Tautological systems and quantum differential equations of homogeneous spaces

同义反复系统和齐次空间的量子微分方程

• 批准号: 527733662
• 负责人: Professor Dr. Christian Sevenheck
• 金额: --
• 依托单位: Professur Algebra
• 依托单位国家: 德国
• 项目类别: Research Grants
• 资助国家: 德国
• 项目状态: 未结题
• 来源: https://gepris.dfg.de/gepris/projekt/527733662?language=en
• 关键词: Tautological; systems; quantum; differential; equations

This project aims at extending our knowledge of the mirror symmetry phenomenon for homogeneous spaces, which are a very important class of algebraic varieties defined by transitive group actions. Mirror symmetry is well understood in various aspects (enumerative, Hodge theoretic, homological) for toric varieties, however, many questions remains unanswered when stepping out of the toric world. The current proposal intends to make considerable progress for various classes of homogeneous spaces using several rather different approaches. On the one hand, we seek to extend the known mirror theorems to larger classes of homogeneous spaces, partly by concrete identification of the quantum-D-modules with the character D-modules of the mirror. This technique shall in particular be used in the case of cominuscule spaces. On the other hand, we plan to make systematic use of the theory of tautological systems, which are a vast generalization of the classical hypergeometric GKZ-D-modules. Building on the recently achieved functorial construction of tautological system, we seek to describe the quantum-D-modules of homogeneous spaces from them via a dimensional reduction, and to study related differential system such as Frenkel-Gross connections and generalized Kloosterman D-modules. We will also use Hodge theoretic results on tautological systems to study the irregular Hodge filtration for those D-modules. When relating the classical, Lie-theoretic approach to the mirror conjecture with these more involved considerations concerning tautological systems, we will re-interpret canonical Landau-Ginzburg models as dimensional reductions of Lefschetz fibrations of the mirror variety. We will also study the representation-theoretic constructions of canonical Landau-Ginzburg models further, in order to generalize this construction beyond the class of cominuscule spaces.

这个项目的目的是扩大我们的知识镜像对称现象的齐性空间，这是一类非常重要的代数簇定义的传递群作用。镜像对称在各个方面（枚举的，霍奇理论的，同调的）都得到了很好的理解，但是，当走出环面世界时，许多问题仍然没有答案。目前的建议打算取得相当大的进展，各类齐性空间使用几个相当不同的方法。一方面，我们试图将已知的镜像定理推广到更大类的齐性空间，部分是通过具体的标识量子D-模与镜像的特征D-模。这种技术应特别用于微丛空间的情况。另一方面，我们计划系统地利用重言式系统的理论，重言式系统是经典超几何GKZ-D-模的一个广泛推广。在重言式系统的函子构造的基础上，我们试图通过降维来描述齐次空间的量子D模，并研究相关的微分系统，例如Frenkel-Gross联络和广义Kloosterman D模。我们还将利用重言式系统的Hodge理论结果来研究这些D-模的非正则Hodge滤子。当把经典的李群理论方法与这些涉及重言式系统的考虑联系起来时，我们将把经典的朗道-金斯堡模型重新解释为镜像簇的莱夫谢茨纤维化的降维。我们还将进一步研究规范Landau-Ginzburg模型的表示论构造，以便将这种构造推广到可余空间类之外。