Helicity line bundles have nonzero Chern number

The vectors ${\mathbf{e}}_{\theta },{\mathbf{e}}_{\phi }$ are real unit vectors orthogonal to each other and to $\mathbf{n}$, and $\mathbf{n}\times {\mathbf{e}}_{\theta }={\mathbf{e}}_{\phi }$, $\mathbf{n}\times {\mathbf{e}}_{\phi }=-{\mathbf{e}}_{\theta }$. Hence $e_{\sigma }$ has unit norm and satisfies $h(\mathbf{n})e_{\sigma }=\sigma e_{\sigma }$, so it is a local frame of ${\mathcal{L}}_{\sigma ,\mathbf{n}}$. Direct differentiation gives ${\mathbf{e}}_{\theta }\cdot d{\mathbf{e}}_{\phi }=-\cos \theta d\phi$ and ${\mathbf{e}}_{\phi }\cdot d{\mathbf{e}}_{\theta }=\cos \theta d\phi$; the diagonal terms vanish. Therefore $e_{\sigma }^{†}d{e}_{\sigma }=\frac{i\sigma }{2}({\mathbf{e}}_{\theta }\cdot d{\mathbf{e}}_{\phi }-{\mathbf{e}}_{\phi }\cdot d{\mathbf{e}}_{\theta })=-i\sigma \cos \theta d\phi \mathrm{.}$ Multiplying by $-i$ gives $A_{\sigma }=-\sigma \cos \theta d\phi$, and differentiating gives $F_{\sigma }=\sigma \sin \theta d\theta \wedge d\phi$.

In Cartesian coordinates, the north and south gauges for $\sigma =+1$ are $e_{+}^N=\frac{1}{\sqrt{2}}\left(\begin{array}{c}(1+n_3)\mathrm{/}2-(n_1+i{n}_2{)}^2\mathrm{/}[2(1+n_3)]\\ i(1+n_3)\mathrm{/}2+i(n_1+i{n}_2{)}^2\mathrm{/}[2(1+n_3)]\\ -(n_1+i{n}_2)\end{array}\right),e_{+}^S=\frac{1}{\sqrt{2}}\left(\begin{array}{c}(n_3-1)\mathrm{/}2+(n_1-i{n}_2{)}^2\mathrm{/}[2(1-n_3)]\\ i(1-n_3)\mathrm{/}2+i(n_1-i{n}_2{)}^2\mathrm{/}[2(1-n_3)]\\ -(n_1-i{n}_2)\end{array}\right)\mathrm{.}$ Their denominators are nonzero on the north and south charts respectively, so they are smooth there. Since $e_{-}^N=\overline{e_{+}^N}$ and $e_{-}^S=\overline{e_{+}^S}$, the gauges for $\sigma =-1$ are smooth as well. Each has unit norm because it differs from $e_{\sigma }$ by a phase. On the overlap, $e_{\sigma }^N=e^{2i\sigma \phi }{e}_{\sigma }^S$, and the transition map $e^{i\phi }\mapsto e^{2i\sigma \phi }$ has degree $2\sigma$.

Since $F_{\sigma }=\sigma d\mathrm{\Omega }$, [F2] gives ${\int }_{S^2}{F}_{\sigma }=4\pi \sigma$, hence the normalized Berry flux is $2\sigma$. The skew-Hermitian curvature is ${\mathrm{\Omega }}_{\sigma }=d{\omega }_{\sigma }=i{F}_{\sigma }$ because ${\omega }_{\sigma }=i{A}_{\sigma }$. Under the stated Chern-Weil convention, $c_1({\mathcal{L}}_{\sigma })[S^2]=\frac{i}{2\pi }{\int }_{S^2}i{F}_{\sigma }=-\frac{1}{2\pi }{\int }_{S^2}{F}_{\sigma }=-2\sigma \mathrm{.}$

Suppose a global smooth unit section $s$ of ${\mathcal{L}}_{\sigma }$ existed. Then $A_s=-i{s}^{†}ds$ would be a globally defined smooth real one-form with compact support because $S^2$ is compact by [F4]. On each local unit frame, writing $s=g{e}_{\sigma }$ with $\mathrm{\mid }g\mathrm{\mid }=1$ gives $A_s=A_{\sigma }-i{g}^{-1}dg$; the correction is closed because $d(g^{-1}dg)=-g^{-2}dg\wedge dg=0$, so $d{A}_s=F_{\sigma }$. The sphere is oriented and boundaryless by the Statement and [F4], so [F3] yields ${\int }_{S^2}{F}_{\sigma }={\int }_{S^2}d{A}_s={\int }_{\mathrm{\partial }{S}^2}{A}_s=0$. This contradicts ${\int }_{S^2}{F}_{\sigma }=4\pi \sigma \mathrm{\ne }0$ from step 2.1. Therefore no global smooth unit helicity eigenvector exists.