Quantcast
  • Register
PhysicsOverflow is a next-generation academic platform for physicists and astronomers, including a community peer review system and a postgraduate-level discussion forum analogous to MathOverflow.

Welcome to PhysicsOverflow! PhysicsOverflow is an open platform for community peer review and graduate-level Physics discussion.

Please help promote PhysicsOverflow ads elsewhere if you like it.

News

New printer friendly PO pages!

Migration to Bielefeld University was successful!

Please vote for this year's PhysicsOverflow ads!

Please do help out in categorising submissions. Submit a paper to PhysicsOverflow!

... see more

Tools for paper authors

Submit paper
Claim Paper Authorship

Tools for SE users

Search User
Reclaim SE Account
Request Account Merger
Nativise imported posts
Claim post (deleted users)
Import SE post

Users whose questions have been imported from Physics Stack Exchange, Theoretical Physics Stack Exchange, or any other Stack Exchange site are kindly requested to reclaim their account and not to register as a new user.

Public \(\beta\) tools

Report a bug with a feature
Request a new functionality
404 page design
Send feedback

Attributions

(propose a free ad)

Site Statistics

145 submissions , 122 unreviewed
3,930 questions , 1,398 unanswered
4,853 answers , 20,624 comments
1,470 users with positive rep
501 active unimported users
More ...

Cohomology and Strings

+ 1 like - 0 dislike
115 views

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
asked Sep 20, 2016 in Theoretical Physics by Jordan (15 points) [ no revision ]
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

2 Answers

+ 2 like - 0 dislike

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.

answered Sep 20, 2016 by Ryan Thorngren (1,605 points) [ no revision ]
+ 2 like - 0 dislike

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
answered Sep 21, 2016 by Urs Schreiber (5,795 points) [ no revision ]

Your answer

Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead.
To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL.
Please consult the FAQ for as to how to format your post.
This is the answer box; if you want to write a comment instead, please use the 'add comment' button.
Live preview (may slow down editor)   Preview
Your name to display (optional):
Privacy: Your email address will only be used for sending these notifications.
Anti-spam verification:
If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:
$\varnothing\hbar$ysicsOverflow
Then drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds).
To avoid this verification in future, please log in or register.




user contributions licensed under cc by-sa 3.0 with attribution required

Your rights
...