Alternative bi-Hamiltonian structures for WDVV equations of associativity

TL;DR

This paper explores alternative bi-Hamiltonian structures for WDVV equations, revealing how different choices of variables lead to various Hamiltonian formulations unified within a covariant symplectic framework.

Contribution

It introduces a covariant approach to understanding multiple Hamiltonian structures of WDVV equations, connecting them through the Witten-Zuckerman symplectic form.

Findings

Different Hamiltonian structures correspond to different variable choices.

All structures can be unified in a covariant symplectic framework.

Variational formulation leads to Hamiltonian operators via Dirac brackets.

Abstract

The WDVV equations of associativity in 2-d topological field theory are completely integrable third order Monge-Amp\`ere equations which admit bi-Hamiltonian structure. The time variable plays a distinguished role in the discussion of Hamiltonian structure whereas in the theory of WDVV equations none of the independent variables merits such a distinction. WDVV equations admit very different alternative Hamiltonian structures under different possible choices of the time variable but all these various Hamiltonian formulations can be brought together in the framework of the covariant theory of symplectic structure. They can be identified as different components of the covariant Witten-Zuckerman symplectic 2-form current density where a variational formulation of the WDVV equation that leads to the Hamiltonian operator through the Dirac bracket is available.