Representation Theory, Calabi-Yau Manifolds, and Mirror Symmetry

该奖项将支持研究人员参加11月28日至2022年12月1日在哈佛大学数学科学与应用中心（CMSA）举行的代表性理论，卡-丘流形和镜像对称研讨会。在过去的半个世纪里，量子场论使许多看似不同的数学领域得到了统一，包括几何学、表示论、代数学、组合学和数论。本次会议的目的是汇集这些领域的活跃研究人员，以加强联系，促进沟通，并在物理学和数学的各个领域的接口产生新的想法。这个提议是一个持续努力的一部分，目的是在严格的数学水平上理解拓扑镜像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单一的，蓬勃发展的学科中的密集互动创造基础设施。 镜像对称性，由物理学家在20世纪90年代发现，揭示了辛几何和复杂几何之间的深刻二重性。它将一般难以计算的Gromov-Witten不变量转化为复分析中的周期积分，这些周期积分是可计算的并且满足一定的偏微分方程。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方案提出了一种Calabi-Yau流形具有一种特殊的拉格朗日环面纤维化，它的镜像可以通过取环面纤维的对偶来构造。顶点算子代数（VOA）是从共形场论中产生的代数结构，由Borcherds公理化。某些VOA可以几何地实现为模空间的上同调环上的算子代数。一个关键的例子是Y-代数的Gaiotto和Rapcak的模空间上的尖峰瞬子的某些复曲面卡-丘三倍。Rapcak、Soibelman、Yang和Zhao提出了这幅图的一个广泛推广，它是基于Hall代数对模空间上同调的作用。另一方面，某些复曲面Calabi-Yau三重及其nc变形的复曲面形式可以通过镜像对称从福谷范畴构造出来。因此Hall代数作用在辛几何中有着有趣的意义。本次会议将为这些领域的研究人员提供一个交流的论坛，希望解决这些问题，并找到镜像对称和表示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NSF的法定使命，并已被认为值得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的支持。

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.