Geometric constructions of two-dimensional (0,2) SUSY theories

We consider the field theories on multiple stacks of D5-branes wrapped on four cycles of resolved/deformed conifold geometries fibered over a two-torus. The central charges of the D5-branes are slightly misaligned when the branes are wrapped on various rigid holomorphic two-cycles or when they have different charges with respect to a magnetic flux turned on the two-torus. The wrapped D5-branes preserve (0, 2) supersymmetry in two dimensions if the Kahler moduli and the magnetic flux are related. Our geometries are T-dual to the brane configurations considered by Kutasov-Lin, and we provide a geometric interpretation for their equality between the field theory D-terms and the magnetic fluxes. We also consider the geometric transitions for rigid holomorphic two-cycles fibered over a two-torus with magnetic flux and discuss the partial breaking of supersymmetry after the geometric transition.