Nonperturbative universal Chern-Simons theory

TL;DR

This paper derives a universal integral representation for the full free energy of Chern-Simons theory on S^3, demonstrating its universality and dualities, and connects it with topological string theory and nonperturbative effects.

Contribution

It provides a universal integral formula for both perturbative and nonperturbative parts of Chern-Simons free energy, establishing universality and dualities, and links with topological string theory.

Findings

Universal integral representation for Chern-Simons free energy.

Manifest N → -N duality for classical groups.

Asymptotic match with Barnes G-function, confirming string duality.

Abstract

Closed simple integral representation through Vogel's universal parameters is found both for perturbative and nonperturbative (which is inverse invariant group volume) parts of free energy of Chern-Simons theory on $S^3$. This proves the universality of that partition function. For classical groups it manifestly satisfy N \rightarrow -N duality, in apparent contradiction with previously used ones. For SU(N) we show that asymptotic of nonperturbative part of our partition function coincides with that of Barnes G-function, recover Chern-Simons/topological string duality in genus expansion and resolve abovementioned contradiction. We discuss few possible directions of development of these results: derivation of representation of free energy through Gopakumar-Vafa invariants, possible appearance of non-perturbative additional terms, 1/N expansion for exceptional groups, duality between string coupling constant and K\"ahler parameters, etc.