Hiroaki Kanno : Chern-Simons theory, enumeration and Macdonald functions

In this talk we present a curious combinatorial identity (a conjecture) which involves a certain specialization of the Macdonald functions and explain how we have arrived at it in our attempt at refining the topological vertex. The topological vertex is a building block for instanton counting problems in topological string/gauge theory. It was obtained from the duality that relates the enumeration problem to the Chern-Simons theory and expressed in terms of an appropriate specialization of the Schur functions. Our refined version of the topological vertex gives a building block of the Nekrasov's partition function and is expected to have some relations to holomogical version of link invariants.