Tuesday, February 05, 2008

Free fermionic heterotic models

Cleaver, Faraggi, Manno, Timirgaziu argue that they have found a free fermionic heterotic model with a Bose-Fermi degeneracy at the classical level (and thus a vanishing one-loop cosmological constant) that nevertheless breaks supersymmetry perturbatively because they seem to be able to show that one can't simultaneously solve the D-flatness and F-flatness conditions at any finite order in the string coupling.

It looks as a step to solve the cosmological constant problem...

Their result is interesting and different from the field-theoretical intuition but it would be even more interesting if they could show that the cosmological constant also vanishes to higher orders - whatever is necessary to make it tiny enough to agree with the observations.

In other words, it would be even more interesting if they had some evidence that the cosmological constant might be smaller than the value expected from the field-theoretical dimensional analysis, namely than the superpartner splitting mass scale to the fourth.

Free fermionic models

However, I want to say a few general things about the free fermionic heterotic models. Fifteen years ago, slightly before the duality revolution in string theory was getting started, I didn't pay much attention to it. Several times a week, I went to the Karlov computer lab that already had the Internet connectivity to read new papers on the arXiv. Incidentally, Karlov is not far from the place where Einstein was working while in Prague. But once again, duality papers were not my main focus. In fact, I only started to systematically learn the dualities - and to be impressed by them - when I was attracted to Matrix theory at the end of 1996.

The Karlov church, a few meters from the department of maths and physics where I had the Internet access since late 1992

During the years 1993-1995, string phenomenology was a clear winner for me and I was impatiently looking for new papers about the heterotic phenomenology.

Normally we think about the heterotic vacua in terms of a ten-dimensional heterotic string theory compactified on a six-dimensional manifold. However, you may choose a non-geometric, free fermionic approach to these compactifications. Whenever a vacuum can be generated in both ways, the two descriptions may be shown to be exactly equivalent. So we are clearly talking about the same theory even though it might be easier to study some vacua in one picture or another.

The fields on the worldsheet

What are the degrees of freedom of a heterotic string? Let us talk in terms of the light-cone gauge degrees of freedom which is useful because all excitations of the strings are physical and we don't have to investigate who is a ghost. The left-moving portion of the heterotic string is inherited from the bosonic string and it has D-2=24 bosons in the light-cone gauge.

The right-moving portion of the heterotic string is inherited from the ten-dimensional type I/II superstring and it has D-2=8 bosons and 8 fermions (either Green-Schwarz or Ramond-Neveu-Schwarz) in the light cone gauge. Keep D-2=2 bosons on each side - corresponding to the transverse part of the four spacetime dimensions that we normally observe - and transform all remaining degrees of freedom into fermions, knowing that one boson is equivalent to two fermions.

In the left-moving sector, you will have 2x22=44 fermions while in the right-moving sector, you will have 2x6+8=20 fermions. In total, there are 64 real chiral fermions living on the worldsheet. They can play different roles and you must impose the proper generalized GSO conditions and include all the sectors where some fields are periodic and others are antiperiodic (or have a different phase as the monodromy).

The GSO projections and sectors

The overall GSO projection must always be there, for modular invariance, but to get semi-realistic four-dimensional vacua, you also include some other projections that give rise to sectors where some groups are periodic and others are antiperiodic. When you choose a certain set of GSO projection operators, the so-called NAHE set - named after Nanopoulos, Antoniadis, Hagelin, Ellis - you will naturally end up with three-generation models similar to supersymmetric grand unified theories.

NAHE means "pretty" in Hebrew while NAHÉ means "naked" in Czech and the characters who invented this gadget are very interesting. For example, John Hagelin became a U.S. presidential candidate representing the spiritual interests of Maharishi Mahesh-Yogi.

Incidentally, right after the Velvet Revolution, when all kinds of world views gained the freedom to penetrate into the former socialist country, disciples of Mahesh-Yogi came to our high school to teach us how to meditate and they talked not only about various bizarre Eastern religious things that I didn't pay too much attention to but also about a unified field theory of everything. It sounded highly conceivable and despite being a canonical scientific skeptic, I wasn't quite sure whether I should have believed them at least for several days. ;-) Today, I think that everything that was so impressively adjusted to influence me was invented by John Hagelin. It can't be that hard for such a bright person to influence a receptive science fan from a high school.

Independently of these bizarre religious things, believe me that the NAHE paper is an extremely serious, high-quality paper that belongs to the best papers about the phenomenology beyond the Standard Model that we have even today, in 2008. The number of generations - three - occurs kind of naturally in this framework, much like the grand unified group (or its Standard Model subgroup) plus the correct representations for the fermions.

If you don't like extra dimensions, this model is the closest thing to a unified model of quantum gravity and particle physics you can have. In the free-fermionic setup, the conditions for the critical dimension are rephrased in such a way that string theory predicts the right value of a certain combination of the number of generations and the rank of the gauge group.

Top-quark mass

If you care about predictions, the free fermionic models were used by Faraggi to predict the top quark mass as 175-180 GeV back in 1991, more than 3 years before the top quark was actually discovered. His calculation was later shown to be inaccurate and the particular model he used should have given about 190 GeV. Moreover, there could have been other ways to guess that the top quark mass was near 175 GeV. But try to honestly evaluate the papers that were predicting a top quark mass before the particle was observed. I think you will agree that the string-theoretical framework naturally leads one close to the correct values, to say the least.

Stringy compactifications

The free fermionic models are sometimes equivalent to compactifications at orbifolds whose radii are simple rational multiples of the self-dual radius under T-duality. They're the true string-size compactifications. Others don't have a well-known geometric dual. Nevertheless, I always found these compactifications natural and promising and I still do.

On the other hand, I would recommend to avoid any kind of simple vs complex bias. The people who are good geometers surely enjoy to study complicated multi-dimensional manifolds and believe that this knowledge will be necessary for the ultimate description of particle physics. Others might think that a theory of free fermions is somewhat easier than complicated manifolds that are not really needed because the simple, free theory can lead to the same observable phenomena in four dimensions while its toolkit is more economical.

Nature treats simple and hard as equal

I would think that Nature doesn't care about these biases. It is not difficult for Her to compute properties of multi-dimensional manifolds but She is also not afraid to deal with childish theories based on the free fermions. So we must leave it up to Her to decide. But I also think that string theory always seems to provide us with some kind of duality between the simple and the hard. When something is consistent but too hard, it will eventually be shown to have an easy dual description. This is a kind of belief that is supported by circumstantial evidence.

This duality, whenever it exists, is also a very specific reason why you should think that the complicated manifolds and the simpler choices of GSO projections are "equally big" and your prior probabilities for both of them should be comparable.

At any rate, we should study everything that looks promising, realistic, and passes non-trivial consistency checks. The free fermionic heterotic models almost certainly belong to this category.

And that's the memo.

1 comment:

  1. I have spent some time recently trying to understand "doubly special relativity" and other related ideas of Lorentz/CPT violation.

    Some of this has been prompted by the recent post at cosmic variance http://cosmicvariance.com/2008/02/06/aether-compactification/

    Again, I am still learning things and I won't pretend to know all of what is being said (yet); but I have come to the conclusion that people in certain communities in physics have become detached from trying to explain observations, and are now joining some new "cargo cult" replete with suicidal dogma.

    I only mention this because it is interesting to watch entire fields of science implode before one's eyes.

    Unification of GR/QM is a worthy goal, since observation has shown that these two theories have been incredibly correct. However, to make new theories based on assumptions that we have observed to be wrong is something entirely different.

    This brings me back to "doubly special relativity"; other than sounding a lot like the "double secret probation" of Animal House fame; I find the current development of such theories as troublesome.

    I am not sophisticated enough to discuss Lorentz and CPT symmetries or violations in a technical way; so I will avoid discussion of such things until I am able.

    I can however see that people developing these theories take them seriously; and not as explorations of the limits of current theories. Of the two choices, the latter is perfectly acceptable; but the former is borderline lunacy.

    I feel confident in that statement simply because viable theories must be consistent with real observations about our universe; otherwise they are complete rubbish. The strength of our current theories is that they actually predict observations in nature.

    To clarify further, some theories will predict things that we have never bothered to look for. The theory is thus enhanced if we find the thing that is predicted. However, the theory also needs to conform with existing observations.

    If a theory is constructed to predict something we have never observed even after we have looked for it really carefully; then that theory is clearly fiction.

    Maybe we haven't looked carefully enough?

    Perhaps, but suppose it is impossible to ever look carefully enough?

    Now we have entered the realm of lunacy.

    Some people might want to lump string theory into the group of theories that can never be falsified; however, from what I have taught myself so far, it is perfectly consistent with observations already made, and is mathematically sound. Perhaps a final theory might not be a string theory "exactly"; but my money is on the strong likelihood that the underlying math will in principle be shared with whatever theory replaces it.

    I find it disheartening to see promising minds warped by clearly unsound thought. I hope scientists actually start practicing science again, because despite current theories to the contrary, nature is not very forgiving.