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In this paper, we discuss the simple current orbifold of a rational Narain CFT (Narain RCFT). This is a method of constructing other rational CFTs from a given rational CFT, by 'orbifolding' the global symmetry formed by a particular primary fields (called the simple current). Our main result is that a Narain RCFT satisfying certain conditions can be described in the form of a simple current orbifold of another Narain RCFT, and we have shown how the discrete torsion in taking that orbifold is obtained. Additionally, the partition function can be considered a simple current orbifold with discrete torsion, which is determined by the lattice and the B-field. We establish that the partition function can be expressed as a polynomial, with the variables substituted by certain q -series. In a specific scenario, this polynomial corresponds to the weight enumerator polynomial of an error-correcting code. Using this correspondence to the code theory, we can relate the B-field, the discrete torsion, and the B-form to each other. [ABSTRACT FROM AUTHOR]

We study the entanglement Hamiltonian for the ground state of one-dimensional free fermions in the presence of an inhomogeneous chemical potential. In particular, we consider a lattice with a linear, as well as a continuum system with a quadratic potential. It is shown that, for both models, conformal field theory predicts a Bisognano–Wichmann form for the entanglement Hamiltonian of a half-infinite system. Furthermore, despite being nonrelativistic, this result is inherited by our models in the form of operators that commute exactly with the entanglement Hamiltonian. After appropriate rescaling, they also yield an excellent approximation of the entanglement spectra, which becomes asymptotically exact in the bulk of the trapped Fermi gas. For the gradient chain, however, the conformal result is recovered only after taking a proper continuum limit. [ABSTRACT FROM AUTHOR]

WZW models based on super Lie algebras play an important role for the description of string theory on AdS spaces. In particular, for the case of AdS 3 × S 3 with pure NS–NS flux the super Lie algebra of psu (1 , 1 | 2) k appears in the hybrid formalism, and higher dimensional AdS spaces can be described in terms of related supergroup cosets. In this paper we study the WZW models based on u (2 | 2) 1 and psu (2 | 2) 1 that may play a role for the worldsheet theory that is dual to free super Yang–Mills in 4D. [ABSTRACT FROM AUTHOR]

The structure of type A and B trace anomalies is reanalyzed in terms of the universal behavior of dimension −2 invariant amplitudes. Based on it a general argument for trace anomaly matching between the unbroken and broken phases of a CFT is given. The structure of moduli trace anomalies and their transformations under source reparametrizations is discussed in detail. [ABSTRACT FROM AUTHOR]

We utilize a holographic construction to compute the entanglement negativity for bipartite mixed state configurations of a single subsystem in CFTd s with a conserved charge dual to bulk A d S d + 1 geometries. In this context, we obtain the holographic entanglement negativity for single subsystems with long rectangular strip geometries in CFTd s with a conserved charge dual to bulk non extremal and extremal Reissner–Nordström (RN)- A d S d + 1 black holes at the leading order in a perturbation theory. We demonstrate that the holographic entanglement negativity computed involves the elimination of thermal contributions at the leading order confirming earlier results in the literature. This also conforms to quantum information theory expectations and constitutes further consistency checks for the holographic construction. [ABSTRACT FROM AUTHOR]

In this paper we study ensembles of finite real spectral triples equipped with a path integral over the space of possible Dirac operators. In the noncommutative geometric setting of spectral triples, Dirac operators take the center stage as a replacement for a metric on a manifold. Thus, this path integral serves as a noncommutative analogue of integration over metrics, a key feature of a theory of quantum gravity. From these integrals in the so-called double scaling limit we derive critical exponents of minimal models from Liouville conformal field theory coupled with gravity. Additionally, the asymptotics of the partition function of these models satisfy differential equations such as Painlevé I, as a reduction of the KDV hierarchy, which is predicted by conformal field theory. This is all proven using well-established and rigorous techniques from random matrix theory. [ABSTRACT FROM AUTHOR]

This article reviews some results of the SAGEX programme that have developed in the understanding of the interplay of supersymmetry and modular covariance of scattering amplitudes in type IIB superstring theory and its holographic image in N = 4 supersymmetric Yang–Mills theory (SYM). The first section includes the determination of exact expressions for BPS interactions in the low-energy expansion of type IIB superstring amplitudes. The second section concerns properties of a certain class of integrated correlators in N = 4 SYM with arbitrary classical gauge group that are exactly determined by supersymmetric localisation. Not only do these reproduce known features of perturbative and non-perturbative N = 4 SYM for any classical gauge group, but they have large- N expansions that are in accord with expectations based on the holographic correspondence with superstring theory. The final section focusses on modular graph functions. These are modular functions that are closely associated with coefficients in the low-energy expansion of superstring perturbation theory and have recently received quite a lot of interest in both the physics and mathematics literature. [ABSTRACT FROM AUTHOR]

The soft limits of scattering amplitudes have been extensively studied due to their essential role in the computation of physical observables in collider physics. The universal factorisation that occurs in these kinematic limits has been shown to be related to conservation laws associated with asymptotic, or large, gauge symmetries. This connection has led to a deeper understanding of the symmetries of gauge and gravitational theories and to a reformulation of scattering amplitudes in a basis of boost eigenstates which makes manifest the two-dimensional global conformal symmetry of the celestial sphere. The recast, or celestial, amplitudes possess many of the properties of conformal field theory (CFT) correlation functions which has suggested a path towards a holographic description of asymptotically flat spacetimes. In this review we consider these interconnected developments in our understanding of soft theorems, asymptotic symmetries and CFT with a focus on the structure and symmetries of the celestial amplitudes and their holographic interpretation. [ABSTRACT FROM AUTHOR]

We study the out-of-equilibrium dynamics of noninteracting fermions in one dimension and in continuum space, in the presence of a delta impurity potential at the origin whose strength g is varied at time t = 0. The system is prepared in its ground state with g = g 0 = +∞, with two different densities and Fermi wave-vectors k L and k R on the two half-spaces x > 0 and x < 0 respectively. It then evolves for t > 0 as an isolated system, with a finite impurity strength g. We compute exactly the time dependent density and current. For a fixed position x and in the large time limit t â†' ∞, the system reaches a non-equilibrium stationary state (NESS). We obtain analytically the correlation kernel, density, particle current, and energy current in the NESS, and characterize their relaxation, which is algebraic in time. In particular, in the NESS, we show that, away from the impurity, the particle density displays oscillations which are the non-equilibrium analog of the Friedel oscillations. In the regime of ‘rays’, x / t = Îľ fixed with x, t â†' ∞, we compute the same quantities and observe the emergence of two light cones, associated to the Fermi velocities k L and k R in the initial state. Interestingly, we find non trivial quantum correlations between two opposite rays with velocities Îľ and âˆ' Îľ which we compute explicitly. We extend to a continuum setting and to a correlated initial state the analytical methods developed in a recent work of Ljubotina, Sotiriadis and Prosen, in the context of a discrete fermionic chain with an impurity. We also generalize our results to an initial state at finite temperature, recovering, via explicit calculations, some predictions of conformal field theory in the low energy limit. [ABSTRACT FROM AUTHOR]

We propose a covariant definition of the fractional Kleinâ€"Gordon equation (KGE) with long-range interactions independent of the metric of the underlying manifold. As an example we consider the fractional KGE on AdS2+1, computing the explicit kernel representation of the fractional Laplaceâ€"Beltrami operator as well as the two-point propagator of the fractional KGE. Our results suggest that the propagator only exists if the mass is small compared to the inverse AdS radius, presumably because the AdS space expands faster with distance as a flat space of the same dimension. Our results are expected to be useful in particular for new applications of the AdS/CFT correspondence within statistical mechanics and quantum information. [ABSTRACT FROM AUTHOR]

We propose a covariant definition of the fractional Kleinâ€"Gordon equation (KGE) with long-range interactions independent of the metric of the underlying manifold. As an example we consider the fractional KGE on AdS2+1, computing the explicit kernel representation of the fractional Laplaceâ€"Beltrami operator as well as the two-point propagator of the fractional KGE. Our results suggest that the propagator only exists if the mass is small compared to the inverse AdS radius, presumably because the AdS space expands faster with distance as a flat space of the same dimension. Our results are expected to be useful in particular for new applications of the AdS/CFT correspondence within statistical mechanics and quantum information. [ABSTRACT FROM AUTHOR]

Starting with a gentle approach to the Aldayâ€"Gaiottoâ€"Tachikawa (AGT) correspondence from its 6d origin, these notes provide a wide (albeit shallow) survey of the literature on numerous extensions of the correspondence up to early 2020. This is an extended writeup of the lectures given at the Winter School ‘YRISW 2020’ to appear in a special issue of J. Phys. A. Class S is a wide class of 4d N = 2 supersymmetric gauge theories (ranging from super-QCD (quantum chromodynamics) to non-Lagrangian theories) obtained by twisted compactification of 6d N = (2 , 0) superconformal theories on a Riemann surface C. This 6d construction yields the Coulomb branch and Seibergâ€"Witten geometry of class S theories, geometrizes S-duality, and leads to the AGT correspondence, which states that many observables of class S theories are equal to 2d conformal field theory (CFT) correlators. For instance, the four-sphere partition function of a 4d N = 2 S U (2) superconformal quiver theory is equal to a Liouville CFT correlator of primary operators. Extensions of the AGT correspondence abound: asymptotically-free gauge theories and Argyresâ€"Douglas theories correspond to irregular CFT operators, quivers with higher-rank gauge groups and non-Lagrangian tinkertoys such as T N correspond to Toda CFT correlators, and nonlocal operators (Wilsonâ€"’t Hooft loops, surface operators, domain walls) correspond to Verlinde networks, degenerate primary operators, braiding and fusion kernels, and Riemann surfaces with boundaries. [ABSTRACT FROM AUTHOR]

The generalized Wilson loop operator interpolating between the supersymmetric and the ordinary Wilson loop in N = 4 SYM theory provides an interesting example of renormalization group flow on a line defect: the scalar coupling parameter ζ has a non-trivial beta function and may be viewed as a running coupling constant in a 1D defect QFT. In this paper we continue the study of this operator, generalizing previous results for the beta function and Wilson loop expectation value to the case of an arbitrary representation of the gauge group and beyond the planar limit. Focusing on the scalar ladder limit where the generalized Wilson loop reduces to a purely scalar line operator in a free adjoint theory, and specializing to the case of the rank k symmetric representation of SU (N), we also consider a certain ‘semiclassical’ limit where k is taken to infinity with the product kζ 2 fixed. This limit can be conveniently studied using a 1D defect QFT representation in terms of N commuting bosons. Using this representation, we compute the beta function and the circular loop expectation value in the large k limit, and use it to derive constraints on the structure of the beta function for general representation. We discuss the corresponding 1D RG flow and comment on the consistency of the results with the 1D defect version of the F-theorem. [ABSTRACT FROM AUTHOR]

We construct a one-parameter family of lattice models starting from a two-dimensional rational conformal field theory on a torus with a regular lattice of holes, each of which is equipped with a conformal boundary condition. The lattice model is obtained by cutting the surface into triangles with clipped-off edges using open channel factorisation. The parameter is given by the hole radius. At finite radius, high energy states are suppressed and the model is effectively finite. In the zero-radius limit, it recovers the CFT amplitude exactly. In the touching hole limit, one obtains a topological field theory. If one chooses a special conformal boundary condition which we call ‘cloaking boundary condition’, then for each value of the radius the fusion category of topological line defects of the CFT is contained in the lattice model. The fact that the full topological symmetry of the initial CFT is realised exactly is a key feature of our lattice models. We provide an explicit recursive procedure to evaluate the interaction vertex on arbitrary states. As an example, we study the lattice model obtained from the Ising CFT on a torus with one hole, decomposed into two lattice cells. We numerically compare the truncated lattice model to the CFT expression obtained from expanding the boundary state in terms of the hole radius and we find good agreement at intermediate values of the radius. [ABSTRACT FROM AUTHOR]

In previous work, two of the authors determined, completely and rigorously, a solution space S N for a homogeneous system of 2 N + 3 linear partial differential equations (PDEs) in 2 N variables that arises in conformal field theory (CFT) and multiple Schrammâ€"Löwner evolution (SLE Îş ). The system comprises 2 N null-state equations and three conformal ward identities that govern CFT correlation functions of 2 N one-leg boundary operators or SLE Îş partition functions. M Bauer et al conjectured a formula, expressed in terms of ‘pure SLE Îş partition functions,’ for the probability that the growing curves of a multiple-SLE Îş process join in a particular connectivity. In a previous article, we rigorously define certain elements of S N , which we call ‘connectivity weights,’ argue that they are in fact pure SLE Îş partition functions, and show how to find explicit formulas for them in terms of Coulomb gas contour integrals. Our formal definition of the connectivity weights immediately leads to a method for finding explicit expressions for them. However, this method gives very complicated formulas where simpler versions may be available, and it is not applicable for certain values of Îş ∈ (0, 8) corresponding to well-known critical lattice models in statistical mechanics. In this article, we determine expressions for all connectivity weights in S N for N ∈ {1, 2, 3, 4} (those with N ∈ {3, 4} are new) and for so-called ‘rainbow connectivity weights’ in S N for all N ∈ Z + + 1. We verify these formulas by explicitly showing that they satisfy the formal definition of a connectivity weight. In appendix, we investigate logarithmic singularities of some of these expressions, appearing for certain values of Îş predicted by logarithmic CFT. [ABSTRACT FROM AUTHOR]

We study the thermodynamic Bethe ansatz (TBA) equations for pure T T ÂŻ perturbations of some simple integrable quantum field theories with a single bosonic or fermionic particle, in particular the massive sinh-Gordon model and its ultraviolet (UV) limit which is a deformation of the conformally invariant free massless boson. Whereas the TBA equations for T T ÂŻ deformations of massive theories are in principle known, the TBA equations we propose for the deformations of conformal field theories are relatively new and require a special factorization in rapidity variables of the CDD factor for the scattering of the massless particles. The latter TBA equations can be solved exactly and reproduce the known results for the ground state energy on a cylinder of circumference R which were previously obtained using different methods based for instance on the Burgers differential equation. Special attention is paid to the c -theorem in this context which is discussed in some detail. For positive infra-red central charge c IR, for flows consistent with the c -theorem the ground state energy develops a (previously known) square-root singularity toward the UV, which strongly suggests the theories are UV incomplete in these physically important cases. We suggest that the singularity indicates a tachyonic vacuum instability. Other cases with c IR < 0 do not have this singularity are interpreted as being UV complete with c UV = 0. We extend our results to a continuously variable c IR by introducing a chemical potential and suggest this as a possible toy model for the T T ÂŻ perturbed Liouville theory. [ABSTRACT FROM AUTHOR]

We construct a conformal field theory dual to string theory on AdS3 with pure NSâ€"NS flux. It is given by a symmetric orbifold of a linear dilaton theory deformed by a marginal operator from the twist-2 sector. We compute two- and three-point functions on the CFT side to 4th order in conformal perturbation theory at large N. They agree with the string computation at genus 0, thus providing ample evidence for a duality. We also show that the full spectra of both short and long strings on the CFT and the string side match. The duality should be understood as perturbative in N âˆ'1. [ABSTRACT FROM AUTHOR]

We consider 5d supersymmetric gauge theories with unitary groups in the Ω-background and study codim-2/4 BPS defects supported on orthogonal planes intersecting at the origin along a circle. The intersecting defects arise upon implementing the most generic Higgsing (geometric transition) to the parent higher dimensional theory, and they are described by pairs of 3d supersymmetric gauge theories with unitary groups interacting through 1d matter at the intersection. We explore the relations between instanton and generalized vortex calculus, pointing out a duality between intersecting defects subject to the Ω-background and a deformation of supergroup gauge theories, the exact supergroup point being achieved in the self-dual or unrefined limit. Embedding our setup into refined topological strings and in the simplest case when the parent 5d theory is Abelian, we are able to identify the supergroup theory dual to the intersecting defects as the supergroup version of refined Chernâ€"Simons theory via open/closed duality. We also discuss the BPS/CFT side of the correspondence, finding an interesting large rank duality with super-instanton counting. [ABSTRACT FROM AUTHOR]

The staircase model (SM) is a simple generalization of the sinh-Gordon model, obtained by complexifying the coupling constant. This produces a new theory with many interesting features. Chief among them is the fact that scaling functions such as Zamolodchikov's c-function display 'roaming' behaviour, that is, they visit the vicinity of an infinite number of conformal fixed points, the unitary minimal models of conformal field theory. This rich structure also makes the model an interesting candidate for study using the generalized hydrodynamic approach to integrable quantum field theory (IQFT). By studying hydrodynamic quantities such as the effective velocities of quasiparticles we can develop a more physical picture of interaction in the theory, both at and away from equilibrium. Indeed, we find that in the SM the effective velocity displays monotonicity features not found in any other IQFT. These admit a natural interpretation when investigated in the context of the massless theories known as models, which provide an effective description of massless flows between consecutive unitary minimal models. Consequently, we find that the effective velocity in the SM can be reconstructed by a 'cut and paste' procedure involving the equivalent functions associated to the massless excitations of suitable models. We also investigate the average currents and densities of higher spin conserved quantities in the partitioning protocol. [ABSTRACT FROM AUTHOR]

We consider the non-equilibrium dynamics of the entanglement entropy of a one-dimensional quantum gas of hard-core particles, initially confined in a box potential at zero temperature. At t = 0 the right edge of the box is suddenly released and the system is let free to expand. During this expansion, the initially correlated region propagates with a non-homogeneous profile, leading to the growth of entanglement entropy. This setting is investigated in the hydrodynamic regime, with tools stemming from semi-classical Wigner function approach and with recent developments of quantum fluctuating hydrodynamics. Within this framework, the entanglement entropy can be associated to a correlation function of chiral twist-fields of the conformal field theory that lives along the Fermi contour and it can be exactly determined. Our predictions for the entanglement evolution are found in agreement with and generalize previous results in literature based on numerical calculations and heuristic arguments. [ABSTRACT FROM AUTHOR]

Heavy–heavy–light–light (HHLL) correlators of pairwise identical scalars in CFTs with a large central charge in any number of dimensions admit a double scaling limit where the ratio of the heavy conformal dimension to the central charge becomes large as the separation between the light operators becomes null. In this limit the stress tensor sector of a generic HHLL correlator receives contributions from the multi stress tensor operators with any number of stress tensors, as long as their twist is not increased by index contractions. We show how one can compute this leading twist stress tensor sector when the conformal dimension of the light operators is large and the stress tensor sector approximates the thermal CFT correlator. In this regime the value of the correlator is related to the length of the spacelike geodesic which approaches the boundary of the dual asymptotically AdS spacetime at the points of light operator insertions. We provide a detailed description of the infinite volume limit. In two spacetime dimensions the HHLL Virasoro vacuum block is reproduced, while in four spacetime dimensions the result is written in terms of elliptic integrals. [ABSTRACT FROM AUTHOR]

We show that the fermionization of the Monster CFT with respect to is the tensor product of a free fermion and the Baby Monster CFT. The chiral fermion parity of the free fermion implies that the Monster CFT is self-dual under the orbifold, i.e. it enjoys the Kramers–Wannier duality. The Kramers–Wannier duality defect extends the Monster group to a larger category of topological defect lines that contains an Ising subcategory. We introduce the defect McKay–Thompson series defined as the Monster partition function twisted by the duality defect, and find that the coefficients can be decomposed into the dimensions of the (projective) irreducible representations of the Baby Monster group. We further prove that the defect McKay–Thompson series is invariant under the genus-zero congruence subgroup 16D0 of. [ABSTRACT FROM AUTHOR]

The quest for extension of holographic correspondence to non-relativistic sectors naturally includes Schrödinger backgrounds and their field theory duals. In this paper we study the holography by probing the correspondence with pulsating strings. The case we consider is pulsating strings in five-dimensional Schrödinger space times five-torus T1,1, which has as field theory dual a dipole CFT. First we find particular pulsating string solutions and then semi-classically quantize the theory. We obtain the wave function of the problem and thoroughly study the corrections to the energy, which by duality are supposed to give anomalous dimensions of certain operators in the dipole CFT. [ABSTRACT FROM AUTHOR]

We present an analytic study of conformal field theories on the real projective space , focusing on the two-point functions of scalar operators. Due to the partially broken conformal symmetry, these are non-trivial functions of a conformal cross ratio and are constrained to obey a crossing equation. After reviewing basic facts about the structure of correlators on , we study a simple holographic setup which captures the essential features of boundary correlators on. The analysis is based on calculations of Witten diagrams on the quotient space , and leads to an analytic approach to two-point functions. In particular, we argue that the structure of the conformal block decomposition of the exchange Witten diagrams suggests a natural basis of analytic functionals, whose action on the conformal blocks turns the crossing equation into certain sum rules. We test this approach in the canonical example of ϕ4 theory in dimension d = 4 − ϵ, extracting the CFT data to order ϵ2. We also check our results by standard field theory methods, both in the large N and ϵ expansions. Finally, we briefly discuss the relation of our analysis to the problem of construction of local bulk operators in AdS/CFT. [ABSTRACT FROM AUTHOR]

Anyon models are algebraic structures that model universal topological properties in topological phases of matter and can be regarded as mathematical characterization of topological order in two spacial dimensions. It is conjectured that every anyon model, or mathematically unitary modular tensor category, can be realized as the representation category of some chiral conformal field theory, or mathematically vertex operator algebra/local conformal net. This conjecture is known to be true for abelian anyon models providing support for the conjecture. We reexamine abelian anyon models from several different angles. First anyon models are algebraic data for both topological quantum field theories and chiral conformal field theories. While it is known that each abelian anyon model can be realized by a quantum abelian Chern–Simons (CS) theory and chiral conformal field theory, the construction is not algorithmic. Our goal is to provide such an explicit algorithm for a K-matrix in CS theory and a positive definite even one for a lattice conformal field theory. Secondly anyon models and chiral conformal field theories underlie the bulk-edge correspondence for topological phases of matter. But there are interesting subtleties in this correspondence when stability of the edge theory and topological symmetry are taken into consideration. Therefore, our focus is on the algorithmic reconstruction of extremal chiral conformal field theories with small central charges. Finally we conjecture that a much stronger reconstruction holds for abelian anyon models: every abelian anyon model can be realized as the representation category of some non-lattice extremal vertex operator algebra generalizing the moonshine realization of the trivial anyon model. [ABSTRACT FROM AUTHOR]

A review of Boundary and defect conformal field theory: open problems and applications, following a workshop held at Chicheley Hall, Buckinghamshire, UK, 7–8 Sept. 2017. We attempt to provide a broad, bird's-eye view of the latest progress in boundary and defect conformal field theory in various sub-fields of theoretical physics, including the renormalization group, integrability, conformal bootstrap, topological field theory, supersymmetry, holographic duality, and more. We also discuss open questions and promising research directions in each of these sub-fields, and combinations thereof. [ABSTRACT FROM AUTHOR]

We consider Lie superalgebras under constraints of Hamiltonian reduction, yielding finite W-superalgebras which provide candidates for quadratic spacetime superalgebras. These have an undeformed bosonic symmetry algebra (even generators) graded by a fermionic sector (supersymmetry generators) with anticommutator brackets which are quadratic in the even generators. We analyze the reduction of several Lie superalgebras of type gl(M|N) or osp(M|2N) at the classical (Poisson bracket) level, and also establish their quantum (Lie bracket) equivalents. Purely bosonic extensions are also considered. As a special case we recover a recently identified quadratic superconformal algebra, certain of whose unitary irreducible massless representations (in four dimensions) are 'zero-step' multiplets, with no attendant superpartners. Other cases studied include a six dimensional quadratic superconformal algebra with vectorial odd generators, and a variant quadratic superalgebra with undeformed osp(1|2N) singleton supersymmetry, and a triplet of spinorial supercharges. [ABSTRACT FROM AUTHOR]

The symmetric orbifold of is the CFT dual of string theory on with minimal NS–NS flux. We study its symmetry algebra and provide evidence that it does not have any deformation parameter. This suggests that the symmetric orbifold is (at least locally) the most symmetrical CFT in its moduli space. [ABSTRACT FROM AUTHOR]

Hecke operators relate characters of rational conformal field theories (RCFTs) with different central charges, and extend the previously studied Galois symmetry of modular representations and fusion algebras. We show that the conductor N of an RCFT and the quadratic residues modulo N play an important role in the computation and classification of Galois permutations. We establish a field correspondence in different theories through the picture of effective central charge, which combines Galois inner automorphisms and the structure of simple currents. We then make a first attempt to extend Hecke operators to the full data of modular tensor categories. The Galois symmetry encountered in the modular data transforms the fusion and the braiding matrices as well, and yields isomorphic structures in theories related by Hecke operators. [ABSTRACT FROM AUTHOR]

In a September 1976 PRL Eguchi and Freund considered two topological invariants: the Pontryagin number and the Euler number and posed the question: to what anomalies do they contribute? They found that P appears in the integrated divergence of the axial fermion number current, thus providing a novel topological interpretation of the anomaly found by Kimura in 1969 and Delbourgo and Salam in 1972. However, they found no analogous role for χ. This provoked my interest and, drawing on my April 1976 paper with Deser and Isham on gravitational Weyl anomalies, I was able to show that for conformal field theories the trace of the stress tensor depends on just two constants: where F is the square of the Weyl tensor and is the Euler number. For free CFTs with Ns massless fields of spin s. [ABSTRACT FROM AUTHOR]

In this study, we examined consequences of unconventional time development of two-dimensional conformal field theory induced by the L1 and L−1 operators, employing the formalism previously developed in a study of sine-square deformation. We discovered that the retainment of the Virasoro algebra requires the presence of a cut-off near the fixed points. The introduction of a scale by the cut-off makes it possible to recapture the formula for entanglement entropy in a natural and straightforward manner. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

We review the status of the two-dimensional Abelian sandpile model as a strong candidate to provide a lattice realization of logarithmic conformal invariance with a central charge c = -2. Evidence supporting this view is collected from various aspects of the model. These include the study of some conformally invariant boundary conditions, and the corresponding boundary condition changing fields, the calculation of correlations of certain bulk and boundary observables (the height variables) as well as a proper account of the necessary dissipation, which allows for a physical understanding of some of the strange but generic features of logarithmic theories. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

We construct chiral algebras that centralize rank-2 Nichols algebras with at least one fermionic generator. This gives 'logarithmic' W-algebra extensions of a fractional-level sℓ(2) algebra. We discuss crucial aspects of the emerging general relation between Nichols algebras and logarithmic conformal field theory (CFT) models: (i) the extra input, beyond the Nichols algebra proper, needed to uniquely specify a conformal model; (ii) a relation between the CFT counterparts of Nichols algebras connected by Weyl groupoid maps; and (iii) the common double bosonization U(X) of such Nichols algebras. For an extended chiral algebra, candidates for its simple modules that are counterparts of the U(X) simple modules are proposed, as a first step toward a functorial relation between U(X) and W-algebra representation categories. [ABSTRACT FROM AUTHOR]

Conformal sigma models and Wess-Zumino-Witten (WZW) models on coset superspaces provide important examples of logarithmic conformal field theories. They possess many applications to problems in string and condensed matter theory. We reviewrecent results and developments, including the general construction of WZW models on type-I supergroups, the classification of conformal sigma models and their embedding into string theory. [ABSTRACT FROM AUTHOR]

Jordan cells in transfer matrices of finite lattice models are a signature of the logarithmic character of the conformal field theories that appear in their thermodynamical limit. The transfer matrix of periodic loop models, TN, is an element of the periodic Temperley-Lieb algebra ꘓPTLN(β, α), where N is the number of sites on a section of the cylinder, and β = -q - q-1 = 2 cos λ and α the weights of contractible and non-contractible loops. The thermodynamic limit of TN is believed to describe a conformal field theory of central charge c = 1-6λ²/(π(λ-π)). The abstract element TN acts naturally on (a sum of) spaces ṼNd, similar to those upon which the standard modules of the (classical) Temperley-Lieb algebra act. These spaces known as sectors are labeled by the numbers of defects d and depend on a twist parameter v that keeps track of the winding of defects around the cylinder. Criteria are given for non-trivial Jordan cells of TN both between sectors with distinct defect numbers and within a given sector. [ABSTRACT FROM AUTHOR]

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 ...(1, 2), related to the triplet model ...(1, 2), symplectic fermions and the fermionic bc ghost system; the fractional level Wess-Zumino-Witten model based on sl (2) at k = -1/2, related to the bosonic ß? ghost system; and the Wess-Zumino-Witten model for the Lie supergroup GL (1|1), related to SL (2|1) at k = -1/2 and 1, the Bershadsky-Polyakov algebra W(2)3 and the Feigin-Semikhatov algebras W(2)n . 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 logarithmicminimal modelsW(q, p), the fractional level Wess-Zumino-Witten models, and the Wess-Zumino- Witten models on Lie supergroups (excluding OSP (1|2n)). 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 ...(1, 2) 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. [ABSTRACT FROM AUTHOR]

In this article, we review some aspects of logarithmic conformal field theories (LCFTs) which can be inferred from the characters of irreducible submodules of indecomposable modules. We will mainly consider the W(2, 2p - 1, 2p - 1, 2p - 1) series of triplet algebras and a bit logarithmic extensions of the minimal Virasoro models. Since in all known examples of LCFTs the vacuum representation of the maximally extended chiral symmetry algebra is an irreducible submodule of a larger, indecomposable module, its character provides a lot of non-trivial information about the theory such as a set of functions which spans the space of all torus amplitudes. Despite such characters being modular forms of inhomogeneous weight, they fit in the ADET-classification of fermionic sum representations. Thus, they show that LCFTs naturally have to be taken into account when attempting to classify rational conformal field theories. [ABSTRACT FROM AUTHOR]

Recent developments on the emergence of logarithmic terms in correlators, or response functions of modelswhich exhibit dynamical symmetries analogous to conformal invariance in not necessarily relativistic systems, are reviewed. The main examples of these are logarithmic Schrödinger invariance and logarithmic conformal Galilean invariance. Some applications of these ideas to statistical physics are described. [ABSTRACT FROM AUTHOR]

We describe an approach to logarithmic conformal field theories as limits of sequences of ordinary conformal field theories with varying central charge c. Logarithmic behaviour arises from degeneracies in the spectrum of scaling dimensions at certain values of c. The theories we consider are all invariant under some internal symmetry group, and logarithmic behaviour occurs when the decomposition of the physical observables into irreducible operators becomes singular. Examples considered are quenched random magnets using the replica formalism, self-avoiding walks as the n → 0 limit of the O(n) model, and percolation as the limit Q → 1 of the Potts model. In these cases we identify logarithmic operators and pay particular attention to how the c → 0 paradox is resolved and how the b-parameter is evaluated. We also show how this approach gives information on logarithmic behaviour in the extended Ising model, uniform spanning trees and the O(-2) model. Most of our results apply to general dimensionality. We also consider massive logarithmic theories and, in two dimensions, derive sum rules for the effective central charge and the b-parameter. [ABSTRACT FROM AUTHOR]

Logarithmic operators and logarithmic conformal field theories are reviewed. Prominent examples considered here include c = -2 and c = 0 logarithmic conformal field theories c = 0 logarithmic conformal field theories are especially interesting since they describe some of the critical points of a variety of longstanding problems involving a two dimensional quantum particle moving in a spatially random potential, as well as critical two dimensional selfavoiding random walks and percolation. Lack of classification of logarithmic conformal field theories remains a major impediment to progress towards finding complete solutions to these problems. [ABSTRACT FROM AUTHOR]

Itwas previously established that the critical staggeredXXZspin chain provides a lattice regularization of the black hole conformal field theory (CFT). We reconsider the continuum limit of this spin chain with the exact method of nonlinear integral equations (NLIEs), paying particular attention to the effects of a singular integration kernel. With the help of the NLIEs, we rederive the continuous black hole spectrum, but also numerically match the density of states of the spin chain with that of the CFT, which is a new result. Finally, we briefly discuss the integrable structure of the black hole CFT and the identification of its massive integrable perturbation on the lattice. [ABSTRACT FROM AUTHOR]

We elaborate on a recently conjectured relation of Painlevé transcendents and 2D conformal field theory. General solutions of Painlevé VI, V and III are expressed in terms of c = 1 conformal blocks and their irregular limits, AGT related to instanton partition functions in N = 2 supersymmetric gauge theories with Nf = 0, 1, 2, 3, 4. The resulting combinatorial series representations of Painlevé 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. [ABSTRACT FROM AUTHOR]

We consider the mutual Rényi information I(n)(A, B) = SA(n) + SB(n) - SA∪B(n) 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 I(n)(A, B) ~ CA(n) CB(n) (RARB/r2)α, where α is the smallest sum of the scaling dimensions of operators whose product has the quantum numbers of the vacuum, and the constants CA,B(n) 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 CA(2) RAd-1 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 Rényi entropies of a single spherical region for a scalar field theory with a small mass. [ABSTRACT FROM AUTHOR]

We study finite loop models on a lattice wrapped around a cylinder. A section of the cylinder has N sites. We use a family of link modules over the periodic Temperley-Lieb algebra ϵPTLN(β, α) introduced by Martin and Saleur, and Graham and Lehrer. These are labeled by the numbers of sites N and of defects d, and extend the standard modules of the original Temperley-Lieb algebra. Besides the defining parameters β = u2 + u-2 with u = eiλ/2 (weight of contractible loops) and a (weight of non-contractible loops), this family also depends on a twist parameter v that keeps track of how the defects wind around the cylinder. The transfer matrix TN (λ, v) depends on the anisotropy v and the spectral parameter λ that fixes the model. (The thermodynamic limit of TN is believed to describe conformal field theory of central charge c = 1 - 6λ2/(π(λ - π)).) The family of periodic XXZ Hamiltonians is extended to depend on this new parameter v, and the relationship between this family and the loop models is established. The Gram determinant for the natural bilinear form on these link modules is shown to factorize in terms of an intertwiner i˜Nd between these link representations and the eigenspaces of Sz of the XXZ models. This map is shown to be an isomorphism for generic values of u and v, and the critical curves in the plane of these parameters for which i˜Nd fails to be an isomorphism are given. [ABSTRACT FROM AUTHOR]

We investigate a non-equilibrium one-dimensional model known as the raise and peel model describing a growing surface which grows locally and has non-local desorption. For specific values of adsorption (ua) and desorption (ud) rates, the model shows interesting features. At ua = ud, the model is described by a conformal field theory (with conformal charge c = 0) and its stationary probability can be mapped onto the ground state of the XXZ quantum chain. Moreover, for the regime ua ⩾ ud, the model shows a phase in which the avalanche distribution is scale-invariant. In this work, we study the surface dynamics by looking at avalanche distributions using a finite-sized scaling formalism and explore the effect of adding a wall to the model. The model shows the same universality for the cases with and without a wall for an odd number of tiles removed, but we find a new exponent in the presence of a wall for an even number of tiles released in an avalanche. New insights into the effect of parity on avalanche distributions are discussed and we provide a new conjecture for the probability distribution of avalanches with a wall obtained by using an exact diagonalization of small lattices and Monte Carlo simulations. [ABSTRACT FROM AUTHOR]

We consider a generalized Lieb-Liniger model, describing a one-dimensional Bose gas with all its conservation laws appearing in the density matrix. This will be the case for the generalized Gibbs ensemble, or when the conserved charges are added to the Hamiltonian. The finite-size corrections are calculated for the energy spectrum. Large-distance asymptotics of correlation functions are then determined using methods from conformal field theory. [ABSTRACT FROM AUTHOR]

I give some scalar field theory calculations on a d-dimensional lune of arbitrary angle, evaluating, numerically, the effective action which is expressed as a simple quadrature, for conformal coupling. Using this, the entanglement and R´enyi entropies are computed. Massive fields are also considered and a renormalization to make the (one-loop) effective action vanish for infinite mass is suggested and used, not entirely successfully. However a universal coefficient is derived from the large mass expansion. From the deformation of the corresponding lune result, I conjecture that the effective action on all odd manifolds with a simple conical singularity has an extremum when the singularity disappears. For the round sphere, I show how to convert the quadrature form of the conformal Laplacian determinant into the more usual sum of Riemann ζ -functions (and log 2). [ABSTRACT FROM AUTHOR]

Totally symmetric, arbitrary spin massless and massive free fields in flat and AdS space and conformal fields in flat space are studied. An extended gauge-invariant Hamiltonian action for such fields is obtained. The action is constructed out of phase space fields and Lagrangian multipliers which are free of algebraic constraints. Gauge transformations of the phase space fields and Lagrangian multipliers are derived. Use of the Poincaré parametrization of AdS space allows us to treat fields in flat space and AdS space on an equal footing. [ABSTRACT FROM AUTHOR]