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 Tr. Mosk. Mat. Obs., 2017, Volume 78, Issue 1, Pages 17–88 (Mi mmo593)

Representations of superconformal algebras and mock theta functions

V. G. Kaca, M. Wakimotob

a Department of Mathematics, M.I.T, Cambridge, MA 02139, USA
b 12–4 Karato-Rokkoudai, Kita-ku, Kobe 651–1334, Japan

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 $\mathrm{SL}_2(\mathbb{Z})$–invariant space. This result extends to admissible $\hat{\mathfrak{g}}$–modules, where $\mathfrak{g}$ is a simple Lie algebra or $\mathrm{osp}_{1|n}$. Applying the quantum Hamiltonian reduction (QHR) to admissible $\hat{\mathfrak{g}}$–modules when $\mathfrak{g} =s\ell_2$ (resp. $=\mathrm{osp}_{1|2}$) 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 $\mathfrak{g}$ is a basic Lie superalgebra. For example, if $\mathfrak{g}=s\ell_{2|1}$ (resp. $=\mathrm{osp}_{3|2}$), 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 $\mathrm{osp}_{1|n}$, 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 $\mathrm{SL}_2(\mathbb{Z})$–invariant space.

Key words and phrases: 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

 Funding Agency Grant Number National Science Foundation The first named author supported in part by an NSF grant. The second named author supported in part by Department of Mathematics, M.I.T

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English version:
Transactions of the Moscow Mathematical Society, 2017, 78, 9–74

Document Type: Article
UDC: 512.554.32, 512.554.38, 517.986.5, 515.178.1, 517.547.582
MSC: 17B67, 17B10, 17B68, 11F50, 33E05
Revised: 01.04.2017
Language: English

Citation: V. G. Kac, M. Wakimoto, “Representations of superconformal algebras and mock theta functions”, Tr. Mosk. Mat. Obs., 78, no. 1, MCCME, M., 2017, 17–88; Trans. Moscow Math. Soc., 78 (2017), 9–74

Citation in format AMSBIB
\Bibitem{KacWak17} \by V.~G.~Kac, M.~Wakimoto \paper Representations of superconformal algebras and mock theta functions \serial Tr. Mosk. Mat. Obs. \yr 2017 \vol 78 \issue 1 \pages 17--88 \publ MCCME \publaddr M. \mathnet{http://mi.mathnet.ru/mmo593} \elib{http://elibrary.ru/item.asp?id=37045053} \transl \jour Trans. Moscow Math. Soc. \yr 2017 \vol 78 \pages 9--74 \crossref{https://doi.org/10.1090/mosc/268} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85037665584}