We present a new expression for the partition function of the refined Chern-Simons theory on S3 with an arbitrary gauge group, which is explicitly equal to 1 when the coupling constant is zero. Using this form of the partition function we show that the previously known Krefl-Schwarz representation of the partition function of the refined Chern-Simons theory on S3 can be generalized to all simply laced algebras.For all non-simply laced gauge algebras, we derive similar representations of that partition function, which makes it possible to transform it into a product of multiple sine functions aiming at the further establishment of duality with the refined topological strings.

We present a universal, in Vogel’s sense, expression for the quantum dimension of the Cartan product of arbitrary powers of the adjoint and X2 representations of simple Lie algebras. The same formula mysteriously yields quantum dimensions of some other representations of the same Lie algebra for permuted universal parameters, provided linear resolvability of singularities is applied. We list these representations for the exceptional algebras and their stable versions for the classical algebras (when the rank of the classical algebra is sufficiently large with regard to the powers of representations). Universal formulas may have singularities at the points in Vogel’s plane, corresponding to some simple Lie algebras. We prove that our formula is linearly resolvable at all those singular points, i.e., yields finite answers when restricted either on the classical or the exceptional lines, and make a conjecture that these answers coincide with (quantum) dimensions of some irreducible representations. In a number of cases, particularly, in the case of so(8) algebra, which belongs both to the orthogonal and the exceptional lines, it is confirmed that both resolutions yield relevant answers. We note that an irreducible representation may have several universal formulas for its (quantum) dimension and discuss the impact of this phenomenon to the method presented by Cohen and de Man [C. R.Acad. Sci., Ser. 1 322(5), 427–432 (1996)] for derivation of universal formulas.

The antisymmetric square of the adjoint representation of any simple Lie algebra is equal to the sum of adjoint and X2 representations. We present universal formulae for quantum dimensions of an arbitrary Cartan power of X2. They are analyzed for singular cases and permuted universal Vogel's parameters. X2 has been the only representation in the decomposition of the square of the adjoint with unknown universal series. Application to universal knot polynomials is discussed.MSC classes: 17B20, 17B37, 57M25

We present the partition function of the refined Chern-Simons theory on S 3 with arbitrary A,B,C,D gauge algebra in terms of multiple sine functions. For B and C cases this representation is novel. It allows us to conjecture duality to some refined and orientifolded versions of the topological string on the resolved conifold, and carry out the detailed identification of different contributions. The free energies for D and C algebras possess the usual halved contribution from the A theory, i.e. orientable surfaces, and contributions of non-orientable surfaces with one cross-cup, with opposite signs, similar as for the non-refined theories. However, in the refined case, both theories possess in addition a non-zero contribution of orientable surfaces with two cross-cups. In particular, we observe a trebling of the Kähler parameter, in the sense of a refinement and world-sheet (i.e. the number of cross-cups) dependent quantum shift. For B algebra the contribution of Klein bottles is zero, as is the case in the non-refined theory, and the one-cross-cup terms differ from the D and C cases. We also calculate some non-perturbative corrections.