Mirror Constructions: Develop, Unify, Apply

镜像结构：开发、统一、应用

• 批准号: EP/S03062X/1
• 负责人: Tyler Kelly
• 金额: $ 29.87万
• 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: 2019
• 资助国家: 英国
• 起止时间: 2019 至 无数据
• 项目状态: 已结题
• 来源: https://gtr.ukri.org/projects?ref=EP%2FS03062X%2F1
• 关键词: Mirror; Constructions; Develop; Unify; Apply

In this project, we research geometric problems inspired by string theory. In string theory, we view subatomic particles as strings, not points, requiring the universe to have six extra small dimensions called a Calabi-Yau shape. If we trace the string as it moves through time, it creates a (Riemann) surface. String theory has predicted amazing mathematics, which we, as mathematicians, prove rigorously.We are mainly focussed on studying shapes that can be viewed as the solution to a set of polynomial equations. Chosen with the correct data, such a system of equations can be used to define a Calabi-Yau shape. String theory predicts a duality that states that, for any Calabi-Yau space, there exists another space called the mirror. Various physical and geometric data between these two shapes is exchanged, creating a relationship that has come to be known as mirror symmetry. A key problem in this field is how one, given the Calabi-Yau space, finds the mirror space that is related to it. Once an explicit construction is developed, we then can check if a mirror relationship holds. There are various constructions in the literature with varying degrees of evidence of mirror symmetry; however, they often disagree! We aim in this project to deal with this discrepancy, unifying their approaches. In the same vein, we aim to potentially create new Calabi-Yau varieties while also giving their mirror shape, adding to the library of mirror pairs that currently exist.While Calabi-Yau spaces are often very difficult to visualize, they often have algebraic descriptions that are easy to study. In this project, we often will deform the Calabi-Yau shape so much that it is no longer even a Calabi-Yau space but some easier algebraic structure, known in the physics literature as a Landau-Ginzburg model. By proving relations between Landau-Ginzburg models, we will often find relations between Calabi-Yau shapes themselves. Thus, we will be able to relate various constructions algebraically in order to create a better overview of mirror proposals. Indeed, this explains the discrepancy above between different constructions for mirrors in the literature.In addition, we will study the algebraic relations to Landau-Ginzburg models in order to create new relations between Fano manifolds. While there is a large project regarding classification of Fano manifolds in low dimension, they often have the same interesting or intrinsic piece of algebraic structure, known as a (fractional) Calabi-Yau category. We aim to apply our intuition from unifying constructions in order to find relations between this fundamental data in order to streamline the relations between potential Fano manifolds. Lastly, we apply our understanding of the geometry of various Calabi-Yau spaces to computational number theory. The one-dimensional case of a Calabi-Yau shape, the elliptic curve, has played a leading role in cryptography in the last few decades; however, there have been recent proposals that have led to needing more understanding of higher dimensions. By interacting with computational number theorists, we will isolate fundamental Calabi-Yau shapes that exhibit interesting explicit number-theoretic phenomena, leading to applications for L-series.

在这个项目中，我们研究受弦理论启发的几何问题。在弦理论中，我们将亚原子粒子视为弦，而不是点，这要求宇宙有六个额外的小维度，称为卡拉比-丘形状。如果我们追踪弦随时间的移动，它会创建一个（黎曼）曲面。弦理论预言了令人惊奇的数学，我们作为数学家，严格地证明了这一点。我们主要专注于研究可以被视为一组多项式方程的解的形状。选择正确的数据后，这样的方程组可用于定义 Calabi-Yau 形状。弦理论预测了一种对偶性，即对于任何卡拉比-丘空间，都存在另一个称为镜子的空间。这两种形状之间的各种物理和几何数据被交换，创建了一种被称为镜像对称的关系。该领域的一个关键问题是，在给定卡拉比-丘空间的情况下，如何找到与其相关的镜像空间。一旦开发出显式构造，我们就可以检查镜像关系是否成立。文献中有多种结构具有不同程度的镜像对称证据。然而，他们常常意见不一致！我们的目标是在这个项目中解决这种差异，统一他们的方法。同样，我们的目标是创造新的 Calabi-Yau 变体，同时赋予它们的镜像形状，添加到现有的镜像对库中。虽然 Calabi-Yau 空间通常很难可视化，但它们通常具有代数描述那些很容易学习的。在这个项目中，我们经常会对卡拉比-丘形状进行如此大的变形，以至于它不再是卡拉比-丘空间，而是一些更简单的代数结构，在物理文献中称为朗道-金兹堡模型。通过证明 Landau-Ginzburg 模型之间的关系，我们经常会发现 Calabi-Yau 形状本身之间的关系。因此，我们将能够以代数方式关联各种构造，以便更好地概述镜像提案。事实上，这解释了文献中不同镜子结构之间的差异。此外，我们将研究朗道-金兹堡模型的代数关系，以便在法诺流形之间创建新的关系。虽然有一个关于低维 Fano 流形分类的大型项目，但它们通常具有相同的有趣或内在的代数结构，称为（分数）Calabi-Yau 类别。我们的目标是运用统一构造的直觉来找到这些基本数据之间的关系，从而简化潜在法诺流形之间的关系。最后，我们将对各种卡拉比-丘空间几何的理解应用到计算数论中。卡拉比-丘形状的一维情况，即椭圆曲线，在过去几十年中在密码学中发挥了主导作用；然而，最近的一些提议导致需要对更高维度有更多的了解。通过与计算数论学家的互动，我们将分离出基本的 Calabi-Yau 形状，这些形状表现出有趣的明确数论现象，从而导致 L 级数的应用。

项目成果

Multiplicative preprojective algebras of Dynkin quivers

Dynkin 箭袋的乘法原射代数

• DOI: http://dx.10.48550/arxiv.2104.08362
• 发表时间: 2021
• 作者: Kaplan D

Multiplicative preprojective algebras of Dynkin quivers

Dynkin 箭袋的乘法原射代数

• DOI: http://dx.10.1016/j.jpaa.2022.107146
• 发表时间: 2023
• 期刊: Journal of Pure and Applied Algebra
• 影响因子: 0.8
• 作者: Kaplan D

Exceptional Collections for Mirrors of Invertible Polynomials

可逆多项式镜像的特殊集合

• DOI: http://dx.10.48550/arxiv.2001.06500
• 发表时间: 2020
• 作者: Favero D

Fano Schemes of Complete Intersections in Toric Varieties

环面簇中完全交集的 Fano 方案

• DOI: http://dx.10.48550/arxiv.1910.05593
• 发表时间: 2019
• 作者: Ilten N

A maximally-graded invertible cubic threefold that does not admit a full exceptional collection of line bundles

最大分级可逆立方三重，不允许线束的完整异常集合

• DOI: http://dx.10.1017/fms.2020.44
• 发表时间: 2020
• 期刊: Forum of Mathematics, Sigma
• 作者: Favero D