Geometry and Arithmetic in Theoretical Physics

这个跨学科的研究项目将理论物理中的方法和激励问题与数论和算术几何的数学方法结合起来。近年来，这两个非常不同的研究领域之间的惊人联系已经被观察到：首席研究员已经参与了高能物理和数论之间关系的研究，她目前正在开发新的数学方法来更好地理解这种神秘的联系。该项目的目标是将这种联系从最初的量子场论扩展到其他领域，包括统计物理、弦理论和量子信息。这个项目包括几个相关的调查。高能物理中用于描述粒子物理实验事件的费曼图计算具有丰富但尚未完全理解的数学结构，涉及代数变化的动机和周期，是代数几何、算术几何和数论的研究对象。本项目旨在开发新技术，以提高对量子场论中这些算术结构的理解。类似的基于周期和代数变量动机的数学结构出现在为应用于宇宙学而开发的修正引力模型中；这一意想不到的新发展将作为研究的一部分进行深入调查。该项目还将研究一个离散化弦理论“全息对应”的框架，在整体空间上的引力和边界上的共形场理论之间，用p进数建立在数论术语中。该项目旨在将AdS/CFT对应的p进形式发展成一个更大的理论框架，以期用算术术语解释从信息中出现的时空几何。这项工作还将继续发展另一种新的方法来研究几何和信息之间的关系，即基于爱因斯坦方程的特殊解（引力瞬子）的拓扑量子计算的任意子模型的构建，这些模型沿着二维表面具有奇点。研究者也在寻求量子统计力学的新数论方法，其中量子系统的性质对应于解析数论中研究的某些类型的zeta函数的性质，并且系统的平衡状态捕获伽罗瓦对称性。最后，研究者正在开发一种在理论计算机科学中分布式计算的新方法，该方法基于物理学中作为量子引力模型的动态三角测量技术。

This interdisciplinary research project combines methods and motivating questions arising in theoretical physics with mathematical methods of number theory and arithmetic geometry. Surprising connections between these two very different areas of research have been observed in recent years: the principal investigator has been involved in the study of these relationships between high-energy physics and number theory, and she is currently developing new mathematical approaches towards a better understanding of this mysterious connection. The goal of this project is to extend this connection beyond its original occurrence in quantum field theory to other areas, including statistical physics, string theory, and quantum information. This project comprises several related investigations. The Feynman diagram calculations used in high-energy physics to describe events in particle physics experiments have a rich and not yet fully understood mathematical structure, involving motives and periods of algebraic varieties, which are objects of study in algebraic geometry, arithmetic geometry, and number theory. This project aims to develop new techniques to improve understanding of these arithmetic structures in quantum field theory. Similar mathematical structures based on periods and motives of algebraic varieties occur in models of modified gravity developed for applications to cosmology; this new, unexpected development will be investigated in depth as part of the research. The project will also study a framework for a discretization of the "holographic correspondence" of string theory, between gravity on a bulk space and conformal field theory on the boundary, built in number-theoretic terms using p-adic numbers. The project aims to develop this p-adic form of the AdS/CFT correspondence into a larger theoretical framework, with a view to explaining in arithmetic terms the emergence of spacetime geometry from information. The work will also continue development of another novel approach to the relation between geometry and information, a construction of anyon models of topological quantum computation based on special solutions (gravitational instantons) of the Einstein equations, with singularities along two-dimensional surfaces. The investigator is also pursuing new number theoretic approaches to quantum statistical mechanics, where the properties of the quantum systems correspond to properties of certain classes of zeta functions studied in analytic number theory, and the equilibrium states of the systems capture Galois symmetries. Finally, the investigator is developing a new approach to distributed computing in theoretical computer science, based on the technique of dynamical triangulations developed in physics as a model of quantum gravity.