We interpret the physical symmetry preserving the lepton number as a shadow of a finite pseudo-Riemannian structure of the standard model. Using the pseudo-Riemannian generalizations of real spectral triples we describe the geometries with indefinite metric over finite-dimensional algebras and their Riemannian shadows. We apply the discussion to the standard model spectral triple, and classify possible time orientations leading to restrictions on the physical parameters and symmetries. I. INTRODUCTIONNoncommutative geometry offers an intriguing possibility of a new insight into the structure of all fundamental interactions, linking purely geometric gravity with the electroweak and strong interactions of elementary fermions [1]. The possibility is explored in one way into extending the geometric notions to describe models that could approximate spacetime [2] and, on the other hand, to gather the information about the structure of geometry underlying experimentally verified models of fundamental constituents of matter [3,4].The crucial role in the understanding of the geometric interpretation of the standard model of particle physics is based on the finite geometry, linked to the finite-dimensional algebra C ⊕ H ⊕ M 3 (C) and the related finite spectral triple. Even though finite spectral triples have been classified some time ago [5,6] and the model has been extensively studied, it still can surprise and shed new light on the structure of fundamental interaction. An example is the recent discovery of unexpected duality [7] in the standard model Clifford algebra (called Hodge duality) that is satisfied only for certain values of physical parameters (bare masses and mixing matrices).

We study the properties of a bosonization procedure based on Clifford algebra valued degrees of freedom, valid for spaces of any dimension. We present its interpretation in terms of fermions in presence of ℤ2 gauge fields satisfying a modified Gauss’ law, resembling Chern-Simons-like theories. Our bosonization prescription involves constraints, which are interpreted as a flatness condition for the gauge field. Solution of the constraints is presented for toroidal geometries of dimension two. Duality between our model and (d − 1)- form ℤ2 gauge theory is derived, which elucidates the relation between the approach taken here with another bosonization map proposed recently.

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