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Three-Dimensional Mirror Symmetry for Characteristic Classes on Bow Varieties

弓品种特征类的三维镜像对称

基本信息

批准号：
2200867
负责人：
Richard Rimanyi
金额：
$ 13万
依托单位：
University of North Carolina at Chapel Hill
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2022
资助国家：
美国
起止时间：
2022-08-15 至 2025-07-31
项目状态：
未结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2200867&HistoricalAwards=false
关键词：
Three Dimensional Mirror Symmetry Characteristic

项目摘要

The geometric manifestation of solving systems of polynomial equations involves studying the "shape" of all solutions of such a system, which is called an algebraic variety. Most of the points of an algebraic variety are smooth in the sense that their neighborhoods look like n-dimensional linear spaces. However, certain points called “singularities” do not have this smooth property, for example the vertex of a cone. There is a plethora of complicated possible singularities, and their understanding has applications in physics, robotics, and economics. A key tool to study singularities is assigning calculable discrete invariants (for example numbers or sequences of numbers) to them in such a way that the discrete invariant encodes some of the geometric properties of the singularity. On type of these discrete invariants are what are called "characteristic classes," which come in many variants and flavors. Remarkably, a recently discovered characteristic class of singularities, called a "stable envelope," also appears in string theory in theoretical physics. The relation to string theory suggests a so-far hidden symmetry of characteristic classes. Namely, it predicts that there are pairs of seemingly unrelated algebraic varieties whose stable envelopes of singularities coincide. The main goal of the project is to prove this statement in its mathematical setting and derive geometric applications. The project will provide research training opportunities for graduate students. The PI will generalize the concept of stable envelopes from Nakajima quiver varieties to a broader class of varieties called Cherkis bow varieties. The algebraic combinatorics underlying the geometric study of bow varieties are NS5-D5 brane configurations, which are more complete than that of quiver varieties or homogeneous spaces. This fact gives rise to new operations on bow varieties: Hanany-Witten transition and combinatorial 3d mirror symmetry. The PI will utilize these operations to build a Hall algebra structure on the elliptic cohomology of bow varieties, generalizing cohomological and K theoretic Hall algebras of quivers. The elliptic Hall algebra structure complements the geometric study of singularities, and the two together will be used to organize an inductive proof of a three-dimensional mirror symmetry statement. The cohomological and K-theoretic limits provide coincidences among motivic invariants of singularities. Dimension count arguments of the same elliptic Hall algebra promise Donaldson-Thomas invariants, as well as identities among elliptic functions that generalize quantum dilogarithm identities.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

求解多项式方程组的几何表现包括研究该方程组所有解的“形状”，这被称为代数变分。代数变量的大多数点都是光滑的，因为它们的邻域看起来像n维线性空间。然而，某些被称为“奇点”的点不具有这种光滑性，例如圆锥的顶点。有太多复杂的可能的奇点，它们的理解在物理学、机器人和经济学中都有应用。研究奇点的一个关键工具是给奇点分配可计算的离散不变量（例如数字或数字序列），这样离散不变量就可以编码奇点的一些几何性质。这些离散不变量中的一类被称为“特征类”，它们有许多变体和风格。值得注意的是，最近发现的奇点特征类，称为“稳定包络”，也出现在理论物理的弦理论中。与弦理论的关系暗示了迄今为止隐藏的特征类的对称性。也就是说，它预测有一对看似无关的代数变异，其稳定奇异包络重合。该项目的主要目标是在数学环境中证明这一说法，并推导出几何应用。该项目将为研究生提供研究培训机会。PI将把稳定包络的概念从中岛箭箭品种推广到更广泛的品种类别，称为切尔基斯弓品种。弓簇几何研究的代数组合学基础是NS5-D5膜构型，它比颤动簇或齐次空间更完备。这一事实引起了对弓变体的新运算：Hanany-Witten跃迁和组合三维镜像对称。PI将利用这些运算在弓的椭圆上同调上建立霍尔代数结构，推广颤振的上同调和K理论霍尔代数。椭圆霍尔代数结构补充了奇点的几何研究，两者将一起用于组织三维镜像对称陈述的归纳证明。上同调极限和k理论极限提供了奇点的动机不变量之间的重合。同一椭圆Hall代数的维数参数保证了Donaldson-Thomas不变量，以及椭圆函数间的恒等式，推广了量子二对数恒等式。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。