Mirror Symmetry for Fibrations and Degenerations

镜像对称性作为一门学科，起源于理论物理和弦理论。弦理论的核心思想是，亚原子粒子是细小的弦环，而不是点。然后，当这些弦的环振动、移动并相互作用时，我们观察到的亚原子物理就会出现。然而，为了产生我们在宇宙中观察到的物理属性，弦需要比我们通常的四个维度(3个空间和1个时间)提供的更多的移动空间。为了解决这个问题，弦理论假设宇宙应该有六个微小的额外维度，这些维度盘绕在一起形成一个称为“Calabi-Yau流形”的形状。有许多不同的Calabi-Yau流形，我们在弦理论中使用哪一个是重要的。正如改变光速会从根本上改变我们宇宙的物理学一样，改变卡拉比-尤流形也应该如此。然而，在弦理论发展的早期，物理学家们注意到了一个奇怪的反常现象：每个Calabi-Yau流形似乎都有一个伙伴Calabi-Yau流形，当穿过弦理论机器时，它给出了相同的物理预测。这种观察到的Calabi-Yau流形配对是已知的第一个镜像对称的例子。在数学上，镜像对称可以被认为是许多几何对象(如Calabi-Yau流形)有一个“镜像伙伴”的想法：第二个几何对象的性质与第一个密切相关。这是一个非常强大的数学工具。通常，关于几何对象的数学难题可以通过镜像对称性转化为关于其镜像对象的简单得多的问题。然而，有一个基本问题限制了它在实践中的使用：给定一个几何对象，我们通常不知道如何为它构造一个镜像伙伴！解决此问题的尝试导致了许多镜像伙伴的特殊定义，其中每一种定义都适用于某些类型的几何对象，而对于其他类型的对象则完全失败。这就引出了镜像对称的第二个基本问题：是否存在一个将所有不同的公式整合到一个一致的框架中的单一的最重要的理论？这项提议旨在通过证明镜像对称的两个最常用的公式实际上是一个更大图景的一部分来解决第二个问题。这两个公式是“Calabi-Yau镜像对称性”和“Fano/LG对应”，前者是上述Calabi-Yau流形的原始公式，后者指出称为“Fano流形”的几何对象的镜像伙伴是“Landau-Ginzburg(LG)模型”。这一理论的一个强有力的应用是构造新的Calabi-Yau流形镜像对，该理论也将作为本提议的一部分进行研究。要做到这一点，首先要从Fano流形及其镜像合作伙伴LG模型开始；此类配对的许多例子都是已知的。利用这一理论，可以将Fano流形粘合在一起得到Calabi-Yau流形，然后将它们的镜像伙伴LG模型粘合在一起得到第二个Calabi-Yau流形，这样得到的两个Calabi-Yau流形就是镜像伙伴。

Mirror symmetry, as a discipline, has its roots in theoretical physics and string theory. The core idea of string theory is that subatomic particles are tiny loops of string, instead of points. The subatomic physics that we observe then arises as these loops of string vibrate, move about, and interact with each other. However, to produce the physical properties that we observe in our universe, the strings need more space to move than is afforded to them by our usual four dimensions (3 space and 1 time). To solve this problem, string theory postulates that the universe should have six tiny extra dimensions, which are coiled up together into a shape called a "Calabi-Yau manifold". There are many different Calabi-Yau manifolds, and which one we use in string theory is important. Just as changing the speed of light would fundamentally alter the physics of our universe, so too should changing the Calabi-Yau manifold. However, early in the development of string theory, physicists noticed a curious anomaly: every Calabi-Yau manifold seems to have a partner Calabi-Yau manifold, which gives identical physical predictions when passed through the string theory machinery.This observed pairing-up of Calabi-Yau manifolds was the first known example of mirror symmetry. Mathematically, mirror symmetry can be thought of as the idea that many geometric objects (such as Calabi-Yau manifolds) have a "mirror partner": a second geometric object whose properties are closely related to the first.This is a tremendously powerful mathematical tool. Often, difficult mathematical questions about a geometric object can be translated, through mirror symmetry, into much simpler questions about its mirror partner. However, there is a fundamental problem that restricts the use of this in practice: given a geometric object, we usually have no idea how to construct a mirror partner for it! Attempts to solve this problem have led to a number of ad-hoc definitions of mirror partners, each of which works for some types of geometric objects and completely fails for others. This leads to the second fundamental problem of mirror symmetry: is there a single overarching theory that combines all of the different formulations into one consistent framework?This proposal aims to address this second question by showing that two of the most frequently used formulations of mirror symmetry are actually parts of one bigger picture. The two formulations in question are "Calabi-Yau mirror symmetry", which is the original formulation for Calabi-Yau manifolds as described above, and the "Fano/LG correspondence", which states that the mirror partner of a geometric object called a "Fano manifold" is a "Landau-Ginzburg (LG) model".A powerful application of this theory, that will also be studied as part of this proposal, is to the construction of new mirror pairs of Calabi-Yau manifolds. To do this, one starts with Fano manifolds and their mirror partner LG models; many examples of such pairs are known. Using the theory developed in this proposal, one may glue together Fano manifolds to get a Calabi-Yau manifold, and glue together their mirror partner LG models to get a second Calabi-Yau manifold, such that the two Calabi-Yau manifolds obtained are mirror partners.