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# Communications in Number Theory and Physics

## Volume 6 (2012)

### Number 2

### BPS invariants of $\CN=4$ gauge theory on Hirzebruch surfaces

Pages: 497 – 516

DOI: http://dx.doi.org/10.4310/CNTP.2012.v6.n2.a4

#### Author

#### Abstract

Generating functions of BPS invariants for $\CN=4$ $U(r)$ gauge theory on aHirzebruch surface with $r\leq 3$ are computed. The BPS invariantsprovide the Betti numbers of moduli spaces of semi-stable sheaves.The generating functions for $r=2$ are expressed in terms of higher level Appellfunctions for a certain polarization of the surface. The level corresponds to the self-intersection of the basecurve of the Hirzebruch surface. The non-holomorphic functions aredetermined, which added to the holomorphic generating functions providefunctions, which transform as a modular form.