# nLab Quadratic Functions in Geometry, Topology, and M-Theory

### Context

#### Differential cohomology

differential cohomology

## Application to gauge theory

This entry is about the article

which discusses differential cohomology and uses it to study quadratic refinements of intersection pairing $(x,y) \mapsto \int_\Sigma x \cup y$ in cohomology. This is motivated from and related to the physics of self-dual higher gauge theory appearing in string theory and the corresponding quantum anomalies.

The article introduces a systematic general definition for the refinement of any generalized (Eilenberg-Steenrod) cohomology theory to differential cohomology (the context for higher gauge fields in physics) in terms of differential function complexes.

In this construction continuous classifying maps from a smooth manifold into a spectrum representing the given cohomology are equipped with smooth differential forms that have under the de Rham theorem the same class in real cohomology as the pullback along the classifying map of a collection of given real cocycles on the spectrum.

The connecton of this work to the physics of the electromagnetic field and of higher gauge fields in string theory was later developed further notably in

• Dan Freed, Dirac charge quantization and generalized differential cohomology (arXiv).

To this date generalized differential cohomology theories keep being studied mostly with motivation from string theory, but the work of Hopkins and Singer has put this subject on solid mathematical ground, and an independent mathematical field of differential cohomology is developing since then. See the list of references at differential cohomology.