NSF-BSF: Derived and quantum corrected structures on arithmetic and geometric moduli

NSF-BSF：算术和几何模量的导出和量子校正结构

基本信息

批准号：
2200914
负责人：
Tony Pantev
金额：
$ 35.91万
依托单位：
University of Pennsylvania
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2022
资助国家：
美国
起止时间：
2022-08-01 至 2025-07-31
项目状态：
未结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2200914&HistoricalAwards=false
关键词：
NSF BSF Derived quantum corrected

项目摘要

This is a research in algebraic geometry. The field of algebraic geometry studies geometric models by distilling and encoding their essential complexity in polynomial equations. The project integrates ideas from quantum physics and the physical study of symmetries of the fundamental laws of nature to extract new and unexpected information about the geometry of spaces and the deep properties of number systems. The project will focus on unraveling the structure of hidden and broken symmetries of quantum fields and to capture this structure in computable invariants. These invariants will give a new mathematical tool for understanding and proving various empirically observed physics dualities, and also mysterious arithmetic dualities, which are expected to identify a priori unrelated quantum theories. The project will unify the analytic and geometric properties of parameter spaces of representations in arbitrary dimension and sets the stage for understanding the basic local symmetries of moduli problems in a way suitable for pragmatic use in a broad spectrum of applications. The work will be immediately relevant to deep questions in symplectic geometry, number theory, string theory and quantum field theory. This project provides research training opportunities for graduate students.Three directions will be studied. The first is to construct geometric realizations of arithmetic duality maps and will use geometry to produce new insights into the complexity of Galois representations. In the second project a new method is proposed for understanding maps between coisotropic branes, parametrizing the deformations of the composition laws for such maps, and extracting new symplectic information from such deformations. The final project uses the geometry of higher stacks to develop a duality and decomposition formalism for canceling anomalies and understanding secondary quantum symmetries in quantum field theory.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.

这是代数几何的研究。代数几何领域通过在多项式方程中提取和编码几何模型的本质复杂性来研究几何模型。该项目整合了量子物理学和自然基本定律对称性的物理研究的思想，以提取有关空间几何和数字系统深层属性的新的和意想不到的信息。该项目将致力于揭开量子场隐藏对称性和破缺对称性的结构，并以可计算的不变量形式捕获这种结构。这些不变量将为理解和证明各种经验观察到的物理对偶性以及神秘的算术对偶性提供新的数学工具，这些对偶性有望识别先验的不相关的量子理论。该项目将统一任意维度表示的参数空间的分析和几何性质，并为理解模问题的基本局部对称性奠定基础，以适合在广泛应用中实际使用的方式。这项工作将与辛几何、数论、弦论和量子场论中的深层问题直接相关。本项目为研究生提供研究训练机会。将研究三个方向。第一个是构建算术对偶图的几何实现，并将使用几何对伽罗瓦表示的复杂性产生新的见解。在第二个项目中，提出了一种新方法来理解各向同性膜之间的图，参数化此类图的合成定律的变形，并从此类变形中提取新的辛信息。最终项目使用更高堆栈的几何结构来开发对偶性和分解形式，以消除量子场论中的异常并理解二次量子对称性。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力优点和知识进行评估，被认为值得支持。更广泛的影响审查标准。