On refined Chern-Simons / topological string duality for classical gauge groups

TL;DR

This paper derives the partition function of refined Chern-Simons theory for classical gauge groups, explores its duality with refined topological strings, and uncovers novel contributions from non-orientable surfaces and cross-cups.

Contribution

It provides a new representation of the partition function for B and C gauge groups and conjectures a duality with refined topological string theories, including detailed contributions from non-orientable surfaces.

Findings

Novel representation for B and C gauge groups.

Identification of contributions from non-orientable surfaces with cross-cups.

Observation of a trebling of the Kähler parameter in the refined case.

Abstract

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 K\"ahler 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. For the (refined) constant maps terms of these theories we suggest a modular-invariant representation, which leads to natural topological string interpretation. We also calculate some non-perturbative corrections.