Module parallel transports in fuzzy gauge theory

TL;DR

This paper introduces a notion of parallel transport for modules over finite matrix algebras in fuzzy gauge theory, leading to gauge-invariant observables that distinguish gauge classes and address the gauge copy problem.

Contribution

It defines module parallel transport in fuzzy gauge theory and constructs gauge-invariant observables that separate gauge equivalence classes.

Findings

Constructed module parallel transport via differential equations.

Defined gauge-invariant observables from parallel transports.

Proved observables separate gauge classes, solving the gauge copy problem.

Abstract

In this article we define and investigate a notion of parallel transport on finite projective modules over finite matrix algebras. Given a derivation-based differential calculus on the algebra and a connection on the module, we construct for every derivation X a module parallel transport, which is a lift to the module of the one-parameter group of algebra automorphisms generated by X. This parallel transport morphism is determined uniquely by an ordinary differential equation depending on the covariant derivative along X. Based on these parallel transport morphisms, we define a basic set of gauge invariant observables, i.e. functions from the space of connections to the complex numbers. For modules equipped with a hermitian structure, we prove that this set of observables is separating on the space of gauge equivalence classes of hermitian connections. This solves the gauge copy problem for fuzzy gauge theories.