$F$-manifolds, multi-flat structures and Painlev\'e transcendents

TL;DR

This paper explores multi-flat structures on $F$-manifolds, linking their existence to integrability conditions, and connects these geometric structures to Painlevé equations, providing explicit classifications and examples in various dimensions.

Contribution

It introduces conditions for the existence of multi-flat $F$-structures, classifies them in low dimensions, and relates their properties to Painlevé transcendents, including non-semisimple cases.

Findings

Multi-flat structures cannot have more than three flat connections in general.

Explicit parametrizations of bi-flat and tri-flat $F$-manifolds in dimensions 2, 3, and 4.

Connections between non-semisimple $F$-manifolds and Painlevé IV, V, and VI equations.

Abstract

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. 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. Using this fact we show that in general there can not be multi-flat structures with more than three flat connections. When the relevant distributions are integrable we construct bi-flat $F$-manifolds in dimension $2$ and $3$, and tri-flat $F$-manifolds in dimensions $3$ and $4$. In particular we obtain a parametrization of three-dimensional bi-flat $F$ in terms of a system of six first order ODEs that can be reduced to the full family of P$_{VI}$ equation and we construct non-trivial examples of four dimensional tri-flat $F$ manifolds that are controlled by hypergeometric functions. In the second part of the paper we extend our analysis to include non-semisimple multi-flat $F$-manifolds. We show that in dimension three, regular non-semisimple bi-flat $F$-manifolds are locally parameterized by solutions of the full P$_{IV}$ and P$_{V}$ equations, according to the Jordan normal form of the endomorphism $L=E\circ$. Combining this result with the local parametrization of $3$-dimensional bi-flat $F$-manifolds we have that confluences of P$_{IV}$, P$_{V}$ and P$_{VI}$ 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.