Those solutions are connected on the configuration space of the theory and may be related by various dualities. However, the known stringy descriptions that are semi-realistic and weakly coupled may still be divided into several major categories:
- Weakly coupled heterotic E8 x E8 strings
- Heterotic M-theory in 11 dimensions; the strongly coupled limit of (1) with two end-of-the-world domain walls
- M-theory on singular 7-dimensional G2-holonomy manifolds
- Type IIA orbifold/orientifold braneworlds
- F-theory on local singularities
The largest group (5) also includes the "generic" models - that one may describe as F-theory on Calabi-Yau four-folds - that have been used by the anthropic people. Because I don't consider the large number of models in a class to be a positive argument in favor of a scenario, this anthropic interpretation of the large class (5) will be ignored.
However, even independently of that, the group (5) contains some subgroups of "simplified" and "more specific" models that may be imagined in various ways and have various interesting properties. In particular, the group (5) includes the Vafa et al. bottom-up F-theory model building with singularities as large as the E8 singularity. Also, (5) includes the simple "moral" T-duals of the category (4) - type IIB braneworlds with D3-branes and similar things occupying a nearly flat space.
Relationships between the scenarios
While one cannot identify each model with "twin city" models in all other groups in a one-to-one fashion, it's still true that for various compactifications, there are dualities which are sometimes many-to-one equivalences. Let me mention a few of the basic ones that are enough to connect all five groups.
The group (1) is obviously related to (2) because (2) is the strong coupling limit of (1). Also, if the Calabi-Yau manifold in (1) is written as a T3-fibration, one may use the heterotic/K3 duality and obtain (3), the M-theory compactification with a K3-fibered G2-holonomy manifold. So (3) is connected to (1), too.
Locally on the compact G2-holonomy manifold, one may also try to shrink some cycles and get to a type IIA description. So (3) is related by the type IIA/M-theory duality to (4). And (4) is at least in some cases a T-dual of (5) - the usual dualities between type IIA and type IIB string theories. Let me just re-emphasize that this simple list is not exhaustive. There are other relationships between the vacua in different groups, too. Chances are that if you consider a vacuum in one group, you may learn a lot if you find a complementary perspective on it using a vacuum from a different group.
Advantages
The group (1) is the most field-theory-like scenario in string theory. It usually agrees with the conventional supersymmetric and grand unified model building in field theory even though there are some extra characteristic string effects modifying the grand unification, too.
The group (2) is similar except that the 11th dimension may be pretty large in which case one gets a higher-dimensional theory well beneath the Planck scale. At any rate, (1) and (2) are naturally exploiting the gauge coupling unification and other beautiful arguments of grand unified theories.
The group (3), much like (2), uses M-theory, the highest-dimensional description (if you don't count the fiber-like 2 extra dimensions in F-theory as dimensions). However, the group (3) doesn't contain the ends-of-the-world. Its singularities are pointlike which some people might view as more natural.
The braneworlds (4) and (5) may break the coupling unification and other advantages but they may have natural explanations for the fermion mass hierarchy and other things.
It's also interesting to look where the spectrum of the Standard Model is located. In (1), it lives everywhere in the 10D bulk (codimension 0). In (2), it lives in the 10D bulk on the end-of-the-world domain walls (codimension 1). In (3), it's on points of a 7-dimensional manifold (codimension 7). In (4), most of the matter depends on D6-branes and their intersections (codimension 3 or 6). In (5), it's mostly D3-branes (codimension 6 but also possibly 4 and 2).
Branewolds may have gotten some evidence
I would still say that (1) and (2) are the most motivated one but the possible observation of the 150 GeV new particle by the CDF, which might be a new U(1) gauge boson, has surely shifted my opinions a little bit, especially after I read a couple of cool articles about the realistic type IIB braneworlds. In fact, I hadn't previously read the following 3 papers:
- Berenstein, Leigh, Vishnu, Krishna, Brahma, Shiva (2001): The Standard Model on a D-brane
- Lüst, Stieberger, Taylor (2008): The LHC String Hunter's Companion
- Anchordoqui, Goldberg, Huang, Lüst, Taylor (2011): Stringy Origin of Tevatron Wjj anomaly
I've seen various braneworld constructions but the Berenstein et al. 2001 construction strikes me as a very natural one. They study D3-branes on a nice orbifold of C^3 - an orbifold singularity that may occur in the 6 compact dimensions.
One orbifolds C^3 by a group known as Delta_{27} which is a "special" element in an infinite collection of finite groups Delta_{3n^2} for n=3. The group is generated by some added third roots of unity to z1,z2,z3 and by the cyclic permutation of z1,z2,z3: note how extremely natural this group is! And this natural orbifold in 6 extra dimensions pretty much produces the Standard Model - with 6 Higgs doublets.
Now, following the general description of Douglas and Moore for quiver diagrams, and particular derivations by Brian Greene et al. and others for the open string spectrum on the Delta orbifolds, Berenstein et al. have found out that one may get a very nice Standard-Model-like construction. It gives three generations - and in some sense, you may say that the number "3" is being linked to the number of complex compactified dimensions, so it is being "explained" here.
The fermionic spectrum, as described also in the 2008 paper mentioned above, looks very natural - and still "qualitatively differently" from the nice embedding in the grand unified theories. One has three stacks, morally U(3) x U(2) x U(1) gauge groups, and the charges of the fermions under them are
(1,1,0)
(2,0,0)
(-1,0,1)
(0,-1,1)
(0,2,0)
(0,0,-2)
Note that one includes all permutations of (2,0,0) and (1,1,0) - imagine that you flip the convention for the third sign - and the negatives of these vectors. Very natural, right? One may actually derive this spectrum from the non-Abelian orbifold of C^3 above.
However, you may define the hypercharge Y as the inner product of the six 3-vectors above with (-2/3,1,0) and one gets 1/3, -4/3, 2/3; -1; 2; 0. This hypercharge allows one to interpret the spectrum - six vectors above - as the left-handed quark doublet (3,2); anti-up-quark singlet (3bar,1); anti-down-quark singlet (3bar,1); left-handed lepton doublet (1,2); positron singlet (1,1); neutrino singlet (1,1) - an unnecessary right-handed neutrino.
Just to be sure, you may also define the right B-L quantum numbers of the spinors as the inner product of the six 3-vectors above with (-1/6,1/2,-1/2). A perfectly valid spectrum.
New U(1) groups
Now, these models have lots of new U(1)s - something that is really natural and generic within the stringy braneworlds. In the 2011 paper, Lüst and collaborators diagonalize the mass matrix for the new Z' and Z'' bosons, imposing various constraints, and they see that they can get the new 140 GeV particle.
When they do adjust the models in this way, they produce spectacular new predictions. In particular, there should be another Z'' boson at mass slightly above 3 TeV or so, potentially still available to the LHC, and the string scale at 5-10 TeV. A new collider would probably be needed for that but it is imaginable.
Imagine that the Tevatron 145 GeV signal is genuine and there will be an accumulating evidence supporting a new U(1) group. I would surely think that this would significantly increase the probability that the braneworlds are right.
If some other predictions of the model above were confirmed, it would be good to build a more powerful collider that would try to search for stringy physics at tens of TeVs because the possibility that the strings are almost behind the corner is just fascinating. It has always been too good to be true but if the experiments provided some evidence for some characteristic signatures of the braneworlds, the model could also be good enough to be true. ;-)
Strangely enough, these braneworld models predict some of the stringy physics to be almost as observable as supersymmetry - the superpartner masses may be a few TeV here. The idea that some more "specifically stringy" phenomena would be seen before supersymmetry used to look foreign or fantastic; however, showing that the expectations have been wrong or too modest is something that the experiments have the right to occasionally do.
Off-topic, Facebook: Today, Facebook made an interesting step in its (not quite) cold war against Google when it enabled the e-mail addresses on Facebook. If you have a Facebook account, you may start to use your.own.email@facebook.com that will be sent and readable on Facebook. Try to log into Facebook and check it.
Google Plus One: If you look at the bottom of any post, below "posted by", you will see the five "share buttons" and there is a new, sixth button, with "+1" in it.
Just like Facebook attacked Gmail today with its competition, Google just attacked the Facebook "like" button by its "+1" competition. See an explanation by Google. If you click at it, it will just add number "+1" for you only, and the people who are known to Google to be your contacts - via Gmail - may see on Google search pages that the page was "plus-oned" by you. Try it if you liked this article a bit. ;-)
So far, it doesn't seem to work reliably. You're more likely to see the "+1" button at the main TRF page, beneath each article's excerpt.
Bousso-Susskind crackpottery in Nude Socialist
I kind of expected it but now I know it. Even though the Nude Socialist journalist called Justin something asked me for a discussion and called me by phone for half an hour, after which he thanked and claimed that he began to understand what many-worlds of quantum mechanics etc. mean, he finally wrote a totally uncritical article promoting the Bousso-Susskind hypermultiverse crackpottery, denying that he had ever talked to me, or anyone else who realizes it is a crackpottery, after all.
This time I hesitated and finally said Yes but next time I will instantly say No to any single person from this crackpot tabloid. On the other hand, I must say that this new piece by Amanda Gefter in the same Nude Socialist is just flawless for a popular summary of the 150 GeV bump - even though it's true that she may have used many blogospherical sources that contain the same insight and facts.
