Communications in Mathematical Physics

, Volume 359, Issue 1, pp 297–346 | Cite as

Multiple D3-Instantons and Mock Modular Forms II

  • Sergei Alexandrov
  • Sibasish Banerjee
  • Jan Manschot
  • Boris Pioline


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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© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Laboratoire Charles Coulomb (L2C)UMR 5221 CNRS-Université de MontpellierMontpellierFrance
  2. 2.Theoretical Physics Department, Case C01600CERNGeneva 23Switzerland
  3. 3.IPhT, CEASaclay, Gif-sur-YvetteFrance
  4. 4.School of MathematicsTrinity CollegeDublin 2Ireland
  5. 5.Hamilton Mathematical InstituteTrinity CollegeDublin 2Ireland
  6. 6.Laboratoire de Physique Théorique et Hautes Energies, CNRS UMR 7589Université Pierre et Marie CurieParis Cedex 05France
  7. 7.UMR 7589Sorbonne Universités, UPMC Université Paris 6ParisFrance

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