Connection 1-Form
Definition. The $\mathfrak{g}$-valued 1-form $\omega$ on the total space $P$ encoding a Connection on a Principal Bundle: it reproduces generators on vertical vectors and transforms by $\mathrm{Ad}$ under the group action. Horizontal = $\ker\omega$.
Upstairs vs. downstairs — the key bookkeeping. - $\omega$ lives upstairs on $P$: global, unique, gauge-independent. The real geometric object. - The potential $A_\alpha = s_\alpha^*\omega$ lives downstairs on $U_\alpha \subseteq M$: the pullback of $\omega$ along a local section $s_\alpha$ (= gauge choice). Gauge-dependent, defined only patchwise.
Changing section $s_\alpha \to s_\alpha \cdot g$ changes the pullback: $$A \mapsto g^{-1} A\, g + g^{-1} dg \quad\xrightarrow{\;U(1),\ g = e^{i\theta}\;}\quad A \mapsto A + i\,d\theta.$$
Resolution of Landau's §18 puzzle. "The potential is not unique" — because $A$ is a shadow of $\omega$, and the shadow depends on the angle of the light (the section). Nothing about $\omega$ itself is ambiguous. Gauge freedom is projection artifact, not physical indeterminacy.
Physics anchor. $A = -iA_\mu dx^\mu$ (factors of $i, e, \hbar, c$ by convention): Landau's four-potential §16. In nonabelian theories $A_\mu$ is matrix-valued and the $g^{-1}A g$ term makes gauge fields self-referential.
Related: Connection on a Principal Bundle · Four-Potential · Gauge Transformation · Curvature 2-Form · Module 08 - Connections and Curvature
Sources: Sternberg Ch. VII.
Linked from
- Connection on a Principal Bundle
- Covariant Derivative
- Curvature 2-Form
- Differential Form
- Four-Potential
- Gauge Transformation
- Landau & Lifshitz Vol. 2 — The Classical Theory of Fields
- Lie Algebra
- Module 04 — Landau Reread: Electromagnetism as Geometry
- Module 08 — Connections and Curvature
- Module 09 — Electromagnetism as a U(1) Gauge Theory
- Section of a Bundle
- Transition Functions