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，旨在建立代数和几何结果，其灵感源自弦理论的对偶性。就在世纪之交之前，理论物理学提供了对几何的见解，从而导致了现代几何研究的许多成功。他们发现弦理论中的对偶性对高维几何的研究具有影响和应用，使得有关某些六维空间几何的经典问题的答案变得容易理解。粗略地说，（经典）弦理论要为宇宙提供潜在的物理理论，就需要宇宙是10维的。其中四个维度是我们在生活中经历的标准的三个空间维度和一个时间维度，另外六个维度是所谓的卡拉比-丘流形。目前尚不清楚卡拉比-丘流形有多少个，我们以多种不同的方式研究它们，但弦理论使我们在几何研究学科之间建立了深厚的联系。特别是，弦理论中的对偶性指出，每个卡拉比-丘流形都有一个“镜子”，它是另一个卡拉比-丘流形，因此一个流形的各种几何和物理属性被封装在其镜子的其他几何和物理属性中。这种数学现象现在被称为镜像对称。特别是，与卡拉比-丘流形相关的辛几何中的硬计算和计算开放问题现在被编码在其镜像的代数几何中。在镜像对称出现时，这些代数几何计算变得容易得多，然后它们被用作我们旨在证明辛几何的指导原则。这使得计数几何中的百年难题得以解决。 1994 年，菲尔兹奖得主 Kontsevich 提供了镜像对称的猜想但完全数学版本，将辛几何编码为所谓的 Fukaya 范畴，并将代数几何编码为相干滑轮的派生范畴。这为这种物理和几何现象提供了强有力的代数表述。在过去的三十年里，镜像对称性得到了扩展，现在我们发现镜像对称性不仅是卡拉比-丘流形之间的关系，而且还涉及更多的几何空间（例如、Fano 流形、log Calabi-Yau 品种）。然而，它也被扩展到奇点的研究。有趣的是，我们可以通过构造一个函数来模拟某些空间的几何形状，使得该函数沿着原始空间是奇异的。然后，我们可以对该模型进行变形，但仍然可以获得弦理论的物理模型。这是 Landau-Ginzburg 模型的示例。在少数情况下，Landau-Ginzburg 模型已经建立了镜像对称性，并且它在研究经典的高维形状（如 Calabi-Yau 流形）中已被证明是强大的。然而，Landau-Ginzburg 模型的镜像对称性研究仍然存在一些基础问题需要处理。理想情况下，我们希望证明朗道-金茨堡模型的康采维奇猜想的一种形式，但在我们这样做之前，我们需要了解朗道-金茨堡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，证明与上述相干滑轮的派生类别（称为（矩阵）因式分解类别）类似的各种结构结果。