Solitons and Nonsmooth Diffeomorphisms in Conformal Nets

We show that any solitonic representation of a conformal (diffeomorphism covariant) net on S1 has positive energy and construct an uncountable family of mutually inequivalent solitonic representations of any conformal net, using nonsmooth diffeomorphisms. On the loop group nets, we show that these representations induce representations of the subgroup of loops compactly supported in S1\ { - 1 } that do not extend to the whole loop group. In the case of the U(1)-current net, we extend the diffeomorphism covariance to the Sobolev diffeomorphisms Ds(S1) , s> 2 , and show that the positive-energy vacuum representations of Diff +(S1) with integer central charges extend to Ds(S1). The solitonic representations constructed above for the U (1) -current net and for Virasoro nets with integral central charge are continuously covariant with respect to the stabilizer subgroup of Diff +(S1) of - 1 of the circle.