(Fermionic)Mass Meets (Intrinsic)Curvature

TL;DR

The paper presents a geometric interpretation of fermion mass matrices as curvature within the framework of gauge theories, linking algebraic properties to geometric structures and deriving a natural Lagrangian density.

Contribution

It introduces a geometric approach using vacuum pairs to interpret fermion masses as curvature, connecting algebraic and geometric aspects in gauge theories.

Findings

Mass matrices are interpreted as curvature in a geometric framework.

A natural class of Lagrangian densities arises from the geometry of Clifford modules.

The spectral invariant Lagrangian density describes free fermions geometrically.

Abstract

Using the notion of vacuum pairs we show how the (square of the) mass matrix of the fermions can be considered geometrically as curvature. This curvature together with the curvature of space-time, defines the total curvature of the Clifford module bundle representing a ``free'' fermion within the geometrical setup of spontaneously broken Yang-Mills-Higgs gauge theories. The geometrical frame discussed here gives rise to a natural class of Lagrangian densities. It is shown that the geometry of the Clifford module bundle representing a free fermion is described by a canonical spectral invariant Lagrangian density.