Left-Right Symmetric Models in Noncommutative Geometries?

In the Standard Model of electro-weak and strong forces, parity is broken explicitly by the choice of inequivalent representations for leftand right-handed fermions. Within the frame work of Yang-Mills-Higgs models this is certainly an aesthetic draw back that physicists have tried to correct by the introduction of left-right symmetric models. These are Yang-Mills-Higgs models where parity is broken spontaneously together with the gauge symmetry. However the price to pay for this aesthetic surgery is high, especially on an aesthetic scale: the simplest left-right symmetric model for electro-weak forces has a SU(2)L × SU(2)R ×U(1) group and four scalar representations transforming like two 2L ⊗ 2R representations, a 3L ⊗ 1R and a 1L ⊗ 3R. The Higgs potential contains some twenty coupling constants [11]. For a tiny class of Yang-Mills-Higgs models, noncommutative geometry derives the Higgs mechanism, i.e. the scalar representation and symmetry breaking Higgs potential, from first principles. Parity violation is crucial here in the sense that vector-like models are not in this tiny class, the Standard Model, however, with its explicit parity violation qualifies for noncommutative geometry. Left-right symmetric models are halfway in between vector-like models and models with explicit parity breaking and its is natural to ask whether they do qualify for noncommutative geometry. In the Connes-Lott models [7], a first try was made but unfortunately did not qualify [2]. This was, however, before the setting of a precise notion of noncommutative geometry (Connes’ Axioms [9]) and the introduction of real structure which gave a very rigid structure to construction of finite spectral triples. A complete classification of these latter was made in [3], enabling to construct some very general finite spectral triples in the framework of these axioms. Here we shall use this general classification in order to study the most general LRS models, with the aim to check if they can be expected to be physical. We shall first define the LRS models in the context of almost commutative geometries, by just specializing the usual LRS models to this approach. However, Connes’Axioms imply a number of conditions that the model has to fulfill in order to be well-defined. In particular, we will see that having 2 groups acting respectively