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

Lagrangian Floer theory for trivalent graphs and homological mirror symmetry for curves