Chiral-Yang-Mills theory, non commutative differential geometry, and the need for a Lie super-algebra

TL;DR

This paper introduces a novel extension of Yang-Mills theory incorporating a Lie-Kac super-algebra, linking chiral fermions via a gauge-chiral 1-form, and naturally explains spontaneous symmetry breaking within a non-commutative geometric framework.

Contribution

It proposes a new gauge structure using Lie-Kac super-algebras that unifies chiral fermions and Higgs fields, with implications for non-commutative geometry.

Findings

Associativity of covariant differential requires Lie-Kac super-algebra

Spontaneous symmetry breaking occurs along an odd generator

Connects Yang-Mills theory with non-commutative geometry

Abstract

In Yang-Mills theory, the charges of the left and right massless Fermions are independent of each other. We propose a new paradigm where we remove this freedom and densify the algebraic structure of Yang-Mills theory by integrating the scalar Higgs field into a new gauge-chiral 1-form which connects Fermions of opposite chiralities. Using the Bianchi identity, we prove that the corresponding covariant differential is associative if and only if we gauge a Lie-Kac super-algebra. In this model, spontaneous symmetry breakdown naturally occurs along an odd generator of the super-algebra and induces a representation of the Connes-Lott non commutative differential geometry of the 2-point finite space.