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Topological defects and smooth excitations determine the properties ofsystems showing collective order. We introduce a generic non-singular fieldtheory that comprehensively describes defects and excitations in systems with$O(n)$ broken rotational symmetry. Within this formalism, we explore fastevents, such as defect nucleation/annihilation and dynamical phase transitionswhere the interplay between topological defects and non-linear excitations isparticularly important. To highlight its versatility, we apply this formalismin the context of Bose-Einstein condensates, active nematics, and crystallattices.

A unified field theory of topological defects and non-linear local excitations