Mirrors of curves and their Fukaya categories

The mirror of a genus g curve can be viewed as a trivalent configuration of $3g-3$ rational curves meeting in $2g-2$ triple points; more precisely, this singular configuration arises as the critical locus of the superpotential in a $3$-dimensional Landau-Ginzburg mirror. In joint work with Alexander Efimov and Ludmil Katzarkov, we introduce a notion of Fukaya category for such a configuration of rational curves, where objects are embedded graphs with trivalent vertices at the triple points, and morphisms are linear combinations of intersection points as in usual Floer theory. We will describe the proposed construction of the structure maps of these Fukaya categories, attempt to provide some motivation, and outline examples of calculations that can be carried out to verify homological mirror symmetry in this setting.