André Henriques (Oxford)

Title:

The Segal-Neretin semigroup of annuli

 

Abstract:

The Lie algebra X(S ¹) of vector fields on S ¹ integrates to the Lie group Diff(S ¹) of diffeomorphisms of S ¹. But it is well known, since the work of G. Segal and Y. Neretin in the 80'es and 90'es, that there is no complex Lie group whose Lie algebra is the complexification of X(S ¹). That role is played by the semigroup of annuli, whose elements are "annuli" : genus zero Riemann surfaces with two boundary circles, parametrised by S ¹.

The group Diff(S ¹) sits at the boundary of the semigroup of annuli, and consists of those annuli which are "completely thin", i.e., have empty interior. In this talk, I will consider an enlargement of the semigroup of annuli, where annuli are allowed to be "partially thin": the two boundary circles are allowed to touch each other along a subinterval. I will explain how to integrate the universal (Virasoro) central extension of the semigroup of annuli to a central extension of the semigroup of annuli, and in what sense that latter is a universal central extension.

I will then explain how to integrate unitary representations of the Virasoro algebra to representations of that central extension of the semigroup of annuli to a central extension of the semigroup of annuli, and in what sense that latter is a universal central extension. I will then explain how to integrate unitary representations of the Virasoro algebra to representations of that central extension of the semigroup of annuli.