Generalized Donaldson-Thomas invariants

Calabi-Yau三折空间是一类特殊的六维曲面空间，具有丰富的几何结构。它们对从事代数和微分几何的数学家以及从事弦理论的物理学家非常感兴趣。基础物理学中最大的问题是找到一种单一的理论，它成功地结合了爱因斯坦的广义相对论和量子理论，广义相对论是指星系等非常大的物体的物理学，量子理论是非常小的物体的物理学，如原子。弦理论是实现这一目标的领先候选者。它预言时空的维度不是4(3个空间加1次)，而是10。额外的6个维度卷成一个3折的Calabi-Yau，半径非常小。因此，根据弦理论，Calabi-Yau三重折叠描述了空间本身的真空。弦理论家利用物理推理做出了关于Calabi-Yau三重折叠的非凡的数学预测，即所谓的镜像对称，这些预测在许多情况下都得到了验证，并在数学家中引起了极大的兴奋。Donaldson-Thomas不变量是与Calabi-Yau三重M相关的数的系统，它计算生活在M上的一些数学对象(半稳定的相干轨道)。定义很复杂。它们在数学上很有趣，因为它们在M的连续变形下是不变的，并编码了关于M的神秘的、不平凡的信息。它们在计算物理上重要的物体(膜、BPS状态)时是物理上有趣的。目前只知道在特殊情况下(半稳定和稳定重合时)如何定义Donaldson-Thomas不变量。我们建议找出如何将定义扩展到一般情况。我们还旨在找到这些扩展的不变量在稳定性条件变化时的变换规律(这对于旧的不变量来说也是未知的)，并通过实例计算它们。我们希望这将导致对稳定性条件空间的更好的理解，它是弦理论真空空间的一部分，这是一个非常重要但知之甚少的空间。

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.