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The SU(N) Wilson Loop Average in 2 Dimensions: We solve explicitly a closed, linear loop equation for the SU(2) Wilson loop
average on a two-dimensional plane and generalize the solution to the case of
the SU(N) Wilson loop average with an arbitrary closed contour. Furthermore,
the flat space solution is generalized to any two-dimensional manifold for the
SU(2) Wilson loop average and to any two-dimensional manifold of genus 0 for
the SU(N) Wilson loop average.
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Graphs and Reflection Groups: It is shown that graphs that generalize the ADE Dynkin diagrams and have
appeared in various contexts of two-dimensional field theory may be regarded in
a natural way as encoding the geometry of a root system. After recalling what
are the conditions satisfied by these graphs, we define a bilinear form on a
root system in terms of the adjacency matrices of these graphs and undertake
the study of the group generated by the reflections in the hyperplanes
orthogonal to these roots. Some ``non integrally laced " graphs are shown to be
associated with subgroups of these reflection groups. The empirical relevance
of these graphs in the classification of conformal field theories or in the
construction of integrable lattice models is recalled, and the connections with
recent developments in the context of ${\cal N}=2$ supersymmetric theories and
topological field theories are discussed.
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Spontaneous Symmetry Breaking from an On-Shell Perspective: We show how the well known patterns of masses and interactions that arise
from spontaneous symmetry breaking can be determined from an entirely on-shell
perspective, that is, without reference to Lagrangians, gauge symmetries, or
fields acquiring a vacuum expectation value. To do this, we review how
consistent factorization of $2\rightarrow 2$ tree level scattering can lead to
the familiar structures of Yang-Mills theories, and extend this to find
structures of Yukawa theories. Considering only spins-$0$, $1/2$ and $1$
particles, we construct all the allowed on-shell UV amplitudes under a symmetry
group $G$, and consider all the possible IR amplitudes. By demanding that
on-shell IR amplitudes match onto on-shell UV amplitudes in the high energy
limit, we reproduce the Higgs mechanism and generate masses for spins-$1/2$ and
$1$, find that there is a subgroup $H \subseteq G$ in the IR, and other
interesting relations. To highlight the results, we show the breaking pattern
of the Standard Model $U(1)_{EM} \subset SU(2)_L \times U(1)_Y $, along with
the generation of the masses and interactions of the particles.
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On the construction of gauge theories from non critical type 0 strings: We investigate Polyakov's proposal of constructing Yang-Mills theories by
using non critical type 0 strings. We break conformal invariance by putting the
system at finite temperature and find that the entropy of the cosmological
solutions for these theories matches that of a gas of weakly interacting
Yang-Mills bosons, up to a numerical constant. The computation of the entropy
using the effective action approach presents some novelties in that the whole
contribution comes from the RR fields. We also find an area law and a mass gap
in the theory and show that such behavior persists for $p>4$. We comment on the
possible physical meaning of this result.
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Internal structure of hairy rotating black holes in three dimensions: We construct hairy rotating black hole solutions in three dimensional
Einstein gravity coupled to a complex scalar field. When we turn on a real and
uniform source on the dual CFT, the black hole is stationary with two Killing
vectors and we show that there is no inner horizon for the black hole and the
system evolves smoothly into a Kasner universe. When we turn on a complex and
periodic driving source on the dual CFT with a phase velocity equal to the
angular velocity of the black hole, we have a time-dependent black hole with
only one Killing vector. We show that inside the black hole, after a rapid
collapse of the Einstein-Rosen bridge, oscillations of the scalar field follow.
Then the system evolves into the Kasner epoch with possible Kasner inversion,
which occurs in most of the parameter regimes. In both cases, one of the metric
fields obeys a simple relation between its value at the horizon and in the
Kasner epoch.
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On conformal higher spin wave operators: We analyze free conformal higher spin actions and the corresponding wave
operators in arbitrary even dimensions and backgrounds. We show that the wave
operators do not factorize in general, and identify the Weyl tensor and its
derivatives as the obstruction to factorization. We give a manifestly
factorized form for them on (A)dS backgrounds for arbitrary spin and on
Einstein backgrounds for spin 2. We are also able to fix the conformal wave
operator in d=4 for s=3 up to linear order in the Riemann tensor on generic
Bach-flat backgrounds.
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Hyperbolic Vacua in Minkowski Space: Families of Lorentz, but not Poincare, invariant vacua are constructed for a
massless scalar field in 4D Minkowski space. These are generalizations of the
Rindler vacuum with a larger symmetry group. Explicit expressions are given as
squeezed excitations of the Poincare vacuum. The effective reduced vacua on the
3D hyperbolic de Sitter slices are the well-known de Sitter $\alpha$-vacua with
antipodal singularities in the Wightman function. Several special interesting
cases are discussed.
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Running of the Spectrum of Cosmological Perturbations in String Gas
Cosmology: We compute the running of the spectrum of cosmological perturbations in
String Gas Cosmology, making use of a smooth parametrization of the transition
between the early Hagedorn phase and the later radiation phase. We find that
the running has the same sign as in simple models of single scalar field
inflation. Its magnitude is proportional to $(1 - n_s)$ ($n_s$ being the slope
index of the spectrum), and it is thus parametrically larger than for
inflationary cosmology, where it is proportional to $(1 - n_s)^2$.
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The ultraviolet Behaviour of Integrable Quantum Field Theories, Affine
Toda Field Theory: We investigate the thermodynamic Bethe ansatz (TBA) equations for a system of
particles which dynamically interacts via the scattering matrix of affine Toda
field theory and whose statistical interaction is of a general Haldane type. Up
to the first leading order, we provide general approximated analytical
expressions for the solutions of these equations from which we derive general
formulae for the ultraviolet scaling functions for theories in which the
underlying Lie algebra is simply laced. For several explicit models we compare
the quality of the approximated analytical solutions against the numerical
solutions. We address the question of existence and uniqueness of the solutions
of the TBA-equations, derive precise error estimates and determine the rate of
convergence for the applied numerical procedure. A general expression for the
Fourier transformed kernels of the TBA-equations allows to derive the related
Y-systems and a reformulation of the equations into a universal form.
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Exact Solution of the Harmonic Oscillator in Arbitrary Dimensions with
Minimal Length Uncertainty Relations: We determine the energy eigenvalues and eigenfunctions of the harmonic
oscillator where the coordinates and momenta are assumed to obey the modified
commutation relations [x_i,p_j]=i hbar[(1+ beta p^2) delta_{ij} + beta' p_i
p_j]. These commutation relations are motivated by the fact they lead to the
minimal length uncertainty relations which appear in perturbative string
theory. Our solutions illustrate how certain features of string theory may
manifest themselves in simple quantum mechanical systems through the
modification of the canonical commutation relations. We discuss whether such
effects are observable in precision measurements on electrons trapped in strong
magnetic fields.
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Auxiliary tensor fields for Sp(2,R) self-duality: The coset Sp(2,R)/U(1) is parametrized by two real scalar fields. We
generalize the formalism of auxiliary tensor (bispinor) fields in U(1)
self-dual nonlinear models of abelian gauge fields to the case of Sp(2,R)
self-duality. In this new formulation, Sp(2,R) duality of the nonlinear
scalar-gauge equations of motion is equivalent to an Sp(2,R) invariance of the
auxiliary interaction. We derive this result in two different ways, aiming at
its further application to supersymmetric theories. We also consider an
extension to interactions with higher derivatives.
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Confinement and Deconfinement in Gauge Theories: A Quantum Field Theory: After a brief discussion of small and large gauge transformations and the
nature of observables, we discuss superselection sectors in gauge theories.
There are an infinity of them, classified by large gauge transformations. Gauge
theory sectors are labelled by the eigenvalues of a complete commuting set
(CCS) of these transformations.
In QED, the standard chemical potential is one such operator generating
global U(1). There are many more given by the moments of the electric field on
the sphere at infinity. In QCD, the CCS are constructed from the two commuting
generators spanning a Cartan subalgebra.
We show that any element of a large gauge transformation can be added to the
standard Hamiltonian as a chemical potential without changing field equations
and that in QCD, they lead to confined and deconfined phases . A speculation
about the physical meaning of these chemical potentials is also made.
Comment: This note is based on seminars by the author. So only a limited
number of references are given, from which further literature can be traced.
A paper is under preparation.
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The physical mechanism of AdS instability and Holographic Thermalization: Gravitational falling in AdS has two characteristic properties \cite{ohsin}:
i) A thick shell becomes a thin shell. ii) Any shape become spherical. Such
focusing character of AdS, for the collapse of dusts, leads to the rapid
thermalization mechanism in strongly interacting system. For the collapse of a
wave, it explains the cascade of energy to UV through repeated bounces, which
has been extensively discussed in recent numerical works. Therefore the
focusing is the physical mechanism of instability of AdS. Such sharp contrast
between the dust and wave in collapse, together with the experimental
observation of rapid thermalization, suggest that the initial condition of
created particles in RHIC is in a state with random character rather than a
coherent one. Two time scales, one for thermalization and the other for
hydro-nization are defined and calculated in terms of the total mass density
and energy distribution of the initial particles. We find $t_{th}\sim
(1-c_1/E^2)^{1/2}/T $ so that softer modes thermalize earlier. However, for
hydro-nization, $t_{hyd} \sim 1/E^{2/3}T^{1/3}$ therefore harder modes come
earlier. We also show that near horizon limit of Dp brane solutions have
similar focusing effect which is enough to guarantee the early thermalization.
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G/H M-branes and AdS_{p+2} Geometries: We prove the existence of a new class of BPS saturated M-branes. They are in
one-to-one correspondence with the Freund--Rubin compactifications of M-theory
on either (AdS_4) x (G/H) or (AdS_7) x (G/H), where G/H is the seven (or four)
dimensional Einstein coset manifolds classified long ago in the context of
Kaluza Klein supergravity. The G/H M-branes are solitons that interpolate
between flat space at infinity and the old Kaluza-Klein compactifications at
the horizon. They preserve N/2 supersymmetries where N is the number of Killing
spinors of the (AdS) x (G/H) vacuum. A crucial ingredient in our discussion is
the identification of a solvable Lie algebra parametrization of the Lorentzian
non compact coset SO(2,p+1)/SO(1,p+1) corresponding to anti de Sitter space
AdS_{p+2} . The solvable coordinates are those naturally emerging from the near
horizon limit of the G/H p-brane and correspond to the Bertotti Robinson form
of the anti-de-Sitter metric. The pull-back of anti-de-Sitter isometries on the
p-brane world-volume contain, in particular, the broken conformal
transformations recently found in the literature.
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Perturbative Vacua from IIB Matrix Model: It has not been clarified whether a matrix model can describe various vacua
of string theory. In this paper, we show that the IIB matrix model includes
type IIA string theory. In the naive large N limit of the IIB matrix model,
configurations consisting of simultaneously diagonalizable matrices form a
moduli space, although the unique vacuum would be determined by complicated
dynamics. This moduli space should correspond to a part of perturbatively
stable vacua of string theory. Actually, one point on the moduli space
represents type IIA string theory. Instead of integrating over the moduli space
in the path-integral, we can consider each of the simultaneously diagonalizable
configurations as a background and set the fluctuations of the diagonal
elements to zero. Such procedure is known as quenching in the context of the
large N reduced models. By quenching the diagonal elements of the matrices to
an appropriate configuration, we show that the quenched IIB matrix model is
equivalent to the two-dimensional large N N=8 super Yang-Mills theory on a
cylinder. This theory is nothing but matrix string theory and is known to be
equivalent to type IIA string theory. As a result, we find the manner to take
the large N limit in the IIB matrix model.
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Twist Deformations of the Supersymmetric Quantum Mechanics: The N-extended Supersymmetric Quantum Mechanics is deformed via an abelian
twist which preserves the super-Hopf algebra structure of its Universal
Enveloping Superalgebra. Two constructions are possible. For even N one can
identify the 1D N-extended superalgebra with the fermionic Heisenberg algebra.
Alternatively, supersymmetry generators can be realized as operators belonging
to the Universal Enveloping Superalgebra of one bosonic and several fermionic
oscillators. The deformed system is described in terms of twisted operators
satisfying twist-deformed (anti)commutators. The main differences between an
abelian twist defined in terms of fermionic operators and an abelian twist
defined in terms of bosonic operators are discussed.
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Entanglement Negativity Transitions in Chaotic Eigenstates: It was recently noted that the entanglement entropy for a subsystem of a
chaotic eigenstate exhibits an enhanced correction when the subsystem
approaches a phase transition at half the total system size. This enhanced
correction was derived for general subsystems by Dong and Wang by summing over
noncrossing permutations, which can be thought of as ``saddles'' either in a
sum emerging from averaging over Wick contractions or in an analogous
gravitational calculation. We extend these results to the case of entanglement
negativity, an entanglement measure defined on a bipartite density matrix. We
focus on a particular transition previously studied in a toy model of JT
gravity, one for which the sum over permutations was found to give similar (or
even stronger) enhanced corrections. We derive and resum the relevant
permutations to give a form for the averaged negativity spectrum, reproducing
the gravitational answer for some quantities and finding tension with other
quantities, namely the partially transposed entropy. Along the way, we extend
the results of Dong and Wang to the case of $n < 1$ R\'enyi entropy, showing
that it always receives volume law corrections.
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Precision test of AdS$_6$/CFT$_5$ in Type IIB: Large classes of warped AdS$_6$ solutions were constructed recently in Type
IIB supergravity, and identified as holographic duals for five-dimensional
superconformal field theories realized by $(p,q)$ five-brane webs. We confront
holographic results for the five sphere partition functions obtained from these
solutions with computations for the putative dual field theories. We obtain the
sphere partition functions and conformal central charges in gauge theory
deformations of the superconformal field theories numerically using
supersymmetric localization, and extrapolate the results to the conformal fixed
points. In the appropriate large $N$ limits, the results match precisely to the
supergravity computations, providing strong support for the proposed dualities.
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On the self-consistency of off-shell Slavnov-Taylor identities in QCD: Using Hopf-algebraic structures as well as diagrammatic techniques for
determining the Slavnov-Taylor identities for QCD we construct the relations
for the triple and quartic gluon vertices at one loop. By making the
longitudinal projection on an external gluon of a Green's function we show that
the gluon self-energy of that leg is consistently replaced by a ghost
self-energy. The resulting identities are then studied by evaluating all the
graphs for an off-shell non-exceptional momentum configuration. In the case of
the 3-point function this is for the most general momentum case and for the
4-point function we consider the fully symmetric point.
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Theoretical Aspects of Quintom Models: Quintom models, with its Equation of State being able to cross the
cosmological constant boundary $w=-1$, turns out to be attractive for
phenomenological study. It can not only be applicable for dark energy model for
current universe, but also lead to a bounce scenario in the early universe.
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The 3d Ising Model represented as Random Surfaces: We consider a random surface representation of the three-dimensional Ising
model.The model exhibit scaling behaviour and a new critical index $\k$ which
relates $\g_{string}$ for the bosonic string to the exponent $\a$ of the
specific heat of the 3d Ising model is introduced. We try to determine $\k$ by
numerical simulations.
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Conformal symmetry of gravity and the cosmological constant problem: In absence of matter Einstein gravity with a cosmological constant $\La$ can
be formulated as a scale-free theory depending only on the dimensionless
coupling constant G \Lambda where G is Newton constant. We derive the conformal
field theory (CFT) and its improved stress-energy tensor that describe the
dynamics of conformally flat perturbations of the metric. The CFT has the form
of a constrained \lambda \phi^{4} field theory. In the cosmological framework
the model describes the usual Friedmann-Robertson-Walker flat universe. The
conformal symmetry of the gravity sector is broken by coupling with matter. The
dimensional coupling constants G and \Lambda are introduced by different terms
in this coupling. If the vacuum of quantum matter fields respects the symmetry
of the gravity sector, the vacuum energy has to be zero and the ``physical''
cosmological constant is generated by the coupling of gravity with matter. This
could explain the tiny value of the observed energy density driving the
accelerating expansion of the universe.
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The Fuzzy Analog of Chiral Diffeomorphisms in higher dimensional Quantum
Field Theories: The well-known fact that classical automorphisms of (compactified) Minkowski
spacetime (Poincare or conformal trandsformations) also allow a natural
derivation/interpretation in the modular setting (in the operator-algebraic
sense of Tomita and Takesaki) of the algebraic formulation of QFT has an
interesting nontrivial chiral generalization to the diffeomorphisms of the
circle. Combined with recent ideas on algebraic (d-1)-dimensional lightfront
holography, these diffeomorphisms turn out to be images of ``fuzzy'' acting
groups in the original d-dimensional (massive) QFT. These actions do not
require any spacetime noncommutativity and are in complete harmony with
causality and localization principles. Their use tightens the relation with
kinematic chiral structures on the causal horizon and makes recent attempts to
explain the required universal structure of a possible future quantum
Bekenstein law in terms of Virasoro algebra structures more palatable.
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What is dimensional reduction really telling us?: Numerous approaches to quantum gravity report a reduction in the number of
spacetime dimensions at the Planck scale. However, accepting the reality of
dimensional reduction also means accepting its consequences, including a
variable speed of light. We provide numerical evidence for a variable speed of
light in the causal dynamical triangulation (CDT) approach to quantum gravity,
showing that it closely matches the superluminality implied by dimensional
reduction. We argue that reconciling the appearance of dimensional reduction
with a constant speed of light may require modifying our understanding of time,
an idea originally proposed in Ref. 1.
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PQ Axiverse: We show that the strong CP problem is solved in a large class of
compactifications of string theory. The Peccei-Quinn mechanism solves the
strong CP problem if the CP-breaking effects of the ultraviolet completion of
gravity and of QCD are small compared to the CP-preserving axion potential
generated by low-energy QCD instantons. We characterize both classes of
effects. To understand quantum gravitational effects, we consider an ensemble
of flux compactifications of type IIB string theory on orientifolds of
Calabi-Yau hypersurfaces in the geometric regime, taking a simple model of QCD
on D7-branes. We show that the D-brane instanton contribution to the neutron
electric dipole moment falls exponentially in $N^4$, with $N$ the number of
axions. In particular, this contribution is negligible in all models in our
ensemble with $N>17$. We interpret this result as a consequence of large $N$
effects in the geometry that create hierarchies in instanton actions and also
suppress the ultraviolet cutoff. We also compute the CP breaking due to
high-energy instantons in QCD. In the absence of vectorlike pairs, we find
contributions to the neutron electric dipole moment that are not excluded, but
that could be accessible to future experiments if the scale of supersymmetry
breaking is sufficiently low. The existence of vectorlike pairs can lead to a
larger dipole moment. Finally, we show that a significant fraction of models
are allowed by standard cosmological and astrophysical constraints.
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A Goldstone Boson Equivalence for Inflation: The effective field theory of single-field inflation characterizes the
inflationary epoch in terms of a pattern of symmetry breaking. An operator
acquires a time-dependent vacuum expectation value, defining a preferred
spatial slicing. In the absence of dynamical gravity, the fluctuations around
the time-dependent background are described by the Goldstone boson associated
with this symmetry breaking process. With gravity, the Goldstone is eaten by
the metric, becoming the scalar metric fluctuation. In this paper, we will show
that in general single-field inflation, the statistics of scalar metric
fluctuations are given by the statistics of this Goldstone boson it decoupled
from gravity up to corrections that are controlled as an expansion in slow-roll
parameters. This even holds in the presence of additional parameters, like the
speed of sound, that naively enhance the impact of the gravitational terms. In
the process, we derive expressions for leading and sub-leading gravitational
corrections to all-orders in the Goldstone boson.
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Impurity-doped scalar fields in arbitrary dimensions: We investigate the presence of localized structures for relativistic scalar
fields coupled to impurities in arbitrary spatial dimensions. Such systems
present spatial inhomogeneity, realized through the inclusion of explicit
coordinate dependence in the Lagrangian. It is shown that, in stark contrast to
the impurity-free scenario, Derrick's argument does not present a strong
hindrance to the existence of stable solutions in this case. Bogomol'nyi
equations giving rise to global minima of the energy are found, and some of the
ensuing BPS configurations are presented.
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Supersymmetry and complexified spectrum on Euclidean AdS$_2$: Quantum study of supersymmetric theories on Euclidean two dimensional anti-de
Sitter space (AdS$_2$) is invalid if we use the standard normalizable
functional basis due to its incompatibility with supersymmetry. We cure this
problem by demonstrating that supersymmetry requires complexified spectrum and
constructing the supersymmetric basis for scalar and spinor fields. Our new
basis is free of fermionic zero modes, delta-function normalizable with respect
to a newly defined inner product, and compatible with the supersymmetric
asymptotic boundary condition. We also explore the one-loop evaluation using
this basis and show that it agrees with the standard nonsupersymmetric basis up
to a global contribution arising from the fermion zero mode.
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Discrete symmetries as automorphisms of the proper Poincare group: We present the consistent approach to finding the discrete transformations in
the representation spaces of the proper Poincar\'e group. To this end we use
the possibility to establish a correspondence between involutory automorphisms
of the proper Poincar\'e group and the discrete transformations. As a result,
we derive rules of the discrete transformations for arbitrary spin-tensor
fields without the use of relativistic wave equations. Besides, we construct
explicitly fields carrying representations of the extended Poincar\'e group,
which includes the discrete transformations as well.
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Partition function on spheres: how (not) to use zeta function
regularization: It is known that not all summation methods are linear and stable. Zeta
function regularization is in general non-linear. However, in some cases formal
manipulations with "zeta function" regularization (assuming linearity of sums)
lead to correct results. We consider several examples and show why this
happens.
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Dilaton Gravity in $2+ε$ Dimensions: Quantum theory of dilaton gravity is studied in $2+\epsilon$ dimensions.
Divergences are computed and renormalized at one-loop order. The mixing between
the Liouville field and the dilaton field eliminates $1/\epsilon$ singularity
in the Liouville-dilaton propagator. This smooth behavior of the dilaton
gravity theory in the $\epsilon \rightarrow 0$ limit solves the oversubtraction
problem which afflicted the higher orders of the Einstein gravity in
$2+\epsilon$ dimensions. As a nontrivial fixed point, we find a dilaton gravity
action which can be transformed to a CGHS type action.
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Black Holes and Abelian Symmetry Breaking: Black hole configurations offer insights on the non-linear aspects of
gravitational theories, and can suggest testable predictions for modifications
of General Relativity. In this work, we examine exact black hole configurations
in vector-tensor theories, originally proposed to explain dark energy by
breaking the Abelian symmetry with a non-minimal coupling of the vector to
gravity. We are able to evade the no-go theorems by Bekenstein on the existence
of regular black holes in vector-tensor theories with Proca mass terms, and
exhibit regular black hole solutions with a profile for the longitudinal vector
polarization, characterised by an additional charge. We analytically find the
most general static, spherically symmetric black hole solutions with and
without a cosmological constant, and study in some detail their features, such
as how the geometry depends on the vector charges. We also include angular
momentum, and find solutions describing slowly-rotating black holes. Finally,
we extend some of these solutions to higher dimensions.
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Bogolyubov Quasiparticles in Constrained Systems: The paper is devoted to the formulation of quantum field theory for an early
universe in General Relativity considered as the Dirac general constrained
system. The main idea is the Hamiltonian reduction of the constrained system in
terms of measurable quantities of the observational cosmology: the world proper
time, cosmic scale factor, and the density of matter. We define " particles" as
field variables in the holomorphic representation which diagonalize the
measurable density. The Bogoliubov quasiparticles are determined by
diagonalization of the equations of motion (but not only of the initial
Hamiltonian) to get the set of integrals of motion (or conserved quantum
numbers, in quantum theory). This approach is applied to describe particle
creation in the models of the early universe where the Hubble parameter goes to
infinity.
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Flux-induced SUSY-breaking soft terms on D7-D3 brane systems: We study the effect of RR and NSNS 3-form fluxes on the effective action of
the worldvolume fields of Type IIB D7/D3-brane configurations. The D7-branes
wrap 4-cycles on a local Calabi-Yau geometry. This is an extension of previous
work on hep-th/0311241, where a similar analysis was applied to the case of
D3-branes. Our present analysis is based on the D7- and D3-brane
Dirac-Born-Infeld and Chern-Simons actions, and makes full use of the
R-symmetries of the system, which allow us to compute explicitly results for
the fields lying at the D3-D7 intersections. A number of interesting new
properties appear as compared to the simpler case of configurations with only
D3-branes. As a general result one finds that fluxes stabilize some or all of
the D7-brane moduli. We argue that this is important for the problem of
stabilizing Kahler moduli through non-perturbative effects in KKLT-like vacua.
We also show that (0,3) imaginary self-dual fluxes, which lead to
compactifications with zero vacuum energy, give rise to SUSY-breaking soft
terms including gaugino and scalar masses, and trilinear terms. Particular
examples of chiral MSSM-like models of this class of vacua, based on D3-D7
brane systems at orbifold singularities are presented.
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Volume modulus inflation and a low scale of SUSY breaking: The relation between the Hubble constant and the scale of supersymmetry
breaking is investigated in models of inflation dominated by a string modulus.
Usually in this kind of models the gravitino mass is of the same order of
magnitude as the Hubble constant which is not desirable from the
phenomenological point of view. It is shown that slow-roll saddle point
inflation may be compatible with a low scale of supersymmetry breaking only if
some corrections to the lowest order Kahler potential are taken into account.
However, choosing an appropriate Kahler potential is not enough. There are also
conditions for the superpotential, and e.g. the popular racetrack
superpotential turns out to be not suitable. A model is proposed in which
slow-roll inflation and a light gravitino are compatible. It is based on a
superpotential with a triple gaugino condensation and the Kahler potential with
the leading string corrections. The problem of fine tuning and experimental
constraints are discussed for that model.
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Excited D-brane decay in Cubic String Field Theory and in Bosonic String
Theory: In the cubic string field theory, using the gauge invariant operators
corresponding to the on-shell closed string vertex operators, we have
explicitly evaluated the decay amplitudes of two open string tachyons or gauge
fields to one closed string tachyon or graviton up to level two. We then
evaluated the same amplitudes in the bosonic string theory, and shown that the
amplitudes in both theories have exactly the same pole structure. We have also
expanded the decay amplitudes in the bosonic string theory around the
Mandelstam variable s=0, and shown that their leading contact terms are fully
consistent with a tachyonic Dirac-Born-Infeld action which includes both open
string and closed string tachyon.
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Black hole thermodynamics with the cosmological constant as independent
variable: Bridge between the enthalpy and the Euclidean path integral
approaches: Viewing the cosmological constant $\Lambda<0$ as an independent variable, we
consider the thermodynamics of the Schwarzschild black hole in an anti-de
Sitter (AdS) background. For this system, there is one approach which regards
the enthalpy as the master thermodynamic variable and makes sense if one
considers the vacuum pressure due to the cosmological constant acting in the
volume inside the horizon and the outer size of the system is not restricted.
From this approach a first law of thermodynamics emerges naturally. There is
yet another approach based on the Euclidean action principle and its path
integral that puts the black hole inside a cavity, defines a quasilocal energy
at the cavity's boundary, and from which a first law of thermodynamics in a
different version also emerges naturally. The first approach has affinities
with critical phenomena in condensed matter physics and the second approach is
an ingredient necessary for the construction of quantum gravity. The bridge
between the two approaches is carried out rigorously, putting thus the
enthalpic thermodynamics with $\Lambda$ as independent variable on the same
footing as the quasilocal energy approach.
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On the construction of renormalized gauge theories using renormalization
group techniques: The aim of these lectures is to describe a construction, as self-contained as
possible, of renormalized gauge theories. Following a suggestion of Polchinski,
we base our analysis on the Wilson renormalization group method. After a
discussion of the infinite cut-off limit, we study the short distance
properties of the Green functions verifying the validity of Wilson short
distance expansion. We also consider the problem of the extension to the
quantum level of the classical symmetries of the theory. With this purpose we
analyze in details the breakings induced by the cut-off in a $SU(2)$ gauge
symmetry and we prove the possibility of compensating these breakings by a
suitable choice of non-gauge invariant counter terms.
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Implicit regularization beyond one loop order: scalar field theories: Implicit regularization (IR) has been shown as an useful momentum space tool
for perturbative calculations in dimension specific theories, such as chiral
gauge, topological and supersymmetric quantum field theoretical models at one
loop level. In this paper, we aim at generalizing systematically IR to be
applicable beyond one loop order. We use a scalar field theory as an example
and pave the way for the extension to quantum field theories which are richer
from the symmetry content viewpoint. Particularly, we show that a natural
(minimal) renormalization scheme emerges, in which the infinities displayed in
terms of integrals in one internal momentum are subtracted, whereas infrared
and ultraviolet modes do not mix and therefore leave no room for ambiguities. A
systematic cancelation of the infrared divergences at any loop order takes
place between the ultraviolet finite and divergent parts of the amplitude for
non-exceptional momenta leaving, as a byproduct, a renormalization group scale.
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Gravity dual D3-braneworld and Open/Closed string duality: A covariantly constant dynamical two-form is exploited on a $D_3$-brane to
obtain its gravity dual action, governing an $S^3$ deformed $AdS_5$ black hole,
in a type IIB string theory on $S^1\times K3$. We invoke the Kaluza-Klein
compactification to work out the open/closed string duality. Interestingly, the
Reissner-Nordstrom black hole is obtained on the "non-Reimannian" braneworld.
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The anisotropic coupling of gravity and electromagnetism in
Hořava-Lifshitz theory: We analyze the electromagnetic-gravity interaction in a pure
Ho\v{r}ava-Lifshitz framework. To do so we formulate the Ho\v{r}ava-Lifshitz
gravity in $4+1$ dimensions and perform a Kaluza-Klein reduction to $3+1$
dimensions. We use this reduction as a mathematical procedure to obtain the
$3+1$ coupled theory, which at the end is considered as a fundamental,
self-consistent, theory. The critical value of the dimensionless coupling
constant in the kinetic term of the action is $\lambda=1/4$. It is the kinetic
conformal point for the non-relativistic electromagnetic-gravity interaction.
In distinction, the corresponding kinetic conformal value for pure
Ho\v{r}ava-Lifshitz gravity in $3+1$ dimensions is $\lambda=1/3$. We analyze
the geometrical structure of the critical and noncritical cases, they
correspond to different theories. The physical degrees of freedom propagated by
the noncritical theory are the transverse traceless graviton, the transverse
gauge vector and two scalar fields. In the critical theory one of the scalars
is absent, only the dilaton scalar field is present. The gravity and vector
excitations propagate with the same speed, which at low energy can be taken to
be the speed of light. The field equations for the gauge vector in the
non-relativistic theory have exactly the same form as the relativistic
electromagnetic field equations arising from the Kaluza-Klein reduction of
General Relativity, and are equal to them for a particular value of one of the
coupling constants. The potential in the Hamiltonian is a polynomial of finite
degree in the gauge vector and its covariant derivatives.
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Strongly Coupled Quantum and Classical Systems and Zeno's Effect: A model interaction between a two-state quantum system and a classical
switching device is analysed and shown to lead to the quantum Zeno effect for
large values of the coupling constant k . A minimal piecewise deterministic
random process compatible with the Liouville equation is described, and it is
shown that 1/k can be interpreted as the jump frequency of the classical device
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Maximally Helicity Violating Disk Amplitudes, Twistors and
Transcendental Integrals: We obtain simple expressions for tree-level maximally helicity violating
amplitudes of N gauge bosons from disk world-sheets of open superstrings. The
amplitudes are written in terms of (N-3)! hypergeometric integrals depending on
kinematic parameters, weighted by certain kinematic factors. The integrals are
transcendental in a strict sense defined in this work. The respective kinematic
factors can be succinctly written in terms of "dual" momentum twistors. The
amplitudes are computed by using the prescription proposed by Berkovits and
Maldacena.
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How to go with an RG Flow: We apply the formalism of holographic renormalization to domain wall
solutions of 5-dimensional supergravity which are dual to deformed conformal
field theories in 4 dimensions. We carefully compute one- and two-point
functions of the energy-momentum tensor and the scalar operator mixing with it
in two specific holographic flows, resolving previous difficulties with these
correlation functions. As expected, two-point functions have a 0-mass dilaton
pole for the Coulomb branch flow in which conformal symmetry is broken
spontaneously but not for the flow dual to a mass deformation in which it is
broken explicitly. A previous puzzle of the energy scale in the Coulomb branch
flow is explained.
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D-brane masses at special fibres of hypergeometric families of
Calabi-Yau threefolds, modular forms, and periods: We consider the fourteen families $W$ of Calabi-Yau threefolds with one
complex structure parameter and Picard-Fuchs equation of hypergeometric type,
like the mirror of the quintic in $\mathbb{P}^4$. Mirror symmetry identifies
the masses of even--dimensional D--branes of the mirror Calabi-Yau $M$ with
four periods of the holomorphic $(3,0)$-form over a symplectic basis of
$H_3(W,\mathbb{Z})$. It was discovered by Chad Schoen that the singular fiber
at the conifold of the quintic gives rise to a Hecke eigenform of weight four
under $\Gamma_0(25)$, whose Hecke eigenvalues are determined by the Hasse-Weil
zeta function which can be obtained by counting points of that fiber over
finite fields. Similar features are known for the thirteen other cases. In two
cases we further find special regular points, so called rank two attractor
points, where the Hasse-Weil zeta function gives rise to modular forms of
weight four and two. We numerically identify entries of the period matrix at
these special fibers as periods and quasiperiods of the associated modular
forms. In one case we prove this by constructing a correspondence between the
conifold fiber and a Kuga-Sato variety. We also comment on simpler applications
to local Calabi-Yau threefolds.
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The Microscopic Approach to N=1 Super Yang-Mills Theories: We give a brief account of the recent progresses in super Yang-Mills theories
based in particular on the application of Nekrasov's instanton technology to
the case of N=1 supersymmetry. We have developed a first-principle formalism
from which any chiral observable in the theory can be computed, including in
strongly coupled confining vacua. The correlators are first expressed in terms
of some external variables as sums over colored partitions. The external
variables are then fixed to their physical values by extremizing the
microscopic quantum superpotential. Remarquably, the results can be shown to
coincide with the Dijkgraaf-Vafa matrix model approach, which uses a totally
different mathematical framework. These results clarify many important
properties of N=1 theories, related in particular to generalized Konishi
anomaly equations and to Veneziano-Yankielowicz terms in the glueball
superpotentials. The proof of the equivalence between the formalisms based on
colored partitions and on matrices is also a proof of the open/closed string
duality in the chiral sector of the theories.
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Comments on cosmological RG flows: We study cosmological backgrounds from the point of view of the dS/CFT
correspondence and its renormalization group flow extension. We focus on the
case where gravity is coupled to a single scalar with a potential. Depending on
the latter, the scalar can drive both inflation and the accelerated expansion
(dS) phase in the far future. We also comment on quintessence scenarios, and
flows familiar from the AdS/CFT correspondence. We finally make a tentative
embedding of this discussion in string theory where the scalar is the dilaton
and the potential is generated at the perturbative level.
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The Googly Amplitudes in Gauge Theory: The googly amplitudes in gauge theory are computed by using the off shell MHV
vertices with the newly proposed rules of Cachazo, Svrcek and Witten. The
result is in agreement with the previously well-known results. In particular we
also obtain a simple result for the all negative but one positive helicity
amplitude when one of the external line is off shell.
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Fusion for AdS/CFT boundary S-matrices: We propose a fusion formula for AdS/CFT worldsheet boundary S-matrices. We
show that, starting from the fundamental Y=0 boundary S-matrix, this formula
correctly reproduces the two-particle bound-state boundary S-matrices.
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Eleven dimensional supergravity in light cone gauge: Light-cone gauge manifestly supersymmetric formulation of eleven dimensional
supergravity is developed. The formulation is given entirely in terms of light
cone scalar superfield, allowing us to treat all component fields on an equal
footing. All higher derivative on mass shell manifestly supersymmetric 4-point
functions invariant with respect to linear supersymmetry transformations and
corresponding (in gravitational bosonic sector) to terms constructed from four
Riemann tensors and derivatives are found. Superspace representation for
4-point scattering amplitudes is also obtained. Superfield representation of
linearized interaction vertex of superparticle and supergravity fields is
presented. All 4-point higher derivative interaction vertices of
ten-dimensional supersymmetric Yang-Mills theory are also determined.
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Space-time uncertainty relation and Lorentz invariance: We discuss a Lorentz covariant space-time uncertainty relation, which agrees
with that of Karolyhazy-Ng-van Dam when an observational time period delta t is
larger than the Planck time lp. At delta t < lp, this uncertainty relation
takes roughly the form delta t delta x > lp^2, which can be derived from the
condition prohibiting the multi-production of probes to a geometry. We show
that there exists a minimal area rather than a minimal length in the
four-dimensional case. We study also a three-dimensional free field theory on a
non-commutative space-time realizing the uncertainty relation. We derive the
algebra among the coordinate and momentum operators and define a
positive-definite norm of the representation space. In four-dimensional
space-time, the Jacobi identity should be violated in the algebraic
representation of the uncertainty relation.
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On N-spike strings in conformal gauge with NS-NS fluxes: The $AdS_3\times S^3$ string sigma model supported both by NS-NS and R-R
fluxes has become a well known integrable model, however a putative dual field
theory description remains incomplete. We study the anomalous dimensions of
twist operators in this theory via semiclassical string methods. We describe
the construction of a multi-cusp closed string in conformal gauge moving in
$AdS_3$ with fluxes, which supposedly is dual to a general higher twist
operator. After analyzing the string profiles and conserved charges for the
string, we find the exact dispersion relation between the charges in the `long'
string limit. This dispersion relation in leading order turns out to be similar
to the case of pure RR flux, with the coupling being scaled by a factor that
depends on the amount of NS-NS flux turned on. We also analyse the case of pure
NS flux, where the dispersion relation simplifies considerably. Furthermore, we
discuss the implications of these results at length.
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Reality, measurement and locality in Quantum Field Theory: It is currently believed that the local causality of Quantum Field Theory
(QFT) is destroyed by the measurement process. This belief is also based on the
Einstein-Podolsky-Rosen (EPR) paradox and on the so-called Bell's theorem, that
are thought to prove the existence of a mysterious, instantaneous action
between distant measurements. However, I have shown recently that the EPR
argument is removed, in an interpretation-independent way, by taking into
account the fact that the Standard Model of Particle Physics prevents the
production of entangled states with a definite number of particles. This result
is used here to argue in favor of a statistical interpretation of QFT and to
show that it allows for a full reconciliation with locality and causality.
Within such an interpretation, as Ballentine and Jarret pointed out long ago,
Bell's theorem does not demonstrate any nonlocality.
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Inflation at the TeV scale with a PNGB curvaton: We investigate a particular type of curvaton mechanism, under which inflation
can occur at Hubble scale of order 1 TeV. The curvaton is a pseudo
Nambu-Goldstone boson, whose order parameter increases after a phase transition
during inflation, triggered by the gradual decrease of the Hubble scale. The
mechanism is studied in the context of modular inflation, where the inflaton is
a string axion. We show that the mechanism is successful for natural values of
the model parameters, provided the phase transition occurs much earlier than
the time when the cosmological scales exit the horizon. Also, it turns our that
the radial mode for our curvaton must be a flaton field.
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Superconformal symmetry and representations: We give an introduction to conformal and superconformal algebras and their
representations in various dimensions. Special emphasis is put on 4d
$\mathcal{N}=2$ superconformal symmetry. This is the writeup of the lectures
given at the Winter School "YRISW 2020" to appear in a special issue of JPhysA.
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Gravity in Randall-Sundrum two D-brane model: We analyse Randall-Sundrum two D-brane model by linear perturbation and then
consider the linearised gravity on the D-brane. The qualitative contribution
from the Kaluza-Klein modes of gauge fields to the coupling to the gravity on
the brane will be addressed. As a consequence, the gauge fields localised on
the brane are shown not to contribute to the gravity on the brane at large
distances. Although the coupling between gauge fields and gravity appears in
the next order, the ordinary coupling cannot be realised.
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Slavnov-Taylor Identities in Coulomb Gauge Yang-Mills Theory: Two aspects of the color charge in Coulomb gauge continuum Yang-Mills theory
are discussed. The first aspect is the existence of a conserved and vanishing
total charge exhibited within the first order functional formalism. The second
aspect is the closure of the set of Slavnov-Taylor identities in the second
order functional formalism, such that the exact solution for temporal Green's
functions is in principle possible and thereby preserving the color charge.
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Magnetized Baryonic layer and a novel BPS bound in the
gauged-Non-Linear-Sigma-Model-Maxwell theory in (3+1)-dimensions through
Hamilton-Jacobi equation: It is show that one can derive a novel BPS bound for the gauged
Non-Linear-Sigma-Model (NLSM) Maxwell theory in (3+1) dimensions which can
actually be saturated. Such novel bound is constructed using Hamilton-Jacobi
equation from classical mechanics. The configurations saturating the bound
represent Hadronic layers possessing both Baryonic charge and magnetic flux.
However, unlike what happens in the more common situations, the topological
charge which appears naturally in the BPS bound is a non-linear function of the
Baryonic charge. This BPS bound can be saturated when the surface area of the
layer is quantized. The far-reaching implications of these results are
discussed. In particular, we determine the exact relation between the magnetic
flux and the Baryonic charge as well as the critical value of the Baryonic
chemical potential beyond which these configurations become thermodynamically
unstable.
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$1/2$ BPS Structure Constants and Random Matrices: We study three point functions of half BPS operators in $\mathcal{N}=4$ super
Yang-Mills theory focusing on correltors of two of the operators with dimension
of order $\Delta\sim N^2$ and a light single trace operator. These describe
vacuum expectation values of type IIB supergravity modes in LLM backgrounds
that do not necessarily preserve the same symmetries as the background
solution. We propose a class of complex matrix models that fully capture the
combinatorics of the problem, and describe their solution in the large $N$
limit. In simple regimes when the dual description is in terms of widely
separated condensates of giant gravitons we find that the models are solvable
in the large $N$ and can be approximated by unitary Jacobi ensembles; we
describe how these distributions are reproduced in the dual bubbling geometry
picture for large droplets. In the case of two eigenvalue droplets the model is
exactly solvable at finite $N$. As a result we compute all half-BPS structure
constants of heavy-heavy-light type, and reproduce the formulas found via
holographic renormalization in the large $N$ limit. We also comment on
structure constants of three heavy operators.
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Pedagogical Introduction to Hamiltonian BRST formalism: Hamiltonian BRST formalism (FV formalism) includes many auxiliary fields
without explanation. Its path-integration has a simple form by using BRST
charge, but its construction is quite mechanically and hard to understand
physical meaning. In this paper we perform the phase space path-integral with
requiring BRST invariance for action and measure, and show that the resultant
form is equivalent to the Hamiltonian BRST (FV) formalism in gravitational
theory. This explains why so many auxiliary fields are necessary to be
introduced. We also find the gauge fixing is automatically done by requiring
the BRST invariance of the path-integral measure. This is a pedagogical
introduction to Hamiltonian BRST formalism.
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Analytical solutions for fermions on a thick brane with a piecewise and
smooth warp factor: In this paper we study analytical solutions for fermion localization in
Randall-Sundrum (RS) models. We show that there exist special couplings between
scalar fields and fermions giving us discrete massive localizable modes.
Besides this we obtain resonances in some models by analytical methods.
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Gravitational Mediation of Supersymmetry Breaking in Superstring Theory: SUSY breaking and its mediation are among the most important problems of
supersymmetric generalizations of the standard model. The idea of
gravity-mediated SUSY breaking, proposed in 1982 by Arnowitt, Chamseddine and
Nath, and independently by Barbieri, Ferrara and Savoy, fits naturally into
superstring theory, where it can be realized at both classical as well as
quantum levels. This talk is dedicated to Pran Nath on his 65th birthday.
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Comments on Double Scaled Little String Theory: We study little string theory in a weak coupling limit defined in \gk.
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N=1/2 Supersymmetric gauge theory in noncommutative space: A formulation of (non-anticommutative) N=1/2 supersymmetric U(N) gauge theory
in noncommutative space is studied. We show that at one loop
UV/IR mixing occurs. A generalization of Seiberg-Witten map to noncommutative
and non-anticommutative superspace is employed to obtain an action in terms of
commuting fields at first order in the noncommutativity parameter tetha. This
leads to abelian and non-abelian gauge theories whose supersymmetry
transformations are local and non-local, respectively.
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Instantons and Conformal Holography: We study a subsector of the AdS_4/CFT_3 correspondence where a class of
solutions in the bulk and on the boundary can be explicitly compared. The bulk
gravitational theory contains a conformally coupled scalar field with a Phi^4
potential, and is holographically related to a massless scalar with a Phi^6
interaction in three dimensions. We consider the scalar sector of the bulk
theory and match bulk and boundary classical solutions of the equations of
motion. Of particular interest is the matching of the bulk and the boundary
instanton solutions which underlies the relationship between bulk and boundary
vacua with broken conformal invariance. Using a form of radial quantization we
show that quantum states in the bulk correspond to multiply-occupied single
particle quantum states in the boundary theory. This allows us to explicitly
identify the boundary composite operator which is dual to the bulk scalar, at
the free theory level as well as in the instanton vacuum. We conclude with a
discussion of possible implications of our results.
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Rainbow gravity corrections to the entropic force: The entropic force attracts a lot of interest for its multifunctional
properties. For instance, Einstein's field equation, Newton's law of
gravitation and the Friedmann equation can be derived from the entropic force.
In this paper, utilizing a new kind of rainbow gravity model that was proposed
by Magueijo and Smolin, we explore the quantum gravity corrections to the
entropic force. First, we derive the modified thermodynamics of a rainbow black
hole via its surface gravity. Then, according to Verlinde's theory, the quantum
corrections to the entropic force are obtained. The result shows that the
modified entropic force is related not only to the properties of the black hole
but also the Planck length $\ell_p$, and the rainbow parameter $\gamma$.
Furthermore, based on the rainbow gravity corrected entropic force, the
modified Einstein's field equation and the modified Friedmann equation are also
derived.
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Supersymmetric Boundary Conditions in N=4 Super Yang-Mills Theory: We study boundary conditions in N=4 super Yang-Mills theory that preserve
one-half the supersymmetry. The obvious Dirichlet boundary conditions can be
modified to allow some of the scalar fields to have a ``pole'' at the boundary.
The obvious Neumann boundary conditions can be modified by coupling to
additional fields supported at the boundary. The obvious boundary conditions
associated with orientifolds can also be generalized. In preparation for a
separate study of how electric-magnetic duality acts on these boundary
conditions, we explore moduli spaces of solutions of Nahm's equations that
appear in the presence of a boundary. Though our main interest is in boundary
conditions that are Lorentz-invariant (to the extent possible in the presence
of a boundary), we also explore non-Lorentz-invariant but half-BPS deformations
of Neumann boundary conditions. We make preliminary comments on the action of
electric-magnetic duality, deferring a more serious study to a later paper.
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Quantum integrability in two-dimensional systems with boundary: In this paper we consider affine Toda systems defined on the half-plane and
study the issue of integrability, i.e. the construction of higher-spin
conserved currents in the presence of a boundary perturbation. First at the
classical level we formulate the problem within a Lax pair approach which
allows to determine the general structure of the boundary perturbation
compatible with integrability. Then we analyze the situation at the quantum
level and compute corrections to the classical conservation laws in specific
examples. We find that, except for the sinh-Gordon model, the existence of
quantum conserved currents requires a finite renormalization of the boundary
potential.
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2d QCD and Integrability, Part I: 't Hooft model: We study analytical properties and integrable structures of the meson
spectrum in large $N_c$ QCD$_2$. We show that the integral equation that
determines the masses of the mesons, often called the 't Hooft equation, is
equivalent to finding solutions to a TQ-Baxter equation. Using the Baxter
equation, we extract systematic expansions of the energy levels as well as
analytic asymptotic expressions for wavefunctions. Our analysis extends
previous results for a special quark mass by Fateev et al. to arbitrary quark
masses. This reformulation, together with its relation to an inhomogeneous
Fredholm equation, is particularly suited for analytical treatments and makes
accessible the analytic structure of the spectrum in the complex plane of the
quark masses. We also comment on applications of our techniques to
non-perturbative topological string partition functions.
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Topological versus Non--Topological Theories and $p-q$ Duality in $c \le
1$ 2d Gravity Models: We discuss the non--perturbative formulation for $c \leq 1$ string theory.
The field theory like formulation of topological and non--topological models is
presented. The integral representation for arbitrary $(p,q)$ solutions is
derived which explicitly obeys $p-q$ duality of these theories. The exact
solutions to string equation and various examples are also discussed.
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Integrable XYZ Spin Chain with Boundaries: We consider a general class of boundary terms of the open XYZ spin-1/2 chain
compatible with integrability. We have obtained the general elliptic solution
of $K$-matrix obeying the boundary Yang-Baxter equation using the $R$-matrix of
the eight vertex model and derived the associated integrable spin-chain
Hamiltonian.
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Study of QED singular properties for variable gyromagnetic ratio
$g\simeq 2$: Using the external field method, {\it i.e.\/} evaluating the effective action
$V_{\mathrm{eff}}$ for an arbitrarily strong constant and homogeneous field, we
explore nonperturbative properties of QED allowing arbitrary gyromagnetic ratio
$g$. We find a cusp at $g = 2$ in: a) The QED $b_0$-renormalization group
coefficient, and in the infinite wavelength limit in b) a subclass containing
the pseudoscalar ${\cal P}^{2n}= (\vec E\cdot\vec B)^{2n} $ of light-light
scattering coefficients. Properties of $b_0$ imply for certain domains of $g$
asymptotic freedom in an Abelian theory.
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Non-static Dimensional Reduction of QED_3 at Finite Temperature: We study an extreme non-static limit of 2+1-dimensional QED obtained by
making a dimensional reduction so that all fields are spatially uniform but
time dependent. This dimensional reduction leads to a 0+1-dimensional field
theory that inherits many of the features of the 2+1-dimensional model, such as
Chern-Simons terms, time-reversal violation, an analogue of parity violation,
and global U(2) flavor symmetry. At one-loop level, interactions induce a
Chern-Simons term at finite T with coefficient tanh(beta m_F/2), where m_F is
the fermion mass. The finite temperature two loop self-energies are also
computed, and are non-zero for all temperatures.
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Axions and Superfluidity in Weyl Semimetals: An effective field theory (EFT) for dynamical axions in Weyl semimetals
(WSMs) is presented. A pseudoscalar axion excitation is predicted in WSMs at
sufficiently low temperatures, independently of the strength of the Weyl
fermion self-coupling. For strong fermion self-coupling the axion is the
gapless Goldstone boson of chiral $U(1)^{\text{ch}}$ spontaneous symmetry
breaking. For weak fermion self-coupling an axion is also generated at non-zero
chiral density for Weyl nodes displaced in energy, as a gapless collective mode
of correlated fermion pair excitations of the Fermi surface. This is an
explicit example of the extension of Goldstone's theorem to symmetry breaking
by the axial anomaly itself. In both cases the axion is a chiral density wave
or phason mode of the superfluid state of the WSM, and the Weyl fermions form a
chiral condensate $\langle\bar{\psi}\psi\rangle$ at low temperatures. In the
presence of an applied magnetic field the axion mode becomes gapped, in analogy
to the Anderson-Higgs mechanism in a superconductor. 't Hooft anomaly matching
from ultraviolet to infrared scales is directly verified in the EFT approach.
WSMs thus provide an interesting quantum system in which superfluid, non-Fermi
liquid behavior, and a dynamical axion are predicted to follow directly from
the axial anomaly in a consistent EFT that may be tested experimentally.
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Low-energy theorem and OPE in the conformal window of massless QCD: We develop a new technique, based on a low-energy theorem (LET) of NSVZ type
derived in arXiv:1701.07833, for the nonperturbative investigation of SU(N) QCD
with N${}_f$ massless quarks - or, more generally, of massless QCD-like
theories - in phases where the beta function, $\beta(g)$, with $g=g(\mu)$ the
renormalized gauge coupling, admits an isolated zero, $g_*$, in the infrared
(IR) or ultraviolet (UV). We point out that the LET sets constraints on 3-point
correlators involving the insertion of $Tr\, F^2$, its anomalous dimension
$\gamma_{F^2}$, and the anomalous dimensions of multiplicatively renormalizable
operators at $g_*$. These constraints intertwine with the exact conformal
scaling for $g(\mu)\rightarrow g_*$ with $\mu\neq 0,+\infty$ fixed and the
IR/UV asymptotics - which may or may not coincide with the IR/UV limit of the
aforementioned conformal scaling - for $\Lambda_{\scriptscriptstyle{IR/UV}}$
fixed. In the conformal case we also discuss how the LET for bare correlators
is the rationale for the existence in massless QCD of the mysterious divergent
contact term in the OPE of $Tr\,F^2$ with itself discovered in perturbation
theory in arXiv:1209.1516, arXiv:1407.6921 and computed to all orders in
arXiv:1601.08094. Specifically, if $\gamma_{F^2}$ does not vanish, the
divergent contact term in the rhs of the LET for the 2-point correlator of
$Tr\,F^2$ has to match - and we verify by direct computation that it actually
does - the divergence in the lhs due to the nontrivial anomalous dimension of
$Tr\,F^2$. Hence, remarkably, the additive renormalization due to the divergent
contact term in the rhs is related by the LET to the multiplicative
renormalization in the lhs, in such a way that a suitably renormalized version
of the LET has no ambiguity for additive renormalization.
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Inflaton in anisotropic higher derivative gravity: Existence and stability analysis of the Kantowski-Sachs type inflationary
universe in a higher derivative scalar-tensor gravity theory is studied in
details. Isotropic de Sitter background solution is shown to be stable against
any anisotropic perturbation during the inflationary era. Stability of the de
Sitter space in the post inflationary era can also be realized with proper
choice of coupling constants.
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Finite Unification and Top Quark Mass: In unified gauge theories there exist renormalization group invariant
relations among gauge and Yukawa couplings that are compatible with
perturbative renormalizability, which could be considered as a Gauge-Yukawa
Unification. Such relations are even necessary to ensure all-loop finiteness in
Finite Unified Theories, which have vanishing $\beta$-functions beyond the
unification point. We elucidate this alternative way of unification, and then
present its phenomenological consequences in $SU(5)$-based models.
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Heterotic-string amplitudes at one loop: modular graph forms and
relations to open strings: We investigate one-loop four-point scattering of non-abelian gauge bosons in
heterotic string theory and identify new connections with the corresponding
open-string amplitude. In the low-energy expansion of the heterotic-string
amplitude, the integrals over torus punctures are systematically evaluated in
terms of modular graph forms, certain non-holomorphic modular forms. For a
specific torus integral, the modular graph forms in the low-energy expansion
are related to the elliptic multiple zeta values from the analogous open-string
integrations over cylinder boundaries. The detailed correspondence between
these modular graph forms and elliptic multiple zeta values supports a recent
proposal for an elliptic generalization of the single-valued map at genus zero.
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Flat deformations of type IIB S-folds: Type IIB S-folds of the form $\textrm{AdS}_{4} \times \textrm{S}^1 \times
\textrm{S}^5$ have been shown to contain axion-like deformations parameterising
flat directions in the 4D scalar potential and corresponding to marginal
deformations of the dual S-fold CFT's. In this note we present a
group-theoretical characterisation of such flat deformations and provide a 5D
interpretation thereof in terms of $\mathfrak{so}(6)$-valued duality twists
inducing a class of Cremmer--Scherk--Schwarz flat gaugings in a reduction from
5D to 4D. In this manner we establish the existence of two flat deformations
for the $\mathcal{N}=4$ and $\textrm{SO}(4)$ symmetric S-fold causing a
symmetry breaking down to its $\textrm{U}(1)^2$ Cartan subgroup. The result is
a new two-parameter family of non-supersymmetric S-folds which are
perturbatively stable at the lower-dimensional supergravity level, thus
providing the first examples of such type IIB backgrounds.
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Operators with Large Quantum Numbers, Spinning Strings, and Giant
Gravitons: We study the behaviour of spinning strings in the background of various
distributions of smeared giant gravitons in supergravity. This gives insights
into the behaviour of operators of high dimension, spin and R-charge. Using a
new coordinate system recently presented in the literature, we find that it is
particularly natural to prepare backgrounds in which the probe operators
develop a variety of interesting new behaviours. Among these are the possession
of orbital angular momentum as well as spin, the breakdown of logarithmic
scaling of dimension with spin in the high spin regime, and novel
splitting/fractionation processes.
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Supersymmetric quantum mechanics and the Riemann hypothesis: We construct a supersymmetric quantum mechanical model in which the energy
eigenvalues of the Hamiltonians are the products of Riemann zeta functions. We
show that the trivial and nontrivial zeros of the Riemann zeta function
naturally correspond to the vanishing ground state energies in this model. The
model provides a natural form of supersymmetry.
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Conservation Laws and Formation of Singularities in Relativistic
Theories of Extended Objects: The dynamics of an M-dimensional extended object whose M+1 dimensional world
volume in M+2 dimensional space-time has vanishing mean curvature is formulated
in term of geometrical variables (the first and second fundamental form of the
time-dependent surface $\sum_M$), and simple relations involving the rate of
change of the total area of $\sum_M$, the enclosed volume as well as the
spatial mean -- and intrinsic scalar curvature, integrated over $\sum_M$, are
derived. It is shown that the non-linear equations of motion for $\sum_M(t)$
can be viewed as consistency conditions of an associated linear system that
gives rise to the existence of non-local conserved quantities (involving the
Christoffel-symbols of the flat M+1 dimensional euclidean submanifold swept out
in ${\Bbb R}^{M+1}$). For M=1 one can show that all motions are necessarily
singular (the curvature of a closed string in the plane can not be everywhere
regular at all times) and for M=2, an explicit solution in terms of elliptic
functions is exhibited, which is neither rotationally nor axially symmetric. As
a by-product, 3-fold-periodic spacelike maximal hypersurfaces in ${\Bbb
R}^{1,3}$ are found.
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Quantum Chrono-Topology of Nuclear and Sub-Nuclear Reactions: A quantum time topological space is developed and applied to solve some
problems about quantum theory. It is disconnected and satifies specific
separation axioms. The degree of disconnectedness of the time-space is a
decreasing function of the number of simultaneous or almost simultaneous
fundamental interactions. In this topology the U+R Penrose dynamics is
implemented by means of a time evolution operator in QFT. This operator is
unitary or non-unitary, depending on the type of quantization of the field
action-integral. The time evolution operator allows to find the Boltzmann
factor in QFT in the above space-time. From an elementary solution of the
Liouville equation the quantization of the time follows and the Planck constant
has been calculated. Compatibility between time-reversal and irreversibility is
spontaneously obtained. The renormalization of the field action-integral
follows from quantization. The solution of the measurement problem and the wave
function reduction have been deduced in the framework of the Schroedinger
theory. The Schroedinger cat's paradoxon and the paradoxon of the wave packet
decay have been resolved.
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Pohlmeyer-reduced form of string theory in AdS_5 x S^5: semiclassical
expansion: We consider the Pohlmeyer-reduced formulation of the AdS_5 x S^5 superstring.
It is constructed by introducing new variables which are algebraically related
to supercoset current components so that the Virasoro conditions are
automatically solved. The reduced theory is a gauged WZW model supplemented
with an integrable potential and fermionic terms that ensure its UV finiteness.
The original superstring theory and its reduced counterpart are closely related
at the classical level, and we conjecture that they remain related at the
quantum level as well, in the sense that their quantum partition functions
evaluated on respective classical solutions are equal. We provide evidence for
the validity of this conjecture at the one-loop level, i.e. at the first
non-trivial order of the semiclassical expansion near several classes of
classical solutions.
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Soft wall model for a holographic superconductor: We apply the soft wall holographic model from hadron physics to a description
of the high-$T_c$ superconductivity. In comparison with the existing bottom-up
holographic superconductors, the proposed approach is more phenomenological. On
the other hand, it is much simpler and has more freedom for fitting the
conductivity properties of the real high-$T_c$ materials. We demonstrate some
examples of emerging models and discuss a possible origin of the approach.
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From matrix models' topological expansion to topological string
theories: counting surfaces with algebraic geometry: The 2-matrix model has been introduced to study Ising model on random
surfaces. Since then, the link between matrix models and combinatorics of
discrete surfaces has strongly tightened. This manuscript aims to investigate
these deep links and extend them beyond the matrix models, following my work's
evolution. First, I take care to define properly the hermitian 2 matrix model
which gives rise to generating functions of discrete surfaces equipped with a
spin structure. Then, I show how to compute all the terms in the topological
expansion of any observable by using algebraic geometry tools. They are
obtained as differential forms on an algebraic curve associated to the model:
the spectral curve. In a second part, I show how to define such differentials
on any algebraic curve even if it does not come from a matrix model. I then
study their numerous symmetry properties under deformations of the algebraic
curve. In particular, I show that these objects coincide with the topological
expansion of the observable of a matrix model if the algebraic curve is the
spectral curve of this model. Finally, I show that fine tuning the parameters
ensure that these objects can be promoted to modular invariants and satisfy the
holomorphic anomaly equation of the Kodaira-Spencer theory. This gives a new
hint that the Dijkgraaf-Vafa conjecture is correct.
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The dual string sigma-model of the SU_q(3) sector: In four-dimensional N=4 super Yang-Mills (SYM) the SU(3) sub-sector spanned
by purely holomorphic fields is isomorphic to the corresponding mixed one
spanned by both holomorphic and antiholomorphic fields. This is no longer the
case when one considers the marginally deformed N=4 SYM. The mixed SU(3) sector
marginally deformed by a complex parameter beta, i.e. SU_q(3) with q=e^{2
i\pi\beta}, has been shown to be integrable at one-loop hep-th/0703150, while
it is not the case for the corresponding purely holomorphic one. Moreover, the
marginally deformed N=4 SYM also has a gravity dual constructed by Lunin and
Maldacena in hep-th/0502086. However, the mixed SU_q(3) sector has not been
studied from the supergravity point of view. Hence in this note, for the case
of purely imaginary marginal $\beta$-deformations, we compute the superstring
SU_q(3) \sigma-model in the fast spinning string limit and show that, for
rational spinning strings, it reproduces the energy computed via Bethe
equations.
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D-brane Potentials in the Warped Resolved Conifold and Natural Inflation: In this paper we obtain a model of Natural Inflation from string theory with
a Planckian decay constant. We investigate D-brane dynamics in the background
of the warped resolved conifold (WRC) throat approximation of Type IIB string
compactifications on Calabi-Yau manifolds. When we glue the throat to a compact
bulk Calabi-Yau, we generate a D-brane potential which is a solution to the
Laplace equation on the resolved conifold. We can exactly solve this equation,
including dependence on the angular coordinates. The solutions are valid down
to the tip of the resolved conifold, which is not the case for the more
commonly used deformed conifold. This allows us to exploit the effect of the
warping, which is strongest at the tip. We inflate near the tip using an
angular coordinate of a D5-brane in the WRC which has a discrete shift
symmetry, and feels a cosine potential, giving us a model of Natural Inflation,
from which it is possible to get a Planckian decay constant whilst maintaining
control over the backreaction. This is because the decay constant for a wrapped
brane contains powers of the warp factor, and so can be made large, while the
wrapping parameter can be kept small enough so that backreaction is under
control.
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Quantum Mechanics of Particle on a torus knot: Curvature and Torsion
Effects: Constraints play an important role in dynamical systems. However, the subtle
effect of constraints in quantum mechanics is not very well studied. In the
present work we concentrate on the quantum dynamics of a point particle moving
on a non-trivial torus knot. We explicitly take into account the role of
curvature and torsion, generated by the constraints that keep the particle on
the knot. We exploit the "Geometry Induced Potential (GIP) approach" to
construct the Schrodinger equation for the dynamical system, obtaining thereby
new results in terms of particle energy eigenvalues and eigenfunctions. We
compare our results with existing literature that completely ignored the
contributions of curvature and torsion. In particular, we explicitly show how
the "knottedness" of the path influences the results. In the process we have
revealed a (possibly un-noticed) "topological invariant".
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Towards a Coulomb gas of instantons in the SO(4)xU(1) Higgs model on R_4: The $SO(4)\times U(1)$ Higgs model on $\R_4$ is extended by a $F^3$ term so
that the action receives a nonvanishing contribution from the interactions of
2-instantons and 3-instantons, and can be expressed as the inverse of the
Laplacian on $\R_4$ in terms of the mutual distances of the instantons. The
one-instanton solutions of both the basic and the extended models have been
studied in detail numerically.
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Cutoff Dependence and Complexity of the CFT$_2$ Ground State: We present the vacuum of a two-dimensional conformal field theory (CFT$_2$)
as a network of Wilson lines in $SL(2,\mathbb{R}) \times SL(2,\mathbb{R})$
Chern-Simons theory, which is conventionally used to study gravity in
three-dimensional anti-de Sitter space (AdS$_3$). The position and shape of the
network encode the cutoff scale at which the ground state density operator is
defined. A general argument suggests identifying the `density of complexity' of
this network with the extrinsic curvature of the cutoff surface in AdS$_3$,
which by the Gauss-Bonnet theorem agrees with the holographic Complexity =
Volume proposal.
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N=1 SCFTs from F-theory on Orbifolds: We study four-dimensional superconformal field theories living on the
worldvolume of $D3$ branes probing minimally-supersymmetric F-theory
backgrounds, focusing on the case of orbi-orientifold setups with and without
7-branes. We observe that these theories are closely related to
compactifications of six-dimensional $\mathcal{N}=(1,0)$ theories on a torus
with flux, where the flux quanta is mapped in Type IIB to the defining data of
the orbifold group. We analyze the cases of class $\mathcal{S}_k$ theories as
well as of compactifications of the E-string and of orbi-instanton theories. We
also classify $\mathcal{S}$-fold configurations in F-theory preserving minimal
supersymmetry in four dimensions and their mass deformations.
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Modified Gravity via Spontaneous Symmetry Breaking: We construct effective field theories in which gravity is modified via
spontaneous breaking of local Lorentz invariance. This is a gravitational
analogue of the Higgs mechanism. These theories possess additional graviton
modes and modified dispersion relations. They are manifestly well-behaved in
the UV and free of discontinuities of the van Dam-Veltman-Zakharov type,
ensuring compatibility with standard tests of gravity. They may have important
phenomenological effects on large distance scales, offering an alternative to
dark energy. For the case in which the symmetry is broken by a vector field
with the wrong sign mass term, we identify four massless graviton modes (all
with positive-definite norm for a suitable choice of a parameter) and show the
absence of the discontinuity.
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Moyal Quantization on Fuzzy Sphere: We study the quantization of compact space on the basis of the Moyal
quantization. We first construct the $su(2)$ algebra that are the functions of
canonical coordinates $a$ and $a^*$. We make use of them to define the adjoint
operators, which is used to define the fuzzy sphere and constitute the algebra.
We show that the vacuum is constructed as the powers of $a^*$, in contrast to
the flat case where the vacuum is defined by the exponential function of $a$
and $a^*$. We present how the analogy of the creation operator acting on the
vacuum is obtaied. The construction does not resort to the ordinary creation
and annihilation operators.
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On Generalized $Q$-systems: We formulate $Q$-systems for the closed XXZ, open XXX and open
quantum-group-invariant XXZ quantum spin chains. Polynomial solutions of these
$Q$-systems can be found efficiently, which in turn lead directly to the
admissible solutions of the corresponding Bethe ansatz equations.
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Stability of the aether: The requirements for stability of a Lorentz violating theory are analyzed. In
particular we conclude that Einstein-aether theory can be stable when its modes
have any phase velocity, rather than only the speed of light as was argued in a
recent paper.
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Monopoles, Vortices and Strings: Confinement and Deconfinement in 2+1
Dimensions at Weak Coupling: We consider, from several complementary perspectives, the physics of
confinement and deconfinement in the 2+1 dimensional Georgi-Glashow model.
Polyakov's monopole plasma and 't Hooft's vortex condensation are discussed
first. We then discuss the physics of confining strings at zero temperature. We
review the Hamiltonian variational approach and show how the linear confining
potential arises in this framework. The second part of this review is devoted
to study of the deconfining phase transition. We show that the mechanism of the
transition is the restoration of 't Hooft's magnetic symmetry in the deconfined
phase. The heavy charged $W$ bosons play a crucial role in the dynamics of the
transition, and we discuss the interplay between the charged $W$ plasma and the
binding of monopoles at high temperature. Finally we discuss the phase
transition from the point of view of confining strings. We show that from this
point of view the transition is not driven by the Hagedorn mechanism
(proliferation of arbitrarily long strings), but rather by the "disintegration"
of the string due to the proliferation of 0 branes.
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W-extended Kac representations and integrable boundary conditions in the
logarithmic minimal models WLM(1,p): We construct new Yang-Baxter integrable boundary conditions in the lattice
approach to the logarithmic minimal model WLM(1,p) giving rise to reducible yet
indecomposable representations of rank 1 in the continuum scaling limit. We
interpret these W-extended Kac representations as finitely-generated W-extended
Feigin-Fuchs modules over the triplet W-algebra W(p). The W-extended fusion
rules of these representations are inferred from the recently conjectured
Virasoro fusion rules of the Kac representations in the underlying logarithmic
minimal model LM(1,p). We also introduce the modules contragredient to the
W-extended Kac modules and work out the correspondingly-extended fusion
algebra. Our results are in accordance with the Kazhdan-Lusztig dual of tensor
products of modules over the restricted quantum universal enveloping algebra
$\bar{U}_q(sl_2)$ at $q=e^{\pi i/p}$. Finally, polynomial fusion rings
isomorphic with the various fusion algebras are determined, and the
corresponding Grothendieck ring of characters is identified.
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Black Hole Complementary Principle and The Noncommutative Membrane: In the spirit of Black Hole Complementary Principle, we have found the
noncommutative membrane of Scharzchild Black Holes. In this paper we extend our
results to Kerr Black Hole and see the same story. Also we make a conjecture
that spacetimes is noncommutative on the stretched membrane of the more general
Kerr-Newman Black Hole.
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Twist Deformation of Rotationally Invariant Quantum Mechanics: Non-commutative Quantum Mechanics in 3D is investigated in the framework of
the abelian Drinfeld twist which deforms a given Hopf algebra while preserving
its Hopf algebra structure. Composite operators (of coordinates and momenta)
entering the Hamiltonian have to be reinterpreted as primitive elements of a
dynamical Lie algebra which could be either finite (for the harmonic
oscillator) or infinite (in the general case). The deformed brackets of the
deformed angular momenta close the so(3) algebra. On the other hand, undeformed
rotationally invariant operators can become, under deformation, anomalous (the
anomaly vanishes when the deformation parameter goes to zero). The deformed
operators, Taylor-expanded in the deformation parameter, can be selected to
minimize the anomaly. We present the deformations (and their anomalies) of
undeformed rotationally-invariant operators corresponding to the harmonic
oscillator (quadratic potential), the anharmonic oscillator (quartic potential)
and the Coulomb potential.
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Elliptic Blowup Equations for 6d SCFTs. IV: Matters: Given the recent geometrical classification of 6d $(1,0)$ SCFTs, a major
question is how to compute for this large class their elliptic genera. The
latter encode the refined BPS spectrum of the SCFTs, which determines geometric
invariants of the associated elliptic non-compact Calabi-Yau threefolds. In
this paper we establish for all 6d $(1,0)$ SCFTs in the atomic classification
blowup equations that fix these elliptic genera to large extent. The latter
fall into two types: the unity- and the vanishing blowup equations. For almost
all rank one theories, we find unity blowup equations which determine the
elliptic genera completely. We develop several techniques to compute elliptic
genera and BPS invariants from the blowup equations, including a recursion
formula with respect to the number of strings, a Weyl orbit expansion, a
refined BPS expansion and an $\epsilon_1,\epsilon_2$ expansion. For higher-rank
theories, we propose a gluing rule to obtain all their blowup equations based
on those of rank one theories. For example, we explicitly give the elliptic
blowup equations for the three higher-rank non-Higgsable clusters, ADE chain of
$-2$ curves and conformal matter theories. We also give the toric construction
for many elliptic non-compact Calabi-Yau threefolds which engineer 6d $(1,0)$
SCFTs with various matter representations.
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Homogeneous Yang-Baxter deformations as generalized diffeomorphisms: Yang-Baxter (YB) deformations of string sigma model provide deformed target
spaces. We propose that homogeneous YB deformations always lead to a certain
class of $\beta$-twisted backgrounds and represent the bosonic part of the
supergravity fields in terms of the classical r-matrix associated with the YB
deformation. We then show that various $\beta$-twisted backgrounds can be
realized by considering generalized diffeomorphisms in the undeformed
background. Our result extends the notable relation between the YB deformations
and (non-commuting) TsT transformations. We also discuss more general
deformations beyond the YB deformations.
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Scattering amplitudes in affine gravity: Affine gravity is a connection-based formulation of gravity that does not
involve a metric. After a review of basic properties of affine gravity, we
compute the tree-level scattering amplitude of scalar particles interacting
gravitationally via the connection in a curved spacetime. We find that, while
classically equivalent to general relativity, affine gravity differs from
metric quantum gravity.
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Equivalent Hamiltonian for Lee Model: Using the techniques of quasi-Hermitian quantum mechanics and quantum field
theory we use a similarity transformation to construct an equivalent Hermitian
Hamiltonian for the Lee model. In the field theory confined to the $V/N\theta$
sector it effectively decouples $V$, replacing the three-point interaction of
the original Lee model by an additional mass term for the $V$ particle and a
four-point interaction between $N$ and $\theta$. While the construction is
originally motivated by the regime where the bare coupling becomes imaginary,
leading to a ghost, it applies equally to the standard Hermitian regime where
the bare coupling is real. In that case the similarity transformation becomes a
unitary transformation.
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Conformal Invariance of Interacting WZNW Models: We consider two level $k$ WZNW models coupled to each other through a
generalized Thirring-like current-current interaction. It is shown that in the
large $k$ limit, this interacting system can be presented as a two-parameter
perturbation around a nonunitary WZNW model. The perturbation operators are the
sigma model kinetic terms with metric related to the Thirring coupling
constants. The renormalizability of the perturbed model leads to an algebraic
equation for couplings. This equation coincides with the master Virasoro
equation. We find that the beta functions of the two-parameter perturbation
have nontrivial zeros depending on the Thirring coupling constants. Thus we
exhibit that solutions to the master equation provide nontrivial conformal
points to the system of two interacting WZNW models.
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Commensurate lock-in in holographic non-homogeneous lattices: We consider the spontaneous formation of striped structures in a holographic
model which possesses explicit translational symmetry breaking, dual to an
ionic lattice with spatially modulated chemical potential. We focus on the
perturbative study of the marginal modes which drive the transition to a phase
exhibiting spontaneous stripes. We study the wave-vectors of the instabilities
with largest critical temperature in a wide range of backgrounds characterized
by the period and the amplitude of the chemical potential modulation.
We report the first holographic observation of the commensurate lock-in
between the spontaneous stripes and the underlying ionic lattice, which takes
place when the amplitude of the lattice is large enough. We also observe an
incommensurate regime in which the amplitude of the lattice is finite, but the
preferred stripe wave-vector is different from that of the lattice.
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Surface Operators and Knot Homologies: Topological gauge theories in four dimensions which admit surface operators
provide a natural framework for realizing homological knot invariants. Every
such theory leads to an action of the braid group on branes on the
corresponding moduli space. This action plays a key role in the construction of
homological knot invariants. We illustrate the general construction with
examples based on surface operators in N=2 and N=4 twisted gauge theories which
lead to a categorification of the Alexander polynomial, the equivariant knot
signature, and certain analogs of the Casson invariant.
This paper is based on a lecture delivered at the International Congress on
Mathematical Physics 2006, Rio de Janeiro, and at the RTN Workshop 2006,
Napoli.
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The Indispensability of Ghost Fields in the Light-Cone Gauge
Quantization of Gauge Fields: We continue McCartor and Robertson's recent demonstration of the
indispensability of ghost fields in the light-cone gauge quantization of gauge
fields. It is shown that the ghost fields are indispensable in deriving
well-defined antiderivatives and in regularizing the most singular component of
gauge field propagator. To this end it is sufficient to confine ourselves to
noninteracting abelian fields. Furthermore to circumvent dealing with
constrained systems, we construct the temporal gauge canonical formulation of
the free electromagnetic field in auxiliary coordinates
$x^{\mu}=(x^-,x^+,x^1,x^2)$ where $x^-=x^0 cos{\theta}-x^3 sin{\theta}, x^+=x^0
sin{\theta}+x^3 cos{\theta}$ and $x^-$ plays the role of time. In so doing we
can quantize the fields canonically without any constraints, unambiguously
introduce "static ghost fields" as residual gauge degrees of freedom and
construct the light-cone gauge solution in the light-cone representation by
simply taking the light-cone limit (${\theta}\to \pi/4$). As a by product we
find that, with a suitable choice of vacuum the Mandelstam-Leibbrandt form of
the propagator can be derived in the ${\theta}=0$ case (the temporal gauge
formulation in the equal-time representation).
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Computation of Critical Exponent Eta at O(1/N_f^2) in Quantum
Electrodynamics in Arbitrary Dimensions: We present a detailed evaluation of $\eta$, the critical exponent
corresponding to the electron anomalous dimension, at $O(1/N^2_{\!f})$ in a
large flavour expansion of QED in arbitrary dimensions in the Landau gauge. The
method involves solving the skeleton Dyson equations with dressed propagators
in the critical region of the theory. Various techniques to compute massless
two loop Feynman diagrams, which are of independent interest, are also given.
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Quantization of charges and fluxes in warped Stenzel geometry: We examine the quantization of fluxes for the warped Stiefel cone and Stenzel
geometries and their orbifolds, and distinguish the roles of three related
notions of charge: Page, Maxwell, and brane. The orbifolds admit discrete
torsion, and we describe the associated quantum numbers which are consistent
with the geometry in its large radius and small radius limits from both the
type IIA and the M-theory perspectives. The discrete torsion, measured by a
Page charge, is related to the number of fractional branes. We relate the
shifts in the Page charges under large gauge transformations to the
Hanany-Witten brane creation effect.
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Arbitrariness in the gravitational Chern-Simons-like term induced
radiatively: Lorentz violation through a radiatively induced Chern-Simons-like term in a
fermionic theory embedded in linearized quantum gravity with a Lorentz- and
CPT-violating axial-vector term in the fermionic sector proportional to a
constant field $b_\mu$ has been recently studied. In a similar fashion as for
the extended-QED model of Carroll-Field-Jackiw, we explicitly show that neither
gauge invariance nor the more stringent momentum routing invariance condition
on underlying Feynman diagrams fix the arbitrariness inherent to such induced
term at one loop order. We present the calculation in a nonperturbative
expansion in $b_\mu$ and within a framework which besides operating in the
physical dimension (and thus avoiding $\gamma_5$ matrix Clifford algebra
ambiguities), judiciously parametrizes regularization dependent arbitrary
parameters usually fixed by symmetries.
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Gauge fields in (A)dS within the unfolded approach: algebraic aspects: It has recently been shown that generalized connections of the (A)dS space
symmetry algebra provide an effective geometric and algebraic framework for all
types of gauge fields in (A)dS, both for massless and partially-massless. The
equations of motion are equipped with a nilpotent operator called $\sigma_-$
whose cohomology groups correspond to the dynamically relevant quantities like
differential gauge parameters, dynamical fields, gauge invariant field
equations, Bianchi identities etc. In the paper the $\sigma_-$-cohomology is
computed for all gauge theories of this type and the field-theoretical
interpretation is discussed. In the simplest cases the $\sigma_-$-cohomology is
equivalent to the ordinary Lie algebra cohomology.
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Dynamical Generation of Solitons in a 1+1 Dimensional Chiral Field
Theory: Non-Perturbative Dirac Operator Resolvent Analysis: We analyze the 1+1 dimensional Nambu-Jona-Lasinio model non-perturbatively.
We study non-trivial saddle points of the effective action in which the
composite fields $\sigx=<\bar\psi\psi>$ and $\pix=<\bar\psii\gam_5\psi>$ form
static space dependent configurations. These configurations may be viewed as
one dimensional chiral bags that trap the original fermions (``quarks'') into
stable extended entities (``hadrons''). We provide explicit expressions for the
profiles of some of these objects and calculate their masses. Our analysis of
these saddle points, and in particular, the proof that the $\sigx, \pix$
condensations must give rise to a reflectionless Dirac operator, appear to us
simpler and more direct than the calculations previously done by Shei, using
the inverse scattering method following Dashen, Hasslacher, and Neveu.
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Higher-loop anomalies in chiral gravities: The one-loop anomalies for chiral $W_{3}$ gravity are derived using the
Fujikawa regularisation method. The expected two-loop anomalies are then
obtained by imposing the Wess-Zumino consistency conditions on the one-loop
results. The anomalies found in this way agree with those already known from
explicit Feynman diagram calculations. We then directly verify that the order
$\hbar^2$ non-local BRST Ward identity anomalies, arising from the ``dressing''
of the one-loop results, satisfy Lam's theorem. It is also shown that in a
rigorous calculation of $Q^2$ anomaly for the BRST charge, one recovers both
the non-local as well as the local anomalies. We further verify that, in chiral
gravities, the non-local anomalies in the BRST Ward identity can be obtained by
the application of the anomalous operator $Q^2$, calculated using operator
products, to an appropriately defined gauge fermion. Finally, we give arguments
to show why this relation should hold generally in reparametrisation-invariant
theories.
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Unitary Networks from the Exact Renormalization of Wave Functionals: The exact renormalization group (ERG) for $O(N)$ vector models (at large $N$)
on flat Euclidean space can be interpreted as the bulk dynamics corresponding
to a holographically dual higher spin gauge theory on $AdS_{d+1}$. This was
established in the sense that at large $N$ the generating functional of
correlation functions of single trace operators is reproduced by the on-shell
action of the bulk higher spin theory, which is most simply presented in a
first-order (phase space) formalism. In this paper, we extend the ERG formalism
to the wave functionals of arbitrary states of the $O(N)$ vector model at the
free fixed point. We find that the ERG flow of the ground state and a specific
class of excited states is implemented by the action of unitary operators which
can be chosen to be local. Consequently, the ERG equations provide a continuum
notion of a tensor network. We compare this tensor network with the
entanglement renormalization networks, MERA, and its continuum version, cMERA,
which have appeared recently in holographic contexts. In particular the ERG
tensor network appears to share the general structure of cMERA but differs in
important ways. We comment on possible holographic implications.
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On Dimensional Extension of Supersymmetry: From Worldlines to
Worldsheets: There exist myriads of off-shell worldline supermultiplets for
(N{\leq}32)-extended supersymmetry in which every supercharge maps a component
field to precisely one other component field or its derivative. A subset of
these extends to off-shell worldsheet (p,q)-supersymmetry and is characterized
by the twin theorems 2.1 and 2.2 in this note. The evasion of the obstruction
defined in these theorems is conjectured to be sufficient for a worldline
supermultiplet to extend to worldsheet supersymmetry; it is also a necessary
filter for dimensional extension to higher-dimensional spacetime. We show
explicitly how to "re-engineer" an Adinkra---if permitted by the twin theorems
2.1 and 2.2---so as to depict an off-shell supermultiplet of worldsheet
(p,q)-supersymmetry.
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Infinite Abelian Subalgebras in Quantum W-Algebras: An Elementary Proof: An elementary proof is given for the existence of infinite dimensional
abelian subalgebras in quantum W-algebras. In suitable realizations these
subalgebras define the conserved charges of various quantum integrable systems.
We consider all principle W-algebras associated with the simple Lie algebras.
The proof is based on the more general result that for a class of vertex
operators the quantum operators are related to their classical counterparts by
an equivalence transformation.
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N-Impurity Superstring Spectra Near the pp-Wave Limit: The complicated non-linear sigma model that characterizes the first
finite-radius curvature correction to the pp-wave limit of IIB superstring
theory on AdS_5 x S^5 has been shown to generate energy spectra that perfectly
match, to two loops in the modified 't Hooft parameter lambda', finite R-charge
corrections to anomalous dimension spectra of large-R N=4 super Yang-Mills
theory in the planar limit. This test of the AdS/CFT correspondence has been
carried out for the specific cases of two and three string excitations, which
are dual to gauge theory R-charge impurities. We generalize this analysis on
the string side by directly computing string energy eigenvalues in certain
protected sectors of the theory for an arbitrary number of worldsheet
excitations with arbitrary mode-number assignments. While our results match all
existing gauge theory predictions to two-loop order in lambda', we again
observe a mismatch at three loops between string and gauge theory. We find
remarkable agreement to all loops in lambda', however, with the near pp-wave
limit of a recently proposed Bethe ansatz for the quantized string Hamiltonian
in the su(2) sector. Based on earlier two- and three-impurity results, we also
infer the full multiplet decomposition of the N-impurity superstring theory
with distinct mode excitations to two loops in lambda'.
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Gravity and the stability of the Higgs sector: We pursued the question of the influence of a strong gravitational field on
the structure of the Higgs effective potential in the gauge-less top-Higgs
sector of the Standard Model with an additional scalar singlet. To this end, we
calculated the one-loop corrected effective potential in an arbitrary curved
spacetime. We have found that the gravity induced terms in the effective
potential may influence its behavior in both small and large field regions.
This result indicated the necessity of a more careful investigation of the
effect of high curvature in the problems concerning the stability of the Higgs
effective potential in the full Standard Model.
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Diagrams for Symmetric Product Orbifolds: We develop a diagrammatic language for symmetric product orbifolds of
two-dimensional conformal field theories. Correlation functions of twist
operators are written as sums of diagrams: each diagram corresponds to a
branched covering map from a surface where the fields are single-valued to the
base sphere where twist operators are inserted. This diagrammatic language
facilitates the study of the large N limit and makes more transparent the
analogy between symmetric product orbifolds and free non-abelian gauge
theories. We give a general algorithm to calculate the leading large N
contribution to four-point correlators of twist fields.
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Truncated Nambu-Poisson Bracket and Entropy Formula for Multiple
Membranes: We show that there exists a cut-off version of Nambu-Poisson bracket which
defines a finite dimensional Lie 3-algebra. The algebra still satisfies the
fundamental identity and thus produces N=8 supersymmetric BLG type equation of
motion for multiple M2 branes. By counting the number of the moduli and the
degree of freedom, we derive an entropy formula which scales as N^{3/2} as
expected for the multiple M2 branes.
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Canonical formulation and conserved charges of double field theory: We provide the canonical formulation of double field theory. It is shown that
this dynamics is subject to primary and secondary constraints. The Poisson
bracket algebra of secondary constraints is shown to close on-shell according
to the C-bracket. A systematic way of writing boundary integrals in doubled
geometry is given. By including appropriate boundary terms in the double field
theory Hamiltonian, expressions for conserved energy and momentum of an
asymptotically flat doubled space-time are obtained and applied to a number of
solutions.
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Concerning a natural compatibility condition between the action and the
renormalized operator product: In this article we note that in a number of situations the operator product
and the classical action satisfy a natural compatibility condition. We consider
the interest of this condition to be twofold: First, the naturality
(functoriality) of the compatibility condition suggests that it be used for
geometrical applications of renormalized functional integration. Second, the
compatibility can be used as the definition of a category; consideration of
this category as the central object of study in quantum field theory seems to
have quite some advantages over previously introduced theories of the type
``S-matrix theory'', ``Vertex operator algebras'', since this seems to be the
only category in which both the action and the expectation values enter, the
two being linked roughly speaking by a combination of the Frobenius property
and the renormalized Schwinger-Dyson equation.
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The Solution of the Relativistic Schrodinger Equation for the
$δ'$-Function Potential in 1-dimension Using Cutoff Regularization: We study the relativistic version of Schr\"odinger equation for a point
particle in 1-d with potential of the first derivative of the delta function.
The momentum cutoff regularization is used to study the bound state and
scattering states. The initial calculations show that the reciprocal of the
bare coupling constant is ultra-violet divergent, and the resultant expression
cannot be renormalized in the usual sense. Therefore a general procedure has
been developed to derive different physical properties of the system. The
procedure is used first on the non-relativistic case for the purpose of
clarification and comparisons. The results from the relativistic case show that
this system behaves exactly like the delta function potential, which means it
also shares the same features with quantum field theories, like being
asymptotically free, and in the massless limit, it undergoes dimensional
transmutation and it possesses an infrared conformal fixed point.
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String-scale Gauge Coupling Relations in the Supersymmetric Pati-Salam
Models from Intersecting D6-branes: We have constructed all the three-family ${\cal N} = 1$ supersymmetric
Pati-Salam models from intersecting D6-branes, and obtained 33 independent
models in total. But how to realize the string-scale gauge coupling relations
in these models is a big challenge. We first discuss how to decouple the exotic
particles in these models. In addition, we consider the adjoint chiral
mulitplets for $SU(4)_C$ and $SU(2)_L$ gauge symmetries, the Standard Model
(SM) vector-like particles from D6-brane intersections, as well as the
vector-like particles from the ${\cal N}=2$ subsector. We show that the gauge
coupling relations at string scale can be achieved via two-loop renormalization
group equation running for all these supersymmetric Pati-Salam models.
Therefore, we propose a concrete way to obtain the string-scale gauge coupling
realtions for the generic intersecting D-brane models.
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Kink-antikink collisions in the phi^6 model: We study kink-antikink collisions in the one-dimensional non-integrable
scalar phi^6 model. Although the single-kink solutions for this model do not
possess an internal vibrational mode, our simulations reveal a resonant
scattering structure, thereby providing a counterexample to the standard belief
that existence of such a mode is a necessary condition for multi-bounce
resonances in general kink-antikink collisions. We investigate the two-bounce
windows in detail, and present evidence that this structure is caused by
existence of bound states in the spectrum of small oscillations about a
combined kink-antikink configuration.
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Explicit construction of the finite dimensional indecomposable
representations of the simple Lie-Kac $SU(2/1)$ superalgebra and their low
level non diagonal super Casimir operators: All finite dimensional irreducible representations of the simple Lie-Kac
super algebra SU(2/1) are explicitly constructed in the Chevalley basis as
complex matrices. For typical representations, the distinguished Dynkin label
is not quantized. We then construct the generic atypical indecomposable quivers
classified by Marcu, Su and Germoni and typical indecomposable N-generations
block triangular extensions for any irreducible module and any integer N. In
addition to the quadratic and cubic super-Casimir operators $C_2$ and $C_3$,
the supercenter of the enveloping algebra contains a chiral ghost super-Casimir
operator T of mixed order (2,4)in the odd generators, proportional to the
superidentity grading operator $\chi$, and satisfying $T = \chi\;C_2$ and we
define a new factorizable chiral-Casimir $T^-=C_2(1-\chi)/2=(UV+WX)(VU+XW)$
where (U,V,W,X) are the odd generators. In most indecomposable cases, the
super-Casimirs are non diagonal. We compute their pseudo-eigenvalues.
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Logarithmic loop corrections, moduli stabilisation and de Sitter vacua
in string theory: We study string loop corrections to the gravity kinetic terms in type IIB
compactifications on Calabi-Yau threefolds or their orbifold limits, in the
presence of $D7$-branes and orientifold planes. We show that they exhibit in
general a logarithmic behaviour in the large volume limit transverse to the
$D7$-branes, induced by a localised four-dimensional Einstein-Hilbert action
that appears at a lower order in the closed string sector, found in the past.
Here, we compute the coefficient of the logarithmic corrections and use them to
provide an explicit realisation of a mechanism for K\"ahler moduli
stabilisation that we have proposed recently, which does not rely on
non-perturbative effects and lead to de Sitter vacua. Our result avoids no-go
theorems of perturbative stabilisation due to runaway potentials, in a way
similar to the Coleman-Weinberg mechanism, and provides a counter example to
one of the swampland conjectures concerning de Sitter vacua in quantum gravity,
once string loop effects are taken into account; it thus paves the way for
embedding the Standard Model of particle physics and cosmology in string
theory.
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On Perturbation Theory and Critical Exponents for Self-Similar Systems: Gravitational critical collapse in the Einstein-axion-dilaton system is known
to lead to continuous self-similar solutions characterized by the Choptuik
critical exponent $\gamma$. We complete the existing literature on the subject
by computing the linear perturbation equations in the case where the
axion-dilaton system assumes a parabolic form. Next, we solve the perturbation
equations in a newly discovered self-similar solution in the hyperbolic case,
which allows us to extract the Choptuik exponent. Our main result is that this
exponent depends not only on the dimensions of spacetime but also the
particular ansatz and the critical solutions that one started with.
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Riemann surfaces, separation of variables and classical and quantum
integrability: We show that Riemann surfaces, and separated variables immediately provide
classical Poisson commuting Hamiltonians. We show that Baxter's equations for
separated variables immediately provide quantum commuting Hamiltonians. The
construction is simple, general, and does not rely on the Yang--Baxter
equation.
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Confinement in Gauge Theories from the Condensation of World-Sheet
Defects in Liouville String: We present a Liouville-string approach to confinement in four-dimensional
gauge theories, which extends previous approaches to include non-conformal
theories. We consider Liouville field theory on world sheets whose boundaries
are the Wilson loops of gauge theory, which exhibit vortex and spike defects.
We show that world-sheet vortex condensation occurs when the Wilson loop is
embedded in four target space-time dimensions, and show that this corresponds
to the condensation of gauge magnetic monopoles in target space. We also show
that vortex condensation generates a effective string tension corresponding to
the confinement of electric degrees of freedom. The tension is independent of
the string length in a gauge theory whose electric coupling varies
logarithmically with the length scale. The Liouville field is naturally
interpreted as an extra target dimension, with an anti-de-Sitter (AdS)
structure induced by recoil effects on the gauge monopoles, interpreted as D
branes of the effective string theory. Black holes in the bulk AdS space
correspond to world-sheet defects, so that phases of the bulk gravitational
system correspond to the different world-sheet phases, and hence to different
phases of the four-dimensional gauge theory. Deconfinement is associated with a
Berezinskii-Kosterlitz-Thouless transition of vortices on the Wilson-loop world
sheet, corresponding in turn to a phase transition of the black holes in the
bulk AdS space.
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Duality Between Dirac Fermions in Curved Spacetime and Optical solitons
in Non-Linear Schrodinger Model: Magic of $1+1$-Dimensional Bosonization: Bosonization in curved spacetime maps massive Thirring model
(self-interacting Dirac fermions) to a generalized sine-Gordon model, both
living in $1+1$-dimensional curved spacetime. Applying this duality we have
shown that the Thirring model fermion, in non-relativistic limit, gets
identified with the soliton of non-linear Scrodinger model with Kerr form of
non-linearity. We discuss one particular optical soliton in the latter model
and relate it with the Thirring model fermion.
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The Hagedorn temperature in a decoupled sector of AdS/CFT: We match the Hagedorn/deconfinement temperature of planar N=4 super
Yang-Mills (SYM) on R x S^3 to the Hagedorn temperature of string theory on
AdS_5 x S^5. The match is done in a near-critical region where both gauge
theory and string theory are weakly coupled. On the gauge theory side we are
taking a decoupling limit found in hep-th/0605234 in which the physics of
planar N=4 SYM is given exactly by the ferromagnetic XXX_{1/2} Heisenberg spin
chain. We find moreover a general relation between the Hagedorn/deconfinement
temperature and the thermodynamics of the Heisenberg spin chain. On the string
theory side, we identify the dual limit which is taken of string theory on a
maximally symmetric pp-wave background with a flat direction, obtained from a
Penrose limit of AdS_5 x S^5. We compute the Hagedorn temperature of the string
theory and find agreement with the Hagedorn/deconfinement temperature computed
on the gauge theory side. Finally, we discuss a modified decoupling limit in
which planar N=4 SYM reduces to the XXX_{1/2} Heisenberg spin chain with an
external magnetic field.
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Exceptionally simple integrated correlators in $\mathcal{N}=4$
supersymmetric Yang-Mills theory: Supersymmetric localisation has led to several modern developments in the
study of integrated correlators in $\mathcal{N}=4$ supersymmetric Yang-Mills
(SYM) theory. In particular, exact results have been derived for certain
integrated four-point functions of superconformal primary operators in the
stress tensor multiplet valid for all classical gauge groups, $SU(N)$, $SO(N)$,
and $USp(2N)$, and for all values of the complex coupling,
$\tau=\theta/(2\pi)+4\pi i/g^2_{_{YM}}$. In this work we extend this analysis
and provide a unified two-dimensional lattice sum representation for all simple
gauge groups, in particular for the exceptional series $E_r$ (with $r=6,7,8$),
$F_4$ and $G_2$. These expressions are manifestly covariant under
Goddard-Nuyts-Olive duality which for $F_4$ and $G_2$ is given by particular
Fuchsian groups. We show that the perturbation expansion of these integrated
correlators is universal in the sense that it can be written as a single
function of three parameters, called Vogel parameters, and a suitable 't
Hooft-like coupling. To obtain the perturbative expansion for the integrated
correlator with a given gauge group we simply need substituting in this
universal expression specific values for the Vogel parameters. At the
non-perturbative level we conjecture a formula for the one-instanton Nekrasov
partition function with simple gauge group and general $\Omega$-deformation
background. We check that our expression reduces in various limits to known
results and that it produces, via supersymmetric localisation, the same
one-instanton contribution to the integrated correlator as the one derived from
the lattice sum. Finally, we consider the action of the hyperbolic Laplace
operator in $\tau$ on the integrated correlators with exceptional gauge groups
and derive inhomogeneous Laplace equations very similar to the ones previously
obtained for classical gauge groups.
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Frustrating and Diluting Dynamical Lattice Ising Spins: We investigate what happens to the third order ferromagnetic phase transition
displayed by the Ising model on various dynamical planar lattices (ie coupled
to 2D quantum gravity) when we introduce annealed bond disorder in the form of
either antiferromagnetic couplings or null couplings. We also look at the
effect of such disordering for the Ising model on general $\phi^3$ and $\phi^4$
Feynman diagrams.
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Conformal Gravity Redux: Ghost-turned-Tachyon: We analyze conformal gravity in translationally invariant approximation,
where the metric is taken to depend on time but not on spatial coordinates. We
find that the field mode which in perturbation theory has a ghostlike kinetic
term, turns into a tachyon when nonlinear interaction is accounted for. The
kinetic term and potential for this mode have opposite signs. Solutions of
nonlinear classical equations of motion develop a singularity in finite time
determined by the initial conditions.
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Parikh-Wilczek Tunneling from Noncommutative Higher Dimensional Black
Holes: We study tunneling of massless and massive particles through the smeared
quantum horizon of the extra-dimensional Schwarzschild black holes. The
emission rate of the particles' tunneling is modified by noncommutativity
effects in a bulk spacetime of dimension $d$. The issues of information loss
and possible correlations between emitted particles are discussed. We show that
even by considering both noncommutativity and braneworld effects, there is no
correlation between different modes of evaporation at least at late-time and
within approximations used in the calculations. However, incorporation of
quantum gravity effects such as modification of the standard dispersion
relation or generalization of the Heisenberg uncertainty principle, leads to
the correlation between emitted particles. Although time-evolution of these
correlations is not trivial, a part of information coming out of the black hole
can be preserved in these correlations. On the other hand, as a well-known
result of spacetime noncommutativity, a part of information may be preserved in
a stable black hole remnant.
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The symmetry structure of the anti-self-dual Einstein hierarchy: An important example of a multi-dimensional integrable system is the
anti-self-dual Einstein equations. By studying the symmetries of these
equations, a recursion operator is found and the associated hierarchy
constructed. Owing to the properties of the recursion operator one may
construct a hierarchy of symmetries and find the algebra generated by them. In
addition, the Lax pair for this hierarchy is constructed.
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Gravity from Entanglement and RG Flow in a Top-down Approach: The duality between a $d$-dimensional conformal field theory with relevant
deformation and a gravity theory on an asymptotically AdS$_{d+1}$ geometry, has
become a suitable tool in the investigation of the emergence of gravity from
quantum entanglement in field theory. Recently, we have tested the duality
between the mass-deformed ABJM theory and asymptotically AdS$_4$ gravity
theory, which is obtained from the KK reduction of the 11-dimensional
supergravity on the LLM geometry. In this paper, we extend the KK reduction
procedure beyond the linear order and establish non-trivial KK maps between
4-dimensional fields and 11-dimensional fluctuations. We rely on this
gauge/gravity duality to calculate the entanglement entropy by using the
Ryu-Takayanagi holographic formula and the path integral method developed by
Faulkner. We show that the entanglement entropies obtained using these two
methods agree when the asymptotically AdS$_4$ metric satisfies the linearized
Einstein equation with nonvanishing energy-momentum tensor for two scalar
fields. These scalar fields encode the information of the relevant deformation
of the ABJM theory. This confirms that the asymptotic limit of LLM geometry is
the emergent gravity of the quantum entanglement in the mass-deformed ABJM
theory with a small mass parameter. We also comment on the issue of the
relative entropy and the Fisher information in our setup.
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Towards unified theory of $2d$ gravity: We introduce a new 1-matrix model with arbitrary potential and the
matrix-valued background field. Its partition function is a $\tau$-function of
KP-hierarchy, subjected to a kind of ${\cal L}_{-1}$-constraint. Moreover,
partition function behaves smoothly in the limit of infinitely large matrices.
If the potential is equal to $X^{K+1}$, this partition function becomes a
$\tau$-function of $K$-reduced KP-hierarchy, obeying a set of ${\cal W}
_K$-algebra constraints identical to those conjectured in \cite{FKN91} for
double-scaling continuum limit of $(K-1)$-matrix model. In the case of $K=2$
the statement reduces to the early established \cite{MMM91b} relation between
Kontsevich model and the ordinary $2d$ quantum gravity . Kontsevich model with
generic potential may be considered as interpolation between all the models of
$2d$ quantum gravity with $c<1$ preserving the property of integrability and
the analogue of string equation.
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Remarks on QCD$_4$ with fundamental and adjoint matter: We study 4-dimensional SU(N) gauge theory with one adjoint Weyl fermion and
fundamental matter - either bosonic or fermionic. Symmetries, their 't Hooft
anomalies, and the Vafa-Witten-Weingarten theorems strongly constrain the
possible bulk phases. The first part of the paper is dedicated to a single
fundamental fermion or boson. As long as the adjoint Weyl fermion is massless,
this theory always possesses a $\mathbb{Z}_{2N}^\chi$ chiral symmetry, which
breaks spontaneously, supporting $N$ vacua and domain walls between them for a
generic mass of the matter fields. We argue, however, that the domain walls
generically undergo a phase transition, and we establish the corresponding 3d
gauge theories on the walls. The rest of the paper is dedicated to studying the
multi-flavor fundamental matter. Here, the phases crucially depend on the ratio
of the number of colors and the number of fundamental flavors. We also discuss
the limiting scenarios of heavy adjoint and fundamentals, which align neatly
with our current understanding of QCD and $\mathcal{N}=1$ super Yang-Mills
theory.
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The Natural TeV Cutoff of the Higgs Field from the Multiplicative
Lagrangian: The various types of the non-standard Lagrangian can be added to the standard
Lagrangian with the invariant of the equation of motion in the low energy
limit. In this paper, we construct the multiplicative Lagrangian of a complex
scalar field giving the approximated Klein-Gordon equation from the inverse
problem of the calculus of variation. Then, this multiplicative Lagrangian with
arbitrary high cutoff is applied to the toy model of the Higgs mechanism in
U(1)-gauge symmetry in order to study the simple effects in the Higgs physics.
We show that, after spontaneous symmetry breaking happens, the Higgs vev is
free from the Fermi-coupling constant and the Higgs field gets the natural
cutoff in TeV scale. The other relevant coupling constants, the UV-sensitivity
of Higgs mass due to the loop correction, some applications on the strong CP
problem as well as anomalous small fermion mass, and the cosmological constant
problem are also discussed.
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Lectures on entanglement in quantum field theory: These notes grew from a series of lectures given by the authors during the
last decade. They will be published in the proceedings of TASI 2021. After a
brief introduction to quantum information theory tools, they are organized in
four chapters covering the following subjects: Entanglement in quantum field
theory, Irreversibility theorems, Energy-entropy bounds, Entanglement and
symmetries.
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Scalar QED $\hbar$-Corrections to the Coulomb Potential: The leading long-distance 1-loop quantum corrections to the Coulomb potential
are derived for scalar QED and their gauge-independence is explicitly checked.
The potential is obtained from the direct calculation of the 2-particle
scattering amplitude, taking into account all relevant 1-loop diagrams. Our
investigation should be regarded as a first step towards the same programme for
effective Quantum Gravity. In particular, with our calculation in the framework
of scalar QED, we are able to demonstrate the incompleteness of some previous
studies concerning the Quantum Gravity counterpart.
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A Gravity Dual of Localized Tachyon Condensation in Intersecting Branes: The method of probe brane is the powerful one to obtain the effective action
living on the probe brane from supergravity. We apply this method to the
unstable brane systems, and understand the tachyon condensation in the context
of the open/closed duality. First, we probe the parallel coincident branes by
the anti-brane. In this case, the mass squared of the stretched string becomes
negative infinite in the decoupling limit. So that the dual open string field
theory is difficult to understand. Next, we probe parallel coincident branes by
a brane intersecting with an angle. In this case, the stretched strings have
the tachyonic modes localized near the intersecting point, and by taking the
appropriate limit for the intersection angle, we can leave mass squared of this
modes negative finite in the decoupling limit. Then we can obtain the
information about the localized tachyon condensation from the probe brane
action obtained using supergravity.
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Strong coupling regime in two-dimensional large-N scalar quantum
chromodynamics: Two-dimensional large-$N$ quantum chromodynamics with scalar quarks is
considered with particular emphasis on its strong coupling regime which has not
been studied so far. Techniques necessary to deal with the infinitely
oscillatory bound state wave functions in the strong coupling regime are
developed. I derive an estimate for the ground state mass and show that (1) the
lightest hadron in the theory is massless and (2) the ground state mass is
continuous across the transition between the weak and the strong coupling.
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An anthology of non-local QFT and QFT on noncommutative spacetime: Ever since the appearance of renormalization theory there have been several
differently motivated attempts at non-localized (in the sense of not generated
by point-like fields) relativistic particle theories, the most recent one being
at QFT on non-commutative Minkowski spacetime. The often conceptually
uncritical and historically forgetful contemporary approach to these problems
calls for a critical review the light of previous results on this subject.
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Dynamical Symmetry Breaking and Magnetic Confinement in QCD: We present a gauge independent method to construct the effective action of
QCD, and calculate the one loop effective action of $SU(2)$ QCD in an arbitrary
constant background field. Our result establishes the existence of a dynamical
symmetry breaking by demonstrating that the effective potential develops a
unique and stable vacuum made of the monopole condensation in one loop
approximation. This provides a strong evidence for the magnetic confinement of
color through the dual Meissner effect in the non-Abelian gauge theory. The
result is obtained by separating the topological degrees which describe the
non-Abelian monopoles from the dynamical degrees of the gauge potential, and
integrating out all the dynamical degrees of QCD. We present three independent
arguments to support our result.
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Deformations of Closed Strings and Topological Open Membranes: We study deformations of topological closed strings. A well-known example is
the perturbation of a topological closed string by itself, where the
associative OPE product is deformed, and which is governed by the WDVV
equations. Our main interest will be closed strings that arise as the boundary
theory for topological open membranes, where the boundary string is deformed by
the bulk membrane operators. The main example is the topological open membrane
theory with a nonzero 3-form field in the bulk. In this case the Lie bracket of
the current algebra is deformed, leading in general to a correction of the
Jacobi identity. We identify these deformations in terms of deformation theory.
To this end we describe the deformation of the algebraic structure of the
closed string, given by the BRST operator, the associative product and the Lie
bracket. Quite remarkably, we find that there are three classes of deformations
for the closed string, two of which are exemplified by the WDVV theory and the
topological open membrane. The third class remains largely mysterious, as we
have no explicit example.
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Conceptual issues in combining general relativity and quantum theory: Points of conflict between the principles of general relativity and quantum
theory are highlighted. I argue that the current language of QFT is inadequete
to deal with gravity and review attempts to identify some of the features which
are likely to present in the correct theory of quantum gravity.
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The Fundamental Need for a SM Higgs and the Weak Gravity Conjecture: Compactifying the SM down to 3D or 2D one may obtain AdS vacua depending on
the neutrino mass spectrum. It has been recently shown that, by insisting in
the absence of these vacua, as suggested by {\it Weak Gravity Conjecture} (WGC)
arguments, intriguing constraints on the value of the lightest neutrino mass
and the 4D cosmological constant are obtained. For fixed Yukawa coupling one
also obtains an upper bound on the EW scale $\left\langle
H\right\rangle\lesssim {\Lambda_4^{1/4}} /{Y_{\nu_{i}}}$,where $\Lambda_4$ is
the 4D cosmological constant and $Y_{\nu_{i}}$ the Yukawa coupling of the
lightest (Dirac) neutrino. This bound may lead to a reassessment of the gauge
hierarchy problem. In this letter, following the same line of arguments, we
point out that the SM without a Higgs field would give rise to new AdS lower
dimensional vacua. Absence of latter would require the very existence of the SM
Higgs. Furthermore one can derive a lower bound on the Higgs vev $\left\langle
H\right\rangle\gtrsim \Lambda_{\text{QCD}}$ which is required by the absence of
AdS vacua in lower dimensions. The lowest number of quark/lepton generations in
which this need for a Higgs applies is three, giving a justification for family
replication. We also reassess the connection between the EW scale, neutrino
masses and the c.c. in this approach. The EW fine-tuning is here related to the
proximity between the c.c. scale $\Lambda_4^{1/4}$ and the lightest neutrino
mass $m_{\nu_i}$ by the expression $ \frac {\Delta H}{H} \lesssim \frac
{(a\Lambda_4^ {1/4} -m_{\nu_i})} {m_{\nu_i}}. $ We emphasize that all the above
results rely on the assumption of the stability of the AdS SM vacua found.
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Resultants and Gravity Amplitudes: Two very different formulations of the tree-level S-matrix of N=8 Einstein
supergravity in terms of rational maps are known to exist. In both
formulations, the computation of a scattering amplitude of n particles in the k
R-charge sector involves an integral over the moduli space of certain
holomorphic maps of degree d=k-1. In this paper we show that both formulations
can be simplified when written in a manifestly parity invariant form as
integrals over holomorphic maps of bi-degree (d,n-d-2). In one formulation the
full integrand becomes directly the product of the resultants of each of the
two maps defining the one of bi-degree (d,n-d-2). In the second formulation, a
very different structure appears. The integrand contains the determinant of a
(n-3)x(n-3) matrix and a 'Jacobian'. We prove that the determinant is a
polynomial in the coefficients of the maps and contains the two resultants as
factors.
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BRST-antifield Quantization: a Short Review: Most of the known models describing the fundamental interactions have a gauge
freedom. In the standard path integral, it is necessary to "fix the gauge" in
order to avoid integrating over unphysical degrees of freedom. Gauge
independence might then become a tricky issue, especially when the structure of
the gauge symmetries is intricate. In the modern approach to this question, it
is BRST invariance that effectively implements gauge invariance. This set of
lectures briefly reviews some key ideas underlying the BRST-antifield
formalism, which yields a systematic procedure to path-integrate any type of
gauge system, while (usually) manifestly preserving spacetime covariance. The
quantized theory possesses a global invariance under the so-called BRST
transformation, which is nilpotent of order two. The cohomology of the BRST
differential is the central element that controls the physics. Its relationship
with the observables is sketched and explained. How anomalies appear in the
"quantum master equation" of the antifield formalism is also discussed. These
notes are based on lectures given by MH at the 10th Saalburg Summer School on
Modern Theoretical Methods from the 30th of August to the 10th of September,
2004 in Wolfersdorf, Germany and were prepared by AF and AM. The exercises
which were discussed at the school are also included.
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Micromanaging de Sitter holography: We develop tools to engineer de Sitter vacua with semi-holographic duals,
using elliptic fibrations and orientifolds to uplift Freund-Rubin
compactifications with CFT duals. The dual brane construction is compact and
constitutes a microscopic realization of the dS/dS correspondence, realizing
d-dimensional de Sitter space as a warped compactification down to
(d-1)-dimensional de Sitter gravity coupled to a pair of large-N matter
sectors. This provides a parametric microscopic interpretation of the
Gibbons-Hawking entropy. We illustrate these ideas with an explicit class of
examples in three dimensions, and describe ongoing work on four-dimensional
constructions.
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Entropy of Extremal Black Holes in Two Dimension: In this paper we apply the entropy function formalism to the two-dimensional
black hole which come from the compactification of the heterotic string theory
with the dilaton coupling function. We find the Bekenstein-Hawking entropy from
the value of the entropy function at its saddle point. Also we consider higher
derivative terms. After that we apply the entropy function formalism to the
Jackiw-Teitelboim (JT) model where we consider the effect of string-loop to
this model.
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Localizing Gravity on a String-Like Defect in Six Dimensions: We present a metric solution in six dimensions where gravity is localized on
a four-dimensional singular string-like defect. The corrections to
four-dimensional gravity from the bulk continuum modes are suppressed by ${\cal
O}(1/r^3)$. No tuning of the bulk cosmological constant to the brane tension is
required in order to cancel the four-dimensional cosmological constant.
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Second Quantization of the Stueckelberg Relativistic Quantum Theory and
Associated Gauge Fields: The gauge compensation fields induced by the differential operators of the
Stueckelberg-Schr\"odinger equation are discussed, as well as the relation
between these fields and the standard Maxwell fields. An action is constructed
and the second quantization of the fields carried out using a constraint
procedure. Some remarks are made on the properties of the second quantized
matter fields.
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Range of novel black hole phase transitions via massive gravity: Triple
points and N-fold reentrant phase transitions: Massive gravities in anti-de Sitter spacetime can be viewed as effective dual
field theories of different phases of condensed matter systems with broken
translational symmetry such as solids, (perfect) fluids, and liquid crystals.
Motivated by this fact, we explore the black hole chemistry (BHC) of these
theories and find a new range of novel phase transitions close to realistic
ones in ordinary physical systems. We find that the equation of state of
topological black holes (TBHs) at their inflection point(s) in $d$-dimensional
spacetime reduces to a polynomial equation of degree $(d-4)$, which yields up
to $n=(d-4)$ critical points. As a result, for (neutral) TBHs, we observe
triple-point phenomena with the associated first-order phase transitions (in $d
\ge 7$), and a new phenomenon we call an $N$-fold reentrant phase transition,
in which several ($N$) regions of thermodynamic phase space exhibit distinct
reentrant phase transitions, with associated virtual triple points and
zeroth-order phase transitions (in $d \ge 8$), as well as Van der Waals
transitions (in $d \ge 5$) and reentrant (in $d \ge 6$) behavior. We conclude
that BHC in higher-dimensional massive gravity is very likely to offer further
new surprises.
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Looking for event horizons using UV/IR relations: A primary goal in holographic theories of gravity is to study the causal
structure of spacetime from the field theory point of view. This is a
particularly difficult problem when the spacetime has a non-trivial causal
structure, such as a black hole. We attempt to study causality through the
UV/IR relation between field theory and spacetime quantities, which encodes
information about bulk position. We study the UV/IR relations for charged black
hole spacetimes in the AdS/CFT correspondence. We find that the UV/IR relations
have a number of interesting features, but find little information about the
presence of a horizon in the bulk. The scale of Wilson loops is simply related
to radial position, whether there is a horizon or not. For time-dependent
probes, the part of the history near the horizon only effects the late-time
behaviour of field theory observables. Static supergravity probes have a finite
scale size related to radial position in generic black holes, but there is an
interesting logarithmic divergence as the temperature approaches zero.
|
Inviolable energy conditions from entanglement inequalities: Via the AdS/CFT correspondence, fundamental constraints on the entanglement
structure of quantum systems translate to constraints on spacetime geometries
that must be satisfied in any consistent theory of quantum gravity. In this
paper, we investigate such constraints arising from strong subadditivity and
from the positivity and monotonicity of relative entropy in examples with
highly-symmetric spacetimes. Our results may be interpreted as a set of energy
conditions restricting the possible form of the stress-energy tensor in
consistent theories of Einstein gravity coupled to matter.
|
A Layman's Guide to M-theory: The best candidate for a fundamental unified theory of all physical phenomena
is no longer ten-dimensional superstring theory but rather eleven-dimensional
{\it M-theory}. In the words of Fields medalist Edward Witten, ``M stands for
`Magical', `Mystery' or `Membrane', according to taste''. New evidence in favor
of this theory is appearing daily on the internet and represents the most
exciting development in the subject since 1984 when the superstring revolution
first burst on the scene. (Talk delivered at the Abdus Salam Memorial Meeting,
ICTP, Trieste, November 1997.)
|
Mass singularity and confining property in $QED_3$: We discuss the properties of the position space fermion propagator in three
dimensional QED which has been found previouly based on Ward-Takahashi-identity
for soft-photon emission vertex and spectral representation.There is a new type
of mass singularity which governs the long distance behaviour.It leads the
propagator vanish at large distance.This term corresponds to dynamical mass in
position space.Our model shows confining property and dynamical mass generation
for arbitrary coupling constant.Since we used dispersion retation in deriving
spectral function there is a physical mass which sets a mass scale.For finite
cut off we obtain the full propagator in the dispersion integral as a
superposition of different massses.Low energy behaviour of the proagator is
modified to decrease by position dependent mass.In the limit of zero infrared
cut-off the propagator vanishes with a new kind of infrared behaviour.
|
The Renormalization-Group Method Applied to Asymptotic Analysis of
Vector Fields: The renormalization group method of Goldenfeld, Oono and their collaborators
is applied to asymptotic analysis of vector fields. The method is formulated on
the basis of the theory of envelopes, as was done for scalar fields. This
formulation actually completes the discussion of the previous work for scalar
equations. It is shown in a generic way that the method applied to equations
with a bifurcation leads to the Landau-Stuart and the (time-dependent)
Ginzburg-Landau equations. It is confirmed that this method is actually a
powerful theory for the reduction of the dynamics as the reductive perturbation
method is. Some examples for ordinary diferential equations, such as the forced
Duffing, the Lotka-Volterra and the Lorenz equations, are worked out in this
method: The time evolution of the solution of the Lotka-Volterra equation is
explicitly given, while the center manifolds of the Lorenz equation are
constructed in a simple way in the RG method.
|
Absence of isolated critical points with nonstandard critical exponents
in the four-dimensional regularization of Lovelock gravity: Hyperbolic vacuum black holes in Lovelock gravity theories of odd order $N$,
in which $N$ denotes the order of higher-curvature corrections, are known to
have the so-called isolated critical points with nonstandard critical exponents
(as $\alpha = 0$, $\beta = 1$, $\gamma = N-1$, and $\delta = N$), different
from those of mean-field critical exponents (with $\alpha = 0$, $\beta = 1/2$,
$\gamma = 1$, and $\delta = 3$). Motivated by this important observation, here,
we explore the consequences of taking the $D \to 4$ limit of Lovelock gravity
and the possibility of finding nonstandard critical exponents associated with
isolated critical points in four-dimensions by use of the four-dimensional
regularization technique, proposed recently by Glavan and Lin
\cite{Glavan2020}. To do so, we first present $\text{AdS}_4$ Einstein-Lovelock
black holes with fine-tuned Lovelock couplings in the regularized theory, which
is needed for our purpose. Next, it is shown that the regularized $4D$
Einstein-Lovelock gravity theories of odd order $N > 3$ do not possess any
physical isolated critical point, unlike the conventional Lovelock gravity. In
fact, the critical (inflection) points of the equation of state always occur
for the branch of black holes with negative entropy. The situation is quite
different for the case of the regularized $4D$ Einstein-Lovelock gravity with
cubic curvature corrections ($N=3$). In this case ($N=3$), although the entropy
is non-negative and the equation of state of hyperbolic vacuum black holes has
a nonstandard Taylor expansion about its inflection point, but there is no
criticality associated with this special point. At the inflection point, the
physical properties of the black hole system change drastically ...
|
Admissible states in quantum phase space: We address the question of which phase space functionals might represent a
quantum state. We derive necessary and sufficient conditions for both pure and
mixed phase space quantum states. From the pure state quantum condition we
obtain a formula for the momentum correlations of arbitrary order and derive
explicit expressions for the wavefunctions in terms of time dependent and
independent Wigner functions. We show that the pure state quantum condition is
preserved by the Moyal (but not by the classical Liouville) time evolution and
is consistent with a generic stargenvalue equation. As a by-product Baker's
converse construction is generalized both to an arbitrary stargenvalue
equation, associated to a generic phase space symbol, as well as to the time
dependent case. These results are properly extended to the mixed state quantum
condition, which is proved to imply the Heisenberg uncertainty relations.
Globally, this formalism yields the complete characterization of the
kinematical structure of Wigner quantum mechanics. The previous results are
then succinctly generalized for various quasi-distributions. Finally, the
formalism is illustrated through the simple examples of the harmonic oscillator
and the free Gaussian wave packet. As a by-product, we obtain in the former
example an integral representation of the Hermite polynomials.
|
The Effective Potential of the Conformal Factor in Asymptotically Safe
Quantum Gravity: The effective potential of the conformal factor in the effective average
action approach to Quantum Einstein Gravity is discussed. It is shown, without
invoking any truncation or other approximations, that if the theory has has a
non-Gaussian ultraviolet fixed point and is asymptotically safe the potential
has a characteristic behavior near the origin. This behavior might be
observable in numerical simulations.
|
Gauge Independent Phase Structure of Gauged Nambu-Jona-Lasinio and
Yukawa Models: We investigate the critical behavior of the gauged NJL model (QED plus
4-fermion interaction) and the gauged Yukawa model by use of the inversion
method.
By calculating the gauge-invariant chiral condensate in the inversion method
to the lowest order, we derive the critical line which separates the
spontaneous chiral-symmetry breaking phase from the chiral symmetric one. The
critical exponent for the chiral order parameter associated with the second
order chiral phase transition is shown to take the mean-field value together
with possible logarithmic correction to the mean-field prediction.
All the above results are gauge-parameter independent and are compared with
the previous results obtained from the Schwinger-Dyson equation for the fermion
propagator.
|
Local conformal symmetry in non-Riemannian geometry and the origin of
physical scales: We introduce an extension of the Standard Model and General Relativity built
upon the principle of local conformal invariance, which represents a
generalization of a previous work by Bars, Steinhardt and Turok. This is
naturally realized by adopting as a geometric framework a particular class of
non-Riemannian geometries, first studied by Weyl. The gravitational sector is
enriched by a scalar and a vector field. The latter has a geometric origin and
represents the novel feature of our approach. We argue that physical scales
could emerge from a theory with no dimensionful parameters, as a result of the
spontaneous breakdown of conformal and electroweak symmetries. We study the
dynamics of matter fields in this modified gravity theory and show that test
particles follow geodesics of the Levi-Civita connection, thus resolving an old
criticism raised by Einstein against Weyl's original proposal.
|
Consistent description of field theories with non-Hermitian mass terms: We review how to describe a field theory that includes a non-Hermitian mass
term in the region of parameter space where the Lagrangian is $PT$-symmetric.
The discrete symmetries of the system are essential for understanding the
consistency of the model, and the link between conserved current and variation
of the Lagrangian has to be revisited in the case of continuous symmetries.
|
The Space-Time Manifold as a Critical Solid: It is argued that the problems of the cosmological constant, stability and
renormalizability of quantum gravity can be solved if the space-time manifold
arises through spontaneous symmetry breaking. A ``pre-manifold" model is
presented in which many points are connected by random bonds. A set of $D$ real
numbers assigned to each point are coupled between points connected by bonds.
It is then found that the dominant configuration of bonds is a flat
$D$-dimensional manifold, on which there is a massless matter field.
Long-wavelength fluctuations can describe quantized massless gravity if
$D=\;4$, $6$, $8...$.
|
Small Q balls: We develop an adequate description of non-topological solitons with a small
charge, for which the thin-wall approximation is not valid. There is no
classical lower limit on the charge of a stable Q-ball. We examine the
parameters of these small-charge solitons and discuss the limits of
applicability of the semiclassical approximation.
|
Fusion algebra and Verlinde's formula: We show that the coefficients of decomposition into an irreducible components
of the tensor powers of level $r$ symmetric algebra of adjoint representation
coincide with the Verlinder numbers. Also we construct (for $sl(2)) the
representations of a general linear group those dimensions are given by
corresponding Verlinde's numbers.
|
From confinement to adjoint zero-modes: Starting from our proposed model of the Yang-Mills vacuum based on fractional
instantons, we review the intellectual itinerary which has guided part of our
scientific activity up to our recent work on adjoint zero-modes for calorons.
|
Berry Phase, Lorentz Covariance, and Anomalous Velocity for Dirac and
Weyl Particles: We consider the relation between spin and the Berry-phase contribution to the
anomalous velocity of massive and massless Dirac particles. We extend the Berry
connection that depends only on the spatial components of the particle momentum
to one that depends on the the space and time components in a covariant manner.
We show that this covariant Berry connection captures the Thomas-precession
part of the Bargmann-Michel-Telegdi spin evolution, and contrast it with the
traditional (unitary, but not naturally covariant) Berry connection that
describes spin-orbit coupling. We then consider how the covariant connection
enters the classical relativistic dynamics of spinning particles due to
Mathisson, Papapetrou and Dixon. We discuss the problems that arise with
Lorentz covariance in the massless case, and trace them mathematically to a
failure of the Wigner-translation part of the massless-particle little group to
be an exact gauge symmetry in the presence of interactions, and physically to
the fact that the measured position of a massless spinning particle is
necessarily observer dependent.
|
Antisymmetric tensor matter fields in a curved space-time: An analysis about the antisymmetric tensor matter fields Avdeev-Chizhov
theory in a curved space-time is performed. We show, in a curved spacetime,
that the Avdeev-Chizhov theory can be seen as a kind of a $\lambda\phi^{4}$
theory for a "complex self-dual" field. This relationship between
Avdeev-Chizhov theory and $\lambda\phi^{4}$ theory simplify the study of tensor
matter fields in a curved space-time. The energy-momentum tensor for matter
fields is computed.
|
On the holomorphic factorization for superconformal fields: For a generic value of the central charge, we prove the holomorphic
factorization of partition functions for free superconformal fields which are
defined on a compact Riemann surface without boundary. The partition functions
are viewed as functionals of the Beltrami coefficients and their fermionic
partners which variables parametrize superconformal classes of metrics.
|
Oscillators from nonlinear realizations: We construct the systems of the harmonic and Pais-Uhlenbeck oscillators,
which are invariant with respect to arbitrary noncompact Lie algebras. The
equations of motion of these systems can be obtained with the help of the
formalism of nonlinear realizations. We prove that it is always possible to
choose time and the fields within this formalism in such a way that the
equations of motion become linear and, therefore, reduce to ones of ordinary
harmonic and Pais-Uhlenbeck oscillators. The first-order actions, that produce
these equations, can also be provided. As particular examples of this
construction, we discuss the $so(2,3)$ and $G_{2(2)}$ algebras.
|
Radiative Classical Gravitational Observables at $\mathcal O(G^3)$ from
Scattering Amplitudes: We compute classical gravitational observables for the scattering of two
spinless black holes in general relativity and $\mathcal N {=} 8$ supergravity
in the formalism of Kosower, Maybee, and O'Connell (KMOC). We focus on the
gravitational impulse with radiation reaction and the radiated momentum in
black hole scattering at $\mathcal O(G^3)$ to all orders in the velocity. These
classical observables require the construction and evaluation of certain
loop-level quantities which are greatly simplified by harnessing recent
advances from scattering amplitudes and collider physics. In particular, we
make use of generalized unitarity to construct the relevant loop integrands,
employ reverse unitarity, the method of regions, integration-by-parts (IBP),
and (canonical) differential equations to simplify and evaluate all loop and
phase-space integrals to obtain the classical gravitational observables of
interest to two-loop order. The KMOC formalism naturally incorporates radiation
effects which enables us to explore these classical quantities beyond the
conservative two-body dynamics. From the impulse and the radiated momentum, we
extract the scattering angle and the radiated energy. Finally, we discuss
universality of the impulse in the high-energy limit and the relation to the
eikonal phase.
|
Generating Geodesic Flows and Supergravity Solutions: We consider the geodesic motion on the symmetric moduli spaces that arise
after timelike and spacelike reductions of supergravity theories. The geodesics
correspond to timelike respectively spacelike $p$-brane solutions when they are
lifted over a $p$-dimensional flat space. In particular, we consider the
problem of constructing \emph{the minimal generating solution}: A geodesic with
the minimal number of free parameters such that all other geodesics are
generated through isometries. We give an intrinsic characterization of this
solution in a wide class of orbits for various supergravities in different
dimensions. We apply our method to three cases: (i) Einstein vacuum solutions,
(ii) extreme and non-extreme D=4 black holes in N=8 supergravity and their
relation to N=2 STU black holes and (iii) Euclidean wormholes in $D\geq 3$. In
case (iii) we present an easy and general criterium for the existence of
regular wormholes for a given scalar coset.
|
The Analytic Bootstrap in Fermionic CFTs: We apply the method of the large spin bootstrap to analyse fermionic
conformal field theories with weakly broken higher spin symmetry. Through the
study of correlators of composite operators, we find the anomalous dimensions
and OPE coefficients in the Gross-Neveu model in $d=2+\varepsilon$ dimensions
and the Gross-Neveu-Yukawa model in $d=4-\varepsilon$ dimensions, based only on
crossing symmetry. Furthermore a non-trivial solution in the $d=2+\varepsilon$
expansion is found for a fermionic theory in which the fundamental field is not
part of the spectrum. The results are perturbative in $\varepsilon$ and valid
to all orders in the spin, reproducing known results for operator dimensions
and providing some new results for operator dimensions and OPE coefficients.
|
Stringy effects in black hole decay: We compute the low energy decay rates of near-extremal three(four) charge
black holes in five(four) dimensional N=4 string theory to sub-leading order in
the large charge approximation. This involves studying stringy corrections to
scattering amplitudes of a scalar field off a black hole. We adapt and use
recently developed techniques to compute such amplitudes as near-horizon
quantities. We then compare this with the corresponding calculation in the
microscopic configuration carrying the same charges as the black hole. We find
perfect agreement between the microscopic and macroscopic calculations; in the
cases we study, the zero energy limit of the scattering cross section is equal
to four times the Wald entropy of the black hole.
|
Large-N Collective Field Theory Applied to Anyons in Magnetic Fields: We present a large-$N$ collective field formalism for anyons in external
magnetic fields interacting with arbitrary two-body potential. We discuss how
the Landau level is reproduced in our framework. We apply it to the soluble
model for anyons proposed by Girvin et al., and obtain the dispersion relation
of collective modes for arbitrary statistical parameters.
|
Some Aspects of Scattering in (Non) Commutative Gauge Theories: We study almost-forward scattering in the context of usual and
non-commutative QED. We study the semi-classical behaviour of particles
undergoing this scattering process in the two theories, and show that the shock
wave picture, effective in QED fails for NCQED. Further, we show that whereas
in QED, there are no leading logarithmic contributions to the amplitude upto
sixth order, uncancelled logarithms appear in NCQED.
|
Phase transitions in Wilson loop correlator from integrability in global
AdS: We directly compute Wilson loop/Wilson loop correlators on ${\mathbb R}\times
$S$^3$ in AdS/CFT by constructing space-like minimal surfaces that connect two
space-like circular contours on the boundary of global AdS that are separated
by a space-like interval. We compare these minimal surfaces to the disconnected
"double cap" solutions both to regulate the area, and show when the
connected/disconnected solution is preferred. We find that for sufficiently
large Wilson loops no transition occurs because the Wilson loops cannot be
sufficiently separated on the sphere. This may be considered an effect similar
to the Hawking-Page transition: the size of the sphere introduces a new scale
into the problem, and so one can expect phase transitions to depend on this
data. To construct the minimal area solutions, we employ a reduction a la
Arutyunov-Russo-Tseytlin (used by them for spinning strings), and rely on the
integrability of the reduced set of equations to write explicit results.
|
Analytic structure of the $n = 7$ scattering amplitude in
$\mathcal{N}=4$ SYM theory in multi-Regge kinematics: Conformal Regge cut
contribution: In this second part of our investigation of the analytic structure of the
$2\to5$ scattering amplitude in the planar limit of $\mathcal{N}=4$ SYM in
multi-Regge kinematics we compute, in all kinematic regions, the Regge cut
contributions in leading order. The results are infrared finite and conformally
invariant.
|
The space-time symmetry group of a spin 1/2 elementary particle: The space-time symmetry group of a model of a relativistic spin 1/2
elementary particle, which satisfies Dirac's equation when quantized, is
analyzed. It is shown that this group, larger than the Poincare group, also
contains space-time dilations and local rotations. It has two Casimir
operators, one is the spin and the other is the spin projection on the body
frame. Its similarities with the standard model are discussed. If we consider
this last spin observable as describing isospin, then, this Dirac particle
represents a massive system of spin 1/2 and isospin 1/2. There are two possible
irreducible representations of this kind of particles, a colourless or a
coloured one, where the colour observable is also another spin contribution
related to the zitterbewegung. It is the spin, with its twofold structure, the
only intrinsic property of this Dirac elementary particle.
|
Klein Bottles and Simple Currents: The standard Klein bottle coefficient in the construction of open descendants
is shown to equal the Frobenius-Schur indicator of a conformal field theory.
Other consistent Klein bottle projections are shown to correspond to simple
currents. These observations enable us to generalize the standard open string
construction from C-diagonal parent theories to include non-standard Klein
bottles. Using (generalizations of) the Frobenius-Schur indicator we prove
positivity and integrality of the resulting open and closed string state
multiplicities for standard as well as non-standard Klein bottles.
|
Non-anticommutative chiral singlet deformation of N=(1,1) gauge theory: We study the SO(4)x SU(2) invariant Q-deformation of Euclidean N=(1,1) gauge
theories in the harmonic superspace formulation. This deformation preserves
chirality and Grassmann harmonic analyticity but breaks N=(1,1) to N=(1,0)
supersymmetry. The action of the deformed gauge theory is an integral over the
chiral superspace, and only the purely chiral part of the covariant superfield
strength contributes to it. We give the component form of the N=(1,0)
supersymmetric action for the gauge groups U(1) and U(n>1). In the U(1) and
U(2) cases, we find the explicit nonlinear field redefinition (Seiberg-Witten
map) relating the deformed N=(1,1) gauge multiplet to the undeformed one. This
map exists in the general U(n) case as well, and we use this fact to argue that
the deformed U(n) gauge theory can be nonlinearly reduced to a theory with the
gauge group SU(n).
|
Chiral Topologically Massive Gravity and Extremal B-F Scalars: At a critical ``chiral'' coupling, topologically massive gravity with a
negative cosmological constant exhibits several unusual features, including the
emergence of a new logarithmic branch of solutions and a linearization
instability for certain boundary conditions. I show that at this coupling, the
linearized theory may be parametrized by a free scalar field at the
Breitenlohner-Freedman bound, and use this description to investigate these
features.
|
New RG Fixed Points and Duality in Supersymmetric SP(N_c) and SO(N_c)
Gauge Theories: We present evidence for new, non-trivial RG fixed points with dual magnetic
descriptions in $N=1$ supersymmetric $SP(N_c)$ and $SO(N_c)$ gauge theories.
The $SP(N_c)$ case involves matter $X$ in the antisymmetric tensor
representation and $N_f$ flavors of quarks $Q$ in the fundamental
representation. The $SO(N_c)$ case involves matter $X$ in the symmetric tensor
representation and $N_f$ flavors of quarks $Q$ in the vector representation of
$SO(N_c)$. Perturbing these theories by superpotentials $W(X)$, we find a
variety of interesting RG fixed points with dual descriptions. The duality in
these theories is similar to that found by Kutasov and by Kutasov and Schwimmer
in $SU(N_c)$ with adjoint $X$ and $N_f$ quarks in the fundamental.
|
Complexification of Gauge Theories: For the case of a first-class constrained system with an equivariant momentum
map, we study the conditions under which the double process of reducing to the
constraint surface and dividing out by the group of gauge transformations $G$
is equivalent to the single process of dividing out the initial phase space by
the complexification $G_C$ of $G$. For the particular case of a phase space
action that is the lift of a configuration space action, conditions are found
under which, in finite dimensions, the physical phase space of a gauge system
with first-class constraints is diffeomorphic to a manifold imbedded in the
physical configuration space of the complexified gauge system. Similar
conditions are shown to hold in the infinite-dimensional example of Yang-Mills
theories. As a physical application we discuss the adequateness of using
holomorphic Wilson loop variables as (generalized) global coordinates on the
physical phase space of Yang-Mills theory.
|
M-Strings: M2 branes suspended between adjacent parallel M5 branes lead to light
strings, the `M-strings'. In this paper we compute the elliptic genus of
M-strings, twisted by maximally allowed symmetries that preserve 2d (2,0)
supersymmetry. In a codimension one subspace of parameters this reduces to the
elliptic genus of the (4,4) supersymmetric A_{n-1} quiver theory in 2d. We
contrast the elliptic genus of N M-strings with the (4,4) sigma model on the
N-fold symmetric product of R^4. For N=1 they are the same, but for N>1 they
are close, but not identical. Instead the elliptic genus of (4,4) N M-strings
is the same as the elliptic genus of (4,0) sigma models on the N-fold symmetric
product of R^4, but where the right-moving fermions couple to a modification of
the tangent bundle. This construction arises from a dual A_{n-1} quiver 6d
gauge theory with U(1) gauge groups. Moreover we compute the elliptic genus of
domain walls which separate different numbers of M2 branes on the two sides of
the wall.
|
Beyond the String Genus: In an earlier work we used a path integral analysis to propose a higher genus
generalization of the elliptic genus. We found a cobordism invariant
parametrized by Teichmuller space. Here we simplify the formula and study the
behavior of our invariant under the action of the mapping class group of the
Riemann surface. We find that our invariant is a modular function with
multiplier just as in genus one.
|
Infinite Dimensional Geometry and Quantum Field Theory of Strings. II.
Infinite Dimensional Noncommutative Geometry of a Self-Interacting String
Field: A geometric interpretation of quantum self-interacting string field theory is
given. Relations between various approaches to the second quantization of an
interacting string are described in terms of the geometric quantization. An
algorithm to construct a quantum nonperturbative interacting string field
theory in the quantum group formalism is proposed. problems of a metric
background (in)dependence are discussed.
|
Boosting some type--D metrics: We are presenting a general solution to the classical
Einstein--Maxwell--dilaton--axion equations starting from a metric of type--D.
Namely, this stringy solution is the result of a transformation on a general
vacuum type--D solution to the Einstein's equations which was studied in detail
some years ago.
|
EFT and the SUSY Index on the 2nd Sheet: The counting of BPS states in four-dimensional ${\cal N}=1$ theories has
attracted a lot of attention in recent years. For superconformal theories,
these states are in one-to-one correspondence with local operators in various
short representations. The generating function for this counting problem has
branch cuts and hence several Cardy-like limits, which are analogous to
high-temperature limits. Particularly interesting is the second sheet, which
has been shown to capture the microstates and phases of supersymmetric black
holes in AdS$_5$. Here we present a 3d Effective Field Theory (EFT) approach to
the high-temperature limit on the second sheet. We use the EFT to derive the
behavior of the index at orders $\beta^{-2},\beta^{-1},\beta^0$. We also make a
conjecture for $O(\beta)$, where we argue that the expansion truncates up to
exponentially small corrections. An important point is the existence of vector
multiplet zero modes, unaccompanied by massless matter fields. The runaway of
Affleck-Harvey-Witten is however avoided by a non-perturbative confinement
mechanism. This confinement mechanism guarantees that our results are robust.
|
N=1/2 gauge theory and its instanton moduli space from open strings in
R-R background: We derive the four dimensional N=1/2 super Yang-Mills theory from tree-level
computations in RNS open string theory with insertions of closed string
Ramond-Ramond vertices. We also study instanton configurations in this gauge
theory and their ADHM moduli space, using systems of D3 and D(-1) branes in a
R-R background.
|
Residual gauge symmetry in light-cone electromagnetism: We analyze the residual gauge freedom in light-cone electromagnetism in four
dimensions. The standard boundary conditions involved in the so-called $lc_2$
formalism, which contains only the two physical degrees of freedom, allow for a
subset of residual gauge transformations. We relax the boundary conditions
imposed on the fields in order to obtain all the residual gauge
transformations. We compute the canonical generators for Poincar\'e and gauge
transformations with these relaxed boundary conditions. This enables us to
distinguish between the trivial (proper) and large (improper) gauge
transformations in light-cone electromagnetism. We then employ the
Newman-Penrose formalism to identify the incoming and outgoing radiation
fields. We comment on the quadratic form structure of light-cone Hamiltonians,
often encountered in $lc_2$ gauge theories.
|
Noncommuting Momenta of Topological Solitons: We show that momentum operators of a topological soliton may not commute
among themselves when the soliton is associated with the second cohomology
$H^2$ of the target space. The commutation relation is proportional to the
winding number, taking a constant value within each topological sector. The
noncommutativity makes it impossible to specify the momentum of a topological
soliton, and induces a Magnus force.
|
Toward a 3d Ising model with a weakly-coupled string theory dual: It has long been expected that the 3d Ising model can be thought of as a
string theory, where one interprets the domain walls that separate up spins
from down spins as two-dimensional string worldsheets. The usual Ising
Hamiltonian measures the area of these domain walls. This theory has string
coupling of unit magnitude. We add new local terms to the Ising Hamiltonian
that further weight each spin configuration by a factor depending on the genus
of the corresponding domain wall, resulting in a new 3d Ising model that has a
tunable bare string coupling $g_s$. We use a combination of analytical and
numerical methods to analyze the phase structure of this model as $g_s$ is
varied. We study statistical properties of the topology of worldsheets and
discuss the prospects of using this new deformation at weak string coupling to
find a worldsheet description of the 3d Ising transition.
|
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