Quantum Gravity in Noncommutative Geometry

理论物理中最具挑战性的问题是量子物理定律与爱因斯坦广义相对论的统一。超弦理论和非对易几何在这个方向上得到了最有力的追寻。这两种方法是密切相关的。非对易几何为研究用标准黎曼几何方法所不能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，是描述普朗克能量几何的一个很有前途的候选者。这一提议的主要焦点是在次普朗克尺度上研究时空的性质，确定动力学，并找到超弦理论和非对易几何之间的联系。找到一个非对易时空是至关重要的，它的极限是一个连续的四维流形乘以一个复制标准模型的有限空间的乘积。此外，还将研究手征Dirac算符在随机矩阵近似下的泛函积分，用连续参数变形时空，寻找光谱作用的边界项的热核方法，添加膨胀、膨胀和轴子，存在奇点时的光谱作用的行为，以及在Planck长度量级的动态场的非微扰演化。将研究拓扑五维重力、具有柱对称性的正则BPS解和重力系的精确解。更广泛的影响：由于它的跨学科性质，这项工作，在非对易几何和弦理论的前沿，将对物理学家和数学家都有好处。贝鲁特美国大学(AUB)在推动美国在中东的教育体系方面发挥着非常重要的作用。从历史上看，重点是通识教育。新的目标是为促进研究提供一个充满活力的环境。2008年10月推出的理论物理博士项目就是朝着这个方向迈出的一步，有望吸引来自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.