The statement in that paragraph is a little vague. What is meant is that:
The B-field fully generally is given by a triple consisting of
- a class χ∈H3(X,Z) (its topological sector)
- together with a differential form in H∈Ω3closed(X) (the field strength)
- and an isomorphism between the images of both H and χ in H3(X,R) -- that's what locally is given by the 2-form B which gives the B-field its name.
In summary this means that the B-field is a cocycle in "degree-3 differential cohomology".
Now in topologically trivial situations, then the integral class is trivial and all the information is in the 2-form. But in topologically non-trivial situations one has to be more precise.
Now discrete torsion orbifolds are such a topologically non-trivial situation of sorts. In fact here everything is in equivariant cohomology, but otherwise the idea is the same. In any case, in such a situation there is in general a non-trivial integer class underlying the B-field, and has to be taken into account.
This post imported from StackExchange Physics at 2016-09-21 15:25 (UTC), posted by SE-user Urs Schreiber