# Positive energy representations of Sobolev diffeomorphism groups of the circle

August 07, 2018

We show that any positive energy projective representation of Diff(S^1)
extends to a strongly continuous projective unitary representation of the
fractional Sobolev diffeomorphisms D^s(S^1) with 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.

Keywords:

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