Higher-level Appell functions, modular transformations, and characters

Semikhatov, Alexei; Taormina, Anne; Tipunin, Ilya

Abstract

We study modular transformation properties of a class of indefinite theta series involved in characters of infinite-dimensional Lie superalgebras. The level- 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-(+1) bundle . We evaluate modular transformations of the functions and construct the action of an SL(2,) 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.

Citation

Semikhatov, A., Taormina, A., & Tipunin, I. (2005). Higher-level Appell functions, modular transformations, and characters. Communications in Mathematical Physics, 255(2), 469-512. https://doi.org/10.1007/s00220-004-1280-7

• Journal Article Type: Article
• Publication Date: Apr 1, 2005
• Deposit Date: Feb 26, 2008
• Publicly Available Date: Apr 16, 2013
• Journal: Communications in Mathematical Physics
• Print ISSN: 0010-3616
• Electronic ISSN: 1432-0916
• Publisher: Springer
• Peer Reviewed: Peer Reviewed
• Volume: 255
• Issue: 2
• Pages: 469-512
• DOI: https://doi.org/10.1007/s00220-004-1280-7
• Public URL: https://durham-repository.worktribe.com/output/1599442