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

PO is now at the Physics Department of Bielefeld University!

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

205 submissions , 163 unreviewed
5,082 questions , 2,232 unanswered
5,353 answers , 22,789 comments
1,470 users with positive rep
820 active unimported users
More ...

  What is the connection between extra dimensions in Kaluza-Klein type theories and those in string theories?

+ 3 like - 0 dislike
895 views

This follows to some extent from a question I asked previously about the flaws of Kaluza-Klein theories.

It appears to me that Kaluza-Klein theories attach additional dimensions to spacetime that are related to the gauge freedoms of field theories. I believe the original model was to attach a $U(1)$ dimension to the usual 4-dimensional spacetime to reproduce electromagnetism. But, as explained in the answers to my previous question, these extra dimensions have all sorts of problems.

String theories also (famously?) require extra dimensions. So, is there a connection between the higher-dimensional descriptions? What do they have in common and how do they differ? For example, the $U(1)\times SU(2)\times SU(3)$ group is 7-dimensional, which, when attached to the 4 dimensions of spacetime, gives 11 dimensions. I hear the same number is bandied about in string theory although there's no obvious reason they should be related at all.


This post has been migrated from (A51.SE)

asked Mar 22, 2012 in Theoretical Physics by Warrick (60 points) [ revision history ]
edited Jan 31, 2015 by dimension10

1 Answer

+ 3 like - 0 dislike

The group manifold $U(1) \times SU(2)\times SU(3)$ is $1+3+8=12$-dimensional, not 7-dimensional.

You probably meant the dimension of a manifold that may have this group as its isometry group. But one may show that no such low-dimensional manifold can be interpreted as the extra dimensions of string theory to produce a realistic model.

The oldest Kaluza-Klein theory had an extra circular dimension whose isometry is $U(1)$. More generally, one may have more complicated manifolds with the isometry group $G$ (isometry is a map of the manifold onto itself, or a diffeomorphism, that preserves the metric at each point, the true "symmetry" of the manifold). The isometry group always becomes the gauge group in the lower-dimensional description. These facts about the Kaluza-Klein theory are fully reproduced as a low-energy feature of some string compactifications.

But as I have mentioned, realistic models with a large enough gauge group to include the Standard Model which would come purely from the original Kaluza-Klein mechanism don't exist in string theory. That's why realistic stringy vacua have a different origin of the gauge symmetries. For example, a stack of $N$ branes has a $U(N)$ gauge group which may become orthogonal or symplectic at the orientifold planes. M-theory and F-theory admit extra gauge groups from singularities. Heterotic string theory or Hořava-Witten heterotic M-theory contain extra $E_8$ gauge groups, already in the maximum dimension (or codimension one boundary, in the M-theory case) that are simply inherited (and partially broken) in four dimensions.

All these possibilities are related by various dualities (non-obvious but exact equivalences) in string theory. And in some sense, all of them are stringy generalizations of the original Kaluza-Klein theory. For example, the $E_8\times E_8$ or $SO(32)$ gauge group of the heterotic string comes from 16 chiral "purely left-moving" spacetime dimensions in the spacetime where the heterotic string may live. In some stringy sense, the gauge group may still be interpreted as the isometry of the manifold. Well, $U(1)^{16}$ arises as the standard isometry of the torus and the remaining generators of the gauge group have a "stringy origin" which may be interpreted as the "string-generalized geometry".

This post has been migrated from (A51.SE)
answered Mar 24, 2012 by Luboš Motl (10,278 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:
p$\hbar$ys$\varnothing$csOverflow
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).
Please complete the anti-spam verification




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

Your rights
...