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

We present the partition function of Chern-Simons theory with the exceptional gauge group on threesphere in the form of a partition function of the refined closed topological string with relation 2 tau = g(s)(1 - b) between single Kahler parameter tau, string coupling constant g(s) and refinement parameter b, where b = 5/3, 5/2, 3, 4, 6 for G(2), F-4, E-6, E-7, E-8, respectively. The non-zero BPS invariants N-JL,JR(d)(d - degree) are N-0,1/2(2) = 1, N-0,1(11) = 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 Excwith Vogel's parameters satisfying gamma= 2(alpha + ss). This line contains points, corresponding to the all exceptional groups. The same results are obtained for Fline gamma = alpha + beta (containing SU(4), SO(10) and E-G groups), with the non-zero N-0,12(2) = 1, N-0,1(7)= 1. In both cases refinement parameter b(= -epsilon(2)/epsilon(1) in terms of Nekrasov's parameters) is given in terms of universal parameters, restricted to the line, by b = -beta/alpha. (C) 2020 The Author. Published by Elsevier B.V.