Berezin Quantization, Conformal Welding and the Bott-Virasoro Group

Following Nag-Sullivan, we study the representation of the group $$ of diffeomorphisms of the circle on the Hilbert space of holomorphic functions. Conformal welding provides triangular decompositions for the corresponding symplectic transformations. We apply Berezin formalism and lift this decomposition to operators acting on the Fock space. This lift provides quantization of conformal welding, gives a new representative of the Bott-Virasoso cocycle class, and leads to a surprising identity for the Takhtajan-Teo energy functional on $$.