Hawking from Catalan

TL;DR

This paper develops a new diagrammatic approach to compute Virasoro conformal blocks in AdS$_3$/CFT$_2$, revealing a recursion relation linked to Catalan numbers and connecting to black hole thermodynamics.

Contribution

It introduces on-shell tree diagrams for Virasoro graviton exchanges and derives a recursion relation generalizing Catalan numbers, enabling all-order summation of the vacuum block.

Findings

Derived a differential recursion relation for the Virasoro vacuum block.

Summed the on-shell diagrams to all orders using the recursion.

Connected the large $h_H/c$ limit to the Hawking temperature of BTZ black holes.

Abstract

The Virasoro algebra determines all `graviton' matrix elements in AdS$_3$/CFT$_2$. We study the explicit exchange of any number of Virasoro gravitons between heavy and light CFT$_2$ operators at large central charge. These graviton exchanges can be written in terms of new on-shell tree diagrams, organized in a perturbative expansion in $h_H/c$, the heavy operator dimension divided by the central charge. The Virasoro vacuum conformal block, which is the sum of all the tree diagrams, obeys a differential recursion relation generalizing that of the Catalan numbers. We use this recursion relation to sum the on-shell diagrams to all orders, computing the Virasoro vacuum block. Extrapolating to large $h_H/c$ determines the Hawking temperature of a BTZ black hole in dual AdS$_3$ theories.