The bihamiltonian structures of the DR/DZ hierarchies at the approximation up to genus one

TL;DR

This paper proves a conjecture about the bihamiltonian structure of the DR hierarchy at genus one, relating it to the DZ hierarchy through explicit transformations, advancing understanding of integrable hierarchies in mathematical physics.

Contribution

It confirms a conjectured second Hamiltonian structure for the DR hierarchy at genus one and links it explicitly to the DZ hierarchy via Miura transformation.

Findings

Proves the conjecture at genus one approximation.

Establishes an explicit relation between the brackets of DR and DZ hierarchies.

Provides a new formula for the second Hamiltonian structure in this context.

Abstract

In a recent paper, giving an arbitrary homogeneous cohomological field theory (CohFT), Rossi, Shadrin, and the first author proposed a simple formula for a bracket on the space local functionals that conjecturally gives a second Hamiltonian structure for the double ramification hierarchy associated to the CohFT. In this paper we prove this conjecture at the approximation up to genus 1 and relate this bracket to the second Poisson bracket of the Dubrovin-Zhang hierarchy by an explicit Miura transformation.