This week the Simons Center is hosting a workshop on Differential Cohomology and its applications in physics. I won’t try and give an explanation of what differential cohomology is here, with a little luck the videos of the talks will soon be on-line. Very briefly, this subject is about an extension of the usual sort of cohomology theory that provides finer information. It was discovered independently by Deligne in an algebraic geometry context (his construction is often called “Deligne Cohomology”) and by Jim Simons and Jeff Cheeger in a differential geometry context. The subject made its appearance in physics first through Wess-Zumino-Witten terms in non-linear sigma models and the Chern-Simons term in gauge theories.
Dan Freed’s first lecture included an extensive discussion of one recent example that uses a generalized cohomology theory, and thus generalized differential cohomology, see here for details. Mike Hopkins discussed his work with Singer which led to this paper, and some ongoing work from a more generalized perspective. He started with some history, explaining that things began with a specific example he noticed in work on topological modular forms that Witten had found around the same time in work on the partition function of the fivebrane. He described this initial impetus as like discovering that they both were looking at the same intriguing specific tropical fish, with attempts to understand it leading to a huge ferocious formalism he characterizes as a shark that lept out of the tank.
In the afternoon, Jim Simons gave a wonderful description of the early history of his work on Chern-Simons invariants and Cheeger-Simons differential characters, leading up to recent work trying to prove that certain properties uniquely characterize this kind of theory. He began his talk by noting that only one small piece of chalk was available and complaining “I paid all this money for this place and all I get to use is one broken piece of chalk?”. The story started when he tried to work out a combinatorial formula for the signature in 4 dimensions, by analogy with what one does starting with the Chern-Weil formula for the Euler characteristic. In the signature case, the evaluation of a 4d Pontryagin class leads to the study of a 3-form on the boundary, which he investigated with Chern, leading to Chern-Simons theory. This is much the same problem as the one that (more than a decade later) I started working on as a graduate student in physics, trying to figure out how to calculate the second Chern number of a lattice gauge field configuration.
Finally Krzysztof Gawedzi gave an interesting talk reviewing the by-now-extensive history of the use of this kind of mathematics in physics, including various incarnations of the notion of a “gerbe”. Unfortunately I’m back in the city now, hope to follow the rest of the workshop via video at some point.