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

TL;DR

This paper explores the Landau-Ginzburg model's boundary conditions preserving A-type N=2 supersymmetry, establishing their connection to boundary states in conformal field theory and providing a microscopic description of special Lagrangian submanifolds.

Contribution

It extends linear boundary conditions to non-linear ones in LG models, linking them to special Lagrangian submanifolds and generalizing to hypersurfaces in projective space.

Findings

Explicit computation of open-string Witten index in LG models.

Boundary conditions must commute with (W(φ)-W̄(φ̄)) for consistency.

Application to T^3 supersymmetric cycle in the quintic.

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(\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.