Yang-Yang functions, Monodromy and knot polynomials

TL;DR

This paper establishes a connection between Yang-Yang functions, monodromy representations, and knot polynomials, revealing how braid group actions relate to quantum group R-matrices in Lie algebra representations.

Contribution

It explicitly constructs a $ obreakbZ[t,t^{-1}]$-module bundle from Yang-Yang functions and proves the equivalence of monodromy and braid group representations for classical Lie algebras.

Findings

Explicit wall-crossing formula for fundamental representations.

Monodromy representation matches braid group representation from quantum R-matrices.

Symmetry breaking and rotation transformations commute on the fiber.

Abstract

We derive a structure of $\mathbb{Z}[t,t^{-1}]$-module bundle from a family of Yang-Yang functions. For the fundamental representation of the complex simple Lie algebra of classical type, we give explicit wall-crossing formula and prove that the monodromy representation of the $\mathbb{Z}[t,t^{-1}]$-module bundle is equivalent to the braid group representation induced by the universal R-matrices of $U_{h}(g)$. We show that two transformations induced on the fiber by the symmetry breaking deformation and respectively the rotation of two complex parameters commute with each other.