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

### Context

#### Differential cohomology

differential cohomology

## Ingredients

• cohomology

• differential geometry

• ## Connections on bundles

• connection on a bundle

• curvature

• Chern-Weil theory

• ## Higher abelian differential cohomology

• differential function complex

• differential orientation

• ordinary differential cohomology

• differential K-theory

• differential elliptic cohomology

• differential cobordism cohomology

• ## Higher nonabelian differential cohomology

• Chern-Weil theory in Smooth∞Grpd

• ∞-Chern-Simons theory

• ## Fiber integration

• higher holonomy

• fiber integration in differential cohomology

• ## Application to gauge theory

• gauge theory

• gauge field

• quantum anomaly

• This entry is about the article

which discusses (ordinary) differential cohomology refinements of generalized (Eilenberg-Steenrod) cohomology and uses it to study quadratic refinements (via characteristic cohomology classes) of intersection pairings $(x,y) \mapsto \int_\Sigma x \cup y$ in ordinary cohomology. Mathematically this refines the construction of Theta characteristics to ordinary differential cohomology and to higher intermediate Jacobians. Physically it is motivated from and related to self-dual higher gauge theory (see there for more) appearing in string theory and the corresponding quantum anomalies. In particular it makes rigorous the construction (Witten 96) of the partition function of the self-dual B-field in the 6d (2,0)-superconformal QFT on the worldvolume of the M5-brane via geometric quantization of abelian 7d Chern-Simons theory.

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.

For a somewhat streamlined account see at at differential cohomology hexagon the section Hopkins-Singer coefficients.

A review talk is recorded at

• Michael Hopkins, Differential cohomology for general cohomology theories and one physical motivation, talk at Simons Center Workshop on Dierential Cohomology (2011) (video)

This states that the article “should have been written” in terms of smooth infinity-stacks. For references that do so see for instance at differential cohomology hexagon the list of references.

The connection of this work to the physics of the electromagnetic field and of higher gauge field in string theory was later further advertized in

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 – References and at differential cohomology hexagon – References .