N= 2 boundary conditions for non-linear sigma models and Landau-Ginzburg models

We study N=2 nonlinear two dimensional sigma models with boundaries and theirmassive generalizations (the Landau-Ginzburg models). These models are definedover either Kahler or bihermitian target space manifolds. We determine the mostgeneral local N=2 superconformal boundary conditions (D-branes) for these sigmamodels. In the Kahler case we reproduce the known results in a systematicfashion including interesting results concerning the coisotropic A-type branes.We further analyse the N=2 superconformal boundary conditions for sigma modelsdefined over a bihermitian manifold with torsion. We interpret the boundaryconditions in terms of different types of submanifolds of the target space. Wepoint out how the open sigma models correspond to new types of target spacegeometry. For the massive Landau-Ginzburg models (both Kahler and bihermitian)we discuss an important class of supersymmetric boundary conditions whichadmits a nice geometrical interpretation.Comment: 48 pages, latex, references and minor comments added, the version to appear in JHE