Number of Calabi-Yau Manifolds in Superstring Theories

[meteor]

I want to know how many Calabi-Yau manifolds there are in each of the 5 superstring theories. Can you point me in the right direction?

 

[notevenwrong]

Calabi Yau manifolds exist independently
of the five string theories. There are
known to be at least 10-100,000 families
of these manifolds. Each family can have
a dimension of order 100 or more, i.e. comes
with hundreds of parameters which in some
sense determine the size and shape of the
Calabi-Yau.

It is quite possible the number of Calabi
Yau families is infinite. This remains an
open problem in algebraic geometry and you
can find some algebraic geometers who
believe the number is finite, some who believe it is infinite.

 

[meteor]

Very interesting. But well, my question is more like to know what's the number of CY manifolds that are supposed to exist in each of these theories. Because I know that one of the proposals of superstring theory is that in the beginning, there were 10 dimensions uncompactified, but suddenly 6 of these dimensions were compactified in one or more CY manifolds. That's my doubt. If they were compactified in an unique CY manifold, or in a set of them. If this is the case, what's the exact number of these manifolds? Very thankful to who can respond
I was thinking also if could be some way of decompactify the dimensions trapped in a CY manifold--maybe manipulating it somehow?

 

[notevenwrong]

It's not true that string theory predicts that
the universe starts off with 10 flat dimensions
then 6 get compactified. As far as anyone
knows, if string theory makes any sense, strings in 10 flat dimensions makes sense,
and there is no mechanism (other than
wishful thinking) for six dimensions to spontaneously compactify.

As far as anyone knows, one Calabi-Yau
is as good as any other. In 1984, people
hoped there was a small number of Calabi-Yaus and maybe some way to rule out
all but a unique one. Nobody believes this
particular piece of wishful thinking anymore.

 

[meteor]

I think that you are not understanding what's the gist of what I'm asking (my blame too, cause I'm not very comfortable writting in english). The question is: Did the dimensions compactified in one (a single, a unique) CY manifold, so they are compactified in THIS manifold that is located somewhere, or did they compactified in more than one? Is like asking: If we have the volume of the universe,say 10¹⁰⁰ km³, and then if there's only ONE CY manifold in this volume then the density of CY manifolds is 1/(10¹⁰⁰ km³), or perhaps, there are 100 CY manifolds then the density is 100/(10¹⁰⁰ km³). This is the type of information that I want to know: the number (nor the different types) of CY manifolds that are predicted to exist

I've read that 6 dimensions were compactified starting from an initial state of 10 dimensions:
http://www.fortunecity.com/emachines/e11/86/dimens.html "
"But, of course, all this takes place in 10 dimensions. Physicists retrieve our more familiar 4-dimensional Universe by assuming that, during the big bang, 6 of the 10 dimensions curled up (or "compactified")"

 

[notevenwrong]

String theory really just has nothing at
all to say about your question. It has nothing
to say about how compactification comes
about, and thus nothing to say about which
CY manifolds or how many you end up with.

 

[rick1138]

There is a C-Y manifold at every point in space, or rather, the 6 extra dimensions have everywhere the shape of a C-Y manifold.

 

[rick1138]

Here is a great paper I just found on the mathematics of C-Y manifolds.:

http://www-thphys.physics.ox.ac.uk/users/PeterAusting/Conference/Talks/candelas.pdf


The best book is Tristan Hubsch's Calabi-Yau Manifolds: A Bestiary for Physicists.

 

[meteor]

Ah, this is great! Now, reading about CY manifolds in mathworld I will quote this interesting paragraph:




"Whatever definition is used, Calabi-Yau manifolds, as well as their moduli spaces, have interesting properties. One is the symmetries in the numbers forming the Hodge diamond of a compact Calabi-Yau manifold. It is surprising that these symmetries, called mirror symmetry, can be realized by another Calabi-Yau manifold, the so-called mirror of the original Calabi-Yau manifold. The two manifolds together form a mirror pair. Some of the symmetries of the geometry of mirror pairs have been the object of recent research."


http://mathworld.wolfram.com/Calabi-YauSpace.html

BTW, the concept of mirror symmetry was introduced by Brian Greene

So, I will try to know what's that Hodge diamond. So, if two Cy manifolds form a mirror pair, I guess that must be two ADJACENT CY manifolds, or perhaps two CY manifolds that are separated certain distance can also form a mirror pair?

 

[meteor]

"the hodge diamond displays the Hodge numbers of a manifold"
from here
http://www2.cs.cmu.edu/~jcl/classnotes/math/geometry/hodge_diamond/hodge_diamond.html
Hodge number: "The Hodge number is the analog of the Betti number on real manifolds for complex manifolds"
This is the definition in this page
http://www2.cs.cmu.edu/~jcl/classnotes/math/geometry/hodge_number/hodge_number.html#hodge number

 

[rick1138]

Tristan Hubsch's Calabi-Yau Manifolds: A Bestiary for Physicists gives explicit methods for constructing Hodge diamonds. It is a great book, in content and in style. One of the few books on math, outside of Conway's, that alsoo qualifies as a work of great literature.