  In this work, we construct a representation of the Virasoro algebra in the
canonical Hilbert space associated to Liouville conformal field theory. The
study of the Virasoro operators is performed through the introduction of a new
family of Markovian dynamics associated to holomorphic vector fields defined in
the disk. As an output, we show that the Hamiltonian of Liouville conformal
field theory can be diagonalized through the action of the Virasoro algebra.
This enables to show that the scattering matrix of the theory is diagonal and
that the family of the so-called primary fields (which are eigenvectors of the
Hamiltonian) admits an analytic extension to the whole complex plane, as
conjectured in the physics literature.
