Speaker
Description
The development of the group $E_{10}$ interpreted as the dark halo group of the visible $(9+1)$ dimensional space-time, allowing it to be integrated with the internal symmetry groups of $E_8\!\times \!E_8$ , $O(32)$ , and $O(16)\!\times \!O(16)$ of the heterotic string, is demonstrated. The representation of the Lorentzian root lattice of $E_{10}$ by integer $2\!\times \!2$ octonionic Jordan matrices that also represent quantized momenta in periodic space-time will be demonstrated as well as the root system of the Conway-Sloane lattice as an integer $3\!\times \!3$ exceptional Jordan matrix, which has $E_{10}\!\times \!E_8\!\times \!E_8$ as sublattices, including their associated root diagrams, also giving the expression of the Weyl reflections for $E_{10}$ and the Conway-Sloane lattice by means of Jordan products. We comment on the prospects of writing right-moving string and left-moving superstring Lagrangians by means of Jordan matrices $J_{L,R}(z)$ in $\bf{27}$ and $\bf{\overline{27}}$ representations of $E_{6,-26}$ and as cosets $G/H$ conformal fields, where $G=E_7$ and $H=E_6\!\times \!U(1)$.