144 Progress of Theoretical Physics Supplement No. 171, 2007 Unified Theories: From Finiteness to Fuzzy Extra Dimensions Myriam Mondrag´ on 1 and George Zoupanos 2 1 Instituto de F´ ısica, Universidad Nacional Aut´ onoma de M´ exico, Apdo. Postal 20-364, M´ exico 01000, D.F., M´ exico 2 Physics Department, National Technical University, GR-157 80 Zografou, Athens, Greece 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 FUTs taking into account all theoretical and experimental constraints. Unified theories defined in more than four dimensions provide after dimensional reduction in four dimensions a further unification of the gauge, Yukawa and Higgs sectors as well as of the soft super- symmetry breaking sector in the case of supersymmetric theories. Furthermore, if the extra dimensions are fuzzy instead of continuous, the theories become renormalizable. Due to lack of space we cannot review the last two subjects here and we suggest consultation of the original papers. §1. Introduction The theoretical efforts to establish a deeper understanding of Nature have led to very interesting frameworks such as String theories and Non-commutative Geometry both of which aim to describe physics at the Planck scale. Looking for the origin of the idea that coordinates might not commute we might have to go back to the days of Heisenberg. In the recent years the birth of such speculations can be found in Refs. 1) and 2). In the spirit of Non-commutative Geometry also particle models with non-commutative gauge theory were explored 3) (see also Refs. 4)–7)). On the other hand the present intensive research has been triggered by the natural realiza- tion of non-commutativity of space in the string theory context of D-branes in the presence of a constant background antisymmetric field. 8) After the work of Seiberg and Witten, 9) where a map (SW map) between non-commutative and commuta- tive gauge theories has been described, there has been a lot of activity also in the construction of non-commutative phenomenological Lagrangians, for example vari- ous non-commutative standard model like Lagrangians have been proposed. 10), 11), ∗) In particular in Ref. 11), following the SW map methods developed in Ref. 12), a non-commutative standard model with SU (3) × SU (2) × U (1) gauge group has been presented. These non-commutative models represent interesting generalizations of the SM and hint at possible new physics. However they do not address the usual *) These SM actions are mainly considered as effective actions because they are not renormaliz- able. The effective action interpretation is consistent with the SM in Ref. 11) being anomaly free 13) Non-commutative phenomenology has been discussed in Ref. 14). Downloaded from https://academic.oup.com/ptps/article-abstract/doi/10.1143/PTPS.171.144/1890045 by guest on 26 May 2020