Homological Algebra of Landau-Ginzburg Mirror Symmetry

Landau-Ginzburg 镜像对称的同调代数

• 批准号: EP/Y033574/1
• 负责人: Tyler Kelly
• 金额: $ 10.45万
• 依托单位: University of Birmingham
• 依托单位国家: 英国
• 项目类别: Research Grant
• 财政年份: 2024
• 资助国家: 英国
• 起止时间: 2024 至 无数据
• 项目状态: 未结题
• 来源: https://gtr.ukri.org/projects?ref=EP%2FY033574%2F1
• 关键词: Homological; Algebra; Landau; Ginzburg; Mirror

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.

这是一个研究项目，旨在建立最初来自弦理论的二元性的代数和几何结果。在世纪之交之前，理论物理学为几何学提供了一个见解，从而在几何研究中取得了许多现代成功。他们发现弦理论中的二元性对研究更高维几何形状的研究具有含义和应用，从而回答了有关某些六维空间的几何形状的经典问题。粗略地说，对于（经典）字符串理论为宇宙提供了潜在的物理理论，它要求宇宙是10维的。这些维度中有四个是标准的3个空间维度和我们在生活中经历的一个时间维度，而其他六个是所谓的Calabi-yau歧管。目前尚不清楚有多少卡拉比（Calabi-Yau）歧管，我们以许多不同的方式研究它们，但是弦理论使我们在几何研究学科之间建立了深厚的联系。尤其是弦理论中的双重性指出，每个calabi-yau歧管都有一个“镜子”，这是另一个calabi-yau歧管，因此一个镜子的各种几何和物理特性都封装在其镜像的其他几何和物理特性中。数学中的这种现象现在称为镜像对称性。尤其是，与卡拉比远流形相关的符号几何形状中的硬计算和计算开放问题现在在其镜像的代数几何形状中编码。在镜像对称性的开始时，这些代数几何计算要容易得多，然后将它们用作指导原理，以使我们旨在证明在符号几何形状中。这使得可以实现的枚举几何形状中的这个百年历史的问题。 1994年，纪念奖得主肯特维奇（Kontsevich）提供了镜子对称性的猜想，但完全数学版本，编码了所谓的福卡亚类别中的符号几何形状，以及在衍生的相干冰期类别中的代数几何形状。这为这种物理和几何现象提供了强大的配方。过去三十年来，镜像对称性已经扩大了，现在可以看到，镜像对称不仅是卡拉比雅歧管之间的关系，而且是更多的几何学空间（例如，fano cormololds，log calabi-yau yau yau calabi-yau varieties）。但是，它也扩展到了奇异性研究。有趣的是，可以通过构造功能来对某些空间的几何形状进行建模，从而使该函数沿原始空间是单数。然后，人们可以变形该模型，并仍然获得字符串理论的物理模型。这是Landau-Ginzburg模型的一个例子。在少数情况下，已经为Landau-Ginzburg模型建立了镜像对称性，并且在研究经典的高维形状（例如Calabi-yau歧管）的研究中已被证明具有强大的功能。但是，在Landau-Ginzburg模型的镜像对称性研究中，仍然存在基本问题。理想情况下，我们想证明Kontsevich对Landau-Ginzburg模型的猜想的一种形式，但是在我们这样做之前，我们将需要了解Landau-Ginzburg模型的代数几何方面。该项目旨在更好地理解Landau-Ginzburg模型的这种分类观点，证明了它们对上述相干滑轮类别类别的类似物的各种结构结果，称为（矩阵）分解类别。