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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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