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Transactions of the Moscow Mathematical Society

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Representations of superconformal algebras and mock theta functions


Authors: V. G. Kac and M. Wakimoto
Original publication: Trudy Moskovskogo Matematicheskogo Obshchestva, tom 78 (2017), vypusk 1.
Journal: Trans. Moscow Math. Soc. 2017, 9-74
MSC (2010): Primary 17B67, 17B10, 17B68, 11F50, 33E05
DOI: https://doi.org/10.1090/mosc/268
Published electronically: December 1, 2017
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Abstract: It is well known that the normalized characters of integrable highest weight modules of given level over an affine Lie algebra $ \hat {\mathfrak{g}}$ span an $ \textup {SL}_2(\mathbb{Z})$-invariant space. This result extends to admissible $ \hat {\mathfrak{g}}$-modules, where $ \mathfrak{g}$ is a simple Lie algebra or $ \textup {osp}_{1\vert n}$. Applying the quantum Hamiltonian reduction (QHR) to admissible $ \hat {\mathfrak{g}}$-modules when $ \mathfrak{g} =s\ell _2$ (resp. $ =\textup {osp}_{1\vert 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 $ \mathfrak{g}$ is a basic Lie superalgebra. For example, if $ \mathfrak{g}=s\ell _{2\vert 1}$ (resp. $ =\textup {osp}_{3\vert 2}$), we thus obtain modular invariant families of $ \hat {\mathfrak{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 $ \mathfrak{g}$ is a basic Lie superalgebra different from a simple Lie algebra or $ \textup {osp}_{1\vert n}$, modular invariance of normalized supercharacters of admissible $ \hat {\mathfrak{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 $ \textup {SL}_2(\mathbb{Z})$-invariant space.


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Additional Information

V. G. Kac
Affiliation: Department of Mathematics, M.I.T, Cambridge, Massachusetts 02139
Email: kac@math.mit.edu

M. Wakimoto
Affiliation: 12–4 Karato-Rokkoudai, Kita-ku, Kobe 651–1334, Japan
Email: wakimoto@r6.dion.ne.jp

DOI: https://doi.org/10.1090/mosc/268
Keywords: Basic Lie superalgebra, affine Lie superalgebra, superconformal algebra, integrable and admissible representations of affine Lie superalgebras, quantum Hamiltonian reduction, theta function, mock theta function and its modification, modular invariant family of characters
Published electronically: December 1, 2017
Additional Notes: The first named author was supported in part by an NSF grant. The second named author was supported in part by Department of Mathematics, M.I.T
Dedicated: To Ernest Borisovich Vinberg on his 80th birthday
Article copyright: © Copyright 2017 V.G.Kac, M.Wakimoto

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