2D CFT Research Articles

Character relations and replication identities in 2d Conformal Field Theory

We study replication identities satisfied by conformal characters of a 2D CFT, providing a natural framework for a physics interpretation of the famous Hauptmodul property of Monstrous Moonshine, and illustrate the underlying ideas in simple cases.

Holographic entanglement entropy for a large class of states in 2D CFT

In this paper, we study the entanglement entropy in a large class of states of two-dimensional conformal field theory in the the large central charge limit. This class of states includes the states created by the insertion of a finite number of local heavy operators. By using the monodromy analysis, we obtain the leading order entanglement entropy for the general state. We show that it is exactly captured by the Ryu-Takayanagi formula, by using the Wilsonian line prescription in the Chern-Simons formulation of the AdS$_3$ gravity.

On 3d bulk geometry of Virasoro coadjoint orbits: orbit invariant charges and Virasoro hair on locally AdS $$_3$$ 3 geometries

Expanding upon [arXiv:1404.4472, 1511.06079], we provide further detailed analysis of Ba\~nados geometries, the most general solutions to the AdS3 Einstein gravity with Brown-Henneaux boundary conditions. We analyze in some detail the causal, horizon and boundary structure, and geodesic motion on these geometries, as well as the two class of symplectic charges one can associate with these geometries: charges associated with the exact symmetries and the Virasoro charges. We elaborate further the one-to-one relation between the coadjoint orbits of two copies of Virasoro group and Ba\~nados geometries. We discuss that the information about the Ba\~nados goemetries fall into two categories: "orbit invariant" information and "Virasoro hairs". The former are geometric quantities while the latter are specified by the non-local surface integrals. We elaborate on multi-BTZ geometries which have some number of disconnected pieces at the horizon bifurcation curve. We study multi-BTZ black hole thermodynamics and discuss that the thermodynamic quantities are orbit invariants. We also comment on the implications of our analysis for a 2d CFT dual which could possibly be dual to AdS3 Einstein gravity.

Non-chiral 2d CFT with integer energy levels

The partition function of 2d conformal field theory is a modular invariant function. It is known that the partition function of a holomorphic CFT whose central charge is a multiple of 24 is a polynomial in the Klein function. In this paper, by using the medium temperature expansion we show that every modular invariant partition function can be mapped to a holomorphic partition function whose structure can be determined similarly. We use this map to study partition function of CFTs with half-integer left and right conformal weights. We show that the corresponding left and right central charges are necessarily multiples of 4. Furthermore, the degree of degeneracy of high-energy levels can be uniquely determined in terms of the degeneracy in the low energy states.

Universal bounds on charged states in 2d CFT and 3d gravity

We derive an explicit bound on the dimension of the lightest charged state in two dimensional conformal field theories with a global abelian symmetry. We find that the bound scales with $c$ and provide examples that parametrically saturate this bound. We also prove than any such theory must contain a state with charge-to-mass ratio above a minimal lower bound. We comment on the implications for charged states in three dimensional theories of gravity.

Magnetized Kerr/CFT correspondence

We extend the conjectured Kerr/CFT correspondence to the case of extremal Kerr black holes immersed by a magnetic field, namely the extremal Melvin–Kerr black holes. We compute the central charge which appears in the associated Virasoro algebra generated by a class of diffeomorphisms that satisfies a set of boundary conditions in the near horizon of an extremal Melvin–Kerr black hole. Our results support the Kerr/CFT conjecture, where the macroscopic Bekenstein–Hawking entropy for an extremal Melvin–Kerr black hole matches the result obtained from a dual 2D CFT microscopic computation using Cardy formula. Interestingly, the dual CFT description could be non-unitary, due to the possibility of negative central charge.

Charged entanglement entropy of local operators

In this work we consider the time evolution of charged Renyi entanglement entropies after exciting the vacuum with local fermionic operators. In order to explore the information contained in charged Renyi entropies, we perform computations of their excess due to the operator excitation in 2d CFT, free fermionic field theories in various dimensions as well as holographically. In the analysis we focus on the dependence on the entanglement charge, the chemical potential and the spacetime dimension. We find that excesses of charged (Renyi) entanglement entropy can be interpreted in terms of charged quasiparticles. Moreover, we show that by appropriately tuning the chemical potential charged Renyi entropies can be used to extract entanglement in a certain charge sector of the excited state.

Emergent space-time and the supersymmetric index

It is of interest to find criteria on a 2d CFT which indicate that it gives rise to emergent gravity in a macroscopic 3d AdS space via holography. Symmetric orbifolds in the large $N$ limit have partition functions which are consistent with an emergent space-time string theory with $L_{\rm string} \sim L_{\rm AdS}$. For supersymmetric CFTs, the elliptic genus can serve as a sensitive probe of whether the SCFT admits a large radius gravity description with $L_{\rm string} \ll L_{\rm AdS}$ after one deforms away from the symmetric orbifold point in moduli space. We discuss several classes of constructions whose elliptic genera strongly hint that gravity with $L_{\rm Planck} \ll L_{\rm string} \ll L_{\rm AdS}$ can emerge at suitable points in moduli space.

Black rings in U(1)3 supergravity and their dual 2d CFT

We study the near-horizon geometry of black ring solutions in five-dimensional U(1)3 supergravity with three electric dipole charges and one angular momentum. We consider the extremal vanishing horizon (EVH) limit of these solutions and show that the near-horizon geometries develop AdS3 throats locally. At the near-EVH near horizon limit, the AdS3 factor turns into a BTZ black hole. By analysing the first law of thermodynamics for black rings we show that at the EVH limit, they reduce to the first law of thermodynamics for BTZ black holes. Using the AdS3/CFT2 duality, we propose a dual CFT to describe the near-horizon low energy dynamics of near-EVH black rings. We also discuss the connection between our CFT proposal and the Kerr/CFT correspondence in the cases where these two overlap.

Deformed SW curve and the null vector decoupling equation in Toda field theory

It is shown that the deformed Seiberg-Witten curve equation after Fourier transform is mapped into a differential equation for the AGT dual 2d CFT cnformal block containing an extra completely degenerate field. We carefully match parameters in two sides of duality thus providing not only a simple independent prove of the AGT correspondence in Nekrasov-Shatashvili limit, but also an extension of AGT to the case when a secondary field is included in the CFT conformal block. Implications of our results in the study of monodromy problems for a large class of n’th order Fuchsian differential equations are discussed.

Seed conformal blocks in 4D CFT

We compute in closed analytical form the minimal set of "seed" conformal blocks associated to the exchange of generic mixed symmetry spinor/tensor operators in an arbitrary representation (l,\bar l) of the Lorentz group in four dimensional conformal field theories. These blocks arise from 4-point functions involving two scalars, one (0,|l-\bar l|) and one (|l-\bar l|,0) spinors or tensors. We directly solve the set of Casimir equations, that can elegantly be written in a compact form for any (l,\bar l), by using an educated ansatz and reducing the problem to an algebraic linear system. Various details on the form of the ansatz have been deduced by using the so called shadow formalism. The complexity of the conformal blocks depends on the value of p=|l-\bar l | and grows with p, in analogy to what happens to scalar conformal blocks in d even space-time dimensions as d increases. These results open the way to bootstrap 4-point functions involving arbitrary spinor/tensor operators in four dimensional conformal field theories.

Classical limit of irregular blocks and Mathieu functions

The Nekrasov-Shatashvili limit of the N=2 SU(2) pure gauge (Omega-deformed) super Yang-Mills theory encodes the information about the spectrum of the Mathieu operator. On the other hand, the Mathieu equation emerges entirely within the frame of two-dimensional conformal field theory (2d CFT) as the classical limit of the null vector decoupling equation for some degenerate irregular block. Therefore, it seems to be possible to investigate the spectrum of the Mathieu operator employing the techniques of 2d CFT. To exploit this strategy, a full correspondence between the Mathieu equation and its realization within 2d CFT has to be established. In our previous paper [1], we have found that the expression of the Mathieu eigenvalue given in terms of the classical irregular block exactly coincides with the well known weak coupling expansion of this eigenvalue in the case in which the auxiliary parameter is the noninteger Floquet exponent. In the present work we verify that the formula for the corresponding eigenfunction obtained from the irregular block reproduces the so-called Mathieu exponent from which the noninteger order elliptic cosine and sine functions may be constructed. The derivation of the Mathieu equation within the formalism of 2d CFT is based on conjectures concerning the asymptotic behaviour of irregular blocks in the classical limit. A proof of these hypotheses is sketched. Finally, we speculate on how it could be possible to use the methods of 2d CFT in order to get from the irregular block the eigenvalues of the Mathieu operator in other regions of the coupling constant.

Irregular blocks, N = 2 gauge theory and Mathieu system

The Alday-Gayotto-Tachikawa (AGT) conjecture relates 4d N = 2, SU(2) SYM theories with Nf matter hypermultiplets to 2d CFT. In case of pure 4d N = 2, SU(2) SYM there is a corresponding irregular conformal block in 2d CFT. The AGT correspondence may be extended within a certain limit (the Nekrasov-Shataschvili limit) to the correspondence between an effective twisted superpotentials of 2d N = 2 SUSY and the Zamolodchikov's “classical” conformal blocks. When narrowed to the pure 4d N = 2 SYM case its limit is related to an irregular classical conformal block. It will be shown that according to the triple correspondence (2dCFT/Gauge/Bethe - c.f. Piatek's talk) the irregular classical conformal block yields spectrum of Mathieu operator. The latter can be obtained as a “classical” limit of the null vector decoupling equation for three-point degenerate irregular block. It will also be shown that the Mathieu spectrum can be also obtained from the limit of the pure gauge theory as a solution of the saddle point equation as well as from the Bohr-Sommerfeld quantization of the Seiberg-Witten theory.

Counting RG flows

Interpreting renormalization group flows as solitons interpolating between different fixed points, we ask various questions that are normally asked in soliton physics but not in renormalization theory. Can one count RG flows? Are there different "topological sectors" for RG flows? What is the moduli space of an RG flow, and how does it compare to familiar moduli spaces of (supersymmetric) dowain walls? Analyzing these questions in a wide variety of contexts --- from counting RG walls to AdS/CFT correspondence --- will not only provide favorable answers, but will also lead us to a unified general framework that is powerful enough to account for peculiar RG flows and predict new physical phenomena. Namely, using Bott's version of Morse theory we relate the topology of conformal manifolds to certain properties of RG flows that can be used as precise diagnostics and "topological obstructions" for the strong form of the C-theorem in any dimension. Moreover, this framework suggests a precise mechanism for how the violation of the strong C-theorem happens and predicts "phase transitions" along the RG flow when the topological obstruction is non-trivial. Along the way, we also find new conformal manifolds in well-known 4d CFT's and point out connections with the superconformal index and classifying spaces of global symmetry groups.

Large interval limit of Rényi entropy at high temperature

In this paper, we propose a novel expansion to compute the large interval limit of the R\'enyi entropy of 2D CFT at high temperature. Via the replica trick, the single interval R\'enyi entropy of 2D CFT at finite temperature could be read from the partition function on $n$-sheeted torus connected with each other along a branch cut. We calculate the partition function by inserting a complete basis across the branch cut. Because of the monodromy condition across the branch cut in the large interval limit, the basis of the states should be the ones in the twist sector. We study the twist sector of a general module of CFT and find that there is an one-to-one correspondence between the twist sector states and the normal sector states. As an application, we revisit the non-compact free scalar theory and discuss the large interval limit of the R\'enyi entropy of this theory by using our proposal. We find complete agreement in the leading and next-leading orders with direct expansion of the exact partition function. Moreover, we prove the relation (\ref{th}) between thermal entropy and the entanglement entropy for a generic CFT with discrete spectrum.

Wilson loop invariants from WN conformal blocks

Knot and link polynomials are topological invariants calculated from the expectation value of loop operators in topological field theories. In 3D Chern–Simons theory, these invariants can be found from crossing and braiding matrices of four-point conformal blocks of the boundary 2D CFT. We calculate crossing and braiding matrices for WN conformal blocks with one component in the fundamental representation and another component in a rectangular representation of SU(N), which can be used to obtain HOMFLY knot and link invariants for these cases. We also discuss how our approach can be generalized to invariants in higher-representations of WN algebra.

Deconstructing conformal blocks in 4D CFT

We show how conformal partial waves (or conformal blocks) of spinor/tensor correlators can be related to each other by means of differential operators in four dimensional conformal field theories. We explicitly construct such differential operators for all possible conformal partial waves associated to four-point functions of arbitrary traceless symmetric operators. Our method allows any conformal partial wave to be extracted from a few “seed” correlators, simplifying dramatically the computation needed to bootstrap tensor correlators.

Worldline approach to semi-classical conformal blocks

We extend recent results on semi-classical conformal blocks in 2d CFT and their relation to 3D gravity via the AdS/CFT correspondence. We consider four-point functions with two heavy and two light external operators, along with the exchange of a light operator. By explicit computation, we establish precise agreement between these CFT objects and a simple picture of particle worldlines joined by cubic vertices propagating in asymptotically AdS3 geometries (conical defects or BTZ black holes). We provide a simple argument that explains this agreement.

Generalized information and entanglement entropy, gravitation and holography

A nonextensive statistical mechanics entropy that depends only on the probability distribution is proposed in the framework of superstatistics. It is based on a Γ(χ2) distribution that depends on β and also on pl. The corresponding modified von Neumann entropy is constructed; it is shown that it can also be obtained from a generalized Replica trick. We further demonstrate a generalized H-theorem. Considering the entropy as a function of the temperature and volume, it is possible to generalize the equation of state of an ideal gas. Moreover, following the entropic force formulation a generalized Newton's law is obtained, and following the proposal that the Einstein equations can be deduced from the Clausius law, we discuss on the structure that a generalized Einstein's theory would have. Lastly, we address the question whether the generalized entanglement entropy can play a role in the gauge/gravity duality. We pay attention to 2d CFT and their gravity duals. The correction terms to the von Neumann entropy result more relevant than the usual UV ones and also than those due to the area dependent AdS3 entropy which result comparable to the UV ones. Then the correction terms due to the new entropy would modify the Ryu–Takayanagi identification between the CFT entanglement entropy and the AdS entropy in a different manner than the UV ones or than the corrections to the AdS3 area dependent entropy.

Modular invariance and entanglement entropy

We study the Rényi and entanglement entropies for free 2d CFT’s at finite temperature and finite size, with emphasis on their properties under modular transformations of the torus. We address the issue of summing over fermion spin structures in the replica trick, and show that the relation between entanglement and thermal entropy determines two different ways to perform this sum in the limits of small and large interval. Both answers are modular covariant, rather than invariant. Our results are compared with those for a free boson at unit radius in the two limits and complete agreement is found, supporting the view that entanglement respects Bose-Fermi duality. We extend our computations to multiple free Dirac fermions having correlated spin structures, dual to free bosons on the Spin(2d) weight lattice.