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

弦理论试图调和二十世纪早期物理学的两大理论--广义相对论和量子力学。它基于这样一种直觉，即我们应该把像电子这样的基本物体想象成一小段振动的弦，而不是更传统的小圆球或粒子的图片。不幸的是，这个理论只有在我们假设宇宙是十维的情况下才能起作用，而且由于我们只观察到四维(三维空间加时间)，物理学家通过假设它们紧紧地卷曲得太小而看不见来解释其余的六维。如果你只有一个维度可以卷曲，那么很简单--你会得到一个圆圈。然而，如果你有两个维度，那么就有很多可能性--球体和环面(环形甜甜圈的表面)就是两个例子--当你达到六个维度时，事情就变得非常复杂了。这使得物理学变得非常困难，因为你会得到不同的结果，这取决于你选择使用的是哪个特定的六维空间。这就是纯数学变得有用的地方，因为在过去的一百年里，数学家们已经开发出了许多技术来研究更高维度的弯曲空间。因此，一个重要的总体目标是获取物理概念，并找到与它们相对应的数学工具(或者如果必要的话，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，数学家们对它的理解仍然很少。然而，在已经完成的情况下，它被证明是令人惊讶的有用的，特别是对数学家来说。这是因为物理学家使用数学家还不了解的直觉和计算，可以对这些空间的性质做出数学家永远不会猜到的预测。这些预测中的许多后来被其他方法证明是正确的。最有趣、最令人印象深刻的这类把戏涉及“对偶性”，即两个明显不相连的空间的物理性质可以被证明是相同的，这意味着这两个空间的许多深层数学性质也必须相同。有时，人们甚至可以从一个空间获取物理，然后不断地改变它，直到它从另一个空间变成物理，即使这两个空间完全不同，不能相互连续变形。我的研究是关于那种“开放”的字符串，这意味着它们有终点(与“闭合”字符串相反，后者是循环)。这些终端不允许在太空中自由漫游，它们必须停留在被称为“膜”的物理物体上。当你考虑一个空间中的所有膜和它们之间的所有字符串时(然后把事情简化一点)，你会得到一个被称为‘dg-ategory’的数学对象。我的目的是理解这些dg范畴的例子，以及将它们联系在一起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.