Mirror Symmetry and Frobenius Manifolds

该项目将研究Frobenius流形框架下的镜像对称猜想。他的猜想有很多层次，从关于霍奇数的纯粹拓扑陈述到关于a -无穷范畴的等价的最先进的表述。Frobenius流形描述了一个中间层次，在这个层次上，被认为与这种对称相关的结构比拓扑图更丰富，但也没有完整的不易处理的分类结构。在物理框架中，有一种利用基本构件和张量积和绕轨两种操作来构造镜像对称伙伴的方法。这两种操作之一，张量积一直是R. Kaufmann先前研究的对象，在Frobenius流形理论中得到了充分的理解。基本构建模块的等效将是具有孤立临界点的函数的奇异性的通用展开的一个版本。然而，最后一步仍有待完成；也就是轨道形成的定义。这相当于研究有限群行为，并定义范畴中正确的商。在过去的几年里，在弦理论的推动下，数学界和物理界之间进行了卓有成效的互动。例如，这使代数几何学者和物理学家联合起来研究镜像对称的问题。这是一种基于物理论证的推测对称，它对代数几何和枚举几何有着惊人的影响。

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.