Yang-Yang functions, Monodromy and knot polynomials

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.