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Selecta Mathematica

, Volume 24, Issue 5, pp 3927–3972 | Cite as

Indefinite theta series and generalized error functions

  • Sergei Alexandrov
  • Sibasish Banerjee
  • Jan Manschot
  • Boris Pioline
Article
  • 19 Downloads

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.

Mathematics Subject Classification

11F27 11F37 11F16 

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Notes

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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Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  • Sergei Alexandrov
    • 1
  • Sibasish Banerjee
    • 2
  • Jan Manschot
    • 3
  • Boris Pioline
    • 4
    • 5
    • 6
  1. 1.Laboratoire Charles Coulomb (L2C), UMR 5221 CNRSUniversité de MontpellierMontpellierFrance
  2. 2.IPhT, CEA, SaclayGif-sur-YvetteFrance
  3. 3.School of MathematicsTrinity CollegeDublin 2Ireland
  4. 4.TH Department, Case C01600CERNGeneva 23Switzerland
  5. 5.Sorbonne Université, LPTHE, UMR 7589Campus Pierre et Marie CurieParisFrance
  6. 6.CNRS, LPTHE, UMR 7589ParisFrance

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