## Wednesday, September 08, 2010 ... //

### Three-generation Gepner heterotic models

One of the most interesting hep-th papers today was written by Gato-Rivera and Schellekens and it is called

Asymmetric Gepner models II. Heterotic weight lifting.
The authors build on their March 2010 paper and analyze realistic heterotic Gepner models. The well-known classes of such models typically "ban" three generations - and they universally produce fractionally charged massless states (relatively to the string scale). These are big problems for a viable phenomenology.

However, in the new paper, a conventional, overly simplistic component of the CFT is replaced by a genuinely N=0 model on the bosonic side. As a result, "3" becomes a possible - and relatively frequent - number of the generations that these models predict although "zero families" seems to be the numerical winner.

Lots of histograms show the distribution of possible gauge groups - SO(10), SU(5), Pati-Salam, and so on. While these arguably "more generic" (2,0) models fix the problem of this class with the number of families, and 250 three-generation moduli spaces are identified, they apparently keep on predicting massless fractionally charged states.

Gepner models in general

I love Gepner models as an example of the inseparability of "geometric" and "non-geometric" concepts in quantum gravity. What are these models?

Perturbative string theory is described by a world sheet theory, a two-dimensional field theory defined on the world sheet, i.e. the 1+1-dimensional history of the splitting and joining strings as embedded into spacetime.

The two-dimensional field theory typically contains scalar fields X(sigma,tau) which tell you where a particular point of the world sheet, one with the world sheet coordinates equal to (sigma, tau), is located in spacetime. There may also be some fermions but in a "maximally geometrized" supersymmetric string theory, the fermions are linked to the bosons X in a one-to-one fashion.

The critical dimension, as you know, is determined by some cancellation of the conformal anomaly - an inconsistency of string models with a wrong number of dimensions (breaking of the essential scale invariance of the world sheet theory by virtual particles, if you wish).

For a purely bosonic string, it is D=26. For the superstring, you obtain D=10 spacetime dimensions as the only consistent option.

Even if you work with curved geometric backgrounds in the spacetime, it may be locally (in spacetime) represented by some scalar bosons X. Their high curvature in spacetime is reflected by their strong interaction terms on the world sheet. But the counting still tells you that D=10, at least in the cases where you may preserve some spacetime or world sheet supersymmetry.

However, you may try to design "Frankenstein" models of string theory. The abstract consistency criteria tell you that the conformal field theory should have the right "central charge" - roughly speaking the number of degrees of freedom (the number of spacetime coordinates if this is the only kind of degrees of freedom you have). And the operators must be mutually local and the toroidal partition sum has to be modular-invariant.

But consistency doesn't really tell you that the degrees of freedom have to include scalar fields X only. It doesn't tell you that the degrees of freedom have to be geometric!

Instead, you may try to remove the "natural" geometric fields X(sigma,tau) from the world sheet and replace them by more general two-dimensional theories such as the (long-distance limit of) the Ising model, the Potts model, or by the more general classes of "minimal models".

They have nothing like the geometric X(sigma,tau) fields that would describe a spacetime geometry. Instead, they boast various "spin fields" - something like Sigma(sigma,tau) - and their generalizations that are utterly non-geometric: there's no spacetime in them. Their central charges are rational numbers. So they carry the same amount of "conformal anomaly" as a rational - but generally fractional - number of spacetime coordinates X(sigma,tau). They do the same job as objects with fractional dimensions but you don't really see any geometry in them.

In fact, you may combine the right number of them so that they may fully replace the D=10 spacetime coordinates (and their fermionic counterparts). If you add the right projections so that the spacetime supersymmetry may be resuscitated, you may believe that you have invented a completely new, "Frankenstein" type of string theory where the natural and possibly curved spacetime coordinates - describing Calabi-Yau shapes etc. - are replaced by some seemingly "man-made" components such as the Ising models.

However, if you study physics of these models in detail, you will find out that despite their looking "man-made", these models are actually fully equivalent to the natural, Calabi-Yau models with 6 compact dimensions! If you pick a Calabi-Yau shape and fix its shape and/or size parameters to particular values (the required sizes are usually comparable to the string scale), you will obtain a model that is absolutely equivalent to the Gepner model!

Even though the number of dimensions in a Gepner model is a problematic question a priori, and you could think that such a model strictly implies that the total dimension is not D=10 because the "man-made" components such as the Ising model can't be counted as dimensions, especially because of their non-integer central charges, the detailed analysis of the physics of these "man-made" components proves that they actually still contain the six dimensions that we know from the "geometric" models.

And many facts about the Calabi-Yau shapes and other things may be determined from the Gepner models, too.

String/M-theory can't resolutely tell you that the total dimension is D=10 or D=11. After all, the M-theoretically models that inherently have D=11 may be dual to models based on type I, type II, or heterotic string theories whose maximum of large spacetime dimensions is D=10. You can't say that one of the descriptions is better than others.

However, when you work hard and attempt to reveal all conceivable ways how the degrees of freedom in your model may get "stretched" in spacetime and tell you something about new dimensions, you will always find out that the spacetime-supersymmetric vacua of all kinds locally (in spacetime) behave according to the rules of M-theory, type I or II string theories, or heterotic string theories.

As far as we can say, there are no other solutions.

So even though your precise algorithm to construct a model may completely obscure the geometric character of your model, the ultimate physics - and precise mathematics - of the model can't obscure it. If you compare the physics with the physics of other models carefully enough, you will see that D=10 or D=11 is always there. So morally speaking, it's always a part of string theory.