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.

这是代数几何形状的研究。代数几何形状领域通过蒸馏和编码其在多项式方程中的基本复杂性来研究几何模型。该项目将量子物理学的思想和对自然基本定律的对称性的物理研究整合在一起，以提取有关空间几何形状和数字系统的深层特性的新的和意外的信息。该项目将着重于阐明量子场隐藏和破碎的对称性的结构，并在可计算中的不变性中捕获这种结构。这些不变的人将提供一种新的数学工具，用于理解和证明各种经验观察到的物理二重性，以及神秘的算术二重性，这些偶性有望确定先验无关的量子理论。该项目将在任意维度中统一表示形式的参数空间的分析和几何特性，并为理解模量问题的基本局部对称性设定了阶段，以适合在广泛应用程序中实用使用的方式。这项工作将与符合性几何学，数理论，弦理论和量子场理论中的深层问题有关。该项目为研究生提供了研究培训机会。将研究三个方向。首先是构建算术偶性图的几何实现，并将使用几何形状对Galois表示的复杂性产生新的见解。在第二个项目中，提出了一种新的方法来理解共同体麸皮之间的地图，对此类地图的组成定律的变形进行了参数，并从此类变形中提取了新的符合性信息。最终项目使用较高堆栈的几何形状来开发二元性和分解形式主义，以取消异常和理解量子场理论中的次要量子对称性。该奖项反映了NSF的法定任务，并被认为是值得通过基金会的知识分子和更广泛影响的评估来通过评估来获得支持的。