In two spacetime dimensions the Virasoro heavy-heavy-light-light (HHLL) vacuum block in a certain limit is governed by the Catalan numbers. The equation for their generating function can be generalized to a differential equation which the logarithm of the block satisfies. We show that a similar story holds for the HHLL   <math> <mi>W</mi> </math>  $$ \mathcal{W} $$   N  vacuum blocks, where a suitable generalization of the Catalan numbers plays the main role. Moreover, the   <math> <mi>W</mi> </math>  $$ \mathcal{W} $$   N  blocks have the same form as the stress tensor sector of HHLL near lightcone conformal correlators in 2(N − 1) spacetime dimensions. In the latter case the Catalan numbers are generalized to the numbers of linear extensions of certain partially ordered sets.

Ireland

Robin Karlsson

Manuela Kulaxizi

Gim Ng

Andrei Parnachev

United States

Petar Tadić

CFT correlators,   <math> <mi>W</mi> </math>  $$ \mathcal{W} $$ -algebras and generalized Catalan numbers

We study warped flat geometries in three-dimensional topologically massive gravity. They are quotients of global warped flat spacetime, whose isometries are given by the 2-dimensional centrally extended Poincaré algebra. The latter can be obtained as a certain scaling limit of Warped AdS3 space with a positive cosmological constant. We discuss the causal structure of the resulting spacetimes using projection diagrams. We study their charges and thermodynamics, together with asymptotic Killing vectors preserving a consistent set of boundary conditions including them. The asymptotic symmetry group is given by a Warped CFT algebra, with a vanishing current level. A generalization of the derivation of the Warped CFT Cardy formula applies in this case, reproducing the entropy of the warped flat cosmological spacetimes.

Belgium

Stéphane Detournay

Wout Merbis

Austria

Raphaela Wutte

Warped flatland

The eikonal phase which determines the Regge limit of the gravitational scat- tering amplitude of a light particle off a heavy one in Minkowski spacetimes admits an expansion in the ratio of the Schwarzschild radius of the heavy particle to the impact parameter. Such an eikonal phase in AdS spacetimes of any dimensionality has been com- puted to all orders and reduces to the corresponding Minkowski result when both the impact parameter and the Schwarzschild radius are much smaller than the AdS radius. The leading term in the AdS eikonal phase can be reproduced in the dual CFT by a single stress tensor conformal block, but the subleading term is a result of an infinite sum of the double stress tensor contributions. We provide a closed form expression for the OPE coef- ficients of the leading twist double stress tensors in four spacetime dimensions and perform the sum to compute the corresponding lightcone behavior of a heavy-heavy-light-light CFT correlator. The resulting compact expression passes a few nontrivial independent checks. In particular, it agrees with the subleading eikonal phase at large impact parameter.

Subleading eikonal, AdS/CFT and double stress tensors

Two dimensional CFTs have an infinite set of commuting conserved charges, known as the quantum KdV charges, built out of the stress tensor. We compute the thermal correlation functions of the these KdV charges on a circle. We show that these correlation functions are given by quasi-modular differential operators acting on the torus partition function. We determine their modular transformation properties, give explicit expressions in a number of cases, and give an expression for an arbitrary correlation function which is determined up to a finite number of functions of the central charge. We show that these modular differential operators annihilate the characters of the (2m + 1, 2) family of non-unitary minimal models. We also show that the distribution of KdV charges becomes sharply peaked at large level.

Canada

Alexander Maloney

United Kingdom

Simon Ross

Ioannis Tsiares

Thermal correlation functions of KdV charges in 2D CFT

Two-dimensional CFTs have an infinite set of commuting conserved charges, known as the quantum KdV charges. We study the Generalized Gibbs Ensemble with chemical potentials for these charges at high temperature. In a large central charge limit, the partition function can be computed in a saddle-point approximation. We compare the ensemble values of the KdV charges to the values in a microstate, and find that they match irrespective of the values of the chemical potentials. We study the partition function at finite central charge perturbatively in the chemical potentials, and find that this degeneracy is broken. We also study the statistics of the KdV charges at high level within a Virasoro representation, and find that they are sharply peaked.

Generalized Gibbs ensemble and the statistics of KdV charges in 2D CFT

We give a holographic description of global conformal blocks in two dimensional conformal field theory on the sphere and on the torus. We show that the conformal blocks for one-point functions on the torus can be written as Witten diagrams in thermal AdS. This is accomplished by deriving a general conformal Casimir equation for global conformal blocks, and showing that Witten diagrams obey the same equation. We study the semi-classical limit of n -point conformal blocks, and show that these equal the action of a network of bulk world-lines obeying appropriate geodesic equations. We give an alternate description in the Chern-Simons formulation of 3D gravity, where the conformal blocks are described by networks of Wilson lines, and argue that these formulations are equivalent.

Per Kraus

Henry Maxfield

China

Jie-qiang Wu

Witten diagrams for torus conformal blocks

We investigate the constraints of crossing symmetry on CFT correlation functions. Four point conformal blocks are naturally viewed as functions on the upper-half plane, on which crossing symmetry acts by PSL(2, ℤ $$ \mathbb{Z} $$ ) modular transformations. This allows us to construct a unique, crossing symmetric function out of a given conformal block by averaging over PSL(2, ℤ $$ \mathbb{Z} $$ ). In some two dimensional CFTs the correlation functions are precisely equal to the modular average of the contributions of a finite number of light states. For example, in the two dimensional Ising and tri-critical Ising model CFTs, the correlation functions of identical operators are equal to the PSL(2, ℤ $$ \mathbb{Z} $$ ) average of the Virasoro vacuum block; this determines the 3 point function coefficients uniquely in terms of the central charge. The sum over PSL(2, ℤ $$ \mathbb{Z} $$ ) in CFT 2 has a natural AdS 3 interpretation as a sum over semi-classical saddle points, which describe particles propagating along rational tangles in the bulk. We demonstrate this explicitly for the correlation function of certain heavy operators, where we compute holographically the semi-classical conformal block with a heavy internal operator.

A conformal block Farey tail

We derive the holographic entanglement entropy contribution from pure and mixed gravitational Chern-Simons(CS) terms in AdS 2 k +1 . This is done through two different methods: first, by a direct evaluation of CS action in a holographic replica geometry and second by a descent of Dong’s derivation applied to the corresponding anomaly polynomial. In lower dimensions ( k = 1 , 2), the formula coincides with the Tachikawa formula for black hole entropy from gravitational CS terms. New extrinsic curvature corrections appear for k ≥ 3: we give explicit and concise expressions for the two pure gravitational CS terms in AdS 7 and present various consistency checks, including agreements with the black hole entropy formula when evaluated at the bifurcation surface.

France

Tatsuo Azeyanagi

R. Loganayagam

Holographic entanglement for Chern-Simons terms

We extend the results of [1], computing one loop partition functions for massive fields with spin half in AdS 2 using the quasinormal mode method proposed by Denef, Hartnoll, and Sachdev [2]. We find the finite representations of SO(2, 1) for spin zero and spin half, consisting of a highest weight state | h 〉 and descendants with non-unitary values of h . These finite representations capture the poles and zeroes of the one loop determinants. Together with the asymptotic behavior of the partition functions (which can be easily computed using a large mass heat kernel expansion), these are sufficient to determine the full answer for the one loop determinants. We also discuss extensions to higher dimensional AdS 2 n and higher spins.

Denmark

Cynthia Keeler

Pedro Lisbão

Partition functions with spin in AdS 2 via quasinormal mode methods

We consider warped AdS 3 black holes in generic higher derivatives gravity theories in 2+1 dimensions. The asymptotic symmetry group of the phase space containing these black holes is the semi-direct product of a centrally extended Virasoro algebra and an affine u (1) Kac-Moody algebra. Previous works have shown that in some specific theories, the entropy of these black holes agrees with a Cardy-like entropy formula derived for warped conformal field theories. In this paper, we show that this entropy matching continues to hold for the most general higher derivative theories of gravity. We also discuss the existence of phase transitions.