The Segal-Neretin semigroup of annuli

Abstract

The Lie algebra of vector fields on $S^1$ integrates to the Lie group of diffeomorphisms of $S^1$. It is well known since the work of Segal and Neretin that there is no Lie group whose Lie algebra is the complexification of vector fields on $S^1$. A substitute for that non-existent group is provided by the complex semigroup whose elements are annuli: genus zero Riemann surfaces with two boundary circles parametrized by $S^1$. The group $\mathrm{Diff}(S^1)$ sits at the boundary of that semigroup, and can be thought of as annuli which are completely thin, i.e. with empty interior. In this paper, we consider an enlargement of the semigroup of annuli, denoted $\mathrm{Ann}$, where the annuli are allowed to be partially thin: their two boundary circles are allowed to touch each other along an arbitrary closed subset. We prove that every (partially thin) annulus $A\in \mathrm{Ann}$ is the time-ordered exponential of a path with values in the cone of inward pointing complexified vector fields on $S^1$, and use that fact to construct a central extension \[ 0\to \mathbb{C} \times \mathbb{Z} \to \tilde{\mathrm{Ann}} \to \mathrm{Ann} \to 0 \] that integrates the universal (Virasoro) central extension of the Lie algebra of vector fields on $S^1$. In later work, we will prove that every unitary positive energy representations of the Virasoro algebra integrates to a holomorphic representation of $\tilde{\mathrm{Ann}}$ by bounded operators on a Hilbert space.