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 . We then proved a bosonization formula in terms of Nakajimaʼs Heisenberg operators (Nakajima, 1997 [23]). In this paper we apply this operator in the special case when S=C2 with a particular action of a torus, and prove that the generating functions of equivariant Chern numbers on Hilbn 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 Nekrasovʼs 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 Haimanʼs character theory of the Bridgeland, King and Reid isomorphism (Haiman, 2003 [11]).