Twisted differential cohomology

TL;DR

This paper develops a framework for twisted differential cohomology theories, combining homotopy theoretic and oo-categorical approaches, and establishes their fundamental properties and existence results.

Contribution

It introduces the concept of differential twists and constructs twisted differential cohomology groups, providing foundational results for various cohomology theories.

Findings

Existence and uniqueness of differential twists for generalized cohomology theories

Construction of twisted differential cohomology groups and spectra

Application potential to K-theory, topological modular forms, and more

Abstract

The main goal of the present paper is the construction of twisted generalized differential cohomology theories and the comprehensive statement of its basic functorial properties. Technically it combines the homotopy theoretic approach to (untwisted) generalized differential cohomology developed by Hopkins-Singer and later by the first author and D. Gepner with the oo-categorical treatement of twisted cohomology by Ando-Blumberg-Gepner. We introduce the notion of a differential twist for a given generalized cohomology theory and construct twisted differential cohomology groups (resp. spectra). The main technical results of the paper are existence and uniqueness statements for differential twists. These results will be applied in a variety of examples, including K-theory, topological modular forms and other cohomology theories.