Loop differential K-theory

Thomas Tradler; Scott O. Wilson; Mahmoud Zeinalian

doi:10.5802/ambp.348

Thomas Tradler ¹ ; Scott O. Wilson ² ; Mahmoud Zeinalian ³

¹ Department of Mathematics College of Technology City University of New York 300 Jay Street Brooklyn, NY 11201 (USA)
² Department of Mathematics Queens College City University of New York 65-30 Kissena Blvd. Flushing, NY 11367 (USA)
³ Department of Mathematics Long Island University LIU Post 720 Northern Boulevard Brookville, NY 11548 (USA)

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

Résumé

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 $L M$ , 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.

DOI : 10.5802/ambp.348

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

Affiliations des auteurs :

Thomas Tradler ¹ ; Scott O. Wilson ² ; Mahmoud Zeinalian ³

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Thomas Tradler; Scott O. Wilson; Mahmoud Zeinalian. Loop differential K-theory. Annales mathématiques Blaise Pascal, Tome 22 (2015) no. 1, pp. 121-163. doi : 10.5802/ambp.348. https://ambp.centre-mersenne.org/articles/10.5802/ambp.348/

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