On deformations of quasi-Miura transformations and the Dubrovin-Zhang bracket

TL;DR

This paper analyzes the deformation structure of the Dubrovin-Zhang Poisson bracket in integrable PDE hierarchies linked to Frobenius manifolds, providing a new proof via weak quasi-Miura transformations.

Contribution

It offers a new proof of the deformation formula for the Poisson bracket using weak quasi-Miura transformations, simplifying previous complex computations.

Findings

Reproved the deformation formula using a new approach.

Connected the hierarchy of PDEs with its dispersionless limit.

Enhanced understanding of the structure of deformations in integrable systems.

Abstract

In our recent paper we proved the polynomiality of a Poisson bracket for a class of infinite-dimensional Hamiltonian systems of PDE's associated to semi-simple Frobenius structures. In the conformal (homogeneous) case, these systems are exactly the hierarchies of Dubrovin-Zhang, and the bracket is the first Poisson structure of their hierarchy. Our approach was based on a very involved computation of a deformation formula for the bracket with respect to the Givental-Y.-P. Lee Lie algebra action. In this paper, we discuss the structure of that deformation formula. In particular, we reprove it using a deformation formula for weak quasi-Miura transformation that relates our hierarchy of PDE's with its dispersionless limit.