Tuesday, November 08, 2005

Calabi-Yaus from MSSM

Last week or so, James Gray has given a very interesting talk about their work on deducing the details of string theory from low-energy physics. Normally, the string theorists start at the top. You pick your favorite compactification (you decide whether it is heterotic or type II, what is the shape of extra dimensions, which fluxes and which branes you allow), figure out what its low-energy spectrum is, and deduce the moduli spaces, couplings, and low-energy effective field theory in general.

They propose the reverse approach. Take one of the simplest low-energy phenomenological models compatible with your observations - such as the MSSM. Define its gauge-invariant monomials to parameterize a moduli space; these are used as F-terms or D-terms. Determine the dimensionality and topology of the resulting moduli space. And find a string model that exactly matches it.

They often mention that they would like to derive that the moduli space is a Calabi-Yau three-fold itself. I find it a bit exaggerated. The Calabi-Yau space can only be a moduli space at low energies if you consider something like one D3-brane on Calabi-Yaus in type IIB - but there are really no phenomenologically viable models of this kind. Note that the 3-fold in the F-theory flux constructions is not a Calabi-Yau manifold.

In reality, the moduli spaces at low energies describe the moduli spaces of shapes of manifolds such as the Calabi-Yaus or moduli spaces of gauge bundles over them. They can have many different dimensionalities and topologies. In the top-down approach, we know very well what is a natural requirement for a model to be phenomenologically appealing: we want to get as close to the Standard Model or MSSM as possible, and remove all exotics.

A corresponding task in their bottom-up approach is not quite determined, as far as I can see. What properties should the low-energy moduli spaces derived by their algorithm have in order to tell us that it looks like it comes from string theory? I think that the idea that it should be a Calabi-Yau manifold is naive, and I don't have any better replacement for this proposed answer.

For example, they seem very excited by having obtained a moduli space whose Hodge numbers coincide with those of a CP^2:
  • h0,0=h1,1=h2,2=1

and otherwise zero. I personally have not understood why this "simple" Hodge diamond is more attractive than other Hodge diamonds that they could have derived.

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