nLab
loop group

Idea

The loop space of a topological group $G$ inherits the structure of a group under pointwise group multiplication of loops. This is called a loop group of $G$.

(Notice that this is a group structure in addition to the infinity-group-structure of any loop space under composition of loops).

If $G$ is a Lie group, then there is a smooth version of the loop group consisting of smooth functions $S^1\to G$. By the discussion at manifold structure of mapping spaces the collection of such smooth maps is itself an infinite-dimensional smooth manifold and so the smooth loop group of a Lie group is an infinite-dimensional Lie group.

Among all infinite-dimensional Lie groups, loop groups are a most well behaved class. In particular their representation theory is similar to that of compact Lie groups.

Some of these nice properties are solely due to the circle $S^1$ being a compact manifold. For $X$ any other compact manifold there is similarly an infinite-dimensional Lie group $[X,G]$ of smooth functions $X\to G$ under pointwise multiplication in $G$.

Such mapping groups appear in physics notably as groups of gauge transformations over a spacetime/worldvolume $X$. Accordingly, loop groups play a prominint role in 1- and 2-dimensional quantum field theory, notably the WZW model describing the propagation of a string on $G$. The current algebras (affine algebras) which arise as Lie algebras of (centrally extended) loop groups derive their name from this relation to physics. Accordingly, as for compact Lie groups, the representation theory of loop groups is naturally understood in terms of their geometric quantization (by a loop variant of the orbit method).

On the other hand, for $X$ of dimension greater that 1 there are very few known results about the properties of the mapping group $[X,G]$.

Properties

Lie algebra

Let $G$ be a compact Lie group. Write $𝔤$ for its Lie algebra.

Proposition

The Lie algebra of $LG$ is the loop Lie algebra?

$$\mathrm{Lie}(LG)\simeq L\mathrm{Lie}(G)=L𝔤.$$Lie(L G) \simeq L Lie(G) = L \mathfrak{g} \,.

Complexification

Let $G$ be a compact Lie group.

Proposition

The complexification? of $LG$ is the loop group of the complexification of $G$

$$(LG{)}_{ℂ}\simeq L(G_{ℂ}).$$(L G)_{\mathbb{C}} \simeq L (G_\mathbb{C}) \,.

Central extensions

Loop groups of compact Lie groups have canonical central extensions, often called Kac-Moody central extensions . A detailed discussion is in (PressleySegal). A review is in (BCSS)

Representations

Positive energy

Write

$$t_{\theta }:LG\to \mathrm{LG}$$t_\theta \colon L G \to LG

for the automorphism which rotates loops by an angle? $\theta$.

The corresponding semidirect product group we write $S^1⋊LG$

Definition

Let $V$ be a topological vector space. A linear representation

$$S^1\to \mathrm{Aut}(V)$$S^1 \to Aut(V)

of the circle group is called positive if $\exp (i\theta )$ acts by $\exp (iA\theta )$ where $A\in \mathrm{End}(V)$ is an operator with positive spectrum.

A linear representation

$$\rho :LG\to \mathrm{Aut}(V)$$\rho : L G \to Aut(V)

is said to have positive energy if it extends to a representation of the semidirect product group $S^1⋊LG$ such that the restriction to $S^1$ is positive.

By geometric quantization (looped orbit method)

Let $G$ be a compact Lie group. Let $T\hookrightarrow G$ be the inclusion of a maximal torus. There is a fiber sequence

$$\begin{array}{ccc}G/T& \to & LG/T\\ & & \downarrow \\ & & LG/G& \simeq \Omega G\end{array}.$$\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 $⟨\lambda ,-⟩\in {𝔤}^{*}$ for which $T\simeq G_{\lambda }$ is the stabilizer subgroup.

If moreover $G$ is simply connected, then the weight lattice ${\Gamma }_{\mathrm{wt}}\subset {𝔱}^{*}\simeq 𝔱$ of the Lie group $G$ is isomorphic to the group of group characters

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

Proposition

The irreducible projective positive energy representations? of $LG$ correspond precisley to the possible geometric quantizations of $LG/T$ (as in the orbit method).

More in detail:

The degree-2 integral cohomology of $LG/T$ is

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

   $$0\le \lambda (h_{\alpha })\le n⟨h_{\alpha },h_{\alpha }⟩.$$0 \leq \lambda(h_\alpha) \leq n \langle h_\alpha, h_\alpha\rangle \,.

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

Related concepts

• Verlinde ring

• string group

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)

A review of some aspects with an eye towards loop groups as part of the crossed module of groups representing a string 2-group is in

• BCSS, From loop groups to 2-groups (web)

The relation between representations of loop groups and twisted K-theory over the group is the topic of

• Dan Freed, Mike Hopkins, Constantin Teleman, Loop Groups and Twisted K-Theory