Vertex Operators and Moduli Spaces of Sheaves

The Nekrasov partition function in supersymmetric quantum gauge theory is mathematically formulated as an equivariant integral over certain moduli spaces of sheaves on a complex surface. In SeibergWitten Theory and Random Partitions, Nekrasov and Okounkov studied these integrals using the representation theory of “vertex operators” and the infinite wedge representation. Many of these operators arise naturally from correspondences on the moduli spaces, such as Nakajima’s Heisenberg operators, and Grojnowski’s vertex operators. In this paper, we build a new vertex operator out of the Chern class of a vector bundle on a pair of moduli spaces. This operator has the advantage that it connects to the partition function by definition. It also incorporates the canonical class of the surface, whereas many other studies assume that the class vanishes. When the moduli space is the Hilbert scheme, we present an explicit expression in the Nakajima operators, and the resulting combinatorial identities. We then apply the vertex operator to the above integrals. In agreeable cases, the commutation properties of the vertex operator result in modularity properties of the partition function and related correlation functions. We present examples in which the integrals are completely determined by their modularity, and their first few values.