Some Mathematical Aspects of the Calabi-Yau/Landau-Ginzburg Correspondence

弦理论是调和二十世纪初物理学的两大理论——广义相对论和量子力学的尝试。它基于这样的直觉：我们应该将电子等基本物体视为振动的弦的小片，而不是更传统的小圆球或粒子的图像。不幸的是，只有我们假设宇宙是十维的，这个理论才能成立，而且由于我们只观察到四个维度（三个空间加时间），物理学家通过假设它们紧紧地卷曲得太小而我们无法看到来解释其余的六个维度。如果你只有一个维度可以卷曲，那么很简单 - 你会得到一个圆。然而，如果你有二维，那么就有很多可能性 - 球体和环面（环形甜甜圈的表面）是两个例子 - 当你达到六维时，它会变得非常复杂。这使得物理学变得非常困难，因为根据您选择使用的特定六维空间，您会得到不同的结果。这就是纯数学发挥作用的地方，因为过去一百年的数学家已经开发出了许多研究高维弯曲空间的技术。因此，一个重要的总体目标是采用物理思想并找到与其相对应的数学工具（或者如有必要，构建新的工具）。这是非常困难的，因为弦理论属于物理学的一个分支，称为量子场论，而数学家对它的理解仍然非常少。然而，在实际情况中，它被证明是非常有用的，特别是对数学家来说。这是因为物理学家利用数学家尚不理解的直觉和计算，可以对这些空间的性质做出数学家永远猜不到的预测。其中许多预测后来被其他方法证明是正确的。最有趣和最令人印象深刻的技巧涉及“对偶性”，其中两个明显不相连的空间上的物理特性可以被证明是相同的，这意味着这些空间的许多深层数学属性也必须是相同的。有时甚至可以从一个空间取出物理，然后不断地改变它，直到变成另一个空间的物理，即使这两个空间完全不同，不能连续变形。我的研究是关于“开放”字符串的类型，这意味着它们有端点（而不是“闭合”字符串，即循环）。这些端点不允许在太空中自由漫游，它们必须停留在称为“膜”的物理物体上。当您考虑空间中的所有膜以及它们之间的所有弦（然后稍微简化一下）时，您会得到一个称为“dg-category”的数学对象。我的目的是了解这些 dg 类别的示例以及将它们联系起来的物理二元性的影响。

String theory is an attempt to reconcile the two great theories of the early twentieth century physics, general relativity and quantum mechanics. It is based on the intuition that we should think of fundamental objects like electrons as being like little pieces of vibrating string, instead of the more traditional picture of little round balls or particles. Unfortunately the theory can only be made to work if we assume that the universe is ten-dimensional, and since we only observe four dimensions (three space plus time) physicists explain away the remaining six by assuming they are tightly curled up too small for us to see. If you only have one dimension to curl up then it's simple - you get a circle. If you have two dimensions however then there are lots of possibilities - a sphere and a torus (the surface of a ring doughnut) are two examples - and by the time you get to six dimensions it gets really complicated. This makes the physics very difficult, because you get different results depending on which particular six-dimensional space you choose to work with. This is where pure mathematics becomes useful, as mathematicians working over the past hundred years have developed lots of techniques for studying curved spaces in higher dimensions. Thus an important general goal is to take physical ideas and find mathematical tools (or if necessary, construct new ones) that correspond to them. This is very difficult, because string theory belongs to a branch of physics, called quantum field theory, which is still very poorly understood by mathematicians. However, in cases where it has been done it has proved amazingly useful, particularly to the mathematicians. This is because the physicists, using intuitions and calculations that mathematicians don't yet understand, can make predictions about the properties of these spaces that mathematicians would never have guessed. Many of these predictions have later been proved correct by other methods. The most interesting and impressive such tricks involve 'dualities', where the physics on two apparently unconnected spaces can be shown to be the same, which means that lots of deep mathematical properties of the spaces must also be the same. Sometimes one can even take the physics from one space and then change it continuously until it turns into the physics from another space, even though the two spaces are completely different and can't be continuously deformed into each other. My research is about the kind of strings which are 'open', which means they have end-points (as opposed to 'closed' strings which are loops). These end-points aren't allowed to roam freely in space, they have to stay on physical objects called 'branes'. When you consider all the branes in a space and all the strings between them (and then simplify things a bit) you get a mathematical object called a 'dg-category'. My aim is to understand examples of these dg-categories and the effects of the physical dualities that link them.