Kac-Wakimoto characters and non-holomorphic Jacobi forms

Bringmann, Kathrin; Olivetto, René

doi:10.1090/tran/6709

Kac-Wakimoto characters and non-holomorphic Jacobi forms

Authors: Kathrin Bringmann and René Olivetto
Journal: Trans. Amer. Math. Soc. 369 (2017), 1163-1184
MSC (2010): Primary 11F03, 11F22, 11F37, 11F50
DOI: https://doi.org/10.1090/tran/6709
Published electronically: April 15, 2016
MathSciNet review: 3572269
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper, we investigate the automorphic properties of certain characters introduced by Kac and Wakimoto pertaining to $s\ell (m,n)^{\wedge }$ highest weight modules. Extending previous work of the first author and Ono, the first author and Folsom, and the second author, we investigate the general case, not specializing the Jacobi variables. We prove that the Kac-Wakimoto characters are essentially holomorphic parts of multivariable mixed H-harmonic Maass-Jacobi forms, which are certain non-holomorphic generalizations of classical holomorphic Jacobi forms. This also gives extra structure to the previous considered cases.

References

Rolf Berndt and Ralf Schmidt, Elements of the representation theory of the Jacobi group, Progress in Mathematics, vol. 163, Birkhäuser Verlag, Basel, 1998. MR 1634977
Richard E. Borcherds, Monstrous moonshine and monstrous Lie superalgebras, Invent. Math. 109 (1992), no. 2, 405–444. MR 1172696, DOI https://doi.org/10.1007/BF01232032
Kathrin Bringmann and Amanda Folsom, On the asymptotic behavior of Kac-Wakimoto characters, Proc. Amer. Math. Soc. 141 (2013), no. 5, 1567–1576. MR 3020844, DOI https://doi.org/10.1090/S0002-9939-2012-11439-X
Kathrin Bringmann and Amanda Folsom, Almost harmonic Maass forms and Kac-Wakimoto characters, J. Reine Angew. Math. 694 (2014), 179–202. MR 3259042, DOI https://doi.org/10.1515/crelle-2012-0102
Kathrin Bringmann, Amanda Folsom, and Karl Mahlburg, Quasimodular forms and $s\ell (m|m)^\wedge $ characters, Ramanujan J. 36 (2015), no. 1-2, 103–116. MR 3296713, DOI https://doi.org/10.1007/s11139-014-9621-4
Kathrin Bringmann and Ken Ono, Some characters of Kac and Wakimoto and nonholomorphic modular functions, Math. Ann. 345 (2009), no. 3, 547–558. MR 2534107, DOI https://doi.org/10.1007/s00208-009-0364-2
Kathrin Bringmann and Olav K. Richter, Zagier-type dualities and lifting maps for harmonic Maass-Jacobi forms, Adv. Math. 225 (2010), no. 4, 2298–2315. MR 2680205, DOI https://doi.org/10.1016/j.aim.2010.03.033
Kathrin Bringmann, Martin Raum, and Olav K. Richter, Harmonic Maass-Jacobi forms with singularities and a theta-like decomposition, Trans. Amer. Math. Soc. 367 (2015), no. 9, 6647–6670. MR 3356950, DOI https://doi.org/10.1090/S0002-9947-2015-06418-6
Kathrin Bringmann and Sander Zwegers, Rank-Crank type PDE’s and non-holomorphic Jacobi forms, Math. Res. Lett. 17 (2010), no. 4, 589–600. MR 2661165, DOI https://doi.org/10.4310/MRL.2010.v17.n4.a1
Jan Hendrik Bruinier and Jens Funke, On two geometric theta lifts, Duke Math. J. 125 (2004), no. 1, 45–90. MR 2097357, DOI https://doi.org/10.1215/S0012-7094-04-12513-8
J. H. Conway and S. P. Norton, Monstrous moonshine, Bull. London Math. Soc. 11 (1979), no. 3, 308–339. MR 554399, DOI https://doi.org/10.1112/blms/11.3.308

A. Dabholkar, S. Murthy, and D. Zagier, Quantum black holes, wall crossing, and mock modular forms, to appear in Cambridge Monographs in Mathematical Physics.

Martin Eichler and Don Zagier, The theory of Jacobi forms, Progress in Mathematics, vol. 55, Birkhäuser Boston, Inc., Boston, MA, 1985. MR 781735

Amanda Folsom, Kac-Wakimoto characters and universal mock theta functions, Trans. Amer. Math. Soc. 363 (2011), no. 1, 439–455. MR 2719689, DOI https://doi.org/10.1090/S0002-9947-2010-05181-5

I. B. Frenkel, J. Lepowsky, and A. Meurman, A natural representation of the Fischer-Griess Monster with the modular function $J$ as character, Proc. Nat. Acad. Sci. U.S.A. 81 (1984), no. 10, , Phys. Sci., 3256–3260. MR 747596, DOI https://doi.org/10.1073/pnas.81.10.3256

F. Hirzebruch and D. Zagier, Intersection numbers of curves on Hilbert modular surfaces and modular forms of Nebentypus, Invent. Math. 36 (1976), 57–113. MR 453649, DOI https://doi.org/10.1007/BF01390005

V. G. Kac, Infinite-dimensional Lie algebras, and the Dedekind $\eta $-function, Funkcional. Anal. i Priložen. 8 (1974), no. 1, 77–78 (Russian). MR 0374210

Victor G. Kac and Minoru Wakimoto, Integrable highest weight modules over affine superalgebras and Appell’s function, Comm. Math. Phys. 215 (2001), no. 3, 631–682. MR 1810948, DOI https://doi.org/10.1007/s002200000315

Masanobu Kaneko and Don Zagier, A generalized Jacobi theta function and quasimodular forms, The moduli space of curves (Texel Island, 1994) Progr. Math., vol. 129, Birkhäuser Boston, Boston, MA, 1995, pp. 165–172. MR 1363056, DOI https://doi.org/10.1007/978-1-4612-4264-2_6

Anatoly Libgober, Elliptic genera, real algebraic varieties and quasi-Jacobi forms, Topology of stratified spaces, Math. Sci. Res. Inst. Publ., vol. 58, Cambridge Univ. Press, Cambridge, 2011, pp. 95–120. MR 2796409

Hans Maass, Lectures on modular functions of one complex variable, Tata Institute of Fundamental Research Lectures on Mathematics, No. 29, Tata Institute of Fundamental Research, Bombay, 1964. Notes by Sunder Lal. MR 0218305

R. Olivetto, Harmonic Maass forms, meromorphic Jacobi forms, and applications to Lie superalgebras, Ph.D. Thesis, Univerity of Cologne, in preparation.

René Olivetto, On the Fourier coefficients of meromorphic Jacobi forms, Int. J. Number Theory 10 (2014), no. 6, 1519–1540. MR 3248169, DOI https://doi.org/10.1142/S1793042114500419

Ameya Pitale, Jacobi Maaßforms, Abh. Math. Semin. Univ. Hambg. 79 (2009), no. 1, 87–111. MR 2541345, DOI https://doi.org/10.1007/s12188-008-0013-9

Hans Rademacher, Topics in analytic number theory, Springer-Verlag, New York-Heidelberg, 1973. Edited by E. Grosswald, J. Lehner and M. Newman; Die Grundlehren der mathematischen Wissenschaften, Band 169. MR 0364103

C. Ziegler, Jacobi forms of higher degree, Abh. Math. Sem. Univ. Hamburg 59 (1989), 191–224. MR 1049896, DOI https://doi.org/10.1007/BF02942329

S. Zwegers, Multivariable Appell functions, preprint (2010).

S. Zwegers, Mock theta functions, Ph.D. Thesis, Universiteit Utrecht (2002).