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February 10, 2009

The Algebra of Grand Unified Theories

Posted by John Baez

John Huerta is a student of mine who’s really interested in particle physics. Pretty soon he’ll plunge into his thesis work on exceptional algebraic structures and their role in physics — especially super-Yang–Mills theory, superstring theory and supergravity, but maybe also grand unified theories. But first he needs to pass his oral. Here are the slides for his talk:

Since most of the oral examiners will be from the math department, Huerta’s talk will start with a presentation of facts that particle physicists already know… but reformulated in a way that mathematicians can enjoy. So, he’ll start by explaining the group representation theory underlying the Standard Model. Then he’ll explain — in deliberately very brief and sketchy terms! — the idea of a Grand Unified Theory, or GUT. Then he’ll review three famous GUTs:

  • The SU(5) theory proposed by Georgi and Glashow. This unifies the three colors of quark (red, green, and blue) with the two weak isospin states (up and down) by stuffing the color symmetry group SU(3) and the weak isospin symmetry group SU(2) into SU(5) in an obvious way. The Standard Model symmetry group is SU(3) ×\times SU(2) ×\times U(1), so the theory also needs to stuff U(1) inside SU(5) as a subgroup that commutes with SU(3) ×\times SU(2). This can be done, at least mod a discrete subgroup — and amazingly, the result naturally ‘explains’ the bizarre pattern of hypercharges (that is, U(1) representations) seen in the Standard Model! Instead of a complicated looking representation of the complicated-looking group SU(3) ×\times SU(2) ×\times U(1), quarks and leptons are now described by the obvious representation of SU(5) on the exterior algebra Λ 5\Lambda \mathbb{C}^5.

  • The Spin(10) theory proposed by Georgi. Using the fact that a 5-dimensional complex vector space can be seen as a 10-dimensional real vector space, we can see SU(5) as a subgroup of Spin(10). The representation of SU(5) on Λ 5\Lambda \mathbb{C}^5 is reducible, but it extends to a representation of Spin(10) that’s almost irreducible: it splits into just two parts, one for particles and the other for antiparticles.

  • The Spin(4) ×\times Spin(6) theory proposed by Pati and Salam. By what I’ve said, the Standard Model symmetry group is a subgroup of Spin(10). But in fact, it’s a subgroup of the smaller group Spin(4) ×\times Spin(6). Since Spin(4) \cong SU(2) ×\times SU(2) and Spin(6) \cong SU(4), we can also think of the symmetry group of the Pati–Salam model as SU(2) ×\times SU(2) ×\times SU(4). One copy of SU(2) corresponds to weak isospin — and it acts nontrivially only on left-handed particles. The other corresponds to a right-handed version of weak isospin. Since we don’t see this, it must be spontaneously broken in this model. Finally, the SU(4) treats the leptons as a fourth color of quark — say, ‘white’.

Then comes the fun part: all three of these models, together with the Standard Model, fit into a unified framework. John Huerta explains this in terms of a commutative cube of groups.

Of course all three of these models have their problems! They predict proton decay at too fast a rate, and so on. So at best, they are just steppingstones towards a better theory. Still, they’re intriguing attempts to fit the messy pack of particles we know into an elegant picture. And, it’s possible for mathematicians who know some group theory but not much physics to have a lot of fun learning this stuff. So, John and I are writing a gentle introduction to the algebra of Grand Unified Theories, suitable for mathematicians, which covers the material in his talk. I’ll announce it on this blog soon. But where should we publish it? Suggestions, anyone?

Posted at February 10, 2009 9:21 PM UTC

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45 Comments & 3 Trackbacks

Re: The Algebra of Grand Unified Theories

Super! This will be a great resource as I’ve finally started reading through Peter Woit’s book. Is there an easy way to put the slides “together” so it is more printable? (If the only way is to go and delete all the partial slides, don’t bother–I can do that myself before printing.)

Posted by: stefan on February 10, 2009 10:39 PM | Permalink | Reply to this

Re: The Algebra of Grand Unified Theories

I’ll ask John Huerta if he knows an easy way to do this.

In a few weeks we’ll put a 65-page-long expository paper about this subject on the nnCafé. Later we’ll put it on the arXiv, and put an html version on my website.

Posted by: John Baez on February 10, 2009 10:48 PM | Permalink | Reply to this

Re: The Algebra of Grand Unified Theories

Please devote many pages to “a presentation of facts that particle physicists already know”.

I look forward to reading it.

Posted by: sirix on February 11, 2009 12:32 AM | Permalink | Reply to this

Re: The Algebra of Grand Unified Theories

Assuming he’s using beamer, change the line that looks like this:

“\documentclass[10pt]{beamer}”,

to a line that looks like this:
“\documentclass[10pt,handout]{beamer}”.

Posted by: chris on February 11, 2009 1:21 AM | Permalink | Reply to this

Re: The Algebra of Grand Unified Theories

Here is the printer-friendly version, with all overlays collapsed.

Posted by: John Huerta on February 11, 2009 4:59 AM | Permalink | Reply to this

Re: The Algebra of Grand Unified Theories

Note to all sliders:

follow Huerta’s example - provide printer friendly collapses

Posted by: jim stasheff on February 11, 2009 1:38 PM | Permalink | Reply to this

Re: The Algebra of Grand Unified Theories

There are a number of representation-theoretic questions in particle physics that are not addressed by these GUTs, as I’m sure you know. I wonder if, in the paper or here, you may touch on them?

(1) All the spatial stuff: spinors, linear momentum, etc. In particular, this part of the gauge group can turn left- and right-hand particles into each other (just fly past the particle faster than it’s going, and look back at it).

(2) Mass. Or maybe this is part of (1). In particular, the mass terms are exactly the interactions between left- and right-handed particles.

(3) Flavors. The neutrino pure-mass states are not the same as the neutrino pure-flavor states (operators Flavor and Mass don’t commute). In particular, since neutrino mass is not zero, the neutrinos oscillate in flavor.

(4) Bosons. E.g. W and Z have charge, and the gluons have color?

A friend of mine was part of the group at MiniBooNE, establishing that there are only three flavors of neutrino. So she’s explained my neutrino questions in physics-y language. But I know more math than physics, so I’d love to hear some of the above explained as clearly as in those great slides.

Posted by: Theo on February 11, 2009 12:56 AM | Permalink | Reply to this

Re: The Algebra of Grand Unified Theories

Hi, Theo! Unfortunately, that stuff is not in our paper, except for a little bit of (4). We do, however, talk a lot more about the Standard Model representation theory than I do in these slides.

I would be happy to talk about it here, insofar as I know it myself. And maybe some other people will join in.

To start off, in (1), it sounds like you’re confusing the particle’s helicity with its chirality.

A particle’s helicity is the projection of its spin vector onto the direction of its momentum. Helicity is usually not Lorentz invariant, for exactly the reason you describe: Boost to a frame where the momentum is going the opposite way, and you reversed helicity.

Helicity is Lorentz invariant for a massless particle, which travels at the speed of light. This makes it impossible to boost to a frame where the momentum reverses direction. (You could apply a spatial reflection, but that’s outside the connected component of the Lorentz group.)

A particle’s chirality, on the other hand, comes from the way the Dirac spinor we use to represent the particle splits up into left- and right-handed Weyl spinors. This is always Lorentz invariant (the left and right projection operators commute with Lorentz transformations), but for a massive particle, it doesn’t commute with the Hamiltonian, so massive particles oscillate from one chirality to another.

As above, however, “Lorentz invariant” means invariant under the connected component of the Lorentz group. A mere spatial reflection is enough to interchange left and right.

For a massless particle, chirality is constant, and is related to the helicity. (That is, the Weyl spinors are the states of definite helicity).

You can read more about this here, on Wikipedia. This really helped clear it up for me.

Posted by: John Huerta on February 11, 2009 10:48 AM | Permalink | Reply to this

Re: The Algebra of Grand Unified Theories

and don’t forget the Higgs of North Carolina

Posted by: jim stasheff on February 11, 2009 1:40 PM | Permalink | Reply to this

Re: The Algebra of Grand Unified Theories

“But where should we publish it? Suggestions, anyone?”

I imagine that your opus could fit well in the collection “Panoramas et Synthèses” of the “Société Mathématiques de France”.

Otherwise I am pretty sure that the collection “Travaux mathématiques” here in Luxembourg would gladly welcome it: http://wwwen.uni.lu/recherche/fstc/unite_de_recherche_en_mathematiques_conferences/journal

Posted by: yael on February 11, 2009 2:00 PM | Permalink | Reply to this

Re: The Algebra of Grand Unified Theories

Thanks for these suggestions! In the Anglosphere there seems to be a shortage of places to publish expository papers in mathematical physics — except for conference proceedings, which don’t work for everything I write. I published my article on the octonions in the Bulletin of the American Mathematical Society, and we’re considering submitting this paper there too… but it may be too ‘physical’ for that venue. So, it’s nice to hear about other options.

Posted by: John Baez on February 11, 2009 6:09 PM | Permalink | Reply to this

anti-proofreading?; Re: The Algebra of Grand Unified Theories

Should “representation of Spin(10) that’s almost irreducible: it splits into just two parts, one for particles and the other for particles.” be “representation of Spin(10) that’s almost irreducible: it splits into just two parts, one for particles and the other for antiparticles.”?

Or is this where ghost particles start haunting the theory?

Posted by: Jonathan Vos Post on February 12, 2009 3:58 AM | Permalink | Reply to this

Re: anti-proofreading?; Re: The Algebra of Grand Unified Theories

anti-particles are ‘observable’ / physical
ghost particles are mathematical
cf BFV and BV theories

Posted by: jim stasheff on February 12, 2009 2:10 PM | Permalink | Reply to this

Re: anti-proofreading?; Re: The Algebra of Grand Unified Theories

Yes,

“a representation of Spin(10) that’s almost irreducible: it splits into just two parts, one for particles and the other for particles”

should have been

“a representation of Spin(10) that’s almost irreducible: it splits into just two parts, one for particles and the other for antiparticles.”

Thanks! Fixed.

Posted by: John Baez on February 12, 2009 5:17 PM | Permalink | Reply to this

Re: The Algebra of Grand Unified Theories

Hi John(s),

I like the “cube of GUTs”; I hadn’t thought of Pati-Salam as a subset of a Spin(10) before. Neat.

It seems straight forward to include gravity as Spin(3,1) acting on fermions as spinors. Why not include that (and expand the review to the algebra of GUTs and gravity)?

If gravity isn’t included, I can see how this cube can grow with the inclusion of the Spin(10) in E6. But I expect you can guess where I think things go if you combine gravity and Pati-Salam.

In any case, I’ll also look forward to reading the review. Will you be using weight diagrams?

It doesn’t matter where you publish it – the readers you care about will find it on the arxiv or linked from here.

Best,
Garrett

Posted by: Garrett on February 12, 2009 9:12 PM | Permalink | Reply to this

Re: The Algebra of Grand Unified Theories

If you were going to grow the cube by including Spin(10) in E6 then which representation of E6 would you use?

Unless I have missed something you would have to enlarge the representation. My limited understanding suggests that this is a prediction of new particles.

Adding gravity, or going further in the direction you might wish to, would surely lead to still larger representations?

Posted by: Bruce Westbury on February 13, 2009 1:35 PM | Permalink | Reply to this

Re: The Algebra of Grand Unified Theories

Bruce W. wrote:

If you were going to grow the cube by including Spin(10)Spin(10) in E 6E_6 then which representation of E 6E_6 would you use?

I’m not sure what Garrett was talking about, but maybe he was talking about something I mentioned back in week253. Namely, that the Lie algebra of E 6E_6 can be nicely built as a direct sum of 𝔰𝔬(10)\mathfrak{so}(10), its Dirac spinor representation on Λ( 5)\Lambda(\mathbb{C}^5), and 𝔲(1)\mathfrak{u}(1). Since Λ( 5)\Lambda(\mathbb{C}^5) describes a single generation of fermions and their antiparticles in the Standard Model, this gives a tantalizing — but murky — relation between the Spin(10)Spin(10) GUT and E 6E_6.

I’m fascinated by this, but I don’t see what to do with it — it could be just a fata morgana. So, it’s not the sort of thing I’d want to say much about in a review article.

Posted by: John Baez on February 14, 2009 3:26 AM | Permalink | Reply to this

Re: The Algebra of Grand Unified Theories

Garrett writes:

I like the “cube of GUTs”; I hadn’t thought of Pati–Salam as a subset of a Spin(10)Spin(10) before. Neat.

Thanks! In our paper, we’ll say some more about how the Standard Model, the SU(5)SU(5) theory, the Spin(10)Spin(10) theory and the Pati–Salam Spin(4)×Spin(6)Spin(4) \times Spin(6) theory form a neat package.

It seems straightforward to include gravity as Spin(3,1)Spin(3,1) acting on fermions as spinors. Why not include that (and expand the review to the algebra of GUTs and gravity)?

Having almost finished a 65-page paper explaining GUTs to mathematicians who know nothing of particle physics beyond the existence of protons and neutrons, we are not at all eager to expand this paper to include gravity!

We’ll leave that as an exercise to the interested reader.

However, we do have plans to study the octonions and exceptional Lie groups and their connections to physics. John Huerta is my only student brave enough to tackle these topics!

Will you be using weight diagrams?

Nope; we wanted to keep it simple and self-contained, so we’re developing the representation theory we need from scratch, and we don’t need very much.

It doesn’t matter where you publish it – the readers you care about will find it on the arxiv or linked from here.

Unlike you and me, John Huerta will soon be looking for a math job, so it really does matter where we publish this. For example, if we publish it in the AMS Bulletin, lots of American mathematicians will see it, since that journal comes free with membership in the AMS. That would be good, since the readers we care about include people on hiring committees throughout this great nation, from sea to shining sea.

Posted by: John Baez on February 13, 2009 8:55 PM | Permalink | Reply to this

Re: The Algebra of Grand Unified Theories

Hi, Garrett.

JB might be able to guess where combining gravity and Pati-Salam goes, but I can’t. Out of curiosity, what did you have in mind? E8?

I see JB beat me to answering your questions about our paper.

Posted by: John Huerta on February 17, 2009 10:41 AM | Permalink | Reply to this

Re: The Algebra of Grand Unified Theories

Hi John,

I think its best if you try combining GR and Pati-Salam and see what you can get. Just write down the Dirac operator in curved spacetime and have a look at how the D2 of the gravitational spin connection and D2 and D3 of Pati-Salam act on fermions (paying attention to left and right-chiral parts). The D2’s combine in a D4 acting on the fermions, which live in an 8x8, with the D3 in another D4 acting on them. Relying on triality, I’m pretty sure this D4+D4+8x8 is D8 (similar to how D5 plus fermions is E6). But I’m curious if you might be led to other conclusions.

Posted by: Garrett on March 10, 2009 11:54 PM | Permalink | Reply to this

Re: The Algebra of Grand Unified Theories

The D2’s combine in a D4 acting on the fermions, which live in an 8x8, with the D3 in another D4 acting on them.

Umh, no they don’t, as was rather patiently explained more than a year ago.

Posted by: Jacques Distler on March 11, 2009 5:27 AM | Permalink | PGP Sig | Reply to this

Re: The Algebra of Grand Unified Theories

Umh, no they don’t, as was rather patiently explained more than a year ago.

I think that analysis was flawed because it uses the C (charge) conjugate states as antiparticles, as is conventional in the GUT literature. But when dealing with gravity, one needs to use the CPT conjugates as antiparticles (which amounts to complex conjugation and the dual representation space). I’m pretty sure this is true, but I’m curious to hear (either) John’s thoughts on it.

Posted by: Garrett on March 11, 2009 9:42 PM | Permalink | Reply to this

Re: The Algebra of Grand Unified Theories

I don’t know what you’re talking about.

Could you explain?

There’s a well-defined formalism for chiral spinors in curved space (and a very large literature on the subject). And what I wrote is perfectly compatible with that formalism.

If you think there’s something wrong with the conventional treatment of chiral spinor in curved space, then we can set aside all this mumbo jumbo about Pati-Salam, and deal with that.

Posted by: Jacques Distler on March 11, 2009 9:57 PM | Permalink | PGP Sig | Reply to this

Re: The Algebra of Grand Unified Theories

OK, sure, as long as John (Baez) doesn’t mind this conversation in his thread. I guess it is relevant to John Huerta’s slides, since he mentions the duals as antiparticles.

This issue of chiral fermions in curved spacetime is clearer if one first considers the toy model of a massless left-chiral neutrino in Riemannian spacetime. The spin(4) spin connection in this case is comprised of two independent su(2) parts, ω L\omega_L and ω R\omega_R. When we write down the action for this neutrino, it doesn’t include ω R\omega_R, only ω L\omega_L. This action has neutrino solutions, which have left handed helicity, and antineutrino solutions, which have right handed helicity – and neither of these interact with the ω R\omega_R part of the spin(4) spin connection. Algebraically, these fields are in the 2 and 2* representation spaces of su(2) Lsu(2)_L. This is consistent with using the CPT conjugates as antiparticles. In your analysis, you have put the antineutrinos in a nontrivial representation space of su(2) Rsu(2)_R, which is incorrect because there is no interaction between these fields and the ω R\omega_R part of spin(4).

Posted by: Garrett on March 12, 2009 3:57 AM | Permalink | Reply to this

Re: The Algebra of Grand Unified Theories

Umh. No.

Since you’ve decided to talk about Spin(4)=SU(2)×SU(2)Spin(4)=SU(2)\times SU(2), then we should talk about spinors in Euclidean-signature. (I’d have be equally happy to talk about Minkowski signature, but you chose Euclidean, so let’s stick with that.)

Let MM be a Riemannian manifold which is oriented and spin. Let S ±MS_\pm \to M be the bundle of positive (respectively, negative) chirality spinors. Let PMP\to M be a principal GG-bundle, for GG a compact Lie group (the “gauge group” of our theory) and VMV\to M a vector bundle (with Hermitian inner product), associated to PP via a unitary representation, RR, of GG.

The Riemannian structure endows S S_- with a skew-bilinear form on the fibers (and similarly for S +S_+). Combining that with the aforementioned Hermitian inner product on VV (which we view as a bilinear form on the fibers of V¯V\overline{V}\otimes V), we get a bilinear form (,):Γ(S V¯)Γ(S V)C (M) (\cdot , \cdot ): \Gamma(S_-\otimes \overline{V})\otimes \Gamma(S_-\otimes V)\to C^\infty(M)

Our “left-handed” fermions will be denoted ψΓ(S +V)\psi \in \Gamma(S_+\otimes V). In Minkowski signature, the right-handed fermions are related to their left-handed cousins by Hermitian conjugation. But in Euclidean signature, they are independent fields, χΓ(S V¯)\chi \in \Gamma(S_-\otimes \overline{V}). (This is the familiar doubling of fermions in Euclidean signature.)

Given a unitary connection, AA, on VV, the chiral Dirac operator, D A:Γ(S +V)Γ(S V) D_A : \Gamma(S_+\otimes V) \to \Gamma(S_- \otimes V) is a first-order elliptic differential operator. The “action” for these fermions is MdVol(χ,D Aψ) \int_M dVol\,\, (\chi, D_A \psi)

Finally, let’s tie that into the representation-theory shorthand. Spin(4)=SU(2)×SU(2)Spin(4) = SU(2)\times SU(2) acts on sections of the total spin bundle. Sections of S +S_+ transform in the (2,1)(2,1) representation; sections of S S_- transform in the (1,2)(1,2) representation. (Since these are independent representations, that’s why χ\chi and ψ\psi are independent fields in Euclidean signature.) Combining that with the action of GG, we say the ψ\psi transforms as a (2,1;R)(2,1; R), and χ\chi transforms as a (1,2;R¯)(1,2; \overline{R}).

Now, in my post, I was working with Minkowski signature spacetimes. the connected component, Spin(3,1) 0=SL(2,)Spin(3,1)_0 = SL(2,\mathbb{C}). Left-handed spinors (sections of S +S_+) transform as the 22 of SL(2,)SL(2,\mathbb{C}). Right-handed spinors (sections of S S_-) transform as the 2¯\overline{2}. In Minkowski signature, χ\chi isn’t an independent field. Rather, it’s the Hermitian conjugate of ψ\psi. But everything else I’ve said, about the Euclidean case, translates straightforwardly. In particular, ψ\psi transforms as a (2;R)(2; R) and χ=ψ *\chi =\psi^* transforms a (2¯;R¯)(\overline{2}; \overline{R}).

Now do you understand my post?

In particular, for G=SU(2)×SU(2)×SU(4)G= SU(2)\times SU(2)\times SU(4) and R=(2,1,4)(1,2,4¯) R = (2,1,4) \oplus (1,2,\overline{4}) it should be trivial to verify that the fermions do not assemble themselves into an (8,8)(8,8) when we embed Spin(4)×GSpin(8)×Spin(8)Spin(4)\times G \subset Spin(8)\times Spin(8).

(Hint: the (8,8)(8,8) is a real representation, so you don’t even need to do a computation.)

Posted by: Jacques Distler on March 12, 2009 5:25 AM | Permalink | PGP Sig | Reply to this

Re: The Algebra of Grand Unified Theories

Hi, Jacques. Pardon me for butting into your discussion with Garrett, but JB and I have been working on the relationship of super-Yang-Mills theory and the division algebras of late, and we’re trying to do it as index-free as possible.

Naturally, bilinear pairings of spinors turn out to be really important, since they show up both in the SYM langrangian and in the SUSY variation. So I’d really like to hear more about this skew-bilinear pairing of spinors, (,)( \cdot, \cdot ) that is induced by the Riemannian stucture. In particular:

1. How is it induced?

If I had to guess, I’d say we’re identifying the Clifford algebra with the exterior algebra, Cl(V)Λ(V)\mathrm{Cl}(V) \cong \Lambda(V), and using the Riemannian structure induced on that, but that ‘s symmetric rather than skew, which clashes with what you’re saying.

2. How is it different in different signatures? Does it work in all dimensions?

3. Does it become symmetric when the spinors are Grassmannian? Is it modified?

4. It pairs spinors of the same chirality. What about spinors of different chirality?

I ask this because sometimes I want to turn pairs of spinors into 1-forms, like this:

(1)A(V)=<χ,Vψ> A(V) = \lt \chi, V \psi \gt

where <,>\lt \cdot, \cdot \gt is some pairing of spinors, but it can’t quite be the one you describe, because multiplying by a vector VV in the Clifford algebra will generally change chirality.

Whew! That’s a lot of questions. Any help you could give (including pointers to good references) would be much appreciated!

Thanks!

Posted by: John Huerta on March 17, 2009 4:47 AM | Permalink | Reply to this

Re: The Algebra of Grand Unified Theories

If I had to guess, I’d say we’re identifying the Clifford algebra with the exterior algebra, Cl(V)(V)Cl(V)\simeq \wedge(V), and using the Riemannian structure induced on that, but that’s symmetric rather than skew, which clashes with what you’re saying.

It’s also not a pairing on spinors.

The bundle of spinors has structure group Spin(4)=SU(2)×SU(2)Spin(4) = SU(2)\times SU(2), and is reducible. Call the irreducible subbundles S ±S_\pm. The fact that the structure group of S +S_+ is SU(2)SU(2) means that 2S +\wedge^2 S_+ is trivializable and (up to a multiplicative constant) the spin connection determines a trivialization (and mutatis mutandis for S S_-).

Does it become symmetric when the spinors are Grassmannian?

Of course, making the spinors Grassmannian turns antisymmetric products into symmetric ones and vice versa.

… but it can’t quite be the one you describe, because multiplying by a vector V in the Clifford algebra will generally change chirality.

That’s exactly how we built the kinetic term for the fermions. But note that we had to start with χ\chi and ψ\psi of opposite chirality, for exactly the reason you mention.

Put another way, the Riemannian structure provides (again, up to scale) an isomorphism S +S T S_+\otimes S_- \simeq T\otimes \mathbb{C}

which is what we use to construct a kinetic term.

How is it different in different signatures? Does it work in all dimensions?

The story is, of course, highly dimension and signature-dependent. To really understand spinors, one should work it out for all dimensions and signatures. This is a finite amount of work, because Bott periodicity identifies Cliff(p+8,q)Cliff(p,q)(16) Cliff(p+1,q+1)Cliff(p,q)(2) \begin{gathered} Cliff(p+8,q) \simeq Cliff(p,q)\otimes \mathbb{R}(16)\\ Cliff(p+1,q+1) \simeq Cliff(p,q)\otimes \mathbb{R}(2) \end{gathered}

For instance, in dimension 8, the bilinear pairing, on S ±S_\pm, is symmetric, rather than anti-symmetric. In dimension 6, the pairing is between spinors of opposite chirality (i.e., S +S S_+\otimes S_- has a trivial subbundle) and the isomorphism needed to construct a kinetic term is 2S + 2S T \wedge^2 S_+ \simeq \wedge^2 S_- \simeq T\otimes \mathbb{C}

Any help you could give (including pointers to good references) would be much appreciated!

I would suggest Atiyah, Bott and Shapiro as the place to start.

Posted by: Jacques Distler on March 17, 2009 7:13 AM | Permalink | PGP Sig | Reply to this

Re: The Algebra of Grand Unified Theories

The story is, of course, highly dimension and signature-dependent.

Let me clarify that.

The symmetry properties of the bilinear (trilinear) forms are dimension-dependent, but signature-independent (i.e., depend on p+qmod8p+q \mod 8). Whether the irreducible spinor representations are real, pseudoreal (quaternionic) or complex depends on pqmod8p-q \mod 8.

Posted by: Jacques Distler on March 17, 2009 2:01 PM | Permalink | PGP Sig | Reply to this

P.S.

Algebraically, these fields are in the 22 and 2 *2^* representation spaces of su(2) Lsu(2)_L.

Just as a point of information, SU(2)SU(2) has, up to equivalence, only one irreducible 2-dimensional representation.

I have no idea what you mean by your distinction between “the 22 and 2 *2^* representation spaces of su(2) Lsu(2)_L.”

Perhaps my previous response was misdirected.

Posted by: Jacques Distler on March 13, 2009 5:15 AM | Permalink | PGP Sig | Reply to this

Re: The Algebra of Grand Unified Theories

Garrett writes:

… as long as John (Baez) doesn’t mind this conversation in his thread.

It’s fine with me! Just try not to grit your teeth too much, Jacques!

Posted by: John Baez on March 13, 2009 2:21 AM | Permalink | Reply to this

Re: The Algebra of Grand Unified Theories

Just try not to grit your teeth too much, Jacques!

I will try to keep that in mind.

It would cheer me greatly if my explanations were to prove useful to someone else, besides their intended audience of one.

Posted by: Jacques Distler on March 13, 2009 6:00 AM | Permalink | PGP Sig | Reply to this

Re: The Algebra of Grand Unified Theories

OK, let’s see if I have this straight. The conjugate of ψ=(2;2,1,4)(2;1,2,4¯) \psi = ( 2 ; 2 , 1 , 4) \oplus (2 ; 1 , 2, \bar{4}) is ψ *=(2¯;2,1,4¯)(2¯;1,2,4¯) \psi^* = ( \bar{2} ; 2 , 1 , \bar{4} ) \oplus ( \bar{2} ; 1 , 2, \bar{4} )

Now, this isn’t the C (charge) conjugate. Is it the CP (charge and parity) conjugate?

Also, this conjugate is not the dual representation space of SL(2,)×SU(2)×SU(2)×SU(4)SL(2,\mathbb{C}) \times SU(2) \times SU(2) \times SU(4), is it?

Posted by: Garrett on March 14, 2009 8:31 PM | Permalink | Reply to this

Minkowski Signature

OK, let’s see if I have this straight. The conjugate of ψ=(2;2,1,4)(2;1,2,4¯)\psi=(2;2,1,4)\oplus(2;1,2,\overline{4}) is ψ *=(2¯;2,1,4¯)(2¯;1,2,4¯)\psi^*=(\overline{2};2,1,\overline{4})\oplus(\overline{2};1,2,\overline{4})

I presume you mean ψ *=(2¯;2,1,4¯)(2¯;1,2,4) \psi^*=(\overline{2};2,1,\overline{4})\oplus(\overline{2};1,2,4) This is the Hermitian conjugate field to ψ\psi. Wess and Bagger (and I, on most days) would denote it by ψ¯\overline{\psi}.

Now, this isn’t the C (charge) conjugate. Is it the CP (charge and parity) conjugate?

In a general chiral gauge theory, there’s no particular reason to expect that C or CP exist as symmetries. (In any case, there’s no spatial inversion in the definition of ψ *\psi^*.)

Also, this conjugate is not the dual representation space of SL(2,)×SU(2)×SU(2)×SU(4)SL(2,\mathbb{C})\times SU(2)\times SU(2)\times SU(4), is it?

No it’s not.

For the defining 2-dimensional representation of SL(2,)SL(2,\mathbb{C}), 22= 22¯=1 \wedge^2 2 = \wedge^2\overline{2} = 1 the trivial representation. On the other hand, 22¯=4 2\otimes \overline{2} =4 the defining 4-dimensional representation of SO(3,1)SO(3,1).

Posted by: Jacques Distler on March 14, 2009 10:46 PM | Permalink | PGP Sig | Reply to this

Re: The Algebra of Grand Unified Theories

So, Garrett, now that you understand Weyl spinors, do you see why the fermions of a Pati-Salam generation do not assemble themselves into an (8,8) of D 4×D 4D_4\times D_4?

Posted by: Jacques Distler on March 15, 2009 7:15 PM | Permalink | PGP Sig | Reply to this

Re: The Algebra of Grand Unified Theories

Hmm, yes, but this brings up something interesting. If one generation of Standard Model fermions is in half of the 8 +×8 +8_+ \times 8_+ of E8E8 and the antifermions are in the 8 ×8 8_- \times 8_-, then the triality transformation, TT,

8 +×8 +8 ×8 8 v×8 v8 +×8 +8_+ \times 8_+ \to 8_- \times 8_- \to 8_v \times 8_v \to 8_+ \times 8_+

can’t transform the fermions and anti-fermions into 8 v×8 v8_v \times 8_v simultaneously. But we can see a different symmetry transformation of E8E8 at work here: H=TH = -T. Guess I’ll call that “hexality.”

If the fermions, l+ql + q, are the “positive” half of one of these blocks, +8 +×8 ++8_+ \times 8^+, then this hexality transformation maps this to the other five positive and negative block halves and back:

+8 +×8 +8 ×8 +8 v×8 v8 +×8 ++8 ×8 8 v×8 v+8 +×8 + +8_+ \times 8_+ \to -8_- \times 8_- \to +8_v \times 8_v \to -8_+ \times 8_+ \to +8_- \times 8_- \to -8_v \times 8_v \to +8_+ \times 8_+

That’s pretty nifty. So, with this in hand, the first generation anti-quarks are q¯=Hq8 ×8 \bar{q} = H q \subset -8_- \times 8_- and the anti-leptons are l¯=H 4l+8 ×8 \bar{l} = H^4 l \subset +8_- \times 8_-. By applying this hexality transformation we can fill up all three 8×88 \times 8 blocks of E8E8 with different versions of these fermions. And, hmm… a complete set of fermions and antifermions ends up in 8 v×8 v8_v \times 8_v, which is what we need for D8D8. But, of course, this isn’t obviously a generation, because it only got here by applying different powers of hexality to different parts of the first generation. It is interesting though. Well, to me anyway – I’m sure your teeth are nearly gone by now.

Posted by: Garrett on March 16, 2009 3:48 AM | Permalink | Reply to this

Triality, Hexality, …

Well, since I have never been able to make heads or tails of your triality idea, I’m unlikely to understand the “hexality” refinement thereof.

If one generation of Standard Model fermions is in half of the 8 +×8 +8_+\times 8_+ of E 8E_8 and the antifermions are in the 8 ×8 8_&#8722;\times 8_-.

I’m not sure what you mean.

A generation of fermions occupies “half” of an (8 +,8 +)(8_+, 8_+) and half of an (8 ,8 )(8_-, 8_-).

The “rest” of the (8 +,8 +)(8 ,8 )(8_+, 8_+) \oplus (8_-, 8_-) is an anti-generation. (No, these are not the “anti-particles” of a generation. In Nature, there are no anti-generations. That’s what it means when we say that the Standard Model is a chiral gauge theory)

Posted by: Jacques Distler on March 16, 2009 4:20 AM | Permalink | PGP Sig | Reply to this

Re: Triality, Hexality, …

I was speaking of the particles (leptons and quarks l+ql+q) as fermions and their antiparticles (l¯+q¯\bar{l}+\bar{q}) as antifermions.

Posted by: Garrett on March 16, 2009 4:38 AM | Permalink | Reply to this

Re: Triality, Hexality, …

My statement remains the same: of the particles seen in Nature, half are found in the (8 +,8 +)(8_+, 8_+) and half in the (8 ,8 )(8_-, 8_-). The other half of the (8 +,8 +)(8 ,8 )(8_+, 8_+)\oplus (8_-, 8_-) consists of particle not found in Nature.

Originally, you were maintaining that this “triality” idea of yours was just a temporary stand-in for some future rewriting of your model, which would actually contain 3 generations of chiral fermions (instead of zero net generations).

For a year now, it’s been clear that, no matter what embedding you use, when one decomposes 248=(2,R)+(2¯,R¯)+"bosons" 248 = (2,R) + (\overline{2},\overline{R}) + \text{"bosons"}

  1. RR is, at most 32-dimensional, which is too small to accommodate 3 generations (that would require dim(R)45dim(R)\geq 45).
  2. In any case, RR is always real or pseudoreal, which means that the theory is nonchiral and the net number of generations is zero.

In light of that, it would seem that “triality” (or “hexality” or ….) is no longer just a temporary stand-in. To go forward, are you just going to abandon the Spin-Statistics Theorem, chiral fermions, and all that “conventional” stuff?

Posted by: Jacques Distler on March 16, 2009 5:32 AM | Permalink | PGP Sig | Reply to this

Re: Triality, Hexality, …

I find it plenty interesting that the structure of one generation of fermions and all the bosons of the Standard Model and gravity are part of the structure of E8. That fact is hardly trivial! I understand that you think the rest of the apparent particles in E8 make any realistic model involving this fact unworkable, but I don’t think so, even if I don’t know how to make it work yet.

Posted by: Garrett on March 16, 2009 6:07 AM | Permalink | Reply to this

Re: Triality, Hexality, …

That fact is hardly trivial!

For at least some embeddings, it’s entirely trivial.

Consider, e.g., SL(2,)SL(2,\mathbb{C}) embedded in E 8(24)E_{8(-24)}, in such a way that the centralizer is Spin(1,11)Spin(1,11). This just comes from embedding in the maximal subgroup Spin(4,12)Spin(3,1)×Spin(1,11) Spin(4,12) \supset Spin(3,1)\times Spin(1,11)

The fermions arise from decomposing the 128 of Spin(4,12)Spin(4,12): 128=(2,32)(2¯,32¯) 128 = (2,32) \oplus (\overline{2}, \overline{32})

That’s all well and good but, as the authors of this post would tell you, Spin(1,11)Spin(10)Spin(1,11)\supset Spin(10), where Spin(10)Spin(10) is one of the classic grand unified groups. And, under this embedding, 32 =1616¯ 32¯ =1616¯ \begin{aligned} 32&= 16 \oplus \overline{16}\\ \overline{32}&= 16 \oplus \overline{16} \end{aligned} That’s to say, we get a generation and an anti-generation.

To summarize:

  1. There’s no surprise whatsoever that, in this case, the centralizer of Spin(3,1)Spin(3,1) contains the grand unified group Spin(10)Spin(10).
  2. There’s no surprise whatsoever that the fermions (which come from the 128) decompose as 1616s and 16¯\overline{16}s of Spin(10)Spin(10).
  3. There’s no surprise whatsoever that the net number of generations is zero. While the 3232 was complex, as a representation of Spin(1,11)Spin(1,11), it’s pseudoreal as a representation of Spin(11)Spin(11). Hence the theory is nonchiral when we gauge any subgroup of Spin(11)Spin(11) (in particular, Spin(10)Spin(10)).

There are other embeddings where Spin(10)Spin(10) doesn’t appear, but Spin(4)×Spin(6)Spin(4)\times Spin(6) does. But, once you realize (as our hosts point out) that a generation of this “Pati-Salam” group arises in the same way as it does in Spin(10)Spin(10) — as a spinor representation — then the result that the fermions consist of a generation and an anti-generation is no more mysterious than it was in the Spin(10)Spin(10) case.

Posted by: Jacques Distler on March 16, 2009 7:46 AM | Permalink | PGP Sig | Reply to this
Read the post The Okubo Algebra
Weblog: The n-Category Café
Excerpt: Okubo invented a mysterious 8-dimensional division algebra while pondering quarks. What's it all about?
Tracked: March 13, 2009 2:22 AM
Read the post The Algebra of Grand Unified Theories II
Weblog: The n-Category Café
Excerpt: If you know a bit of group representation theory and you've always wanted to understand some particle physics, now is your chance: read a gentle expository account of the algebraic patterns lurking behind three famous Grand Unified Theories!
Tracked: March 16, 2009 2:45 AM
Read the post Synchronicity
Weblog: Musings
Excerpt: Like Freddy Krueger ...
Tracked: June 19, 2009 6:07 AM

Re: The Algebra of Grand Unified Theories

Congratulations to John and John! (see p. 9)

Posted by: Mark Meckes on December 20, 2012 4:36 PM | Permalink | Reply to this

Re: The Algebra of Grand Unified Theories

Hear hear!

Posted by: Tom Leinster on December 21, 2012 2:18 AM | Permalink | Reply to this

Re: The Algebra of Grand Unified Theories

Thanks! I’m going to meet John Huerta at the Joint Mathematics Meeting in San Diego on Thursday January 10th when we pick up that prize. He’s coming from Lisbon. Alissa Crans will also be at that event… anyone else here going to that math conference?

Posted by: John Baez on December 21, 2012 3:24 AM | Permalink | Reply to this

Re: The Algebra of Grand Unified Theories

I ought to be there, since I’ll be in San Diego at the time, but I haven’t worked up the enthusiasm to register for it yet. Is there anything mathematical happening there that I ought to be interested in, aside from you receiving an awesome prize? (Congratulations!) In any case, we should all get together sometime while you’re in town.

Posted by: Mike Shulman on December 21, 2012 3:58 AM | Permalink | Reply to this

Re: The Algebra of Grand Unified Theories

I’ll be there on Thursday afternoon, Friday, and Saturday. To me the big events will be Emily Shuckburgh’s invited address on ‘Using Mathematics to Better Understand the Earth’s Climate’ on Wednesday, January 9, 11:10–12:00 (which I’ll miss), Kenneth Golden’s public lecture on ‘Mathematics and the Melting Polar Ice Caps’ on on Saturday, January 12, 3:00–4:00, and a big bunch of other talks that form the kickoff of the 2013 special year on ‘The Mathematics of Planet Earth’.

Posted by: John Baez on December 21, 2012 5:09 PM | Permalink | Reply to this

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