Semigroup of annuli in Liouville CFT

Understand

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 $n\geq 0$.