Exact Path Integral for 3D Quantum Gravity

TL;DR

This paper computes the exact partition function of 3D Euclidean quantum gravity with negative cosmological constant using localization in Chern-Simons theory, revealing a modular invariant result matching Witten's predictions.

Contribution

It provides the first exact, non-perturbative partition function for 3D quantum gravity via localization of Chern-Simons theory, confirming the J-function in specific cases.

Findings

Partition function is modular invariant.

Matches Witten's J-function for central charge 24.

Supports non-perturbative quantum gravity formulations.

Abstract

Three dimensional Euclidean pure gravity with a negative cosmological constant can be formulated in terms of the Chern-Simons theory, classically. This theory can be written in a supersymmetric way by introducing auxiliary gauginos and scalars. We calculate the exact partition function of this Chern-Simons theory by using the localization technique. Thus, we obtain the quantum gravity partition function, assuming that it can be obtained non-perturbatively by summing over partition functions of the Chern-Simons theory on topologically different manifolds. The resultant partition function is modular invariant, and in the case in which the central charge is expected to be 24, it is the J-function, predicted by Witten.