Virasoro Conformal Blocks and Thermality from Classical Background Fields

TL;DR

This work shows that in 2d CFTs at large central charge, the coupling of the stress tensor to heavy operators can be absorbed into a non-trivial background metric, allowing heavy-light Virasoro blocks to be computed as global blocks in a transformed coordinate system. It provides a precise semi-classical block formula, extends the construction to U(1) currents, and furnishes a Zamolodchikov-style recursion to access finite-$c$ corrections. The authors interpret these results in AdS$_3$/CFT$_2$, linking heavy states to deficit/BTZ geometries and deriving bulk-boundary consistency for three-point and entanglement-entropy observables, including thermal behavior in appropriate limits. These findings illuminate the universality of black hole thermality in AdS$_3$ and offer a concrete, background-field-based framework for studying Virasoro blocks, thermality, and holographic locality at large central charge.

Abstract

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 $AdS_3$, or equivalently, the eigenstate thermalization hypothesis for $CFT_2$ at large central charge.

Figures (3)

• Figure 1: This figure suggests an AdS picture for the heavy-light CFT correlator -- a light probe interacting with a heavy particle or black hole. We will show that the Virasoro blocks can be computed in this limit by placing the CFT$_2$ in an appropriately chosen 2d metric, which is related to the flat metric by a conformal transformation.
• Figure 2: This figure suggests how the exchange of all Virasoro descendants of a state $| h \rangle$ corresponds to the exchange of $| h \rangle$ plus any number of gravitons in AdS$_3$. This is sufficient to build the full, non-perturbative AdS$_3$ gravitational field entirely from the CFT$_2$.
• Figure 3: This figure suggests the conformal mapping from $z$ to $w$ coordinates, so that stress-tensor exchange between heavy and light operators has been absorbed by the new background metric. Note that we have placed the heavy operators at $0$ and $\infty$ for simplicity, although we place one of them at $1$ to emphasize a particular OPE limit in section \ref{['sec:HeavyOperatorsClassicalBackgrounds']}.