On the ring structure of spark characters

TL;DR

This paper introduces a new approach to understanding the ring structure of differential characters on smooth manifolds using the smooth hyperspark complex, providing explicit formulas and applications to the unit circle.

Contribution

It offers a novel description of the ring structure via the hyperspark complex, explicit product formulas, and a direct proof of the isomorphism with smooth Deligne cohomology rings.

Findings

Explicit product formula for differential characters

Calculation of the product for characters of the unit circle

Construction of an isomorphism between spark classes and Deligne cohomology

Abstract

We give a new description of the ring structure on the differential characters of a smooth manifold via the smooth hyperspark complex. We show the explicit product formula, and as an application, calculate the product for differential characters of the unit circle. Applying the presentation of spark classes by smooth hypersparks, we give an explicit construction of the isomorphism between groups of spark classes and the $(p,p)$ part of smooth Deligne cohomology groups associated to a smooth manifold. We then give a new direct proof that this is an isomorphism of ring structures.