Twisted smooth Deligne cohomology

TL;DR

This paper develops a theory of twisted Deligne cohomology, extending its classical form by incorporating degree one twists, and explores its properties, examples, and potential applications.

Contribution

It introduces the first explicit construction of twisted Deligne cohomology, expanding the scope of differential cohomology theories with practical examples.

Findings

Defined degree one twists for Deligne cohomology

Established main properties of the twisted theory

Provided illustrative examples and applications

Abstract

Deligne cohomology can be viewed as a differential refinement of integral cohomology, hence captures both topological and geometric information. On the other hand, it can be viewed as the simplest nontrivial version of a differential cohomology theory. While more involved differential cohomology theories have been explicitly twisted, the same has not been done to Deligne cohomology, although existence is known at a general abstract level. We work out what it means to twist Deligne cohomology, by taking degree one twists of both integral cohomology and de Rham cohomology. We present the main properties of the new theory and illustrate its use with examples and applications. Given how versatile Deligne cohomology has proven to be, we believe that this explicit and utilizable treatment of its twisted version will be useful.