The Gravitational Sector in the Connes-Lott Formulation of the Standard Model

TL;DR

This paper explores the non-commutative geometric framework underlying the Standard Model, revealing conditions for isotropic metrics and linking the Higgs field to the geometry's distance function.

Contribution

It demonstrates that the Riemannian metrics on the two sheets must be identical under certain conditions and connects the Higgs field's vacuum expectation value to the inter-sheet distance.

Findings

Metrics on the two sheets are identical when space of forms is isotropic.

The inter-sheet distance is governed by a scalar field.

The Higgs VEV determines the weak scale.

Abstract

We study the Riemannian aspect and the Hilbert-Einstein gravitational action of the non-commutative geometry underlying the Connes-Lott construction of the action functional of the standard model. This geometry involves a two-sheeted, Euclidian space-time. We show that if we require the space of forms to be locally isotropic and the Higgs scalar to be dynamical, then the Riemannian metrics on the two sheets of Euclidian space-time must be identical. We also show that the distance function between the two sheets is determined by a single, real scalar field whose VEV sets the weak scale.