Baez on the Geometry of the Standard Model

John Baez has a very interesting new paper on the arXiv this evening entitled Calabi-Yau Manifolds and the Standard Model. In it he points out that the standard model gauge group (which he carefully defines as SU(3)xSU(2)xU(1)/N, where N is a six-element subgroup that acts trivially on the standard model particles) is the subgroup of SU(5) that preserves a splitting of C5 into orthogonal 2 and 3 dimensional complex subspaces. Furthermore, if you think of SU(5) as a subgroup of SO(10), then the spinor representation of SO(10) on restriction to the standard model group has exactly the properties of a single generation of the standard model.

Baez would like to think of SO(10) as the frame rotations in the Riemannian geometry of a 10d manifold X. The SU(5) is then the holonomy subgroup picked out by a choice of Calabi-Yau complex structure on the manifold. One way to get such an X is as the product of R4 and a compact 6-manifold M6, picking Calabi-Yau structures on both manifolds in the product. What is happening here is related to an old idea I wrote a paper about a very long time ago (see Nuclear Physics B, vol. 303, pgs. 329-342, from 1988). By picking an orthogonal complex structure on R4, one picks out a U(2) in SO(4) (the Euclideanized Lorentz group), and it is tempting to identify this with the electroweak U(2). This is one part of what is happening in Baez’s construction. It’s very hard though to see what to do with this within the standard gauge theory framework; this is true both for my old idea and for Baez’s newer one. Maybe string theorists can come up with some way of implementing this idea of thinking of the standard model gauge group in terms of the Riemannian geometry of the target space of a string. If so I might even get interested in string theory…..

I don’t immediately see from Baez’s paper why the hypercharge assignments come out right. I need to sit down and work that out, but it’s getting late this evening. There are some other issues his paper raises that I’d like to think about, and maybe I’ll finally get around to doing some work to see whether what I’ve learned about spin geometry in recent years has any use in this context.

I also noticed today that Baez is advertising for students to come to UC Riverside to study Quantum Mathematics. I like the term, and for many students who really care about mathematics and fundamental physics, this would be worth thinking about.

Please, commenters who want to write about their favorite ideas about standard model geometry, try and stick to any aspects of this directly related to Baez’s paper.

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66 Responses to Baez on the Geometry of the Standard Model

  1. a says:

    JB wrote: a writes:

    Just to quickly answer the question above: there are no 10-dimensional CY manifolds with the standard group as holonomy group. Once you reduce as far as that, you have to go right down to SU(2) x SU(3).

    Well, that would be a bit sad. Could you point me towards a proof?

    OK let me back off a moment and say that I was talking about the simply connected case. Back to that in a minute. S[U2 x U3] is a subgroup of U2 x U3, so we [as you know] are talking about a *Riemannian product* of Kaehler manifolds. But for a product of two manifolds to be Ricci-flat, each one separately has to be Ricci-flat. Hence the holonomy group has to be contained in SU2 x SU3. Somehow writing it out like this makes it look trivial, but there is something highly non-trivial going on here; see Besse’s book.
    All this is in the simply connected case. In the more general case you might, by cleverly taking a Riemannian quotient that affects both factors non-trivially, get a holonomy group which is a *disconnected* group which sort of interpolates between SU2 x SU3 and the standard group. That might be interesting but it still wouldn’t be the standard group itself of course.
    Let me conclude by saying that although I have no clue as to how your idea can be used, at least I opened the paper and looked at it; which is more than I can say about the dry technical exercises which, particularly over the last couple of years, have clogged the arxiv. And several of the responses in this thread made my jaw drop after the manner of one of the skeletons in “Corpse Bride”. I do hope that people won’t be discouraged from posting papers like yours. God knows, we don’t exactly have an oversupply of interesting papers these days……

  2. Ruadhan says:

    I think Nick’s question is answered by the minimum description length criterion. Essentially, the log probability that a particular conjecture, C, which happens to explain some data, D, is just a coincidence, is given (more or less) by the amount of information needed to specify the conjecture (the complexity of the conjecture) minus the amount of information needed to specify the data. To apply this in practice you need a way to “encode” conjectures, or rather, you need to make a list of all conjectures, and then the complexity of a conjecture is the log of its position in the list. (Be careful, though – you can’t simply choose an ordering which puts your conjecture at the top; you need to choose a way of generating the list which doesn’t require much information to specify).

    I’ve never seen anybody apply this procedure to the problem of classifying physical theories, but I think it would be a great idea for somebody who isn’t me to work on. It’s exactly the kind of procedure that will tell us how surprised we should be that this apparently simple group mod that one looks like the standard model. It’s numerology done properly, so to speak.

  3. Tony Smith says:

    “a’ refers to Besse’s book and says “… In the more general case …[than]… the simply connected case. you might, by cleverly taking a Riemannian quotient that affects both factors non-trivially, get a holonomy group which is a *disconnected* group which sort of interpolates between SU2 x SU3 and the standard group. …”.

    However, Besse says at pages 323-324:
    “… 11.21 Theorem. Let M be a compact Kahlerian manifold with vanishing real first Chern class. Then M admits as a finite holomorphic covering the product of a complex torus by a simply-connected Kahlerian manifold, again with vanishing real first Chern class.

    Theorem 11.21 allows us to restrict the study of the compact Kahlerian manifolds with vanishing real first Chern class to the study of simply-connected ones. …”.

    In light of that, it may not be feasible for John to evade “a”‘s statement that “… there are no 10-dimensional CY manifolds with the standard group as holonomy group. Once you reduce as far as that, you have to go right down to SU(2) x SU(3). …”, unless somehow by using noncompact things Besse’s Theorem 11.21 might be evaded.

    However, please do NOT take this as being critical of John for having written the subject paper. It is, as “a” says, a truly “interesting paper” and such things are indeed all too rare nowadays.
    To me, even if John’s paper’s construction does not apply to Ricci-flat Kahler manifolds, it is doubtless applicable to some non-Ricci-Flat Kahler manifolds,
    since Ricci-flatness is means (in string theory context) unbroken supersymmetry,
    maybe the lesson to be learned from John’s paper is that the standard model, and probably nature itself, does not like currently fashionable string supersymmetry (no matter how attached to it some physicists may be).

    Tony Smith

  4. Quantum_Ranger says:


    Surely Riemannian Geometry is the crux of General Relativity?.. Relativity is Observer dependant? ..Stringtheory needs a continuious string field background as stated by TS above ‘ricci-flatness’? is exactly what you get when observers are removed, an observer is nothing more than a curveture imposed by the presence of an ‘observer’ being that observers are composed of tangable matter?

    Baez paper is really a very complex “Hand-Wavy” investigation into discrete notions of continued or, broken dimensions that energy and fields exist within.

    In fact, it is really the bases of how matter and energy are contructed and manupulated, using the techniques of “seperation” or “extended” continuations of inquiry?

  5. Quantum_Ranger says:

    Should have stipulated:Reductionism v Separation
    as the standard definition:continuous = Reductionism
    “a continued decomposition” from a prefered or real location.

  6. Urs says:

    since Ricci-flatness is means (in string theory context) unbroken supersymmetry,

    That’s just the simplest case. When for instance (p-form-)fluxes are turned on susy no longer comes with plain Ricci-flatness in general.

    Heuristically this is because further fields have further energy-density which back-reacts on the geometry by its gravitational field.

    More technically it is because the susy one is talking about here is the existence of a spinor which is constant with respect to a given covariant derivative. That covariant derivative receives contributions from essentially all of the background fields. For instance Kalb-Ramond flux gives rise to a torsion term, etc.

  7. MathPhys says:

    Agressive behaviour has always been part of the theoretical high energy physics scene in the US. Motl’s behaviour is silly and childish. He’s so predictible, that by now he’s boring.

  8. Tony Smith says:

    Urs says that “Ricci-flatness is means (in string theory context) unbroken supersymmetry” is “just the simplest case. When for instance (p-form-)fluxes are turned on susy no longer comes with plain Ricci-flatness in general. Heuristically this is because further fields have further energy-density which back-reacts on the geometry by its gravitational field. …”.

    Then, could John’s construction be seen as indicating
    that “simplest case” Ricci-flat Calabi-Yau superstring theory may be physically unrealistic
    that perhaps realistic string theory models should be based on non-Calabi-Yau, non-Ricci-flat, Kahler manifolds for 6+4=10 dimensional string “background” (maybe even including my favorite pair , CP3 and CP2) ?

    If so, might that mean that the “landscape” based on Calabi-Yau manifolds may have a flawed basis, and that work like John’s might be a good way to search for a physically realistic string theory model ?

    Tony Smith

    Tony Smith

  9. Urs says:

    could John’s construction be seen as indicating
    that ‘simplest case’ Ricci-flat Calabi-Yau superstring theory may be physically unrealistic

    Isn’t that essentially asking if any spacetime which is not a manifold of S(U(2)xU(3))-holonomy may be physically unrealistic? Sure it may. But unless you find some mechanism why spacetime should have S(U(2)xU(3)) holonomy it’s hard to see a relation between one and the other.

    might that mean that the ‘landscape’ based on Calabi-Yau manifolds may have a flawed basis

    The landscape arises precisely due to the freedom of turning on all sorts of discrete values for these fluxes.

    But I am no expert on landscape issues, so that’s all I am going to say on that point.

    There is little point, I think, in trying to see any relations between standard string phenomenology and an approach that tries to find particles in spinor reps.

    In ordinary string phenomenology, the fermions arise from the R-NS and the NS-R sector of the string. This means that they are really just spacetime spinors. Now, if you identify the components of that spinor with particles, you are left with something like scalars, no?

    That’s why I said above that – if you wanted to identitfy spacetime spinor components with spinning particles, you’d need to supply another set of ordinary spinor degrees of freedom. For instance by using “bispinors” or whatever you’d call them.

    That’s just a very vague observation. It is not new, but has been investigated in papers like the one I mentioned above. Being totally untrained in phenomenology, I cannot judge the viability of such ideas. All I can do is make the general observavtion that such bilinears in spacetime spinors appear in the string’s spectrum, too. Namely in the RR sector, where they are known as RR p-form fields.

    Make of that what you wish.

  10. Juan R. says:

    What is your opinion on

    the relationship between Cosmology and quantum electrodynamics claimed on Hoyle F. and Narlikar, J. V. Reviews of modern Physics 1995, 67(1) 113-155?

    Juan R.

    Center for CANONICAL |SCIENCE)

  11. Nick Warner says:


    From my comments about Lubos you can only reasonably infer that I consider him to be “somewhat undiplomatic”. I have no wish to be drawn into a more personal discussion. The purpose of my posts was try to explain, more diplomatically, some of the substance underlying Lubos’ post and why his particular post here, but *not* his mode of expression, probably represents the views of many people who work on physics from higher dimensions.

    I read DMS’s comment. It is hard to determine whether his comments were directed at me or Lubos (perhaps this ambiguity was your point, Peter). At any rate, I agree that there is a lot of nonsense under the “string inspired” banner. Indeed, the worst of it is actually fundamentally inconsistent with string theory. However, this is beside the point here. Holding me responsible for the excesses of the “string inspired” is comparable to holding me (a US resident) responsible for George Bush’s supreme court nominations ….. . While you may be justified in disliking the messenger and his other messages, one should keep focussed on the actual content of a particular post within a thread. In this instance the subtantial content of Lubos’ post is correct, but the way he chose to express it was not.

  12. woit says:

    Hi Nick,

    Yes, my point was that the way in which you choose to describe and deal with Lubos’s comments here gives the strong impression that you think his behavior is acceptable (if “undiplomatic’), and this will influence people and cause them to draw conclusions you’re not going to be happy with.

    I understand that there was a substantive point in what Lubos wrote that may have been worth discussing. There also were substantive points in his comment (his claims that John thought he had discovered all sorts of well-known facts), that were stupid, dishonest and offensive, not just undiplomatic.

  13. D R Lunsford says:

    two words: Charm School!


  14. Nigel says:

    Luboš Motl of Harvard University once wrote something that was sensible: ‘… quantum mechanics is perhaps the deepest idea we know. It is once again a deformation of a conceptually simpler picture of classical physics.’ –

    I can’t understand why he seems to think string theory fits into this. String theory is not a deformation of conceptually simpler ideas, it is a real mess with no empirical basis at all. I wish he would grasp that you need to go back and review the evidence, discard obsolete speculations (ST), before making progress.

    I know that Motl’s being paid some grant by Uncle Sam or whoever to pontificate on crank ST, or he wouldn’t be under such pressure. He complains about Baez rediscovering things, but this is what happens in real science! Darwin rediscovered the evolution theory of Anaximander while Copernicus rediscovered the solar system of Aristarchus. Cranks who deny rediscoveries are just creationists believing in an earth-centred world. You have to feel sorry for Motl.

  15. Juan R. says:


    In some sense i agree with Motl at this point.

    String theory ‘born’ (of course after its failure on strong-force) from a very simple idea: a string vibrating could offer us a two-spin vibration mode resembling the hypotetical graviton.

    Since then string theorists unable to understand gravity (or any other part of real science) have claimed that string theory was “important”, “fundamental” or even “elegant”. Since 40 years ago, the topic had been modified, radicalized, and today ‘string’ theory is a ‘deformation’ of conceptually simpler ideas.

    Today, there is not simplicity, there is not elegance, there is not empirical basis, there is not fundamental point of view, there is not unified theory. Is there nothing? Well i think that ego remain and this is the reason that string theorists -specially smart ones -are unable to recognize that they have failed 🙂

    In fact, today there is not theory or ‘strings’. Now, the conceptually simple idea of reducing ‘all’ (of course, this was always an exageration from arrogant people as Schwartz or Greene) to a simple vibrating string in 4 dimensions is abandoned.

    Now, one finds 11D that reduce to 10D in weak coupling limit, those 10 contain 16 ‘inner’ dimensions in that unelegant mixture of bosonic and fermionic stuff, all aderezed with zero mass and none connection with real particles. The breaking into 4×6 is done by hand, it is not well defined, etc.

    The initial single parameter transformed into a mixture of more than 10000 parameters in the best of cases. It is true that initially the posibility for reducing the near 20 of standard model to 1 was really good, but if my math is correct 10000 >> 20 and the situation now is worse.

    The theory of everything turned into the theory of nothing.

    From the posibility for understand ‘all’ of our universe they are passing to the unability for understand even one of multiple (posibly infinite) predicted ‘universes’.

    What is more, from the elegant reduction of dozens and dozens of different particles to a single item. Now there are p-branes (0, 1, 2, 3, 4, 5, 6, 7, 8, and 9). All the theory (so say) offering towers and towers and towers of unobserved effects.


    NO, the standard model is incorrect. There is not pointlike objects on Nature. The basic item of nature is a small unidimensional object

    Now some of them are claiming

    NO, string theory is incorrect. There is not real unidimensional object on Nature. The basic item of nature is a small pointlike object: the D0-brane

    Etc, Etc.

    Well i could agree that was a ‘distorsion’. One very very great 🙂

    Juan R.

    Center for CANONICAL |SCIENCE)

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