What Shapes Can Extra Dimensions Take in String Theory?

[Trollfaz]

I read Brian Greenes book "The Elegant Universe".
In it there is one chapter talking about extra dimensions. The first idea was proposed by Kaluza and Klein. They suggested that the extra fifth dimension exists as a circle, is it right. Is this called a KK circle?
Now Greene also mentioned that mathematics of string theory requires the extra dimensions to take what we call a Calabi Yau shape so does this mean that circle is just one of the many possibilities of the shapes of extra dimensions? Greene mentioned that there are 10^500 types of shapes that's why we live in a string landscape according to string theory.

 

[martinbn]

I read Brian Greenes book "The Elegant Universe".
In it there is one chapter talking about extra dimensions. The first idea was proposed by Kaluza and Klein. They suggested that the extra fifth dimension exists as a circle, is it right. Is this called a KK circle?

I don't think it has a name.

Now Greene also mentioned that mathematics of string theory requires the extra dimensions to take what we call a Calabi Yau shape so does this mean that circle is just one of the many possibilities of the shapes of extra dimensions?

Yes, but in string theory you need higher dimensional spaces, not just circles.

Greene mentioned that there are 10^500 types of shapes that's why we live in a string landscape according to string theory.

 

[ohwilleke]

The first idea was proposed by Kaluza and Klein. They suggested that the extra fifth dimension exists as a circle, is it right. Is this called a KK circle?

There are lots of terms that are used in connection with Kaluza Klein theory and variants of it, most of which are contained in this article that have subtly different technical meanings. I'm not sure which term would best fit the concept that you are trying to described briefly in layman's terms.

Now Greene also mentioned that mathematics of string theory requires the extra dimensions to take what we call a Calabi Yau shape so does this mean that circle is just one of the many possibilities of the shapes of extra dimensions?

There are many possible topologies of extra dimensions. The biggest division is between "compactified" extra dimensions, and large extra dimensions.

In theories with large extra dimensions, there are other dimensions that are not "curled up", and the universe appears four dimensional, instead, because important classes of interactions happen only on a "brane" and thus can't spread to all parts of some of the "extra" dimensions.

Greene mentioned that there are 10^500 types of shapes that's why we live in a string landscape according to string theory.

I would say that to limit the sources of the string theory landscape to differences arising from different extra dimension topologies is an oversimplification, although that is one important factor.

"The large number of possibilities arises from choices of Calabi–Yau manifolds and choices of generalized magnetic fluxes over various homology cycles, found in F-theory."