  Following Nag-Sullivan, we study the representation of the group ${\rm
Diff}^+(S^1)$ 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 ${\rm Diff}^+(S^1)$.
