Loop differential K-theory

Tradler, Thomas; Wilson, Scott O.; Zeinalian, Mahmoud

Annales mathématiques Blaise Pascal, Tome 22 (2015) no. 1, pp. 121-163.

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 [14]. 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.

Classification : 58J28, 19A99, 55P35
Mots clés : Differential $K$-Theory, Bismut-Chern-Simons forms, Loop spaces

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