Five dimensional gauge theories and vertex operators

We study supersymmetric gauge theories in five dimensions, using their relation to the $K$-theory of the moduli spaces of torsion free sheaves. In the spirit of the BPS/CFT correspondence the partition function and the expectation values of the chiral, BPS protected observables are given by the matrix elements and more generally by the correlation functions in some $q$-deformed conformal field theory in two dimensions. We show that the coupling of the gauge theory to the bi-fundamental matter hypermultiplet inserts a particular vertex operator in this theory. In this way we get a generalization of the main result of a paper by E.C. and A.O. to $K$-theory. The theory of interpolating Macdonald polynomials is an important tool in our construction.