Floer cohomology in the mirror of the projective plane and a binodal cubic curve

We construct a family of Lagrangian submanifolds in the Landau--Ginzburg mirror to the projective plane equipped with a binodal cubic curve as anticanonical divisor. These objects correspond under mirror symmetry to the powers of the twisting sheaf O(1), and hence their Floer cohomology groups form an algebra isomorphic to the homogeneous coordinate ring. An interesting feature is the presence of a singular torus fibration on the mirror, of which the Lagrangians are sections. The algebra structure on the Floer cohomology is computed by counting sections of Lefschetz fibrations. Our results agree with the tropical analog proposed by Abouzaid--Gross--Siebert. An extension to mirrors of the complements of components of the anticanonical divisor is discussed.