  We consider Landau-Ginzburg (LG) models with boundary conditions preserving
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
corresponding 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 boundary conditions and determine their consistency with A-type N=2
supersymmetry. This enables us to provide a microscopic description of special
Lagrangian submanifolds in C^n due to Harvey and Lawson. We generalise this
construction to the case of hypersurfaces in P^n. We find that the boundary
conditions must necessarily have vanishing Poisson bracket with the combination
(W(\phi)-\bar{W}(\bar{\phi})), where W(\phi) is 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.
