Poincar\'e Series, 3d Gravity and Averages of Rational CFT

TL;DR

This paper explores the Poincaré series approach to 3d gravity duals of Rational CFTs, showing that certain models can be viewed as averages over multiple CFTs and analyzing their properties across different models and boundary conditions.

Contribution

It demonstrates that for specific WZW models, the Poincaré series yields a positive combination of modular-invariant partition functions, supporting an average CFT interpretation of the bulk gravity theory.

Findings

SU(2)$_k$ models provide unitary examples with positive Poincaré series.

The weights of the average CFTs are computed for all seed primaries and relevant levels.

Different WZW models exhibit distinct features in their Poincaré sums and boundary contributions.

Abstract

We investigate the Poincar\'e 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 Poincar\'e 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.