Noncommutative Geometry, background independence, and $B-L$ extension of the Standard Model

In these notes we present a new framework, called "algebraic backgrounds", which is a slight modification of Connes' Spectral Triples theory which allows for a transparent representation of diffeomorphisms and spin structure equivalences. In an algebraic background there is no fixed Dirac operator but, instead, a bimodule of noncommutative 1-forms which plays the role of the differential structure of the manifold. The configuration space is the space of Dirac operators which are compatible with this bimodule. They play the role of metrics compatible with the differential structure. We present the cases of manifolds and finite graphs as motivations. The symmetry group of the algebraic background associated to a manifold turns out to be exactly that of the tetradic formulation of General Relativity, but the configuration space contains extra fields, not associated with a metric. However the projection on metric fields is diffeomorphism invariant so that they can be safely removed. The same framework adapted to the Standard Model automatically gauge the B-L symmetry. The resulting Kaluza-Klein theory contains many non-SM fields, most of them only acting on flavour space. They can be removed like in the GR case. What remains is a B-L extension of the Standard Model with a complex scalar breaking the new symmetry.