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The geometry of loop groups. (English) Zbl 0619.58003
The space $$\Omega$$ G of based loops on a compact Lie group admits a Kähler metric. We find a formula for its curvature in terms of Toeplitz operators. By analogy with Chern-Weil theory in finite dimensions we use the curvature form to define Chern classes of $$\Omega$$ G. These geometric Chern classes do not directly come from topology, as they do in finite dimensions. Extra geometric structure - a Fredholm structure - must be imposed before characteristic classes are defined topologically. There is a natural Fredholm structure on $$\Omega$$ G induced from the family of Toeplitz operators. We use an index theorem for families of Fredholms parametrized by a group to show that the Chern classes of the Toeplitz family agree with the Chern classes defined by curvature. Explicit formulas for $$\Omega$$ SU(n) are obtained. Extensions to more general groups of gauge transformations are considered.

##### MSC:
 58B20 Riemannian, Finsler and other geometric structures on infinite-dimensional manifolds 57R20 Characteristic classes and numbers in differential topology
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