F-manifolds, multi-flat structures and painleve transcendents

In this paper we study F-manifolds equipped with multiple flat connections and multiple F-products, that are required to be compatible in a suitable sense. Multi-flat F-manifolds are the analogue for F-manifolds of Frobenius manifolds with multi-Hamiltonian structures. In the semisimple case, we show that a necessary condition for the existence of such multiple flat connections can be expressed in terms of the integrability of a distribution of vector fields that are related to the eventual identities for the multiple products involved. These vector fields satisfy the commutation relations of the centerless Virasoro algebra. We prove that the distributions associated to bi-flat and tri-flat F-manifolds are integrable, while in other cases they are maximally non-integrable. Using this fact we show that there can not be non-trivial semisimple multi-flat structures with more than three flat connections. When the relevant distributions are integrable, coupling the invariants of the foliations they determine with Tsarev's conditions, we construct biflat and tri-flat semisimple F-manifolds in dimension 3. In particular we obtain a parameterization of three-dimensional bi-flat F-manifolds in terms of a system of six first order ODEs that can be reduced to the full family of PV I equations. In the second part of the paper we study the non-semisimple case. We show that threedimensional regular non-semisimple bi-flat F-manifolds are locally parameterized by solutions of the full PIV and PV equations, according to the Jordan normal form of the endomorphism L = E. As a consequence, combining this result with the result of the first part on the semisimple case we have that confluences of PIV, PV and PV I correspond to collisions of eigenvalues of L preserving the regularity. Furthermore, we show that, contrary to the semisimple situation, it is possible to construct regular non-semisimple multi-flat F-manifolds, with any number of compatible flat connections.