Mathematics

We study F-manifolds equipped with multiple flat connections (and multiple F-products), that are required to be compatible in a suitable sense. In the first part of the talk we consider bi-flat F-manifolds and we show that in dimension three, they are locally parameterized by solutions of the full Painlevé IV,V and VI equations, according to the Jordan normal form of the operator of multiplication by the Euler vector field. In the second part of the talk we discuss conditions for the existence of multi-flat structures. In the semisimple case we show that a necessary condition can be expressed in terms of the integrability of a distribution of vector fields that are related to the unit vector fields for the multiple products involved. Using this fact we show that in general there can not be multi-flat structures with more than three flat connections. On the contrary, in the non-semisimple case, it is possible to construct multi-flat F-manifolds, with any number of compatible flat connections. Based on joint works with Alessandro Arsie.