Delocalized equivariant coholomogy of symmetric products

Abstract

For any closed complex manifold $X$, we calculate the Poincaré and Hodge polynomials of the delocalized equivariant cohomology $H^{\ast }(X^n,S_n)$ with a grading specified by physicists. As a consequence, we recover a special case of a formula for the elliptic genera of symmetric products in Dijkgraaf-Moore-Verlinde-Verlinde \cite{Dij-Moo-Ver-Ver}. For a projective surface X, our results matches with the corresponding formulas for the Hilbert scheme of X^[n]. We also give geometric construction of an action of a Heisenberg superalgebra on ${\sum }_{n\ge 0}{H}^{\ast ,\ast }(X^n,S_n)$, imitating the constructions for equivariant K-theory by Segal \cite{Seg} and Wang \cite{Wan}. There is a corresponding version for $H^{-\ast ,\ast }$.