Mirror Symmetry and Frobenius Manifolds

镜像对称和弗罗贝尼乌斯流形

• 批准号: 0070681
• 负责人: Ralph Kaufmann
• 金额: $ 8.23万
• 依托单位: University of Southern California
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2000
• 资助国家: 美国
• 起止时间: 2000-06-01 至 2003-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0070681&HistoricalAwards=false
• 关键词: Mirror; Symmetry; Frobenius; Manifolds

The project will study the mirror-symmetry conjecture in the framework of Frobenius manifolds. his conjecture has many levels, from a purely topological statement about Hodge numbers to the most advanced formulation in terms of equivalences of A-infinity categories. There is an intermediate level described by Frobenius manifolds, where the structures which are supposed to be related by this symmetry are richer than the topological picture, but also do not have the full not easily handleable categorical structure. In the physical framework there is a way to construct mirror-symmetric partners by using elementary building blocks and the two operations of tensor product and orbifolding. One of the two operations, the tensor product has been the object of previous research of R. Kaufmann and is fully understood within the theory Frobenius manifolds. The equivalent of the elementary building blocks will be a version of the miniversal unfoldings of singularities of functions with isolated critical points. The last step, however, remains to be completed; i.e. the definition of orbifolding. This amounts to studying finite group actions and defining the correct quotient in the category.In the past few years a fruitful interaction between the mathematics and physics communities has developed driven by the subject of string theory. This has for instance united algebraic geometers and physicists in looking into questions about mirror symmetry. This is a conjectural symmetry based on physical arguments, which has striking implications for algebraic and enumerative geometry.

该项目将在Frobenius流形的框架内研究镜像对称猜想。他的猜想有很多层次，从纯粹的关于Hodge数的拓扑陈述到A-无穷大范畴的等价性方面的最高级表述。有一个由Frobenius流形描述的中间层次，其中被认为与这种对称性相关的结构比拓扑图更丰富，但也不具有完整的、不易处理的范畴结构。在物理框架中，有一种方法可以通过使用基本的积木块以及张量积和orbilolding两种运算来构造镜像对称的伙伴。作为两种运算之一，张量积一直是R.Kaufmann以前研究的对象，并且在Frobenius流形理论中得到了充分的理解。基本构件的等价物将是具有孤立临界点的函数奇点的极小展开的一个版本。然而，最后一步仍有待完成；即，对ORBORBOLDING的定义。这相当于研究有限群作用和定义范畴中的正确商。在过去的几年里，在弦理论的推动下，数学和物理界之间进行了卓有成效的互动。例如，这使得代数几何学家和物理学家联合起来研究关于镜面对称性的问题。这是一种基于物理论证的猜想对称性，它对代数几何和计数几何有着惊人的影响。