**Reduction of Couplings in Unified Theories: From Finiteness to
Fuzzy Extra Dimensions**

**Abstract:**

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 present the predictions of the best, so far, FUT. 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 G 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. This scheme represents so far an excellent
example in which classical reduction of couplings takes place. However
since it leads to renormalizable theories, has the ingredients to become a
framework for quantum reduction of couplings too.