Homological Mirror Symmetry for toric stacks

弦理论是描述普朗克尺度时空几何的量子引力理论的候选理论。它不仅统一了粒子物理学中所有的力和物质类型，而且它也是数学家们将各种看似遥远的学科联系起来的巨大灵感来源。镜像对称是弦理论中关于辛几何和复几何猜想的一个领域。这是对两种几何结构的研究，辛结构和复杂结构，它们有着非常不同的风格：辛几何是非常灵活的，具有无限维的对称性，完全不是代数的，但复杂几何是非常刚性和代数的。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Kontsevich在1994年提出的一个框架，它给出了对围绕镜像对称的奥秘的概念性理解。Calabi-Yau 3-fold是弦理论中重要的一类具有丰富几何结构的六维弯曲空间，包括复杂结构和辛结构。粗略地说，同调镜像对称猜想认为Calabi-Yau 3-fold是成对出现的，因此在某种精确意义上，一个Calabi-Yau 3-fold的辛几何等价于另一个的复几何。在经典几何中，这两个空间是相当不同的（它们有不同的拓扑或形状），但弦理论预测，如果考虑到量子效应，它们是等价的。这表明我们应该彻底改变我们对几何的观点：我们不应该区分这两个空间，就像我们不应该区分欧几里得几何中的全等图形一样。镜像对称的一个重要方面是它没有保留问题的难度：它经常将一个空间的辛几何中的困难问题转换为另一个空间的复杂（或代数）几何中的较容易问题，从而导致许多令人惊讶的应用。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换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来说很难证明镜像对称，事实上，只有少数情况下，已知同调镜像对称是成立的。我们将解决在更一般的情况下证明它的问题，使用弦理论中的另一个思想叫做膜平铺。膜砖是弦理论家发明的组合物体，它被期望在一个空间的辛几何和其镜像的复几何之间插入，平等地对待它们，因此很好地符合人们不应该区分镜像对称的两面的哲学。我们对膜铺和辛几何之间的关系知之甚少，我们希望澄清和证明它们。我们期望膜铺层和复几何之间的关系更易于处理，因此对膜铺层和辛几何之间关系的证明将引导我们对这些例子的同调镜像对称的证明。在我们的例子中，我们还将研究特殊拉格朗日3-褶皱的几何，它是Calabi-Yau 3-褶皱的一种特殊的子空间，具有最小的体积。这些子空间的存在不仅对辛几何和微分几何很重要，而且对弦理论家也很重要，因为它们是被称为d膜的物理对象的经典极限。我们希望证明这些子空间的存在与否是由一个叫做布里奇兰稳定性的代数准则决定的，这个准则最初来自弦理论。

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.