Chern-Simons theory, matrix integrals, and perturbative three-manifold invariants

TL;DR

This paper develops a method to compute universal perturbative invariants of rational homology spheres using Chern-Simons theory, matrix integrals, and Gaussian ensembles, with explicit results for Seifert spaces.

Contribution

It extends previous work by expressing Chern-Simons partition functions for Seifert spaces in terms of matrix integrals, enabling explicit calculation of invariants.

Findings

Explicit formulas for invariants up to order five

Extension of Lawrence and Rozansky's results

Connection between Chern-Simons theory and random matrix averages

Abstract

The universal perturbative invariants of rational homology spheres can be extracted from the Chern-Simons partition function by combining perturbative and nonperturbative results. We spell out the general procedure to compute these invariants, and we work out in detail the case of Seifert spaces. By extending some previous results of Lawrence and Rozansky, the Chern-Simons partition function with arbitrary simply-laced group for these spaces is written in terms of matrix integrals. The analysis of the perturbative expansion amounts to the evaluation of averages in a Gaussian ensemble of random matrices. As a result, explicit expressions for the universal perturbative invariants of Seifert homology spheres up to order five are presented.