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Higher-Level Appell Functions, Modular Transformations, and Characters

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We study modular transformation properties of a class of indefinite theta series involved in characters of infinite-dimensional Lie superalgebras. The level-ℓ Appell functions satisfy open quasiperiodicity relations with additive theta-function terms emerging in translating by the “period.” Generalizing the well-known interpretation of theta functions as sections of line bundles, the function enters the construction of a section of a rank-(ℓ+1) bundle . We evaluate modular transformations of the functions and construct the action of an SL(2,ℤ) subgroup that leaves the section of constructed from invariant.

Modular transformation properties of are applied to the affine Lie superalgebra at a rational level k>−1 and to the N=2 super-Virasoro algebra, to derive modular transformations of “admissible” characters, which are not periodic under the spectral flow and cannot therefore be rationally expressed through theta functions. This gives an example where constructing a modular group action involves extensions among representations in a nonrational conformal model.

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Authors and Affiliations

  1. Tamm Theory Department, Lebedev Physics Institute, Russian Academy of Sciences, 53 Leninsky prospect, Moscow, 119991, Russia

    A.M. Semikhatov & I.Yu. Tipunin

  2. Department of Mathematical Sciences, University of Durham, Science Laboratories, South Road, Durham, DH13LE, United Kingdom

    A. Taorimina

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  1. A.M. Semikhatov
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  2. A. Taorimina
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  3. I.Yu. Tipunin
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Communicated by L. Takhtajan

Acknowledgement We are grateful to B.L. Feigin for interesting discussions, to J. Fuchs for a useful suggestion, and to V.I. Ritus for his help with the small-t asymptotic expansion. AMS acknowledges support from the Royal Society through a grant RCM/ExAgr and the kind hospitality in Durham. AT acknowledges support from a Small Collaborative Grant of the London Mathematical Society that made a trip to Moscow possible, and the warm welcome extended to her during her visit. AMS & IYuT were supported in part by the grant LSS-1578.2003.2, by the Foundation for Support of Russian Science, and by the RFBR Grant 04-01-00303. IYuT was also supported in part by the RFBR Grant 03-01-06135 and the INTAS Grant 00-01-254.

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Semikhatov, A., Taorimina, A. & Tipunin, I. Higher-Level Appell Functions, Modular Transformations, and Characters. Commun. Math. Phys. 255, 469–512 (2005). https://doi.org/10.1007/s00220-004-1280-7

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