## Wednesday, April 04, 2007 ... //

### Three families and Higgsing of F-theoretical GUTs

Jacob Bourjaily has a pair of extremely interesting papers:

I am using the correct numbers that I expect to be associated with the preprints after the monstrous April 2007 identifier bug is fixed.

Recall that F-theory where F stands for "father" or "vaFa" (the father of F-theory) is a formal description of vacua of 10-dimensional type IIB string theory with a non-constant dilaton-axion in terms of a 12-dimensional geometry whose two dimensions are compactified on a two-torus whose complex structure corresponds to the dilaton-axion complex field of type IIB string theory. The ten-dimensional base of this elliptic fiber bundle is the type IIB spacetime.

Just like type IIA string theory is interpreted as M(other)-theory on a circle, type IIB string theory is interpreted as F(ather)-theory on a two-torus. Both male and female feminists should notice that M-theory and F-theory are not equivalent in any way. They have, in fact, a different number of dimensions and play different roles in the structure of string theory.

If viewed as a twelve-dimensional theory, F-theory has the maximum dimension you can ever have in supersymmetric string theory. The more dimensions you have, the more geometric description of physics you get. In this sense, F-theory is as geometric as possible (M-theory is the second most geometric way of describing vacua of string theory).

All kinds of matter fields, gauge groups etc. are interpreted as features of a twelve-dimensional geometry. Designing the right shape of the twelve dimensions that reproduces a given low-energy physics is referred to as geometric engineering.

In the first paper, Bourjaily engineers SU(5) grand unified theories and deforms the corresponding singular F-theory geometry in such a way that the deformation is indistinguishable from breaking of the SU(5) gauge group down to the Standard Model. The geometric options for this Higgsing are much more restricted than they would be in the low-energy effective field theory.

In the second paper, he breaks an SO(10) GUT with the spinorial 16-dimensional representation to the Standard Model which is analogous to the first paper. What I find even more fascinating is the last half-sentence of the abstract: he claims that an E8 singularity in F-theory can be unfolded and it uniquely leads to a three-generation model!

If it makes sense, it's truly fascinating. Let me spell this statement once again:
F-theory with all matter coming from an isolated E8 singularity predicts three generations of fermions because of purely group-theoretical reasons!
Recall that heterotic strings with E8 gauge groups - the beautiful models that give you the right gauge groups and representations for the fermions, together with a gauge coupling unification - may have an F-theory dual but the number of generations in the heterotic vacuum depends on the precise geometry of the Calabi-Yau manifold (and perhaps also all the gauge bundles if they're non-trivial). If the F-theory picture makes three generations more natural, that's really fascinating.

I am an E8 lover - a stringy GUT fan, if you wish - which is why an E8 starting point looks more attractive to me than various intersecting braneworlds and Standard Model singularities. All big shots who happen to read this note should spend an hour with these two papers!