Quantum Symmetries in String Theory

如果地球两边的两个人都扔了一个球，同样的事情也会发生：球以恒定的加速度落向地面。这是地球引力旋转对称性的简单结果。更广泛地说，对称性的力量在于它能够告诉我们，在可能看起来非常不同的情况下，物理定律是相关的。通过这种方式，对称性对物理学产生了深远的影响，经常使不可能的问题成为可能。我的研究目的是开发新的基于对称性的方法来解决理论物理中具有挑战性的公开问题。在20世纪初，发现了现代物理学中最成功的两个理论。第一个是广义相对论，描述了大距离的经典引力，第二个是量子力学，是原子尺度物理的基础。然而，试图用标准方法统一它们会导致无法控制的无穷大。这个长期存在的问题表明，发展量子引力理论需要一个新的框架。从考虑在空间中延伸的基本物体的基本思想出发，弦理论已经成为量子引力理论的主要候选者。携带引力的粒子--引力子--是由弦组成的振荡的闭合环路。弦理论最广为人知的事实之一是它需要10个时空维度。恢复我们宇宙的4维空间的尝试已经取得了显著的成果，导致了丰富的新数学，并从根本上改变了我们对物理理论的看法。将弦理论与4维物理联系起来的一种方法是通过ADS/CFT通信。这种对应是两个模型之间显著的二元性。第一种是在高度对称的弯曲时空中用闭弦描述的引力量子理论。第二个是4维的量子场论，它与量子色动力学有重要的联系：原子核中质子和中子相互作用的规范理论。对偶性的力量是弦理论中的解析计算给我们提供了关于量子场论的强耦合区域的新信息。强耦合系统通常很难研究，并且是科学中许多重要的开放问题的基础。事实上，严格地证明杨-米尔斯理论中存在质量差距是千禧年奖的七个问题之一。因此，ADS/CFT对应将两个长期存在的理论物理问题联系在一起：描述引力的量子理论和求解强耦合量子场理论。ADS/CFT对应的弦不仅在弯曲的时空上运动，而且在背景场中运动。这可以比作电子在背景磁场中运动：电子与磁场相互作用，使其沿着弯曲的路径运动。许多重要的弦理论都是这种类型的，它们通常很难研究。对称性可以为我们提供一个解决方案。对称性最显著的表现之一是可积性，这是某些物理模型的丰富的数学特性。可积性可以被认为是一个大的隐藏对称性的存在，可以用来得到准确的结果。在特殊情况下，ADS/CFT对应的弦具有这种显着的性质，利用相关的方法，这些理论有可能被求解。我的研究目的是发展新的基于对称性的方法，促进我们对可积性在弦理论和ADS/CFT对应中所起的作用的认识。这将使我们能够解决非平凡背景下的弦理论，并构建规范/重力二元性的新例子，从而为了解现实世界中量子引力和强耦合物理的基本性质提供新的见解。

If two people on opposite sides of the planet both drop a ball the same thing happens: the ball falls towards the ground with constant acceleration. This is a simple consequence of the rotational symmetry of the Earth's gravity. More generally, the power of symmetry lies in its ability to tell us that the laws of physics are related in situations that may look very different. In this way symmetry has had a profound impact on physics, often making impossible problems possible. The aim of my research is to develop novel symmetry-based methods tackling challenging open problems in theoretical physics.In the early 20th century two of the most successful theories of modern physics were discovered. The first, general relativity, describes classical gravity at large distances, while the second, quantum mechanics, underlies atomic-scale physics. However, attempting to unify them using standard methods leads to uncontrollable infinities. This long-standing problem suggests that a new framework is needed to develop a quantum theory of gravity.Starting from the elementary idea of considering fundamental objects that are extended in space, string theory has become a leading candidate for the quantum theory of gravity. Gravitons, the particles that carry the gravitational force, are oscillating closed loops of string. One of the most well-known facts about string theory is that it requires 10 space-time dimensions. Attempts to recover the 4 dimensions of our universe have been remarkably fruitful, leading to a wealth of new mathematics and fundamentally changing how we think about physical theories.One way to relate string theory to physics in 4 dimensions is through the AdS/CFT correspondence. This correspondence is a remarkable duality between two models. The first is a quantum theory of gravity described by closed strings on a highly symmetric curved space-time. The second is a quantum field theory in 4 dimensions, which has important connections to quantum chromodynamics: the gauge theory of the strong interaction holding protons and neutrons together in the nuclei of atoms.The power of the duality is that analytic computations in string theory give us new information about the strongly-coupled regime of quantum field theory. Strongly-coupled systems are typically hard to study and underlie many important open questions in science. Indeed, rigorously establishing the existence of a mass gap in Yang-Mills theory is one of the seven Millennium Prize problems. The AdS/CFT correspondence therefore ties together two long-standing problems of theoretical physics: describing a quantum theory of gravity and solving strongly-coupled quantum field theories.The strings of the AdS/CFT correspondence are not just moving on a curved space-time, but also in background fields. This can be compared to an electron moving in a background magnetic field: the electron interacts with the magnetic field causing it to follow a curved path. Many important string theories are of this type and they are typically difficult to study. Symmetry can provide us with a solution. One of the most striking manifestations of symmetry is integrability, a rich mathematical property of certain physical models. Integrability can be thought of as the presence of a large hidden symmetry that can be used to derive exact results. In special cases the strings of the AdS/CFT correspondence have this remarkable property and using the associated methods these theories can potentially be solved.The aim of my research is to develop novel symmetry-based methods advancing our knowledge of the role integrability plays in string theory and the AdS/CFT correspondence. This would allow us to solve string theories on non-trivial backgrounds and to construct new examples of gauge/gravity duality, thereby providing new insights into the fundamental nature of quantum gravity and strongly-coupled physics in the real world.