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 $\chi \in H^3(X,\mathbb{Z})$ (its topological sector)
- together with a differential form in $H \in \Omega^3_{closed}(X)$ (the field strength)
- and an isomorphism between the images of both $H$ and $\chi$ in $H^3(X,\mathbb{R})$ -- 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