Dimensional Reduction of Dirac Operator

TL;DR

This paper presents an explicit example of reducing a twelve-dimensional Dirac operator with SU(3) symmetry to a four-dimensional form, resulting in a complex interacting spinor and gauge field system.

Contribution

It provides a detailed construction of dimensional reduction for a Dirac operator on a high-dimensional manifold with SU(3) symmetry, illustrating features beyond simpler models.

Findings

Reduced Dirac equation describes massive SU(3)-octet spinors

Interaction with SU(3)-gauge field includes a curvature-dependent source

Demonstrates complexity of dimensional reduction in nontrivial geometries

Abstract

We construct an explicit example of dimensional reduction of the free massless Dirac operator with an internal SU(3) symmetry, defined on a twelve-dimensional manifold that is the total space of a principal SU(3)-bundle over a four-dimensional (nonflat) pseudo-Riemannian manifold. Upon dimensional reduction the free twelve-dimensional Dirac equation is transformed into a rather nontrivial four-dimensional one: a pair of massive Lorentz spinor SU(3)-octets interacting with an SU(3)-gauge field with a source term depending on the curvature tensor of the gauge field. The SU(3) group is complicated enough to illustrate features of the general case. It should not be confused with the color SU}(3) of quantum chromodynamics where the fundamental spinors, the quark fields, are SU(3) triplets rather than octets.