Covariant derivatives - Lec 25 - Frederic Schuller

The video begins with an introduction to covariant derivatives on associated vector bundles. It discusses the concept starting from a principal bundle P with fiber G, a connection one form Omega, and the construction of an associated vector bundle PF. The role of a left action on the fibers F is emphasized, which is necessary for the construction of the associated bundle. The connection one form is shown to enable parallel transport between fibers of the principal bundle, which is then used to define the covariant derivative for sections of the associated bundle, assuming F is a vector space and the left action is linear.

The presenter acknowledges the technical difficulty in constructing covariant derivatives directly on associated bundles and introduces a neater method. This involves associating a unique function Phi Sigma on the principal bundle P with any section Sigma of the associated bundle. The function Phi Sigma must be G-equivariant, which encodes the information about the left action. The video outlines the process of constructing a covariant derivative for a section by first defining an exterior covariant derivative on the principal bundle and then relating it back to the section on the associated bundle.

The video presents a key theorem stating that there is a one-to-one correspondence between sections of an associated bundle and G-equivariant functions on the principal bundle. This bijection is established by defining a G-equivariant function Phi on the principal bundle that corresponds to a section Sigma. The G-equivariance property ensures compatibility with the group actions. The proof involves constructing a section from a G-equivariant function and vice versa, showing that this process is well-defined and preserves information.

The video script details the process of defining a covariant derivative on the principal bundle by first inducing a function from a section and then using the exterior covariant derivative on this function. It is shown that the covariant derivative of the induced function is equal to the exterior derivative plus a correction term involving the connection one form Omega. The script also demonstrates that this derivative has the required properties, such as linearity and additivity.

The process continues with the localization of the covariant derivative by introducing a section of the principal bundle, which allows for the pullback of the covariant derivative to the base manifold. This results in a covariant derivative that acts on local sections of the associated bundle. The script explains how the pullback of the connection one form Omega, known as the Yang-Mills field, and the induced function Phi are used to define the covariant derivative on the base manifold.

The video explores the implications of G-equivariance, particularly in the context of linear actions. It shows how the connection one form Omega appears when considering the action of a group element close to the identity. The script derives a corollary that relates the action of a vertical vector field to the connection one form, which simplifies the expression for the G-equivariant condition. This corollary is then used to facilitate the construction of the covariant derivative.

The final paragraph discusses the implications of the local nature of the covariant derivative and the connection coefficients in the context of quantum mechanics. It highlights that while the connection one form and the left action can be chosen independently, certain factors, such as a factor of two in two dimensions, must be absorbed into the left action rather than the connection. This choice becomes significant when discussing quantum mechanics on curved spaces, as it affects the transformation behavior of the Yang-Mills field.