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In a recent paper, giving an arbitrary homogeneous cohomological field theory(CohFT), Rossi, Shadrin, and the first author proposed a simple formula for abracket on the space local functionals that conjecturally gives a secondHamiltonian structure for the double ramification hierarchy associated to theCohFT. In this paper we prove this conjecture at the approximation up to genus1 and relate this bracket to the second Poisson bracket of the Dubrovin-Zhanghierarchy by an explicit Miura transformation.

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