2D Conformal Field Theory Research Articles

Classical codes and chiral CFTs at higher genus

Higher genus modular invariance of two-dimensional conformal field theories (CFTs) is a largely unexplored area. In this paper, we derive explicit expressions for the higher genus partition functions of a specific class of CFTs: code CFTs, which are constructed using classical error-correcting codes. In this setting, the Sp(2g, ℤ) modular transformations of genus g Riemann surfaces can be recast as a simple set of linear maps acting on 2g polynomial variables, which comprise an object called the code enumerator polynomial. The CFT partition function is directly related to the enumerator polynomial, meaning that solutions of the linear constraints from modular invariance immediately give a set of seemingly consistent partition functions at a given genus. We then find that higher genus constraints, plus consistency under degeneration limits of the Riemann surface, greatly reduces the number of possible code CFTs. This work provides a step towards a full understanding of the constraints from higher genus modular invariance on 2d CFTs.

Solving Conformal Field Theories with Artificial Intelligence

In this Letter, we deploy for the first time reinforcement-learning algorithms in the context of the conformal-bootstrap program to obtain numerical solutions of conformal field theories (CFTs). As an illustration, we use a soft actor-critic algorithm and find approximate solutions to the truncated crossing equations of two-dimensional CFTs, successfully identifying well-known theories like the 2D Ising model and the 2D CFT of a compactified scalar. Our methods can perform efficient high-dimensional searches that can be used to study arbitrary (unitary or nonunitary) CFTs in any spacetime dimension.

Replica wormholes from Liouville theory

The replica wormholes are a key to the existence of the islands that play a central role in a recent proposal for the resolution of the black hole information paradox. In this paper, we study the replica wormholes in the JT gravity, a model of two-dimensional quantum gravity coupled to a non-dynamical dilaton, by making use of the 2d conformal field theory (CFT) description, namely, the Liouville theory coupled to the (2, p) minimal matter in the p → ∞ limit. In the Liouville CFT description, the replica wormholes are created by the twist operators and the gravitational part of the bulk entanglement entropy can be reproduced from the twist operator correlators. We propose the precise dictionary and show how this correspondence works in detail.

Restricted Boltzmann machine representation for the groundstate and excited states of Kitaev Honeycomb model

In this work, the capability of restricted Boltzmann machines (RBMs) to find solutions for the Kitaev honeycomb model with periodic boundary conditions is investigated. The measured groundstate energy of the system is compared and, for small lattice sizes (e.g. with 18 spinors), shown to agree with the analytically derived value of the energy up to a deviation of . Moreover, the wave-functions we find have overlap with the exact ground state wave-functions. Furthermore, the possibility of realizing anyons in the RBM is discussed and an algorithm is given to build these anyonic excitations and braid them for possible future applications in quantum computation. Using the correspondence between topological field theories in (2 + 1)d and 2d conformal field theories, we propose an identification between our RBM states with the Moore-Read state and conformal blocks of the 2d Ising model.

Surface defect, anomalies and b-extremization

Quantum field theories (QFT) in the presence of defects exhibit new types of anomalies which play an important role in constraining the defect dynamics and defect renormalization group (RG) flows. Here we study surface defects and their anomalies in conformal field theories (CFT) of general spacetime dimensions. When the defect is conformal, it is characterized by a conformal b-anomaly analogous to the c-anomaly of 2d CFTs. The b-theorem states that b must monotonically decrease under defect RG flows and was proven by coupling to a spurious defect dilaton. We revisit the proof by deriving explicitly the dilaton effective action for defect RG flow in the free scalar theory. For conformal surface defects preserving mathcal{N} = (0, 2) supersymmetry, we prove a universal relation between the b-anomaly and the ’t Hooft anomaly for the U(1)r symmetry. We also establish the b-extremization principle that identifies the superconformal U(1)r symmetry from mathcal{N} = (0, 2) preserving RG flows. Together they provide a powerful tool to extract the b-anomaly of strongly coupled surface defects. To illustrate our method, we determine the b-anomalies for a number of surface defects in 3d, 4d and 6d SCFTs. We also comment on manifestations of these defect conformal and ’t Hooft anomalies in defect correlation functions.

Entanglement branes and factorization in conformal field theory

In this work, we consider the question of local Hilbert space factorization in 2D conformal field theory. Generalizing previous work on entanglement and open-closed TQFT, we interpret the factorization of CFT states in terms of path integral processes that split and join the Hilbert spaces of circles and intervals. More abstractly, these processes are cobordisms of an extended CFT which are defined purely in terms of the OPE data. In addition to the usual sewing axioms, we impose an entanglement boundary condition that is satisfied by the vacuum Ishibashi state. This choice of entanglement boundary state leads to reduced density matrices that sum over super-selection sectors, which we identify as the CFT edge modes. Finally, we relate our factorization map to the co-product formula for the CFT symmetry algebra, which we show is equivalent to a Boguliubov transformation in the case of a free boson.

Generalized Gibbs Ensemble of 2D CFTs with U(1) charge from the AGT correspondence

The Generalized Gibbs Ensemble (GGE) is relevant to understand the thermalization of quantum systems with an infinite set of conserved charges. In this work, we analyze the GGE partition function of 2D Conformal Field Theories (CFTs) with a U(1) charge and quantum Benjamin-Ono2 (qBO2) hierarchy charges. We use the Alday-Gaiotto-Tachikawa (AGT) correspondence to express the thermal trace in terms of the Alba-Fateev-Litvinov-Tarnopolskiy (AFLT) basis of descendants, which diagonalizes all charges. We analyze the GGE partition function in the thermodynamic semiclassical limit, including the first order quantum correction. We find that the equality between GGE averages and primary eigenvalues of the qBO2 charges is attainable in the strict large c limit and potentially violated at the subleading 1/c order. We also obtain the finite c partition function when only the first non-trivial charge is turned on, expressed in terms of partial theta functions. Our results should be relevant to the eigenstate thermalization hypothesis for charged CFTs, Warped CFTs and effective field theory descriptions of condensed matter systems.

The quasi-particle picture and its breakdown after local quenches: mutual information, negativity, and reflected entropy

We study the dynamics of (Rényi) mutual information, logarithmic negativity, and (Rényi) reflected entropy after exciting the ground state by a local operator. Together with recent results from ref. [1], we are able to conjecture a close-knit structure between the three quantities that emerges in states excited above the vacuum, including both local and global quantum quenches. This structure intimately depends on the chaoticity of the theory i.e. there exist distinct sets of equivalences for integrable and chaotic theories. For rational conformal field theories (RCFT), we find all quantities to compute the quantum dimension of the primary operator inserted. In contrast, we find the correlation measures to grow (logarithmically) without bound in all c > 1 conformal field theories with a finite twist gap. In comparing the calculations in the two classes of theories, we are able to identify the dynamical mechanism for the breakdown of the quasi-particle picture in 2D conformal field theories. Intriguingly, we also find preliminary evidence that our general lessons apply to quantum systems considerably distinct from conformal field theories, such as integrable and chaotic spin chains, suggesting a universality of entanglement dynamics in non-equilibrium systems.

Virasoro Hair and Entropy for Axisymmetric Killing Horizons.

We show that the gravitational phase space for the near-horizon region of a bifurcate, axisymmetric Killing horizon in any dimension admits a 2D conformal symmetry algebra with central charges proportional to the area. This extends the construction of Haco et. al. [J. High Energy Phys. 12 (2018) 098JHEPFG1029-847910.1007/JHEP12(2018)098] to generic Killing horizons appearing in solutions of Einstein's equations and motivates a holographic description in terms of a 2D conformal field theory. The Cardy entropy in such a field theory agrees with the Bekenstein-Hawking entropy of the horizon, suggesting a microscopic interpretation. A set of appendixes is included in the Supplemental Material that provides examples and further details of the calculations presented in the main text.

Random boundary geometry and gravity dual of Toverline{T} deformation

We study the random geometry approach to the Toverline{T} deformation of 2d conformal field theory developed by Cardy and discuss its realization in a gravity dual. In this representation, the gravity dual of the Toverline{T} deformation becomes a straightforward translation of the field theory language. Namely, the dual geometry is an ensemble of AdS3 spaces or BTZ black holes, without a finite cutoff, but instead with randomly fluctuating boundary diffeomorphisms. This reflects an increase in degrees of freedom in the renormalization group flow to the UV by the irrelevant Toverline{T} operator. We streamline the method of computation and calculate the energy spectrum and the thermal free energy in a manner that can be directly translated into the gravity dual language. We further generalize this approach to correlation functions and reproduce the all-order result with universal logarithmic corrections computed by Cardy in a different method. In contrast to earlier proposals, this version of the gravity dual of the Toverline{T} deformation works not only for the energy spectrum and the thermal free energy but also for correlation functions.

Formula omitted] WZW conformal blocks from SU(N) instanton partition functions on [formula omitted

Generalizations of the AGT correspondence between 4D N=2SU(2) supersymmetric gauge theory on C2 with Ω-deformation and 2D Liouville conformal field theory include a correspondence between 4D N=2SU(N) supersymmetric gauge theories, N=2,3,…, on C2/Zn, n=2,3,…, with Ω-deformation and 2D conformal field theories with WN,npara (n-th parafermion WN) symmetry and slˆ(n)N symmetry. In this work, we trivialize the factor with WN,npara symmetry in the 4D SU(N) instanton partition functions on C2/Zn (by using specific choices of parameters and imposing specific conditions on the N-tuples of Young diagrams that label the states), and extract the 2D slˆ(n)N WZW conformal blocks, n=2,3,…, N=1,2,….

Perturbative 4D conformal field theories and representation theory of diagram algebras

The correlators of free four dimensional conformal field theories (CFT4) have been shown to be given by amplitudes in two-dimensional so(4, 2) equivariant topological field theories (TFT2), by using a vertex operator formalism for the correlators. We show that this can be extended to perturbative interacting conformal field theories, using two representation theoretic constructions. A co-product deformation for the conformal algebra accommodates the equivariant construction of composite operators in the presence of non-additive anomalous dimensions. Explicit expressions for the co-product deformation are given within a sector of mathcal{N} = 4 SYM and for the Wilson-Fischer fixed point near four dimensions. The extension of conformal equivariance beyond integer dimensions (relevant for the Wilson-Fischer fixed point) leads to the definition of an associative diagram algebra U★, abstracted from Uso(d) in the limit of large integer d, which admits extension of Uso(d) representation theory to general real (or complex) d. The algebra is related, via oscillator realisations, to so(d) equivariant maps and Brauer category diagrams. Tensor representations are constructed where the diagram algebra acts on tensor products of a fundamental diagram representation. A similar diagrammatic algebra U★, 2, related to a general d extension for Uso(d, 2) is defined, and some of its lowest weight representations relevant to the Wilson-Fischer fixed point are described.

Boundary correlators in WZW model on AdS2

Boundary correlators of elementary fields in some 2d conformal field theories defined on AdS2 have a particularly simple structure. For example, the correlators of the Liouville scalar happen to be the same as the correlators of the chiral component of the stress tensor on a plane restricted to the real line. Here we show that an analogous relation is true also in the WZW model: boundary correlators of the WZW scalars have the same structure as the correlators of chiral Kac-Moody currents. This is checked at the level of the tree and one-loop Witten diagrams in AdS2. We also compute some tree-level correlators in a generic σ-model defined on AdS2 and show that they simplify only in the WZW case where an extra Kac-Moody symmetry appears. In particular, the terms in 4- point correlators having logarithmic dependence on 1d cross-ratio cancel only at the WZW point. One motivation behind this work is to learn how to compute AdS2 loop corrections in 2d models with derivative interactions related to the study of correlators of operators on Wilson loops in string theory in AdS.

Coloured refined topological vertices and parafermion conformal field theories**To Professor Mikio Sato on his 90th birthday.

We extend the definition of the refined topological vertex to an n-coloured refined topological vertex that depends on n free bosons, and compute the 5D strip partition function made of N pairs of vertices and conjugate vertices. Using geometric engineering and the AGT correspondence, the 4D limit of this strip partition function is identified with a (normalized) matrix element of a (primary state) vertex operator that intertwines two (arbitrary descendant) states in a (generically non-rational) 2D conformal field theory with parafermion primary states.

DBI in the IR

The infra-red limit of a planar static D3-brane in AdS is a tensionless D3-brane at the anti-de Sitter horizon with dynamics governed by a strong-field limit of the Dirac–Born–Infeld action, analogous to that found from the Born–Infeld action by Bialynicki-Birula. As in that case, the field equations are those of an interacting 4D conformal invariant field theory with an electromagnetic duality invariance, but the D3-brane origin makes these properties manifest. We also find an -invariant action for these equations.

Hidden conformal symmetry of the rotating charged AdS black holes in quadratic f(T) gravity

The nonextremal Kerr black holes have been considered to be holographically dual to two-dimensional (2D) conformal field theories (CFTs). In this paper, we extend the holography to the case of an asymptotically anti--de Sitter (AdS) rotating charged black holes in $f$($T$) gravity, where $f(T) = T + \alpha T^2$, where $\alpha$ is a constant. We find that the scalar wave radial equation at the near-horizon region implies the existence of the 2D conformal symmetries. We note that the $2\pi$ identification of the azimuthal angle $\phi$ in the black hole line element, corresponds to a spontaneous breaking of the conformal symmetry by left and right temperatures $T_{L}$ and $T_{R}$, respectively. We show that choosing proper central charges for the dual CFT, we produce exactly the macroscopic Bekenstein-Hawking entropy from the microscopic Cardy entropy for the dual CFT. These observations suggest that the rotating charged AdS black hole in $f$($T$) gravity is dual to a 2D CFT at finite temperatures $T_{L}$ and $T_{R}$ for a specific value of mass $M$, rotational, charge, and $f$($T$) parameters, $\Omega$, $Q$, and $|\alpha|$, respectively.

Generalized Eigenstate Thermalization Hypothesis in 2D Conformal Field Theories.

Infinite-dimensional conformal symmetry in two dimensions leads to integrability of 2D conformal field theories (CFTs) by giving rise to an infinite tower of local conserved quantum Korteweg-de Vries (qKdV) charges in involution. We discuss how the presence of conserved charges constrains equilibration in 2D CFTs. We propose that in the thermodynamic limit large central charge 2D CFTs satisfy generalized eigenstate thermalization, with the values of qKdV charges forming a complete set of thermodynamically relevant quantities, which unambiguously determine expectation values of all local observables from the vacuum family. Equivalence of ensembles further provides that local properties of an eigenstate can be described by the generalized Gibbs ensemble that includes only qKdV charges. In the case of a general initial state, upon equilibration, the emerging generalized Gibbs ensemble will necessarily include negative chemical potentials and holographically will be described by a quasiclassical black hole with quantum soft hair.

Signature of quantum chaos in operator entanglement in 2d CFTs

We study operator entanglement measures of the unitary evolution operators of (1 + 1)-dimensional conformal field theories (CFTs), aiming to uncover their scrambling and chaotic behaviors. In particular, we compute the bi-partite and tri-partite mutual information for various configurations of input and output subsystems, and as a function of time. We contrast three different CFTs: the free fermion theory, compactified free boson theory at various radii, and CFTs with holographic dual. We find that the bi-partite mutual information exhibits distinct behaviors for these different CFTs, reflecting the different information scrambling capabilities of these unitary operators; while a quasi-particle picture can describe the case of the free fermion and free boson CFTs well, it completely fails for the case of holographic CFTs. Similarly, the tri-partite mutual information also distinguishes the unitary evolution operators of different CFTs. In particular, its late-time behaviors, when the output subsystems are semi-infinite, are quite distinct for these theories. We speculate that for holographic theories the late-time saturation value of the tri-partite mutual information takes the largest possible negative value and saturates the lower bound among quantum field theories.

Quantum statistics and spacetime topology: Quantum surgery formulas

To formulate the universal constraints of quantum statistics data of generic long-range entangled quantum systems, we introduce the geometric-topology surgery theory on spacetime manifolds where quantum systems reside, cutting and gluing the associated quantum amplitudes, specifically in 2+1 and 3+1 spacetime dimensions. First, we introduce the fusion data for worldline and worldsheet operators capable of creating anyonic excitations of particles and strings, well-defined in gapped states of matter with intrinsic topological orders. Second, we introduce the braiding statistics data of particles and strings, such as the geometric Berry matrices for particle–string Aharonov–Bohm, 3-string, 4-string, or multi-string adiabatic loop braiding process, encoded by submanifold links, in the closed spacetime 3-manifolds and 4-manifolds. Third, we derive new “quantum surgery” formulas and constraints, analogous to Verlinde formula associating fusion and braiding statistics data via spacetime surgery, essential for defining the theory of topological orders, 3d and 4d TQFTs and potentially correlated to bootstrap boundary physics such as gapless modes, extended defects, 2d and 3d conformal field theories or quantum anomalies.This article is meant to be an extended and further detailed elaboration of our previous work Wang, Wen and Yau (2016) [1] and Chapter 6 of Wang (2015) [2]. Our theory applies to general quantum theories and quantum mechanical systems, also applicable to, but not necessarily requiring the quantum field theory description.

From the Weyl Anomaly to Entropy of Two-Dimensional Boundaries and Defects.

We study whether the relations between the Weyl anomaly, entanglement entropy (EE), and thermal entropy of a two-dimensional (2D) conformal field theory (CFT) extend to 2D boundaries of 3D CFTs, or 2D defects of D≥3 CFTs. The Weyl anomaly of a 2D boundary or defect defines two or three central charges, respectively. One of these, b, obeys a c theorem, as in 2D CFT. For a 2D defect, we show that another, d_{2}, interpreted as the defect's "conformal dimension," must be non-negative if the averaged null energy condition holds in the presence of the defect. We show that the EE of a sphere centered on a planar defect has a logarithmic contribution from the defect fixed by b and d_{2}. Using this and known holographic results, we compute b and d_{2} for 1/2-Bogomol'nyi-Prasad-Sommerfield surface operators in the maximally supersymmetric (SUSY) 4D and 6D CFTs. The results are consistent with b's c theorem. Via free field and holographic examples we show that no universal "Cardy formula" relates the central charges to thermal entropy.