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 [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.
Highlights
Much attention has been given recently to differential cohomology theories, as they play an increasingly important role in geometry, topology and mathematical physics
For the purposes of this paper we focus on the even degree part of K-theory, denoted by K 0, and the model of even differential K-theory presented by Simons-Sullivan in [14], which proceeds by defining an equivalence relation on the set of a connections on a bundle, by requiring that the Chern-Simons form, associated to a path of connections, is exact
Proceeding in much the same way as in [14], we prove that the condition of BCS(∇s ) being exact defines an equivalence relation on the set of connections on a bundle, and use this to define a functor from manifolds to rings, which we call loop differential K-theory
Summary
Much attention has been given recently to differential cohomology theories, as they play an increasingly important role in geometry, topology and mathematical physics. Proceeding in much the same way as in [14], we prove that the condition of BCS(∇s ) being exact defines an equivalence relation on the set of connections on a bundle, and use this to define a functor from manifolds to rings, which we call loop differential K-theory Elements in this ring contain the additional information of the trace of holonomy of a connection, and the entire extension of the trace of holonomy to a co-cycle on the free loop space known as the Bismut-Chern form, which is an equivariantly closed form on the free loop space that restricts to the classical Chern character [1], [8], [10], [16]. We are optimistic that the extension of the Chern-Simons form to the free loop space, referred to here as the Bismut-Chern-Simons form, will have a field theoretic interpretation, and may be of interest in other mathematical discussions that begin with the Chern-Simons form, such as 3-dimensional TFT’s, quantum computation, and knot invariants
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