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Meromorphic Jacobi Forms of Half-Integral Index and Umbral Moonshine Modules

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Abstract

In this work we consider an association of meromorphic Jacobi forms of half-integral index to the pure D-type cases of umbral moonshine, and solve the module problem for four of these cases by constructing vertex operator superalgebras that realise the corresponding meromorphic Jacobi forms as graded traces. We also present a general discussion of meromorphic Jacobi forms with half-integral index and their relationship to mock modular forms.

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Notes

  1. Our nomenclature follows [15], except that elliptic forms are assumed to be holomorphic in that work, and the focus there is on integral index.

References

  1. Eguchi, T., Ooguri, H., Tachikawa, Y.: Notes on the K3 Surface and the Mathieu group \(M_{24}\). Exp. Math. 20, 91–96 (2011). arXiv:1004.0956 [hep-th]

  2. Cheng, M.C.N., Duncan, J.F.R., Harvey, J.A.: Umbral Moonshine. Commun. Number Theory Phys. 8(2), 101–242 (2014). https://doi.org/10.4310/CNTP.2014.v8.n2.a1. arXiv:1204.2779 [math.RT]

  3. Cheng, M.C.N., Duncan, J.F.R., Harvey, J.A.: Umbral moonshine and the Niemeier lattices. Res. Math. Sci. 1(3), 1–81 (2014). arXiv:1307.5793 [math.RT]

  4. Cheng, M.C.N., Duncan, J.F.R.: Optimal Mock Jacobi Theta Functions, ’ArXiv e-prints (May, 2016) , arXiv:1605.04480 [math.NT]

  5. Gannon, T.: Much ado about Mathieu. Adv. Math. 301, 322–358 (2016). https://doi.org/10.1016/j.aim.2016.06.014

    Article  MathSciNet  MATH  Google Scholar 

  6. Duncan, J.F.R., Griffin, M.J., Ono, K.: Proof of the Umbral Moonshine Conjecture, Research in the Mathematical Sciences 2 (2015) , arXiv:1503.01472 [math.RT]

  7. Conway, J.H., Norton, S.P.: Monstrous moonshine. Bull. Lond. Math. Soc. 11(3), 308–339 (1979). https://doi.org/10.1112/blms/11.3.308

    Article  MathSciNet  MATH  Google Scholar 

  8. Borcherds, R.E.: Monstrous moonshine and monstrous Lie superalgebras. Invent. Math. 109(2), 405–444 (1992). https://doi.org/10.1007/BF01232032

    Article  ADS  MathSciNet  MATH  Google Scholar 

  9. Frenkel, I. B., Lepowsky, J., Meurman, A.: A moonshine module for the Monster. In: Vertex Operators in Mathematics and Physics (Berkeley, Calif., 1983), vol. 3 of Math. Sci. Res. Inst. Publ., pp. 231–273. Springer, New York (1985)

  10. Frenkel, I.B., Lepowsky, J., Meurman, A.: A natural representation of the Fischer-Griess Monster with the modular function \(J\) as character. Proc. Natl. Acad. Sci. U.S.A. 81(10), 3256–3260 (1984)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  11. Borcherds, R.: Vertex algebras, Kac-Moody algebras, and the Monster. Proc. Natl. Acad. Sci. U.S.A 83(10), 3068–3071 (1986)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  12. Frenkel, I.B., Lepowsky, J., Meurman, A.: Vertex Operator Algebras and the Monster. Pure and Applied Mathematics, vol. 134. Academic Press Inc., Boston, MA (1988)

    MATH  Google Scholar 

  13. Duncan, J., Harvey, J.: The umbral moonshine module for the unique unimodular Niemeier root system. Algebra Number Theory 11(3), 505–535 (2017). https://doi.org/10.2140/ant.2017.11.505

    Article  MathSciNet  MATH  Google Scholar 

  14. Duncan, J.F.R., O’Desky, A.: Super vertex algebras, meromorphic Jacobi forms and umbral moonshine. J. Algebra 515, 389–407 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  15. Dabholkar, A., Murthy, S., Zagier, D.: Quantum Black Holes, Wall Crossing, and Mock Modular Forms, arXiv:1208.4074 [hep-th]

  16. Johnson-Freyd, T., Treumann, D.: \({ H}^4({\rm Co}_0;{\mathbf{Z}})={ Z}/24\). International Mathematics Research Notices 219, (2018)

  17. Anagiannis, V., Cheng, M.C.N., Harrison, S.M.: K3 Elliptic Genus and an Umbral Moonshine Module, ArXiv e-prints (September, 2017), arXiv:1709.01952 [hep-th]

  18. Zwegers, S.: Mock Theta Functions. Ph.D. thesis, Utrecht University, http://igitur-archive.library.uu.nl/dissertations/2003-0127-094324/inhoud.htm. (2002)

  19. Eichler, M., Zagier, D.: The Theory of Jacobi Forms. Birkhäuser, New York (1985)

    Book  MATH  Google Scholar 

  20. Zwegers, S.: Multivariable Appell Functions (2010) (Preprint)

  21. Cheng, M.C.N., Dong, X., Duncan, J.F.R., Harrison, S., Kachru, S., Wrase, T.: Mock modular Mathieu moonshine modules. Res. Math. Sci. 2, 13 (2015). arXiv:1406.5502 [hep-th]

  22. Malikov, F., Schechtman, V., Vaintrob, A.: Chiral de Rham complex. Commun. Math. Phys. 204(2), 439–473 (1999). https://doi.org/10.1007/s002200050653

    Article  ADS  MathSciNet  MATH  Google Scholar 

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Acknowledgements

We thank Andrew O’Desky for discussions on closely related topics, and we thank the anonymous referees for helpful comments and suggestions. The work of M.C. was supported by ERC starting grant H2020 ERC StG #640159. J.D. acknowledges support from the Simons Foundation (#316779), and the U.S. National Science Foundation (DMS 1203162, DMS 1601306).

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Authors and Affiliations

  1. Institute of Physics and Korteweg-de Vries Institute for Mathematics, University of Amsterdam, Amsterdam, The Netherlands

    Miranda C. N. Cheng

  2. Department of Mathematics and Computer Science, Emory University, Atlanta, GA, 30322, USA

    John F. R. Duncan

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  1. Miranda C. N. Cheng
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  2. John F. R. Duncan
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Correspondence to John F. R. Duncan.

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Communicated by Y. Kawahigashi

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Cheng, M.C.N., Duncan, J.F.R. Meromorphic Jacobi Forms of Half-Integral Index and Umbral Moonshine Modules. Commun. Math. Phys. 370, 759–780 (2019). https://doi.org/10.1007/s00220-019-03540-2

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