Chern-Simons theory with the exceptional gauge group as a refined topological string

R.L Mkrtchyan

doi:10.1016/j.physletb.2020.135692

Abstract

We present the partition function of Chern-Simons theory with the exceptional gauge group on three-sphere in the form of a partition function of the refined closed topological string with relation 2τ=gs(1−b) between single Kähler parameter τ, string coupling constant gs and refinement parameter b, where b=53,52,3,4,6 for G2,F4,E6,E7,E8, respectively. The non-zero BPS invariants NJL,JRd (d - degree) are N0,122=1,N0,111=1. Besides these terms, partition function of Chern-Simons theory contains term corresponding to the refined constant maps of string theory.Derivation is based on the universal (in Vogel's sense) form of a Chern-Simons partition function on three-sphere, restricted to exceptional line Exc with Vogel's parameters satisfying γ=2(α+β). This line contains points, corresponding to the all exceptional groups. The same results are obtained for F line γ=α+β (containing SU(4),SO(10) and E6 groups), with the non-zero N0,122=1,N0,17=1.In both cases refinement parameter b (=−ϵ2/ϵ1 in terms of Nekrasov's parameters) is given in terms of universal parameters, restricted to the line, by b=−β/α.

Highlights

Partition function Z of Chern-Simons theory with an arbitrary gauge group on 3d sphere was calculated exactly by Witten [1]
It is interesting to consider Chern-Simons theory with the SO(8) and SU (3) gauge groups. Their points on Vogel’s plane both belong to Exc line, so they are assumed to be dual to refined topological strings with parameters z = −1, −2 (i.e. b = 2, 3/2), respectively
Both already have their duals among unrefined topological string theories - the one on conifold, in SU (3) case, and on the orientifold of conifold in the case of SO(8)

Summary

Introduction

Partition function Z of Chern-Simons theory with an arbitrary gauge group on 3d sphere was calculated exactly by Witten [1]. In the papers [2, 3] we have transformed this partition function into the universal form, i.e. expressed it in the terms of Vogel’s parameters α, β, γ, which are homogeneous coordinates of Vogel’s plane (i.e. they are relevant up to rescaling and permutations) [4, 5] Their correspondence with the simple Lie algebras is given in Vogel’s table 1. This form has an advantage that it can be transformed further into the sum over the residues of poles in the integral representation of the multiple sine functions (see [13] for definition and properties of multiple sine functions) One series of these poles (”perturbative” ones) gives Gopakumar-Vafa expression for partition function of topological string, whereas the remaining poles give non-perturbative corrections.

Chern-Simons theories with exceptional groups

Conclusion