Purely non-local Hamiltonian formalism, Kohno connections and $\vee$-systems

TL;DR

This paper extends non-local Hamiltonian formalism to Riemannian F-manifolds, relates Kohno connections and $ abla$-systems, and constructs Hamiltonian structures for $ vee$-systems, broadening the mathematical framework for integrable systems.

Contribution

It generalizes non-local Hamiltonian formalism to non-semisimple cases, links Kohno property with $ vee$-systems, and constructs Hamiltonian structures for degenerate $ vee$-systems.

Findings

Recurrence relations split into two Lenard-Magri chains in flat case.

Kohno property and $ vee$-system condition are equivalent under certain conditions.

Purely non-local Hamiltonian structures can be associated with any $ vee$-system.

Abstract

In this paper, we extend purely non-local Hamiltonian formalism to a class of Riemannian F-manifolds, without assumptions on the semisimplicity of the product $\circ$ or on the flatness of the connection $\nabla$. In the flat case we show that the recurrence relations for the principal hierarchy can be re-interpreted using a local and purely non-local Hamiltonian operators and in this case they split into two Lenard-Magri chains, one involving the even terms, the other involving the odd terms. Furthermore, we give an elementary proof that the Kohno property and the $\vee$-system condition are equivalent under suitable conditions and we show how to associate a purely non-local Hamiltonian structure to any $\vee$-system, including degenerate ones.