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 $\mathbb{R}\rightarrow\mathbb{Z}$)

Answer 1

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.

Answer 2

A discrete torsion basically introduce phases to give relative weight to Euler characteristic of subspaces to alter the number of generations to obtain a new theory according to relation*, $$2n=\frac{1}{|G|}\Sigma\epsilon(g,h)x(g,h)$$ where $G$ is generally abelian and $\epsilon(g,h)$ are the phases. These phases obviously correspond to the circle group as $U(1)$ elements, and break cohomology class coefficients from $\mathbb{R}$ to $\mathbb{Z}$ as quotient $\mathbb{R}/\mathbb{Z}$ is circle group.

*C. Vafa, nucl. phys. B273 (1986), 592-606.