# Two-fold refinement of non simply laced Chern-Simons theories

## TL;DR

This paper introduces a two-fold refinement of non simply laced Chern-Simons theories inspired by Macdonald-Cherednik deformation, providing explicit formulas and exploring potential dualities with refined topological string theories.

## Contribution

It proposes a novel two-fold refinement of Chern-Simons partition functions for non simply laced Lie algebras, extending existing formulas and deriving explicit integral representations.

## Key findings

- Derived explicit integral representations of the refined partition functions
- Presented generalized universal-like expressions for the two-fold refined theories
- Suggested potential dualities with refined topological string theories

## Abstract

Inspired by the two-parameter Macdonald-Cherednik 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.