String Compactifications: From Geometry to Effective Field Theory

该奖项资助弗吉尼亚理工大学的劳拉·安德森教授、詹姆斯·格雷教授和埃里克·夏普教授的研究活动。弦理论是关于自然如何运行的基本理论的建议，在该理论中，物理学和几何学的作用本质上是相互交织的。虽然弦理论试图回答的问题是物理问题，但通向这些答案的道路往往涉及现代数学中的前沿挑战。该奖项资助了一个探索弦理论产生的物理学的合作研究计划。由于弦理论预测了长度、宽度和高度之外的额外不可见维度，这些额外维度必须以一种使它们太小而无法用当前实验检测到的方式来“卷曲”。然而，弦理论所预测的物理特性，在很大程度上取决于这些卷曲维度的几何性质。这项工作的目标包括加强弦理论与粒子物理学当前进展之间的联系，部分是通过界定和表征与这些卷曲的维度相关的几何图形。经验表明，当强烈的物理要求用几何学的语言来表达时，它们可以为物理和数学领域的新的和意想不到的结果打开大门。因此，这一领域的研究为基础物理学提供了新的见解，从而促进了国家利益。安德森、格雷和夏普教授也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。他们的努力将包括组织会议和研讨会，增加物理学家和数学家之间关于两个领域之间紧迫问题的对话。在学生培训和专业对话的所有这些方面，安德森、格雷和夏普教授致力于积极鼓励将代表不足的群体的成员纳入科学进步的第一线。更具体地说，安德森、格雷和夏普教授将研究一类新的对称(称为更高形式的对称)，以及弦论四维紧凑化的两个最灵活的框架，即杂交弦理论和F理论。他们还将研究一种名为分解的引人注目的应用。这是一种观察，即具有某些更高形式对称性的量子场论等价于其他量子场论的不相交并。在杂化弦理论中，新的几何工具将被用来计算先前未确定的有效理论方面，包括N=1物质场Kahler势和物理归一化Yukawa耦合(包括非微扰贡献)。这项研究的目标将是了解绳子紧凑过程中粒子的质量和相互作用。此外，还将开发新的工具来研究杂化弦理论中的拓扑变化转变的物理学。在F理论的背景下，椭圆原纤化几何的新结果将被用来研究光滑的Calabi-Yau变种集的可能有界性、杂化/F理论对偶以及通量对超势贡献的显式四维场相关形式。最后，安德森、格雷和夏普教授还将应用他们对SCFT模空间的全局结构的一些最新见解来研究可能的沼泽猜想。这一奖项反映了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。

This award funds the research activities of Professors Lara Anderson, James Gray, and Eric Sharpe at Virginia Tech.String theory is a proposal for a fundamental theory of how nature operates in which the roles of physics and 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 funds a collaborative program of research to explore the physics that arises from string theory. Because string theory predicts extra unseen dimensions beyond length, width, and height, these extra dimensions must be "curled up" in ways that render them too small to be detected with current experiments. However, the physics that string theory predicts depends crucially on the geometric properties of these curled-up dimensions. The goals of this work include strengthening the links between string theory and current progress in particle physics, in part by bounding and characterizing the geometries associated with these curled-up dimensions. Experience shows that when strong physical requirements are expressed in the language of geometry, 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 providing new insights into fundamental physics. Professors Anderson, Gray and Sharpe will also involve junior scientists in this project, including a postdoctoral researcher and several graduate students. Their efforts will include the organizing of conferences and workshops that will increase dialog between physicists and mathematicians on pressing problems at the boundary between both fields. In all of these aspects of student training and professional dialog, Professors Anderson, Gray and Sharpe are committed to actively encouraging the inclusion of members of under-represented groups into the frontline of progress in the sciences.More specifically, Professors Anderson, Gray, and Sharpe will investigate a new class of symmetries (known as higher-form symmetries), and two of the most flexible frameworks for four-dimensional compactifications of string theory, namely heterotic string theory and F-theory. They will also investigate a striking application known as decomposition. This is an observation that quantum field theories with certain higher-form symmetries are equivalent to disjoint unions of other quantum field theories. Within heterotic string theory, novel geometric tools will be used to compute previously undetermined aspects of the effective theory, including the N=1 matter field Kahler potential and physically normalized Yukawa couplings (including non-perturbative contributions). The goal of this study will be to understand the masses and interactions of particles within string compactification. Furthermore, new tools will be developed to study the physics of topology-changing transitions within heterotic string theory. In the context of F-theory, new results in the geometry of elliptic fibrations will be used to study the possible boundedness of the set of smooth Calabi-Yau varieties, heterotic/F-theory duality and 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.