A polynomial bracket for the Dubrovin--Zhang hierarchies
A polynomial bracket for the Dubrovin--Zhang hierarchies
复制标题
DOI：
10.4310/jdg/1352211225
复制
发表时间：
2010-09
期刊：
影响因子：
--
通讯作者：
中科院分区：
文献类型：
--
作者：
我们定义了与 Frobenius 流形的属展开上的 Givental 群作用的半简单轨道上的任意 tau 函数相关的哈密顿偏微分层次。我们证明方程、哈密顿量和括号是因变量相对于空间变量的导数的加权齐次多项式。在共形（同质）Frobenius 结构的特定情况下，我们的层次结构与 Dubrovin-Zhang 层次结构一致，Dubrovin-Zhang 层次结构与底层 Frobenius 结构规范相关。因此，我们的方法可以证明方程、哈密顿量和这些层次结构的泊松括号之一的多项式，正如杜布罗文和张猜想的那样。
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.