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Exact Path Integral for 3D Quantum Gravity.

Norihiro Iizuka¹, Akinori Tanaka^1,2, Seiji Terashima³

¹Department of Physics, Osaka University, Toyonaka, Osaka 560-0043, Japan.

Physical Review Letters

|November 10, 2015

Summary

We calculated the quantum gravity partition function using Chern-Simons theory and localization techniques. The resulting modular-invariant partition function matches Witten

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Area of Science:

Theoretical physics
Quantum gravity
String theory

Background:

Three-dimensional Euclidean pure gravity with a negative cosmological constant is classically described by Chern-Simons theory.
This formulation can be extended to a supersymmetric framework by incorporating auxiliary gauginos and scalars.

Purpose of the Study:

To compute the exact partition function of the supersymmetric Chern-Simons theory.
To derive the quantum gravity partition function by summing over topologically distinct manifolds.

Main Methods:

Utilizing the localization technique to calculate the exact partition function of the Chern-Simons theory.
Nonperturbative summation of partition functions over different topological manifolds.

Main Results:

The exact partition function of the Chern-Simons theory was successfully computed.
The resulting quantum gravity partition function is modular invariant.
For a central charge of 24, the partition function corresponds to the J function predicted by Witten.

Conclusions:

The study provides a nonperturbative method for calculating quantum gravity partition functions.
The findings connect Chern-Simons theory, localization, and modular forms in the context of quantum gravity.
The results support Witten's predictions regarding the J function in quantum gravity.