# local equivalence of loop group representations

Let $G$ be a compact, simple, connected, simply connected (cscsc) Lie group, and let its smooth loop group $LG:=C^\infty(S^1,G)$. Given an interval $I\subset S^1$, we have the local loop group $$L_IG := \{\gamma\in LG \ | \ \forall z\not\in I \ \gamma(z)=e\}$$ which is a subgroup of $LG$. Let $k\ge 1$ be an integer. The level $k$ central extension of $LG$ is denoted $\mathcal{L}G_k$. It restricts to a central extension of the local loop group that we denote $\mathcal{L}_IG_k$.

A representation of $\mathcal{L}G_k$ on a Hilbert space is called positive energy if it admits a covariant action of $S^1$ (i.e., the action should extend to $S^1\ltimes \mathcal{L}G_k$) whose infinitesimal generator has positive spectrum. Here, the center of $\mathcal{L}G_k$ 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 $\mathcal{L}_IG_k$.

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 $\mathcal{L}G_k$. 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 $\mathcal{L}G_k$. 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 $\mathcal{L}G_1$ that is the identity on $\mathcal{L}_IG_1$.
• Every level $k$ rep appears in the restriction of a level 1 rep under the map $\mathcal{L}G_k\to \mathcal{L}G_1$ induced by the $k$-fold cover of $S^1\to S^1$.

There are proofs in the literature, due to A. Wassermann (here p23) and V. Toledano-Laredo (here p82) 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?

• probably not really helpful, but there seems to be a paper in preparation by Toledano Laredo which maybe could shed some light about this question in the B and C case – Marcel Bischoff Jun 27 '11 at 21:02
• There seems to be a proof of Claim 2 in some unpublished notes, namely in Böckenhauer and Evans "Modular Invariants, Graphs and $\alpha$-induction for Nets of Subfactors. II" on page 66 is claimed that local equivalence hold for any compact connected simple $G$ with reference to the unplished notes of Wassermann "Subfactors arising from positive energy representations of some infinte dimensional groups", 1999. – Marcel Bischoff Aug 9 '11 at 9:36
• Many thanks to the person who sent me Wassermann's preprint. – André Henriques May 28 '12 at 21:12
• After having a further look at his preprint, it is now my opinion that Antony Wassermann does not provide a complete proof of the desired claim. So the question is open. – André Henriques Sep 23 '14 at 19:21