Unless you don't want it, may I reproduce the transcript here for our convenience?
garrett said:
==quote from G.L. website, transcript of talk==
Thank you, Jorge, for the introduction.
1.
OK, this is a periodic table of the standard model and gravity -- it's a schematic diagram of all the elementary fields we know of and the algebraic relationships between them.
The gluons are su(3) valued 1-forms interacting with the red green and blue quarks. The electroweak W (which I've colored yellow) is an su(2) interacting with the Higgs and left-chiral fermion doublets, which I've colored yellowish. The electroweak B is a u(1) interacting with the fermions and Higgs according to their weak hypercharges. The gravitational spin connection, omega, is an so(3,1) interacting with the left and right chiral fermions, and with the gravitational frame, e. The frame, and Higgs, phi, interact with the left and right-chiral parts of the fermions, giving them Dirac masses. And this structure is repeated over the three generations of fermions.
This large collection of fields and interactions is pretty complicated. It sure doesn't look like the structure of just one mathematical object, which is what we want in a theory of everything. But today I will describe a uniquely beautiful mathematical object with precisely this structure, and further details are available in this conservatively titled paper.
We will proceed by starting with the most straightforward unification we could hope for.
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2.
We join all of these fields as parts of one superconnection, over a four dimensional base manifold. This general idea should be familiar from grand unified theories, which combine the gauge fields into a single, larger connection. We're proceeding in the same spirit, but going further by using two unusual tricks. First, we're including gravity -- the connection AND the frame -- as parts of this connection. This reproduces general relativity through the MacDowell-Mansouri approach to gravity, discovered in the late seventies, which I first learned about in Smolin, Freidel, and Starodubtsev's quantum gravity papers. The second trick is that we're also including all the fermions in this superconnection, as Lie algebra valued Grassmann numbers. Now, at first look, this second trick shouldn't work. When we calculate the dynamics of this connection by taking its curvature, the interactions between fields will come from their Lie bracket. But we know gravity and the gauge fields interact with the fermions in fundamental representations. The fermions, such as this Dirac spinor column of spin up and spin down left and right chiral fields, live in a fundamental representation space, and these certainly don't appear to be Lie algebra elements. So how can this possibly work? Well, it turns out that for all five exceptional Lie groups, there are Lie brackets that act like the fundamental action. The structure of these algebras is such that some Lie algebra elements ARE fundamental representation space elements. This fact makes it possible to include the fermions in the connection as Lie algebra valued fields.
Using these two tricks, the unification is very straightforward. But in order to understand it we'll need to review some representation theory.
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3.
The root system of a Lie algebra is a way of describing its structure, independent of the choice of generators. We can start with whatever representation we want, and pick R generators that mutually commute. The R dimensional space, built using these generators, is called a Cartan subalgebra. We get a linear operator on the rest of the Lie algebra by putting the Cartan subalgebra elements in one side of the Lie bracket. This gives a set of eigenvectors, called root vectors, spanning the Lie algebra -- each with a unique eigenvalue, called a root. The root coordinates determine points, the root system, in R dimensional space. Now, the cool thing that happens here is that the Lie bracket between two root vectors gives a third only if their corresponding roots add to give the third. In this way, the pattern of the root system in R dimensions (which can be quite pretty) corresponds to the structure of the Lie algebra -- and it's independent of our choice of generators, which just corresponds to a rotation of the root system.
We can also use our Cartan subalgebra to calculate eigenvectors and eigenvalues, called weight vectors and weights, corresponding to the Lie algebra action on any representation space. The pattern of weights describes the representation. For example, the roots are just the weights in the adjoint representation.
Now, all of this connects to physics because these weight vectors, algebraicly, are the particle states, and their weights are their quantum numbers. This gets a lot clearer with an example.
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4.
The gluons live in su(3), shown here with the eight Gell-Mann matrices as generators. The Cartan subalgebra, on the diagonal of the matrix, is two dimensional -- producing a root space with coordinates along the g3 and g8 axes. Here's a typical root vector, the green anti-blue gluon, and the eigenequation gives its root coordinates, minus one-half and square root of three over two. These gluons act on the quarks in the fundamenal three representation space, with a red quark represented by this canonical unit vector, with weight coordinates of one half and one over two root three. The anti-quarks are in the dual representation, with the opposite weights.
Plotting these weights for the gluons and quarks makes a nice pattern.
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5.
These are the six gluon root vectors and roots of su(3), which are plotted as blue circles, and the three quark and three anti-quark weights of three colors, which are plotted as triangles and inverted triangles.
The interactions correspond to Lie brackets, which correspond to vector addition of the weights in this pattern. If we take the red-anti-green gluon there on the far right, and add it (as a vector) to the green quark, we get a red quark. This is the result of a typical quark-gluon interaction.
Now, this is all standard representation theory. But there's something unusual here. This weight system is not just a weight system -- it's also a root system. It's the root system for the smallest simple exceptional Lie group, G2. We can take the green quark and the red-anti-green gluon to be the simple roots, and build this Dynkin diagram for this root system, corresponding to G2.
So this diagram shows that we don't have to think of quarks as being in the fundamental three representation space, we can treat them as Lie algebra elements of G2.
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6.
The G2 Lie algebra breaks up into an su(3) subalgebra, which are the gluons, and the three and dual three, identified as quarks and anti-quarks. The Lie bracket between gluons and quarks, as root vectors of g2, gives all the interactions -- the Feynman vertices for these particles.
This is a remarkable development -- it implies that use of a particular representation for these elementary particles is not prescribed by nature, as we previously thought. Instead, the quarks and gluons live in the G2 Lie group, a nice symmetric manifold with a geometry described by the relations between its diffeomorphisms. And this doesn't only work for the quarks and gluons in G2 -- it's going to work like this for all other particles in the standard model, which may be identified as Lie algebra elements of the other exceptional Lie groups.
But first I want to show you how we can describe Lie algebra root systems as projections of larger root systems, so we can spot subalgebras.
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7.
The root system of SO(7) can be described as all edge and face midpoints of a cube in three dimensions, here in coordinates x, y, and z. The eight weights of an so(7) spinor are at the vertices of a cube half this size. By rotating to new coordinates, g3 g8 and B2, and projecting along the B2 axis, all of these weights project to the roots of G2 -- a subalgebra of so(7) -- and two project to the origin. This is a graphical representation of the well know embedding of su(n) in so(2n), and is a typical example of how to see a group as a subgroup of a larger one.
These overlapping G2 roots exist at different levels of B2, which we can see by spinning the cube I just described.
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After rotating this cube, we can see there's a grading of this root system along the B2 direction. The B2 coordinate for these weights is related to the weak hypercharge for quarks and leptons.
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12.
If we go back to our periodic table, we can see the B2 coordinates are the weak hypercharges for the left-chiral leptons and quarks, along the top two rows of the table. In order to get the correct hypercharges for all the fermions, we need another U(1) field, part of B1, that acts to add and subtract one to the weak hypercharges of right chiral fermions, along the bottom two rows of the table. By adding these two B's to get our electroweak U(1) field, we get the correct hypercharges. This is actually an old idea -- it's part of the Pati-Salam grand unified theory.
The Pati-Salam model is a left-right symmetric theory. It adds a right SU(2) partner, B1, to the left SU(2) electroweak W. The third Pauli matrix in this su(2)R is the B13 field that combines with B2. The other two fields in B1 must get large masses somehow, breaking this left-right symmetry, because we haven't seen them.
So, the Pati-Salam model implies the hypercharge comes from two independent sets of quantum numbers, B13 and B2. Once we've made this split, we can see from this pattern of interactions that the electroweak, Higgs, and gravitational fields on the left hand side of this diagram are parts of some larger algebra. The biggest clue is that if the Higgs is going to give Dirac masses to the fermions, it's going to need to interact with their left and right chiral parts. Combining the Higgs with the gravitational frame will do this for us. Let me talk about gravity a bit so we can see how this works.
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13.
The so(3,1) valued spin connection, with generators written here as Clifford algebra bivectors, can be broken up into left and right chiral parts, built with Pauli matrices -- more familiar as the self-dual and anti-self-dual parts of the connection. The Cartan subalgebra is the diagonal of this four by four matrix, and it gives up and down root vectors for the left and right chiral spin connection fields, with corresponding root coordinates as shown in the table. The gravitational frame is a Clifford algebra vector, and breaks up into these weight vectors, with the weights shown in the table.
When we write out the Dirac operator in curved spacetime, the spin connection acts on the fermions in the fundamental four representation space -- they're Dirac spinors, built from stacking two Vile spinors, with spin-up and spin-down parts as the weight vectors, and the spin quantum numbers shown in the table.
When the frame interacts with the fermions it mixes the left and right-chiral parts, which we can see by multiplying the frame matrix times the Dirac spinor, or by adding their weights in the table. For example, we could add the weights of the spatial-up frame field to those of a left-chiral-down fermion to get a right-chiral-up fermion, the result of this interaction.
The structure of the electroweak interactions is very similar.
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14.
We can identify the left-chiral acting su(2), the W, and right-chiral acting su(2), the B1, of the Pati-Salam model as parts of an SO(4) electroweak connection. If we put the four Higgs fields in the vector of this SO(4) we get the correct weights for the standard model Higgs. And the Pati-Salam model gives us the correct weights for the fermions.
When we rotate the B2 coordinates for the fermions in with the B13 coordinate (the equation there at the bottom), we get the correct standard model hypercharges, with the electroweak, U(1), coupling constant equal to the square root of three fifths. This gives the same prediction for the Weinberg angle as almost all grand unified theories. The electric charge quantum numbers come from adding W3 with half Y, and are shown in the table.
Since we have gravity described by the SO(3,1) spin connection acting on the frame as a vector, and we have the electroweak connection as an SO(4) acting on the four Higgs in a vector, it's very natural to combine all these as parts of a unified, SO(7,1) graviweak connection.
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15.
This is the combined, SO(7,1) graviweak connection, with the frame and Higgs combined as a simple bivector. This is a very unusual feature, the frame and Higgs are parts of one composite field in this theory, with this frame-Higgs field occupying sixteen Lie algebra elements.
We can pick a chiral Clifford matrix representation for Cl(7,1) and use that (there on the bottom) to calculate the roots for the SO(7,1) fields and the weights for the fermions, here as elements of the positive-chiral eight dimensional spinor. But that's the hard way to do it. The easy way is to just combine the two pairs of coordinates we already found to get the roots in four dimensions, as shown in the table.
Now, these 24 roots of the D4 root system that we end up with have a nice set of symmetries called triality. There are many planes through this root system such that if we rotate in one of these planes by two pi over three, it gives us the same set of roots. But this doesn't work if we include this eight-spinor at the bottom of the table -- the same triality rotation of these weights gives us eight new weights, and another application of the same rotation gives eight more. But if we include these two sets of triality partners in a larger group, we do have a root system with triality -- it's the root system of the rank four exceptional Lie group, F4.
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16.
This particular triality rotation permutes three of the F4 coordinates. Applying it three times to any root gets us back where we started. Applying this triality rotation to the positive-chiral eight-spinor gives the triality-equivalent eight vector, and applying this triality rotation to that gives the negative-chiral eight-spinor. Since these new fields have the same quantum numbers under this triality symmetry, we label them as the three generations of fermions -- with smaller triangles for their their symbols in the tables and plots. This assignment of particles to triality partners is tentative, and I expect the three physical generations of fermions to have some more complicated relationship to these triality partners.
But for now, we can plot the 48 roots of the F4 root system, rotated and projected down to two dimensions.
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17.
This shows the graviweak gauge fields and three generations of leptons, with lines connecting the triality partners. Since projection is a linear operation, we can still compute interactions by adding the roots visually. For example, the yellow W+, on the far right, interacts with any of the left-chiral electrons (the yellow triangles on the left) to give left-chiral electron-neutrinos (the darker yellow triangles on the right).
We're limited in what we can do in this diagram, because there are no anti-fermions here, and no quarks or gluons. To get the whole picture, we have to combine F4 and G2.
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