AdS_3 Partition Functions Reconstructed

TL;DR

The paper reframes holomorphically factorized AdS3 gravity partition functions as modular sums (Poincaré series) over cosets of SL(2,Z), giving a physical interpretation as sums over geometries. It shows how the $J(\tau)$ partition function for $k=1$ and its higher-$k$ generalizations can be constructed from polar data, and introduces the Farey transform $DJ(\tau)$ as a convergent weight-2 series with a gravity-like measure, aligning the math with gravity path integrals. Extensions to larger central charges follow by differentiation and Laurent-expansion of polar parts. The work also highlights puzzles about geometries without classical realizations and non-geometric contributions arising in holomorphic factorization, pointing to remaining conceptual questions about the gravity/CFT dictionary in AdS3.

Abstract

For pure gravity in AdS_3, Witten has given a recipe for the construction of holomorphically factorizable partition functions of pure gravity theories with central charge c=24k. The partition function was found to be a polynomial in the modular invariant j-function. We show that the partition function can be obtained instead as a modular sum which has a more physical interpretation as a sum over geometries. We express both the j-function and its derivative in terms of such a sum.