group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
For $n \in \mathbb{N}$ the Lie group $Spin^c(n)$ is a central extension
of the special orthogonal group by the circle group. This comes with a long fiber sequence
where $W_3$ is the third integral Stiefel-Whitney class .
By the definition at twisted cohomology, for a given class $[c] \in H^3(X, \mathbb{Z})$, a $c$-twisted $spin^c$-structure is a choice of homotopy
The space/∞-groupoid of all twisted $Spin^c$-structures on $X$ is the homotopy fiber $W_3 Struc_{tw}(T X)$ in the pasting diagram of homotopy pullbacks
where the right vertical morphism is the canonical effective epimorphism that picks one point in each connected component.
Since $U(1) \to Spin^c \to SO$ is a sequence of Lie groups, the above may be lifted from the (∞,1)-topos Top $\simeq$ ∞Grpd to Smooth∞Grpd.
More precisely, by the discussion at Lie group cohomology (and smooth ∞-groupoid -- structures) the characteristic map $W_3 : B SO \to B^2 U(1)$ in $\infty Grpd$ has, up to equivalence, a unique lift
to Smooth∞Grpd, where on the right we have the delooping of the smooth circle 2-group.
By the general definition at twisted differential c-structure , the 2-groupoid of smooth twisted $spin^c$-structures $\mathbf{W}_3 Struc_{tw}(X)$ is the joint (∞,1)-pullback
The existence of an ordinary spin structure on a space $X$ is, as discussed there, the condition for $X$ to serve as the target space for the spinning particle sigma-model, in that the existence of this structure is precisely the condition that the corresponding fermionic quantum anomaly on the worldline vanishes.
Twisted $spin^c$-structures appear similarly as the conditions for the analogous quantum anomaly cancellation, but now of the open type II superstring ending on a D-brane. This is also called the Freed-Witten anomaly cancellation.
More precisely, in these applications the class of $W_3(TX) - H$ need not vanish, it only needs to be $n$-torsion if there is moreover a twisted bundle of rank $n$ on the $D$-brane.
See the references below for details.
The notion of twisted $Spin^c$-structures as such were apparently first discussed in section 5 of
More discussion appears in section 3 of
The refinement to smooth twisted structures is discussed in section 4.1 of
The need for twisted $Spin^c$-structures as Freed-Witten anomaly cancellation condition on the worldvolume of D-branes in string theory was first discussed in
More details are in
A clean formulation and review is provided in
Loriano Bonora, Fabio Ferrari Ruffino, Raffaele Savelli, Classifying A-field and B-field configurations in the presence of D-branes (arXiv:0810.4291)
Fabio Ferrari Ruffino, Classifying A-field and B-field configurations in the presence of D-branes - Part II: Stacks of D-branes (arXiv:1104.2798)
Fabio Ferrari Ruffino, Topics on topology and superstring theory (arXiv:0910.4524)
and
In (Laine) the discussion of FW-anomaly cancellation with finite-rank gauge bundles is towards the very end, culminating in equation (3.41).