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Report number	arXiv:1702.05497 ; L2C:17-011 ; CERN-TH-2017-040 ; IPHT-T17-020 ; L2C:17-011 ; CERN-TH-2017-040 ; IPhT-T17/020
Title	Multiple D3-instantons and mock modular forms II
Related title	Multiple D3-instantons and mock modular forms II
Author(s)	Alexandrov, Sergei (U. Montpellier, L2C ; CERN) ; Banerjee, Sibasish (IPhT, Saclay) ; Manschot, Jan (Trinity Coll., Dublin ; Hamilton Math. Inst., Dublin) ; Pioline, Boris (CERN ; Paris, LPTHE)
Publication	2018-03-10
Imprint	2017-02-17
Number of pages	50
Note	24+24 pages
In:	Commun. Math. Phys. 359 (2018) 297-346
DOI	10.1007/s00220-018-3114-z
Subject category	math.NT ; Particle Physics - Theory ; math.MP ; Mathematical Physics and Mathematics ; Mathematical Physics and Mathematics ; math.AG ; math-ph ; hep-th
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,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.
Copyright/License	arXiv nonexclusive-distrib. 1.0 Preprint: © 2016-2026 CERN (License: CC-BY-4.0)

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