Quasimaps to Nakajima Varieties

给定空间中曲线的计数是计数几何的一个基本问题。这个问题的起源可以追溯到量子物理学，特别是弦理论，其中曲线计数为基本粒子提供了跃迁幅度。在这个项目中，PI将研究随着Nakajima颤动变化而出现的空间的这个问题。这些空间配备了以量子环群表示编码的内部对称性。由于这些对称性，中岛箭图簇的计数几何极其丰富，并与许多数学领域联系在一起。更好地理解Nakajima箭图簇的计数几何将在表示论、代数几何、数论、组合学和理论物理中产生新的结果。这一领域的许多开放问题适合研究生研究项目，并将为学生快速介绍当代数学的许多高级领域提供理想的机会。更具体地说，这个项目将研究和计算具有各种边界条件的Nakajima箭图簇的拟映射的生成函数，揭示这些函数之间的新的对偶，并证明由3D镜像对称性启发的开放猜想。该项目还将通过分析p-add场上的量子微分方程来揭示生成函数的新的算术性质。将使用的主要技术工具包括拟映射模空间的(代数)几何、等变椭圆上同调、量子循环群的表示理论以及量子Knizhnik-Zamolodchikov方程的解的积分表示。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。

Counting curves in a given space is a fundamental problem of enumerative geometry. The origin of this problem can be traced back to quantum physics, and especially string theory, where the curve counting provides transition amplitudes for elementary particles. In this project the PI will study this problem for spaces that arise as Nakajima quiver varieties. These spaces are equipped with internal symmetries encoded in representations of quantum loop groups. Thanks to these symmetries, the enumerative geometry of Nakajima quiver varieties is extremely rich and connected with many areas of mathematics. A better understanding of the enumerative geometry of Nakajima quiver varieties will lead to new results in representation theory, algebraic geometry, number theory, combinatorics and theoretical physics. Many open questions in this field are suitable for graduate research projects and will provide ideal opportunities for students' rapid introduction to many advanced areas of contemporary mathematics. More specifically, this project will investigate and compute the generating functions of quasimaps to Nakajima quiver varieties with various boundary conditions, uncover new dualities between these functions, and prove open conjectures inspired by 3D-mirror symmetry. The project will also reveal new arithmetic properties of the generating functions via the analysis of quantum differential equations over p-adic fields. The main technical tools to be used include the (algebraic) geometry of quasimap moduli spaces, equivariant elliptic cohomology, representation theory of quantum loop groups, and integral representations of solutions of the quantum Knizhnik-Zamolodchikov equations.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.