Multiple D3 Instantons and Mock Modular Forms I
We study D3 instanton corrections to the hypermultiplet moduli space in type IIB string theory compactified on a Calabi Yau threefold. In a previous work, consistency of D3 instantons with S duality was established at first order in the instanton expansion, using the modular properties of the M5 brane elliptic genus. We extend this analysis to the two instanton level, where wall crossing phenomena start playing a role. We focus on the contact potential, an analogue of the Kahler potential which must transform as a modular form under S duality. We show that it can be expressed in terms of a suitable modification of the partition function of D4 D2 D0 BPS black holes, constructed out of the generating function of MSW invariants (the latter coincide with Donaldson Thomas invariants in a particular chamber) Modular invariance of the contact potential then requires that, in the case where the D3 brane wraps a reducible divisor, the generating function of MSW invariants must transform as a vector valued mock modular form, with a specific modular completion built from the MSW invariants of the constituents. Physically, this gives a powerful constraint on the degeneracies of BPS black holes. Mathematically, our result gives a universal prediction for the modular properties of Donaldson Thomas invariants of pure two dimensional sheaves.