Three-Dimensional Mirror Symmetry for Characteristic Classes on Bow Varieties

求解多项式方程组的几何表现涉及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，例如圆锥的顶点。有太多复杂的可能奇点，对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（例如数字或数字序列）分配给它们，使得离散不变量编码奇点的一些几何性质。这些离散不变量的一种类型是所谓的“特征类”，它有许多变体和风味。值得注意的是，最近发现的一类特殊奇点，称为“稳定包络”，也出现在理论物理学的弦理论中。它与弦理论的关系暗示了迄今为止特征类的隐藏对称性。也就是说，它预言有对看似无关的代数簇，其稳定的奇点包络重合。该项目的主要目标是在其数学设置中证明这一陈述并导出几何应用。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Nakajima bow品种推广到更广泛的一类品种，称为Cherkis bow品种。弓簇的几何研究的代数组合学基础是NS 5-D5膜配置，它比弓簇或齐性空间更完整。这一事实引起了新的操作弓品种：Hanany-Witten过渡和组合三维镜像对称。PI将利用这些操作建立一个霍尔代数结构上的椭圆上同调的弓品种，推广上同调和K理论霍尔代数的箭图。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motivic不变量之间的一致性。同一椭圆霍尔代数的维数计算参数承诺唐纳森-托马斯不变量，以及椭圆函数之间的恒等式，推广量子双对数恒等式。该奖项反映了NSF的法定使命，并已被认为值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估的支持。

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.