B-type Landau-Ginzburg models with one-dimensional target

We summarize the main results of our investigation of B-type topological Landau-Ginzburg models whose target is an arbitrary open Riemann surface. Such a Riemann surface need not be affine algebraic and in particular it may have infinite genus or an infinite number of Freudenthal ends. Under mild conditions on the Landau-Ginzburg superpotential, we give a complete description of the triangulated structure of the category of topological D-branes in such models as well as counting formulas for the number of topological D-branes considered up to relevant equivalence relations.