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Multiple D3-Instantons and Mock Modular Forms I

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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 Kähler 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 the 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.

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Authors and Affiliations

  1. Laboratoire Charles Coulomb (L2C), UMR 5221 CNRS-Université de Montpellier, 34095, Montpellier, France

    Sergei Alexandrov

  2. IPhT, CEA, Saclay, 91191, Gif-sur-Yvette, France

    Sibasish Banerjee

  3. School of Mathematics, Trinity College, Dublin 2, Ireland

    Jan Manschot

  4. CERN PH-TH, Case C01600, CERN, 1211, Geneva 23, Switzerland

    Boris Pioline

  5. Laboratoire de Physique Théorique et Hautes Energies, CNRS UMR 7589, Université Pierre et Marie Curie, 4 place Jussieu, 75252, Paris cedex 05, France

    Boris Pioline

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  1. Sergei Alexandrov
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  2. Sibasish Banerjee
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  3. Jan Manschot
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  4. Boris Pioline
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Correspondence to Jan Manschot.

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Communicated by X. Yin

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Alexandrov, S., Banerjee, S., Manschot, J. et al. Multiple D3-Instantons and Mock Modular Forms I. Commun. Math. Phys. 353, 379–411 (2017). https://doi.org/10.1007/s00220-016-2799-0

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