Conformal Field Theories in Holography and String Theory

宇宙中有四种基本的力。电磁力和两种核力用量子场论（QFT）描述；引力是由广义相对论（GR）描述的。弦理论旨在统一这四种力，同时提供引力的量子力学描述。在这一努力中，尺度不变的量子场论，即所谓的共形场论（CFTs）至关重要：一方面，它们描述了弦如何运动和相互作用；另一方面，通过全息原理，他们可以描述量子引力。该项目旨在通过研究弦理论和量子傅里叶变换及其应用，加深对弦理论和量子傅里叶变换的理解，重点关注它们的数学结构及其与数学的联系。这项工作旨在构建可以用于弦理论的cft的新例子。此外，该项目将对教育产生更广泛的影响，为本科生和研究生提供合适的活动和研究项目，从而使他们了解当代研究。PI还将与亚利桑那大学数学教师招聘和保留中心合作，开发模块，向高中生介绍物理和其他STEM学科的定量思维和数学推理。在更技术性的层面上，PI将发展二维共形微扰理论，重点关注轨道CFTs。他将利用这一点来研究Calabi-Yau sigma模型模空间中有理cft和非理性cft之间的相互作用，并研究全息二维cft（如对称轨道）的光谱如何在其模空间上变化。他还将使用置换轨道和晶格轨道构造新的全息CFT，这可能会导致AdS/CFT对偶性的新例子。他将特别关注具有稀疏光谱的cft和极值或近极值cft。这项研究将依赖于最近的数学成果，如顶点算子代数的非阿贝尔轨道理论。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。

There are four fundamental forces in the universe. The electromagnetic force and the two nuclear forces are described by Quantum Field Theory (QFT); the gravitational force is described by the theory of General Relativity (GR). String Theory aims to unify these four forces, while at the same time providing a quantum mechanical description of gravity. In this endeavor, quantum field theories that are scale invariant, so called conformal field theories (CFTs), are crucial: On the one hand they describe how strings move and interact; on the other hand, through the principle of holography, they can describe quantum gravity. The PI aims to deepen the understanding of string theory and QFTs by investigating such CFTs and their applications, with a focus on their mathematical structure and their connection to mathematics. The work aims to construct new examples of CFTs that can be used in string theory. In addition, this project will have a broader impact on education by offering suitable activities and research projects for undergraduate and graduate students, thereby introducing them to contemporary research. The PI will also work with the Center for Recruitment and Retention of Mathematics Teachers at the University of Arizona to develop modules that introduce high school students to quantitative thinking and mathematical reasoning in physics and other STEM disciplines.At a more technical level, the PI will develop 2D conformal perturbation theory with a focus on orbifold CFTs. He will use this to investigate the interplay between rational and irrational CFTs in moduli spaces of Calabi-Yau sigma models, and to investigate how the spectrum of holographic 2D CFTs such as symmetric orbifolds changes over their moduli space. He will also construct new holographic CFTs using permutation orbifolds and lattice orbifolds, which may lead to new examples of AdS/CFT dualities. He will particularly focus on CFTs with sparse light spectrum and extremal or near extremal CFTs. This research will rely on recent results from mathematics such as the theory of non-abelian orbifolds of vertex operator algebras.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.