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Hello, my name is Garrett Lisi, and today I will be describing how to combine all fields of the standard model and gravity in a purely geometric Yang-Mills theory with a single connection. 

The standard model is kind of a mess. It has a collection of Lie algebra valued connection 1-forms describing the electroweak and strong fields (shown here with arrows under them designating their form grade) as well as Higgs scalar fields and Grassmann valued spinor fields describing the fermions. Gravity is described by the spin connection, a Clifford bivector valued 1-form, and the frame, a Clifford vector valued 1-form.

All these fields interact. The three ostensibly different types of algebraic elements, Clifford, Lie, and spinor, may be represented by matrices. This hints at the fact that these different types are actually interchangeable. A spinor, usually represented as a matrix column, may be recast as a Clifford algebra element, and a Clifford algebra is just a type of Lie algebra, which may be represented by matrices.

In fact, it's possible to write the entire standard model and gravity as a big matrix, representing a Lie algebra valued connection.

So, without further ado, here it is.


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All the fields of the standard model and gravity are in this sixteen by sixteen matrix of 1-forms and Grassmann numbers. The matrix breaks down into blocks, H, G, and Psi, with the 1-forms of Psi replaced by their Grassmann valued gauge ghosts via the BRST gauge fixing technique -- giving one generation of fermions as shown here.

All interactions of the standard model come from the curvature of this connection. Basically, we square the big matrix. The weak su(2) field, represented by a Pauli matrix block of W's, acts from the left on itself, on the Higgs fields, and on the left chiral fermions. (Each element of this big matrix is actually a two by two block.) The strong su(3) field, represented by a Gell-Mann matrix block, acts from the right on itself and on the colored quarks. The weak u(1) B field acts from the left and the right on the Higgs and on the fermions, giving the correct hypercharge assignments. The left and right chiral parts of the spin connection act on the frame and act from the left on the chiral fermions. The Higgs also acts on the fermions, as in the standard model, giving them Dirac masses.

This matrix really is a beautiful thing -- I think it would look nice on a T-shirt. :) And it doesn't just come from nowhere. The H and G blocks are representations of the so(1,7) Lie algebra, which acts on the Psi block as Clifford bivectors acting on spinors. Let me show how this matrix is built, in more detail, by starting with the gravitational part -- the blocks containing the left and right chiral parts of the spin connection, omega, and the frame, e.


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We use the four by four, Vile representation of the four spacetime Dirac matrices. It's a chiral representation, built using the Kronecker product of Pauli matrices. This gives six block-diagonal matrices representing the Clifford bivectors.

The spin connection is a Clifford bivector valued 1-form, and the frame is a Clifford vector valued 1-form, and both may be written out in terms of coefficients and basis elements -- if you like that sort of thing. When we use the Vile representation, these Clifford valued 1-forms break up into their left and right chiral parts -- each a two by two block defined by the spin connection and frame coefficients multiplying Pauli matrices.

It's worth noting that the algebra of vectors and bivectors is the same as the algebra of bivectors in one higher dimension. This implies that the MacDowell-Mansouri approach of describing gravity as an so(1,4) bivector connection is the same as describing gravity in terms of a combined spacetime vector and bivector connection.

What we do next is extend the MacDowell-Mansouri trick by combining gravity, Higgs, and electroweak fields in an so(1,7) connection.


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An eight dimensional Clifford algebra may be represented with matrices that are sixteen-by-sixteen, built using the Kronecker product of four Pauli matrices. Using a chiral representation, the twenty-eight Clifford bivectors -- serving as the generators of so(1,7) -- may be represented by eight-by-eight sub-matrices.

These twenty-eight algebraic elements are assigned to the spin connection, frame, Higgs, and electroweak fields. The spin connection is built from the six spacetime bivectors. The electroweak fields are built from four of the higher bivectors. And the frame and Higgs are included in a very interesting way.

The frame is a spacetime Clifford vector valued 1-form, and the Higgs is a higher Clifford vector valued scalar field. These two are combined, by Clifford multiplication, in the frame-Higgs bivector, which is the mixed part of the so(1,7) connection. Since the Higgs and frame share a conformal degree of freedom, one or the other should be normalized. We normalize the Higgs vector, restricting it to be of some constant length, M.

These various parts are added together to give the bosonic connection, H -- an so(1,7) connection, with Clifford bivectors identified as the Lie algebra generators. The correct dynamics come from computing the curvature of this connection.


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The curvature of H is an so(1,7) valued 2-form, with bivector parts breaking up into the three sets of bivectors: The spacetime bivector part includes the Riemann curvature 2-form, plus an area 2-form. The mixed bivector part includes the torsion 2-form, and the covariant derivative of the Higgs. The higher bivector part is the curvature of the electroweak gauge fields. Everything works nicely. And we don't need to use indices anymore -- just algebra.

It is probably best to define the action for this connection as a modified BF theory action over a four dimensional base manifold. Varying B and substituting it back in, this gives the familiar quadratic action terms for the gauge fields and for the Higgs. There is also a term quadratic in the torsion. And it will take some convincing that the remaining term, quadratic in the spacetime curvature, is the Einstein-Hilbert action.
 
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The gravitational part of the action is the same as for MacDowell-Mansouri gravity. This is formulated as a topological BF theory modified by an extra quadratic term involving the spacetime Clifford psuedoscalar, gamma. The angle brackets in this expression return the Clifford scalar part, or, if you like, the trace.

Varying B and plugging it back in gives the action quadratic in the spacetime curvature. Expanding this out results in three terms: The term quadratic in the Riemann curvature is the Chern-Simons boundary term. The four frame term is the volume element. And the last term, with the area bivector 2-form multiplying the Riemann curvature bivector 2-form, gives the curvature scalar.

Tossing out the boundary term, this is the Einstein-Hilbert action, with a positive cosmological constant equal to three quarters times M squared. The cosmological constant relates directly to the Higgs normalization constant, which serves as its vacuum expectation value.

This calculation is more or less the same as is done for MacDowell-Mansouri gravity. And I need to thank Lee Smolin and Artem Starodubtsev for reviving this approach to gravity. If it wasn't for their work, I never would have thought it possible to include all the fields in one big connection. And I wouldn't be here talking with you today.

So, back to that big connection...


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Now that we've discussed the H block of this connection, you should be more willing to believe that this big matrix really does represent the standard model and gravity. The G block is constructed similarly to the H, using some so(1,7) bivectors to build a su(3) sub-algebra. And recall that the Psi block, the fermions, are the BRST ghosts replacing a block of 1-forms that were gauged away.

So, the big question corresponding to this big matrix is: where the heck does it come from? A year ago I knew what the matrix was, and even published it, but I had no idea where it was from -- just that it seemed to work. But it doesn't work perfectly. There is only one generation of fermions in the matrix, and the masses don't work quite right. To have three generations, we need to find two more copies of that Psi. Over the past year I tried all sorts of ways to justify getting two more copies. Really, I tried every ridiculous thing you could imagine. Then, one month ago, I tried the obvious thing.

I asked the question: "Maybe this big matrix, with three blocks of fermions instead of one, represents the Lie algebra of some particular Lie group?" This seemed a farfetched thing to hope for, and I had no reason to believe it was true. It just seemed so unlikely. But, you know, I was trying everything so I figured I'd give it a shot.

So I calculated how many Lie algebra generators it would take to build this matrix -- 248. Then I went looking for a group with a dimension something like that.


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This is a complete list of the simple compact Lie groups. So, which one matches...

I think the first thing I tried was SO(23). (pause) You know, I'm a physicist, so I hadn't had much exposure to the exceptional groups. We just don't use them much. But the mathematicians seem to think they're very pretty. And, sure enough, after a little wandering I bumped into E8, the largest exceptional group, with a dimension of 248.

But just because a Lie group has the right dimension, that doesn't mean its Lie algebra has the same structure as the one we want. But this was encouraging enough that I had to see if it did, as a long shot. So I went looking for descriptions of E8.

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I found this description by John Baez, in This Weeks Finds.

He says E8 is composed of two blocks of so(8), acting on three eight by eight blocks as matrix multiplication from the left and from the right! Exactly what I was looking for.

At this point, I wanted very badly to get my hands on the structure constants of E8 corresponding to this description. Good luck finding them. Apparently, high flying mathematicians are above anything as lowly as Lie algebra generators and structure constants.

I was kind of desperate to find the actual structure constants, so I turned to the dark arts...



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I unearthed this dusty tome, with an appendix on E8 written by He Who Shall Not Be Named.

These are the explicit structure constants, but not quite in the form we want. They're interesting on their own though. It turns out the Lie algebra of E8 can be described as a block of the 120 Clifford bivectors for so(16) acting from the left on a positive chiral spinor -- a column of 128 numbers.

This in itself is pretty cool. Did you ever wonder why spinors exist, or why Clifford algebra seems to pop up everywhere? Now, if this E8 gauge theory turns out to be the theory of everything, it answers these questions. Clifford algebra is part of the structure of E8, and the algebraic structure of spinors is a big part of physics because they are part of E8. The algebra of bivectors multiplying spinors, the one we're presented with in graduate school as coming out of a clear blue sky, is sitting right there in the Lie algebra of E8. Neat, eh?

Anyway, this isn't exactly the set of generators we want -- we need to break them up into smaller pieces.


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The 120 so(16) generators are reassigned into two sets of 28 so(8) generators and the remaining block of 64. The 128 so(16) spinor breaks up into two blocks of 64. This way, we get the generators we wanted, corresponding to our H, G, and three Psi's.

Next we use the structure constants between the old generators to calculate the new ones.


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These are the E8 Lie brackets between elements of H, G, and Psi. The brackets there on the right won't matter much -- it's the ones on the left we're really interested in. And, sure enough, they turn out to be as John Baez described, with the H's acting on the three Psi's from the left, and the G's acting on the Psi's from the right. With each acting in a different representation. 

This is what we had thought was too good to hope for. It matches the algebra of the blocks in the big matrix, but with three generations. We can use this to build a theory of everything that's just a Yang-Mills theory with E8 as the gauge group.

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We don't want to use the compact real form of E8, since the Lorentz subgroup isn't compact. (Although it might be fun to play with the compact real form for practice.) It's easy enough to build non-compact forms of E8 -- all we have to do is change the signatures or representations of the Clifford algebra used in the E8 structure constants.

Using this, we can match up with the so(1,7)'s we used in our big matrix -- with the H and the G acting from the left and right on the blocks of three generations of fermions, all determined by the Lie algebra of E8.

This is exactly what we would hope for in a theory of everything, and it's a purely geometric theory. A Yang-Mills gauge theory like this, which mathematicians call a principal bundle, is described as a big manifold with vector fields on it -- and that's it. The algebraic structure -- the metric, the spinors, the Lie algebra operators, everything -- all comes from the Lie derivatives between these vector fields on a big manifold. And E8 is regarded as one of the prettiest Lie group manifolds around. And, right there, sitting in the structure of E8, is the entire standard model.

OK, so, is it perfect right now, as I've written it? No, not quite. The H and G do act on one of those Psi's correctly, exactly as they do in the big matrix. But they act on the other two Psi's in a slightly different representation. So the different generations of fermions are going to need to get mixed up in a special way if they're going to be eigenvectors of the standard model gauge fields. This isn't bad, and in fact it's exactly how it's going to have to work in order to get the CKM mass mixing matrices for the fermions. But I'm not sure precisely how it's going to work just yet. I plan to spend the next year working on it. And, of course, beautiful as this E8 theory is, nature may just turn out not to work this way.

OK, so what did I really just do in the last twenty minutes, and what does this theory mean for the Loop Quantum Gravity community?


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I showed that all the standard model fields and gravity may be described by a single E8 connection, with three generations of fermions emerging as gauge ghosts.

It's looking good, but the fermion assignments aren't working perfectly -- I still have work to do. If things go well, I might get the fermion masses. Or, it might just be wrong.

And there are lots of things I don't know. I don't know why the action is what it is. And I don't know how the full E8 symmetry gets broken down to the standard model. It would also be nice to get some hints of where quantum mechanics comes from.

But, even at its current stage, there are lots of things this theory implies for the Quantum Gravity community.

The theory favors the MacDowell-Mansouri description of gravity as a modified BF theory, so I hope people continue with that approach. Also, this E8 theory has an intimately intertwined gravitational frame and Higgs scalar multiplet, which may be an important model building tool. Also, all I'm offering is a connection -- in this theory, it's still up in the air why a 4D base manifold emerges. I'm sure some people here have much more to say about that than I do.

For the big picture... You guys are already adept at building quantized, background independent models with a connection -- doing this for E8 is just a small matter of increasing the scale a bit. :) If this works, Loop Quantum Gravity with E8 could very well be a complete and fully successful theory of everything.

That's about all I have to say -- but I'm sure there are questions.