Chiral spinors and gauge fields in noncommutative curved space-time

TL;DR

This paper generalizes core geometric concepts to non-commutative curved space-time, constructing a gravitational action and modeling chiral fermions coupled to gauge fields, revealing potential parity violation effects.

Contribution

It introduces a framework for non-commutative geometry incorporating gravity and gauge fields, extending the Standard Model to curved non-commutative space-time.

Findings

Construction of Einstein-Hilbert-Cartan terms in non-commutative geometry

Model of chiral spinors on a two-sheeted space-time with gauge fields

Potential parity violation effects due to gravity

Abstract

The fundamental concepts of Riemannian geometry, such as differential forms, vielbein, metric, connection, torsion and curvature, are generalized in the context of non-commutative geometry. This allows us to construct the Einstein-Hilbert-Cartan terms, in addition to the bosonic and fermionic ones in the Lagrangian of an action functional on non-commutative spaces. As an example, and also as a prelude to the Standard Model that includes gravitational interactions, we present a model of chiral spinor fields on a curved two-sheeted space-time with two distinct abelian gauge fields. In this model, the full spectrum of the generalized metric consists of pairs of tensor, vector and scalar fields. They are coupled to the chiral fermions and the gauge fields leading to possible parity violation effects triggered by gravity.