Quantum Gravity in Noncommutative Geometry

理论物理学中最具挑战性的问题是如何将量子物理定律与爱因斯坦的广义相对论统一起来。在这个方向上，最受关注的思想是超弦理论和非交换几何。这两种方法密切相关。非交换几何为研究黎曼几何标准方法无法处理的空间提供了一个新的框架。它为规范理论的研究提供了一个新的视角，是描述普朗克能量下几何形状的一个有希望的候选者。该提案的主要重点是在亚普朗克尺度上研究时空的本质，确定动力学并寻找超弦理论与非对易几何之间的联系。寻找其极限为连续四维流形与复制标准模型的有限空间乘积的非交换时空是至关重要的。还将研究手性狄拉克算子在随机矩阵近似中的泛函积分，用连续参数变形时空，寻找谱作用边界项的热核方法，添加膨胀子、膨胀子和轴子，谱作用在奇点存在下的行为以及在普朗克长度数阶尺度上的动力学场的非摄动演化。将研究拓扑五维重力，具有圆柱对称的规则BPS解，以及重力系统的精确解。更广泛的影响：由于其跨学科的性质，这项工作，在非交换几何和弦理论的前沿，将有利于物理学家和数学家。贝鲁特美国大学（AUB）在中东地区推广美国教育体系方面发挥着非常重要的作用。从历史上看，重点是自由教育。新的目标是为促进研究提供一个充满活力的环境。2008年10月启动的理论物理博士项目就是朝这个方向迈出的一步，并有望吸引中东其他大学的高素质学生。

The most challenging problem in theoretical physics is the unification of the laws of quantum physics with Einstein's general theory of relativity. The ideas in this direction that have been pursued most forcefully are superstring theory and noncommutative geometry. These two approaches are intimately related. Noncommutative geometry has provided a new framework to study spaces which could not be handled by the standard methods of Riemannian geometry. It provides a new perspective in the study of gauge theories and is a promising candidate to describe geometries at Planckian energies. The main focus of this proposal is to study the nature of space-time at subplanckian scales, determining the dynamics and finding connections between superstring theory and noncommutative geometry. It is essential to find the noncommutative space-time whose limit is the product of a continuous four-dimensional manifold times a finite space that reproduces the standard model. Also to be invesigated will be the functional integral for chiral Dirac operators in the random matrix approximation, deforming space-time with continuous parameters, heat kernel methods to find boundary terms for the spectral action, adding an inflaton, dilaton and axion, the behavior of spectral actions in presence of singularities and nonperturbative evolution of the dynamical fields at scales of the order of the planck length. Topological five-dimensional gravity, regular BPS solutions with cylindrical symmetry, and exact solutions for the bigravity system will be studied. Broader impact : Due to its interdisciplinary nature, this work, on the frontiers of noncommutative geometry and string theory, will be of benefit to both physicists and mathematicians. The American University of Beirut (AUB) plays a very important role in promoting the American educational system in the Middle East. Historically, the emphasis was on liberal education. The new aim is to provide in addition a vibrant environment for the promotion of research. The launching of the Ph.D. program in theoretical physics in October 2008 is a step in this direction and will hopefully attract students of high caliber from other universities in the Middle East.