Differential equations for periods and flat coordinates in two-dimensional topological matter theories

Abstract We consider two-dimensional topological Landau-Ginzburg models. In order to obtain the free energy of these models, and to determine the Kahler potential for the marginal perturbations, one needs to determine flat or “special” coordinates that can be used to parametrize the perturbations of the superpotentials. This paper describes the relationship between the natural Landau-Ginzburg parametrization and these flat coordinates. In particular we show how one can explicitly obtain the differential equations that relate the two. We discuss the problem for both Calabi-Yau manifolds and for general topological matter models (with arbitrary central charges) with relevant and marginal perturbations. We also give a number of examples.