Knot Invariants From Generalized Yang-Yang Functions

Starting from the free field realization of Kac-Moody Lie algebra, we define a gen-eralized Yang-Yang function. Then for the Lie algebra of type An, we derive braiding and fusion matrix by braiding the thimble from the generalized Yang-Yang function. One can construct a knots invariant H(K) from the braiding and fusion matrix. It is an isotopy invariant and obeys a skein relation. From them, we show that the correspond-ing knots invariant is HOMFLY polynomial. For the fundamental representations of the Lie algebras of type Bn, Cn and Dn, we derive the corresponding knot invariants and prove that they are Kauffman polynomial.