A Sheaf-Theoretic Approach to M5-Brane Geometry

该项目旨在建立和加强数学和理论物理之间的丰富联系，这将有助于推进这两个领域。 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：引力和粒子理论。 PI最近的数学进步将提供严格的计算和预测物理量的兴趣。 此外，PI将继续监督年轻的数学家，举办研讨会，传播和广泛谈论研究。 除此之外，PI将服务于更广泛的社区，继续运行一个数学圈，并继续发挥核心作用的夏桥计划的西北大学。PI Zaslow，在一些工程，率先使用层理论辛几何和镜像对称。他最近的工作不变量的勒让德结和拉格朗日曲面发现新的连接到集群理论。 在目前的工作中，Zaslow将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。在这个关键维度上出现了新的有趣的现象。Zaslow将执行以下操作：1.找到编码勒让德曲面的拉格朗日填充边界的全纯盘的计数的超势，并在所有框架中建立Ooguri-Vafa积分;2.解释这些计数是如何由来自簇理论的（Seiberg样）突变确定的（在每个属中，Donaldson-Thomas变换将它们与简单的构建块相关联）;3.建立“框架二元性”（即，证明亏格-g模空间通过不同的框架计算具有g个节点的所有对称箭图的DT不变量）;4.通过扭曲层将Q-变形引入层理论，将这些结果在复三维空间中推广到解析锥的非精确设置;5.应用这一机制结conormal解释大N对偶使用层理论;6.证明Zaslow-Treumann发现的拉格朗日函数具有特殊的拉格朗日嵌入;7.用镜像对称解释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.