James Pascaleff

This paper explores an example of homological mirror symmetry, relating Lagrangian branes in a Landau-Ginzburg model (X^∨, W) to coherent sheaves on the projective plane. The Landau-Ginzburg model is regarded as the mirror to the pair consisting of the projective plane along with a chosen anticanonical divisor, which we take to be the union of a conic and a line. This is the binodal cubic curve of the title. The topology of X^∨ and the superpotential W reflect this particular choice, whereas, for example, taking the anticanonical divisor to be the union of three transverse lines gives rise to the the "Hori-Vafa" mirror.

This Landau-Ginzburg model is interesting from the point of view of the Strominger-Yau-Zaslow (SYZ) picture of mirror symmetry, because the total space X^∨ carries a special Lagrangian torus fibration with a singularity. The Lagrangian submanifolds we consider are sections of this torus fibration, meaning that they correspond to line bundles on the mirror projective plane, in line with general expectations from the SYZ picture.

The algebra structure on the Floer cohomology of these Lagrangian submanifolds is computed, and matched with the algebra structure on the hom-spaces of the corresponding line bundles. The tools here are symplectic techniques developed by Seidel for counting pseudoholomorphic sections of Lefschetz fibrations. One of the later sections describes the "tropical" interpretation of the result: the Lagrangian Floer cohomology has a natural basis in bijection with integral points in an affine manifold, and the algebra structure counts particular tropical curves in this manifold. In the last section we show how our computation can be used to describe the wrapped Floer cohomology of the Lagrangian submanifolds as well.