Differential cohomology and topological actions in physics

We use differential cohomology to systematically construct a large class of topological actions in physics, including Chern–Simons terms,Wess–Zumino–Novikov–Witten terms, and theta terms (continuous or discrete). We introduce a notion of invariant differential cohomology and use it to describe theories with global symmetries and we use equivariant differential cohomology to describe theories with gauge symmetries. There is a natural map from equivariant to invariant differential cohomology whose failure to surject detects ’t Hooft anomalies, i.e. global symmetries which cannot be gauged. We describe a number of simple examples from quantum mechanics, such as a rigid body or an electric charge coupled to a magnetic monopole. We also describe examples of sigma models, such as those describing non-abelian bosonization in two dimensions, for which we offer an intrinsically bosonic description of the $\operatorname{mod}-2$-valued ’t Hooft anomaly that is traditionally seen by passing to the dual theory of Majorana fermions. Along the way, we describe a smooth structure on equivariant differential cohomology and prove various exactness and splitting properties that help with the characterization of both the equivariant and invariant theories.

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J.D. and B.G. are supported by STFC consolidated grant ST/P000681/1, and B.G. is supported by King’s College, Cambridge.

Published 14 August 2024