The Mathematics of Real and Open Topological Strings

弦理论是一种通过振动弦来表示基本粒子的模型，其目的是统一自然界的四种基本力。虽然弦理论是当今物理学的主要范式之一，但它还没有做出实验上可检验的预测。然而，它产生了许多数学预测，导致了代数几何和辛拓扑的基本发展，特别是在全纯曲线方面。该项目的两个方向将进一步在所谓的真实的和开放的扇区中对弦理论进行数学测试，这些扇区长期以来落后于标准的封闭扇区，并发展相关的数学框架和不同数学领域之间的联系，包括枚举代数几何，纽结理论，这个项目将建立在建立弦理论真实的部分背后的数学基础的工作上，推进数学理解的预测所产生的这一部门，并建立数学基础的密切相关的开放部门弦理论。该项目还将利用较早的工作，建立了BCOV镜像对称预测，用于五次三倍中的亏格曲线的计数，并将继续当前博士生的最近工作，该工作引入了一种强大的拓扑技术，用于将稳定曲线的Deligne-Mumford模空间的同调关系提升到稳定（伪）该奖项反映了NSF的法定使命，并被认为是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。

String theory is a model that represents elementary particles by vibrating strings with the aim of unifying the four fundamental forces of nature. While string theory is one of the main paradigms in physics today, it has yet to make experimentally testable predictions. However, it has generated many mathematical predictions that have led to fundamental developments in algebraic geometry and symplectic topology, especially in relation to holomorphic curves. This project's two directions will further test string theory mathematically in the so-called real and open sectors, which have long lagged behind the standard closed sector, and develop the associated mathematical framework and connections between different fields of mathematics, including enumerative algebraic geometry, knot theory, and representation theory.This project will build on work that established the mathematical foundations behind the real sector of string theory, to advance the mathematical understanding of the predictions arising from this sector and to establish the mathematical foundations of the closely related open sector of string theory. The project will also utilize older work that established the BCOV mirror symmetry prediction for counts of genus one curves in a quintic threefold and it will continue the recent work of a current doctoral student that introduced a powerful topological technique for lifting homology relations from Deligne-Mumford moduli spaces of stable curves to moduli spaces of stable (pseudo-) holomorphic maps.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.