Quadratic Functions in Geometry, Topology, and M-Theory

This entry is about the article

- Michael Hopkins, Isadore Singer,
*Quadratic Functions in Geometry, Topology,and M-Theory*J. Differential Geom. Volume 70, Number 3 (2005), 329-452. (journal, arXiv:math.AT/0211216)

which discusses (ordinary) differential cohomology refinements of generalized (Eilenberg-Steenrod) cohomology and uses it to study quadratic refinements (via characteristic cohomology classes) of intersection pairing $(x,y) \mapsto \int_\Sigma x \cup y$ 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.

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 maybe streamlined discussion of this construction of differential cohomology see at *differential cohomology hexagon* the section *Hopkins-Singer coefficients*.

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

- Daniel 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 – References* and at *differential cohomology hexagon – References* .

category: reference

Revised on June 2, 2014 07:01:46
by Urs Schreiber
(82.113.99.242)