Multiple D3-Instantons and Mock Modular Forms I

• DOI: 10.1007/s00220-016-2799-0
• Publication Date: 2016-11-11T14:22:10Z
• Authors: Sergei Alexandrov; Sibasish Banerjee; Jan Manschot; Boris Pioline
• Keywords: Instanton; Modular invariance; Duality (order theory)

24+24 pages<br/>We analyze the modular properties of D3-brane instanton corrections to the hypermultiplet 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 isometric action of S-duality on this moduli space: the generating function of DT invariants in the large volume attractor chamber must be a vector-valued mock modular form with specified modular properties. In this work, we prove that this condition is also sufficient at two-instanton 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 transformation, 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 conditions for their convergence, which may be of independent mathematical interest.<br/>