Differential Cohomology: Categories, Characteristic Classes, and Connections

TL;DR

This paper provides a modern, homotopy-theoretic overview of differential cohomology, connecting classical theories like Chern-Weil with contemporary approaches, and explores various applications in geometry and physics.

Contribution

It introduces a homotopy-theoretic framework for differential cohomology, including differential characteristic classes and lifts of classical invariants, with accessible background and diverse applications.

Findings

Unified perspective on differential cohomology using sheaves

Construction of differential lifts of characteristic classes

Applications to configuration spaces and field theories

Abstract

We give an overview of differential cohomology from a modern, homotopy-theoretic perspective in terms of sheaves on manifolds. Although modern techniques are used, we base our discussion in the classical precursors to this modern approach, such as Chern-Weil theory and differential characters, and include the necessary background to increase accessibility. Special treatment is given to differential characteristic classes, including a differential lift of the first Pontryagin class. Multiple applications, including to configuration spaces, invertible field theories, and conformal immersions, are also discussed. This book is based on talks given at MIT's Juvitop seminar jointly with UT Austin in the Fall of 2019.