Solitons and Nonsmooth Diffeomorphisms in Conformal Nets

Abstract

We show that any solitonic representation of a conformal (diffeomorphism covariant) net on S 1 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 S 1 \ {−1} which 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 D s (S 1 ), s > 2, and show that the positive-energy vacuum representations of Diff + (S 1 ) with integer central charges extend to D s (S 1 ). 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 + (S 1 ) of −1 of the circle.

Explicit examples of probability distributions for the energy density in two-dimensional conformal field theory

“…To close we note that our argument depended on an extension of the Virasoro relations (50) to non-smooth test functions, so some care is needed. The argument is rigorously valid for a ≥ 3 at least, because then g, h are at least four times continuously differentiable (see [14] for rigorous proofs; more recent work in a similar direction is presented in [15,16]). It is a reasonable conjecture that the overall result holds for a = 2 (and perhaps even a = 1 as well) and this is supported by numerical evidence to be reported elsewhere.…”

“…thanks Sebastiano Carpi for a useful conversation relating to Refs. [14][15][16] and Bradley Lang for conducting the numerical calculations mentioned in Sec. V.…”

Relative entropy for coherent states in chiral CFT

“…The composition formula (2.10) needs to be justified for non-smooth diffeomorphims. Such questions have been dealt with in great detail by [38,39], and all the relevant arguments are in fact available in the literature. We will therefore be brief.…”

“…Next, we recall that it has been shown in lemma A.3 of [39] that any γ P B 0 can be decomposed as Exppgq˝γ, with g P b 0 and with γ P Diff 8 . The construction of g depends on certain arbitrary choices, but by going through the proof of lemma A.3 of [39], one can see that locally near a given γ, the assignment of g (hence of γ) is C 1 in the topology of B 0 (and the induced topology on b 0 . One can then define a unitary transformation implementing γ by Upγq :" e iT pgq V pγq, and [39] have shown that this gives a projective representation of B 0 (prop.…”

“…5. To go to the second line, we have used the transformation formula (2.14), justified in the present case by lemma 3.3 of [39] (which thereby in particular shows that the subspace tUpγq|Ψy | Ψ P V, γ P B 0 u is in the domain of any T pf q, when f P b 0 ). Next, we want to change the representers Upγq to new representers V pγq by assigning additional phases in such a way that (2.10) holds.…”