AdS_3 Partition Functions Reconstructed

TL;DR

This paper reinterprets the partition functions of pure AdS_3 gravity as sums over geometries, providing a more physical perspective on their modular properties and expressing key functions as geometric sums.

Contribution

It introduces a new modular sum representation of the AdS_3 gravity partition function, linking it to a sum over geometries rather than purely algebraic constructions.

Findings

Partition function expressed as a modular sum over geometries

Reformulation of the j-function and its derivative in geometric terms

Enhanced physical understanding of AdS_3 gravity partition functions

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.