Soliton representations and Sobolev diffeomorphism symmetry in CFT

We show that any positive energy representation of the group of (orientation preserving) smooth diffeomorphisms of the circle Diff(S^1) can be extended to a strongly continuous unitary projective representation of the group of (orientation preserving) fractional Sobolev diffeomorphisms D^s(S^1) with real Sobolev exponent s>3. For some positive energy representations, i.e for the positive energy vacuum representations of Diff(S^1) with positive integer central charge, we can improve the implementation to the group D^s(S^1) with s>2. We show that a conformal net of von Neumann algebras on the circle is always D^s(S^1)-covariant, s>3. Furthermore, we show that a given positive energy representation U of Diff(S^1) cannot be extended to some less-smooth diffeomorphisms, and from this fact we obtain an uncountable family of proper soliton representations. From these soliton representations we construct irreducible unitary projective positive energy representations of the group ΛSU(N) consisting of loops with support not containing the point -1 (resp. B_0, the stabilizer subgroup of -1) which do not extend to LSU(N) (resp. Diff(S^1)).