Connections on bundles
Higher abelian differential cohomology
Higher nonabelian differential cohomology
Application to gauge theory
For any characteristic class, its homotopy fibers on cocycle ∞-groupoids represent -twisted cohomology (for instance twisted bundles, twisted spin structures, etc.).
If is refined to a characteristic class in Smooth∞Grpd there may exist further refinements to ordinary differential cohomology. The twisted cohomology of these differential characteristic classes may be called twisted differential structures . For instance differential string structures . See below for more examples.
These structures have a natural interpretation and play a natural roles as physical fields (see there for a comprehensive discussion).
Let be a cohesive (∞,1)-topos, usually Smooth∞Grpd or SynthDiff∞Grpd or the like.
Let be ∞-group objects in and let
be a morphism of their delooping objects / moduli stacks.
For any object and an -principal ∞-bundle over , the ∞-groupoid
hence the (∞,1)-pullback
we may call equivalently
The following definition looks at a differential refinement of this situation.
For a characteristic map in and its differential refinement, sending connections on ∞-bundles to circle n-bundles with connection (see ∞-Chern-Weil homomorphism, we may think of this also as an extended Lagrangian for a higher gauge theory).
We write for the corresponding twisted cohomology,
Twisted differential -structures appear in various guises in the background gauge fields of string theory application.
The notion was introduced in
and expanded on in
An exposition is in
Lecture notes include
A general account is in section 5.2 of
it is proposed to call such twisted structures “relative fields”.