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

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