A Puzzle About General Covariance and Gauge

TL;DR

This paper examines the concept of general covariance, arguing that Yang-Mills theory does not meet certain criteria for being generally covariant due to the gauge-natural nature of its bundles, and relates this to the role of solder forms.

Contribution

It introduces criteria for general covariance, shows Yang-Mills theory's bundles are gauge-natural rather than natural, and links this to the significance of solder forms in the theories.

Findings

Yang-Mills bundles are gauge-natural, not natural.

Criteria for general covariance are not satisfied by Yang-Mills theory.

General covariance is better understood as functoriality.

Abstract

We consider two simple criteria for when a physical theory should be said to be "generally covariant", and we argue that these criteria are not met by Yang-Mills theory, even on geometric formulations of that theory. The reason, we show, is that the bundles encountered in Yang-Mills theory are not natural bundles; instead, they are gauge-natural. We then show how these observations relate to previous arguments about the significance of solder forms in assessing disanalogies between general relativity and Yang-Mills theory. We conclude by suggesting that general covariance is really about functoriality.