Open Access
September 2012 A polynomial bracket for the Dubrovin-Zhang hierarchies
Alexandr Buryak, Hessel Posthuma, Sergey Shadrin
J. Differential Geom. 92(1): 153-185 (September 2012). DOI: 10.4310/jdg/1352211225


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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Alexandr Buryak. Hessel Posthuma. Sergey Shadrin. "A polynomial bracket for the Dubrovin-Zhang hierarchies." J. Differential Geom. 92 (1) 153 - 185, September 2012.


Published: September 2012
First available in Project Euclid: 6 November 2012

zbMATH: 1259.53079
MathSciNet: MR2998900
Digital Object Identifier: 10.4310/jdg/1352211225

Rights: Copyright © 2012 Lehigh University

Vol.92 • No. 1 • September 2012
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