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

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.