Generalized Donaldson-Thomas invariants

Calabi-Yau三褶是一种特殊的六维弯曲空间，具有大量的几何结构。它们对研究代数和微分几何的数学家以及研究弦理论的物理学家都很有兴趣。基础物理学的最大问题是找到一种理论，它能成功地把爱因斯坦的广义相对论和量子论结合起来。广义相对论是关于非常大的物体，如星系的物理学，量子理论是关于非常小的物体，如原子的物理学。弦理论是解决这个问题的主要候选理论。它预言时空的维度不是4（3个空间加1个时间），而是10。额外的6个维度卷成一个卡拉比-丘3折叠，半径非常小。所以根据弦理论，卡拉比-丘三折描述了空间本身的真空。利用物理推理，弦理论家对卡拉比-丘三折做出了非凡的数学预测，被称为镜像对称，这些预测在许多情况下得到了验证，并在数学家中引起了极大的兴奋。Donaldson-Thomas不变量是与Calabi-Yau 3-fold M相关的数系统，它计算M上存在的一些数学对象（半稳定相干束）。它们在数学上是有趣的，因为它们在M的连续变形下是不变的，并且编码关于M的神秘的、不平凡的信息。它们在物理上是有趣的，因为它们计算物理上重要的对象（膜、BPS状态）。目前只知道在一种特殊情况下（当半稳定和稳定重合时）如何定义Donaldson-Thomas不变量。我们打算找出如何把这个定义推广到一般情况。我们还旨在找出这些扩展不变量在稳定性变化条件下的变换规律（即使是旧不变量也不知道），并通过实例计算它们。我们希望这将导致对稳定条件空间的更好理解，这是弦理论真空空间的一部分，一个非常重要但却知之甚少的空间。

Calabi-Yau 3-folds are a special kind of 6-dimensional curved space, with a lot of geometrical structure. They are of great interest to mathematicians working in Algebraic and Differential Geometry, and to physicists working in String Theory. The greatest problem in fundamental physics is to find a single theory which successfully combines Einstein's General Relativity -- the physics of very large things, such as galaxies -- and Quantum Theory -- the physics of very small things, such as atoms. String Theory is the leading candidate for doing this. It predicts that the dimension of space-time is not 4 (3 space plus one time), but 10. The extra 6 dimensions are rolled up in a Calabi-Yau 3-fold, with very small radius. So according to String Theory, Calabi-Yau 3-folds describe the vacuum of space itself. Using physical reasoning, String Theorists made extraordinary mathematical predictions about Calabi-Yau 3-folds, known as Mirror Symmetry , which have been verified in many cases, and cause much excitement among mathematicians. Donaldson-Thomas invariants are systems of numbers associated to a Calabi-Yau 3-fold M which count some mathematical objects ( semistable coherent sheaves ) which live on M. The definition is complicated. They are mathematically interesting because they are unchanged under continuous deformations of M, and encode mysterious, nontrivial information about M. They are physically interesting as they count physically important objects (branes, BPS states). It is at present only known how to define Donaldson-Thomas invariants in a special case (when semistable and stable coincide). We propose to find out how to extend the definition to the general case. We also aim to find the transformation laws for these extended invariants under change of stability condition (this is not known even for the old invariants), and to compute them in examples. We hope this will lead to a better understanding of the space of stability conditions, which is part of the space of String Theory vacua, a very important but poorly understood space.