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Representations of affine superalgebras and mock theta functions II (1402.0727v1)

Published 4 Feb 2014 in math.RT

Abstract: We show that the normalized supercharacters of principal admissible modules, associated to each integrable atypical module over the affine Lie superalgebra $\widehat{sl}{2|1}$ can be modified, using Zwegers' real analytic corrections, to form an $SL_2(\mathbf{Z})$-invariant family of functions. Using a variation of Zwegers' correction, we obtain a similar result for $\widehat{osp}{3|2}$. Applying the quantum Hamiltonian reduction, this leads to new families of positive energy modules over the $N=2$ (resp. $N=3$) superconformal algebras with central charge $c=3 (1-\frac{2m+2}{M})$, where $m \in \mathbf{Z}{\geq 0}, M \in \mathbf{Z}{\geq 2}$, gcd$(2m+2,M)=1$ if $m>0$ (resp. $c=-3\frac{2m+1}{M}$, where $m \in \mathbf{Z}{\geq 0}, M \in \mathbf{Z}{\geq 2}$ gcd$(4m +2, M) =1)$, whose modified supercharacters form an $SL_2(\mathbf{Z})$-invariant family of functions.

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