Conncections and connection 1-forms - Lec 21 - Frederic Schuller
Why Mathematics Stack Exchange recommends it
Mathematics Stack Exchange users cite it as a reference for the definition and defining properties of the connection 1-form on a principal bundle, often with timestamps.
Recommended by 1 people across 1 different months, 2023 to 2024.
What it is
Lecture 21 of Frederic Schuller's 'Lectures on the Geometric Anatomy of Theoretical Physics' series, covering connections on principal bundles and connection 1-forms. It is a university-style blackboard lecture by Dr. Schuller, about an hour long.
What people said
Recommends CBBAM · 2023-08 · Mathematics Stack Exchange · on “Intuition behind connection 1-forms and Ehresmann connections”
I am learning about mathematical gauge theory and so far I have been able to develop an intuition behind all the objects I've read about such as principal bundles, associated bundles, and vertical/horizontal tangent spaces. However I have recently come across Ehresmann connections and connection 1-forms and I have been having trouble seeing the motivation behind them. I understand the technical definition for both but I cannot picture what idea they're trying to capture. Given that there is a one-to-one correspondence between connection 1-forms and Ehresmann connections I assume they both represent the same idea, but I am missing what that idea is. This lack of intuition has made it hard for
Cites falamiw · 2024-08 · Mathematics Stack Exchange · on “Connection $1$-form on principal $G$ bundle”
Okay, I get it. Meanwhile, I was watching Frederic Schuller's video where at 56:00 [this video] he defined $\omega_{p\triangleleft g}\left((\triangleleft g)_* X^A_p\right)\stackrel{?}{=}\omega_{p\triangleleft g}\left(X_p^{(\operatorname{Ad}_{g^{-1}})_* A}\right)$ where for each $A\in T_eG$ induced a vector field $X_p^A f:=f(p\triangleleft \exp(tA))'(0),\forall p\in P,f\in C^\infty(P)$ which I suspect the same condition (2) of my definition. [1]
Cites Antoni Parellada · 2024-09 · Mathematics Stack Exchange · on “What is the second property of the connection one-form?”
According to this lecture [this video] @min 43:58, the connection one-form $\omega$ in a principal bundle has the properties: $\omega_p(X^A_p)=A$ $\left((\lhd g)^{*}\omega\right)_p(X_p)=\left(\mathop{Ad}_{(g^{-1})}\right)_{*}\omega_p(X_p)$ Which I think is saying on the LHS that the pullback of the right action by a $g \in G$ on a vector at point $p$ is on the RHS the push forward of the adjoint (a vector in the Lie algebra) acting on the result of the one-form after this one form has already acted on a vector field, i.e. $\omega_p(X_p)$ , yielding Lie algebra elements. So do we have two vectors in the Lie algebra interacting on the RHS, $\left(\mathop{Ad}_{(g^{-1})}\right)_{*}$ and $\omega_p(X_p