What is the significance of Calabi-Yau spaces in modern string theory?

[The hermit]

Hello,

Physicists have formulated a "Calabi-Yau space", to show the supplementary dimensions in the string theory. But this space have six dimensions, and the M theory, the presentest string theory, have seven supplementary dimensions...

So, what is this Calabi-Yau space in the string theory today ? Is she still using ?


The hermit

 

[starkind]

I'll take a quick shot at this, altho I am by no means best qualified. C-Y manifolds are very small highly folded spaces which are thought to fill all of space, and are supposed to explain the missing energy in some particle reactions.

String theorists come in all sorts, and I am sure C-Y manifolds are still being explored, but Marcus is probably best able to report on the current status in the literature.

 

[Sauron]

The M-theory si belived to be some strong coupling limit of some of the best known string theorys. In particular it first appeared as an strong coupling limit of type II A. The spectrum of D0-branes of type IIA seemed to behae as if theoy would be Kaluza-Klein modes of another theory. But KK modes inidcate an aditional dimension so that theory was in an upper dimension, i.e. it was a theory in 11 dimensions. The aditional dimension should be compactified in a circle of a certain radius. The radius of that circle was related to the coupling of the type II A theory by a relation typically of the form R=\lambda^{2/3}.

This implies that the size of the aditional dimension would be diferent that the size of the rest of the aditional dimensions of type II A theory which would be tipically of Planck size.

Later it was realized that M-theory would also be an strong coupling limit of heterotic E(8)xE(8) string theory. That limit is known as "heterotic M-theory". One realization of this scenarios is the Horava-Witten model. Ther the 11th dimension is "orbifolded" to S^1/Z(2). That orbifold can be viewed as asegment with the extrems being distinguised points because they are invariant under the action of Z(2).

Well, with this preliminars I can give ou the answer. The orbifolded 11th dimension together with the other 10 dimensions acts as a bulk space.The extremse of the orbifold are 10 dimensinal hyperplanes. Each o this hyperplanes has associated an E(8)xE(8) matter hypermultiplet. One is the "visible" sector and the other is the "hidden" sector. I guess that this could sound to many of you. Yes, it is very similar to the Randall-Sundrum brane cosmologies. All what remains to do is to compactify the hyperplanes. They are compactified to Calaby-Yaus, or whatever, to a Planckian size.

This is grosso modo one possible way to deal with the 11 dimensions of M-theory.

 

[Demystifier]

Calabi-Yau are complex manifolds, which means that their dimension is even. Therefore, our Universe cannot be obtained just from a Calabi-Yau compactification of the 11-dimensional M-theory. For the details, see the post of Sauron above.

 

[AlphaNumeric2]

There's a lot of work done on G2 manifolds. They are sort of like the 7d version of Calabi Yau spaces, in that they have a lot of the interesting properties CYs do (Ricci flat, N=1 SUSY). If books like Joyce are anything to go by though, they are even more complicated than CYs, which aren't exactly straight forward themselves!

 

[BenTheMan]

I think it's pretty obvious that M-Theory is an eleven dimensional theory (that's what people tell me, at least), and Witten proved various ways to get String Theory out of M-Theory. As far as I understand, you have to compactify M-Theory on something like a line segment first, to get a ten dimensional theory. Then you compactify six of the ten dimensions on a Calabi-Yau.

So the short answer is that string theorists still use Calabi-Yaus to do physics. Most string theorists (like myself, if I can be called such) work in ten dimensions. There are some people who work in eleven dimensions, and think about compactifying on a seven dimensional manifold, like G2. These people are much smarter than I am:) We must keep in our minds that the ten dimensional picture is only a limit of an eleven dimensional theory, which we don't know much about.