Flat F-manifolds, F-CohFTs, and integrable hierarchies

TL;DR

This paper introduces a new hierarchy associated with F-cohomological field theories, proving the existence of integrable dispersive deformations for semisimple flat F-manifolds and providing explicit classifications in specific cases.

Contribution

It constructs the double ramification hierarchy for F-cohomological field theories and proves the existence of integrable dispersive deformations for semisimple flat F-manifolds.

Findings

Proved existence of dispersive deformations at all orders in the dispersion parameter.

Classified rank 1 hierarchies of DR type at order 9 in dispersion.

Classified homogeneous DR hierarchies for 2D flat F-manifolds at genus 1.

Abstract

We define the double ramification hierarchy associated to an F-cohomological field theory and use this construction to prove that the principal hierarchy of any semisimple (homogeneous) flat F-manifold possesses a (homogeneous) integrable dispersive deformation at all orders in the dispersion parameter. The proof is based on the reconstruction of an F-CohFT starting from a semisimple flat F-manifold and additional data in genus $1$, obtained in our previous work. Our construction of these dispersive deformations is quite explicit and we compute several examples. In particular, we provide a complete classification of rank $1$ hierarchies of DR type at the order $9$ approximation in the dispersion parameter and of homogeneous DR hierarchies associated with all $2$-dimensional homogeneous flat F-manifolds at genus $1$ approximation.