Higher-level Appell functions, modular transformations, and characters

Semikhatov, Alexei; Taormina, Anne; Tipunin, Ilya

doi:10.1007/s00220-004-1280-7

Higher-level Appell functions, modular transformations, and characters

Semikhatov, Alexei; Taormina, Anne; Tipunin, Ilya

Authors

Alexei Semikhatov

Professor Anne Taormina anne.taormina@durham.ac.uk
Professor

Ilya Tipunin

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