= Pullback-volume representation of a topological current
{title2=$d(\phi^*\omega)=0,\quad Q=\int_{\mathrm{space}}\phi^*\omega$}
A closed target $d$-form $\omega$ pulls back under a spacetime field $\phi$ to a closed form $\alpha=\phi^*\omega$. Its dual defines an identically conserved <topological current>. For a compact oriented spatial $d$-manifold and an equally dimensional oriented target with $\int\omega=1$, the spatial integral is the <topological degree>. Conservation follows without the field equations; changing boundary flux can change the charge.
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