Topological closed-string interpretation of Chern-Simons theory

TL;DR

This paper provides a string-theoretic interpretation of the perturbative expansion of SU(N) Chern-Simons theory's free energy, revealing connections to moduli spaces of surfaces and spacetime topologies.

Contribution

It introduces a topological closed-string perspective on Chern-Simons theory, linking the free energy expansion to moduli space Euler characteristics and spacetime topology contributions.

Findings

Free energy expansion manifests rank-level duality.

String interpretation relates to moduli spaces of surfaces.

Differences in expansions on three-sphere and three-torus illuminate topology effects.

Abstract

The exact free energy of SU($N$) Chern-Simons theory at level $k$ is expanded in powers of $(N+k)^{-2}.$ This expansion keeps rank-level duality manifest, and simplifies as $k$ becomes large, keeping $N$ fixed (or vice versa)---this is the weak-coupling (strong-coupling) limit. With the standard normalization, the free energy on the three-sphere in this limit is shown to be the generating function of the Euler characteristics of the moduli spaces of surfaces of genus $g,$ providing a string interpretation for the perturbative expansion. A similar expansion is found for the three-torus, with differences that shed light on contributions from different spacetime topologies in string theory.