Vertex operators and quasimodularity of Chern numbers on the Hilbert scheme

In Carlsson and Okounkov (preprint) [7], Okounkov and the author defined a family of vertex operators on the equivariant cohomology groups of the Hilbert scheme of points on a smooth quasi-projective surface as a characteristic class of certain canonical bundles on Hilb(*) S x Hilb(*) S. We then proved a bosonization formula in terms of Nakajima's Heisenberg operators (Nakajima, 1997 [23]). In this paper we apply this operator in the special case when S = C-2 with a particular action of a torus, and prove that the generating functions of equivariant Chern numbers on Hilb(n) over n, are quasimodular forms in the generating variable. This property determines the answer up to a finite-dimensional vector space of functions of the generating variable, q. These generating functions can be thought of as the analogous correlation functions to Nekrasov's partition function in rank 1. We present a different proof of the bosonization formula which is based on the proof of a more general formula in K-theory given in an upcoming paper by Nekrasov, Okounkov and the author (in preparation) [6], but specialized to the surface of interest. By altering a certain bundle that appears in Carlsson et al. (in preparation) [6], and specializing the surface, the proof actually reduces to a much simpler self-contained application of the infinite wedge representation. This picture is consistent with both the original introduction of this operator in Nekrasov and Okounkov (2006) [26] and with Haiman's character theory of the Bridgeland, King and Reid isomorphism (Haiman, 2003 [11]). (C) 2011 Published by Elsevier Inc.