A Notion To Ponder

The experiment. Electrons pass on both sides of a shielded solenoid ($\vec B \neq 0$ inside, $\vec B = 0$ everywhere the electrons go). The interference pattern shifts by $$\Delta\varphi = \frac{e}{\hbar c}\oint_\gamma A_\mu\,dx^\mu = \frac{e}{\hbar c}\,\Phi,$$ where $\Phi$ is the enclosed flux. Physics happens where the field strength vanishes.

Geometric diagnosis. Outside the solenoid the connection is flat ($F = 0$) but not gauge-trivial: the region is not simply connected, and the holonomy around the solenoid is a nontrivial element of $U(1)$. In form language: $A$ is closed but not exact on $\mathbb{R}^3 \setminus \text{solenoid}$ — the physical realization of the de Rham Cohomology class generating $H^1$.

What it settles. - $F$ is not the complete physical content of electromagnetism; holonomy is the gauge-invariant completion. (Four-Potential status ladder, rung 3.) - Landau's local formalism cannot see this: it's invisible to any statement about $F$ at points. - Yet $A$ itself is still not physical — only $e^{i\frac{e}{\hbar c}\oint A}$ is. The observable is the holonomy, exactly the connection-theoretic prediction.

Contrast with the monopole. AB: flat connection, nontrivial $\pi_1$ of the base, $H^1$ story. Dirac Monopole: curvature concentrated, nontrivial bundle, $H^2$ story. The two independent ways topology enters gauge theory.

Related: Parallel Transport and Holonomy · de Rham Cohomology · Stokes Theorem · Module 11 - Topology - Monopoles and Chern Classes

Sources: Aharonov & Bohm (1959); Woit; not in Landau — that absence is the point.