A Note on Symmetries of WDVV Equations

TL;DR

This paper explores the symmetries of WDVV equations through bi-hamiltonian structures, characterizing the moduli space of models and analyzing genus-one free energy transformations.

Contribution

It introduces a bi-hamiltonian perspective on WDVV symmetries, linking them to Miura transformations and characterizing the moduli space via the polytropic exponent.

Findings

Moduli space characterized by polytropic exponent h

Symmetries act as canonical Miura transformations

Transformation properties of genus-one free energy analyzed

Abstract

We investigate symmetries of Witten-Dijkgraaf-E.Verlinde-H.Verlinde (WDVV) equations proposed by Dubrovin from bi-hamiltonian point of view. These symmetries can be viewed as canonical Miura transformations between genus-zero bi-hamiltonian systems of hydrodynamic type. In particular,we show that the moduli space of two-primary models under symmetries of WDVV can be characterized by the polytropic exponent $h$. Furthermore, we also discuss the transformation properties of free energy at genus-one level.