Nonperturbative universal Chern-Simons theory

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.Comment: 18 pages, Final journal version, references added and refine

Refined Chern–Simons versus Vogel universality

“…If we expect that there exist knot invariants depending on Vogel parameters, we should expect that the partition functions of ChernSimons theory for all simple Lie algebras can be expressed in an universal fashion in terms of these parameters. This has been proved for Chern-Simons theory on S 3 in [13,14], where a closed integral expression in terms of the Vogel parameters for the partition function has been derived.…”

“…The partition function of universal Chern-Simons on S 3 has been shown in [13,14] to be factorizable as…”

“…Rather unrelated to these developments, very recently there has been a proposal for a three parameter generalization of Chern-Simons theory, dubbed universal Chern- Simons [13,14]. The original motivation for such a universal Chern-Simons theory goes back to the work of Vogel [15,16].…”

“…This extension is given by the formulas of [14] where t is considered as an independent parameter (i.e., we drop Vogel's condition t = h = a + b + c, with h the dual Coxeter number). We will find that the refined Chern-Simons partition functions on S 3 with refinement parameter β of [10,11] can be obtained by choosing the parameters in the SU(N) case as (−2, 2β, βN, βN, κ) and in the SO(2N) case as (−2, 4β, 2βN − 4β, 2βN − 2β, κ).…”

On universal knot polynomials

“…As shown in [9,22], in the Chern-Simons gauge theory [23] on the 3d sphere universal are such quantities as central charge, perturbative and non-perturbative partition functions, etc., and particularly unknotted Wilson average in adjoint representation. In the present paper we discuss the problem of universality of adjoint Wilson averages for some knotted curves.…”