Where does the D4-D5-E6 model come from? What does it look like?
Start with the Standard Model. Now add Gravity. Try to Unify the Standard Model with Gravity. Supergravity. Superstrings. The D4-D5-E6 Model. Dimensional Reduction of Spacetime. Why is it called the D4-D5-E6 Model? What is a Shilov BoundaryY? How do you Visualize the D4-D5-E6 Model?
The D4-D5-E6 model contains both the Standard Model and Gravity.
This is a non-technical crude history of how it came to be and how I visualize it. Here, I have not included technical references or references to credit who did what and when. They can be found in the other pages and papers dealing with the D4-D5-E6 model.In the early 1970s, just before the Standard Model was established, Armand Wyler noticed something: the electromagnetic fine structure constant seemed to be related to the ratios of volumes of bounded complex homogeneous domains and their Shilov boundaries. He did not give clear physical reasons for the relationship, and his results were dismissed by most physicists as mere coincidence.
Start with the Standard Model.
The Standard Model has 3 basic parts: 1. a spacetime (You can think of spacetime as being fundamentally a lattice, with the continuous manifold being just a convenient way to think of it for doing calculations involving calculus, Lie groups, and such things.); 2. fermion particles and antiparticles, such as electrons and quarks, that roughly speaking are the matter in the model (You can think of them as being things located at points in the spacetime, or vertices of the spacetime lattice); and 3. gauge bosons, or particles that carry the forces between particles (You can think of them as being "located" on lines in spacetime connecting the particles that are acting on each other, or as being on links of the spacetime lattice). The gauge group of a given force is the symmetry group of the gauge bosons that carry the given force. The Standard Model is made by taking the total gauge group to be the Cartesian product of the 3 gauge groups of the 3 forces (electromagnetic, weak, and color). In the Standard Model, there is NO FUNDAMENTAL RELATION among the three parts. The spacetime, fermion particles and antiparticles, and gauge bosons are just put in by hand, so to speak.Now add Gravity.
Since Gravity can be formulated in terms of curvature of spacetime, the force of Gravity DOES have a FUNDAMENTAL RELATION to spacetime.Try to Unify the Standard Model with Gravity.
Physicists tried to combine Gravity with the Standard Model by treating Gravity like the other forces, that is, as a force described by a gauge group. Their choice of gauge group for Gravity was the 10-dimensional Poincare group of spacetime translations (4-dim), spacetime rotations (3-dim), and spacetime Lorentz boosts (3-dim). HOWEVER, the models whose total gauge group was the Cartesian product of the Poincare group and the groups of the Standard Model did not work well mathematically, and soon there was a theorem saying that there is NO WAY you can make a Cartesian product of the Poincare group and the Standard Model groups into the total gauge group of a mathematically consistent gauge field theory.Supergravity.
Physicists did not just give up, they tried another way: SUPERGRAVITY. It avoided the Cartesian product problem by using Lie Superalgebras of Lie Supergroups instead of Lie algebras of Lie groups. Lie superalgebras are combinations of two Lie algebras NOT by Cartesian product, but by taking one of the Lie algebras to be "fermionic" and the other to be "bosonic", and defining their interactions with each other by using a SUPERSYMMETRY between fermions and bosons. The fermionic Lie algebra should contain the Lie algebra of the total Standard Model Lie group. The bosonic Lie algebra was used for gravity. After playing around some, physicists realized that the best Lie algebra for Supergravity theories was not the 10-dim Poincare group, but was the 10-dim deSitter group Spin(5), or its noncompact version Spin(2,3). Spin(5) is the covering group of 5-dim rotations, and is equal to 10-dim group Sp(2) related to 2x2 matrices of quaternions. Supergravity was very good because now gravity was included, and there was a direct relation between at leat one force and spacetime. BUT AGAIN, A MATHEMATICAL PROBLEM CAME UP: You could classify all the Lie Superalgebras, and see which ones had Spin(5)=Sp(2) as the bosonic part, which is what you needed to get gravity. The ones that worked had SO(N) as the fermionic part. SO(N) is the group of N-dim rotations. Therefore, you could write down the equations for the theory with fermionic part SO(N) and bosonic part Spin(5)=Sp(2). You would get gravity just fine from the bosonic part Spin(5)=Sp(2), BUT you had to decide which N to use for SO(N) and how to get the Standard Model groups from the fermionic part SO(N). The mathematics of the model clearly showed that the mathematically nicest N was N=8, so that the fermionic part should be SO(8), BECAUSE OF THE OCTONIONIC STRUCTURE OF SO(8). This N=8 Supergravity was also called 11-dimensional Supergravity, because it could be formulated in 11-dimensional spacetime. Since 11 = 4+7, our physical 4-dimensional spacetime could be seen as the result of compactifying, or curling up into very small things, the other 7-dimensions of 11-dim spacetime. Since the compactified 7-dim things could be thought of as 7-dimensional spheres, and since 7-dim spheres have a Malcev algebra structure that is related to the Lie algebra of SO(8) and since the 7-dim sphere lives in 8-dim Octonion space and is parallelizable by the 7-dim imaginary Octonions, the underlying beauty of Supergravity is DUE TO OCTONIONIC STRUCTURE. HOWEVER, the Supergravity model was constructed using Lie Superalgebras in such a way that SO(8) (interpreted that way) WAS NOT BIG ENOUGH TO INCLUDE THE STANDARD MODEL GROUPS. There was also another problem with Supergravity: The Supersymmetry of Supergravity was a naive 1-1 correspondence between fermions and bosons, which said "For every fermion there is a corresponding boson, and vice versa." SUCH NAIVE SUPERSYMMETRY HAS NEVER BEEN OBSERVED: The fermions - neutrinos, electrons, quarks ARE NOT in 1-1 correspondence with the gauge bosons - photon, weak bosons, gluons, gravitons.Superstrings.
Although Supergravity had fatal flaws, it had a nice underlying OCTONIONIC unity, so the question was How to use it as a basis for a better model? Most physicists decided to: 1. Keep NAIVE 1-1 fermion-boson supersymmetry; 2. Give up SO(8) as a gauge group; 3. Generalize point-particle Spin(5)=Sp(2) gravity to gravity based on String theory, the theory of vibrating strings in N-dim spacetime. Why vibrating strings? In the conventional quantum mechanics courses of the early education of most physicists, they are drilled to think of the example of the 1-dim harmonic oscillator - the vibrating string - as the fundamental model in terms of which physics theories should be visualized. The result was Superstring theory. In Superstring theory, the math of vibrating strings showed that the nicest dimension for spacetime was N=26. I think that the nice 26-dimensional structures are due to the residue of OCTONIONIC structure left over from SO(8) Supergravity. The residual Octonionic structures in Supersting theory give it some very nice mathematical structures BUT it still has the NAIVE 1-1 fermion-boson supersymmetry that has NEVER been observed experimentally.The D4-D5-E6 Model
I did NOT go from Supergravity to Superstring theory. I decided to go from Supergravity as follows: 1. Give up NAIVE 1-1 fermion-boson supersymmetry; 2. Keep SO(8) as a gauge group with Octonionic structure; 3. Give up Lie superalgebras and try to put gravity into the SO(8). To get everything from SO(8), recall that the finite (Weyl group) reflection group that generates SO(8) is the group of the 4-dimensional 24-cell.
SO(8) has 28 generators:
24 for each vertex of the 24-cell
plus 4 for the dimensions of the 4-dim space of the 24-cell.
I let the 28 generators of SO(8) be the
generators of the gauge bosons:
8+3+1 = 12 of them for the Standard Model;
the other 16 for a U(4),
which contains 15-dim SU(4)=Spin(6),
which contains 10-dim Spin(5)=Sp(2) for gravity.
The part of the U(4) not used for gravity
gives the Higgs mechanism and a complex phase for
quantum propagators.
If the 28-dim adjoint representation of SO(8)
gives the gauge bosons, then
WHAT GIVES SPACETIME?
SO(8) has an 8-dim vector representation,
as the 8-dim space on which the rotations act.
I let that space be an 8-dimensional spacetime.
Next,
WHAT GIVES FERMION PARTICLES AND ANTIPARTICLES?
Fermions should come from spinor representations,
but SO(8) has no spinor representations,
BUT its 2-fold covering group Spin(8)
DOES have TWO 8-dim half-spinor representations,
so I let one be 8 fermion particles
and the other be 8 fermion anti-particles.
Subtle Triality Supersymmetry.
Here is the Dynkin diagram for Spin(8). Each vertex represents a representation of Spin(8), with the center vertex (Spin(8)) corresponding to the 28-dimensional adjoint representation that I identified with gauge bosons.
The three representations for
spacetime (blue dot),
fermion particles (red dot), and
fermion antiparticles (green dot)
are EACH 8-dimensional with Octonionic structure.
They are ALL isomorphic by the Spin(8) Triality Automorphism,
which can be represented by rotating or interchanging
the 3 arms of the Dynkin diagram of Spin(8).
The Triality isomorphism between spacetime
and fermion particles and fermion antiparticles
constitutes a
SUBTLE SUPERSYMMETRY between fermions and spacetime.
Dimensional Reduction of Spacetime.
At this stage my D4-D5-E6 model has a problem: Spacetime is 8-dim, not 4-dim. HOWEVER, if I REQUIRE that physical processes take place in an ASSOCIATIVE QUATERNIONIC SUBSPACE of Octonionic 8-dim spacetime, the resulting math gives me not only a 4-dim spacetime, but also 3 generations of fermions and the decomposition of the 28 gauge bosons into the 12 gauge bosons of the Standard Model plus the 16-dim U(4) that gives Gravity, and the Higgs mechanism.Why is it called the D4-D5-E6 Model?
D4 is a label for the 28-dim Spin(8) Lie algebra, whose 28-dim adjoint representation gives gauge bosons. D5 is a label for the 45-dim Spin(10) Lie algebra, which contains both 28-dim Spin(8) gauge bosons AND a 16-dim complexification of 8-dim spacetime, along with a 1-dim complex phase. E6 is a label for the 78-dim E6 Lie algebra, which contains both the 45-dim Spin(10) Lie algebra AND a 32-dim complexification of 8+8 = 16 fermion particles and anti-particles, along with a 1-dim complex phase. The following diagram shows how E6 contains D5 = Spin(10) which contains D4 = Spin(8):
The fermion particle-antiparticle part of E6
that is added to D5 = Spin(10) is
the coset space E6 / (D5 x U(1)).
It is a complex space corresponding to
a bounded complex homogeneous domain
whose Shilov boundary is 8+8 = 16-dimensional.
The part of D5 = Spin(10)
that is added to D4 = Spin(8) is
the coset space D5 / (D4 x U(1)).
It is a complex space corresponding to
a bounded complex homogeneous domain
whose Shilov boundary is 8-dimensional.
In the D4-D5-E6 model,
particle masses and force strengths
are determined in part by ratios of volumes
of bounded complex homogeneous domains
and their Shilov boundaries.
WHAT IS A SHILOV BOUNDARY?
Such boundaries have been called Silov and Shilov
(two different transliterations from Russian),
and are also called characteristic manifolds,
Bergman-Silov boundaries, and (I think)
distinguished boundaries, and perhaps other things.
They are generally subsets of the topological
boundaries of bounded complex domains.
For example, the bounded complex bidisk, or
Cartesian product of two disks, each disk in C,
is a bounded complex domain in C2.
Its Silov boundary is S1 x S1 , the Cartesian
product of two circles.
The Silov boundary can be described as the
set (a,b) such that both a and b are on a
bounding circle of one of the disks.
Its topological boundary is more complicated:
the set (a,b) such that either a or b is on a
bounding circle of one of the disks.
What the Silov boundary is good for is that it is
the minimal subset of the topological boundary
over which you can integrate an analytic function
to reproduce its interior values by the Poisson kernel.
Therefore the Silov boundary is closely related to
harmonic functions, Green's functions, etc.
There ought to be a good book that discusses
lots of examples and is introductory,
but if there is, I don't know it.
Harmonic Analysis of Functions of Several
Complex Variables in the Classical Domains by Hua
(Am. Math. Soc., 1979)
discusses a lot of examples, but is far from introductory.
Also advanced is Geometric Analysis on Symmetric Spaces
by Helgason (Am. Math. Soc., 1994).
Harmonic Function Theory by Axler, Bourdon, and Ramey
(Springer-Verlag, 1992)
is introductory, but does not do a lot of examples.
The Shilov boundaries and related structures are continuum objects.
As such, they are not truly fundamental,
but they are very useful to enable limited minds like mine
to do calculations
whose results can be compared with experiments.
How do you Visualize the D4-D5-E6 Model?
Since each of the 3 parts of it, gauge bosons, spacetime, and fermions, comes from a representation of Spin(8) you can visualize any part of it as being created from any other part of it, spacetime and fermions being related to each other by Subtle Triality Supersymmetry:
and
the (24+4)=28-dim space of gauge bosons being
related to
the 8 + (8+8) = 24-dim space of spacetime + fermions
by the duality between
the 24 vertices of a 24-cell
and
the 24 cells of a 24-cell (or equivalently
the 24 vertices of a dual 24-cell):
Therefore everything can be pictured as being
in flux, so to speak,
with every part being at the same time
both fundamental and derived.
If you add to that model-in-flux picture
the superpositions of all worlds,
or histories, of the Many-Worlds quantum theory,
you have the way it should be visualized:
a huge superposition of models-in-flux,
constrained only by maintaining
the relationships among the parts
that come from the underlying Octonionic structure.
My mind has trouble comprehending that,
and
it helps my limited mind to have a fixed reference point.
How I do that is to FIX (in my mind) the
8-dimensional octonionic spacetime lattice,
and therefore the 4-dimensional spacetime lattice
that results from dimensional reduction.
Then I visualize all the model-in-flux structures as
taking place on the spacetime lattice,
or really a superposition
of as many 8-dim lattices as there are Many-Worlds.
Using the FIXED lattice as a reference point
lets my limited mind see enough detail so that
I can work out calculations of particle masses
and force strengths. That is useful and good.
HOWEVER, it has lead me sometimes to speak of
the FIXED lattice as being THE FIXED SPACETIME LATTICE.
That is fundamentally incorrect and bad.
The model is better than that.