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

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.

arXiv comment34 pages

References in corpus (14)

• The structure of 2D semi-simple field theories
• Double ramification cycles on the moduli spaces of curves
• Double ramification cycles and integrable hierarchies
• Integrals of psi-classes over double ramification cycles
• Tau-structure for the Double Ramification Hierarchies
• Complex reflection groups, logarithmic connections and bi-flat F-manifolds
• From Darboux-Egorov system to bi-flat $F$-manifolds
• Deformations of semisimple Poisson pencils of hydrodynamic type are unobstructed
• $F$-manifolds, multi-flat structures and Painlevé transcendents
• Integrable systems of double ramification type
• DR/DZ equivalence conjecture and tautological relations
• Darboux-Egorov system, bi-flat $F$-manifolds and Painlevé VI
• A group action on Losev-Manin cohomological field theories
• Towards a bihamiltonian structure for the double ramification hierarchy

Cited by in corpus (6)

• Super tau-covers of bihamiltonian integrable hierarchies
• Integrable systems, Nijenhuis geometry and Lauricella bi-flat structures
• Stable tree expressions with Omega-classes and Double Ramification cycles
• Integrable systems of finite type from F-cohomological field theories without unit
• WDVV equations: Hamiltonian operators and symbolic computations
• Moduli spaces of residueless meromorphic differentials and the KP hierarchy