Mirror Symmetry for Fibrations and Degenerations

镜像对称作为一门学科，其根源在于理论物理学和弦理论。弦理论的核心思想是，亚原子粒子是弦的微小环，而不是点。我们所观察到的亚原子物理学就产生于这些弦环的振动、移动和相互作用。然而，为了产生我们在宇宙中观察到的物理性质，弦需要比我们通常的四维（3个空间和1个时间）提供更多的空间来移动。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，它们盘绕在一起形成一个称为“卡-丘流形”的形状。有许多不同的卡-丘流形，我们在弦理论中使用哪一个很重要。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-丘流形也应该如此。然而，在弦理论发展的早期，物理学家注意到一个奇怪的异常：每个卡-丘流形似乎都有一个伙伴卡-丘流形，当通过弦理论机器时，它给出了相同的物理预言，这种观察到的卡-丘流形配对是第一个已知的镜像对称的例子。在数学上，镜像对称可以被认为是许多几何对象（如卡-丘流形）都有一个“镜像伙伴”的想法：第二个几何对象的性质与第一个几何对象密切相关。这是一个非常强大的数学工具。通常，关于几何对象的困难数学问题可以通过镜像对称转化为关于其镜像伙伴的简单得多的问题。然而，有一个基本的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！为了解决这个问题，人们提出了许多镜像伙伴的特别定义，每一种定义都适用于某些类型的几何对象，而对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：是否存在一个单一的总体理论，将所有不同的表述结合到一个一致的框架中？这个建议旨在解决第二个问题，通过展示两个最常用的镜像对称公式实际上是一个更大的图片的一部分。有问题的两个公式是“Calabi-Yau镜像对称”，这是上述Calabi-Yau流形的原始公式，以及“Fano/LG对应”，它指出称为“Fano流形”的几何对象的镜像伙伴是“Landau-Ginzburg（LG）模型”。这个理论的一个强大的应用，也将作为这个建议的一部分进行研究，就是构造新的镜像对。要做到这一点，首先从Fano流形和它们的镜像伙伴LG模型开始;这种对的许多例子都是已知的。利用这个理论，我们可以将Fano流形粘在一起得到一个Calabi-Yau流形，并将它们的镜像伙伴LG模型粘在一起得到第二个Calabi-Yau流形，这样得到的两个Calabi-Yau流形是镜像伙伴。

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.