Homological Mirror Symmetry for Homogeneous Spaces

主要研究者的主要研究兴趣在于辛拓扑。辛拓扑关注辛流形的整体结构，辛流形是一类首先出现在经典力学中的相空间。格罗莫夫发展了一个伪全纯曲线的理论，这导致了辛拓扑学的许多进步，包括一类辛不变量，现在被称为格罗莫夫-威滕不变量。更复杂的不变量，即福谷范畴，是主流研究的主题。这些研究大部分来自物理学家在镜像对称的名义下所做的预测，镜像对称是数学和物理学的一个领域，在很大程度上仍然是理论性的。这些结构是惊人的，因为他们已经被理解为预测一个相当普遍的对应关系辛几何和代数几何-一个巨大的数学领域，其根源可以追溯到古代。人们可能会问的各种问题的几何辛流形可以回答的研究，而不是代数几何的一个不同的流形（它的“镜像”），反之亦然。这个项目将关注的镜像理论的研究，以经典的Bott-Borel-Weil建设代数几何。Bott-Borel-Weil给出了半单李群的所有有限维不可约表示在相应齐性空间上的等变向量丛的几何描述。后者是法诺品种，其中Kontsevich的同调镜像对称性已被广泛研究在过去的十年。特别是，这些品种的预测镜像合作伙伴是一个明确已知的朗道-金兹伯格模型。 同调镜像对称表明应该存在拉格朗日子流形，镜像等变向量丛，并且这样的拉格朗日子流形对的弗洛尔上同调应该形成李群表示的基础向量空间。在最近的一项工作中（与J. Pascaleff合作），PI定义了等变拉格朗日膜的概念，其中等变可以理解为镜像簇上李代数的代数作用。此外，还给出了上述镜像理论到Bott-Borel-Weil构造的最简单的非平凡例子。目前的项目将扩展这些建设的情况下，任意半单李代数的主要动机是确定一个典型的基础表示来自拉格朗日子流形的交叉点。

The principal investigator's main research interests lie in symplectic topology. Symplectic topology is concerned with global structures of symplectic manifolds, a class of spaces that appeared first as the phase space in classical mechanics. Gromov developed a theory of pseudoholomorphic curves, which has led to a number of advancements in symplectic topology, including a class of symplectic invariants now known as Gromov-Witten invariants. More sophisticated invariants, namely Fukaya categories, are the subject of mainstream research. Much of this research arises from predictions made by physicists under the name of mirror symmetry, an area of both mathematics and physics that remains largely conjectural. These conjectures are striking as they have been understood to predict a rather general correspondence between symplectic geometry and algebraic geometry - a huge field of mathematics whose roots go back to ancient times. Various questions one may ask about the geometry of a symplectic manifold can be answered by studying instead the algebraic geometry of a different manifold (its "mirror"), and vice-versa.This project will be concerned with a study of a mirror theory to the classical Bott-Borel-Weil construction in algebraic geometry. Bott-Borel-Weil gives a geometric description of all finite dimensional, irreducible representations of semi-simple Lie groups in terms of equivariant vector bundles on the corresponding homogeneous spaces. The latter are Fano varieties, for which Kontsevich's homological mirror symmetry has been studied extensively over the past decade. In particular, the predicted mirror partner to such varieties is an explicitly known Landau-Ginzburg model. Homological mirror symmetry suggests that there should be Lagrangian submanifolds, mirroring the equivariant vector bundles, and Floer cohomology of pairs of such Lagrangians should form the underlying vector space of a representation of the Lie group. In a recent work (jointly with J. Pascaleff), PI defined the notion of an equivariant Lagrangian brane where equivariance is to be understood with respect to an algebraic action of a Lie algebra on the mirror variety. In addition, the simplest non-trivial example of the aforementioned mirror theory to Bott-Borel-Weil construction was worked out. The current project will extend these constructions to the case of an arbitrary semisimple Lie algebra with the main motivation being the identification of a canonical basis of representations coming from intersections of Lagrangian submanifolds.