Search

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]

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 2 τ = g s (1 − b) between single Kähler parameter τ , 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 J L , J R 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 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 S U (4) , S O (10) and E 6 groups), with the non-zero N 0 , 1 2 2 = 1 , N 0 , 1 7 = 1. In both cases refinement parameter b (= − ϵ 2 / ϵ 1 in terms of Nekrasov's parameters) is given in terms of universal parameters, restricted to the line, by b = − β / α. [ABSTRACT FROM AUTHOR]