Differential Cohomology: Categories, Characteristic Classes, and Connections

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.
Comment: Comments very welcome! Edited by: A. Amabel, A. Debray, P. Haine. Talks by: A. Amabel, D. Chua, A. Debray, S. Devalapurkar, D. Freed, P. Haine, M. Hopkins, G. Parker, C. Reid, and A. Zhang. v2. Added Chapter 13 on classifying spaces for G-bundles. Improved notation. Corrected numerous typos and made a number of minor additions. v1. 232 pages