Journal of Differential Geometry

A polynomial bracket for the Dubrovin-Zhang hierarchies

Alexandr Buryak, Hessel Posthuma, and Sergey Shadrin

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We define a hierarchy of Hamiltonian PDEs associated to an arbitrary tau-function in the semi-simple orbit of the Givental group action on genus expansions of Frobenius manifolds. We prove that the equations, the Hamiltonians, and the bracket are weighted-homogeneous polynomials in the derivatives of the dependent variables with respect to the space variable.

In the particular case of a conformal (homogeneous) Frobenius structure, our hierarchy coincides with the Dubrovin–Zhang hierarchy that is canonically associated to the underlying Frobenius structure. Therefore, our approach allows to prove the polynomiality of the equations, Hamiltonians, and one of the Poisson brackets of these hierarchies, as conjectured by Dubrovin and Zhang.

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J. Differential Geom., Volume 92, Number 1 (2012), 153-185.

First available in Project Euclid: 6 November 2012

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Buryak, Alexandr; Posthuma, Hessel; Shadrin, Sergey. A polynomial bracket for the Dubrovin-Zhang hierarchies. J. Differential Geom. 92 (2012), no. 1, 153--185. doi:10.4310/jdg/1352211225.

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