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Register a new userRepresentations of affine superalgebras and mock theta functions III
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We study modular invariance of normalized supercharacters of tame integrable modules over an affine Lie superalgebra, associated to an arbitrary basic Lie superalgebra $ mathfrak{g}. $ For this we develop a several step modification process of multivariable mock theta functions, where at each step a Zwegers type modifier is used. We show that the span of the resulting modified normalized supercharacters is $ SL_2(mathbb{Z}) $-invariant, with the transformation matrix equal, in the case the Killing form on $mathfrak{g}$ is non-degenerate, to that for the subalgebra $ mathfrak{g}^! $ of $ mathfrak{g}, $ orthogonal to a maximal isotropic set of roots of $ mathfrak{g}. $
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We show that the normalized supercharacters of principal admissible modules over the affine Lie superalgebra $hat{sell}_{2|1}$ (resp. $hat{psell}_{2|2}$) can be modified, using Zwegers real analytic corrections, to form a modular (resp. $S$-) invariant family of functions. Applying the quantum Hamiltonian reduction, this leads to a new family of positive energy modules over the N=2 (resp. N=4) superconformal algebras with central charge $3(1-frac{2m+2}{M})$, where $m in mathbb{Z}_{geq 0}$, $Min mathbb{Z}_{geq 2}$, $gcd(2m+2,M)=1$ if $m>0$ (resp. $6(frac{m}{M}-1)$, where $m in mathbb{Z}_{geq 1}, Min mathbb{Z}_{geq 2}$, $gcd(2m,M)=1$ if $m>1$), whose modified characters and supercharacters form a modular invariant family.
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. $c=-3frac{2m+1}{M}$, 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.
It is well known that the normaized characters of integrable highest weight modules of given level over an affine Lie algebra $hat{frak{g}}$ span an $SL_2(mathbf{Z})$-invariant space. This result extends to admissible $hat{frak{g}}$-modules, where $frak{g}$ is a simple Lie algebra or $osp_{1|n}$. Applying the quantum Hamiltonian reduction (QHR) to admissible $hat{frak{g}}$-modules when $frak{g} =sl_2$ (resp. $=osp_{1|2}$) 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 $frak{g}$ is a basic Lie superalgebra. For example, if $frak{g}=sl_{2|1}$ (resp. $=osp_{3|2}$), we thus obtain modular invariant families of $hat{frak{g}}$-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 $frak{g}$ is a basic Lie superalgebra different from a simple Lie algebra or $osp_{1|n}$, modular invariance of normalized supercharacters of admissible $hat{frak{g}}$-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 $SL_2(mathbf{Z})$-invariant space.
We give a characterization of modified (in the sense of Zwegers) mock theta functions, parallel to that of ordinary theta functions. Namely, modified mock theta functions are characterized by their analyticity properties, elliptic transformation properties, and by being annihilated by certain second order differential operators.
For all almost affine (hyperbolic) Lie superalgebras, the defining relations are computed in terms of their Chevalley generators.
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