We consider systems in which a continuous symmetry G is spontaneously broken to a subgroup H such that the effective action for the Goldstone modes contains topologically non-trivial terms. The effect of such terms is for example that skyrmions of the symmetry-breaking order parameter carry a quantized charge, or that vortices of the order parameter carry topologically protected gapless modes. When does spontaneous symmetry breaking lead to a particular Goldstone topological term? To answer this, we consider the role of the 'anomaly' of the G symmetry. We argue that in general, the appropriate concept to consider is a 'compatibility relation' between the Goldstone invariants and the G anomaly. We illustrate this through several examples such as the canonical Thouless pump, the quantum Hall ferromagnet, and symmetry breaking on the boundary of topological insulators.

Reference: ArXiv 2410.05268