Differential Characters and Geometric Chains

TL;DR

This paper explores Cheeger-Simons differential characters, offering geometric insights into their structure, fiber integration, and applications like holonomy and transgression, establishing uniqueness and extending to boundary cases.

Contribution

It provides a geometric description of the ring structure and fiber integration for differential characters, proving their uniqueness and extending fiber integration to boundary cases.

Findings

Geometric descriptions of ring structure and fiber integration

Proof of uniqueness of differential cohomology theories

Extension of fiber integration to fibers with boundary

Abstract

We study Cheeger-Simons differential characters and provide geometric descriptions of the ring structure and of the fiber integration map. The uniqueness of differential cohomology (up to unique natural transformation) is proved by deriving an explicit formula for any natural transformation between a differential cohomology theory and the model given by differential characters. Fiber integration for fibers with boundary is treated in the context of relative differential characters. As applications we treat higher-dimensional holonomy, parallel transport, and transgression.