Higher-Dimensional Unified Theories with continuous and fuzzy coset spaces as extra dimensions

Abstract

We first briefly review the Coset Space Dimensional Reduction (CSDR) programme and present the results of the best model so far, based on the $\mathcal{N} = 1$, $d = 10$, $E_8$ gauge theory reduced over the nearly-K\"ahler manifold $SU(3)/U(1)\times U(1)$. Then, we present the adjustment of the CSDR programme in the case that the extra dimensions are considered to be fuzzy coset spaces and then, the best model constructed in this framework, too, which is the trinification GUT, $SU(3)^3$.