In Garrett Lisi’s original paper on “An exceptionally simple theory of everything” from 2007, one of the loose ends he left was a full description of the three generations of fermions, in particular the three different masses of negatively-charged free particles (electron, muon and tau particle). He expressed a vague hope that the generation “symmetry” should be related to the “triality” automorphism of D4 + D4, embedded in E8, but wasn’t able to fill in enough details to convince many people that this was a viable option.
Let’s take a step back, and ask what we are really looking for here. Suppose that we, like Lisi, are convinced that E8 is so rich and beautiful that it must hold the secret of the universe. Suspend your disbelief in this leap of faith, and just ask, where exactly in E8 is the generation symmetry? Lisi said he thought it was a triality symmetry, but Distler and Garibaldi said no, it isn’t. Actually Distler and Garibaldi said there is no generation symmetry in E8 at all. But they didn’t prove that, because they didn’t consider all the cases. We must consider all the cases.
There are exactly three generations, of that there is no doubt. Therefore the generation symmetry is a symmetry of order 3. In E8 there are exactly four types of elements of order 3, which I shall call types 1, 2, 3 and 5, since this is the traditional way to count in quantum field theory. Actually, there is a good mathematical reason for these names, but we needn’t go into it here. Type 2 elements come from the “roots” of E8, and have centralizers of type U(1) x E7. Type 1 elements include the triality symmetries, and have centralizers of type U(1) x Spin(14). In both cases you will notice that the discrete symmetry of order 3 has turned into a continuous symmetry U(1), which is a rather worrying sign that we might have lost the fundamental property that the generation symmetry must have order 3. As far as E8 is concerned, it could have any order at all.
Type 3 elements have centralizers of type SU(3) x E6, while type 5 have centralizers of type SU(9). In both cases you will notice that there is no U(1) component, and therefore the discrete generation symmetry remains discrete. This is important, because it means that we should eventually be able to prove that there are exactly 3 generations, directly from the fundamental properties of E8. Now it is well-known to physicists that the scalars in SU(n) have order n, so you see there is a scalar of order 3 in SU(3) available for a generation symmetry. But in the Standard Model, SU(3) isn’t used like this, and the scalar is a “colour phase” rather than a “generation symmetry”, so maybe this won’t work. So let’s consider SU(9), which has scalars of order 9. Now the remarkable thing is, that SU(9) is not actually a subgroup of E8 – only the quotient group SU(9)/Z_3 is in E8. So the scalars have been reduced from 9 (which has no plausible physical interpretation) to 3 (which is eminently suitable for a generation symmetry).
Are you with me so far? A priori, the generation symmetry could be of any of the types 1, 2, 3 or 5. Type 1 has been tried and found wanting. Type 2 is deprecated, because the discreteness is not fundamental. Type 3 is being used for something else, which leaves type 5 as the most likely candidate.
Now let us introduce the spin, so that we know when we are talking about fermions, and when we are talking about bosons. That means introducing Spin(16), which contains U(8). Our elements of order 3 can have any number of non-trivial eigenvalues in U(8), from 1 up to 8. If they have 1, 4 or 7, then they are type 1. If they have 3 or 6, they are type 3. If they have 2, they are type 2. If they have 5, they are type 5. If they have 8, then they are either type 2 or type 5, and experts on E8 know how to tell which is which. So to study type 5 elements of order 3, we can embed them as scalars in either U(5) or U(8), as subgroups of Spin(16). But these two differ profoundly in their relationship to spin, so they are physically completely different, and it is important to choose the right one. Well, we surely want the generation symmetry to be Lorentz-invariant, which means we have to choose the U(5) case.
At this stage we have to get a bit more technical (sorry if you thought it was already too technical) and consider the possible real forms. Everyone agrees that the semi-split real form of E8 is the only plausible option, and hence the spin group is Spin(12,4), containing U(6,2). The latter extends to two different real forms of SU(9), namely SU(7,2) and SU(6,3), both of which occur in this real form of E8. We can choose either U(5) or U(4,1) or U(3,2) in U(6,2), which gives the various possibilities SU(5) x SU(2,2), SU(5) x SU(1,3), SU(4,1) x SU(3,1), SU(4,1) x SU(2,2), SU(3,2) x SU(3,1) and SU(3,2) x SU(4). I am not sure if all of these actually occur in E8(-24), and the U(3,2) cases are anyway ruled out by Lorentz invariance. The Standard Model SU(3) x SU(2) gauge group rules out the U(4,1) cases, assuming the gauge group has to be compact, which leaves the two U(5) cases as the only plausible ones.
Do they both occur in E8(-24)? I think they do, because the full centralizer of SU(5) is SU(2,3), which contains both SU(2,2) and SU(1,3), but I couldn’t say for sure at this moment. I’ve written a whole paper about SU(5) x SU(2,3), so perhaps I should consult that. But which is more plausible as a model of physics: SU(7,2)/Z_3 or SU(6,3)/Z_3? I vote for the latter, but we really need to try both. If we want the generation symmetry to be Lorentz invariant, then we need the Lorentz group to be inside the appropriate form of SU(9), and therefore inside SU(2,2) or SU(1,3). The former contains SL(2,C), the latter contains SO(1,3). Which Lorentz group do we want? Or do we want both? We can’t have both at the same time, but we can have one or the other.
Perhaps particle physics, with SL(2,C), embeds in SU(7,2)/Z_3, and gravitational physics, with SO(1,3), embeds in SU(6,3)/Z_3? Perhaps these two copies of “the” Lorentz group are fundamentally different? Well, yes, they are fundamentally different, for very basic and unavoidable mathematical reasons. Virtually the whole of 20th century theoretical physics has been devoted to the impossible task of trying to square this circle. I fail to understand why physicists spend so much time trying to hammer a square peg into a round hole. But when I point out that the peg is square and the hole is round, they call me stupid and ignorant, crackpot and lunatic.
So let’s not try to have a theory of everything just yet, let’s just try to get a generation symmetry into particle physics. I have argued that in any E8 model of particle physics, the generation symmetry must be of type 5, and its centralizer must be exactly SU(7,2)/Z_3. The symmetry breaks under the Lorentz group to (SU(5) x SL(2,C) x U(1) x U(1))/Z_6 but various other breakings of symmetry are also interesting, such as (SU(3) x SU(2) x SU(2,2) x U(1) x U(1))/Z_12. But this isn’t the interesting part of physics, this is just the gauge group – i.e. the choice of coordinates. The physics happens in the rest of E8, that is the other 168 real dimensions, which is the 9x8x7/3x2x1 = 84 complex dimensions of the anti-symmetric cube of the 9-dimensional representation of SU(7,2). That is the space where particles change from one generation to another, move through spacetime, hit each other, and generally make life complicated.
So, what have we learnt today? We have learnt that if we start from the assumption (“physical intuition” or “habitual use of familiar mathematics” or “wild guess” or “wishful thinking” according to your preference) that there must be an E8 theory of everything, and we add to it the experimental fact that there are three generations of leptons, then the gauge group is SU(7,2)/Z_3, with adjoint representation 9 x 9* – 1, and the physics is in the antisymmetric cube of 9. Since this is not the same as the antisymmetric cube of 9*, the theory is chiral. Contrary to the Distler-Garibaldi Theorem that purports to show that there is no chiral E8 theory, we have shown that every (reasonable, i.e. gauge invariant, Lorentz covariant) E8 theory is chiral.