Let $G$ be a compact Lie group and write $L G$ for its loop group. See there for details and notation.
We discuss the quantization of loop groups in the sense of geometric quantization of their canonical prequantum bundle.
Let $G$ be a compact Lie group. Let $T \hookrightarrow G$ be the inclusion of a maximal torus. There is a fiber sequence
By the discussion at orbit method, if $G$ is a semisimple Lie group, then $G/T$ is isomorphic to the coadjoint orbit of an element $\langle \lambda , -\rangle \in \mathfrak{g}^*$ for which $T \simeq G_\lambda$ is the stabilizer subgroup.
If moreover $G$ is simply connected, then the weight lattice $\Gamma_{wt} \subset \mathfrak{t}^* \simeq \mathfrak{t}$ of the Lie group $G$ is isomorphic to the group of group characters
The irreducible projective positive energy representations of $L G$ correspond precisley to the possible geometric quantizations of $L G / T$ (as in the orbit method).
More in detail:
The degree-2 integral cohomology of $L G / T$ is
Writing $L_{n,\lambda}$ for the corresponding complex line bundle with level $n \in \mathbb{Z}$ and weight $\lambda \in \hat T$ we have that
the space of holomorphic sections of $L_{n,\lambda}$ is either zero or is an irreducible positive energy representation;
every such arises this way;
and is non-zero precisely if $(n,\lambda)$ is positive in the sense that for each positive coroot? $h_\alpha$ of $G$
This appears for instance as (Segal, prop. 4.2).
The standard textbook on loop groups is
A review talk is
Created on November 6, 2013 at 03:05:42. See the history of this page for a list of all contributions to it.