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

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.

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Published 19 October 2012