A Notion To Ponder

The setup. A magnetic charge $g$ at the origin: $\vec B = g\,\hat r / r^2$. Then $\nabla \cdot \vec B \neq 0$ at the origin, so no global vector potential exists on any sphere enclosing it ($\int_{S^2} F = 4\pi g \neq 0$, but if $F = dA$ globally, Stokes would force $0$).

The bundle resolution. Use two patches on $S^2$: $$A_N = g(1 - \cos\theta)\,d\phi \quad (\text{regular except south pole}), \qquad A_S = -g(1 + \cos\theta)\,d\phi.$$ On the equatorial overlap, $A_N - A_S = 2g\,d\phi$ — a Gauge Transformation with $g_{NS} = e^{2ieg\phi/\hbar c}$. Single-valuedness of this transition function requires $$\frac{2eg}{\hbar c} \in \mathbb{Z}$$ — Dirac quantization: one monopole anywhere quantizes all electric charge everywhere.

Geometric summary. A monopole of charge $n$ = the principal $U(1)$-bundle over $S^2$ with first Chern number $n$; the $n = 1$ bundle is the Hopf Fibration. The "Dirac string" of the old literature is a gauge artifact — the singular attempt to use one patch where two are required.

What Landau can't say. In Landau's formalism, $\nabla\cdot\vec B = 0$ is baked in via $F = dA$ globally (§26). Monopoles require upgrading $A$ from a global 1-form to a connection on a nontrivial bundle — the cleanest demonstration that the bundle language adds physical, not just aesthetic, content.

Related: Hopf Fibration · Characteristic Classes · Transition Functions · U(1) · Module 11 - Topology - Monopoles and Chern Classes

Sources: Dirac (1931); Wu–Yang (1975); Nakahara Ch. 10.