The relationship between the 26 and 10-11 dimensional cases is subtle. The bosonic string has Virasoro cancellation of central anomaly for D = 26, and for supersymmetric theory it is 11 dimensional. The 11 dimensional case involves M^4xS^7, where the S^7, or its related S^6 under the light cone gauge, contains the Calabi-Yau data under compactification. The 7-sphere here plays a central role in this matter, for it defines a holonomy which is crucial in defining a cubic action. I will say that this is holographic, and there are ways of working a matrix theory this way. The best approach is the Jordan exceptional algebra. This is the algebra of the octonions, which extends the E_8 octonions into a triality. The J^2(O) = R () V [() = oplus] is of the form
[math]
J^2(O) = \left(\matrix{z_0 & {\cal O}\cr
{\bar{\cal O}} & z_1}\right)
[/math]
which is extended to the J^3(O) matrix algebra. The triality in J^3(O) includes an E_8 matrix of vectors and two spinor matrices. These do correspond to a Feynman diagram where a vector (boson) decays into a spinor and its conjugate. The Feynman diagram is a three-way spoke, with V ~ O going in and O' and O" going out. The J^3(O) is R () J^2(O) () θ () bar-θ,
[math]
J^3(O) = \left(\matrix{z_1 & {\cal O}_0 & {\bar{\cal O}}_2\cr
{\bar{\cal O}}_0 & z_2 & {\cal O}_1\cr
{\cal O}_2 & {\bar{\cal O}}_1 & z_0}\right)
[/math]
for the spinorial (fermionic fields) octonions in O and O". The vector term given by the ocotinion O, V ~ J^2(O) decays into V --> ψ bar-ψ. This is a manifestation of the automorphism of G_2, and is an elementary Feynman diagram for a supersymmetric gauge interaction. So the two spinorial octonions are the O^2 or two E_8's which you refer to.
The space here is 27 dimensional. The O's are each 8-dimensional, which gives a general span of 3x8 3 = 27 dimensional. The above triality condition, along with some anomaly cancellations of vertex algebras, defines a space of reduced dimension of 8 3 = 11 dimensional. On the light cone frame (infinite momentum frame) the space in 27 dimensions is reduced to 26 dimensions and the 11 dimensional space to 10. These are the corresponding bosonic string Lorentizian spacetime and the supersymmetric space of supergravity respectively. The diagonal elements of this matrix define a Chern-Simons lagrangian of scalar terms for x_i --> p_i A_i (the cubic nature of this is apparent) and a general Lagrangian defined as the determinant of J^3(O) under all triality transformations defines a cubic action. This then determines an equivalency between a field theory term and a dual boundary field. This field ~ boundary of dual field is a cornerstone of AdS.CFT. The AdS/CFT can be found for the case where two scalars in J^3(O) define timelike directions.
Cheers LC
