$F$-manifolds, multi-flat structures and Painlevé transcendents

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.

69 pages, appendix added

References in corpus (3)

• $F$-manifolds with eventual identities, bidifferential calculus and twisted Lenard-Magri chains
• Exact Poisson pencils, $\tau$-structures and topological hierarchies
• Critical points, Lauricella functions and Whitham-type equations

Cited by in corpus (10)

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• Applications of Nijenhuis geometry II: maximal pencils of multihamiltonian structures of hydrodynamic type
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• Degenerate Riemann-Hilbert-Birkhoff problems, semisimplicity, and convergence of WDVV-potentials
• Bi-Hamiltonian structure of the Oriented Associativity Equation
• Regular non-semisimple Dubrovin-Frobenius manifolds
• Integrable systems, Nijenhuis geometry and Lauricella bi-flat structures
• 3-dimensional F-manifolds
• Riemann-Hilbert-Birkhoff inverse problem for semisimple flat $F$-manifolds, and convergence of oriented associativity potentials
• Regular flat structure and generalized Okubo system