Representation Theory, Calabi-Yau Manifolds, and Mirror Symmetry

该奖项将支持参加2022年11月28日至12月1日在哈佛大学数学科学与应用中心(CMSA)举行的表示理论、Calabi-Yau流形和镜像对称研讨会的研究人员。在过去的半个世纪里，量子场论导致了许多看似不同的数学领域的统一，包括几何、表示论、代数、组合学和数论。这次会议的目的是将这些领域的活跃研究人员聚集在一起，目的是加强联系，促进交流，并在物理学和数学的各个领域之间产生新的想法。这项提议是一项持续努力的一部分，目的是在严格的数学水平上理解拓扑镜对称和弦理论的所有方面，将这种理解传递给数学家和物理学家，并为这两个群体在一个蓬勃发展的学科中的密集互动创建基础设施。镜像对称是物理学家在20世纪90年代发现的，揭示了辛几何和复杂几何之间的深刻二元性。它将一般难以计算的Gromov-Witten不变量转化为复分析中可计算且满足某些偏微分方程组的周期积分。对于给定的Calabi-Yau流形，构造它的镜像是一个长期存在的问题。著名的Strominger-Yau-Zaslow程序提出，Calabi-Yau流形具有特殊的拉格朗日环面纤维，其镜像可以通过取环面纤维的对偶来构造。顶点算符代数(VOAS)是由共形场理论发展而来的一种代数结构，并被BorCherds公理化。某些VOAs可以几何地实现为模空间上同调环上的算子的代数。一个关键的例子是Gaiotto和Rapcak的Y-代数在某些环Calabi-Yau三重环的尖峰瞬时子模空间上的作用。Rapcak，Soibelman，Yang和赵已经提出了这一图景的广泛推广，并且是基于Hall代数在模空间的上同调上的作用。另一方面，某些环状Calabi-Yau三重环的箭图形式及其NC形变可以通过镜像对称从Fukaya范畴构造出来。因此，霍尔代数作用在辛几何中具有有趣的意义。这次会议将为这些领域的研究人员提供一个交流的论坛，希望解决这些猜想，并找到镜像对称性和表象理论之间的进一步联系。研讨会的网页是https://cmsa.fas.harvard.edu/homological-mirror-symmetry-workshop/.This奖，反映了国家科学基金会的法定使命，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。

This award will support the participation of researchers at the workshop Representation Theory, Calabi-Yau Manifolds, and Mirror Symmetry to he held November 28-December 1, 2022 at the Center of Mathematical Sciences and Applications (CMSA) at Harvard University. Quantum field theory has led to a unification of many seemingly disparate areas of mathematics during the past half century, including geometry, representation theory, algebra, combinatorics, and number theory. The purpose of this conference is to bring together active researchers in these areas with the goal of strengthening ties, fostering communication, and generating new ideas at the interface of physics and various areas of mathematics. This proposal is part of an ongoing effort directed at understanding all aspects of topological mirror symmetry and string theory at mathematical levels of rigor, transmitting this understanding to both mathematicians and physicists, and creating the infrastructure for intensive interactions between these two groups within a single, flourishing subject. Mirror symmetry, discovered by physicists in the 1990s, reveals a deep duality between symplectic and complex geometries. It transforms Gromov-Witten invariants, which are difficult to compute in general, to period integrals in complex analysis, which are computable and satisfy certain PDEs. For a given Calabi-Yau manifold, it is a longstanding problem to construct its mirror. The celebrated Strominger-Yau-Zaslow program proposed that a Calabi-Yau manifold has a special Lagrangian torus fibration, and its mirror can be constructed by taking the dual of the torus fibers. Vertex operator algebra (VOAs) are algebraic structures that arose from conformal field theory, and were axiomatized by Borcherds. Certain VOAs can be realized geometrically as algebras of operators on the cohomology rings of moduli spaces. A key example is the action of the Y-algebras of Gaiotto and Rapcak on the moduli space of spiked instantons of certain toric Calabi-Yau threefolds. A vast generalization of this picture has been proposed by Rapcak, Soibelman, Yang, and Zhao, and is based on the action of the Hall algebra on the cohomology of moduli spaces. On the other hand, the quiver formulation of some toric Calabi-Yau threefolds and their nc deformations can be constructed from the Fukaya category via mirror symmetry. Thus the Hall algebra action has an interesting meaning in symplectic geometry. This conference will provide a forum for researchers in these fields to communicate in the hope of resolving these conjectures and finding further connections between mirror symmetry and representation theory. The workshop web page is https://cmsa.fas.harvard.edu/homological-mirror-symmetry-workshop/.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.