Loop Groups and QNEC

We construct and study solitonic representations of the conformal net associated to some vacuum Positive Energy Representation (PER) of a loop group LG. For the corresponding solitonic states, we prove the Quantum Null Energy Condition (QNEC) and the Bekenstein Bound. As an intermediate result, we show that a Positive Energy Representation of a loop group LG can be extended to a PER of Hs(S1, G) for s> 3 / 2 , where G is any compact, simple and simply connected Lie group. We also show the existence of the exponential map of the semidirect product LG⋊ R, with R a one-parameter subgroup of Diff +(S1) , and we compute the adjoint action of Hs+1(S1, G) on the stress energy tensor.