Special and general types
For the Lie group is a central extension
of the special orthogonal group by the circle group. This comes with a long fiber sequence
where is the third integral Stiefel-Whitney class .
By the definition at twisted cohomology, for a given class , a -twisted -structure is a choice of homotopy
The space/∞-groupoid of all twisted -structures on is the homotopy fiber 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 is a sequence of Lie groups, the above may be lifted from the (∞,1)-topos Top ∞Grpd to Smooth∞Grpd.
More precisely, by the discussion at Lie group cohomology (and smooth ∞-groupoid -- structures) the characteristic map in 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 -structures is the joint (∞,1)-pullback
Anomaly cancellation in physics
The existence of an ordinary spin structure on a space is, as discussed there, the condition for 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 -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 need not vanish, it only needs to be -torsion if there is moreover a twisted bundle of rank on the -brane.
See the references below for details.
The notion of twisted -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 -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)
- Kim Laine, Geometric and topological aspects of Type IIB D-branes (arXiv:0912.0460)
In (Laine) the discussion of FW-anomaly cancellation with finite-rank gauge bundles is towards the very end, culminating in equation (3.41).