Semigroup of annuli in Liouville CFT

In conformal field theory, the semigroup of annuli with boundary parametrisation plays a special role, in that it generates the whole algebra of local conformal symmetries, the so-called Virasoro algebra. The subgroup of elements $\mathbb A_f=\mathbb D\setminus f(\mathbb D^\circ)$ for contracting biholomorphisms $f:\mathbb D\to f(\mathbb D)\subset \mathbb D^\circ$ with $f(0)=0$ is called the holomorphic semigroup of annuli. In this article, we construct a differentiable representation of the holomorphic semigroup on the space of bounded operators on the Hilbert space $\mathcal H$ of Liouville Conformal Field Theory. We show that it generates under differentiation the positive Virasoro elements $\mathbf L_n,\tilde{\mathbf L}_n$ for $n\geq 0$. We also construct a projective representation of the semigroup of annuli in the space of bounded operators on $\mathcal H$ in terms of Segal amplitudes and show that all Virasoro elements $\mathbf L_n,\tilde{\mathbf L}_n$ for $n\in\mathbb Z$ are generated by differentiation of these annuli amplitudes. Finally, we use this to show that the Segal amplitudes for Liouville theory are differentiable with respect to their boundary parametrisations, and the differential is computed in terms of Virasoro generators. This paper will serve, in a forthcoming work, as a fundamental tool in the construction of conformal blocks as globally defined holomorphic sections of a holomorphic line bundle on Teichm\"uller space and satisfying the Ward identities.