A Sheaf-Theoretic Approach to M5-Brane Geometry

该项目旨在建立和加强数学与理论物理之间的丰富联系，从而促进这两个领域的发展。在物理学中，m理论是一种统一不同弦理论的方法，这些弦理论本身为统一自然力提供了一个理论框架：重力和粒子理论。PI最近在数学上的进步将提供对物理量的严格计算和预测。此外，PI将继续监督年轻数学家，举办研讨会，并广泛传播和发表研究成果。除此之外，PI还将继续运营一个数学圈，并继续在西北大学的夏季桥梁项目中发挥核心作用，从而为更广泛的社区服务。皮·扎斯洛（PI Zaslow）在许多作品中率先将束理论应用于辛几何和镜像对称中。他最近关于legendrin结和拉格朗日曲面的不变量的研究为聚类理论找到了新的联系。在目前的工作中，Zaslow将把这些联系扩展到弦理论中对m5膜物理学非常感兴趣的维度：三维拉格朗日和二维勒让德曲面。新的和有趣的现象发生在这个关键的维度。Zaslow将做以下工作：1 .寻找包含拉格朗日填充的Legendrian曲面的全纯盘的超势编码计数，并在所有坐标系中建立Ooguri-Vafa完整性；2 .解释这些计数是如何由聚类理论（在每个属中，Donaldson-Thomas转换将它们与简单的构建块联系起来）中的（Seiberg-like）突变决定的；3 .建立“分幅对偶性”（即证明属-g模空间通过不同的分幅计算所有具有g节点的对称颤振的DT不变量）；4 .通过将q -变形通过扭曲的轴束纳入轴束理论，将这些结果在复三维空间中的非精确设置推广到已解形褶；5 .将此机制应用于结共法线，用束理论解释大n对偶性；6 .证明Zaslow-Treumann发现的拉格朗日量具有特殊的拉格朗日嵌入；通过镜像对称解释Goncharov-Kenyon-Beauville可积系统。

The project aims to establish and strengthen rich connections between mathematics and theoretical physics which will serve to advance both fields. In physics, M-theory is an approach to unifying the different string theories, which themselves provide a theoretical framework for unifying the forces of nature: gravity and particle theory. The PI's recent mathematical advances will provide rigorous calculations and predictions of quantities of physical interest. In addition, the PI will continue to supervise young mathematicians, run seminars and disseminate and speak widely about the research. Beyond this, the PI will serve the broader community by continuing to run a math circle and by continuing to play an central role in the summer Bridge program of Northwestern University.PI Zaslow, in a number of works, spearheaded the use of sheaf theory in symplectic geometry and mirror symmetry. His recent work on invariants of Legendrian knots and Lagrangian surfaces found new connections to cluster theory. In the present work, Zaslow will extend these linkages to the dimension of great interest to the physics of M5-branes in string theory: three-dimensional Lagrangians and two-dimensional Legendrian surfaces. New and interesting phenomena occur at this critical dimension. Zaslow will do the following: 1. find superpotentials encoding counts of holomorphic disks bounding Lagrangian fillings of Legendrian surfaces and establish Ooguri-Vafa integrality in all framings;2. explain how such counts are determined by (Seiberg-like) mutations from cluster theory (at each genus, Donaldson-Thomas transformations relate them to simple building blocks);3. establish "framing duality" (i.e., prove that the genus-g moduli space computes, via different framings, DT invariants of all symmetric quivers with g nodes);4. extend these results in complex three-space to the nonexact setting of the resolved conifold by incorporating Q-deformations into the sheaf theory via twisted sheaves;5. apply this machinery to knot conormals to explain large-N duality using sheaf theory;6. prove the Lagrangians found by Zaslow-Treumann have special-Lagrangian embeddings;7. explain the Goncharov-Kenyon-Beauville integrable system via mirror symmetry.