Tony Smith said:
I don't understand exactly how an E8 Cartan matrix in the Kummer surface intersection form would produce an E8 principal bundle symmetry group over the 4-real-dim Kummer surface as base manifold,
or
how it would explicitly correspond to the root vectors of the E8 Lie algebra of the E8 principal bundle symmetry group.
Tony Smith
This is of course a bit of an open question, and is something which I have been attempting to address. This leads into an issue of compactification, Calabi-Yau spaces and orbifolds. What I am about to write is a sketch of one possibility I am considering. This has some suggestive possibilities.
A Kummer surface is a specific case of a K3 surface (K-cubed Kummer, Kahler & Kodiara). The 2-surface given by z_0^4~+~z_1^4~+~z_2^4~+~z_3^4~=~0 is a two-dim C surface in CP^3 and is an exception to most K3 manifolds which are not embedded in a projective space, or defined by this sort of polynomial. K3 manifolds are diffeomorphic to each other, so one specific example translates to another.A general Kummer surface obeys a quartic equation of the sort
<br />
(x_0^2~+~x_1^2~+~x_2^2~+~mx_3^2)^2~+~\lambda abcd~=~0<br />
for the abcd functions of the x_i's. For the first and second pairs of these coordinates the real and imaginary parts of a complex variable then this is invariant under an abelian reparameterization z~\rightarrow~e^{i\theta}z. This then defines a fanning of the projective space and a form of algebraic variety called a Toric variety. These are sometimes called weighted projective spaces.
The projective space CP^2 the weighted projective space defines the equivalence class on the complex coordinates in { C { P}^2 .} by the map CP^2~\rightarrow~C{P^2}_w} defined by the action on the coordinates,
<br />
[z_1,~z_2]~\mapsto~[{z_1}^{a_1},~{z_2}^{a_2}],<br />
or
<br />
[z_1,~z_2]~\mapsto~[{r_1}^{a_1}e^{ia_1 \theta_1},<br />
~{r_2}^{a_2}e^{ia_2 \theta_2}].<br />
This establishes an identification between the points in the {[0,~2 \pi r/a]} "pie slices" or fan sections of each complex line.
Now consider two maps:
<br />
f: CP^2~ \rightarrow~CP^2(a)~=~CP^2_w <br />
<br />
g: CP^2~\rightarrow~CP^2(b)~=~CP^2_{w'},<br />
so that the weights for the two maps are unequal. dz_j and dz'_j are differential basis one-forms in CP}^2_w and CP}^2_{w'} respectively, which are easily computed. The dual vectors, V_j, V'_jare easily computed. The vectors defines as L^{a_j}~=~a_j V_j are easily found and these obey a Witt algebra commutator which with the central extension may be extended to the Virasoro algebra.
<br />
[L^{a_j},~L^{b_j}]~=~(a_j~-~b_j) L^{a_j + b_j}~+~c(a_j ,b_j)<br />
For the Virasoro algebra without center
<br />
[L^a,~L^b]~=~{C^{ab}}_cL^{c}<br />
write the vector,\xi^\alpha~=~{\xi^\alpha}_a L^a where {\xi^\alpha} is an element of the Lie algebra \cal G. The commutator in of {\xi^\alpha}_a~\in~\cal G can be found as
<br />
[{\xi^\alpha}_a,~{\xi^\beta}_b]~=~{{C_g}^{\alpha \beta}}_\gamma {\xi^\gamma}_{a+b}<br />
associated with the Lie algebra \cal G.
Within a local trivialization connection coefficients may be defined as,
<br />
{(\eta^{-1}\partial_{\mu} \eta)_{\alpha}}^{\beta}<br />
~=~\eta_{\alpha \gamma} \partial_{\mu}\eta^{\gamma \beta}<br />
~=~{\xi^{\dagger}}_{\alpha a}{\xi^{\gamma}}_a<br />
({\partial_{\mu} \xi^{\dagger}}_{\gamma}{\xi_b}^{\beta}~+~<br />
{\xi^{\dagger}}_{\gamma}\partial_{\mu}{\xi_b}^{\beta}),<br />
which are the conjugate terms {{A^\dagger}_\alpha}^\beta}_\mu~+~{{A_\alpha}^\beta}_\mu~=~{{\cal A}_{\alpha}}^\beta}_\mu. The curvature tensors {{\cal F}_{\alpha}}^\beta~=~{F_{\alpha}}^{\beta\dagger}~+~ {{F}_{\alpha}}^\beta consists of holomorphic and antiholomorphic curvatures,
<br />
{{{\cal F }_{\alpha}}^{\beta}}_{\mu \nu}~=<br />
\partial_{[\nu}(\eta_{\alpha \gamma} \partial_{\mu ]}\eta^{\beta \gamma})<br />
~=~\partial_{[\nu}{{{{\cal A}}^{\beta \gamma}}_{\mu ]}}<br />
~+~{{\cal A}^\dagger}_{\alpha \beta [ \mu}{{\cal A}^{\beta \gamma}}_{\nu ]}. <br />
It is possible to demonstrate that this obeys transformation properties of a gauge theory.
So this suggests a possible way in which the C(E_8)\oplus C(E_8)\oplus\sigma_x and the intersection form are associated with a fibration. I think the set of these K3 spaces and compactifications is assigned to the particles or maybe SUSY pairs of fields. The algebraic geometric definition of a surface S is according to the sheaf cohomology of a group G_s. In this way I think this might be related to sheaf structure similar to twistor theory.
Anyway this is where my "frontier" on this lies at the time. It will take some time to work this out, if I can. I am just one guy here, and I have had this idea cooking for not that long.
Lawrence B. Crowell