本文证明齐次半单 CohFT 的 DR 层级具有显式双 Hamilton 结构，关键是利用 DR/DZ 层级的 Miura 等价及 DZ 第二 Poisson 括号的多项式性。
摘要：
在与半单上同调场论（CohFT）相关的可积系统的两种研究方法中，一种由 Dubrovin 和 Zhang 提出，另一种是较新的、利用双分歧（DR）循环几何的方法；第二种方法具有非常显式的优势。DR 层级的 Poisson 算子是 $η^{-1} \partial_x$，其中 $η$ 是 CohFT 的度量，而 Hamilton 量被显式定义为 CohFT 与 DR 循环、最高 Hodge 类 $λ_g$ 以及 psi 类的幂的相交数的生成函数。DR 层级是否被赋予双 Hamilton 结构这一问题看起来要困难得多。在我们此前与 S. Shadrin 合作的工作中，当 CohFT 是齐次的时候，我们提出了一个微分算子的显式公式，并猜想它会给出所需的双 Hamilton 结构。本文证明了这一猜想。我们的证明基于两个最近得到证明的结果：在 Miura 变换下 DR 层级与半单 CohFT 的 Dubrovin–Zhang 层级等价，以及齐次半单 CohFT 的 DZ 层级的第二 Poisson 括号具有多项式性。特别地，我们的第二 Poisson 括号通过 DR/DZ 等价与 DZ 层级的第二 Poisson 括号一致，从而为它们的双 Hamilton 结构提供了一种显著显式的方法。
英文摘要：
Of the two approaches to integrable systems associated to semisimple cohomological field theories (CohFTs), the one suggested by Dubrovin and Zhang and the more recent one using the geometry of the double ramification (DR) cycle, the second has the advantage of being very explicit. The Poisson operator of the DR hierarchy is $η^{-1} \partial_x$, where $η$ is the metric of the CohFT, and the Hamiltonians are explicitly defined as generating functions of intersection numbers of the CohFT with the DR cycle, the top Hodge class $λ_g$, and powers of a psi-class. The question whether the DR hierarchy is endowed with a bihamiltonian structure appeared to be much harder. In our previous work in collaboration with S. Shadrin, when the CohFT is homogeneous, we proposed an explicit formula for a differential operator and conjectured that it would provide the required bihamiltonian structure. In this paper, we prove this conjecture. Our proof is based on two recently proved results: the equivalence of the DR hierarchy and the Dubrovin-Zhang hierarchy of a semisimple CohFT under Miura transformation and the polynomiality of the second Poisson bracket of the DZ hierarchy of a homogeneous semisimple CohFT. In particular, our second Poisson bracket coincides through the DR/DZ equivalence with the second Poisson bracket of the DZ hierarchy, hence providing a remarkably explicit approach to their bihamiltonian structure.
Bihamiltonian tests for integrable systems associated to rank-$1$ F-CohFTs
National Research University Higher School of Economics（国立高等经济大学）；Skolkovo Institute of Science and Technology（斯科尔科沃科学技术研究所）；University of Science and Technology of China（中国科学技术大学）
