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

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.

Quantum construction of a unitary SU(2/1) model of the electro-weak interactions with 2 Higgs doublets

“…In section 9, we show that the chiral decomposition of the Higgs scalars, in conjunction with the Dirac 1-forms Γ of [TM06], naturally constructs a covariant differential corre-sponding to the SU(2/1) Hermitian algebra first defined by Sternberg and Wolf [SW78]. The chirality operator and the complex Higgs structure conspire so that the corresponding curvature 2-forms is well defined and indeed valued in the adjoint representation of the SU (2)U (1) even Lie algebra.…”

“…Our new result is to show that this structure exactly fits the chiral decomposition (5.3) of the Φ fields, allowing us to construct a classical curvature 2-forms (9.12), where the unexpected apparition of the chirality operator χ in the definition of the Hermitian Lie bracket (9.9) exactly compensates the signs implied when adapting our chiral connection 1-form A [TM06] to the doubling of the Φ fields. There is also a probable relation with non commutative differential geometry, as discussed in [TM06].…”

“…Since the scalar fields (5.3) play the role of a gauge field associated to the odd generators of the superalgebra, it seems natural to introduce them as part of the covariant differential and curvature 2-form. Such a construction was proposed in [TM06], where we found that the covariant differential is associative if and only if the Fermions sit in a representation of the superalgebra graded by the chirality. As shown in sections 2 and 3, this condition is met by the observed leptons and quarks.…”

“…We would like to construct a quantum field theory, based on the SU (2/1) superalgebra, which would extend the Yang-Mills theory [YM54] associated to its maximal SU (2)U (1) Lie sub-algebra and incorporate in some way the odd matrices. In [TM06], we have shown that we can construct an associative covariant exterior differential, mixing left and right chiral Fermions, if and only if the Fermion form a representation of a Lie superalgebra, graded by the chirality. As we have just seen, these conditions are met exactly by the existing fundamental Fermions: the leptons and the quarks.…”

“…Since the new bilinear scalar term in the H-curvature (9.13) coincides with (6.3), the scalar potential is proportional to (6.4) which can be rewritten as (6.9). It is curious to see that the combined effect of the leptons and quarks in (6.9) is mimicked in the classical construction (10.1) by combined effect of the 2 chiral ways (9.11-12) of extending the superalgebra SU (2/1) curvature of [TM06] to the H-algebra.…”