Abstract
Mirror symmetry for higher genus curves is usually formulated and studied in
terms of Landau-Ginzburg models; however the critical locus of the
superpotential is arguably of greater intrinsic relevance to mirror symmetry
than the whole Landau-Ginzburg model. Accordingly, we propose a new approach to
the A-model of the mirror, viewed as a trivalent configuration of rational
curves together with some extra data at the nodal points. In this context, we
introduce a version of Lagrangian Floer theory and the Fukaya category for
trivalent graphs, and show that homological mirror symmetry holds, namely, that
the Fukaya category of a trivalent configuration of rational curves is
equivalent to the derived category of a non-Archimedean generalized Tate curve.