On the Landau-Ginzburg description of boundary CFTs and special Lagrangian submanifolds

We consider Landau-Ginzburg (LG) models with boundary conditions pre- serving A-type N = 2 supersymmetry. We show the equivalence of a linear class of boundary conditions in the LG model to a particular class of boundary states in the cor- responding CFT by an explicit computation of the open-string Witten index in the LG model. We extend the linear class of boundary conditions to general non-linear bound- ary conditions and determine their consistency with A-type N = 2 supersymmetry. This enables us to provide a microscopic description of special Lagrangian submani- folds in C n due to Harvey and Lawson. We generalise this construction to the case of hypersurfaces in P n . We nd that the boundary conditions must necessarily have vanishing Poisson bracket with the combination (W () W ()), where W ( )i s the appropriate superpotential for the hypersurface. An interesting application considered is the T 3 supersymmetric cycle of the quintic in the large complex structure limit.