For $c$ any characteristic class, its homotopy fibers on cocycle ∞-groupoids represent $c$-twisted cohomology (for instance twisted bundles, twisted spin structures, etc.).
If $c$ is refined to a characteristic class $\mathbf{c}$ in Smooth∞Grpd there may exist further refinements $\hat {\mathbf{c}}$ 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 $\mathbf{H}$ be a cohesive (∞,1)-topos, usually $\mathbf{H} =$ Smooth∞Grpd or SynthDiff∞Grpd or the like.
Let $K, G$ be ∞-group objects in $\mathbf{H}$ and let
be a morphism of their delooping objects / moduli stacks.
For $X \in \mathbf{H}$ any object and $P \to X$ an $K$-principal ∞-bundle over $X$, the ∞-groupoid
hence the (∞,1)-pullback
we may call equivalently
the $\infty$-groupoid of $K$-structures on $P$ (with respect to the given $\mathbf{c}$);
the $\infty$-groupoid of $[P]$-twisted $\mathbf{c}$-structures.
As discussed at twisted cohomology, we may think of an object in $\mathbf{c}Struc_{[P]}(X)$ as a section (up to homotopy) $\sigma$
where we think of $\mathbf{c}$ as being the universal twisting $\infty$-bundle and where $g : X \to \mathbf{B}K$ is a morphism presenting $P$.
The following definition looks at a differential refinement of this situation.
For $\mathbf{c} : \mathbf{B}G \to \mathbf{B}^n U(1)$ a characteristic map in $\mathbf{H}$ and $\hat {\mathbf{c}} : \mathbf{B}G_{\mathrm{conn}} \to \mathbf{B}^n U(1)_{\mathrm{conn}}$ 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 $\hat {\mathbf{c}}\mathrm{Struc}_{\mathrm{tw}}(X)$ for the corresponding twisted cohomology,
Twisted differential $\mathbf{c}$-structures appear in various guises in the background gauge fields of string theory application.
orthogonal structure / Riemannian metric; see the discussion at vielbein .
Higher differential spin structures
The notion was introduced in
and expanded on in
An exposition is in
Lecture notes include
A general account is in section 4.2 of
In
it is proposed to call such twisted structures “relative fields”.