Three-Dimensional Mirror Symmetry for Characteristic Classes on Bow Varieties

求解多项式方程组的几何表现包括研究这样一个系统的所有解的“形状”，这被称为代数簇。代数簇中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，某些称为“奇点”的点不具有这种光滑特性，例如圆锥体的顶点。有太多复杂的可能奇点，它们的理解在物理学、机器人学和经济学中都有应用。研究奇点的一个关键工具是将可计算的离散不变量(例如数或数序列)赋给它们，使离散不变量编码奇点的一些几何属性。这些离散不变量的类型是所谓的“特征类”，它有许多变体和风格。值得注意的是，最近发现的一类特征奇点，称为“稳定包络”，也出现在理论物理的弦理论中。与弦理论的关系表明，到目前为止，特征类具有一种隐藏的对称性。也就是说，它预测存在一对看似无关的代数簇，它们的稳定奇点包络重合。该项目的主要目标是在其数学背景下证明这一说法，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袋品种推广到更广泛的称为切尔基斯弓形品种的品种。弓形簇几何研究的代数组合是NS5-D5膜组态，它比箭形簇或齐次空间更为完备。这一事实引发了对弓变体的新运算：Hanany-Witten转变和组合3D镜面对称性。PI将利用这些运算在弓簇的椭圆上同调上建立Hall代数结构，推广箭图的上同调和K理论Hall代数。椭圆霍尔代数结构补充了奇点的几何研究，这两者将一起用来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。同样的椭圆霍尔代数的维计参数承诺了Donaldson-Thomas不变量，以及推广了量子双对数恒等式的椭圆函数之间的恒等式。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.