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Homological Mirror Symmetry for toric stacks

复曲面堆叠的同调镜像对称

基本信息

批准号：
EP/F055366/1
负责人：
Dominic Joyce
金额：
$ 6.14万
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：
2008
资助国家：
英国
起止时间：
2008 至无数据
项目状态：
已结题

来源：
https://gtr.ukri.org/projects?ref=EP%2FF055366%2F1
关键词：
Homological Mirror Symmetry toric stacks

项目摘要

String Theory is a candidate for the theory of quantum gravity describing the geometry of space-time at the Planck scale. It not only unifies all forces and types of matter in particle physics, but it has also been an enormous source of inspiration for mathematicians to relate various seemingly distant subjects.Mirror Symmetry is an area of conjectures from String Theory relating Symplectic Geometry and Complex Geometry . These are the study of two kinds of geometric structures, symplectic structures and complex structures , and have very different flavours: symplectic geometry is very flexible, with an infinite-dimensional amount of symmetry, and not at all algebraic, but complex geometry is very rigid and algebraic. It had a profound impact on various fields of mathematics, gave birth to a number of new theories and stimulated the development of many existing subjects.Homological Mirror Symmetry is a framework which gives a conceptual understanding of the mysteries surrounding Mirror Symmetry, proposed by Kontsevich in 1994. It concerns Calabi-Yau 3-folds , a class of six-dimensional curved spaces with rich geometrical structures including a complex structure and a symplectic structure, which are important in String Theory. Roughly speaking, the Homological Mirror Symmetry Conjecture says that Calabi-Yau 3-folds come in pairs such that the Symplectic Geometry of one Calabi-Yau 3-fold is equivalent, in a certain precise sense, to the Complex Geometry of the other.In classical geometry, these two spaces are rather different (they have different topologies , or shapes), but String Theory predicts that they become equivalent if one takes quantum effects into account. This suggests we should drastically change our point view on geometry: one should not distinguish these two spaces, just as one should not distinguish congruent figures in Euclidean geometry.One important aspect of Mirror Symmetry is that it does not preserve the difficulty of problems: it often transforms difficult problems in the Symplectic Geometry of one space to easier problems in the Complex (or Algebraic) Geometry of the other, thus leading to many astonishing applications. The situation is similar to classical Fourier analysis, which allows one to transform difficult differential equations to easier algebraic equations.The fact that Mirror Symmetry transforms difficult problems into easier ones implies that it is difficult to prove Mirror Symmetry in general, and indeed there are only a few cases where Homological Mirror Symmetry is known to hold. We will tackle the problem of proving it in more general cases, using another idea from String Theory called brane tilings .Brane tilings are combinatorial objects invented by String Theorists, which are expected to interpolate between the Symplectic Geometry of one space and the Complex Geometry of its mirror, treating both of them on an equal footing, and hence fit nicely with the philosophy that one should not distinguish the two sides of Mirror Symmetry.Little is known about the relations between brane tilings and Symplectic Geometry, and we hope to clarify and prove them. We expect that the relations between brane tilings and Complex Geometry are more manageable, so that a proof of the relation between brane tilings and Symplectic Geometry will lead us to a proof of Homological Mirror Symmetry for these examples.In our examples we will also study the geometry of special Lagrangian 3-folds , which are a special kind of subspace of a Calabi-Yau 3-fold, with minimal volume. The existence of such subspaces is important not only for Symplectic and Differential Geometers, but also for String Theorists, since they are the classical limits of physical objects called D-branes . We hope to prove the conjecture that existence or not of these subspaces is governed by an algebraic criterion called Bridgeland stability , which came originally from String Theory.

弦理论是描述普朗克尺度下时空几何的量子引力理论的候选者。它不仅统一了粒子物理学中所有的力和物质类型，而且也是数学家们联系各种看似遥远的主题的巨大灵感来源。镜像对称是弦论的一个领域，涉及辛几何和复几何。这些是两种几何结构的研究，辛结构和复杂的结构，并有非常不同的味道：辛几何是非常灵活的，具有无限维的对称性，而不是在所有代数，但复杂的几何是非常严格和代数。它对数学的各个领域产生了深远的影响，产生了许多新的理论，并刺激了许多现有学科的发展。同调镜像对称是一个框架，它给出了一个概念性的理解周围的奥秘镜像对称，由Kontsevich在1994年提出。它涉及Calabi-Yau 3-folds，一类六维弯曲空间，具有丰富的几何结构，包括复结构和辛结构，这在弦论中很重要。粗略地说，同调镜像对称猜想说，卡-丘3重是成对的，使得一个卡-丘3重的辛几何在某种精确意义上等价于另一个卡-丘3重的复几何。在经典几何中，这两个空间是相当不同的（它们有不同的拓扑结构或形状），但弦论预测，如果考虑到量子效应，它们将变得等价。这表明我们应该彻底改变我们的观点，对几何：一个不应该区分这两个空间，就像一个不应该区分全等的数字在欧几里德几何。一个重要方面的镜像对称是，它不保留困难的问题：它经常把一个空间的辛几何中的困难问题转化为复空间中的较容易的问题，（或代数学）几何的其他，从而导致许多惊人的应用。这种情况类似于经典的傅立叶分析，它允许人们将困难的微分方程转化为更容易的代数方程。镜像对称将困难的问题转化为更容易的问题这一事实意味着一般情况下很难证明镜像对称，实际上只有少数情况下同调镜像对称是已知的。我们将在更一般的情况下解决证明它的问题，使用弦论中的另一个概念，称为膜镶嵌。膜镶嵌是弦论家发明的组合对象，它被期望在一个空间的辛几何和它的镜像的复几何之间插值，平等地对待两者，因此很好地符合人们不应该区分镜像对称的两面的哲学。关于膜镶嵌和辛几何之间的关系知之甚少，我们希望澄清和证明它们。我们期望膜镶嵌和复几何之间的关系更易于处理，因此膜镶嵌和辛几何之间关系的证明将引导我们证明这些例子的同调镜像对称。在我们的例子中，我们还将研究特殊拉格朗日3-折叠的几何，这是Calabi-Yau 3-折叠的一种特殊的子空间，具有最小的体积。这种子空间的存在不仅对辛几何和微分几何很重要，而且对弦论也很重要，因为它们是被称为D-膜的物理对象的经典极限。我们希望证明这样一个猜想，即这些子空间的存在与否取决于一个叫做Bridgeland稳定性的代数判据，这个判据最初来自弦理论。