Quantum interference is studied for a charged particle on a multiply connected configuration space. The holes of the space are assumed to accommodate independent magnetic fluxes. As follows from an instanton calculation, tunneling of the particle is suppressed completely in such a geometry for appropriately chosen field strengths. This set of values defines an algebraic variety in parameter space, not a manifold. The quenching persists even if the geometric symmetry of the system is broken. Aharonov-Bohm-type experiments and mesoscopic tunneling devices are natural candidates to observe many-path interference
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