On a super-Virasoro group, a semigroup of annuli, and Gauss--Berezin integral operators

Abstract

Denote by $\mathcal{A}$ the Grassmann algebra with a countable number of generators, by ${\mathcal{A}}_{\overline{0}}$, ${\mathcal{A}}_{\overline{1}}$ its even and odd parts. We consider supergroups as groups over $\mathcal{A}$. Consider the Neveu--Schwarz Lie superalgebra $\mathfrak{ns}={\mathfrak{ns}}_{\overline{0}}\oplus {\mathfrak{ns}}_{\overline{1}}$. Consider its Grassmannization $\mathfrak{ns}(\mathcal{A}):=({\mathfrak{ns}}_{\overline{0}}\otimes {\mathcal{A}}_{\overline{0}})\oplus ({\mathfrak{ns}}_{\overline{1}}\otimes {\mathcal{A}}_{\overline{1}})$ and the corresponding supergroup $\mathrm{NS}(\mathcal{A})$. We describe this group explicitly in different ways. Next, we define a complexification $\Gamma (\mathcal{A})$ of $\mathrm{NS}(\mathcal{A})$. It is a semigroup, whose elements are superannuli of dimension $1|1$ equipped with contact structures; the multiplication is gluing of such superannuli. Under some inequalities for parameters, we show that unitary representations of $\mathrm{NS}(\mathcal{A})$ admit extensions to representations of $\Gamma (\mathcal{A})$. We do not assume that the reader has prior knowledge of Lie superalgebras and supergroups.

67p