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BPS invariants of N=4 gauge theory on a surface

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.