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arXiv:2409.13766

Study of Information Restoring Effects in Virasoro Blocks

Shivrat Sachdeva

Published Sep 18, 2024• 1 citation
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Abstract

It is well-known that in the large central charge $c$ limit, the Virasoro blocks in CFT$_2$ suffer from information loss when studying the BTZ black holes in the dual AdS$_3$ theory. Recent studies have proved that non-perturbative $e^{-c}$ corrections resolves information loss. Here, we study universal features of the spectrum of heavy states from general properties of finite $c$ Virasoro conformal blocks. Furthermore, we investigate the behavior of correlators beyond "Page" time $ t \sim S_{BH}$, entropy of the black hole. In the Lorentzian regime, we wish to better understand the transition from an exponential decay behaviour of Virasoro blocks in the heavy-light limit to a power-law decay at late times.

DOI
10.48550/arxiv.2409.13766

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It is well-known that in the large central charge ccc limit, the Virasoro blocks in CFT2_22​ suffer from information loss when studying the BTZ black holes in the dual AdS3_33​ theory. Recent studies have proved that non-perturbative e−ce^{-c}e−c corrections resolves information loss. Here, we study universal features of the spectrum of heavy states from general properties of finite ccc Virasoro conformal blocks. Furthermore, we investigate the behavior of correlators beyond "Page" time t∼SBH t \sim S_{BH}t∼SBH​, entropy of the black hole. In the Lorentzian regime, we wish to better understand the transition from an exponential decay behaviour of Virasoro blocks in the heavy-light limit to a power-law decay at late times.
We show that in 2d CFTs at large central charge, the coupling of the stress tensor to heavy operators can be re-absorbed by placing the CFT in a non-trivial background metric. This leads to a more precise computation of the Virasoro conformal blocks between heavy and light operators, which are shown to be equivalent to global conformal blocks evaluated in the new background. We also generalize to the case where the operators carry U(1) charges. The refined Virasoro blocks can be used as the seed for a new Virasoro block recursion relation expanded in the heavy-light limit. We comment on the implications of our results for the universality of black hole thermality in AdS3, or equivalently, the eigenstate thermalization hypothesis for CFT2 at large central charge.
We consider the semiclassical limit of the vacuum Virasoro block describing the diagonal 4-point correlation functions on the sphere. At large central charge c, after exponentiation, it depends on two fixed ratios h H /c and h L /c, where h H,L are the conformal dimensions of the 4-point function operators. The semiclassical block may be expanded in powers of the light ratio h L /c and the leading non-trivial (linear) order is known in closed form as a function of h H /c. Recently, this contribution has been matched against AdS3 gravity calculations where heavy operators build up a classical geometry corresponding to a BTZ black hole, while the light operators are described by a geodesic in this background. Here, we compute for the first time the next-to-leading quadratic correction O \mathcal{O} O ((h L /c)2), again in closed form for generic heavy operator ratio h H /c. The result is a highly non-trivial extension of the leading order and may be relevant for further refined AdS3/CFT2 tests. Applications to the two-interval Rényi entropy are also presented.
Abstract We derive a universal asymptotic formula for generic boundary conditions for the average value of the bulk-to-boundary and boundary Operator Product Expansion coefficients of any unitary, compact two-dimensional Boundary CFT (BCFT) with c > 1. The asymptotic limit consists of taking one or more boundary primary operators — which transform under a single copy of the Virasoro algebra — to have parametrically large conformal dimension for fixed central charge. In particular, we find a single universal expression that interpolates between distinct heavy regimes, exactly as in the case of bulk OPE asymptotics [1]. The expression depends universally on the boundary entropy and the central charge, and not on any other details of the theory. We derive these asymptotics by studying crossing symmetry of various correlation functions on higher genus Riemann surfaces with open boundaries. Essential in the derivation is the use of the irrational versions of the crossing kernels that relate holomorphic Virasoro blocks in different channels. Our results strongly suggest an extended version of the Eigenstate Thermalization Hypothesis for boundary OPE coefficients, where the hierarchy between the diagonal and non-diagonal term in the ansatz is further controlled by the boundary entropy. We finally comment on the applications of our results in the context of AdS3/BCFT2, as well as on the recent relation of BCFTs with lower dimensional models of evaporating black holes.
Abstract We explore the structures of light cone and Regge limit singularities of n-point Virasoro conformal blocks in c > 1 two-dimensional conformal field theories with no chiral primaries, using fusion matrix approach. These CFTs include not only holographic CFTs dual to classical gravity, but also their full quantum corrections, since this approach allows us to explore full 1/c corrections. As the important applications, we study time dependence of Renyi entropy after a local quench and out-of-time ordered correlator (OTOC) at late time. We first show that, the n-th (n > 2) Renyi entropy after a local quench in our CFT grows logarithmically at late time, for any c and any conformal dimensions of excited primary. In particular, we find that this behavior is independent of c, contrary to the expectation that the finite c correction fixes the late time Renyi entropy to be constant. We also show that the constant part of the late time Renyi entropy is given by a monodromy matrix. We also investigate OTOCs by using the monodromy matrix. We first rewrite the monodromy matrix in terms of fusion matrix explicitly. By this expression, we find that the OTOC decays exponentially in time, and the decay rates are divided into three patterns, depending on the dimensions of external operators. We note that our result is valid for any c > 1 and any external operator dimensions. Our monodromy matrix approach can be generalized to the Liouville theory and we show that the Liouville OTOC approaches constant in the late time regime. We emphasize that, there is a number of other applications of the fusion and the monodromy matrix approaches, such as solving the conformal bootstrap equation. Therefore, it is tempting to believe that the fusion and monodromy matrix approaches provide a key to understanding the AdS/CFT correspondence.
A bstract The light cone OPE limit provides a significant amount of information regarding the conformal field theory (CFT), like the high-low temperature limit of the partition function. We started with the light cone bootstrap in the general CFT 2 with c > 1. For this purpose, we needed an explicit asymptotic form of the Virasoro conformal blocks in the limit z → 1, which was unknown until now. In this study, we computed it in general by studying the pole structure of the fusion matrix (or the crossing kernel). Applying this result to the light cone bootstrap, we obtained the universal total twist (or equivalently, the universal binding energy) of two particles at a large angular momentum. In particular, we found that the total twist is saturated by the value c−112 \frac{c-1}{12} 12c−1​ c − 1 12 if the total Liouville momentum exceeds beyond the BTZ threshold . This might be interpreted as a black hole formation in AdS 3 . As another application of our light cone singularity, we studied the dynamics of entanglement after a global quench and found a Renyi phase transition as the replica number was varied. We also investigated the dynamics of the 2nd Renyi entropy after a local quench. We also provide a universal form of the Regge limit of the Virasoro conformal blocks from the analysis of the light cone singularity. This Regge limit is related to the general n -th Renyi entropy after a local quench and out of time ordered correlators.
We consider Virasoro conformal blocks in the large central charge limit. There are different regimes depending on the behavior of the conformal dimensions. The most simple regime is reduced to the global sl(2,C) sl\left(2,\mathrm{\mathbb{C}}\right) sl(2,C) conformal blocks while the most complicated one is known as the classical conformal blocks. Recently, Fitzpatrick, Kaplan, and Walters showed that the two regimes are related through the intermediate stage of the so-called heavy-light semiclassical limit. We study this idea in the particular case of the 5-point conformal block. To find the 5-point global block we use the projector technique and the Casimir operator approach. Furthermore, we discuss the relation between the global and the heavy-light limits and construct the heavy-light block from the global block. In this way we reproduce our previous results for the 5-point perturbative classical block obtained by means of the monodromy method.