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SIGMA 4 (2008), 026, 26 pages arXiv:0802.3454
https://doi.org/10.3842/SIGMA.2008.026
Contribution to the Proceedings of the Seventh International Conference Symmetry in Nonlinear Mathematical Physics
Unified Gauge Theories and Reduction of Couplings: from Finiteness to Fuzzy Extra Dimensions
Myriam Mondragón a and George Zoupanos b
a) Inst. de Física, Universidad Nacional Autónoma de México, México
b) Physics Department, National Technical University, Athens, Greece
Received November 01, 2007, in final form January 31, 2008; Published online February 23, 2008
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
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 SN2. We explicitly find the tower
of massive Kaluza-Klein modes consistent with an interpretation as
gauge theory on M4 × S2, the scalars being interpreted as
gauge fields on S2. Depending on the parameters of the model the
low-energy gauge group can be SU(n), or broken further to SU(n1)
× SU(n2) × 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.
Key words:
unification; gauge theories; finiteness; higher dimensions; fuzzy sphere; non-commutative gauge theories; renormalizability.
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