Frobenius流形理论是现代数学物理中非常重要的研究课题, 在 Gromov--Witten理论、奇点理论、镜像对称、可积系统理论等的研究中有着非常重要的应用. Frobe-nius流形理论的一个重要内容是由Dubrovin和张友金建立的关于半单Frobenius流形与一类双哈密顿可积方程簇之间联系的Dubrovin--Zhang理论.在有关 Hodge积分、等变 Gromov--Witten不变量等的研究中, 人们也发现某些著名的可积系统与一类广义Frobenius流形有着密切的联系. 与通常的 Frobenius流形相比, 这类广义Frobenius流形的单位向量场关于Frobenius流形的度量不再平坦, 故我们称其为具有非平坦单位的广义Frobenius 流形,也简称其为广义 Frobenius流形.本文的目的是将Dubrovin--Zhang理论推广到这类广义Frobenius流形.本文的主要结果是发展了广义Frobenius流形的主方程簇的拓扑形变理论. 我们首先造了一簇具有双哈密顿结构的流体力学型可积方程簇,证明了这一可积方程簇具有tau结构与Virasoro 对称, 并且这些Virasoro对称可以由通过广义Frobenius流形的单值数据所构造而来的一族Virasoro算子的系数来表示. 然后利用Virasoro对称在tau函数上的作用的线性化条件研究了主方程簇的形变,导出了用于构造主方程簇形变的拟 Miura变换所满足的一组线性方程, 即广义Frobenius流形的圈方程.圈方程的解所对应的拟Miura变换给出了其主方程簇的一个形变, 我们称之为主方程簇的拓扑形变. 由于广义Frobenius流形的单位向量场不平坦, 上述理论的建立与通常Frobenius流形的 Dubrovin--Zhang 理论的建立有若干很不平凡的不同之处. 例如, 与通常的Frobenius流形的主方程簇相比, 在一个广义Frobenius 流形的主方程簇中我们必须引入无穷多额外的流? 再比如, 在 Virasoro算子的构造、圈方程的推导等过程中, 我们需要从n维广义Frobenius流形出发构造一个n+2维非半单的Frobenius 流形, 并利用该n+2维流形的几何结构? 最后,我们将上述理论应用到两个重要的广义Frobenius流形的例子, 通过具体求解它们的圈方程给出了其主方程簇的拓扑形变的低亏格近似, 并提出了这两个主方程簇的拓扑形变与可积系统理论中三个重要的可积方程簇之间联系的猜想,它们分别是Volterra 方程簇,q-形变KdV方程簇以及Ablowitz--Ladik方程簇.
The theory of Frobenius manifolds is an important research subject in modern mathematical physics, which has very important applications in the study of Gromov-Witten theory, singularity theory, mirror symmetry and integrable systems. One of the important parts of this theory is the Dubrovin-Zhang theory on the relationship between semisimple Frobenius manifolds and bihamiltonian integrable hierarchies. In the study of Hodge integrals and equivariant Gromov-Witten invariants, people also found that certain well-known integrable hierarchies are closely related to a class of generalized Frobenius manifolds. Compare with usual Frobenius manifolds, the unit vector fields of such generalized Frobenius manifolds are no longer flat with respect to the metrics of the generalized Frobenius manifolds, so we call them generalized Frobenius manifolds with non-flat unity, and we also call them generalized Frobenius for simplicity. The purpose of this thesis is to generalized the Dubrovin-Zhang theory to such a class of generalized Frobenius manifolds.The main result of this thesis is the development of the theory of topological deformations of the Principal Hierarchies of generalized Frobenius manifolds. We first construct a bihamiltonian integrable hierarchy of hydrodynamic type by using the geometric structure of a generalized Frobenius manifold, and we call it the Principal Hierarchy of the generalized Frobenius manifold. We prove that this integrable hierarchy possesses a tau structure and an infinite number of Virasoro symmetries, and these Virasoro symmetries can be represented by the coefficients of a set of Virasoro operators which can be constructed from the monodromy data of the generalized Frobenius manifold. Then we study deformations of the Principal Hierarchy of a generalized Frobenius manifold by using the condition of linearization of the action of the Virasoro symmetries on the tau function of the deformed integrable hierarchy, and we derive a system of linear equations, called the loop equations of the generalized Frobenius manifold, for the quasi-Miura transformation which is used to construct a deformation the Principal Hierarchy. For a generalized semisimple Frobenius manifold, the quasi-Miura transformation given by a solution of the loop equations yields a deformation of the Principal Hierarchy, which is called the topological deformation of the Principal Hierarchy. Due to the non-flatness of the unit vector field the establishment of the above theory is quite nontrivially different from that of the Dubrovin-Zhang theory for a usual Frobenius manifold in several steps. For example, compare to the Principal Hierarchy of a usual Frobenius manifold, we must introduce infinitely many additional flows to the Principal Hierarchy of a generalized Frobenius manifold. For another example, in order to construct the Virasoro operators and to derive the loop equations of an n-dimensional generalized Frobenius manifold, we need to introduce an n+2 dimensional non-semisimple Frobenius manifold and to use its geometric structure. Finally, we apply the above theory to two important examples of generalized Frobenius manifolds, and obtain the low-genus approximations of the topological deformations of their Principal Hierarchies by explicitly solving their loop equations. We propose some conjectures on the relationship between the topological deformations of the Principal Hierarchies of these two generalized Frobenius manifolds and three bihamiltonian integrable hierarchies which are well known in the theory of integrable systems, they are the Volterra hierarchy (also called the discrete KdV hierarchy), the q-deformed KdV hierarchy and the Ablowitz-Ladik hierarchy.
