Loop group

In mathematics, a loop group is, in the most common Lie-theoretic sense, the group Template:Math of smooth maps from the circle Template:Math to a Lie group Template:Math, with multiplication defined pointwise. When Template:Math is a compact Lie group, Template:Math is a basic example of an infinite-dimensional Lie group, with Lie algebra Template:Math.

The subgroup Template:Math of based loops is fundamental in homotopy theory, while central extensions of loop groups and their projective representations are closely related to affine Kac–Moody algebras, conformal field theory, and the Verlinde formula. In algebraic geometry one also studies algebraic loop groups, defined by Template:Math, together with their associated affine Grassmannians and affine flag varieties.

Definition

Let Template:Math be a topological group. The set Template:Math of continuous maps from the circle to Template:Math becomes a topological group under pointwise multiplication when equipped with the compact-open topology. Since Template:Math is compact, this is the same as the topology of uniform convergence.

In Lie theory one usually considers the group

$LG=C^{\mathrm{\infty }}(S^1,G)$

of smooth loops in a finite-dimensional Lie group Template:Math. It is endowed with the smooth compact-open topology, namely the initial topology induced by the iterated tangent maps

$C^{\mathrm{\infty }}(S^1,G)\to \prod _{k\ge 0}C(T^k{S}^1,T^{k}G).$

With this topology, Template:Math is an infinite-dimensional Lie group.

Its Lie algebra is

$L𝔤=C^{\mathrm{\infty }}(S^1,𝔤),$

with pointwise bracket. Since Template:Math is compact, the smooth compact-open topology on $L𝔤$ is the Fréchet topology of uniform convergence of all derivatives on Template:Math; equivalently, after choosing an angular coordinate on Template:Math and a norm on $𝔤$, it is defined by the seminorms

$p_n(X)=\underset{\theta \in S^1}{sup}\Vert X^{(n)}(\theta )\Vert (n\ge 0).$

For compact Template:Math, smooth loop groups are modeled on nuclear Fréchet spaces.

Basic constructions

Free and based loop groups

The free loop group of Template:Math is Template:Math itself. The based loop group is

$\mathrm{\Omega }G=\{\gamma \in LG:\gamma (1)=e\},$

the kernel of the evaluation map

${\mathrm{ev}}_1:LG\to G,\gamma \mapsto \gamma (1).$

Thus Template:Math is a closed normal subgroup of Template:Math. The inclusion of constant loops gives a splitting of Template:Math, so there is a split exact sequence

$1\to \mathrm{\Omega }G\to LG\stackrel{{\mathrm{ev}}_1}{\to }G\to 1,$

and hence a semidirect product decomposition

$LG\cong \mathrm{\Omega }G⋊G.$

Relation with loop spaces

As a topological space, Template:Math is the based loop space of Template:Math. Its pointwise product and the usual concatenation of based loops are different operations, but they induce the same multiplication up to homotopy; this is a manifestation of the Eckmann–Hilton argument.

Basic topology

The splitting of the evaluation map

${\mathrm{ev}}_1:LG\to G$

by constant loops identifies Template:Math with Template:Math as a topological space:

$LG\cong G\times \mathrm{\Omega }G,\gamma \mapsto (\gamma (1),\gamma (1{)}^{-1}\gamma ).$

Thus the topology of Template:Math is determined by that of Template:Math together with the based loop space Template:Math.

In particular, the connected components of Template:Math are classified by

${\pi }_0(LG)\cong {\pi }_0(G)\times {\pi }_0(\mathrm{\Omega }G).$

If Template:Math is connected, then ${\pi }_0(\mathrm{\Omega }G)\cong {\pi }_1(G)$, so

${\pi }_0(LG)\cong {\pi }_1(G).$

Hence Template:Math is connected whenever Template:Math is simply connected.

More generally, for each Template:Math,

${\pi }_k(LG)\cong {\pi }_k(G)\oplus {\pi }_k(\mathrm{\Omega }G)\cong {\pi }_k(G)\oplus {\pi }_{k+1}(G).$

Thus the homotopy groups of a loop group are determined by those of Template:Math shifted by one degree together with those of Template:Math itself.

These elementary identifications are one reason loop groups are important in algebraic topology. In the unitary case they are closely related to Bott periodicity, and Segal's Grassmannian model of the homogeneous space Template:Math makes this relation explicit.

Rotation action and positive energy

Besides pointwise multiplication, loop groups carry a natural action of the circle by rotating the parameter:

$(R_{\varphi }\gamma )(e^{i\theta })=\gamma (e^{i(\theta +\varphi )}).$

This allows one to form the semidirect product

$T_{\mathrm{r}\mathrm{o}\mathrm{t}}⋉LG,$

where Template:Math acts on Template:Math by rotation.

This action is the starting point for the theory of positive-energy representations. A representation of Template:Math is said to have positive energy if it is equipped with a positive action of Template:Math making it a representation of $T_{\mathrm{r}\mathrm{o}\mathrm{t}}⋉LG$. In Segal's formulation, the action of $e^{i\varphi }\in T_{\mathrm{r}\mathrm{o}\mathrm{t}}$ is positive if it is given by $e^{iA\varphi }$ for an operator Template:Math whose spectrum is bounded below.

For compact Template:Math, irreducible positive-energy representations are a distinguished class of projective representations of loop groups. They extend holomorphically to the complexified loop group and decompose into finite-dimensional energy eigenspaces. For this reason, the representation theory of loop groups of a compact Template:Math often resembles that of compact groups.

Infinite-dimensional Lie group structure

If Template:Math is a finite-dimensional Lie group, then suitable spaces of loops in Template:Math inherit infinite-dimensional manifold structures. For smooth loops, Template:Math is a Fréchet Lie group, and its Lie algebra is the loop algebra

$L𝔤=C^{\mathrm{\infty }}(S^1,𝔤),$

with pointwise bracket

$[X,Y](z)=[X(z),Y(z)].$

The exponential map is induced pointwise from that of Template:Math:

${\exp }_{LG}(X)(z)={\exp }_G(X(z)).$

For compact Template:Math, loop groups are among the simplest and most studied examples of infinite-dimensional Lie groups.

To develop differential geometry on loop groups one often uses Sobolev completions Template:Math. In particular, based loop groups of compact, connected, simply connected, simple Lie groups carry natural geometric structures, including Kähler metrics in the Hilbert-manifold setting.

Homogeneous spaces and factorization

The quotient Template:Math, where Template:Math is embedded as the subgroup of constant loops, can be identified with Template:Math. This homogeneous-space viewpoint is central in the geometry of loop groups.

If Template:Math is compact and Template:Math is its complexification, then the complexified loop group Template:Math admits factorization phenomena analogous to Birkhoff factorization and Bruhat decomposition. These decompositions play a major role in the geometry of loop groups, the theory of Toeplitz operators, and the construction of solutions to integrable systems.

Central extensions and representation theory

Many natural representations attached to the loop group are not honest representations of Template:Math, but of a central extension of Template:Math by the circle group Template:Math.

For a compact Lie group Template:Math, integral classes in Template:Math give rise, by transgression, to central extensions of the loop group. Such extensions are often described by a level. The corresponding projective unitary representations include the integrable highest-weight or positive-energy representations, which are closely related to representations of the associated affine Kac–Moody algebra.

The representation theory of loop groups is also linked to the Verlinde ring and to twisted equivariant K-theory. In work of Freed, Hopkins, and Teleman, the Verlinde ring of positive-energy representations is identified with an appropriate twisted equivariant K-group of Template:Math.

Twisted loop groups

Let Template:Math be an automorphism of Template:Math of finite order Template:Math. The corresponding twisted loop group consists of smooth maps Template:Math satisfying

$\gamma (\theta +2\pi )=\sigma (\gamma (\theta )).$

Equivalently, after passing to the circle, one may regard twisted loops as sections of a bundle over Template:Math with monodromy Template:Math.

Twisted loop groups occur naturally in the theory of affine Dynkin diagrams, in representation theory, and in the theory of affine flag varieties. They include the twisted affine Kac–Moody types as Lie-algebraic counterparts.

Algebraic loop groups

In algebraic geometry and arithmetic geometry, one replaces smooth loops by Laurent series. If Template:Math is an algebraic group over a field Template:Math, its algebraic loop group is the functor

$LG(R)=G(R((t)))$

on Template:Math-algebras Template:Math. The associated positive loop group is

$L^{+}G(R)=G(R[[t]]).$

These objects underlie the theory of the affine Grassmannian

${\mathrm{Gr}}_G=LG/L^{+}G$

and affine flag varieties, which are central in geometric representation theory, the geometric Langlands program, and the theory of local models of Shimura varieties.

Complex and holomorphic loop groups

If Template:Math is a compact Lie group with complexification Template:Math, then the smooth loop group Template:Math has a complexification

$L{G}_{𝐂}=C^{\mathrm{\infty }}(S^1,G_{𝐂}).$

This is one of the special features of loop groups among infinite-dimensional Lie groups.

A closely related role is played by subgroups of loops that extend holomorphically across one of the discs bounded by Template:Math. Writing Template:Math for the unit disc and Template:Math for its exterior, one defines subgroups Template:Math and Template:Math consisting of boundary values of holomorphic maps Template:Math and Template:Math, respectively.

These holomorphic subgroups enter the Birkhoff factorization theorem, according to which a loop in Template:Math can, on suitable strata, be written in the form

$\gamma ={\gamma }_{-}\lambda {\gamma }_{+},$

where Template:Math, Template:Math, and Template:Math is a homomorphism into a maximal torus. This factorization is the infinite-dimensional analogue of Bruhat decomposition and underlies much of the geometry of homogeneous spaces of loop groups.

Holomorphic methods also enter representation theory. In Segal's Borel–Weil picture, positive-energy representations are realized as spaces of holomorphic sections of line bundles over homogeneous spaces attached to loop groups, and every irreducible positive-energy representation extends to a holomorphic representation of the complexified loop group.

Examples

The simplest nontrivial example is Template:Math. In this case, smooth loops are classified up to connected component by their winding number. More generally, if Template:Math is a torus with Lie algebra Template:Math and cocharacter lattice

$\mathrm{\Pi }={\exp }^{-1}(1)/(2\pi )\subset 𝔱,$

then the loop group has a canonical decomposition

$LT\cong T\times \mathrm{\Pi }\times U,$

where

$U=\exp (V),V=\{\beta :S^1\to 𝔱|{\int }_{S^1}\beta (s)ds=0\}.$

In particular, the connected components of Template:Math are indexed by $\mathrm{\Pi }\cong {\pi }_1(T)$. For Template:Math, this gives

$L{S}^1\cong S^1\times 𝐙\times U,$

so the components of Template:Math are indexed by the integers.

Index theory

Loop groups enter index theory in several related ways. One of the earliest is through the determinant line bundle on the infinite-dimensional Grassmannian associated with a loop group. In the Grassmannian model used by Pressley, Segal, and others, this line bundle is tied to the basic central extension of the loop group and to geometric realizations of its representations.

Analytic index theory also appears through families of Toeplitz operators and through spectral flow. In the torus case, Freed, Hopkins, and Teleman describe a family of Dirac operators whose spectral flow gives the basic topological class needed for their construction of twisted K-theory classes associated with loop-group representations.

A deeper connection comes from the Dirac family attached to a positive-energy representation of a loop group. In the work of Freed, Hopkins, and Teleman, such families of Fredholm operators produce classes in twisted equivariant K-theory, and this construction is one of the ingredients in their identification of the Verlinde ring with twisted equivariant K-theory of Template:Math.

These index-theoretic constructions link loop-group representation theory with geometric quantization, central extensions, and the topology of the group Template:Math itself.

Applications

Loop groups arise in several areas of mathematics and mathematical physics.

• In algebraic topology, the based loop group Template:Math is a basic example of an H-space and is closely related to the homotopy type of Template:Math.
• In representation theory, loop groups and their central extensions lead to affine Kac–Moody algebras and positive-energy representations.
• In algebraic geometry, algebraic loop groups give rise to affine Grassmannians and affine flag varieties.
• In mathematical physics, loop groups occur in conformal field theory, Chern–Simons theory, and twisted equivariant K-theory.

Integrable systems

Loop groups play a central role in the modern theory of integrable systems. A large class of nonlinear evolution equations can be written in Lax pair or [[ Korteweg–De Vries equation#Zero-curvature representation|zero-curvature]] form with a complex spectral parameter Template:Math. When the coefficients depend rationally or Laurent-polynomially on Template:Math and take values in a Lie algebra Template:Math, they may be viewed as elements of a loop algebra Template:Math. Splittings of loop algebras into positive and negative parts, together with factorization in the corresponding loop groups, then produce commuting hierarchies of flows and solution-generating procedures.

A basic example is the KdV hierarchy. In their study of equations of KdV type, Segal and Wilson showed that a large class of solutions can be constructed from points of an infinite-dimensional Grassmannian associated with a loop group. In this picture the commuting flows are induced by the action of a positive loop subgroup, and the corresponding solutions are described in terms of Baker functions and tau functionTemplate:Dns.

Loop-group factorization also underlies dressing transformations and Bäcklund transformations. Terng and Uhlenbeck formulated conservation laws, scattering theory, hierarchies, and Bäcklund transformations within a common framework of loop-group actions, particularly for the ZS–AKNS hierarchy, which includes the nonlinear Schrödinger equation, modified KdV, and the Template:Math-wave equation.

The same methods occur in differential geometry. Loop-group constructions are used for harmonic maps into Lie groups and symmetric spaces, the chiral model, and a range of geometric integrable systems such as constant-mean-curvature and isothermic surfaces. In compact cases, global Birkhoff- and Iwasawa-type decompositions strengthen the dressing method and lead to global Weierstrass-type representations for some geometric integrable systems.

Hodge theory

Loop groups also appear in a more specialized interaction with Hodge theory. In work of Jeremy Daniel, a loop Hodge structure is defined as an infinite-dimensional analogue of a Hodge structure incorporating features of loop-group geometry, and variations of loop Hodge structures are shown to be equivalent to harmonic bundles.

From this point of view, non-abelian Hodge theory can be expressed in terms of period maps with values in infinite-dimensional period domains related to loop-group constructions.

Current groups

A current group generalizes a loop group, replacing the circle with a smooth manifold Template:Math. Thus a current group is the group of smooth mappings from Template:Math into Template:Math, with multiplication defined pointwise:

$C^{\mathrm{\infty }}(M,G)=\{f:M\to G\mid f\text{ is smooth}\},(f_1{f}_2)(x)=f_1(x)f_2(x).$

More generally, Segal described the mapping groups $\mathrm{Map}(X,G)$ for compact manifolds Template:Math as higher-dimensional analogues of loop groups, noting that in mathematical physics they occur as current groups and gauge groups.

If Template:Math is a topological group, the continuous mapping space Template:Math becomes a topological group with the compact-open topology. For smooth maps one usually uses the smooth compact-open topology. Under suitable hypotheses on Template:Math and Template:Math, this gives Template:Math a natural infinite-dimensional Lie group structure. Its Lie algebra is the corresponding current algebra

$C^{\mathrm{\infty }}(M,𝔤),$

with pointwise Lie bracket.

For non-compact manifolds one often studies the compactly supported current group Template:Math, or more generally section groups Template:Math of bundles of Lie groups over Template:Math. Gauge groups of principal bundles are of this form: if Template:Math is a principal Template:Math-bundle, then its gauge group is isomorphic to the section group Template:Math, and the compactly supported gauge group to Template:Math.

Current groups occur naturally in quantum field theory and gauge theory. Compared with loop groups, however, their general representation theory is much less fully developed; much of the recent work has focused on central extensions and on special classes of representations, such as bounded or positive-energy representations of gauge groups.

See also

• Affine Grassmannian
• Affine Kac–Moody algebra
• Loop algebra
• Loop space
• Topological group
• Virasoro group

References

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