Conformal Field Theories in Holography and String Theory

宇宙中有四种基本力量。电磁力和两个核力用量子场论来描述，引力用广义相对论来描述。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(CFT)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，它们可以描述量子引力。PI旨在通过研究弦理论和QFT及其应用来加深对弦理论和QFT的理解，重点是它们的数学结构和它们与数学的联系。这项工作旨在构建可用于弦理论的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适当的活动和研究项目，从而向他们介绍当代研究。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学科中的定量思维和数学推理。在更技术的层面上，PI将开发2D共形微扰理论，重点是两个或两个CFT。他将利用这一点来研究Calabi-Yau sigma模型的模空间中有理和无理CFT之间的相互作用，并研究全息2D CFT(如对称的orbilold)在其模空间中的谱如何变化。他还将利用置换和晶格排列来构造新的全息CFT，这可能会导致ADS/CFT对偶的新例子。他将特别关注光谱稀少的CFT和极端或接近极端的CFT。这项研究将依赖于数学的最新结果，如顶点算子代数的非交换或二叉形理论。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。

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.