Bridging Frameworks via Mirror Symmetry

通过镜像对称桥接框架

• 批准号: EP/N004922/2
• 负责人: Tyler Kelly
• 金额: $ 9.87万
• 依托单位: University of Birmingham
• 依托单位国家: 英国
• 项目类别: Fellowship
• 财政年份: 2018
• 资助国家: 英国
• 起止时间: 2018 至 无数据
• 项目状态: 已结题
• 来源: https://gtr.ukri.org/projects?ref=EP%2FN004922%2F2
• 关键词: Bridging; Frameworks; via; Mirror; Symmetry

Stand in one place. Ask the question "What are the possible ways you could face while standing there?'' One answer is from zero degrees to 360 degrees, but that is not a fully-satisfying answer. The most intuitive answer is you can turn around in a circle. This answer is an example of a geometric classification of possible solutions, or a moduli space. Moduli spaces are ubiquitous in geometry. From conic sections to the range of motion of a robot, one is studying moduli spaces. In algebraic geometry, we study the geometry of the solutions of polynomials and associated geometric classification problems. When one has many variables and uses higher degrees, such questions become difficult. Such shapes formed by Typically there are three ways to study varieties: looking at other objects that sit inside them, finding ways that they sit inside other objects, and finding invariants that help classify them.In the last 25 years, string theory has giving intuitive frameworks for studying certain classical algebro-geometric objects, Calabi-Yau shapes. In string theory, Calabi-Yau shapes are added to the space-time continuum in order to get physical models for the universe. In mathematics, this led to a geometric duality called mirror symmetry which focuses on the duality between Type IIA and IIB string theory. This rich framework allows many connections between mathematical fields, typically symplectic geometry and algebraic geometry.Many of the connections made have to do with enumerative geometry, studying how many curves of a certain type sit inside higher dimensional objects. Mirror symmetry turned this problem in symplectic geometry into an algebro-geometric problem, making it easier to compute the answer. Some of the connections sit in number theory. Varieties have number-theoretic analogues where one can study them over a finite field, providing geometric analogues to the Riemann zeta function. The proposed research plan focuses on finding bridges amongst fields motivated by mirror symmetry. The proposal involves the following projects:1.) Providing a method to compute the FJRW-invariants in symplectic geometry by linking the invariants to an algebro-geometric setting then using tropical geometry. These invariants describe how many curves of a certain type sit in a generalized version of a Calabi-Yau shape, called a Landau-Ginzburg model.2.) Studying the number theoretic properties of Calabi-Yau shapes when viewed under mirror symmetry, harnessing properties of the zeta function associated to these shapes.3.) Classify a certain class of higher-dimensional analogues to polygons by using their correspondence to algebraic objects by using geometric quotients, consequently giving a classification of certain types of Calabi-Yau shapes.4.) Codify what mirror symmetry means for another type of string theory, heterotic mirror symmetry.The work presented here will provide more links amongst mathematical fields, creating a more cohesive mathematical community. Each project takes two fields and connects them in a way so that both fields can contribute to the understanding of Calabi-Yau shapes.

站在一处。问“你站在那里可能面对的方式有哪些？”一个答案是从0度到360度，但这并不是一个完全令人满意的答案。最直观的答案是你可以转一圈这个答案是可能解决方案的几何分类的一个例子，模空间在几何中无处不在，从圆锥曲线到机器人的运动范围，人们正在研究代数空间。几何，我们研究多项式解的几何以及相关的几何分类问题，当一个变量有很多并且使用更高的次数时，此类问题变得很困难。通常有三种方法来研究品种：观察其他物体。在过去 25 年里，弦理论为研究某些经典代数几何对象（卡拉比-丘形状）提供了直观的框架。在弦理论中，卡拉比-丘形状被添加到时空连续体中以获得宇宙的物理模型。在数学中，这导致了一种称为镜像对称的几何对偶性，它侧重于 IIA 型和 IIB 型弦理论之间的对偶性。这个丰富的框架允许数学领域之间的许多联系，通常是辛几何和代数几何。许多联系都与枚举几何有关，研究某种类型的曲线有多少条位于更高维度的对象内。镜像对称将辛几何中的这个问题转化为代数几何问题，使得计算答案变得更加容易。有些联系存在于数论中。品种具有数论类似物，人们可以在有限域上研究它们，为黎曼 zeta 函数提供几何类似物。拟议的研究计划侧重于寻找镜像对称驱动的领域之间的桥梁。该提案涉及以下项目：1.) 通过将不变量链接到代数几何设置，然后使用热带几何，提供一种计算辛几何中 FJRW 不变量的方法。这些不变量描述了卡拉比-丘形状的广义版本（称为朗道-金兹堡模型）中有多少某种类型的曲线。2.) 研究在镜面对称下观察时卡拉比-丘形状的数论性质，利用性质与这些形状相关的 zeta 函数。3.) 通过使用几何商，利用它们与代数对象的对应关系，对多边形的某一类高维类似物进行分类，从而给出分类某些类型的卡拉比-丘形状。4.) 整理出镜像对称对于另一种类型的弦理论（异质镜像对称）意味着什么。这里介绍的工作将在数学领域之间提供更多联系，创建一个更有凝聚力的数学社区。每个项目都采用两个领域，并以某种方式将它们连接起来，以便这两个领域都有助于理解卡拉比-丘形状。

项目成果

Derived categories of BHK mirrors

BHK镜子的衍生类别

• DOI: 10.1016/j.aim.2019.06.013
• 发表时间: 2016-02-18
• 期刊: Advances in Mathematics
• 影响因子: 1.7
• 作者: David Favero;Tyler L. Kelly
• 通讯作者: Tyler L. Kelly

Exceptional collections for mirrors of invertible polynomials

可逆多项式镜像的特殊集合

• DOI: http://dx.10.1007/s00209-023-03258-x
• 发表时间: 2023
• 期刊: Mathematische Zeitschrift
• 影响因子: 0.8
• 作者: Favero D

2017 MATRIX Annals

2017 年矩阵年鉴

• 发表时间: 2019
• 作者: Doran C.F.

Mirror Symmetry for open r-spin invariants

开放 r-自旋不变量的镜像对称

• 发表时间: 2022-03-04
• 作者: M. Gross;Tyler L. Kelly;Ran J. Tessler
• 通讯作者: Ran J. Tessler

Open FJRW Theory and Mirror Symmetry

开放式 FJRW 理论和镜像对称

• DOI: http://dx.10.48550/arxiv.2203.02435
• 发表时间: 2022
• 作者: Gross M