Four-Potential
Landau's definition (§16). The interaction of a charge with the field enters the action through a 4-vector $A_i = (\phi, \vec{A})$: $$S_{int} = -\frac{e}{c}\int A_i\, dx^i.$$
Geometric identity. $A = A_i\,dx^i$ is the local expression — the pullback along a gauge choice — of a Connection 1-Form on a U(1) Principal Bundle over spacetime. Landau's interaction term is the integral of a 1-form along the worldline: exactly the data needed to define parallel transport of the particle's phase.
Status ladder (each rung was historically controversial): 1. Computational trick (19th c.): potentials as calculational aids for $\vec E, \vec B$. 2. Necessary for dynamics (Landau §16–17): no Lagrangian formulation without $A$. 3. Necessary for QM (Aharonov-Bohm Effect): holonomy of $A$ observable where $F=0$. 4. A connection (this study): $A$ is the local shadow of a global geometric object $\omega$; its gauge ambiguity is the shadow's frame-dependence, and in monopole sectors no single $A$ can even exist globally.
Related: Connection 1-Form · Gauge Transformation · Electromagnetic Field Tensor · Minimal Coupling · Module 09 - Electromagnetism as a U(1) Gauge Theory
Sources: Landau §16–§18.