Topological charge ($Q$)

A topological charge is an integer or other discrete invariant determined by the homotopy class or characteristic class of a field configuration. Continuous finite-energy deformations cannot change it without crossing a singular or forbidden configuration.

A normalized vacuum field on a sphere surrounding a core can define $S^{d-1}\to S^{d-1}$ with integer topological degree. Nonzero degree obstructs its extension through the enclosed ball while staying in the vacuum manifold. This accounts for the winding number of a vortex and the spherical degree of a magnetic monopole, with the appropriate vacuum targets.

A nonzero topological degree obstructs smooth unwinding with fixed boundary data, but does not by itself give a stable finite size. In three dimensions a two-derivative energy scales as $R$ and can decrease as a configuration shrinks at fixed degree. The four-derivative Skyrme term scales as $R^{-1}$ and can balance this tendency. Derrick theorem addresses the energetic issue; a singular limiting configuration can evade smooth homotopy conservation.

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.

A connected component, or homotopy class, of admissible field configurations with fixed asymptotic conditions. Distinct topological sectors cannot be joined by a continuous deformation within that configuration space. A nonzero topological charge labels a sector, but a stable finite-size energy minimum also depends on the energy functional.