MTd2 said:
I understand what you say, and I agree with that. I thought that you could interpret this problem in terms of fibre buddle. The total space containing E(8), while the fibre and the base, each one, containing a kind of SU(5), although with different signatures. It would also make a map betten gravity and the other fields, without coupling them. The coupling would be done at the total space.
E_8 is a lattice of roots, which exists in 8 dimensions. The lattice is defined by the set of Weyl chamber reflections
<br />
x~\rightarrow~x~-~2r\frac{r\cdot x}{|r|^2}<br />
on any vector x by the root vector r. These reflections define a set of angles, which for complex groups include dihedral angles and angles between higher dimensional sublattice structures. For E_8 the set of roots, 240 in all, defines the Grosset polytope which exists in 8 dimensions.
The roots may correspond to roots for some subgroups, and this can be broken out in a number of ways. As you said Baez indicated that a representation of a group does not give automatically a representation of all its subgroups. .
What you indicate with respect to the "bulk" is not far off the mark from what people want to do. We have groups of interior symmetries [A_i,~A_j]~=~C^k_{ij}A_k, such as found in gauge fields, and there are then exterior symmetries given by the Lorentz-Poincare generators P_a and M_{ab} (and the Pauli-Ljubanski vector), with possible symmetries on the (0, 1/2)-(1/2, 0) spinor representations of the theory (supersymmetry) and finally the discrete symmetries on C-P-T. One central distinction between the internal symmetries and exterior symmetries (spacetime) is that internal symmetries are compact such as SO(n) while exterior symmetries are noncompact such as SO(3,1). Now in the E_8 root paper by Garrett there is the group SO(7,1) = SO(3,1)xSO(4) (plus on the algebra level) and of course the three "8's" framed on this. In this way a noncompact group can have a compact subgroup.
A simple example is the the Lorentz group which consists of three ordinary rotation in space plus three boosts, which are hyperbolic. This is SL(2,C) ~ SU(2)xSU(1,1), and so we might think of the embedding of gauge groups with compact group structure with general relativity as analogous to this.
If we think of gravity as a gauge-like theory with F~=~dA~+~A\wedge A for nonabelian gauge fields the DE's for these on the classical level are nonlinear. Yet we can quantize these, but renormalization is a bit complicated. We can well enough quantize a SO(4) theory obtained in euclideanization. But gravity is a strangely different. Why? The gauge group SU(1,1) is hyperbolic. In the Pauli matrix representation we have that \tau_z~=~i\sigma_z. So we cha form a gauge connection
<br />
A~=~A^{\pm}\sigma_{\pm}~+~iA^3 \sigma_3<br />
and for the group element g~=~e^{ix\tau_3}~=~e^{-x\sigma_3} the connection term transforms as
<br />
A'~=~g^{-1}Ag~+~g^{-1}dg~=~e^{-2x}A^{\pm}\sigma_{\pm}~+~iA^3\sigma_3<br />
and for x~\rightarrow~\infty this gives A~\rightarrow~iA^3\sigma_3. Now A^{\pm}\sigma_{\pm} and A^3\sigma_3 have distinct holonomy groups and are thus distinct points (moduli) in the moduli space. But this limit has a curious implication that the field F~=~dA~+~A\wedge A for these two are the same and the moduli are not separable. In other words the moduli space for gravity is not Hausdorff. This is the most serious problem for quantum gravity.
I have written some on this, and later I might illustrate how this requires some interplay between Golay codes and Goppa codes. Goppa codes are a very different domain, where here the Hamming distance is computed from algebraic varieties, such as projective varieties or elliptic curves. The point set topolology is non-Hausdorff or Zariski in this system. This is a crucial element to quantum gravity, unless you want to work completely in an elliptical domain, but this physically would mean the universe has not tunnelled out of the vacuum with imaginary time into a real state with real time. So there is a lot more to this physics than finding representations of groups --- though that is an important part.
Lawrence B. Crowell