Thursday, May 14, 2009

Prehistory of the F-theory GUTs (mini?)- revolution

After an intensive training in algebraic geometry and the reading of the use of type II-B/F-theory in cosmology (KKLT, moduli stabilization and all that) I tried to do a direct attack to the two papers that initiated the F-theory revolution.

I had read generic aspects of them in the Lubos and Distler's blogs (well, Distler is actually disappeared and only put an entry about the first paper, this). I also found useful an entry in U-duality blog linking a paper of Schwartzabout the status of superstring theory, this entry

Well, my attack flailed miserably. I remotely understood some of the statements but I didn´t understand where they come from. I needed to go to the bibliography of the Vafa et all papers. I also have found clarifying some hints in follow up papers.

The first thing that I needed to understand properly is what a local model is. The proposal of local models seems to be born in a paper by (mostly) Spanish string theorist´s: G. Aldazabal1, L. E. Ib´a˜nez2, F. Quevedo3 and A. M. Uranga in the paper D-Branes at Singularities : A Bottom-Up Approach to the String Embedding of the Standard Model

The idea is to do instead of a top-down approach, that is, choose a compactification, study the resulting physic and see how well it matches the MSSM (minimal supersymmetric standard model) or something resembling the known physic one does a bottom up approach. It consists of two steps (I cite form the paper):

i) Look for local configurations of D-branes with worldvolume theories resembling
the SM as much as possible. In particular we should search for a gauge group SU(3) ×
SU(2) × U(1) but also for the presence of three chiral quark-lepton generations. Asking
also for D = 4 N = 1 unbroken supersymmetry may be optional, depending on what
our assumptions about what solves the hierarchy problem are. At this level the theory
needs no compactification and the D-branes may be embedded in the full 10-dimensional
Minkowski space. On the other hand, gravity still remains ten-dimensional, and hence
this cannot be the whole story.
ii) The above local D-brane configuration may in general be part of a larger global
model. In particular, if the six transverse dimensions are now compactified, one can in
general obtain appropriate four-dimensional gravity with Planck mass proportional to the
compactification radii.

In that paper the fields come form D-bran physics, i.e. open strings ending on the branes. They consider the branes suited at singularities of an orbifold. In F-Theory (and also M-theory) that approach takes a much more difficult form, but the idea is the same.

By the way, the idea of local models has had some development outside of the F-theory revolution, see for example Building the Standard Model on a D3-brane, by the Verlinde brothers. In general that attempts were influenced by the paradigm of D-brane intersections (of which I don´t know too much neither).

Almost at the same time that the now famous Vafa papers it appeared in arxiv another paper on model building with F-theory by Ron Donagi and Martijn Wijnholt (arXiv:0802.2969v2). I find that paper very illuminating. It explains in a very accessible way many facts of F-theory. For example the difference of the 7-brane of F-theory, which is not necessarily an D-brane but instead is a brane where a (p,q) string can end. It also clarifies the geometric idea behind the local model approach and gives intuitions on how gauge matter can appears, based in considerations of the supersymetry limit of F-theory.

I still haven't read all the paper. But in some point it talks about some of the other hard to follow questions of the F-theory revolution, the subject of ADE groups and it's relation to algebraic geometry and, in particular, the Kodaira classification of singularities.

That topic is covered in some early (mid nineties) papers of Vafa, for example Geometric Singularities and Enhanced Gauge Symmetries or Matter From Geometry

I am still trying to catch many aspects of how that works, but I believe that the essence of the argument is that F-theory and M-theory are related by dualities. Compactifiying F-theory in K3 surfaces (complex two dimensional Calabi-Yaus) is equivalent to compactify M-theory in a torus. The spectra of M-theory is easy to obtain and, by duality, one gets an idea of the particle spectra in F-theory and how it relates to the structure of the singularities. actually it is more complicated that that, and one must see how the idea holds in realistic compactifications. By the way, that work was previous to the local model engineering approach.

In this approach of F-theory the local model idea morphs into what is known as gravity decoupling. It is perfectly explained in the U-duality blog entry whcich I cite:

The criterion is that it should be possible to make the dimensions transverse to the 4-cycles wrapped by the 7-branes arbitrarily large. Equivalently, it should be possible to contract the 4-cycles to points while holding the six-dimensional volume fixed. Such contractible 4-cycles must be positive curvature Kahler manifolds. These are fully classified and are given by manifolds called del Pezzo manifolds (or del Pezzo surfaces), which are denoted dP_n. The integer n takes the values 0 ≤ n ≤ 8.9 The del Pezzos have a close relationship with the exceptional Lie algebras E_n. The basic idea is that they contain 2-cycles whose intersections are characterized by the E_n Dynkin diagram. By this type of F-theory construction, one can construct an SU(5) or SO(10) SUSY-GUT model. Constructions that involve 7-branes of various types are much more subtle – and also more interesting than ones that only involve D7-branes. D7-branes are mutually local. A stack of N of them gives U(N) gauge symmetry. Matter fields at intersections (due to stretched open strings) are bifundamental. However, different kinds of 7-branes are mutually nonlocal. As a result, there are stacks (corresponding to the ADE classification of singularities) that can give U(N), SO(2N) or even E_N gauge symmetry."

Well, this is just the beginning of the history. One must consider how the GUT groups are broken, this is achieved by means of hypercharge U(1) fluxes. I still must understand many points so I will stop here before misguiding to the possible readers.

I am finding very usefull this papers: F-theory Compactifications for Supersymmetric GUTs by Joseph Marsano, Natalia Saulina and Sakura Sch¨afer-Nameki and E ffective Field Theories for Local Models in F-Theory and M-Theory by Jacob L. Bourjaily.

This last one explains that, actually, the technology of the F-theory revolution also applies to M-theory. In fact both theories seem to have shared part of the development as is seem in the paper Chiral Fermions from Manifolds Of G2 Holonomy by Bobby Acharya and Edward Witten.

By the way, in the improbable case the reader wouldn't know what Vafa papers I am talking about here are the links: First paper and second paper

When I would gain a better understanding of the subject I'll try to do more posts on the subject, but certainly it would a good idea for my readers to see the Lubos blog entries on the same subject (and the Distler ones if he returns to the blogosphere). Well, surely there are more people out there who also could do a fine work bloging about those topics, certainly better than what can be reasonably expected form me .

By the way, a last note. This works are getting string theory very, very near of the phenomenology of LHC particle physics and cosmology testable effects. In fact it gets many pieces of actual physic by separate. Seemingly it "only" remains to join them. For example it must be addressed in deep the relevance of gin from local to global and the role that moduli stabilization plays there. Some work is on the way, but I will not give the links now. After all I wouldn´t like to fall in the category of "linker not thinker" ;-).


Matti Pitkanen said...


I admire you determination to learn about the mathematical background of string theories - or M or F theories or whatever they might be called at the next decade;-). Also your honesty and ability to admit that you have not understood something. There are frustratingly many linkers-not-thinkers among the bloggers.

Personally I see as the basic problem that people begin from a theoretical framework involving large number of ad hoc assumptions rather than from basic physical questions. This means that GUT type extension of standard model - certainly the least imaginative manner to generalize standard model that one can imagine - is taken as the cherished starting point. One can make a long list of questions challenging this belief. For instance:

Does standard model gauge group have some deeper meaning? What is the real nature of color symmetry? Are baryon and lepton numbers perhaps conserved separately so that proton would be stable? What is the origin of CP breaking? What is behind family replication phenomenon? Elementary particles appear in widely different mass scales: does it really make sense to put them into same multiplet as is done in GUTs or could the notion of mass scale has some deeper meaning? Does Higgs provide the only manner to understand particle massivation? Is the existing realization of supersymmetry the only possible one?

The sad fact is that even the most refined mathematical methods are useless if the theoretical framework relies on wrong physical assumptions

Javier said...

Hey Matti, sorry for the delay in answering.

In the prospect of searching an answer for some of your questions I had ideate a toy scheme of how string theory could answer the question of why there are exactly three generations.

Certainly the idea will not got too far, but I am trying to elaborate that a bit before exposing it. I´ll try to do that in a separate post where I´ll also mention some of the objections you pose here.

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