Non-abelian Structures in String Theory

爱因斯坦的广义相对论描述了大型物体之间的引力：地球和太阳之间的引力，使地球保持在围绕太阳的轨道上，或者是自大爆炸以来一直决定着宇宙膨胀速度的力。广义相对论是一个优雅而准确的理论，它的预言已经被观测彻底检验了。它的中心思想是空间和时间是动力学实体，而重力只不过是时空被变形。量子力学是关于原子以及原子的组成成分的小的理论。从超导性到化学反应的速度，从电子的行为到最近发现的希格斯粒子，它控制着一切。量子理论的预测与实验结果的一致性达到了前所未有的准确性。广义相对论和量子力学是我们描述宇宙的两种范式；然而，人们并不知道如何将它们结合到一个单一的框架中。我们缺乏一个统一的方法来描述物理现象，这限制了我们研究宇宙中一些最迷人的物体(如黑洞)的能力，这些物体同时具有量子力学和引力性质。弦理论目前是我们将这两种理论结合在一起的框架的最佳候选者。它对我们认为在我们的宇宙中可能发生的事情有一些深远的影响，例如额外维度的存在。弦理论如何让广义相对论和量子力学结合在一起的细节还没有被很好地理解：一种观点是，空间和时间不是基本概念，相反，时空及其动力学(即引力)可能来自某些微观自由度，类似于水的运动来自大量水分子的集体行为。因此，广义相对论应该与时空一起从一些比空间和时间本身更基本的物体的量子力学模型中出现。这样的表述将成为量子引力的终极理论。为了实现这一目标，我的工作是通过弦理论工具来探索涌现几何及其性质。*在这个提议中，我概述了一个研究计划，该计划汇集了量子引力研究中的几个最新进展，如全息术、纠缠熵、非对易几何和矩阵模型。这里的中心主题是非阿贝尔对象：x乘以y不等于y乘以x的数学实体，其中最简单的例子是方阵。这种非阿贝尔实体在弦理论中以称为D膜的基本物体的坐标出现，为我们提供了了解时空真实性质的线索。解开这些线索是我工作的主要目的。

Einstein's theory of General Relativity describes gravitational forces between large objects: the attraction between the Earth and the Sun that holds the Earth in its orbit around the Sun, or the forces that have been determining the rate of expansion of the Universe since the Big Bang. General Relativity is an elegant and accurate theory and its predictions have been thoroughly tested by observations. Its central idea is that space and time are dynamical entities and gravity is nothing more than space-time being deformed.******Quantum Mechanics is the theory of the small: the atom, as well as the atom's constituents. It governs everything from superconductivity to the speed of chemical reactions, from the behaviour of electrons to that of the recently discovered Higgs particle. Quantum theory predictions agree with experimental results to an unprecedented accuracy.******General Relativity and Quantum Mechanics are the two paradigms through which we describe our Universe; however, it is not understood how to combine them into a single framework. We lack a unified approach to describing physical phenomena, and this limits our ability to study some of the most fascinating objects in the Universe (such as Black Holes) which are at once quantum mechanical and gravitational in nature.******String Theory is currently our best candidate for a framework through which these two theories can be brought together. It has some profound consequences for what we think might be possible in our Universe, such as existence of extra dimensions. The details of how String Theory allows General Relativity and Quantum Mechanics to come together are not well understood: one idea is that space and time are not fundamental concepts, instead, space-time and its dynamics (i.e., gravity) might emerge from some microscopic degrees of freedom in a way similar to that in which movement of water emerges from the collective behaviour of a large number of water molecules. General Relativity, then, should emerge together with space-time from a Quantum Mechanical model of some objects more fundamental than space and time themselves. Such a formulation would be the ultimate theory of Quantum Gravity. Towards this goal, my work explores emergent geometries and their properties through String Theoretic tools.******In this Proposal, I outline a research program bringing together several recent advancements in Quantum Gravity research such as holography, entanglement entropy, noncommutative geometry and matrix models. The central theme here is non-abelian objects: mathematical entities for which x times y is not the same as y times x, of which simplest example is square matrices. Such non-abelian entities appear in String Theory as coordinates of fundamental objects called D-branes, providing us with clues towards the true nature of space-time. Unravelling these clues is the main purpose of my work.