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。在弦理论——量子引力基础理论的提议——中，物理学和几何学的角色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会带来现代数学中的前沿挑战。该奖项将资助一个合作研究项目，以探索弦紧化产生的物理学。这项工作的目标包括通过为弦现象学主题开发新的基础工具来加强弦理论与当前粒子物理学进展之间的联系。此外，安德森、格雷和夏普教授的目标是进一步限定和表征弦紧化中产生的几何形状。经验表明，当强烈的物理要求用几何语言表达时，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。他们的努力将包括组织会议和研讨会，以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。在学生培训和专业对话的所有这些方面，安德森、夏普和格雷教授都致力于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，pi将研究弦理论的两种最灵活的四维紧化框架：杂散弦理论和f理论。在异质弦理论中，对物理和几何模空间的新颖描述将用于计算有效理论中先前未确定的方面，包括N=1物质场Kahler势和物理归一化汤川耦合。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。这项工作将探索新的对偶性，包括（0,2）镜像对称，以及SCFT的模空间的整体结构。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和有效理论的影响。几何学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。

This award funds the research activities of Professors Lara Anderson, James Gray, and Eric Sharpe at Virginia Tech.In string theory --- a proposal for a fundamental theory of quantum gravity --- 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 leads to cutting-edge challenges in modern mathematics. This award will fund a collaborative program of research to explore the physics that arises from string compactifications. The goals of this work include strengthening the links between string theory and current progress in particle physics by developing new foundational tools for the subject of string phenomenology. In addition, Professors Anderson, Gray and Sharpe aim to further bound and characterize the geometries arising in string compactifications. 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 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 include the organizing of conferences and workshops that will increase dialog between physicists and mathematicians on pressing problems at the boundary of both fields. In all of these aspects of student training and professional dialog, Professors Anderson, Sharpe and Gray are committed to actively encouraging the inclusion of under-represented groups into the frontline of progress in the sciences. More specifically, the PIs will study two of the most flexible frameworks for four-dimensional compactifications of string theory: Heterotic string theory and F-theory. Within heterotic string theory, novel descriptions of the physical and geometric moduli spaces 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. The nonperturbative contributions to Yukawa couplings will also be computed via quantum sheaf cohomology, a generalization of ordinary quantum cohomology. This work will explore new dualities including (0,2) mirror symmetry, as well as the global structure of the moduli space of SCFT's. 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 and effective theories. Recent progress in geometry 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.