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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.

Inspired by the two-parameter Macdonald's deformation of the formulae for non simply laced simple Lie algebras, we propose a two-fold refinement of the partition function of the corresponding Chern-Simons theory on S 3. It is based on a two-fold refinement of the Kac-Peterson formula for the volume of the fundamental domain of the coroot lattice of non simply laced Lie algebras. We further derive explicit integral representations of the two-fold refined Chern-Simons partition functions. We also present the corresponding generalized universal-like expressions for them. With these formulae in hand, one can try to investigate a possible duality of the corresponding Chern-Simons theories with hypothetical two-fold refined topological string theories. [ABSTRACT FROM AUTHOR]