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.

过去的四十年给人带来了不断增强的欣赏，即量子颗粒的相互作用受根本上的几何和代数结构控制。一方面，这使代数和几何形状在粒子物理学中极为有价值。另一方面，可以将强大的物理直觉用于纯数学的发展。在这种传统中，该项目的研究目标符合更广泛的努力，以将现代的数学思想和方法纳入量子场理论中，反之亦然。 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几何表示理论。 PI和他的合作者已经建立了3D超对称仪表理论和符号二元性中的镜像对称性之间建立了深厚关系的开始。 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).在这个项目中，PI和他的合作者将使用规格理论中的新技术来建立与符号双重性相关的类别中对象的系统构造，并在它们之间建立明确的二元性图。他还将定义和调查另一个类别，即福卡 - 塞德尔类型，仪表理论预测在符号双重性中等同于通常的类别，但似乎使二元性表现出许多微妙的方面。最后，他将将开发的方法扩展到研究3D量表理论到4D超对称理论，从而导致构建线路运营商类别的构建 - 对分类墙壁缝制的fomulas和cluster代数的分类具有深远的影响，这些奖项通过NSF的法定宣传和宽广的构建构成的构建构成的构建构成，这是众所周知的范围。 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.

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

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