Let $G$ be a compact, simple, connected, simply connected (cscsc) Lie group, and let $LG:=C^\infty(S^1,G)$ be its smooth loop group. Given an interval $I\subset S^1$, we have the local loop group $$ L_IG := \{\gamma\in LG\,\, |\,\, \gamma(z)=e\quad \forall z\not\in I\,\, \}\,, $$ which is a subgroup of $LG$. Let $k\ge 1$ be an integer. The level $k$ central extension of $LG$ is denoted $\widetilde{LG}_k$. It restricts to a central extension of the local loop group that we denote $\widetilde{L_IG}_k$.
A representation of $\widetilde{LG}_k$ on a Hilbert space is called positive energy if it admits a covariant action of $S^1$ (i.e., the action should extend to $S^1\ltimes \widetilde{LG}_k$) whose infinitesimal generator has positive spectrum. Here, the center of $\widetilde{LG}_k$ is required to act by scalar multiplication.
Definition 1:
Two level $k$ positive energy representation of the loop group are called locally equivalent if they become equivalent when restricted to $\widetilde{L_IG}_k$.
The follows is believed to be true:
Claim 2:
Let $G$ be a cscsc group and let $V$ and $W$ be any two positive energy representations of $\widetilde{LG}_k$. Then $V$ and $W$ are locally equivalent.
I know a paper that proves the following:
Theorem 3:
Let $G$ be a simply laced cscsc group and let $V$ and $W$ be two positive energy representations of $\widetilde{LG}_k$. Then $V$ and $W$ are locally equivalent.
Edit: The argument in [GF] seems to contain a mistake (on lines -4 and -3 of page 600)
The basic ingredients that are needed
(see page 599 of [Gabbiani & Fröhlich Operator algebras and conformal field theory] for the proof) are the following two facts about positive energy representations of simply laced loop groups:
• Every level 1 rep can be obtained from the vacuum rep by precomposing the action by an outer automorphism of $\widetilde{LG}_1$ that is the identity on $\widetilde{L_IG}_1$.
• Every level $k$ rep appears in the restriction of a level 1 rep under the map $\widetilde{LG}_k\to \widetilde{LG}_1$ induced by the $k$-fold cover of $S^1\to S^1$.
There are proofs in the literature (due to Wassermann and Toledano, respectively) for the cases $LSU(n)$ and $LSpin(2n)$, that are based on the theory of free fermions -- actually, Toledano only treats half of the representations of $LSpin(2n)$.
Is there a proof of Claim 2 in the literature?
How does one prove Claim 2?
