Virasoro vacuum block at next-to-leading order in the heavy-light limit

TL;DR

The paper advances the semiclassical analysis of the Virasoro vacuum block in the heavy-light limit by computing the next-to-leading quadratic correction in the light-to-central-charge ratio, $O((h_L/c)^2)$, for generic heavy-to-light configurations. It employs monodromy techniques and Zamolodchikov recursion to derive a closed expression for the derivative $\partial_z f^{(2)}(\lambda,z)$ and provides explicit forms for the two subleading components $f^{(2,2)}(z)$ and $f^{(2,3)}(z)$ through α-expansions, supplemented by checks against higher-order perturbative data. The results feed into a refined understanding of two-interval Rényi entropy at large $c$, yielding closed-form contributions and verified consistency with known limits. These findings strengthen tests of AdS$_3$/CFT$_2$ through back-reaction effects in BTZ backgrounds and set the stage for further extensions to extended symmetries and nontrivial intermediate states.

Abstract

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 AdS_3 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((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 AdS_{3}/CFT_{2} tests. Applications to the two-interval Rényi entropy are also presented.