On a Gopakumar-Vafa form of partition function of Chern-Simons theory on classical and exceptional lines

TL;DR

This paper demonstrates that the partition function of Chern-Simons theory on three-sphere for classical and exceptional groups can be expressed using sine functions, revealing a Gopakumar-Vafa structure and suggesting an extension of gauge/string duality.

Contribution

It introduces a sine function-based representation of Chern-Simons partition functions for classical and exceptional groups, extending gauge/string duality to exceptional groups.

Findings

Partition functions expressed as ratios of sine functions.

Identification of Gopakumar-Vafa structure in these functions.

Presence of non-perturbative terms in the full partition function.

Abstract

We show that partition function of Chern-Simons theory on three-sphere with classical and exceptional groups (actually on the whole corresponding lines in Vogel's plane) can be represented as ratio of respectively triple and double sine functions (last function is essentially a modular quantum dilogarithm). The product representation of sine functions gives Gopakumar-Vafa structure form of partition function, which in turn gives a corresponding integer invariants of manifold after geometrical transition. In this way we suggest to extend gauge/string duality to exceptional groups, although one still have to resolve few problems. In both classical and exceptional cases an additional terms, non-perturbative w.r.t. the string coupling constant, appear. The full universal partition function of Chern-Simons theory on three-sphere is shown to be the ratio of quadruple sine functions. We also briefly discuss the matrix model for exceptional line.