Ringel-Hall algebras of Calabi-Yau 3-folds and Donaldson-Thomas theory

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。镜像对称说两个完全不同的Calabi-Yau 3-fold X， X*可以有相同的量子理论（到目前为止，这还不是一个定义明确的想法），当这种情况发生时，我们可以在X和X*的几何方面之间建立对应关系，从而影响它们的量子理论。通常，这些对应关系涉及的对象看起来完全不同——一个非数学的类比是，肯尼亚的长颈鹿和赞比亚的香蕉之间存在一对一的对应关系。镜像对称故事的一章仍在进行中涉及X和X*上的两种不变量：X的Donaldson-Thomas不变量，它们是计数X上被称为相干束的代数对象的数，应该等于计数X*上特殊拉格朗日三折的其他不变量。特殊拉格朗日三折是非代数对象，表面上看，它们与相干束的区别就像长颈鹿与香蕉的区别一样。当数学家谈论不变量时，他们指的是一个数字，比如42，是通过计算某种几何物体而计算出来的，它有一个重要的性质，就是你可以对底层的几何结构做出很大的改变，但由于神秘的原因，这个数字保持不变。这种不变性让数学家们非常兴奋（也许我们应该做更多的研究？）因为它表明，存在一些独立于这些大变化之外的潜在数学现实，我们还不理解，我们想知道它是什么。Donaldson-Thomas不变量具有这种不变性。在EPSRC的另一项资助下，首席研究员最近证明，如果我们变形Calabi-Yau 3-fold的不同几何部分，Donaldson-Thomas不变量不是固定的，而是通过刚性的过壁公式改变的。也就是说，当我们穿过Calabi-Yau三折空间中的一堵墙时，墙一侧的Donaldson-Thomas不变量可以写成另一侧Donaldson-Thomas不变量积的和。这个项目的目标是证明一些猜想，这些猜想首先有助于解释这个过墙公式，其次允许我们将Donaldson-Thomas不变量推广到包含更多信息的更大类别的新不变量，这也将满足类似形状的过墙公式。事实证明，理解不变量的乘法性质的一个很好的方法，比如Donaldson-Thomas不变量，就是把它们编码在代数态射中，从一个非常大的通用代数，它太大了，无法理解或计算，到一个小得多的显式代数，在这个显式代数中，不变量取其值。在此之前的工作中，主要研究者从通用代数的子空间构造了一个李代数的态射。我们想把它推广到全全称代数上的代数态射，并推广到更大的显式代数上的态射。

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. Mirror Symmetry says that two quite different Calabi-Yau 3-folds X, X* can have identical Quantum Theories (so far, this is not a well-defined idea), and when this happens, we can set up a correspondence between aspects of the geometry of X and X* which affect their Quantum Theories. Often these correspondences relate objects which seem quite different -- a non-mathematical analogue would be to conjecture a one-to-one correspondence between giraffes in Kenya and bananas in Zambia. One chapter of the Mirror Symmetry story which is still work in progress relates two kinds of invariants on X and X*: the Donaldson-Thomas invariants of X, which are numbers counting algebraic objects called coherent sheaves on X, should be equal to other invariants counting special Lagrangian 3-folds on X*. Special Lagrangian 3-folds are non-algebraic objects, superficially as different from coherent sheaves as giraffes are from bananas. When mathematicians talk about invariants they mean a number, such as 42, computed by counting some kind of geometric object, which has the important property that you can make big changes to the underlying geometry, but for mysterious reasons, the number remains the same. This invariance property makes mathematicians very excited (perhaps we should get out more?) as it suggests there is some underlying mathematical reality which is independent of these big changes, which we don't yet understand, and we want to know what it is. Donaldson-Thomas invariants have this kind of invariance property. Funded by another EPSRC grant, the Principal Investigator has recently proved that if we deform a different part of the geometry of the Calabi-Yau 3-fold, Donaldson-Thomas invariants are not fixed, but change by a rigid wall-crossing formula . That is, when we cross a wall in the space of Calabi-Yau 3-folds, the Donaldson-Thomas invariants on one side of the wall can be written as sums of products of Donaldson-Thomas invariants on the other side. The goal of this project is to prove some conjectures which will first help to explain this wall-crossing formula, and secondly allow us to generalize Donaldson-Thomas invariants to a larger class of new invariants containing much more information, which will also satisfy a wall-crossing formula of a similar shape. It turns out that a very nice way of understanding multiplicative properties of invariants, such as Donaldson-Thomas invariants, is to encode them in an algebra morphism from a very large universal algebra , which is far too big to understand or compute, to a much smaller, explicit algebra, where the invariants take their values. Previous work by the Principal Investigator constructed a Lie algebra morphism from a subspace of the universal algebra. We want to extend this to an algebra morphism on the full universal algebra, and generalize it to morphisms to some larger explicit algebras.