Loop group

There are different spaces known as "loop groups" . One of the most studied is defined by considering, for a compact semi-simple Lie group (cf. also Lie group, semi-simple), the loop group ; here, is the circle of complex numbers .

One can take spaces of polynomial, rational, real-analytic, smooth, or loops, in decreasing order of regularity. The metric is actually a Kähler metric. The complex structure is defined by considering as -forms on the Lie algebra of , those with positive Fourier coefficients. Equivalently, is a homogeneous space of the loop group , with isotropy

This is shown by considering the "Grassmannian model" of D. Quillen for . For simplicity, take ; then is naturally identified with the Hilbert–Schmidt Grassmannian of certain Hilbert subspaces of , stable under multiplication by . This also shows that has a holomorphic embedding into a projectivized Hilbert space, by generalized Plücker coordinates; and a determinant line bundle.

While the smooth loop group is an infinite-dimensional Fréchet manifold, the rational and polynomial loop groups have natural filtrations by finite-dimensional algebraic subvarieties. Moreover, by a result of G. Segal, any holomorphic mapping from a compact manifold into actually goes into the rational loops (up to multiplication by a constant). This has application in gauge theory, because a theorem of M.F. Atiyah and S.K. Donaldson identifies, for a classical group , the moduli space of charge- framed -instantons with the moduli spaces of based holomorphic -spheres in , of topological degree . Another application to gauge theory and to the theory of completely integrable systems is given by a construction of K. Uhlenbeck, refining earlier work by V.E. Zakharov and others: this identifies, modulo basepoints, harmonic mappings with certain holomorphic mappings . The degree of now gives the energy of , by a formula of G. Valli, generalizing earlier work of E. Calabi.

The representation theory of has also been studied: the key point is the construction of a universal central extension, which makes it possible to define infinite-dimensional projective representations.

Other spaces commonly known as "loop groups" are the group of diffeomorphism of the circle, , and the loop space of a manifold, .

The space has been studied as reparametrization groups for string theory. It has two natural homogeneous spaces: and , which are infinite-dimensional Kähler manifolds (the Kähler metric is the metric ). The space has been studied in the theory of universal Teichmüller spaces (cf. also Teichmüller space).

When is a manifold, is defined as the space of loops in starting at a fixed basepoint. Here, the group operation is given by composition of loops. The space has been considered in connection with the problem of characterizing parallel transport operators, as configuration space in string theory, and in probability theory.

References

[a1]	A. Pressley, G. Segal, "Loop groups" , Oxford Univ. Press (1986)