A loop group extension of the odd Chern character

2013/11/25 by Scott O. Wilson, Wilson, Scott O.
Mathematics · #Algebraic Topology (math.AT) #Differential Geometry (math.DG) #FOS: Mathematics #K-Theory and Homology (math.KT) #math.AT #math.DG #math.KT

paper · pdf · doi:10.48550/arxiv.1311.6393

12 pages, comments welcome

arxiv created 2013/11/26 · arxiv updated 2013/11/27

Abstract

We show that the universal odd Chern form, defined on the stable unitary group U, extends to the loop group LU in a way that is closed with respect to an equivariant-type differential. This provides an odd analogue to the Bismut-Chern form. We also describe the associated transgression form, the so-called Bismut-Chern-Simons form, and explicate some properties it inherits as a differential form on the space of maps of a cylinder into the stable unitary group. As a corollary, we obtain the Chern character homomorphism from odd K-theory to the periodic cohomology of the free loop space, represented geometrically on the level of differential forms.

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