On the bi-Hamiltonian Geometry of WDVV Equations

TL;DR

This paper investigates the bi-Hamiltonian structure of WDVV associativity equations in four dimensions, revealing that these nonlinear equations can be expressed as commuting hydrodynamic systems with compatible Hamiltonian structures.

Contribution

It demonstrates that the four-dimensional WDVV equations admit a compatible pair of local homogeneous Hamiltonian structures of different orders.

Findings

Existence of a compatible pair of Hamiltonian structures for the systems.

Representation of the equations as commuting hydrodynamic systems.

Identification of the Hamiltonian structures as of first and third order.

Abstract

We consider the WDVV associativity equations in the four dimensional case. These nonlinear equations of third order can be written as a pair of six component commuting two-dimensional non-diagonalizable hydrodynamic type systems. We prove that these systems possess a compatible pair of local homogeneous Hamiltonian structures of Dubrovin--Novikov type (of first and third order, respectively).