Homological Mirror Symmetry for Homogeneous Spaces

主要研究者的主要研究兴趣在于辛拓扑。辛拓扑研究辛流形的整体结构，辛流形是经典力学中最早作为相空间出现的一类空间。Gromov发展了伪全纯曲线理论，这导致了辛拓扑的许多进展，包括一类现在被称为Gromov-Witten不变量的辛不变量。更复杂的不变量，即Fukaya范畴，是主流研究的主题。这项研究的大部分来自物理学家以镜像对称的名义做出的预测，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猜测。这些猜想令人震惊，因为它们被理解为预测了辛几何和代数几何之间相当普遍的对应关系-这是一个巨大的数学领域，其根源可以追溯到古代。关于辛流形的几何的各种问题，可以通过研究不同流形的代数几何(它的“镜像”)来回答，反之亦然。这个项目将涉及到对代数几何中经典的Bott-Borel-Weil构造的镜像理论的研究。Bott-Borel-Weil用等变向量丛的形式给出了半单李群在相应齐次空间上的所有有限维不可约表示的几何刻画。后者是Fano变种，在过去的十年中，Kontsevich的同调镜像对称性已被广泛研究。特别是，预测这些品种的镜像伙伴是一个明确已知的朗道-金兹堡模型。同调镜像对称性表明，应该存在拉格朗日子流形，映射等变向量丛，而这样的拉格朗日对的Floer上同调应该形成李群表示的基本向量空间。在最近的一项工作中(与J·帕斯卡尔夫联合)，Pi定义了等变拉格朗日膜的概念，其中关于镜像簇上的李代数的代数作用的等变是被理解的。此外，还给出了上述镜像理论对Bott-Borel-Weil结构的最简单的非平凡例子。本项目将把这些构造推广到任意半单李代数的情形，其主要动机是识别来自拉格朗日子流形的交集的表示的典范基。

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.