String Compactifications: From Geometry to Effective Field Theory

该奖项资助了弗吉尼亚理工大学的劳拉·安德森、詹姆斯·格雷和埃里克·夏普教授的研究活动。弦理论是一种被提议的物理学框架，包括量子力学（描述亚原子粒子行为的理论）和引力（决定宇宙大尺度行为的力）。在弦理论中，物理学和几何学这一数学学科本质上是交织在一起的。虽然弦理论试图回答的问题是物理的，但通往这些答案的道路往往涉及现代数学中的前沿挑战。该奖项将资助一个合作研究项目，以探索从研究弦理论的各个方面产生的物理学。这项工作的目标包括为弦现象学的主题开发新的基础工具，这是提取实验物理弦理论的可观察结果的尝试。作为他们研究的一部分，安德森、格雷和夏普教授的目标是进一步界定和描述弦理论中出现的几何形状的类型。经验表明，当强烈的物理要求用几何术语表达时，它们可以在物理和数学中打开通往新的和意想不到的结果的大门。因此，该领域的研究通过促进基础科学的进步来促进国家利益。安德森教授、格雷教授和夏普教授将让初级科学家参与这个项目，其中包括一名博士后研究员和几名研究生，他们将参与合作研究。他们的努力还将包括组织会议和研讨会，这些会议和研讨会将增加物理学家和数学家之间在这两个领域边界的紧迫问题上的对话，同时积极鼓励将代表性不足的群体纳入科学进步的前沿。更具体地说，将研究弦理论的两个最灵活的四维紧化框架——即杂质弦理论和f理论。在异质弦理论中，新的几何工具将用于计算有效理论中先前未确定的方面。这些包括N=1物质场Kahler势、物理归一化汤川耦合和微扰超势的真空值。汤川耦合的非微扰贡献也将通过量子束上同调计算，这是普通量子上同调的一种推广。在f理论中，椭圆纤维几何的新结果将用于研究奇异Calabi-Yau流形的性质及其与Hitchin系统的联系，以及研究多纤维流形的普遍性对弦对偶性、有效理论和光滑Calabi-Yau变集的可能有界性的影响。在杂质/ f -理论对偶方面的最新进展将用于提取描述f -理论紧化的有效理论的新特征，包括通量对超势贡献的显式四维场依赖形式。最后，Anderson、Gray和Sharpe教授还将把他们最近的一些见解应用到SCFTs的模空间的整体结构中，以研究可能的沼泽猜想。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。

This award funds the research activities of Professors Lara Anderson, James Gray, and Eric Sharpe at Virginia Tech.String theory is a proposed framework of physics that encompasses both quantum mechanics (the theory describing the behavior of subatomic particles) and gravity (the force dictating the large-scale behavior of the Universe). In string theory, physics and the mathematical subject of geometry are intrinsically intertwined. While the questions that string theory attempts to answer are physical, the path to those answers frequently involves cutting-edge challenges in modern mathematics. This award will fund a collaborative program of research to explore the physics that arises from studying aspects of string theory. The goals of this work include developing new foundational tools for the subject of string phenomenology, which is the attempt to extract the observable consequences of string theory for experimental physics. As part of their research, Professors Anderson, Gray, and Sharpe aim to further bound and characterize the types of geometrical shapes that arise in string theory. Experience shows that when strong physical requirements are expressed in geometrical terms, they can open the door to new and unexpected results in both physics and mathematics. As a result, research in this area advances the national interest by promoting the progress of basic science. Professors Anderson, Gray, and Sharpe will involve junior scientists in this project, including a postdoctoral researcher and several graduate students who will take part in the collaborative research. Their efforts will also include the organizing of conferences and workshops that will increase dialog between physicists and mathematicians on pressing problems at the boundary of both fields, all while actively encouraging the inclusion of under-represented groups into the frontline of progress in the sciences.More specifically, two of the most flexible frameworks for four-dimensional compactifications of string theory --- namely heterotic string theory and F-theory --- will be investigated. Within heterotic string theory, novel geometric tools will be used to compute previously undetermined aspects of the effective theory. These include the N=1 matter field Kahler potential, physically normalized Yukawa couplings, and vacuum values of the perturbative superpotential. The non-perturbative contributions to Yukawa couplings will also be computed via quantum sheaf cohomology, a generalization of ordinary quantum cohomology. Within F-theory, new results in the geometry of elliptic fibrations will be used to study the properties of singular Calabi-Yau manifolds and their links to Hitchin systems, as well as to study the implications of the ubiquity of multiply fibered manifolds for string dualities, effective theories, and possible boundedness of the set of smooth Calabi-Yau varieties. Recent progress in heterotic/F-theory duality will be used to extract new features of the effective theories describing F-theory compactifications, including the explicit four-dimensional field-dependent form of flux contributions to the superpotential. Finally, Professors Anderson, Gray, and Sharpe will also apply some of their recent insights into the global structure of moduli spaces of SCFTs to study possible swampland conjectures.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.