nLab nonabelian differential cohomology




What may be called nonabelian differential cohomology – in combination of nonabelian cohomology and differential cohomology – is a notion of connections on higher bundles with higher gauge group being a possibly non-abelian n-group.

Early motivation was the observation (SSS12) that with the Green-Schwarz mechanism imposed, the B-field in heterotic string theory combines with the ordinary non-abelian gauge field into a non-abelian higher connection (a twisted differential string structure), and analogously so after lifting the situation to the C-field in Hořava-Witten theory (FSS15).

Later, the desire to more accurately model more of the expected subtle topological properties of the C-field (such as the shifted C-field flux quantization combined with Page charge-quantization) led to the hypothesis (“Hypothesis H”) that it ought to be flux quantized in unstable differential 4-Cohomotopy, hence with homotopy type of its higher gauge group being the loop \infty -group of the 4-sphere. This Hypothesis H turns out to subtly reproduce the Green-Schwarz mechanism in its lift to Hořava-Witten theory (FSS22) and also that on M5-branes (SS20, FSS21).

Generally, one may understand differential nonabelian cohomology as modelling flux quantized higher gauge fields (SS23).

For example, also the “Hypothesis K” of D-brane charge quantization in topological K-theory is, at face value, a flux quantization (of the RR-field combined with the B-field) in a non-abelian differential cohomology (see the overview here) which however may be and commonly is regarded as twisted abelian namely as twisted differential K-theory, by regarding the B-field as a “background field” and considering the RR-field in its dependence.


Via higher Cartan connections from (adjusted) Weil algebras

Early explorative notes:

The local structure of L L_\infty -algebra valued differential forms, via dg-algebra homomorphisms out of (“adjusted”) Weil algebras in the de Rham complex of the base manifold:

The Lie integration of this local structure to globally possibly nontrivial actual connections on higher bundles:

For the application of this construction to modelling the Green-Schwarz mechanism see below.

Via the nonabelian character map

A more axiomatic formulation of differential non-abelian cohomology (not yet proven to subsume the above construction) in non-abelian generalization of the original Hopkins-Singer construction of abelian (namely Whitehead-generalized cohomology) differential cohomology, now using a nonabelian generalization of the Chern-Dold character map:

On how this generally serves to reflect flux quantization in higher gauge theories:

reviewed in:

Differential cohomotopy for the 11d SuGra C-field

The example of unstable (and as such non-abelian) differential Cohomotopy (in the context of modeling the supergravity C-field via Hypothesis H):

Higher gauge theory of the Green-Schwarz mechanism

Discussion of higher gauge theory modeling the Green-Schwarz mechanisms for anomaly cancellation in heterotic string theory, on M5-branes, and in related systems in terms of some kind of nonabelian differential cohomology (ordered by arXiv time-stamp):

Last revised on January 11, 2024 at 01:40:38. See the history of this page for a list of all contributions to it.