Representations of superconformal algebras and mock theta functions
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S0077-1554-2017-00268-1.pdf
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Author(s)
Wakimoto, M.
Kac, Victor
Date Issued
December 2017
Journal
Transactions of the Moscow Mathematical Society
Publisher
American Mathematical Society (AMS)
Citation
Kac, V. G., and M. Wakimoto. “Representations of Superconformal Algebras and Mock Theta Functions.” Transactions of the Moscow Mathematical Society, vol. 78, Dec. 2017, pp. 9–74.
Version
Final published version
Abstract
It is well known that the normalized characters of integrable highest
weight modules of given level over an affine Lie algebra [care over g] span an SL[subscript 2](Z)-invariant space. This result extends to admissible [caret over g]-modules, where g is a simple Lie algebra or osp[subscript 1|n]. Applying the quantum Hamiltonian reduction (QHR) to admissible [caret over g]-modules when g = sl[subscript 2] (resp. = osp[subscript 1|2]) one obtains minimal series modules over the Virasoro (resp. N = 1 superconformal algebras), which form modular invariant
families. Another instance of modular invariance occurs for boundary level admissible modules, including when g is a basic Lie superalgebra. For example, if g = sl[subscript 2|1] (resp. = osp[subscript 3|2]), we thus obtain modular invariant families of g-modules, whose QHR produces the minimal series modules for the N = 2 superconformal algebras (resp. a modular invariant family of N = 3 superconformal algebra modules).
However, in the case when g is a basic Lie superalgebra different from a simple Lie algebra or osp[subscript 1|n], modular invariance of normalized supercharacters of admissible [caret over g]-modules holds outside of boundary levels only after their modification in the spirit
of Zwegers’ modification of mock theta functions. Applying the QHR, we obtain families of representations of N = 2, 3, 4 and big N = 4 superconformal algebras, whose modified (super)characters span an SL[subscript 2](Z)-invariant space.
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
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DOI of Published Version
https://doi.org/10.1090/MOSC/268
