ 1
833
+ s(Q)τ
834
+ 2s(R(j))m+τ
835
+ 2.
836
+ SinceR(j)is transit, we have /integraldisplay
837
+ R(j)dµ/lessorsimilarµ(R(j))/lessorsimilars(R(j))m.
838
+ Thus,
839
+ Ij/lessorsimilars(Q)τ
840
+ 2s(R(j))τ
841
+ 2=κ−(j−1)ε
842
+ 2s(Q)τ
843
+ 2s(R)τ
844
+ 2.
845
+ Summing over j≥1, we get
846
+ /integraldisplay
847
+ X\RQ|b(y)|dµ(y)
848
+ ρ(xQ,y)m+τ=N/summationdisplay
849
+ j=1Ij/lessorsimilar1−κ−τ
850
+ 21
851
+ s(Q)τ
852
+ 2s(R)τ
853
+ 2.
854
+ Now let us note that
855
+ |cRQ| ≤/bardbl∆Rg/bardblL1(RQ,µ)
856
+ µ(RQ)≤/bardbl∆Rg/bardblL2(RQ,µ)/radicalbig
857
+ µ(RQ). (5.4)
858
+ We finally obtain
859
+ |(∆Qf,T∗(χX\RQ))|
860
+ /lessorsimilar1
861
+ η(1−κ−τ
862
+ 2)/bracketleftbiggs(Q)
863
+ s(R)/bracketrightbiggτ
864
+ 2/radicalBigg
865
+ µ(Q)
866
+ µ(RQ)/bardbl∆Qf/bardblL2(X;µ)/bardbl∆Rg/bardblL2(X;µ)
867
+ and
868
+ /summationdisplay
869
+ Q,R:Q⊂R,s(Q)<κ−rs(R),RQis transit|cR,Q|·|(∆Qf,T∗(χX\RQ))|
870
+ /lessorsimilar1
871
+ η(1−κ−τ
872
+ 2)/summationdisplay
873
+ j/summationdisplay
874
+ Q,R:Q⊂Rj/bracketleftbiggs(Q)
875
+ s(R)/bracketrightbiggτ
876
+ 2/radicalBigg
877
+ µ(Q)
878
+ µ(Rj)/bardbl∆Qf/bardblL2(X;µ)/bardbl∆Rg/bardblL2(X;µ).
879
+ Lemma 17. For every two families {aQ}Q∈Dtr
880
+ 1and{bR}R∈Dtr
881
+ 2of nonnegative numbers, one has
882
+ /summationdisplay
883
+ Q,R:Q⊂R1TQ,RaQbR≤1
884
+ 1−κ−τ
885
+ 2/bracketleftBig/summationdisplay
886
+ Qa2
887
+ Q/bracketrightBig1
888
+ 2/bracketleftBig/summationdisplay
889
+ Rb2
890
+ R/bracketrightBig1
891
+ 2.
892
+ Proof.Let us “slice” the matrix TQ,Raccording to the ratios(Q)
893
+ s(R). Namely, let
894
+ T(k)
895
+ Q,R=/braceleftBigg
896
+ TQ,R,ifQ⊂R1, s(Q) =κ−ks(R);
897
+ 0,otherwise
898
+ (k= 1,2,...). It is enough to show that for every k≥0,
899
+ /summationdisplay
900
+ Q,RT(k)
901
+ Q,RaQbR≤κ−τ
902
+ 2k/bracketleftBig/summationdisplay
903
+ Qa2
904
+ Q/bracketrightBig1
905
+ 2/bracketleftBig/summationdisplay
906
+ Rb2
907
+ R/bracketrightBig1
908
+ 2.
909
+ The matrix {T(k)
910
+ Q,R}has a very good “block” structure: every aQcan interact with only onebR. So,
911
+ it is enough to estimate each block separately, i.e., to show that for every fixed R∈ Dtran
912
+ 2,
913
+ /summationdisplay
914
+ Q:Q⊂R1,ℓ(Q)=κ−kℓ(R)κ−τ
915
+ 2k/radicalBigg
916
+ µ(Q)
917
+ µ(R1)aQbR≤κ−τ
918
+ 2k/bracketleftBig/summationdisplay
919
+ Qa2
920
+ Q/bracketrightBig1
921
+ 2bR.BERGMAN-TYPE SINGULAR INTEGRAL OPERATORS ON METRIC SPACES 17
922
+ But, reducing both parts by the non-essential factor κ−τ
923
+ 2kbR, we see that this estimate is equivalent
924
+ to the trivial estimate
925
+ /summationdisplay
926
+ Q:Q⊂R1,s(Q)=κ−ks(R)/radicalBigg
927
+ µ(Q)
928
+ µ(R1)aQ
929
+ ≤/bracketleftBig/summationdisplay
930
+ Q:Q⊂R1,s(Q)=κ−ks(R)µ(Q)
931
+ µ(R1)/bracketrightBig1
932
+ 2/bracketleftBig/summationdisplay
933
+ Qa2
934
+ Q/bracketrightBig1
935
+ 2≤/bracketleftBig/summationdisplay
936
+ Qa2
937
+ Q/bracketrightBig1
938
+ 2,
939
+ (since cubes Q∈ D1of fixed size do not intersect,/summationtext
940
+ Q:Q⊂R1,s(Q)=κ−ks(R)µ(Q)≤µ(R1)).
941
+ /square
942
+ Remark 18.We did not use here the fact that {aQ},{bR}are supported on transit cubes. We
943
+ actually proved
944
+ Lemma19. The matrix {TQ,R}defined by
945
+ TQ,R:=/bracketleftbiggs(Q)
946
+ s(R)/bracketrightbiggτ
947
+ 2/radicalBigg
948
+ µ(Q)
949
+ µ(R1)(Q⊂R1),
950
+ generates a bounded operator in l2.
951
+ We just finished estimating an extra term which appeared when we extendχRQto the whole
952
+ 1. So, the extension of χRQto the function 1 does not cause much harm, and we are left with
953
+ estimating the sum/summationdisplay
954
+ Q,R:Q⊂R,s(Q)<κ−rs(R),RQis transitcRQ(∆Qf,T∗1).
955
+ Note that the inner product (∆Qf,T∗1)does not depend onRat all, so it seems to be a good idea
956
+ to sum over Rfor fixedQfirst.
957
+ Recalling that
958
+ cRQ=/a\}bracketle{tg/a\}bracketri}htRQ−/a\}bracketle{tg/a\}bracketri}htR
959
+ and that Λψ= 0⇐⇒ /a\}bracketle{tψ/a\}bracketri}htR0= 0, we conclude that for every Q∈ Dtran
960
+ 1that really appears in the
961
+ above sum,/summationdisplay
962
+ R:R⊃Q,s(R)>κms(Q),RQis transitcRQ=/a\}bracketle{tg/a\}bracketri}htRQ.
963
+ Definition. LetR(Q) be the smallest transitcubeR∈ D2containing Qand such that s(R)≥
964
+ κrs(Q).
965
+ So, we obtain the sum
966
+ /summationdisplay
967
+ Q:s(Q)<κ−rs(R)/a\}bracketle{tg/a\}bracketri}htR(Q)(∆Qf,T∗1)
968
+ to take care of.
969
+ Remark. Let us recall that we had the convention that says that the cub esQconsidered are
970
+ only good ones (and of course they are only transit cubes). Th e range of summation should be
971
+ Q∈ Dtran
972
+ 1,Qis good (default); there exists a cube R∈ Dtran
973
+ 2such thats(Q)<κ−rs(R),Q⊂R
974
+ and the child RQ(the one containing Q) ofRis transit. In other words, in fact, the sum is written18 A. VOLBERG AND B. D. WICK
975
+ formally incorrectly. We have to replace R(Q) byRQin the summation. However, the smallest
976
+ transit cube containing Q(this isR(Q)) and the smallest transit child (containing Q) of a certain
977
+ subcubeRofR0(this child is RQ) are of course the same cube, unless R(Q) =R0. Thus the sum
978
+ formally has some extra terms corresponding to R(Q) =R0. But, they all are zeros! In one of the
979
+ first reductions, we were allowed to work only with with gsuch that Λ g= 0 (recall that Λ gmeans
980
+ the average of gwith respect to µ), so/a\}bracketle{tg/a\}bracketri}htR(Q)= 0 ifR(Q) =R0.
981
+ 5.3.Pseudo-BMOand special paraproduct. Tointroducetheparaproductoperator, werewrite
982
+ our sum as follows/summationdisplay
983
+ Q:s(Q)<κ−rs(R)/a\}bracketle{tg/a\}bracketri}htR(Q)(∆Qf,T∗1) =/summationdisplay
984
+ Q:s(Q)<κ−rs(R)/a\}bracketle{tg/a\}bracketri}htR(Q)(f,∆∗
985
+ QT∗1)
986
+ =
987
+ f,/summationdisplay
988
+ Q:s(Q)<κ−rs(R)/a\}bracketle{tg/a\}bracketri}htR(Q)∆QT∗1
989
+ .
990
+ We use the fact that ∆∗
991
+ Q= ∆Q. We now introduce the paraproduct operator, which will allo w
992
+ us to control term σtran
993
+ 3.
994
+ Definition 20. Given a function F, theparaproduct with symbol Fis the function
995
+ ΠFg(x) :=/summationdisplay
996
+ R∈D2,R⊂R0/a\}bracketle{tg/a\}bracketri}htR/summationdisplay
997
+ Q∈D1,Qgood and transit ,s(Q)=κ−rs(R)∆QF(x).
998
+ As in the case when the metric space is Rd, the behavior of the paraproduct operators will be
999
+ governed by “BMO” conditions on the symbol F. In the case of a metric space though, we face an
1000
+ additional wrinkle since we have to overcome the challenge o f dealing with the dyadic cubes, and
1001
+ we need an appropriate notion of “dilation” in the metric spa ce.
1002
+ Recall that we defined the dilation by the parameter λ≥1 of a setS⊂Xby
1003
+ λ·S:={x∈X: dist(x,S)≤(λ−1)diamS}
1004
+ Note thatS⊂λ·S.
1005
+ Definition 21. A function F∈L2(X;µ) will be called a “pseudo- BMOfunction” if there exists
1006
+ Λ>1 such that for any cube Qwithµ(sQ)≤Ksmdiam(Q)m,s≥1, we have
1007
+ /integraldisplay
1008
+ Q|F(x)−/a\}bracketle{tF/a\}bracketri}htQ|2dµ(x)≤Cµ(ΛQ).
1009
+ Lemma 22. Letµ,Tsatisfy the assumptins of Theorem 1. Then
1010
+ T∗1∈pseudo-BMO . (5.5)
1011
+ HereCdepends only on the constants of Theorem 1.
1012
+ Proof.Forx∈Qwe writeT∗1(x) = (T∗χΛQ)(x)+(T∗χX\ΛQ)(x) =:ϕ(x)+ψ(x). First, we notice
1013
+ that
1014
+ x,y∈Q⇒ |ψ(x)−ψ(y)| ≤C(K,Λ),
1015
+ whereKis the constant form our definition above. This is easy:
1016
+ |ψ(x)−ψ(y)| ≤/integraldisplay
1017
+ X\ΛQ|k(x,t)−k(y,t)|dµ(t) =∞/summationdisplay
1018
+ j=1/integraldisplay
1019
+ Λj+1Q\ΛjQ|k(x,t)−k(y,t)|dµ(t)≤
1020
+ ∞/summationdisplay
1021
+ j=1diam(Q)τ
1022
+ (Λjdiam(Q))m+τK(Λjdiam(Q))m=∞/summationdisplay
1023
+ j=1K
1024
+ Λjτ≤C(K,Λ,τ).BERGMAN-TYPE SINGULAR INTEGRAL OPERATORS ON METRIC SPACES 19
1025
+ Therefore,/integraldisplay
1026
+ Q|ψ(x)−/a\}bracketle{tψ/a\}bracketri}htQ|2dµ(x)/lessorsimilarµ(Q)≤µ(ΛQ).
1027
+ But,/integraldisplay
1028
+ Q|ϕ(x)−/a\}bracketle{tϕ/a\}bracketri}htQ|2dµ(x)/lessorsimilar/integraldisplay
1029
+ Q|T∗χΛQ|2dµ≤Aµ(ΛQ)
1030
+ by theT1 assumption of Theorem 1.
1031
+ /square
1032
+ Lemma 23. Letµ,Tsatisfy the assumptins of Theorem 1. Then
1033
+ /bardblΠT∗1/bardblL2(X;µ)→L2(X;µ)≤C. (5.6)
1034
+ HereCdepends only on the constants of Theorem 1.
1035
+ Proof.LetF=T∗1. In the definition of Π Fall ∆Qare mutually orthogonal. So it is easy to see
1036
+ that
1037
+ /bardblΠFg/bardbl2
1038
+ L2(X;µ)=/summationdisplay
1039
+ R∈D2,R⊂R0|/a\}bracketle{tg/a\}bracketri}htR|2/summationdisplay
1040
+ Q∈D1,Qgood and transit ,s(Q)=κ−rs(R)/bardbl∆QF/bardblL2(X;µ).
1041
+ Put
1042
+ aR:=/summationdisplay
1043
+ Q∈D1,Qgood and transit ,s(Q)=κ−rs(R)/bardbl∆QF/bardblL2(X;µ).
1044
+ By Carleson Embedding Theorem, it is enough to prove that for everyS∈ D2
1045
+ /summationdisplay
1046
+ R∈D2,R⊂SaR≤Cµ(S). (5.7)
1047
+ This is the same as
1048
+ /summationdisplay
1049
+ Q∈D1,Qtransit,s(Q)≤κ−rs(R),dist(Q,∂R)≥s(Q)αs(R)1−α/bardbl∆QF/bardblL2(X;µ)≤Cµ(R). (5.8)
1050
+ Let us consider a Whitney decomposition of Rinto disjoint cubes P, such that 1 .5P⊂R, 1.4P
1051
+ have only bounded multiplicity C(d) of intersection. This can be accomplished by modifying the
1052
+ arguments found in Section 7 of [ 2].
1053
+ Consider the sums
1054
+ sP:=/summationdisplay
1055
+ Q∈D1,Qtransit,s(Q)≤κ−rs(R),Q∪P/\egatio\slash=∅,dist(Q,∂R)≥s(Q)αs(R)1−α/bardbl∆QF/bardblL2(X;µ).(5.9)
1056
+ ThissPcan be zero if there is no transit cubes as above intersecting it. But ifsP/\e}atio\slash= 0 then
1057
+ necessarily
1058
+ µ(P)≤A(d)s(P)m,
1059
+ and moreover
1060
+ µ(sP)≤A(d)sms(P)m,∀s≥1.
1061
+ In fact, in this case Pintersects a transit cube Q, which by elementary geometry is “smaller”’
1062
+ thanP:s(Q)≤c(r,d)s(P). But then the above inequalities follow from the definition oftransit.
1063
+ It is also clear that for large rand forQ,Pas above
1064
+ Q∩P/\e}atio\slash=∅ ⇒Q⊂1.2P .
1065
+ Therefore,
1066
+ sP/\e}atio\slash= 0⇒sP≤/summationdisplay
1067
+ Q∈D1,Qtransit,s(Q)≤κ−rs(R),Q⊂1.2Pdist(Q,∂R)≥s(Q)αs(R)1−α/bardbl∆QF/bardblL2(X;µ).20 A. VOLBERG AND B. D. WICK
1068
+ So
1069
+ sP/\e}atio\slash= 0⇒sP≤/integraldisplay
1070
+ 1.2P|F−/a\}bracketle{tF/a\}bracketri}ht1.2P|2dµ≤Cµ(1.4P).
1071
+ The last inequality follows from Lemma 22.
1072
+ Now we add all sP’s. We get ≤C/summationtextµ(1.4P). This is smaller than C1µ(R) as 1.4P’s have
1073
+ multiplicity C(d)<∞. /square
1074
+ 6.The Diagonal Sum: Controlling Term σ1.
1075
+ To complete the estimate of |(Tfgood,ggood)|in only remains to estimate σ1. But notice that
1076
+ /bardbl∆Qf/bardblL1(X;µ)≤ /bardbl∆Qf/bardblL2(X;µ)/radicalbig
1077
+ µ(Q) and/bardbl∆Rg/bardblL1(X;µ)≤ /bardbl∆Rg/bardblL2(X;µ)/radicalbig
1078
+ µ(R).
1079
+ Remember that all cubes QandRin the sums considered at this point are transit cubes. In
1080
+ particular, in σ1we have that QandRare close and of the almost same size. If a son of Q,S(Q),
1081
+ is terminal, then by Lemma 3
1082
+ |(TχS(Q)∆Qf,∆Rg)| ≤/radicalbig
1083
+ µ(Q)/radicalbig
1084
+ µ(R)
1085
+ s(Q)m/bardbl∆Qf/bardblL2(X;µ)/bardbl∆Rg/bardblL2(X;µ).
1086
+ The sons are terminal, but QandRare transit, so µ(Q)/lessorsimilars(Q)m≈s(R)m. Summing such pairs
1087
+ (and symmetric ones, where a son of Ris terminal) we get C(r)/bardblf/bardblL2(X;µ)/bardblg/bardblL2(X;µ).
1088
+ We are left with the part of σ1, where we sum over QandRsuch that their sons are transit.
1089
+ Then we use pairing
1090
+ |(TχS(Q)∆Qf,χS(R)∆Rg)| ≤ |cS(Q)||cS(R)|/radicalbig
1091
+ µ(S(Q))µ(S(R)).
1092
+ The estimate above follows from our T1 assumption in Theorem 1. Now using ( 5.4), again we
1093
+ obtain
1094
+ |(TχS(Q)∆Qf,χS(R)∆Rg)| ≤C/bardbl∆Qf/bardblL2(X;µ)/bardbl∆Rg/bardblL2(X;µ).
1095
+ This completes the proof of Theorem 1.
1096
+ References
1097
+ [1] Michael Christ, AT(b)theorem with remarks on analytic capacity and the Cauchy int egral, Colloq. Math. 60/61
1098
+ (1990), no. 2, 601–628. ↑1,3
1099
+ [2] T. Hyt¨ onen and H. Martikainen, Non-Homogeneous Tb Theorem on Metric Spaces , preprint. ↑3,4,6,7,19
1100
+ [3] F. Nazarov, S. Treil, and A. Volberg, TheTb-theorem on non-homogeneous spaces , Acta Math. 190(2003), no. 2,
1101
+ 151–239. ↑3
1102
+ [4] ,Accretive system Tb-theorems on nonhomogeneous spaces , Duke Math. J. 113(2002), no. 2, 259–312. ↑3
1103
+ [5] ,Weak type estimates and Cotlar inequalities for Calder´ on- Zygmund operators on nonhomogeneous spaces ,
1104
+ Internat. Math. Res. Notices (1998), no. 9, 463–487. ↑3
1105
+ [6] ,Cauchy integral and Calder´ on-Zygmund operators on nonhomo geneous spaces , Internat. Math. Res. No-
1106
+ tices (1997), no. 15, 703–726. ↑3
1107
+ [7] E. Sawyer and R. L. Wheeden, Weighted inequalities for fractional integrals on Euclide an and homogeneous spaces ,
1108
+ Amer. J. Math. 114(1992), no. 4, 813–874. ↑3
1109
+ [8] A. Volberg, Calder´ on-Zygmund capacities and operators on nonhomogene ous spaces , CBMS Regional Conference
1110
+ Series in Mathematics, vol. 100, Published for the Conferen ce Board of the Mathematical Sciences, Washington,
1111
+ DC, 2003. ↑3
1112
+ [9] A. Volberg and B. D. Wick, Bergman-type Singular Operators and the Characterization of Carleson Measures for
1113
+ Besov–Sobolev Spaces on the Complex Ball (2009), preprint. ↑2,3
1114
+ Alexander Volberg, Department of Mathematics, Michigan St ate University, East Lansing, MI
1115
+ USA 48824
1116
+ E-mail address :[email protected] SINGULAR INTEGRAL OPERATORS ON METRIC SPACES 21
1117
+ Alexander Volberg, Department of Mathematics, University of Edinburgh, James Clerk Maxwell
1118
+ Building, The King’s Buildings, Mayfield Road, Edinburgh S cotland EH9 3JZ
1119
+ E-mail address :[email protected]
1120
+ Brett D. Wick, School of Mathematics, Georgia Institute of T echnology, 686 Cherry Street,
1121
+ Atlanta, GA 30332-1060 USA
1122
+ E-mail address :[email protected]
1123
+ URL:http://people.math.gatech.edu/~bwick6/
1001.0039.txt ADDED
@@ -0,0 +1,166 @@
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
1
+ arXiv:1001.0039v1 [astro-ph.IM] 30 Dec 2009Astronomical Data Analysis Software and Systems XIX P45
2
+ ASP Conference Series, Vol. XXX, 2009
3
+ Y. Mizumoto, K.-I. Morita, and M. Ohishi, eds.
4
+ TGCat, The Chandra Transmission Grating Catalog and
5
+ Archive: Systems, Design and Accessibility
6
+ Arik W. Mitschang1, David P. Huenemoerder2, Joy S. Nichols1
7
+ Abstract. The recently released Chandra Transmission Grating Catalog and
8
+ Archive, TGCat, presents a fully dynamic on-line catalog allowing users to
9
+ browse and categorize Chandra gratings observations quickly and easily, gen-
10
+ erate custom plots of resulting response corrected spectra on- line without the
11
+ need for special software and to download analysis ready product s from multi-
12
+ ple observations in one convenient operation. TGCathas been registered as
13
+ a VO resource with the NVO providing direct access to the catalogs in terface.
14
+ The catalog is supported by a back-end designed to automatically fe tch newly
15
+ public data, process , archive and catalog them, At the same time ut ilizing an
16
+ advanced queue system integrated into the archive’s MySQL datab ase allowing
17
+ large processing projects to take advantage of an unlimited numbe r of CPUs
18
+ across a network for rapid completion. A unique feature of the cat alog is that
19
+ all of the high level functions used to retrieve inputs from the Chan dra archive
20
+ and to generate the final data products are available to the user in an ISIS writ-
21
+ ten library with detailed documentation. Here we present a structu ral overview
22
+ of the Systems, Design, and Accessibility features of the catalog a nd archive.
23
+ 1. Introduction
24
+ TGCataims to be the definitive end-user source for all Chandra HETG S and
25
+ LETGS observations. In order to achieve this goal the catalo g must both have
26
+ the absolute best processed and calibrated data, as well as h ave an interface
27
+ which makes it easy for users to find, review and download thei r observations
28
+ of choice. The science requirements for accurately process ing gratings data are
29
+ discussed elsewhere ( Huenemoerder et. al. 2010 ). This writ ing will focus on
30
+ the automated system that collects and processes new public data, the interfaces
31
+ used for administrative review, and the interfaces and syst ems provided for user
32
+ access.
33
+ 2. Database, Archive, and Subsystems
34
+ TheTGCatprocessing system is comprised of three major components:
35
+ •MySQL database storing meta-data
36
+ •File archive
37
+ •Processing Software
38
+ 1Smithsonian Astrophysical Observatory, Cambridge, MA, US A
39
+ 2MIT Kavli Institute for Space Research, Cambridge, MA
40
+ 12 MITSCHANG, HUENEMOERDER, NICHOLS
41
+ Figure 1. TGCatdata flow
42
+ These components are described in the context of TGCatprocessing in this
43
+ section.
44
+ Tables: Thetables relevant toprocessingincludetheextractions, source, spec-
45
+ tral properties, files, and queue tables. The extractions ta ble has one entry per
46
+ processed extraction, where extraction is taken to mean a si ngle source in a
47
+ uniqueChandra observation ID (ObsID). Any one ObsIDcan have many sources
48
+ and any one source can be in many ObsIDs, the extractions tabl e will store one
49
+ entry for each combination thereof. In order to consolidate all extractions of a
50
+ single source, there is a source table indexed on SIMBAD1identifier, a TGCat
51
+ identifier, and coordinates, that associates entries in the extractions table. The
52
+ files table tracks processing output products and summary im ages allowing the
53
+ webpagestoeasily displayinformation onfilesavailable fo rdownload. Thespec-
54
+ tral properties table stores and indexes spectral properti es in several different
55
+ wavebands, these data are also available in a fits table for do wnload.
56
+ Data Flow: The flow of data through the system is rather simple, as illus-
57
+ trated in Figure 1. A bash script run via cronat regular intervals downloads a
58
+ list of gratings observations from the public chandra archi ve2and compares with
59
+ the list of ObsIDs that have been submitted to TGCat. Any not in TGCat
60
+ will be added to the queue table to be processed in line. Daemo n processes,
61
+ written in python, run on any number of network connected hos ts continuously
62
+ requesting entries from the queue in a FIFOmanner, the first process to ask
63
+ entries from the queue table will retrieve the first entry ( by time of creation
64
+ ). They then parse the queue entry which can contain a number o f custom
65
+ processing parameters, setup work spaces and logging, fork off the processing
66
+ which is implemented in ISIS ( Houck & Denicola 2000 ) interac tive library
67
+ functions, which are available for download3and custom use, and finally ingest
68
+ 1http://simbad.u-strasbg.fr/simbad
69
+ 2http://cda.harvard.edu/chaser/
70
+ 3http://space.mit.edu/cxc/analysis/tgcatTGCat 3
71
+ data into the database and archive. During the ingest step, s everal checks are
72
+ done to evaluate whether or not the resultant processed data are a complete
73
+ set and worth being manually reviewed by TGCatscientists. If so, meta data
74
+ are added to the database’s main extractions table which ret urns a unique iden-
75
+ tifier that is then used to tag the file based products. At this t ime linking is
76
+ done between the source table and extractions table. If not, the queue entry is
77
+ marked as an error and notification sent. TGCatoperators have the choice to
78
+ investigate the existing processing workspace or simply re -evaluate parameters
79
+ and re-queue the extraction as new.
80
+ Validation and Verification: Each extraction available for browsing has
81
+ been reviewed by a member of the TGCatscience team in order to confirm
82
+ zeroth order placement, proper masking, etc. This is done vi a an internal web-
83
+ site nearly mimicking the public interface for reviewing ex tractions but with
84
+ the addition of forms for queueing processing and for markin g the extraction as
85
+ “good”, “bad”, or “warning” and optional comment fields whic h are available
86
+ for review by the end user. “bad” extractions are never shown on the public
87
+ site and administrators have the option of rejecting any oth er extraction for
88
+ whatever reason. Because each extraction is tagged with the date of processing,
89
+ the version of TGCatused for processing, and a so called group ID which is
90
+ unique per group of extractions intended to be of the same obj ect for a single
91
+ ObsID, keeping track of accepted sources and avoiding dupli cates is made easy.
92
+ 3. Access
93
+ TGCathas several interfaces for data access. Currently three are operational
94
+ includingthewebbrowserinterface, andtwoVirtualObserv atoryaccessprotocol
95
+ interfaces. Each oftheseaccess interfaces areimplemente dasa“plug-in”written
96
+ in php. Plug-in functions to create output are called by a cat alog independent
97
+ generalized query library written in php under TGCatdevelopment dubbed
98
+ “queryLib”. This modular approach makes it trivial to add ac cess interfaces to
99
+ TGCat.
100
+ queryLib: queryLib stores state and type information of an individual query
101
+ in a database table. Each time a query is performed, fields in t his table such as
102
+ awhereclause, sort fields, indexed IDs and columns are populated wi th infor-
103
+ mation specific to that query. As a user interacts with the par ticular interface,
104
+ information in this table entry are updated to reflect the new state of the query
105
+ through object method calls. queryLib initialized a query b y creating a new en-
106
+ try in the table with two unique IDs: one is the primary key ide ntifier assigned
107
+ at creation and can be used to reference the query, the other i s created on the fly
108
+ using a combination of the primary key and a random hash strin g that is very
109
+ unlikely to be guessed. This is done so that a malicious user c annot change the
110
+ state of anyone’s query by simply guessing an ID. The queryLi b speeds up the
111
+ process of redrawing a table of results by storing the indexe d primary keys of
112
+ all entries selected rather than having to rerun a potential ly complicated where
113
+ clause.4 MITSCHANG, HUENEMOERDER, NICHOLS
114
+ Web Catalog Interface: The web interface4toTGCatallows users browse
115
+ extraction and object entries in a way that is most suitable t o the catalog data
116
+ structure. Each extraction has an associated preview page w here a set of stan-
117
+ dard plots produced at the time of processing is available fo r viewing. A table
118
+ of spectral properties providing quick general categoriza tion, and a table of each
119
+ individual file available for the extraction along with a lin k to download that file
120
+ are provided. Perhaps one of the most important features of t he web interface
121
+ is the interactive plotting of individual or combined spect ra directly on the site.
122
+ This is implemented server side by taking the POST request fo r plotting param-
123
+ eters along with a unique file name, creating a small script of the commands for
124
+ loading data and plotting, and piping this script into an ISI S process. Malicious
125
+ useisprevented by checking all parameter inputsfor approp riatevalues, running
126
+ the ISIS process as an unprivileged user, and checking the te mp file name for
127
+ validity before piping. Since the file name is known at the tim e of the request,
128
+ the page simply needs to reload to show the new plot. The comma nds, any error
129
+ output, and an ASCII dump of the plot data are available for do wnload as well.
130
+ Virtual Observatory: TGCatis a registered VO service providing both
131
+ the Simple Cone Search5and Simple Image Access6Protocols. These are both
132
+ implemented as an XML output plug-ins taking input from a php script that
133
+ provides appropriate GET parameters, then creates a query o bject after parsing
134
+ the input data, finally running the query in exactly the same w ay as for the web
135
+ and ASCII interfaces. Error handling is done by the calling s cript and meta
136
+ data is provided by running a query with parameters known to h ave no catalog
137
+ entries ( indexed ID=0 ). VO queries, and any other type, can b e tracked in
138
+ the query table using the type column. In this way we can track statistics on
139
+ requests coming from services such as datascope7.
140
+ Data Package Downloads: Generally, users will want to download more
141
+ than one file at a time for analysis, such as such as spectra and responses or
142
+ productsfrom multiple extractions of the same target. To th is endTGCatruns
143
+ apackager processthatparsesaqueuetablemuchlikethepro cessingqueuetable
144
+ described above. Requested packages are added to the packag e queue table,
145
+ validated and read by the packager which then fetches data to a temp space
146
+ placing them in a directory hierarchy tagged with ObsID and TGCatID. The
147
+ entire hierarchy is then tarred, compressed and provided to the user along with
148
+ file checksums via HTTP.
149
+ Acknowledgements. We would like to thank Dan Dewey for extensive input
150
+ on data organization and interface layout, and Mike Nowak fo r ISIS plotting
151
+ routines and advice on the interactive plotting interface. This work is supported
152
+ 4http://tgcat.mit.edu
153
+ 5http://tgcat.mit.edu/tgCli.php?OUTPUT=V
154
+ 6http://tgcat.mit.edu/tgSia.php
155
+ 7http://heasarc.gsfc.nasa.gov/cgi-bin/vo/datascope/i nit.plTGCat 5
156
+ by the Chandra X-ray Center (CXC) NASA contract NAS8-03060. DPH was
157
+ supportedby NASA through the Smithsonian Astrophysical Ob servatory (SAO)
158
+ contract SV3-73016 for the Chandra X-Ray Center and Science Instruments.
159
+ References
160
+ Huenemoerder, et. al. 2010, ( in preparation )
161
+ Houck, J. C., Denicola, L. A. 2000, in ASP Conf. Ser. 216, ADASS IX, ed. N. Manset,
162
+ C. Veillet, & D. Crabtree (San Francisco: ASP), 591
163
+ Williams, Roy, Hanisch, Robert, Szalay, Alex, Plante, Ray 2009, Simple Cone Search
164
+ Version1.03,http://www.ivoa.net/Documents/REC/DAL/ConeSea rch-20080222.html
165
+ Tody, Doug & Plante, Ray 2009, Simple Image Access Specification Ve rsion 1.0,
166
+ http://www.ivoa.net/Documents/SIA/20091008/PR-SIA-1.0-20 091008.html
1001.0040.txt ADDED
@@ -0,0 +1,725 @@
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
1
+ arXiv:1001.0040v2 [math-ph] 16 Sep 2010COURANT ALGEBROIDS FROM CATEGORIFIED
2
+ SYMPLECTIC GEOMETRY
3
+ CHRISTOPHER L. ROGERS
4
+ Abstract. In categorified symplectic geometry, one studies the cate-
5
+ gorified algebraic and geometric structures that naturally arise on man-
6
+ ifolds equipped with a closed nondegenerate ( n+ 1)-form. The case
7
+ relevant to classical string theory is when n= 2 and is called ‘2-plectic
8
+ geometry’. Just as the Poisson bracket makes the smooth func tions on
9
+ a symplectic manifold into a Lie algebra, there is a Lie 2-alg ebra of
10
+ observables associated to any 2-plectic manifold. String t heory, closed
11
+ 3-forms and Lie 2-algebras also play important roles in the t heory of
12
+ Courant algebroids. Courant algebroids are vector bundles which gen-
13
+ eralize the structures found in tangent bundles and quadrat ic Lie alge-
14
+ bras. It is known that a particular kind of Courant algebroid (called an
15
+ exact Courant algebroid) naturally arises in string theory , and that such
16
+ an algebroid is classified up to isomorphism by a closed 3-for m on the
17
+ base space, which then induces a Lie 2-algebra structure on t he space of
18
+ global sections. In this paper we begin to establish precise connections
19
+ between 2-plectic manifolds and Courant algebroids. We pro ve that any
20
+ manifold Mequipped with a 2-plectic form ωgives an exact Courant
21
+ algebroid EωoverMwithˇSevera class [ ω], and we construct an embed-
22
+ ding of the Lie 2-algebra of observables into the Lie 2-algeb ra of sections
23
+ ofEω. We then show that this embedding identifies the observables as
24
+ particular infinitesimal symmetries of Eωwhich preserve the 2-plectic
25
+ structure on M.
26
+ 1.Introduction
27
+ The underlying geometric structures of interest in categor ified symplectic
28
+ geometry are multisymplectic manifolds: manifolds equipp ed with a closed,
29
+ nondegenerate form of degree ≥2 [8]. This kind of geometry originated in
30
+ the work of DeDonder [10] and Weyl [24] on the calculus of vari ations, and
31
+ more recently has been used as a formalism to investigate cla ssical field the-
32
+ ories [11, 12, 13]. In this paper, we call a manifold ‘ n-plectic’ if it is equipped
33
+ with a closed nondegenerate ( n+ 1)-form. Hence ordinary symplectic ge-
34
+ ometry corresponds to the n= 1 case, and the corresponding 1-dimensional
35
+ field theory is just the classical mechanics of point particl es. In general,
36
+ examples of n-plectic manifolds include phase spaces suitable for descr ibing
37
+ n-dimensional classical field theories. We will be primarily concerned with
38
+ Date: October 29, 2018.
39
+ This work was partially supported by a grant from The Foundat ional Questions
40
+ Institute.
41
+ 12 CHRISTOPHER L. ROGERS
42
+ then= 2case. Thisis the firstreally new case of n-plectic geometry andthe
43
+ corresponding 2-dimensional field theories of interest inc lude bosonic string
44
+ theory. Indeed, just as the phase space of the classical part icle is a mani-
45
+ fold equipped with a closed, nondegenerate 2-form, the phas e space of the
46
+ classical string is a finite-dimensional manifold equipped with a closed non-
47
+ degenerate 3-form. This phase space is often called the ‘mul tiphase space’
48
+ of the string [11] in order to distinguish it from the infinite -dimensional
49
+ symplectic manifolds that are used as phase spaces in string field theory [6].
50
+ In classical mechanics, the relevant mathematical structu res are not just
51
+ geometric, butalsoalgebraic. Thesymplecticformgives th espaceofsmooth
52
+ functions the structure of Poisson algebra. Analogously, i n classical string
53
+ theory, the 2-plectic form induces a bilinear skew-symmetr ic bracket on
54
+ a particular subspace of differential 1-forms, which we call H amiltonian.
55
+ The Hamiltonian 1-forms and smooth functions form the under lying chain
56
+ complex of an algebraic structure known as a semistrict Lie 2 -algebra. A
57
+ semistrict Lie 2-algebra can be viewed as a categorified Lie a lgebra in which
58
+ the Jacobi identity is weakened and is required to hold only u p to isomor-
59
+ phism. Equivalently, it can be described as a 2-term L∞-algebra, i.e. a
60
+ generalization of a 2-term differential graded Lie algebra in which the Ja-
61
+ cobi identity is only satisfied up to chain homotopy [1, 14]. J ust as the
62
+ Poisson algebra of smooth functions represents the observa bles of a system
63
+ of particles, it has been shown that the Lie 2-algebra of Hami ltonian 1-forms
64
+ contains the observables of the classical string [2]. In gen eral, ann-plectic
65
+ structure will give rise to a L∞-algebra on an n-term chain complex of dif-
66
+ ferential forms in which the ( n−1)-forms correspond to the observables of
67
+ ann-dimensional classical field theory [15].
68
+ Many of the ingredients found in 2-plectic geometry are also found in
69
+ the theory of Courant algebroids, which was also developed b y generalizing
70
+ structures found in symplectic geometry. Courant algebroi ds were first used
71
+ by Courant [9] to study generalizations of pre-symplectic a nd Poisson struc-
72
+ tures in the theory of constrained mechanical systems. Roug hly, a Courant
73
+ algebroid is a vector bundle that generalizes the structure of a tangent bun-
74
+ dle equipped with a symmetric nondegenerate bilinear form o n the fibers.
75
+ In particular, the underlying vector bundle of a Courant alg ebroid comes
76
+ equipped with a skew-symmetric bracket on the space of globa l sections.
77
+ However, unlike the Lie bracket of vector fields, the bracket need not satisfy
78
+ the Jacobi identity.
79
+ In a letter to Weinstein, ˇSevera [20] described how a certain type of
80
+ Courant algebroid, known as an exact Courant algebroid, app ears naturally
81
+ when studying 2-dimensional variational problems. In clas sical string the-
82
+ ory, the string can be represented as a map φ: Σ→Mfrom a 2-dimensional
83
+ parameter space Σ into a manifold Mcorresponding to space-time. The
84
+ imageφ(Σ) is called the string world-sheet. The map φextremizes the inte-
85
+ gral of a 2-form θ∈Ω2(M) over its world-sheet. Hence the classical string
86
+ is a solution to a 2-dimensional variational problem. The 2- formθis calledCOURANT ALGEBROIDS FROM CATEGORIFIED SYMPLECTIC GEOMETRY 3
87
+ the Lagrangian and depends on elements of the first jet bundle of the trivial
88
+ bundleΣ ×M. TheLagrangian isnotunique. A solution φremainsinvariant
89
+ if an exact 1-form or ‘divergence’ is added to θ. It is, in fact, the 3-form dθ
90
+ that is relevant. Inthiscontext, ˇSeveraobserved that the3-form dθuniquely
91
+ specifies (up to isomorphism) the structure of an exact Coura nt algebroid
92
+ overM. The general correspondence between exact Courant algebro ids and
93
+ closed 3-forms on the base space was further developed by ˇSevera, and also
94
+ by Bressler and Chervov [4], to give a complete classificatio n. An exact
95
+ Courant algebroid over Mis determined up to isomorphism by its ˇSevera
96
+ class: an element [ ω] in the third de Rham cohomology of M.
97
+ Just as in 2-plectic geometry, the underlying geometric str ucture of a
98
+ Courant algebroid has an algebraic manifestation. Roytenb erg and Wein-
99
+ stein [16] showed that the bracket on the space of global sect ions induces
100
+ anL∞structure. If we are considering an exact Courant algebroid , then
101
+ the global sections can be identified with ordered pairs of ve ctor fields and
102
+ 1-forms on the base space. Roytenberg and Weinstein’s resul ts imply that
103
+ these sections, when combined with the smooth functions on t he base space,
104
+ form a semistrict Lie 2-algebra [23]. Moreover, the bracket of the Lie 2-
105
+ algebra is determined by a closed 3-form corresponding to a r epresentative
106
+ of theˇSevera class [21].
107
+ Thus there are striking similarities between 2-plectic man ifolds and ex-
108
+ act Courant algebroids. Both originate from attempts to gen eralize certain
109
+ aspects of symplectic geometry. Both come equipped with a cl osed 3-form
110
+ that gives rise to a Lie 2-algebra structure on a chain comple x consisting
111
+ of smooth functions and differential 1-forms. In this paper, w e prove that
112
+ there is indeed a connection between the two. We show that any manifold
113
+ Mequipped with a 2-plectic form ωgives an exact Courant algebroid Eω
114
+ withˇSevera class [ ω], and that there is an embedding of the Lie 2-algebra
115
+ of observables into the Lie 2-algebra corresponding to Eω. Moreover, this
116
+ embedding allows us to characterize the Hamiltonian 1-form s as particular
117
+ infinitesimal symmetries of Eωwhich preserve the 2-plectic structure on M.
118
+ 2.Courant algebroids
119
+ Here we recall some basic facts and examples of Courant algeb roids and
120
+ then we proceed to describe ˇSevera’s classification of exact Courant alge-
121
+ broids. There are several equivalent definitions of a Couran t algebroid found
122
+ in the literature. In this paper we use the definition given by Roytenberg
123
+ [17].
124
+ Definition 2.1. ACourant algebroid is a vector bundle E→Mequipped
125
+ with a nondegenerate symmetric bilinear form /an}bracketle{t·,·/an}bracketri}hton the bundle, a skew-
126
+ symmetric bracket /llbracket·,·/rrbracketonΓ(E), and a bundle map (called the anchor)
127
+ ρ:E→TMsuch that for all e1,e2,e3∈Γ(E)and for all f,g∈C∞(M)the
128
+ following properties hold:
129
+ (1)/llbrackete1,/llbrackete2,e3/rrbracket/rrbracket−/llbracket/llbrackete1,e2/rrbracket,e3/rrbracket−/llbrackete2,/llbrackete1,e3/rrbracket/rrbracket=−DT(e1,e2,e3),4 CHRISTOPHER L. ROGERS
130
+ (2)ρ([e1,e2]) = [ρ(e1),ρ(e2)],
131
+ (3) [e1,fe2] =f[e1,e2]+ρ(e1)(f)e2−1
132
+ 2/an}bracketle{te1,e2/an}bracketri}htDf,
133
+ (4)/an}bracketle{tDf,Dg/an}bracketri}ht= 0,
134
+ (5)ρ(e1)(/an}bracketle{te2,e3/an}bracketri}ht) =/an}bracketle{t[e1,e2]+1
135
+ 2D/an}bracketle{te1,e2/an}bracketri}ht,e3/an}bracketri}ht+/an}bracketle{te2,[e1,e3]+1
136
+ 2D/an}bracketle{te1,e3/an}bracketri}ht/an}bracketri}ht,
137
+ where[·,·]is the Lie bracket of vector fields, D:C∞(M)→Γ(E)is the map
138
+ defined by /an}bracketle{tDf,e/an}bracketri}ht=ρ(e)f, and
139
+ T(e1,e2,e3) =1
140
+ 6(/an}bracketle{t/llbrackete1,e2/rrbracket,e3/an}bracketri}ht+/an}bracketle{t/llbrackete3,e1/rrbracket,e2/an}bracketri}ht+/an}bracketle{t/llbrackete2,e3/rrbracket,e1/an}bracketri}ht).
141
+ The bracket in Definition 2.1 is skew-symmetric, but the first property
142
+ implies that it needs only to satisfy the Jacobi identity “up toDT”. (The
143
+ notation suggests we think of this as a boundary.) The functi onTis often
144
+ referred to as the Jacobiator . (When there is no risk of confusion, we shall
145
+ refer to the Courant algebroid with underlying vector bundl eE→MasE.)
146
+ Note that the vector bundle Emay be identified with E∗via the bilinear
147
+ form/an}bracketle{t·,·/an}bracketri}htand therefore we have the dual map
148
+ ρ∗:T∗M→E.
149
+ Hence the map Dis simply the pullback of the de Rham differential by ρ∗.
150
+ Thereisanalternatedefinitiongiven by ˇSevera[20]forCourantalgebroids
151
+ which uses a bilinear operation on sections that satisfies a J acobi identity
152
+ but is not skew-symmetric.
153
+ Definition 2.2. ACourant algebroid is a vector bundle E→Mtogether
154
+ with a nondegenerate symmetric bilinear form /an}bracketle{t·,·/an}bracketri}hton the bundle, a bilinear
155
+ operation ◦onΓ(E), and a bundle map ρ:E→TMsuch that for all
156
+ e1,e2,e3∈Γ(E)and for all f∈C∞(M)the following properties hold:
157
+ (1)e1◦(e2◦e3) = (e1◦e2)◦e3+e2◦(e1◦e3),
158
+ (2)ρ(e1◦e2) = [ρ(e1),ρ(e2)],
159
+ (3)e1◦fe2=f(e1◦e2)+ρ(e1)(f)e2,
160
+ (4)e1◦e1=1
161
+ 2D/an}bracketle{te1,e1/an}bracketri}ht,
162
+ (5)ρ(e1)(/an}bracketle{te2,e3/an}bracketri}ht) =/an}bracketle{te1◦e2,e3/an}bracketri}ht+/an}bracketle{te2,e1◦e3/an}bracketri}ht,
163
+ where[·,·]is the Lie bracket of vector fields, and D:C∞(M)→Γ(E)is the
164
+ map defined by /an}bracketle{tDf,e/an}bracketri}ht=ρ(e)f.
165
+ The “bracket” ◦is related to the bracket given in Definition 2.1 by:
166
+ x◦y=/llbracketx,y/rrbracket+1
167
+ 2D/an}bracketle{tx,y/an}bracketri}ht. (1)
168
+ Roytenberg [17] showed that if Eis a Courant algebroid in the sense of
169
+ Definition 2.1 with bracket /llbracket·,·/rrbracket, bilinear form /an}bracketle{t·,·/an}bracketri}htand anchor ρ, thenEis
170
+ a Courant algebroid in the sense of Definition 2.2 with the sam e anchor and
171
+ bilinear form but with bracket ◦given by Eq. 1. Unless otherwise stated, all
172
+ Courant algebroids mentioned in this paper are Courant alge broids in the
173
+ sense of Definition 2.1. We introduced Definition 2.2 mainly t o connect our
174
+ results here with previous results in the literature.COURANT ALGEBROIDS FROM CATEGORIFIED SYMPLECTIC GEOMETRY 5
175
+ Example 1.An important example of a Courant algebroid is the standard
176
+ Courant algebroid E0=TM⊕T∗Mover any manifold Mwith bracket
177
+ /llbracket(v1,α1),(v2,α2)/rrbracket0=/parenleftbigg
178
+ [v1,v2],Lv1α2−Lv2α1−1
179
+ 2d(ιv1α2−ιv2α1)/parenrightbigg
180
+ ,(2)
181
+ and bilinear form
182
+ /an}bracketle{t(v1,α1),(v2,α2)/an}bracketri}ht=ιv1α2+ιv2α1. (3)
183
+ In this case the anchor ρ:E0→TMis the projection map, and for a
184
+ function f∈C∞(M),Df= (0,df).
185
+ The standard Courant algebroid is the prototypical example of anexact
186
+ Courant algebroid [4].
187
+ Definition 2.3. A Courant algebroid E→Mwith anchor map ρ:E→TM
188
+ isexactiff
189
+ 0→T∗Mρ∗
190
+ →Eρ→TM→0
191
+ is an exact sequence of vector bundles.
192
+ 2.1.TheˇSevera class of an exact Courant algebroid. ˇSevera’s clas-
193
+ sification originates in the idea that choosing a splitting o f the above short
194
+ exact sequence corresponds to defining a kind of connection.
195
+ Definition 2.4. Aconnection on an exact Courant algebroid Eover a
196
+ manifold Mis a map of vector bundles A:TM→Esuch that
197
+ (1)ρ◦A= idTM,
198
+ (2)/an}bracketle{tA(v1),A(v2)/an}bracketri}ht= 0for allv1,v2∈TM,
199
+ whereρ:E→TMand/an}bracketle{t·,·/an}bracketri}htare the anchor and bilinear form, respectively.
200
+ IfAis a connection and θ∈Ω2(M) is a 2-form then one can construct a
201
+ new connection:
202
+ (A+θ)(v) =A(v)+ρ∗θ(v,·). (4)
203
+ (A+θ) satisfies the first condition of Definition 2.4 since ker ρ= imρ∗. The
204
+ second condition follows from the fact that we have by definit ion ofρ∗:
205
+ /an}bracketle{tρ∗(α),e/an}bracketri}ht=α(ρ(e)) (5)
206
+ for alle∈Γ(E) andα∈Ω1(M). Furthermore, one can show that any two
207
+ connections on an exact Courant algebroid must differ (as in Eq . 4) by a
208
+ 2-form on M. Hence the space of connections on an exact Courant algebroi d
209
+ is an affine space modeled on the vector space of 2-forms Ω2(M) [4].
210
+ The failure of a connection to preserve the bracket gives a su itable notion
211
+ of curvature:
212
+ Definition 2.5. IfEis an exact Courant algebroid over Mwith bracket /llbracket·,·/rrbracket
213
+ andA:TM→Eis a connection then the curvature is a map F:TM×
214
+ TM→Edefined by
215
+ F(v1,v2) =/llbracketA(v1),A(v2)/rrbracket−A([v1,v2]).6 CHRISTOPHER L. ROGERS
216
+ IfFis the curvature of a connection Athen given v1,v2∈TM, it follows
217
+ from exactness and axiom 2 in Definition 2.1 that there exists a 1-form
218
+ αv1,v2∈Ω1(M) such that F(v1,v2) =ρ∗(αv1,v2). Since Ais a connection,
219
+ its image is isotropic in E. Therefore for any v3∈TMwe have:
220
+ /an}bracketle{tF(v1,v2),A(v3)/an}bracketri}ht=/an}bracketle{t/llbracketA(v1),A(v2)/rrbracket,A(v3)/an}bracketri}ht.
221
+ The above formula allows one to associate the curvature Fto a 3-form on
222
+ M:
223
+ Proposition 2.6. LetEbe an exact Courant algebroid over a manifold M
224
+ with bracket /llbracket·,·/rrbracketand bilinear form /an}bracketle{t·,·/an}bracketri}ht. LetA:TM→Ebe a connection
225
+ onE. Then given vector fields v1,v2,v3onM:
226
+ (1)The function
227
+ ω(v1,v2,v3) =/an}bracketle{t/llbracketA(v1),A(v2)/rrbracket,A(v3)/an}bracketri}ht
228
+ defines a closed 3-form on M.
229
+ (2)Ifθ∈Ω2(M)is a 2-form and ˜A=A+θthen
230
+ ˜ω(v1,v2,v3) =/an}bracketle{t/llbracket˜A(v1),˜A(v2)/rrbracket,˜A(v3)/an}bracketri}ht
231
+ =ω(v1,v2,v3)+dθ(v1,v2,v3).
232
+ Proof.The statements in the proposition are proven in Lemmas 4.2.6 , 4.2.7,
233
+ and 4.3.4 in the paper by Bressler and Chervov [4]. In their wo rk they
234
+ define a Courant algebroid using Definition 2.2, and therefor e their bracket
235
+ satisfies the Jacobi identity, but is not skew-symmetric. In our notation,
236
+ their definition of the curvature 3-form is:
237
+ ω′(v1,v2,v3) =/an}bracketle{tA(v1)◦A(v2),A(v3)/an}bracketri}ht.
238
+ In particular they show that ◦satisfying the Jacobi identity implies ω′is
239
+ closed. The Jacobiator corresponding to the Courant bracke t is non-trivial
240
+ in general. However the isotropicity of the connection and E q. 1 imply
241
+ A(v1)◦A(v2) =/llbracketA(v1),A(v2)/rrbracket∀v1,v2∈TM.
242
+ Henceω′=ω, so all the needed results in [4] apply here. /square
243
+ Thus the above proposition implies that the curvature 3-for m of an exact
244
+ Courantalgebroid over Mgives awell-defined cohomology class in H3
245
+ DR(M),
246
+ independent of the choice of connection.
247
+ 2.2.Twisting the Courant bracket. The previous section describes how
248
+ to go from exact Courant algebroids to closed 3-forms. Now we describe the
249
+ reverse process. In Example 1 we showed that one can define the standard
250
+ Courant algebroid E0over any manifold M. The total space is the direct
251
+ sumTM⊕T∗M, the bracket and bilinear form are given in Eqs. 2 and 3,
252
+ and the anchor is simply the projection. The inclusion A(v) = (v,0) of the
253
+ tangent bundle into the direct sum is obviously a connection onE0and it
254
+ is easy to see that the standard Courant algebroid has zero cu rvature.COURANT ALGEBROIDS FROM CATEGORIFIED SYMPLECTIC GEOMETRY 7
255
+ ˇSevera and Weinstein [20, 21] observed that the bracket on E0could be
256
+ twisted by a closed 3-form ω∈Ω3(M) on the base:
257
+ /llbracket(v1,α1),(v2,α2)/rrbracketω=/llbracket(v1,α1),(v2,α2)/rrbracket0+ω(v1,v2,·).
258
+ This gives a new Courant algebroid Eωwith the same anchor and bilinear
259
+ form. Using Eqs. 2 and 3 we can compute the curvature 3-form of this new
260
+ Courant algebroid:
261
+ /an}bracketle{t/llbracketA(v1),A(v2)/rrbracket,A(v3)/an}bracketri}ht=/an}bracketle{t/llbracket(v1,0),(v2,0)/rrbracket,(v3,0)/an}bracketri}ht
262
+ =/an}bracketle{t([v1,v2],ω(v1,v2,·)),(v3,0)/an}bracketri}ht
263
+ =ω(v1,v2,v3),
264
+ and we see that Eωis an exact Courant algebroid over MwithˇSevera class
265
+ [ω].
266
+ 3. 2-plectic geometry
267
+ We nowgive abriefoverview of 2-plectic geometry. Moredeta ils including
268
+ motivation for several of the definitions presented here can be found in our
269
+ previous work with Baez and Hoffnung [2, 3].
270
+ Definition 3.1. A3-formωon aC∞manifold Mis2-plectic , or more
271
+ specifically a 2-plectic structure , if it is both closed:
272
+ dω= 0,
273
+ and nondegenerate:
274
+ ∀v∈TxM, ιvω= 0⇒v= 0
275
+ Ifωis a2-plectic form on Mwe call the pair (M,ω)a2-plectic manifold .
276
+ The 2-plectic structure induces an injective map from the sp ace of vector
277
+ fields on Mto the space of 2-forms on M. This leads us to the following
278
+ definition:
279
+ Definition 3.2. Let(M,ω)be a2-plectic manifold. A 1-form αonMis
280
+ Hamiltonian if there exists a vector field vαonMsuch that
281
+ dα=−ιvαω.
282
+ We sayvαis theHamiltonian vector field corresponding to α. The set
283
+ of Hamiltonian 1-forms and the set of Hamiltonian vector fiel ds on a 2-
284
+ plectic manifold are both vector spaces and are denoted as Ham(M)and
285
+ VectH(M), respectively.
286
+ The Hamiltonian vector field vαis unique if it exists, but note there may
287
+ be 1-forms αhaving no Hamiltonian vector field. Furthermore, two distin ct
288
+ Hamiltonian 1-forms may differ by a closed 1-form and therefor e share the
289
+ same Hamiltonian vector field.
290
+ We can generalize thePoisson bracket of functionsin symple ctic geometry
291
+ by defining a bracket of Hamiltonian 1-forms.8 CHRISTOPHER L. ROGERS
292
+ Definition 3.3. Givenα,β∈Ham(M), thebracket {α,β}is the
293
+ 1-form given by
294
+ {α,β}=ιvβιvαω.
295
+ Proposition 3.4. Letα,β,γ∈Ham(M)and letvα,vβ,vγbe the respective
296
+ Hamiltonian vector fields. The bracket {·,·}has the following properties:
297
+ (1)The bracket of Hamiltonian forms is Hamiltonian:
298
+ d{α,β}=−ι[vα,vβ]ω, (6)
299
+ so in particular we have
300
+ v{α,β}= [vα,vβ].
301
+ (2)The bracket is skew-symmetric:
302
+ {α,β}=−{β,α} (7)
303
+ (3)The bracket satisfies the Jacobi identity up to an exact 1-form :
304
+ {α,{β,γ}}−{{α,β},γ}−{β,{α,γ}}=dJα,β,γ (8)
305
+ withJα,β,γ=ιvαιvβιvγω.
306
+ Proof.See Proposition 3.7 in [2]. /square
307
+ 4.Lie2-algebras
308
+ Both the Courant bracket and the bracket on Hamiltonian 1-fo rms are,
309
+ roughly, Lie brackets which satisfy the Jacobi identity up t o an exact 1-
310
+ form. This leads us to the notion of a Lie 2-algebra: a categor y equipped
311
+ with structures analogous to those of a Lie algebra, for whic h the usual laws
312
+ involving skew-symmetry and the Jacobi identity hold up to i somorphism
313
+ [1, 19]. A Lie 2-algebra in which the isomorphisms are actual equalities
314
+ is called a strict Lie 2-algebra. A Lie 2-algebra in which the laws govern-
315
+ ing skew-symmetry are equalities but the Jacobi identity ho lds only up to
316
+ isomorphism is called a semistrict Lie 2-algebra.
317
+ Here we define a semistrict Lie 2-algebra to be a 2-term chain c omplex
318
+ of vector spaces equipped with structures analogous to thos e of a Lie al-
319
+ gebra, for which the usual laws hold up to chain homotopy. In t his guise,
320
+ a semistrict Lie 2-algebra is nothing more than a 2-term L∞-algebra. For
321
+ more details, we refer the reader to the work of Lada and Stash eff [14], and
322
+ the work of Baez and Crans [1].
323
+ Definition 4.1. Asemistrict Lie 2-algebra is a2-term chain complex
324
+ of vector spaces L= (L1d→L0)equipped with:
325
+ •a chain map [·,·]:L⊗L→Lcalled the bracket;
326
+ •an antisymmetric chain homotopy J:L⊗L⊗L→Lfrom the chain
327
+ map
328
+ L⊗L⊗L→L
329
+ x⊗y⊗z/mapsto−→[x,[y,z]],COURANT ALGEBROIDS FROM CATEGORIFIED SYMPLECTIC GEOMETRY 9
330
+ to the chain map
331
+ L⊗L⊗L→L
332
+ x⊗y⊗z/mapsto−→[[x,y],z]+[y,[x,z]]
333
+ called the Jacobiator ,
334
+ such that the following equation holds:
335
+ [x,J(y,z,w)] +J(x,[y,z],w) +J(x,z,[y,w]) +[J(x,y,z),w]
336
+ +[z,J(x,y,w)] =J(x,y,[z,w]) +J([x,y],z,w)
337
+ +[y,J(x,z,w)] +J(y,[x,z],w) +J(y,z,[x,w]).(9)
338
+ We will also need a suitable notion of morphism:
339
+ Definition 4.2. Given semistrict Lie 2-algebras LandL′with bracket and
340
+ Jacobiator [·,·],Jand[·,·]′,J′respectively, a homomorphism fromLto
341
+ L′consists of:
342
+ •a chain map φ= (φ0,φ1) :L→L′, and
343
+ •a chain homotopy φ2:L⊗L→Lfrom the chain map
344
+ L⊗L→L
345
+ x⊗y/mapsto−→[φ(x),φ(y)]′,
346
+ to the chain map
347
+ L⊗L→L
348
+ x⊗y/mapsto−→φ([x,y])
349
+ such that the following equation holds:
350
+ J′(φ0(x),φ0(y),φ0(z))−φ1(J(x,y,z)) =
351
+ φ2(x,[y,z])−φ2([x,y],z)−φ2(y,[x,z])−[φ2(x,y),φ0(z)]′
352
+ +[φ0(x),φ2(y,z)]′−[φ0(y),φ2(x,z)]′.(10)
353
+ This definition is equivalent to the definition of a morphism b etween 2-
354
+ termL∞-algebras. (The same definition is given in [1], but it contai ns a
355
+ typographical error.)
356
+ 4.1.The Lie 2-algebra from a 2-plectic manifold. Given a 2-plectic
357
+ manifold( M,ω), wecanconstructasemistrictLie2-algebra. Theunderlyi ng
358
+ 2-term chain complex is namely:
359
+ L=C∞(M)d→Ham(M)
360
+ wheredis the usual exterior derivative of functions. This chain co mplex is
361
+ well-defined, since any exact form is Hamiltonian, with 0 as i ts Hamiltonian
362
+ vector field. We can construct a chain map
363
+ {·,·}:L⊗L→L,
364
+ by extending the bracket on Ham( M) trivially to L. In other words, in
365
+ degree 0, the chain map is given as in Definition 3.3:
366
+ {α,β}=ιvβιvαω,10 CHRISTOPHER L. ROGERS
367
+ and in degrees 1 and 2, we set it equal to zero:
368
+ {α,f}= 0,{f,α}= 0,{f,g}= 0.
369
+ The precise construction of this Lie 2-algebra is given in th e following the-
370
+ orem:
371
+ Theorem 4.3. If(M,ω)is a2-plectic manifold, there is a semistrict Lie
372
+ 2-algebraL(M,ω)where:
373
+ •the space of 0-chains is Ham(M),
374
+ •the space of 1-chains is C∞,
375
+ •the differential is the exterior derivative d:C∞→Ham(M),
376
+ •the bracket is {·,·},
377
+ •the Jacobiator is the linear map JL: Ham(M)⊗Ham(M)⊗Ham(M)→
378
+ C∞defined by JL(α,β,γ) =ιvαιvβιvγω.
379
+ Proof.See Theorem 4.4 in [2]. /square
380
+ 4.2.The Lie 2-algebra from a Courant algebroid. Given any Courant
381
+ algebroid E→Mwith bilinear form /an}bracketle{t·,·/an}bracketri}ht, bracket /llbracket·,·/rrbracket, and anchor ρ:E→
382
+ TM, we can construct a 2-term chain complex
383
+ C=C∞(M)D→Γ(E),
384
+ with differential D=ρ∗d. The bracket /llbracket·,·/rrbracketon global sections can be ex-
385
+ tended to a chain map /llbracket·,·/rrbracket:C⊗C→C. Ife1,e2are degree 0 chains then
386
+ /llbrackete1,e2/rrbracketis the original bracket. If eis a degree 0 chain and f,gare degree 1
387
+ chains, then we define:
388
+ /llbrackete,f/rrbracket=−/llbracketf,e/rrbracket=1
389
+ 2/an}bracketle{te,Df/an}bracketri}ht
390
+ /llbracketf,g/rrbracket= 0.
391
+ This extended bracket gives a semistrict Lie 2-algebra on th e complex C:
392
+ Theorem 4.4. IfEis a Courant algebroid, there is a semistrict Lie 2-
393
+ algebraC(E)where:
394
+ •the space of 0-chains is Γ(E),
395
+ •the space of 1-chains is C∞(M),
396
+ •the differential the map D:C∞(M)→Γ(M),
397
+ •the bracket is /llbracket·,·/rrbracket,
398
+ •the Jacobiator is the linear map JC: Γ(M)⊗Γ(M)⊗Γ(M)→C∞(M)
399
+ defined by
400
+ JC(e1,e2,e3) =−T(e1,e2,e3)
401
+ =−1
402
+ 6(/an}bracketle{t/llbrackete1,e2/rrbracket,e3/an}bracketri}ht+/an}bracketle{t/llbrackete3,e1/rrbracket,e2/an}bracketri}ht+/an}bracketle{t/llbrackete2,e3/rrbracket,e1/an}bracketri}ht).
403
+ Proof.Theproofthat aCourantalgebroid inthesenseofDefinition 2 .1gives
404
+ rise to a semistrict Lie 2-algebra follows from the work done by Roytenberg
405
+ on graded symplectic supermanifolds [18] and Lie 2-algebra s [19]. In partic-
406
+ ular we refer the reader to Example 5.4 of [19] and Section 4 of [18].COURANT ALGEBROIDS FROM CATEGORIFIED SYMPLECTIC GEOMETRY 11
407
+ Onthe other hand, theoriginal construction of Roytenberg a nd Weinstein
408
+ [16] gives a L∞-algebra on the complex:
409
+ 0→kerDι→C∞(M)D→Γ(E),
410
+ with trivial structure maps lnforn≥3. Moreover, the map l2(correspond-
411
+ ing to the bracket /llbracket·,·/rrbracketgiven above) is trivial in degree >1 and the map
412
+ l3(corresponding to the Jacobiator JC) is trivial in degree >0. Hence we
413
+ can restrict this L∞-algebra to our complex Cand use the results in [1] that
414
+ relateL∞-algebras with semistrict Lie 2-algebras. /square
415
+ 5.The Courant algebroid associated to a 2-plectic manifold
416
+ Now we have the necessary machinery in place to describe how C ourant
417
+ algebroids connect with 2-plectic geometry. First, recall the discussion in
418
+ Section 2.2 on twisting the bracket of the standard Courant a lgebroid E0by
419
+ a closed 3-form. From Definition 3.1, we immediately have the following:
420
+ Proposition 5.1. Let(M,ω)be a2-plectic manifold. There exists an exact
421
+ Courant algebroid EωwithˇSevera class [ω]overMwith underlying vector
422
+ bundleTM⊕T∗M→M, anchor ρ(v,α) =v, and bracket and bilinear form
423
+ given by:
424
+ /llbracket(v1,α1),(v2,α2)/rrbracketω=/parenleftbigg
425
+ [v1,v2],Lv1α2−Lv2α1−1
426
+ 2d(ιv1α2−ιv2α1)+ιv2ιv1ω/parenrightbigg
427
+ ,
428
+ /an}bracketle{t(v1,α1),(v2,α2)/an}bracketri}ht=ιv1α2+ιv2α1.
429
+ More importantly, the Courant algebroid constructed in Pro position 5.1
430
+ not only encodes the 2-plectic structure ω, but also the corresponding Lie
431
+ 2-algebra constructed in Theorem 4.3:
432
+ Theorem 5.2. Let(M,ω)be a2-plectic manifold and let Eωbe its corre-
433
+ sponding Courant algebroid. Let L(M,ω)andC(Eω)be the semistrict Lie
434
+ 2-algebras corresponding to (M,ω)andEω, respectively. Then there exists
435
+ a homomorphism embedding L(M,ω)intoC(Eω).
436
+ Before we prove the theorem, we introduce some lemmas to ease the
437
+ calculations. In the notation that follows, if α,βare Hamiltonian 1-forms
438
+ with corresponding vector fields vα,vβ, then
439
+ B(α,β) =1
440
+ 2(ιvαβ−ιvβα). (11)
441
+ Also by the symbol /anticlockwisewe mean cyclic permutations of the symbols α,β,γ.
442
+ Lemma 5.3. Ifα,β∈Ham(M)with corresponding Hamiltonian vector
443
+ fieldsvα,vβ, thenLvαβ={α,β}+dιvαβ.
444
+ Proof.SinceLv=ιvd+dιv,
445
+ Lvαβ=ιvαdβ+dιvαβ=−ιvαιvβω+dιvαβ={α,β}+dιvαβ.
446
+ /square12 CHRISTOPHER L. ROGERS
447
+ Lemma 5.4. Ifα,β,γ∈Ham(M)with corresponding Hamiltonian vector
448
+ fieldsvα,vβ,vγ, then
449
+ ι[vα,vβ]γ+/anticlockwise=−3ιvαιvβιvγω+2/parenleftbig
450
+ ιvαdB(β,γ)+ιvγdB(α,β)+ιvβdB(γ,α)/parenrightbig
451
+ .
452
+ Proof.The identity ι[v1,v2]=Lv1ιv2−ιv2Lv1and Lemma 5.3 imply:
453
+ ι[vα,vβ]γ=Lvαιvβγ−ιvβLvαγ
454
+ =Lvαιvβγ−ιvβ({α,γ}+dιvαγ)
455
+ =ιvαdιvβγ−ιvβιvγιvαω−ιvβdιvαγ,
456
+ where the last equality follows from the definition of the bra cket.
457
+ Hence it follows that:
458
+ ι[vγ,vα]β=ιvγdιvαβ−ιvαιvβιvγω−ιvαdιvγβ,
459
+ ι[vβ,vγ]α=ιvβdιvγα−ιvγιvαιvβω−ιvγdιvβα.
460
+ Finally, note 2 ιvαdB(β,γ) =ιvαdιvβγ−ιvαdιvγβ. /square
461
+ Lemma 5.5. Ifα,β∈Ham(M)with corresponding Hamiltonian vector
462
+ fieldsvα,vβ, then
463
+ Lvβα−Lvαβ=−2({α,β}+dB(α,β)).
464
+ Proof.Follows immediately from Lemma 5.3 and the definition of B(α,β).
465
+ /square
466
+ Proof of Theorem 5.2. We will construct a homomorphism from L(M,ω) to
467
+ C(Eω). LetLbe the underlying chain complex of L(M,ω) consisting of
468
+ Hamiltonian 1-forms in degree 0 and smooth functions in degr ee 1. Let
469
+ Cbe the underlying chain of C(Eω) consisting of global sections of Eωin
470
+ degree 0 and smooth functions in degree 1. The bracket /llbracket·,·/rrbracketωdenotes the
471
+ extension of the bracket on Γ( Eω) to the complex Cin the sense of Theorem
472
+ 4.4. Let φ0:L0→C0be given by
473
+ φ0(α) = (vα,−α),
474
+ wherevαis the Hamiltonian vector field corresponding to α. Letφ1:L1→
475
+ C1be given by
476
+ φ1(f) =−f.
477
+ Finally let φ2:L0⊗L0→C1be given by
478
+ φ2(α,β) =−B(α,β) =−1
479
+ 2(ιvαβ−ιvβα).
480
+ Nowweshow φ2isawell-definedchainhomotopyinthesenseofDefinition
481
+ 4.2. For degree 0 we have:
482
+ /llbracketφ0(α),φ0(β)/rrbracketω=/parenleftbigg
483
+ [vα,vβ],Lvα(−β)−Lvβ(−α)+1
484
+ 2d/parenleftbig
485
+ ιvαβ−ιvβα/parenrightbig
486
+ +ιvβιvαω/parenrightbigg
487
+ = ([vα,vβ],−{α,β}+dφ2(α,β)),COURANT ALGEBROIDS FROM CATEGORIFIED SYMPLECTIC GEOMETRY 13
488
+ where the last equality above follows from Lemma 5.5. By Prop osition 3.4,
489
+ the Hamiltonian vector field of {α,β}is [vα,vβ]. Hence we have:
490
+ /llbracketφ0(α),φ0(β)/rrbracketω−φ0({α,β}) =dφ2(α,β).
491
+ Indegree1, thebracket {·,·}istrivial. Henceitfollows fromthedefinition
492
+ of/llbracket·,·/rrbracketωand the bilinear form on Eω(given in Proposition 5.1 ) that
493
+ /llbracketφ0(α),φ1(f)/rrbracketω=−/llbracketφ1(f),φ0(α)/rrbracketω=1
494
+ 2/an}bracketle{t(vα,−α),(0,−df)/an}bracketri}ht=φ2(α,df).
495
+ Therefore φ2is a chain homotopy.
496
+ It remains to show the coherence condition (Eq. 10 in Definiti on 4.2) is
497
+ satisfied. We rewrite the Jacobiator JCas:
498
+ JC(φ0(α),φ0(β),φ0(γ)) =−1
499
+ 6/an}bracketle{t/llbracketφ0(α),φ0(β)/rrbracket,φ0(γ)/an}bracketri}ht+/anticlockwise
500
+ =−1
501
+ 6/an}bracketle{t([vα,vβ],−{α,β}−dB(α,β)),(vγ,−γ)/an}bracketri}ht+/anticlockwise
502
+ =1
503
+ 6/parenleftBig
504
+ ι[vα,vβ]γ+ιvγιvβιvαω+ιvγdB(α,β)/parenrightBig
505
+ +/anticlockwise
506
+ =−JL(α,β,γ)+1
507
+ 2/parenleftbig
508
+ ιvγdB(α,β)+/anticlockwise/parenrightbig
509
+ .
510
+ The last equality above follows from Lemma 5.4 and the definit ion of the
511
+ Jacobiator JL. Therefore the left-hand side of Eq. 10 is
512
+ JC(φ0(α),φ0(β),φ0(γ))−φ1(JL(α,β,γ)) =1
513
+ 2/parenleftbig
514
+ ιvγdB(α,β)+/anticlockwise/parenrightbig
515
+ .
516
+ By the skew-symmetry of the brackets, the right-hand side of Eq. 10 can
517
+ be rewritten as:
518
+ (/llbracketφ0(α),φ2(β,γ)/rrbracketω+/anticlockwise)−(φ2({α,β},γ)+/anticlockwise).
519
+ From the definitions of the bracket, bilinear form and φ2we have:
520
+ /llbracketφ0(α),φ2(β,γ)/rrbracketω+/anticlockwise=1
521
+ 2/an}bracketle{t(vα,−α),(0,dφ2(β,γ)/an}bracketri}ht+/anticlockwise
522
+ =−1
523
+ 2ιvαdB(β,γ)+/anticlockwise,
524
+ and:
525
+ φ2({α,β},γ)+/anticlockwise=−1
526
+ 2/parenleftBig
527
+ ι[vα,vβ]γ−ιvγιvβιvαω/parenrightBig
528
+ =−(ιvαdB(β,γ)+/anticlockwise).
529
+ The last equality above follows again from Lemma 5.4. Theref ore the right-
530
+ hand side of Eq. 10 is
531
+ 1
532
+ 2/parenleftbig
533
+ ιvγdB(α,β)+/anticlockwise/parenrightbig
534
+ .
535
+ Hencethemaps φ0,φ1,φ2give ahomomorphismofsemistrictLie2-algebras.
536
+ /square14 CHRISTOPHER L. ROGERS
537
+ We note that Roytenberg [19] has shown that a Courant algebro id defined
538
+ using Definition 2.2 with the bilinear operation ◦induces a hemistrict Lie
539
+ 2-algebra on the complex Cdescribed in Theorem 4.4 above. A hemistrict
540
+ Lie 2-algebra is a Lie 2-algebra in which the skew-symmetry h olds up to
541
+ isomorphism, while the Jacobi identity holds as an equality . We have proven
542
+ in previous work [2] that a 2-plectic structure also gives ri se to a hemistrict
543
+ Lie 2-algebra on the complex described in Theorem 4.3. One ca n show that
544
+ all results presented above, in particular Theorem 5.2, car ry over to the
545
+ hemistrict case.
546
+ 6.Hamiltonian 1-forms as infinitesimal symmetries of the
547
+ Courant algebroid
548
+ Givena2-plecticmanifold( M,ω), theLie2-algebraofobservables L(M,ω)
549
+ identifies particular infinitesimal symmetries of the corre sponding Courant
550
+ algebroid Eωvia the embedding described in the proof of Theorem 5.2. To
551
+ see this, we first recall some basic facts concerning automor phisms of exact
552
+ Courant algebroids. The presentation here follows the work of Bursztyn,
553
+ Cavalcanti, and Gualtieri [7].
554
+ Definition 6.1. LetE→Mbe a Courant algebroid with bilinear form /an}bracketle{t·,·/an}bracketri}ht,
555
+ bracket /llbracket·,·/rrbracket, and anchor ρ:E→TM. Anautomorphism is a bundle
556
+ isomorphism F:E→Ecovering a diffeomorphism ϕ:M→Msuch that
557
+ (1)ϕ∗/an}bracketle{tF(e1),F(e2)/an}bracketri}ht=/an}bracketle{te1,e2/an}bracketri}ht,
558
+ (2)F(/llbrackete1,e2/rrbracket) =/llbracketF(e1),F(e2)/rrbracket,
559
+ (3)ρ(F(e1)) =ϕ∗(ρ(e1)).
560
+ Consider the exact Courant algebroid Eωdescribed in Section 2.2 with
561
+ underlyingvector bundle TM⊕T∗M→MandˇSevera class [ ω]∈H3
562
+ DR(M).
563
+ Given a 2-form B∈Ω2(M), one can define a bundle isomorphism
564
+ expB:TM⊕T∗M→TM⊕T∗M
565
+ by
566
+ expB(v,α) = (v,α+ιvB).
567
+ The map exp Bis known as a ‘gauge transformation’. It covers the identity
568
+ id:M→Mand therefore is compatible (in the sense of Definition 6.1) w ith
569
+ the anchor ρ(v,α) =v. Since Bis skew-symmetric, exp Bpreserves the
570
+ bilinear form /an}bracketle{t(v1,α1),(v2,α2)/an}bracketri}ht=ιv1α2+ιv2α1. However a simple compu-
571
+ tation shows that exp Bpreserves the bracket /llbracket·,·/rrbracketω(defined in Eq. 2.2) if
572
+ and only if Bis a closed 2-form:
573
+ /llbracketexpB(v1,α1),expB(v2,α2)/rrbracketω= expB/parenleftbig
574
+ /llbracket(v1,α1),(v2,α2)/rrbracketω+dB/parenrightbig
575
+ .
576
+ Given a diffeomorphism ϕ:M→Mof the base space, one can define a
577
+ bundle isomorphism Φ: TM⊕T∗M→TM⊕T∗Mby
578
+ Φ(v,α) =/parenleftBig
579
+ ϕ∗v,(ϕ∗)−1α/parenrightBig
580
+ .COURANT ALGEBROIDS FROM CATEGORIFIED SYMPLECTIC GEOMETRY 15
581
+ ThemapΦsatisfiesconditions1and3ofDefinition6.1butdoes notpreserve
582
+ the bracket in general:
583
+ /llbracketΦ(v1,α1),Φ(v2,α2)/rrbracketω= Φ/parenleftBig
584
+ /llbracket(v1,α1),(v2,α2)/rrbracketϕ∗ω/parenrightBig
585
+ .
586
+ Bursztyn, Cavalcanti, and Gualtieri [7] showed that any aut omorphism F
587
+ of the exact Courant algebroid Eωmust be of the form
588
+ F= ΦexpB, (12)
589
+ where Φ is constructed from a diffeomorphism ϕ:M→Msuch that
590
+ ω−ϕ∗ω=dB. (13)
591
+ This classification of automorphisms allows one to classify the infinitesimal
592
+ symmetries as well. Let
593
+ Ft= ΦtexptB=/parenleftBig
594
+ ϕt∗exptB,(ϕ∗
595
+ t)−1exptB/parenrightBig
596
+ bea 1-parameter family of automorphismsof the Courant alge broidEωwith
597
+ F0= idEω. Letu∈Vect(M) be the vector field that generates the flow ϕ−t.
598
+ Then differentiation gives:
599
+ dFt
600
+ dt(v,α)/vextendsingle/vextendsingle/vextendsingle/vextendsingle
601
+ t=0= ([u,v],Luα+ιvB).
602
+ Sinceω−ϕ∗
603
+ tω=tdB, it follows that uandBmust satisfy the equality:
604
+ Luω=dB. (14)
605
+ Theseinfinitesimaltransformationsarecalled derivations [7]oftheCourant
606
+ algebroid Eω, sincetheycorrespondtolinearfirstorderdifferential oper ators
607
+ which act as derivations of the non-skew-symmetric bracket :
608
+ (v1,α1)◦ω(v2,α2) =/llbracket(v1,α1),(v2,α2)/rrbracketω+1
609
+ 2d/an}bracketle{t(v1,α1),(v2,α2)/an}bracketri}ht.(15)
610
+ = ([v1,v2],Lv1α2−ιv2dα1+ιv2ιv1ω). (16)
611
+ In general, derivations are pairs ( u,B)∈Vect(M)⊕Ω2(M) satisfying Eq.
612
+ 14. They act on global sections ( v,α)∈Γ(Eω) by:
613
+ (u,B)·(v,α) = ([u,v],Luα+ιvB).
614
+ Global sections themselves naturally act as derivations vi a anadjoint
615
+ action[7]. Given ( u,β)∈Γ(Eω) letBbe the 2-form
616
+ B=−dβ+ιuω. (17)
617
+ Define ad (u,β): Γ(Eω)→Γ(Eω) by
618
+ ad(u,β)(v,α) = (u,B)·(v,α) = ([u,v],Luα+ιv(−dβ+ιuω)).(18)
619
+ One can see this is indeed the adjoint action in the usual sens e if one con-
620
+ siders the non-skew-symmetric bracket given in Eq. 15:
621
+ ad(u,β)(v,α) = (u,β)◦ω(v,α).16 CHRISTOPHER L. ROGERS
622
+ Recall that in the proof of Theorem 5.2 we constructed a homom orphism
623
+ of Lie 2-algebras using the map φ0: Ham(M)→Γ(Eω) defined by
624
+ φ0(α) = (vα,−α),
625
+ wherevαis the Hamiltonian vector field correspondingto α. Comparing Eq.
626
+ 17 to Definition 3.2 of a Hamiltonian 1-form, we see that a sect ion (u,β)∈
627
+ Γ(Eω) is in the image of the map φ0if and only if its adjoint action ad (u,β)
628
+ corresponds to the pair ( u,0)∈Vect(M)⊕Ω2(M). This implies that ad (u,β)
629
+ preserves the 2-plectic structure on Mand that −βis a Hamiltonian 1-form
630
+ with Hamiltonian vector field u. Also if uis complete, then Eqs. 12 and 13
631
+ imply that the 1-parameter family Ftof Courant algebroid automorphisms
632
+ generated by ad (u,β)correspondsto a1-parameter family of diffeomorphisms
633
+ ϕt:M→Mwhich preserve the 2-plectic structure:
634
+ ϕ∗
635
+ tω=ω.
636
+ In analogy with symplectic geometry, we call such automorph ismsHamil-
637
+ tonian 2-plectomorphisms .
638
+ We provide the following proposition as a summary of the disc ussion
639
+ presented in this section:
640
+ Proposition 6.2. Let(M,ω)be a 2-plectic manifold and let Eωbe its cor-
641
+ responding Courant algebroid. There is a one-to-one correspo ndence be-
642
+ tween the Hamiltonian 1-forms Ham(M)on(M,ω)and sections (u,β)of
643
+ Eωwhose adjoint action satisfies the equality
644
+ ad(u,β)(v,α) = (Luv,Luα).
645
+ 7.Conclusions
646
+ We suspect that the results presented here are preliminary a nd indicate
647
+ a deeper relationship between 2-plectic geometry and the th eory of Courant
648
+ algebroids. For example, the discussion of connections and curvature in
649
+ Section 2.1 is reminiscent of the theory of gerbes with conne ction [5], whose
650
+ relationship with Courant algebroids has been already stud ied [4, 20]. In 2-
651
+ plecticgeometry, gerbeshavebeenconjecturedtoplayarol einthegeometric
652
+ quantization ofa2-plecticmanifold[2]. Itwillbeinteres tingtoseehowthese
653
+ different points of view complement each other.
654
+ In general, much work has been done on studying the geometric struc-
655
+ tures induced by Courant algebroids (e.g. Dirac structures , twisted Dirac
656
+ structures). Perhaps this work can aid 2-plectic geometry s ince many geo-
657
+ metric structures in this context are somewhat less underst ood or remain
658
+ ill-defined (e.g. the notion of a 2-Lagrangian submanifold o r 2-polarization).
659
+ On the other hand, n-plectic manifolds are well understood in the role
660
+ they play in classical field theory [11], and are also underst ood algebraically
661
+ in the sense that an n-plectic structure gives an n-termL∞-algebra on a
662
+ chain complex of differential forms [15]. Perhaps these insig hts can aid inCOURANT ALGEBROIDS FROM CATEGORIFIED SYMPLECTIC GEOMETRY 17
663
+ understanding ‘higher’ analogs of Courant algebroids (e.g . Lien-algebroids)
664
+ and complement similar ideas discussed by ˇSevera in [22].
665
+ 8.Acknowledgments
666
+ We thank John Baez, Yael Fregier, Dmitry Roytenberg, Urs Sch rieber,
667
+ James Stasheff and Marco Zambon for helpful comments, questi ons, and
668
+ conversations.
669
+ References
670
+ [1] J. Baez and A. Crans, Higher-dimensional algebra VI: Lie 2-algebras, TAC12(2004),
671
+ 492–528. Also available as arXiv:math/0307263.
672
+ [2] J. Baez, A. Hoffnung, and C. Rogers, Categorified symplect ic geometry and
673
+ the classical string, Comm. Math. Phys. 293(2010), 701–715. Also available as
674
+ arXiv:0808.0246.
675
+ [3] J. Baez and C. Rogers, Categorified symplectic geometry a nd the string Lie 2-algebra,
676
+ available as arXiv:0901.4721.
677
+ [4] P. Bressler and A. Chervov, Courant algebroids, J. Math. Sci. (N.Y.) 128(2005),
678
+ 3030–3053. Also available as arXiv:hep-th/0212195.
679
+ [5] J.-L. Brylinski, Loop Spaces, Characteristic Classes and Geometric Quantiz ation,
680
+ Birkhauser, Boston, 1993.
681
+ [6] M.J. Bowick and S.G. Rajeev, Closed bosonic string theor y,Nuc. Phys. B 293(1987),
682
+ 348–384.
683
+ [7] H. Bursztyn, G.R. Cavalcanti, and M. Gualtieri, Reducti on of Courant algebroids
684
+ and generalized complex structures, Adv. Math. 211(2007), 726–765. Also available
685
+ as arXiv:math/0509640.
686
+ [8] F. Cantrijn, A.Ibort, andM. DeLeon, Onthegeometryofmu ltisymplectic manifolds,
687
+ J. Austral. Math. Soc. (Series A) 66(1999), 303–330.
688
+ [9] T. Courant, Dirac manifolds, Trans. Amer. Math. Soc. 319(1990), 631���661.
689
+ [10] T. DeDonder, Theorie Invariantive du Calcul des Variations , Gauthier–Villars, Paris,
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+ 1935.
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+ [11] M. Gotay, J. Isenberg, J. Marsden, and R. Montgomery, Mo mentum maps and classi-
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+ cal relativistic fields.Part I:covariantfieldtheory, avai lable as arXiv:physics/9801019.
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+ [12] F. H´ elein, Hamiltonian formalisms for multidimensio nal calculus of variations and
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+ perturbation theory, in Noncompact Problems at the Intersection of Geometry , eds.
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+ A. Bahri et al, AMS, Providence, Rhode Island, 2001, pp. 127–148. Also ava ilable as
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+ arXiv:math-ph/0212036.
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+ [13] J. Kijowski, A finite-dimensional canonical formalism in the classical field theory,
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+ Commun. Math. Phys. 30(1973), 99–128.
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+ [14] T. Lada and J. Stasheff, Introduction to sh Lie algebras f or physicists, Int. Jour.
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+ Theor. Phys. 32(7) (1993), 1087–1103. Also available as hep-th/9209099.
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+ [15] C. Rogers, L∞-algebras from multisymplectic geometry, in preparation.
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+ [16] D. Roytenberg and A. Weinstein, Courant algebroids and strongly homotopy Lie
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+ algebras, Lett. Math. Phys. 46(1998), 81–93. Also available as arXiv:math/9802118.
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+ [17] D. Roytenberg, Courant Algebroids, Derived Brackets and Even Symplectic S uper-
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+ manifolds , Ph.D. thesis, UC Berkeley, 1999. Also available as arXiv:m ath/9910078.
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+ [18] D. Roytenberg, On the structure of graded symplectic su permanifolds and Courant
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+ algebroids, in Quantization, Poisson Brackets and Beyond , ed. T. Voronov, Con-
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+ temp. Math. ,315, AMS, Providence, RI, 2002, pp. 169–185. Also available as
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+ arXiv:math/0203110.18 CHRISTOPHER L. ROGERS
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+ [19] D. Roytenberg, On weak Lie 2-algebras, in: P. Kielanows kiet al(eds.) XXVI Work-
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+ shop on Geometrical Methods in Physics. AIP Conference Proc eedings956, pp. 180-
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+ 198. American InstituteofPhysics, Melville (2007). Alsoa vailable as arXiv:0712.3461.
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+ [20] P. ˇSevera, Letter to Alan Weinstein, available at
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+ http://sophia.dtp.fmph.uniba.sk/ ~severa/letters/
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+ [21] P.ˇSevera, and A. Weinstein, Poisson geometry with a 3-form bac kground, Prog.
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+ Theor. Phys. Suppl. 144(2001), 145–154. Also available as arXiv:math/0107133.
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+ able as arXiv:0910.2147.
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+ Department of Mathematics, University of California, Rive rside, Califor-
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+ nia 92521, USA
1001.0041.txt ADDED
@@ -0,0 +1,497 @@
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
1
+ arXiv:1001.0041v2 [math.MG] 2 Jul 2010Almost-Euclidean subspaces of ℓN
2
+ 1via tensor
3
+ products: a simple approach to randomness
4
+ reduction
5
+ Piotr Indyk1⋆and Stanislaw Szarek2⋆⋆
6
7
+ 2CWRU & Paris 6 [email protected]
8
+ Abstract. It has been known since 1970’s that the N-dimensional ℓ1-
9
+ space contains nearly Euclidean subspaces whose dimension isΩ(N).
10
+ However, proofs of existence of such subspaces were probabi listic, hence
11
+ non-constructive, which made the results not-quite-suita ble for subse-
12
+ quently discovered applications to high-dimensional near est neighbor
13
+ search, error-correcting codes over the reals, compressiv e sensing and
14
+ other computational problems. In this paper we present a “lo w-tech”
15
+ scheme which, for any γ >0, allows us toexhibitnearly Euclidean Ω(N)-
16
+ dimensional subspaces of ℓN
17
+ 1while using only Nγrandom bits. Our re-
18
+ sults extend and complement (particularly) recent work by G uruswami-
19
+ Lee-Wigderson. Characteristic features of our approach in clude (1) sim-
20
+ plicity (we use only tensor products) and (2) yielding almos t Euclidean
21
+ subspaces with arbitrarily small distortions.
22
+ 1 Introduction
23
+ It is a well-known fact that for any vector x∈RN, itsℓ2andℓ1norms are
24
+ related by the (optimal) inequality /⌊ar⌈⌊lx/⌊ar⌈⌊l2≤ /⌊ar⌈⌊lx/⌊ar⌈⌊l1≤√
25
+ N/⌊ar⌈⌊lx/⌊ar⌈⌊l2. However, classical
26
+ results in geometric functional analysis show that for a “substant ial fraction” of
27
+ vectors , the relation between its 1-norm and 2-norm can be made m uch tighter.
28
+ Specifically, [FLM77,Kas77,GG84] show that there exists a subspace E⊂RNof
29
+ dimensionm=αN, and a scaling constant Ssuch that for all x∈E
30
+ 1/D·√
31
+ N/⌊ar⌈⌊lx/⌊ar⌈⌊l2≤S/⌊ar⌈⌊lx/⌊ar⌈⌊l1≤√
32
+ N/⌊ar⌈⌊lx/⌊ar⌈⌊l2 (1)
33
+ whereα∈(0,1) andD=D(α), called the distortion ofE, are absolute (notably
34
+ dimension-free) constants.Overthe last few years,such “almos t-Euclidean”sub-
35
+ spaces ofℓN
36
+ 1have found numerous applications, to high-dimensional nearest
37
+ neighbor search [Ind00], error-correcting codes over reals and c ompressive sens-
38
+ ing [KT07,GLR08,GLW08], vector quantization [LV06], oblivious dimension ality
39
+ ⋆This research has been supported in part by David and Lucille Packard Fellowship,
40
+ MADALGO (Center for Massive Data Algorithmics, funded by th e Danish National
41
+ Research Association) and NSF grant CCF-0728645.
42
+ ⋆⋆Supported in part by grants from the National Science Founda tion (U.S.A.) and the
43
+ U.S.-Israel BSF.reduction and ǫ-samples for high-dimensional half-spaces [KRS09], and to other
44
+ problems.
45
+ For the above applications, it is convenient and sometimes crucial th at the
46
+ subspaceEis defined in an explicit manner3. However, the aforementioned re-
47
+ sults do not providemuch guidance in this regard,since they use the probabilistic
48
+ method. Specifically, either the vectors spanning E, or the vectors spanning the
49
+ space dual to E, are i.i.d. random variables from some distribution. As a result,
50
+ the constructionsrequire Ω(N2)independent randomvariablesasstartingpoint.
51
+ Until recently, the largest explicitly constructible almost-Euclidean subspace of
52
+ ℓN
53
+ 1, due to Rudin [Rud60] (cf. [LLR94]), had only a dimension of Θ(√
54
+ N).
55
+ During the last few years, there has been a renewed interest in the prob-
56
+ lem[AM06,Sza06,Ind07,LS07,GLR08,GLW08],withresearchersusingide asgained
57
+ from the study of expanders, extractorsand error-correctin gcodes to obtain sev-
58
+ eral explicit constructions. The work progressed on two fronts, focusing on (a)
59
+ fully explicit constructions of subspaces attempting to maximize the dimension
60
+ and minimize the distortion [Ind07,GLR08], as well as (b) construction s using
61
+ limited randomness, with dimension and distortion matching (at least q ualita-
62
+ tively)theexistentialdimensionanddistortionbounds[Ind00,AM06,L S07,GLW08].
63
+ The parameters of the constructions are depicted in Figure 1. Qua litatively,
64
+ they show that in the fully explicit case, one can achieve either arbitr arily low
65
+ distortion or arbitrarily high subspace dimension, but not (yet?) bo th. In the
66
+ low-randomness case, one can achieve arbitrarily high subspace dim ension and
67
+ constant distortion while using randomness that is sub-linear in N; achieving
68
+ arbitrarily low distortion was possible as well, albeit at a price of (super )-linear
69
+ randomness.
70
+ Reference Distortion Subspace dimension Randomness
71
+ [Ind07] 1+ǫ N1−oǫ(1)explicit
72
+ [GLR08] (logN)Oη(logloglog N)(1−η)N explicit
73
+ [Ind00] 1+ǫ Ω(ǫ2/log(1/ǫ))NO(Nlog2N)
74
+ [AM06,LS07] Oη(1) (1−η)N O(N)
75
+ [GLW08] 2Oη(1/γ)(1−η)N O(Nγ)
76
+ This paper 1+ǫ (γǫ)O(1/γ)N O(Nγ)
77
+ Fig.1.The best known results for constructing almost-Euclidean s ubspaces of ℓN
78
+ 1. The
79
+ parameters ǫ,η,γ∈(0,1) are assumed to be constants, although we explicitly point
80
+ out when the dependence on them is subsumed by the big-Oh nota tion.
81
+ 3For the purpose of this paper “explicit” means “the basis of Ecan be generated
82
+ by a deterministic algorithm with running time polynomial i nN.” However, the
83
+ individual constructions can be even “more explicit” than t hat.Our result In this paper we show that, using sub-linear randomness, one can
84
+ constructasubspacewitharbitrarilysmalldistortionwhilekeepingit sdimension
85
+ proportional to N. More precisely, we have:
86
+ Theorem 1 Letǫ,γ∈(0,1). GivenN∈N, assume that we have at our dis-
87
+ posal a sequence of random bits of length max{Nγ,C(ǫ,γ)}log(N/(ǫγ)). Then,
88
+ in deterministic polynomial (in N) time, we can generate numbers M >0,
89
+ m≥c(ǫ,γ)Nand anm-dimensional subspace of ℓN
90
+ 1E, for which we have
91
+ ∀x∈E,(1−ǫ)M/⌊ar⌈⌊lx/⌊ar⌈⌊l2≤ /⌊ar⌈⌊lx/⌊ar⌈⌊l1≤(1+ǫ)M/⌊ar⌈⌊lx/⌊ar⌈⌊l2
92
+ with probability greater than 98%.
93
+ In a sense, this complements the result of [GLW08], optimizing the dist ortion
94
+ of the subspace at the expense of its dimension. Our approach also allows to
95
+ retrieve – using a simpler and low-tech approach – the results of [GLW 08] (see
96
+ the comments at the end of the Introduction).
97
+ Overview of techniques The ideas behind many of the prior constructions as
98
+ well as this work can be viewed as variants of the related developmen ts in the
99
+ context of error-correcting codes. Specifically, the construct ion of [Ind07] resem-
100
+ bles the approach of amplifying minimum distance of a code using expan ders
101
+ developed in [ABN+92], while the constructions of [GLR08,GLW08] were in-
102
+ spired by low-density parity check codes. The reason for this stat e of affairs is
103
+ that a vector whose ℓ1norms and ℓ2norms are very different must be “well-
104
+ spread”, i.e., a small subset of its coordinates cannot contain most of itsℓ2
105
+ mass (cf. [Ind07,GLR08]). This is akin to a property required from a g ood error-
106
+ correcting code, where the weight (a.k.a. the ℓ0norm) of each codeword cannot
107
+ be concentrated on a small subset of its coordinates.
108
+ In this vein, our construction utilizes a tool frequently used for (lin ear) error-
109
+ correcting codes, namely the tensor product . Recall that, for two linear codes
110
+ C1⊂ {0,1}n1andC2⊂ {0,1}n2, their tensor product is a code C⊂ {0,1}n1n2,
111
+ such that for any codeword c∈C(viewed as an n1×n2matrix), each column of
112
+ cbelongs toC1and each row of cbelongs toC2. It is known that the dimension
113
+ ofCis a product of the dimensions of C1andC2, and that the same holds
114
+ for the minimum distance. This enables constructing a code of “large ” block-
115
+ lengthNkby starting from a code of “small” block-length Nand tensoring it k
116
+ times. Here, we roughly show that the tensor product of two subs paces yields a
117
+ subspace whose distortion is a product of the distortions of the su bspaces. Thus,
118
+ we can randomly choose an initial small low-distortion subspace, and tensor it
119
+ with itself to yield the desired dimension.
120
+ However, tensoring alone does not seem sufficient to give a subspac e with
121
+ distortionarbitrarilycloseto1.Thisisbecausewecanonlyanalyzeth edistortion
122
+ of the product space for the case when the scaling factor Sin Equation 1 is
123
+ equal to 1 (technically, we only prove the left inequality, and rely on t he general
124
+ relation between the ℓ2andℓ1for the upper bound). For S= 1, however, the
125
+ best achievable distortion is strictly greater than 1, and tensoring can make itonly larger. To avoid this problem, instead of the ℓN
126
+ 1norm we use the ℓN/B
127
+ 1(ℓB
128
+ 2)
129
+ norm, for a “small” value of B. The latter norm (say, denoted by /⌊ar⌈⌊l · /⌊ar⌈⌊l) treats
130
+ the vector as a sequence of N/B“blocks” of length B, and returns the sum of
131
+ theℓ2norms of the blocks. We show that there exist subspaces E⊂ℓN/B
132
+ 1(ℓB
133
+ 2)
134
+ such that for any x∈Ewe have
135
+ 1/D·/radicalbig
136
+ N/B/⌊ar⌈⌊lx/⌊ar⌈⌊l2≤ /⌊ar⌈⌊lx/⌊ar⌈⌊l ≤/radicalbig
137
+ N/B/⌊ar⌈⌊lx/⌊ar⌈⌊l2
138
+ forDthat is arbitrarily close to 1. Thus, we can construct almost-Euclide an
139
+ subspaces of ℓ1(ℓ2) of desired dimensions using tensoring, and get rid of the
140
+ “inner”ℓ2norm at the end of the process.
141
+ We point out that if we do not insist on distortion arbitrarily close to 1,
142
+ the “blocks” are not needed and the argument simplifies substantia lly. In par-
143
+ ticular, to retrieve the results of [GLW08], it is enough to combine the scalar-
144
+ valued version of Proposition 1 below with “off-the-shelf” random co nstructions
145
+ [Kas77,GG84] yielding – in the notation of Equation 1 – a subspace E, for which
146
+ the parameter αis close to 1.
147
+ 2 Tensoring subspaces of L1
148
+ We start by defining some basic notions and notation used in this sect ion.
149
+ Norms and distortion In this section we adopt the “continuous” notation for
150
+ vectorsand norms. Specifically, considera real Hilbert space Hand a probability
151
+ measureµover [0,1]. Forp∈[1,∞] consider the space Lp(H) ofH-valuedp-
152
+ integrable functions fendowed with the norm
153
+ /⌊ar⌈⌊lf/⌊ar⌈⌊lp=/⌊ar⌈⌊lf/⌊ar⌈⌊lLp(H)=/parenleftbigg/integraldisplay
154
+ /⌊ar⌈⌊lf(x)/⌊ar⌈⌊lp
155
+ Hdµ(x)/parenrightbigg1/p
156
+ In what follows we will omit µfrom the formulae since the measure will be
157
+ clear from the context (and largely irrelevant). As our main result c oncerns
158
+ finite dimensional spaces, it suffices to focus on the case where µis simply the
159
+ normalizedcountingmeasureoverthe discreteset {0,1/n,...(n−1)/n}for some
160
+ fixedn∈N(although the statements hold in full generality). In this setting, t he
161
+ functionsffromLp(H) areequivalent to n-dimensional vectorswith coordinates
162
+ inH.4The advantage of using the Lpnorms as opposed to the ℓpnorms that
163
+ the relation between the 1-norm and the 2-norm does not involve sc aling factors
164
+ that depend on dimension, i.e., we have /⌊ar⌈⌊lf/⌊ar⌈⌊l2≥ /⌊ar⌈⌊lf/⌊ar⌈⌊l1for allf∈L2(H) (note
165
+ that, for the Lpnorms, the “trivial” inequality goes in the other direction than
166
+ for theℓpnorms). This simplifies the notation considerably.
167
+ 4The values from Hroughly correspond to the finite-dimensional “blocks” in th e
168
+ construction sketched in the introduction. Note that Hcan be discretized similarly
169
+ as theLp-spaces; alternatively, functions that are constant on int ervals of the type/parenleftBig
170
+ (k−1)/N,k/N/parenrightBig
171
+ can be considered in lieu of discrete measures.We will be interested in lialmost subspaces E⊂L2(H) on which the 1-norm
172
+ and 2-norm uniformly agree, i.e., for some c∈(0,1],
173
+ /⌊ar⌈⌊lf/⌊ar⌈⌊l2≥ /⌊ar⌈⌊lf/⌊ar⌈⌊l1≥c/⌊ar⌈⌊lf/⌊ar⌈⌊l2 (2)
174
+ for allf∈E. The best (the largest) constant cthat works in (2) will be denoted
175
+ Λ1(E). For completeness, we also define Λ1(E) = 0 if noc>0 works.
176
+ Tensor products IfH,Kare Hilbert spaces, H⊗2Kis their (Hilbertian) tensor
177
+ product, which may be (for example) described by the following prop erty: if (ej)
178
+ is an orthonormal sequence in Hand (fk) is an orthonormal sequence in K,
179
+ then (ej⊗fk) is an orthonormal sequence in H ⊗2K(a basis if ( ej) and (fk)
180
+ were bases). Next, any element of L2(H)⊗ Kis canonically identified with a
181
+ function in the space L2(H ⊗2K); note that such functions are H ⊗K-valued,
182
+ but are defined on the same probability space as their counterpart s fromL2(H).
183
+ IfE⊂L2(H) is a linear subspace, E⊗Kis – under this identification – a linear
184
+ subspace of L2(H⊗2K).
185
+ As hinted in the Introduction, our argument depends (roughly) on the fact
186
+ that the property expressed by (1) or (2) “passes” to tensor p roducts of sub-
187
+ spaces, and that it “survives” replacing scalar-valued functions b y ones that
188
+ have values in a Hilbert space. Statements to similar effect of various degrees
189
+ of generality and precision are widely available in the mathematical liter ature,
190
+ see for example [MZ39,Bec75,And80,FJ80]. However, we are not awar e of a ref-
191
+ erence that subsumes all the facts needed here and so we presen t an elementary
192
+ self-contained proof.
193
+ We start with two preliminary lemmas.
194
+ Lemma 1 Ifg1,g2,...∈E⊂L2(H), then
195
+ /integraldisplay/parenleftbig/summationdisplay
196
+ k/⌊ar⌈⌊lgk(x)/⌊ar⌈⌊l2
197
+ H/parenrightbig1/2dx≥Λ1(E)/parenleftBig/integraldisplay/summationdisplay
198
+ k/⌊ar⌈⌊lgk(x)/⌊ar⌈⌊l2
199
+ Hdx/parenrightBig1/2
200
+ .
201
+ ProofLetKbe an auxiliary Hilbert space and ( ek) an orthonormal sequence
202
+ (O.N.S.) in K. We will apply Minkowski inequality – a continuous version of
203
+ the triangle inequality, which says that for vector valued functions /⌊ar⌈⌊l/integraltext
204
+ h/⌊ar⌈⌊l ≤/integraltext/⌊ar⌈⌊lh/⌊ar⌈⌊l– to the K-valued function h(x) =/summationtext
205
+ k/⌊ar⌈⌊lgk(x)/⌊ar⌈⌊lHek. As is easily seen,
206
+ /⌊ar⌈⌊l/integraltext
207
+ h/⌊ar⌈⌊lK=/⌊ar⌈⌊l/summationtext
208
+ k/parenleftbig/integraltext
209
+ /⌊ar⌈⌊lgk(x)/⌊ar⌈⌊lHdx/parenrightbig
210
+ ek/⌊ar⌈⌊lK=/parenleftbig/summationtext
211
+ k/⌊ar⌈⌊lgk/⌊ar⌈⌊l2
212
+ L1(H)/parenrightbig1/2. Given that gk∈
213
+ E,/⌊ar⌈⌊lgk/⌊ar⌈⌊lL1(H)≥Λ1(E)/⌊ar⌈⌊lgk/⌊ar⌈⌊lL2(H)and so
214
+ /vextenddouble/vextenddouble/vextenddouble/integraldisplay
215
+ h/vextenddouble/vextenddouble/vextenddouble
216
+ K≥Λ1(E)/parenleftBig/integraldisplay/summationdisplay
217
+ k/⌊ar⌈⌊lgk(x)/⌊ar⌈⌊l2
218
+ Hdx/parenrightBig1/2
219
+ On the other hand, the left hand side of the inequality in Lemma 1 is exa ctly/integraltext
220
+ /⌊ar⌈⌊lh/⌊ar⌈⌊lK, so the Minkowski inequality yields the required estimate.
221
+ We are now ready to state the next lemma. Recall that Eis a linear subspace
222
+ ofL2(H), andKis a Hilbert space.Lemma 2 Λ1(E⊗K) =Λ1(E)
223
+ IfE⊂L2=L2(R), the lemma says that any estimate of type (2) for scalar
224
+ functionsf∈Ecarries overto their linear combinations with vector coefficients,
225
+ namely to functions of the type/summationtext
226
+ jvjfj,fj∈E,vj∈ K. In the general case,
227
+ any estimate for H-valued functions f∈E⊂L2(H) carries over to functions of
228
+ the form/summationtext
229
+ jfj⊗vj∈L2(H⊗2K), withfj∈E,vj∈ K.
230
+ Proof of Lemma 2 Let (ek) be an orthonormalbasis of K. In fact w.l.o.g. we may
231
+ assume that K=ℓ2and that (ek) is the canonical orthonormal basis. Consider
232
+ g=/summationtext
233
+ jfj⊗vj, wherefj∈Eandvj∈ K. Then also g=/summationtext
234
+ kgk⊗ekfor some
235
+ gk∈Eand hence (pointwise) /⌊ar⌈⌊lg(x)/⌊ar⌈⌊lH⊗2K=/parenleftbig/summationtext
236
+ k/⌊ar⌈⌊lgk(x)/⌊ar⌈⌊l2
237
+ H/parenrightbig1/2. Accordingly,
238
+ /⌊ar⌈⌊lg/⌊ar⌈⌊lL2(H⊗2K)=/parenleftbig/integraltext/summationtext
239
+ k/⌊ar⌈⌊lgk(x)/⌊ar⌈⌊l2
240
+ Hdx/parenrightbig1/2,while/⌊ar⌈⌊lg/⌊ar⌈⌊lL1(H⊗2K)=/integraltext/parenleftbig/summationtext
241
+ k/⌊ar⌈⌊lgk(x)/⌊ar⌈⌊l2
242
+ H/parenrightbig1/2dx.
243
+ Comparing such quantities is exactly the object of Lemma 1, which imp lies that
244
+ /⌊ar⌈⌊lg/⌊ar⌈⌊lL1(H⊗2K)≥Λ1(E)/⌊ar⌈⌊lg/⌊ar⌈⌊lL2(H⊗2K).Sinceg∈E⊗Kwasarbitrary,it follows that
245
+ Λ1(E⊗K)≥Λ1(E). The reverse inequality is automatic (except in the trivial
246
+ case dim K= 0, which we will ignore).
247
+ IfE⊂L2(H) andF⊂L2(K) are subspaces, E⊗Fis the subspace of
248
+ L2(H ⊗2K) spanned by f⊗gwithf∈E,g∈F. (For clarity, f⊗gis a
249
+ function on the product of the underlying probability spaces and is defined by
250
+ (x,y)→f(x)⊗g(y)∈ H⊗K .)
251
+ The next proposition shows the key property of tensoring almost- Euclidean
252
+ spaces.
253
+ Proposition 1. Λ1(E⊗F)≥Λ1(E)Λ1(F)
254
+ ProofLet (ϕj) and (ψk) be orthonormal bases of respectively EandFand let
255
+ g=/summationtext
256
+ j,ktjkϕj⊗ψk.Weneedtoshowthat /⌊ar⌈⌊lg/⌊ar⌈⌊lL1(H⊗2K)≥Λ1(E)Λ1(F)/⌊ar⌈⌊lg/⌊ar⌈⌊lL2(H⊗2K),
257
+ where thep-norms refer to the product probability space, for example
258
+ /⌊ar⌈⌊lg/⌊ar⌈⌊lL1(H⊗2K)=/integraldisplay /integraldisplay/vextenddouble/vextenddouble/summationdisplay
259
+ j,ktjkϕj(x)⊗ψk(y)/vextenddouble/vextenddouble
260
+ H⊗2Kdxdy.
261
+ Rewriting the expression under the sum and subsequently applying L emma 2 to
262
+ the inner integral for fixed ygives
263
+ /integraldisplay/vextenddouble/vextenddouble/summationdisplay
264
+ j,ktjkϕj(x)⊗ψk(y)/vextenddouble/vextenddouble
265
+ H⊗2Kdx=/integraldisplay/vextenddouble/vextenddouble/summationdisplay
266
+ jϕj(x)⊗/parenleftBig/summationdisplay
267
+ ktjkψk(y)/parenrightBig/vextenddouble/vextenddouble
268
+ H⊗2Kdx
269
+ ≥Λ1(E)/parenleftBig/integraldisplay/vextenddouble/vextenddouble/summationdisplay
270
+ jϕj(x)⊗/parenleftBig/summationdisplay
271
+ ktjkψk(y)/parenrightBig/vextenddouble/vextenddouble2
272
+ H⊗2Kdx/parenrightBig1/2
273
+ =Λ1(E)/parenleftBig/summationdisplay
274
+ j/vextenddouble/vextenddouble/summationdisplay
275
+ ktjkψk(y)/vextenddouble/vextenddouble2
276
+ K/parenrightBig1/2In turn,/summationtext
277
+ ktjkψk∈F(for allj) and so, by Lemma 1,
278
+ /integraldisplay/parenleftBig/summationdisplay
279
+ j/vextenddouble/vextenddouble/summationdisplay
280
+ ktjkψk(y)/vextenddouble/vextenddouble2
281
+ K/parenrightBig1/2
282
+ dy≥Λ1(F)/parenleftBig/integraldisplay/summationdisplay
283
+ j/vextenddouble/vextenddouble/summationdisplay
284
+ ktjkψk(y)/vextenddouble/vextenddouble2
285
+ Kdy/parenrightBig1/2
286
+ =Λ1(F)/⌊ar⌈⌊lg/⌊ar⌈⌊lL2(H⊗2K).
287
+ Combining the above formulae yields the conclusion of the Proposition .
288
+ 3 The construction
289
+ In this section we describe our low-randomness construction. We s tart from a
290
+ recap of the probabilistic construction, since we use it as a building blo ck.
291
+ 3.1 Dvoretzky’s theorem, and its “tangible” version
292
+ For general normed spaces, the following is one possible statement of the well-
293
+ known Dvoretzky’s theorem [Dvo61]:
294
+ Givenm∈Nandε>0there isN=N(m,ε)such that, for any norm on RN
295
+ there is an m-dimensional subspace on which the ratio of ℓ1andℓ2norms is
296
+ (approximately) constant, up to a multiplicative factor 1+ε.
297
+ For specific norms this statement can be made more precise, both in describing
298
+ the dependence N=N(m,ε) and in identifying the constant of (approximate)
299
+ proportionality of norms. The following version is (essentially) due to Milman
300
+ [Mil71].
301
+ Dvoretzky’s theorem (Tangible version) Consider the N-dimensional Eu-
302
+ clidean space (real orcomplex) endowed with the Euclidean norm /⌊ar⌈⌊l·/⌊ar⌈⌊l2and some
303
+ other norm /⌊ar⌈⌊l·/⌊ar⌈⌊lsuch that, for some b>0,/⌊ar⌈⌊l·/⌊ar⌈⌊l ≤b/⌊ar⌈⌊l·/⌊ar⌈⌊l2. LetM=E/⌊ar⌈⌊lX/⌊ar⌈⌊l, whereX
304
+ is a random variable uniformly distributed on the unit Euclidean sphere . Then
305
+ there exists a computable universal constant c >0, so that if 0< ε <1and
306
+ m≤cε2(M/b)2N, then for more than 99% (with respect to the Haar measure)
307
+ m-dimensional subspaces Ewe have
308
+ ∀x∈E,(1−ε)M/⌊ar⌈⌊lx/⌊ar⌈⌊l2≤ /⌊ar⌈⌊lx/⌊ar⌈⌊l ≤(1+ε)M/⌊ar⌈⌊lx/⌊ar⌈⌊l2. (3)
309
+ Alternative good expositions of the theorem are in, e.g., [FLM77], [MS8 6] and
310
+ [Pis89]. We point out that standard and most elementary proofs yield m≤
311
+ cε2/log(1/ε)(M/b)2N; the dependence on εof orderε2was obtained in the
312
+ important papers [Gor85,Sch89], see also [ASW10].
313
+ 3.2 The case of ℓn
314
+ 1(ℓB
315
+ 2)
316
+ OurobjectivenowistoapplyDvoretzky’stheoremandsubsequent lyProposition
317
+ 1 to spaces of the form ℓn
318
+ 1(ℓB
319
+ 2) for some n,B∈N, so from now on we set/⌊ar⌈⌊l·/⌊ar⌈⌊l:=/⌊ar⌈⌊l·/⌊ar⌈⌊lℓn
320
+ 1(ℓB
321
+ 2)To that end, we need to determine the values of the parameter
322
+ Mthat appears in the theorem. (The optimal value of bis clearly√n, as in
323
+ the scalar case, i.e., when B= 1.) We have the following standard (cf. [Bal97],
324
+ Lecture 9)
325
+ Lemma 3
326
+ M(n,B) :=Ex∈SnB−1/⌊ar⌈⌊lx/⌊ar⌈⌊l=Γ(B+1
327
+ 2)
328
+ Γ(B
329
+ 2)Γ(nB
330
+ 2)
331
+ Γ(nB+1
332
+ 2)n.
333
+ In particular,/radicalBig
334
+ 1+1
335
+ n−1/radicalBig
336
+ 2
337
+ π√n>M(n,1)>/radicalBig
338
+ 2
339
+ π√nfor alln∈N(the scalar
340
+ case) andM(n,B)>/radicalBig
341
+ 1−1
342
+ B√nfor alln,B∈N.
343
+ The equality is shown by relating (via passing to polar coordinates) sp heri-
344
+ cal averages of norms to Gaussian means: if Xis a random variable uniformly
345
+ distributed on the Euclidean sphere SN−1andYhas the standard Gaussian
346
+ distribution on RN, then, for any norm /⌊ar⌈⌊l·/⌊ar⌈⌊l,
347
+ E/⌊ar⌈⌊lY/⌊ar⌈⌊l=√
348
+ 2Γ(N+1
349
+ 2)
350
+ Γ(N
351
+ 2)E/⌊ar⌈⌊lX/⌊ar⌈⌊l
352
+ The inequalities follow from the estimates/radicalBig
353
+ x−1
354
+ 2<Γ(x+1
355
+ 2)
356
+ Γ(x)<√x(forx≥1
357
+ 2),
358
+ which in turn are consequences of log-convexity of Γand its functional equation
359
+ Γ(y+1) =yΓ(y). (Alternatively, Stirling’s formula may be used to arrive at a
360
+ similar conclusion.)
361
+ Combining Dvoretzky’s theorem with Lemma 3 yields
362
+ Corollary 1 If0< ε <1andm≤c1ε2n, then for more than 99%of the
363
+ m-dimensional subspaces E⊂ℓn
364
+ 1we have
365
+ ∀x∈E(1−ε)/radicalbigg
366
+ 2
367
+ π√n/⌊ar⌈⌊lx/⌊ar⌈⌊l2≤ /⌊ar⌈⌊lx/⌊ar⌈⌊l1≤(1+ε)/radicalbigg
368
+ 1+1
369
+ n−1/radicalbigg
370
+ 2
371
+ π√n/⌊ar⌈⌊lx/⌊ar⌈⌊l2(4)
372
+ Similarly, if B >1andm≤c2ε2nB, then for more than 99%of them-
373
+ dimensional subspaces E⊂ℓn
374
+ 1(ℓB
375
+ 2)we have
376
+ ∀x∈E(1−ε)/radicalbigg
377
+ 1−1
378
+ B√n/⌊ar⌈⌊lx/⌊ar⌈⌊l2≤ /⌊ar⌈⌊lx/⌊ar⌈⌊l ≤√n/⌊ar⌈⌊lx/⌊ar⌈⌊l2 (5)
379
+ We point out that the upper estimate on /⌊ar⌈⌊lx/⌊ar⌈⌊lin the second inequality is valid
380
+ for allx∈ℓn
381
+ 1(ℓB
382
+ 2) and, like the estimate M(n,B)≤√n, follows just from the
383
+ Cauchy-Schwarz inequality.
384
+ Since a random subspace chosen uniformly according to the Haar me asure
385
+ on the manifold of m-dimensional subspaces of RN(orCN) can be constructed
386
+ from anN×mrandom Gaussian matrix, we may apply standard discretization
387
+ techniques to obtain the followingCorollary 2 There is a deterministic algorithm that, given ε,B,m,n as in
388
+ Corollary 1 and a sequence of O(mnlog(mn/ǫ))random bits, generates sub-
389
+ spacesEas in Corollary 1 with probability greater than 98%, in time polynomial
390
+ in1/ε+B+m+n.
391
+ We point out that in the literature on the “randomness-reduction” , one typ-
392
+ ically uses Bernoulli matrices in lieu of Gaussian ones. This enables avoid ing the
393
+ discretization issue, since the problem is phrased directly in terms of random
394
+ bits. Still, since proofs of Dvoretzky type theorems for Bernoulli m atrices are
395
+ often much harder than for their Gaussian counterparts, we pre fer to appeal in-
396
+ stead to a simple discretization ofGaussian random variables.We not e, however,
397
+ that the early approach of [Kas77] was based on Bernoulli matrices .
398
+ We are now ready to conclude the proof of Theorem 1. Given ε∈(0,1)
399
+ andn∈N, chooseB=⌈ε−1⌉andm=⌊cε2(1−1
400
+ B)nB⌋ ≥c0ε2nB. Corollary
401
+ 2 (Equation 5) and repeated application of Proposition 1 give us a sub space
402
+ F⊂ℓν
403
+ 1(ℓβ
404
+ 2) (whereν=nkandβ=Bk) of dimension mk≥(c0ε2)kνβsuch that
405
+ ∀x∈F(1−ε)3k/2nk/2/⌊ar⌈⌊lx/⌊ar⌈⌊l2≤ /⌊ar⌈⌊lx/⌊ar⌈⌊l ≤nk/2/⌊ar⌈⌊lx/⌊ar⌈⌊l2.
406
+ Moreover,F=E⊗E⊗...⊗E, whereE⊂ℓn
407
+ 1(ℓB
408
+ 2) is a typical m-dimensional
409
+ subspace.Thusin ordertoproduce E, henceF,weonlyneed togeneratea“typi-
410
+ cal”m≈c0ε2(νβ))1/ksubspace of the nB= (νβ))1/k-dimensional space ℓn
411
+ 1(ℓB
412
+ 2).
413
+ Note that for fixed εandk >1,nBandmare asymptotically (substantially)
414
+ smaller than dim F. Further, in order to efficiently represent Fas a subspace of
415
+ anℓ1-space, we only need to find a good embedding of ℓβ
416
+ 2intoℓ1. This can be
417
+ done using Corollary 2 (Equation 4); note that βdepends only on εandk. Thus
418
+ we reduced the problem of finding “large” almost Euclidean subspace s ofℓN
419
+ 1to
420
+ similar problems for much smaller dimensions.
421
+ Theorem 1 now follows from the above discussion. The argument give s, e.g.,
422
+ c(ε,γ) = (cεγ)3/γandC(ε,γ) =c(ε,γ)−1.
423
+ References
424
+ [ABN+92] Noga Alon, Jehoshua Bruck, Joseph Naor, Moni Naor, and Ro nny Roth.
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+ Construction of asymptotically good low-rate error-corre cting codes through
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+ pseudo-random graphs. IEEE Transactions on Information Theory , 38:509–516,
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+ 1992.
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+ [And80] K. F. Andersen, Inequalities for Scalar-Valued Lin ear Operators That Extend
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+ to Their Vector-Valued Analogues. J. Math. Anal. Appl. 77 (1980), 264–269
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+ [AM06] S. Artstein-Avidan and V. D. Milman. Logarithmic red uction of the level of
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+ randomness in some probabilistic geometric constructions .Journal of Functional
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+ Analysis, 235:297–329, 2006.
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+ [ASW10] G. Aubrun, S. Szarek and E. Werner. Hastings’s addit ivity counterexample
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+ via Dvoretzky’s theorem. Arxiv.org eprint 1003.4925
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+ [Bal97] K. Ball. An elementary introduction to modern conve x geometry. Flavors of
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+ geometry , edited by S. Levy, Math. Sci. Res. Inst. Publ. 31, 1–58, Cambridge Univ.
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+ Press, Cambridge, 1997.[Bec75] W. Beckner. Inequalities in Fourier analysis. Annals of Math. 102 (1975),
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+ 159–182.
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+ [DS01] K. R. Davidson and S. J. Szarek. Local operator theory , random matrices
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+ and Banach spaces. In Handbook of the geometry of Banach spaces, edited by W.
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+ B. Johnson and J. Lindenstrauss (North-Holland, Amsterdam , 2001), Vol. 1, pp.
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+ 317–366; (North-Holland, Amsterdam, 2003), Vol. 2, pp. 181 9–1820.
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+ [Dvo61] A. Dvoretzky. Some Results on Convex Bodies and Bana ch Spaces. In Proc.
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+ Internat. Sympos. Linear Spaces (Jerusalem, 1960) . Jerusalem Academic Press,
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+ Jerusalem; Pergamon, Oxford,1961, pp. 123–160.
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+ [FJ80] T. Figiel and W. B. Johnson. Large subspaces of ℓn
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+ ∞and estimates of the
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+ Gordon-Lewis constant. Israel J. Math. 37 (1980), 92–112.
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+ [FLM77] T. Figiel, J. Lindenstrauss, and V. D. Milman. The di mension of almost
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+ spherical sections of convex bodies. Acta Math. 139 (1977), no. 1-2, 53–94.
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+ [GG84] A. Garnaev and E. Gluskin. The widths of a Euclidean ba ll.Soviet Math.
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+ Dokl.30 (1984), 200–204 (English translation)
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+ [Gor85] Y. Gordon. Some inequalities for Gaussian processe s and applications. Israel
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+ J. Math. 50 (1985), 265–289.
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+ [GLR08] V. Guruswami, J. Lee, and A. Razborov. Almost euclid ean subspaces of l1
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+ via expander codes. SODA, 2008.
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+ [GLW08] V. Guruswami, J. Lee, and A. Wigderson. Euclidean se ctions with sublinear
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+ randomness and error-correction over the reals. RANDOM , 2008.
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+ [Ind00] P. Indyk. Dimensionality reduction techniques for proximity problems. Pro-
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+ ceedings of the Ninth ACM-SIAM Symposium on Discrete Algori thms, 2000.
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+ [Ind07] P. Indyk. Uncertainty principles, extractors and e xplicit embedding of l2 into
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+ l1.STOC, 2007.
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+ [KRS09] Z. Karnin, Y. Rabani, and A. Shpilka. Explicit Dimen sion Reduction and Its
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+ Applications ECCC TR09-121 , 2009.
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+ [Kas77] B. S. Kashin. The widths of certain finite-dimension al sets and classes of
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+ smooth functions. Izv. Akad. Nauk SSSR Ser. Mat. , 41(2):334351, 1977.
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+ [KT07] B. S. Kashin and V. N. Temlyakov. A remark on compresse d sensing. Mathe-
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+ matical Notes 82 (2007), 748–755.
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+ [LLR94] N. Linial, E. London, and Y. Rabinovich. The geometr y of graphs and some
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+ of its algorithmic applications. FOCS, pages 577–591, 1994.
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+ [LPRTV05] A. Litvak, A. Pajor, M. Rudelson, N. Tomczak-Jaeg ermann, R. Vershynin.
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+ Euclidean embeddings in spaces of finite volume ratio via ran dom matrices. J.
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+ Reine Angew. Math. 589 (2005), 1–19
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+ [LS07] S. Lovett and S. Sodin. Almost euclidean sections of t he n-dimensional cross-
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+ polytope using O(n) random bits. ECCC Report TR07-012 , 2007.
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+ [LV06] Yu. Lyubarskii and R. Vershynin. Uncertainty princi ples and vector quantiza-
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+ tion. Arxiv.org eprint math.NA/0611343
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+ lin´ eaires. Fund. Math. 32 (1939), 113–121.
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+ ics. Towards 2000. (Tel Aviv, 1999). Geom. Funct. Anal. 2000, Special Volume,
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+ Part II, 792–815.
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+ [MS86] V. D. Milman and G. Schechtman. Asymptotic theory of fi nite-dimensional
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+ normed spaces. With an appendix by M. Gromov. Lecture Notes in Math. 1200,
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+ Springer-Verlag, Berlin 1986.[Pis89] G. Pisier. The volume of convex bodies and Banach spa ce geometry. Cambridge
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+ Tracts in Mathematics, 94. Cambridge University Press, Cambridge, 1989.
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+ [Rud60] W. Rudin. Trigonometric series with gaps. J. Math. Mech. , 9:203–227, 1960.
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+ theorem. In Geometric aspects of functional analysis (1987–88), 274–277, Lecture
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+ Notes in Math. 1376, Springer, Berlin, 1989.
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+ ropean Math. Soc. 2006. Available online from icm2006.org
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1001.0042.txt ADDED
@@ -0,0 +1,629 @@
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
1
+ arXiv:1001.0042v2 [hep-th] 16 Apr 2010Topological Gravity in Seven Dimensions
2
+ H. L¨ u†‡and Yi Pang⋆
3
+ †China Economics and Management Academy
4
+ Central University of Finance and Economics, Beijing 100081
5
+ ‡Institute for Advanced Study, Shenzhen University, Nanhai A ve 3688, Shenzhen 518060
6
+ ⋆Key Laboratory of Frontiers in Theoretical Physics
7
+ Institute of Theoretical Physics, Chinese Academy of Sciences , Beijing 100190
8
+ ABSTRACT
9
+ We obtain new topological gravity in seven dimensions by add ing two topological terms
10
+ to the Einstein-Hilbert action. For certain choice of the co upling constants, these terms
11
+ may have an origin as the R4correction to the 3-form field equation of eleven-dimension al
12
+ supergravity. We derive the full set of the equations of moti on, and obtain large classes of
13
+ solutions including static AdS black holes, squashed seven spheres and Q111spaces.1 Introduction
14
+ There has been considerable interest in topological gauge t heories [1] because of their wide
15
+ application in physics. The most studied example is the thre e-dimensional one. In addition
16
+ to the Einstein-Hilbert term, the theory has the Chern-Simo ns term, given by
17
+ S=1
18
+ µ/integraldisplay
19
+ d3xTr(dω∧ω−2
20
+ 3ω∧ω∧ω), (1)
21
+ whereωcan be either a Yang-Mills gauge potential or the connection for gravity. Topo-
22
+ logical Yang-Mills theory can provide a fundamental interp retation for anyons [2]; it can
23
+ also generate Lorentz violation dynamically [3]. Topologi cal gravity [4] becomes dynamical
24
+ with a propagating massive particle, with the mass proporti onal to the coupling constant
25
+ µ. Recently, a cosmological constant is added and the corresp onding boundary conformal
26
+ field theory (CFT) is discussed [5]. The three-dimensional m assive topological gravity is
27
+ conjectured to be unitary for certain parameter region even though the theory has higher
28
+ derivatives in time [6].
29
+ The attention on higher dimensional generalizations is con siderably less. The five di-
30
+ mensional Yang-Mills Chern-Simons term was discussed in [7 ], but there is no gravity coun-
31
+ terpart dueto the fact that the holonomy group SO(1,4) has no invariant rank-3 symmetric
32
+ tensor. In seven dimensions, Yang-Mills Chern-Simons term s arise naturally from N= 4
33
+ supergravity [8]. As in the case of three dimensions, we find t hat such terms in the grav-
34
+ ity sector can be obtained directly from those in the Yang-Mi lls sector by replacing the
35
+ gauge potential Ato the connection Γ. Moreover, as we shall see later, seven-d imensional
36
+ topological gravity has a direct origin in eleven-dimensio nal supergravity, while any higher-
37
+ dimensional origin of the three-dimensional theory remain s unknown.
38
+ In section 2, we present the two topological terms in seven di mensions, and discuss their
39
+ properties. Sincethey arenot manifestly invariant underg eneral coordinate transformation,
40
+ we find it is more convenient to lift the system to eight dimens ions in order to derive the
41
+ equations of motion (EOMs). We obtain the full set. In sectio n 3, we construct large
42
+ classes of solutions including static Anti-de Sitter (AdS) black holes, squashed S7andQ111.
43
+ We emphasize that all the previously-known static (AdS) bla ck holes remain to be solutions
44
+ whenthetopological termsare addedinto theaction. Thisis analogous to threedimensions,
45
+ where the BTZ black hole remains to be a solution in topologic al massive gravity. We
46
+ conclude in section 4.
47
+ 22 The theory
48
+ In seven dimensions, there are two topological terms; they a re given by
49
+ S1=µ/integraldisplay
50
+ Ω(7)
51
+ 1=µ/integraldisplay
52
+ Tr(Γ∧Θ+1
53
+ 3Γ3)∧Tr(Θ2) =µ/integraldisplay
54
+ Ω(3)∧dΩ(3), (2)
55
+ S2=ν/integraldisplay
56
+ Ω(7)
57
+ 2=ν/integraldisplay
58
+ Tr(Θ3∧Γ+2
59
+ 5Θ2∧Γ3+1
60
+ 5Θ∧Γ2∧Θ∧Γ+1
61
+ 5Θ∧Γ5+1
62
+ 35Γ7),
63
+ with Ω(3)= Tr(dΓ∧Γ−2
64
+ 3Γ3). Here, Θ is the curvature 2-form, defined as Θ ≡dΓ−Γ∧Γ,
65
+ andµ,νare two parameters of length dimension 5. (We rescale the tot al action by the
66
+ seven-dimensional Newton constant.) The 3-form Ω(3)has the same structure as the Chern-
67
+ Simons term in D= 3, except that now Γ depends on seven coordinates. Ω(7)
68
+ 1and Ω(7)
69
+ 2are
70
+ topological in the same sense as Ω(3)being topological in D= 3. We can lift the system to
71
+ D= 8, with the seven-dimensional spacetime as the boundary. T hen, we have
72
+ dΩ(7)
73
+ 1=Y(8)
74
+ 1≡Tr(Θ∧Θ)∧Tr(Θ∧Θ), dΩ(7)
75
+ 2=Y(8)
76
+ 2≡Tr(Θ∧Θ∧Θ∧Θ).(3)
77
+ As we have mentioned earlier, these terms can be derived from the Yang-Mills Chern-
78
+ Simons terms in [8] by changing the gauge potential to the con nection.1Note that the
79
+ Pontryagin term is proportional to Y(8)
80
+ 1−2Y(8)
81
+ 2, corresponding to ν=−2µ. In eleven-
82
+ dimensional supergravity, there is an R4correction to the field equation, namely d∗F(4)=
83
+ 1
84
+ 2F(4)∧F(4)+X(8), whereX(8)is given by
85
+ X(8)∝Y(8)
86
+ 1−4Y(8)
87
+ 2. (4)
88
+ Thus forν=−4µ, the topological terms can be obtained from the S4reduction of super-
89
+ gravity inD= 11, and the coupling constant is proportional to the 4-form M5-brane fluxes.
90
+ For large fluxes, this topological term dominates the higher -order corrections.
91
+ To derive the contribution to the EOMs from the Chern-Simons terms, it is necessary
92
+ to perform their variation with respect to the metric. These topological terms are not
93
+ manifestly invariant under the general coordinate transfo rmation, but Y(8)
94
+ 1andY(8)
95
+ 2are.
96
+ Wefindthataconvenient waytoderivethevariationistolift thesystemtoeightdimensions.
97
+ Let us first consider the variation of S1. In terms of coordinate components, we have
98
+ /integraldisplay
99
+ dΩ(7)
100
+ 1=1
101
+ 16/integraldisplay
102
+ d8xǫν1ν2ν3ν4ν5ν6ν7ν8Rµ1µ2ν1ν2Rµ2µ1ν3ν4Rµ3µ4ν5ν6Rµ4µ3ν7ν8. (5)
103
+ 1In [8], the field strength 2-form is defined by F=dB+gB∧B, with gauge coupling g= 2. Then by
104
+ rescaling the field B→B/gandF→F/gand setting g=−2, one can obtain the same expressions as the
105
+ ones given here.
106
+ 3Here we use Greek letters to denote the eight-dimensional co ordinates and Latin letters to
107
+ represent the seven-dimensional ones hereafter. We adopt t he convention ǫ12345678= 1.
108
+ For an infinitesimal variation of the metric δg, using the Bianchi identity and the fol-
109
+ lowing relation
110
+ δRµ
111
+ ναβ=δΓµ
112
+ νβ;α−δΓµ
113
+ να;β, (6)
114
+ we find that
115
+ /integraldisplay
116
+ dδΩ(7)
117
+ 1=−1
118
+ 2/integraldisplay
119
+ d8x√g/parenleftBig1√gǫν1ν2ν3ν4ν5ν6ν7ν8Rµ1µ2ν1ν2Rµ2µ1ν3ν4Rµ3µ4ν5ν6δΓµ4µ3ν7/parenrightBig
120
+ ;ν8
121
+ ≡1
122
+ 2/integraldisplay
123
+ d∗J, (7)
124
+ where “;” denotes a covariant derivative and ∗is the Hodge dual. For simplicity, we have
125
+ introduced a 1-form current J=Jαdxα. Its components are given by
126
+ Jα=1√gǫν1ν2ν3ν4ν5ν6ν7αRµ1µ2ν1ν2Rµ2µ1ν3ν4Rµ3µ4ν5ν6δΓµ4µ3ν7. (8)
127
+ Clearly, we have d∗J=−√gJα;αd8x, Thus we obtain
128
+ δΩ(7)
129
+ 1=1
130
+ 2∗J, (9)
131
+ up to a total derivative term. Now restricting the coordinat e indices to seven dimensions
132
+ only, we have
133
+ δS1= 4µ/integraldisplay
134
+ Tr(Θ∧Θ)∧Tr(Θ∧δΓ). (10)
135
+ The variation of S2can be obtained in the same manner, given by
136
+ δS2= 4ν/integraldisplay
137
+ Tr(Θ∧Θ∧Θ∧δΓ). (11)
138
+ Finally, we make use of the variation of the connection
139
+ δΓi
140
+ mj=1
141
+ 2gin(δgnm;j+δgnj;m−δgml;n), (12)
142
+ and after integrating by parts, we obtain thecontributions to EOMs from the Chern-Simons
143
+ terms, given by
144
+ Cij
145
+ 1=δS1√gδgij=µ
146
+ 4√g[ǫij1j2j3j4j5j6(Ri1
147
+ i2j1j2Ri2
148
+ i1j3j4Rjk
149
+ j5j6);k+i↔j],
150
+ Cij
151
+ 2=δS2√gδgij=ν
152
+ 4√g[ǫij1j2j3j4j5j6(Rk
153
+ i1j1j2Ri1
154
+ i2j3j4Rji2
155
+ j5j6);k+i↔j].(13)
156
+ For the total action S, which is the sum of the Einstein-Hilbert action, cosmologi cal
157
+ constant Λ and S1+S2, the corresponding full set of EOMs is given by
158
+ Rij−1
159
+ 2gijR+Λgij+Cij
160
+ 1+Cij
161
+ 2= 0. (14)
162
+ 4It should be remarked that under a large gauge transformatio n Γ→ O−1ΓO−O−1dO,
163
+ the action transforms as S→S+µv(O)+νw(O), where
164
+ v(O) =/integraldisplay
165
+ −1
166
+ 3d/parenleftBig
167
+ Tr(O−1dO)3∧Ω(3)/parenrightBig
168
+ ;w(O) =−1
169
+ 35/integraldisplay
170
+ Tr(O−1dO)7.(15)
171
+ Thevterm is trivial and gives no restriction to the parameter µ, while the wterm should
172
+ be classified by the seventh homotopy group of SO(1,6)
173
+ π7[SO(1,6)]≃π7[SO(6)]≃Z. (16)
174
+ The invariance of eiSrequires that
175
+ ν= 2πn, n = 0,±1,±2.... (17)
176
+ This quantization condition is clearly consistent with the M5-brane quantization, since
177
+ it has a direct origin in D= 11. This result is completely different from that in three
178
+ dimensions, where the SO(1,2) is homotoplically trivial and the mass parameter is not
179
+ quantized. Moreover, since νis quantized, S2will not be renormalized in the quantum
180
+ theory. This suggests some intriguing properties in the cor responding CFT dual.
181
+ 3 Solutions
182
+ Spherically-symmetric solutions:
183
+ Having obtained the full set of EOMs for topological gravity in seven dimensions, we are
184
+ in the position to construct solutions. It is clear that the m aximally-symmetric space(time)
185
+ is unmodifiedby theinclusion of the topological terms. Then extsimplest case is to consider
186
+ the spherically-symmetric ansatz, given by
187
+ ds2=−F(r)dt2+dr2
188
+ G(r)+r2dΩ2
189
+ 5. (18)
190
+ We find that for this ansatz, the contributions from the topol ogical terms Cij
191
+ 1andCij
192
+ 2
193
+ vanish identically. This implies that the previously-know n static (AdS) black holes, charged
194
+ or neutral, are still solutions when the topological terms a re added to the action. This is
195
+ analogous to three dimensions, where the BTZ black hole is st ill a solution in massive
196
+ topological gravity. However the thermodynamic quantitie s such as the mass and entropy
197
+ will acquire modifications [9, 10].
198
+ S3bundle over S4:
199
+ 5We now turn our attention to the Euclidean theory. In three di mensions, there exists a
200
+ large class of squashed S3or AdS 3[11]. We expect the same in seven dimensions. Without
201
+ loss of generality, we set Λ = 30 so that it can give rise to a uni t roundS7. We first consider
202
+ the squashed S7that can be viewed as an S3bundle over S4. The metric ansatz is given
203
+ by
204
+ ds2=α3/summationdisplay
205
+ i=1(σi−cos2(1
206
+ 2θ)˜σi)2+β/parenleftBig
207
+ dθ2+1
208
+ 4sin2θ3/summationdisplay
209
+ i=1˜σ2
210
+ i/parenrightBig
211
+ . (19)
212
+ whereσiand ˜σiare theSU(2) left-invariant 1-forms, satisfying dσi=1
213
+ 2ǫijkσj∧σkand
214
+ d˜σi=1
215
+ 2ǫijk˜σj∧˜σk. The metric is Einstein provided that either α=β=1
216
+ 4orα=1
217
+ 5β=9
218
+ 100.
219
+ The first case corresponds to the round S7and the second is a squashed S7that is also
220
+ Einstein. Now with the contribution from the topological te rms, the EOMs can be reduced
221
+ to
222
+ 2α2+4αβ(7β−2)−β2= 0, (20)
223
+ together with
224
+ √α(α−β)3(4(10α+β)µ−(55α+7β)ν)+2β6(20αβ−4α−β) = 0.(21)
225
+ It is clear from (20) that there exists one and only one positi veαfor any positive β. The
226
+ squashing parameter γ≡α/βlies in the range 0 <γ <2+3√
227
+ 2. Note that when 2 µ= 3ν,
228
+ the squashed S7that is Eisntein remains Einstein.
229
+ S1bundle over CP3:
230
+ There is another way of squashing an S7, which can be viewed as an S1bundle over
231
+ CP3. The metric ansatz is given by
232
+ ds2=α(dτ+sin2θ(dψ+B))2+βds2
233
+ CP3,
234
+ ds2
235
+ CP3=dθ2+sin2θcos2θ(dψ+B)2+sin2θ/parenleftBig
236
+ d˜θ2+1
237
+ 4sin2˜θcos2˜θσ2
238
+ 3
239
+ +1
240
+ 4sin2˜θ(σ2
241
+ 1+σ2
242
+ 2)/parenrightBig
243
+ ,
244
+ B=1
245
+ 2sin2˜θσ3. (22)
246
+ It is of a round S7whenα=β= 1. In general, the EOMs imply that
247
+ α=β(8−7β),8µ+ν+β3
248
+ 10976(β−1)2√α= 0. (23)
249
+ The squashing parameter γ≡α/βlies in the range (0 ,8).
250
+ Squashed Q111spaces:
251
+ 6TheQ111space is an Einstein-Sasaki space of U(1) bundle over S2×S2×S2. We
252
+ consider the following ansatz
253
+ ds2=α/parenleftBig
254
+ dψ+3/summationdisplay
255
+ i=1cosθidφi/parenrightBig2
256
+ +β3/summationdisplay
257
+ i=1(dθ2
258
+ i+sinθ2
259
+ idφ2
260
+ i). (24)
261
+ It is ofQ111provided that α=1
262
+ 2β= 1/16, and it remains so for ν= 0. In general, we have
263
+ α= 4β(1−7β),8(α−β)(2α−β)µ+α(2α−3β)ν+β5(α−8β+60β2)
264
+ 4α3/2= 0.(25)
265
+ Thus the squashing parameter γ≡α/βlies in the range (0 ,4). We expect that many of
266
+ the squashed homogeneous spaces in seven dimensions are now solutions in this new gravity
267
+ theory, and we shall not enumerate them further.
268
+ 4 Conclusions
269
+ This work is motivated by studying the classical solutions o f Einstein-Chern-Simons gravity
270
+ with asymptotic AdS structure. In seven dimensions, there a re two topological Chern-
271
+ Simons terms, and we obtain the full set of equations of motio n. We find that spherically-
272
+ symmetric solutions are unmodified by the inclusion of these topological terms. We also
273
+ obtain squashed S7andQ111spaces, where the squashing parameter is related to the cou-
274
+ pling constants of the topological terms. It is intriguing t o see that these known squashed
275
+ homogeneous spaces which appear to have no connection can no w be unified under our new
276
+ gravity theory.
277
+ As in three dimensions, our topological gravity should play an important role in explor-
278
+ ing the AdS 7/CFT6correspondence. The CFT 6that describes the world-volume theory of
279
+ multiple M5-branes is yet to be known, and our solutions prov ide many new gravity dual
280
+ backgrounds. The quantization condition for one of the coup ling constant suggests an un-
281
+ usual property of the CFT 6that is absent in lower dimensions. Additional future direc tions
282
+ include a classification of all topological gravities in (4 k+3) dimensions, investigating the
283
+ linearization of D= 7topological gravity and obtaining the propagating degre es of freedom.
284
+ Acknowledgement
285
+ We are grateful to Chris Pope for useful discussions. Y.P. is supported in part by the NSFC
286
+ grant No.1053060/A050207 and the NSFC group grant No.10821 504.
287
+ 7References
288
+ [1] S. Deser, R. Jackiw and S. Templeton, “Topologically mas sive gauge theories,” An-
289
+ nals Phys. 140, 372 (1982) [Erratum-ibid. 185, 406.1988 APNYA,281,409 (1988
290
+ APNYA,281,409-449.2000)].
291
+ [2] F. Wilczek and A. Zee, “Linking numbers, spin, and statis tics of solitons,” Phys. Rev.
292
+ Lett.51, 2250 (1983).
293
+ [3] S.M. Carroll, G.B. Field and R. Jackiw, “Limits on a Loren tz and parity violating
294
+ modification of electrodynamics,” Phys. Rev. D 41, 1231 (1990).
295
+ [4] S. Deser, R. Jackiw and G. ’t Hooft, “Three-Dimensional E instein gravity: dynamics
296
+ of flat space,” Annals Phys. 152, 220 (1984).
297
+ [5] E. Witten, “Three-dimensional gravity revisited,” arX iv:0706.3359 [hep-th].
298
+ [6] W. Li, W. Song and A. Strominger, “Chiral gravity in three dimensions,” JHEP 0804,
299
+ 082 (2008) [arXiv:0801.4566 [hep-th]]. E.A. Bergshoeff, O. H ohm and P.K. Townsend,
300
+ “Massive gravity in three dimensions,” Phys. Rev. Lett. 102, 201301 (2009)
301
+ [arXiv:0901.1766 [hep-th]].
302
+ [7] M. Gunaydin, G. Sierra and P.K. Townsend, “Quantization of the gauge coupling
303
+ constant in a five-dimensional Yang-Mills/Einstein superg ravity theory,” Phys. Rev.
304
+ Lett.53, 322 (1984).
305
+ [8] M. Pernici, K. Pilch and P. van Nieuwenhuizen, “Gauged ma ximally extended super-
306
+ gravity in seven-dimensions,” Phys. Lett. B 143, 103 (1984).
307
+ [9] S. Deser and B. Tekin, “Energy in topologically massive g ravity,” Class. Quant. Grav.
308
+ 20, L259 (2003) [arXiv:gr-qc/0307073].
309
+ [10] Y. Tachikawa, “Black hole entropy in the presence of Che rn-Simons terms,” Class.
310
+ Quant. Grav. 24, 737 (2007) [arXiv:hep-th/0611141].
311
+ [11] D.D.K. Chow, C.N. Pope and E. Sezgin, “Exact solutions o f topologically massive
312
+ gravity,” arXiv:0906.3559 [hep-th].
313
+ 8arXiv:1001.0042v2 [hep-th] 16 Apr 2010Seven-Dimensional Gravity with Topological Terms
314
+ H. L¨ u†‡and Yi Pang⋆
315
+ †China Economics and Management Academy
316
+ Central University of Finance and Economics, Beijing 100081
317
+ ‡Institute for Advanced Study, Shenzhen University, Nanhai A ve 3688, Shenzhen 518060
318
+ ⋆Key Laboratory of Frontiers in Theoretical Physics
319
+ Institute of Theoretical Physics, Chinese Academy of Sciences , Beijing 100190
320
+ ABSTRACT
321
+ We construct new seven-dimensional gravity by adding two to pological terms to the
322
+ Einstein-Hilbert action. For certain choice of the couplin g constants, these terms may be
323
+ related to the R4correction to the 3-form field equation of eleven-dimension al supergravity.
324
+ We derive the full set of the equations of motion. We find that t he static spherically-
325
+ symmetric black holes are unmodified by the topological term s. We obtain squashed AdS 7,
326
+ and also squashed seven spheres and Q111spaces in Euclidean signature.1 Introduction
327
+ There has been considerable interest in topological gauge t heories [1] because of their wide
328
+ application in physics. The most studied example is the thre e-dimensional one. In addition
329
+ to the Einstein-Hilbert term, the theory has the Chern-Simo ns term, given by
330
+ S=1
331
+ µ/integraldisplay
332
+ d3xTr(dω∧ω+2
333
+ 3ω∧ω∧ω), (1)
334
+ whereωcan be either a Yang-Mills gauge potential or the connection for gravity. Topo-
335
+ logical Yang-Mills theory can provide a fundamental interp retation for anyons [2]; it can
336
+ also generate Lorentz violation dynamically [3]. Topologi cally massive gravity [4] becomes
337
+ dynamical with a propagating massive particle, with the mas s proportional to the coupling
338
+ constantµ. Recently, a cosmological constant is added and the corresp onding boundary
339
+ conformal field theory (CFT) is discussed [5]. The three-dim ensional massive topological
340
+ gravity is conjectured to be unitary for certain parameter r egion even though the theory
341
+ has higher derivatives in time [6].
342
+ The attention on higher dimensional generalizations is con siderably less. The five di-
343
+ mensional Yang-Mills Chern-Simons term was discussed in [7 ], but there is no gravity coun-
344
+ terpart dueto the fact that the holonomy group SO(1,4) has no invariant rank-3 symmetric
345
+ tensor. In seven dimensions, Yang-Mills Chern-Simons term s arise naturally from N= 4
346
+ supergravity [8]. As in the case of three dimensions, we find t hat such terms in the grav-
347
+ ity sector can be obtained directly from those in the Yang-Mi lls sector by replacing the
348
+ gauge potential Ato the connection Γ. As we shall see later, these topological terms in
349
+ seven dimensions may be related to the anomaly cancelation t erms in eleven-dimensional
350
+ supergravity.
351
+ In section 2, we present the two topological terms in seven di mensions, and discuss their
352
+ properties. Sincethey arenot manifestly invariant underg eneral coordinate transformation,
353
+ we find it is more convenient to lift the system to eight dimens ions in order to derive the
354
+ equations of motion (EOMs). We obtain the full set. In sectio n 3, we construct large classes
355
+ of solutions. We find that the static spherically-symmetric black holes are unmodified by
356
+ the topological terms. This is analogous to three dimension s, where the BTZ black hole
357
+ remains to be a solution in topologically massive gravity. I n Euclidean signature, we obtain
358
+ squashedS7andQ111spaces. In particular, one of the squashed seven sphere can b e Wick
359
+ rotated to become squashed AdS 7. We conclude in section 4.
360
+ 22 The theory
361
+ In seven dimensions, there are two topological terms; they a re given by
362
+ S1= ˜µ/integraldisplay
363
+ Ω(7)
364
+ 1= ˜µ/integraldisplay
365
+ Tr(Γ∧Θ−1
366
+ 3Γ3)∧Tr(Θ2) = ˜µ/integraldisplay
367
+ Ω(3)∧dΩ(3), (2)
368
+ S2= ˜ν/integraldisplay
369
+ Ω(7)
370
+ 2= ˜ν/integraldisplay
371
+ Tr(Θ3∧Γ−2
372
+ 5Θ2∧Γ3−1
373
+ 5Θ∧Γ2∧Θ∧Γ+1
374
+ 5Θ∧Γ5−1
375
+ 35Γ7),
376
+ with Ω(3)= Tr(dΓ∧Γ+2
377
+ 3Γ3). Here, Θ is the curvature 2-form, defined as Θ ≡dΓ+Γ∧Γ,
378
+ and ˜µ,˜νare two parameters of length dimension 5. (We rescale the tot al action by the
379
+ seven-dimensional Newton constant.) The 3-form Ω(3)has the same structure as the Chern-
380
+ Simons term in D= 3, except that now Γ depends on seven coordinates. Ω(7)
381
+ 1and Ω(7)
382
+ 2are
383
+ topological in the same sense as Ω(3)being topological in D= 3. We can lift the system to
384
+ D= 8, with the seven-dimensional spacetime as the boundary. T hen, we have
385
+ dΩ(7)
386
+ 1=Y(8)
387
+ 1≡Tr(Θ∧Θ)∧Tr(Θ∧Θ), dΩ(7)
388
+ 2=Y(8)
389
+ 2≡Tr(Θ∧Θ∧Θ∧Θ).(3)
390
+ As we have mentioned earlier, these terms can be derived from the Yang-Mills Chern-
391
+ Simons terms in [8] by changing the gauge potential to the con nection.1Note that the
392
+ Pontryagin term is proportional to Y(8)
393
+ 1−2Y(8)
394
+ 2, corresponding to ˜ ν=−2˜µ. In eleven-
395
+ dimensional supergravity, there is an R4correction to the field equation, namely d∗F(4)=
396
+ 1
397
+ 2F(4)∧F(4)+X(8), whereX(8)is given by
398
+ X(8)∝Y(8)
399
+ 1−4Y(8)
400
+ 2. (4)
401
+ Thus for ˜ν=−4˜µ, the topological terms can be obtained from the S4reduction of super-
402
+ gravity inD= 11, and the coupling constant is proportional to the 4-form M5-brane fluxes.
403
+ For large fluxes, this topological term dominates the higher -order corrections.
404
+ To derive the contribution to the EOMs from the Chern-Simons terms, it is necessary
405
+ to perform their variation with respect to the metric. These topological terms are not
406
+ manifestly invariant under the general coordinate transfo rmation, but Y(8)
407
+ 1andY(8)
408
+ 2are.
409
+ Wefindthataconvenient waytoderivethevariationistolift thesystemtoeightdimensions.
410
+ Let us first consider the variation of S1. In terms of coordinate components, we have
411
+ /integraldisplay
412
+ dΩ(7)
413
+ 1=1
414
+ 16/integraldisplay
415
+ d8xǫν1ν2ν3ν4ν5ν6ν7ν8Rµ1µ2ν1ν2Rµ2µ1ν3ν4Rµ3µ4ν5ν6Rµ4µ3ν7ν8. (5)
416
+ 1In [8], the field strength 2-form is defined by F=dB+gB∧B, with gauge coupling g= 2. Then by
417
+ rescaling the field B→B/gandF→F/gand setting g= 2, one can obtain the same expressions as the
418
+ ones given here.
419
+ 3Here we use Greek letters to denote the eight-dimensional co ordinates and Latin letters to
420
+ represent the seven-dimensional ones hereafter. We adopt t he convention ǫ12345678= 1.
421
+ For an infinitesimal variation of the metric δg, using the Bianchi identity and the fol-
422
+ lowing relation
423
+ δRµ
424
+ ναβ=δΓµ
425
+ νβ;α−δΓµ
426
+ να;β, (6)
427
+ we find that
428
+ /integraldisplay
429
+ dδΩ(7)
430
+ 1=−1
431
+ 2/integraldisplay
432
+ d8x√g/parenleftBig1√gǫν1ν2ν3ν4ν5ν6ν7ν8Rµ1µ2ν1ν2Rµ2µ1ν3ν4Rµ3µ4ν5ν6δΓµ4µ3ν7/parenrightBig
433
+ ;ν8
434
+ ≡1
435
+ 2/integraldisplay
436
+ d∗J, (7)
437
+ where “;” denotes a covariant derivative and ∗is the Hodge dual. For simplicity, we have
438
+ introduced a 1-form current J=Jαdxα. Its components are given by
439
+ Jα=1√gǫν1ν2ν3ν4ν5ν6ν7αRµ1µ2ν1ν2Rµ2µ1ν3ν4Rµ3µ4ν5ν6δΓµ4µ3ν7. (8)
440
+ Clearly, we have d∗J=−√gJα;αd8x, Thus we obtain
441
+ δΩ(7)
442
+ 1=1
443
+ 2∗J, (9)
444
+ up to a total derivative term. Now restricting the coordinat e indices to seven dimensions
445
+ only, we have
446
+ δS1= 4˜µ/integraldisplay
447
+ Tr(Θ∧Θ)∧Tr(Θ∧δΓ). (10)
448
+ The variation of S2can be obtained in the same manner, given by
449
+ δS2= 4˜ν/integraldisplay
450
+ Tr(Θ∧Θ∧Θ∧δΓ). (11)
451
+ Finally, we make use of the variation of the connection
452
+ δΓi
453
+ mj=1
454
+ 2gin(δgnm;j+δgnj;m−δgml;n), (12)
455
+ and after integrating by parts, we obtain thecontributions to EOMs from the Chern-Simons
456
+ terms, given by
457
+ Cij
458
+ 1=δS1√gδgij=µ
459
+ 4√g[ǫij1j2j3j4j5j6(Ri1
460
+ i2j1j2Ri2
461
+ i1j3j4Rjk
462
+ j5j6);k+i↔j],
463
+ Cij
464
+ 2=δS2√gδgij=ν
465
+ 4√g[ǫij1j2j3j4j5j6(Rk
466
+ i1j1j2Ri1
467
+ i2j3j4Rji2
468
+ j5j6);k+i↔j].(13)
469
+ For the total action S, which is the sum of the Einstein-Hilbert action, cosmologi cal
470
+ constant Λ and S1+S2, the corresponding full set of EOMs is given by
471
+ Rij−1
472
+ 2gijR+Λgij+Cij
473
+ 1+Cij
474
+ 2= 0. (14)
475
+ 4It should be remarked that under a large gauge transformatio n Γ→ OΓO−1−dOO−1,
476
+ the action transforms as S→S+ ˜µv(O)+ ˜νw(O), where
477
+ v(O) =/integraldisplay
478
+ 1
479
+ 3d/parenleftBig
480
+ Tr(dOO−1)3∧Ω(3)/parenrightBig
481
+ ;w(O) =1
482
+ 35/integraldisplay
483
+ Tr(dOO−1)7.(15)
484
+ Thevterm is trivial and gives no restriction to the parameter ˜ µ, while the wterm should
485
+ be classified by the seventh homotopy group of SO(1,6)
486
+ π7[SO(1,6)]≃π7[SO(6)]≃Z. (16)
487
+ The invariance of eiSrequires that
488
+ 64π4˜ν= 2πn, n = 0,±1,±2.... (17)
489
+ This result is completely different from that in three dimensi ons, where the SO(1,2) is
490
+ homotoplically trivial and the mass parameter is not quanti zed. Moreover, since ˜ νis quan-
491
+ tized,S2will not be renormalized in the quantum theory. This suggest s some intriguing
492
+ properties in the corresponding CFT dual.
493
+ 3 Solutions
494
+ Spherically-symmetric solutions:
495
+ Having obtained the full set of EOMs for topological gravity in seven dimensions, we are
496
+ in the position to construct solutions. It is clear that the m aximally-symmetric space(time)
497
+ is unmodifiedby theinclusion of the topological terms. Then extsimplest case is to consider
498
+ the spherically-symmetric ansatz, given by
499
+ ds2=−F(r)dt2+dr2
500
+ G(r)+r2dΩ2
501
+ 5. (18)
502
+ We find that for this ansatz, the contributions from the topol ogical terms Cij
503
+ 1andCij
504
+ 2
505
+ vanish identically. This implies that the previously-know n static (AdS) black holes, charged
506
+ or neutral, are still solutions when the topological terms a re added to the action. This is
507
+ analogous to three dimensions, where the BTZ black hole is st ill a solution in massive
508
+ topological gravity. However the thermodynamic quantitie s such as the mass and entropy
509
+ will acquire modifications [9, 10].
510
+ As we shall discuss presently, there also exist squashed AdS 7solutions.
511
+ S3bundle over S4:
512
+ 5We now turn our attention to the Euclidean theory. In three di mensions, there exists a
513
+ large class of squashed S3or AdS 3[11]. We expect the same in seven dimensions. Without
514
+ loss of generality, we set Λ = 30 so that it can give rise to a uni t roundS7. We first consider
515
+ the squashed S7that can be viewed as an S3bundle over S4. The metric ansatz is given
516
+ by
517
+ ds2=α3/summationdisplay
518
+ i=1(σi−cos2(1
519
+ 2θ)˜σi)2+β/parenleftBig
520
+ dθ2+1
521
+ 4sin2θ3/summationdisplay
522
+ i=1˜σ2
523
+ i/parenrightBig
524
+ . (19)
525
+ whereσiand ˜σiare theSU(2) left-invariant 1-forms, satisfying dσi=1
526
+ 2ǫijkσj∧σkand
527
+ d˜σi=1
528
+ 2ǫijk˜σj∧˜σk. The metric is Einstein provided that either α=β=1
529
+ 4orα=1
530
+ 5β=9
531
+ 100.
532
+ The first case corresponds to the round S7and the second is a squashed S7that is also
533
+ Einstein. Now with the contribution from the topological te rms, the EOMs can be reduced
534
+ to
535
+ 2α2+4αβ(7β−2)−β2= 0, (20)
536
+ together with
537
+ √α(α−β)3(4(10α+β)˜µ−(55α+7β)ν)+2β6(20αβ−4α−β) = 0.(21)
538
+ It is clear from (20) that there exists one and only one positi veαfor any positive β. The
539
+ squashing parameter γ≡α/βlies in the range 0 <γ <2+3√
540
+ 2. Note that when 2˜ µ= 3˜ν,
541
+ the squashed S7that is Eisntein remains Einstein.
542
+ S1bundle over CP3:
543
+ There is another way of squashing an S7, which can be viewed as an S1bundle over
544
+ CP3. This example can be generalized to Minkowskian signature t o give rise to squashed
545
+ AdS7[12]. The metric ansatz is given by
546
+ ds2=α(dτ+sin2θ(dψ+B))2+βds2
547
+ CP3,
548
+ ds2
549
+ CP3=dθ2+sin2θcos2θ(dψ+B)2+sin2θ/parenleftBig
550
+ d˜θ2+1
551
+ 4sin2˜θcos2˜θσ2
552
+ 3
553
+ +1
554
+ 4sin2˜θ(σ2
555
+ 1+σ2
556
+ 2)/parenrightBig
557
+ ,
558
+ B=1
559
+ 2sin2˜θσ3. (22)
560
+ It is of a round S7whenα=β= 1. In general, the EOMs imply that
561
+ α=β(8−7β),8˜µ+ ˜ν+β3
562
+ 10976(β−1)2√α= 0. (23)
563
+ The squashing parameter γ≡α/βlies in the range (0 ,8).
564
+ Squashed Q111spaces:
565
+ 6TheQ111space is an Einstein-Sasaki space of U(1) bundle over S2×S2×S2. We
566
+ consider the following ansatz
567
+ ds2=α/parenleftBig
568
+ dψ+3/summationdisplay
569
+ i=1cosθidφi/parenrightBig2
570
+ +β3/summationdisplay
571
+ i=1(dθ2
572
+ i+sinθ2
573
+ idφ2
574
+ i). (24)
575
+ It is ofQ111provided that α=1
576
+ 2β= 1/16, and it remains so for ˜ ν= 0. In general, we have
577
+ α= 4β(1−7β),8(α−β)(2α−β)˜µ+α(2α−3β)˜ν+β5(α−8β+60β2)
578
+ 4α3/2= 0.(25)
579
+ Thus the squashing parameter γ≡α/βlies in the range (0 ,4). We expect that many of
580
+ the squashed homogeneous spaces in seven dimensions are now solutions in this new gravity
581
+ theory, and we shall not enumerate them further.
582
+ 4 Conclusions
583
+ This work is motivated by studying the classical solutions o f Einstein-Chern-Simons gravity
584
+ with asymptotic AdS structure. In seven dimensions, there a re two topological Chern-
585
+ Simons terms, and we obtain the full set of equations of motio n. We find that spherically-
586
+ symmetric solutions are unmodified by the inclusion of these topological terms. We also
587
+ obtain squashed AdS 7, and squashed S7andQ111spaces in Euclidean signature, where
588
+ the squashing parameter is related to the coupling constant s of the topological terms. It is
589
+ intriguing to see that these known squashed homogeneous spa ces which appear to have no
590
+ connection can now be unified under our new gravity theory.
591
+ As in three dimensions, our topological gravity should play an important role in explor-
592
+ ing the AdS 7/CFT6correspondence. The CFT 6that describes the world-volume theory of
593
+ multiple M5-branes is yet to be known, and our solutions prov ide many new gravity dual
594
+ backgrounds. The quantization condition for one of the coup ling constant suggests an un-
595
+ usual property of the CFT 6that is absent in lower dimensions. Additional future direc tions
596
+ include a classification of all topological gravities in (4 k+3) dimensions, investigating the
597
+ linearization of D= 7topological gravity and obtaining the propagating degre es of freedom.
598
+ Acknowledgement
599
+ We are grateful to Chris Pope for useful discussions. Y.P. is supported in part by the NSFC
600
+ grant No.1053060/A050207 and the NSFC group grant No.10821 504.
601
+ 7References
602
+ [1] S. Deser, R. Jackiw and S. Templeton, “Topologically mas sive gauge theories,” An-
603
+ nals Phys. 140, 372 (1982) [Erratum-ibid. 185, 406.1988 APNYA,281,409 (1988
604
+ APNYA,281,409-449.2000)].
605
+ [2] F. Wilczek and A. Zee, “Linking numbers, spin, and statis tics of solitons,” Phys. Rev.
606
+ Lett.51, 2250 (1983).
607
+ [3] S.M. Carroll, G.B. Field and R. Jackiw, “Limits on a Loren tz and parity violating
608
+ modification of electrodynamics,” Phys. Rev. D 41, 1231 (1990).
609
+ [4] S. Deser, R. Jackiw and G. ’t Hooft, “Three-Dimensional E instein gravity: dynamics
610
+ of flat space,” Annals Phys. 152, 220 (1984).
611
+ [5] E. Witten, “Three-dimensional gravity revisited,” arX iv:0706.3359 [hep-th].
612
+ [6] W. Li, W. Song and A. Strominger, “Chiral gravity in three dimensions,” JHEP 0804,
613
+ 082 (2008) [arXiv:0801.4566 [hep-th]]. E.A. Bergshoeff, O. H ohm and P.K. Townsend,
614
+ “Massive gravity in three dimensions,” Phys. Rev. Lett. 102, 201301 (2009) arXiv:
615
+ 0901.1766 [hep-th].
616
+ [7] M. Gunaydin, G. Sierra and P.K. Townsend, “Quantization of the gauge coupling
617
+ constant in a five-dimensional Yang-Mills/Einstein superg ravity theory,” Phys. Rev.
618
+ Lett.53, 322 (1984).
619
+ [8] M. Pernici, K. Pilch and P. van Nieuwenhuizen, “Gauged ma ximally extended super-
620
+ gravity in seven-dimensions,” Phys. Lett. B 143, 103 (1984).
621
+ [9] S. Deser and B. Tekin, “Energy in topologically massive g ravity,” Class. Quant. Grav.
622
+ 20, L259 (2003), gr-qc/0307073.
623
+ [10] Y. Tachikawa, “Black hole entropy in the presence of Che rn-Simons terms,” Class.
624
+ Quant. Grav. 24, 737 (2007), hep-th/0611141.
625
+ [11] D.D.K. Chow, C.N. Pope and E. Sezgin, “Exact solutions o f topologically massive
626
+ gravity,” arXiv:0906.3559 [hep-th].
627
+ [12] P. Hoxha, R.R. Martinez-Acosta and C.N. Pope, “Kaluza- Klein consistency, Killing
628
+ vectors, and Kaehler spaces,” Class. Quant. Grav. 17, 4207 (2000), hep-th/0005172.
629
+ 8
1001.0043.txt ADDED
@@ -0,0 +1,399 @@
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
1
+ arXiv:1001.0043v2 [astro-ph.EP] 13 Jan 2010Strong Constraints to the Putative Planet Candidate around VB 10 using
2
+ Doppler spectroscopy1
3
+ Guillem Anglada-Escud´ e
4
+ Department of Terrestrial Magnetism, Carnegie Institution o f Washington
5
+ 5241 Broad Branch Road, NW, Washington, DC 20015 USA
6
7
+ Evgenya Shkolnik
8
+ Department of Terrestrial Magnetism, Carnegie Institution o f Washington
9
+ 5241 Broad Branch Road, NW, Washington, DC 20015 USA
10
11
+ Alycia J. Weinberger
12
+ Department of Terrestrial Magnetism, Carnegie Institution o f Washington
13
+ 5241 Broad Branch Road, NW, Washington, DC 20015 USA
14
15
+ Ian B. Thompson
16
+ The Observatories of the Carnegie Institution of Washington
17
+ 813 Santa Barbara Street, Pasadena, CA 91101 USA
18
19
+ David J. Osip
20
+ Las Campanas Observatory, Carnegie Institution of Washington
21
+ Colina El Pino Casilla 601, La Serena, Chile
22
23
+ John H. Debes
24
+ Goddard Space Flight Center, NASA Postdoctoral Program
25
+ 8463 Greenbelt Rd, Greenbelt, MD 20770, USA
26
27
+ ABSTRACT– 2 –
28
+ We present new radial velocity measurements of the ultra-co ol dwarf VB 10,
29
+ which was recently announced to host a giant planet detected with astrometry. The
30
+ new observations were obtained using optical spectrograph s(MIKE/Magellan and ES-
31
+ PaDOnS/CHFT) and cover a 63% of the reported period of 270 day s. We apply Least-
32
+ squares periodograms to identify the most significant signa ls and evaluate their corre-
33
+ spondingFalse Alarm Probabilities. We show that this metho d is the proper generaliza-
34
+ tion to astrometric data because (1) it mitigates the coupli ng of the orbital parameters
35
+ with the parallax and proper motion, and (2) it permits a dire ct generalization to
36
+ include non-linear Keplerian parameters in a combined fit to astrometry and radial ve-
37
+ locity data. In fact, our analysis of the astrometry alone un covers the reported 270 d
38
+ period and an even stronger signal at ∼50 days. We estimate the uncertainties in the
39
+ parameters using a Markov Chain Monte Carlo approach. The no minal precision of the
40
+ new Doppler measurements is about 150 s−1while their standard deviation is 250 ms−1.
41
+ However, the best fit solutions still have RMS of 200 ms−1indicating that the excess
42
+ in variability is due to uncontrolled systematic errors rat her than the candidate com-
43
+ panions detected in the astrometry. Although the new data al one cannot rule-out the
44
+ presence of a candidate, when combined with published radia l velocity measurements,
45
+ the False Alarm Probabilities of the best solutions grow to u nacceptable levels strongly
46
+ suggesting that the observed astrometric wobble is not due t o an unseen companion.
47
+ Subject headings: astrometry, methods: statistical, stars: individual (VB 1 0), tech-
48
+ niques: radial velocities
49
+ 1. Introduction
50
+ Pravdo & Shaklan (2009) recently announced the discovery of an astrometric companion to
51
+ VB 10, an ultra-cool dwarf with a mass of ≈0.08 M ⊙. From a Keplerian fit to the motion, they
52
+ determined a mass of 6 M Jand a period of 270 d. Thus VB 10 became the lowest mass star
53
+ known to harbor a planetary companion. The mass ratio betwee n VB 10 and its companion, ∼13,
54
+ also is intriguing. A similar mass ratio for a Solar-type sta r would make the companion a brown
55
+ dwarf, but brown dwarfs as small separation companions to st ars are quite rare. VB 10 is itself the
56
+ secondary in a wide binary with V1428 Aql, a M2.5 star (van Bie sbroeck 1961). At a distance of
57
+ 1Based on observations collected with the 6.5 meter Magellan Telescopes located at Las Campanas Observatory,
58
+ Chile, at the W. M. Keck Observatory and the Canada-France-H awaii Telescope (CFHT). The Keck Observatory is
59
+ operated as a scientific partnership between the California Institute of Technology, the University of California, and
60
+ NASA, and was made possible by the generous financial support of the W. M. Keck Foundation. CFHT is operated
61
+ by the National Research Council of Canada, the Institut Nat ional des Sciences de l’Univers of the Centre National
62
+ de la Recherche Scientique of France, and the University of H awaii.– 3 –
63
+ 5.8 pc from the Sun, the 74′′separation of this proper motion binary corresponds to a pro jected
64
+ separation of 430 AU.
65
+ Such low-mass stars have not been the target of intensive pre cision radial velocity (PRV)
66
+ monitoring because they have low visual fluxes and high stell ar activity. For example, the dedicated
67
+ HARPS M-dwarf planet search observes stars2only brighter than V=14 and of moderate to low
68
+ activity levels (Bonfils et al. 2007). VB 10 has V mag = 17.3 and is known to be a flare star
69
+ (Berger et al. 2008). PRV and lensing planet searches have so far found only 13 stars under 0.5
70
+ M⊙hosting 18 planets, and of these, more than half have masses b elow 0.1 M J.
71
+ Despite the challenges, searches for planetary companions to low mass stars are of continuing
72
+ interest. Low-mass stars appear less likely to have lower ma ss stellar companions and less likely to
73
+ harbor planets than Solar-mass stars (Cumming et al. 2008). When they do have companions, they
74
+ tend to be stars of nearly equal mass to the primary (Burgasse r et al. 2007). The mass function of
75
+ planets orbiting M dwarfs, and how it differs from the planet ma ss function for higher-mass stars,
76
+ provides a constraint on the planet formation mechanism(s) in general. Disks sufficiently massive
77
+ to form Jupiter-mass planets appear to be rare around brown d warfs, whose disks generally look
78
+ like lower mass versions of T Tauri disks (Scholz et al. 2006) . High mass companions would have
79
+ to form via a binary-like fragmentation mechanism (e.g. Fon t-Ribera et al. 2009). Thus how a 6
80
+ MJplanet could form around an ∼80 MJstar and how common such high-mass ratio companions
81
+ remain important questions (Boss et al. 2009).
82
+ The reported planet’s astrometric orbit predicts a radial v elocity (RV) amplitude of at least 1
83
+ km s−1for a circular orbit and up to several km s−1for an eccentric orbit. This magnitude signal is
84
+ detectable with ordinary RV measurements without requirin g the adoption of precision techniques
85
+ such as an iodine cell or simultaneous thorium reference.
86
+ Although several RV measurements of VB 10 exist in the litera ture before 2009, it is difficult
87
+ to combine the historical RVs (see list in Table 4 of Pravdo & S haklan (2009)), as each observation
88
+ used different calibration techniques and/or RV standards th at introduce zero-point offsets and the
89
+ typical uncertainties are also large ( ∼1.5 km s−1). The most precise measurements in the literature
90
+ were recently published by Zapatero Osorio et al. (2009) (he reafter Z09), but provide only a “hint
91
+ of variability.” These data did little to constrain the orbi tal parameters of the planet beyond what
92
+ the astrometry had already done (Anglada-Escud´ e et al. 200 9).
93
+ Here, we present a more precise set of RV observations over 17 5 days (or 65% of the reported
94
+ orbital period). We also present general techniques for joi nt fitting of astrometric and RV data and
95
+ show how they can be used to constrain the orbit of the candida te planet.
96
+ 2http://www.eso.org/sci/observing/proposals/77/gto/h arps/3.txt– 4 –
97
+ 2. New Data
98
+ We acquired spectra at 7 epochs in 2009 with the MIKE spectrog raph at the Magellan Clay
99
+ telescope at Las CampanasObservatory(Chile). We usedthe0 .35′′andthe0.5′′slits whichproduce
100
+ a spectral resolution of ≈45,000 and 35,000, respectively, across the 4900 – 10000 ˚A range of the
101
+ red chip. The seeing was in the range from 0 .5 to 1.1′′. These data were reduced using the facility
102
+ pipeline (Kelson 2003).
103
+ We also have in hand a single spectrum of VB 10 taken in 2006 usi ng the HIRES (Vogt et al.
104
+ 1994)) on the Keck I 10-m telescope. We used the 0.861′′slit to obtain a spectral resolution of
105
+ λ/∆λ≈58,000 at λ∼7000˚A. We used the GG475 order-blocking filter and the red cross-di sperser
106
+ to maximize throughput in the red orders.
107
+ To increase the phase coverage, an additional spectrum was o btained using the ESPaDOnS on
108
+ the CFHT 3.6-m telescope. ESPaDOnS is fiber fed from the Casse grain to Coud´ e focus where the
109
+ fiber image is projected onto a Bowen-Walraven slicer at the s pectrograph entrance. ESPaDOnS’
110
+ ‘star+sky’ mode records the full spectrum over 40 grating or ders covering 3700 to 10400 ˚A at
111
+ a spectral resolution of λ/∆λ≈68,000. The data were reduced using Libre Esprit described in
112
+ Donati et al. (1997, 2007).
113
+ Each stellar exposure is bias-subtracted and flat-fielded fo r pixel-to-pixel sensitivity variations.
114
+ After optimal extraction, the 1-D spectra are wavelength ca librated with a thorium-argon arc. To
115
+ correct for instrumental drifts, we used the telluric molec ular oxygen A band (from 7620 – 7660
116
+ ˚A) which aligns the MIKE spectra to 40 m s−1, after which we corrected for the heliocentric
117
+ velocity. Consistency tests with the bluer Oxygen band show s comparable values but with larger
118
+ measurement error.
119
+ The final spectra are of moderate S/N reaching ≈25 per pixel at 8000 ˚A. Each night, spectra
120
+ were also taken of a M-dwarf RV standard, namely GJ 699 (Barna rd’s star; SpT = M4V) and/or
121
+ GJ 908 (SpT = M1V).
122
+ To measure VB 10’s RV, we cross-correlated each of 9 orders be tween 7000 and 9000 ˚A (ex-
123
+ cluding those with strong telluric absorption) where VB 10 e mits most of its optical light, with the
124
+ spectrumofGJ699 and/orGJ908taken onthesamenight usingI RAF’s1fxcorroutine(Fitzpatrick
125
+ 1993). Both GJ 699 and GJ 908 have been monitored for planets f or years and none has been found
126
+ within the RV stability level of 0.1 km s−1. Here we use the systemic RVs published by a planet-
127
+ search team (Nidever et al. 2002): RV(GJ 699) = –110.506 km s−1and RV(GJ 908) = –71.147
128
+ km s−1. The zero-point of the absolute RVs is uncertain at the 0.4 km s−1level. We measured
129
+ the RVs from the gaussian peak fitted to the cross-correlatio n function (CCF) of each order and
130
+ adopt the average RV of all orders with a mean standard deviat ion of the individual measurements
131
+ of 0.150 km s−1. The average of all our measurements is 36.02 km s−1with a standard deviation
132
+ 1IRAF (Image Reduction and Analysis Facility), http://iraf .noao.edu/– 5 –
133
+ of 0.25 km s−1. An observing log with the measured RVs and uncertainties fo r VB 10 is shown in
134
+ Table 1.
135
+ 3. Data Analysis: Combining Astrometry and Radial Velocities
136
+ In this section, we reanalyze the original astrometric data to calculate the likelihood of astro-
137
+ metrically allowed solutions, and then combine the astrome try and RV data sets in a consistent
138
+ framework to quantify how the new RV measurements constrain the possibleorbits of the candidate
139
+ signals observed in the astrometry of VB 10b.
140
+ 3.1. Least-squares periodograms
141
+ The most popular method to look for periodicities in data is t he so-called Lomb-Scargle
142
+ periodogram. A version adapted to deal with astrometric two -dimensional data developed by
143
+ Catanzarite et al. (2006) (Joint Lomb Scargle periodogram) was implemented in the discovery pa-
144
+ per of VB 10b (Pravdo & Shaklan 2009). Any method based on the L omb-Scargle periodogram
145
+ performs optimally only under an important implicit assump tion: all other signals (e.g. linear
146
+ trend, an average offset, etc.) can be subtracted from the data without affecting the significance of
147
+ the signal under investigation. This assumption does not ho ld for astrometry because the proper
148
+ motion and the parallax are also a significant part of the sign al and they typically correlate with
149
+ the periodic motion of a planet (see Black & Scargle 1982).
150
+ We use instead a Least-squares periodogram. The weighted Le ast-squares solution is obtained
151
+ by fitting all the free parameters in the model for a given peri od. The sum of the weighted residuals
152
+ divided by Nis the so-called χ2statistic. Then, each χ2
153
+ Pof a given model with kPparameters, can
154
+ be compared to the χ2
155
+ 0of the null hypothesis with k0free parameters by computing the power, z,
156
+ as
157
+ z(P) =(χ2
158
+ 0−χ2
159
+ P)/(kP−k0)
160
+ χ2
161
+ P/(Nobs−kP)(1)
162
+ where a large zis interpreted as a very significant solution. The values of zfollow a Fisher F-
163
+ distribution with kP−k0andNobs−kPdegrees of freedom (Scargle 1982; Cumming 2004). Even
164
+ if only noise is present, a periodogram will contain several peaks (see Scargle 1982, as an example)
165
+ whoseexistencehavetobeconsideredinobtainingtheproba bilityofaspuriousdetection. Assuming
166
+ Gaussian noise, the probability that a peak in the periodogr am has a power higher than z(P) by
167
+ chance is the so-called False Alarm Probability (FAP) :
168
+ FAP = 1 −(1−Prob[z > z(P)])M(2)
169
+ whereMis the number of independent frequencies. In the case of unev en sampling, Mcan be quite
170
+ large and is roughly the number of periodogram peaks one coul d expect from a data set with only– 6 –
171
+ Gaussian noise and thesame cadence as thereal observations . We adopt the recipe M≈2∆T/Pmin
172
+ given in Cumming (2004, Sec 2.2), where ∆ Tis the time-span of the observations and Pminis the
173
+ minimum period searched. One still has to select Pminarbitrarily. Assuming a Pmin= 20 days, the
174
+ astrometric data alone has M∼300, and the combination of astrometry and RVs has M∼360.
175
+ In our particular problem, the null hypothesis is the basic k inematic model with k0= 6 param-
176
+ eters: 2 coordinates, 2 proper motions, parallax and system ic RV. As a first approach, our simplest
177
+ non-null hypothesis considers circular orbits, astrometr ic data only and one RV measurement. For
178
+ a given period, the number of free parameters is then kP= 10: the 6 kinematic ones plus the four
179
+ Thiele Innes elements A,B,FandG(e.g. Wright & Howard 2009). Since the model is linear in all
180
+ 10 parameters, the power can be efficiently computed for many p eriods between 20 days and 4000
181
+ days to obtain a familiar representation of the periodogram that we call a Circular Least-squares
182
+ Periodogram (CLP). The CLP of the astrometric data, shown at top in Figure 1, displays two
183
+ obvious peaks: the reported one at 270 days (Pravdo & Shaklan 2009) and a more significant one
184
+ at 49.9 days, both with high power and very low FAPs.
185
+ To find the full Keplerian solution for both periods and estim ate their FAPs, we perform a
186
+ Least-squares periodogram sampling a grid of fixed eccentri city-period (eP) pairs and fitting all
187
+ other parameters. For each eP pair kPis 11: the null-hypothesis ( X0,Y0,µX,µY,πandv0)
188
+ plus all the other Keplerian elements: Mass of the planet, Ω, ω,i, and the Mean anomaly at the
189
+ initial epoch M0(see Wright & Howard 2009, for a recent review). We analyze bo thastrometry
190
+ onlyandastrometry+RVs . Theχ2of the best fit solution is then used to obtain each FAP as
191
+ previously described. Figure 1 shows the resulting color-c oded FAPs for each eccentricity–period
192
+ pair (eP-map).
193
+ 3.1.1. Astrometry only
194
+ A value of M= 300 has been used to obtain the FAP, and our result at 270 d qua litatively
195
+ agrees with Pravdo & Shaklan (2009), however the more signifi cant period is at ∼50 d. For both
196
+ periods, there are regions with FAP <1% spanning all possible eccentricities (second row in Fig-
197
+ ure 1). The best fits and their χ2per degree of freedom (¯ χ2) are summarized in Table 2. The
198
+ obtained results for the 270 d period are in agreement with th ose reported in the discovery paper
199
+ by Pravdo & Shaklan (2009). The best fit solution for the 50 d pe riod has mass ∼15 MJ, which
200
+ would be a very low mass brown dwarf. It is important to point o ut that the best fit inclination is
201
+ close to 90 (edge on) for both solutions. The uncertainties o n the orbital parameters are quantified
202
+ in Sec 3.2.– 7 –
203
+ 3.1.2. Astrometry+RVs
204
+ We now fit jointly for the best orbital solution to the astrome try and RVs. Our campaign
205
+ covered about 65% of the 270 d orbit. The standard deviation o f all our RVs measurements is
206
+ 250 m s−1(null hypothesis) which is larger than the individual uncer tainties in Table 4. When we
207
+ cross-correlate our standards, we measure a similar RMS of 2 00 m s−1, which indicates that the
208
+ difference is due to an uncontrolled or unmeasured systematic . The RMS of the RVs for the best
209
+ fit solution is 200 m s−1, which is not statistically different from the RMS of the null h ypothesis.
210
+ This is another indication that our measurements contain sy stematic errors at the level of 100 −200
211
+ m/s. Despite of that, we use the nominal errors in the Least-s quares solution as the best estimates
212
+ for the individual uncertainties we can provide. In Figure 2 , we show the best solutions to both
213
+ signals including all the data.
214
+ For the 270 d period, our RV non detection cannot exclude a small region of orbital solutions
215
+ arounde∼0.8 with a FAP between 1%–5% – see Figure 1, third row right panel . We now add
216
+ the RVs measurements by Z09 and solve for a joint solution. A z ero-point offset between datasets
217
+ is added as an additional free parameter. The combined RV mea surements force the eccentricity
218
+ to large values which apparently still provides a reasonabl e fit to the astrometry (see top panels in
219
+ Figure 2). However, the FAPs are now all higher than 10% (Figu re 1, bottom right panel), which
220
+ indicates that the signal can be barely distinguished from t he noise fluctuations. The “hint” of
221
+ detection in Z09 based on one discrepant value at 3 .1−σout of five can be due to random errors
222
+ with a non negligible probability.
223
+ For the 50 d period, there are still several orbits that provi de a decent fit to the combined
224
+ astrometry and the new RV data with a FAP lower than 1%. These o ccupy a small space around
225
+ the best joint solution, with e= 0.90 (see Figure 1, 3rd row, left panel) and an inclination clos e
226
+ to 0. Large eccentricity causes the duration of fast RV varia tion to be very short (and difficult to
227
+ catch); an inclination close to 0 tends to suppress any RVs si gnal. Such inclination is in apparent
228
+ contradiction with the one obtained using the astrometry al one (∼90 deg). The reason is the
229
+ following: while the new fit to the astrometry forced by the RV s is much worse than the one
230
+ obtained from the astrometry alone, such a solution still re presents an improvement compared to
231
+ the null hypothesis. Adding Z09 data to the fit increases the F AP of the most likely solution to 2%,
232
+ an eccentricity of 0 .91 and the inclination close to 0 (see Table 2). This suggests that the signal at
233
+ 50 d is also spurious, even though it has slightly better chan ces of survival than the one at 270 d.
234
+ 3.2.A Posteriori Probability Distributions
235
+ We adapt the method developed by Ford (2005, 2006) to assess u ncertainties in orbit determi-
236
+ nations by obtaining the a posteriori probability distribution for the parameters using a Markov
237
+ Chain with a Gibbs sampler strategy. Our problem is identica l to the one described by Ford (2005),
238
+ where now the χ2contains both RV and astrometric observations and the model has a few more– 8 –
239
+ free parameters. Several properly adjusted MCMC with 106steps have been computed obtaining
240
+ compatible results. The step sizes of the Gibbs sampler are i nitialized with the formal errors from
241
+ the best fit Least-squares solution, and adjusted to obtain a transition probability between 10%
242
+ and 20%. The first 105steps of each chain are rejected. The final distributions mat ch very well the
243
+ areas of low FAPs in the eP-maps (see Figure 3 as an example) gi ving further proof that the chains
244
+ have converged to the equilibrium distributions. The MCMC c ontains 13 free parameters – the 11
245
+ from the Least-squares periodogram plus eccentricity and p eriod. When the RVs measurements
246
+ from Z09 are included, and additional offset parameter is incl uded.
247
+ Table 2 presents the standard deviations obtained via the MC MC for both the 50 d and 270 d
248
+ periods using astrometry alone and astrometry + all RV data. As an example, we show the two
249
+ dimensional density of states in period-eccentricity spac e in Figure 3 (left) obtained in both cases
250
+ aroundthe 270 d signal. Themarginalized distributionsfor ein theform of histograms are shownin
251
+ Figure 3 (right). For the astrometry-alone case, the distri bution of eis almost uniform. It becomes
252
+ strongly peaked towards high eccentricities when all the RV data are included. Since the best fit
253
+ solution at 270 d is poor (¯ χ2= 1.76), the corresponding χ2minimum is not very deep which is
254
+ reflected in a significant increase in the derived uncertaint ies (See Table 2). The same happens to
255
+ the signal at 50 d with the exception of the inclination that h as a small uncertainty (4 deg) close
256
+ to 0. Even though this solution has a low FAP, the inclination has to be coincidentally very small
257
+ to suppress any RV signal and very different from using astrome try only (94 deg), rasing serious
258
+ doubts of its reality.
259
+ 4. Discussion and Conclusions
260
+ The non-detection of a significant RV variation in our data se t already discards most orbital
261
+ configurations allowed by the astrometry. When combined wit h Z09 RVs measurements, there are
262
+ no remaining solutions with a FAP lower than 10% around the 27 0 d period, so the presence of
263
+ a planet candidate at that period is not supported by the obse rvations. For the 50 d period, the
264
+ constraints arealso strongandbecome almost definitive whe ntheZ09data is included. Even highly
265
+ eccentric solutions have a relatively large FAP ( >2%). We find that particular combinations of
266
+ eccentricity, inclination and ωcan fit an almost flat RV curve indicating that the analytic met hods
267
+ applied to estimate FAPs for high eccentricities tend to giv e over optimistic results and that this
268
+ issue should be studied in more detail.
269
+ We have developed and implemented useful tools for detailed analysis of combined astrometric
270
+ and RV data: Circular Least-squares periodogram as the prop er generalization of the classic Lomb-
271
+ Scargle periodogram to deal with astrometric data, eP-maps to visualize the most likely period–
272
+ eccentricity combinations and a Bayesian characterizatio n of the parameter uncertainties based on
273
+ a MCMC approach.
274
+ VB 10 is also part of the Carnegie Astrometric Planet Search p rogram (Boss et al. 2009). RV– 9 –
275
+ measurements with precision techniques in the near-infrar ed (Bean et al. 2009) may provide the
276
+ required accuracy to put even stronger limits to the existen ce of VB10b or find other planets in the
277
+ system. VB 10 will certainly be observed by the space astrome try mission Gaia (Perryman et al.
278
+ 2001), which would be capable of finding a planet with a period of 270 d and as small as 0 .2 MJ.
279
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280
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319
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320
+ This preprint was prepared with the AAS L ATEX macros v5.2.– 11 –
321
+ Table 1. Log of RVs
322
+ Telescope UT Date HJD Slit Width RV (w/ GJ 699)aRV (w/ GJ 908)a
323
+ +Instrument –2450000′′km s−1km s−1
324
+ Keck I + HIRES 2006 Aug 12 3959.57 0.86 – 35.59 ±0.15
325
+ Clay+MIKE 2009 Jun 06 4988.74 0.35 36.23 ±0.13 35.99 ±0.15
326
+ Clay+MIKE 2009 Jun 07 4989.82 0.50 36.22 ±0.13 36.09 ±0.20
327
+ Clay+MIKE 2009 Jun 08 4990.75 0.50 36.15 ±0.12 36.10 ±0.22
328
+ Clay+MIKE 2009 Jun 30 5012.72 0.35 35.72 ±0.11 –
329
+ Clay+MIKE 2009 Jul 25 5037.66 0.50 35.96 ±0.11 36.03 ±0.11
330
+ Clay+MIKE 2009 Sep 04 5078.58 0.50 35.96 ±0.09 36.37 ±0.24
331
+ Clay+MIKE 2009 Oct 15 5119.54 0.50 36.30 ±0.14 36.41 ±0.13
332
+ Clay+MIKE 2009 Oct 26 5130.51 0.50 36.41 ±0.16 36.27 ±0.18
333
+ CFHT+ESPaDOnS 2009 Nov 29 5164.69 –b– 35.74 ±0.20
334
+ aUncertainties are the standard deviation of the 9 orders of t he cross correlation and do not include the 40
335
+ m s−1systematic uncertainty from the telluric wavelength corre ction. Absolute radial velocity determination
336
+ has an uncertainty of 0 .4 km/s but it is not relevant for orbital fitting purposes.
337
+ bESPaDOnS is a fiber fed spectrograph with an effective resolut ion of R ∼68000 in the wavelength range
338
+ of interest
339
+ Table 2. Best fitting valuesa. Uncertainties obtained from a MCMC with 106steps.
340
+ Parameter Astrometry 50 d Astrometry 270 d Astro+ all RV 50 d A stro+ all RV 270 d
341
+ X0(mas) -16.6 ±1.6 -14.1d±3.2 -21.15±2.3 -17.9 ±4.7
342
+ Y0(mas) -408.0 ±1.9 -406.1d±3.5 409.52±2.8 -410.5 ±5.51
343
+ µR.A.(mas/yr) -588.98 ±0.25 -589.08 ±0.25 -588.66±0.29 -589.21 ±0.26
344
+ µDec(mas/yr) -1360.95 ±0.25 -1361.08 ±0.24 -1361.02 ±0.25 -1361.36 ±0.20
345
+ π(mas) 168.3 ±1.51 169.5 ±1.4 169.95±1.37 169.24 ±1.30
346
+ v0(km/s) 35.2 ±1.4 35.4d±1.050 36.06±0.11 36.05 ±0.08
347
+ voffset(km/s) - - 1.5±0.42 1.5 ±0.36
348
+ P(days) 49.7 ±0.5 272.1 ±4.1 49.84±0.11 278.5 ±2.7
349
+ Mass(MJ) 17.5 ±4.4 7.1 ±2.7 13.7±6.4 5.0 ±2.9
350
+ e 0.22c±0.30 0.48c±0.31 0.91±0.13 0.90 ±0.16
351
+ i(deg) 93 ±5 90 ±15 4±5 110c±50
352
+ Ω(deg) 40 ±20 220 ±25 13c±100 40c±66
353
+ ω(deg) 20 ±40 30c±80 122c±60 17c±90
354
+ M0(deg) 270 ±0 170c±108 340c±70 156c±80
355
+ a(AU)d0.12 0.36 0.12 0.36
356
+ ¯χ2
357
+ 02.28 2.28 2.75 2.75
358
+ ¯χ20.87 0.93 1.62 1.76
359
+ aThe mass of VB 10 is assumed to be 0 .078 M ⊙according to Pravdo & Shaklan (2009)
360
+ bLarge uncertainty due to correlation with the eccentricity
361
+ cUnconstrained or poorly constrained
362
+ dDerived quantity using Kepler equations– 12 –
363
+ Fig. 1.— Top panel. Circular Least-squares periodogramshowingthe two most si gnificant periods
364
+ with their corresponding False Alarm Probabilities (FAP). Second row. FAPs obtained for a grid
365
+ of Eccentricity–Period pairs around the 50 d (left) and the 2 70 d (right) when only astrometry is
366
+ considered. Third row. FAPs obtained when our new RV are included to the fit. Bottom row.
367
+ Final FAPs obtained when all published RV data are combined i n a joint fit.– 13 –
368
+ 0 0.2 0.4 0.6 0.8 1
369
+ Phase-4-20246810R.A.(mas)
370
+ 0 0.2 0.4 0.6 0.8 1-4-20246810
371
+ 0 0.2 0.4 0.6 0.8 1
372
+ Phase-4-20246810 Dec (mas)
373
+ 0 0.2 0.4 0.6 0.8 1-4-20246810
374
+ 0 0.2 0.4 0.6 0.8 1
375
+ Phase-4-20246810R.A.(mas)
376
+ 0 0.2 0.4 0.6 0.8 1-4-20246810
377
+ 0 0.2 0.4 0.6 0.8 1
378
+ Phase-4-20246810 Dec (mas)
379
+ 0 0.2 0.4 0.6 0.8 1-4-20246810
380
+ 0 0.2 0.4 0.6 0.8 1
381
+ Phase32333435363738RV (km/s)
382
+ 0 0.2 0.4 0.6 0.8 132333435363738
383
+ 0 0.2 0.4 0.6 0.8 1
384
+ Phase32333435363738
385
+ MIKE
386
+ HIRES
387
+ CFHT
388
+ NIRSPEC Z09P = 49.84 days P = 278.5 days
389
+ Fig. 2.— The best fit (lowest χ2) joint solutions to the Pravdo & Shaklan (2009) astrometry a nd
390
+ used RVs for the two signals. Top panels contain the astromet ric offsets after the removal of the
391
+ corresponding parallax and the proper motion. The lower pan els contain all RVs used. Each RV
392
+ point represents the weighted average of the values obtaine d using both reference stars if available.
393
+ The best fit offset has been applied to Z09 data (Green triangles ). Phase 0 corresponds to the first
394
+ astrometric epoch at JD 2451438 .64 and the corresponding folding periods are given on the top .
395
+ Fig. 3.— Left: Steps in period-eccentricity space of a Marko v Chain of 106elements applied to the
396
+ astrometry only (black) and to the astrometry+all RV data (b rown). The distribution resembles
397
+ the FAP contours on the eP-maps around 270 days indicating th at the chain has successfully
398
+ converged to the equilibrium distribution. Right: Histogr am reproducing the marginalized density
399
+ distributions in efor the astrometry only and astrometry+RV around the 270 d so lution.
1001.0044.txt ADDED
@@ -0,0 +1,1486 @@
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
1
+ arXiv:1001.0044v3 [math.PR] 31 Mar 2011A law of large numbers approximation for
2
+ Markov population processes with countably
3
+ many types
4
+ A. D. Barbour∗and M. J. Luczak†
5
+ Universit¨ at Z¨ urich and London School of Economics
6
+ Abstract
7
+ When modelling metapopulation dynamics, the influence of a s in-
8
+ gle patch on the metapopulation depends on the number of indi vidu-
9
+ als in the patch. Since the population size has no natural upp er limit,
10
+ this leads to systems in which there are countably infinitely many
11
+ possible types of individual. Analogous considerations ap ply in the
12
+ transmission of parasitic diseases. In this paper, we prove a law of
13
+ large numbers for quite general systems of this kind, togeth er with
14
+ a rather sharp bound on the rate of convergence in an appropri ately
15
+ chosen weighted ℓ1norm.
16
+ Keywords: Epidemic models, metapopulation processes, countably many
17
+ types, quantitative law of large numbers, Markov population proce sses
18
+ AMS subject classification: 92D30, 60J27, 60B12
19
+ Running head: A law of large numbers approximation
20
+ 1 Introduction
21
+ There are many biological systems that consist of entities that diffe r in their
22
+ influence according to the number of active elements associated wit h them,
23
+ ∗Angewandte Mathematik, Universit¨ at Z¨ urich, Winterthurertra sse 190, CH-8057
24
+ Z¨URICH; ADB was supported in part by Schweizerischer Nationalfond s Projekt Nr. 20–
25
+ 107935/1.
26
+ †London School of Economics; MJL was supported in part by a STICE RD grant.
27
+ 1and can be divided into types accordingly. In parasitic diseases (Bar bour &
28
+ Kafetzaki 1993, Luchsinger 2001a,b, Kretzschmar 1993), the in fectivity of a
29
+ host depends on the number of parasites that it carries; in metapo pulations,
30
+ the migration pressure exerted by a patch is related to the number of its
31
+ inhabitants (Arrigoni 2003); the behaviour of a cell may depend on the num-
32
+ ber of copies of a particular gene that it contains (Kimmel & Axelrod 2 002,
33
+ Chapter 7); and so on. In none of these examples is there a natura l upper
34
+ limit to the number of associated elements, so that the natural set ting for
35
+ a mathematical model is one in which there are countably infinitely man y
36
+ possible types of individual. In addition, transition rates typically incr ease
37
+ with the number of associated elements in the system — for instance , each
38
+ parasite has an individual death rate, so that the overall death ra te of par-
39
+ asites grows at least as fast as the number of parasites — and this le ads
40
+ to processes with unbounded transition rates. This paper is conce rned with
41
+ approximations to density dependent Markov models of this kind, wh en the
42
+ typical population size Nbecomes large.
43
+ In density dependent Markov population processes with only finitely
44
+ many types of individual, a law of large numbers approximation, in the f orm
45
+ ofasystemofordinarydifferentialequations, wasestablishedbyK urtz(1970),
46
+ together with a diffusion approximation (Kurtz, 1971). In the infinit e di-
47
+ mensional case, the law of large numbers was proved for some spec ific mod-
48
+ els (Barbour & Kafetzaki 1993, Luchsinger 2001b, Arrigoni 2003 , see also
49
+ L´ eonard 1990), using individually tailored methods. A more general result
50
+ was then given by Eibeck & Wagner (2003). In Barbour & Luczak (20 08),
51
+ the law of large numbers was strengthened by the addition of an err or bound
52
+ inℓ1that is close to optimal order in N. Their argument makes use of an
53
+ intermediate approximation involving an independent particles proce ss, for
54
+ which the law of large numbers is relatively easy to analyse. This proce ss is
55
+ then shown to be sufficiently close to the interacting process of act ual inter-
56
+ est, by means of a coupling argument. However, the generality of t he results
57
+ obtained is limited by the simple structure of the intermediate proces s, and
58
+ the model of Arrigoni (2003), for instance, lies outside their scop e.
59
+ In this paper, we develop an entirely different approach, which circu m-
60
+ vents the need for an intermediate approximation, enabling a much w ider
61
+ class of models to be addressed. The setting is that of families of Mar kov
62
+ population processes XN:= (XN(t), t≥0),N≥1, taking values in the
63
+ countable space X+:={X∈ZZ+
64
+ +;/summationtext
65
+ m≥0Xm<∞}. Each component repre-
66
+ 2sents the number of individuals of a particular type, and there are c ountably
67
+ many types possible; however, at any given time, there are only finit ely
68
+ many individuals in the system. The process evolves as a Markov proc ess
69
+ with state-dependent transitions
70
+ X→X+Jat rate NαJ(N−1X), X∈ X+, J∈ J,(1.1)
71
+ where each jump is of bounded influence, in the sense that
72
+ J ⊂ {X∈ZZ+;/summationdisplay
73
+ m≥0|Xm| ≤J∗<∞},for some fixed J∗<∞,(1.2)
74
+ so that the number of individuals affected is uniformly bounded. Dens ity
75
+ dependence is reflected in the fact that the arguments of the fun ctionsαJ
76
+ are counts normalised by the ‘typical size’ N. Writing R:=RZ+
77
+ +, the func-
78
+ tionsαJ:R →R+are assumed to satisfy
79
+ /summationdisplay
80
+ J∈JαJ(ξ)<∞, ξ∈ R0, (1.3)
81
+ whereR0:={ξ∈ R:ξi= 0 for all but finitely many i}; this assumption
82
+ implies that the processes XNare pure jump processes, at least for some
83
+ non-zero length of time. To prevent the paths leaving X+, we also assume
84
+ thatJl≥ −1 for each l, and that αJ(ξ) = 0 ifξl= 0 for any J∈ Jsuch
85
+ thatJl=−1. Some remarks on the consequences of allowing transitions J
86
+ withJl≤ −2 for some lare made at the end of Section 4.
87
+ Thelawoflargenumbersisthenformallyexpressed intermsofthesy stem
88
+ ofdeterministic equations
89
+
90
+ dt=/summationdisplay
91
+ J∈JJαJ(ξ) =:F0(ξ), (1.4)
92
+ to be understood componentwise for those ξ∈ Rsuch that
93
+ /summationdisplay
94
+ J∈J|Jl|αJ(ξ)<∞,for alll≥0,
95
+ thus by assumption including R0. Here, the quantity F0represents the in-
96
+ finitesimal average drift of the components of the random proces s. However,
97
+ in this generality, it is not even immediately clear that equations (1.4) h ave
98
+ a solution.
99
+ 3In order to make progress, it is assumed that the unbounded comp onents
100
+ in the transition rates can be assimilated into a linear part, in the sens e
101
+ thatF0can be written in the form
102
+ F0(ξ) =Aξ+F(ξ), (1.5)
103
+ again to be understood componentwise, where Ais a constant Z+×Z+
104
+ matrix. These equations are then treated as a perturbed linear sy stem
105
+ (Pazy 1983, Chapter 6). Under suitable assumptions on A, there exists a
106
+ measure µonZ+, defining a weighted ℓ1norm/⌊ard⌊l · /⌊ard⌊lµonR, and a strongly
107
+ /⌊ard⌊l·/⌊ard⌊lµ–continuoussemigroup {R(t), t≥0}oftransitionmatriceshaving point-
108
+ wise derivative R′(0) =A. IfFis locally /⌊ard⌊l·/⌊ard⌊lµ–Lipschitz and /⌊ard⌊lx(0)/⌊ard⌊lµ<∞,
109
+ this suggests using the solution xof the integral equation
110
+ x(t) =R(t)x(0)+/integraldisplayt
111
+ 0R(t−s)F(x(s))ds (1.6)
112
+ as an approximation to xN:=N−1XN, instead of solving the deterministic
113
+ equations (1.4) directly. We go on to show that the solution XNof the
114
+ stochastic system can be expressed using a formula similar to (1.6), which
115
+ has an additional stochastic component in the perturbation:
116
+ xN(t) =R(t)xN(0)+/integraldisplayt
117
+ 0R(t−s)F(xN(s))ds+/tildewidemN(t),(1.7)
118
+ where
119
+ /tildewidemN(t) :=/integraldisplayt
120
+ 0R(t−s)dmN(s), (1.8)
121
+ andmNis the local martingale given by
122
+ mN(t) :=xN(t)−xN(0)−/integraldisplayt
123
+ 0F0(xN(s))ds. (1.9)
124
+ The quantity mNcanbe expected to be small, at least componentwise, under
125
+ reasonable conditions.
126
+ To obtain tight control over /tildewidemNin all components simultaneously, suf-
127
+ ficient to ensure that sup0≤s≤t/⌊ard⌊l/tildewidemN(s)/⌊ard⌊lµis small, we derive Chernoff–like
128
+ boundsonthedeviations ofthemost significant components, witht hehelpof
129
+ a family of exponential martingales. The remaining components are t reated
130
+ usingsomegeneral a prioriboundsonthebehaviourofthestochasticsystem.
131
+ 4This allows us to take the difference between the stochastic and det erministic
132
+ equations (1.7) and (1.6), after which a Gronwall argument can be c arried
133
+ through, leading to the desired approximation.
134
+ The main result, Theorem 4.7, guarantees an approximation error o f or-
135
+ derO(N−1/2√logN) in the weighted ℓ1metric/⌊ard⌊l·/⌊ard⌊lµ, except on an event of
136
+ probability of order O(N−1logN). More precisely, for each T >0, there
137
+ exist constants K(1)
138
+ T,K(2)
139
+ T,K(3)
140
+ Tsuch that, for Nlarge enough, if
141
+ /⌊ard⌊lN−1XN(0)−x(0)/⌊ard⌊lµ≤K(1)
142
+ T/radicalbigg
143
+ logN
144
+ N,
145
+ then
146
+ P/parenleftBig
147
+ sup
148
+ 0≤t≤T/⌊ard⌊lN−1XN(t)−x(t)/⌊ard⌊lµ> K(2)
149
+ T/radicalbigg
150
+ logN
151
+ N/parenrightBig
152
+ ≤K(3)
153
+ TlogN
154
+ N.(1.10)
155
+ Theerrorboundissharper, byafactoroflog N, thanthatgiveninBarbour&
156
+ Luczak(2008),andthetheoremisapplicabletoamuch widerclassof models.
157
+ However, the method of proof involves moment arguments, which r equire
158
+ somewhat stronger assumptions on the initial state of the system , and, in
159
+ models such as that of Barbour & Kafetzaki (1993), onthe choice of infection
160
+ distributions allowed. The conditions under which the theorem holds c an be
161
+ divided into three categories: growth conditions on the transition r ates, so
162
+ that the a prioribounds, which have the character of moment bounds, can
163
+ be established; conditions on the matrix A, sufficient to limit the growth of
164
+ the semigroup R, and (together with the properties of F) to determine the
165
+ weights defining the metric in which the approximation is to be carried o ut;
166
+ and conditions on the initial state of the system. The a priori bounds are
167
+ derived in Section 2, the semigroup analysis is conducted in Section 3, and
168
+ the approximation proper is carried out in Section 4. The paper conc ludes
169
+ in Section 5 with some examples.
170
+ The form (1.8) of the stochastic component /tildewidemN(t) in (1.7) is very simi-
171
+ lar to that of a key element in the analysis of stochastic partial differ ential
172
+ equations; see, for example, Chow (2007, Section 6.6). The SPDE a rguments
173
+ used for its control are however typically conducted in a Hilbert spa ce con-
174
+ text. Our setting is quite different in nature, and it does not seem cle ar how
175
+ to translate the SPDE methods into our context.
176
+ 52 A priori bounds
177
+ We begin by imposing further conditions on the transition rates of th e pro-
178
+ cessXN, sufficient to constrain its paths to bounded subsets of X+dur-
179
+ ing finite time intervals, and in particular to ensure that only finitely ma ny
180
+ jumps can occur in finite time. The conditions that follow have the flav our
181
+ of moment conditions on the jump distributions. Since the index j∈Z+is
182
+ symbolic in nature, we start by fixing an ν∈ R, such that ν(j) reflects in
183
+ some sense the ‘size’ of j, with most indices being ‘large’:
184
+ ν(j)≥1 for allj≥0 and lim
185
+ j→∞ν(j) =∞. (2.1)
186
+ We then define the analogues of higher empirical moments using the q uanti-
187
+ tiesνr∈ R, defined by νr(j) :=ν(j)r,r≥0, setting
188
+ Sr(x) :=/summationdisplay
189
+ j≥0νr(j)xj=xTνr, x∈ R0, (2.2)
190
+ where, for x∈ R0andy∈ R,xTy:=/summationtext
191
+ l≥0xlyl. In particular, for X∈ X+,
192
+ S0(X) =/⌊ard⌊lX/⌊ard⌊l1. Note that, because of (2.1), for any r≥1,
193
+ #{X∈ X+:Sr(X)≤K}<∞for allK >0. (2.3)
194
+ To formulate the conditions that limit the growth of the empirical mom ents
195
+ ofXN(t) witht, we also define
196
+ Ur(x) :=/summationdisplay
197
+ J∈JαJ(x)JTνr;Vr(x) :=/summationdisplay
198
+ J∈JαJ(x)(JTνr)2, x∈ R.(2.4)
199
+ The assumptions that we shall need are then as follows.
200
+ Assumption 2.1 There exists a νsatisfying (2.1)andr(1)
201
+ max,r(2)
202
+ max≥1such
203
+ that, for all X∈ X+,
204
+ /summationdisplay
205
+ J∈JαJ(N−1X)|JTνr|<∞,0≤r≤r(1)
206
+ max,(2.5)
207
+ the case r= 0following from (1.2)and(1.3); furthermore, for some non-
208
+ negative constants krl, the inequalities
209
+ U0(x)≤k01S0(x)+k04,
210
+ U1(x)≤k11S1(x)+k14, (2.6)
211
+ Ur(x)≤ {kr1+kr2S0(x)}Sr(x)+kr4,2≤r≤r(1)
212
+ max;
213
+ 6and
214
+ V0(x)≤k03S1(x)+k05,
215
+ Vr(x)≤kr3Sp(r)(x)+kr5,1≤r≤r(2)
216
+ max, (2.7)
217
+ are satisfied, where 1≤p(r)≤r(1)
218
+ maxfor1≤r≤r(2)
219
+ max.
220
+ The quantities r(1)
221
+ maxandr(2)
222
+ maxusually need to be reasonably large, if Assump-
223
+ tion 4.2 below is to be satisfied.
224
+ Now, for XNas in the introduction, we let tXNndenote the time of its
225
+ n-th jump, with tXN
226
+ 0= 0, and set tXN∞:= lim n→∞tXNn, possibly infinite. For
227
+ 0≤t < tXN∞, we define
228
+ S(N)
229
+ r(t) :=Sr(XN(t));U(N)
230
+ r(t) :=Ur(xN(t));V(N)
231
+ r(t) :=Vr(xN(t)),
232
+ (2.8)
233
+ once again with xN(t) :=N−1XN(t), and also
234
+ τ(N)
235
+ r(C) := inf {t < tXN
236
+ ∞:S(N)
237
+ r(t)≥NC}, r≥0,(2.9)
238
+ where the infimum of the empty set is taken to be ∞. Our first result shows
239
+ thattXN∞=∞a.s., and limits the expectations of S(N)
240
+ 0(t) andS(N)
241
+ 1(t) for any
242
+ fixedt.
243
+ In what follows, we shall write F(N)
244
+ s=σ(XN(u),0≤u≤s), so that
245
+ (F(N)
246
+ s:s≥0) is the natural filtration of the process XN.
247
+ Lemma 2.2 Under Assumptions 2.1, tXN∞=∞a.s. Furthermore, for any
248
+ t≥0,
249
+ E{S(N)
250
+ 0(t)} ≤(S(N)
251
+ 0(0)+Nk04t)ek01t;
252
+ E{S(N)
253
+ 1(t)} ≤(S(N)
254
+ 1(0)+Nk14t)ek11t.
255
+ Proof. Introducing the formal generator ANassociated with (1.1),
256
+ ANf(X) :=N/summationdisplay
257
+ J∈JαJ(N−1X){f(X+J)−f(X)}, X∈ X+,(2.10)
258
+ we note that NUl(x) =ANSl(Nx). Hence, if we define M(N)
259
+ lby
260
+ M(N)
261
+ l(t) :=S(N)
262
+ l(t)−S(N)
263
+ l(0)−N/integraldisplayt
264
+ 0U(N)
265
+ l(u)du, t ≥0,(2.11)
266
+ 7for 0≤l≤r(1)
267
+ max, it is immediate from (2.3), (2.5) and (2.6) that the process
268
+ (M(N)
269
+ l(t∧τ(N)
270
+ 1(C)), t≥0) is a zero mean F(N)–martingale for each C >0.
271
+ In particular, considering M(N)
272
+ 1(t∧τ(N)
273
+ 1(C)), it follows in view of (2.6) that
274
+ E{S(N)
275
+ 1(t∧τ(N)
276
+ 1(C))} ≤S(N)
277
+ 1(0)+E/braceleftBigg/integraldisplayt∧τ(N)
278
+ 1(C)
279
+ 0{k11S(N)
280
+ 1(u)+Nk14}du/bracerightBigg
281
+ ≤S(N)
282
+ 1(0)+/integraldisplayt
283
+ 0(k11E{S(N)
284
+ 1(u∧τ(N)
285
+ 1(C))}+Nk14)du.
286
+ Using Gronwall’s inequality, we deduce that
287
+ E{S(N)
288
+ 1(t∧τ(N)
289
+ 1(C))} ≤(S(N)
290
+ 1(0)+Nk14t)ek11t,(2.12)
291
+ uniformly in C >0, and hence that
292
+ P/bracketleftBig
293
+ sup
294
+ 0≤s≤tS1(XN(s))≥NC/bracketrightBig
295
+ ≤C−1(S1(xN(0))+k14t)ek11t(2.13)
296
+ also. Hence sup0≤s≤tS1(XN(s))<∞a.s. for any t, limC→∞τ(N)
297
+ 1(C) =∞
298
+ a.s., and, from (2.3) and (1.3), it thus follows that tXN∞=∞a.s. The bound
299
+ onE{S(N)
300
+ 1(t)}is now immediate, and that on E{S(N)
301
+ 0(t)}follows by applying
302
+ the same Gronwall argument to M(N)
303
+ 0(t∧τ(N)
304
+ 1(C)).
305
+ The next lemma shows that, if any T >0 is fixed and Cis chosen large
306
+ enough, then, with high probability, N−1S(N)
307
+ 0(t)≤Cholds for all 0 ≤t≤T.
308
+ Lemma 2.3 Assume that Assumptions 2.1 are satisfied, and that S(N)
309
+ 0(0)≤
310
+ NC0andS(N)
311
+ 1(0)≤NC1. Then, for any C≥2(C0+k04T)ek01T, we have
312
+ P[{τ(N)
313
+ 0(C)≤T}]≤(C1∨1)K00/(NC2),
314
+ whereK00depends on Tand the parameters of the model.
315
+ Proof. It is immediate from (2.11) and (2.6) that
316
+ S(N)
317
+ 0(t) =S(N)
318
+ 0(0)+N/integraldisplayt
319
+ 0U(N)
320
+ 0(u)du+M(N)
321
+ 0(t)
322
+ ≤S(N)
323
+ 0(0)+/integraldisplayt
324
+ 0(k01S(N)
325
+ 0(u)+Nk04)du+ sup
326
+ 0≤u≤tM(N)
327
+ 0(u).(2.14)
328
+ 8Hence, from Gronwall’s inequality, if S(N)
329
+ 0(0)≤NC0, then
330
+ S(N)
331
+ 0(t)≤/braceleftbigg
332
+ N(C0+k04T)+ sup
333
+ 0≤u≤tM(N)
334
+ 0(u)/bracerightbigg
335
+ ek01t.(2.15)
336
+ Now, considering the quadratic variation of M(N)
337
+ 0, we have
338
+ E/braceleftBigg
339
+ {M(N)
340
+ 0(t∧τ(N)
341
+ 1(C′))}2−N/integraldisplayt∧τ(N)
342
+ 1(C′)
343
+ 0V(N)
344
+ 0(u)du/bracerightBigg
345
+ = 0 (2.16)
346
+ for anyC′>0, from which it follows, much as above, that
347
+ E/parenleftBig
348
+ {M(N)
349
+ 0(t∧τ(N)
350
+ 1(C′))}2/parenrightBig
351
+ ≤E/braceleftbigg
352
+ N/integraldisplayt
353
+ 0V(N)
354
+ 0(u∧τ(N)
355
+ 1(C′))du/bracerightbigg
356
+ ≤/integraldisplayt
357
+ 0{k03ES(N)
358
+ 1(u∧τ(N)
359
+ 1(C′))+Nk05}du.
360
+ Using (2.12), we thus find that
361
+ E/parenleftBig
362
+ {M(N)
363
+ 0(t∧τ(N)
364
+ 1(C′))}2/parenrightBig
365
+ ≤k03
366
+ k11N(C1+k14T)(ek11t−1)+Nk05t,(2.17)
367
+ uniformlyforall C′. Doob’smaximal inequality appliedto M(N)
368
+ 0(t∧τ(N)
369
+ 1(C′))
370
+ now allows us to deduce that, for any C′,a >0,
371
+ P/bracketleftBig
372
+ sup
373
+ 0≤u≤TM(N)
374
+ 0(u∧τ(N)
375
+ 1(C′))> aN/bracketrightBig
376
+ ≤1
377
+ Na2/braceleftbiggk03
378
+ k11(C1+k14T){ek11T−1}+k05T/bracerightbigg
379
+ =:C1K01+K02
380
+ Na2,
381
+ say, so that, letting C′→ ∞,
382
+ P/bracketleftBig
383
+ sup
384
+ 0≤u≤TM(N)
385
+ 0(u)> aN/bracketrightBig
386
+ ≤C1K01+K02
387
+ Na2
388
+ also. Taking a=1
389
+ 2Ce−k01Tand putting the result into (2.15), the lemma
390
+ follows.
391
+ In the next theorem, we control the ‘higher ν-moments’ S(N)
392
+ r(t) ofXN(t).
393
+ 9Theorem 2.4 Assume thatAssumptions 2.1are satisfied, andthat S(N)
394
+ 1(0)≤
395
+ NC1andS(N)
396
+ p(1)(0)≤NC′
397
+ 1. Then, for 2≤r≤r(1)
398
+ maxand for any C >0, we
399
+ have
400
+ E{S(N)
401
+ r(t∧τ(N)
402
+ 0(C))} ≤(S(N)
403
+ r(0)+Nkr4t)e(kr1+Ckr2)t,0≤t≤T.(2.18)
404
+ Furthermore, if for 1≤r≤r(2)
405
+ max,S(N)
406
+ r(0)≤NCrandS(N)
407
+ p(r)(0)≤NC′
408
+ r,
409
+ then, for any γ≥1,
410
+ P[ sup
411
+ 0≤t≤TS(N)
412
+ r(t∧τ(N)
413
+ 0(C))≥NγC′′
414
+ rT]≤Kr0γ−2N−1, (2.19)
415
+ where
416
+ C′′
417
+ rT:= (Cr+kr4T+/radicalbig
418
+ (C′
419
+ r∨1))e(kr1+Ckr2)T
420
+ andKr0depends on C,Tand the parameters of the model.
421
+ Proof. Recalling (2.11), use the argument leading to (2.12) with the martin-
422
+ galesM(N)
423
+ r(t∧τ(N)
424
+ 1(C′)∧τ(N)
425
+ 0(C)), for any C′>0, to deduce that
426
+ ES(N)
427
+ r(t∧τ(N)
428
+ 1(C′)∧τ(N)
429
+ 0(C))
430
+ ≤S(N)
431
+ r(0)+/integraldisplayt
432
+ 0/parenleftBig
433
+ {kr1+Ckr2}E/braceleftBig
434
+ S(N)
435
+ r(u∧τ(N)
436
+ 1(C′)∧τ(N)
437
+ 0(C))/bracerightBig
438
+ +Nkr4/parenrightBig
439
+ du,
440
+ for 1≤r≤r(1)
441
+ max, sinceN−1S(N)
442
+ 0(u)≤Cwhenu≤τ(N)
443
+ 0(C): define k12= 0.
444
+ Gronwall’s inequality now implies that
445
+ ES(N)
446
+ r(t∧τ(N)
447
+ 1(C′)∧τ(N)
448
+ 0(C))≤(S(N)
449
+ r(0)+Nkr4t)e(kr1+Ckr2)t,(2.20)
450
+ for 1≤r≤r(1)
451
+ max, and (2.18) follows by Fatou’s lemma, on letting C′→ ∞.
452
+ Now, also from (2.11) and (2.6), we have, for t≥0 and each r≤r(1)
453
+ max,
454
+ S(N)
455
+ r(t∧τ(N)
456
+ 0(C))
457
+ =S(N)
458
+ r(0)+N/integraldisplayt∧τ(N)
459
+ 0(C)
460
+ 0U(N)
461
+ r(u)du+M(N)
462
+ r(t∧τ(N)
463
+ 0(C))
464
+ ≤S(N)
465
+ r(0)+/integraldisplayt
466
+ 0/parenleftBig
467
+ {kr1+Ckr2}S(N)
468
+ r(u∧τ(N)
469
+ 0(C))+Nkr4/parenrightBig
470
+ du
471
+ + sup
472
+ 0≤u≤tM(N)
473
+ r(u∧τ(N)
474
+ 0(C)).
475
+ 10Hence, from Gronwall’s inequality, for all t≥0 andr≤r(1)
476
+ max,
477
+ S(N)
478
+ r(t∧τ(N)
479
+ 0(C))≤/braceleftBig
480
+ N(Cr+kr4t)+ sup
481
+ 0≤u≤tM(N)
482
+ r(u∧τ(N)
483
+ 0(C))/bracerightBig
484
+ e(kr1+Ckr2)t.
485
+ (2.21)
486
+ Now, as in (2.16), we have
487
+ E/braceleftBigg
488
+ {M(N)
489
+ r(t∧τ(N)
490
+ 1(C′)∧τ(N)
491
+ 0(C))}2−N/integraldisplayt∧τ(N)
492
+ 1(C′)∧τ(N)
493
+ 0(C)
494
+ 0V(N)
495
+ r(u)du/bracerightBigg
496
+ = 0,
497
+ (2.22)
498
+ from which it follows, using (2.7), that, for 1 ≤r≤r(2)
499
+ max,
500
+ E/parenleftBig
501
+ {M(N)
502
+ r(t∧τ(N)
503
+ 1(C′)∧τ(N)
504
+ 0(C))}2/parenrightBig
505
+ ≤E/braceleftBigg
506
+ N/integraldisplayt∧τ(N)
507
+ 1(C′)∧τ(N)
508
+ 0(C))
509
+ 0V(N)
510
+ r(u)du/bracerightBigg
511
+ ≤/integraldisplayt
512
+ 0{kr3ES(N)
513
+ p(r)(u∧τ(N)
514
+ 1(C′)∧τ(N)
515
+ 0(C))+Nkr5}du
516
+ ≤N(C′
517
+ r+kp(r),4T)kr3
518
+ kp(r),1+Ckp(r),2(e(kp(r),1+Ckp(r),2t)−1)+Nkr5T,
519
+ this last by (2.20), since p(r)≤r(1)
520
+ maxfor 1≤r≤r(2)
521
+ max. Using Doob’s
522
+ inequality, it follows that, for any a >0,
523
+ P/bracketleftBig
524
+ sup
525
+ 0≤u≤TM(N)
526
+ r(u∧τ(N)
527
+ 0(C))> aN/bracketrightBig
528
+ ≤1
529
+ Na2/braceleftbiggkr3(C′
530
+ r+kp(r),4T)
531
+ kp(r),1+Ckp(r),2(e(kp(r),1+Ckp(r),2T)−1)+kr5T/bracerightbigg
532
+ =:C′
533
+ rKr1+Kr2
534
+ Na2.
535
+ Takinga=γ/radicalbig
536
+ (C′
537
+ r∨1) and putting the result into (2.21) gives (2.19), with
538
+ Kr0= (C′
539
+ rKr1+Kr2)/(C′
540
+ r∨1).
541
+ Note also that sup0≤t≤TS(N)
542
+ r(t)<∞a.s. for all 0 ≤r≤r(2)
543
+ max, in view of
544
+ Lemma 2.3 and Theorem 2.4.
545
+ In what follows, we shall particularly need to control quantities of t he
546
+ form/summationtext
547
+ J∈JαJ(xN(s))d(J,ζ), where xN:=N−1XNand
548
+ d(J,ζ) :=/summationdisplay
549
+ j≥0|Jj|ζ(j), (2.23)
550
+ 11forζ∈ Rchosen such that ζ(j)≥1 grows fast enough with j: see (4.12).
551
+ Defining
552
+ τ(N)(a,ζ) := inf/braceleftBigg
553
+ s:/summationdisplay
554
+ J∈JαJ(xN(s))d(J,ζ)≥a/bracerightBigg
555
+ ,(2.24)
556
+ infinite if there is no such s, we show in the following corollary that, under
557
+ suitable assumptions, τ(N)(a,ζ) is rarely less than T.
558
+ Corollary 2.5 Assume that Assumptions 2.1 hold, and that ζis such that
559
+ /summationdisplay
560
+ J∈JαJ(N−1X)d(J,ζ)≤ {k1N−1Sr(X)+k2}b(2.25)
561
+ for some 1≤r:=r(ζ)≤r(2)
562
+ maxand some b=b(ζ)≥1. For this value
563
+ ofr, assume that S(N)
564
+ r(0)≤NCrandS(N)
565
+ p(r)(0)≤NC′
566
+ rfor some constants
567
+ CrandC′
568
+ r. Assume further that S(N)
569
+ 0(0)≤NC0,S(N)
570
+ 1(0)≤NC1for some
571
+ constants C0,C1, and define C:= 2(C0+k04T)ek01T. Then
572
+ P[τ(N)(a,ζ)≤T]≤N−1{Kr0γ−2
573
+ a+K00(C1∨1)C−2},
574
+ for anya≥ {k2+k1C′′
575
+ rT}b, whereγa:= (a1/b−k2)/{k1C′′
576
+ rT},Kr0andC′′
577
+ rT
578
+ are as in Theorem 2.4, and K00is as in Lemma 2.3.
579
+ Proof. In view of (2.25), it is enough to bound the probability
580
+ P[ sup
581
+ 0≤t≤TS(N)
582
+ r(t)≥N(a1/b−k2)/k1].
583
+ However, Lemma 2.3 and Theorem 2.4 together bound this probability by
584
+ N−1/braceleftbig
585
+ Kr0γ−2
586
+ a+K00(C1∨1)C−2/bracerightbig
587
+ ,
588
+ whereγais as defined above, as long as a1/b−k2≥k1C′′
589
+ rT.
590
+ If (2.25) is satisfied,/summationtext
591
+ J∈JαJ(xN(s))d(J,ζ)isa.s. bounded on0 ≤s≤T,
592
+ becauseS(N)
593
+ r(s) is. The corollary shows that the sum is then bounded by
594
+ {k2+k1C′′
595
+ r,T}b, except on an event of probability of order O(N−1). Usually,
596
+ one can choose b= 1.
597
+ 123 Semigroup properties
598
+ We make the following initial assumptions about the matrix A: first, that
599
+ Aij≥0 for alli/ne}ationslash=j≥0;/summationdisplay
600
+ j/negationslash=iAji<∞for alli≥0,(3.1)
601
+ and then that, for some µ∈RZ+
602
+ +such that µ(m)≥1 for each m≥0, and
603
+ for some w≥0,
604
+ ATµ≤wµ. (3.2)
605
+ We then use µto define the µ-norm
606
+ /⌊ard⌊lξ/⌊ard⌊lµ:=/summationdisplay
607
+ m≥0µ(m)|ξm|onRµ:={ξ∈ R:/⌊ard⌊lξ/⌊ard⌊lµ<∞}.(3.3)
608
+ Note that there may be many possible choices for µ. In what follows, it is
609
+ important that Fbe a Lipschitz operator with respect to the µ-norm, and
610
+ this has to be borne in mind when choosing µ.
611
+ Setting
612
+ Qij:=AT
613
+ ijµ(j)/µ(i)−wδij, (3.4)
614
+ whereδis the Kronecker delta, we note that Qij≥0 fori/ne}ationslash=j, and that
615
+ 0≤/summationdisplay
616
+ j/negationslash=iQij=/summationdisplay
617
+ j/negationslash=iAT
618
+ ijµ(j)/µ(i)≤w−Aii=−Qii,
619
+ using (3.2) for the inequality, so that Qii≤0. Hence Qcan be augmented to
620
+ a conservative Q–matrix, in the sense of Markov jump processes, by adding a
621
+ coffin state ∂, and setting Qi∂:=−/summationtext
622
+ j≥0Qij≥0. LetP(·) denote the semi-
623
+ group of Markov transition matrices corresponding to the minimal p rocess
624
+ associated with Q; then, in particular,
625
+ Q=P′(0) and P′(t) =QP(t) for all t≥0 (3.5)
626
+ (Reuter 1957, Theorem 3). Set
627
+ RT
628
+ ij(t) :=ewtµ(i)Pij(t)/µ(j). (3.6)
629
+ 13Theorem 3.1 LetAsatisfy Assumptions (3.1)and(3.2). Then, with the
630
+ above definitions, Ris a strongly continuous semigroup on Rµ, and
631
+ /summationdisplay
632
+ i≥0µ(i)Rij(t)≤µ(j)ewtfor alljandt. (3.7)
633
+ Furthermore, the sums/summationtext
634
+ j≥0Rij(t)Ajk= (R(t)A)ikare well defined for all
635
+ i,k, and
636
+ A=R′(0)andR′(t) =R(t)Afor allt≥0.(3.8)
637
+ Proof. We note first that, for x∈ Rµ,
638
+ /⌊ard⌊lR(t)x/⌊ard⌊lµ≤/summationdisplay
639
+ i≥0µ(i)/summationdisplay
640
+ j≥0Rij(t)|xj|=ewt/summationdisplay
641
+ i≥0/summationdisplay
642
+ j≥0µ(j)Pji(t)|xj|
643
+ ≤ewt/summationdisplay
644
+ j≥0µ(j)|xj|=ewt/⌊ard⌊lx/⌊ard⌊lµ, (3.9)
645
+ sinceP(t) is substochastic on Z+; henceR:Rµ→ Rµ. To show strong
646
+ continuity, we take x∈ Rµ, and consider
647
+ /⌊ard⌊lR(t)x−x/⌊ard⌊lµ=/summationdisplay
648
+ i≥0µ(i)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/summationdisplay
649
+ j≥0Rij(t)xj−xi/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle=/summationdisplay
650
+ i≥0/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleewt/summationdisplay
651
+ j≥0µ(j)Pji(t)xj−µ(i)xi/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle
652
+ ≤(ewt−1)/summationdisplay
653
+ i≥0/summationdisplay
654
+ j≥0µ(j)Pji(t)xj+/summationdisplay
655
+ i≥0/summationdisplay
656
+ j/negationslash=iµ(j)Pji(t)xj+/summationdisplay
657
+ i≥0µ(i)xi(1−Pii(t))
658
+ ≤(ewt−1)/summationdisplay
659
+ j≥0µ(j)xj+2/summationdisplay
660
+ i≥0µ(i)xi(1−Pii(t)),
661
+ from which it follows that lim t→0/⌊ard⌊lR(t)x−x/⌊ard⌊lµ= 0, by dominated conver-
662
+ gence, since lim t→0Pii(t) = 1 for each i≥0.
663
+ The inequality (3.7) follows from the definition of Rand the fact that P
664
+ is substochastic on Z+. Then
665
+ (ATRT(t))ij=/summationdisplay
666
+ k/negationslash=iQikµ(i)
667
+ µ(k)ewtµ(k)
668
+ µ(j)Pkj(t)+(Qii+w)ewtµ(i)
669
+ µ(j)Pij(t)
670
+ =µ(i)
671
+ µ(j)[(QP(t))ij+wPij(t)]ewt,
672
+ 14with (QP(t))ij=/summationtext
673
+ k≥0QikPkj(t) well defined because P(t) is sub-stochastic
674
+ andQis conservative. Using (3.5), this gives
675
+ (ATRT(t))ij=µ(i)
676
+ µ(j)d
677
+ dt[Pij(t)ewt] =d
678
+ dtRT
679
+ ij(t),
680
+ and this establishes (3.8).
681
+ 4 Main approximation
682
+ LetXN,N≥1, beasequence ofpure jumpMarkov processes asinSection 1,
683
+ withAandFdefined as in (1.4) and (1.5), and suppose that F:Rµ→ Rµ,
684
+ withRµas defined in (3.3), for some µsuch that Assumption (3.2) holds.
685
+ Suppose also that Fis locally Lipschitz in the µ-norm: for any z >0,
686
+ sup
687
+ x/negationslash=y:/bardblx/bardblµ,/bardbly/bardblµ≤z/⌊ard⌊lF(x)−F(y)/⌊ard⌊lµ//⌊ard⌊lx−y/⌊ard⌊lµ≤K(µ,F;z)<∞.(4.1)
688
+ Then, for x(0)∈ RµandRas in (3.6), the integral equation
689
+ x(t) =R(t)x(0)+/integraldisplayt
690
+ 0R(t−s)F(x(s))ds. (4.2)
691
+ has a unique continuous solution xinRµon some non-empty time interval
692
+ [0,tmax), such that, if tmax<∞, then/⌊ard⌊lx(t)/⌊ard⌊lµ→ ∞ast→tmax(Pazy 1983,
693
+ Theorem 1.4, Chapter 6). Thus, if Awere the generator of R, the function x
694
+ would be a mild solution of the deterministic equations (1.4). We now wish
695
+ to show that the process xN:=N−1XNis close to x. To do so, we need a
696
+ corresponding representation for XN.
697
+ To find such a representation, let W(t),t≥0, be a pure jump path on X+
698
+ that has only finitely many jumps up to time T. Then we can write
699
+ W(t) =W(0)+/summationdisplay
700
+ j:σj≤t∆W(σj),0≤t≤T, (4.3)
701
+ where ∆W(s) :=W(s)−W(s−)andσj,j≥1, denote thetimes when Whas
702
+ its jumps. Now let Asatisfy (3.1) and (3.2), and let R(·) be the associated
703
+ semigroup, as defined in (3.6). Define the path W∗(t), 0≤t≤T, from the
704
+ equation
705
+ W∗(t) :=R(t)W(0)+/summationtext
706
+ j:σj≤tR(t−σj)∆j−/integraltextt
707
+ 0R(t−s)AW(s)ds,
708
+ (4.4)
709
+ 15where ∆ j:= ∆W(σj). Note that the latter integral makes sense, because
710
+ each of the sums/summationtext
711
+ j≥0Rij(t)Ajkis well defined, from Theorem 3.1, and
712
+ because only finitely many of the coordinates of Ware non-zero.
713
+ Lemma 4.1 W∗(t) =W(t)for all0≤t≤T.
714
+ Proof. Fix any t, and suppose that W∗(s) =W(s) for alls≤t. This is
715
+ clearly the case for t= 0. Let σ(t)> tdenote the time of the first jump
716
+ ofWaftert. Then, for any 0 < h < σ(t)−t, using the semigroup property
717
+ forRand (4.4),
718
+ W∗(t+h)−W∗(t)
719
+ = (R(h)−I)R(t)W(0)+/summationdisplay
720
+ j:σj≤t(R(h)−I)R(t−σj)∆j (4.5)
721
+ −/integraldisplayt
722
+ 0(R(h)−I)R(t−s)AW(s)ds−/integraldisplayt+h
723
+ tR(t+h−s)AW(t)ds,
724
+ where, in the last integral, we use the fact that there are no jumps ofW
725
+ between tandt+h. Thus we have
726
+ W∗(t+h)−W∗(t)
727
+ = (R(h)−I)
728
+
729
+ R(t)W(0)+/summationdisplay
730
+ j:σj≤tR(t−σj)∆j−/integraldisplayt
731
+ 0R(t−s)AW(s)ds
732
+
733
+
734
+ −/integraldisplayt+h
735
+ tR(t+h−s)AW(t)ds
736
+ = (R(h)−I)W(t)−/integraldisplayt+h
737
+ tR(t+h−s)AW(t)ds. (4.6)
738
+ But now, for x∈ X+,
739
+ /integraldisplayt+h
740
+ tR(t+h−s)Axds= (R(h)−I)x,
741
+ from (3.8), so that W∗(t+h) =W∗(t) for all t+h < σ(t), implying that
742
+ W∗(s) =W(s) for all s < σ(t). On the other hand, from (4.4), we have
743
+ W∗(σ(t))−W∗(σ(t)−) = ∆W(σ(t)), so that W∗(s) =W(s) for alls≤σ(t).
744
+ Thus we can prove equality over the interval [0 ,σ1], and then successively
745
+ over the intervals [ σj,σj+1], until [0 ,T] is covered.
746
+ 16Now suppose that Warises as a realization of XN. ThenXNhas transi-
747
+ tion rates such that
748
+ MN(t) :=/summationdisplay
749
+ j:σj≤t∆XN(σj)−/integraldisplayt
750
+ 0AXN(s)ds−/integraldisplayt
751
+ 0NF(xN(s))ds(4.7)
752
+ is a zero mean local martingale. In view of Lemma 4.1, we can use (4.4) t o
753
+ write
754
+ XN(t) =R(t)XN(0)+/tildewiderMN(t)+N/integraldisplayt
755
+ 0R(t−s)F(xN(s))ds,(4.8)
756
+ where
757
+ /tildewiderMN(t) :=/summationdisplay
758
+ j:σj≤tR(t−σj)∆XN(σj)
759
+ −/integraldisplayt
760
+ 0R(t−s)AXN(s)ds−/integraldisplayt
761
+ 0R(t−s)NF(xN(s))ds.(4.9)
762
+ Thus, comparing (4.8) and (4.2), we expect xNandxto be close, for
763
+ 0≤t≤T < tmax, provided that we can show that supt≤T/⌊ard⌊l/tildewidemN(t)/⌊ard⌊lµis small,
764
+ where/tildewidemN(t) :=N−1/tildewiderMN(t). Indeed, if xN(0) andx(0) are close, then
765
+ /⌊ard⌊lxN(t)−x(t)/⌊ard⌊lµ
766
+ ≤ /⌊ard⌊lR(t)(xN(0)−x(0))/⌊ard⌊lµ
767
+ +/integraldisplayt
768
+ 0/⌊ard⌊lR(t−s)[F(xN(s))−F(x(s))]/⌊ard⌊lµds+/⌊ard⌊l/tildewidemN(t)/⌊ard⌊lµ
769
+ ≤ewt/⌊ard⌊lxN(0)−x(0)/⌊ard⌊lµ
770
+ +/integraldisplayt
771
+ 0ew(t−s)K(µ,F;2ΞT)/⌊ard⌊lxN(s)−x(s)/⌊ard⌊lµds+/⌊ard⌊l/tildewidemN(t)/⌊ard⌊lµ,(4.10)
772
+ by (3.9), with the stage apparently set for Gronwall’s inequality, ass uming
773
+ that/⌊ard⌊lxN(0)−x(0)/⌊ard⌊lµand sup0≤t≤T/⌊ard⌊l/tildewidemN(t)/⌊ard⌊lµare small enough that then
774
+ /⌊ard⌊lxN(t)/⌊ard⌊lµ≤2ΞTfor 0≤t≤T, where Ξ T:= sup0≤t≤T/⌊ard⌊lx(t)/⌊ard⌊lµ.
775
+ Bounding sup0≤t≤T/⌊ard⌊l/tildewidemN(t)/⌊ard⌊lµis, however, not so easy. Since /tildewiderMNis not
776
+ itselfamartingale, wecannotdirectlyapplymartingaleinequalitiestoc ontrol
777
+ its fluctuations. However, since
778
+ /tildewiderMN(t) =/integraldisplayt
779
+ 0R(t−s)dMN(s), (4.11)
780
+ 17we can hope to use control over the local martingale MNinstead. For this
781
+ and the subsequent argument, we introduce some further assum ptions.
782
+ Assumption 4.2
783
+ 1. There exists r=rµ≤r(2)
784
+ maxsuch that supj≥0{µ(j)/νr(j)}<∞.
785
+ 2. There exists ζ∈ Rwithζ(j)≥1for alljsuch that (2.25)is satisfied
786
+ for some b=b(ζ)≥1andr=r(ζ)such that 1≤r(ζ)≤r(2)
787
+ max, and that
788
+ Z:=/summationdisplay
789
+ k≥0µ(k)(|Akk|+1)/radicalbig
790
+ ζ(k)<∞. (4.12)
791
+ The requirement that ζsatisfies (4.12) as well as satisfying (2.25) for some
792
+ r≤r(2)
793
+ maximplies in practice that it must be possible to take r(1)
794
+ maxandr(2)
795
+ max
796
+ to be quite large in Assumption 2.1; see the examples in Section 5.
797
+ Note that part 1 of Assumption 4.2 implies that lim j→∞{µ(j)/νr(j)}= 0
798
+ for some r= ˜rµ≤rµ+1. We define
799
+ ρ(ζ,µ) := max {r(ζ),p(r(ζ)),˜rµ}, (4.13)
800
+ wherep(·) is as in Assumptions 2.1. We can now prove the following lemma,
801
+ which enables us to control the paths of /tildewiderMNby using fluctuation bounds for
802
+ the martingale MN.
803
+ Lemma 4.3 Under Assumption 4.2,
804
+ /tildewiderMN(t) =MN(t)+/integraldisplayt
805
+ 0R(t−s)AMN(s)ds.
806
+ Proof. From (3.8), we have
807
+ R(t−s) =I+/integraldisplayt−s
808
+ 0R(v)Adv.
809
+ Substituting this into (4.11), we obtain
810
+ /tildewiderMN(t) =/integraldisplayt
811
+ 0R(t���s)dMN(s)
812
+ 18=MN(t)+/integraldisplayt
813
+ 0/braceleftbigg/integraldisplayt
814
+ 0R(v)A1[0,t−s](v)dv/bracerightbigg
815
+ dMN(s)
816
+ =MN(t)+/integraldisplayt
817
+ 0/braceleftbigg/integraldisplayt
818
+ 0R(v)A1[0,t−s](v)dv/bracerightbigg
819
+ dXN(s)
820
+ −/integraldisplayt
821
+ 0/braceleftbigg/integraldisplayt
822
+ 0R(v)A1[0,t−s](v)dv/bracerightbigg
823
+ F0(xN(s))ds.
824
+ It remains to change the order of integration in the double integrals , for
825
+ which we use Fubini’s theorem.
826
+ In the first, the outer integral is almost surely a finite sum, and at e ach
827
+ jump time tXN
828
+ lwe havedXN(tXN
829
+ l)∈ J. Hence it is enough that, for each i,
830
+ mandt,/summationtext
831
+ j≥0Rij(t)Ajmis absolutely summable, which follows from Theo-
832
+ rem 3.1. Thus we have
833
+ /integraldisplayt
834
+ 0/braceleftbigg/integraldisplayt
835
+ 0R(v)A1[0,t−s](v)dv/bracerightbigg
836
+ dXN(s) =/integraldisplayt
837
+ 0R(v)A{XN(t−v)−XN(0)}dv.
838
+ (4.14)
839
+ For the second, the k-th component of R(v)AF0(xN(s)) is just
840
+ /summationdisplay
841
+ j≥0Rkj(v)/summationdisplay
842
+ l≥0Ajl/summationdisplay
843
+ J∈JJlαJ(xN(s)). (4.15)
844
+ Now, from (3.7), we have 0 ≤Rkj(v)≤µ(j)ewv/µ(k), and
845
+ /summationdisplay
846
+ j≥0µ(j)|Ajl| ≤µ(l)(2|All|+w), (4.16)
847
+ becauseATµ≤wµ. Hence, puttingabsolutevaluesinthesummandsin(4.15)
848
+ yields at most
849
+ ewv
850
+ µ(k)/summationdisplay
851
+ J∈JαJ(xN(s))/summationdisplay
852
+ l≥0|Jl|µ(l)(2|All|+w).
853
+ Now, in view of (4.12) and since ζ(j)≥1 for allj, there is a constant K <∞
854
+ such that µ(l)(2|All|+w)≤Kζ(l). Furthermore, ζsatisfies (2.25), so that,
855
+ by Corollary 2.5,/summationtext
856
+ J∈JαJ(xN(s))/summationtext
857
+ l≥0|Jl|ζ(l) is a.s. uniformly bounded in
858
+ 0≤s≤T. Hence we can apply Fubini’s theorem, obtaining
859
+ /integraldisplayt
860
+ 0/braceleftbigg/integraldisplayt
861
+ 0R(v)A1[0,t−s](v)dv/bracerightbigg
862
+ F0(xN(s))ds=/integraldisplayt
863
+ 0R(v)A/braceleftbigg/integraldisplayt−v
864
+ 0F0(xN(s))ds/bracerightbigg
865
+ dv,
866
+ 19and combining this with (4.14) proves the lemma.
867
+ We now introduce the exponential martingales that we use to bound the
868
+ fluctuations of MN. Forθ∈RZ+bounded and x∈ Rµ,
869
+ ZN,θ(t) :=eθTxN(t)exp/braceleftBig
870
+ −/integraltextt
871
+ 0gNθ(xN(s−))ds/bracerightBig
872
+ , t≥0,
873
+ is a non-negative finite variation local martingale, where
874
+ gNθ(ξ) :=/summationdisplay
875
+ J∈JNαJ(ξ)/parenleftBig
876
+ eN−1θTJ−1/parenrightBig
877
+ .
878
+ Fort≥0, we have
879
+ logZN,θ(t) =θTxN(t)−/integraldisplayt
880
+ 0gNθ(xN(s−))ds
881
+ =θTmN(t)−/integraldisplayt
882
+ 0ϕN,θ(xN(s−),s)ds, (4.17)
883
+ where
884
+ ϕN,θ(ξ) :=/summationdisplay
885
+ J∈JNαJ(ξ)/parenleftBig
886
+ eN−1θTJ−1−N−1θTJ/parenrightBig
887
+ ,(4.18)
888
+ andmN(t) :=N−1MN(t). Note also that we can write
889
+ ϕN,θ(ξ) =N/integraldisplay1
890
+ 0(1−r)D2vN(ξ,rθ)[θ,θ]dr, (4.19)
891
+ where
892
+ vN(ξ,θ′) :=/summationdisplay
893
+ J∈JαJ(ξ)eN−1(θ′)TJ,
894
+ andD2vNdenotes thematrixofsecond derivatives withrespect totheseco nd
895
+ argument:
896
+ D2vN(ξ,θ′)[ζ1,ζ2] :=N−2/summationdisplay
897
+ J∈JαJ(ξ)eN−1(θ′)TJζT
898
+ 1JJTζ2(4.20)
899
+ for anyζ1,ζ2∈ Rµ.
900
+ Now choose any B:= (Bk, k≥0)∈ R, and define ˜ τ(N)
901
+ k(B) by
902
+ ˜τ(N)
903
+ k(B) := inf/braceleftBigg
904
+ t≥0:/summationdisplay
905
+ J:Jk/negationslash=0αJ(xN(t−))> Bk/bracerightBigg
906
+ .
907
+ Our exponential bound is as follows.
908
+ 20Lemma 4.4 For anyk≥0,
909
+ P
910
+ sup
911
+ 0≤t≤T∧˜τ(N)
912
+ k(B)|mk
913
+ N(t)| ≥δ
914
+ ≤2exp(−δ2N/2BkK∗T).
915
+ for all0< δ≤BkK∗T, whereK∗:=J2
916
+ ∗eJ∗, andJ∗is as in(1.2).
917
+ Proof. Takeθ=e(k)β, forβto be chosen later. We shall argue by stopping
918
+ the local martingale ZN,θat timeσ(N)(k,δ), where
919
+ σ(N)(k,δ) :=T∧˜τ(N)
920
+ k(B)∧inf{t:mk
921
+ N(t)≥δ}.
922
+ Note that eN−1θTJ≤eJ∗, so long as |β| ≤N, so that
923
+ D2vN(ξ,rθ)[θ,θ]≤N−2/parenleftBigg/summationdisplay
924
+ J:Jk/negationslash=0αJ(ξ)/parenrightBigg
925
+ β2K∗.
926
+ Thus, from (4.19), we have
927
+ ϕN,θ(xN(u−))≤1
928
+ 2N−1Bkβ2K∗, u≤˜τ(N)
929
+ k(B),
930
+ and hence, on the event that σ(N)(k,δ) = inf{t:mk
931
+ N(t)≥δ} ≤(T∧˜τ(N)
932
+ k(B)),
933
+ we have
934
+ ZN,θ(σ(k,δ))≥exp{βδ−1
935
+ 2N−1Bkβ2K∗T}.
936
+ But since ZN,θ(0) = 1, it now follows from the optional stopping theorem
937
+ and Fatou’s lemma that
938
+ 1≥E{ZN,θ(σ(N)(k,δ))}
939
+ ≥P/bracketleftBig
940
+ sup
941
+ 0≤t≤T∧˜τ(N)
942
+ k(B)mk
943
+ N(t)≥δ/bracketrightBig
944
+ exp{βδ−1
945
+ 2N−1Bkβ2K∗T}.
946
+ We can choose β=δN/B kK∗T, as long as δ/BkK∗T≤1, obtaining
947
+ P
948
+ sup
949
+ 0≤t≤T∧˜τ(N)
950
+ k(B)mk
951
+ N(t)≥δ
952
+ ≤exp(−δ2N/2BkK∗T).
953
+ Repeating with
954
+ ˜σ(N)(k,δ) :=T∧˜τ(N)
955
+ k(B)∧inf{t:−mk
956
+ N(t)≥δ},
957
+ 21and choosing β=δN/B kK∗T, gives the lemma.
958
+ Theprecedinglemmagivesaboundforeachindividualcomponentof MN.
959
+ We need first to translate this into a statement for all components simulta-
960
+ neously. For ζas in Assumption 4.2, we start by writing
961
+ Z(1)
962
+ ∗:= max
963
+ k≥1k−1#{m:ζ(m)≤k};Z(2)
964
+ ∗:= sup
965
+ k≥0µ(k)(|Akk|+1)/radicalbig
966
+ ζ(k).(4.21)
967
+ Z(2)
968
+ ∗is clearly finite, because of Assumption 4.2, and the same is true for Z(1)
969
+
970
+ also, since Zof Assumption 4.2 is at least # {m:ζ(m)≤k}/√
971
+ k, for each k.
972
+ Then, using the definition (2.24) of τ(N)(a,ζ), note that, for every k,
973
+ /summationdisplay
974
+ J:Jk/negationslash=0αJ(xN(t))h(k)≤/summationdisplay
975
+ J:Jk/negationslash=0αJ(xN(t))h(k)d(J,ζ)
976
+ |Jk|ζ(k)≤ah(k)
977
+ ζ(k),(4.22)
978
+ for anyt < τ(N)(a,ζ) and any h∈ R, and that, for any K ⊆Z+,
979
+ /summationdisplay
980
+ k∈K/summationdisplay
981
+ J:Jk/negationslash=0αJ(xN(t))h(k)≤/summationdisplay
982
+ k∈K/summationdisplay
983
+ J:Jk/negationslash=0αJ(xN(t))h(k)d(J,ζ)
984
+ |Jk|ζ(k)
985
+ ≤a
986
+ mink∈K(ζ(k)/h(k)). (4.23)
987
+ From (4.22) with h(k) = 1 for all k, if we choose B:= (a/ζ(k), k≥0), then
988
+ τ(N)(a,ζ)≤˜τ(N)
989
+ k(B) for allk. For this choice of B, we can take
990
+ δ2
991
+ k:=δ2
992
+ k(a) :=4aK∗TlogN
993
+ Nζ(k)=4BkK∗TlogN
994
+ N(4.24)
995
+ in Lemma 4.4 for k∈κN(a), where
996
+ κN(a) :=/braceleftbig
997
+ k:ζ(k)≤1
998
+ 4aK∗TN/logN/bracerightbig
999
+ ={k:Bk≥4logN/K∗TN},
1000
+ (4.25)
1001
+ since then δk(a)≤BkK∗T. Note that then, from (4.12),
1002
+ /summationdisplay
1003
+ k∈κN(a)µ(k)δk(a)≤2Z/radicalbig
1004
+ aK∗TN−1logN, (4.26)
1005
+ withZas defined in Assumption 4.2, and that
1006
+ |κN(a)| ≤1
1007
+ 4aZ(1)
1008
+ ∗K∗TN/logN. (4.27)
1009
+ 22Lemma 4.5 If Assumptions 4.2 are satisfied, taking δk(a)andκN(a)as
1010
+ defined in (4.24)and(4.25), and for any η∈ R, we have
1011
+ 1.P
1012
+ /uniondisplay
1013
+ k∈κN(a)/braceleftBig
1014
+ sup
1015
+ 0≤t≤T∧τ(N)(a,ζ)|mN(t)| ≥δk(a)/bracerightBig
1016
+ ≤aZ(1)
1017
+ ∗K∗T
1018
+ 2NlogN;
1019
+ 2.P
1020
+ /summationdisplay
1021
+ k/∈κN(a)Xk
1022
+ N(t) = 0for all0≤t≤T∧τ(N)(a,ζ)
1023
+ ≥1−4logN
1024
+ K∗N;
1025
+ 3. sup
1026
+ 0≤t≤T∧τ(N)(a,ζ)
1027
+
1028
+ /summationdisplay
1029
+ k/∈κN(a)η(k)|Fk(xN(t))|
1030
+
1031
+ ≤aJ∗
1032
+ mink/∈κN(a)(ζ(k)/η(k)).
1033
+ Proof. For part 1, use Lemma 4.4 together with (4.24) and (4.27) to give
1034
+ the bound. For part 2, the total rate of jumps into coordinates w ith indices
1035
+ k /∈κN(a) is
1036
+ /summationdisplay
1037
+ k/∈κN(a)/summationdisplay
1038
+ J:Jk/negationslash=0αJ(xN(t))≤a
1039
+ mink/∈κN(a)ζ(k),
1040
+ ift≤τ(N)(a,ζ),using(4.23)with K= (κN(a))c,which, combinedwith(4.25),
1041
+ proves the claim. For the final part, if t≤τ(N)(a,ζ),
1042
+ /summationdisplay
1043
+ k/∈κN(a)η(k)|Fk(xN(t))| ≤/summationdisplay
1044
+ k/∈κN(a)η(k)/summationdisplay
1045
+ J:Jk/negationslash=0αJ(xN(t))J∗,
1046
+ and the inequality follows once more from (4.23).
1047
+ LetB(1)
1048
+ N(a) andB(2)
1049
+ N(a) denote the events
1050
+ B(1)
1051
+ N(a) :=
1052
+
1053
+ /summationdisplay
1054
+ k/∈κN(a)Xk
1055
+ N(t) = 0 for all 0 ≤t≤T∧τ(N)(a,ζ)
1056
+
1057
+ ;
1058
+ B(2)
1059
+ N(a) :=
1060
+ /intersectiondisplay
1061
+ k∈κN(a)/braceleftBig
1062
+ sup
1063
+ 0≤t≤T∧τ(N)(a,ζ)|mN(t)| ≤δk(a)/bracerightBig
1064
+ ,(4.28)
1065
+ and setBN(a) :=B(1)
1066
+ N(a)∩B(2)
1067
+ N(a). Then, by Lemma 4.5, we deduce that
1068
+ P[BN(a)c]≤aZ(1)
1069
+ ∗K∗T
1070
+ 2NlogN+4logN
1071
+ K∗N, (4.29)
1072
+ 23of order O(N−1logN) for each fixed a. Thus we have all the components
1073
+ ofMNsimultaneously controlled, except on a set of small probability. We
1074
+ now translate this into the desired assertion about the fluctuation s of/tildewidemN.
1075
+ Lemma 4.6 If Assumptions 4.2 are satisfied, then, on the event BN(a),
1076
+ sup
1077
+ 0≤t≤T∧τ(N)(a,ζ)/⌊ard⌊l/tildewidemN(t)/⌊ard⌊lµ≤√aK4.6/radicalbigg
1078
+ logN
1079
+ N,
1080
+ where the constant K4.6depends on Tand the parameters of the process.
1081
+ Proof. From Lemma 4.3, it follows that
1082
+ sup
1083
+ 0≤t≤T∧τ(N)(a,ζ)/⌊ard⌊l/tildewidemN(t)/⌊ard⌊lµ (4.30)
1084
+ ≤sup
1085
+ 0≤t≤T∧τ(N)(a,ζ)/⌊ard⌊lmN(t)/⌊ard⌊lµ+ sup
1086
+ 0≤t≤T∧τ(N)(a,ζ)/integraldisplayt
1087
+ 0/⌊ard⌊lR(t−s)AmN(s)/⌊ard⌊lµds.
1088
+ For the first term, on BN(a) and for 0 ≤t≤T∧τ(N)(a,ζ), we have
1089
+ /⌊ard⌊lmN(t)/⌊ard⌊lµ≤/summationdisplay
1090
+ k∈κN(a)µ(k)δk(a)+/integraldisplayt
1091
+ 0/summationdisplay
1092
+ k/∈κN(a)µ(k)|Fk(xN(u))|du.
1093
+ The first sum is bounded using (4.26) by 2 Z√aK∗T N−1/2√logN, the sec-
1094
+ ond, from Lemma 4.5 and (4.25), by
1095
+ TaJ∗
1096
+ mink/∈κN(a)(ζ(k)/µ(k))≤Z(2)
1097
+ ∗2J∗/radicalbigg
1098
+ Ta
1099
+ K∗/radicalbigg
1100
+ logN
1101
+ N.
1102
+ For the second term in (4.30), from (3.7) and (4.16), we note that
1103
+ /⌊ard⌊lR(t−s)AmN(s)/⌊ard⌊lµ≤/summationdisplay
1104
+ k≥0µ(k)/summationdisplay
1105
+ l≥0Rkl(t−s)/summationdisplay
1106
+ r≥0|Alr||mr
1107
+ N(s)|
1108
+ ≤ew(t−s)/summationdisplay
1109
+ l≥0µ(l)/summationdisplay
1110
+ r≥0|Alr||mr
1111
+ N(s)|
1112
+ ≤ew(t−s)/summationdisplay
1113
+ r≥0µ(r){2|Arr|+w}|mr
1114
+ N(s)|.
1115
+ 24OnBN(a) and for 0 ≤s≤T∧τ(N)(a,ζ), from (4.12), the sum for r∈κN(a)
1116
+ is bounded using
1117
+ /summationdisplay
1118
+ r∈κN(a)µ(r){2|Arr|+w}|mr
1119
+ N(s)|
1120
+ ≤/summationdisplay
1121
+ r∈κN(a)µ(r){2|Arr|+w}δr(a)
1122
+ ≤/summationdisplay
1123
+ r∈κN(a)µ(r){2|Arr|+w}/radicalBigg
1124
+ 4aK∗TlogN
1125
+ Nζ(r)
1126
+ ≤(2∨w)Z/radicalbig
1127
+ 4aK∗T/radicalbigg
1128
+ logN
1129
+ N.
1130
+ The remaining sum is then bounded by Lemma 4.5, on the set BN(a) and
1131
+ for 0≤s≤T∧τ(N)(a,ζ), giving at most
1132
+ /summationdisplay
1133
+ r/∈κN(a)µ(r){2|Arr|+w}|mr
1134
+ N(s)|
1135
+ ≤/summationdisplay
1136
+ r/∈κN(a)µ(r){2|Arr|+w}/integraldisplays
1137
+ 0|Fr(xN(t))|dt
1138
+ ≤(2∨w)saJ∗
1139
+ mink/∈κN(a)(ζ(k)/µ(k){|Akk|+1})
1140
+ ≤(2∨w)Z(2)
1141
+ ∗2J∗/radicalbigg
1142
+ Ta
1143
+ K∗/radicalbigg
1144
+ logN
1145
+ N.
1146
+ Integrating, it follows that
1147
+ sup
1148
+ 0≤t≤T∧τ(N)(a,ζ)/integraldisplayt
1149
+ 0/⌊ard⌊lR(t−s)AmN(s)/⌊ard⌊lµds
1150
+ ≤(2T∨1)ewT/braceleftBigg/radicalbig
1151
+ 4aK∗TZ+Z(2)
1152
+ ∗J2J∗/radicalbigg
1153
+ Ta
1154
+ K∗/bracerightBigg/radicalbigg
1155
+ logN
1156
+ N,
1157
+ and the lemma follows.
1158
+ This has now established the control on sup0≤t≤T/⌊ard⌊l/tildewidemN(t)/⌊ard⌊lµthat we need,
1159
+ in order to translate (4.10) into a proof of the main theorem.
1160
+ 25Theorem 4.7 Suppose that (1.2),(1.3),(3.1),(3.2)and(4.1)are all satis-
1161
+ fied, and that Assumptions 2.1 and 4.2 hold. Recalling the defi nition(4.13)
1162
+ ofρ(ζ,µ), forζas given in Assumption 4.2, suppose that S(N)
1163
+ ρ(ζ,µ)(0)≤NC∗
1164
+ for some C∗<∞.
1165
+ Letxdenote the solution to (4.2)with initial condition x(0)satisfying
1166
+ Sρ(ζ,µ)(x(0))<∞. Thentmax=∞.
1167
+ Fix any T, and define ΞT:= sup0≤t≤T/⌊ard⌊lx(t)/⌊ard⌊lµ. If/⌊ard⌊lxN(0)−x(0)/⌊ard⌊lµ≤
1168
+ 1
1169
+ 2ΞTe−(w+k∗)T, wherek∗:=ewTK(µ,F;2ΞT), then there exist constants c1,c2
1170
+ depending on C∗,Tand the parameters of the process, such that for all N
1171
+ large enough
1172
+ P/parenleftBigg
1173
+ sup
1174
+ 0≤t≤T/⌊ard⌊lxN(t)−x(t)/⌊ard⌊lµ>/parenleftBigg
1175
+ ewT/⌊ard⌊lxN(0)−x(0)/⌊ard⌊lµ+c1/radicalbigg
1176
+ logN
1177
+ N/parenrightBigg
1178
+ ek∗T/parenrightBigg
1179
+ ≤c2logN
1180
+ N. (4.31)
1181
+ Proof. AsS(N)
1182
+ ρ(ζ,µ)(0)≤NC∗, it follows also that S(N)
1183
+ r(0)≤NC∗for all
1184
+ 0≤r≤ρ(ζ,µ). Fix any T < tmax, takeC:= 2(C∗+k04T)ek01T, and observe
1185
+ that, for r≤ρ(ζ,µ)∧r(2)
1186
+ max, and such that p(r)≤ρ(ζ,µ), we can take
1187
+ C′′
1188
+ rT≤/tildewideCrT:={2(C∗∨1)+kr4T}e(kr1+Ckr2)T, (4.32)
1189
+ in Theorem 2.4, since we can take C∗to bound CrandC′
1190
+ r. In particular,
1191
+ r=r(ζ) as defined in Assumption 4.2 satisfies both the conditions on r
1192
+ for (4.32) to hold. Then, taking a:={k2+k1/tildewideCr(ζ)T}b(ζ)in Corollary 2.5, it
1193
+ follows that for some constant c3>0, on the event BN(a),
1194
+ P[τ(N)(a,ζ)≤T]≤c3N−1.
1195
+ Then, from (4.29), for some constant c4,P[BN(a)c]≤c4N−1logN. Here,
1196
+ the constants c3,c4depend on C∗,Tand the parameters of the process.
1197
+ We now use Lemma 4.6 to bound the martingale term in (4.10). It fol-
1198
+ lows that, on the event BN(a)∩ {τ(N)(a,ζ)> T}and on the event that
1199
+ /⌊ard⌊lxN(s)−x(s)/⌊ard⌊lµ≤ΞTfor all 0≤s≤t,
1200
+ /⌊ard⌊lxN(t)−x(t)/⌊ard⌊lµ≤/parenleftBigg
1201
+ ewT/⌊ard⌊lxN(0)−x(0)/⌊ard⌊lµ+√aK4.6/radicalbigg
1202
+ logN
1203
+ N/parenrightBigg
1204
+ +k∗/integraldisplayt
1205
+ 0/⌊ard⌊lxN(s)−x(s)/⌊ard⌊lµds,
1206
+ 26wherek∗:=ewTK(µ,F;2ΞT). Then from Gronwall’s inequality, on the
1207
+ eventBN(a)∩{τ(N)(a,ζ)> T},
1208
+ /⌊ard⌊lxN(t)−x(t)/⌊ard⌊lµ≤/parenleftBigg
1209
+ ewT/⌊ard⌊lxN(0)−x(0)/⌊ard⌊lµ+√aK4.6/radicalbigg
1210
+ logN
1211
+ N/parenrightBigg
1212
+ ek∗t,
1213
+ (4.33)
1214
+ for all 0≤t≤T, provided that
1215
+ /parenleftBigg
1216
+ ewT/⌊ard⌊lxN(0)−x(0)/⌊ard⌊lµ+√aK4.6/radicalbigg
1217
+ logN
1218
+ N/parenrightBigg
1219
+ ≤ΞTe−k∗T.
1220
+ This is true for all Nsufficiently large, if /⌊ard⌊lxN(0)−x(0)/⌊ard⌊lµ≤1
1221
+ 2ΞTe−(w+k∗)T,
1222
+ which we have assumed. We have thus proved (4.31), since, as show n above,
1223
+ P(BN(a)c∪{τ(N)(a,ζ)> T}c) =O(N−1logN).
1224
+ We now use this to show that in fact tmax=∞. Forx(0) as above, we
1225
+ can take xj
1226
+ N(0) :=N−1⌊Nxj(0)⌋ ≤xj(0), so that S(N)
1227
+ ρ(ζ,µ)(0)≤NC∗forC∗:=
1228
+ Sρ(ζ,µ)(x(0))<∞. Then, by (4.13), lim j→∞{µ(j)/νρ(ζ,µ)(j)}= 0, so it fol-
1229
+ lowseasilyusing boundedconvergence that /⌊ard⌊lxN(0)−x(0)/⌊ard⌊lµ→0asN→ ∞.
1230
+ Hence, for any T < t max, it follows from (4.31) that /⌊ard⌊lxN(t)−x(t)/⌊ard⌊lµ→D0
1231
+ asN→ ∞, fort≤T, with uniform bounds over the interval, where ‘ →D’
1232
+ denotes convergence in distribution. Also, by Assumption 4.2, ther e is a con-
1233
+ stantc5such that /⌊ard⌊lxN(t)/⌊ard⌊lµ≤c5N−1S(N)
1234
+ rµ(t) for each t, whererµ≤r(2)
1235
+ maxand
1236
+ rµ≤ρ(ζ,µ). Hence, using Lemma 2.3 and Theorem 2.4, sup0≤t≤2T/⌊ard⌊lxN(t)/⌊ard⌊lµ
1237
+ remains bounded in probability as N→ ∞. Hence it is impossible that
1238
+ /⌊ard⌊lx(t)/⌊ard⌊lµ→ ∞asT→tmax<∞,implyingthatinfact tmax=∞forsuchx(0).
1239
+ Remark . The dependence on the initial conditions is considerably compli-
1240
+ cated by the way the constant Cappears in the exponent, for instance in the
1241
+ expression for /tildewideCrTin the proof of Theorem 4.7. However, if kr2in Assump-
1242
+ tions 2.1 can be chosen to be zero, as for instance in the examples be low, the
1243
+ dependence simplifies correspondingly.
1244
+ Therearebiologicallyplausiblemodelsinwhichtherestrictionto Jl≥ −1
1245
+ is irksome. In populations in which members of a given type lcan fight one
1246
+ another, a natural possibility is to have a transition J=−2e(l)at a rate
1247
+ proportional to Xl(Xl−1), which translates to αJ=α(N)
1248
+ J=γxl(xl−N−1),
1249
+ a function depending on N. Replacing this with αJ=γ(xl)2removes the
1250
+ 27N-dependence, but yields a process that can jump to negative value s ofXl.
1251
+ For this reason, it is useful to be able to allow the transition rates αJto
1252
+ depend on N.
1253
+ Since the arguments inthis paper are not limiting arguments for N→ ∞,
1254
+ it does not require many changes to derive the corresponding resu lts. Quan-
1255
+ tities such as A,F,Ur(x) andVr(x) now depend on N; however, Theorem 4.7
1256
+ continues toholdwithconstants c1andc2thatdo notdepend on N, provided
1257
+ thatµ,w,ν, theklmfrom Assumption 2.1 and ζfrom Assumption 4.2 can
1258
+ be chosen to be independent of N, and that the quantities Z(l)
1259
+ ���from (4.21)
1260
+ can be bounded uniformly in N. On the other hand, the solution x=x(N)
1261
+ of (4.2) that acts as approximation to xNin Theorem 4.7 now itself depends
1262
+ onN, through R=R(N)andF=F(N). IfA(and hence R) can be taken
1263
+ to be independent of N, and lim N→∞/⌊ard⌊lF(N)−F/⌊ard⌊lµ= 0 for some fixed µ–
1264
+ Lipschitz function F, a Gronwall argument can be used to derive a bound
1265
+ for the difference between x(N)and the (fixed) solution xto equation (4.2)
1266
+ withN-independent RandF. IfAhas to depend on N, the situation is
1267
+ more delicate.
1268
+ 5 Examples
1269
+ We begin with some general remarks, to show that the assumptions are sat-
1270
+ isfied in many practical contexts. We then discuss two particular ex amples,
1271
+ those of Kretzschmar (1993) and of Arrigoni (2003), that fitte d poorly or
1272
+ not at all into the general setting of Barbour & Luczak (2008), th ough the
1273
+ other systems referred to in the introduction could also be treate d similarly.
1274
+ In both of our chosen examples, the index jrepresents a number of individ-
1275
+ uals — parasites in a host in the first, animals in a patch in the second —
1276
+ and we shall for now use the former terminology for the preliminary, general
1277
+ discussion.
1278
+ Transitions that can typically be envisaged are: births of a few para sites,
1279
+ which may occur either in the same host, or in another, if infection is b eing
1280
+ represented; births and immigration of hosts, with or without para sites; mi-
1281
+ gration of parasites between hosts; deaths of parasites; death s of hosts; and
1282
+ treatment of hosts, leading to the deaths of many of the host’s pa rasites. For
1283
+ births of parasites, there is a transition X→X+J, whereJtakes the form
1284
+ Jl= 1;Jm=−1;Jj= 0, j/ne}ationslash=l,m, (5.1)
1285
+ 28indicating that one m-host has become an l-host. For births of parasites
1286
+ within a host, a transition rate of the form bl−mmXmcould be envisaged,
1287
+ withl > m, the interpretation being that there are Xmhosts with parasite
1288
+ burdenm, each of which gives birth to soffspring at rate bs, for some small
1289
+ values of s. For infection of an m-host, a possible transition rate would be
1290
+ of the form
1291
+ Xm/summationdisplay
1292
+ j≥0N−1Xjλpj,l−m,
1293
+ since an m-host comes into contact with j-hosts at a rate proportional to
1294
+ their density in the host population, and pjrrepresents the probability of a
1295
+ j-host transferring rparasites to the infected host during the contact. The
1296
+ probability distributions pj·can be expected to be stochastically increasing
1297
+ inj. Deaths of parasites also give rise to transitions of the form (5.1),
1298
+ but now with l < m, the simplest form of rate being just dmXmforl=
1299
+ m−1, though d=dmcould also be chosen to increase with parasite burden.
1300
+ Treatment of a host would lead to values of lmuch smaller than m, and
1301
+ a rate of the form κXmfor the transition with l= 0 would represent fully
1302
+ successful treatment of randomly chosen individuals. Births and d eaths of
1303
+ hosts and immigration all lead to transitions of the form
1304
+ Jl=±1;Jj= 0, j/ne}ationslash=l. (5.2)
1305
+ Fordeaths, Jl=−1, anda typical ratewould be d′Xl. Forbirths, Jl= 1, and
1306
+ a possible rate would be/summationtext
1307
+ j≥0Xjb′
1308
+ jl(withl= 0 only, if new-born individuals
1309
+ are free of parasites). For immigration, constant rates λlcould be supposed.
1310
+ Finally, for migration of individual parasites between hosts, transit ions are
1311
+ of the form
1312
+ Jl=Jm=−1;Jl+1= 1;Jm−1= 1;Jj= 0, j/ne}ationslash=l,m,l+1,m−1,
1313
+ (5.3)
1314
+ a possible rate being γmXmN−1Xl.
1315
+ For all the above transitions, we can take J∗= 2 in (1.2), and (1.3) is
1316
+ satisfied in biologically sensible models. (3.1) and (3.2) depend on the wa y in
1317
+ which the matrix Acan be defined, which is more model specific; in practice,
1318
+ (3.1) is very simple to check. The choice of µin (3.2) is influenced by the
1319
+ need to have (4.1) satisfied. For Assumptions 2.1, a possible choice o fνis to
1320
+ takeν(j) = (j+1) for each j≥0, withS1(X) then representing the num-
1321
+ ber of hosts plus the number of parasites. Satisfying (2.5) is then e asy for
1322
+ 29transitions only involving the movement of a single parasite, but in gen eral
1323
+ requires assumptions as to the existence of the r-th moments of the distri-
1324
+ butions of the numbers of parasites introduced at birth, immigratio n and
1325
+ infection events. For (2.6), in which transitions involving a net reduc tion
1326
+ in the total number of parasites and hosts can be disregarded, th e parasite
1327
+ birth events are those in which the rates typically have a factor mXmfor
1328
+ transitions with Jm=−1, withmin principle unbounded. However, at such
1329
+ events, an m-individual changes to an m+sindividual, with the number s
1330
+ of offspring of the parasite being typically small, so that the value of JTνr
1331
+ associated with this rate has magnitude mr−1; the product mXmmr−1, when
1332
+ summed over m, then yields a contribution of magnitude Sr(X), which is al-
1333
+ lowable in(2.6). Similar considerations showthat theterms N−1S0(X)Sr(X)
1334
+ accommodate the migration rates suggested above. Finally, in orde r to have
1335
+ Assumptions 4.2 satisfied, it is in practice necessary that Assumptio ns 2.1
1336
+ are satisfied for large values of r, thereby imposing restrictions on the dis-
1337
+ tributions of the numbers of parasites introduced at birth, immigra tion and
1338
+ infection events, as above.
1339
+ 5.1 Kretzschmar’s model
1340
+ Kretzschmar (1993) introduced a model of a parasitic infection, in which the
1341
+ transitions from state Xare as follows:
1342
+ J=e(i−1)−e(i)at rate Niµxi, i ≥1;
1343
+ J=−e(i)at rate N(κ+iα)xi, i≥0;
1344
+ J=e(0)at rate Nβ/summationtext
1345
+ i≥0xiθi;
1346
+ J=e(i+1)−e(i)at rate Nλxiϕ(x), i ≥0,
1347
+ wherex:=N−1X,ϕ(x) :=/⌊ard⌊lx/⌊ard⌊l11{c+/⌊ard⌊lx/⌊ard⌊l1}−1withc >0, and/⌊ard⌊lx/⌊ard⌊l11:=/summationtext
1348
+ j≥1j|x|j; here, 0≤θ≤1, andθidenotes its i-th power (our θcorresponds
1349
+ to the constant ξin [7]). Both (1.2) and (1.3) are obviously satisfied. For
1350
+ Assumptions (3.1), (3.2) and (4.1), we note that equation corresp onding
1351
+ to (1.5) has
1352
+ Aii=−{κ+i(α+µ)};AT
1353
+ i,i−1=iµandAT
1354
+ i0=βθi, i≥2;
1355
+ A11=−{κ+α+µ};AT
1356
+ 10=µ+βθ;
1357
+ A00=−κ+β, i≥1,
1358
+ 30with all other elements of the matrix equal to zero, and
1359
+ Fi(x) =λ(xi−1−xi)ϕ(x), i≥1;F0(x) =−λx0ϕ(x).
1360
+ Hence Assumption (3.1) isimmediate, andAssumption (3.2)holds for µ(j) =
1361
+ (j+1)s, for any s≥0, withw= (β−κ)+. For the choice µ(j) =j+1,F
1362
+ maps elements of RµtoRµ, and is also locally Lipschitz in the µ-norm, with
1363
+ K(µ,F;Ξ) =c−2λΞ(2c+Ξ).
1364
+ For Assumptions 2.1, choose ν=µ; then (2.5) is a finite sum for each
1365
+ r≥0. Turning to (2.6), it is immediate that U0(x)≤βS0(x). Then, for
1366
+ r≥1,
1367
+ /summationdisplay
1368
+ i≥0λϕ(N−1X)Xi{(i+2)r−(i+1)r} ≤λS1(X)
1369
+ S0(X)/summationdisplay
1370
+ i≥0rXi(i+2)r−1
1371
+ ≤r2r−1λSr(X),
1372
+ since, by Jensen’s inequality, S1(X)Sr−1(X)≤S0(X)Sr(X). Hence we can
1373
+ takekr2=kr4= 0 and kr1=β+r2r−1λin (2.6), for any r≥1, so that
1374
+ r(1)
1375
+ max=∞. Finally, for (2.7),
1376
+ V0(x)≤(κ+β)S0(x)+αS1(x),
1377
+ so thatk03=κ+β+αandk05= 0, and
1378
+ Vr(x)≤r2(κS2r(x)+αS2r+1(x)+µS2r−1(x)+22(r−1)λS2r−1(x))+βS0(x),
1379
+ so that we can take p(r) = 2r+1,kr3=β+r2{κ+α+µ+22(r−1)λ}, and
1380
+ kr5= 0 for any r≥1, and so r(2)
1381
+ max=∞. In Assumptions 4.2, we can clearly
1382
+ takerµ= 1 and ζ(k) = (k+1)7, givingr(ζ) = 8,b(ζ) = 1 and ρ(ζ,µ) = 17.
1383
+ 5.2 Arrigoni’s model
1384
+ Inthemetapopulation model ofArrigoni (2003), thetransitions f romstate X
1385
+ are as follows:
1386
+ J=e(i−1)−e(i)at rateNixi(di+γ(1−ρ)), i ≥2;
1387
+ J=e(0)−e(1)at rateNx1(d1+γ(1−ρ)+κ);
1388
+ J=e(i+1)−e(i)at rateNibixi, i ≥1;
1389
+ J=e(0)−e(i)at rateNxiκ, i ≥2;
1390
+ J=e(k+1)−e(k)+e(i−1)−e(i)at rateNixixkργ, k ≥0, i≥1;
1391
+ 31as before, x:=N−1X. Here, the total number N=/summationtext
1392
+ j≥0Xj=S0(X) of
1393
+ patches remains constant throughout, and the number of animals in any one
1394
+ patch changes by at most one at each transition; in the final (migra tion)
1395
+ transition, however, the numbers in two patches change simultane ously. In
1396
+ the above transitions, γ,ρ,κare non-negative, and ( di),(bi) are sequences of
1397
+ non-negative numbers.
1398
+ Once again, both (1.2) and (1.3) are obviously satisfied. The equatio n
1399
+ corresponding to (1.4) can now be expressed by taking
1400
+ Aii=−{κ+i(bi+di+γ)};AT
1401
+ i,i−1=i(di+γ);AT
1402
+ i,i+1=ibi, i≥1;
1403
+ A00=−κ,
1404
+ with all other elements of Aequal to zero, and
1405
+ Fi(x) =ργ/⌊ard⌊lx/⌊ard⌊l11(xi−1−xi), i≥1;F0(x) =−ργx0/⌊ard⌊lx/⌊ard⌊l11+κ,
1406
+ where we have used the fact that N−1/summationtext
1407
+ j≥0Xj= 1. Hence Assumption (3.1)
1408
+ is again immediate, and Assumption (3.2) holds for µ(j) = 1 with w= 0,
1409
+ forµ(j) =j+ 1 with w= max i(bi−di−γ−κ)+(assuming ( bi) and (di)
1410
+ to be such that this is finite), or indeed for µ(j) = (j+1)swith any s≥2,
1411
+ with appropriate choice of w. With the choice µ(j) =j+1,Fagain maps
1412
+ elements of RµtoRµ, and is also locally Lipschitz in the µ-norm, with
1413
+ K(µ,F;Ξ) = 3ργΞ.
1414
+ To check Assumptions 2.1, take ν=µ; once again, (2.5) is a finite sum
1415
+ for each r. Then, for (2.6), it is immediate that U0(x) = 0. For any r≥1,
1416
+ using arguments from the previous example,
1417
+ Ur(x)≤r2r−1/braceleftBigg/summationdisplay
1418
+ i≥1ibixi(i+1)r−1+/summationdisplay
1419
+ i≥1/summationdisplay
1420
+ k≥0iργxixk(k+1)r−1/bracerightBigg
1421
+ ≤r2r−1{max
1422
+ ibiSr(x)+ργS1(x)Sr−1(x)}
1423
+ ≤r2r−1{max
1424
+ ibiSr(x)+ργS0(x)Sr(x)},
1425
+ so that, since S0(x) = 1, we can take kr1=r2r−1(maxibi+ργ) andkr2=
1426
+ kr4= 0 in (2.6), and r(1)
1427
+ max=∞. Finally, for (2.7), V0(x) = 0 and, for r≥1,
1428
+ Vr(x)
1429
+ ≤r2/braceleftBig
1430
+ 22(r−1)max
1431
+ ibiS2r−1(x)+max
1432
+ i(i−1di)S2r(x)+γ(1−ρ)S2r−1(x)
1433
+ +ργ(22(r−1)S1(x)S2r−2(x)+S0(x)S2r−1(x))/bracerightBig
1434
+ +κS2r(x),
1435
+ 32so that we can take p(r) = 2r, and (assuming i−1dito be finite)
1436
+ kr3=κ+r2{22(r−1)(max
1437
+ ibi+ργ)+max
1438
+ i(i−1di)+γ},
1439
+ andkr5= 0 for any r≥1, andr(2)
1440
+ max=∞. In Assumptions 4.2, we can again
1441
+ takerµ= 1 and ζ(k) = (k+1)7, givingr(ζ) = 8,b(ζ) = 1 and ρ(ζ,µ) = 16.
1442
+ Acknowledgement
1443
+ We wish to thank a referee for recommendations that have substa ntially
1444
+ streamlined our arguments. ADB wishes to thank both the Institut e for
1445
+ MathematicalSciencesoftheNationalUniversityofSingaporeand theMittag–
1446
+ Leffler Institute for providing a welcoming environment while part of t his
1447
+ work was accomplished. MJL thanks the University of Z¨ urich for th eir hos-
1448
+ pitality on a number of visits.
1449
+ References
1450
+ [1]Arrigoni, F. (2003). Deterministic approximation of a stochastic
1451
+ metapopulation model. Adv. Appl. Prob. 35691–720.
1452
+ [2]Barbour, A. D. andKafetzaki, M. (1993). A host–parasite model
1453
+ yielding heterogeneous parasite loads. J. Math. Biology 31157–176.
1454
+ [3]Barbour, A. D. andLuczak, M. J. (2008). Laws of large numbers
1455
+ for epidemic models with countably many types. Ann. Appl. Probab. 18
1456
+ 2208–2238.
1457
+ [4]Chow, P.-L. (2007).Stochastic partial differential equations. Chapman
1458
+ and Hall, Boca Raton.
1459
+ [5]Eibeck, A. andWagner, W. (2003). Stochastic interacting particle
1460
+ systems and non-linear kinetic equations. Ann. Appl. Probab. 13845–
1461
+ 889.
1462
+ [6]Kimmel, M. andAxelrod, D. E. (2002).Branching processes in biol-
1463
+ ogy.Springer, Berlin.
1464
+ 33[7]Kretzschmar, M. (1993).Comparison ofaninfinite dimensional model
1465
+ for parasitic diseases with a related 2-dimensional system. J. Math. Anal-
1466
+ ysis Applics 176235–260.
1467
+ [8]Kurtz, T. G. (1970). Solutions of ordinary differential equations as
1468
+ limits of pure jump Markov processes. J. Appl. Probab. 749–58.
1469
+ [9]Kurtz, T. G. (1971).Limit theorems forsequences ofjumpMarkov pro-
1470
+ cesses approximating ordinary differential processes. J. Appl. Probab. 8
1471
+ 344–356.
1472
+ [10]L´eonard, C. (1990).Some epidemic systems are long range interacting
1473
+ particle systems. In: Stochastic Processes in Epidemic Theory , Eds J.-P.
1474
+ Gabriel, C. Lef` evre & P. Picard, Lecture Notes in Biomathematics 86
1475
+ 170–183: Springer, New York.
1476
+ [11]Luchsinger, C. J. (1999).MathematicalModelsofaParasiticDisease,
1477
+ Ph.D. thesis, University of Z¨ urich.
1478
+ [12]Luchsinger, C. J. (2001a). Stochastic models of a parasitic infection,
1479
+ exhibiting three basic reproduction ratios. J. Math. Biol. 42, 532–554.
1480
+ [13]Luchsinger, C. J. (2001b). Approximating the long term behaviour
1481
+ of a model for parasitic infection. J. Math. Biol. 42, 555–581.
1482
+ [14]Pazy, A. (1983).Semigroups of Linear Operators and Applications to
1483
+ Partial Differential Equations. Springer, Berlin.
1484
+ [15]Reuter, G. E. H. (1957). Denumerable Markov processes and the
1485
+ associated contraction semigroups on l.Acta Math. 97, 1–46.
1486
+ 34
1001.0045.txt ADDED
@@ -0,0 +1,454 @@
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
1
+ arXiv:1001.0045v1 [hep-ph] 31 Dec 2009EPJ manuscript No.
2
+ (will be inserted by the editor)
3
+ Axial Anomaly and Mixing Parameters of
4
+ Pseudoscalar Mesons
5
+ Yaroslav N. Klopot1,a, Armen G. Oganesian2,b, and Oleg V. Teryaev1,c
6
+ 1Joint Institute for Nuclear Research, Bogoliubov Laborato ry of Theoretical Physics, Dubna 141980,
7
+ Russia
8
+ 2Institute of Theoretical and Experimental Physics, B.Cher emushkinskaya 25, Moscow 117218, Russia
9
+ Abstract. In this work the analysis of mixing parameters of the system i nvolving
10
+ η,η′mesons and some third massive state Gis carried out. We use the generalized
11
+ mixing scheme with three angles. The framework of the disper sive approach to
12
+ Abelian axial anomaly of isoscalar non-singlet current and the analysis of exper-
13
+ imental data of charmonium radiative decays ratio allow us t o get a number of
14
+ quite strict constraints for the mixing parameters. The ana lysis shows that the
15
+ equal values of axial current coupling constants f8andf0are preferable which
16
+ may be considered as a manifestation of SU(3) and chiral symmetry.
17
+ 1 Introduction
18
+ This work is developing the approach of the papers [1,2] and is devot ed to the significant
19
+ problem of mixing of pseudoscalar mesons. It is especially important w ith a number of current
20
+ and planned experiments.
21
+ The problem of η-η′mixing has been studied for many years. The usual approach with on e
22
+ mixingangledominatedfordecades,butintherecentyearsthemor eelaboratedschemesappear
23
+ to be unavoidable [3–8]. In particular, the theoretical ground of th is was based on the recent
24
+ progress in the ChPT [9–11]. On the other hand, it was shown, that t he current experimental
25
+ data cannot satisfactory describe the whole set of experiments w ithin the one-angle mixing
26
+ scheme.
27
+ The mixing schemes are usually enunciated either in terms of SU(3) or quark basis. In
28
+ our paper [2] we construct and use the generalization of SU(3) basis similar to the mixing of
29
+ massive neutrinos. This is because we use the dispersive approach t o axial anomaly ( [12], [13]
30
+ for a review) to find some model-independent and precise restrictio n on the mixing parameters.
31
+ It was shown that any scheme with more than one angle unavoidably d emands an additional
32
+ admixture of higher mass state. If we restrict ourselves to only on e additional state G(denoted
33
+ as a glueball without really specifying its nature) then the general m ixing scheme can be
34
+ described in terms of 3 angles. In particular cases the number of an gles can be reduced to two.
35
+ In the paper [2] the analysis of different conventional (and most ph ysically interesting)
36
+ particular cases was performed (including two-angle mixing schemes ) basing on the dispersive
37
+ representation of axial anomaly from one side and charmonium deca ys ratio from the other
38
+ side.
39
+ ae-mail:[email protected]
40
+ be-mail:[email protected]
41
+ ce-mail:[email protected] Will be inserted by the editor
42
+ The main conclusion of the paper [2] is that in all considered cases the only reasonable
43
+ solutions appear at f8=f0≃fπ. The main aim of this work is to check whether this relation
44
+ remains valid in the most general case with some specific constraints imposed.
45
+ This paper is organized as follows. In the Sec. 2 we introduce our not ation and the general
46
+ approach to the mixing. In Sec. 3 we derive the basic equations relyin g on the dispersive
47
+ approach to Abelian axial anomaly of isoscalar non-singlet current J8
48
+ µ5and the charmonium
49
+ radiativedecayratio RJ/Ψ, while in Sec. 4 we performthe numerical analysisofthese equations .
50
+ Finally, in Sec. 5 we present the conclusion.
51
+ 2 Mixing scheme
52
+ We start with a ( N-component) vector of physical pseudoscalar fields consisting of the fields of
53
+ the lightest pseudoscalar mesons and other fields:
54
+ /tildewideΦ≡
55
+ π0
56
+ η
57
+ η′
58
+ G
59
+ ...
60
+ . (1)
61
+ We are not able to specify the physical nature of the other compon ents with higher masses,
62
+ the lowest of which Gcan be either a glueball or some excited state1. Let us also introduce,
63
+ following [16,17], a set of SU(3) fields ϕ3,ϕ8,ϕ0(Φ1,Φ2,Φ3) and complement them with other
64
+ (sterile) fields gi(Φi,i= 4..N)
65
+ Φ=
66
+ ϕ3
67
+ ϕ8
68
+ ϕ0
69
+ g
70
+ ...
71
+ . (2)
72
+ The three upper fields ϕ3,ϕ8,ϕ0are the only ones which define the generalized PCAC
73
+ relationforaxialcurrent Ja
74
+ µ5=qγµγ5λa
75
+
76
+ 2q(nosummationover acontraryto jandkisassumed):
77
+ ∂µJa
78
+ µ5=faδ∆L
79
+ δΦa=FajMjkΦk, a= 3,8,0, j,k= 1..N, (3)
80
+ where∆Lis the mass term in the effective Lagrangian with a non-diagonal mass matrixM(as
81
+ fieldsΦkare not orthogonal to each other):
82
+ ∆L=1
83
+ 2ΦTMΦ, (4)
84
+ andFis a matrix of decay constants2:
85
+ F≡
86
+ f30 0 0...0
87
+ 0f80 0...0
88
+ 0 0f00...0
89
+ . (5)
90
+ In order to proceed from initial SU(3) fields Φto physical mass fields /tildewideΦthe unitary (real,
91
+ as the CP-violating effects are negligible) matrix Uis introduced
92
+ 1Note, that the mixing with the excited states is usually(e.g . [14,15]) supposed to be suppressed.
93
+ 2Note, that matrix of decay constants Fis non-square expressing the fact that generally the number
94
+ ofSU(3) currents is less then the number of all possible states in volved in mixing. The similar situation
95
+ takes place (see e.g. [18]) in one of the extensions of the Sta ndard Model – neutrino mixing scenario
96
+ involving sterile neutrinos.Will be inserted by the editor 3
97
+ /tildewideΦ=UΦ (6)
98
+ that diagonalizes the mass matrix
99
+ UMUT=/tildewiderM≡diag(m2
100
+ π0,m2
101
+ η,m2
102
+ η′,m2
103
+ G,...), (7)
104
+ wheremπ,mη,mη′andmGare the masses of the π,η,η′mesons and glueball state G,
105
+ respectively.
106
+ Simple transformations of Eq.(3) read:
107
+ ∂µJµ5=FUT/tildewiderM/tildewideΦ (8)
108
+ This formula is close to those obtained in [16,17] (in the limit of small mix ing). When the decay
109
+ constants are equal, it is reduced to formula (3.40) in [19].
110
+ The matrix elements of ∂µJµ5between vacuum state and physical states |/tildewiderΦk∝angbracketright
111
+ ∝angbracketleft0|∂µJa
112
+ µ5|/tildewiderΦk∝angbracketright=Fa
113
+ i(UT/tildewiderM)i
114
+ k (9)
115
+ can be compared to the standard definition of the ”physical” couplin g constants of axial cur-
116
+ rents:
117
+ ∝angbracketleft0|Ja
118
+ µ5|/tildewiderΦk∝angbracketright=ifa
119
+ kqµ. (10)
120
+ From (9) and (10) follows the relation
121
+ fa
122
+ k=Fa
123
+ i(UT)i
124
+ k=fa(UT)a
125
+ k. (11)
126
+ This expression (recall, that there is no summation over a) clearly shows that fa
127
+ kare obtained
128
+ bymultiplicationofeachlineof UTbyrespectivecoupling faandformanon-diagonal(contrary
129
+ toF) matrix.
130
+ Taking into account the well-known smallness of π0mixing with the η,η′sector [16,17,20]
131
+ and neglecting all higher contributions we restrict our consideratio n to three physical states
132
+ η,η′,Gand two currents J8
133
+ µ5,J0
134
+ µ5. Then the divergencies of the axial currents (recall, that Gis
135
+ a first mass state heavier than η′):
136
+ /parenleftbigg∂µJ8
137
+ µ5
138
+ ∂µJ0
139
+ µ5/parenrightbigg
140
+ =/parenleftbigg
141
+ f80 0
142
+ 0f00/parenrightbigg
143
+ UT
144
+ m2
145
+ η0 0
146
+ 0m2
147
+ η′0
148
+ 0 0m2
149
+ G
150
+ 
151
+ η
152
+ η′
153
+ G
154
+ . (12)
155
+ Exploring the mentioned similarity of the meson and lepton mixing, we us e the Euler
156
+ parametrization for the mixing matrix U(we use notation ci≡cosθi,si≡sinθi):
157
+ U=
158
+ c8c3−c0s3s8−c3s8−c8c0s3s3s0
159
+ s3c8+c3c0s8−s3s8+c3c8c0−c3s0
160
+ s8s0 c8s0 c0
161
+ . (13)
162
+ In the following consideration we will need the divergency of the octe t current ∂µJ8
163
+ µ5, so let
164
+ us write it out explicitly:
165
+ ∂µJ8
166
+ µ5=f8(m2
167
+ ηη(c8c3−c0s3s8)+m2
168
+ η′η′(s3c8+c3c0s8)+m2
169
+ GG(s8s0)). (14)
170
+ As soon as in the chiral limit J8
171
+ µ5should be conserved, from Eq.(14) follows that coefficients
172
+ of the terms m2
173
+ η′,m2
174
+ Gmust decrease at least as ( mη/mη′,G)2. More specifically, we expect the
175
+ following limits for the terms of Eq.(14):
176
+ |s8s0|
177
+ |s3c8+c3c0s8|/lessorsimilar/parenleftbiggmη
178
+ mG/parenrightbigg2
179
+ . (15)4 Will be inserted by the editor
180
+ 3 Abelian axial anomaly and charmonium decays ratio
181
+ In our paper the dispersive form of the anomaly sum rule will be exten sively used, so we remind
182
+ briefly the main points of this approach (see e.g. review [13] for det ails).
183
+ Consider a matrix element of a transition of the axial current to two photons with momenta
184
+ pandp′
185
+ Tµαβ(p,p′) =∝angbracketleftp,p′|Jµ5|0∝angbracketright. (16)
186
+ The general form of Tµαβfor a case p2=p′2can be represented in terms of structure
187
+ functions (form factors):
188
+ Tµαβ(p,p′) =F1(q2)qµǫαβρσpρp′
189
+ σ+
190
+ 1
191
+ 2F2(q2)[pα
192
+ p2ǫµβρσpρp′
193
+ σ−p′
194
+ β
195
+ p2ǫµαρσpρp′
196
+ σ−ǫµαβσ(p−p′)σ],(17)
197
+ whereq=p+p′. The functions F1(q2),F2(q2) can be described by dispersion relations with
198
+ no subtractions and anomaly condition in QCD results in the sum rule:
199
+ ∞/integraldisplay
200
+ 0Im F1(q2)dq2= 2αNc/summationdisplay
201
+ e2
202
+ q, (18)
203
+ whereeqare quark electric charges and Ncis the number of colors. This sum rule [21] was
204
+ developed by Jiˇ r´ ı Hoˇ rejˇ s´ ı [22], and later generalized [23]. Not ice that in QCD this equation
205
+ does not have any perturbative corrections [24], and it is expected that it does not have any
206
+ non-perturbative corrections as well due to the ’t Hooft’s consist ency principle [25]. It will be
207
+ important for us that as q2→ ∞the function ImF1(q2) decreases as 1 /q4(see discussion
208
+ in Ref. [2]). Note also that the relation (18) contains only mass-indep endent terms, which is
209
+ especially important for the 8th component of the axial current J8
210
+ µ5containing strange quarks:
211
+ J8
212
+ µ5=1√
213
+ 6(¯uγµγ5u+¯dγµγ5d−2¯sγµγ5s). (19)
214
+ The general sum rule (18) takes the form:
215
+ ∞/integraldisplay
216
+ 0Im F1(q2)dq2=2√
217
+ 6α(e2
218
+ u+e2
219
+ d−2e2
220
+ s)Nc=/radicalbigg
221
+ 2
222
+ 3α , (20)
223
+ whereeu= 2/3,ed=es=−1/3,Nc= 3.
224
+ In order to separate the form factor F1(q2), multiply Tµαβ(p,p′) byqµ/q2. Then, taking the
225
+ imaginary part of F1(q2), using the expression for ∂µJ8
226
+ µ5from Eq.(12) and unitarity we get:
227
+ ImF1(q2) =Im qµ1
228
+ q2∝angbracketleft2γ|J(8)
229
+ µ5|0∝angbracketright=
230
+ −f8
231
+ q2∝angbracketleft2γ|[m2
232
+ ηη(c8c3−c0s3s8)+m2
233
+ η′η′(s3c8+c3c0s8)+m2
234
+ GGs8s0]|0∝angbracketright=
235
+ πf8[Aηδ(q2−m2
236
+ η)(c8c3−c0s3s8)+Aη′δ(q2−m2
237
+ η′)(s3c8+c3c0s8)+AGδ(q2−m2
238
+ G)(s8s0)].
239
+ (21)
240
+ If we employ the sum rule (20), we obtain a simple equation:
241
+ (c8c3−c0s3s8)+β(s3c8+c3c0s8)+γ(s8s0) =ξ, (22)
242
+ whereWill be inserted by the editor 5
243
+ β≡Aη′
244
+ Aη=/radicaligg
245
+ Γη′→2γ
246
+ Γη→2γm3η
247
+ m3
248
+ η′, γ≡AG
249
+ Aη=/radicaligg
250
+ ΓG→2γ
251
+ Γη→2γm3η
252
+ m3
253
+ G, (23)
254
+ ξ≡/radicaligg
255
+ α2m3η
256
+ 96π3Γη→2γ1
257
+ f2
258
+ 8, Γη→2γ=m3
259
+ η
260
+ 64πA2
261
+ η. (24)
262
+ Note that if we include higher resonancesin this equation, they will be suppressed as 1 /m2
263
+ res
264
+ by virtue of the mentioned above asymptotic behavior of F1(q2)∝1/q4. For the last two terms
265
+ in (22) we can specify this constraint as follows:
266
+ |s8s0|
267
+ |s3c8+c3c0s8|/lessorsimilarβ
268
+ γ/parenleftbiggmη′
269
+ mG/parenrightbigg2
270
+ . (25)
271
+ As an additional experimental constraint we use, following [26,27], t he data of the decay
272
+ ratioRJ/Ψ= (Γ(J/Ψ)→η′γ)/(Γ(J/Ψ)→ηγ).
273
+ As it was pointed out in [28], the radiative decays J/Ψ→η(η′)γare dominated by non-
274
+ perturbative gluonic matrix elements, and the ratio of the decay ra tesRJ/Ψ= (Γ(J/Ψ)→
275
+ η′γ)/(Γ(J/Ψ)→ηγ) can be expressed as follows:
276
+ RJ/Ψ=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle∝angbracketleft0|G/tildewideG|η′∝angbracketright
277
+ ∝angbracketleft0|G/tildewideG|η∝angbracketright/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2/parenleftbiggpη′
278
+ pη/parenrightbigg3
279
+ , (26)
280
+ wherepη(η′)=MJ/Ψ(1−m2
281
+ η(η′)/M2
282
+ J/Ψ)/2. The advantage of this ratio is expected smallness of
283
+ perturbative and non-perturbative corrections.
284
+ The divergencies of singlet and octet components of the axial curr ent in terms of quark
285
+ fields can be written as:
286
+ ∂µJ8
287
+ µ5=1√
288
+ 6(muuγ5u+mddγ5d−2mssγ5s), (27)
289
+ ∂µJ0
290
+ µ5=1√
291
+ 3(muuγ5u+mddγ5d+mssγ5s)+1
292
+ 2√
293
+ 33αs
294
+ 4πG/tildewideG. (28)
295
+ Following [26], neglect the contribution of u- and d- quark masses, th en the matrix elements
296
+ of the anomaly term between the vacuum and η,η′states are:
297
+
298
+ 3αs
299
+ 8π∝angbracketleft0|G/tildewideG|η∝angbracketright=∝angbracketleft0|∂µJ(0)
300
+ µ5|η∝angbracketright+1√
301
+ 2∝angbracketleft0|∂µJ(8)
302
+ µ5|η∝angbracketright, (29)
303
+
304
+ 3αs
305
+ 8π∝angbracketleft0|G/tildewideG|η′∝angbracketright=∝angbracketleft0|∂µJ(0)
306
+ µ5|η′∝angbracketright+1√
307
+ 2∝angbracketleft0|∂µJ(8)
308
+ µ5|η′∝angbracketright. (30)
309
+ Using Eq. (12), (26), (29), (30) we deduce:
310
+ RJ/Ψ=/bracketleftiggf0(−s3s8+c3c8c0)+1√
311
+ 2f8(s3c8+c3c0s8)
312
+ f0(−c3s8−c8c0s3)+1√
313
+ 2f8(c8c3−c0s3s8)/bracketrightigg2
314
+ ×/parenleftbiggmη′
315
+ mη/parenrightbigg4/parenleftbiggpη′
316
+ pη/parenrightbigg3
317
+ .(31)
318
+ 4 Analysis
319
+ For further analysis it is convenient to rewrite the equations (22), (31) in terms of angles
320
+ θ1≡θ8+θ3,θ8andθ0:
321
+ 1
322
+ 2(c1+c2−c0(c2−c1))+β
323
+ 2(s1−s2+c0(s1+s2))+γ(s8s0) =ξ. (32)6 Will be inserted by the editor
324
+ RJ/Ψ=/bracketleftiggf0(c1−c2+c0(c1+c2))+1√
325
+ 2f8(s1−s2+c0(s1+s2))
326
+ f0(−s1−s2−c0(s1−s2))+1√
327
+ 2f8(c1+c2−c0(c2−c1))/bracketrightigg2/parenleftbiggmη′
328
+ mη/parenrightbigg4/parenleftbiggpη′
329
+ pη/parenrightbigg3
330
+ ,(33)
331
+ whereθ2≡2θ8−θ1.
332
+ The angles θ1,θ8,θ0have the explicit physical meaning. From the definition (13) of the
333
+ mixing matrix Uone can see that the angle θ1describes the overlap in the η−η′system with
334
+ an accuracy ∼θ2
335
+ 0/2 and coincides with their mixing angle as θ0→0. At the same time θ0is
336
+ responsible for the glueball admixture to η−η′system, and s8s0describes the contribution of
337
+ the glueball state Gto the octet component of axial current ∂J8
338
+ µ5only.
339
+ In the further analysis we will use the following assumptions:
340
+ I) As we discussed in Sec. 2, the last term in (14) should be suppress ed as (mη/mG)2. So
341
+ we impose the following constraint:
342
+ |s8s0|
343
+ |s3c8+c3c0s8|/lessorsimilar/parenleftbiggmη
344
+ mG/parenrightbigg2
345
+ . (34)
346
+ II) In sec 3 we found another constraint, which follows from the as ymptotic behavior of
347
+ ImF1(see 25):
348
+ |s8s0|
349
+ |s3c8+c3c0s8|/lessorsimilarβ
350
+ γ/parenleftbiggmη′
351
+ mG/parenrightbigg2
352
+ . (35)
353
+ III) In our numerical analysis we suppose that γcannot exceed 1 (i.e. ΓG→2γ/m3
354
+ G/lessorsimilar
355
+ Γη→2γ/m3
356
+ η). This restriction corresponds to the assumption that 2-photon decay widths of
357
+ pseudoscalar mesons grow like the third power of their masses, or in other words, the glueball
358
+ coupling to quarks is of the same order as for the meson octet stat es.
359
+ IV) We accept that the decay constants obey the relation f8/greaterorsimilarf0/greaterorsimilarfπ(for various kinds
360
+ of justification see, e.g., [3,9]).
361
+ For the purposes of numerical analysis, the values of RJ/Ψ(RJ/Ψ= 4.8±0.6), masses
362
+ and two-photon decay widths of η,η′mesons are taken from PDG [29]. Using the values
363
+ mη,mη′,Γη→2γ,Γη′→2γ, we see that the relation for the constraint (34) is more strict tha n the
364
+ constraint (35). Supposing the minimal mass of the glueball to be of ordermG≃3mη≃1.5
365
+ GeV, we get the estimation:
366
+ |s8s0|/|s3c8+c3c0s8|/lessorsimilar0.1. (36)
367
+ On Fig. 1 the plots of the equations (32) and (33) in the parameter s pace (θ8,θ1) are shown
368
+ for different values of decay constants f8,f0and mixing angle θ0. The dashed curves denote
369
+ experimental uncertainties. The intersection points of the curve s represent the solutions of both
370
+ equations (32),(33). The filled area indicates the region, where the constraint (36) is valid. The
371
+ plotted range of angle θ1is limited to the physically interesting region, where the solution for
372
+ relatively small angles θ0exists. Let us note for completeness, that there is another solut ion for
373
+ θ1∼90◦,θ0/greaterorsimilar50◦which does not seem to have a physical sense.
374
+ The numerical analysis shows, that the solution of the equations (3 2) and (33) satisfying
375
+ the mentioned above constraint is possible only for rather small mixin g angleθ0and for decay
376
+ constants f8,f0closetoeachotherandcloseto fπ:forf8/fπ=f0/fπ= 1.0the possiblerangeof
377
+ mixing angle θ0isθ0= (0÷25)◦(see Fig. 1(a)-1(c) for demonstration), for f8/fπ=f0/fπ= 1.1
378
+ the possible range of mixing angle θ0isθ0= (0÷20)◦.
379
+ There is no solutions for decay constant values f8/fπ= 1.1,f0/fπ= 1.0 for any θ0(see Fig.
380
+ 1(d)-1(f) for demonstration), and for any f0/lessorsimilarf8in case of f8/fπ≥1.2. The obtained results
381
+ are quite stable: even if we relax the constraint (36) making its r.h.s. several times larger, all
382
+ the conclusions are preserved.
383
+ Note finally, that this result is in contradiction with the prediction for the decay constant
384
+ f8/fπ= 1.34 [10] obtained in the Large NcChPT.Will be inserted by the editor 7
385
+ /Minus150/Minus100/Minus50050100150/Minus40/Minus30/Minus20/Minus100
386
+ Θ8/LBracket1degrees/RBracket1Θ1/LBracket1degrees/RBracket1
387
+ (a) (f8,f0) = (1.0,1.0)fπ,θ0=
388
+ 0◦/Minus150/Minus100/Minus50050100150/Minus40/Minus30/Minus20/Minus100
389
+ Θ8/LBracket1degrees/RBracket1Θ1/LBracket1degrees/RBracket1
390
+ (b) (f8,f0) = (1.0,1.0)fπ,θ0=
391
+ 5◦/Minus150/Minus100/Minus50050100150/Minus40/Minus30/Minus20/Minus100
392
+ Θ8/LBracket1degrees/RBracket1Θ1/LBracket1degrees/RBracket1
393
+ (c) (f8,f0) = (1.0,1.0)fπ,θ0=
394
+ 30◦
395
+ /Minus150/Minus100/Minus50050100150/Minus40/Minus30/Minus20/Minus100
396
+ Θ8/LBracket1degrees/RBracket1Θ1/LBracket1degrees/RBracket1
397
+ (d) (f8,f0) = (1.1,1.0)fπ,θ0=
398
+ 0◦/Minus150/Minus100/Minus50050100150/Minus40/Minus30/Minus20/Minus100
399
+ Θ8/LBracket1degrees/RBracket1Θ1/LBracket1degrees/RBracket1
400
+ (e) (f8,f0) = (1.1,1.0)fπ,θ0=
401
+ 5◦/Minus150/Minus100/Minus50050100150/Minus40/Minus30/Minus20/Minus100
402
+ Θ8/LBracket1degrees/RBracket1Θ1/LBracket1degrees/RBracket1
403
+ (f) (f8,f0) = (1.1,1.0)fπ,θ0=
404
+ 30◦
405
+ Fig. 1.The solutions of the Eq. (32) (thin curves, blue online) and ( 33)(thick curves, red online) with
406
+ the experimental uncertainties (dashed curves) for differe nt values of the parameters f8,f0andθ0. The
407
+ shaded area indicates the region, where the relation (36) is valid.
408
+ 5 Conclusion
409
+ In this paper we studied what can be learnt about the mixing in the pse udoscalar sector from
410
+ the dispersive approach to axial anomaly.
411
+ Our analysis shows that the equal values of axial current coupling c onstants f8andf0are
412
+ favorablewhich may be considered as a manifestation of SU(3) and chiral symmetry. Moreover,
413
+ with a less definiteness the relation fπ≈f8≈f0[2] is also supported.
414
+ Theanalysisdemands f8<1.2fπwhich deviatesat 10%levelfrom theresultsofcalculations
415
+ within the chiral perturbation theory ( f8= 1.34fπ) [10].
416
+ Thevalueofthe mixingangle θ0,whichisresponsibleforthe glueballadmixturetothe η−η′,
417
+ is limited to θ0<25◦for (f8,f0) = (1.0,1.0)fπand toθ0<20◦for (f8,f0) = (1.0,1.0)fπ.
418
+ The improvement of the experimental data of RJ/Ψcan significantly limit the constraints
419
+ for the parameters θ0,θ8andf8,f0.
420
+ We thank J. Hoˇ rejˇ s´ ı, B. L. Ioffe and M. A. Ivanov for useful co mments and discussions.
421
+ Y. K. and O. T. gratefully acknowledge the organizers of the works hop for hospitality and
422
+ support. This work was supported in part by RFBR (Grants 09-02- 00732,09-02-01149),by the
423
+ funds from EC to the project ”Study of the Strong Interacting M atter” under contract N0.
424
+ R113-CT-2004-506078 and by CRDF Project RUP2-2961-MO-09.
425
+ References
426
+ 1. Y. N. Klopot, A. G. Oganesian, O. V. Teryaev, arXiv:0810.1 217 [hep-ph] (2008).8 Will be inserted by the editor
427
+ 2. Y. N. Klopot, A. G. Oganesian, O. V. Teryaev, arXiv:0911.0 180 [hep-ph] (2009).
428
+ 3. T. Feldmann, P. Kroll, B. Stech, Phys. Lett. B449, 339 (1999).
429
+ 4. T. Feldmann, P. Kroll, B. Stech, Phys. Rev. D58, 114006 (1998).
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+ 5. F. De Fazio, M. R. Pennington, JHEP07, 051 (2000).
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+ 6. P. Kroll, Mod. Phys. Lett. A20, 2667 (2005).
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+ 7. R. Escribano, J.-M. Frere, JHEP06, 029 (2005).
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+ 8. V. Mathieu, V. Vento arXiv:0910.0212 (2009).
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+ 9. H. Leutwyler, Nucl. Phys. Proc. Suppl. 64, 223 (1998).
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+ 10. R. Kaiser, H. Leutwyler, hep-ph/9806336 (1998).
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+ 11. R. Kaiser, H. Leutwyler, Eur. Phys. J. C17, 623 (2000).
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+ 12. A. D. Dolgov, V. I. Zakharov, Nucl. Phys. B27, 525 (1971).
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+ 13. B. L. Ioffe, Int. J. Mod. Phys. A21, 6249 (2006).
439
+ 14. R. Escribano, arXiv:0712.1814 [hep-ph] (2007).
440
+ 15. H.-Y. Cheng, H.-n. Li, K.-F. Liu, Phys. Rev. D79, 014024 (2009).
441
+ 16. B. L. Ioffe, Yad. Fiz. 29, 1611 (1979).
442
+ 17. B. L. Ioffe, M. A. Shifman, Phys. Lett. B95, 99 (1980).
443
+ 18. S. M. Bilenky, C. Giunti, W. Grimus, Prog. Part. Nucl. Phys. 43, 1 (1999).
444
+ 19. D. Diakonov, M. V. Polyakov, C. Weiss, Nucl. Phys. B461, 539 (1996).
445
+ 20. B. L. Ioffe, A. G. Oganesian, Phys. Lett. B647, 389 (2007).
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+ 21. Y. Frishman, A. Schwimmer, T. Banks, S. Yankielowicz, Nucl. Phys. B177, 157 (1981).
447
+ 22. J. Horejsi, Phys. Rev. D32, 1029 (1985).
448
+ 23. O. L. Veretin, O. V. Teryaev, Phys. Atom. Nucl. 58, 2150 (1995).
449
+ 24. S. L. Adler, W. A. Bardeen, Phys. Rev. 182, 1517 (1969).
450
+ 25. J. Horejsi, O. Teryaev, Z. Phys. C65, 691 (1995).
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+ 26. R. Akhoury, J. M. Frere, Phys. Lett. B220, 258 (1989).
452
+ 27. P. Ball, J. M. Frere, M. Tytgat, Phys. Lett. B365, 367 (1996).
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+ 28. V. A. Novikov, M. A. Shifman, A. I. Vainshtein, V. I. Zakha rov,Nucl. Phys. B165, 55 (1980).
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+ 29. C. Amsler, et al.,Phys. Lett. B667, 1 (2008).
1001.0046.txt ADDED
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1
+ arXiv:1001.0046v1 [math.CO] 31 Dec 2009THE CAUCHY-SCHWARZ INEQUALITY IN CAYLEY GRAPH
2
+ AND TOURNAMENT STRUCTURES ON FINITE FIELDS
3
+ STEPHAN FOLDES AND L ´ASZL´O MAJOR
4
+ Abstract. The Cayley graph construction provides a natural grid struc ture
5
+ on a finite vector space over a field of prime or prime square car dinality, where
6
+ the characteristic is congruent to 3 modulo 4, in addition to the quadratic
7
+ residue tournament structure on the prime subfield. Distanc e from the null
8
+ vector in the grid graph defines a Manhattan norm. The Hermiti an inner prod-
9
+ uct on these spaces over finite fields behaves in some respects similarly to the
10
+ real and complex case. An analogue of the Cauchy-Schwarz ine quality is valid
11
+ with respect to the Manhattan norm. With respect to the non-t ransitive order
12
+ provided by the quadratic residue tournament, an analogue o f the Cauchy-
13
+ Schwarz inequality holds in arbitrarily large neighborhoo ds of the null vector,
14
+ when the characteristic is an appropriate large prime.
15
+ 1.Manhattan norms and grid graphs
16
+ We consider the finite fields FpandFp2of prime and prime square cardinality,
17
+ wherep≡3 mod 4. The field Fp2has a natural graph structure with the field
18
+ elements as vertices, two distinct vertices u,zbeing adjacent if ( z−u)4= 1. The
19
+ subfieldFpofFp2then induces a subgraph in which xandyare adjacent if and only
20
+ if (y−x)2= 1.The graph Fp2is isomorphic to the Cartesian square C2
21
+ p=Cp/squareCp,
22
+ whereCpis ap-cycle and within Fp2the induced subgraph Fpis itself a p-cycle.
23
+ Clearly the graph Fp2is not planar, but can be drawn as a grid on the torus.
24
+ For any connected graph whose vertex set is a group, the distanc e of any vertex
25
+ zfrom the identity element of the group is called the normofz, denoted N(z).
26
+ In general, distances and norms measured in connected subgraph s induced by sub-
27
+ groups can be larger than distances and norms measured with refe rence to the
28
+ whole graph. However, with respect to the distance-preserving s ubgraph induced
29
+ byFpinFp2, the norm of any z∈Fpis the same as its norm with respect to the
30
+ whole graph Fp2: this is simply the length of the shortest path from 0 to zin the
31
+ cycle induced by Fp.
32
+ Forq=porq=p2, then-dimensional vector space Fn
33
+ qis also endowed with
34
+ the Cartesian product graph structure Fq/square···/squareFqisomorphic to Cn
35
+ porC2n
36
+ p. The
37
+ norm of a vector v= (v1,...,v n) inFn
38
+ qis then equal to the sum N(v1)+···+N(vn)
39
+ and we also write N(v) for this vector norm.
40
+ The Gaussian integers Z[i] also constitute a graph in which uandzare adjacent
41
+ if and only if ( z−u)4= 1.
42
+ Date: Dec 24, 2009.
43
+ 1991Mathematics Subject Classification. Primary 05C12, 05C20, 05C25; Secondary 06F99,
44
+ 11T99.
45
+ Key words and phrases. Cauchy-Schwarz inequality, triangle inequality, submult iplicativity,
46
+ finite field, quadratic field extension, quadratic residue to urnament, grid graph, Manhattan dis-
47
+ tance, discrete norm, Gaussian integers, graph product, gr aph quotient, Cayley graph.
48
+ 12 STEPHAN FOLDES AND L ´ASZL´O MAJOR
49
+ It iseasytoseethatthenorminthis infinite Manhattan grid satisfiesthetriangle
50
+ and submultiplicative inequalities
51
+ N(u+z)≤N(u)+N(z)
52
+ N(uz)≤N(u)N(z)
53
+ To emphasize that the norms on Fp2,Fn
54
+ p2andZ[i] are understood with reference
55
+ to the specific grid graphs defined above, we call these norms Manhattan norms .
56
+ Throughout this paper we think of Fp2as the ring quotient Z[i]/(p).
57
+ 2.Graph quotients and Cayley graphs
58
+ Given a graph G(undirected, with possible loops) on vertex set Vand an equi-
59
+ valence relation ≡onV, thequotient graph G/≡is defined as follows: the vertices
60
+ ofG/≡are the equivalence classes of ≡, and classes A,Bare adjacent if for some
61
+ a∈A,b∈B, the elements a,bare adjacent in G. Note that the distance of Ato
62
+ Bin the quotient graph is at most equal to, but possibly less than the m inimum of
63
+ the distances atobfor alla∈A,b∈B. Note also that G/≡can have loops even
64
+ ifGhas not.
65
+ Given a group Gwith identity element eand a set Γ of group elements that
66
+ generates G, the(left) Cayley graph C(G,Γ) ofGwith respect to Γ has vertex set
67
+ G, elements a,b∈Gbeing considered adjacent if ab−1orba−1belongs to Γ. For
68
+ each congruence ≡of the group G, corresponding to some normal subgroup H, Γ
69
+ yields a generating set Γ ≡ofG/≡consisting with those classes of ≡that intersect
70
+ Γ. The graph quotient of C(G,Γ) by the equivalence ≡coincides with the Cayley
71
+ graph of the quotient graph G/≡with respect to Γ ≡. ForR⊆Ginducing a
72
+ connected subgraph [ R] inC(G,Γ), denote by dR(x,y) the distance function of the
73
+ subgraph [ R]. Denoting by xHtheH-coset of any x∈G, this relates to norms in
74
+ C(G,Γ) andC(G,Γ)/≡as follows: for all x∈R,
75
+ dR(x,e)≥N(x)≥N(xH)
76
+ Both inequalities can be strict. However, we have:
77
+ Cayley Graph Quotient Lemma. Let a group Gwith identity ebe generated
78
+ byΓ⊆G, and consider any normal subgroup Hwith corresponding congruence ≡.
79
+ There is a set R⊆Ghaving exactly one element in common with each congruence
80
+ class modulo H, and such that for every x∈R
81
+ dR(x,e) =N(x) =N(xH)
82
+ Proof.We can define the unique (representative) element r(A)∈R∩Afor each
83
+ cosetAby induction on the distance d(H,A) ofAfromHinC(G,Γ)/≡. Let
84
+ r(H) =e. Assuming r(A) defined for all Awithd(H,A)≤m, let a coset Bhave
85
+ distance m+1 from H. Choose any coset Aadjacent to Bwithd(H,A) =mand
86
+ elements a∈A,b∈Bthat are adjacent in C(G,Γ). Letr(B) =ba−1r(A)./square
87
+ We can apply the above lemma in the case where G=Z[i], Γ ={1,i}and
88
+ H=pZ[i] ={pa+pbi:a,b∈Z}foraprimeinteger p≡3 mod 4. Now C(G,Γ)and
89
+ C(G,Γ)/≡are the Manhattan grid graphs on Z[i] andZ[i]/H=Fp2, respectively.
90
+ Referringtothe set Rofrepresentativesinthe lemma, forany H-cosetsX,Yletx,y
91
+ be the unique elements in X∩R,Y∩R. Asxy∈XY, we have N(XY)≤N(xy).
92
+ By the submultiplicative inequality in Z[i] we have N(xy)≤N(x)N(y). Using the3
93
+ lemmawehave N(x)N(y) =N(X)N(Y). Thisyieldsasubmultiplicativeinequality
94
+ inFp2and a similar reasoning on the coset X+Yyields a triangle inequality:
95
+ Triangle and Submultiplicative Inequalities in Fp2.For allu,zinFp2
96
+ N(u+z)≤N(u)+N(z)
97
+ N(uz)≤N(u)N(z)
98
+ /square
99
+ This indicates that Manhattan distance provides a well-behaved not ion of neigh-
100
+ borhood of 0 in the finite fields Fp2.
101
+ 3.Squares in Fpand non-transitive order
102
+ For each prime p≡3 mod 4 the quadratic residue tournament onFpis the
103
+ directed graph with vertex set Fpin which there is an arrowfrom vertex xto vertex
104
+ yify−xis a non-zero square in Fp, in which case we write x <py. We write x≤py
105
+ ifx <pyorx=y. The relation ≤pis reflexive, anti-symmetric but not transitive,
106
+ and for every x/ne}ationslash=yexactly one of x≤pyory≤pxholds. Using Dirichlet’s
107
+ theorem on primes in arithmetic progressions, Kustaanheimo showe d [4] that for
108
+ every positive integer k, there is a prime p≡3 mod 4, such that ≤pis a transitive
109
+ (and linear) order relation on {0,1,...,k} ⊆Fp, that is, all positive integers up to
110
+ kare quadratic residues mod p. Obviously kcannot exceed ( p−1)/2. Implications
111
+ of [4] and related questions were investigated by J¨ arnefelt, Kust aanheimo, Quist
112
+ [3, 5], in particular with a view to discrete models in physics, also in subse quent
113
+ application-oriented work between the 1950’s (Coish [1]) and the 198 0’s (Nambu
114
+ [6]). For further references see [2]. In particular [4] implies that for every positive
115
+ integerk, there is a prime p≡3 mod 4, such that all z∈Fp2withN(z)≤kare
116
+ squares in Fp2. (Note that all elements of the prime subfield Fpare squares in Fp2.)
117
+ To emphasise the analogy of the relation ≤pwith the ordinary inequality relation
118
+ ≤among numbers, we say that a non-zero z∈Fp2ispositiveifz∈Fpand 0≤pz.
119
+ 4.Inner products compared in non-transitive order
120
+ The only non-trivial automorphism of the field Fp2associates to each z∈Fp2
121
+ itsconjugate z. Theinner product v·wof vectors v= (v1,...,v n) andw=
122
+ (w1,...,w n) inFn
123
+ p2is defined as the scalar v1w1+···+vnwn∈Fp2. This inner
124
+ product is left and right distributive over vector addition, satisfies v·w=w·v,
125
+ c(v·w) = (cv)·w=v·(cw) for allc∈Fp2. However, while v·vbelongs to the
126
+ prime subfield Fp,v·vis not necessarily positive, and can be 0 even if v/ne}ationslash=0. Still,
127
+ a conditional version of positive definiteness holds locally:
128
+ Conditional Positive Definiteness. For every k≥1there is a prime p≡3
129
+ mod 4, such that for all n≥1and for all vectors v∈Fn
130
+ p2of Manhattan norm
131
+ N(v)≤k,we have 0≤pv·vwith equality if and only if v=0.
132
+ Proof.By Kustaanheimo’s result in [4] there is a prime integer p≡3 mod 4 such
133
+ that 0,1,...,2k3are all quadratic residues mod p. Forv= (v1,...,v n) inFn
134
+ p2, let
135
+ vj=aj+bji, wherei2=−1. IfN(v)≤kthen for all j,N(aj)≤kandN(bj)≤k,
136
+ vjvj=a2
137
+ j+b2
138
+ jbelongs to the set of squares {0,...,2k2}. Sincevjcan be non-zero
139
+ for at most kindices 1 ≤j≤nonly, the sum of the corresponding terms a2
140
+ j+b2
141
+ j
142
+ belongs to the set of squares {0,1,...,2k3}. /square4 STEPHAN FOLDES AND L ´ASZL´O MAJOR
143
+ Note that for all vectors v,w∈Fn
144
+ p2
145
+ (v·w)(w·v) = (v·w)(v·w)∈Fpand
146
+ (v·v)(w·w)∈Fp
147
+ Ifvandwareproportional , i.e. if there exists a scalar cinFp2such that v=cw
148
+ orw=cv, then the above two products are equal. Generally, they are relat ed in
149
+ the quadratic residue tournament of Fpas follows.
150
+ Cauchy-Schwarz Inequality for Quadratic Residue Tourname nts.For eve-
151
+ ryk≥1there is a prime p≡3 mod 4 , such that for all n≥1and for all vectors
152
+ v,w∈Fn
153
+ p2of Manhattan norm at most k,
154
+ (v·w)(w·v)≤p(v·v)(w·w)
155
+ Proof.Forn= 1 the inequality holds trivially as the two sides are equal. Assume
156
+ n≥2,v= (v1,...,v n),w= (w1,...,w n). Forall1 ≤i≤n,N(vi)≤k,N(wi)≤k.
157
+ By Kustaanheimo’s result [4] there is a prime p≡3 mod 4 such that all positive
158
+ integers up to 4 k6are quadratic residues modulo p. For each of the/parenleftbign
159
+ 2/parenrightbig
160
+ pairs
161
+ {i,j} ⊆ {1,...,n},i/ne}ationslash=j, by the triangle and submultiplicative inequalities in Fp2
162
+ N[(viwj−vjwi)(viwj−vjwj)]≤(k2+k2)2= 4k4
163
+ Thus the element
164
+ (viwjviwj+vjwivjwi)−(viwjvjwi+vjwiviwj) = (viwj−vjwi)(viwj−vjwj)
165
+ is a square of Manhattan norm at most 4 k4inFp, and it is non-zero for at most/parenleftbigk
166
+ 2/parenrightbig
167
+ ≤k2pairs{i,j}. Summing over all pairs {i,j}, all but at most/parenleftbigk
168
+ 2/parenrightbig
169
+ ≤k2terms
170
+ vanish in the sum/summationdisplay
171
+ [(viwjviwj+vjwivjwi)−(viwjvjwi+vjwiviwj)]
172
+ which therefore has Manhattan norm at most 4 k6and it must also be a square in
173
+ Fp. But this sum is equal to the difference of products
174
+ n/summationdisplay
175
+ i=1vivin/summationdisplay
176
+ j=1wjwj−n/summationdisplay
177
+ i=1viwin/summationdisplay
178
+ j=1vjwj= (v·v)(w·w)−(v·w)(w·v)
179
+ which is consequently a square in Fp. /square
180
+ Remark. From the proof it is clear that, in analogy with the classical Cauchy-
181
+ Schwarz inequality, for vectors v,wof norm not exceeding kinFn
182
+ p2, wherepis
183
+ related to kas stipulated above, the Cauchy-Schwarz inequality with respect t o≤p
184
+ holds with equality if and only if viwj−vjwi= 0 for all i,j, i.e. if and only if v,w
185
+ are proportional.
186
+ We note that the inequality established above is conditional, it holds on ly in a
187
+ specified Manhattan neighborhood of the null vector. Every non- zero element of
188
+ Fpcan be written as a sum of two squares, in particular there are a,b∈Fp, such
189
+ thata2+b2=−1. Forz=a+biwe have zz=−1. As soon as n≥2, inFn
190
+ p2let
191
+ v= (a,b,0,...,0) and w= (bz,−az,0,...,0)
192
+ The inequality ( v·w)(w·v)≤p(v·v)(w·w) fails because the left-hand side is 0
193
+ and the right-hand side is −1. In fact if n≥3, the inequality can be invalidated
194
+ with vectors v,winFn
195
+ pas follows. Taking again a,b∈Fpwitha2+b2=−1, let
196
+ v= (1,a,b,0,...,0) and w= (1,0,0,0,...,0)5
197
+ However,the Cauchy-Schwarzinequalityholdsunconditionallyinthe 2-dimensional
198
+ case for vectors with components in Fp:
199
+ Special case of F2
200
+ p.Letpbe a prime congruent 3modulo4. For all vectors v,w
201
+ inF2
202
+ p
203
+ (v·w)(w·v)≤p(v·v)(w·w)
204
+ Proof.Nowtheconjugationappearinginthe innerproductsisthe identity. Written
205
+ in components,
206
+ (v·v)(w·w)−(v·w)(w·v) = (v2
207
+ 1+v2
208
+ 2)(w2
209
+ 1+w2
210
+ 2)−(v1w1+v2w2)2=
211
+ =v2
212
+ 1w2
213
+ 2+v2
214
+ 2w2
215
+ 1−2v1w1v2w2= (v1w2−v2w1)2
216
+ /square
217
+ 5.Manhattan norm of inner product
218
+ The Manhattan norm can be seen to be submultiplicative not only on th e ring
219
+ Z[i] and its quotient field Fp2, but on all vector spaces Fn
220
+ p2, with respect to the
221
+ inner product:
222
+ Cauchy-Schwarz Inequality for Manhattan Norm on Fn
223
+ p2.Consider any
224
+ primep≡3 mod 4 and letn≥1. For all v,w∈Fn
225
+ p2
226
+ N(v·w)≤N(v)N(w)
227
+ Proof.Letv= (v1,...,v n),w= (w1,...,w n)∈Fn
228
+ p2. Thenv·w=/summationtextvjwj. Clearly
229
+ N(z) =N(z) for any z∈Fp2. By the triangle and submultiplicative inequalities in
230
+ Fp2we have
231
+ N(v·w) =N/parenleftbig/summationtextvjwj/parenrightbig
232
+ ≤/summationtextN/parenleftbig
233
+ vjwj/parenrightbig
234
+ ≤/summationtextN(vj)N(wj)≤
235
+ ≤/summationtextN(vj)/summationtextN(wj) =N(v)N(w)
236
+ /square
237
+ Remark. The inequality N(v·w)≤N(v)N(w) is easily interpreted and continues
238
+ to hold for v,win the module ( Z[i]/mZ[i])nfor any positive integer m. As soon as
239
+ mis composite, or a prime not congruent to 3 modulo 4, the ring Z[i]/mZ[i] fails
240
+ to be an integral domain.
241
+ References
242
+ [1] H.R. Coish, Elementary particles in a finite world geomet ry, Phys. Rev. 114 - 1 (1959) 383-388
243
+ [2] S. Foldes, The Lorentz group and its finite field analogues : local isomorphism and approxima-
244
+ tion, J. Math. Phys. 49, 093512 (2008)
245
+ [3] G. J¨ arnefelt, P. Kustaanheimo, An observation on finite geometries, in Proc. Skandinaviske
246
+ Matematikerkongress i Trondheim 1949, 166-182
247
+ [4] P. Kustaanheimo, A note on a finite approximation of the eu clidean plane geometry, Comment.
248
+ Phys.-Math. Soc. Sc. Fenn. XV. 19 (1950) 1-11
249
+ [5] P. Kustaanheimo, B. Qvist, On differentiation in Galois fi elds. Ann. Acad. Sci. Fennicae. Ser.
250
+ A. I. Math.-Phys. 1952, (1952). no. 137, 12 pp.
251
+ [6] Y. Nambu, Field theory of Galois fields, in Field Theory an d Quantum Statistics, eds. J.A.
252
+ Batalin et.al., Institute of Physics Publishing 1987, pp. 6 25-6366 STEPHAN FOLDES AND L ´ASZL´O MAJOR
253
+ Stephan Foldes
254
+ Institute of Mathematics,
255
+ Tampere University of Technology,
256
+ PL 553, 33101 Tampere, Finland
257
+ E-mail address :[email protected]
258
+ L´aszl´o Major
259
+ Institute of Mathematics,
260
+ Tampere University of Technology,
261
+ PL 553, 33101 Tampere, Finland
262
+ E-mail address :[email protected]
1001.0047.txt ADDED
@@ -0,0 +1,953 @@
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
1
+ Coulomb interaction and transient charging of excited states in open nanosystems
2
+ Valeriu Moldoveanu,1Andrei Manolescu,2Chi-Shung Tang,3and Vidar Gudmundsson4
3
+ 1National Institute of Materials Physics, P.O. Box MG-7, Bucharest-Magurele, Romania
4
+ 2Reykjavik University, School of Science and Engineering, Kringlan 1, IS-103 Reykjavik, Iceland
5
+ 3Department of Mechanical Engineering, National United University, Lienda, Miaoli 36003, Taiwan
6
+ 4Science Institute, University of Iceland, Dunhaga 3, IS-107 Reykjavik, Iceland
7
+ We obtain and analyze the e ect of electron-electron Coulomb interaction on the time dependent
8
+ current
9
+ the contact is gradually switched on in time and we calculate the time dependent reduced density
10
+ operator of the sample using the generalized master equation. The many-electron states (MES) of
11
+ the isolated sample are derived with the exact diagonalization method. The chemical potentials of
12
+ the two leads create a bias window which determines which MES are relevant to the charging and
13
+ discharging of the sample and to the currents, during the transient or steady states. We discuss the
14
+ contribution of the MES with xed number of electrons Nand we nd that in the transient regime
15
+ there are excited states more active than the ground state even for N= 1. This is a dynamical
16
+ signature of the Coulomb blockade phenomenon. We discuss numerical results for three sample
17
+ models: short 1D chain, 2D lattice, and 2D parabolic quantum wire.
18
+ PACS numbers: 73.23.Hk, 85.35.Ds, 85.35.Be, 73.21.La
19
+ I. INTRODUCTION
20
+ Due to the increasing interest in ultra-fast electron
21
+ dynamics considerable progress occurred recently in the
22
+ theoretical description of time dependent mesoscopic
23
+ transport. New methods and numerical implementations
24
+ are rapidly evolving. Transient currents in open nanos-
25
+ tructures are studied with Green-Keldysh formalism,1,2,3
26
+ scattering theory,4and quantum master equation.5,6,7,8
27
+ Most of the results were obtained for noninteracting elec-
28
+ trons due to the well known computational diculties to
29
+ include time-dependent Coulomb e ects.
30
+ It is nevertheless clear that the electron-electron inter-
31
+ action is important in such problems. An e ort to incor-
32
+ porate it has been recently done by Kurth et al.9followed
33
+ by My oh anen et al.10who have described correlated time-
34
+ dependent transport in a short 1D chain de ned by a
35
+ lattice Hamiltonian. The 1D sample was connected to
36
+ external leads and the current was driven by a time-
37
+ dependent bias. Those authors used a method based
38
+ on the Kadano -Baym equation for the non-equilibrium
39
+ Green's function combined with the time-dependent den-
40
+ sity functional theory to include the Coulomb interac-
41
+ tion in the sample. Once the Green's functions were
42
+ calculated total average quantities of interest could be
43
+ obtained, like charge density or current, both in the tran-
44
+ sitory and in the steady state. However this method does
45
+ not say much about the dynamics of speci c internal
46
+ states of the sample system. In view of the spectroscopy
47
+ of excited states11it is important to have a theoretical
48
+ tool for understanding separately the charging and re-
49
+ laxation of the ground states and excited states in meso-
50
+ scopic systems in time-dependent conditions.
51
+ Our alternative is to use the statistical, or density op-
52
+ erator. The complete information about the time evo-
53
+ lution of each quantum state of the sample is captured
54
+ in the reduced density operator (RDO), which is the so-lution of the generalized master equation (GME). Once
55
+ the RDO is de ned in the Fock space the inclusion of
56
+ the Coulomb interaction becomes a known computational
57
+ problem: obtaining the many-electron states (MES) of
58
+ the sample. The RDO matrix is then calculated in the
59
+ basis of the interacting MES.
60
+ Let us enumerate some of the previous theoretical
61
+ schemes to treat transport and electron-electron inter-
62
+ action with the master equation. One of the rst at-
63
+ tempts to derive a master equation for an interacting sys-
64
+ tem with time-dependent perturbations belongs to Lan-
65
+ greth and Nordlander for the Anderson model.12Gurvitz
66
+ and Prager started from the time-dependent Shr odinger
67
+ equation for the MES wave functions and ended up with
68
+ Bloch-like rate equations for the density matrix of a quan-
69
+ tum dot.13The electronic currents were calculated in the
70
+ steady state and it was shown that the Coulomb interac-
71
+ tion renormalizes the tunneling rates between the leads
72
+ and the system. In the same context K onig et al.14de-
73
+ veloped a powerful diagrammatic technique by expand-
74
+ ing the RDO of a mesoscopic system in powers of the
75
+ tunneling Hamiltonian. The time-dependence of the sta-
76
+ tistical operator of the coupled and interacting system
77
+ implies a quantum master equation for the so called pop-
78
+ ulations. In this method the Coulomb interactions are
79
+ treated exactly, which makes it appealing for studying
80
+ various correlation e ects like cotunneling.15The con-
81
+ nection between the real-time diagrammatic approach of
82
+ K onig et al.14and the Nakajima-Zwanzig approach16,17
83
+ to the generalized master equation (GME) approach was
84
+ made transparent by Timm.18
85
+ More recently Li and Yan19combined the n-resolved
86
+ master equation and the time dependent density-
87
+ functional method to write down a Kohn-Sham master
88
+ equation for the reduced single-particle density matrix.
89
+ Also, Esposito and Galperin,20using the equation of mo-
90
+ tion for the Hubbard operators, have obtained a many-arXiv:1001.0047v1 [cond-mat.mes-hall] 30 Dec 20092
91
+ body description of quantum transport in an open sys-
92
+ tem and established a connection between the GME and
93
+ non-equilibrium Greeen's functions. They studied simple
94
+ systems in the steady state regime: a resonant level cou-
95
+ pled to a a single vibration mode, an interacting dot with
96
+ two spins, and a two-level bridge. Another recent work
97
+ by Darau et al.21implemented the GME for a benzene
98
+ single-electron transistor and used exact MES to compute
99
+ steady state currents within the Markov approximation.21
100
+ The stability diagram and the conductance peaks were
101
+ obtained and a current blocking due to interferences be-
102
+ tween degenerated orbitals was noticed.
103
+ In our previous papers7,8we considered the GME
104
+ method for the RDO of independent electrons in the Fock
105
+ space. We discussed the transient transport through
106
+ quantum dots and quantum wires. The contact between
107
+ the leads and the sample was switched on at a certain ini-
108
+ tial moment t0. We discussed extensively the occupation
109
+ of the states within the bias window and the geometrical
110
+ e ects on the transient currents. We described the cou-
111
+ pling between the sample and the leads via a tunneling
112
+ Hamiltonian in which we took into account the spatial
113
+ extension of the wave functions of both subsystems in
114
+ the contact region.
115
+ In spite of earlier or more recent attempts a complete
116
+ description of the Coulomb e ects in the time-dependent
117
+ transport is still missing, especially in sample models
118
+ larger than a few sites. In the present work we com-
119
+ bine the GME method with the Coulomb interaction in
120
+ the sample and we analyze the dynamics of the electrons
121
+ starting with the moment when the leads are coupled to
122
+ the sample until a steady state is reached. The Coulomb
123
+ interaction is included in the Hamiltonian of the isolated
124
+ sample and the interacting MES are calculated with the
125
+ exact diagonalization method. This means the Coulomb
126
+ interaction is fully included with no mean eld assump-
127
+ tion or density-functional model. The number of single-
128
+ electron states (SES) used to de ne the matrix elements
129
+ of the Hamiltonian of interacting electrons is suciently
130
+ large such that the MES of interest are convergent. Due
131
+ to the nite bias window only a limited number of MES
132
+ participate to the charge transport through the sample,
133
+ i. e.only those energetically compatible with the elec-
134
+ trons in the leads. Hence the MES of interest are selected
135
+ by the chemical potentials in the leads. We calculate the
136
+ RDO matrix elements in the subspace of these MES using
137
+ the GME. The electron-electron interaction in the leads
138
+ is neglected.
139
+ It is well known that the Fock space increases expo-
140
+ nentially with the number of SES. In addition the time
141
+ dependent numerical solution of the GME is also com-
142
+ putational expensive. So at this stage we are limited to
143
+ describe only few electrons in the system: up to ve in a
144
+ small system, but only up to three in a larger one.
145
+ The paper is organized as follows. In Section 2 we
146
+ brie
147
+ interaction, and the selection of the MES. Next, in Sec-
148
+ tion 3, we show results for three models: a short 1Dchain, a 2D lattice of 12 10 sites, and a nite quantum
149
+ wire with parabolic lateral con nement. Conclusions and
150
+ discussions are presented in Section 4.
151
+ II. GME METHOD AND COULOMB
152
+ INTERACTION
153
+ In this section we summarize the main lines of our
154
+ method. The equations apply both to the lattice and
155
+ continuous models. The time-dependent transport prob-
156
+ lem is considered within the partitioning approach which
157
+ is known both from the pioneering work of Caroli22and
158
+ from the derivation of the GME. Prior to an initial time
159
+ t0the left lead (L) having a \source" role, and the right
160
+ lead (R) having a \drain" role, are not connected to the
161
+ sample and therefore can be characterized by equilibrium
162
+ states with chemical potentials LandRrespectively.
163
+ Our aim is to compute the time dependent currents
164
+ ing through the sample and leads starting at moment t0,
165
+ when the three subsystems are connected, until a station-
166
+ ary state is reached.
167
+ The generic Hamiltonian of the total system consisting
168
+ of the sample plus the leads is:
169
+ H(t) =HL+HR+HS+HT(t): (1)
170
+ Hlwithl=L;R are the Hamiltonians of the leads. We
171
+ denote by "qland qlthe single-particle energies and
172
+ wave functions respectively, for each lead. Using the cre-
173
+ ation and annihilation operators associated to the single-
174
+ particle states, cy
175
+ qlandcql, we can write
176
+ Hl=Z
177
+ dq"qlcy
178
+ qlcql: (2)
179
+ HSis the Hamiltonian of the sample. In the absence
180
+ of the interaction the SES have discrete energies denoted
181
+ asEnand corresponding one-body wave functions n(r).
182
+ Using now the creation and annihilation operators for the
183
+ sample SES, dy
184
+ nanddn, we can write
185
+ HS=X
186
+ nEndy
187
+ ndn+1
188
+ 2X
189
+ nm
190
+ n0m0Vnm;n0m0dy
191
+ ndy
192
+ mdm0dn0:(3)
193
+ The second term in Eq. (3) is the Coulomb interaction.
194
+ In the SES basis the two-body matrix elements are given
195
+ by:
196
+ Vnm;n0m0=Z
197
+ drdr0
198
+ n(r)
199
+ m(r0)u(rr0)n0(r)m0(r0);
200
+ (4)
201
+ whereu(rr0) is the Coulomb potential.
202
+ The third term of Eq. (1) is the so-called tunneling
203
+ Hamiltonian describing the transfer of particles between
204
+ the leads and the sample:
205
+ HT(t) =X
206
+ l=L;RX
207
+ nZ
208
+ dql(t)(Tl
209
+ qncy
210
+ qldn+h:c:):(5)3
211
+ HTcontains two important elements: (1) The time de-
212
+ pendent switching functions l(t) which open the contact
213
+ between the leads and the sample; these functions mimic
214
+ the presence of a time dependent potential barrier. (2)
215
+ The coupling Tl
216
+ qnbetween a state with momentum qof
217
+ the leadland the state nof the isolated sample, with
218
+ wave function n. The coupling coecients Tl
219
+ qndepend
220
+ on the energies of the coupled states and, maybe more
221
+ important, on the amplitude of the wave functions in the
222
+ contact region. As we have shown in our previous work7,8
223
+ this construction allows us to capture geometrical e ects
224
+ in the electronic transfer. A precise de nition of the cou-
225
+ pling coecients is however model speci c, and will be
226
+ mentioned in the next section.
227
+ The evolution of our system is completely determined
228
+ by the statistical operator W(t) associated to the total
229
+ Hamiltonian H(t) de ned in Eq.(1). W(t) is the solution
230
+ of the quantum Liouville equation with a known initial
231
+ value, prior to the coupling of the sample and leads:
232
+ i~_W(t) = [H(t);W(t)]; W (tt0) =LRS;(6)
233
+ The isolated leads are described by equilibrium distribu-
234
+ tions,
235
+ l=e (HllNl)
236
+ Trlfe (HllNl)g; l=L;R; (7)
237
+ and the isolated sample by the density operator S. Af-
238
+ ter the coupling moment the dynamics of the sample is
239
+ conveniently described by the RDO which is de ned by
240
+ averaging the total statistical operator over those degrees
241
+ of freedom belonging to the leads:
242
+ (t) = TrLTrRW(t); (t0) =S: (8)
243
+ In the absence of the electron-electron interaction the
244
+ MES eigenvectors of HSare bit-strings of the form ji=
245
+ ji
246
+ 1;i
247
+ 2;::;i
248
+ n:::i, wherei
249
+ n= 0;1 is the occupation number
250
+ of then-th SES. The set fgis a basis in the Fock space
251
+ of the isolated sample and the RDO can be seen as a
252
+ matrix in this basis. From Eqs. (6)-(8) we obtain in the
253
+ lowest (2-nd) order in the coupling parameters Tl
254
+ qnthe
255
+ GME (see Ref. 7 for details):
256
+ _(t) =i
257
+ ~[HS;(t)]
258
+ 1
259
+ ~2X
260
+ l=L;RZ
261
+ dql(t)([Tql;
262
+ ql(t)] +h:c:);(9)
263
+ where the coupling operator Tqlhas matrix elements
264
+ (Tql)=X
265
+ nTl
266
+ qnhjdyji: (10)
267
+ The operators
268
+ qland qlare de ned as
269
+
270
+ ql(t) =eitHSZt
271
+ t0dsl(s)ql(s)ei(st)"qleitHS;
272
+ ql(s) =eisHS
273
+ Ty
274
+ ql(s)(1fl)(s)Ty
275
+ qlfl
276
+ eisHSandflis the Fermi function of the lead l.
277
+ In the presence of the electron-electron interaction in
278
+ the sample the MES which are eigenstates of HSare lin-
279
+ ear combinations of bit-strings: HSj ) =E j ), where
280
+ j ) =P
281
+ C ji,C being the mixing coecients
282
+ which can be found together with the energies E by
283
+ diagonalizing HS. (To distinguish better between the
284
+ noninteracting and the interacting MES we use the right
285
+ angular bracket for the former and the regular curved
286
+ bracket for the later.) Using now the set f gas a basis,
287
+ i. e.theinteracting MES, the GME has the same form
288
+ as Eq. (9), where the matrix elements of all operators
289
+ are now de ned in the interacting basis and the matrix
290
+ elements of the coupling operators are
291
+ (Tql) =X
292
+ nTl
293
+ qn( jdyj ): (11)
294
+ Because the sample is open the number of electrons N
295
+ contained in the sample is not xed. The Hamiltonian
296
+ HSgiven in Eq. (3) commutes with the total \number"
297
+ operatorP
298
+ ndy
299
+ ndn. ThusNis a \good quantum number"
300
+ such that any state j ) has a xed number of electrons.
301
+ So the MES can also be labeled as j ) =jN;i) with
302
+ i= 0;1;2;:::an index for the ground and excited states
303
+ of the MES subset with Nelectrons. The many-body
304
+ energies can also be written as E =E(i)
305
+ N. In the practical
306
+ calculations Nvaries between 0 (the vacuum state) and
307
+ Nmaxwhich is the total number of SES considered in the
308
+ numerical diagonalization of HS. The total number of
309
+ MES is thus 2Nmax.
310
+ If the coupling between the leads and the sample is
311
+ not too strong we expect that only a limited number of
312
+ MES participate e ectively to the electronic transport.
313
+ These states are naturally selected by the bias window
314
+ [R;L]. In the following examples, by selecting suitable
315
+ values of the chemical potentials in the leads, we will
316
+ truncate the basis of interacting MES to a reasonably
317
+ small subset such that we can solve numerically Eq. (9)
318
+ with our available computing resources. To relate the
319
+ bias window with the e ective MES we need to consider
320
+ the chemical potential of the isolated sample containing
321
+ Nelectrons,
322
+ (i)
323
+ N=E(i)
324
+ NE(0)
325
+ N1; (12)
326
+ which is the energy required to add the N-th electron on
327
+ top of the ground state with N1 to obtain the i-th MES
328
+ withNparticles.23We expect the current associated to
329
+ the MESjN;i) to depend on the location of the chemical
330
+ potential(i)
331
+ Nrelatively to the bias window. In particular
332
+ it is clear that if at the coupling moment t0the sample is
333
+ empty all MES with (i)
334
+ NLwill remain empty both
335
+ during the transient and the steady states, so they can
336
+ be safely ignored when solving the GME.4
337
+ III. MODELS AND RESULTS
338
+ We have numerically implemented the GME method
339
+ both for lattice and continuous models. The sample mod-
340
+ els are: a short 1D chain with 5 sites, a 2D rectangular
341
+ lattice with 1210 = 120 sites, and a short quantum
342
+ wire withe parabolic lateral con nement. In all cases the
343
+ coupling functions have the form
344
+ l(t) = 12
345
+ e
346
+ with
347
+ ment, which is t0= 0, we have l(0) = 0 (no coupling),
348
+ and in the steady state, for t!1 ,l= 1 (full coupling).
349
+ A. A toy model: short 1D chain
350
+ In this model the two semi-in nite leads are attached
351
+ to the ends of a 1D chain with 5 sites. The coupling
352
+ between a lead state with wave function qland a sam-
353
+ ple state with wave function nis given by the product
354
+ between the wave functions at the contact site:
355
+ Tl
356
+ qn=Vl 
357
+ ql(0)n(il); (14)
358
+ where 0 is the contact site of the lead l=L;R, the end
359
+ sites of the sample being iL= 1 andiR= 5.
360
+ FIG. 1: (Color online) The equilibrium chemical potentials
361
+ (0)
362
+ Nfor 1N5 as a function of the interaction strength
363
+ U. The dotted lines mark the chemical potentials of the leads
364
+ selected in the transport simulations shown in the next gure,
365
+ i. e.L= 5:25 andR= 4:75.
366
+ The reason to call this a toy model is that we can ob-
367
+ tain the complete set of 25= 32 MES, i. e.we do not
368
+ need to cut the basis of the 5 SES. We also do not need
369
+ to cut the MES basis, all matrix elements of the statisti-
370
+ cal operator can be numerically calculated, even if not all
371
+ of them might be important for the currents. In addition
372
+ we will consider the strength of the Coulomb interaction
373
+ as a free parameter U, whereas in a realistic systems this
374
+ is xed by the electron charge and the dielectric con-
375
+ stant of the material. Our goal is to have a qualitativeunderstanding of the underlying physics, and in particu-
376
+ lar to show the presence of the Coulomb blocking e ects
377
+ at certain values of Uor of the chemical potentials of the
378
+ leads. The Coulomb matrix elements de ned in Eq. (4)
379
+ are calculated as
380
+ Vnm;n0m0=X
381
+ i6=i0
382
+ n(i)
383
+ m(i0)U
384
+ jii0jn0(i)m0(i0):(15)
385
+ In Fig.1 we show the equilibrium chemical potentials
386
+ (0)
387
+ Ncorresponding to ground states with 1 N5 par-
388
+ ticles against the interaction strength U. One observes
389
+ a linear dependence of (0)
390
+ NonU, with slope increasing
391
+ withN. Obviously the total Coulomb energy increases
392
+ both withUandN.
393
+ Let us now brie
394
+ scenario.24Suppose the isolated sample contains Nelec-
395
+ trons and the chemical potentials of the leads are cho-
396
+ sen such that (0)
397
+ N< R< L< (0)
398
+ N+1. Then the bias
399
+ window [R;L] may include one or more of the excited
400
+ con gurations with Nparticles. In general some states
401
+ withNelectrons may have excitation energies exceed-
402
+ ingLor even(0)
403
+ N+1. This situation corresponds to the
404
+ Coulomb blockade phenomenon. Indeed, the addition of
405
+ the (N+ 1)-th electron is energetically forbidden. Con-
406
+ sequently the current in the steady state should vanish.
407
+ However, shorter or longer transient currents are gener-
408
+ ated by all many-body con gurations in the vicinity of
409
+ the bias window.
410
+ Fig. 2(a) and 2(b) show the total currents in the left
411
+ lead and the total charge residing in the sample for sev-
412
+ eral values of the interaction strength. Uis measured
413
+ in units of tS, the hopping parameter in the sample,7
414
+ and the time is expressed in units of ~=tSwhile the cur-
415
+ rent is in units of etS=~. The coupling constant in Eq.
416
+ (13) is
417
+ (0) =j00000ih00000j.
418
+ The chemical potentials of the leads, L= 5:25 and
419
+ R= 4:75, are chosen such that in the absence of
420
+ Coulomb interaction, i. e. forU= 0,(0)
421
+ 4is located
422
+ within the bias window. In this case we obtain in the
423
+ steady state the mean number of electrons about 3.6 and
424
+ a non-vanishing current in the leads. This is understand-
425
+ able, since (0)
426
+ 4=E4= 5, which is the 4-th level of
427
+ the isolated sample. The occupation of this level in the
428
+ steady state is about 0.6, the other states being either full
429
+ or empty. Also in this case, the excited states have small
430
+ contributions to the steady state current as the system
431
+ tends to be in the ground state with N= 3 electrons.
432
+ Those contributions may also depend on the coupling
433
+ strength of individual states with the leads, but in gen-
434
+ eral remain small.25
435
+ The situation may change for U6= 0. For the inter-
436
+ acting system, e. g. forU= 0:3, the system settles down
437
+ in the Coulomb blockade regime, the total current be-
438
+ ing almost suppressed in the steady state. This happens
439
+ because the interaction pushes the chemical potentials
440
+ upwards such that for U= 0:3 both ground states with5
441
+ FIG. 2: (Color online) The total current entering the 5 1
442
+ sample from the left lead as a function of time for the di erent
443
+ values of the interaction strength U. The chemical potentials
444
+ of the leads L= 5:25 andR= 4:75.
445
+ N= 3 andN= 4 electrons are outside the bias win-
446
+ dow and cannot produce steady currents. When the in-
447
+ teraction strength is further increased to U= 0:5 and
448
+ U= 1 the steady state currents are gradually restored.
449
+ This could look surprising, but one can see in Fig.1 that
450
+ by increasing Uthe ground state con guration with 3
451
+ electrons approaches and enters the bias window. Con-
452
+ sequently the transport becomes again possible. Note
453
+ that while the steady state currents are not monotonous
454
+ w.r.t.Uthe charge absorbed in the system continuously
455
+ decreases, Fig. 2(b).
456
+ In transport experiments the strength of the electron-
457
+ electron interaction is indeed xed. The usual way to
458
+ obtain the Coulomb blockade is to vary the chemical po-
459
+ tentials of the leads relatively to the energy levels of the
460
+ sample, or vice versa. In Fig. 3 we show the currents
461
+ in both leads for di erent values of the chemical poten-
462
+ tialR, while keeping xed L= 7. The strength of
463
+ the Coulomb interaction is U= 1 and(0)
464
+ 4almost equals
465
+ L. The steady state value of the current decreases as
466
+ Rincreases, because fewer states are included in the
467
+ bias window. The Coulomb blockade onset occurs for
468
+ R>5, when(0)
469
+ 3drops below R. We observe that the
470
+ FIG. 3: The time-dependent total currents in the left and
471
+ right leads at di erent values of the chemical potential R.
472
+ The current in the right lead starts at negative values. Other
473
+ parameters: VL=VR= 0:750,U= 1:0.
474
+ maximum value of the total current in the left lead does
475
+ not change much when Rvaries. In contrast, the tran-
476
+ sient current in the right lead is negative and increases
477
+ in magnitude as Rincreases. This means that the right
478
+ lead feeds the many-body con gurations that fall below
479
+ R.
480
+ The contribution of the excited states to the transient
481
+ and steady state currents depends strongly on the bias
482
+ window. In Fig. 4 we show the currents entering the sam-
483
+ ple from the left lead, carried by the states with N= 2
484
+ andN= 3 electrons, for R= 3;4;5 (the cases with non-
485
+ vanishing current in the steady state). We also show sep-
486
+ arately the contribution to the currents associated to the
487
+ ground state con gurations, related to (0)
488
+ 2and(0)
489
+ 3, and
490
+ the complementary contribution of all the excited states
491
+ with 2 and 3 particles. In this case the wave vectors of
492
+ the ground states are mostly given by the non-interacting
493
+ wave vectors:j11000iwith weight 97% and j11100iwith
494
+ 98% forN= 2 andN= 3 respectively.
495
+ ForR= 3 the steady state current of the ground
496
+ state con guration is vanishingly small and so the total
497
+ negative current associated to two-particle states comes
498
+ mostly from the excited states. In the many-body energy
499
+ spectrum of the isolated sample we obtain 5 excited con-
500
+ gurations with (i)
501
+ 22[R;L] = [3;7]. AsRmoves up
502
+ the steady state current of the ground state with N= 2
503
+ becomes also negative. The combined contributions of
504
+ the excited states vanishes at R= 5. As can be seen
505
+ from Fig. 1 R= 5 is well above (0)
506
+ 2, but very close to
507
+ (0)
508
+ 3. Consequently, the ground con guration with N= 2
509
+ is heavily populated in the steady state, whereas the ex-
510
+ cited states have low probability and thus weak current.
511
+ Actually, as we have checked, all the currents associated
512
+ to each excited state with N= 2 vanish individually. In
513
+ the transient regime however the N= 2 currents in all
514
+ three cases are dominated by the excites states.
515
+ The currents of the excited states having N= 3 elec-6
516
+ FIG. 4: The separate contributions to the current of the
517
+ ground state with Nparticles and of allexcited states with
518
+ Nparticles, for di erent values of R. For completeness we
519
+ also include the total currents JLfor the same con gura-
520
+ tions. The discussion is made in the text. Other parameters:
521
+ VL=VR= 0:750,U= 1:0.
522
+ trons are positive at R= 3, but change sign at R= 4.
523
+ ForR= 5 their magnitude exceeds the contribution of
524
+ the ground state which is always positive. A more de-
525
+ tailed analysis of the currents carried by speci c excited
526
+ states will be given for the 2D model.
527
+ Finally, both in the transient and in the steady states
528
+ the currents have small periodic
529
+ by the permanent transitions of electrons between the
530
+ states in the sample and the states in the leads and
531
+ back.25They are best seen in Fig. 2(a). Such
532
+ have also been obtained very recently by Kurth et al. us-
533
+ ing combination of the non-equilibrium Green's functions
534
+ and the time dependent density-functional theory of the
535
+ Coulomb interaction.26
536
+ B. 2D lattice
537
+ We show now results for a 2D rectangular lattice with
538
+ 1210 sites. For a lattice constant of a= 5 nm this
539
+ sample can be seen as a discrete version of a quantum
540
+ dot of 60 nm50 nm. We used the lowest 10 SES of the
541
+ non-interacting sample in the numerical diagonalization
542
+ of the interacting Hamiltonian. This number is sucient
543
+ to produce convergent results for the rst 50 MES for
544
+ an interaction strength U= 0:8. The Coulomb matrix
545
+ elements are calculated in the same way as for the 1Dcase, Eq. (15), except that now the site indices are two-
546
+ dimensional, i. e.i= (ix;iy) andi0= (i0
547
+ x;i0
548
+ y).
549
+ The two contact sites are chosen at diagonally opposite
550
+ corners of the sample. The coupling coecients are cal-
551
+ culated with Eq. (14), like for the 1D chain, and depend
552
+ on the wave function of the particular SES at the con-
553
+ tact sites. These coecients are illustrated in Fig. 5(a).
554
+ The reduced density matrix is calculated using the rst
555
+ 50 MES. This allowed us to take into account many-body
556
+ con gurations with up to 3 electrons.
557
+ In Fig. 5(b) we show the chemical potentials (i)
558
+ Nfor
559
+ the ground and excited states with N= 1;2;and 3 par-
560
+ ticles. At the initial moment t0= 0 the system is empty.
561
+ Based on the previous example, the main contribution to
562
+ the currents in the steady states is expected from those
563
+ MES with ground state chemical potentials located inside
564
+ the bias window [ R;L]. One also observes excited con-
565
+ gurations with Nparticles having chemical potentials
566
+ larger than (0)
567
+ N+1.
568
+ FIG. 5: (Color online) (a) The coupling amplitudes jTqnj2
569
+ forn= 1;::;5 between single-particle states in the leads with
570
+ momentum qand the lowest 5 single-particle states of the
571
+ isolated dot. (b) The generalized chemical potentials for N-
572
+ particle interacting con gurations. The red crosses mark (0)
573
+ N
574
+ while the other ones correspond to generalized potentials (i)
575
+ N
576
+ related to the i-th excited state of the Nparticle system.
577
+ In the following we discuss the currents carried by the7
578
+ various many-body states involved in transport. In a rst
579
+ series of calculations we selected the chemical potential
580
+ R= 0:2 and used two values of the chemical potential
581
+ of the left lead L= 0:4 andL= 0:6. ForR= 0:2 and
582
+ L= 0:4 the bias window contains only the 1-st and the
583
+ 2-nd excited con gurations with N= 1, Fig. 5(b). The
584
+ ground states for N= 1 andN= 2 are instead located
585
+ below and above the bias window, respectively. Conse-
586
+ quently the steady state current is very small. When
587
+ Lincreases to 0.6 the ground state con guration with
588
+ N= 2 enters the bias window and the current increases,
589
+ Fig. 6(a).
590
+ To analyze the transient regime we split the current
591
+ into contributions given by the ground state and excited
592
+ states with 1 electron (see Fig. 6(b)). When L= 0:4 the
593
+ 1-st and 2-nd excited state carry currents exceeding the
594
+ current associated to the ground state, which survive all
595
+ the way to the steady state. The current corresponding
596
+ to the 2-nd excited state is smaller than the current of the
597
+ 1-st excited state, but comparable to that of the ground
598
+ state. This is explained by the strength of the coupling
599
+ coecients shown in Fig. 5(a), the 2-nd single-particle
600
+ state being stronger coupled to the leads. The remaining
601
+ higher excited states give oscillating and fast decaying
602
+ transient currents. In Fig. 6(c) L= 0:6 and therefore
603
+ higher excited states enter the bias window; their tran-
604
+ sient currents are still decaying but at a smaller rate.
605
+ Comparing with Fig. 6(a) it in clear that the transient
606
+ regime is dominated by excited states.
607
+ Next we discuss currents associated with states having
608
+ 2 and 3 electrons. We keep now xed R= 0:35 and
609
+ again increase Lstarting with 0.6. Fig. 7(a) shows the
610
+ total currents in the left lead for N= 2 andN= 3. As
611
+ the bias increases the transient currents are enhanced,
612
+ but they become comparable as the system approaches
613
+ the steady state. In Fig. 7(b) the total current on three
614
+ particle states shows a di erent behavior: the steady
615
+ states value increases drastically when Lmoves up. To
616
+ explain this one can look again at the diagram of the
617
+ chemical potentials, Fig. 5(b). At L= 0:6 the 3-particle
618
+ con gurations are above the bias window and as such
619
+ they contribute less to the current. In contrast, as L
620
+ increases the ground state con guration with N= 3 en-
621
+ ters the bias window, the window is closer to the excited
622
+ states, and thus the total current increases. Actually, for
623
+ L= 0:8 and 0:9 the current for N= 3 does not reach
624
+ the steady state in the time interval considered.
625
+ Now we look at the contribution of the excited states
626
+ withN= 2 for two cases, L= 0:6 andL= 0:9. Again,
627
+ the inspection of the diagram in Fig. 5(b) predicts the
628
+ results of Fig. 8. When L= 0:6 there is just one excited
629
+ con guration within the bias window, in addition to the
630
+ ground state. In Fig. 8(a) we see that in the steady state
631
+ these two con gurations give signi cant contributions to
632
+ the current, whereas the higher excited states play a role
633
+ only in the transient regime. Fig. 8(b) shows that at
634
+ L= 0:9 the currents of the excited states and of the
635
+ ground state are decreasing, some of them reaching even
636
+ FIG. 6: (a) The total currents in the left and right leads for
637
+ L= 0:6 andL= 0:4, while keeping R= 0:2. (b) The
638
+ partial currents in the left lead for single-particle states when
639
+ L= 0:4 andR= 0:2. (c) The partial currents in the left
640
+ lead for single-particle states when L= 0:6 andR= 0:2
641
+ negative values towards the steady state. This happens
642
+ because the bias window includes now the ground state
643
+ withN= 3 and the excited states with N= 2 deplete in
644
+ the favor of the ground state.
645
+ The sign of the current carried by states with Npar-
646
+ ticles depends on the placement of the corresponding
647
+ ground state chemical potential relatively to the bias win-
648
+ dow. For example if we x L= 1:5 andR= 0:65 we8
649
+ FIG. 7: (a) The total current in the left lead carried by all
650
+ many-body con gurations with N= 2, for increasing values
651
+ ofL(i. e.0.6,0.7,0.8 and 0.9) and R= 0:2. (b) The same
652
+ forN= 3.
653
+ obtain(0)
654
+ 2< L. Fig. 9(a) shows the N-particle cur-
655
+ rents when the sample initially contains two electrons in
656
+ the ground state. This initial state evolves faster to the
657
+ steady state than the empty system. While for N= 3
658
+ the current in the left lead is positive, for both N= 2
659
+ andN= 1 the currents are negative. The charge re-
660
+ siding on each N-particle state and the total charge are
661
+ shown in Fig. 9(b). Since single-particle con gurations
662
+ are unlikely their occupation vanishes. The total charge
663
+ accumulated on the N= 3 states increases up to 2, while
664
+ the total charge on the N= 2 states decreases from 2 to
665
+ 0.75. The sign of the current for N= 2 becomes pos-
666
+ itive when Ris lowered to 0.2, Fig. 9(c), and exceeds
667
+ the current carried by the states with N= 3. This is
668
+ because the 1-st and the 2-nd SES practically determine
669
+ the ground state with two electrons and thus (0)
670
+ 2, and
671
+ also because the 1-st SES is strongly coupled to the leads.
672
+ However, the current with N= 1 is still negative.
673
+ FIG. 8: (a) The total current in the left lead carried by all
674
+ many-body con gurations with N= 2 atL= 0:6. (b) The
675
+ same forL= 0:9. Other parameters R= 0:35.
676
+ C. Parabolic quantum wire
677
+ In this subsection we apply the GME with Coulomb
678
+ interaction to describe the transport through a short
679
+ quantum wire of length Lx= 300 nm with a parabolic
680
+ con nement in the y-direction perpendicular to the di-
681
+ rection of transport. The contact ends of the isolated
682
+ wire atLx=2 are described by hard walls. This is
683
+ now a continuous model, where a large functional ba-
684
+ sis is used to expand the eigenfunctions of the system
685
+ in. In a similar manner we use a functional basis with
686
+ complete truncated sets of continuous and discrete func-
687
+ tions to expand the eigenfunctions of the semi-in nite
688
+ parabolic leads in. To show that we can describe the com-
689
+ bined geometrical e ects imposed on the system by it's
690
+ geometry and an external perpendicular magnetic eld
691
+ we place the quantum wire is in an external magnetic
692
+ eld of strength 1 :0 T. The characteristic con nement
693
+ energy is given by ~
694
+ 0= 1:0 meV. We assume GaAs
695
+ parameters with m= 0:067me,= 12:4 meV. The
696
+ magnetic length modi ed by the parabolic con nement
697
+ isaw=p
698
+ ~=(m
699
+ w), with
700
+ 2
701
+ w=
702
+ 2
703
+ 0+!2
704
+ c. and the
705
+ cyclotron frequency !c=eB=(mc). AtB= 1:0 T,
706
+ aw= 23:87 nm. The semi-in nite leads having the same9
707
+ FIG. 9: (a) The total current in the left lead carried by N-
708
+ particle states and the total charge. for L= 1:5 and for
709
+ R= 0:65. (b) The occupation number of the N-particle
710
+ states. (c) The total current in the left lead carried by N-
711
+ particle states for L= 1:2 andR= 0:2. (d) The occupation
712
+ number of the N-particle states and the total charge.parabolic con nement and being subject to the same ex-
713
+ ternal perpendicular magnetic eld have a continuous en-
714
+ ergy spectrum with discrete Landau sub-bands.
715
+ The Coulomb potential in Eq. (4) in the 2D wire is
716
+ described by
717
+ u(rr0) =e2
718
+ p
719
+ (xx0)2+ (yy0)2+2; (16)
720
+ with the small convergence parameter ( =aw) = 0:01
721
+ to facilitate the two-dimensional numerical integration
722
+ needed for the matrix elements (4).
723
+ After the GME (9) has been transformed to the in-
724
+ teracting many-electron basis by the unitary transfor-
725
+ mation obtained by the diagonalization of HS(3) we
726
+ truncate the RDO (8) to 32 MES. For the bias range
727
+ 0:0=LR1:7 meV used here 10 SES are
728
+ sucient to obtain these lowest 32 states with good accu-
729
+ racy. We will be omitting singly occupied states of high
730
+ energy that should not be relavant for the parameters
731
+ here. The natural strength of the Coulomb interaction
732
+ will only give us MES that are occupied by one or two
733
+ electrons in the energy range 0 to 6 meV covered by the
734
+ 32 MES.
735
+ Since in the partitioning approach [ HS;HL] = 0 we
736
+ have to construct Tl
737
+ qnas a non-local overlap ofnand
738
+ L;R
739
+ qon the contact regions Cl; l=L;R:8
740
+ Tl
741
+ qn=Z
742
+ Cldrdr0
743
+ 
744
+ ql(r)n(r)gl
745
+ qn(r;r0) +h:c:
746
+ :(17)
747
+ gl
748
+ qn(r;r0) =gl
749
+ 0exp
750
+ l
751
+ 1(xx0)2l
752
+ 2(yy0)2
753
+ expjEn"qlj
754
+ l
755
+ E
756
+ : (18)
757
+ As before"qlis the energy spectrum of lead l, andEn
758
+ is the energy of the SES numbered by nin the quan-
759
+ tum wire. The quantum number qfor the states in leads
760
+ represents both the discrete Landau band number and a
761
+ continuous quantum number that can be related to the
762
+ momentum of a particular state. Here we use the pa-
763
+ rameters1a2
764
+ w=2a2
765
+ w= 0:25, LR
766
+ E= 0:25 meV, and
767
+ gLR
768
+ 0= 40 meV for B= 1:0 T. The domain of the over-
769
+ lap integral for the leads is 2awinto the lead or the
770
+ system forxandx0from each end of the wire at Lx=2
771
+ and between4awforyandy0, see Ref. (8) for an exact
772
+ de nition. All the SES will be coupled to the leads, but
773
+ the coupling strength will depend on the character of the
774
+ SES, whether it is an edge- or bulk state and other ner
775
+ geometrical details that is brought about by the magnetic
776
+ eld.
777
+ The right chemical potential Ris held at 1 :4 meV
778
+ and the transport properties are calculated for di erent
779
+ values of the bias  by varyingL. Figure 10 compares
780
+ the total occupation of all one- electron and two-electron
781
+ MES for the interacting system at two di erent values of
782
+ the bias. At,  = 0:2 meV we see that almost solely10
783
+ FIG. 10: (Color online) The total charge residing in one- and
784
+ two-electron states as a function of time for two di erent val-
785
+ ues of the bias  .B= 1:0 T,Lx= 300 nm, ~
786
+ 0= 1:0
787
+ meV.
788
+ one-electron states are occupied, while for  = 1:2 meV
789
+ initially it is likely to have one-electron states occupied,
790
+ but very soon the occupation of the two-electron states
791
+ becomes as probable with the likelihood of the occupa-
792
+ tion of the one-electron states fast reducing with time.
793
+ We also have to admit here that even though the steady
794
+ state value of the total current through the system can
795
+ be deduced by the values of the current at 270 ps, the
796
+ charging of the system takes much longer time, since we
797
+ are using here a very weak coupling to the leads that
798
+ mimics a tunneling regime.
799
+ If we now use the average value of the current in the left
800
+ and right leads at t= 270 ps as a measure of the steady
801
+ state current we get the information displayed in Fig.
802
+ 11, where the steady state value of the current is shown
803
+ for the interacting system as a function of the bias and
804
+ compared to the charge in the system. We have a clear
805
+ Coulomb blocking in the interacting system. In the case
806
+ of a non-interacting system the lack of a gap between the
807
+ one- and two electron MES and a strong mixing of the
808
+ energy regimes of two- and three-electron states the two-
809
+ electron plateau only appears as a small shoulder. The
810
+ FIG. 11: The total steady state current for interacting 10
811
+ SES, and the total charge at t= 270 ps. for di erent values
812
+ of the bias  .B= 1:0 T,Lx= 300 nm, ~
813
+ 0= 1:0 meV.
814
+ 32 MES selected here include no three-electron or MES
815
+ with higher number of electrons. It should be mentioned
816
+ here that a di erent choice of the right bias Rcan result
817
+ in the system charging faster and thus at the same time
818
+ the total current through it being smaller. This comes
819
+ from the fact that the states have a di erent coupling to
820
+ the leads and the time range shown here is very much in
821
+ the transient- or it's long exponential decay regime.
822
+ Figure 12 displaying the current in the right lead gives
823
+ an idea how the Coulomb blocking plateau appears after
824
+ the transition regime. The transition regime where the
825
+ FIG. 12: The total current in the right lead for interacting
826
+ and non-interacting 10 SES as a function of the bias  and
827
+ time.B= 1:0 T,Lx= 300 nm, ~
828
+ 0= 1:0 meV.
829
+ right current goes negative, i. e.where it supplies charge
830
+ to the system is partially truncated from the gure.11
831
+ IV. SUMMARY AND CONCLUSIONS
832
+ We calculated time-dependent currents in open meso-
833
+ scopic systems composed by a sample attached to two
834
+ semi-in nite leads, by solving the generalized master
835
+ equation for the reduced density operator acting in the
836
+ Fock space of the sample. This is the natural frame-
837
+ work for including the Coulomb electron-electron inter-
838
+ action in the sample, which is the main achievement of
839
+ this work. The Coulomb interaction is treated in the
840
+ spirit of the exact diagonalization method, i. e.in a pure
841
+ many-body manner. The interacting many-body states
842
+ of the sample are expanded in the basis of non-interacting
843
+ \bit-string" states with unspeci ed number of electrons.
844
+ We believe our method is a viable alternative to a recent
845
+ approach based on a time-dependent density-functional
846
+ model.9,10,26We used three sample models, a short 1D
847
+ wire with 5 sites, but also a larger 2D lattice with 120
848
+ sites and a continuous model, whereas the cited group
849
+ used much smaller samples even with no structure.26
850
+ Indeed, due to computational limitations we could use
851
+ only a restricted, e ective number of many-body states in
852
+ the GME, between 30-50 depending on the model, from
853
+ the bottom of the energy spectrum. We chose the bias
854
+ window [R;L] and the strength of the sample-leads
855
+ coupling parameters VR;Lsuch that only the e ective
856
+ states contribute to the transport of electrons through
857
+ the sample, whereas the states with higher energy are
858
+ unreachable by the electrons. Consequently the number
859
+ of electrons in the sample can be only up to 3 or 4.
860
+ We could calculate the contribution to the charge and
861
+ currents in the sample and in the leads respectively, cor-
862
+ responding to any particular many-body state. We use
863
+ the 1D chain as a toy model to emphasize the dominant
864
+ role of the excited states in the transient regime and the
865
+ onset of the Coulomb blockade in the steady state. A
866
+ similar 1D model with 4 sites 1D has been considered
867
+ recently by My oh anen et al.10
868
+ As shown also in our previous works on time-dependenttransport in non-interacting systems the GME method
869
+ includes information on the energy structure of the sam-
870
+ ple, but also on the geometrical properties re
871
+ the wave functions and sample-lead contacts.7,8,25Here
872
+ we illustrate these aspects, in the interacting case, for two
873
+ nanosystems: a two-dimensional quantum dot described
874
+ by a lattice Hamiltonian and a short parabolic quantum
875
+ wire. The time-dependent occupation of speci c many-
876
+ body states was thoroughly analyzed, for di erent values
877
+ of the chemical potentials of the leads. It turned out
878
+ that the excited states with Nelectrons contribute to
879
+ the steady state currents if the ground state con guration
880
+ withN+ 1 particles is not available for transport. How-
881
+ ever, if(0)
882
+ N<Rand at the same time (0)
883
+ N+1lies within
884
+ the bias window the excited states with Nparticles
885
+ are active only in the transient regime and become de-
886
+ populated in the steady state regime. This behavior is of
887
+ interest in the excited-state spectroscopy experiments.11
888
+ To our knowledge the time-dependent currents associated
889
+ to excited states have not been discussed theoretically so
890
+ far.
891
+ Acknowledgments
892
+ The authors acknowledge nancial support from the
893
+ Development Fund of the Reykjavik University Grant
894
+ No. T09001, the Research and Instruments Funds of the
895
+ Icelandic State, the Research Fund of the University of
896
+ Iceland, the Icelandic Science and Technology Research
897
+ Programme for Postgenomic Biomedicine, Nanoscience
898
+ and Nanotechnology, the National Science Council of Tai-
899
+ wan under contract No. NSC97-2112-M-239-003-MY3.
900
+ V.M. also acknowledges the hospitality of the Reykjavik
901
+ University, Science Institute and the partial nancial sup-
902
+ port from PNCDI2 program (grant No. 515/2009) and
903
+ grant No. 45N/2009.
904
+ 1G. Stefanucci, S. Kurth, A. Rubio and E. K. U. Gross,
905
+ Phys. Rev. B 77, 075339 (2008).
906
+ 2V. Moldoveanu, A. Manolescu and V. Gudmundsson, Phys.
907
+ Rev. B 76, 085330 (2007)
908
+ 3X. Zheng, F. Wang, C. Y. Yam, Y. Mo, and G.H. Chen,
909
+ Phys. Rev. B 75, 195127 (2007).
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+ 4V. Gudmundsson, G. Thorgilsson, C-S Tang, and V.
911
+ Moldoveanu, Phys. Rev. B 77, 035329 (2008).
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+ 5U. Harbola, M. Esposito, and S. Mukamel, Phys. Rev. B
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+ 74, 235309 (2006).
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+ 6S. Welack, M. Schreiber, and U. Kleinekath ofer, J. Chem.
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+ Phys. 124, 044712 (2006)
916
+ 7V. Moldoveanu, A. Manolescu and V. Gudmundsson, New
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+ J. Phys. 11, 073019 (2009).
918
+ 8V. Gudmundsson, C. Gainar, C-S Tang, V. Moldoveanu,
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+ A. Manolescu, to appear in New J. Phys.
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+ 9S. Kurth, G. Stefanucci, C.-O. Almbladh, A. Rubio, andE. K. U. Gross, Phys. Rev. B 72, 035308 (2005).
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+ 10P. My oh anen, A. Stan, G. Stefanucci, and R. van Leeuwen,
922
+ Phys. Rev. B 80, 115107 (2009), Europhys. Lett. 84, 67001
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+ (2008).
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+ 11T. Fujisawa, D. G. Austing, Y. Tokura, Y. Hirayama, and
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+ S. Tarucha, J. Phys.: Condens. Matter 15, R1395 (2003).
926
+ 12D. C. Langreth and P. Nordlander, Phys. Rev. B 43, 2541
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+ (1991)
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+ 13S. A. Gurvitz and Ya. S. Prager, Phys. Rev. B 53, 15932
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+ (1996).
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+ 14J. K onig, H. Schoeller, and G. Sch on, Phys. Rev. Lett. 76,
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+ 1715 (1996); J. K onig, J. Schmid, H. Schoeller, and G.
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+ Sch on, Phys. Rev. B 54, 16 820 (1996).
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+ 15D. Becker and D. Pfannkuche, Physical Review B 77
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+ 205307 (2008).
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+ 16S. Nakajima, Prog. Theor. Phys. 20, 948 (1958).
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+ 17R. Zwanzig, J. Chem. Phys. 33, 1338 (1960).12
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+ 18C. Timm, Phys. Rev. B 77, 195416 (2008).
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+ 19X. Q. Li and Y. J. Yan, Phys. Rev. B 75, 075114 (2007).
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+ 20M.Esposito and M. Galperin, Phys. Rev. B 79, 205303
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+ (2009).
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+ 21D. Darau, G. Begemann, A. Donarini, and M. Grifoni,
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+ Physical Review B 79235404 (2009).
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+ 22C. Caroli, R. Combescot, P. Nozieres, and D. Saint-James,
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+ J. Phys. C 4, 916 (1971).
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+ 23E. Vaz, J. Kyriakidis, J. Chem. Phys. 129, 024703 (2008)
946
+ 24L.P. Kouwenhouven et al., in Mesoscopic Electron Trans-port, edited by L.L. Sohn, L.P. Kouwenhouven, and G.
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+ Schn, NATO Advanced Study Institute, Series E, Vol. 345
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+ (Kluwer, Dordrecht, 1997).
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+ 25V. Moldoveanu, A. Manolescu and V. Gudmundsson, Phys.
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+ Rev. B 80, 205325 (2009).
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+ 26S. Kurth, G. Stefanucci, E. Khosravi, C. Verdozzi, ad E.
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+ K. U. Gross, e-print arXiv:0911.3870 [cond-mat.mes-hall]
953
+ (2009).
1001.0048.txt ADDED
@@ -0,0 +1,1038 @@
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
1
+ arXiv:1001.0048v2 [math.AP] 7 Jan 2010Nonlinear stability of periodic traveling wave solutions o f
2
+ viscous conservation laws in dimensions one and two
3
+ Mathew A. Johnson∗Kevin Zumbrun†
4
+ November 12, 2018
5
+ Keywords : Periodic traveling waves; Bloch decomposition; modulate d waves.
6
+ 2000 MR Subject Classification : 35B35.
7
+ Abstract
8
+ Extending results of Oh and Zumbrun in dimensions d≥3, we establish nonlin-
9
+ ear stability and asymptotic behavior of spatially-periodic traveling- wave solutions of
10
+ viscous systems of conservation laws in critical dimensions d= 1,2, under a natural
11
+ set of spectral stability assumptions introduced by Schneider in th e setting of reaction
12
+ diffusion equations. The key new steps in the analysis beyond that in d imensionsd≥3
13
+ are a refined Green function estimate separating off translation as the slowest decaying
14
+ linear mode and a novel scheme for detecting cancellation at the leve l of the nonlinear
15
+ iteration in the Duhamel representation of a modulated periodic wav e.
16
+ 1 Introduction
17
+ Nonclassical viscous conservation laws arising in multiph ase fluid and solid mechanics ex-
18
+ hibit a rich variety of traveling wave phenomena, including homoclinic (pulse-type) and
19
+ periodic solutions along with the standard heteroclinic (s hock, or front-type) solutions
20
+ [GZ, Z6, OZ1, OZ2]. Here, we investigate stability of period ic traveling waves: specifi-
21
+ cally, sufficient conditions for stability of the wave. Our ma in result, generalizing results of
22
+ Oh and Zumbrun [OZ4] in dimensions d≥3, is to show that strong spectral stability in the
23
+ sense of Schneider [S1, S2, S3] implies linearized and nonli nearL1∩HK→L∞bounded
24
+ stability, for all dimensions d≥1, andasymptotic stability for dimensions d≥2.
25
+ ∗Indiana University, Bloomington, IN 47405; matjohn@india na.edu: Research of M.J. was partially sup-
26
+ ported by an NSF Postdoctoral Fellowship under NSF grant DMS -0902192.
27
+ †Indiana University, Bloomington, IN 47405; kzumbrun@indi ana.edu: Research of K.Z. was partially
28
+ supported under NSF grants no. DMS-0300487 and DMS-0801745 .
29
+ 11 INTRODUCTION 2
30
+ More precisely, we show that small L1∩Hsperturbations of a planar periodic solution
31
+ u(x,t)≡¯u(x1) (without loss of generality taken stationary) converge at Gaussian rate in
32
+ Lp,p≥2 to a modulation
33
+ (1.1) ¯ u(x1−ψ(x,t)),
34
+ of the unperturbed wave, where x= (x1,˜x), ˜x= (x2,...,xd), andψis a scalar function
35
+ whosex- andt-gradients likewise decay at least at Gaussian rate in all Lp,p≥2, but which
36
+ itself decays more slowly by a factor t1/2; in particular, ψis merely bounded in L∞for
37
+ dimensiond= 1.
38
+ The one-dimensional study of spectral stability of spatial ly periodic traveling waves of
39
+ systems of viscous conservation laws was initiated by Oh and Zumbrun [OZ1] in the “quasi-
40
+ Hamiltonian” case that the traveling-wave equation posses ses an integral of motion, and in
41
+ the general case by Serre[Se1]. An important contribution o f Serre was to point out a larger
42
+ connection between the linearized dispersion relation (th e functionλ(ξ) relating spectra to
43
+ wave number of the linearized operator about the wave) near z ero and the formal Whitham
44
+ averaged system obtained by slow modulation, or WKB, approx imation.
45
+ In [OZ3], this was extended to multi-dimensions, relating t he linearized dispersion rela-
46
+ tion near zero to
47
+ (1.2)∂tM+/summationdisplay
48
+ j∂xjFj= 0,
49
+ ∂t(ΩN)+∇x(ΩS) = 0,
50
+ whereM∈Rndenotes the average over one period, Fjthe average of an associated flux,
51
+ Ω =|∇xΨ| ∈R1the frequency, S=−Ψt/|∇xΨ| ∈R1the speeds, andN=∇xΨ/|∇xΨ| ∈
52
+ Rdthe normal νassociated with nearby periodic waves, with an additional c onstraint
53
+ (1.3) curl (Ω N) = curl ∇xΨ≡0.
54
+ As an immediate corollary, similarly as in [OZ1], [Se1] in th e one-dimensional case, this
55
+ yieldedas anecessary condition formulti-dimensional sta bility hyperbolicityoftheaveraged
56
+ system (1.2)–(1.3).
57
+ The present study is informed by but does not directly rely on this observation relating
58
+ Whitham averaging and spectral stability properties. Like wise, the Evans function tech-
59
+ niquesusedin[Se1,OZ3]toestablishthisconnection play n oroleinouranalysis; indeed, the
60
+ Evans function makes no appearance here. Rather, we rely on a direct Bloch-decomposition
61
+ argument in the spirit of Schneider [S1, S2, S3], combining s harp linearized estimates with
62
+ subtle cancellation in nonlinear source terms arising from the modulated wave approxima-
63
+ tion. The analytical techniques used to realize this progra m are somewhat different from
64
+ those of [S1, S2, S3], however, coming instead from the theor y of stability of viscous shock
65
+ fronts through a line of investigation carried out in [OZ1, O Z2, OZ3, OZ4, HoZ]. In partic-
66
+ ular, the nonsmooth dispersion relation at ξ= 0 typical for convection-diffusion equations1 INTRODUCTION 3
67
+ requires different treatment from that of [S1, S2, S3] in the re action diffusion case; see Re-
68
+ mark 2.4. Moreover, we detect nonlinear cancellation in the physicalx-tdomain rather than
69
+ the frequency domain as in [S1, S2, S3]. The main difference bet ween the present analysis
70
+ and that of [OZ4] is the systematic incorporation of modulat ion approximation (1.1).
71
+ 1.1 Equations and assumptions
72
+ Consider a parabolic system of conservation laws
73
+ (1.4) ut+/summationdisplay
74
+ jfj(u)xj= ∆xu,
75
+ u∈ U(open)∈Rn,fj∈Rn,x∈Rd,d≥1,t∈R+, and a periodic traveling wave solution
76
+ (1.5) u= ¯u(x·ν−st),
77
+ of periodX, satisfying the traveling-wave ODE ¯ u′′= (/summationtext
78
+ jνjfj(¯u))′−s¯u′with boundary
79
+ conditions ¯u(0) = ¯u(X) =:u0.Integrating, we obtain a first-order profile equation
80
+ (1.6) ¯ u′=/summationdisplay
81
+ jνjfj(¯u)−s¯u−q,
82
+ where (u0,q,s,ν,X )≡constant. Without loss of generality take ν=e1,s= 0, so that
83
+ ¯u= ¯u(x1) represents a stationary solution depending only on x1.
84
+ Following [Se1, OZ3, OZ4], we assume:
85
+ (H1)fj∈CK+1,K≥[d/2]+4.
86
+ (H2) Themap H:R×U×R×Sd−1×Rn→Rntaking (X;a,s,ν,q)/mapsto→u(X;a,s,ν,q)−a
87
+ is a submersion at point ( ¯X;¯u(0),0,e1,¯q), whereu(·;·) is the solution operator for (1.6).
88
+ Conditions (H1)–(H2) imply that the set of periodic solutio ns in the vicinity of ¯ uform
89
+ a smooth (n+d+1)-dimensional manifold {¯ua(x·ν(a)−α−s(a)t)}, withα∈R,a∈Rn+d.
90
+ 1.1.1 Linearized equations
91
+ Linearizing (1.4) about ¯ u(·), we obtain
92
+ (1.7) vt=Lv:= ∆xv−/summationdisplay
93
+ (Ajv)xj,
94
+ where coefficients Aj:=Dfj(¯u) are now periodic functions of x1. Taking the Fourier
95
+ transform in the transverse coordinate ˜ x= (x2,···,xd), we obtain
96
+ (1.8)ˆvt=L˜ξˆv= ˆvx1,x1−(A1ˆv)x1−i/summationdisplay
97
+ j/negationslash=1Ajξjˆv−/summationdisplay
98
+ j/negationslash=1ξ2
99
+ jˆv,
100
+ where˜ξ= (ξ2,···,ξd) is the transverse frequency vector.1 INTRODUCTION 4
101
+ 1.1.2 Bloch–Fourier decomposition and stability conditions
102
+ Following [G, S1, S2, S3], we define the family of operators
103
+ (1.9) Lξ=e−iξ1x1L˜ξeiξ1x1
104
+ operating on the class of L2periodic functions on [0 ,X]; the (L2) spectrum of L˜ξis equal
105
+ to the union of the spectra of all Lξwithξ1real with associated eigenfunctions
106
+ (1.10) w(x1,˜ξ,λ) :=eiξ1x1q(x1,ξ1,˜ξ,λ),
107
+ whereq, periodic, is an eigenfunction of Lξ. By continuity of spectrum, and discreteness of
108
+ the spectrum of the elliptic operators Lξon the compact domain [0 ,X], we have that the
109
+ spectra ofLξmay be described as the union of countably many continuous su rfacesλj(ξ).
110
+ Without loss of generality taking X= 1, recall now the Bloch–Fourier representation
111
+ (1.11) u(x) =/parenleftBig1
112
+ 2π/parenrightBigd/integraldisplayπ
113
+ −π/integraldisplay
114
+ Rd−1eiξ·xˆu(ξ,x1)dξ1d˜ξ
115
+ of anL2functionu, where ˆu(ξ,x1) :=/summationtext
116
+ ke2πikx1ˆu(ξ1+ 2πk,˜ξ) are periodic functions of
117
+ periodX= 1, ˆu(˜ξ) denoting with slight abuse of notation the Fourier transfo rm ofuin the
118
+ full variable x. By Parseval’s identity, the Bloch–Fourier transform u(x)→ˆu(ξ,x1) is an
119
+ isometry in L2:
120
+ (1.12) /ba∇dblu/ba∇dblL2(x)=/ba∇dblˆu/ba∇dblL2(ξ;L2(x1)),
121
+ whereL2(x1) is taken on [0 ,1] andL2(ξ) on [−π,π]×Rd−1. Moreover, it diagonalizes the
122
+ periodic-coefficient operator L, yielding the inverse Bloch–Fourier transform representation
123
+ (1.13) eLtu0=/parenleftBig1
124
+ 2π/parenrightBigd/integraldisplayπ
125
+ −π/integraldisplay
126
+ Rd−1eiξ·xeLξtˆu0(ξ,x1)dξ1d˜ξ
127
+ relating behavior of the linearized system to that of the dia gonal operators Lξ.
128
+ Following [OZ4], weassumealongwith(H1)–(H2) the strong spectral stability conditions:
129
+ (D1)σ(Lξ)⊂ {Reλ<0}forξ/ne}ationslash= 0.
130
+ (D2) Reσ(Lξ)≤ −θ|ξ|2,θ>0, forξ∈Rdand|ξ|sufficiently small.
131
+ (D3)λ= 0 is a semisimple eigenvalue of L0of multiplicity exactly n+1.1
132
+ For each fixed angle ˆξ:=ξ/|ξ|, expandLξ=L0+|ξ|L1+|ξ|2L2. By assumption (D3)
133
+ and standard spectral perturbation theory, there exist n+1 smooth eigenvalues
134
+ (1.14) λj(ξ) =−iaj(ξ)+o(|ξ|)
135
+ 1The zero eigenspace of L0is at least ( n+1)-dimensional by the linearized existence theory and (H2 ),
136
+ and hence n+ 1 is the minimal multiplicity; see [Se1, OZ3]. As noted in [O Z1, OZ3], minimal dimension
137
+ of this zero eigenspace implies that ( M,NΩ) of (1.2) gives a nonsingular coordinatization of the fami ly of
138
+ periodic traveling-wave solutions near ¯ u.1 INTRODUCTION 5
139
+ ofLξbifurcating from λ= 0 atξ= 0, where −iajare homogeneous degree one functions
140
+ given by |ξ|times the eigenvalues of Π 0L1|KerL0, with Π 0the zero eigenprojection of L0.
141
+ Conditions(D1)–(D3) areexactly thespectralassumptions of[S1,S2,S3], corresponding
142
+ to “dissipativity” of the large-time behavior of the linear ized system. As in [OZ4], we make
143
+ the further nondegeneracy hypothesis:
144
+ (H3) The eigenvalues λ=−iaj(ξ)/|ξ|of Π0L1
145
+ KerL0are simple.
146
+ The functions ajmay be seen to be the characteristics associated with the Whi tham av-
147
+ eraged system (1.2)–(1.3) linearized about the values of M,S,N, Ω associated with the
148
+ background wave ¯ u; see [OZ3, OZ4]. Thus, (D1) implies weak hyperbolicity of (1 .2)–(1.3)
149
+ (reality ofaj), while (H1) corresponds to strict hyperbolicity.
150
+ 1.2 Main results
151
+ With these preliminaries, we can now state our main results.
152
+ Theorem 1.1. Assuming (H1)–(H3) and (D1)–(D3), for some C >0andψ∈WK,∞(x,t),
153
+ (1.15)|˜u−¯u(·−ψ)|Lp(t)≤C(1+t)−d
154
+ 2(1−1/p)|˜u−¯u|L1∩HK|t=0,
155
+ |˜u−¯u(·−ψ)|HK(t)≤C(1+t)−d
156
+ 4|˜u−¯u|L1∩HK|t=0,
157
+ |(ψt,ψx)|WK+1,p≤C(1+t)−d
158
+ 2(1−1/p)|˜u−¯u|L1∩HK|t=0,
159
+ and
160
+ (1.16) |˜u−¯u|Lp(t),|ψ(t)|Lp≤C(1+t)−d
161
+ 2(1−1
162
+ p)+1
163
+ 2|˜u−¯u|L1∩HK|t=0
164
+ for allt≥0,p≥2,d= 1, for solutions ˜uof(1.4)with|˜u−¯u|L1∩HK|t=0sufficiently small.
165
+ In particular, ¯uis nonlinearly bounded L1∩HK→L∞stable for dimension d= 1.
166
+ Theorem 1.2. Assuming (H1)–(H3) and (D1)–(D3), for any ε >0, someC >0and
167
+ ψ∈WK,∞(x,t),
168
+ (1.17)|˜u−¯u(·−ψ)|Lp(t)≤C(1+t)−d
169
+ 2(1−1/p)|˜u−¯u|L1∩HK|t=0,
170
+ |˜u−¯u(·−ψ)|HK(t)≤C(1+t)−d
171
+ 4|˜u−¯u|L1∩HK|t=0,
172
+ |(ψt,ψx)|WK+1,p≤C(1+t)−d
173
+ 2(1−1/p)+ε−1
174
+ 2|˜u−¯u|L1∩HK|t=0,
175
+ and
176
+ (1.18)|˜u−¯u|Lp(t),|ψ(t)|Lp≤C(1+t)−d
177
+ 2(1−1
178
+ p)+ε|˜u−¯u|L1∩HK|t=0,
179
+ |˜u−¯u|HK(t),|ψ(t)|HK≤C(1+t)−d
180
+ 4+ε|˜u−¯u|L1∩HK|t=0,
181
+ for allt≥0,p≥2,d= 2, for solutions ˜uof(1.4)with|˜u−¯u|L1∩HK|t=0sufficiently small.
182
+ In particular, ¯uis nonlinearly asymptotically L1∩HK→HKstable for dimension d= 2.1 INTRODUCTION 6
183
+ Remark 1.1. In Theorem 1.2, derivatives in x∈R2refer to total derivatives. Moreover,
184
+ unless specified by an appropriate index, throughout this pa per derivatives in spatial variable
185
+ xwill always refer to the total derivative of the function.
186
+ In dimension one, Theorem 1.1 asserts only bounded L1∩HK→L∞stability, a very
187
+ weak notion of stability. The absence of decay in perturbati on ˜u−¯uindicates the delicacy
188
+ of the nonlinear analysis in this case. In particular, it is c rucial to separate off the slower-
189
+ decaying modulated behavior (1.1) in order to close the nonl inear iteration argument.
190
+ Remark 1.2. In dimension d= 1, it is straightforward to show that the results of Theorem
191
+ 1.1 extend to all 1 ≤p≤ ∞using the pointwise techniques of [OZ2]; see Remark 3.3.
192
+ Remark 1.3. The slow decay of |˜u−¯u|Lp(t)∼ |ψ(t)|Lpin (1.16) is due to nonlinear
193
+ interactions; as shown in [OZ2, OZ4], the linearized decay r ate is faster by factor (1+ t)−1/2
194
+ (Proposition 2.1). In [OZ4], it was shown that for d≥3, where linear effects dominate
195
+ behavior, (1.16) may be replaced by the stronger estimate |˜u−¯u|Lp(t),|ψ(t)|Lp≤C(1+
196
+ t)−d
197
+ 2(1−1
198
+ p)|˜u−¯u|L1∩HK|t=0.These distinctions reflect fine details of both linearized es timates
199
+ (Section 3) and nonlinear structure (Sections 4.1–4.2) tha t are not immediately apparent
200
+ from the formal Whitham approximation (1.2)–(1.3).
201
+ 1.3 Discussion and open problems
202
+ Linearized stability under the same assumptions, with shar p rates of decay, was established
203
+ ford= 1 [OZ2] and for d≥1 in [OZ4], along with nonlinear stability for d≥3. Theorem
204
+ 1.1 completes this line of investigation by establishing no nlinear stability in the critical
205
+ dimensions d= 1,2, a fundamental open problem cited in [OZ1, OZ4].
206
+ This gives a generalization of the work of [S1, S2, S3] for rea ction diffusion equations
207
+ to the case of viscous conservation laws. Recall that the ana lysis of [S1, S2, S3] concerns
208
+ also multiply periodic waves, i.e., waves that are either pe riodic or else constant in each
209
+ coordinate direction. It is straightforward to verify that the methods of this paper apply
210
+ essentially unchanged to this case, to give a corresponding stability result under the analog
211
+ of (H1)–(H3), (D1)–(D3), as we intend to report further in a f uture work. Likewise, the
212
+ extension from the semilinear parabolic case treated here t o the general quasilinear case is
213
+ straightforward, following the treatment of [OZ4].
214
+ On the other hand, as noted in [OZ2], condition (D3) is in the c onservation law setting
215
+ nongeneric, corresponding to the special “quasi-Hamilton ian” situation studied there; in
216
+ particular, it implies that speed is to first order constant a mong the family of spatially pe-
217
+ riodic traveling-wave solutions nearby ¯ u. In the generic case that (D3) is violated, behavior
218
+ is essentially different [OZ1, OZ2], and perturbations decay more slowly at the linearized
219
+ level. Nonlinear stability remains an interesting open pro blem in this setting.
220
+ Our approach to stability in the critical dimensions d= 1,2, as suggested in [OZ4], is,
221
+ loosely following the approach of [S1, S2, S3], to subtract o ut a slower-decaying part of the
222
+ solution describedby anappropriatemodulation equation a ndshowthat theresidualdecays2 BASIC LINEARIZED STABILITY ESTIMATES 7
223
+ sufficiently rapidly to close a nonlinear iteration. It is wor th noting that the modulated
224
+ approximation ¯ u(x1−ψ(x,t)) of (1.1) is not the full Ansatz
225
+ (1.19) ¯ ua(Ψ(x,t)),
226
+ Ψ(x,t) :=x1−ψ(x,t), associated with the Whitham averaged system (1.2)–(1.3) , where ¯ua
227
+ isthemanifoldofperiodicsolutions near ¯ uintroducedbelow(H2), butonlythetranslational
228
+ part not involving perturbations ain the profile. (See [OZ3] for the derivation of (1.2)–
229
+ (1.3) and (1.19).) That is, we don’t need to separate out all v ariations along the manifold
230
+ of periodic solutions, but only the special variations conn ected with translation invariance.
231
+ The technical reason is an asymmetry in y-derivative estimates in the parts of the Green
232
+ function associated with these various modes, something th at is not apparent without a
233
+ detailed study of linearized behavior as carried out here. T his also makes sense formally,
234
+ if one considers that (1.2) indicates that variables a,∇xΨ are roughly comparable, which
235
+ would suggest, by the diffusive behavior Ψ >>∇xΨ, thatais neglible with respect to Ψ.
236
+ However, note that in the case that (D3) holds, hence wave spe ed is stationary along the
237
+ manifold of periodic solutions, the final equation of (1.2) d ecouples to (Ψ x)t= (ΩN)t= 0,
238
+ and could be written as Ψ t= 0 in terms of Ψ alone. Hence, there is some ambiguity in this
239
+ degenerate case which of Ψ, Ψ xis the primary variable, and in terms of linear behavior, the
240
+ decay of variations aand Ψ are in fact comparable [OZ4]; in the generic case, aand Ψxare
241
+ comparable at the linearized level [OZ2]. It would be very in teresting to better understand
242
+ the connection between the Whitham averaged system (or suit able higher-order correction)
243
+ and behavior at the nonlinear level, as explored at the linea r level in [OZ3, OZ4, JZ1, JZB].
244
+ 2 Basic linearized stability estimates
245
+ We begin by recalling the basic linearized stability estima tes derived in [OZ4]. We will
246
+ sharpen these afterward in Section 3. By standard spectral p erturbation theory [K], the
247
+ total eigenprojection P(ξ) onto the eigenspace of Lξassociated with the eigenvalues λj(ξ),
248
+ j= 1,...,n+1describedintheintroductioniswell-definedandanalyti cinξforξsufficiently
249
+ small, since these (by discreteness of the spectra of Lξ) are separated at ξ= 0 from the rest
250
+ of the spectrum of L0. Introducing a smooth cutoff function φ(ξ) that is identically one
251
+ for|ξ| ≤εand identically zero for |ξ| ≥2ε,ε >0 sufficiently small, we split the solution
252
+ operatorS(t) :=eLtinto low- and high-frequency parts
253
+ (2.1) SI(t)u0:=/parenleftBig1
254
+ 2π/parenrightBigd/integraldisplayπ
255
+ −π/integraldisplay
256
+ Rd−1eiξ·xφ(ξ)P(ξ)eLξtˆu0(ξ,x1)dξ1d˜ξ
257
+ and
258
+ (2.2) SII(t)u0:=/parenleftBig1
259
+ 2π/parenrightBigd/integraldisplayπ
260
+ −π/integraldisplay
261
+ Rd−1eiξ·x/parenleftbig
262
+ I−φP(ξ)/parenrightbig
263
+ eLξtˆu0(ξ,x1)dξ1d˜ξ.2 BASIC LINEARIZED STABILITY ESTIMATES 8
264
+ 2.1 High-frequency bounds
265
+ By standard sectorial bounds [He, Pa] and spectral separati on ofλj(ξ) from the remaining
266
+ spectra ofLξ, we have trivially the exponential decay bounds
267
+ (2.3)/ba∇dbleLξt(I−φP(ξ))f/ba∇dblL2([0,X])≤Ce−θt/ba∇dblf/ba∇dblL2([0,X]),
268
+ /ba∇dbleLξt(I−φP(ξ))∂l
269
+ x1f/ba∇dblL2([0,X])≤Ct−l
270
+ 2e−θt/ba∇dblf/ba∇dblL2([0,X]),
271
+ /ba∇dbl∂l
272
+ x1eLξt(I−φP(ξ))f/ba∇dblL2([0,X])≤Ct−l
273
+ 2e−θt/ba∇dblf/ba∇dblL2([0,X]),
274
+ forθ,C >0, and 0 ≤m≤K(Kas in (H1)). Together with (1.12), these give immediately
275
+ the following estimates.
276
+ Proposition 2.1 ([OZ4]).Under assumptions (H1)–(H3) and (D1)–(D2), for some θ,
277
+ C >0, and allt>0,2≤p≤ ∞,0≤l≤K+1,0≤m≤K,
278
+ (2.4)/ba∇dbl∂l
279
+ xSII(t)f/ba∇dblL2(x),/ba∇dblSII(t)∂l
280
+ xf/ba∇dblL2(x)≤Ct−l
281
+ 2e−θt/ba∇dblf/ba∇dblL2(x),
282
+ /ba∇dbl∂m
283
+ xSII(t)f/ba∇dblLp(x),/ba∇dblSII(t)∂m
284
+ xf/ba∇dblLp(x)≤Ct−d
285
+ 2(1
286
+ 2−1
287
+ p)−m
288
+ 2e−θt/ba∇dblf/ba∇dblL2(x),
289
+ where, again, derivatives in the variable x∈Rdrefer to total derivatives.
290
+ Proof.The first inequalities follow immediately by (1.12) and (2.3 ). The second follows for
291
+ x1derivatives in the case p=∞,m= 0 by Sobolev embedding from
292
+ /ba∇dblSII(t)f/ba∇dblL∞(˜x;L2(x1))≤Ct−d−1
293
+ 4e−θt/ba∇dblf/ba∇dblL2([0,X])
294
+ and
295
+ /ba∇dbl∂x1SII(t)f/ba∇dblL∞(˜x;L2(x1))≤Ct−d−1
296
+ 4−1
297
+ 2e−θt/ba∇dblf/ba∇dblL2([0,X]),
298
+ which follow by an application of (1.12) in the x1variable and the Hausdorff–Young in-
299
+ equality /ba∇dblf/ba∇dblL∞(˜x)≤ /ba∇dblˆf/ba∇dblL1(˜ξ)in the variable ˜ x. The result for derivatives in x1and general
300
+ 2≤p≤ ∞then follows by Lpinterpolation. Finally, the result for derivatives in ˜ xfollows
301
+ from the inverse Fourier transform, equation (2.2), and the large|ξ|bound
302
+ |eLtf|L2(x1)≤e−θ|˜ξ|2t|f|L2(x1),|ξ|sufficiently large ,
303
+ which easily follows from Parseval and the fact that Lξis a relatively compact perturbation
304
+ of∂2
305
+ x−|ξ|2. Thus, by the above estimate we have
306
+ /ba∇dbleLt∂˜xf/ba∇dblL2(x)≤C/ba∇dbleLξt|˜ξ|ˆf/ba∇dblL2(x1,ξ)
307
+ ≤Csup/parenleftBig
308
+ e−θ|˜ξ|2t|ξ|/parenrightBig
309
+ /ba∇dblˆf/ba∇dblL2(x1,ξ)
310
+ ≤Ct−1/2/ba∇dblf/ba∇dblL2(x).
311
+ A similar argument applies for 1 ≤m≤K.2 BASIC LINEARIZED STABILITY ESTIMATES 9
312
+ 2.2 Low-frequency bounds
313
+ Denote by
314
+ (2.5) GI(x,t;y) :=SI(t)δy(x)
315
+ the Green kernel associated with SI, and
316
+ (2.6) [ GI
317
+ ξ(x1,t;y1)] :=φ(ξ)P(ξ)eLξt[δy1(x1)]
318
+ the corresponding kernel appearing within the Bloch–Fouri er representation of GI, where
319
+ the brackets on [ Gξ] and [δy] denote the periodic extensions of these functions onto the
320
+ whole line. Then, we have the following descriptions of GI, [GI
321
+ ξ], deriving from the spectral
322
+ expansion (1.14) of Lξnearξ= 0.
323
+ Proposition 2.2 ([OZ4]).Under assumptions (H1)–(H3) and (D1)–(D3),
324
+ (2.7)[GI
325
+ ξ(x1,t;y1)] =φ(ξ)n+1/summationdisplay
326
+ j=1eλj(ξ)tqj(ξ,x1)˜qj(ξ,y1)∗,
327
+ GI(x,t;y) =/parenleftBig1
328
+ 2π/parenrightBigd/integraldisplay
329
+ Rdeiξ·(x−y)[GI
330
+ ξ(x1,t;y1)]dξ
331
+ =/parenleftBig1
332
+ 2π/parenrightBigd/integraldisplay
333
+ Rdeiξ·(x−y)φ(ξ)n+1/summationdisplay
334
+ j=1eλj(ξ)tqj(ξ,x1)˜qj(ξ,y1)∗dξ,
335
+ where∗denotes matrix adjoint, or complex conjugate transpose, qj(ξ,·)and˜qj(ξ,·)are right
336
+ and left eigenfunctions of Lξassociated with eigenvalues λj(ξ)defined in (1.14), normalized
337
+ so that/an}b∇acketle{t˜qj,qj/an}b∇acket∇i}ht ≡1, whereλj/|ξ|is a smooth function of |ξ|andˆξ:=ξ/|ξ|andqjand˜qj
338
+ are smooth functions of |ξ|,ˆξ:=ξ/|ξ|, andx1ory1, withℜλj(ξ)≤ −θ|ξ|2.
339
+ Proof.Smooth dependence of λjand ofq, ˜qas functions in L2[0,X] follow from standard
340
+ spectral perturbation theory [K] using the fact that λjsplit to first order in |ξ|asξis varied
341
+ along rays through the origin, and that Lξvaries smoothly with angle ˆξ. Smoothness of
342
+ qj, ˜qjinx1,y1then follow from the fact that they satisfy the eigenvalue eq uation forLξ,
343
+ which has smooth, periodic coefficients. Likewise, (2.7)(i) is immediate from the spectral
344
+ decomposition of elliptic operators on finite domains. Subs tituting (2.5) into (2.1) and
345
+ computing
346
+ (2.8) /hatwideδy(ξ,x1) =/summationdisplay
347
+ ke2πikx1/hatwideδy(ξ+2πke1) =/summationdisplay
348
+ ke2πikx1e−iξ·y−2πiky1=e−iξ·y[δy1(x1)],
349
+ where the second and third equalities follow from the fact th at the Fourier transform either
350
+ continuous or discrete of the delta-function is unity, we ob tain
351
+ GI(x,t;y) =/parenleftBig1
352
+ 2π/parenrightBigd/integraldisplayπ
353
+ −π/integraldisplay
354
+ Rd−1eiξ·xφP(ξ)eLξt/hatwideδy(ξ,x1)dξ
355
+ =/parenleftBig1
356
+ 2π/parenrightBigd/integraldisplayπ
357
+ −π/integraldisplay
358
+ Rd−1eiξ·(x−y)φP(ξ)eLξt[δy1(x1)]dξ,2 BASIC LINEARIZED STABILITY ESTIMATES 10
359
+ yielding (2.7)(ii) by (2.6)(i) and the fact that φis supported on [ −π,π].
360
+ Proposition 2.3 ([OZ4]).Under assumptions (H1)-(H3) and (D1)-(D3),
361
+ (2.9) sup
362
+ y/ba∇dblGI(·,t,;y)/ba∇dblLp(x),sup
363
+ y/ba∇dbl∂x,yGI(·,t,;y)/ba∇dblLp(x)≤C(1+t)−d
364
+ 2(1−1
365
+ p)
366
+ for all2≤p≤ ∞,t≥0, whereC >0is independent of p.
367
+ Proof.From representation (2.7)(ii) and ℜλj(ξ)≤ −θ|ξ|2, we obtain by the triangle in-
368
+ equality
369
+ (2.10) /ba∇dblGI/ba∇dblL∞(x,y)≤C/ba∇dble−θ|ξ|2tφ(ξ)/ba∇dblL1(ξ)≤C(1+t)−d
370
+ 2,
371
+ verifying the bounds for p=∞. Derivative bounds follow similarly, since derivatives fa lling
372
+ onqjor ˜qjare harmless, whereas derivatives falling on eiξ·(x−y)bring down a factor of ξ,
373
+ again harmless because of the cutoff function φ.
374
+ To obtain bounds for p= 2, we note that (2.7)(ii) may be viewed itself as a Bloch–
375
+ Fourier decomposition with respect to variable z:=x−y, withyappearing as a parameter.
376
+ Recalling (1.12), we may thus estimate
377
+ (2.11)sup
378
+ y/ba∇dblGI(x,t;y)/ba∇dblL2(x)=/summationdisplay
379
+ jsup
380
+ y/ba∇dblφ(ξ)eλj(ξ)tqj(·,z1)˜q∗
381
+ j(·,y1)/ba∇dblL2(ξ;L2(z1∈[0,X]))
382
+ ≤C/summationdisplay
383
+ jsup
384
+ y/ba∇dblφ(ξ)e−θ|ξ|2t/ba∇dblL2(ξ)/ba∇dblqj/ba∇dblL2(0,X)/ba∇dbl˜qj/ba∇dblL∞(0,X)
385
+ ≤C(1+t)−d
386
+ 4,
387
+ where we have used in a crucial way the boundedness of ˜ qj; derivative bounds follow simi-
388
+ larly. Finally, bounds for 2 ≤p≤ ∞follow byLp-interpolation.
389
+ Remark 2.4. In obtaining the key L2-estimate, we have used in an essential way the
390
+ periodic structure of qj, ˜qj. For, viewing GIas a general pseudodifferential expression
391
+ rather than a Bloch–Fourier decomposition, we find that the s moothness of qj, ˜qjis not
392
+ sufficient to apply standard L2→L2bounds of H¨ ormander, which require blowup in ξ
393
+ derivatives at less than the critical rate |ξ|−1found here; see, e.g., [H] for further discussion.
394
+ Nor do the weighted energy estimate techniques used in [S1, S 2, S3] apply here, as these also
395
+ rely on the property of smoothness of λj,qj, ˜qjwith respect to ξat the origin ξ= 0. The
396
+ lack of smoothness of the linearized dispersion relation at the origin is an essential technical
397
+ difference separating the conservation law from the reaction diffusion case; see [OZ4] for
398
+ further discussion.
399
+ Remark 2.5. Underlying the above analysis, and also the technically rat her different
400
+ approach of [OZ2], is the fundamental relation
401
+ (2.12) G(x,t;y) =/parenleftBig1
402
+ 2π/parenrightBigd/integraldisplayπ
403
+ −π/integraldisplay
404
+ Rd−1eiξ·(x−y)[Gξ(x1,t;y1)]dξ2 BASIC LINEARIZED STABILITY ESTIMATES 11
405
+ which, provided σ(Lξ) is semisimple, yields the simple formula
406
+ G(x,t;y) =/parenleftBig1
407
+ 2π/parenrightBigd/integraldisplayπ
408
+ −π/integraldisplay
409
+ Rd−1eiξ·(x−y)/summationdisplay
410
+ jeλj(ξ)tqj(ξ,x1)˜qj(ξ,y1)∗dξ
411
+ resembling that of the constant-coefficient case, where λjruns through the spectrum of Lξ.
412
+ The basic idea in both cases is to separate off the principal pa rt of the series involving small
413
+ λj(ξ) and estimate the remainder as a faster-decaying residual.
414
+ Corollary 2.6 ([OZ4]).Under assumptions (H1)–(H3) and (D1)–(D3), for all p≥2,t≥0,
415
+ (2.13) /ba∇dblSI(t)f/ba∇dblLp,/ba∇dbl∂xSI(t)f/ba∇dblLp,/ba∇dblSI(t)∂xf/ba∇dblLp≤C(1+t)−d
416
+ 2(1−1
417
+ p)/ba∇dblf/ba∇dblL1.
418
+ Proof.Immediate, from (2.9) and the triangle inequality, as, for e xample,
419
+ /ba∇dblSI(t)f(·)/ba∇dblLp=/vextenddouble/vextenddouble/vextenddouble/integraldisplay
420
+ RdGI(x,t;y)f(y)dy/vextenddouble/vextenddouble/vextenddouble
421
+ Lp(x)≤/integraldisplay
422
+ Rdsup
423
+ y/ba∇dblGI(·,t;y)/ba∇dblLp|f(y)|dy.
424
+ Proposition 2.1 ([OZ4]).Assuming (H1)-(H3), (D1)-(D3), for some C >0, allt≥0,
425
+ p≥2,0≤l≤K,
426
+ (2.14) /ba∇dblS(t)∂l
427
+ xu0/ba∇dblLp≤Ct−l
428
+ 2(1+t)−d
429
+ 2(1
430
+ 2−1
431
+ p)+l
432
+ 2t−d
433
+ 4−l
434
+ 2/ba∇dblu0/ba∇dblL1∩L2.
435
+ Proof.Immediate, from (2.4) and (2.13).
436
+ 2.3 Additional estimates
437
+ Lemma 2.7. Assuming (H1)–(H3), (D1)–(D3), for all t≥0,0≤l≤K,
438
+ (2.15) /ba∇dbl∂l
439
+ xSI(t)f/ba∇dblLp(x),/ba∇dblSI(t)∂l
440
+ xf/ba∇dblLp(x)≤C(1+t)−d
441
+ 2(1/2−1/p)/ba∇dblf/ba∇dblL2(x).
442
+ Proof.From boundedness of the spectral projections Pj(ξ) =qj/an}b∇acketle{t˜qj,·/an}b∇acket∇i}htinL2[0,X] and their
443
+ derivatives, another consequence of first-order splitting of eigenvalues λj(ξ) at the origin,
444
+ we obtain boundedness of φ(ξ)P(ξ)eLξtand thus, by (1.12), the global bounds
445
+ (2.16) /ba∇dbl∂l
446
+ xSI(t)f/ba∇dblL2(x),/ba∇dblSI(t)∂l
447
+ xf/ba∇dblL2(x)≤C/ba∇dblf/ba∇dblL2(x),
448
+ for allt≥0, yielding the result for p= 2. Moreover, by boundedness of ˜ q,qin allLp(x1),
449
+ we have
450
+ |φ(ξ)P(ξ)eLξtˆf(ξ,·)|L∞(x1)≤Ce−θ|ξ|2t|P(ξ)ˆf(ξ,·)|L∞(x1)≤Ce−θ|ξ|2t|ˆf(ξ,·)|L2(x1),3 REFINED LINEARIZED ESTIMATES 12
451
+ C, θ>0, yielding by SIf=/parenleftBig
452
+ 1
453
+ 2π/parenrightBigd/integraltextπ
454
+ −π/integraltext
455
+ Rd−1eiξ·xφ(ξ)P(ξ)eLξtˆf(ξ,x1)dξ1d˜ξthe bound
456
+ (2.17)/ba∇dblSI(t)f/ba∇dblL∞(x)≤/parenleftBig1
457
+ 2π/parenrightBigd/integraldisplayπ
458
+ −π/integraldisplay
459
+ Rd−1|φ(ξ)P(ξ)eLξtˆf(ξ,·)|L∞(x1)dξ1d˜ξ
460
+ ≤/parenleftBig1
461
+ 2π/parenrightBigd/integraldisplayπ
462
+ −π/integraldisplay
463
+ Rd−1Cφ(ξ)e−θ|ξ|2t|ˆf(ξ,·)|L2(x1)dξ1d˜ξ
464
+ ≤C|φ(ξ)e−θ|ξ|2t|L2(ξ)|ˆf|L2(ξ,x1)
465
+ =C(1+t)−d
466
+ 4/ba∇dblf/ba∇dblL2([0,X]),
467
+ yielding the result for p=∞,l= 0. The result for p=∞, 1≤l≤Kfollows by a similar
468
+ argument. The result for general 2 ≤p≤ ∞then follows by Lpinterpolation between p= 2
469
+ andp=∞.
470
+ By Riesz–Thorin interpolation between (2.15) and (2.13), w e obtain the following, ap-
471
+ parently sharp bounds between various LqandLp.2
472
+ Corollary 2.8. Assuming (H0)–(H3) and (D1)–(D3), for all 1≤q≤2≤p,t≥0,
473
+ 0≤l≤K,
474
+ (2.18) /ba∇dbl∂l
475
+ xSI(t)f/ba∇dblLp,/ba∇dblSI(t)∂l
476
+ xf/ba∇dblLp≤C(1+t)−d
477
+ 2(1
478
+ q−1
479
+ p)/ba∇dblf/ba∇dblLq.
480
+ Proposition 2.2. Assuming (H1)-(H3), (D1)-(D3), for some C >0, allt≥0,1≤q≤
481
+ 2≤p, and0≤l≤K,
482
+ (2.19) /ba∇dblS(t)∂l
483
+ xu0/ba∇dblLp≤C(1+t)−d
484
+ 2(1
485
+ 2−1
486
+ p)+l
487
+ 2t−d
488
+ 2(1
489
+ q−1
490
+ 2)−l
491
+ 2/ba∇dblu0/ba∇dblLq∩L2.
492
+ Proof.Immediate, from (2.4) and (2.8).
493
+ 3 Refined linearized estimates
494
+ The bounds of Proposition 2.1 are sufficient to establish nonl inear stability and asymptotic
495
+ behavior in dimensions d≥3, as shown in [OZ4]. However, they are not sufficient in the
496
+ critical dimensions d= 1,2; see Remark 1, Section 7 of [OZ4]. Comparison with standard
497
+ diffusive stability arguments as in [Z7] show that this is due t o the fact that the full solution
498
+ operator |S(t)∂x|decays no faster than S(t), or, equivalently, Gyno faster than G.
499
+ Following the basic strategy introduced in [ZH, Z1, MaZ2, Ma Z4] in the context of vis-
500
+ cous shock waves, we now perform a refined linearized estimat e separating slower-decaying
501
+ translational modes from a faster-decaying “good” part of t he solution operator. This will
502
+ be used in Section 4 in combination with certain nonlinear ca ncellation estimates to show
503
+ convergence to the modulated approximation (1.1) at a faste r rate sufficient to close the
504
+ nonlinear iteration.
505
+ The key to this decomposition is the following observation.
506
+ 2The inclusion of general p≥2 in Lemma 2.7 repairs an omission in [OZ4], where the bounds ( 2.8) were
507
+ stated but not used.3 REFINED LINEARIZED ESTIMATES 13
508
+ Lemma 3.1. Assuming (H1)–(H3), (D1)–(D3), let λj(ξ/|ξ|,ξ),qj(ξ/|ξ|,ξ,·),˜qj(ξ/|ξ|,ξ,·)
509
+ denote the eigenvalues and associated right and left eigenf unctions of Lξ, withqj,˜qjsmooth
510
+ functions of ξ/|ξ|and|ξ|as noted in Prop. 2.2. Then, without loss of generality, q1(ω,0,·)≡
511
+ ¯u′, while˜qj(ω,0,·)forj/ne}ationslash= 1are constant functions depending only on angle ω=ξ/|ξ|.
512
+ Proof.Expanding Lξ=L0+|ξ|L1
513
+ ξ/|ξ|+|ξ|2L2
514
+ ξ/|ξ|as in the introduction, consider the con-
515
+ tinuous family of spectral perturbation problems in |ξ|indexed by angle ω=ξ/|ξ|. Then,
516
+ both facts follow by standard perturbation theory [K] using the observations that ¯ u′is in
517
+ the right kernel of L0and constant functions care in the left kernel of L0, with
518
+ /an}b∇acketle{tc,L1¯u′/an}b∇acket∇i}ht=/an}b∇acketle{tc,(ω1(2∂x1−A1)−/summationdisplay
519
+ j/negationslash=1ωjAj))¯u′/an}b∇acket∇i}ht=/an}b∇acketle{tc,ω1∂2
520
+ x1¯u−/summationdisplay
521
+ j/negationslash=1ωj∂x1fj(¯u)/an}b∇acket∇i}ht ≡0,
522
+ where/an}b∇acketle{t·,·/an}b∇acket∇i}htdenotesL2(x1) inner product on the interval x1∈[0,X], that the dimension
523
+ of kerL0by assumption is ( n+ 1), so that the orthogonal complement of ¯ u′in KerL0
524
+ is dimension nso exactly the set of constant functions, and that by (H3) the functions
525
+ qj(ω,0,·) and ˜qj(ω,0) are right and left eigenfunctions of Π 0L1|kerL0(Π0as earlier denoting
526
+ the zero eigenprojection associated with L0).
527
+ Remark 3.2. The key observation of Lemma 3.1 can be motivated by the form o f the
528
+ Whitham averaged system (1.2). For, recalling (Section 1.3 ) that (D3) implies that speed
529
+ sis stationary to first order at ¯ ualong the manifold of nearby periodic solutions, we find
530
+ that the last equation of (1.2) reduces to ( ∇xΨ)t= 0, i.e., the equation for the translational
531
+ variation Ψ decouples from the equations for variations in o ther modes. This corresponds
532
+ heuristically to the fact derived above that the translatio nal mode ¯u′(x1) decouples in the
533
+ first-order eigenfunction expansion.
534
+ Corollary 3.1. Under assumptions (H1)–(H3), (D1)–(D3), the Green functio nG(x,t;y)
535
+ of(1.7)decomposes as G=E+˜G,
536
+ (3.1) E= ¯u′(x)e(x,t;y),
537
+ where, for some C >0, allt>0,1≤q≤2≤p≤ ∞,0≤j,k,l,j+l≤K,1≤r≤2,
538
+ (3.2)/vextendsingle/vextendsingle/vextendsingle/integraldisplay+∞
539
+ −∞˜G(x,t;y)f(y)dy/vextendsingle/vextendsingle/vextendsingle
540
+ Lp(x)≤C(1+t)−d
541
+ 2(1/2−1/p)t−1
542
+ 2(1/q−1/2)|f|Lq∩L2,
543
+ /vextendsingle/vextendsingle/vextendsingle/integraldisplay+∞
544
+ −∞∂r
545
+ y˜G(x,t;y)f(y)dy/vextendsingle/vextendsingle/vextendsingle
546
+ Lp(x)≤C(1+t)−d
547
+ 2(1/2−1/p)−1
548
+ 2+r
549
+ 2
550
+ ×t−d
551
+ 2(1/q−1/2)−r
552
+ 2|f|Lq∩L2,
553
+ /vextendsingle/vextendsingle/vextendsingle/integraldisplay+∞
554
+ −∞∂r
555
+ t˜G(x,t;y)f(y)dy/vextendsingle/vextendsingle/vextendsingle
556
+ Lp(x)≤C(1+t)−d
557
+ 2(1/2−1/p)−1
558
+ 2+r
559
+ ×t−d
560
+ 2(1/q−1/2)−r|f|Lq∩L2.3 REFINED LINEARIZED ESTIMATES 14
561
+ (3.3)/vextendsingle/vextendsingle/vextendsingle/integraldisplay+∞
562
+ −∞∂j
563
+ x∂k
564
+ t∂l
565
+ ye(x,t;y)f(y)dy/vextendsingle/vextendsingle/vextendsingle
566
+ Lp≤(1+t)−d
567
+ 2(1/q−1/p)−(j+k)
568
+ 2|f|Lq.
569
+ Moreover,e(x,t;y)≡0fort≤1.
570
+ Proof.We first treat the simpler case q= 1. Recalling that
571
+ (3.4) GI(x,t;y) =/parenleftBig1
572
+ 2π/parenrightBigd/integraldisplay
573
+ Rdeiξ·(x−y)φ(ξ)n+1/summationdisplay
574
+ j=1eλj(ξ)tqj(ξ,x1)˜qj(ξ,y1)∗dξ,
575
+ define
576
+ (3.5) ˜e(x,t;y) =/parenleftBig1
577
+ 2π/parenrightBigd/integraldisplay
578
+ Rdeiξ·(x−y)φ(ξ)eλ1(ξ)t˜q1(ξ,y1)∗dξ,
579
+ so that
580
+ (3.6)
581
+ GI(x,t;y)−¯u′(x1)˜e(x,t;y) =/parenleftBig1
582
+ 2π/parenrightBigd/integraldisplay
583
+ Rdeiξ·(x−y)φ(ξ)n+1/summationdisplay
584
+ j=2eλj(ξ)tqj(ξ/|ξ|,0,x1)˜qj(ξ,y1)∗dξ
585
+ +/parenleftBig1
586
+ 2π/parenrightBigd/integraldisplay
587
+ Rdn+1/summationdisplay
588
+ j=1eiξ·(x−y)φ(ξ)eλj(ξ)tO(|ξ|)dξ.
589
+ Noting, by Lemma 3.1, that ∂y˜q(ω,0,y)≡constant for j/ne}ationslash= 1, we have therefore
590
+ (3.7)∂r
591
+ y(GI(x,t;y)−¯u′(x1)˜e(x,t;y)) =/parenleftBig1
592
+ 2π/parenrightBigd/integraldisplay
593
+ Rdeiξ·(x−y)φ(ξ)n+1/summationdisplay
594
+ j=1eλj(ξ)tO(|ξ|)dξ,
595
+ which readily gives
596
+ (3.8) |∂r
597
+ y(GI(x,t;y)−¯u′(x1)˜e(x,t;y))|Lp≤C(1+t)−d
598
+ 2(1−1/p)−1
599
+ 2,
600
+ p≥2, by the same argument used to prove (2.9), and similarly
601
+ (3.9) |∂r
602
+ t(GI(x,t;y)−¯u′(x1)˜e(x,t;y))|Lp≤c(1+t)−d
603
+ 2(1−1/p)−1
604
+ 2.
605
+ These yield (3.2) by the triangle inequality.
606
+ Defininge(x,t;y) :=χ(t)˜e(x,t;y), whereχisasmoothcutofffunctionsuchthat χ(t)≡1
607
+ fort≥2 andχ(t)≡0 fort≤1, and setting ˜G:=G−¯u′(x1)e(x,t;y), we readily obtain the
608
+ estimates (3.2) by combining (3.9) with bound (2.4) on GII. Bounds (3.3) follow from (3.5)
609
+ by the argument used to prove (2.9), together with the observ ation thatx- ort-derivatives
610
+ bring down factors of |ξ|, followed again by an application of the triangle inequalit y.
611
+ Thecases1 ≤q≤2followsimilarly, bytheargumentsusedtoprove(2.15)and (2.8).4 NONLINEAR STABILITY IN DIMENSION ONE 15
612
+ Remark 3.3. Despite their apparent complexity, the above bounds may be r ecognized
613
+ as essentially just the standard diffusive bounds satisfied fo r the heat equation [Z7]. For
614
+ dimensiond= 1, it may be shown using pointwise techniques as in [OZ2] tha t the bounds
615
+ of Corollary 3.1 extend to all 1 ≤q≤p≤ ∞.
616
+ Note the strong analogy between the Green function decompos ition of Corollary 3.1
617
+ and that of [MaZ3, Z4] in the viscous shock case. We pursue thi s analogy further in the
618
+ nonlinearanalysisofthefollowingsections, combiningth e“instantaneous tracking” strategy
619
+ of [ZH, Z1, Z4, Z7, MaZ2, MaZ4] with a type of cancellation est imate introduced in [HoZ].
620
+ 4 Nonlinear stability in dimension one
621
+ For clarity, we carry out the nonlinear stability analysis i n detail in the most difficult,
622
+ one-dimensional, case, indicating afterward by a few brief remarks the extension to d= 2.
623
+ Hereafter, take x∈R1, dropping the indices on fjandxjand writing ut+f(u)x=uxx.
624
+ 4.1 Nonlinear perturbation equations
625
+ Given a solution ˜ u(x,t) of (1.4), define the nonlinear perturbation variable
626
+ (4.1) v=u−¯u= ˜u(x+ψ(x,t))−¯u(x),
627
+ where
628
+ (4.2) u(x,t) := ˜u(x+ψ(x,t))
629
+ andψ:R×R→Ris to be chosen later.
630
+ Lemma 4.1. Forv,uas in(4.1),(4.2),
631
+ (4.3) ut+f(u)x−uxx= (∂t−L)¯u′(x1)ψ(x,t)+∂xR+(∂t+∂2
632
+ x)S,
633
+ where
634
+ R:=vψt+vψxx+(¯ux+vx)ψ2
635
+ x
636
+ 1+ψx=O(|v|(|ψt|+|ψxx|)+/parenleftBig|¯ux|+|vx|
637
+ 1−|ψx|/parenrightBig
638
+ |ψx|2)
639
+ and
640
+ S:=−vψx=O(|v|(|ψx|).
641
+ Proof.To begin, notice from the definition of uin (4.2) we have by a straightforward
642
+ computation
643
+ ut(x,t) = ˜ux(x+ψ(x,t),t)ψt(x,t)+ ˜ut(x+ψ,t)
644
+ f(u(x,t))x=df(˜u(x+ψ(x,t),t))˜ux(x+ψ,t)·(1+ψx(x,t))4 NONLINEAR STABILITY IN DIMENSION ONE 16
645
+ and
646
+ uxx(x,t) = (˜ux(x+ψ(x,t),t)·(1+ψx(x,t)))x
647
+ = ˜uxx(x+ψ(x,t),t)·(1+ψx(x,t))+(˜ux(x+ψ(x,t),t)·ψx(x,t))x.
648
+ Using the fact that ˜ ut+df(˜u)˜ux−˜uxx= 0, it follows that
649
+ (4.4)ut+f(u)x−uxx= ˜uxψt+df(˜u)˜uxψx−˜uxxψx−(˜uxψx)x
650
+ = ˜uxψt−˜utψx−(˜uxψx)x
651
+ where it is understood that derivatives of ˜ uappearing on the righthand side are evaluated
652
+ at (x+ψ(x,t),t). Moreover, by another direct calculation, using the fact t hatL(¯u′(x)) = 0
653
+ by translation invariance, we have
654
+ (∂t−L)¯u′(x)ψ= ¯uxψt−¯utψx−(¯uxψx)x.
655
+ Subtracting, and using the facts that, by differentiation of ( ¯u+v)(x,t) = ˜u(x+ψ,t),
656
+ (4.5)¯ux+vx= ˜ux(1+ψx),
657
+ ¯ut+vt= ˜ut+ ˜uxψt,
658
+ so that
659
+ (4.6)˜ux−¯ux−vx=−(¯ux+vx)ψx
660
+ 1+ψx,
661
+ ˜ut−¯ut−vt=−(¯ux+vx)ψt
662
+ 1+ψx,
663
+ we obtain
664
+ ut+f(u)x−uxx= (∂t−L)¯u′(x)ψ+vxψt−vtψx−(vxψx)x+/parenleftBig
665
+ (¯ux+vx)ψ2
666
+ x
667
+ 1+ψx/parenrightBig
668
+ x,
669
+ yielding (4.3) by vxψt−vtψx= (vψt)x−(vψx)tand (vxψx)x= (vψx)xx−(vψxx)x.
670
+ Corollary 4.2. The nonlinear residual vdefined in (4.1)satisfies
671
+ (4.7) vt−Lv= (∂t−L)¯u′(x1)ψ−Qx+Rx+(∂t+∂2
672
+ x)S,
673
+ where
674
+ (4.8) Q:=f(˜u(x+ψ(x,t),t))−f(¯u(x))−df(¯u(x))v=O(|v|2),
675
+ (4.9) R:=vψt+vψxx+(¯ux+vx)ψ2
676
+ x
677
+ 1+ψx,
678
+ and
679
+ (4.10) S:=−vψx=O(|v|(|ψx|).
680
+ Proof.Taylor expansion comparing (4.3) and ¯ ut+f(¯u)x−¯uxx= 0.4 NONLINEAR STABILITY IN DIMENSION ONE 17
681
+ 4.2 Cancellation estimate
682
+ Our strategy in writing (4.7) is motivated by the following b asic cancellation principle.
683
+ Proposition 4.3 ([HoZ]).For anyf(y,s)∈Lp∩C2withf(y,0)≡0, there holds
684
+ (4.11)/integraldisplayt
685
+ 0/integraldisplay
686
+ G(x,t−s;y)(∂s−Ly)f(y,s)dyds=f(x,t).
687
+ Proof.Integrating the left hand side by parts, we obtain
688
+ (4.12)/integraldisplay
689
+ G(x,0;y)f(y,t)dy−/integraldisplay
690
+ G(x,t;y)f(y,0)dy+/integraldisplayt
691
+ 0/integraldisplay
692
+ (∂t−Ly)∗G(x,t−s;y)f(y,s)dyds.
693
+ Noting that, by duality,
694
+ (∂t−Ly)∗G(x,t−s;y) =δ(x−y)δ(t−s),
695
+ δ(·) here denoting the Dirac delta-distribution, we find that th e third term on the righthand
696
+ side vanishes in (4.12), while, because G(x,0;y) =δ(x−y), the first term is simply f(x,t).
697
+ The second term vanishes by f(y,0)≡0.
698
+ Remark 4.1. Forψ=ψ(t), term (∂t−L)¯u′ψin (4.7) reduces to the term ˙ψ(t)¯u′(x)
699
+ appearing in the shock wave case [ZH, Z1, Z4, Z7, MaZ2, MaZ4].
700
+ 4.3 Nonlinear damping estimate
701
+ Proposition 4.2. Letv0∈HK(Kas in (H1)), and suppose that for 0≤t≤T, theHK
702
+ norm ofvand theHK(x,t)norms ofψtandψxremain bounded by a sufficiently small
703
+ constant. There are then constants θ1,2>0so that, for all 0≤t≤T,
704
+ (4.13) |v(t)|2
705
+ HK≤Ce−θ1t|v(0)|2
706
+ HK+C/integraldisplayt
707
+ 0e−θ2(t−s)/parenleftBig
708
+ |v|2
709
+ L2+|(ψt,ψx)|2
710
+ HK(x,t)/parenrightBig
711
+ (s)ds.
712
+ Proof.Subtracting from the equation (4.4) for uthe equation for ¯ u, we may write the
713
+ nonlinear perturbation equation as
714
+ (4.14) vt+(df(¯u)v)x−vxx=Q(v)x+ ˜uxψt−˜utψx−(˜uxψx)x,
715
+ where it is understood that derivatives of ˜ uappearing on the righthand side are evaluated
716
+ at (x+ψ(x,t),t). Using (4.6) to replace ˜ uxand ˜utrespectively by ¯ ux+vx−(¯ux+vx)ψx
717
+ 1+ψx
718
+ and ¯ut+vt−(¯ux+vx)ψt
719
+ 1+ψx, and moving the resulting vtψxterm to the lefthand side of
720
+ (4.14), we obtain
721
+ (4.15)(1+ψx)vt−vxx=−(df(¯u)v)x+Q(v)x+ ¯uxψt
722
+ −((¯ux+vx)ψx)x+/parenleftBig
723
+ (¯ux+vx)ψ2
724
+ x
725
+ 1+ψx/parenrightBig
726
+ x.4 NONLINEAR STABILITY IN DIMENSION ONE 18
727
+ Taking the L2inner product in xof/summationtextK
728
+ j=0∂2j
729
+ xv
730
+ 1+ψxagainst (4.15), integrating by parts, and
731
+ rearranging the resulting terms, we arrive at the inequalit y
732
+ ∂t|v|2
733
+ HK(t)≤ −θ|∂K+1
734
+ xv|2
735
+ L2+C/parenleftBig
736
+ |v|2
737
+ HK+|(ψt,ψx)|2
738
+ HK(x,t)/parenrightBig
739
+ ,
740
+ for someθ >0,C >0, so long as |˜u|HKremains bounded, and |v|HKand|(ψt,ψx)|HK(x,t)
741
+ remain sufficiently small. Using the Sobolev interpolation |v|2
742
+ HK≤ |∂K+1
743
+ xv|2
744
+ L2+˜C|v|2
745
+ L2for
746
+ ˜C >0 sufficiently large, we obtain ∂t|v|2
747
+ HK(t)≤ −˜θ|v|2
748
+ HK+C/parenleftBig
749
+ |v|2
750
+ L2+|(ψt,ψx)|2
751
+ HK(x,t)/parenrightBig
752
+ from which (4.13) follows by Gronwall’s inequality.
753
+ 4.4 Integral representation/ ψ-evolution scheme
754
+ By Proposition 4.3, we have, applying Duhamel’s principle t o (4.7),
755
+ (4.16)v(x,t) =/integraldisplay∞
756
+ −∞G(x,t;y)v0(y)dy
757
+ +/integraldisplayt
758
+ 0/integraldisplay∞
759
+ −∞G(x,t−s;y)(−Qy+Rx+St+Syy)(y,s)dyds+ψ(t)¯u′(x).
760
+ Definingψimplicitly as
761
+ (4.17)ψ(x,t) =−/integraldisplay∞
762
+ −∞e(x,t;y)u0(y)dy
763
+ −/integraldisplayt
764
+ 0/integraldisplay+∞
765
+ −∞e(x,t−s;y)(−Qy+Rx+St+Syy)(y,s)dyds,
766
+ following [ZH, Z4, MaZ2, MaZ3], where eis defined as in (3.1), and substituting in (4.16)
767
+ the decomposition G= ¯u′(x)e+˜Gof Corollary 3.1, we obtain the integral representation
768
+ (4.18)v(x,t) =/integraldisplay∞
769
+ −∞˜G(x,t;y)v0(y)dy
770
+ +/integraldisplayt
771
+ 0/integraldisplay∞
772
+ −∞˜G(x,t−s;y)(−Qy+Rx+St+Syy)(y,s)dyds,
773
+ and, differentiating (4.17) with respect to t, and recalling that e(x,s;y)≡0 fors≤1,
774
+ (4.19)∂j
775
+ t∂k
776
+ xψ(x,t) =−/integraldisplay∞
777
+ −∞∂j
778
+ t∂k
779
+ xe(x,t;y)u0(y)dy
780
+ −/integraldisplayt
781
+ 0/integraldisplay+∞
782
+ −∞∂j
783
+ t∂k
784
+ xe(x,t−s;y)(−Qy+Rx+St+Syy)(y,s)dyds.
785
+ Equations (4.18), (4.19) together form a complete system in the variables ( v,∂j
786
+ tψ,∂k
787
+ xψ),
788
+ 0≤j≤1, 0≤k≤K, from the solution of which we may afterward recover the shif tψvia
789
+ (4.17). From the original differential equation (4.7) togeth er with (4.19), we readily obtain
790
+ short-time existence and continuity with respect to tof solutions ( v,ψt,ψx)∈HKby a
791
+ standard contraction-mapping argument based on (4.13), (4 .17), and and (3.3).4 NONLINEAR STABILITY IN DIMENSION ONE 19
792
+ 4.5 Nonlinear iteration
793
+ Associated with the solution ( u,ψt,ψx) of integral system (4.18)–(4.19), define
794
+ (4.20)ζ(t) := sup
795
+ 0≤s≤t|(v,ψt,ψx)|HK(s)(1+s)1/4.
796
+ Lemma 4.3. For allt≥0for whichζ(t)is finite, some C >0, andE0:=|u0|L1∩HK,
797
+ (4.21) ζ(t)≤C(E0+ζ(t)2).
798
+ Proof.By (4.9)–(4.10) and definition (4.20),
799
+ (4.22) |(Q,R,S)|L1∩L∞≤ |(v,vx,ψt,ψx)|2
800
+ L2+|(v,vx,ψt,ψx)|2
801
+ L∞≤Cζ(t)2(1+t)−1
802
+ 2,
803
+ so long as |ψx| ≤ |ψx|HK≤ζ(t) remains small, and likewise (using the equation to bound t
804
+ derivatives in terms of x-derivatives of up to two orders)
805
+ (4.23) |(∂t+∂2
806
+ x)S|L1∩L∞≤ |(v,ψx)|2
807
+ H2+|(v,ψx)|2
808
+ W2,∞≤Cζ(t)2(1+t)−1
809
+ 2.
810
+ Applying Corollary 3.1 with q= 1,d= 1 to representations (4.18)–(4.19), we obtain for
811
+ any 2≤p<∞
812
+ (4.24)|v(·,t)|Lp(x)≤C(1+t)−1
813
+ 2(1−1/p)E0
814
+ +Cζ(t)2/integraldisplayt
815
+ 0(1+t−s)−1
816
+ 2(1/2−1/p)(t−s)−3
817
+ 4(1+s)−1
818
+ 2ds
819
+ ≤C(E0+ζ(t)2)(1+t)−1
820
+ 2(1−1/p)
821
+ and
822
+ (4.25)
823
+ |(ψt,ψx)(·,t)|WK,p≤C(1+t)−1
824
+ 2E0+Cζ(t)2/integraldisplayt
825
+ 0(1+t−s)−1
826
+ 2(1−1/p)−1/2(1+s)−1
827
+ 2ds
828
+ ≤C(E0+ζ(t)2)(1+t)−1
829
+ 2(1−1/p).
830
+ Using (4.13) and(4.24)–(4.25), we obtain |v(·,t)|HK(x)≤C(E0+ζ(t)2)(1+t)−1
831
+ 4. Combining
832
+ this with (4.25), p= 2, rearranging, and recalling definition (4.20), we obtain (4.3).
833
+ Proof of Theorem 1.1. By short-time HKexistence theory, /ba∇dbl(v,ψt,ψx)/ba∇dblHKis continuous
834
+ so long as it remains small, hence ηremains continuous so long as it remains small. By
835
+ (4.3), therefore, it follows by continuous induction that η(t)≤2Cη0fort≥0, ifη0<1/4C,
836
+ yielding by (4.20) the result (1.15) for p= 2. Applying (4.24)–(4.25), we obtain (1.15) for
837
+ 2≤p≤p∗for anyp∗<∞, with uniform constant C. Takingp∗>4 and estimating
838
+ |Q|L2,|R|L2,|S|L2(t)≤ |(v,ψt,ψx)|2
839
+ L4≤CE0(1+t)−3
840
+ 45 NONLINEAR STABILITY IN DIMENSION TWO 20
841
+ in place of the weaker (4.22), then applying Corollary 3.1 wi thq= 2,d= 1, we obtain
842
+ finally (1.15) for 2 ≤p≤ ∞, by a computation similar (4.24)–(4.25); we omit the detail s of
843
+ this final bootstrap argument. Estimate (1.16) then follows using (3.3) with q=d= 1, by
844
+ (4.26)
845
+ |ψ(t)|Lp≤CE0+Cζ(t)2/integraldisplayt
846
+ 0(1+t−s)−1
847
+ 2(1−1/p)(1+s)−1
848
+ 2ds≤C(1+t)1
849
+ 2p(E0+ζ(t)2),
850
+ together with the fact that ˜ u(x,t)−¯u(x) =v(x−ψ,t)+(¯u(x)−¯u(x−ψ),so that|˜u(·,t)−¯u|
851
+ is controlled by the sum of |v|and|¯u(x)−¯u(x−ψ)| ∼ |ψ|. This yields stability for
852
+ |u−¯u|L1∩HK|t=0sufficiently small, as described in the final line of the theore m.
853
+ 5 Nonlinear stability in dimension two
854
+ We now briefly sketch the extension to dimension d= 2. Given a solution ˜ u(x,t) of (1.4),
855
+ define the nonlinear perturbation variable
856
+ (5.1) v=u−¯u= ˜u(x1+ψ(x,t),x2,t)−¯u(x1),
857
+ where
858
+ (5.2) u(x,t) := ˜u(x1+ψ(x,t),t)
859
+ andψ:Rd×R→Ris to be chosen later.
860
+ Lemma 5.1. Forv,uas in(5.2),
861
+ (5.3)ut+d/summationdisplay
862
+ j=1fj(u)xj−d/summationdisplay
863
+ j=1uxjxj= (∂t−L)¯u′(x1)ψ(x,t)+d/summationdisplay
864
+ j=1∂xjRj+∂tS+T,
865
+ where
866
+ Rj=O((|v,ψt,ψx)||(v,vx,ψt,ψx)|), S:=−vψx1= (|v|(|ψx|), T:=O(|ψx|3+|(v,ψx)||ψxx|).
867
+ Proof.Similarly as in the proof of Lemma 4.1, it follows by a straigh tforward computation
868
+ Using the fact that ˜ ut+/summationtext
869
+ jdfj(˜u)˜uxj−/summationtext
870
+ j˜uxjxj= 0, it follows that
871
+ (5.4)ut+/summationdisplay
872
+ jdfj(u)uxj−/summationdisplay
873
+ juxjxj= ˜ux1ψt−˜utψx1+/summationdisplay
874
+ j/negationslash=1dfj(˜u)˜ux1ψxj
875
+ −/summationdisplay
876
+ j/negationslash=1˜uxjx1ψxj−/summationdisplay
877
+ j(˜ux1ψxj)xj,
878
+ where it is understood that derivatives of ˜ uappearing on the righthand side are evaluated
879
+ at (x+ψ(x,t),t). Moreover, by another direct calculation, using the fact t hatL(¯u′(x1)) = 0
880
+ by translation invariance, we have
881
+ (∂t−L)¯u′(x1)ψ= ¯ux1ψt−¯utψx1+/summationdisplay
882
+ j/negationslash=1dfj(¯u)¯ux1ψxj−/summationdisplay
883
+ j/negationslash=1¯uxjx1ψxj−/summationdisplay
884
+ j(¯ux1ψxj)xj.5 NONLINEAR STABILITY IN DIMENSION TWO 21
885
+ Subtracting, and using (4.5) and
886
+ (5.5)¯uxj+vxj= ˜uxj+ ˜ux1ψxj,
887
+ ¯ut+vt= ˜ut+ ˜ux1ψt,
888
+ so that
889
+ (5.6)˜uxj−¯uxj−vxj=−(¯ux1+vx1)ψxj
890
+ 1+ψx1,
891
+ ˜ut−¯ut−vt=−(¯ux1+vx1)ψt
892
+ 1+ψx1,
893
+ we obtain
894
+ ut+/summationdisplay
895
+ jdfj(u)uxj−/summationdisplay
896
+ juxjxj= (∂t−L)¯u′(x1)ψ+vx1ψt−vtψx1
897
+ +/summationdisplay
898
+ j/negationslash=1(dfj(˜u)˜ux1−dfj(¯u)¯ux1)ψxj
899
+ −/summationdisplay
900
+ j/negationslash=1(˜uxjx1−¯uxjx1)ψxj−/summationdisplay
901
+ j((˜ux1−¯ux1)ψxj)xj.
902
+ Usingvx1ψt−vtψx1= (vψt)x1−(vψx1)t,
903
+ dfj(˜u)˜ux1=f(u)x1−dfj(˜u)˜ux1ψx1=f(u)x1(1−ψx)−dfj(˜u)˜ux1ψ2
904
+ x1,
905
+ and ˜uxjx1= (˜uxj)x1−˜uxjx1ψx1= (˜uxj)x1(1−ψx1)+ ˜uxjx1ψ2
906
+ x1,and rearranging, we obtain
907
+ ut+/summationdisplay
908
+ jdfj(u)uxj−/summationdisplay
909
+ juxjxj= (∂t−L)¯u′(x1)ψ+(vψt)x1−(vψx1)t
910
+ +/summationdisplay
911
+ j/negationslash=1(fj(u)−fj(¯u))x1)ψxj
912
+ −/summationdisplay
913
+ j/negationslash=1f(u)x1ψx1ψxj−/summationdisplay
914
+ j/negationslash=1dfj(˜u)˜ux1ψ2
915
+ x1ψxj
916
+ −/summationdisplay
917
+ j/negationslash=1(˜uxj−¯uxj)x1ψxj+/summationdisplay
918
+ j/negationslash=1(˜uxj)x1ψx1ψxj
919
+ +/summationdisplay
920
+ j/negationslash=1˜uxjx1ψ2
921
+ x1ψxj
922
+ −/summationdisplay
923
+ j(vx1ψx1)xj−/summationdisplay
924
+ j/parenleftBig
925
+ (¯ux1+vx1)ψxjψx1
926
+ 1+ψx1/parenrightBig
927
+ xj.
928
+ Noting that
929
+ (fj(u)−fj(¯u))x1)ψxj= ((fj(u)−fj(¯u)ψxj)x1−(fj(u)−fj(¯u))ψxjx1,5 NONLINEAR STABILITY IN DIMENSION TWO 22
930
+ f(u)x1ψx1ψxj= (f(u)ψx1ψxj)x1−f(u)(ψx1ψxj)x1,
931
+ and
932
+ (˜uxj−¯uxj)x1ψxj= ((˜uxj−¯uxj)ψxj)x1−(˜uxj−¯uxj)ψxjx1,
933
+ with|fj(u)−fj(¯u)|=O(|v|) and|˜uxj−¯uxj|=O(|v|),we obtain the result
934
+ Proof of Theorem 1.2. The result of Lemma 5.1 is the only part of the analysis that di ffers
935
+ essentially from that of the one-dimensional case. The canc ellation and nonlinear damping
936
+ arguments go through exactly as before to yield the analogs o f Propositions 4.3 and (4.2).
937
+ Likewise, we obtain a Duhamel representation analogous to ( 4.18)–(4.19), forming a closed
938
+ system in variables ( v,ψx,ψt).
939
+ To obtain the analog of Lemma 4.3, completing the proof of non linear stability, we can
940
+ carry out a somewhat simpler argument than in the one-dimens ional case, using Corollary
941
+ 3.1 withd= 2,q= 2 for all estimates, not only the final bootstrap argument, g iving in
942
+ place of (4.24) the estimate
943
+ (5.7)
944
+ |v(·,t)|Lp(x)≤C(1+t)−(1−1/p)E0+Cζ(t)2/integraldisplayt
945
+ 0(1+t−s)−(1/2−1/p)(t−s)−1
946
+ 2(1+s)−1ds
947
+ ≤C(E0+ζ(t)2)(1+t)−(1−1/p),
948
+ (5.8)|(ψx,ψt)(·,t)|Lp(x)≤C(1+t)−(1−1/p)−1
949
+ 2E0
950
+ +Cζ(t)2/integraldisplayt
951
+ 0(1+t−s)−(1/2−1/p)(t−s)−1
952
+ 2(1+s)−1ds
953
+ ≤C(E0+ζ(t)2)(1+t)ε−(1−1/p)−1
954
+ 2
955
+ for divergence-form source terms, and
956
+ (5.9)|v(·,t)|Lp(x)≤Cζ(t)2/integraldisplayt
957
+ 0(1+t−s)−(1/2−1/p)(1+s)−3
958
+ 2ds
959
+ ≤C(E0+ζ(t)2)(1+t)−(1−1/p),
960
+ (5.10)|(ψx,ψt)(·,t)|Lp(x)≤C(1+t)−(1−1/p)−1
961
+ 2E0
962
+ +Cζ(t)2/integraldisplayt
963
+ 0(1+t−s)−(1/2−1/p)(t−s)−1
964
+ 2(1+s)−3
965
+ 2ds
966
+ ≤C(E0+ζ(t)2)(1+t)ε−(1−1/p)−1
967
+ 2
968
+ for faster-decaying nondivergence-form source terms.
969
+ We omit the details, which are entirely similar to, but subst antially simpler than, those
970
+ of the one-dimensional case.REFERENCES 23
971
+ References
972
+ [G] R. Gardner, On the structure of the spectra of periodic traveling waves , J. Math.
973
+ Pures Appl. 72 (1993), 415-439.
974
+ [GZ] R. Gardner and K. Zumbrun, The Gap Lemma and geometric criteria for in-
975
+ stability of viscous shock profiles , Comm. Pure Appl. Math. 51 (1998), no. 7,
976
+ 797–85.
977
+ [He] D. Henry, Geometric theory of semilinear parabolic equations , Lecture Notes in
978
+ Mathematics, Springer–Verlag, Berlin (1981).
979
+ [HoZ] D. Hoff and K. Zumbrun Asymptotic behavior of multidimensional scalar viscous
980
+ shock fronts , Indiana Univ. Math. Journal, Vol. 49, No. 2 (2000).
981
+ [H] I.L. Hwang, TheL2-boundedness of pseudodifferential operators, Trans. Amer.
982
+ Math. Soc. 302 (1987) 55–76.
983
+ [JZ1] M. Johnson and K. Zumbrun, Rigorous Justification of the Whitham Modulation
984
+ Equations for the Generalized Korteweg-de Vries Equation, preprint (2009).
985
+ [JZB] M. Johnson, K. Zumbrun, and J. Bronski, Bloch wave expansion vs. Whitham
986
+ Modulation Equations for the Generalized Korteweg-de Vrie s Equation, inprepa-
987
+ ration.
988
+ [K] T. Kato, Perturbation theory for linear operators , Springer–Verlag, Berlin Hei-
989
+ delberg (1985).
990
+ [MaZ2] C. Mascia and K. Zumbrun, Stability of small-amplitude shock profiles of sym-
991
+ metric hyperbolic-parabolic systems, Comm. Pure Appl. Math. 57 (2004), no. 7,
992
+ 841–876.
993
+ [MaZ3] C. Mascia and K. Zumbrun, Pointwise Green function bounds for shock profiles
994
+ of systems with real viscosity. Arch. Ration. Mech. Anal. 169 (2003), no. 3, 177–
995
+ 263.
996
+ [MaZ4] C. Mascia and K. Zumbrun, Stability of large-amplitude viscous shock profiles
997
+ of hyperbolic-parabolic systems, Arch. Ration. Mech. Anal. 172 (2004), no. 1,
998
+ 93–131.
999
+ [OZ1] M. Oh and K. Zumbrun, Stability of periodic solutions of viscous conservation
1000
+ laws with viscosity- 1. Analysis of the Evans function , Arch. Ration. Mech. Anal.
1001
+ 166 (2003), no. 2, 99–166.
1002
+ [OZ2] M. Oh and K. Zumbrun, Stability of periodic solutions of viscous conservation
1003
+ laws with viscosity- Pointwise bounds on the Green function , Arch.Ration. Mech.
1004
+ Anal. 166 (2003), no. 2, 167–196.REFERENCES 24
1005
+ [OZ3] M. Oh, and K. Zumbrun, Low-frequency stability analysis of periodic traveling-
1006
+ wave solutions of viscous conservation laws in several dime nsions, Journal for
1007
+ Analysis and its Applications, 25 (2006), 1–21.
1008
+ [OZ4] M. Oh, and K. Zumbrun, Stability and asymptotic behavior of traveling-wave
1009
+ solutions of viscous conservation laws in several dimensio ns, to appear, Arch.
1010
+ Ration. Mech. Anal.
1011
+ [Pa] A. Pazy, Semigroups of linear operators and applications to partial differen-
1012
+ tial equations, Applied Mathematical Sciences, 44, Springer-Verlag, New Y ork-
1013
+ Berlin, (1983) viii+279 pp. ISBN: 0-387-90845-5.
1014
+ [S1] G.Schneider, Nonlinear diffusive stability of spatially periodic soluti ons– abstract
1015
+ theorem and higher space dimensions , Proceedings of the International Confer-
1016
+ ence on Asymptotics in Nonlinear Diffusive Systems (Sendai, 1 997), 159–167,
1017
+ Tohoku Math. Publ., 8, Tohoku Univ., Sendai, 1998.
1018
+ [S2] G. Schneider, Diffusive stability of spatial periodic solutions of the Swi ft-
1019
+ Hohenberg equation, (English. English summary) Comm. Math. Phys. 178
1020
+ (1996), no. 3, 679–702.
1021
+ [S3] G. Schneider, Nonlinear stability of Taylor vortices in infinite cylinder s,Arch.
1022
+ Rat. Mech. Anal. 144 (1998) no. 2, 121–200.
1023
+ [Se1] D. Serre, Spectral stability of periodic solutions of viscous conser vation laws:
1024
+ Large wavelength analysis , Comm. Partial Differential Equations 30 (2005), no.
1025
+ 1-3, 259–282.
1026
+ [Z1] K. Zumbrun, Refined wave–tracking and stability of viscous Lax shocks , Methods
1027
+ Appl. Anal. 7 (2000) 747–768.
1028
+ [Z4] K. Zumbrun, Stability of large-amplitude shock waves of compressible N avier–
1029
+ Stokes equations, withanappendixbyHelge KristianJenssenandGregoryLyng,
1030
+ in Handbook of mathematical fluid dynamics. Vol. III, 311–53 3, North-Holland,
1031
+ Amsterdam, (2004).
1032
+ [Z6] K. Zumbrun, Dynamical stability of phase transitions in the p-system wi th
1033
+ viscosity-capillarity , SIAM J. Appl. Math. 60 (2000), 1913-1929.
1034
+ [Z7] K. Zumbrun, Instantaneous shock location and one-dimensional nonlinea r sta-
1035
+ bility of viscous shock waves, preprint (2009).
1036
+ [ZH] K. Zumbrun and P. Howard, Pointwise semigroup methods and stability of vis-
1037
+ cous shock waves . Indiana Mathematics Journal V47 (1998), 741–871; Errata,
1038
+ Indiana Univ. Math. J. 51 (2002), no. 4, 1017–1021.
1001.0049.txt ADDED
The diff for this file is too large to render. See raw diff
 
1001.0050.txt ADDED
@@ -0,0 +1,509 @@
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
1
+ arXiv:1001.0050v1 [physics.hist-ph] 30 Dec 2009Fundamental times, lengths and physical constants:
2
+ some unknown contributions by Ettore Majorana
3
+ S. Esposito
4
+ Dipartimento di Scienze Fisiche, Universit` a di Napoli “Fe derico II” and I.N.F.N. Sezione di Napoli,
5
+ Complesso Universitario di Monte S. Angelo, Via Cinthia, 80 126 Naples, Italy∗
6
+ G. Salesi
7
+ Facolt` a di Ingegneria, Universit` a di Bergamo, viale Marc oni 5, 24044 Dalmine (BG) Italy
8
+ and I.N.F.N. Sezione di Milano, via Celoria 16, 20133 Milan, Italy†
9
+ We review the introduction in physics of the concepts of an el ementary space length and of a
10
+ fundamental time scale, analyzing some related unknown con tributions by Ettore Majorana. In
11
+ particular, we discuss the quasi-Coulombian scattering in presence of a finite length scale, as well
12
+ as the introduction of an intrinsic (universal) time delay i n the expressions for the retarded elec-
13
+ tromagnetic potentials. Finally, we also review a special m odel considered by Majorana in order to
14
+ deduce the value of the elementary charge, in such a way antic ipating key ideas later introduced in
15
+ quantum electrodynamics.
16
+ INTRODUCTION: THE KNOWN STORY
17
+ The idea of elementary space or time intervals has been quite a recur ring one in the philosophical and scientific
18
+ literature of every time and, as almost all the fundamental physica l ideas, from time to time it has been recovered
19
+ from oblivion. Typical examples range from the ancient Greek period to our days, namely from Zeno’s paradox on
20
+ Achilles and the tortoise to Poincar´ e and Mach, who regarded the c oncept of continuum as a consequence of our
21
+ physiological limitations.1
22
+ As often shown in the widespread literature on the structure of pa rticles (e.g., see below, the “chronon theory”),
23
+ a time-discretization for an elementary object involves an “interna l” (that is, referred to the center-of-mass frame)
24
+ motion associated with microscopic space distances. For such an ele mentary particle, an “extended-like” structure,
25
+ ratherthanapointlikeone,isexpectedandthishasimportantcons equencesevenforthefundamentaltheorydescribing
26
+ thatparticlessince, indeed, theclassicaltheoryofapointlikechar gedparticleleadstowell-knowndivergencyproblems,
27
+ which are only apparently solved by the renormalization group theor y.
28
+ In the realm of particle physics, the concepts of an elementary time and length are substantially equivalent for
29
+ several reasons: in most cases one can pass from time to space ju st by introducing the speed of light. Already in
30
+ 1930’s (see below), it was early realized that a fundamental length is directly related to the existence of a “cut-off” in
31
+ the momentum transferred during the particle-detector interac tion, which is required in order to avoid the emergence
32
+ of the so-called ultraviolet catastrophe in quantum field theories. A nother way to provide a small distance cut-off in
33
+ field theory is to formulate it on a discrete lattice: this approach was introduced in 1940 by Wentzel [2], but only
34
+ later it went into detail, becoming a standard technique in modern num erical approach to high energy interactions,
35
+ as e.g. in quantum chromodynamics.
36
+ Space-time discretizations, or fundamental scales, are introduc ed in classical and quantum theories mainly for two
37
+ reasons: to prevent undesired infinities in some physical quantities , or (though less known) to explain the emergence
38
+ of an intrinsic angular momentum for spinning particles.
39
+ The first appearance in physics of a fundamental length was sugge sted by H.A. Lorentz, who introduced the so-called
40
+ classical electron radius Rcl≡e2/mc2≃2.82·10−13cm in his famous classical relativistic model of the electron;2it is
41
+ equivalently known as “Lorentz radius”or “Thomsonscattering len gth”. Roughly speaking, such a radius corresponds
42
+ to the electron size required for obtaining its mass from the electro static potential energy, assuming that quantum
43
+ mechanicaleffects are irrelevant. Actually, such effects playa non -negligiblerolein the behaviorofchargedparticlesat
44
+ very short distances, and the classical electron radius may be con sidered as the length scale at which renormalization
45
+ effects become relevant in quantum electrodynamics. In other wor ds,Rcldenotes the watershed between classical and
46
+ 1“...le temps et l’espace ne repr` esentent, au point de vue ph ysiologique, qu’un continue apparent, qu’ils se composent tr` es vraisemblable-
47
+ ment d’elements discontinus, mais qu’on ne peut distinguer nettement les uns des autres.” [1]
48
+ 2Lorentz considered electrons just as rigid spheres, albeit contracting when moving in the ether [3].2
49
+ quantum electrodynamics; it appears in the semiclassical theory fo r the non-relativistic Thomson scattering, as well
50
+ as in the relativistic Klein-Nishina formula [4].
51
+ In early semiclassical models, the electron was substantially regard ed as a charged sphere but, unfortunately, the
52
+ electromagnetic self-energy of such a sphere diverges in the limit of pointlike charge. This was, then, the basic
53
+ motivation that led M.Abraham, Lorentz and, later, P.A.M.Dirac, to derive an equation of motion for the electron
54
+ where a finite, non-vanishing size is fully taken into account [5]. In suc h a way a fundamental length was effectively
55
+ introduced in order to properly describe the motion of charged par ticles. Particularly extensive work on extended,
56
+ not pointlike, charges was carried out by A.Sommerfeld [6]. In stud ying the motion of a sphere of radius Rwith
57
+ a uniform surface charge, he obtained the following “retarded-like ” first-order differential-finite difference equation
58
+ (T≡2R/c):
59
+ ma=e(E+v×B)+2
60
+ 3e2
61
+ Rc2v(t−T)−v(t)
62
+ T, (1)
63
+ where the electromagnetic field of the charge itself causes a self-f orce, which has a delayed effect on its motion. The
64
+ “delay time” 2 R/cappearing in the Sommerfeld equation, corresponding to the time th e light takes to go across
65
+ the sphere, was one of the earlier elementary times introduced in mic rophysics. Later in the 1920s, J.J.Thomson [7]
66
+ suggested that the electric force may act in a discontinuous way, p roducing finite increments of momentum separated
67
+ by finite intervals of time.
68
+ The relativistic version of Sommerfeld’s non-relativistic approach wa s developed by P.Caldirola [8]. His theory of
69
+ the electron is one of the first and simplest theories which assumes a prioria minimum time interval: it is based on
70
+ the existence of an elementary proper-time interval, the so-called chronon. When applied to electrons in an external
71
+ electromagnetic field, such a finite-difference theory succeeds (a lready at the classical level) in overcoming all the
72
+ known difficulties met by the Abraham-Lorentz-Dirac (ALD) theory , like the so-called pre-acceleration problem and
73
+ the emergence of run-away solutions [9, 10]. It is also able to give a c lear answer to the problem related to some
74
+ ambiguities associated with the hyperbolic motion of the electron [11], as well as to the question of whether a free
75
+ falling charged particle does or does not emit radiation [12]. Furtherm ore, at the quantum level, Caldirola’s chronon
76
+ theory is seemingly able to explain the origin of the “classical (Schwing er’s) part”, e/planckover2pi1/2mc·α/2π=e3/4πmc2, of
77
+ the anomalous magnetic momentum of the electron as well as the mas s spectrum of the charged leptons [8]. In this
78
+ theory, time flows continuously, but, when an external force act s on the electron, the reaction of the particle to the
79
+ applied force is not continuous: the value of the electron velocity uµis supposed to jump from uµ(τ−τ0) touµ(τ)
80
+ only at certain positions snalong its world line. These “discrete positions” are such that the elec tron takes a time τ0
81
+ for traveling from one position sn−1to the next one sn. In principle, the electron is still considered as pointlike, but
82
+ the ALD equation for the relativistic radiating electron is replaced: ( i) by a corresponding finite–difference (retarded)
83
+ equation in the velocity uµ(τ)
84
+ m
85
+ τ0/braceleftbigg
86
+ uµ(τ)−uµ(τ−τ0)+uµ(τ)uν(τ)
87
+ c2[uν(τ)−uν(τ−τ0)]/bracerightbigg
88
+ =e
89
+ cFµν(τ)uν(τ) (2)
90
+ (which reduces to the ALD equation when τ0≪τ) and (ii) by a second equation (the so-called transmission law ),
91
+ connecting the discrete positions xµ(τ) along the world line of the particle among themselves:
92
+ xµ(nτ0)−xµ((n−1)τ0) =τ0
93
+ 2{uµ(nτ0)−uµ[(n−1)τ0]}, (3)
94
+ which is valid inside each discrete interval τ0and describes the “internal” motion of the electron. In such equat ions
95
+ the chronon associated with the electron results to beτ0
96
+ 2≡θ0=2
97
+ 3ke2
98
+ mc3≃6.266·10−24s (k≡1/4πε0), depending,
99
+ therefore, on the particle properties (namely, on its charge eand on its rest mass m).
100
+ A different elementary length appearing in modern physics is the Compton wavelength , introduced in 1923 by
101
+ A.H.Compton [13] in his explanation of the scattering of photons by e lectrons. It is defined as λC≡h/mc, and
102
+ corresponds to the wavelength of a photon whose energy equals t he particle rest mass; for the electron, the Compton
103
+ wavelength is about 2 .42·10−10cm. The meaning of this length is quite clear: it indicates the watershe d between
104
+ classical mechanics and quantum wave-mechanics, since for spatia l distances below h/mca microsystem behaves as
105
+ a very quantum object. This was promptly realized by E.Schr¨ oding er in 1930-31 [14] in his study of the so-called3
106
+ Zitterbewegung , which emerges in the Dirac theory by means of the Darwin term appe aring in the (decomposed)
107
+ Hamiltonian of the Dirac equation, as found by C.G.Darwin [15] already in 1928. Indeed, because of the Zitterbewe-
108
+ gung (see also below) the electron undergoes extremely rapid fluct uations on scales just of the order of the Compton
109
+ wavelength, causing, for example, the electrons moving inside an at om to experience a smeared nuclear Coulomb
110
+ potential (this peculiar form of microscopic kinetic energy is presen tly better known as “quantum potential”of the
111
+ Schr¨ odinger equation [16]).
112
+ The Compton wavelength plays also the role of the spatial scale which naturally arise in any semi-classical model of
113
+ spinning particles, when attempting to formulate a kinematical pictu re of the intrinsic angular momentum. Already in
114
+ the 1920s, another extended-like structure of particles (in addit ion to the electrodynamical one mentioned above) was
115
+ discovered, namely the mechanical (or, rather, kinematical) configuration in the particle’s center-of -mass frame due
116
+ to the spin. Indeed, in 1925 R. Kronig, G. Uhlenbeck and S. Goudsmit [17] introduced the celebrated self-rotating
117
+ electron hypothesis, suggesting a physical interpretation for particles sp inning around their own axis. Now, it is
118
+ interesting to notice that any mechanical model of a self-rotating electron fatally leads to a fundamental length for
119
+ this very peculiar rotator, resulting to be just the Compton wavele ngthλC. Anyway, models with a self-rotating
120
+ spinning charge are very naive and problematic.3In the subsequent literature, these problems were overcome by
121
+ considering kinematical theories of the spin based on the Zitterbew egung [19, 20], where spinning particles, though
122
+ not effectively extended, can be considered as something which is ha lf-way between a point and a rotating body (as,
123
+ e.g., a top). In these models the center of the (even pointlike) char ge is spatially distinct from the center-of-mass, so
124
+ thatvelocityandmomentumarenolongerparallelvectors,andanin ternal(microscopic), rotationalspin motionofthe
125
+ particle arises in addition to the external (macroscopic) translatio nal motion of the center-of-mass. As a consequence,
126
+ in the classical limit the global inertial motion of a free spinning particle turns out to be no longer uniform and
127
+ rectilinear, but rather accelerated and helical, with a radius once ag ain equal to the Compton wavelength.
128
+ A different insight was provided by quantum mechanics with its discret e energy levels and the uncertainty principle.
129
+ This led physicists to speculate about space-time discreteness as e arly as the 1930’s. W. Heisenberg himself [21], for
130
+ example, in 1938 noted that physics musthave a fundamental length scale which, together with handc, might allow
131
+ the derivation of the mass of particles. In Heisenberg’s view, this len gth scale would have been around 10−13cm,
132
+ thus corresponding to the classical electron radius. Later, H.S.S nyder [22] in 1947 introduced a minimum length by
133
+ “quantizing” the spacetime, thus anticipating of more than half a ce ntury the so-called “non-commutative geome-
134
+ try” models (see below). Spacetime coordinates are represented by quantum operators and, in Snyder’s approach,
135
+ these operators have a discrete spectrum, thus entailing a discre te interpretation of spacetime. Subsequently, also
136
+ T.D.Lee [23] introduced an effective time discretization on the basis o f the necessarily finite number of measurements
137
+ performable in any finite interval of time.
138
+ The concept of non-continuous spacetime has recently returned into fashion in GUT’s [24], in string theories [25], in
139
+ quantum gravity [26] and in the approaches regarding spacetime eit her as a sort of quantum ether or as a spacetime
140
+ foam [27], or even as endowed with a non-commutative geometry (like in Deformed or Double Special Relativity
141
+ [28]4). In particular, M-Theory, Loop Quantum Gravity [29, 30] and Ca usal Dynamical Triangulation [31] lead
142
+ to postulate an essentially discrete and quantum spacetime, where (as expected from the uncertainty relations)
143
+ fundamental momentum and mass-energy scales naturally arise, in addition to /planckover2pi1andc. In contemporary physics the
144
+ most important case of an elementary space scale is doubtless the “ Planck length” ℓP=/radicalbigg
145
+ /planckover2pi1G
146
+ c3≃1.62·10−33cm; for
147
+ example, it plays a fundamental role in string theory, where it is defin ed as the minimum length of a typical string.
148
+ As a consequence, any space distance smaller than ℓPis deprived of any physical meaning or possibility of being
149
+ measured. M.Planck himself, early in 1899 [32] proposed a system of units to be used in fundamental physics (a
150
+ “System of Natural Units”), where fundamental units are the Pla nck length, the Planck time, and the Planck mass.
151
+ Although Planck did not know yet quantum mechanics and general re lativity, such scales mark out the regions where
152
+ these theories become, in a sense, reciprocally in contrast, so tha t a proper quantum gravity theory is required. At
153
+ the present the Planck length is the minimum spacetime scale present in physics, at which we expect a substantial
154
+ 3It is quite famous a statement by A.O. Barut: “If a spinning pa rticle is not quite a point particle, nor a solid three dimens ional top,
155
+ what can it be?” [18]
156
+ 4It is interesting to point out that in the first of Refs.[28] it is stated that “the special role of the time coordinate in the structure of
157
+ k-Minkowski spacetime forces one to introduce an element of d iscretization in the time direction”.4
158
+ “unification” of high energy microphysics and early cosmology.
159
+ This is, in short terms and until now, the known story of the introdu ction in physics of “fundamental” constants
160
+ with the meaning of elementary lengths or times. However, with the t horough study, started in recent years, of the
161
+ copious set of unpublished notes left by another protagonist of th eoretical physics [33], Ettore Majorana [34], we
162
+ have been acquainted with some unknown contributions pointing out his own elaboration about the arguments just
163
+ described above. In the remaining part of this paper, we will then de scribe and comment such contributions, present
164
+ in the so-called Volumetti [35] and Quaderni [36], containing study and research notes which were not published b y
165
+ Majorana.
166
+ In these booklets, among many different topics the author studied extensively several arguments of electrodynamics
167
+ (both at a classical and at a quantum level). Many of these regarde d topics and methods “usually” discussed in
168
+ the scientific workplaces of 1930s, although they were dealt with in a very original manner and, sometimes, with
169
+ extraordinary results (see, for example, Ref. [37]). Here, howev er, we will focus just on three of these notes, which
170
+ result to be particularly relevant for the topics considered here. A characteristic of the Majorana notes is that they
171
+ always concern specific examples rather than generic theoretical issues. In the following section, then, we report
172
+ the study performed by Majorana about the scattering of partic les by a quasi-Coulombian potential, where a non
173
+ vanishing radius for the scatterer is considered. In Sect.3 we inst ead discuss the introduction of an intrinsic time delay
174
+ in the propagation of the (classical) electromagnetic field, consider ed by Majorana in order to take into account the
175
+ possible effect on the electromagnetic potentials of a fundamental constant length. Finally, in Sect.4 we will describe
176
+ an interesting (though leading to incorrect results) attempt to fin d a relation between the fundamental physical
177
+ constants e,/planckover2pi1andc. Our conclusions and outlook then follow.
178
+ QUASI-COULOMBIAN SCATTERING
179
+ The set of Majorana’s Quaderni opens with the study of the problem of the scattering of particles f rom a quasi-
180
+ Coulombian potential of the form
181
+ V(r) =k√
182
+ r2+a2, (4)
183
+ wherekis a positive constant (related to the electric charge of the potent ial source) and ais, according to Majorana,
184
+ the “magnitude of the radius of the scatterer”. We do not know pr ecisely the motivations for such a study (likely,
185
+ for applications to atomic and nuclear physics problems; see below), but, as a matter of fact, the original intention
186
+ of the author was to extend the well-known Rutherford formula fo r the scattering of a beam of particles (with charge
187
+ Z′eend mass m) from a given body (of charge Ze). In this case we have a pure Coulomb scattering, and the cross
188
+ section (number of scattering particles at an angle θper unit time and solid angle) is given by the classical formula
189
+ f(θ) =Z2Z′2e4
190
+ 4m2v4sin4θ/2=Z2Z′2e4
191
+ 16T2sin4θ/2, (5)
192
+ wherevis the velocity of the incident particles and Tis their kinetic energy. Majorana deduced the above Rutherford
193
+ formulain his Volumetti , usingboth classicalmechanicsargumentsandthe quantum Borna pproximationmethod [35].
194
+ As well-known, Eq.(5) was employed for the first time by E. Rutherf ord in his experiments on αparticles impinging
195
+ on a gold target, aimed at explaining the structure of the atom (with or without a compact nucleus at its center).
196
+ The modification of the Coulomb scattering potential, considered by Majorana in Eq.(4), certainly introduces an
197
+ improvement in the description of the physical phenomenon, with th e introduction of the dimensions of the scattering
198
+ center (in the Rutherford formula assumed to be pointlike, a= 0), but the story does not end here. In fact, the most
199
+ simple approximation in this line of thinking is to consider the scattering center as a uniformly charged sphere of
200
+ radiusabut, in such a case, the potential outside the sphere is strictly Cou lombian (thanks to the Gauss law), and
201
+ Eq.(4) would not apply. Of course, at the time when Majorana perf ormed his calculations, it was known that the
202
+ nucleus of a given atom is not uniformly charged, being formed by indiv idual particles, but, again, such a situation is
203
+ not described by the simple formula in Eq.(4). An example is the Yukaw a potential [38], where the Coulomb potential
204
+ acquires a screening factor with an exponential form ruled by one m ore parameter, giving the range of nuclear forces.
205
+ In any case, it is striking that the only application of Eq.(4) reporte d in the Quaderni [36] was to the hydrogen
206
+ atom, where just one proton forms its nucleus, thus being conside red effectively as a uniformly charged particle. Thus
207
+ the reasoning of Majorana behind Eq.(4) has a different starting p oint, upon which we will briefly speculate below.5
208
+ For the moment, we only note that, from a strictly mathematical po int of view, the introduction of a non-vanishing
209
+ radius for the scattering center has the effect of regularizing the Coulomb potential which, otherwise, would diverge
210
+ forr→0 (however, by contrast to other cases, Majorana did not resto re the full Coulomb potential by taking the
211
+ limita→0 at the end of his calculations, but always maintained in the present c ase a finite value for a).
212
+ In the problem proposed, Majorana studied the deviations from pu re Coulomb scattering by parameterizing it with
213
+ the ratio i/iRof the effective scattering intensity under an angle θ(i.e. the flux of scattered particles per unit surface
214
+ (normal to the incident direction) and per unit time) with respect to that deduced in the Rutherford approximation.
215
+ Besides the radius a, this ratio depends also on the energy and momentum of the incident particles, but Majorana
216
+ preferred to parameterize these by means of the scattering par ameter (or, as denoted by himself, the “minimum
217
+ approach distance”) bin the Coulomb limit, defined by K/b=T, and from the wavelength λof the free particle. By
218
+ settingα=a/(λ/2π),β=b/(λ/2π), he obtained:
219
+ i=f(α,β,θ)iR. (6)
220
+ Of course, in the pure Coulomb limit, the Rutherford formula should b e recovered, so that f(0,β,θ) = 1. The
221
+ calculations then proceed at evaluating the function f(α,β,θ) by keeping αfixed (that is, for fixed scatterer size)
222
+ and considering the limit β→0 (for scattering particles with increasing momentum, approaching nearer and nearer
223
+ the center). The wavefunction of the system is, then, evaluated perturbatively by expanding it in a series ruled by
224
+ the experimental parameter β, at zeroth order ( β= 0) being used the WKB method [39]. After some passages5. He
225
+ obtains the following analytic result in the approximation considered ( ρ=r/(λ/2π))
226
+ f(α,β,θ) = 2sinθ
227
+ 2/integraldisplay∞
228
+ 0ρ/radicalbig
229
+ ρ2+α2sin/bracketleftbigg
230
+ 2ρsinθ
231
+ 2/bracketrightbigg
232
+ dρ, (7)
233
+ which can be expressed in term of the Bessel kfunction k(αsinθ/2) [40]. In this approximation Majorana then finds
234
+ that the actual scattering intensity may be substantially different from that of the Rutherford formula, for backward
235
+ scattering and provided that the radius ais appreciably different from zero (the ratio i/iRapproaching zero for θ=π
236
+ and increasing α).
237
+ This remarkable result is not explicitly reported in the Quaderni , but Majorana gives an attempt to tabulate the
238
+ radial wavefunction uℓ, satisfying the equation
239
+ u′
240
+ ℓ+/parenleftBigg
241
+ 1−β/radicalbig
242
+ ρ2+α2−ℓ(ℓ+1)
243
+ r2/parenrightBigg
244
+ ue= 0, (8)
245
+ obtained numerically with the method of the particular solutions. Alth ough numerical solutions are searched only for
246
+ the particular case of ℓ= 0 andβ= 0.4 (and not β= 0), here the interesting point is that Majorana explicitly reports
247
+ that “for the hydrogen atom we consider the values β= 0.4,0.5,0.6,0.7andα= 0,0.2,0.4,0.6,0.8,1”. Unfortunately,
248
+ no further discussion is given in the Quaderni ; the key point is, however, the fact that Majorana uses λ/2πas
249
+ the length scale for boththe lengths aandb. Now, it is very natural to measure the scattering parameter (or “the
250
+ minimum approachdistance”) in terms ofthe free particlewavelengt hor, equivalently, (the inverseof) the free particle
251
+ momentum ( λ=h/p), since it is obvious that increasing the momentum (or decreasing th e probe wavelength), the
252
+ incident particle approaches nearer and nearer the scattering ce nter, thus decreasing b. The same would not apply,
253
+ instead, to the radius of the scattering center which should be inde pendent of the incident particle properties, unless
254
+ Majoranaconsiders effective, momentum-dependent, dimensionsforthe scatteringcenter. I n suchacase, theattention
255
+ is evidently shifted from the investigation of an actual, particular ph ysical system considered (particles in a scattering
256
+ potential) to the study of more general properties of the backgr ound field or space. We will come back later on this
257
+ point.
258
+ Finally, we conclude this section by mentioning also another modificatio n of the Coulomb potential considered by
259
+ 5In order to avoid convergence problems in the usage of the Gre en method, Majorana assumes that the scattering force acts o nly for
260
+ distances less then a quantity R, and than let R→ ∞at the end of calculations. The potential in Eq.(4) is thus re placed during
261
+ calculations, by k„1√
262
+ r2+a2−1
263
+ R«6
264
+ Majorana [36], reminiscent of the Gamow potential for the descript ion ofα-decay process, namely
265
+ V=
266
+ 
267
+ V0,forr < R,
268
+ k
269
+ r,forr > R.(9)
270
+ Here the role of the scattering center radius is played by the range R,α=R/(λ/2π), and one more parameter enters,
271
+ that is the depth V0of the potential, measured with respect to the kinetic energy of th e incident particles, A=V0/T.
272
+ Although, in this case, the calculations are only sketched, the key p oints envisaged above are as well present. Now
273
+ the correction function in Eq.(6) is replaced by
274
+ i
275
+ iR=f/parenleftbiggV0
276
+ T,R
277
+ λ/2π,b
278
+ λ/2π,θ/parenrightbigg
279
+ (10)
280
+ where “for the hydrogen” Majorana considers the same values as before for the α,βparameters (except, obviously,
281
+ α= 0,0.2), while A=2,1.5,1,0.5,0,-0.5,-1,-1.5,-2,-2.5,-3,...,-8. Interestingly, besides negative values for V0(as in the
282
+ Gamow model), Majorana also considers few positive values for the d epth of the potential. As above, however, the
283
+ very notable point is that both the potential parameters V0andRappear to be strictly related to the energy and
284
+ momentum of the probe particles.
285
+ For both the cases studied, in the presence of continuum states ( Eq.(4)) or bound states (Eq.(9)), the reasoning
286
+ is thus the same. This is even more intriguing if compared to the compr ehensive study of the scattering from a pure
287
+ Coulomb potential, performed by Majorana [36] some pages after those discussed here up to now. Indeed, in this case
288
+ the length unit is no more the free particle wavelength, the Bohr rad ius being introduced (and the energy unit is the
289
+ Rydberg), evidently denoting possible applications to atomic problem s. From what discussed above, it is thus clear
290
+ that Majorana’s original motivations for the studies of quasi-Coulo mbian scattering are different from those standard
291
+ related to atomic and nuclear physics.
292
+ RETARDED ELECTROMAGNETIC FIELDS AND THE INTRODUCTION OF A F UNDAMENTAL
293
+ LENGTH
294
+ Some light on this issue may come from other pages of Majorana’s Quaderni [36], where the possibility is considered
295
+ of introducing an intrinsic constant time delay (or, equivalently, an in trinsic space constant) in the expression for the
296
+ retarded electromagnetic fields. Here the treatment is fully classic al, and goes as follows.
297
+ The starting point is the wave equation satisfied by any component o f the electromagnetic potentials, denoted
298
+ generically with f(x,y,z,t), and then the evaluation of the D’Alembert operator for the stan dard retarded field
299
+ denoted by
300
+ ϕ(x,y,x,t) =f/parenleftBig
301
+ x,y,z,t−r
302
+ c/parenrightBig
303
+ ≡f(x,y,z,t). (11)
304
+ The known result is, of course, the following:
305
+ /squaref=∇2ϕ+2
306
+ rc˙ϕ+2
307
+ c∂2ϕ
308
+ ∂r∂t, (12)
309
+ where a dot indicates time differentiation. Majorana then introduce s an intrinsic space constant ε, so that Eq.(11) is
310
+ replaced by
311
+ ϕ(x,y,z,t) =f/parenleftBigg
312
+ x,y,z,t−√
313
+ r2+ε2
314
+ c/parenrightBigg
315
+ =˜f(x,y,z,t). (13)
316
+ The D’Alembertian term entering into the wave equation is thus
317
+ /tildewider/squaref=∇2ϕ−ε2
318
+ c2(r2+ε2)¨ϕ+2r2+3ε2
319
+ c(r2+ε2)3/2˙ϕ+2r
320
+ c√
321
+ r2+ε2∂2ϕ
322
+ ∂r∂t, (14)
323
+ which replacesEq.(12). Majoranaalsointroducesexplicitly atime de layτwhich, in his view, is a“universalconstant”
324
+ taking the value τ= 0 classically; in his calculations, however, he always uses the length
325
+ ε=τc (15)7
326
+ (already introduced), so that we here follow this line of reasoning.
327
+ The wave equation is apparently solved by using the Green method; d enoting with S′the generic source function
328
+ (charge density ρor current density J), the modified generic potential Φ, solution of the wave equation, a ssumes now
329
+ the following from:
330
+ Φ =/integraldisplay1√
331
+ R2+ε2S/parenleftBigg
332
+ x′,y′,z′,t−√
333
+ R2+ε2
334
+ c/parenrightBigg
335
+ dx′dy′dz′(16)
336
+ (withR=|r−r′|, as usual). Majorana then ends his explicit calculations with the writin g of Eq.(16) expanded up
337
+ to second order in εforε→0 (thus approaching the classical limit, as of course coming from exp eriments):
338
+ Φ =/integraldisplay1
339
+ RS/parenleftbigg
340
+ x′,y′,z′,t−R
341
+ c/parenrightbigg
342
+ dx′dy′dz′
343
+ −ε2/braceleftbigg/integraldisplay1
344
+ 2R3S/parenleftbigg
345
+ x′,y′,z′,t−R
346
+ c/parenrightbigg
347
+ dx′dy′dz′
348
+ +/integraldisplay1
349
+ 2R2cS′/parenleftbigg
350
+ x′,y′,z′,t−R
351
+ c/parenrightbigg
352
+ dx′dy′dz′/bracerightbigg
353
+ +... . (17)
354
+ Some comments are now in order. Unfortunately, again we do not kn ow the motivations for such a study, but
355
+ it is intriguing the following observation. By taking the ordinary electr ic monopole limit [41] into Eq.(16), with
356
+ S(r,t−r
357
+ c) =ρ(r),|r−r′|∼=|r|=rand/integraldisplay
358
+ ρ(r′)dx′dy′dz′=Qbeing the electric charge, one can easily obtain
359
+ Φ =Q√
360
+ r2+ε2, (18)
361
+ that is exactly the potential considered in Eq.(4).
362
+ Although Eq.(18) is not explicitly reported in the Quaderni , it seems very likely that the topics discussed here
363
+ and in the previous section (and considered in no other place in the Quaderni ) are related. If so, the “magnitude of
364
+ the radius of the scatterer” in the scattering problem considered above, whose interpretation in terms of the physical
365
+ dimensions of the scattering center led to difficulties, should be inter preted instead as the “universal” space constant
366
+ introduced here. In this case Majorana’s original intention for the study of quasi-Coulombian scattering was not that
367
+ of exploring the properties of the scattering particle (an atom or it s nucleus or even other elementary particles) but
368
+ rather that of investigating the underlying properties of the phys ical space. The intrinsic space constant then plays
369
+ the role of a fundamental length, like the one introduced by Planck in 1899 or that conjectured by Heisenberg in
370
+ 1938 (see the the Introduction). We do not know if Majorana effec tively thought to a given particular value but, in
371
+ any case, to the best of our knowledge, with Majorana it is the first time that such a detailed study was effectively
372
+ undertaken, as early as in the 1930s.
373
+ A RELATION AMONG THE FUNDAMENTAL CONSTANTS
374
+ The great interest of Majorana for electrodynamics [37], in its class ical or (especially) quantum form, is very well
375
+ documented in his writings (see mainly Refs.[35] and [36]), so that ev en the particular studies described above do
376
+ not seem at all surprising. Nevertheless, the line of thinking of such studies goes well beyond mere applications or
377
+ generalizations of standard electrodynamics, as shown in the prev ious pages, involving questions about fundamental
378
+ constants. It is intriguing that yet another time Majorana reason ed about such fundamental questions, though in an
379
+ (apparently) different context, when he was still a student, as ea rly as in 1928. Indeed, in a page of his Volumetti (see
380
+ section 31 of Volumetto II, in Ref.[35]), Majorana attempted to fin d a relation between the fundamental constants
381
+ e,handc, or, more explicitly, an expression for the elementary charge eappearing in the Coulomb law, describing,
382
+ for example, the electrostatic force between two electrons. The reasoning of Majorana goes as follows. In the region
383
+ of space surrounding the two electrons, which are at a distance ℓapart, an electromagnetic field is present that, “in
384
+ some sense” (according to Majorana’s wording), is quantized. Her e we should point out the first interesting thing
385
+ since, in what follows (see below), it is evident that what is considered “quantized” by Majorana is undoubtedly the
386
+ electromagnetic field acting among the two electrons. Nevertheles s, differently from what appearing elsewhere in his
387
+ notes, Majorana does not refer explicitly to an electromagnetic field, but rather to the ’ether’ surrounding the two8
388
+ electrons, that is (apparently) the physical space itself (in the imp roper language of that time). The two electrons are
389
+ assumed to interact amongthem by means of a point particle (what w e could term the quantum of the electromagnetic
390
+ field) moving with a group velocity equal to the speed of light c. Such particle is further assumed to propagate freely
391
+ from one electron to the other, whereas it inverts its motion when “ colliding” with the electrons. The momentum of
392
+ the quantum particle is deduced by a sort of Sommerfeld quantizatio n rule
393
+ |p|=nh
394
+ 2ℓ,
395
+ withn= 1. The “kick” received by the electron is therefore 2 |p|=h/ℓ, and since the number of collisions per unit
396
+ time is 1 /T=c/2ℓ, the magnitude of the force acting on each electron is then F= 2|p|/T=hc/2ℓ2. By equating
397
+ this expression to the Coulomb law, Majorana finally deduces the exp ression he searched for the elementary charge
398
+ e=/radicalbig
399
+ hc/2 which, however, as pointed out explicitly by himself, gives a value “21 times greater than the real one”.
400
+ Though leading to results far from the truth, what discussed by Ma jorana is interesting for many reasons, one of
401
+ which already mentioned above. Indeed, first of all, the mechanical model considered here is the first one to have
402
+ been proposed as an attempt to deduce the value of the electric ch arge (the famous paper by Dirac in Ref.[42] was
403
+ published few years after). However, as it should be evident from w hat reported above, the interpretation given by
404
+ Majorana of the quantized electromagnetic field substantially coincides with that introduced mo re than a decade later
405
+ by R.P. Feynman in quantum electrodynamics [43], the point particles e xchanged by electrons being assumed to be
406
+ the photons. It is then remarkable that fundamental questions r egarding the constants of Nature discussed properly
407
+ only in more recent times, were addressed in pioneering works by Maj orana as early as at the end of 1920s, where,
408
+ on the other hand, the key ideas of quantum field theory were alrea dy present.
409
+ CONCLUSIONS
410
+ The problem of the existence and of the effects of a fundamental le ngth dates back to the early years of the last
411
+ century. As reviewed in the introduction, this concept was introdu ced and rediscovered in physics several times since
412
+ then, in order to solve difficulties of different kind as, e.g., in regularizin g some divergent results. It is, however, quite
413
+ notable the presence of “side” effects associated to (or, rather , postulated about) the existence of such elementary
414
+ space intervals, which could allow, in Heisenberg’s view for example, th e derivation of the mass of the particles, or
415
+ even play a key role in the physical interpretation of the spin variable .
416
+ Strictly related to the introduction of a fundamental length is, on t he other hand, the assumption of the existence
417
+ of a fundamental time, although such hypothesis is less fashioned a mong physicists than the first one.
418
+ In more recent times, the present subject is embedded into the mo re general framework of the quantization of the
419
+ spacetime, where attempts to provide a well-defined quantum theo ry of gravity manifest into different theoretical
420
+ approaches, with the common denominator of a fundamental lengt h scale.
421
+ In this paper we have shown that the known picture about this intrig uing concept has been further enriched by
422
+ several, yet unknown, contributions by Ettore Majorana. As ear ly as the beginning of 1930s, indeed, the Italian
423
+ physicist not only conjectured (as already Planck and, later, othe rs did) the existence of both an elementary length
424
+ and of a fundamental time scale, but also investigated in some detail the immediate physical consequences of such
425
+ hypothesis. In this respect, his contributions once again [33] reve al a farsighted intuition, and result to be quite
426
+ precious pieces of research both on the historical point of view and for present day theoretical investigations.
427
+ Majorana considered, for example, the scattering from electrica lly charged particles when the underlying properties
428
+ of the space are such that an intrinsic length exists, resulting in the modification of the pure Coulomb potential as
429
+ in Eq.(4) or (18). Even just at the classical level, he obtained the s ame effect by introducing an intrinsic time delay,
430
+ which is considered a “universal constant”, in the expression for t he retarded electromagnetic fields.
431
+ At the quantum level, instead, it is quite interesting still another con tribution by Majorana about fundamental
432
+ constants, aimed at finding a relation between e,/planckover2pi1,cor, rather, aimed at deducing the value of the elementary
433
+ charge. Such a scope was pursued, few years later, in a famous pa per by Dirac [42], although based on a different
434
+ theoretical scheme and with different results. Indeed, the partic ularnumerical prediction by Majorana (as opposed to
435
+ the particular theoretical prediction of the existence of magnetic monopoles by Dirac) is far fr om the truth, but the key
436
+ ideabehind it anticipatedofmorethan adecadethe assumption, intr oducedbyFeynmanin quantumelectrodynamics,
437
+ of photon exchange during electromagnetic processes.
438
+ We expect that Majorana’s important ideas and applications recove red from the oblivion through this paper will
439
+ bring some benefit also to the present day abstract theoretical r esearch on the fundamental properties of the physical
440
+ spacetime.9
441
+ Acknowledgments
442
+ The authors warmly thank E.Recami for interesting discussions an d suggestions, while acknowledge the very kind
443
+ collaboration by M.G.Cammarota and M.Longo.
444
+ ∗Electronic address: [email protected]
445
+ †Electronic address: [email protected]
446
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