# Covariance of the extended holonomy

## TL;DR

This paper addresses gauge covariance issues in extended loop holonomies and proposes conditions for constructing covariant extended loops, enabling a viable quantum representation for Yang-Mills theories and gravity.

## Contribution

It introduces a family of extended loops satisfying conditions for gauge covariance, overcoming previous limitations in the quantum loop representation.

## Key findings

- Identified gauge covariance problems in generic extended loops.
- Proposed a family of extended loops with covariant holonomies.
- Established sufficient conditions for gauge covariant extended loops.

## Abstract

It has been pointed out that the holonomy of generic extended loops is not gauge covariant. We show how to define a family of extended loops for which previous criticism does not apply. We also give sufficient conditions that extended loops must satisfy in order to yield covariant holonomies. This makes a quantum representation for Yang--Mills theories and gravity based on extended loops viable.