Bridging Frameworks via Mirror Symmetry

通过镜像对称桥接框架

• 批准号: EP/N004922/1
• 负责人: Tyler Kelly
• 金额: $ 28.37万
• 依托单位: University of Cambridge
• 依托单位国家: 英国
• 项目类别: Fellowship
• 财政年份: 2015
• 资助国家: 英国
• 起止时间: 2015 至 无数据
• 项目状态: 已结题
• 来源: https://gtr.ukri.org/projects?ref=EP%2FN004922%2F1
• 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.

站在一个地方。问这样一个问题：“当你站在那里的时候，你可能会面对什么样的情况？”一个答案是从零度到360度，但这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个圈。这个答案是可能解的几何分类或模空间的一个例子。模空间在几何学中无处不在。从圆锥曲线到机器人的运动范围，人们正在研究模空间。在代数几何中，我们研究多项式的解的几何和相关的几何分类问题。当一个人有许多变量并使用更高的度时，这样的问题变得困难。通常有三种方法来研究多样性：观察位于它们内部的其他物体，找到它们位于其他物体内部的方式，以及找到有助于分类它们的不变量。在过去的25年里，弦理论为研究某些经典的代数几何物体，卡-丘形状，提供了直观的框架。在弦理论中，卡-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，它专注于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，典型的是辛几何和代数几何。许多联系都与枚举几何有关，研究有多少特定类型的曲线位于高维对象中。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容易。有些联系存在于数论中。变种有数论类似物，人们可以在有限域上研究它们，提供黎曼zeta函数的几何类似物。拟议的研究计划的重点是寻找由镜像对称激发的领域之间的桥梁。建议涉及以下项目：（1）给出了一种在辛几何中计算FJRW-不变量的方法，将不变量与代数几何相联系，然后使用热带几何。这些不变量描述了有多少特定类型的曲线位于卡-丘形状的广义版本中，称为Landau-Ginzburg模型。研究镜像对称下卡-丘形状的数论性质，利用与这些形状相关的zeta函数的性质。通过使用几何等价物，利用它们与代数对象的对应关系，对多边形的某类高维类似物进行分类，从而对某些类型的卡-丘形状进行分类。把镜像对称对另一种类型的弦理论--杂合镜像对称--的意义整理出来。这里介绍的工作将在数学领域之间提供更多的联系，创造一个更有凝聚力的数学社区。每个项目都有两个领域，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-丘形状。

项目成果

2017 MATRIX Annals

2017 年矩阵年鉴

• 发表时间: 2019
• 作者: Doran C.F.

Hypergeometric decomposition of symmetric K3 quartic pencils

对称 K3 四次铅笔的超几何分解

• DOI: 10.48550/arxiv.1810.06254
• 发表时间: 2018
• 作者: Doran C

Hypergeometric decomposition of symmetric K3 quartic pencils.

对称 K3 四次铅笔的超几何分解。

• DOI: 10.1007/s40687-020-0203-3
• 发表时间: 2020
• 期刊: Research in the mathematical sciences
• 影响因子: 1.2
• 作者: Doran CF

Fractional Calabi?Yau categories from Landau?Ginzburg models

Landau?Ginzburg 模型中的分数 Calabi?Yau 类别

• DOI: 10.14231/ag-2018-016
• 发表时间: 2018
• 期刊: Algebraic Geometry
• 影响因子: 1.5
• 作者: Favero D

Equivalences of families of stacky toric Calabi-Yau hypersurfaces

堆叠复曲面 Calabi-Yau 超曲面族的等价

• DOI: 10.1090/proc/14154
• 发表时间: 2018
• 期刊: Proceedings of the American Mathematical Society
• 影响因子: 1
• 作者: Doran C