Semigroup of annuli in Liouville CFT

In conformal field theory, the semigroup of annuli with boundary parametrization 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 into the space of bounded operators on the Hilbert space $\mathcal{H}$ of Liouville Conformal Field Theory and show it generates under differentiation the positive Virasoro elements ${\bf L}_n,\tilde{{\bf L}}_n$ 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 ${\bf L}_n,\tilde{{\bf L}}_n$ 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 respetc to their boundary parametrizations, 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 the conformal blocks as globally defined holomorphic sections of a holomorphic line bundle on Teichmuller space and satisfying the Ward identities.