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arXiv:1612.05891 · Journal of High Energy Physics

Various semiclassical limits of torus conformal blocks

Konstantin Alkalaev, Roman Geiko, Vladimir Rappoport

Published Apr 1, 2017• 24 citations
arXivHTMLPDF

Abstract

We study four types of one-point torus blocks arising in the large central charge regime. There are the global block, the light block, the heavy-light block, and the linearized classical block, according to different regimes of conformal dimensions. It is shown that the blocks are not independent being connected to each other by various links. We find that the global, light, and heavy-light blocks correspond to three different contractions of the Virasoro algebra. Also, we formulate the c-recursive representation of the one-point torus blocks which is relevant in the semiclassical approximation.

DOI
10.1007/jhep04(2017)070

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We study four types of one-point torus blocks arising in the large central charge regime. There are the global block, the light block, the heavy-light block, and the linearized classical block, according to different regimes of conformal dimensions. It is shown that the blocks are not independent being connected to each other by various links. We find that the global, light, and heavy-light blocks correspond to three different contractions of the Virasoro algebra. Also, we formulate the c-recursive representation of the one-point torus blocks which is relevant in the semiclassical approximation.
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.
Virasoro conformal blocks are fixed in principle by symmetry, but a closed-form expression is unknown in the general case. In this work, we provide three closed-form expansions for the four-point Virasoro blocks on the sphere, for arbitrary operator dimensions and central charge c. We do so by solving known recursion relations. One representation is a sum over hypergeometric global blocks, whose coefficients we provide at arbitrary level. Another is a sum over semiclassical Virasoro blocks obtained in the limit in which two external operator dimensions scale linearly with large c. In both cases, the 1/c expansion of the Virasoro blocks is easily extracted. We discuss applications of these expansions to entanglement and thermality in conformal field theories and particle scattering in three-dimensional quantum gravity.
Virasoro conformal blocks are fixed in principle by symmetry, but a closed-form expression is unknown in the general case. In this work, we provide two new closed-form expansions for the four-point Virasoro blocks on the sphere, for arbitrary operator dimensions and central charge c. We do so by solving known recursion relations. One representation is a sum over hypergeometric global blocks, whose coefficients we provide at arbitrary level. The other is a sum over semiclassical Virasoro blocks obtained in the limit in which two external operator dimensions scale linearly with large c. In both cases, the 1/c expansion of the Virasoro blocks is easily extracted. We discuss applications of these expansions to entanglement and thermality in conformal field theories and particle scattering in three-dimensional quantum gravity.
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.
A bstract A long-standing conjecture claims that Virasoro conformal blocks exponentiate in the semiclassical limit c → ∞ with h / c finite. However, this has been proven only for four-point blocks on the sphere and one-point blocks on the torus. Here we extend the proof to general conformal blocks for higher-point functions and higher-genus backgrounds in arbitrary channels. The statement is to be understood at the level of a formal power series. Our proof builds upon a novel extension of the oscillator method for the computation of conformal blocks to cases where three internal lines meet at a vertex. This extension also gives a new constructive method to compute global conformal blocks in 2d CFTs at general genus.
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.