CAREER: The Algebraic Structures of Three-Dimensional Gauge Theory

职业：三维规范理论的代数结构

基本信息

批准号：
1753077
负责人：
Tudor Dan Dimofte
金额：
$ 40万
依托单位：
University of California-Davis
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2018
资助国家：
美国
起止时间：
2018-07-01 至 2022-08-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1753077&HistoricalAwards=false
关键词：
CAREER Algebraic Structures Three Dimensional

项目摘要

The past forty years have brought an ever-increasing appreciation that the interactions of quantum particles are controlled by fundamentally geometric and algebraic structures. This has on one hand made algebra and geometry extremely valuable in particle physics; and on the other hand has allowed powerful physical intuition to be used in the development of pure mathematics. Continuing in this tradition, the research goals of this project fit into a broader effort to incorporate modern mathematical ideas and methodology in quantum field theory and vice versa. In educational components of the project, the PI will host a summer school on this theme for graduate students in theoretical physics and mathematics and he will develop a series of lectures for undergraduate and graduate students, freely available online, that delineate various connections between modern algebra/geometry and quantum field theory.In more detail, the PI will combine physical and mathematical approaches to uncover new structures in supersymmetric gauge theory and geometric representation theory. The PI and his collaborators have established the beginnings of a deep relation between mirror symmetry in 3d supersymmetric gauge theories and symplectic duality. Symplectic duality is an equivalence of geometric categories associated to pairs of algebraic-symplectic varieties that was conjectured by Braden, Licata, Proudfoot, and Webster, and that generalizes many classic results in geometric representation theory (such as the Koszul duality of Beilinson, Ginzburg, and Soergel relating categories of modules for simple Lie algebras). In this project, the PI and his collaborators will use new techniques in gauge theory to establish a systematic construction of objects in the categories relevant for symplectic duality, and an explicit duality map between them. He will also define and investigate yet another category, of Fukaya-Seidel type, that gauge theory predicts to be equivalent to the usual categories in Symplectic Duality, but which seems to make many subtle aspects of the duality manifest. Finally, he will extend the methods developed to study 3d gauge theories to 4d supersymmetric gauge theories, where they lead to a construction of the categories of line operators -- with far-reaching implications for categorification of wall-crossing fomulas and of cluster algebras.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.

在过去的四十年里，人们越来越认识到量子粒子的相互作用是由基本的几何和代数结构控制的。一方面，这使得代数和几何在粒子物理学中极其有价值；另一方面，另一方面使得强大的物理直觉能够用于纯数学的发展。延续这一传统，该项目的研究目标符合更广泛的努力，将现代数学思想和方法论纳入量子场论，反之亦然。在该项目的教育部分，PI将为理论物理和数学研究生举办一个以此主题为主题的暑期学校，他将为本科生和研究生开发一系列在线免费讲座，描述现代代数之间的各种联系/几何和量子场论。更详细地说，PI将结合物理和数学方法来揭示超对称规范理论和几何表示理论中的新结构。 PI 和他的合作者已经开始建立三维超对称规范理论中的镜像对称性与辛对偶性之间的深层关系。辛对偶性是与代数辛簇对相关的几何范畴的等价，由 Braden、Licata、Proudfoot 和 Webster 猜想，它概括了几何表示论中的许多经典结果（例如 Beilinson、Ginzburg 的 Koszul 对偶性、和 Soergel 相关的简单李代数模块类别）。在这个项目中，PI和他的合作者将使用规范理论中的新技术来建立与辛对偶相关的类别中的对象的系统构造，以及它们之间的显式对偶图。他还将定义和研究深谷-赛德尔类型的另一个类别，规范理论预测该类别相当于辛对偶中的常见类别，但这似乎使对偶的许多微妙方面显现出来。最后，他将把为研究 3d 规范理论而开发的方法扩展到 4d 超对称规范理论，从而构建线算子的类别，这对穿墙公式和簇代数的分类具有深远的影响。该奖项反映了 NSF 的法定使命，并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，被认为值得支持。