Poincare series, 3d gravity and averages of rational CFT

We investigate the Poincare series 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 Poincare 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.

Automated summary

In this study, the Poincare series approach is used to compute 3d gravity partition functions dual to Rational CFT. The SU(2)(k) WZW models provide unitary examples for certain infinite sets of levels, supporting the interpretation of the bulk gravity theory as dual to an average of distinct CFT's. Different features are found in other WZW models, such as SU(N)(1) and SU(3)(k). Multiple genus-1 boundaries are considered, revealing a class of seed functions for the Poincare sum that accurately reproduces both disconnected and connected contributions.