The Bi-Hamiltonian Structures of the DR and DZ Hierarchies in the Approximation up to Genus One

Abstract

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