Poincare polynomial of moduli spaces of framed sheaves on (stacky) Hirzebruch surfaces

TL;DR

This paper computes the Poincare polynomial of moduli spaces of framed sheaves on stacky Hirzebruch surfaces using localization, revealing their irreducibility and connecting to physics via Vafa-Witten theory and black hole entropy.

Contribution

It extends the computation of Poincare polynomials to stacky Hirzebruch surfaces and classifies fixed points under toric action, linking geometric results to physical theories.

Findings

Computed Poincare polynomial of moduli spaces.

Proved irreducibility of these moduli spaces.

Connected geometric results to Vafa-Witten partition functions.

Abstract

We perform a study of the moduli space of framed torsion-free sheaves on Hirzebruch surfaces by using localization techniques. We discuss some general properties of this moduli space by studying it in the framework of Huybrechts-Lehn theory of framed modules. We classify the fixed points under a toric action on the moduli space, and use this to compute the Poincare polynomial of the latter. This will imply that the moduli spaces we are considering are irreducible. We also consider fractional first Chern classes, which means that we are extending our computation to a stacky deformation of a Hirzebruch surface. From the physical viewpoint, our results provide the partition function of N=4 Vafa-Witten theory on total spaces of line bundles on P1, which is relevant in black hole entropy counting problems according to a conjecture due to Ooguri, Strominger and Vafa.