Poisson Geometry, Quantum Moduli, and Geometric Dualities

泊松几何、量子模和几何对偶

基本信息

批准号：
1901876
负责人：
Tony Pantev
金额：
$ 34.44万
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：
2019
资助国家：
美国
起止时间：
2019-08-01 至 2024-07-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1901876&HistoricalAwards=false
关键词：
Poisson Geometry Quantum Moduli Geometric

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. The project will focus on unravelling the hidden structure of multi-dimensional geometric models of quantum fields and to capture this structure in a cascade of polynomial invariants. These invariants will give a new mathematical tool for understanding and proving various empirically observed physics dualities which are expected to identify a priori unrelated quantum theories. The project aims to unify the analytic and geometric properties of parameter spaces of representations in arbitrary dimension and sets the stage for understanding the basic structure of moduli problems in a way suitable for pragmatic use in a broad spectrum of applications. The proposed work will be immediately relevant to deep questions in symplectic geometry, geometric representation theory, string theory and quantum field theory.Three directions will be studied. The first is to investigate how the Hodge theory of varieties with potentials and the associated perverse and weight filtrations are exchanged by T-duality. In the second project a new method will be developed for constructing moduli of irregular connections on non-compact algebraic manifolds, for building explicit symplectic foliations, and for computing their leaves. This will involve a new de Rham theory of formal boundaries of varieties, and a new construction of sheaf theoretic invariants from the formal geometry. The final project will build a spectral coverformalism for constructing Hecke eigensheaves on moduli of bundles related to intersections of quadrics.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的思想和自然基本定律对称性的物理研究，以提取有关空间几何的新的和意想不到的信息。该项目将专注于揭开量子场多维几何模型的隐藏结构，并在一系列多项式不变量中捕获该结构。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。该项目旨在统一任意维度表示的参数空间的分析和几何性质，并为以适合在广泛应用中实际使用的方式理解模问题的基本结构奠定基础。所提出的工作将直接与辛几何、几何表示论、弦理论和量子场论中的深层问题相关。将研究三个方向。第一个是研究具有潜力的品种霍奇理论以及相关的反常和权重过滤如何通过 T 对偶性进行交换。在第二个项目中，将开发一种新方法，用于构造非紧代数流形上的不规则连接模，用于构建显式辛叶状结构，并计算它们的叶状结构。这将涉及新的品种形式边界的德拉姆理论，以及形式几何中束理论不变量的新构造。最终项目将建立一个谱覆盖形式主义，用于在与二次曲面相交相关的束模上构造赫克本征滑轮。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。