Poincaré Series, 3d Gravity and Averages of Rational CFT

TL;DR

The article investigates whether 3d gravity partition functions with AdS$_3$ boundaries can be understood as averages over RCFTs that share a Kac–Moody algebra, using Poincaré sums over modular group cosets. By analyzing seeds from various RCFTs (notably SU(2)$_k$, SU($N$)$_k$, and Virasoro minimal models) and classifying the resulting modular invariants, the authors identify infinite families where the gravity partition function is a nonnegative linear combination of physical invariants (an ensemble average), as well as cases with unphysical contributions. Extending the framework to multiple genus-1 boundaries reveals connected (wormhole-like) contributions that factorize into products of seed differences $(Z_A-Z_D)$, enabling a consistent multi-boundary averaging picture in several models (notably SU(2)$_k$). The work illuminates how ensemble interpretations of AdS$_3$ gravity can emerge from RCFT data, provides explicit weight formulas, and suggests robust seeds to reproduce multi-boundary correlations, with potential implications for holography beyond semi-classical gravity.

Abstract

We investigate the Poincaré approach to computing 3d gravity partition functions dual to Rational CFT. For a single genus-1 boundary, we show that for certain infinite sets of levels, the SU(2)$_k$ WZW models provide unitary examples for which the Poincare series is a positive linear combination of two modular-invariant partition functions. This supports the interpretation that the bulk gravity theory (a topological Chern-Simons theory in this case) is dual to an average of distinct CFT's sharing the same Kac-Moody algebra. We compute the weights of this average for all seed primaries and all relevant values of k. We then study other WZW models, notably SU($N$)$_1$ and SU(3)$_k$, and find that each class presents rather different features. Finally we consider multiple genus-1 boundaries, where we find a class of seed functions for the Poincaré sum that reproduces both disconnected and connected contributions -- the latter corresponding to analogues of 3-manifold "wormholes" -- such that the expected average is correctly reproduced.