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，目的是统一自然的四种基本力量。虽然弦理论是当今物理学的主要范式之一，但它还没有做出可通过实验检验的预测。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，特别是与全纯曲线有关的发展。这个项目的两个方向将进一步在长期落后于标准封闭部门的所谓实部门和开放部门中对弦理论进行数学测试，并建立相关的数学框架和不同数学领域之间的联系，包括计数代数几何、纽结理论和表示理论。该项目将建立在建立实部门背后的数学基础的工作的基础上，促进对该部门产生的预测的数学理解，并建立密切相关的开放部门弦理论的数学基础。该项目还将利用为五次三次中的亏格一曲线的计数建立BCOV镜像对称性预测的旧工作，它将继续一位当前博士生最近的工作，他引入了一种强大的拓扑技术，将同调关系从稳定曲线的Deligne-Mumford模空间提升到稳定(伪)全纯映射的模空间。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势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.