Conformal Field Theories in Holography and String Theory

宇宙中有四种基本力。电磁力和两种核力由量子场论（QFT）描述;引力由广义相对论（GR）描述。弦理论的目标是统一这四种力，同时提供引力的量子力学描述。在这一奋进中，尺度不变的量子场论，即所谓的共形场论（Conformal Field Theory，CFTs），是至关重要的：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旨在通过调查此类CFTs及其应用来加深对弦理论和QFT的理解，重点关注它们的数学结构及其与数学的联系。这项工作的目的是构建可以用于弦理论的CFTs的新例子。此外，该项目将通过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，对教育产生更广泛的影响。PI还将与亚利桑那大学的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。在更技术的层面上，PI将开发二维共形扰动理论，重点是轨道CFTs。他将利用这一点来研究卡-丘sigma模型的模空间中理性和非理性CFTs之间的相互作用，并研究全息2D CFTs的光谱如何在其模空间中变化，如对称orbifolds。他还将使用置换orbifolds和晶格orbifolds构建新的全息CFTs，这可能会导致AdS/CFT对偶的新例子。他将特别关注稀疏光谱和极值或接近极值的CFTs。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NSF的法定使命，并已被认为是值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估的支持。

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.