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FRG: Collaborative Research: Complex Lagrangians, Integrable Systems, and Quantization

FRG：协作研究：复杂拉格朗日量、可积系统和量化

基本信息

批准号：
2152130
负责人：
Justin Sawon
金额：
$ 74.03万
依托单位：
University of North Carolina at Chapel Hill
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2022
资助国家：
美国
起止时间：
2022-07-15 至 2025-06-30
项目状态：
未结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2152130&HistoricalAwards=false
关键词：
FRG Collaborative Research Complex Lagrangians

项目摘要

Mirror symmetry was discovered over three decades ago in theoretical physics. Since then, it has been a deep mystery appearing in many frontiers of mathematics. In general, it appears to be a hidden relation between symplectic geometry, which is the natural framework for describing classical mechanics, and complex algebraic geometry. More recent discoveries link it with several other areas of mathematics and physics. Still, as of today, there is no systematic understanding of mirror symmetry. The main goal of this focused research group is to establish a universal tool to study mirror symmetry and related geometric questions arising in modern frontiers of geometry and topology. In particular, the project aims at constructing a large class of concrete models that would demonstrate all aspects of mirror symmetry. The investigators bring expertise from different areas of mathematics and will use a variety of techniques. They will also organize workshops, summer schools, and conferences, aimed at training early career researchers in this area, disseminating recent results and facilitating further advances.In the original context, mirror symmetry relates counting problems in one space with complex differential equations on its mirror symmetric space. As understanding has evolved, mirror symmetry has been identified as the Laplace transform in certain cases, which provides an effective mechanism of computing quantum invariants of a space from the complex geometry of its mirror. In general, it is a higher categorical relation between symplectic geometry and complex algebraic geometry. In a gauge theoretic context, mirror symmetry appears in the form of the Langlands duality among algebraic groups. More recent discoveries link it with quantum knot invariants. The main goal of the project is to establish complex Lagrangian geometry as a universal tool to study geometric questions and quantization arising in modern frontiers of geometry and topology and to construct models in which we can see all aspects of mirror symmetry, including the real symplectic-complex algebraic duality, the Langlands duality, the Laplace transform, and a relation to three-manifold invariants. In Lagrange's formulation many non-linear problems can be solved via integrable systems, an approach which is widely applicable, extending far beyond its classical origins to high-energy particle physics and string theory. This project aims to advance understanding of the complex integrable systems that arise in such situations, together with their quantum mechanical counterparts, by considering concrete models constructed on Hitchin integrable systems. The work will employ a generalization of the Laplace transform known as topological recursion that is expected to provide a conjectural new mechanism for holomorphic quantization of these spaces and shed new light on mirror symmetry.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.

镜像对称性是三十多年前在理论物理学中发现的。从那时起，它一直是一个深刻的谜出现在许多数学的前沿。一般来说，它似乎是辛几何（描述经典力学的自然框架）和复代数几何之间的隐藏关系。最近的发现将它与数学和物理学的其他几个领域联系起来。然而，到今天为止，还没有对镜像对称的系统理解。这个重点研究小组的主要目标是建立一个通用的工具来研究镜像对称和相关的几何问题，这些问题出现在几何和拓扑学的现代前沿。特别是，该项目旨在构建一个大型的混凝土模型，以展示镜像对称的各个方面。调查人员带来了来自不同数学领域的专业知识，并将使用各种技术。他们还将组织讲习班，暑期学校和会议，旨在培训这一领域的早期职业研究人员，传播最近的成果，促进进一步的进展。在原来的背景下，镜像对称涉及计数问题在一个空间与复杂的微分方程的镜像对称空间。随着理解的发展，镜像对称在某些情况下被认为是拉普拉斯变换，它提供了一种有效的机制，可以从空间的镜像的复杂几何形状计算空间的量子不变量。一般说来，辛几何与复代数几何是一种更高的范畴关系。在规范理论中，镜像对称以代数群之间的朗兰兹对偶的形式出现。最近的发现将其与量子结不变量联系起来。该项目的主要目标是建立复杂的拉格朗日几何作为一个通用的工具来研究几何问题和量子化所产生的现代前沿的几何和拓扑结构，并构建模型，其中我们可以看到所有方面的镜像对称，包括真实的辛复代数对偶，朗兰兹对偶，拉普拉斯变换，和一个关系到三流形不变量。在拉格朗日的公式中，许多非线性问题可以通过可积系统来解决，这是一种广泛适用的方法，远远超出了其经典起源，延伸到高能粒子物理学和弦理论。该项目旨在通过考虑在希钦可积系统上构建的具体模型，促进对这种情况下出现的复杂可积系统及其量子力学对应系统的理解。这项工作将采用一个推广的拉普拉斯变换称为拓扑递归，预计将提供一个新的机制，这些空间的全纯量子化，并揭示了新的光mirror symmetrical.This奖项反映了NSF的法定使命，并已被认为是值得通过评估使用基金会的智力价值和更广泛的影响审查标准的支持。