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Report number	arXiv:1605.05945 ; L2C:16-056 ; IPHT-T16-037 ; CERN-TH-2016-121 ; TCDMATH-16-08 ; L2C:16-056 ; IPhT-T16/037 ; TCDMATH 16-08 ; CERN-TH-2016-121
Title	Multiple D3-instantons and mock modular forms I
Related title	Multiple D3-instantons and mock modular forms I
Author(s)	Alexandrov, Sergei (U. Montpellier, L2C) ; Banerjee, Sibasish (IPhT, Saclay) ; Manschot, Jan (Trinity Coll., Dublin) ; Pioline, Boris (CERN ; Paris, LPTHE)
Publication	2016-11-11
Imprint	19 May 2016
Number of pages	33
Note	33 pages; v2: minor corrections and clarifications
In:	Commun. Math. Phys. 353 (2017) 379-411
DOI	10.1007/s00220-016-2799-0
Subject category	math.NT ; Mathematical Physics and Mathematics ; math.MP ; math.AG ; math-ph ; hep-th ; Particle Physics - Theory
Abstract	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 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.
Copyright/License	arXiv nonexclusive-distrib. 1.0 Preprint: © 2016-2025 CERN (License: CC-BY-4.0)

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