String Theory, Geometry and Particle Physics

该奖项资助弗吉尼亚理工学院和州立大学的詹姆斯·格雷教授的研究活动。我们对现代物理学的理解有两个理论基础：量子力学和爱因斯坦的引力理论。无论电子有多轻，它们都会受到引力的作用，因此这些量子力学物体必须符合引力理论。然而，如果把量子力学的一般规则应用于包含引力的理论，就会得到无意义的结果。例如，某些事件发生的概率，用这种方法计算出来，结果是无限的。许多理论被提出来试图解决这个问题。到目前为止，最成功的是一个叫做弦理论的理论。这个想法是，自然界的基本组成部分不是粒子，而是微小的一维弦。上面提到的麻烦的概率可以计算为与1除以这些字符串的长度成正比。因此，我们看到，当我们把长度缩小到零并恢复为一个点状粒子时，我们发现除以零--一个无限的答案。然而，如果弦的长度不为零，这个理论就有意义。弦理论有一个很大的问题。从数学上讲，这个理论只有在弦存在于不止三维空间的情况下才有意义。如果一些维度被包裹在一些非常小的形状中，我们就不一定能看到它们。想象一张纸。这显然是一个二维的薄片。现在把纸包成一个圆筒，想象把它卷得越来越紧，变成一个越来越小的管子。如果你能继续这样做（如果纸足够薄），那么最终这张纸看起来就像一条线-一个一维物体。弦论的额外维度的形状决定了生活在三维宇宙中的人所看到的物理学。例如，形状的体积决定了重力的强度，这取决于牛顿常数。额外维度更微妙的几何性质决定了最终三维宇宙的所有其他物理性质--例如，电子是否存在，它们是否携带电荷。鉴于此，一个明显的问题出现了。有没有一种形状，如果用来隐藏弦论的额外维度，会产生一个像我们自己的三维宇宙？这个问题的答案还不知道。拟议中的研究是利用计算机来研究弦理论额外维度的数亿种不同的可能形状。通过这种方式，PI将获得一个关于哪些三维宇宙可以用弦理论描述的全面调查。PI将搜索这个数据库，看看是否有任何形状引起了物理学，就像我们在我们周围看到的那样。该项目的更广泛影响将主要通过对研究生、本科生研究人员和博士后研究助理进行技术培训来实现。资助六个月的博士后研究员的任期包括，与其他年份和人员参与由弗吉尼亚理工大学的资源提供。PI计划在他的家乡举办一个关于计算代数几何和弦紧化的跨学科研讨会，以帮助发展这两个不同领域之间的联系。该项目建议研究F理论和杂合弦理论的紧化，这是目前最有希望重现已知粒子物理的候选者。利用现代计算代数几何的形式化方法，PI将非常详细地研究这些理论的大量Calabi-Yau四重和三重紧化。此外，PI将发展形式主义来描述杂种优势理论的非Kahler紧化的模空间和物质含量。这项工作将试图回答两个问题。首先，我们能否找到弦理论的紧化，不仅再现标准模型的规范群和粒子内容，而且再现一组现实的软超对称破缺项？第二，如果这种紧化确实可以被发现，那么它们对那些尚未被测量的实验观测量的共同预测是什么？这项研究有可能直接促进人类对自然的认识。它可以帮助我们理解弦理论是否只是研究数学和量子场论某些领域的技术工具，或者它是否是我们宇宙的基本物理理论。所提出的工作将发展我们的理解弦紧化的卡-丘和非Kahler流形的情况下，超引力近似是有效的。卡-丘三重理论的工作旨在使理论尽可能接近实验。对非Kahler紧化的研究将为未来从弦理论获得粒子物理模型的更雄心勃勃的方法奠定基础。

This award funds the research activities of Professor James Gray at the Virginia Polytechnic Institute and State University.Two theories underlie our understanding of modern physics: quantum mechanics and Einstein's theory of gravity. Electrons experience gravity, however light they may be, and as such these quantum mechanical objects must fit within a gravitational theory. However, if one applies the usual rules of quantum mechanics to theories containing gravity, one gets nonsense. For example, the probability of certain events occurring, that is calculated in this way, turns out to be infinite. Many theories have been proposed to try and solve this problem. To date, by far the most successful is a theory called string theory. The idea is that instead of the fundamental building blocks of nature being particles, they are tiny, one dimensional, pieces of string. The troublesome probabilities mentioned above can be calculated to be proportional to one divided by the length of these strings. Thus one sees that, as one shrinks the length to zero and recovers a point like particle, one finds division by zero - an infinite answer. If the string length does not vanish, however, the theory can make sense. String theory comes with a very big catch. Mathematically, the theory only makes sense if the strings live in more than just three spatial dimensions. If some of the dimensions are wrapped up in some very small shape, we would not necessarily see them. Think of a piece of paper. This is clearly a two dimensional sheet. Now wrap the paper up into a cylinder and imagine furling it up more and more to make a smaller and smaller tube. If you could keep doing this (and if the paper was thin enough) then eventually the sheet would simply look like a line - a one dimensional object. The shape that the extra dimensions of string theory take determines the physics that someone living in the resulting three-dimensional universe would see. As an example, the volume of the shape determines the strength of gravity, as governed by Newton's constant. More subtle geometrical properties of the extra dimensions determine all of the rest of the physics of the resulting three-dimensional universe - for example, whether electrons exist and whether they carry electric charge. Given this, an obvious question arises. Is there a shape, which if used to hide the extra dimensions of string theory, gives rise to a three dimensional universe like our own? The answer to the question is still unknown. The proposed research is to use computers to study hundreds of millions of different possible shapes for the extra dimensions of string theory. In this way the PI will obtain a comprehensive survey of which three dimensional universes can be described by string theory. The PI will search this database to see if any of the shapes give rise to physics which is like that we see around us. The broader impact of this project will largely be through the technical training of graduate students, undergraduate researchers and a postdoctoral research assistant. Funding for six months of a postdoctoral researcher's tenure is included, with the other years and personnel involved being provided by resources from Virginia Tech. The PI plans to host an interdisciplinary workshop on computational algebraic geometry and string compactification at his home institution, to help develop links between these two different fields.The project proposes to study compactifications of F-theory and heterotic string theory, which are currently the most promising candidates for reproducing known particle physics. Using modern formal methods of computational algebraic geometry, the PI will study large numbers of Calabi-Yau four- and three-fold compactifications of these theories in very fine detail. In addition, the PI will develop formalism to describe the moduli space and matter content of non-Kahler compactifications of heterotic theories. This work will try to answer two questions. First, can one find compactifications of string theory that reproduce not only the gauge group and particle content of the standard model, but also a realistic set of soft supersymmetry breaking terms? Second, if such compactifications can indeed be found, what are their common predictions for experimental observables that are yet to be measured? This research has the potential to advance mankind's knowledge of nature directly. It could help us understand whether string theory is simply a technical tool for studying some areas of mathematics and quantum field theory, or whether it is a fundamental physical theory of our universe. The work proposed will develop our understanding of string compactification on Calabi-Yau and non-Kahler manifolds in situations where the supergravity approximation is valid. The work on Calabi-Yau threefolds is designed to push the theory as close as possible to confrontation with experiment. The research on non-Kahler compactifications will lay the groundwork for more ambitious approaches to obtaining particle physics models from string theory in the future.