I already tried this some month ago: Is there a list (paper, web-site, ...) of SUGRA theories containing the following information: - able to reproduce the standard model (particle content = gauge fields, fermion generations, chiral structure, higgs, ...) - finiteness (I think there are no rigorous proofs but ongoing research programs) - low-energy limit of some string theory
yes and no. Most modern textbooks on string theory include an appendix where the higher dimensional theories are listed, related to its string limit. No mention is done about compactifications to build the standard model, and reviews in that sense are limited to one kind of approach, usually some variant of a D-brane scenario. Paraphrasing Verlinde, I'd say that there is an entropic force acting against the formulation of an unique theory.
Most of the books I am looking now are on 11D KK, and then they refer to 4D sugra as the result of "dimensional reduction" from 11D SUGRA. I guess the trick is to start reading Cremmer Julia Scherk, then download its references, then look for some review citing both CJS and some fundamental reference. In textbooks, the only one I know to avoid strings is Weinberg vol III.
In the old days if you were interested in phenomenology where gravity was present, you studied N=1 4d Sugra period! Thats the only theory that at face value, has any hope of being applicable to the real world. Higher dimensions are not observed, and extended susy possess unrealistic low energy spectrums. Then they figured out dimensional reduction, other various stringy breaking and compactification schemes, dualities and other tricks that enlarged the scope of possibly realistic effective field theories. Of course they typically (but not always) limit in some way to mSugra at lower energies + maybe a few relics. I do not know of a simple way to classify the models, other than to label them by the parent scheme (heterotic compactifications vs brane vs Ftheory etc etc).
Yes but the whole point of reduction is to produce a particle spectrum via the breaking of the N>1 supersymmetries, so for a classification it is reasonable to start from N=8 in 4D, or from N=1 in 11D. Take care with the notation, they changed it from time to time. I'd add, it is vital to be near a university library. Some material you can download from KEK+Spires, but not all. And to start, the appendix of some book, Weinberg or similar, is useful.
That's the point. If I should start from N=8 (N>1) in D=4, somebody should tell me why 8 (and not 4, or 6 ...). If I should start from D=11, somebody should tell me why 11 (I know that in D=11 SUGRA is somehow unique). So we come back to the classification of all SUGRAs.
And 11 is the dimension for the standard model (see elsewhere). Plus, 11 stabilizes to 7+4. Plus, all the m-theory stuff. Plus, 11 is less that 12 and greater than 10 :-D
Re: why D=11 Let G be some (finite-dimensional) Lie group. In order to find the lowest-dimensional manifold which can have G as a symmetry, we need to construct the coset space G/H, where H is a maximal (not necessarily the largest) subgroup of G. dim(G/H) = dim(G) - dim(H) Examples: G = the Poicare' group P(1,3), H = the Lorentz group SL(2,C) P(1,3)/SL(2,C) = [itex]M^{4}[/itex] is the (Poincare-symmetric) 4-dimensional Minkowski space-time. G = SU(2) , H = U(1) SU(2)/U(1) = [itex]S^{2}[/itex] is the (SU(2)-symmetric) 2-dimensional sphere. G = SU(3), H = SU(2) X U(1) SU(3)/SU(2)XU(1) = [itex]CP^{2}[/itex] is the (SU(3)-symmetric) 4-dimensional (complex) projective space. We also note that the circle [itex]S^{1}[/itex] is the U(1)-symmetric 1-dimensional space. Therefore, the manifold [itex]C^{7} = CP^{2}\times S^{2} \times S^{1}[/itex] has 4+2+1=7 dimensions, and it has SU(3)XSU(2)XU(1) symmetry. Clearly [itex]M^{4}\times C^{7}[/itex] is 11-dimensional space with Poincare' and SU(3)XSU(2)XU(1) symmetries. Thus, if you want to constract a (K-K) theory in which su(3)xsu(2)xu(1) gauge fields arise as components of the metric tensor in more than 4 (non-compact) space-time dimensions, you must have at least 7 extra dimensions, i.e., D = 11 is the minimum number with which you can obtain SU(3)XSU(2)XU(1) gauge fields by Kaluza-Klein method. We have good reasons to believe that consistent field theory with gravity coupled to massless particles of spin > 2 does not exist. Since, D>11 supergravity contains such massless, spin>2 particles, we "conclude" that 11 dimensions is the maximum for consistent supergravity. It is remarkable coincidence that D=11, which is the minimum number required by K-K procedure, is the maximum number required by consistent supergravity.
Thanks for this summary; Spin>2 in D>11 was clear to me, but the KK consideration for D>10 is new. So the idea is basically to look for a manifold which could produce the SM via KK. This explains, why D=11 SUGRA is somehow unique, GIVEN the SM.
Indeed this was one of the reasons people in the early 80s thought that D=11 SUGRA was a TOE. Then it turned out that it was nonrenormalizable (a conclusion that has been revisited recently in N=8, d=4 for apparently mysterious reasons --see the modern Twistor program). Can it regain its place as a potential fundamental TOE? Probably not, b/c even if it is perturbatively finite, we now know it has extended objects in its nonperturbative spectrum, and begs the question for a stringy completion.
Does it? Why should I care about string- / M-theory at all if I am able to quantize the theory perturbatively? I agree that one has to look for a non-perturbative quantization, but that does not automatically imply that this must be a string-like approach. As far as I know there is no consistent definition of the perturbation series in string theory beyond two (three?) loops. So we do not even know whether string- / M-theory is the completion of SUGRA in this regime. On the other hand as far as I know the "renormalization" or "finiteness" of SUGRA which is proposed today uses on-shell amplitudes. So again it is unclear how this can be related to string- / M-theory. That's basically the reason why I think that it's interesting to study SUGRA w/o any reference to string- / M-theory. Now I understand the mathematical reason why N=8, D=11 is so interesting. I guess the next step is to understand the KK compactification and realistic SUSY breaking down to the MSSM / SM. Besides the fact that U(1)*SU(2)*SU(3) should be the result, are there any other ideas why one should compactify this theory in a certain why?
Yes. The only fermion in the basic representation of SUGRA in 11D is a 3/2 particle with 128 degrees of freedom. Its bosonic partners are the graviton, which has 44 degrees of freedom, and them a (antisymmetric, iirc) 3-form, A_{\MNR} with 84 degrees degrees of freedom. Now this A_{\MNR}, having three indexes, generates a force tensor with four indexes, which allows to singularize a four dimensional manifold when solving Einstein equations to get compact+big dimensions. This is called the Freund-Rubin solution or something so. You need some extra handwaving to explain why it is 4+7 instead of 7+4. Haelfix, I think that this tensor A relates to the extended object you are asking for. It is because of it that you can tell that 11D SUGRA is a limit of M-theory. My opinion is that the stringy completion provides the flavour. Back in the 80s, the flavour (the number of generations) was searched in the topology of the manifold. So for the dimension. As for why the standard model, where, once you are in seven, it is either that or the sphere. It a sense, the manifold with the standard model is "bigger" than the one of the sphere.
There were to big problems with 11D KK: 1) It was not able to represent chirality 2) It was not able to produce the standard model quantum numbers (the representations of the group). The second problem was adressed by Bailin and Love who shoved that the correct quantum numbers do appear if you add an infinitesimal 12th dimension, providing the U(1) for B-L. (It was not stressed in the original work that it needs to be infinitesimal because B-L is not gauged at all, so fully broken. But "casually", the fact of not having a B-L "photon" is the point that keep us in eleven dimensions) The first problem is not fully solved, but there is a workable approach, by Salam et al: go go down one dimension and then reintroducing by hand the lost U(1), as a field in a non trivial topological configuration. It works in 6 and 10 dimensions. The process is sort of ad-hoc because you do not know how the non trivial configuration appears, you add it in dimension 10 (or 6) instead of going to 11. It is reminiscent of (actually, it anticipated) the links between 10 and 11 in strings.
I mangled my previous post, apologies its late. The story as I understand it is: The maximal supergravity theory in d=4 arises from the dimensional reduction of the D=11 theory (there is only 1 classically), done in the most naive Kaluza-Klein way. Alternatively, you can proceed via the compactification of the type IIA or type IIB 10 dimensional theory. Now, it is this maximal theory in d=4 that is conjectured to be finite (and actually d=3 also is), the higher dimension supergravity theories are most assuredly *not*. However, the fact is there are charged BPS Pbrane solutions that exist for these theories, whether you want them too or not. They essentially show up as solitonic objects, and are extended. Now, the story is well understood in string theory, where M theories classical limit is the aforementioned d=11 supergravity. The M2 and M5 branes that exist there, correspond to the pbrane solutions of the supergravity theory exactly in that classical limit. Compactifying these theories on a 7 torus yields (or a 6 torus if you go via the type II route) the 4d limit where the charged black hole solutions persist. The point is, it really begs the question. If you want to consider perturbatively finite maximal sugra, you still have to take into account solutions that are manifestly stringy in nature if you wish to study the nonpertubative sector. There is no obvious way to get string theory d.o.f to decouple from the theory.
Ok, here is where superstring lore and KK ansatz starts to separate, then: you can not put a 7 torus as a solution of KK, it is a too flat object. You can not even put a 2 torus times something. Most cases, the U(1)s in KK appear inside of fiber bundles. I think it should be not a problem because the G/H compactifications do not produce, I believe -need to review the 12th dimension paper-, generations, so the superstring sector is still needed. What happens, to my regret, is that it is easier to produce papers by getting most or all of the gauge group from the string dof. On other hand, a fermion of 128 dof should be enough to put all the standard model inside.
Yea, I think thats right (at least the first part). For acceptable phenomenology I think you need to compactify on a G2 manifold or something like that as opposed to a torus. Afaik, there is no obvious way to break the SuSY generators down into a N=1, d=4 theory from the d=11 theory and still get plausible phenomenology without appealing to the stringy d.o.f, at least not done in the simple KK way.
Does that mean that D=11, N=8 SUGRA has the potential for the standard model to emerge, but that up to now now natural and complete approach has been found? Or does it mean that one is sure that not even the potential exists and one definitly NEEDS some completion, e.g. string / M? Is this SUGRA still a candidate for a TOE - assuming that on-shell finiteness can be proven?