Stongly Coupled Field Theories, String Theory and Gravity

本项目涉及弦理论和量子场论。有两个广泛的目标。第一部分是使用受弦理论启发的工具来描述原本难以处理的量化宽松制度。第二部分是使用弦理论中的新几何工具来描述重力和我们可观察的宇宙的各个方面。QFT非常好地描述了我们宇宙中亚原子水平上的相互作用，支撑了所有经过实验验证的粒子和相互作用。除非在特殊情况下，QFT的方程式是笨拙的，不能直接分析。标准的方法是用一种称为微扰理论的方法来近似方程。这要求粒子之间的相互作用很弱，而不是普遍存在的情况。通常，相互作用很强，这种情况称为强耦合，微扰理论的近似就失效了。这个问题是理解粒子物理许多方面的一个主要限制。此外，QFT没有描述宏观的相互作用，特别是重力。如果我们正在处理密度非常高的物体，比如黑洞，那么我们需要找到一种将爱因斯坦的广义相对论和量子傅立叶变换相结合的理论。最有可能做到这一点的是弦理论。为了发挥作用，它需要施加严格的数学条件。例如，除了我们观察到的三个维度之外，肯定还有另外六个维度，它们的几何形状非常小，因此对当今的实验来说是不可见的。一个粗略的类比是软管：从远处看，它看起来是一维的，但仔细观察，它有一个额外的圆形方向。描述可见宇宙的物理成为一个与某些空间的几何密切相关的问题，相反，要求将可感知的物理作为弦理论的输出导致了新的几何技术。然后，人们可以将量子修正理解为来自弦理论本身。弦理论为我们理解量子力学和引力提供了新的思路。我们从一个强耦合的QFT开始。全息术是两个理论的等价物，我们称它们为理论A和理论B。理论A是d+1维引力理论，而理论B是在平坦(无重力)d维空间中的QFT。全息术意味着理论A可以被用来学习理论B的强耦合方面，反之亦然。例如，理论B可以是可积的，这意味着它是完全可解的，而关于强耦合体系的信息纯粹由对称性决定。全息术意味着我们可以确定(也许模糊的)理论A的性质，也就是引力理论。相反，通过经典引力，理论A可以用来描述理论B中的强耦合状态。即使理论A和理论B都不是现实的模型--人们通常会简化假设，使二元性发挥作用--人们可能会希望由此产生的特征是通用的，这给我们上了一些关于粒子物理中其他困难问题的新课。这项建议的第一部分涉及在一个新的范例中发展全息学，以及使用可积性来探索QFT的性质。接下来，在引力的背景下，在理解黑洞方面出现了新的想法。源于弦理论的对称性，例如超对称性，已经导致了获得新型黑洞以及理解它们的几何和物理性质的新技术。出现了许多有趣的问题：量子修正对这些黑洞解决方案的作用是什么？它们稳定吗？另一个不同但相关的问题是，我们如何使用弦理论来描述准现实的现象学模型？要做到这一点，需要了解空间的几何形状。什么类型的几何图形导致了我们的宇宙的真实模型？量子修正的作用是什么？这些是构成本提案第二部分的问题类型。

This project is concerned with string theory and quantum field theory (QFT). There are two broad aims. Part I is to use tools inspired from string theory to describe otherwise intractable regimes of QFT. Part II is to use new geometric tools within string theory to describe aspects of gravity and our observable universe. QFT describes interactions at the sub-atomic level of our universe exceptionally well, underpinning all experimentally verified particles and interactions. The equations of QFT, except in special circumstances are unwieldy and not amenable to a direct analysis. The standard approach is to approximate the equations in a manner known as perturbation theory. This requires the interactions between particles be weak, not a universal situation. Often, the interactions are strong, a situation known as strong coupling, and the perturbation theory approximation breaks down. This problem is a major limitation for understanding many aspects of particle physics. Moreover, QFT does not describe macroscopic interactions, in particular gravity. If we are dealing with very dense objects such as black holes, then we need to find a theory that incorporates Einstein's theory of general relativity and QFT. The leading candidate that does this is string theory. In order to work, it requires stringent mathematical conditions be imposed. For example, in addition to the three dimensions we observe, there must exist six additional dimensions, whose geometry is very small and so not visible to present day experiment. A rough analogy is with a hose: from a distance it looks one-dimensional, but on closer inspection there is an additional circular direction. Describing the physics of the observable universe becomes a problem closely tied to the geometry of certain spaces, and conversely, demanding sensible physics as an output of string theory leads to new geometric techniques. One can then understand quantum corrections as coming from the string theory itself. String theory has led to new ideas in our understanding of QFT and gravity. We start with a strongly coupled QFT. Holography is an equivalence between two theories, let us call them theory A and theory B. Theory A is a d+1-dimensional gravity theory while theory B is QFT in flat (without gravity) d-dimensional space. Holography means that theory A can be utilised to learn about strong coupling aspects of theory B and vice versa. For example, theory B can be integrable, meaning it is completely soluble, and information about the strong coupling regime is determined purely by symmetry. Holography means we can determine (perhaps obscure) properties of theory A, the gravity theory. Conversely, via classical gravity, theory A can be used to describe regimes of strong coupling in theory B. Even if theory A nor theory B are not realistic models - one typically makes simplifying assumptions for the dualities to work -- one might hope the resulting features are universal, teaching us some new lessons on otherwise difficult problems in particle physics. Part I of this proposal is concerned with developing holography in a new paradigm of examples, as well as using integrability to explore properties of QFT. Next, in the context of gravity, new ideas have arisen in understanding black holes. Symmetries that derive from string theory, e.g. supersymmetry, have led to novel techniques for obtaining new types of black holes, as well as understanding their geometric and physical properties. Many interesting questions arise: what is the role of quantum corrections to these black hole solutions? Are they stable? A different but related question is how can we use string theory to describe quasi-realistic phenomenological models? Doing so requires understanding the geometry of spaces. What types of geometries lead to realistic models of our universe? What is the role of quantum corrections? These are the types of questions that form part II of this proposal.