Cohomology and Strings

Question

I am going through a paper by Witten and I got confused in the point where the topology of the $B$-field is discussed.

In the first paragraph of page 11, it is explained that when discrete torsion is taken into account, the cohomology class of the $B$-field changes from $H^{3}(\mathcal{M},\mathbb{R})$ to $H^{3}(\mathcal{M},\mathbb{Z})$. I understand the cohomology classes and more or less what is the effect of discrete torsion, but I cannot realize why the cohomology changes in this way under discrete torsion. (namely why $\mathbb{R}\rightarrow\mathbb{Z}$)

This post imported from StackExchange Physics at 2016-09-20 21:45 (UTC), posted by SE-user Jordan

Comments

I am working on orientifolds of the type IIB string theory. I posted the question here in case a specialist can give me some insight. Not necessarily an answer, but even some reference that will help me continue. If I start describing the orientifold action and the discrete torsion effects, it will be never ending. If you have a good background in this, you are more than welcome to give me your lights.

This post imported from StackExchange Physics at 2016-09-21 15:25 (UTC), posted by SE-user Jordan

Answer 1

It's possible to have flat but non-trivial 2-form gauge fields just as it is with 1-form gauge fields. These have trivial curvature forms in $H^3(X,\mathbb{R})$ of course, but non-trivial Chern classes in $H^3(X,\mathbb{Z})$. It turns out that the kernel of the map from the latter to former is precisely the torsion classes, as Witten says.

Answer 2

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

1. a class $\chi \in H^3(X,\mathbb{Z})$ (its topological sector)
2. together with a differential form in $H \in \Omega^3_{closed}(X)$ (the field strength)
3. 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