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Indefinite theta series and generalized error functions

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Abstract

Theta series for lattices with indefinite signature \((n_+,n_-)\) arise in many areas of mathematics including representation theory and enumerative algebraic geometry. Their modular properties are well understood in the Lorentzian case (\(n_+=1\)), but have remained obscure when \(n_+\ge 2\). Using a higher-dimensional generalization of the usual (complementary) error function, discovered in an independent physics project, we construct the modular completion of a class of ‘conformal’ holomorphic theta series (\(n_+=2\)). As an application, we determine the modular properties of a generalized Appell–Lerch sum attached to the lattice \({{\text {A}}}_2\), which arose in the study of rank 3 vector bundles on \(\mathbb {P}^2\). The extension of our method to \(n_+>2\) is outlined.

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Acknowledgements

J. M. thanks Kathrin Bringmann, Thomas Creutzig, Robert Osburn, Larry Rolen, Martin Westerholt-Raum, Don Zagier and Sander Zwegers for discussions about the generating functions derived in [30, 31]. B. P. is grateful to Trinity College Dublin for hospitality during part of this work.

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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, Gif-sur-Yvette, 91191, France

    Sibasish Banerjee

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

    Jan Manschot

  4. TH Department, Case C01600, CERN, 1211, Geneva 23, Switzerland

    Boris Pioline

  5. Sorbonne Université, LPTHE, UMR 7589, Campus Pierre et Marie Curie, 75005, Paris, France

    Boris Pioline

  6. CNRS, LPTHE, UMR 7589, 4 place Jussieu, 75005, Paris, 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.

Additional information

Preprint: L2C:16-078, IPHT-T16/058, TCDMATH 16-09, CERN-TH-2016-142, arXiv:1606.05495.

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Alexandrov, S., Banerjee, S., Manschot, J. et al. Indefinite theta series and generalized error functions. Sel. Math. New Ser. 24, 3927–3972 (2018). https://doi.org/10.1007/s00029-018-0444-9

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