Berezin quantization, conformal welding and the Bott-Virasoro group

Following Nag-Sullivan, we study the representation of the group ${\mathrm{D}\mathrm{i}\mathrm{f}\mathrm{f}}^{+}(S^1)$ of diffeomorphisms of the circle on the Hilbert space of holomorphic functions. Conformal welding provides a triangular decompositions for the corresponding symplectic transformations. We apply Berezin formalism and lift this decomposition to operators acting on the Fock space. This lift provides quantization of conformal welding, gives a new representative of the Bott-Virasoso cocylce class, and leads to a surprising identity for the Takhtajan-Teo energy functional on ${\mathrm{D}\mathrm{i}\mathrm{f}\mathrm{f}}^{+}(S^1)$.