Geodesic
The Bi-Hamiltonian Structures of the DR and DZ Hierarchies in the Approximation up to Genus One
Funkcionalʹnyj analiz i ego priloženiâ, Tome 55 (2021) no. 4, pp. 22-39

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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.
Keywords: moduli space of curves, cohomology ring, partial differential equation. 
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O. Brauer; A. Yu. Buryak. The Bi-Hamiltonian Structures of the DR and DZ Hierarchies in the Approximation up to Genus One. Funkcionalʹnyj analiz i ego priloženiâ, Tome 55 (2021) no. 4, pp. 22-39. http://geodesic.mathdoc.fr/item/FAA_2021_55_4_a1/