Class. Quantum Grav. 17 (2000) 919–927. Printed in the UK PII: S0264-9381(00)08792-X Calabi–Yau mirror symmetry as a gauge theory duality

Abstract. Using brane set-ups we construct dual gauge theories in two dimensions with N D.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. PACS numbers: 1125, 1110L, 1115 1