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

镜像对称是三十多年前在理论物理学中发现的。从那时起，它一直是出现在许多数学前沿的一个深刻的谜。一般来说，辛几何是描述经典力学的自然框架，它与复杂的代数几何之间似乎有一种隐藏的关系。最近的发现将它与数学和物理的其他几个领域联系起来。然而，直到今天，人们对镜像对称还没有系统的理解。本课题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，这些问题出现在现代几何和拓扑学的前沿。特别是，该项目旨在构建一大类具体模型，以展示镜像对称的各个方面。研究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，传播最近的成果并促进进一步的进展。在最初的语境中，镜像对称将一个空间中的计数问题与其镜像对称空间上的复微分方程联系起来。随着理解的发展，镜像对称在某些情况下被确定为拉普拉斯变换，它提供了一种有效的机制，从其镜像的复杂几何计算空间的量子不变量。一般来说，辛几何与复代数几何是一种较高的范畴关系。在规范理论的背景下，镜像对称在代数群中以朗兰兹对偶的形式出现。最近的发现将它与量子结不变量联系起来。该项目的主要目标是建立复拉格朗日几何作为一个通用工具来研究几何问题和量化在现代几何和拓扑学的前沿，并建立模型，我们可以看到镜像对称的所有方面，包括真正的辛-复代数对偶，朗兰兹对偶，拉普拉斯变换，以及与三流形不变量的关系。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，远远超出了它的经典起源，延伸到高能粒子物理和弦理论。该项目旨在通过考虑在希钦可积系统上构建的具体模型，促进对在这种情况下出现的复杂可积系统及其量子力学对应物的理解。这项工作将采用一种被称为拓扑递归的拉普拉斯变换的推广，有望为这些空间的全纯量化提供一种推测性的新机制，并为镜像对称性提供新的线索。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。

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.