nLab quantization of loop groups

Definition

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

$$\array{ G/T &\to& L G / T \\ && \downarrow \\ && L G / G & \simeq \Omega G } \,.$$

Remark

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

$$\Gamma_{wt} \stackrel{\simeq}{\to} Hom_{LieGrp}(G,U(1)) \,.$$

Proposition

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

$$H^2(L G / T) \simeq \mathbb{Z} \oplus H^2(G / T, \mathbb{Z}) \simeq \mathbb{Z} \oplus \hat T \,.$$

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

1. the space of holomorphic sections of $L_{n,\lambda}$ is either zero or is an irreducible positive energy representation;

2. every such arises this way;

3. and is non-zero precisely if $(n,\lambda)$ is positive in the sense that for each positive coroot? $h_\alpha$ of $G$

   $$0 \leq \lambda(h_\alpha) \leq n \langle h_\alpha, h_\alpha\rangle \,.$$

This appears for instance as (Segal, prop. 4.2).

References

The standard textbook on loop groups is

• Andrew Pressley, Graeme Segal, Loop groups Oxford University Press (1988)

A review talk is

• Graeme Segal, Loop groups (pdf)