Complementarity between local and nonlocal topological effects

In certain topological effects the accumulation of a quantum phase shift is accompanied by a local observable effect. We show that such effects manifest a complementarity between nonlocal and local attributes of the topology, which is reminiscent but different from the usual wave-particle complementarity. This complementarity is not a consequence of noncommutativity, rather it is due to the noncanonical nature of the observables. We suggest that a local/nonlocal complementarity is a general feature of topological effects that are "dual" to the Aharonov-Bohm effect.