Topological current (${\partial }_{\mu }{j}_{\mathrm{top}}^{\mu }=0$)

A local current whose conservation follows from field geometry rather than requiring the field equation. For Skyrmions, pulling back the closed volume form on the target three-sphere gives the current associated with topological baryon number in the Skyrme model. Its spatial integral is a topological charge, unchanged by smooth evolution with the prescribed boundary behaviour.

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.