# Cohomology and Strings

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

• It is much better for your question to be self-contained: Currently, without reading the paper, one has to guess what B-field exactly you're talking about. A short description of the context (string theory on orbifolds) would make your question more accessible. Furthermore, it might be helpful to know what exactly your current understanding of "discrete torsion" is. – ACuriousMind Sep 20 '16 at 15:21
• 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. – Jordan Sep 21 '16 at 8:07
• another answer is at physicsoverflow.org/37147 – Arnold Neumaier Nov 17 '16 at 20:11

## 2 Answers

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.

• M. Schreiber, thank you very much for your helpful and rigorous answer,I would like to study in detail your explanation and I will come back for any potential question. – Jordan Sep 23 '16 at 8:13

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.

• Thank you ved for the help, I will check the reference and the above material to get a clear picture. Thanks again. – Jordan Sep 23 '16 at 8:15