We consider D-branes wrapped around supersymmetric cycles of Calabi-Yau manifolds from the viewpoint of N=2 Landau-Ginzburg models with boundary as well as by consideration of boundary states in the corresponding Gepner models. The Landau-Ginzburg approach enables us to provide a target space interpretation for the boundary states. The boundary states are obtained by applying Cardy's procedure to combinations of characters in the Gepner models which are invariant under spectral flow. We are able to relate the two descriptions using the common discrete symmetries of the two descriptions. We are thus able to provide an extension to the boundary of the bulk correspondence between Landau-Ginzburg orbifolds and the corresponding Gepner models.Comment: 28 pages, LaTeX with revtex; (v2) Condition involving superpotential in the boundary LG model imposed, references included ; (v3) final version to appear in journa

We study both A-type and B-type D-branes in the gauged linear sigma model by considering worldsheets with boundary. The boundary conditions on the matter and vector multiplet fields are first considered in the largevolume phase/non-linear sigma model limit of the corresponding CalabiYau manifold, where we also find that we need to add a contact term on the boundary. These considerations enable to us to derive the boundary conditions in the full gauged linear sigma model, including the addition of the appropriate boundary contact terms, such that these boundary conditions have the correct non-linear sigma model limit. Most of our results are derived for the quintic Calabi-Yau manifold, though we comment on possible generalisations.

We show how to compute terms in an expansion of the world-volume superpotential for fairly general D-branes on the quintic Calabi-Yau using linear sigma model techniques, and show in examples that this superpotential captures the geometry and obstruction theory of bundles and sheaves on this Calabi-Yau.

We present a method based on mutations of helices which leads to the construction (in the large volume limit) of exceptional coherent sheaves associated with the ( a l a = 0) orbits in Gepner models. This is explicitly verified for a few examples including some cases where the ambient weighted projective space has singularities not inherited by the CalabiYau hypersurface. The method is based on two conjectures which lead to the analog,in the general case, of the Beilinson quiver for P n . We discuss how one recovers the McKay quiver using the gauged linear sigma model (GLSM) near the orbifold or Gepner point in Kähler moduli space.

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 (φ) − W (φ)), where W (φ) 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.