An article that we once wrote:
Čech cocycles for differential characteristic classes
Advances in Theoretical and Mathematical Physics, Volume 16 Issue 1 (2012), pages 149-250
on connections on smooth principal ∞-bundles and their higher Chern-Weil secondary invariants.
This is part of the story discussed at differential cohomology in a cohesive topos .
What is called secondary characteristic classes in Chern-Weil theory is a refinement of ordinary characteristic classes of principal bundles from cohomology to differential cohomology: to bundles and higher gerbes with smooth connection.
We consider the problem of refining the construction of secondary characteristic classes from cohomology sets to cocycle spaces and from Lie groups to higher connected covers of Lie groups by smooth ∞-groups: by smooth groupal A-∞ spaces.
This allows us to study the homotopy fibers of the differential characteristic maps thus obtained and to show how these describe differential obstruction problems. This applies in particular to higher twisted differential spin structures called twisted differential string structures and twisted differential fivebrane structures.
To that end we define for every L-∞ algebra $\mathfrak{g}$ a smooth ∞-group $G$ integrating it, and define smooth $G$-principal ∞-bundles with connection. For every L-∞-algebra cocycle of suitable degree, we give a refined ∞-Chern-Weil homomorphism that sends these $\infty$-bundles to cocycles in ordinary differential cohomology that lift the corresponding curvature characteristic classes.
When applied to the canonical 3-cocycle of the Lie algebra of a simple and simply connected Lie group $G$ this construction gives a refinement of the secondary first fractional Pontryagin class of $G$-principal bundles to cocycle space. Its homotopy fiber is the 2-groupoid of smooth String(G)-principal 2-bundles with 2-connection, where String(G) is a smooth 2-group refinement of the topological string group. Its homotopy fibers over non-trivial classes we identify with the 2-groupoid of twisted differential string structures that appears in the Green-Schwarz anomaly cancellation mechanism of heterotic string theory.
Finally, when our construction is applied to the canonical 7-cocycle on the Lie 2-algebra of the string 2-group, it produces a secondary characteristic map for String-principal 2-bundles which refines the second fractional Pontryagin class. Its homotopy fiber is the 6-groupoid of principal 6-bundles with 6-connection over the fivebrane 6-group. Its homotopy fibers over nontrivial classes are accordingly twisted differential fivebrane structures that have beeen argued in (SSS09) to control the anomaly cancellation mechanism in magnetic dual heterotic string theory.
differential cohomology in a cohesive topos
Čech cocycles for differential characteristic classes
See differential cohomology in a cohesive topos -- references .