The Mathematics of Real and Open Topological Strings

弦理论是一种模型，它通过振动弦来表示基本粒子，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，但它尚未做出可实验验证的预测。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长期落后于标准封闭领域的所谓实领域和开放领域对弦理论进行数学测试，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、结论和表示理论。该项目将建立在建立弦理论真实部分背后的数学基础的基础上，推进对该部分产生的预测的数学理解，并建立密切相关的弦理论开放部分的数学基础。该项目还将利用以前的工作，即建立五次三重曲线中1属曲线计数的BCOV镜像对称预测，并将继续当前博士生最近的工作，该工作引入了一种强大的拓扑技术，用于将稳定曲线的delignee - mumford模空间提升到稳定（伪）全纯映射的模空间的同调关系。该奖项反映了美国国家科学基金会的法定使命，并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，被认为值得支持。

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.