Calabi-Yau mirror symmetry as a gauge theory duality

Using brane set-ups we construct dual gauge theories in two dimensions with N = (2, 2) supersymmetry. Two different dualities are realized. One is bas ically a consequence of three-dimensional mirror symmetry. The nonlinear si gma model with a Calabi-Yau target space on the Higgs branch of the gauge t heory is mapped into an equivalent non-linear sigma model on the Coulomb br anch on the dual, realizing a T-dual target :;pace with torsion. The second dual is genuine to two dimensions. In addition to swapping Higgs and Coulo mb branches it trades twisted for untwisted multiplets, implying a sign fli p of the left-moving U(1)(R) charge. Successive application of both dualiti es leads to geometric mirror symmetry for the Calabi-Yau target space.