Abstract
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 \({SL(2,\mathbb{Z})}\) 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.
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Alexandrov, S., Banerjee, S., Manschot, J. et al. Multiple D3-Instantons and Mock Modular Forms II. Commun. Math. Phys. 359, 297–346 (2018). https://doi.org/10.1007/s00220-018-3114-z
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DOI: https://doi.org/10.1007/s00220-018-3114-z