Wilson loops in gauge theories pose a fundamental challenge for dualities.
Wilson loops are labeled by a representation of the gauge group and should map
under duality to loop operators labeled by the same data, yet generically, dual
theories have completely different gauge groups. In this paper we resolve this
conundrum for three dimensional mirror symmetry. We show that Wilson loops are
exchanged under mirror symmetry with Vortex loop operators, whose microscopic
definition in terms of a supersymmetric quantum mechanics coupled to the theory
encode in a non-trivial way a representation of the original gauge group,
despite that the gauge groups of mirror theories can be radically different.
Our predictions for the mirror map, which we derive guided by branes in string
theory, are confirmed by the computation of the exact expectation value of
Wilson and Vortex loop operators on the three-sphere.Comment: 92 pages, v2: minor clarifications in the introduction, to be
published in JHE