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John Baez

June 27, 2007

This Week's Finds in Mathematical Physics (Week 253)

John Baez

Yay! Classes are over! Soon I'm going to Paris for three weeks, to

talk to Paul-Andre Mellies about logic, games and category theory.

But right now I'm in a vacation mood. So, I want to take a break from

the Tale of Groupoidification, and mention some thoughts prompted by

the work of Garrett Lisi:

1) Garrett Lisi, Deferential Geometry, http://deferentialgeometry.org/

Garrett is a cool dude who likes to ponder physics while living a

low-budget, high-fun lifestyle: hanging out in Hawaii, surfing, and

stuff like that. He recently won a Foundational Questions Institute

award to think about ways to unify particle physics and gravity. This

is an institute devoted precisely to risky endeavors like this.

Lately he's been visiting California. So, before giving a talk at

Loops '07 - a loop quantum gravity conference taking place in Mexico

this week - he stopped by Riverside to explain what he's been up to.

Briefly, he's been trying to explain the 3 generations of elementary

particles using some math called "triality", which is related to the

octonions and the exceptional Lie groups. In fact, he's trying to use

the exceptional Lie group E8 to describe all the particles in the

Standard Model, together with gravity.

I'd like to know if these ideas hold water. So, I should try to

explain them! But as usual, in this Week's Finds I'll wind up

explaining not what Garrett actually did, but what it made me think

about.

For a long time, people have been seeking connections between the

messy pack of particles that populate the Standard Model and

structures that seem beautiful and "inevitable".

A fascinating step in this direction was the SU(5) grand unified

theory proposed in 1975 by Georgi and Glashow. So, I'll start by

summarizing that... and then explain how exceptional Lie groups might

get involved in this game.

What people usually call the gauge group of the Standard Model:

SU(3) x SU(2) x U(1)

actually has a bit of flab in it: there's a normal subgroup that acts

trivially on all known particles. This subgroup is isomorphic to Z/6.

If we mod out by this, we get the "true" gauge group of the Standard

Model:

G = (SU(3) x SU(2) x U(1))/(Z/6)

And, this turns out to have a neat description. It's isomorphic to the

subgroup of SU(5) consisting of matrices like this:

(g 0)

(0 h)

where g is a 3x3 block and h is a 2x2 block. For obvious reasons, I call

this group

S(U(3) x U(2))

If you want some intuition for this, think of the 3x3 block as related

to the strong force, and the 2x2 block as related to the electroweak

force. A 3x3 matrix can mix up the 3 "colors" that quarks come in -

red, green, and blue - and that's what the strong force is all about.

Similarly, a 2x2 matrix can mix up the 2 "isospins" that quarks and

leptons come in - up and down - and that's part of what the electroweak

force is about.

If this isn't enough to make you happy, go back to "week119", where

I reviewed the Standard Model and its relation to the SU(5) grand

unified theory. If even that isn't enough to make you happy, try this:

2) John Baez, Elementary particles,

http://math.ucr.edu/home/baez/qg-spring2003/elementary/

Okay - I'll assume that one way or another, you're happy with the

idea of S(U(3) x U(2)) as the true gauge group of the Standard Model!

Maybe you understand it, maybe you're just willing to nod your head

and accept it.

Now, the fermions of the Standard Model form a very nice representation

of this group. SU(5) has an obvious representation on C^5, via matrix

multiplication. So, it gets a representation on the exterior algebra

Lambda(C^5). If we restrict this from SU(5) to S(U(3) x U(2)), we get

precisely the representation of the true gauge group of the Standard Model

on one generation of fermions and their antiparticles!

This really seems like a miracle when you first see it. All sorts of

weird numbers need to work out exactly right for this trick to succeed.

For example, it's crucial that quarks have charges 2/3 and -1/3, while

leptons have charges 0 and -1. One gets the feeling, pondering this stuff,

that there really is some truth to the SU(5) grand unified theory.

To give you just a little taste of what's going on, let me show you

how the exterior algebra Lambda(C^5) corresponds to one generation of

fermions and their antiparticles. For simplicity I'll use the first

generation, since the other two work just the same:

Lambda^0(C^5) = <left-handed antineutrino>

Lambda^1(C^5) = <right-handed down quark> +

<right-handed positron, right-handed antineutrino>

Lambda^2(C^5) = <left-handed up antiquark> +

<left-handed up quark, left-handed down quark> +

<left-handed positron>

Lambda^3(C^5) = <right-handed electron> +

<right-handed up antiquark, right-handed down antiquark> +

<right-handed up quark>

Lambda^4(C^5) = <left-handed up antiquark> +

<left-handed electron, left-handed neutrino>

Lambda^5(C^5) = <right-handed neutrino>

All the quarks and antiquarks come in 3 colors, which I haven't bothered

to list here. Each space Lambda^p(C^5) is an irreducible representation of

SU(5), but most of these break up into several different irreducible

representations of S(U(3) x U(2)), which are listed as separate rows in

the chart above.

If you're curious how this "breaking up" works, let me explain - it's

sort of pretty. We just use the splitting

C^5 = C^3 + C^2

to chop the spaces Lambda^p(C^5) into pieces.

To see how this works, remember that Lambda^p(C^5) is just the vector

space analogue of the binomial coefficient "5 choose p". A basis of

C^5 consists of 5 things, and the p-element subsets give a basis for

Lambda^p(C^5).

In our application to physics, these 5 things consist of 3 "colors"

- red, green and blue - and 2 "isospins" - up and down. This gives

various possible options.

For example, suppose we want a basis of Lambda^3(C^5). Then we need to

pick 3 things out of 5. We can do this in various ways:

* We can pick 3 colors and no isospins - there's just one way to do that.

* We can pick 2 colors and 1 isospin - there are six ways to do that.

* Or, we can pick 1 color and 2 isospins - there are three ways to do that.

So, in terms of binomial coefficients, we have

(5 choose 3) = (3 choose 3)(2 choose 0) +

(3 choose 2)(2 choose 1) +

(3 choose 1)(2 choose 2)

= 1 + 6 + 3

= 10

In terms of vector spaces we have:

Lambda^3(C^5) = Lambda^3(C^3) tensor Lambda^0(C^2) +

Lambda^2(C^3) tensor Lambda^1(C^2) +

Lambda^1(C^3) tensor Lambda^2(C^2)

Taking dimensions of these vector spaces, we get 10 = 1 + 6 + 3. Finally,

in terms of the SU(5) grand unified theory, we get this:

Lambda^3(C^5) = <right-handed electron> +

<right-handed up antiquark, right-handed down antiquark> +

<right-handed up quark>

If we play this game for all the spaces Lambda^p(C^5), here's what we get:

Lambda^0(C^5) = Lambda^0(C^3) tensor Lambda^0(C^2)

Lambda^1(C^5) = Lambda^1(C^3) tensor Lambda^0(C^2) +

Lambda^0(C^3) tensor Lambda^1(C^2)

Lambda^2(C^5) = Lambda^2(C^3) tensor Lambda^0(C^2) +

Lambda^1(C^3) tensor Lambda^1(C^2) +

Lambda^0(C^3) tensor Lambda^2(C^2)

Lambda^3(C^5) = Lambda^3(C^3) tensor Lambda^0(C^2) +

Lambda^2(C^3) tensor Lambda^1(C^2) +

Lambda^1(C^3) tensor Lambda^2(C^2)

Lambda^4(C^5) = Lambda^3(C^3) tensor Lambda^1(C^2) +

Lambda^2(C^2) tensor Lambda^2(C^2)

Lambda^5(C^5) = Lambda^3(C^3) tensor Lambda^2(C^2)

If we interpret this in terms of physics, we get back our previous chart:

Lambda^0(C^5) = <left-handed antineutrino>

Lambda^1(C^5) = <right-handed down quark> +

<right-handed positron, right-handed antineutrino>

Lambda^2(C^5) = <left-handed up antiquark> +

<left-handed up quark, left-handed down quark> +

<left-handed positron>

Lambda^3(C^5) = <right-handed electron> +

<right-handed up antiquark, right-handed down antiquark> +

<right-handed up quark>

Lambda^4(C^5) = <left-handed up antiquark> +

<left-handed electron, left-handed neutrino>

Lambda^5(C^5) = <right-handed neutrino>

Now, all this is really cool - but in fact, even before inventing the

SU(5) theory, Georgi went a bit further, and unified all the left-handed

fermions above into one irreducible representation of a somewhat bigger

group: Spin(10). This is the double cover of the group SO(10), which

describes rotations in 10 dimensions.

If you look at the chart above, you'll see the left-handed fermions

live in the even grades of the exterior algebra of C^5:

Lambda^{even}(C^5) = Lambda^0(C^5) + Lambda^2(C^5) + Lambda^4(C^5)

This big space forms something called the left-handed Weyl spinor

representation of Spin(10). It's an irreducible representation.

Similarly, the right-handed fermions live in the odd grades:

Lambda^{odd}(C^5) = Lambda^1(C^5) + Lambda^3(C^5) + Lambda^5(C^5)

and this big space forms the right-handed Weyl spinor representation

of Spin(10). It's also irreducible.

I can't resist mentioning that there's also a gadget called the Hodge

star operator that maps Lambda^{even}(C^5) to Lambda^{odd}(C^5), and

vice versa. In terms of physics, this sends left-handed particles

into their right-handed antiparticles, and vice versa!

But if I get into digressions like these, it'll take forever to tackle the

main question: how does this "Weyl spinor" stuff work?

It takes advantage of some very nice general facts. First, C^n is

just another name for R^{2n} equipped with the structure of a complex

vector space. This makes SU(n) into a subgroup of SO(2n). So, it

makes the Lie algebra su(n) into a Lie subalgebra of so(2n).

The group SU(n) acts on the exterior algebra Lambda(C^n). So, its Lie

algebra su(n) also acts on this space. The really cool part is that

this action extends to all of so(2n). This is something you learn

about when you study Clifford algebras, spinors and the like. I don't

know how to explain it without writing down some formulas. So, for

now, please take my word for it!

This business doesn't give a representation of SO(2n) on Lambda(C^n),

but it gives a representation of the double cover, Spin(2n). This is

called the "Dirac spinor" representation. It breaks up into two

irreducible parts:

Lambda(C^n) = Lambda^{even}(C^n) + Lambda^{odd}(C^n)

and these are called the left- and right-handed "Weyl spinor"

representations.

Perhaps it's time for an executive summary of what I've said so

far:

The Dirac spinor representation of Spin(10) neatly encodes everything

about how one generation of fermions interacts with the gauge bosons

in the Standard Model, as long as we remember how S(U(2) x U(3)) sits

inside SO(10), which is double covered by Spin(10).

Of course, there's more to the Standard Model than this. There's also

the Higgs boson, which spontaneously breaks electroweak symmetry and

gives the fermions their masses. And, if we want to use this same

trick to break the symmetry from Spin(10) down to S(U(3) x U(2)), we'd

need to introduce *more* Higgs bosons. This is the ugly part of the

story, it seems. Since I'm on vacation, I'll avoid it for now.

Next: how might exceptional Lie groups get involved in this game?

When Cartan classified compact simple Lie groups, he found 3 infinite

families related to rotations in real, complex and quaternionic vector

spaces: the SO(n)'s, SU(n)'s and Sp(n)'s. He also found 5 exceptions,

which have the charming names G2, F4, E6, E7, and E8. These are all

related to the octonions. G2 is just the automorphism group of the

octonions. The other 4 are closely related to each other - thanks to

the "magic square" of Rosenfeld, Freudenthal and Tits.

I talked about the magic square a bit in "week106" and "week145", and

much more here:

3) John Baez, The magic square,

http://math.ucr.edu/home/baez/octonions/node16.html

Instead of repeating all that, let me just summarize. The magic

square gives vector space isomorphisms as follows:

F4 = so(R + O) + (R tensor O)^2

E6 = so(C + O) + (C tensor O)^2 + Im(C)

E7 = so(H + O) + (H tensor O)^2 + Im(H)

E8 = so(O + O) + (O tensor O)^2

Here F4, E6, E7 and E8 stand for the compact real forms of these

Lie algebras. R, C, H, and O are the usual suspects - the real numbers,

complex numbers, quaternions and octonions. For any real inner product

space V, so(V) stands for the Lie algebra of the rotation group of V.

And, for each of the isomorphisms above, we must equip the vector space

on the right side with a cleverly (but not perversely!) defined Lie

bracket to get the Lie algebra on the left side.

Here's another way to say the same thing, which may ring more bells:

F4 = so(9) + S_9

E6 = so(10) + S_{10}^+ + u(1)

E7 = so(12) + S_{12}^+ + su(2)

E8 = so(16) + S_{16}^+

Here S_9 means the unique irreducible real spinor representation of

so(9). In the other 3 cases, the little plus signs mean that there are

two choices of irreducible real spinor representation, and we take the

left-handed choice.

All this must seem like black magic of the foulest sort if you haven't

wasted months thinking about the octonions and exceptional groups! Be

grateful: I did it so you wouldn't have to.

Anyway: the case of E6 should remind you of something! After all, we've

just been talking about so(10) and its left-handed spinor representation.

These describe the gauge bosons and one generation of left-handed fermions

in the Spin(10) grand unified theory. But now we're seeing this stuff

neatly packed into the Lie algebra of E6!

More precisely, the Lie algebra of E6 can be chopped into 3 pieces

in a noncanonical way:

A) so(10),

B) the left-handed real spinor representation of so(10), which by now

we've given three different names:

S_{10}^+ = Lambda^{even}(C^5) = (C tensor O)^2

and

C) a copy of u(1).

The first part contains all the gauge bosons in the SO(10) grand unified

theory. The second contains one generation of left-handed fermions.

But what about the third?

Well, S_{10}^+ is defined to be a real representation of so(10). But,

it just so happens that the action of so(10) preserves a complex

structure on this space. It's just the obvious complex structure on

(C tensor O)^2. So, there's an action of the unit complex numbers,

U(1), on S_{10}^+ which commutes with the action of so(10).

Differentiating this, we get an action of the Lie algebra u(1):

u(1) x S_{10}^+ -> S_{10}^+

And this map gives part of the cleverly defined Lie bracket operation in

E6 = so(10) + S_{10}^+ + u(1)

All this stuff is mysterious, but suggestive. It could be mere

coincidence, or it could be the tip of an iceberg. It's more fun to

assume the latter. So, let me say some more about it...

The copy of u(1) in here:

E6 = so(10) + S_{10}^+ + u(1)

is pretty amusing from a physics viewpoint. It's if besides the gauge

bosons in so(10), there were one extra gauge boson whose sole role is

to describe the fact that the fermions form a *complex* representation

of so(10). This is funny, since one of the naive ideas you sometimes

hear is that you can take the obvious U(1) symmetry every complex Hilbert

space has and "gauge" it to get electromagnetism.

That's not really the right way to understand electromagnetism! There

are lots of different irreducible representations of U(1), corresponding

to different charges, and in physics we should think about *all* of

these, not just the obvious one that we automatically get from any

complex Hilbert space. If we only used the obvious one, all particles

would have charge 1.

But in the Spin(10) grand unified theory, the electromagnetic u(1)

Lie algebra is sitting inside so(10); it's not the u(1) you see above.

The u(1) you see above is the "obvious" one that the spinor

representation S_{10}^+ gets merely from being a complex Hilbert space.

The splitting

E6 = so(10) + S_{10}^+ + u(1)

also hints at a weird unification of bosons and fermions, something

different from supersymmetry. We're seeing E6 as a Z/2-graded Lie

algebra with so(10) + u(1) as its "bosonic" part and S_{10}^+ as its

"fermionic" part. But, this is not a Lie superalgebra, just an ordinary

Lie algebra with a Z/2 grading!

Furthermore, an ordinary Lie algebra with a Z/2 grading is precisely

what we need to build a "symmetric space". This is really cool, since

it explains what I meant by saying that the split of E6 into bosonic

and fermionic parts is "noncanonical". We'll get a space, and each

point in this space will give a different way of splitting E6 as

E6 = so(10) + S_{10}^+ + u(1)

It's also cool because it gives me an excuse to talk about symmetric

spaces... a topic that deserves a whole week of its own!

This gives me an excuse to say a word or two about symmetric spaces...

a topic that deserves a whole week of its own! Symmetric spaces are

the epitome of symmetry. A "homogeneous space" is a manifold with

enough symmetry that any point looks any other. A "symmetric space"

is a homogeneous space with an extra property: the view from any point

in any direction is the same as the view in the opposite direction!

Euclidean spaces and spheres are the most famous examples of symmetric

spaces. If an ant decides to set up residence on a sphere, any point

is just as good any other. And, if sits anywhere and looks in any

direction, the view is the same as the view in the opposite direction.

The symmetric space we get from the above Z/2-graded Lie algebra is

sort of similar, but more exotic: it's the complexified version of the

octonionic projective plane!

But let's start with the basics:

Suppose someone hands you a Lie algebra g with a Lie subalgebra h.

Then you can form the simply-connected Lie group G whose Lie algebra

is g. Sitting inside G, there's a connected Lie group H whose Lie

algebra is h. The space

G/H

is called a "homogeneous space". Such things are studied in Klein

geometry, and I've been talking about them a lot lately.

But now, suppose g is a Z/2-graded Lie algebra. Its even part will be

a Lie subalgebra; call this h. This gives a specially nice sort of

homogeneous space G/H, called a "symmetric space". This is better

than your average homogeneous space.

Why? First of all, for each point p in G/H there's a map from G/H to

itself called "reflection through p", which fixes the point p and acts

as -1 on the tangent space of p. When our point p comes from the identity

element of G, this reflection map corresponds to the Z/2 grading of the

Lie algebra, which fixes the even part and acts as -1 on the odd part.

This is what I meant by saying that in a symmetric space, "the view in

any direction is the same as the view in the opposite direction".

Second, these reflection maps satisfy some nice equations. Write p>q

for the the result of reflecting q through p. Then we have:

p>(p>q) = q

p>p = p

and

p>(q>r) = (p>q) > (p>r)

A set with an operation satisfying these equations is called an

"involutory quandle".

Let me summarize with a few theorems - I hope they're all true, because

I don't know a book containing all this stuff. We can define a "symmetric

space" to be an involutory quandle that's a manifold, where the operation

> is smooth and the reflection map

x |-> p>x

has derivative -1 at p. Any Z/2-graded Lie algebra gives a symmetric

space. Conversely, any symmetric space has a universal cover that's a

symmetric space coming from a Z/2-graded Lie algebra!

Using this correspondence, the Lie algebra E6 with the Z/2-grading I

described gives a symmetric space, roughly:

E6/(Spin(10) x U(1))

But, this guy is a lot better than your average symmetric space!

For starters, it's a "Riemannian symmetric space". This is a symmetric

space with a Riemannian metric that's preserved by all the operations

of reflection through points.

Compact Riemannian symmetric spaces were classified by Cartan, and you

can see the classification here, in a big chart:

4) Riemannian symmetric spaces, Wikipedia,

http://en.wikipedia.org/wiki/Riemannian_symmetric_space

As you'll see, there are 7 infinite families and 12 exceptional cases.

The symmetric space I'm talking about now, namely E6/(Spin(10) x U(1)),

is called EIII - it's the third exceptional case. And, as you can see

from the chart in this article, it's the complexified version of the

octonionic projective plane! For this reason, I sometimes call it

(C tensor O)P^2

In fact, this space is better than your average Riemannian symmetric

space. It's a Kaehler manifold, thanks to that copy of U(1), which

makes each tangent space complex. Moreover, the Kaehler structure is

preserved by all the operations of reflection through points. So,

it's a "hermitian symmetric space".

You're probably drowning under all this terminology unless you already

know this stuff. I guess it's time for another executive summary:

Each point in the complexified octonionic projective plane gives a

different way of splitting the Lie algebra of E6 into a bosonic part

and a fermionic part. The fermionic part is just what we need to

describe one generation of left-handed Standard Model fermions. The

bosonic part is just what we need for the gauge bosons of the Spin(10)

grand unified theory, together with a copy of u(1), which describes

the *complex structure* of the left-handed Standard Model fermions.

Another nice fact is that (C tensor O)P^2 is one of the Grassmannians

for E6. I explained this general notion of "Grassmannian" back in

"week181", and you can see this 16-dimensional one in the list near

the end of that Week.

Even better, if you geometrically quantize this Grassmannian using the

smallest possible symplectic structure, you get the 27-dimensional

representation of E6 on the exceptional Jordan algebra!

So, there's a lot of seriously cool math going on here... but since

the basic idea of relating the Standard model to E6 is only

half-baked, all the ideas I'm mentioning now are at best

quarter-baked. They're mathematically correct, but I can't tell

if they're leading somewhere interesting.

In fact, I would have kept them in the oven longer had not Garrett

Lisi brought E6's big brother E8 into the game in a tantalizing way.

I'll conclude by summarizing this... and you can look at his website for

more details. But first, let me emphasize that this E8 business is the

most recent and most speculative thing Garrett has done. So, if you

think the following idea is nuts, please don't jump to conclusions and

decide *everything* he's doing is nuts!

Briefly, his idea involves taking the description of E8 I already mentioned:

E8 = so(O + O) + (O tensor O)^2

and writing the linear transformations in so(O + O) as two 8x8 blocks

living in so(O), together with an off-diagonal block living in O tensor O.

This gives

E8 = so(O) + so(O) + (O tensor O)^3

Then, he wants to use each of the three copies of O tensor O to

describe one of the three generations of fermions, while using the

so(O) + so(O) stuff to describe bosons (including gravity).

For this, he builds on some earlier work where he sought to combine

gravity, the Standard Model gauge bosons, the Higgs and *one*

generation of Standard Model fermions in an so(8) version of

MacDowell-Mansouri gravity.

If I were really being responsible, I would describe and assess this

earlier work. But, it's summer and I just want to have fun...

In fact, the above alternate description of E8 is the one Bertram

Kostant told me about back in 1996. He said it a different way, which

is equivalent:

E8 = so(8) + so(8) + End(V_8) + End(S_8^+) + End(S_8^-)

Here V_8, S_8^+ and S_8^- are the vector, left-handed spinor, and

right-handed spinor representations of Spin(8). All three are

8-dimensional, and all are related by outer automorphisms of Spin(8).

That's what "triality" is all about. You can see more details in

"week90".

The idea of relating the three generations to triality is cute. Of

course, even if it worked, you'd need something to give the fermions

in different generations different masses - which is what happens

already in the Standard Model, thanks to the Higgs boson. It's the

bane of all post-Standard Model physics: symmetry looks nice, but the

more symmetry your model has, the more symmetries you need to explain

away! As the White Knight said to Alice:

But I was thinking of a plan

To dye one's whiskers green,

And always use so large a fan

That they could not be seen.

Someday we may think of a way around this problem. But for now, I've got

a more pressing worry. This splitting of E6:

E6 = so(10) + S_{10}^+ + u(1)

corresponds to a Z/2-grading where so(10) + u(1) is the "bosonic" or "even"

part and S_{10}^+ is the "fermionic" or "odd" part. This nicely matches the

way so(10) describes gauge bosons and S_{10}^+ describes fermions in Georgi's

grand unified theory. But, this splitting of E8:

E8 = so(8) + so(8) + End(V_8) + End(S_8^+) + End(S_8^-)

does not correspond to any Z/2-grading where so(8) + so(8) is the bosonic

part and End(V) + End(S^+) + End(S^-) is the fermionic part. There is a

closely related Z/2-grading of E8, but it's this:

E8 = so(16) + S_{16}^+

So, right now I don't feel it's mathematically natural to use this method

to combine bosons and fermions.

But, only time will tell.

Here are some more references. The SU(5) grand unified theory was published

here:

5) Howard Georgi and Sheldon Glashow, Unity of all elementary-particle

forces, Phys. Rev. Lett. 32 (1974), 438.

For a great introduction to the Spin(10) grand unified theory - which

is usually called the SO(10) GUT - try this:

6) Anthony Zee, Quantum Field Theory in a Nutshell, Chapter VII: SO(10)

unification, Princeton U. Press, Princeton, 2003.

Then, try these more advanced review articles:

7) Jogesh C. Pati, Proton decay: a must for theory, a challenge for

experiment, available as hep-ph/0005095.

8) Jogesh C. Pati, Probing grand unification through neutrino oscillations,

leptogenesis, and proton decay, available as hep-ph/0305221.

The last two also consider the gauge group "G(224)", meaning SU(2) x SU(2)

x SU(4).

By the way, there's also a cute relation between the SO(10) grand

unified theory and 10-dimensional Calabi-Yau manifolds, discussed here:

9) John Baez, Calabi-Yau manifolds and the Standard Model, available as

hep-th/0511086

This is an easy consequence of the stuff I've explained this week.

To see what string theorists are doing to understand the Standard Model

these days, see the following papers. Amusingly, they *also* use E8 -

but in a quite different way:

10) Volker Braun, Yang-Hui He, Burt A. Ovrut and Tony Pantev,

A heterotic Standard Model, available as hep-th/0501070.

A Standard Model from the E8 x E8 heterotic superstring,

hep-th/0502155.

Vector bundle extensions, sheaf cohomology, and the heterotic

Standard Model, available as hep-th/0505041.

Heterotic Standard Model moduli, available as hep-th/0509051.

The exact MSSM spectrum from string theory, available as

hep-th/0512177.

All this stuff is really cool - but alas, they get the "minimal

supersymmetric Standard Model", or MSSM, which has a lot more

particles than the Standard Model, and a lot more undetermined

parameters. Of course, these flaws could become advantages if the

next big particle accelerator, the Large Hadron Collider, sees

signs of supersymmetry.

For more on symmetric spaces, try these:

11) Sigurdur Helgason, Differential Geometry, Lie Groups, and

Symmetric Spaces, AMS, Providence, Rhode Island, 2001.

12) Audrey Terras, Harmonic Analysis on Symmetric Spaces and Applications

I, Springer, Berlin, 1985. Harmonic Analysis on Symmetric Spaces

and Applications II, Springer, Berlin, 1988.

13) Arthur Besse, Einstein Manifolds, Springer, Berlin, 1986.

They're all classics. Helgason's book will teach you differential

geometry and Lie groups before doing Cartan's classification of symmetric

spaces. Terras' books are full of fun connections to other branches of

math. Besse's book has lots of nice charts, and goes much deeper into

the Riemannian geometry of symmetric spaces.

These dig deeper into the algebraic aspects of symmetric spaces:

14) W. Bertram, The Geometry of Jordan and Lie structures,

Lecture Notes in Mathematics 1754, Springer, Berlin, 2001.

15) Ottmar Loos, Jordan triple systems, R-spaces and bounded

symmetric domains, Bull. AMS 77 (1971), 558-561.

16) Ottmar Loos, Symmetric Spaces I: General Theory, W. A. Benjamin,

New York, 1969. Symmetric Spaces II: Compact Spaces and Classification,

W. A. Benjamin, New York, 1969.

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