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. Luboš Motl says:

    As far as I understand, John:

    * rediscovered that the Standard Model group is SU(3) x SU(2) x U(1) divided by a certain Z_6 group

    * rediscovered that 16 of spin(10) is a good representation for a single generation of quarks and leptons – i.e. rediscovered one reason behind grand unified theories

    * rediscovered that manifolds with SU(5) holonomy are called Calabi-Yau five-folds

    * wants to study, for a very incomprehensible reason, manifolds whose holonomy coincides with the Standard Model gauge group

    The last point seems completely crazy because holonomy is exactly the symmetry – a part of the tangential group – that is broken by the manifold’s shape, while the gauge group of the Standard Model is a group that must be, on the contrary, completely unbroken.

    Comparing the dimensions 4+6 of the large and hidden dimensions in string theory with the (doubled – real) dimensions of the fundamental reps of SU(2) and SU(3) is pure numerology. The four dimensions of the space we know do not transform under the electroweak SU(2), and the six hidden dimensions do not transform under SU(3).

  2. John Baez says:

    Sometimes it takes work to ignore Motl, but it always pays off. If anyone has anything interesting to say about my paper, I’ll be glad to discuss it here.

  3. D R Lunsford says:

    I’m going to read it now – this sounds fascinating. It’s great to get results 🙂


  4. Thomas Larsson says:

    There is one thing about GUTs that has bothered me for some time. We know that SU(5) is out because it predicts too fast proton decay, whereas SO(10) does not. However, it seems to me that two-step breaking, SO(10) -> SU(5) -> SU(3)xSU(2)xU(1) is also ruled out because of the SU(5) in the middle. Doesn’t this mean that any kind of GUT where SU(5) plays an important role, apparently including John’s, has trouble with proton decay?

    Alas, I have heard people talk about non-minimal SU(5), which apparently evades this problem.

  5. Quantoken says:


    I am not interested in discussing your paper, but in your description of QM in UCR, some dangling words really seem sexy:

    What exactly is the relationship between number theory and quantum mechanics? I have no idea. Can you explain? For example, is there anything particular that the inverse of the fine structure constant necessarily happen to be very close to the 26th odd prime number?:-)

  6. Robert says:

    The fact that the hypercharges work out has been checked already many years ago by people working on SO(10) GUTs.

    I like the observation about 10=4+6 although I don’t see an obvious way to turn it into something that resembles a model.

    And note (besides Lubos’ point about the holonomy being the part of the isometry that is broken which I think is valid) that the CY2 x CY3 is not what we have in nature: At least the large four dimensions are very likely to have full SO(3,1) holonomy as there seem to me arbitrary curvatures around. Of course one could make the approximation (or mumble something about ‘groundstate’ or whatever) that one should take 4D Minkowski space (or (A)dS if one worries about the cosmological constant) for the large dimensions but those in turn have too small a holonomy group.

  7. Urs says:

    the holonomy being the part of the isometry that is broken which I think is valid

    I you wanted to, you could argue that the standard model has only been tested in flat space. The proposed model would then presumeably predict deformations of the standard model group in strong gravitational fields.

    I’d like to note that people interested in what they call ‘geometric algebra’ have played around with similar ideas (I’ll look up concrete papers when I find the time).

    In particular, note that if one wants to adopt the idea that the standard model particles arise as components of a spinor, one has to address the question what then happens to the ‘usual’ spinor degrees of these fermions.

    The only natural answer to that which I am aware of is “Dirac-Kaehler” formalism. Assume the particles are not spinors, but “bispinors”, i.e. inhomogeneous differential form, with the Dirac operator really being the Dirac/Kaehler/deRham operator d + del. The form bundle is Lamda M = S x bar S the product of two spinor bundles. One can interpret the left factor as the observed spinor degrees of freedom, while using the right factor to model families of fermions. That has been proposed several times.

    I could remark that the Dirac operators arising in string theory actually are Dirac/Kaehler operators (on loop space). The reason this is usually not seen is chiral splitting of left- and rightmovers in WZW backgrounds.

  8. X says:

    Can this neat geometrical picture provide any insight into electroweak symmetry breaking? Or the existence of 3 generations of quarks and leptons?

  9. Tony Smith says:

    John Baez, in hep-th/0511086, says:
    “… a G-manifold where G is the Standard Model gauge group is precisely a Calabi–Yau manifold of 10 real dimensions whose tangent spaces split into orthogonal 4- and 6-dimensional subspaces, each preserved by the complex structure and parallel transport. In particular, the product of Calabi–Yau manifolds of dimensions 4 and 6 gives such a G-manifold. Moreover, any such G-manifold is naturally a spin manifold, and Dirac spinors on this manifold transform in the representation of G corresponding to one generation of Standard Model fermions and their antiparticles. … For example, we can take M to be Euclidean R4 and K to be any 6-dimensional Calabi–Yau manifold. …[and then]… M x K is a G-manifold….”.

    Penrose and Rindler, in their book Spinors and Spacetime, v.2 (Cambridge 1986) say (at pages 307-308):
    “… projective twistor space PT … has the structure of a complex projective 3-space CP3 … the points of CM# [compactified complexified Minkowski space] represent the lines of PT. Such a correspondence, in which the lines of a projective 3-space are represented as the points of a quadric in a projective 5-space, is known as the Klein representation …
    Even more relevant to twistor theory was the observation by Sophus Lie in 1869 … that oriented spheres in (complex) Euclidean 3-space could be represented by lines in CP3, with (consistently oriented) contact between spheres represented by meeting lines in CP3. These spheres can be regarded as the intersections of light cones, of points in (complex) Minkowski space, with a constant time …”.

    Is it consistent with John Baez’s physical model picture to use for Calabi–Yau manifolds of dimensions 4 and 6:

    6-dimensional CP3 for a twistor picture of space-time
    4-dimensional CP2 for the (10-6) dimensional space, related to SU(3) in the sense that CP2 = SU(3)/S(U(2)xU(1)) ?

    Would the Klein correspondence allow you to use a conformal spacetime based on the 6-dimensional space of signature (2,4) upon which the Conformal Group SU(2,2) = Spin(2,4) acts ?

    Could the CP2 then be related to the standard model groups in a way similar to that described by Batakis in his paper at Class. Quantum Grav. 3 (1986) L99-L105 in which the CP2 plays the role of the compact extra-dimensional space of a Kaluza-Klein model ?

    Tony Smith

    PS – Thomas Larsson, in this same blog thread, said: “… We know that SU(5) is out because it predicts too fast proton decay …”. Although that is clearly the conventional view as of now, there are alternative definitions of background so that the experimental results may be interpretable as consistent with proton lifetimes not so far from SU(5) predictions. See hep-ex/0008074 by Adarkar, Krishnaswamy, Menon, Sreekantan, Hayashi, Ito, Kawakami, Miyake, and Uchihori, entitled Experimental evidence for G.U.T. Proton Decay.

  10. Kasper Olsen says:

    Personally, I don’t find Baez’ paper that interesting,
    – that the SM gauge group is not SU(3) x SU(2) x U(1), but rather this group divided by Z_6 has been observed many years ago, for example by H.B. Nielsen like 20 years ago (but it is anyhow, a rather trivial observation, since for most purposes we are just interested in the Lie algebra)
    -as Lubos also observed, the inclusion of SU(5) in SO(10) has been known for ages – sure enough, SO(10) has some nice features, but if he’s looking for beauty he could as well have chosen E_8 x (something), or whatever else that thas his SM group as subgroup. Spin(10) is the double cover of SO(10) (which is known for anybody who have heard about Clifford Algebras) and there is basically nothing new in noting, that the Dirac representation is single-valued in SU(5); this is basically just a mathematical ‘accident’
    – his comments about representations ‘on’ the exterior algebra Lambda(C^5) which he observes includes d-quarks, u-quarks etc. is also rather trivial since it is – as far as I can see – just another ‘accident’ related to how Clifford Algebras, Spin-groups and Exterior Algebras of C^n are related (or more precisely, representations of those)
    – his comments about relating this to Calabi-Yau manifolds is also rather fictitious since his observation about the ‘real” SM group preserves the splitting of C^5 into a three- and a two-dimensional subspace is just another ‘accident’ related to the structure of Clifford Algebras, for which he — out of the blue — chose R^10 = C^5
    – and in the same vain – the observation, that if M^4 and K^6 are Calabi-Yau manifolds, then M^4 x K^6 is a G-manifold (where G is his SM group) follows trivially as above. I guess he want’s it to be conneted to string theory (by choosing R^10 = C^5, i.e. the magic number 10 coming out of string theory), but actually it is not (at least not in any non-trivial way) . Sorry, I have to agree with Lubos, he is just playing with numbers 😉


  11. Urs says:

    The paper I had in mind is this. Not that I have looked at it closely enough, but it does build (see the references) on the idea of embedding standard model particles into spinor reps. These people tend to use a slightly non-standard language when talking about spinors, but it’s mostly easy to sort it out.

    I guess if you took these ideas and tried to embed them into something string-motivated in 4+6 dimension it would amount to trying to interpret the closed type II string’s RR states (the p-form fields) collectively as spinor bilinears (what is called ‘algebraic spinors in papers of the above sort’) with interpreting one spinor factor as the observed spin degree of freedoms and decomposing the other in terms of standard model particles of one generation.

    I don’t know if that could work out to a sensible model. But it seems to be a natural way to try to implement the idea of realizing particles species as components of a spinor representation.

  12. Quantum_Ranger says:

    6 Quarks , 4 Forces, = 1 Gravity in 10 dimensonal space?

    Great paper.

  13. Urs says:

    Sorry, it’s me once again. Might be that the link I gave above does not work. The paper it was supposed to lead to is

    G Trayling and W. Baylis
    A geometric basis for the standard model gauge group
    J. Phys. A: Math. Gen. 34 (2001) 3309-3324

  14. Tony Smith says:

    Urs, it may be that the Trayling and Baylis paper you are mentioning can be found at hep-th/0103137.

    Note, for fun, that the arXiv number decodes as number = 137 (the rest of the code is only year and month).

    Tony Smith

  15. “the inclusion of SU(5) in SO(10) has been known for ages – sure enough, SO(10) has some nice features, but if he’s looking for beauty he could as well have chosen E_8 x (something)”

    The idea of someone well know bringing up ideas prematurely pushed to the sidelines and suggesting nice new things to do with them is not so bad. Funny, I pretty much said the same thing a couple days ago in the Weinberg thread. I even mentioned SU(5) GUT and Baez but I had Baez mentioned in relation to the exceptional algebras… so yes why not start at E8, there’s those E6 orbifolded fermions just waiting to suck up the SO(10) bosons. For details of the algebraic glue see Tony Smith, that’s his forte.

  16. Kasper Olsen says:

    John – the fact, that Baez’ observation – that SU(5) is a subgroup of SO(10) – is quite old, does not in itself logically make it “bad” (or irrelevant, unjustified, or whatever). The thing is this – virtually all his assumptions are not substantiated in any way, and this is why he could have decided to consider E_8 or E_8 x E_8 or U(1)^496, or ….

    And I don’t think he is suggesting any nice “new things” with this. Basically, he is only expressing well-known fact in terms of Clifford Algebras, which might seem fancy and ‘new’ to some people; talking about Clifford Algebras is roughly just the same as talking about Dirac matrices, which for obvious reasons are related to Spin-groups, as for example Spin(n) is a certain subgroup of Cl(n,0) – but I guess you know all this.. 😉


  17. Nick Warner says:

    While Lubos is sometimes not as diplomatic as one might like, this does not mean you should ignore his comments – which are essentially correct.

    It may be useful to explain the last one in more detail because the Baez article makes an elementary mistake in the physics underlying compactifications from higher dimensions. Unbroken gauge groups *do not* come from holonomy, and generally the exact opposite is true. The point is that if a field transforms under the holonomy then it is responding to the curvature tensor of the compactifying manifold and as a result fields that transform under the holonomy group will have masses that are set by the compactification scale. This means that a compact manifold, no matter how big it is, will not give rise massless fields that transform under the holonomy group.

    In compactification one finds the low mass/massless fields in the sectors with trivial (generalized) holonomy, and as Lubos says, the holonomy group is precisely what is broken at the compactification scale.

  18. woit says:


    Lubos’s first four comments are just offensive and stupid. John was obviously not claiming to have discovered any of these well-known facts, he was just stating them. Today seems to be my day for pointing this out repeatedly, but I’ll do this here again: you can’t imagine how much damage Lubos has done to the cause of string theory by this kind of behavior. He’s convinced large numbers of people that string theorists are arrogant, ill-informed fanatics. One reason for this is not just his behavior, but the fact that other string theorists seem to think it is all right to behave this way towards non-string theorists, just perhaps “not as diplomatic as one would like”.

    As for the last point, John wasn’t making “an elementary mistake in the physics underlying compactifications from higher dimensions”, since his only comment about the relation of the two mathematical propositions he states to physics was that “it would be nice to find a use for these results”. He was pointing out a characterization of the standard model group that isn’t hard to see but that I hadn’t thought about before, as the subgroup of SO(10) preserving certain geometrical structures.

    There are lots of problems with coming up with any sort of dynamics that make use of this, from the ones I pointed out about the U(2) piece, to the ones you and Lubos note If you try and do standard dynamics on a compactification. John isn’t getting very far, but what he is addressing are two of the biggest problems the standard model leaves unanswered:

    1. Why SU(3)xSU(2)xU(1)/N?

    2. Why the certain specific representations of this group that occur in the standard model?

    What he’s saying is that if you start with 10 dimensional Riemannian geometry and certain specific geometrical structures, from the distinguished 10-d spinor rep you get an answer to 1. and 2. This just changes the problem to

    Why 10 dimensions, and why these specific geometrical structures?

    I’m not at all convinced it’s much of an advance, but it seems to me more worth thinking about than what currently seems to be the preferred answer to questions 1. and 2.: “because of some horrifically complicated lowest energy state of some ill defined infinite dimensional minimization problem we know hardly anything about”. At least it’s different and has a bit of mathematical beauty.

  19. Kasper Olsen says:

    Dear Peter,
    I agree with you, that 1. and 2. should be answered in a more ‘fundamental’ theory. But I don’t actually think that Baez answered these questions. He starts with SO(10), but for no obvious reason, except that of course SO(10) has SU(5) as subgroup which has the SM group (= G) as subgroup. But of course SO(10) has many other subgroups which as well could be ‘interesting’ for the same reasons. The group G has the nice feature observed by Baez, that representations of Spin(10) restricted to G includes the fermion representations given by the exterior algebra Lambda(C^5). But this fact seem (at least to me) to be implicitly included in the old grand unified SU(5) theories. I don’t think this really answered the question #2 about the representations.

  20. woit says:

    Hi Kasper,

    As I said, he leaves unanswered the question “Why 10 dimensions?” If you decide that 10 dimensions is fundamental and you want to do Riemannian geometry in it, you’ve got an SO(10) structure group.

    For a mathematician, if you’re doing geometry in an even number of dimensions, picking an almost complex structure is a very natural thing to do (i.e. you’re identifying the tangent spaces with complex vector spaces). Doing this reduces the structure group to U(5). The Calabi-Yau structure that further reduces to SU(5) is a bit less of an obvious thing to do.

    If your’re working with SO(10) (or, better, its double cover, Spin(10)), the spin representation is a very special one, and you can build all other reps out of it (i.e. you can build all tensors out of spinors, not vice versa). The fact that this spinor rep gives you a generation of standard model particles is well-known. In general, for SO(2n), once you pick a complex structure and reduce to U(n), the spinor rep becomes the complex exterior algebra rep, up to a certain projective factor, one which is trivial if you just look at SU(n). So that’s why John can explicitly construct his spinor rep as an exterior algebra.

  21. Kasper Olsen says:

    Dear Peter – It’s of course correct what you are saying. If you just start by assuming D=10, then the structure group is SO(10), and since D is even, it reduces naturally to U(D/2)=U(5). And then – out of the blue – comes the assumption that the manifold should be a Calabi-Yau which is unique in the sense of having a SU(5) holonomy (while in string theory, as you know, a SU(n) holonomy comes from the requirement of the existence of a covariantly constant spinor). But here, the assumption is not justified – you could as well have chosen the manifold to be Kahler with first Chern class c_1 different from zero, and therefore not CY. But again, only with SU(5) holonomy here can the spinor representation be identified (up to isomorphisms) with the exterior algebra Lambda(C^5), which (roughly) is identified with a generation of fermions. How did that “explain” the #1 on your list — that our Standard Model gauge group should be exactly SU(3)xSU(2)xU(1)/Z_6? (Which has some nice geometrical features observed by Baez and many others) And how did the coincidence, that the fermions ‘fit into’ the exterior algebra here “explain” that our Standard Model fermions should transform as they do? (I.e. answer your question #2, except from the trivial observation, that the representation of Spin(10) decomposes as determined by the exterior algebra mentioned above). If I’m missing something here, please explain 😉

    Best, Kasper

  22. somebody says:

    I’m simply amazed at all LM’s ignorant stuff about holonomy groups and “symmetries”. Holonomy groups are not symmetries and have absolutely nothing to do with isometries….jeez….

    The only problem I see with JB’s stuff is this: is there a manifold with holonomy that is not just *contained* in G, but actually *isomorphic* to it? I think the answer is no, so the actual holonomy group would be a subgroup of the standard group. And that is much less interesting. Still a stimulating paper though.

  23. woit says:

    Hi Kasper,

    To me the not very well-motivated stuff is the extra structure needed to get from U(5) to the standard model group. This is the Calabi-Yau structure on the 10-manifold and the splitting of the tangent space into 2 and 3 d complex spaces. That the standard model particles fit precisely into an SO(10) spinor rep has always been perhaps the most appealing feature of SO(10) grand unification. This is just representation theory. Given any irreducible rep of G, you can ask how it decomposes into H irreducibles for H a subgroup of G. Here G=SO(10), H the standard model group. It’s quite striking that the simplest SO(10) irreducible, the spinor, decomposes as precisely the sum of irreducibles that makes up a standard model generation.

  24. john baez says:

    Peter has tried his best to clarify certain issues, but I can’t resist adding my two cents, since it’s my paper…. 🙂

    Thomas Larsson said:

    However, it seems to me that two-step breaking, SO(10) -> SU(5) -> SU(3)xSU(2)xU(1) is also ruled out because of the SU(5) in the middle. Doesn’t this mean that any kind of GUT where SU(5) plays an important role, apparently including John’s, has trouble with proton decay?

    You raise an important question here, and I think we should talk about it. But for now, I just want to make something clear:

    My paper does not propose a grand unified theory, or any sort of physical theory. It simply points out some facts relating the Standard Model and Calabi-Yau manifolds.

    There’s a reason I’m doing this. So far most approaches to deriving the Standard Model from a more “beautiful” or “unified” theory seem rather artificial. One reason is that the math of the Standard Model looks, to our eyes, ugly and arbitrary. To address this problem, I think some of us should play around the math of the Standard Model… until maybe we see that it’s not so ugly and arbitrary after all!

    In other words, some of us should try to connect the Standard Model to ideas that seem natural and inevitable. (Other people should explore other ideas, including the “landscape” scenario which has the Standard Model as a completely undistinguished inhabitant of a vast panoply of theories.)

    Connecting the details of the Standard Model to truly beautiful mathematics is a tough challenge. Georgi and Glashow’s SU(5) and SO(10) grand unified theories make a promising start. So does Alain Connes’ ideas on the Higgs boson. So does Victor Kac’s work on the exceptional Lie superalgebra E(3,8). So does work on the heterotic Standard Model and the recent attempt to get the
    minimal supersymmetric Standard Model from the heterotic string. Various people who post here have also done their bit, with greater or lesser success.

    I think all these efforts are praiseworthy, even if it turns out that none of them are “the right theory”.

    Indeed, some of this work, like Victor Kac’s work and my own little paper, does not even propose a theory of physics! It still might come in handy.

    We need to explore. We need to play around. We need to stop being so grandiose that we think we’re gonna write a paper that proposes The Theory of Everything and gets it right the first time.

    Robert wrote:

    the CY2 x CY3 is not what we have in nature: : At least the large four dimensions are very likely to have full SO(3,1) holonomy as there seem to me arbitrary curvatures around.

    Right: had I been proposing a theory, I would not have proposed one in which the visible 4 dimensions of spacetime are modelled by a Calabi-Yau manifold.

    The connection on the Calabi-Yau 2-fold (=CY2) in my paper is clearly related to the electroweak force, not gravity.

    Kaspar Olsen writes:

    […] the Baez article makes an elementary mistake in the physics underlying compactifications from higher dimensions. Unbroken gauge groups *do not* come from holonomy, and generally the exact opposite is true.

    Since I wasn’t proposing a physical theory, I was not in the position to make this elementary mistake. You may have thought I was proposing something like a Kaluza-Klein theory, or perhaps planning to use my CY2 x CY3 as a string theory background. I wasn’t. I was doing exactly what I said I was doing in the first sentence of the paper. I wrote:

    The purpose of this note is to point out a curious relation
    between the mathematics of the Standard Model and the geometry of Calabi–Yau manifolds.

    No more, no less.

  25. D R Lunsford says:

    Does this have anything to say about parity in Minkowski space? (i.e. a direct statement about weak interaction)


  26. john baez says:

    Whoops! I just said that Kaspar Olsen wrote:

    “… the Baez article makes an elementary mistake…”

    It was Nick Warner who wrote this.

  27. Who says:

    in case anyone’s interested, this posted today:
    A Polynomial Quantum Algorithm for Approximating the Jones Polynomial

    Dorit Aharonov, Vaughan Jones, Zeph Landau
    26 pages

    “The Jones polynmial, discovered in 1984, is an important knot invariant in topology, which is intimately connected to Topological Quantum Field Theory (TQFT). The works of Freedman, Kitaev, Larsen and Wang provide an efficient simulation of TQFT by a quantum computer, and vice versa. These results implicitly imply the existence of an efficient quantum algorithm … Unfortunately, this important algorithm was never explicitly formulated. Moreover, the results of Freedman et. al are heavily based on deep knowledge of TQFT, which makes the algorithm essentially inaccessible for computer scientists.
    We provide an explicit and simple polynomial algorithm to approximate the Jones polynomial …Our algorithm does not use TQFT at all. By the results of Freedman et. al, our algorithm solves a BQP complete problem.
    The algorithm we provide exhibits a structure which we hope is generalizable to other quantum algorithmic problems…”

  28. john baez says:

    Quantoken writes:

    What exactly is the relationship between number theory and quantum mechanics? I have no idea. Can you explain?

    There are lots of relationships. I think it would be quite a digression to explain them here, but I’ve begun explaining some in various issues of This Week’s Finds, especially week199, week216, week217 and week218.

    I’m just getting started – in part because I’m just getting started on learning number theory. The really interesting ideas on the border of quantum mechanics and number theory, like the Langlands program, the relation between zeta zeroes and quantum chaos, and Connes’ work on the Riemann hypothesis, are pretty darn deep.

    Every week I spend a few hours studying this stuff in the UCR coffee shop with James Dolan. Right now we’re trying to understand the Taniyama-Shimura hypothesis. As a start, we’re trying to see why automorphic forms that are eigenvectors of Hecke operators give Dirichlet series with Euler product expansions. It’s great fun!!!

  29. Kasper Olsen says:

    Peter – it’s correct what you are saying. But I’ll have to ask again: 1) how did the coincidence between the standard model group H and the SU(5) holonomy explain that the interesting group is actually H? – it seems this is what you claimed had been answered.(Except from the obvious thing that you get a splitting of T(M) as C^2 x C^3?)
    2) And how did the representation of H in G=SO(10) actually explain why the standard model fermions are in certain representations? Apart from the fact, that SO(10) potentially could be a good grand unified theory group since the standard model fermions fit into representation of SO(10), which has been known for decades…

    Actually I think John himself answered above: he is just observing some ‘nice’ mathematical ‘coincidences’ between the standard model group H and the Calabi-Yau geometry. And this is of course completely legitimate.

    You can always find thousands of reasons for ‘why’ our SM gauge group should be exactly H, when you actually know what it is. You can also find millions of reasons for why our spacetime seem to have 3+1 dimensions. 4 is an interesting number – there are for example the exotic R^4’s – on which you have an uncountable number of differentiable structures. And actually, we should look at not R^4, but rather R^{3,1}. But, that’s also interesting; the Clifford Algebra Cl(3,1) is isomorphic to H(2), and the Spin group Spin(3,1) is even more beautiful….

    John – no reason to worry; I don’t mind you confused me with Nick

    Best, Kasper

  30. woit says:

    Hi Kasper,

    I should leave this to John, especially since I’m having trouble explaining this, but here’s a last stab at this:

    Given 10d, John is not explaining “why H”, he’s showing that “why H” is equivalent to “why certain specific geometrical structures. He has just reformulated the problem, not solved it.

    The important point about SO(10) representations is that you get precisely the standard model multiplet, no more no less, from choosing a single SO(10) irreducible, the most basic one, the spinor. That choice is very simple and compelling, whereas the standard model multiplet looks complicated. This idea isn’t John’s.

  31. john baez says:

    I could spend all night commenting on comments that criticize me for not achieving grandiose goals that I never claimed to achieve, but I won’t.

    I’ll just say this: particle physicists are a touchy bunch! Give ’em a 4-page paper with two propositions and a corollary – I save the word “theorem” for things that are hard to prove – and they’ll complain you didn’t explain why the universe must be the way it is.

    Reminds me of the guy on sci.physics who’s been viciously attacking me for… not being the next Einstein! Hard to argue with that. But there’s some kind of “I’ve gotta save the world today or it ain’t worth getting out of bed” attitude going on here, and I don’ t like it.

    Anyway, back to physics:

    Everyone always says how the Georgi-Glashow SU(5) theory predicts too high a rate of proton decay, and that kills it. This is one reason my paper is not about the SU(5) theory: it uses some math from the SU(5) theory, but it’s really just about G-manifolds where G is the Standard Model gauge group.

    However: I am ashamed to admit, but I must admit here, that I’ve never carefully gone through the calculation that predicts the rate of proton decay in the SU(5) theory. This is bad, because it’s dangerous to take the conventional wisdom on faith: the conventional wisdom tends to leave out the assumptions behind the results. For example: compare how many people know the conclusions of the Coleman-Mandula theorem to how many people remember its assumptions!

    So, I should strap myself to a chair and work through this calculation. I should also see exactly how supersymmetry helps. I should see how flipped SU(5) changes things. And, I should see how things are different for SO(10). I know what people say about this stuff. But, I should check it for myself.

    Unfortunately, I always seem to have something more fun to do.

    In case anyone here is even worse at this stuff than me, I recommend the Wikipedia articles on the Georgi-Glashow SU(5) GUT and other grand unified theories including SO(10), the Pati-Salam model and flipped SU(5). They don’t compute proton decay, but they’re really not bad for starters!

    For more detail while keeping things pretty easy, there’s chapter VII of Zee’s book Quantum Field Theory in a Nutshell.

    For even more detail, I recommend Ross’ Grand Unification and Mohapatra’s Unification and Supersymmetry: The Frontiers of Quark-Lepton Physics.

    These old review articles are also good:

    D. V. Nanopoulos, Tales of the GUT age, in Grand Unified Theories and Related Topics, proceedings of the 4th Kyoto Summer Institute, World Scientific, Singapore, 1981.

    P. Ramond, Grand unification, in Grand Unified Theories and Related Topics, proceedings of the 4th Kyoto Summer Institute, World Scientific, Singapore, 1981.

    But, given how fundamental the topic is, I’m surprised there are not more books on grand unified theories – books that cover every popular theory in excruciating detail, computing the proton lifetime and the running of coupling constants in each one, etcetera.

    Maybe I’m overlooking some?

  32. “virtually all his assumptions are not substantiated in any way, and this is why he could have decided to consider E_8 or E_8 x E_8 or U(1)^496, or …And I don’t think he is suggesting any nice “new things? with this. Basically, he is only expressing well-known fact in terms of Clifford Algebras”

    Yes I know a Clifford Algebra bivector when I see one so that’s not the new I was thinking of. It’s kind of as you’ve been discussing, just looking at spacetime geometry and standard model bosons and having this be a potential area for “new”. Also as you’ve been discussing, if you want spacetime geometry, SO(10) is kind of on the short list of known places to look and Baez is not suggesting it’s the only place to look. Smith by the way has Clifford Algebra reasons for starting with a single E8.

  33. john baez says:

    Kasper Olson writes:

    […] the inclusion of SU(5) in SO(10) has been known for ages – sure enough, SO(10) has some nice features, but if he’s looking for beauty he could as well have chosen E_8 x (something), or whatever else that [has the] SM group as subgroup.

    Of course for my “G-manifold” idea one wants a nice small orthogonal group O(n) containing the given group G, to get some reasonably simple description of Riemannian n-manifolds whose holonomy group lies in G. When G is the Standard Model gauge group, O(10) is the smallest one that will work. So, I consider 10-dimensional G-manifolds. And, as you know, I prove a couple of things about them: 1) they’re 10d Calabi-Yau manifolds whose tangent spaces split invariantly into 4d and 6d parts, and 2) spinors on them form one generation of the Standard Model fermions.

    But speaking of E8….

    My most recent spell of thinking about this stuff was last summer, when I was stuck in the Beijing Friendship Hotel. My wife was attending an international congress on the history of science. Whenever I got sick of talks on that subject – held in small, miserably hot and stuffy rooms – I went back to my nicely air-conditioned hotel room, took a shower, turned on the BBC news, and worked on finding patterns relating the Standard Model to the octonions and exceptional Lie groups. It was especially fun because I hadn’t brought any notes with me, so I had to rework everything from scratch. Endless hours of amusement!

    I found some neat stuff which I will develop further when I have more time, and eventually write up. The quickest bit to describe was this thing about G-manifolds, so that’s what I wrote up first. Exceptional groups will make their appearance later.

  34. Quantoken says:


    I’d be interested in knowing what you come up with, after you strip yourself to a chair and calculate the proton decay lifetime.

    Mean while, I am making this observation from what I know. All particle decaying phenomena has a decay life time spanning a great range of order of magnitude, from as short as femto second to as long as billions of years. However the longest decay lifetime observed do not exceed a norminal value times the age of the universe scale. We have not seen anything that decays at a time scale like say a thousand times the age of the universe, or a million times, or 10^10 times. A few billion years seems to be an upper limit.

    Any one know a counter example that the decay lifetime is actually observed to be significantly higher than a norminal multiply of the age of the universe? Please let it be known if such counter example exist. If not, then there is a reason why it can go from femto second to billions of years, a dynamic span of more than 40 orders of manitude. But it doesn’t go beyond that. I have a perfect reasonable explanation why that is the case. But I shall refrain from discussing it here.


  35. Passby says:

    Quite interesting topic and discussions, although I don’t understand what your guys talked about!

  36. a says:

    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 SU2 x SU3. There is an obvious 12-dimensional non-CY manifold with a holonomy group very much like the standard group [namely SU5/G] but I’m not sure what the exact holonomy group is after you factor out the center of SU5. I guess it’s ok.

  37. Quantum_Ranger says:

    Did not Ed Witten:

    get pretty close to moving the ‘proton-decay’ goalposts?

    One major problem of a precise proton-decay rate, is that you first have to have a precise “vacuum”.

    I do recall that Witten tried his best a number of years ago, and I recall that I could only conclude that the/our PROTON must have its decay mode from another Universe, this is to say that the appearence of the standard model ‘proton’, must have decayed into that fundemental state “prior” to it appearing at the finite Vacuum in the early Universe.

    This other “Universe” may for all intensive purpose be contained at Blackhole singularity ‘terminal’ junction, as the primordial/first particle in certain cosmological bounce models, aka Turok?

  38. Haelfix says:

    As far as model building in this scenario, I really don’t think it would buy you much. You can’t deform the CY2 or CY3 at all obviously since you would break the all fundamental gauge group.

    Perhaps it might be useful *somehow* to analyze various subtelties of gauge theory using CY machinery, but that seems to me to be somewhat backward. We tend to know more about gauge theory than we do about CYs, and obviously those two specific calabi Yaus picked out aren’t terribly illuminating on the general case.

    Maybe im missing something.

  39. Tony Smith says:

    “a” says “… 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 SU2 x SU3. …”.

    Arthur L. Besse (pseudonym for a group) says as author of the book Einstein Manifolds (Springer 1987) at page 284: “… Hol(g) = U(m) for the canonical Kahler metric on the complex projective space CPm. …”.

    Green, Schwarz, and Witten, in v. 2 of Superstring Theory (Cambridge 1987) say at page 440: “… a metric of SU(N) holonomy is the same thing as a Ricci-flat metric. This … is our first taste of he fact that in finding states of unbroken supersymmetry we actually are finding vacuum states that obey the equations of motion of string field theory. …”.

    So, if the superstring theory requirement of unbroken supersymmetry were removed, could John Baez find the 10-dimensional standard model holonomy model that his paper describes ?

    If the unbroken supersymmetry requirement of Ricci flatness were removed, could holonomy U(3)xU(2) (which includes the standard model) be achieved by using 6-dimensional twistor-and-conformal-related CP3 along with 4-dimensional CP2 ?

    Tony Smith

  40. Nick Warner says:

    Dear All

    This discussion raises an interesting question about when one should pay attention to numerical coincidences. I get a large number of e-mails claiming to have solve the mysteries of the universe based upon numerology. For my own part I have stared for hours at tables of counts of rational curves in Calabi-Yau manifolds in the hope of relating them to characters of Lie Algebras and thereby see interesting dualities. I did this with the odd success until a colleague told me that he just found his complete eight-digit birthday in a particular divisor set ….

    So here is the question: When is numerology pointless and when is it valuable? (There are lots of examples of the latter…. so I am not asking for a catalog.) When do we say “enough!”? There are some obvious criteria for when it is useful:
    (i) When the number of numerical coincidences is outrageously large
    (ii) When there is a conjectured underlying mechanism for which one is trying to gather evidence
    (iii) When the setting suggests many possible mechanisms and the task is to find the correct one ….
    (iv) When your numerology satisfies (i), (ii) or (iii) and does not have an elementary explanation that is already well understood.
    Conversely, numerology is probably pointless if you satisfy none of the conditions above.

    Put more practically, Physicists and Mathematicians have lots of demands on their time and if you are going to attract their attention you have to get beyond the “crank-mail filter” which means that your numerical coincidence better have more than a few matching numbers.

    So we come to my earlier post. John’s article (from my perspective and, I think, Lubos’) fails to meet (i), (ii) and (iv). One might hope that it meets (iii), and the burden of my original post was to examine this possibility in more detail, and expalin why it actually fails (iii). That is, based upon standard approaches and mechanisms of compactification the expressed hope behind the numerology cannot work. Given this, it is especially important to postulate, even very vaguely or roughly, some new proposed mechanism by which the numerology could be given meaning. To hide behind statements like

    “Since I wasn’t proposing a physical theory, I was not in the position to make this elementary mistake.”

    simply makes me (and probably most other physicists) move the idea to the pile of amusing but insubstantial numerical coincidences that arrive in the mail each week. It is possible that this approach will indeed cause me to throw out the solution to life, the universe and everything, but the practical fact is that we all have to make judgements of what will be profitable to pursue and for me John’s paper does not cross this threshhold.

    More constructively, it might be useful to have a discussion on what kinds of conjectures and what kind of evidence is needed in order to engage the average theorist more deeply ….. The criteria will have some interesting universal properties, and probably a number of really quirky individualistic ones …. I suspect that the ones I have given above are pretty universal (but that might just be the arrogance of a particle physicist….)

  41. Michael says:

    Hi Nick,

    nice post. I would like to add one ‘negative’ criterion: If the mechanism suggested by numerology is obviously not operating in nature, it shouldn’t be taken seriously.

    In the present example, quoting Lubos, the “four dimensions of the space we know do not transform under the electroweak SU(2), and the six hidden dimensions do not transform under SU(3).” This seems good enough to dismiss John’s idea as not relevant to physics, doesn’t it?

  42. DMS says:


    What you say is interesting and thoughtful. It is a matter of personal taste what one works on, of course.

    However, there is a LOT of string-inspired “numerology” (think “realistic brane models”)that is out there, that do not explain ANYTHING about the physics beyond the standard model(and in my opinion, never will). So for some string theorists to be going it bullying and insulting anyone coming up with any idea that is not completely string-inspired is a bit rich.


    Seems like you had a busy day yesterday 🙂 Thanks to the arrogant attitude of some(I think they believe they are the next Pauli; well, they are, minus the contribution to physics), I and others I know have changed our attitude from:

    “(Ok, some)String theorists are arrogant, but they may have a point.”
    “(Ok, some)String theorists are arrogant and liars, who thrive on delibrately distorting other people’s work, likeley because they have nothing useful to say.”

    It is amusing to note that Clifford was upset about Krauss’ opinions. Many physicists I know have a much worse opinion of the field(unfair, in my opinion).


  43. Kasper Olsen says:

    Dear John,

    I guess we can all agree that the standard model is not the final story – it is basically just a low-energy effective field theory. So, some of its mathematical ‘coincidences’ might be important and some might not. Therefore, as long as you do not have a more ‘fundamental’ theory, I doubt you will be able to tell which of these coincidences are important and which are not.

    Best, Kasper

  44. X says:

    Modern physics seems to be mostly about piling up beautiful symmetries; with the symmetry breaking required to match the real world as an ugly and arbitrary and an afterthought – this is my subjective impression. Is there some framework in which symmetry breaking is more elegant?

  45. Chris Oakley says:

    Hi X,

    I completely agree. I remember asking “Is there anything better than the Higgs mechanism?” when studying QFT for the first time in 1980/1981 in Cambridge, UK. The lecturer – Peter Goddard, now running the IAS – answered with a forceful “No”. He could justifiably give the same answer today. Whilst I can see the attraction of spending most of one’s time doing classical field theory – it is easier and more elegant, after all – ultimately quantum field theory is the thing that actually counts and I have no doubt that more focus here would quickly lead to constructs greatly superior to the Higgs mechanism.

  46. john baez says:

    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?

  47. Who says:

    the recorded talks from Loops ’05 conference are now available

    in many cases the notes are also available for download as PDF or ppt.
    just click on the speaker’s name

  48. John, I am unsure it is related to your paper, but if you are going towards using GUT SU(5) or SO(10) groups without running up to GUT scale as traditional, it could be worth to bring into attention my unpublished note on extracting GUT angles from Z0 decay only. It is at

    (Referee told that it could do a good footnote to a more important paper. I am a writer of footnotes, you see)

  49. andy.s says:

    Summing up all the contributions, it looks as it John B’s first comment in the thread was right on the money.

    It took a lot of effort, but it was worth it.

  50. woit says:


    John’s idea isn’t really numerology, more just a geometrical characterization of the standard model group. The only real numerology is the standard model multiplet as SO(10) spinor, which isn’t his idea.

    From the way you refer to Lubos, I take it you don’t see much wrong with his behavior. Suit yourself. But do read DMS’s comment. I don’t think you or most string theorists have a clue how much damage Lubos and Susskind together are doing to how string theory is now viewed by other physicists.

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