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

Calabi-Yau 3 重的 Ringel-Hall 代数和 Donaldson-Thomas 理论

基本信息

批准号：
EP/G068798/1
负责人：
Dominic Joyce
金额：
$ 10.71万
依托单位：
University of Oxford
依托单位国家：
英国
项目类别：
Research Grant
财政年份：
2009
资助国家：
英国
起止时间：
2009 至无数据
项目状态：
已结题

来源：
https://gtr.ukri.org/projects?ref=EP%2FG068798%2F1
关键词：
Ringel Hall algebras Calabi Yau

项目摘要

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.

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。镜像对称说，两个完全不同的Calabi-Yau三重折叠X，X*可以拥有相同的量子理论(到目前为止，这还不是一个定义良好的概念)，当这种情况发生时，我们可以在X和X*的几何方面建立起影响他们量子理论的对应关系。通常，这些对应关系涉及的对象似乎完全不同--非数学类比可以推测肯尼亚的长颈鹿和赞比亚的香蕉之间存在一一对应。镜面对称故事中仍在进行中的一章涉及到X和X*上的两种不变量：X的Donaldson-Thomas不变量，它们是计数X上称为相干层的代数对象的数目，应该等于计算X*上特殊拉格朗日3重的其他不变量。特殊的拉格朗日3-折叠是非代数对象，表面上不同于相干的束，就像长颈鹿不同于香蕉一样。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，即你可以对底层的几何对象进行大的改变，但出于神秘的原因，这个数字保持不变。这种不变性让数学家们非常兴奋(也许我们应该更多地了解情况？)正如它所暗示的，存在一些与这些重大变化无关的潜在的数学现实，我们还不了解这些变化，我们想知道它是什么。Donaldson-Thomas不变量具有这种不变性。在另一笔EPSRC拨款的资助下，首席研究员最近证明，如果我们将Calabi-Yau几何的不同部分变形3倍，Donaldson-Thomas不变量不是固定的，而是通过一个严格的跨越墙的公式改变。也就是说，当我们穿过Calabi-Yau三折空间中的一面墙时，墙一侧的Donaldson-Thomas不变量可以写成另一侧Donaldson-Thomas不变量乘积的和。这个项目的目的是证明一些猜想，这些猜想首先将有助于解释这个穿越墙公式，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新不变量，它也将满足类似形状的穿越墙公式。事实证明，理解不变量的乘法性质的一个非常好的方法，例如Donaldson-Thomas不变量，是将它们编码在代数态射中，从一个非常大的通用代数到一个小得多的显式代数，在这个代数态射中，不变量取它们的值。这位主要研究者以前的工作是从泛代数的一个子空间中构造一个李代数态射。我们想把它推广到全泛代数上的代数态射，并把它推广到一些更大的显式代数上的态射。