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.
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Funding
The work of A. B. has been funded within the framework of the HSE University Basic Research Program. The work of O. B. was supported by Becas CONACYT para estudios de Doctoradoen el extranjero awarded by the Mexican government, Ref.: 2020-000000-01EXTF-00096.
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Translated from Funktsional'nyi Analiz i ego Prilozheniya, 2021, Vol. 55, pp. 22-39 https://doi.org/10.4213/faa3933.
Translated by O. Brauer and A. Yu. Buryak
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Brauer, O., Buryak, A.Y. The Bi-Hamiltonian Structures of the DR and DZ Hierarchies in the Approximation up to Genus One. Funct Anal Its Appl 55, 272–285 (2021). https://doi.org/10.1134/S001626632104002X
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DOI: https://doi.org/10.1134/S001626632104002X