Pullback-volume representation of a topological current ($d({\varphi }^{\ast }\omega )=0,Q={\int }_{\mathrm{space}}{\varphi }^{\ast }\omega$)

A closed target $d$-form $\omega$ pulls back under a spacetime field $\varphi$ to a closed form $\alpha ={\varphi }^{\ast }\omega$. 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.