Calabi-Yau mirror symmetry as a gauge theory duality

Using brane set-ups we construct dual gauge theories in two dimensions with calN = (2,2) supersymmetry. Two different dualities are realized. One is basically a consequence of three-dimensional mirror symmetry. The nonlinear sigma model with a Calabi-Yau target space on the Higgs branch of the gauge theory is mapped into an equivalent non-linear sigma model on the Coulomb branch on the dual, realizing a T-dual target space with torsion. The second dual is genuine to two dimensions. In addition to swapping Higgs and Coulomb branches it trades twisted for untwisted multiplets, implying a sign flip of the left-moving U (1)R charge. Successive application of both dualities leads to geometric mirror symmetry for the Calabi-Yau target space.