Every generalized (Eilenberg-Steenrod) cohomology-theory has a refinement to differential cohomology. By ordinary differential cohomology one refers, for emphasis, to the differential refinement of ordinary integral cohomology , hence of the cohomology theory represented by the Eilenberg-MacLane spectrum . To the extent that integral cohomology is often just called cohomology if the context is clear, ordinary differential cohomology is often called just differential cohomology .
Ordinary differential cohomology classifies circle n-bundles with connection. In low degree this are ordinary circle bundles with connection. In the next degree this are circle 2-group principal 2-bundles / bundle gerbes with 2-connection.
Here we write for the ordinary differential cohomology groups of a smooth manifold .
There are two natural morphism
underlying characteristic class
produces the class in integral cohomology that underlies a differential cocycle;
The following is either a definition, if regarded as an axiomatic characterization of ordinary differential cohomology, or it is an proposition, if regarded as a property of one of the models.
Let be a smooth manifold and with Write
for the ordinary differential cohomology of in degree ;
for the collection of differential forms of degree ;
All of these sets are abelian groups: the forms under addition of forms, and the differential cohomology classes are defined or proven (depending on the approach, see above) to have abelian group structure such that the maps to curvatures and characteristic classes, from above are homomorphisms of abelian groups.
The differential cohomology of fits into short exact sequences of abelian groups
curvature exact sequence
characteristic class exact sequence
The second sequence (2) says in words: two connections on the same circle -bundle differ by a globally defined connection -form, well defined up to addition of a form with integral periods.
More is true: both these sequences interlock to form the hexagonal differential cohomology diagram of ordinary differential cohomology. For more see at differential cohomology diagram – Examples – Deligne coefficients.
There are various equivalent cocycle-models for ordinary differential cohomology. They include
Cocycles in degree 2 ordinary differential cohomology are represented by ordinary circle group-principal bundles with connection on . The class is the Chern class of the underlying circle bundle and the form is the curvature 2-form of the connection .
is the integral characteristic class corresponding to the invariant polynomial and the projection
the electromagnetic field is a cocycle in degree 2 ordinary differential cohomology
the Kalb-Ramond field is a cocycle in degree 3;
the supergravity C-field is a cocyce in degree 4.
In abelian higher dimensional Chern-Simons theory in dimension a field configuration is a cocycle in ordinary differential geometry of degree , for .
A good discussion is in
A pedestrian introduction of ordinary differential cocycles is in section 2.3 there.
The systematic construction and definition via a homotopy pullback is in section 3.2.
The relation to Chern-Weil theory is in section 3.3.
A characterization by the two characteristic exact sequences is discussed in
In the general abstract context of cohesive (∞,1)-toposes differential cohomology is discussed in