The geometry of loop groups

Abstract

The space ΩG of based loops on a compact Lie group admits a Kähler metric. Its curvature is expressed in terms of Toeplitz operators, and we define Chern classes by analogy with Chern-Weil theory in finite dimensions. In infinite dimensions extra geometric structure-a Fredholm structure-must be imposed before characteristic classes are defined. There is a natural Fredholm structure on ΩG induced from the family of Toeplitz operators. We use the index theorem for families of Fredholms parametrized by a group (proved in [20]) to show that the Chern classes of the Toeplitz family agree with the Chern classes defined by curvature. Explicit formulas for ΩSU(n) are obtained. We also prove that the real characteristic classes of ΩG vanish for any group G. Extensions to more general groups of gauge transformations are considered. © 1988, International Press of Boston, Inc. All Rights Reserved.