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

这是一项代数几何的研究。代数几何领域通过提取和编码多项式方程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特性的新的和意想不到的信息。该项目将专注于揭示量子场的隐藏对称性和破缺对称性的结构，并在可计算不变量中捕捉这种结构。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无关的量子理论。该项目将统一任意维表示的参数空间的解析和几何性质，并为理解模问题的基本局部对称性奠定基础，以一种适合于广泛应用的实用方式。这项工作将直接与辛几何、数论、弦理论和量子场论的深层问题相关。本项目为研究生提供研究训练机会。将研究三个方向。首先是构建算术对偶映射的几何实现，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，提出了一种新的方法来理解各向同性膜之间的映射，参数化这些映射的组成规律的变形，并从这些变形中提取新的辛信息。最后的项目使用更高堆栈的几何来发展对偶和分解形式，以消除量子场论中的异常和理解次级量子对称性。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.