Multiple D3-instantons and mock modular forms II

Abstract

We analyze the modular properties of D3-brane instanton correct ions to the hy- permultiplet moduli space in type IIB string theory compactified on a Calabi-Yau threefold. In Part I, we found a necessary condition for the existence of an is ometric action of S-duality on this moduli space: the generating function of DT invariants in the large volume attrac- tor chamber must be a vector-valued mock modular form with specifi ed modular properties. In this work, we prove that this condition is also sufficient at two-inst anton order. This is achieved by producing a holomorphic action of SL (2 , Z ) on the twistor space which preserves the holomorphic contact structure. The key step is to cancel the anomalous modular variation of the Darboux coordinates by a local holomorphic contact transf ormation, which is generated by a suitable indefinite theta series. For this purpose we introduce a new family of theta series of signature (2 ,n − 2), find their modular completion, and conjecture sufficient conditio ns for their convergence, which may be of independent mathematical inte rest.