On the coadjoint Virasoro action

The set of coadjoint orbits of the Virasoro algebra at level 1 is in bijection with the set of conjugacy classes in a certain open subset $\widetilde{\mathrm{S}\mathrm{L}}(2,\mathbb{R}{)}_{+}$ of the universal cover of $\mathrm{S}\mathrm{L}(2,\mathbb{R})$. We strengthen this bijection to a Morita equivalence of quasi-symplectic groupoids, integrating the Poisson structure on ${\mathfrak{v}\mathfrak{i}\mathfrak{r}}_{\mathsf{1}}^{\ast }(S^1)$ and the Cartan-Dirac structure on $\widetilde{\mathrm{S}\mathrm{L}}(2,\mathbb{R}{)}_{+}$, respectively.