Flat F-Manifolds, F-CohFTs, and Integrable Hierarchies

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