Topological modes, non-locality and the double copy

The double copy connects scattering amplitudes and other objects in gauge and gravity theories. Open conceptual issues include whether non-local information in gravity theories can be generated from the double copy, and how the double copy should be practically implemented in unseen cases. In this paper, we consider topological theories (with and without mass) in 2+1 dimensions, and argue that this makes a useful playground for exploring non-locality. In particular, topological modes of the gauge field arise, themselves associated with non-trivial global behaviour, and it is not clear {\it a priori} how to double copy them. We settle this issue, and clarify the role of BCJ shifts in modifying how topological modes contribute. We show how our conclusions apply to four- and five-point scattering amplitudes in topological gauge / gravity theories, as a by-product obtaining greatly simplified analytic expressions for the four-point amplitudes.