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 , 22,786 comments
1,470 users with positive rep
820 active unimported users
More ...

  Calabi-Yau manifolds and compactification of extra dimensions in M-theory

+ 1 like - 0 dislike
1456 views

I just finished learning M(atrix) theory and the basics of the compactification of extra dimensions.

The extra 6 dimensions of superstring theory can be compactified on 3 Calabi-Yau manifolds (because 6 real dimensions means 3 complex dimensions).

However, when it comes to M-theory, one cannot compactify on 3.5 Calabi-Yau manifolds, so after compactifying 6 dimensions, where does the extra 1 dimension go? Is it just compactified on a circle, or something like that?

asked May 26, 2013 in Theoretical Physics by dimension10 (1,985 points) [ revision history ]
edited Apr 25, 2014 by dimension10
For clarity, there are CY's of complex dimension two, three, etc., not just one.

This post imported from StackExchange Physics at 2014-03-07 16:42 (UCT), posted by SE-user Vibert
Keyword in this context: $G_2$-manifolds.

This post imported from StackExchange Physics at 2014-03-07 16:42 (UCT), posted by SE-user Qmechanic
@Vibert: That was what I meant. When I said 3 Calabi Yau manifolds, I meant Calabi Yau manifolds of complex dimension 3.

This post imported from StackExchange Physics at 2014-03-07 16:42 (UCT), posted by SE-user Dimensio1n0
@Qmechanic: Thanks a lot!

This post imported from StackExchange Physics at 2014-03-07 16:42 (UCT), posted by SE-user Dimensio1n0
See on the nLab at ncatlab.org/nlab/show/M-theory+on+G2-manifolds

This post imported from StackExchange Physics at 2014-03-07 16:42 (UCT), posted by SE-user Urs Schreiber

1 Answer

+ 1 like - 0 dislike

$$\newcommand{\holonomy}{[\mathcal{H\mathbb{O} \ell}]}$$

This answer is an expansion of Qmechanic's comment

Holonomy

Holonomy can be imagined as the integral, or global version, of the Riemann Curvature Tensor. The Riemann Curvature tensor, indeed is

$$R_{\mu\nu\rho}^\sigma=\mbox{d}\holonomy$$

Where $\mathcal{\holonomy}$ is the Holonomy.

Holonomy Groups

Now, this holonomy is the group action of the Holonomy group of the manifold. So, in other words, the holonomy of the identity of the holonomy group (not doing any sort of a transport) doesn't do anything to a point on the manifold, and that holonomies are hand - wavily, sort - of "associative" (use this statement with caution!), i.e., instead of writerighteing $\phi(g,x)$ or something, if we choose to write something like, say, $g\dagger x$, then:

$$g\dagger\left(h\dagger x\right)=\left(gh\right)\dagger x$$

Oh, and the first statpement becomes, :

$$e\dagger x =x $$

Now, this is not as trivial as it looks. $e$ is the identity of the holonomy group, NOT of the manifold! .

So, where does $G(2)$ come in?

Now, where in the world does $G(2)$ come from? $G(2)$ is a holonomy group of $\bf{\mathbf{\it{7}}}$-dimensional manifolds, called $G(2)$ manifolds. This means that it is possible to use this as a compactification manifold for M-theory. M-theory has a supersymmetry of $\mathcal N=8$. But, if we waNt a supersymmetry of $\mathcal{N}=1$ (accessible at lower energies), n the compactificaqtion manifoldk must get rid of $\frac78$ of the supersymmetry, i.e. retain only $\frac18$.

It so happens to be that $G(2)$ manifolds do indeed satisfy this criterion.

answered Aug 31, 2013 by dimension10 (1,985 points) [ revision history ]
edited Apr 25, 2014 by dimension10

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$\hbar$ysicsOve$\varnothing$flow
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
...