Homological Algebra of Landau-Ginzburg Mirror Symmetry

这是一个研究项目，旨在建立代数和几何结果，灵感来自弦理论的对偶性。就在世纪之交之前，理论物理提供了对几何的洞察，导致了许多现代几何研究的成功。他们发现弦理论中的对偶性对高维几何的研究有影响和应用，使得关于某些六维空间的几何的经典问题的答案变得容易理解。粗略地说，要让(经典)弦理论为宇宙提供潜在的物理理论，它需要宇宙是10维的。其中四个维度是我们在生活中经历的标准的3个空间维度和1个时间维度，另外6个维度是所谓的Calabi-Yau流形。目前尚不清楚有多少Calabi-Yau流形，我们用许多不同的方法来研究它们，但弦理论给了我们几何研究学科之间的深层次联系。特别是，弦理论中的对偶理论指出，每个Calabi-Yau流形都有一个“镜像”，这是另一个Calabi-Yau流形，因此一个流形的各种几何和物理性质都封装在它的镜像的其他几何和物理性质中。数学中的这种现象现在被称为镜像对称。特别是，与Calabi-Yau流形相关的辛几何中的困难计算和计算开放问题现在被编码到它的镜像的代数几何中。在镜像对称开始时，这些代数几何计算要容易得多，然后它们被用作我们在辛几何中要证明的东西的指导原则。这使得枚举几何中的百年问题成为可能。1994年，菲尔兹奖章获得者康采维奇提出了镜像对称的猜想但完全数学的版本，将辛几何编码为Fukaya范畴，将代数几何编码为相干层的派生范畴。在过去的三十年里，镜像对称性得到了扩展，现在发现镜像对称不仅是Calabi-Yau流形之间的关系，而且还有更多的几何空间(例如，Fano流形，log Calabi-Yau簇)。然而，它也被扩展到奇点的研究。有趣的是，人们可以通过构造一个函数来模拟某些空间的几何形状，从而使该函数在原始空间中是奇异的。然后，人们可以变形这个模型，仍然可以得到弦理论的物理模型。这是Landau-Ginzburg模型的一个例子。在少数情况下，已建立了Landau-Ginzburg模型的镜像对称性，并已证明它在研究经典高维形状如Calabi-Yau流形方面具有很强的能力。然而，在研究Landau-Ginzburg模型的镜面对称性时，仍有一些基础性问题需要解决。理想情况下，我们希望证明一种形式的康采维奇猜想的Landau-Ginzburg模型，但在我们这样做之前，我们需要了解Landau-Ginzburg模型的代数几何方面。这个项目旨在更好地理解Landau-Ginzburg模型的这一范畴观点，证明其类似于上面推导出的相干滑轮范畴(称为(矩阵)因式分解范畴)的各种结构结果。

This is a research project to establish algebraic and geometric results inspired by a duality originally from string theory.Right before the turn of the century, theoretical physics provided an insight to geometry that led to many modern successes in geometric research. They discovered a duality in string theory had implications and applications to the study of higher dimensional geometry, making answers to classical questions about the geometry of certain six-dimensional spaces accessible. Roughly speaking, for (classical) string theory to provide a potential physical theory for the universe, it requires the universe to be 10-dimensional. Four of these dimensions are the standard 3 space dimensions and one time dimension we experience in our lives, and the other six are a so-called Calabi-Yau manifold. It is still unclear how many Calabi-Yau manifolds there are, and we study them in many different ways, but string theory has given us a deep connection between geometric research disciplines. In particular, a duality in string theory states that each Calabi-Yau manifold has a "mirror" which is another Calabi-Yau manifold so that various geometric and physical properties of one are encapsulated in other geometric and physical properties of its mirror. This phenomenon in mathematics is now known as mirror symmetry. In particular, hard computations and computational open questions in symplectic geometry associated to a Calabi-Yau manifold were now encoded in the algebraic geometry of its mirror. At the onset of mirror symmetry, these algebro-geometric computations were much easier and then they were then used as a guiding principle for what we aim to prove in symplectic geometry. This made century-old problems in enumerative geometry achievable. In 1994, the Fields Medallist Kontsevich provided a conjectural but fully mathematical version of mirror symmetry, encoding the symplectic geometry in what is called a Fukaya category and the algebraic geometry in a derived category of coherent sheaves. This provided a robust formulation in algebra of this physical and geometric phenomenon.Throughout the past three decades, mirror symmetry has expanded and it is now seen that mirror symmetry is not just a relationship amongst Calabi-Yau manifolds, but many more geometric spaces (e.g., Fano manifolds, log Calabi-Yau varieties). However, it has also been extended to the study of singularities. Interestingly, one can model the geometry of certain spaces by constructing a function so that the function is singular along the original space. Then one can deform this model and still obtain a physical model for string theory. This is an example of a Landau-Ginzburg model. Mirror symmetry has been established for Landau-Ginzburg models in a few cases, and it has been shown to be powerful in the study of classical higher-dimensional shapes such as Calabi-Yau manifolds. However, there are still foundational issues to be handled in the study of mirror symmetry for Landau-Ginzburg models. Ideally, we would like to prove a form of Kontsevich's conjecture for Landau-Ginzburg models, but before we do so in general, we will need to understand the algebro-geometric aspects of Landau-Ginzburg models. This project aims to better understand this categorical point of view for Landau-Ginzburg models, proving various structural results on their analogue of the derived category of coherent sheaves above, known as the (matrix) factorisation category.