A basic introduction to Ehresmann connections and gauge theory

Abstract: Connections are among the most fundamental objects in differential geometry and modern physics. Because of this, students are often introduced to them early in their studies, usually by identifying them with Christoffel symbols under the vague premise that they serve as "corrective terms". While this approach is perfect for working with covariant derivatives and performing computations on smooth manifolds, it obscures the beautiful geometric intuition underlying connections by relegating them to a "black box" computational tool with an inexplicably non-tensorial transformation law.

In this introductory talk, we will begin by recalling the basics of fiber bundles, principal bundles, and associated bundles, and will follow up by defining the general notion of an Ehresmann connection on a fiber bundle. Finally, we will conclude by specializing this concept to principal bundles and revealing how this elegant geometric framework recovers the familiar Christoffel symbols on their associated bundles.

Together, these concepts constitute the foundation of what is known in modern mathematics and physics as gauge theory. Our hope is that this more geometrically well-motivated approach will spark an interest in this fascinating topic for the more curious members of the audience.