Phenomenology and Geometry in Heterotic String Compactifications

位于瑞士日内瓦的大型强子对撞机（Large Hadron Collider）以极高的能量对粒子进行对撞，预示着粒子物理学进入了一个新时代。在过去的四十年里，粒子物理学的传统范式被称为标准模型。它描述了一个不拘一格的粒子动物园（如质子和电子），以及一组控制它们相互作用的方程。然而，标准模型远不是故事的全部。在一定的能量范围以上，它对物理的描述就失效了，留给我们的是一个令人着迷的谜题，到底发生了什么，以及如何解释它——如果答案不在标准模型中，那么我们该去哪里找呢？许多物理学家认为这个问题的答案就在弦理论中。与标准模型（其基本对象是粒子（电子和质子））相反，控制弦理论的基本对象是微小的弦。弦振动，很像吉他弦，它们的谐波产生了类似于标准模型中描述的物体。从这个意义上说，弦理论有可能成为万物理论。然而，弦理论虽然很有前途，但也不是没有问题。例如，它预测我们生活在一个十维宇宙中。这就产生了一个明显的问题——我们只观察到四维（三个空间维度和一个时间方向），那么剩下的六个维度在哪里呢？解决方案在于紧化的概念。这个想法很简单，但很难在脑海中描绘出来：人们想象十维中的六个是卷曲的，包围着一种特殊的空间，这种空间非常小，以至于额外的维度基本上是看不见的。这类似于从很高的高度看花园软管：从这个角度来看，软管似乎是一条一维的蛇。额外的圆形尺寸只有在仔细观察时才会变得明显。同样的现象——只有从特定角度才能感知到的额外维度的存在——也被认为发生在弦理论中。控制弦理论的方程对六维空间的形状和大小施加了严格的限制：只有某些类型的空间是允许的。另一方面，空间的几何形状精确地决定了我们将在四维时空中观察到的实验预测。一个粗略的类比是这样的：万有引力定律表明，要使水滑梯起作用，它必须向下。另一方面，滑梯的形状、曲折和转弯（即它的几何形状）精确地决定了人们滑下滑梯时的感受。这是几何和物理之间相互作用的一个很好的例子——这个项目的一个关键主题。弦理论的物理学做出了一些大胆的数学预测。其中一个预测被称为镜像对称，它意味着六维空间是成对的，从我们四维的角度来看，它们看起来是一样的。关于镜像对称的数学还有很多东西要学，这是一个有趣的问题：弦理论能教给我们什么？在这个项目中，我们有兴趣回答两个互补但相关的问题：1。对于大范围可能的紧化空间，四维实验预测是什么？这样的预测有望在大型强子对撞机中实现。关于紧化空间的数学和几何，弦理论告诉了我们什么？对于每一个可能的紧化空间是否存在镜像对称的一般概念？通过回答这些问题，我们将进一步了解我们的物理宇宙的基本结构，以及我们对数学基本结构的理解。如果从物理学的历史来看，这两个方向是相辅相成的。

The Large Hadron Collider, an experiment in Geneva, Switzerland, that collides particles at extremely high energies, is heralding a new era in particle physics. The conventional paradigm for particle physics throughout the past forty years has become known as the Standard Model . This describes an eclectic zoo of particles (like protons and electrons) together with a set of equations governing their interactions. However, the Standard Model is far from being the whole story. Above a certain energy range, its description of physics breaks down, and we are left with a fascinating puzzle of what is going on, and how to explain it - if the answer doesn't lie in the Standard Model, then where do we look?Many physicists believe the answer to this question lies within string theory. As opposed to the Standard Model, whose fundamental objects are particles (electrons and protons), the fundamental objects governing string theory are tiny strings. The strings vibrate, much like guitar strings, and their harmonics give rise to objects resembling those described in the Standard Model. In this sense, string theory has the potential to be a theory of everything . However, although promising, string theory is not without problems. For example, it predicts that we live in a ten-dimensional universe. This poses an obvious problem - we observe only four-dimensions (three space dimensions and one time direction), so where are the remaining six? The resolution lies in the notion of compactification . The idea is simple, though difficult to picture mentally: one imagines that six of the ten-dimensions are curled up, encompassing a special type of space so small that the extra dimensions are essentially invisible. This is analogous to looking at a garden hose from a great height: from this perspective, the hose appears to be a one-dimensional snake. The extra, circular dimension becomes apparent only upon closer inspection. The same phenomenon - the presence of additional dimensions perceptible only from particular perspectives - is thought to occur in string theory. The equations that govern string theory impose stringent constraints on the shape and size of the six-dimensional spaces: only certain types of spaces are allowed. On the other hand, the geometry of the space dictates precisely which experimental predictions we will observe in our four-dimensional spacetime. A loose analogy is the following: the laws of gravity imply that for a water slide to work, it must point down. On the other hand, the shape, the twists and the turns of the slide (i.e. its geometry) dictate precisely what people feel as they go down the slide. This is a wonderful example of the interplay between geometry and physics - a key theme of this project. The physics of string theory makes some bold mathematical predictions. One such prediction, known as mirror symmetry , implies that the six-dimensional spaces come in pairs, and that, from our four-dimensional point of view, they look identical. There is still much to learn about the mathematics of mirror symmetry, and it is an interesting question to ask: what can string theory teach us?In this project, we are interested in answering two complementary but related questions:1. What are the four-dimensional experimental predictions for a wide range of possible compactification spaces? Such predictions are expected to take place in the Large Hadron Collider. 2. What does string theory tell us about the mathematics and geometry of compactification spaces? Is there a general notion of mirror symmetry for every possible compactification space?By answering such questions, we will have moved further in our understanding of the fundamental structure of our physical universe, as well as in our understanding of a fundamental structure in mathematics. If the history of physics is anything to go by, these two directions go hand-in-hand.