Dirac monopole connection on the Hopf bundle

Consider the Hopf fibration $\pi:S^3\to S^2$, a principal $U(1)$-bundle.

Definition (Dirac monopole connection)

A Dirac monopole connection is a principal connection on $\pi:S^3\to S^2$ whose curvature 2-form on the base represents the generator of $H^2(S^2;\mathbb Z)$ under the usual normalization for circle bundles.

One standard description uses local gauge potentials on the usual two-chart cover of $S^2$:

• On $U_N=S^2\setminus\{\text{south pole}\}$, choose a local section and define a 1-form $A_N$.
• On $U_S=S^2\setminus\{\text{north pole}\}$, choose a local section and define a 1-form $A_S$.

These satisfy on the overlap $U_N\cap U_S$ the gauge relation

for a transition function $e^{i\chi}:U_N\cap U_S\to U(1)$ of winding number $1$.

A convenient explicit choice in spherical coordinates $(\theta,\varphi)$ on the overlap is

which differ by $A_S=A_N-d\varphi$ on the overlap (so $\chi=\varphi$).

Curvature

The curvature 2-form is globally well-defined on $S^2$ by

This is a nonzero multiple of the standard area form. In particular, $F$ is not exact as a globally defined 2-form with the integrality normalization for $U(1)$-bundles; geometrically, this encodes the nontriviality of the Hopf bundle.

Examples

1. Holonomy around latitude circles.
   Fix $\theta=\theta_0$. Parallel transport around the loop $\varphi\in[0,2\pi]$ produces a phase in $U(1)$ whose argument is the integral of $A_N$ (or $A_S$) along the loop; this equals half the solid angle enclosed, matching the curvature-area interpretation.

2. Higher charge monopoles.
   Replacing the transition function $e^{i\varphi}$ by $e^{in\varphi}$ yields a family of principal $U(1)$-bundles and connections with curvature $nF$, corresponding to Chern number $n$.

3. Intrinsic description on the total space.
   There is a canonical $U(1)$-invariant 1-form on $S^3$ (a contact form) whose kernel defines the horizontal distribution; pushing this connection down recovers the above local potentials on the base.