Exts and vertex operators
The cohomology of the Hilbert scheme of points \(\mathrm{Hilb}_n(S)\) of non-singular quasi-projective surfaces \(S\) has been described in terms of vertex operators on \(\mathcal{F}:=\bigoplus_nH^*(\mathrm{Hilb}_n(S,\mathbb{Q}))\) introduced by Nakajima and Grojnowski. In this paper the authors introduce another natural set of vertex operators on \(\mathcal{F}\) that depends on a line bundle \(\mathcal{L}\) over \(S\). These operators are defined in terms of the Chern classes of the alternating Ext-groups between two ideal sheaves twisted by \(\mathcal{L}\), and can be characterized by their commutation relations with the Nakajima operators. Moreover, the authors give an explicit formula for them in terms of the latter.NEWLINENEWLINEFor the \((\mathbb{C}^*)^2\)--equivariant cohomology of \(\mathrm{Hilb}_n(\mathbb{C}^2)\) the trace of these vertex operators becomes the Nekrasov instanton partition function with matter in the adjoint representation. Equivariant localization reduces the above mentioned formula to a Pieri-type formula for the Jack symmetric functions.
scientific article; zbMATH DE number 6053750
scientific article