BPS invariants of N=4 gauge theory on a surface

TL;DR

This paper computes generating functions of BPS invariants for N=4 U(r) gauge theory on a Hirzebruch surface, revealing their modular properties and expressing them via higher level Appell functions for specific polarizations.

Contribution

It introduces explicit calculations of BPS invariants for r=2 and 3 on Hirzebruch surfaces and relates them to higher level Appell functions and modular forms.

Findings

Generating functions for r=2 are expressed with higher level Appell functions.

The functions transform as modular forms when non-holomorphic parts are included.

Explicit BPS invariants are computed for specific gauge groups.

Abstract

Generating functions of BPS invariants for N=4 U(r) gauge theory on a Hirzebruch surface with r=2 and 3 are computed. The BPS invariants provide the Betti numbers of moduli spaces of semi-stable sheaves. The generating functions for r=2 are expressed in terms of higher level Appell functions for a certain polarization of the surface. The level corresponds to the self-intersection of the base curve of the Hirzebruch surface. The non-holomorphic functions are determined, which added to the holomorphic generating functions provide functions which transform as a modular form.