Cheshire $\theta$ terms, Aharonov-Bohm effects, and axions

We discuss unusual $\theta$ terms that can appear in field theories that allow global vortices. These `Cheshire $\theta$ terms’ induce Aharonov-Bohm effects for some particles that move around vortices. For example, a Cheshire $\theta$ term can appear in QCD coupled to an axion and induces Aharonov-Bohm effects for baryons and leptons moving around axion strings.