Hawking from Catalan

TL;DR

The paper develops a novel on-shell diagrammatic framework for summing Virasoro graviton exchanges in AdS3/CFT2 within a heavy–light limit, revealing that the Virasoro vacuum block satisfies a Catalan-like differential recursion. By orthonormalizing Virasoro descendants and organizing exchanges into binary tree diagrams, the authors compute the vacuum block to all orders and show it takes a thermal-like form, linking to the Hawking temperature of BTZ black holes. This approach provides a non-perturbative window into gravitational dynamics from CFT data and suggests deep connections between Virasoro symmetry, thermodynamics, and potential bulk interpretations that could extend to higher dimensions. The work also points to broad future directions, including bulk derivations, connections to established Virasoro techniques, and universal thermodynamic behavior at large central charge.

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.

Figures (5)

• Figure 1: This figure provides a suggestive depiction of how graviton exchanges in AdS build up the classical field experienced by a light probe. We will compute the 'graviton' exchanges in the CFT$_2$ by explicitly summing over the exchange of all multi-stress tensor operators. Note that our computation is not equivalent to the calculation of a sum of Witten diagrams; decomposed in terms of the exchange of states, Witten diagrams include both graviton and double-trace operator exchanges (as first noted in Liu, and later substantially developed and systematized in DO1DO2).
• Figure 2: This figure summarizes the diagrammatic rules used to construct the Virasoro vacuum block. The semi-classical Virasoro block for $\langle{\cal O}_H {\cal O}_H {\cal O}_L {\cal O}_L\rangle$ is constructed by exponentiating all diagrams with an arbitrary number of initial gravitons coupled to the heavy operator, which cascade into one final graviton that couples to the light operator. An example of one such diagram is shown in figure \ref{['fig:VirasoroDiagramExample']}.
• Figure 3: This figure indicates how the Virasoro diagrammatic rules work for a 5-to-1 diagram. To complete the rules, we multiply all the displayed propagators by the vertex rules displayed in figure \ref{['fig:VirasoroRules']} and an overall factor of $2 \left( \frac{6^5 z^s}{s(s-1)} \right)$, where $s$ is the sum of all $a_i$.
• Figure 4: This figure indicates what kinds of structures can arise when we add a $k^{th}$ graviton to a $k-1$ graviton diagram. Stars indicate a sum over possible sub-diagrams. The diagram on the left is of the 'trunk' type, while the one on the right is the opposite extreme, where $k$ attaches to a single other graviton. The central diagram lies between these two extremes. In essence, the algebraic manipulations of appendix \ref{['sec:AppendixDetailsDerivationRules']} massage diagrams of the first and last types (which are accompanied by various prefactors) into a sum over diagrams of all types, redistributing the $k$th graviton into all possible configurations.
• Figure 5: This figure depicts the high-temperature limit of a large AdS$_{d+1}$ black hole. By studying short-distance physics in the CFT, with distances measured in units of the inverse temperature, we can recover the physics of a CFT in $R^d$ at finite temperature, which is dual to a flat black brane in Poincaré patch coordinates.