Dirac monopole connection on the Hopf bundle

On the Hopf bundle $S^3\subset\mathbb C^2\to\mathbb{CP}^1\cong S^2$ , with right action $z\cdot e^{it}=e^{it}z$ , the standard Dirac monopole connection is the $i\mathbb R$ -valued connection form

\omega=\bar z_1\,dz_1+\bar z_2\,dz_2\quad\text{on }S^3.

It is purely imaginary because $d(|z_1|^2+|z_2|^2)=0$ on $S^3$ , is invariant under the circle action, and evaluates to $it$ on the fundamental vector generated by $it\in i\mathbb R$ . Its horizontal spaces are $\ker\omega$ .

Local gauge potentials

Identify the base with spherical coordinates $(\theta,\varphi)$ so that local sections away from the south and north poles are

s_N=(\cos(\theta/2),e^{i\varphi}\sin(\theta/2)),\qquad s_S=(e^{-i\varphi}\cos(\theta/2),\sin(\theta/2)).

Write $s_N^*\omega=iA_N$ and $s_S^*\omega=iA_S$ , with real-valued potentials. Then

A_N=\frac{1-\cos\theta}{2}\,d\varphi,\qquad A_S=-\frac{1+\cos\theta}{2}\,d\varphi.

On the overlap, $s_S=s_Ne^{-i\varphi}$ , so the connection transformation law gives $A_S=A_N-d\varphi$ . The phase $e^{-i\varphi}$ is a globally defined transition function there; $\varphi$ itself is only a local angular coordinate. These potentials extend smoothly to their respective poles.

Curvature and normalization

The curvature on the base is $iF$ , where

F=dA_N=dA_S=\frac12\sin\theta\,d\theta\wedge d\varphi,\qquad \frac1{2\pi}\int_{S^2}F=1.

The nonzero flux proves that $F$ is not exact. With the Chern–Weil convention $c_1=\frac{i}{2\pi}F_\nabla$ , the associated tautological line bundle has $c_1=-[F/(2\pi)]$ ; its dual has first Chern number $+1$ . Calling the monopole charge $+1$ here refers to the displayed real-flux normalization.

The curvature class alone does not specify this standard connection: every connection on the same Hopf bundle has the same normalized cohomology class, while its curvature form can vary by an exact form.

Holonomy and higher charges

For a positively oriented latitude loop at $\theta=\theta_0$ , the right-action horizontal-lift convention above gives the holonomy phase

\exp\!\left(-i\oint A_N\right) =\exp\!\left(-i\pi(1-\cos\theta_0)\right).

Its exponent is minus half the enclosed solid angle. Reversing the loop orientation reverses this sign.

Taking the $n$ th tensor power of the associated line bundle gives local potentials $nA_N,nA_S$ , transition $e^{-in\varphi}$ , and real curvature $nF$ . Its real flux is $n$ , and its first Chern number under the stated convention is $-n$ .