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Pat Harrington science forum beginner
Joined: 16 Apr 2005
Posts: 5

Posted: Sat Apr 16, 2005 9:06 am Post subject:
From a novice: re: size of CalabiYau dimensions...



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

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Urs Schreiber science forum Guru Wannabe
Joined: 04 May 2005
Posts: 127

Posted: Mon Apr 18, 2005 11:16 am Post subject:
Re: From a novice: re: size of CalabiYau dimensions...



"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
CalabiYau 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 sigmamodel. But it can still be shown to describe
strings on 'small' and even pointlike CalabiYau spaces.
This is reviewed for instance in section 5 of
B. Greene,
String Theory on CalabiYau Manifolds
hepth/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

Posted: Mon Apr 18, 2005 11:32 pm Post subject:
Re: From a novice: re: size of CalabiYau dimensions...



On Sat, 16 Apr 2005, Pat Harrington wrote:
Quote:  Is there a good mathematical argument for or against just one of the
CalabiYau dimensions being of exactly the Planck length?

First of all, it is pretty difficult to separate the dimensions of a
CalabiYau threefold in such a way that you could talk about these six
dimensions separately (which is what you could do for a sixtorus). 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 CalabiYau 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 twodimensional torus, these parameters
would include the angle of the generating rectangle and the ratio
of the edges)
* Kahler moduli (the size of 2cycles, i.e. topologically nontrivial
twodimensional submanifolds inside the CalabiYau; for the
twotorus, 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 Bfield over the cycle) that can be changed in such a way
that the space remains Ricciflat (and SU(3) holonomy).
For a CalabiYau threefold (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 comoderator), as a Tduality
performed on all three directions of a threedimensional torus that is
thought to be attached to every point of a threedimensional base space if
you visualize the CalabiYau space in the right way (as a
"T^3fibration", 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 CalabiYau threefold, 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 CalabiYau
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 nongeometric 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 GukovVafaWitten superpotential int(Omega/\H) where H
is a field strength and Omega is the holomorphic 3form. This term does
not depend on the Kahler moduli (the sizes of the twocycles); the latter
must be stabilized differently, by nonperturbative contributions such as
wrapped Dbranes or gaugino condensation. (The Kahler moduli also remain
unstabilized in heterotic strings compactified on a CalabiYau, which is
why its lowenergy limit is known as noscale supergravity.)
The goal to stabilize the Kahler moduli is the essence of the now
wellknown KKLT (KachruKalloshLindeTrivedi) paper whose authors also
add D3branes 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
twocycles must be bigger than roughly l_{string}^2 which is a nontrivial
condition on the Kahler moduli. When the CalabiYau space is large in this
sense, supergravity is a good approximation. For stringy values of the
areas, the full complicated nonlinear 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 CalabiYau". 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
______________________________________________________________________________
Email: lumo@matfyz.cz fax: +1617/4960110 Web: http://lumo.matfyz.cz/
eFax: +1801/4541858 work: +1617/3849488 home: +1617/8684487 (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

Posted: Tue Apr 19, 2005 9:59 am Post subject:
Re: From a novice: re: size of CalabiYau dimensions...



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
CalabiYau dimensions being of exactly the Planck length?
First of all, it is pretty difficult to separate the dimensions of a
CalabiYau threefold in such a way that you could talk about these six
dimensions separately (which is what you could do for a sixtorus). 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 2cycles, 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

Posted: Mon Apr 25, 2005 4:28 pm Post subject:
Re: From a novice: re: size of CalabiYau dimensions...



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 CalabiYau 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

Posted: Mon Apr 25, 2005 8:44 pm Post subject:
Re: From a novice: re: size of CalabiYau dimensions...



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 CalabiYau space" is illdefined  well, unless the
CalabiYau space is a torus. But others, please, don't get discouraged or
manipulated.
______________________________________________________________________________
Email: lumo@matfyz.cz fax: +1617/4960110 Web: http://lumo.matfyz.cz/
eFax: +1801/4541858 work: +1617/3849488 home: +1617/8684487 (call)
Webs: http://schwinger.harvard.edu/~motl/ http://motls.blogspot.com/
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ 

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