under construction
group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
spin geometry, string geometry, fivebrane geometry …
In analogy to how the Lie group spin^c is obtained by twisting the lift through the second stage of the Whitehead tower of $\mathbf{B}O$ by the first Chern class
there is a similar twist by the second Chern class of the lift through the next stage of the Whitehead tower
Accordingly a lift of the structure group to $String^c$ is a $String^c$-structure.
For the moment see at twisted smooth cohomology in string theory for more.
Topological $string^c$-structures were introduced
and shown to induce a twisted Witten genus in
Qingtao Chen, Fei Han, Weiping Zhang, Generalized Witten Genus and Vanishing Theorems, Journal of Differential Geometry 88.1 (2011): 1-39. (arXiv:1003.2325)
Jianqing Yu, Bo Liu, On the Witten Rigidity Theorem for $String^c$ Manifolds, Pacific Journal of Mathematics 266.2 (2013): 477-508. (arXiv:1206.5955)
The push-forward in twisted tmf induced by a $string^c$-structure is discussed in section 11 of
A discussion explicitly in the context of string theory is in
Their smooth refinement and their smooth moduli 2-stacks were introduced in
A general discussion is in section 5.2 of