Poincaré Series, 3d Gravity and Averages of Rational CFT

Viraj Meruliya; Sunil Mukhi; Palash Singh

doi:10.1007/JHEP04(2021)267

paper

Poincaré Series, 3d Gravity and Averages of Rational CFT

arXiv:2102.03136 · doi:10.1007/JHEP04(2021)267

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.

52 pages, 8 tables, no chairs

Poincaré Series, 3d Gravity and Averages of Rational CFT

Abstract

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