Differential Cohomology: Categories, Characteristic Classes, and Connections
Araminta Amabel, Arun Debray, Peter J. Haine

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.
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Taxonomy
TopicsHomotopy and Cohomology in Algebraic Topology · Algebraic structures and combinatorial models · Algebraic Geometry and Number Theory
