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A polynomial bracket for the Dubrovin--Zhang hierarchies

A. Buryak, H. Posthuma, S. Shadrin

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.

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.

Paper Structure

This paper contains 29 sections, 14 theorems, 93 equations.

Table of Contents

  1. Introduction
  2. Dubrovin-Zhang construction and polynomiality
  3. Givental theory
  4. Group action on ingredients of the hierarchy
  5. Organization of the paper
  6. Acknowledgements
  7. The action of the Givental group
  8. Tame partition functions
  9. Genus $0$
  10. Higher genera
  11. The Givental group and Lie algebra
  12. Convention on signs
  13. The hierarchy associated to a potential
  14. The principal hierarchy
  15. Notations for the calculus of variations
  16. ...and 14 more sections

Key Result

Proposition \oldthetheorem

For a tame potential $F_0$, the vector fields $O_{\alpha,k}$, $k\geq 1$, $\alpha=1,\dots,s$, are in involution: $[O_{\alpha,k},O_{\beta,l}]=0$.

Theorems & Definitions (34)

  • Definition \oldthetheorem
  • Proposition \oldthetheorem
  • proof
  • Proposition \oldthetheorem
  • proof
  • Proposition \oldthetheorem
  • proof
  • Proposition \oldthetheorem
  • proof
  • Example \oldthetheorem: The KdV hierarchy
  • ...and 24 more