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

Calabi-Yau 3-folds是一类特殊的6维弯曲空间，具有丰富的几何结构。他们是非常感兴趣的数学家工作在代数和微分几何，并以物理学家工作在弦理论。基础物理学中最大的问题是找到一个单一的理论，它成功地结合了爱因斯坦的广义相对论--非常大的东西，如星系的物理学--和量子理论--非常小的东西，如原子的物理学。弦论是最适合做这件事的理论。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个维度被卷在一个半径非常小的卡-丘3重中。所以根据弦论，卡-丘三重描述了空间本身的真空。利用物理推理，弦论家们对卡-丘三重（被称为镜像对称）做出了非凡的数学预言，这些预言在许多情况下都得到了验证，并引起了数学家们的极大兴趣。镜像对称说，两个完全不同的卡-丘3重X，X* 可以有相同的量子理论（到目前为止，这还不是一个明确定义的想法），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一个对应关系，这会影响它们的量子理论。这些对应关系通常与看起来完全不同的物体有关--一个非数学的类比是推测肯尼亚的长颈鹿和赞比亚的香蕉之间的一一对应关系。镜像对称的故事中有一章仍在进行中，这一章涉及X和X* 上的两种不变量：X的唐纳森-托马斯不变量，这是计算X上称为相干层的代数对象的数，应该等于计算X* 上特殊拉格朗日3重的其他不变量。特殊拉格朗日3重是非代数对象，表面上与相干层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，通过计算某种几何对象来计算，它具有一个重要的属性，即你可以对底层几何进行重大改变，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（也许我们应该多出去走走？）它暗示着存在着一些与这些巨大变化无关的潜在的数学现实，我们还不了解，我们想知道它是什么。唐纳森-托马斯不变量具有这种不变性。由另一个EPSRC资助，首席研究员最近证明，如果我们变形的卡拉比-丘3倍的几何形状的不同部分，唐纳森-托马斯不变量不是固定的，但改变了刚性的跨壁公式。也就是说，当我们穿过卡-丘3重空间中的一堵墙时，墙一侧的唐纳森-托马斯不变量可以写成另一侧的唐纳森-托马斯不变量的乘积和。这个项目的目标是证明一些定理，这些定理首先将有助于解释这个跨壁公式，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的新不变量，这也将满足类似形状的跨壁公式。事实证明，理解不变量（如唐纳森-托马斯不变量）的乘法性质的一个非常好的方法是将它们编码为一个代数态射，从一个非常大的泛代数（太大而无法理解或计算）到一个小得多的显式代数（不变量取其值）。主要研究者以前的工作从泛代数的子空间构造了一个李代数态射。我们想把它推广到全泛代数上的一个代数态射，并把它推广到一些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.