Annales Mathématiques
Blaise Pascal
no. 1
Loop differential K-theory
Thomas Tradler1 ; Scott O. Wilson2 ; Mahmoud Zeinalian3
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.

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.

DOI : 10.5802/ambp.348
Classification : 58J28, 19A99, 55P35
Mots clés : Differential $K$-Theory, Bismut-Chern-Simons forms, Loop spaces

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) 
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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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