We discuss unusual ? terms that can appear in field theories that allow global vortices. These "Cheshire ? terms" induce Aharonov-Bohm effects for some particles that move around vortices. For example, a Cheshire ? 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 ? term.