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 ${\mathcal A}_{\overline 0}$, ${\mathcal A}_{\overline 1}$ 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.