Two-dimensional Conformal Field Theory Research Articles

Elements of celestial conformal field theory

In celestial holography, four-dimensional scattering amplitudes are considered as two-dimensional conformal correlators of a putative two-dimensional celestial conformal field theory (CCFT). The simplest way of converting momentum space amplitudes into CCFT correlators is by taking their Mellin transforms with respect to light-cone energies. For massless particles, like gluons, however, such a construction leads to three-point and four-point correlators that vanish everywhere except for a measure zero hypersurface of celestial coordinates. This is due to the four-dimensional momentum conservation law that constrains the insertion points of the operators associated with massless particles. These correlators are reminiscent of Coulomb gas correlators that, in the absence of background charges, vanish due to charge conservation. We supply the background momentum by coupling Yang-Mills theory to a background dilaton field, with the (complex) dilaton source localized on the celestial sphere. This picture emerges from the physical interpretation of the solutions of the system of differential equations discovered by Banerjee and Ghosh. We show that the solutions can be written as Mellin transforms of the amplitudes evaluated in such a dilaton background. The resultant three-gluon and four-gluon amplitudes are single-valued functions of celestial coordinates enjoying crossing symmetry and all other properties expected from standard CFT correlators. We use them to extract OPEs and compare them with the OPEs extracted from multi-gluon celestial amplitudes without a dilaton background. We perform the conformal block decomposition of the four-gluon single-valued correlator and determine the dimensions, spin and group representations of the entire primary field spectrum of the Yang-Mills sector of CCFT.

Generalized monodromy method in gauge/gravity duality

The method of monodromy is an important tool for computing Virasoro conformal blocks in a two-dimensional Conformal Field Theory (2d CFT) at large central charge and external dimensions. In deriving the form of the monodromy problem, which defines the method, one needs to insert a degenerate operator, usually a level-two operator, into the corresponding correlation function. It can be observed that the choice of which degenerate operator to insert is arbitrary, and they shall reveal the same physical principles underlying the method. In this paper, we exploit this freedom and generalize the method of monodromy by inserting higher-level degenerate operators. We illustrate the case with a level-three operator, and derive the corresponding form of the monodromy problem. We solve the monodromy problem perturbatively and numerically; and check that it agrees with the standard monodromy method, despite the fact that the two versions of the monodromy problem do not seem to be related in any obvious way. The forms corresponding to other higher-level degenerate operators are also discussed. We explain the physical origin of the coincidence and discuss its implication from a mathematical perspective.

Entanglement entropy and negativity in the Ising model with defects

Defects in two-dimensional conformal field theories (CFTs) contain signatures of their characteristics. In this work, we analyze entanglement properties of subsystems in the presence of energy and duality defects in the Ising CFT using the density matrix renormalization group (DMRG) technique. In particular, we compute the entanglement entropy (EE) and the entanglement negativity (EN) in the presence of defects. For the EE, we consider the cases when the defect lies within the subsystem and at the edge of the subsystem. We show that the EE for the duality defect exhibits fundamentally different characteristics compared to the energy defect due to the existence of localized and delocalized zero energy modes. Of special interest is the nontrivial ‘finite-size correction’ in the EE obtained recently using free fermion computations [1]. These corrections arise when the subsystem size is appreciable compared to the total system size and lead to a deviation from the usual logarithmic scaling characteristic of one-dimensional quantum-critical systems. Using matrix product states with open and infinite boundary conditions, we numerically demonstrate the disappearance of the zero mode contribution for finite subsystem sizes in the thermodynamic limit. Our results provide further support to the recent free fermion computations, but clearly contradict earlier analytical field theory calculations based on twisted torus partition functions. Subsequently, we compute the logarithm of the EN (log-EN) between two disjoint subsystems separated by a defect. We show that the log-EN scales logarithmically with the separation of the subsystems. However, the coefficient of this logarithmic scaling yields a continuously-varying effective central charge that is different from that obtained from analogous computations of the EE. The defects leave their fingerprints in the subleading term of the scaling of the log-EN. Furthermore, the log-EN receives similar ‘finite size corrections’ like the EE which leads to deviations from its characteristic logarithmic scaling.

Information scrambling versus quantum revival through the lens of operator entanglement

In this paper, we look for signatures of quantum revivals in two-dimensional conformal field theories (2d CFTs) on a spatially compact manifold by using operator entanglement. It is believed that thermalization does not occur on spatially compact manifolds as the quantum state returns to its initial state which is a phenomenon known as quantum revival. We find that in CFTs such as the free fermion CFT, the operator mutual information exhibits quantum revival in accordance with the relativistic propagation of quasiparticles while in holographic CFTs, the operator mutual information does not exhibit this revival and the quasiparticle picture breaks down. Furthermore, by computing the tripartite operator mutual information, we find that the information scrambling ability of holographic CFTs can be weakened by the finite size effect. We propose a modification of an effective model known as the line tension picture to explain the entanglement dynamics due to the strong scrambling effect and find a close relationship between this model and the wormhole (Einstein-Rosen Bridge) in the holographic bulk dual.

Non-Gaussianities in the statistical distribution of heavy OPE coefficients and wormholes

The Eigenstate Thermalization Hypothesis makes a prediction for the statistical distribution of matrix elements of simple operators in energy eigenstates of chaotic quantum systems. As a leading approximation, off-diagonal matrix elements are described by Gaussian random variables but higher-point correlation functions enforce non-Gaussian corrections which are further exponentially suppressed in the entropy. In this paper, we investigate non- Gaussian corrections to the statistical distribution of heavy-heavy-heavy OPE coefficients in chaotic two-dimensional conformal field theories. Using the Virasoro crossing kernels, we provide asymptotic formulas involving arbitrary numbers of OPE coefficients from modular invariance on genus-g surfaces. We find that the non-Gaussianities are further exponentially suppressed in the entropy, much like the ETH. We discuss the implication of these results for products of CFT partition functions in gravity and Euclidean wormholes. Our results suggest that there are new connected wormhole geometries that dominate over the genus-two wormhole.

Charged moments in W3 higher spin holography

We consider the charged moments in SL(3, ℝ) higher spin holography, as well as in the dual two-dimensional conformal field theory with W3 symmetry. For the vacuum state and a single entangling interval, we show that the W3 algebra of the conformal field theory induces an entanglement W3 algebra acting on the quantum state in the entangling interval. The algebra contains a spin 3 modular charge which commutes with the modular Hamiltonian. The reduced density matrix is characterized by the modular energy and modular charge, hence our definition of the charged moments is also with respect to these conserved quantities. We evaluate the logarithm of the charged moments perturbatively in the spin 3 modular chemical potential, by computing the corresponding connected correlation functions of the modular charge operator up to quartic order in the chemical potential. This method provides access to the charged moments without using charged twist fields. Our result matches known results for the charged moment obtained from the charged topological black hole picture in SL(3, ℝ) higher spin gravity. Since our charged moments are not Gaussian in the chemical potential any longer, we conclude that the dual W3 conformal field theories must feature breakdown of equipartition of entanglement to leading order in the large c expansion.

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.

Generalized entanglement entropies in two-dimensional conformal field theory

We introduce and study generalized Rényi entropies defined through the traces of products of TrB(| Ψi⟩⟨Ψj| ) where ∣Ψi⟩ are eigenstates of a two-dimensional conformal field theory (CFT). When ∣Ψi⟩ = ∣Ψj⟩ these objects reduce to the standard Rényi entropies of the eigenstates of the CFT. Exploiting the path integral formalism, we show that the second generalized Rényi entropies are equivalent to four point correlators. We then focus on a free bosonic theory for which the mode expansion of the fields allows us to develop an efficient strategy to compute the second generalized Rényi entropy for all eigenstates. As a byproduct, our approach also leads to new results for the standard Rényi and relative entropies involving arbitrary descendent states of the bosonic CFT.

Lattice models from CFT on surfaces with holes: I. Torus partition function via two lattice cells

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.

2D CFTs: Large charge is not enough to control the dynamics

In this note we study two-dimensional conformal field theories at large global charge. Since the large-charge sector decouples from the dynamics, it does not control the dynamics and an effective field theory construction that works in higher-dimensional theories fails. It is however possible to use large charge in a double-scaling limit when another controlling parameter is present. We find some general features of the spectrum of models that admit a nonlinear sigma model description in a Wentzel-Kramers-Brillouin approximation and use the large-charge sector of the solvable $SU(2{)}_{k}$ Wess-Zumino-Witten model to argue the regimes of applicability of both the large-$Q$ expansion and the double-scaling limit.

Complexity in a moving mirror model

In a two-dimensional conformal field theory with a moving mirror, known as a moving mirror model, the time evolution of the entanglement entropy shows a Page like curve. This implies that the moving mirror model is useful to understand the island formula. In this paper, we study the time evolution of the subregion CV complexity in the moving mirror model for a better understanding of the island formula of the complexity. In contrast to the entanglement entropy, the subregion CV complexity shows a peculiar behaviour. We discuss this behaviour in more detail.

Classical conformal blocks, Coulomb gas integrals and Richardson-Gaudin models

Virasoro conformal blocks are universal ingredients of correlation functions of two-dimensional conformal field theories (2d CFTs) with Virasoro symmetry. It is acknowledged that in the (classical) limit of large central charge of the Virasoro algebra and large external, and intermediate conformal weights with fixed ratios of these parameters Virasoro blocks exponentiate to functions known as Zamolodchikovs’ classical blocks. The latter are special functions which have awesome mathematical and physical applications. Uniformization, monodromy problems, black holes physics, quantum gravity, entanglement, quantum chaos, holography, mathcal{N} = 2 gauge theory and quantum integrable systems (QIS) are just some of contexts, where classical Virasoro blocks are in use. In this paper, exploiting known connections between power series and integral representations of (quantum) Virasoro blocks, we propose new finite closed formulae for certain multi-point classical Virasoro blocks on the sphere. Indeed, combining classical limit of Virasoro blocks expansions with a saddle point asymptotics of Dotsenko-Fateev (DF) integrals one can relate classical Virasoro blocks with a critical value of the “Dotsenko-Fateev matrix model action”. The latter is the “DF action” evaluated on a solution of saddle point equations which take the form of Bethe equations for certain QIS (Gaudin spin models). A link with integrable models is our main motivation for this research line. For instance, analogous quantities as the “DF on-shell action” appear in 2d CFT realization of the so-called Richardson’s solution of the reduced BCS model describing physics of ultra-small superconducting grains. Precisely, the Richardson solution and its particular limit characterizing the rational Gaudin model have known implementation in 2d CFT in terms of a free field representation of certain (perturbed and unperturbed) WZW blocks. The WZW analogue of the “DF on-shell action” appears here and plays a crucial role, it is a generating function for eigenvalues of quantum integrals of motion. An exploration of relationships between power expansions and Coulomb gas representations of the aforementioned WZW blocks could pave a way for new analytical tools useful in the study of models of the Richardson-Gaudin type.

Surface defects in gauge theory and KZ equation

We study the regular surface defect in the \(\Omega \)-deformed four-dimensional supersymmetric gauge theory with gauge group SU(N) with 2N hypermultiplets in fundamental representation. We prove its vacuum expectation value obeys the Knizhnik–Zamolodchikov equation for the 4-point conformal block of the \(\widehat{{\mathfrak {sl}}}_{N}\)-current algebra, originally introduced in the context of two-dimensional conformal field theory. The level and the vertex operators are determined by the parameters of the \(\Omega \)-background and the masses of the hypermultiplets; the cross-ratio of the 4 points is determined by the complexified gauge coupling. We clarify that in a somewhat subtle way the branching rule is parametrized by the Coulomb moduli. This is an example of the BPS/CFT relation.

Shaping contours of entanglement islands in BCFT

In this paper, we study the fine structure of entanglement in holographic two-dimensional boundary conformal field theories (BCFT) in terms of the spatially resolved quasilocal extension of entanglement entropy — entanglement contour. We find that the boundary induces discontinuities in the contour revealing hidden localization-delocalization patterns of the entanglement degrees of freedom. Moreover, we observe the formation of “islands” where the entanglement contour vanishes identically implying that these regions do not contribute to the entanglement at all. We argue that these phenomena are the manifestation of the entanglement islands recently discussed in the literature. We apply the entanglement contour proposal to the recently discussed BCFT black hole models reproducing the Page curve — moving mirror model and the pair of BCFT in the thermofield double state. From the viewpoint of entanglement contour, the Page curve also carries the imprint of strong delocalization caused by dynamical entanglement islands.

Cluster algebraic description of entanglement patterns for the BTZ black hole

We study the thermal state of a two-dimensional conformal field theory, which is dual to the static BTZ black hole in the high temperature limit. After partitioning the boundary of the static BTZ slice into $N$ subsystems, we show that there is an underlying ${C}_{N\ensuremath{-}1}$ cluster algebra encoding entanglement patterns of the thermal state. We also demonstrate that the polytope encapsulating such patterns in a geometric manner for a fixed $N$ is the cyclohedron ${\mathcal{C}}_{N\ensuremath{-}1}$. Alternatively these patterns of entanglement can be represented in the space of geodesics (kinematic space) in terms of a Zamolodchikov $Y$ system of ${C}_{N\ensuremath{-}1}$ type. The boundary condition for such an $Y$ system is featuring the entropy of the BTZ black hole.

Action complexity in the presence of defects and boundaries

The holographic complexity of formation for the AdS3 2-sided Randall-Sundrum model and the AdS3/BCFT2 models is logarithmically divergent according to the volume conjecture, while it is finite using the action proposal. One might be tempted to conclude that the UV divergences of the volume and action conjectures are always different for defects and boundaries in two-dimensional conformal field theories. We show that this is not the case. In fact, in Janus AdS3 we find that both volume and action proposals provide the same kind of logarithmic divergences.

On the light-ray algebra in conformal field theories

We analyze the commutation relations of light-ray operators in conformal field theories. We first establish the algebra of light-ray operators built out of higher spin currents in free CFTs and find explicit expressions for the corresponding structure constants. The resulting algebras are remarkably similar to the generalized Zamolodchikov’s W∞ algebra in a two-dimensional conformal field theory. We then compute the commutator of generalized energy flow operators in a generic, interacting CFTs in d > 2. We show that it receives contribution from the energy flow operator itself, as well as from the light-ray operators built out of scalar primary operators of dimension ∆ ≤ d − 2, that are present in the OPE of two stress-energy tensors. Commutators of light-ray operators considered in the present paper lead to CFT sum rules which generalize the superconvergence relations and naturally connect to the dispersive sum rules, both of which have been studied recently.

Calabi-Yau CFTs and random matrices

Using numerical methods for finding Ricci-flat metrics, we explore the spectrum of local operators in two-dimensional conformal field theories defined by sigma models on Calabi-Yau targets at large volume. Focusing on the examples of K3 and the quintic, we show that the spectrum, averaged over a region in complex structure moduli space, possesses the same statistical properties as the Gaussian orthogonal ensemble of random matrix theory.

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.

Local Energy Bounds and Strong Locality in Chiral CFT

A family of quantum fields is said to be strongly local if it generates a local net of von Neumann algebras. There are few methods of showing directly strong locality of a quantum field. Among them, linear energy bounds are the most widely used, yet a chiral conformal field of conformal weight $d>2$ cannot admit linear energy bounds. In this paper we give a new direct method to prove strong locality in two-dimensional conformal field theory. We prove that if a chiral conformal field satisfies an energy bound of degree $d-1$, then it also satisfies a certain local version of the energy bound, and this in turn implies strong locality. A central role in our proof is played by diffeomorphism symmetry. As a concrete application, we show that the vertex operator algebra given by a unitary vacuum representation of the $\mathcal{W}_3$-algebra is strongly local. For central charge $c > 2$, this yields a new conformal net. We further prove that these nets do not satisfy strong additivity, and hence are not completely rational.