1 An Exceptionally Simple Theory of Everything Thank you ... for the introduction. And I'd like to thank Sabine for inviting me to visit this wonderful nerd paradise -- I've been greatly enjoying my time here. 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 accordingly. The electroweak B is a u(1) interacting with the fermions and Higgs according to their weak hypercharges. The gravitational spin connection, om, is an so(3,1) interacting with the left and right chiral fermions and with the gravitational frame. The frame and Higgs, e and phi, interact with the left and right-chiral parts of the fermions, giving them Dirac masses. And this structure is repeated over three generations of fermions. This collection of fields and interactions is pretty complicated. It's not at all clear that it's the structure of just one mathematical object, which is what we want in a theory of everything. But today it is my pleasure to show you a uniquely beautiful mathematical object that gives precisely this structure. We will proceed by starting with the best, most straightforward unification we could hope for. 2 Everything as a principal bundle connection We put every field we know of into one connection, 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 gauge group. We're proceeding in the same spirit, but going much further by using two unusual tricks. First, we're including gravity -- the connection and the frame -- as parts of this connection. This works to reproduce general relativity and is called the MacDowell-Mansouri approach to gravity, discovered in the late seventies, which I first learned about in Laurent and Lee's papers. The second trick is that we're also including all the fermions in this connection, 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, like this Dirac spinor column of spin-up and down left and right-chiral fields, and these don't look like 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. This is what makes this second trick possible. This unification is very straightforward. There's nothing fancy going on, and it's all going to work beautifully. In order to understand it we'll need to review some basic representation theory. 3 Review of some representation theory The root system of a Lie algebra is a way of understanding its structure independent of the choice of generators or representation. 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 any of our choices for generators, which correspond to rotations of the root system. We can also use our Cartan subalgebra to get eigenvectors and eigenvalues, called weight vectors and weights, corresponding to the Lie algebra action on any representation space. The roots are just the weights in the adjoint representation. Now, all of this connects to physics because these weight vectors, algebraicly, are particle states, and their weights are their quantum numbers. This gets a lot clearer with an example. 4 Gluon and quark weights The gluons live in su(3), shown here with the eight Gell-Mann matrices as generators. The Cartan subalgebra, here on the diagonal, 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 eigen-equation gives its root coordinates, minus one-half and square root of three over two. The gluons act on the quarks in the fundamenal three representation space, with a red quark represented by this canonical unit vector, with weight coordinate 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. 5 Strong G2 These are the six gluon root vectors and roots of su(3), which I've plotted as blue circles, and the three quark and three anti-quark weights of three colors, which I've plotted as triangles and inverted triangles. The interactions, such as between this red-anti-green gluon and green quark, correspond to Lie brackets, which correspond to addition of the weights -- in this case adding this gluon root to this quark weight gives a red quark. This is all standard and well known. But there's something unusual here. This weight system is not just a weight system -- it's a root system! It's the root system for the smallest simple exceptional group, G2. We can take the green quark and the red-anti-green gluon to be the simple roots, and build the Dynkin diagram for this root system, corresponding to G2. So we don't have to think of quarks as being in the fundamental three representation, we can treat them as Lie algebra elements of G2. 6 Exceptional Lie brackets The G2 Lie algebra breaks up into a su(3) subalgebra, identified with 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 just work for the quarks and gluons in G2 -- it's going to work like this for everything. 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. 7 G2 in SO(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 a 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 roots of G2 -- a subalgebra of so(7) -- and the leptons 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 you can see better if I spin this cube for you. Here you can see there's a "nine grading" of this root system along the B2 direction. The B2 coordinate for these weights -- you can see the plus-minus one-third and one here -- is related to the weak hypercharge for quarks and leptons. 8 Pati-Salam model plus gravity If go back to our periodic table, we can see the B2 coordinates here as the weak hypercharges for the left-chiral leptons and quarks. In order to get the correct hypercharges for all fermions we need another U(1) field, part of B1, that acts to add and subtract one to the weak hypercharge of right chiral fermions. Then, 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-Slam model is a left-right symmetric theory. It has an SU(2)R partner, B1, to the SU(2)L 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 need to get large masses somehow, breaking this left-right symmetry, because we haven't seen them. So the Pati-Salam model (which is part of the structure I'm working towards) 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 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. 9 Gravitational SO(3,1) The so(3,1) valued spin connection, with generators written here as Clifford algebra bivectors, can be broken up into these left and right chiral parts, built with Pauli matrices. 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, and these root coordinates. The gravitational frame is a Clifford algebra vector, and breaks into these weight vectors, with these roots. 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 these quantum numbers. When the frame interacts with the fermions it mixes up the left and right-chiral parts, which we can see by multiplying this matrix times the Dirac spinor, or by adding their weights. Such as this spatial-up frame field interacting with a left-chiral-down fermion to give a right-chiral-up fermion. The structure of the electroweak interactions is very similar. 10 Electroweak SU(2) and U(1) We can identify the left-chiral acting su(2), W, and right-chiral acting su(2), 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, we get the correct standard model hypercharges if we insist that the electroweak U(1) coupling constant equals the square root of three fifths. This gives the same prediction for the Weinberg angle as almost all grand unified theories -- it comes from that cube rotation I showed you. The electric charge quantum numbers for all the fields comes from adding W^3 with half Y. 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 an SO(7,1) graviweak connection. 11 Graviweak SO(7,1) This is our combined, SO(7,1) graviweak connection, with the frame and Higgs combined as a simple bivector element. We can pick a chiral Clifford matrix representation for Cl(7,1) and use that 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 these four dimensional roots. 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 -- the same triality rotation of these roots gives us eight new roots, 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 rank four exceptional group, F4. 12 Graviweak F4 Here's a particular triality rotation that permutes three of our coordinates. Applying it three times gets us back where we started. Applying this triality rotation to our positive-chiral eight-spinor gives us the triality-equivalent eight vector, and applying our triality rotation to that gives us the negative-chiral eight-spinor. Since these new fields have the same quantum numbers under this triality symmetry, it seems natural to label them as the three generations of fermions -- with smaller triangles as their symbols. This is a tentative assignment, and I suspect the physical fermions will be a linear superposition of these -- I need to understand this better. We can plot the 48 roots of the F4 root system, rotated and projected down to two dimensions. 13 F4 root system This shows the gravi-weak 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. The W+ here interacts with the spin-up left-chiral electron to give a spin-up left-chiral electron-neutrino. We're limited though, because there are no anti-fermions in this root system, and no quarks or gluons. To get the whole picture, we have to combine F4 and G2. 14 F4 and G2 together If we fit F4 and G2 together they give the root system of what is perhaps the most mathematically and aesthetically beautiful Lie group -- the largest simple exceptional group, E8. Physically, F4 has the quantum numbers for the graviweak interactions, and the three generation structure, while G2 has the strong interactions, a hypercharge piece, and anti-particles. Combining these gives... everything. In order to match them up, we need to introduce a new quantum number, w, which is determined by the values of the others. Knowing the quantum numbers of the standard model allows us to identify each elementary particle as one of the roots of E8. If we start with E8, we can break its Lie algebra down into the subalgebras of the standard model. First it breaks up into f4 and g2, which operate on their smallest irreducible representations: the 26 is the traceless exceptional Jordan algebra and the smallest irreducible representation space of f4, and the 7 is the smallest irreducible representation of g2. These break up further to give the algebras of the standard model gauge fields, gravity, three generations of fermions, and a handful of new fields. The fit of the standard model and gravity to the 240 roots of E8, is perfect. And we can plot it. 15 E8 root system This is a projection of the E8 root system from eight dimensions down to two, with each root labeled by an elementary particle symbol. We can see every interaction in the standard model by adding these roots together. That's kind of hard to do in this plot though, which shows the roots of E8 spread out into eight concentric circles, as drawn by Gosset by hand in 1900. To see the standard model structure, we'll want to rotate our viewing plane in eight dimensions until we can see the F4 and G2 subalgebras. This is rotating until we can see the F4 subalgebra of E8, with lines drawn between triality partners. Now I'll rotate the viewing plane so we can see the G2 root system, which is orthogonal to the F4 system in E8. The gluons emerge here from the center, and the quarks and anti quarks spread out from the leptons. Now we continue rotating from the plane of F4 to that of G2, and we'll see the 27 elements of the exceptional Jordan algebra converge for each color and anti-color. Here's the G2 subalgebra in E8. Now back to where we started. We couldn't of asked for a better fit of the standard model and gravity into a Lie algebra. We can assemble the periodic table from E8, and compare it to the first slide. 16 E8 periodic table The match of E8 to the standard model is perfect, with the addition of only a few new fields, which I've labeled here as w and x-Phi. These new fields are a lot like the gravitational frame and Higgs fields, but the x interacts differently with different generations, and the Phi has color, and the w is a new U(1) field that interacts differently with different generations. We haven't seen these fields in nature yet, but I don't think they're bad, since we need a new set of Higgs fields like this in order to get the CKM matrix to come out. I'm not sure it's going to work -- it may end up just being wrong -- but it certainly feels promising. This periodic table of the structure of E8 shows how the bosons split into this SO(7,1) or D4 collection of graviweak fields, and this other collection of D4 fields that includes the gluons and the new stuff. These two sets of fields act on the fermions as a collection of three eight by eight matrices, producing exactly the dynamics we want for the standard model. Let me turn away from the pretty pictures and diagrams and get into the nitty gritty of the dynamics, which is also very pretty in its own way. 17 E8 connection The E8 connection breaks into these H1 and H2 D4 parts, which consist of the graviweak fields, the strong fields, and these new ones. The new fields have three generations, color, and anti-color. The H1 and H2 interact with these three generations of fermions, related by triality. When we compute the curvature of this connection it will break up into these same algebraic parts. 18 E8 curvature The calculation of the curvature is straightforward. The SO(3,1) part is the Riemann curvature plus this area term, with this phi the amplitude of the Higgs. In the mixed electrweak part we get non-dynamical torsion coupled to our Higgs, and the covariant derivative of the Higgs. And we get the curvatures of the electroweak gauge fields. The su(3) part has the curvature for the gluons, and the rest of this is the curvature of these new fields I don't completely understand yet. The coolest thing that happens, due to the exceptional structure of E8, is that we get the bosons acting on the fermions as left and right multiplication of matrices in the fundamental representation space. This gives us the correct covariant Dirac operator on the fermions in curved spacetime. Using this curvature we can write the action for everything in a very nice form. 19 Action for everything The most natural way to express this is as a modified BF action, introducing this two form plus anti-grassman three form field as a kind of momentum. In this theory we only have to introduce these two terms in order to recover the standard model and gravity. Varying this new B field and plugging it back in gives us the correct Dirac action, the Einstein-Hilbert action for gravity, and the action for the rest of the bosons. #[[Gravitational part of the action]] #[[Fermionic part of the action]] #[[E.S.T.o.E. summary]] #[[E.S.T.o.E. discussion]]