Unified Gauge Theories and Reduction of Couplings: from Finiteness to Fuzzy Extra Dimensions

Finite Unified Theories (FUTs) are $N=1$ supersymmetric Grand Unified Theories, which can be made all-loop finite, both in the dimensionless (gauge and Yukawa couplings) and dimensionful (soft supersymmetry breaking terms) sectors. This remarkable property, based on the reduction of couplings at the quantum level, provides a drastic reduction in the number of free parameters, which in turn leads to an accurate prediction of the top quark mass in the dimensionless sector, and predictions for the Higgs boson mass and the supersymmetric spectrum in the dimensionful sector. Here we examine the predictions of two such FUTs. Next we consider gauge theories defined in higher dimensions, where the extra dimensions form a fuzzy space (a finite matrix manifold). We reinterpret these gauge theories as four-dimensional theories with Kaluza–Klein modes. We then perform a generalized à la Forgacs–Manton dimensional reduction. We emphasize some striking features emerging such as (i) the appearance of non-Abelian gauge theories in four dimensions starting from an Abelian gauge theory in higher dimensions, (ii) the fact that the spontaneous symmetry breaking of the theory takes place entirely in the extra dimensions and (iii) the renormalizability of the theory both in higher as well as in four dimensions. Then reversing the above approach we present a renormalizable four dimensional $SU(N)$ gauge theory with a suitable multiplet of scalar fields, which via spontaneous symmetry breaking dynamically develops extra dimensions in the form of a fuzzy sphere $S^2_N$. We explicitly find the tower of massive Kaluza–Klein modes consistent with an interpretation as gauge theory on $M^4 \times S^2$, the scalars being interpreted as gauge fields on $S^2$. Depending on the parameters of the model the low-energy gauge group can be $SU(n)$, or broken further to $SU(n_1)\times SU(n_2) \times U(1)$. Therefore the second picture justifies the first one in a renormalizable framework but in addition has the potential to reveal new aspects of the theory.