CAREER: The Algebraic Structures of Three-Dimensional Gauge Theory

职业：三维规范理论的代数结构

• 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

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。这一方面使得代数和几何在粒子物理学中极其有价值;另一方面也使得强有力的物理直觉被用于纯数学的发展。继续这一传统，该项目的研究目标适合更广泛的努力，将现代数学思想和方法纳入量子场论，反之亦然。在该项目的教育部分，PI将举办一个暑期学校在这个主题的研究生在理论物理和数学，他将开发一系列讲座，为本科生和研究生，免费提供在线，描绘现代代数/几何和量子场论之间的各种联系。更详细地说，PI将结合联合收割机物理和数学方法来揭示超对称规范理论和几何表示理论的新结构。PI和他的合作者已经建立了3d超对称规范理论中的镜像对称和辛对偶之间的深刻关系的开端。辛对偶是与代数-辛簇对相关的几何范畴的等价，由Braden、Licata、Proudfoot和韦伯斯特提出，推广了几何表示论中的许多经典结果（如Beilinson、Ginzburg和Soergel的Koszul对偶，涉及简单李代数的模范畴）。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结构，以及它们之间的显式对偶映射。他还将定义和研究另一个范畴，即萨雅-赛德尔类型，规范理论预测它与辛对偶中的通常范畴等价，但它似乎使对偶的许多微妙方面显现出来。最后，他将把研究3d规范理论的方法扩展到4d超对称规范理论，在那里，它们导致了线路操作员类别的构建-对墙的分类具有深远的影响-该奖项反映了NSF的法定使命，并被认为值得通过使用基金会的智力价值和更广泛的影响进行评估来支持审查标准。

项目成果

Coulomb branches of star-shaped quivers

星形箭袋的库仑支

• DOI: 10.1007/jhep02(2019)004
• 发表时间: 2019
• 期刊: Journal of High Energy Physics
• 影响因子: 5.4
• 作者: Dimofte, Tudor;Garner, Niklas
• 通讯作者: Garner, Niklas

(0,2) dualities and the 4-simplex

(0,2) 对偶性和 4-单纯形

• DOI: 10.1007/jhep08(2019)132
• 发表时间: 2019
• 期刊: Journal of High Energy Physics
• 影响因子: 5.4
• 作者: Dimofte, Tudor;Paquette, Natalie M.
• 通讯作者: Paquette, Natalie M.

Mirror symmetry and line operators

• DOI: 10.1007/jhep02(2020)075
• 发表时间: 2019-07
• 期刊: Journal of High Energy Physics
• 影响因子: 5.4
• 作者: Tudor Dimofte;Niklas Garner;M. Geracie;J. Hilburn

Secondary Products in Supersymmetric Field Theory

超对称场论中的二次积

• DOI: 10.1007/s00023-020-00888-3
• 发表时间: 2020
• 期刊: Annales Henri Poincaré
• 作者: Beem, Christopher;Ben-Zvi, David;Bullimore, Mathew;Dimofte, Tudor;Neitzke, Andrew
• 通讯作者: Neitzke, Andrew