Semisimple flat F-manifolds in higher genus

Communications in Mathematical Physics. 2023. Vol. 397. P. 141–197.

Arsie A., Buryak A., Lorenzoni P., Rossi P.

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.

Keywords: риманова поверхность, пространство модулей, moduli space, Riemann surface, Integrable system, интегрируемая система, flat F-manifold, плоское F-многообразие

Publication based on the results of: Representation theory of W-algebras affine and toroidal quantum groups and its applications to geometry, topology and combinatorics (2023)