Homogeneous components in the moduli space of sheaves and Virasoro characters

TL;DR

This paper studies the fixed points of torus actions on the moduli space of sheaves, revealing a connection between irreducible components and Virasoro characters, with a conjecture extending these results.

Contribution

It establishes a decomposition of the generating series of irreducible components into infinite products and links these to Virasoro characters for odd ranks.

Findings

Generating series decompose into infinite products in the homogeneous case.

For odd r, these products match certain Virasoro characters.

Proposes a conjecture for the quasihomogeneous case.

Abstract

The moduli space $\mathcal M(r,n)$ of framed torsion free sheaves on the projective plane with rank $r$ and second Chern class equal to $n$ has the natural action of the $(r+2)$-dimensional torus. In this paper, we look at the fixed point set of different one-dimensional subtori in this torus. We prove that in the homogeneous case the generating series of the numbers of the irreducible components has a beautiful decomposition into an infinite product. In the case of odd $r$ these infinite products coincide with certain Virasoro characters. We also propose a conjecture in a general quasihomogeneous case.