Poincaré series, 3d gravity and averages of rational CFT

<jats:title>A<jats:sc>bstract</jats:sc>
</jats:title><jats:p>We investigate the Poincaré 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)<jats:sub><jats:italic>k</jats:italic></jats:sub> WZW models provide unitary examples for which the Poincaré 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 <jats:italic>k</jats:italic>. We then study other WZW models, notably SU(<jats:italic>N</jats:italic>)<jats:sub>1</jats:sub> and SU(3)<jats:sub><jats:italic>k</jats:italic></jats:sub>, 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.</jats:p>