Loop Groups and QNEC

TL;DR

This paper explores the analytical properties of loop group models, extending positive energy representations, and proves the Quantum Null Energy Condition and Bekenstein Bound for certain states, with applications to conformal nets.

Contribution

It extends PERs of loop groups to Sobolev spaces, explicitly computes the stress tensor action, and proves QNEC and Bekenstein Bound for Sobolev loop states, including a simplified proof for SU(n).

Findings

Extended PERs to Sobolev spaces for loop groups.

Proved QNEC and Bekenstein Bound for Sobolev loop states.

Constructed solitonic representations with discontinuities.

Abstract

We investigate some analytical properties of loop group models, showing that a Positive Energy Representation (PER) of a loop group $LG$ can be extended to a PER of $H^{3/2}(S^1,G)$ for any compact, simple and simply connected Lie group $G$. We then explicitly compute the adjoint action of $H^{5/2}(S^1,G)$ on the stress energy tensor and we use these results to prove the Quantum Null Energy Condition (QNEC) and the Bekenstein Bound for states obtained by applying a Sobolev loop to the vacuum. We also give a simpler proof of these last results in the case $G=SU(n)$. Finally, we construct and study solitonic representations of the loop group conformal nets induced by the conjugation by a loop with a discontinuity in $-1$.