Quantum Gravity Partition Functions in Three Dimensions

TL;DR

This paper investigates the partition function of three-dimensional quantum gravity with negative cosmological constant, exploring quantum corrections, potential additional contributions, and phase transitions, with implications for supergravity.

Contribution

It provides an exact computation of the partition function including quantum corrections and explores the necessity of additional contributions like long strings or complex geometries.

Findings

Quantum corrections can be exactly computed.

Holomorphic factorization reproduces subleading entropy corrections.

The Hawking-Page transition is a Lee-Yang type phase transition.

Abstract

We consider pure three-dimensional quantum gravity with a negative cosmological constant. The sum of known contributions to the partition function from classical geometries can be computed exactly, including quantum corrections. However, the result is not physically sensible, and if the model does exist, there are some additional contributions. One possibility is that the theory may have long strings and a continuous spectrum. Another possibility is that complex geometries need to be included, possibly leading to a holomorphically factorized partition function. We analyze the subleading corrections to the Bekenstein-Hawking entropy and show that these can be correctly reproduced in such a holomorphically factorized theory. We also consider the Hawking-Page phase transition between a thermal gas and a black hole and show that it is a phase transition of Lee-Yang type, associated with a condensation of zeros in the complex temperature plane. Finally, we analyze pure three-dimensional supergravity, with similar results.