Cheshire θ terms, Aharonov-Bohm effects, and axions

We discuss unusual θ terms that can appear in field theories that allow global vortices. These "Cheshire θ terms" induce Aharonov-Bohm effects for some particles that move around vortices. For example, a Cheshire θ term can appear in QCD coupled to an axion and induces Aharonov-Bohm effects for baryons and leptons moving around axion strings. We point out a potential experimental signature left on the spectrum of gravitational waves from axion cosmic string network by the Cheshire θ term.