Showing papers in "Journal of Physics A in 2013"

Constraining conformal field theories with a higher spin symmetry

Abstract: We study the constraints imposed by the existence of a single higher spin conserved current on a three-dimensional conformal field theory (CFT). A single higher spin conserved current implies the existence of an infinite number of higher spin conserved currents. The correlation functions of the stress tensor and the conserved currents are then shown to be equal to those of a free field theory. Namely a theory of N free bosons or free fermions. This is an extension of the Coleman–Mandula theorem to CFT’s, which do not have a conventional S-matrix. We also briefly discuss the case where the higher spin symmetries are ‘slightly’ broken.This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to ‘Higher spin theories and holography’.

Black holes in three dimensional higher spin gravity: a review

Abstract: We review recent progress in the construction of black holes in three dimensional higher spin gravity theories. Starting from spin-3 gravity and working our way toward the theory of an infinite tower of higher spins coupled to matter, we show how to harness higher spin gauge invariance to consistently generalize familiar notions of black holes. We review the construction of black holes with conserved higher spin charges and the computation of their partition functions to leading asymptotic order. In view of the anti-de Sitter/conformal field theory (CFT) correspondence as applied to certain vector-like conformal field theories with extended conformal symmetry, we successfully compare to CFT calculations in a generalized Cardy regime. A brief recollection of pertinent aspects of ordinary gravity is also given.This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to ‘Higher spin theories and holography’.

Minimal model holography

Abstract: We review the duality relating 2D WN minimal model conformal field theories, in a large-N ’t Hooft like limit, to higher spin gravitational theories on AdS3. This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to ‘Higher spin theories and holography’.

Biaxially symmetric solutions to 4D higher spin gravity

Abstract: We review some aspects of biaxially symmetric solutions to Vasiliev?s equations in four-dimensional spacetime with a negative cosmological constant. The solutions, which activate bosonic fields of all spins, are constructed using gauge functions, projectors and deformed oscillators. The deformation parameters, which are formally gauge invariant, are related to generalized electric and magnetic charges in asymptotic weak-field regions. Alternatively, the solutions can be characterized in a dual fashion using 0-form charges which are higher spin Casimir invariants built from combinations of curvatures and all their derivatives that are constant on shell and well-defined everywhere.This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to ?Higher spin theories and holography?.

The higher spin/vector model duality

Abstract: This paper is mainly a review of the dualities between Vasiliev’s higher spin gauge theories in AdS4 and three dimensional large N vector models, with focus on the holographic calculation of correlation functions of higher spin currents. We also present some new results in the computation of parity odd structures in the three point functions in parity violating Vasiliev theories.This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to ‘Higher spin theories and holography’.

Classical and quantum superintegrability with applications

Abstract: A superintegrable system is, roughly speaking, a system that allows more integrals of motion than degrees of freedom. This review is devoted to finite dimensional classical and quantum superintegrable systems with scalar potentials and integrals of motion that are polynomials in the momenta. We present a classification of second-order superintegrable systems in two-dimensional Riemannian and pseudo-Riemannian spaces. It is based on the study of the quadratic algebras of the integrals of motion and on the equivalence of different systems under coupling constant metamorphosis. The determining equations for the existence of integrals of motion of arbitrary order in real Euclidean space E2 are presented and partially solved for the case of third-order integrals. A systematic exposition is given of systems in two and higher dimensional space that allow integrals of arbitrary order. The algebras of integrals of motions are not necessarily quadratic but close polynomially or rationally. The relation between superintegrability and the classification of orthogonal polynomials is analyzed.

ABJ triality: from higher spin fields to strings

Abstract: We demonstrate that a supersymmetric and parity violating version of Vasiliev’s higher spin gauge theory in AdS4 admits boundary conditions that preserve or 6 supersymmetries. In particular, we argue that the Vasiliev theory with U(M) Chan–Paton and boundary condition is holographically dual to the 2+1 dimensional U(N)k × U(M)−k ABJ theory in the limit of large N, k and finite M. In this system all bulk higher spin fields transform in the adjoint of the U(M) gauge group, whose bulk t’Hooft coupling is M/N. Analysis of boundary conditions in Vasiliev theory allows us to determine exact relations between the parity breaking phase of Vasiliev theory and the coefficients of two and three point functions in Chern–Simons vector models at large N. Our picture suggests that the supersymmetric Vasiliev theory can be obtained as a limit of type IIA string theory in AdS, and that the non-Abelian Vasiliev theory at strong bulk ’t Hooft coupling smoothly turn into a string field theory. The fundamental string is a singlet bound state of Vasiliev’s higher spin particles held together by U(M) gauge interactions. This is illustrated by the thermal partition function of free ABJ theory on a two sphere at large M and N even in the analytically tractable free limit. In this system the traces or strings of the low temperature phase break up into their Vasiliev particulate constituents at a U(M) deconfinement phase transition of order unity. At a higher temperature of order Vasiliev’s higher spin fields themselves break up into more elementary constituents at a U(N) deconfinement temperature, in a process described in the bulk as black hole nucleation.This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to ‘Higher spin theories and holography’.

Motivic Multiple Zeta Values and Superstring Amplitudes

Abstract: The structure of tree-level open and closed superstring amplitudes is analyzed. For the open superstring amplitude we find a striking and elegant form, which allows one to disentangle its α′-expansion into several contributions accounting for different classes of multiple zeta values. This form is bolstered by the decomposition of motivic multiple zeta values, i.e. the latter encapsulate the α′-expansion of the superstring amplitude. Moreover, a morphism induced by the coproduct maps the α′-expansion onto a non-commutative Hopf algebra. This map represents a generalization of the symbol of a transcendental function. In terms of elements of this Hopf algebra the α′-expansion assumes a very simple and symmetric form, which carries all the relevant information. Equipped with these results we can also cast the closed superstring amplitude into a very elegant form.

Dynamics of rogue waves in the Davey–Stewartson II equation

Abstract: General rogue waves in the Davey–Stewartson (DS)II equation are derived by the bilinear method, and the solutions are given through determinants. It is shown that the simplest (fundamental) rogue waves are line rogue waves which arise from the constant background in a line profile and then retreat back to the constant background again. It is also shown that multi-rogue waves describe the interaction between several fundamental rogue waves, and higher order rogue waves exhibit different dynamics (such as rising from the constant background but not retreating back to it). Under certain parameter conditions, these rogue waves can blow up to infinity in finite time at isolated spatial points, i.e. exploding rogue waves exist in the DSII equation.

Energy–time uncertainty relation for driven quantum systems

Abstract: We derive generalizations of the energy–time uncertainty relation for driven quantum systems. Using a geometric approach based on the Bures length between mixed quantum states, we obtain explicit expressions for the quantum speed limit time, valid for arbitrary initial and final quantum states and arbitrary unitary driving protocols. Our results establish the fundamental limit on the rate of evolution of closed quantum systems.

Holography, unfolding and higher spin theory

Abstract: Holographic duality is argued to relate classes of models that have equivalent unfolded formulation, hence exhibiting different space-time visualizations for the same theory. This general phenomenon is illustrated by the AdS4 higher spin gauge theory shown to be dual to the theory of 3d conformal currents of all spins interacting with 3d conformal higher spin fields of Chern?Simons type. Generally, the resulting 3d boundary conformal theory is nonlinear, providing an interacting version of the 3d boundary sigma model conjectured by Klebanov and Polyakov to be dual to the AdS4 higher spin theory in the large N limit. Being a gauge theory, it escapes the conditions of the theorem of Maldacena and Zhiboedov, which force a 3d boundary conformal theory to be free. Two reductions of particular higher spin gauge theories where boundary higher spin gauge fields decouple from the currents and which have free-boundary duals are identified. Higher spin holographic duality is also discussed for the cases of AdS3/CFT2 and duality between higher spin theories and nonrelativistic quantum mechanics. In the latter case, it is shown in particular that (dS) AdS geometry in the higher spin setup is dual to the (inverted) harmonic potential in the quantum-mechanical setup.This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to ?Higher spin theories and holography?.

The role of the Riemann–Silberstein vector in classical and quantum theories of electromagnetism

Abstract: It is shown that the use of the Riemann–Silberstein (RS) vector greatly simplifies the description of the electromagnetic field both in the classical domain and in the quantum domain. In this review, we describe many specific examples where this vector enables one to significantly shorten the derivations and make them more transparent. We also argue why the RS vector may be considered as the best possible choice for the photon wavefunction.

Logarithmic conformal field theory: beyond an introduction

Abstract: This article aims to review a selection of central topics and examples in logarithmic conformal field theory. It begins with the remarkable observation of Cardy that the horizontal crossing probability of critical percolation may be computed analytically within the formalism of boundary conformal field theory. Cardy’s derivation relies on certain implicit assumptions which are shown to lead inexorably to indecomposable modules and logarithmic singularities in correlators. For this, a short introduction to the fusion algorithm of Nahm, Gaberdiel and Kausch is provided. While the percolation logarithmic conformal field theory is still not completely understood, there are several examples for which the formalism familiar from rational conformal field theory, including bulk partition functions, correlation functions, modular transformations, fusion rules and the Verlinde formula, has been successfully generalized. This is illustrated for three examples: the singlet model , related to the triplet model , symplectic fermions and the fermionic bc ghost system; the fractional level Wess–Zumino–Witten model based on at , related to the bosonic βγ ghost system; and the Wess–Zumino–Witten model for the Lie supergroup , related to at and 1, the Bershadsky–Polyakov algebra and the Feigin–Semikhatov algebras . These examples have been chosen because they represent the most accessible, and most useful, members of the three best-understood families of logarithmic conformal field theories. The logarithmic minimal models , the fractional level Wess–Zumino–Witten models, and the Wess–Zumino–Witten models on Lie supergroups (excluding ). In this review, the emphasis lies on the representation theory of the underlying chiral algebra and the modular data pertaining to the characters of the representations. Each of the archetypal logarithmic conformal field theories is studied here by first determining its irreducible spectrum, which turns out to be continuous, as well as a selection of natural reducible, but indecomposable, modules. This is followed by a detailed description of how to obtain character formulae for each irreducible, a derivation of the action of the modular group on the characters, and an application of the Verlinde formula to compute the Grothendieck fusion rules. In each case, the (genuine) fusion rules are known, so comparisons can be made and favourable conclusions drawn. In addition, each example admits an infinite set of simple currents, hence extended symmetry algebras may be constructed and a series of bulk modular invariants computed. The spectrum of such an extended theory is typically discrete and this is how the triplet model arises, for example. Moreover, simple current technology admits a derivation of the extended algebra fusion rules from those of its continuous parent theory. Finally, each example is concluded by a brief description of the computation of some bulk correlators, a discussion of the structure of the bulk state space, and remarks concerning more advanced developments and generalizations. The final part gives a very short account of the theory of staggered modules, the (simplest class of) representations that are responsible for the logarithmic singularities that distinguish logarithmic theories from their rational cousins. These modules are discussed in a generality suitable to encompass all the examples met in this review and some of the very basic structure theory is proven. Then, the important quantities known as logarithmic couplings are reviewed for Virasoro staggered modules and their role as fundamentally important parameters, akin to the three-point constants of rational conformal field theory, is discussed. An appendix is also provided in order to introduce some of the necessary, but perhaps unfamiliar, language of homological algebra.

Some results on the mutual information of disjoint regions in higher dimensions

Abstract: We consider the mutual Renyi information of disjoint compact spatial regions A and B in the ground state of a d + 1-dimensional conformal field theory (CFT), in the limit when the separation r between A and B is much greater than their sizes RA, B. We show that in general , where α is the smallest sum of the scaling dimensions of operators whose product has the quantum numbers of the vacuum, and the constants depend only on the shape of the regions and universal data of the CFT. For a free massless scalar field, where α = d − 1, we show that is proportional to the capacitance of a thin conducting slab in the shape of A in d + 1-dimensional electrostatics, and give explicit formulae for this when A is the interior of a sphere Sd − 1 or an ellipsoid. For spherical regions in d = 2 and 3 we obtain explicit results for C(n) for all n and hence for the leading term in the mutual information by taking n → 1. We also compute a universal logarithmic correction to the area law for the Renyi entropies of a single spherical region for a scalar field theory with a small mass.

Notes on Strings and Higher Spins

Abstract: This review is devoted to the intriguing and still largely unexplored links between string theory and higher spins, the types of excitations that lie behind their most cherished properties. A closer look at higher spin fields provides some further clues that string theory describes a broken phase of a higher spin gauge theory. Conversely, string amplitudes contain a wealth of information on higher spin interactions that can clarify long-standing issues related to their infrared behavior.This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to ‘Higher spin theories and holography’.

How instanton combinatorics solves Painlevé VI, V and IIIs

Abstract: We elaborate on a recently conjectured relation of Painlev? transcendents and 2D conformal field theory. General solutions of Painlev? VI, V and III are expressed in terms of c = 1 conformal blocks and their irregular limits, AGT related to instanton partition functions in supersymmetric gauge theories with Nf = 0, 1, 2, 3, 4. The resulting combinatorial series representations of Painlev? functions provide an efficient tool for their numerical computation at finite values of the argument. The series involves sums over bipartitions which, in the simplest cases, coincide with Gessel expansions of certain Toeplitz determinants. Considered applications include Fredholm determinants of classical integrable kernels, scaled gap probability in the bulk of the Gaussian Unitary Ensemble, and all-order conformal perturbation theory expansions of correlation functions in the sine?Gordon field theory at the free-fermion point.

Singular value correlation functions for products of Wishart random matrices

Abstract: We consider the product of M quadratic random matrices with complex elements and no further symmetry, where all matrix elements of each factor have a Gaussian distribution. This generalizes the classical Wishart-Laguerre Gaussian unitary ensemble with M = 1. In this paper, we first compute the joint probability distribution for the singular values of the product matrix when the matrix size N and the number M are fixed but arbitrary. This leads to a determinantal point process which can be realized in two different ways. First, it can be written as a one-matrix singular value model with a nonstandard Jacobian, or second, for M >= 2, as a two-matrix singular value model with a set of auxiliary singular values and a weight proportional to the Meijer G-function. For both formulations, we determine all singular value correlation functions in terms of the kernels of biorthogonal polynomials which we explicitly construct. They are given in terms of the hypergeometric and Meijer G-functions, generalizing the Laguerre polynomials for M = 1. Our investigation was motivated from applications in telecommunication of multi-layered scattering multiple-input and multiple-output channels. We present the ergodic mutual information for finite-N for such a channel model with M - 1 layers of scatterers as an example.

Majorization entropic uncertainty relations

Abstract: Entropic uncertainty relations in a finite-dimensional Hilbert space are investigated. Making use of the majorization technique we derive explicit lower bounds for the sum of Renyi entropies describing probability distributions associated with a given pure state expanded in eigenbases of two observables. Obtained bounds are expressed in terms of the largest singular values of submatrices of the unitary rotation matrix. Numerical simulations show that for a generic unitary matrix of size N = 5, our bound is stronger than the well-known result of Maassen and Uffink (MU) with a probability larger than 98%. We also show that the bounds investigated are invariant under the dephasing and permutation operations. Finally, we derive a classical analogue of the MU uncertainty relation, which is formulated for stochastic transition matrices.

Optimal diffusive search: nonequilibrium resetting versus equilibrium dynamics

Abstract: We study rst-passage time problems for a diusive particle with stochastic resetting with a nite rate r. The optimal search time is compared quantitatively with that of an eective equilibrium Langevin process with the same stationary distribution. It is shown that the intermittent, nonequilibrium strategy with non-vanishing resetting rate is more ecient than the equilibrium dynamics. Our results are extended to multiparticle systems where a team of independent searchers, initially uniformly distributed with a given density, looks for a single immobile target. Both the average and the typical survival probability of the target are smaller in the case of nonequilibrium dynamics.

On the integrability of zero-range chipping models with factorized steady states

Abstract: The conditions of the integrability of general zero range chipping models with factorized steady states, which were proposed in Evans et al (2004 J. Phys. A: Math. Gen. 37 L275), are examined. We find a three-parametric family of hopping probabilities for the models solvable by the Bethe ansatz, which includes most of known integrable stochastic particle models as limiting cases. The solution is based on the quantum binomial formula for two elements of an associative algebra obeying generic homogeneous quadratic relations, which is proved as a byproduct. We use the Bethe ansatz to solve an eigenproblem for the transition matrix of the Markov process. On its basis, we conjecture an integral formula for the Green function of the evolution operator for the model on an infinite lattice and derive the Bethe equations for the spectrum of the model on a ring.

Records in stochastic processes?theory and applications

Abstract: In recent years there has been a surge of interest in the statistics of record-breaking events in stochastic processes. Along with that, many new and interesting applications of the theory of records were discovered and explored. The record statistics of uncorrelated random variables sampled from time-dependent distributions was studied extensively. The findings were applied in various areas to model and explain record-breaking events in observational data. Particularly interesting and fruitful was the study of record-breaking temperatures and their connection with global warming, but also records in sports, biology and some areas in physics were considered in the last years. Similarly, researchers have recently started to understand the record statistics of correlated processes such as random walks, which can be helpful to model record events in financial time series. This review is an attempt to summarize and evaluate the progress that has been made in the field of record statistics throughout recent years.

Towards metric-like higher-spin gauge theories in three dimensions

Abstract: We consider the coupling of a symmetric spin-3 gauge field ϕµνρ to three-dimensional gravity in a second order metric-like formulation. The action that corresponds to an SL(3,R) × SL(3,R) Chern-Simons theory in the frame-like formulation is identified to quadratic order in the spin-3 field. We apply our result to compute corrections to the area law for higher-spin black holes using Wald’s entropy formula.

Analytical eigenstates for the quantum Rabi model

Abstract: We develop a method to find analytical solutions for the eigenstates of the quantum Rabi model. These include symmetric, anti-symmetric and asymmetric analytic solutions given in terms of the confluent Heun functions. Both regular and exceptional solutions are given in a unified form. In addition, the analytic conditions for determining the energy spectrum are obtained. Our results show that conditions proposed by Braak (2011 Phys. Rev. Lett. 107 100401) are a type of sufficiency condition for determining the regular solutions. The well-known Judd isolated exact solutions appear naturally as truncations of the confluent Heun functions.

Holographic applications of logarithmic conformal field theories

Abstract: We review the relations between Jordan cells in various branches of physics, ranging from quantum mechanics to massive gravity theories. Our main focus is on holographic correspondences between critically tuned gravity theories in anti-de Sitter space and logarithmic conformal field theories in various dimensions. We summarize the developments in the past five years, include some novel generalizations and provide an outlook on possible future developments.

Quantum secret sharing with classical Bobs

Abstract: Boyer et al (2007 Phys. Rev. Lett. 99 140501) proposed a novel idea of semi-quantum key distribution, where a key can be securely distributed between Alice, who can perform any quantum operation, and Bob, who is classical. Extending the ?semi-quantum? idea to other tasks of quantum information processing is of interest and worth considering. In this paper, we consider the issue of semi-quantum secret sharing, where a quantum participant Alice can share a secret key with two classical participants, Bobs. After analyzing the existing protocol, we propose a new protocol of semi-quantum secret sharing. Our protocol is more realistic, since it utilizes product states instead of entangled states. We prove that any attempt of an adversary to obtain information necessarily induces some errors that the legitimate users could notice.

Partial Masslessness and Conformal Gravity

Abstract: We use conformal, but ghostful, Weyl gravity to study its ghost-free, second derivative, partially massless (PM) spin-2 component in the presence of Einstein gravity with positive cosmological constant. Specifically, we consider both gravitational- and self-interactions of PM via the fully nonlinear factorization of conformal gravity's Bach tensor into Einstein times Schouten operators. We find that extending PM beyond linear order suffers from familiar higher spin consistency obstructions: it propagates only in Einstein backgrounds, and the conformal gravity route generates only the usual safe, Noether, cubic order vertices.

Logarithmic conformal field theory: a lattice approach

Abstract: Logarithmic conformal field theories (LCFT) play a key role, for instance, in the description of critical geometrical problems (percolation, self-avoiding walks, etc), or of critical points in several classes of disordered systems (transition between plateaux in the integer and spin quantum Hall effects). Much progress in their understanding has been obtained by studying algebraic features of their lattice regularizations. For reasons which are not entirely understood, the non-semi-simple associative algebras underlying these lattice models—such as the Temperley–Lieb algebra or the blob algebra—indeed exhibit, in finite size, properties that are in full correspondence with those of their continuum limits. This applies not only to the structure of indecomposable modules, but also to fusion rules, and provides an ‘experimental’ way of measuring couplings, such as the ‘number b’ quantifying the logarithmic coupling of the stress–energy tensor with its partner. Most results obtained so far have concerned boundary LCFTs and the associated indecomposability in the chiral sector. While the bulk case is considerably more involved (mixing in general left and right moving sectors), progress has also recently been made in this direction, uncovering fascinating structures. This study provides a short general review of our work in this area.

The tensor structure on the representation category of the $\mathcal {W}_p$ triplet algebra

Abstract: We study the braided monoidal structure that the fusion product induces on the Abelian category -mod, the category of representations of the triplet W-algebra . The -algebras are a family of vertex operator algebras that form the simplest known examples of symmetry algebras of logarithmic conformal field theories. We formalize the methods for computing fusion products, developed by Nahm, Gaberdiel and Kausch, that are widely used in the physics literature and illustrate a systematic approach to calculating fusion products in non-semi-simple representation categories. We apply these methods to the braided monoidal structure of -mod, previously constructed by Huang, Lepowsky and Zhang, to prove that this braided monoidal structure is rigid. The rigidity of -mod allows us to prove explicit formulae for the fusion product on the set of all simple and all projective -modules, which were first conjectured by Fuchs, Hwang, Semikhatov and Tipunin; and Gaberdiel and Runkel.

Covariant Lyapunov vectors

Abstract: Recent years have witnessed a growing interest in covariant Lyapunov vectors (CLVs) which span local intrinsic directions in the phase space of chaotic systems. Here, we review the basic results of ergodic theory, with a specific reference to the implications of Oseledets' theorem for the properties of the CLVs. We then present a detailed description of a 'dynamical' algorithm to compute the CLVs and show that it generically converges exponentially in time. We also discuss its numerical performance and compare it with other algorithms presented in the literature. We finally illustrate how CLVs can be used to quantify deviations from hyperbolicity with reference to a dissipative system (a chain of Henon maps) and a Hamiltonian model (a Fermi–Pasta–Ulam chain).This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to 'Lyapunov analysis: from dynamical systems theory to applications'.

5D super Yang–Mills theory and the correspondence to AdS7/CFT6

Abstract: We study the relation between 5D super Yang–Mills theory and the holographic description of 6D (2, 0) superconformal theory. We start by clarifying some issues related to the localization of SYM with matter on S5. We concentrate on the case of a single adjoint hypermultiplet with a mass term and argue that the theory has a symmetry enlargement at mass M = 1/(2r), where r is the S5 radius. However, in order to have a well-defined localization locus it is necessary to rotate M onto the imaginary axis, breaking the enlarged symmetry. Based on our prescription, the imaginary mass values are physical and we show how the localized path integral is consistent with earlier results for 5D SYM in flat space. We then compute the free energy and the expectation value for a circular Wilson loop in the large N limit. The Wilson loop calculation shows a mass dependent constant rescaling between weak and strong coupling. The Wilson loop continued back to to the enlarged symmetry point is consistent with a supergravity computation for an M2 brane using the standard identification of the compactification radius and the 5D coupling. If we continue back to the physical regime and use this value of the mass to determine the compactification radius, then we find agreement between the SYM free energy and the corresponding supergravity calculation. We also verify numerically some of our analytic approximations.