Positive energy representations of Sobolev diffeomorphism groups of the circle

TL;DR

This paper demonstrates that positive energy projective unitary representations of the diffeomorphism group of the circle extend to fractional Sobolev diffeomorphisms for s>3, impacting conformal nets and their covariance properties.

Contribution

It establishes the extension of positive energy representations from Diff(S^1) to D^s(S^1) for s>3, including for universal covers and direct sums of irreducibles.

Findings

Representations extend to D^s(S^1) for s>3

Conformal nets are covariant under D^s(S^1) for s>3

Extensions hold for direct sums of irreducible representations

Abstract

We show that any positive energy projective unitary representation of Diff(S^1) extends to a strongly continuous projective unitary representation of the fractional Sobolev diffeomorphisms D^s(S^1) for any real s>3, and in particular to C^k-diffeomorphisms Diff^k(S^1) with k>=4. A similar result holds for the universal covering groups provided that the representation is assumed to be a direct sum of irreducibles. As an application we show that a conformal net of von Neumann algebras on S^1 is covariant with respect to D^s(S^1), s > 3. Moreover every direct sum of irreducible representations of a conformal net is also D^s(S^1)-covariant.