Let G be a compact, simple, connected, simply connected (cscsc) Lie group, and let its smooth loop group LG:=C∞(S1,G). Given an interval I⊂S1, we have the local loop group
LIG:={γ∈LG | ∀z∉I γ(z)=e}
which is a subgroup of
LG. Let
k≥1 be an integer.
The level
k central extension of
LG is denoted
LGk.
It restricts to a central extension of the local loop group that we denote
LIGk.
A representation of LGk on a Hilbert space is called positive energy if it admits a covariant action of S1 (i.e., the action should extend to S1⋉LGk) whose infinitesimal generator has positive spectrum.
Here, the center of LGk 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
LIGk.
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 LGk. 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 LGk. 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 LG1 that is the identity on LIG1.
• Every level k rep appears in the restriction of a level 1 rep under the map LGk→LG1 induced by the k-fold cover of S1→S1.
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?
This post imported from StackExchange MathOverflow at 2014-09-14 08:22 (UCT), posted by SE-user André Henriques