Instanton counting on Hirzebruch surfaces

TL;DR

This paper studies the moduli space of sheaves on Hirzebruch surfaces using localization, classifies fixed points, computes topological invariants, and connects results to N=4 Vafa-Witten theory relevant for black hole entropy.

Contribution

It provides a detailed classification of fixed points and computes the partition function for N=4 Vafa-Witten theory on Hirzebruch surfaces, linking mathematical and physical insights.

Findings

Classification of fixed points under toric action

Computation of the Poincaré polynomial of the moduli space

Partition function results relevant for black hole entropy

Abstract

We perform a study of the moduli space of framed torsion free sheaves on Hirzebruch surfaces by using localization techniques. After discussing general properties of this moduli space, we classify its fixed points under the appropriate toric action and compute its Poincare' polynomial. From the physical viewpoint, our results provide the partition function of N=4 Vafa-Witten theory on Hirzebruch surfaces, which is relevant in black hole entropy counting problems according to a conjecture due to Ooguri, Strominger and Vafa.