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

TL;DR

The paper develops a microscopic LG-based framework to describe D-branes on supersymmetric cycles, establishing a precise map between linear A-type LG boundary conditions and a class of boundary states in N=2 CFTs through open-string Witten index calculations. It then generalizes to nonlinear A-type boundaries, showing that preserved supersymmetry requires the boundary submanifolds to be Lagrangian and, in the presence of a superpotential, to have vanishing Poisson brackets with W−W̄, effectively tying D-brane geometry to special Lagrangian conditions. The authors apply these insights to CY hypersurfaces, including the quintic, where boundary conditions correspond to straight lines in the W-plane and to SLag submanifolds like the T^3 cycle in the large complex structure limit, offering a microscopic description of Harvey–Lawson SLags and extending to LG orbifolds. The work clarifies how LG data encodes CFT boundary states, provides explicit index-based identifications, and suggests broader applications to Gepner models, mirror symmetry, and LSM descriptions of branes in CY geometries.

Abstract

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)-\bar{W}(\barφ)), 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.