Loop Differential K-theory

2012/01/22 by Thomas Tradler, Tradler, Thomas, Scott O. Wilson +3
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.1201.4593

30 pages; new last section, appendix. To appear in Annales Mathematiques Blaise Pascal

arxiv created 2015/01/13 · arxiv updated 2015/01/14

Abstract

In this paper we introduce an equivariant extension of the Chern-Simons form, associated to a path of connections on a bundle over a manifold M, to the free loop space LM, and show it determines an equivalence relation on the set of connections on a bundle. We use this to define a ring, loop differential K-theory of M, in much the same way that differential K-theory can be defined using the Chern-Simons form [SS]. We show loop differential K-theory yields a refinement of differential K-theory, and in particular incorporates holonomy information into its classes. Additionally, loop differential K-theory is shown to be strictly coarser than the Grothendieck group of bundles with connection up to gauge equivalence. Finally, we calculate loop differential K-theory of the circle.

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