WDVV equations and invariant bi-Hamiltonian formalism

TL;DR

This paper demonstrates that WDVV equations are bi-Hamiltonian in low dimensions, proves invariance for three dimensions, and suggests this property may extend generally, highlighting the role of projective transformations.

Contribution

It establishes the bi-Hamiltonian structure of WDVV equations in low dimensions and explores their invariance under projective transformations, supported by computational tools.

Findings

WDVV equations are bi-Hamiltonian in low dimensions

Invariance under projective transformations is proven for N=3

Examples suggest the property may hold in higher dimensions

Abstract

The purpose of the paper is to show that, in low dimensions, the WDVV equations are bi-Hamiltonian. The invariance of the bi-Hamiltonian formalism is proved for $N=3$. More examples in higher dimensions show that the result might hold in general. The invariance group of the bi-Hamiltonian pairs that we find for WDVV equations is the group of projective transformations. The significance of projective invariance of WDVV equations is discussed in detail. The computer algebra programs that were used for calculations throughout the paper are provided in a GitHub repository.