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From a novice: re: size of Calabi-Yau dimensions...
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Pat Harrington
science forum beginner


Joined: 16 Apr 2005
Posts: 5

PostPosted: Sat Apr 16, 2005 9:06 am    Post subject: From a novice: re: size of Calabi-Yau dimensions... Reply with quote

Is there a good mathematical argument for or against just one of the
Calabi-Yau dimensions being of exactly the Planck length?
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Urs Schreiber
science forum Guru Wannabe


Joined: 04 May 2005
Posts: 127

PostPosted: Mon Apr 18, 2005 11:16 am    Post subject: Re: From a novice: re: size of Calabi-Yau dimensions... Reply with quote

"Pat Harrington" <PatrickDHarrington@hotmail.com> schrieb im Newsbeitrag
news:7517a083.0504160223.575b7acd@posting.google.com...

Quote:
Is there a good mathematical argument for or against just one of the
Calabi-Yau dimensions being of exactly the Planck length?

When the extension of the CY becomes of string size its description in terms
of classical geometry breaks down. Hence one needs to use a worldsheet
conformal field theory wich is not a sigma model (does not come from
embedding the worldsheet into a classical manifold) but something else, like
a Gepner model. A Gepner model is an exactly solvable conformal field theory
whose worldsheet fields cannot straightforwardly be interpreted as embedding
fields, hence it is not a sigma-model. But it can still be shown to describe
strings on 'small' and even pointlike Calabi-Yau spaces.

This is reviewed for instance in section 5 of

B. Greene,
String Theory on Calabi-Yau Manifolds
hep-th/9702155

See in particular the discussion starting on top of . 64.

Searching around I see that this is also reviewed for instance in this
thesis:

http://www.aei.mpg.de/pdf/doctoral/EScheidegger_01.pdf
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Lubos Motl
science forum beginner


Joined: 02 May 2005
Posts: 38

PostPosted: Mon Apr 18, 2005 11:32 pm    Post subject: Re: From a novice: re: size of Calabi-Yau dimensions... Reply with quote

On Sat, 16 Apr 2005, Pat Harrington wrote:

Quote:
Is there a good mathematical argument for or against just one of the
Calabi-Yau dimensions being of exactly the Planck length?

First of all, it is pretty difficult to separate the dimensions of a
Calabi-Yau three-fold in such a way that you could talk about these six
dimensions separately (which is what you could do for a six-torus). Second
of all, in string theory it is more usual to compare the size of the
dimensions to the string length l_{string} which may be much longer than
the Planck length if the coupling is very weak.

The shape and size of a Calabi-Yau manifold is controlled by the following
moduli (parameters that are upgraded to massless scalar fields, depending
on the remaining "large" dimensions, by string theory):

* complex structure moduli (this is more like the shape, as Brian
Greene would call them: for a two-dimensional torus, these parameters
would include the angle of the generating rectangle and the ratio
of the edges)
* Kahler moduli (the size of 2-cycles, i.e. topologically nontrivial
two-dimensional submanifolds inside the Calabi-Yau; for the
two-torus, it's simply its area)

It's a finite (h^{2,1} + h^{1,1}) number of complex parameters (the
Kahler moduli are complexified if the other part is represented by the
integral of the B-field over the cycle) that can be changed in such a way
that the space remains Ricci-flat (and SU(3) holonomy).

For a Calabi-Yau three-fold (six real dimensions), these two classes are
related by mirror symmetry. If you switch from a type IIA description to
type IIB or vice versa, the role of the two types of moduli is
interchanged. Mirror symmetry may be interpreted, following Strominger,
Yau, and Zaslow (the latter being a fellow co-moderator), as a T-duality
performed on all three directions of a three-dimensional torus that is
thought to be attached to every point of a three-dimensional base space if
you visualize the Calabi-Yau space in the right way (as a
"T^3-fibration", i.e. a locally Cartesian product of the base space and a
T^3 whose shape can however depend on the base space).

Mathematically, any value of the moduli is equally good as long as there
are no fluxes i.e. if the field strengths of various types are set to
zero. Nevertheless, there are special values of the moduli where something
nice happens. For example, the quintic hypersurface, the most popular
compact example of a Calabi-Yau three-fold, has 3 special values of the
complex structure moduli - the infinite complex structure (the mirror of a
large manifold); the conifold (where the manifold develops a conical
singularity at one place); and the Gepner point (strong on Calabi-Yau
spaces at this point admit a description in terms of the Gepner models,
which are nice combinations of the minimal models - the minimal models
are simple non-geometric integrable models analogous to the Ising model).

Physically, the equally good character of all values of the moduli implies
exactly massless scalar fields that cause new long range forces -
something that is experimentally implausible by the tests of the
equivalence principle (and by the fact that we don't produce such massless
particles). This means that the value of all moduli must be
fixed/stabilized around some values.

The usual assumption in type II models is that the complex structure
moduli (together with the dilaton) are stabilized by the fluxes, more
precisely by the Gukov-Vafa-Witten superpotential int(Omega/\H) where H
is a field strength and Omega is the holomorphic 3-form. This term does
not depend on the Kahler moduli (the sizes of the two-cycles); the latter
must be stabilized differently, by non-perturbative contributions such as
wrapped D-branes or gaugino condensation. (The Kahler moduli also remain
un-stabilized in heterotic strings compactified on a Calabi-Yau, which is
why its low-energy limit is known as no-scale supergravity.)

The goal to stabilize the Kahler moduli is the essence of the now
well-known KKLT (Kachru-Kallosh-Linde-Trivedi) paper whose authors also
add D3-branes to switch from a large number of possible AdS universes to a
large number of metastable de Sitter Universes, forming a huge "landscape"
of possibilities - the main technical result used to justify the relevance
of the "anthropic principle" for string theory.

For the geometric intuition to be applicable, the size/area of all
two-cycles must be bigger than roughly l_{string}^2 which is a non-trivial
condition on the Kahler moduli. When the Calabi-Yau space is large in this
sense, supergravity is a good approximation. For stringy values of the
areas, the full complicated non-linear sigma model (the description of
strings propagating on a curved space that is comparably big to the
strings) is required, and at special points, it is equivalent to models
such as the Gepner models.

If I return to your original question: you would have to define what you
exactly mean by "the size of one dimension of the Calabi-Yau". If anything
is comparable to l_{string}, it's too short and the geometric intuition
may fail. There's nothing physically wrong with the failing geometric
intuition, but it makes the calculations more difficult.

All the best
Lubos
______________________________________________________________________________
E-mail: lumo@matfyz.cz fax: +1-617/496-0110 Web: http://lumo.matfyz.cz/
eFax: +1-801/454-1858 work: +1-617/384-9488 home: +1-617/868-4487 (call)
Webs: http://schwinger.harvard.edu/~motl/ http://motls.blogspot.com/
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Pat Harrington
science forum beginner


Joined: 16 Apr 2005
Posts: 5

PostPosted: Tue Apr 19, 2005 9:59 am    Post subject: Re: From a novice: re: size of Calabi-Yau dimensions... Reply with quote

Lubos Motl <motl@feynman.harvard.edu> wrote in message news:<Pine.LNX.4.62.0504182102070.8893@feynman.harvard.edu>...
Quote:
On Sat, 16 Apr 2005, Pat Harrington wrote:

Is there a good mathematical argument for or against just one of the
Calabi-Yau dimensions being of exactly the Planck length?

First of all, it is pretty difficult to separate the dimensions of a
Calabi-Yau three-fold in such a way that you could talk about these six
dimensions separately (which is what you could do for a six-torus). Second
of all, in string theory it is more usual to compare the size of the
dimensions to the string length l_{string} which may be much longer than
the Planck length if the coupling is very weak.


Lubos,
Thank you for your reply. I suppose, based on your descriptions
above, what I was trying to ascertain was whether there was a known
set of values (or if one could be derived) for the Kahler moduli
wherein, in one (or more!) of the 2-cycles, i.e., in one of the 2D
submanifolds, one of the dimensions was only the Planck length and the
rest of the values were of varying lengths such that the overall
solution would still account for the laws of physics that we observe.
To my way of thinking, which may be clouded by a lack of expertise in
the area, given such a scenario, along this dimension, there would
effectively only be one "place" for strings to occupy. This would
effectively "tie" them together in this space, making the entire set
of all strings throughout the universe one large stringy object. As
strings vibrate and move through other dimensions, effectively, the
one large stringy object would be pivoting about this particular
space.
I wasn't so much concerned about the likelihood of such a
scenario just whether it was mathematically feasible. Thanks Again!!
Cheers,
Pat
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PatrickDHarrington@hotmai
science forum beginner


Joined: 25 Apr 2005
Posts: 1

PostPosted: Mon Apr 25, 2005 4:28 pm    Post subject: Re: From a novice: re: size of Calabi-Yau dimensions... Reply with quote

Pat Harrington wrote:
Quote:
Lubos Motl <motl@feynman.harvard.edu> wrote in message
news:<Pine.LNX.4.62.0504182102070.8893@feynman.harvard.edu>...
On Sat, 16 Apr 2005, Pat Harrington wrote:

Is there a good mathematical argument for or against just one of
the Calabi-Yau dimensions being of exactly the Planck length?



I was wondering if anyone would be so kind as to verify (or vilify!)
my hypothesis. Thanks!!
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Lubos Motl
science forum beginner


Joined: 02 May 2005
Posts: 38

PostPosted: Mon Apr 25, 2005 8:44 pm    Post subject: Re: From a novice: re: size of Calabi-Yau dimensions... Reply with quote

On Mon, 25 Apr 2005, PatrickDHarrington@hotmail.com wrote:

Quote:
I was wondering if anyone would be so kind as to verify (or vilify!)
my hypothesis. Thanks!!

I am afraid that I did not express too clearly my opinion that the
hypothesis is probably not decidable because the notion of "the size of
*one* dimension of a Calabi-Yau space" is ill-defined - well, unless the
Calabi-Yau space is a torus. But others, please, don't get discouraged or
manipulated.
______________________________________________________________________________
E-mail: lumo@matfyz.cz fax: +1-617/496-0110 Web: http://lumo.matfyz.cz/
eFax: +1-801/454-1858 work: +1-617/384-9488 home: +1-617/868-4487 (call)
Webs: http://schwinger.harvard.edu/~motl/ http://motls.blogspot.com/
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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