*Post by Pat Harrington*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

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