A polynomial bracket for the Dubrovin--Zhang hierarchies

Abstract

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.

31 pages

References in corpus (5)

• On deformations of quasi-Miura transformations and the Dubrovin-Zhang bracket
• Givental graphs and inversion symmetry
• Algebraic structures on the topology of moduli spaces of curves and maps
• Notes on axiomatic Gromov--Witten theory and applications
• Jacobi Structures of Evolutionary Partial Differential Equations

Cited by in corpus (31)

• Double ramification cycles and integrable hierarchies
• On deformations of quasi-Miura transformations and the Dubrovin-Zhang bracket
• W-Constraints for the Total Descendant Potential of a Simple Singularity
• Tau-structure for the Double Ramification Hierarchies
• Givental graphs and inversion symmetry
• Gromov--Witten theory of Fano orbifold curves, Gamma integral structures and ADE-Toda Hierarchies
• DR/DZ equivalence conjecture and tautological relations
• Bihamiltonian Cohomologies and Integrable Hierarchies II: the Tau Structures
• Bi-Hamiltonian recursion, Liu-Pandharipande relations, and vanishing terms of the second Dubrovin-Zhang bracket
• Integrability, Quantization and Moduli Spaces of Curves
• Towards a bihamiltonian structure for the double ramification hierarchy
• Higher genera Catalan numbers and Hirota equations for extended nonlinear Schroedinger hierarchy
• Super tau-covers of bihamiltonian integrable hierarchies
• Extended $r$-spin theory in all genera and the discrete KdV hierarchy
• Dubrovin-Zhang hierarchy for the Hodge integrals
• Tautological relations and integrable systems
• Variational Bihamiltonian Cohomologies and Integrable Hierarchies I: Foundations
• Remarks on intersection numbers and integrable hierarchies. I. Quasi-triviality
• Miura-reciprocal transformations and localizable Poisson pencils
• A conjectural formula for $D{R}_g(a,-a){\lambda }_g$
• Quantum $D_4$ Drinfeld-Sokolov hierarchy and quantum singularity theory
• On the spectral problem of the quantum KdV hierarchy
• Geometry and arithmetic of integrable hierarchies of KdV type. I. Integrality
• Hodge integrals and tau-symmetric integrable hierarchies of Hamiltonian evolutionary PDEs
• BCFG Drinfeld-Sokolov Hierarchies and FJRW-Theory
• Towards Lax formulation of integrable hierarchies of topological type
• Linearization of Virasoro symmetries associated with semisimple Frobenius manifolds
• The bihamiltonian structures of the DR/DZ hierarchies at the approximation up to genus one
• Integrable systems of finite type from F-cohomological field theories without unit
• The quantum integrable hierarchy for the Gromov-Witten theory of elliptic curves
• ILW equation for the Hodge integrals revisited