标题： Frobenius流形和Frobenius代数值可积系统 显示英文标题

标题： Frobenius manifolds and Frobenius algebra-valued integrable systems

摘要： 可积性的概念通常会从具有标量值场的系统扩展到具有代数值场的系统。 在这些扩展中，代数的性质和结构在确保可积性得到保持方面起着核心作用。 在本文中，发展了一种新的Frobenius代数值可积系统的理论。 这是通过利用由Kaufmann、Kontsevich和Manin \cite{Kaufmann,KMK}开发的此类流形的张量积理论，针对来自Frobenius流形的系统实现的。 通过专门化这种构造，使用一个固定的Frobenius代数 $\mathcal{A},$，可以得到这样的理论。 更一般地，可以应用相同的思想来构建一个 $\mathcal{A}$-值的拓扑量子场论。 然后研究了两类可积演化方程的哈密顿性质：无色散和色散演化方程。 讨论了这些思想的应用，并以一个 $\mathcal{A}$-值的修改后的Camassa-Holm方程为例进行了构造。 显示英文摘要

摘要： The notion of integrability will often extend from systems with scalar-valued fields to systems with algebra-valued fields. In such extensions the properties of, and structures on, the algebra play a central role in ensuring integrability is preserved. In this paper a new theory of Frobenius-algebra valued integrable systems is developed. This is achieved for systems derived from Frobenius manifolds by utilizing the theory of tensor products for such manifolds, as developed by Kaufmann, Kontsevich and Manin \cite{Kaufmann,KMK}. By specializing this construction, using a fixed Frobenius algebra $\mathcal{A},$ one can arrive at such a theory. More generally one can apply the same idea to construct an $\mathcal{A}$-valued Topological Quantum Field Theory. The Hamiltonian properties of two classes of integrable evolution equations are then studied: dispersionless and dispersive evolution equations. Application of these ideas are discussed and, as an example, an $\mathcal{A}$-valued modified Camassa-Holm equation is constructed.