Semisimple Flat F-Manifolds in Higher Genus

In this paper, we generalize the Givental theory for Frobenius manifolds and cohomological field theories to flat F-manifolds and F-cohomological field theories. In particular, we define a notion of Givental cone for flat F-manifolds, and we provide a generalization of the Givental group as a matrix loop group acting on them. We show that this action is transitive on semisimple flat F-manifolds. We then extend this action to F-cohomological field theories in all genera. We show that, given a semisimple flat F-manifold and a Givental group element connecting it to the constant flat F-manifold at its origin, one can construct a family of F-CohFTs in all genera, parameterized by a vector in the associative algebra at the origin, whose genus 0 part is the given flat F-manifold. If the flat F-manifold is homogeneous, then the associated family of F-CohFTs contains a subfamily of homogeneous F-CohFTs. However, unlike in the case of Frobenius manifolds and CohFTs, these homogeneous F-CohFTs can have different conformal dimensions, which are determined by the properties of a certain metric associated to the flat F-manifold.