1
Hello, today I will be describing how to combine (unify?) all fields of the standard model and gravity in a purely geometric Yang-Mills theory with a single connection.

(Hello, today I will be describing how to unify all fields of the standard model and gravity as the connection of a purely geometric Yang-Mills theory.

This is a periodic table of the standard model.)
Here I've drawn a periodic table of the standard model.

It has a collection of Lie algebra valued connection 1-forms describing the electroweak and strong fields, 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 -- which is kind of strange since I just described them as living in different algebras. But these three different algebras are in fact 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. Using this fact, we can combine all these fields as parts of a single connection.


2
The two tricks that make this possible were invented back in the sixties and seventies. I'm a very conservative guy, and everything I talk about today could have been done thirty years ago. The framework is just that of Yang-Mills theory -- a non-abelian gauge theory described by a Lie algebra valued connection field over a four dimensional base manifold.

When Faddeev and Popov were working on quantizing Yang-Mills theory they invented the trick of introducing Grassmann valued fields to properly account for the purely gauge parts of a connection. These "ghost" fields take the place of the gauge degrees of freedom of the connection 1-form, and a "ghost extended" connection can be written and its curvature calculated to get the dynamics. This gauge fixing technique, using ghosts, is essential in modern quantum field theory.

A less well known trick was discovered by MacDowell and Monsouri in 1977. They found that the fields and dynamics of gravity could be described purely in terms of a connection -- by combining the spin connection and frame in an so(1,4) connection, computing its curvature, and writing an action for this connection equivalent to the Einstein-Hilbert action.

These two tricks gave me a couple of ideas. In 2002 I realized one could cook up a Lie algebra such that the gauge ghosts would be mathematically equivalent to the physical fermions -- providing a way to include the fermions as parts of a ghost extended connection. In 2005 I learned about the MacDowell-Mansouri trick and discovered that gravity could also be included as part of this large connection by combining it with a Higgs multiplet. In this way, a connection can be built that describes everything in the standard model and gravity.

Since we're used to seeing the algebraic elements of the standard model written out in terms of a matrix representation, I'll go ahead and write this connection for you as a big matrix of 1-forms and Grassmann numbers.


3
All the fields of the standard model and gravity are in this sixteen by sixteen matrix. The matrix breaks down into blocks, H, G, and Psi, with the Psi block consisting of Grassmann valued gauge ghosts -- giving one generation of fermions as shown here.

All interactions come from the curvature of this connection. Basically, we square this 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. The strong su(3) field, represented by a Gell-Mann matrix block, acts from the right and mixes 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 and electroweak bosons, as in the standard model, giving them their masses.

This matrix really is a nice thing -- I think it would look good on a T-shirt. :) Now let me show how it's built, piece by piece, starting with the gravitational part.


4
We use the four by four, chiral representation of the four spacetime Dirac matrices, built using Pauli matrices. These give 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. When we use this 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.

What we do next is use larger Dirac matrices to combine gravity, Higgs, and electroweak fields in an so(1,7) connection.


5
Using a chiral representation for an eight dimensional Clifford algebra, the twenty-eight Clifford bivector generators of so(1,7) may be represented by eight-by-eight sub-matrices. These bivectors split into three different categories.

The spin connection is built from the six spacetime bivectors. The electroweak fields are built from the higher bivectors. And the frame and Higgs multiply the remaining mixed bivectors.

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. 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 fields add to give the bosonic part of our connection. The dynamics come from computing its curvature.


6
The curvature also breaks up into the three categories of bivectors: The spacetime part of the curvature includes the Riemann curvature 2-form plus an area 2-form. The mixed 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.

The action for this connection is best written as a modified BF theory action over a four dimensional base manifold. I pick this action because it agrees with the standard model -- I don't yet know why it is what it is. But it gives the familiar quadratic action terms for the gauge fields and for the Higgs. It will take some convincing that the remaining term, quadratic in the spacetime curvature, is the Einstein-Hilbert action.

 
7
The gravitational part of the action is the same as for MacDowell-Mansouri gravity. It gives three terms: The term quadratic in the Riemann curvature is the Chern-Simons boundary term, which we toss out. The frame term is the volume element. And the last term gives the curvature scalar.

So we see that this is equivalent to the Einstein-Hilbert action, and the cosmological constant relates directly to the Higgs normalization constant -- which serves as its vacuum expectation value.

The only thing left is to write down the action for the whole connection, and confirm that it gives the Dirac action for the fermions.

8
The action for everything turns out to be the standard Faddeev-Popov effective action after including ghosts and their anti-ghosts. When we plug in the bosonic parts of the connection and write things out in components this gives the correct Dirac action in curved spacetime. The "miracle" that happens here is that the inverse-frame in the Dirac action contracts with the frame and the correct Higgs-fermion interaction pops out. That's very nice!

So, yes, the dynamics all work, but where does this big Lie algebra come from?


9
This matrix has an unusual block structure, with H and G multiplying Psi from the left and right. Also, there is only one generation of fermions here -- we need to find the other two. Over the past year I tried everything I could think of to get two more generations of fermions. Nothing seemed to work. And then, two months ago, I tried the obvious thing. If this big matrix, but with two more generations, is the representation of some Lie algebra, then it ought to correspond to some very large Lie group. This block structure didn't look like any group I was familiar with, but I counted up the number of generators needed -- 248 -- and went looking.


10
This is a complete list of the simple compact Lie groups. I think the first thing I tried was SO(23), which didn't work so well. Then I bumped into E8, the largest exceptional group, which I didn't know much about. People seemed to think it was pretty, and it had the right number of generators. But I had no reason to think it would have the structure I was looking for -- it seemed like a long shot.

But it does!


11
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.

This structure of E8 means the strange looking matrix I had built -- with H, G, and Psi blocks -- is actually part of a representation of E8. And the other E8 generators can be assigned to the remaining two generations of fermions to give a complete description of the standard model and gravity as an E8 connection.  


13
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 did 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 CKMPMNS mixing matrices for the fermions. But I'm not sure precisely how it's going to turn out. I plan to spend the next year working on it. And, of course, beautiful as this E8 theory is, nature may just decide to not work this way.

But I want to talk about just how nice this picture will be if it does work.

14
Mathematicians call a Yang-Mills field a principal bundle connection. In fact, the entire subject of differential geometry is mostly about principal bundles -- so this is a very rich area, mathematically. And the modern geometric description of principal bundles is quite nice. A Lie group is understood geometrically as a large manifold with vector fields on it. These vector fields correspond to the Lie algebra generators, and their Lie brackets give the structure constants of the group. For E8, the Lie group manifold is a very beautiful 248 dimensional shape. For a principal bundle, the connection and its curvature describes how the Lie group twists around the base manifold.

The interesting thing here is that all the algebra in the standard model and gravity emerges from the pure geometry of this shape. And there's a well known way of describing the shape mathematically.


15
The structure of a Lie group is described by its root system. The roots are eigenvalues of the Cartan subalgebra. The Cartan subalgebra is spanned by a maximal set of commuting generators, acting on the rest of the Lie algebra through the Lie bracket. The dimension of the Cartan subalgebra is the "rank" of the group -- 8 in the case of E8 -- and the roots are points in this eight dimensional space.

For the standard model, the six generators of the Cartan subalgebra live in the diagonal part of the big matrix. Every particle then corresponds to an eigenvector of these generators, and the eigenvalues -- the roots -- are the charges of the particles. These charges are Cartesian coordinates in six dimensional space -- the charges of a left chiral electron are shown here as an example.

For E8, each of the 240 roots corresponds to a point in eight dimensional space. Connecting these roots to their nearest neighbors gives a beautiful polytope.


14
These roots of E8 can be projected down to two dimensions to make some pretty pictures. You can find this picture in Wikipedia -- here a particular projection was chosen to make the roots spread out in a nice symmetric way in two dimensions. To find the standard model in here, what we want to do is find a projection from eight dimensions down to six such that the coordinates of the points match up with the charges of the standard model. We can try to get a feel for this by looking at the projections into two dimensions. All we have to do is spin this set of points in eight dimensions and look for pretty patterns.

So lets go ahead and spin this thing. You won't find this in Wikipedia yet.

When a pattern pops out -- and there are many -- it means something about E8, its subgroups, and its representations. I'm not an expert at this yet -- I just put this together last week, but I think this nice pattern here has to do with E7 as a subgroup. But it's not the projection we want for the standard model, since you can see these roots overlap in sets of two, four, eight, etc -- when what we want is triplets of points corresponding to three sets of particles having the same charges. So lets rotate around more and see what we can find.

When points line up in parallel planes like that it corresponds to a "grading" of the algebra -- I'm not sure what the physical significance of that is yet.

Ah. Here we go. This pattern is very pretty -- and you can see it has the tripling we're looking for. Also, this hexagonal pattern is good since it corresponds to the root system of su(3). I'm pretty sure this pattern relates to E6 as a subgroup of E8. So let me talk about that.


15
Here are some nice ways the E8 Lie algebra can be broken up into subalgebras corresponding to the algebraic elements of the standard model. I'm just becoming familiar with these, so there's not much I can say about them, but I'll be getting more familiar with these in the coming months.


16
So, what did I just do? I showed how all fields of the standard model and gravity can be combined in a single connection that matches a particularly beautiful Lie group. There's still quite a bit of work to do -- I need to find the exact particle assignments, if I can. And the theory needs to be quantized -- probably using the methods of loop quantum gravity. Also, it would be nice to find a description of why the action is what it is, probably from the theory of characteristic classes, and how symmetry breaking happens. And of course it would be nice to get quantum mechanics to drop out as a bonus.

But, if it all works out -- and nature isn't just playing a big joke on me -- then this model may lead to a nice theory of everything, with the universe described as a very pretty shape.

And that's about all I have to say -- but I'm sure you have some questions.