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

镜面对称是三十多年前在理论物理学中发现的。从那时起，在数学的许多前沿领域，这一直是一个深层次的谜团。一般说来，它似乎是辛几何和复杂代数几何之间的隐藏关系，辛几何是描述经典力学的自然框架。更新的发现将它与数学和物理的其他几个领域联系起来。尽管如此，到今天为止，对于镜面对称还没有系统的理解。这个重点研究小组的主要目标是建立一个通用工具来研究现代几何和拓扑学前沿中出现的镜像对称和相关的几何问题。特别是，该项目的目标是构建一大类混凝土模型，以展示镜像对称的所有方面。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。他们还将组织研讨会、暑期班和会议，旨在培训这一领域的早期职业研究人员，传播最新成果并促进进一步发展。在最初的背景下，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。随着认识的发展，镜像对称性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，可以根据空间镜像的复杂几何来计算空间的量子不变量。一般说来，它是辛几何和复代数几何之间的一种较高的范畴关系。在规范理论的背景下，镜像对称以代数群之间的朗兰兹对偶的形式出现。最近的发现将它与量子结不变量联系在一起。该项目的主要目标是建立复拉格朗日几何作为研究现代几何和拓扑前沿中出现的几何问题和量子化的通用工具，并建立模型，在其中我们可以看到镜像对称的所有方面，包括实辛-复代数对偶、朗兰兹对偶、拉普拉斯变换以及与三维流形不变量的关系。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，这种方法广泛适用，远远超出了其经典起源，扩展到高能粒子物理和弦理论。这个项目旨在通过考虑建立在希钦可积系统上的具体模型来促进对这种情况下出现的复杂可积系统的理解，以及它们的量子力学对应物。这项工作将采用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子化提供一种猜想的新机制，并揭示镜像对称的新曙光。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势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.