Holonomy of a principal composite bundle connection, non-abelian geometric phases and gauge theory of gravity

TL;DR

This paper explores the holonomy of principal composite bundle connections, linking it to non-abelian geometric phases and applying the formalism to gauge theory of gravity with spinor fields, revealing new phase separation insights.

Contribution

It introduces a formalism connecting composite bundle holonomies with non-abelian phases and applies it to gravity gauge theories, offering novel phase separation methods.

Findings

Holonomy relates to non-abelian Stokes theorem for composite bundles.

Describes non-abelian geometric phases when generators do not commute.

Decomposes Lorentz connection holonomy into linear and Cartan parts.

Abstract

We show that the holonomy of a connection defined on a principal composite bundle is related by a non-abelian Stokes theorem to the composition of the holonomies associated with the connections of the component bundles of the composite. We apply this formalism to describe the non-abelian geometric phase (when the geometric phase generator does not commute with the dynamical phase generator). We find then an assumption to obtain a new kind of separation between the dynamical and the geometric phases. We also apply this formalism to the gauge theory of gravity in the presence of a Dirac spinor field in order to decompose the holonomy of the Lorentz connection into holonomies of the linear connection and of the Cartan connection.