Chiral Conformal Field Theory Research Articles

Relative entropy for coherent states in chiral CFT

We consider the relative entropy between the vacuum state and a state obtained by applying an exponentiated stress tensor to the vacuum of a chiral conformal field theory on the lightray. The smearing function of the stress tensor can be viewed as a vector field on the real line generating a diffeomorphism. We show that the relative entropy is equal to $c$ times the so-called Schwarzian action of the diffeomorphism. As an application of this result, we obtain a formula for the relative entropy between the vacuum and a solitonic state.

Representation theory in chiral conformal field theory: from fields to observables

This article develops new techniques for understanding the relationship between the three different mathematical formulations of two-dimensional chiral conformal field theory: conformal nets (axiomatizing local observables), vertex operator algebras (axiomatizing fields), and Segal CFTs. It builds upon previous work (Tener in Adv Math 349:488–563, 2019), which introduced a geometric interpolation procedure for constructing conformal nets from VOAs via Segal CFT, simultaneously relating all three frameworks. In this article, we extend this construction to study the relationship between the representation theory of conformal nets and the representation theory of vertex operator algebras. We define a correspondence between representations in the two contexts, and show how to construct representations of conformal nets from VOAs. We also show that this correspondence is rich enough to relate the respective ‘fusion product’ theories for conformal nets and VOAs, by constructing local intertwiners (in the sense of conformal nets) from intertwining operators (in the sense of VOAs). We use these techniques to show that all WZW conformal nets can be constructed using our geometric interpolation procedure.

Chiral edge states in (2+1)-dimensional topological phases

Chiral edge states of [Formula: see text]-dimensional Abelian and non-Abelian topological phases can be represented by chiral conformal field theories with integer and noninteger values of central charge, respectively. In this work we describe certain edge states in terms of constrained fermionic fields that realize chiral coset CFT structures. This construction arises naturally in the so-called quantum wires approach for topological phases and allows for representing fractionalized edge states directly in terms of fermionic degrees of freedom. At the same time, the constrained fermions description introduces some subtleties concerning gauge anomalies since it involves the coupling of chiral fermions to gauge fields. We describe in this paper how to handle these issues.

A Borcherds–Kac–Moody Superalgebra with Conway Symmetry

We construct a Borcherds Kac-Moody (BKM) superalgebra on which the Conway group Co$_0$ acts faithfully. We show that the BKM algebra is generated by the BRST-closed states in a chiral superstring theory. We use this construction to produce denominator identities for the chiral partition functions of the Conway module $V^{s\natural}$, a supersymmetric $c=12$ chiral conformal field theory whose (twisted) partition functions enjoy moonshine properties and which has automorphism group isomorphic to Co$_0$. In particular, these functions satisfy a genus zero property analogous to that of monstrous moonshine. Finally, we suggest how one may promote the denominators to spacetime BPS indices in type II string theory, which might thus furnish a physical explanation of the genus zero property of Conway moonshine.

Geometric realization of algebraic conformal field theories

We explore new connections between the fields and local observables in two dimensional chiral conformal field theory. We show that in a broad class of examples, the von Neumann algebras of local observables (a conformal net) can be obtained from the fields (a unitary vertex operator algebra) via a continuous geometric interpolation procedure involving Graeme Segal's functorial definition of conformal field theory, and that the conformal net may be thought of as a boundary value of the Segal CFT. In particular, we construct conformal nets from these unitary vertex operator algebras by showing that ‘geometrically mollified’ versions of the fields yield bounded, local observables on the Hilbert space completion of the vertex algebra. These are the first results which unite the three major definitions of chiral conformal field theory. This work is inspired by Henriques' picture of conformal nets arising from degenerate Riemann surfaces.

More on vacua of exotic massive 3D gravity

In a recent paper [arXiv:1806.06254], we explored the space of solutions of the exotic massive 3D gravity theory proposed in [arXiv:1806.04179]. We showed that the theory admits a rich space of vacua, including asymptotically Anti de-Sitter (AdS) geometries obeying different types of boundary conditions. The examples include black holes dressed with low decaying gravitons. Based on what happens in other theories of massive gravity, we conjectured that such geometries appear on a curve of the parameter space (chiral curve) where the exotic massive gravity on AdS with sufficiently strong boundary conditions results dual to a 2D chiral conformal field theory. Here, we show that this expectation is consistent with the conserved charges and thermodynamical properties of the black holes of the theory, which have recently been computed [arXiv:1812.09525]. When the boundary conditions are relaxed relative to the standard Brown-Henneaux boundary conditions, the theory exhibits solutions consistent with the definition of the so-called Log-gravity. The asymptotic behavior of these solutions presents a logarithmic term in the Fefferman-Graham expansion that, nonetheless, is compatible with the AdS asymptotic symmetries. This long range interaction is due to a mode of the massive gravity that becomes massless precisely on the chiral curve. We construct exact solutions exhibiting this behavior, which admit to be interpreted as fully backreacting gravitational waves propagating on an extremal black hole, and carrying non-vanishing gravitational energy. We also discuss other vacua of the theory, such as Warped-AdS$_3$ black holes, gravitational waves on such backgrounds, AdS$_2\times S^1$ spaces, and black holes in dS$_3$ space.

Some Results On Relative Entropy in Quantum Field Theory

We prove that the mutual information for vacuum state as defined by Araki is finite for quantum field theory of free fermions on a Minkowski spacetime of any dimension. In the case of two dimensional chiral conformal field theory (CFT) we use our previous results for the free fermions to show that for a large class of chiral CFT the mutual information is finite. We also provide a direct relation between relative entropy and the index of a representation of conformal net.

Rationalizing CFTs and anyonic imprints on Higgs branches

We continue our program of mapping data of 4D mathcal{N}=2 superconformal field theories (SCFTs) onto observables of 2D chiral rational conformal field theories (RCFTs) by revisiting an infinite set of strongly coupled Argyres-Douglas (AD) SCFTs and their associated logarithmic 2D chiral algebras. First, we turn on discrete flavor fugacities (for continuous flavor symmetries) in a known correspondence between certain unrefined characters of these logarithmic theories and unrefined characters of a set of unitary 2D chiral RCFTs. Motivated by this discussion, we then study 4D Higgs branch renormalization group flows (i.e., flows activated by vevs for which only su(2)R ⊂ su(2)R × u(1)R is spontaneously broken) emanating from our AD theories from the point of view of the unitary 2D theories and find some surprises. In particular, we argue that certain universal pieces of the topological data underlying the 2D chiral algebra representations associated with the 4D infrared (IR) theory can be computed, via Galois conjugation, in the topological quantum field theory (TQFT) underlying the unitary ultraviolet (UV) chiral RCFT. The mapping of this topological data from UV to IR agrees with the fact that, in our theories, the moduli spaces we study consist of free hypermultiplets at generic points if and only if the UV TQFT is a theory of abelian anyons.

Bubbling the NHEK

We build the first family of smooth bubbling microstate geometries that are asymptotic to the near-horizon region of extremal five-dimensional Kerr black holes (NHEK). These black holes arise as extremal non-supersymmetric highly-rotating D1-D5- P solutions in type IIB string theory on T4×S1. Our solutions are asymptotically NHEK in the UV and end in the IR with a smooth cap. In the context of the Kerr/CFT correspondence, these bubbling geometries are dual to pure states of the 1+1 dimensional chiral conformal field theory dual to NHEK. Since our solutions have a bubbling structure in the IR, they correspond to an IR phase of broken conformal symmetry, and their existence supports the possibility that all the pure states whose counting gives the Kerr black hole entropy correspond to horizonless bulk configurations.

Near Horizon Extreme Kerr Geometry

The conjectured Kerr/CFT correspondence states that the quantum theory of gravity in the near horizon of the Near Horizon Extreme Kerr (NHEK) black holes, is holographic dual to a two-dimensional chiral conformal field theory. To obtain NHEK geometry, we take the extreme limit a = M of the Kerr metric and continue by transforming the coordinates. Consequently, we can obtain the new Ernst potentials for that geometry by using the same coordinates transformation. Finally, we can find the central charge as a function of the Papapetrou-Ernst potentials from stretched horizon formalism.

Interplay of SU(2), point group, and translational symmetry for projected entangled pair states: Application to a chiral spin liquid

Projected entangled pair states (PEPS) provide exact representations for many non-chiral topologically ordered states whereas their range of applicability to interacting chiral topological phases remains largely unsettled. In this context, the symmetries of the local PEPS tensors are crucial for determining the characteristic topological features of the state. In this article we examine the constraints that arise when different symmetries are imposed simultaneously on the local tensor such as internal $SU(2)$, point group and translation symmetry. We show how the interplay of these symmetries manifests in the entanglement spectrum which is the main diagnostic tool for chiral topological order. We apply our results to a spin liquid PEPS introduced previously as a chiral generalization of the resonating valence bond state. Our findings explain the discrepancies observed between the entanglement spectrum of this state and the expected edge spectrum described by a chiral conformal field theory. Finally, in a certain parameter region where this PEPS possesses an additional $U(1)$ symmetry we are able to resolve these discrepancies and obtain an entanglement spectrum with the expected state countings and conformal weight.

Vacua of exotic massive 3D gravity

We consider the recently proposed exotic 3D massive gravity. We show that this theory has a rich space of vacua, including asymptotically Anti de-Sitter (AdS) geometries obeying either the standard Brown-Henneaux boundary conditions or the weakened asymptotic behavior of the so-called Log-gravity. Both sectors contain non-Einstein spaces with SO(2) × mathbb{R} isometry group, showing that the Birkhoff theorem does not hold all over the parameter space, even if strong AdS boundary conditions are imposed. Some of these geometries correspond to 3D black holes dressed with a Log-gravity graviton. We conjecture that such geometries appear in a curve of the parameter space where the exotic 3D massive gravity on AdS3 is dual to a chiral conformal field theory. The theory also contains other interesting vacua, including different families of non-AdS black holes.

The Relative Drinfeld Commutant of a Fusion Category and α-Induction

Abstract We establish a correspondence among simple objects of the relative commutant of a full fusion subcategory in a larger fusion category in the sense of Drinfeld, irreducible half-braidings of objects in the larger fusion category with respect to the fusion subcategory, and minimal central projections in the relative tube algebra. Based on this, we explicitly compute certain relative Drinfeld commutants of fusion categories arising from α-induction for braided subfactors. We present examples arising from chiral conformal field theory.

Loop groups and noncommutative geometry

We describe the representation theory of loop groups in terms of K-theory and noncommutative geometry. This is done by constructing suitable spectral triples associated with the level [Formula: see text] projective unitary positive-energy representations of any given loop group [Formula: see text]. The construction is based on certain supersymmetric conformal field theory models associated with [Formula: see text] in the setting of conformal nets. We then generalize the construction to many other rational chiral conformal field theory models including coset models and the moonshine conformal net.

Braided Categories of Endomorphisms as Invariants for Local Quantum Field Theories

We want to establish the "braided action" (defined in the paper) of the DHR category on a universal environment algebra as a complete invariant for completely rational chiral conformal quantum field theories. The environment algebra can either be a single local algebra, or the quasilocal algebra, both of which are model-independent up to isomorphism. The DHR category as an abstract structure is captured by finitely many data (superselection sectors, fusion, and braiding), whereas its braided action encodes the full dynamical information that distinguishes models with isomorphic DHR categories. We show some geometric properties of the "duality pairing" between local algebras and the DHR category which are valid in general (completely rational) chiral CFTs. Under some additional assumptions whose status remains to be settled, the braided action of its DHR category completely classifies a (prime) CFT. The approach does not refer to the vacuum representation, or the knowledge of the vacuum state.

A No-Go Theorem for the Continuum Limit of a Periodic Quantum Spin Chain

We show that the Hilbert space formed from a block spin renormalization construction of a cyclic quantum spin chain (based on the Temperley-Lieb algebra) does not support a chiral conformal field theory whose Hamiltonian generates translation on the circle as a continuous limit of the rotations on the lattice.

{{SO(d,1)}}-Invariant Yang\u2013Baxter Operators and the dS/CFT Correspondence

We propose a model for the dS/CFT correspondence. The model is constructed in terms of a “Yang–Baxter operator” R for unitary representations of the de Sitter group {SO(d,1)}. This R-operator is shown to satisfy the Yang–Baxter equation, unitarity, as well as certain analyticity relations, including in particular a crossing symmetry. With the aid of this operator we construct: (a) a chiral (light-ray) conformal quantum field theory whose internal degrees of freedom transform under the given unitary representation of {SO(d,1)}. By analogy with the O(N) non-linear sigma model, this chiral CFT can be viewed as propagating in a de Sitter spacetime. (b) A (non-unitary) Euclidean conformal quantum field theory on {mathbb{R}^{d-1}}, where SO(d, 1) now acts by conformal transformations in (Euclidean) spacetime. These two theories can be viewed as dual to each other if we interpret {mathbb{R}^{d-1}} as conformal infinity of de Sitter spacetime. Our constructions use semi-local generator fields defined in terms of R and abstract methods from operator algebras.

Bootstrapping chiral CFTs at genus two

Genus two partition functions of 2d chiral conformal field theories are given by Siegel modular forms. We compute their conformal blocks and use them to perform the conformal bootstrap. The advantage of this approach is that it imposes crossing symmetry of an infinite family of four point functions and also modular invariance at the same time. Since for a fixed central charge the ring of Siegel modular forms is finite dimensional, we can perform this analytically. In this way we derive bounds on three point functions and on the spectrum of such theories.

Matrix product approximations to conformal field theories

We establish rigorous error bounds for approximating correlation functions of conformal field theories (CFTs) by certain finite-dimensional tensor networks. For chiral CFTs, the approximation takes the form of a matrix product state. For full CFTs consisting of a chiral and an anti-chiral part, the approximation is given by a finitely correlated state. We show that the bond dimension scales polynomially in the inverse of the approximation error and sub-exponentially in inverse of the minimal distance between insertion points. We illustrate our findings using Wess–Zumino–Witten models, and show that there is a one-to-one correspondence between group-covariant MPS and our approximation.

On classification of extremal non-holomorphic conformal field theories

Rational chiral conformal field theories are organized according to their genus, which consists of a modular tensor category and a central charge c. A long-term goal is to classify unitary rational conformal field theories based on a classification of unitary modular tensor categories. We conjecture that for any unitary modular tensor category , there exists a unitary chiral conformal field theory so that its modular tensor category is . In this paper, we initiate a mathematical program in and around this conjecture. We define a class of extremal vertex operator algebras with minimal conformal dimensions as large as possible for their central charge, and non-trivial representation theory. We show that there are finitely many different characters of extremal vertex operator algebras possessing at most three different irreducible modules. Moreover, we list all of the possible characters for such vertex operator algebras with .