Frobenius流形的W-约束与零亏格类Virasoro约束

W-Constraints and Genus Zero Virasoro-like Constraints for Frobenius Manifolds

作者: 专业:数学 导师:张友金 年度:2013 学位:博士 

关键词
Frobenius流形 可积方程簇 W-约束 类Virasoro约束 拓扑场论

Keywords
Frobenius manifold, integrable hierarchy, W-constraints, Virasoro-likeconstraints, topological field theory
        Frobenius流形是二维拓扑场论的零亏格部分的几何抽象.它将二维拓扑场论与Gromov-Witten不变量理论、奇点理论、可积系统理论等不同的数学分支联系起来.发展Frobenius流形理论成为现代数学物理的一个重要的研究课题.在Frobenius流形理论中,最重要的研究对象之一是从Frobenius流形的零亏格primary自由能重构相应二维拓扑场论的全亏格自由能.本文我们利用Frobenius流形的几何结构,构造与Frobenius流形相联系的自由能满足的两类约束W-约束和零亏格“类Virasoro约束”,这些约束由Fock空间上的线性微分算子给出.对于某些Frobenius流形,我们证明这些约束结合已知的其它约束唯一地确定相应的全亏格自由能.具体地说,对于G-矩阵非退化的半单Frobenius流形,我们利用周期梯度的自然量子化构造其Heisenberg顶点代数上的自然扭模.然后用顶点代数的方法,通过对有限个屏蔽算子的核的取交集,我们定义Heisenberg顶点代数的一个子顶点代数,称为Frobenius流形的W-代数.我们证明W-代数中的元素在自然的扭的态-场映射下的像提供Givental量子化公式给出的完全的descendant势(全亏格自由能取指数)上的约束,称为W-约束.该结果的一个直接推论是完全的descendant势满足Virasoro约束,并且相应的Virasoro算子与Dubrovin-Zhang理论中使用的Virasoro算子相同.特别地,对于单奇点对应的半单Frobenius流形,我们证明相应W-约束在不计一个常数因子的意义下唯一地确定其完全的descendant势;同时我们给出这个唯一性定理在具体计算相交数上的应用.本文的另一个主要结果是对任意的Frobenius流形,构造并证明了其零亏格自由能上的一族新的约束.我们称之为零亏格类Virasoro约束,它的一个子集是零亏格Virasoro约束.在耦合引力的拓扑σ-模型中,零亏格类Virasoro约束可以视为W-约束的类比.特别地,对于耦合引力的P1拓扑σ-模型,我们证明零亏格类Virasoro约束与零亏格dilaton方程唯一地确定其零亏格自由能;从而,根据Dubrovin-Zhang理论,这些约束与高亏格Virasoro约束共同确定了以P1为靶空间的Gromov-Witten不变量.
    The notion of Frobenius manifold is a geometric abstraction of the genus zero partsof2D topological field theories (TFT). It connects2D TFT with several diferent researchbranches of mathematics, including the theory of Gromov-Witten invariants, singularitytheory and the theory of integrable systems, and the development of the theory of Frobe-nius manifolds is an important research subject in modern mathematical physics.One of the most important research activities, that are connected to the theory ofFrobenius manifolds, is to reconstruct the full genera free energy of a2D TFT from itsgenus zero primary free energy. In this thesis we construct, in terms of the geomet-ric structures of Frobenius manifolds, two families of constraints for the free energiesassociated to the Frobenius manifolds. These constraints, obtained from certain lineardiferential operators, are calledW-constraints and genus zero Virasoro-like constraints.We show, for certain classes of Frobenius manifolds, that these constraints themselves ortogether with other known constraints uniquely determine the associated full genera freeenergies.More specifically, for a semisimple Frobenius manifold with a non-degenerate G-matrix, we construct the natural twisted module over the Heisenberg vertex algebra as-sociated to it by employing the natural quantization of the gradients of periods. Then byusing a vertex algebra approach, we define a vertex subalgebra of the Heisenberg vertexalgebra as the intersection of kernels of certain screening operators. We call this vertexsubalgebra theW-algebra. One of our main results is that the image of theW-algebraunder the natural twisted state-field map provides constraints for the total descendantpotential (the exponential of the full genera free energy) given by Givental in terms ofcertain quantization formulas. These constraints are then called theW-constraints. Thedirect corollary of this result is that the total descendant potential satisfies the Virasoroconstraints, where the associated Virasoro operators coincide with those introduced inthe Dubrovin-Zhang theory on the relationship between Frobenius manifolds and inte-grable systems. In particular, for the class of semisimple Frobenius manifolds associatedto simple singularities, we prove that theW-constraints uniquely determine the total de-scendant potential up to a constant factor. We also calculate in several examples someintersection numbers by applying this uniqueness theorem. Another main result of the thesis is the construction of a new family of constraintsfor the genus zero free energy of an arbitrary Frobenius manifold. These constraintscontain the genus zero Virasoro constraints as a subset and are called the genus zeroVirasoro-like constraints. For the topologicalσ-model coupled to gravity, Virasoro-likeconstraints can be considered as an analogue of theW-constraints. In particular, fortheP1topologicalσ-model coupled to gravity, we show that the genus zero Virasoro-like constraints together with the genus zero dilaton equation uniquely determine thecorresponding genus zero free energy. As a result, according to the Dubrovin-Zhangtheory, these constraints together with the higher genus Virasoro constraints uniquelydetermine the Gromov-Witten invariants withP1as the target.
        

Frobenius流形的W-约束与零亏格类Virasoro约束

摘要3-4
Abstract4-5
第1章 引言7-18
第2章 预备知识18-28
    2.1 Frobenius流形18-26
    2.2 Givental辛空间26-28
第3章 半单Frobenius流形的顶点代数及其扭模28-54
    3.1 半单Frobenius流形的Heisenberg顶点代数28-31
    3.2 周期梯度的两类量子化31-38
    3.3 Heisenberg顶点代数上的扭模38-49
        3.3.1 广义δ-分布38-40
        3.3.2 L*-扭的W-模的两个构造40-49
    3.4 Givental相因子与格子顶点代数上的扭模49-54
第4章 一类半单Frobenius流形的W-代数与W-约束54-71
    4.1 W-代数的定义54-56
    4.2 W-约束定理及其证明56-58
    4.3 单奇点 约束的唯一性定理58-64
    4.4 一些例子 A_1,A_2,A_3,D_4等64-71
第5章 Frobenius流形的零亏格类Virasoro约束71-93
    5.1 Virasoro代数的ν-形变与零亏格类Virasoro约束定理72-76
    5.2 交换子的计算76-81
    5.3 R = 0 情形下定理 5.1 的证明81-84
    5.4 一般情形下定理5.1的证明84-88
    5.5 例子 类Virasoro约束在P~1拓扑σ-模型上的应用88-93
第6章 结论93-95
参考文献95-100
致谢100-102
个人简历、在学期间发表的学术论文与研究成果102


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