Loop differential K-theory
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 : https://doi.org/10.5802/ambp.348
Classification : 58J28,  19A99,  55P35
Mots clés : Differential $K$-Theory, Bismut-Chern-Simons forms, Loop spaces
@article{AMBP_2015__22_1_121_0,
author = {Tradler, Thomas and Wilson, Scott O. and Zeinalian, Mahmoud},
title = {Loop differential K-theory},
journal = {Annales Math\'ematiques Blaise Pascal},
pages = {121--163},
publisher = {Annales math\'ematiques Blaise Pascal},
volume = {22},
number = {1},
year = {2015},
doi = {10.5802/ambp.348},
language = {en},
url = {www.numdam.org/item/AMBP_2015__22_1_121_0/}
}
Tradler, Thomas; Wilson, Scott O.; Zeinalian, Mahmoud. Loop differential K-theory. Annales Mathématiques Blaise Pascal, Tome 22 (2015) no. 1, pp. 121-163. doi : 10.5802/ambp.348. http://www.numdam.org/item/AMBP_2015__22_1_121_0/

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