Geometry and Arithmetic in Theoretical Physics

这个跨学科的研究项目结合了方法和激励问题，在理论物理与数论和算术几何的数学方法。近年来，这两个非常不同的研究领域之间出现了令人惊讶的联系：首席研究员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，以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。该项目包括若干相关调查。高能物理中用来描述粒子物理实验中事件的费曼图计算具有丰富但尚未完全理解的数学结构，涉及代数簇的动机和周期，这是代数几何，算术几何和数论的研究对象。 该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。类似的数学结构的基础上的周期和动机的代数簇出现在模型的修改重力应用宇宙学;这一新的，意想不到的发展将深入调查的一部分，研究。该项目还将研究弦理论的“全息对应”的离散化框架，在体空间上的引力和边界上的共形场论之间，使用p-adic数建立在数论术语中。该项目旨在将AdS/CFT对应的这种p-adic形式发展成一个更大的理论框架，以期用算术术语解释时空几何从信息中的出现。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）构建拓扑量子计算的任意子模型，沿着二维表面具有奇点。研究人员还在追求量子统计力学的新数论方法，其中量子系统的属性对应于解析数论中研究的某些类zeta函数的属性，并且系统的平衡态捕获伽罗瓦对称性。最后，研究人员正在开发一种新的方法来分布式计算在理论计算机科学的基础上，在物理学中开发的动态三角测量技术作为量子引力的模型。

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.