Geometric structures and twisted supersymmetry

几何结构和扭曲超对称

• 批准号: EP/X014959/1
• 负责人: Charles Strickland-Constable
• 金额: $ 40.71万
• 依托单位: University of Hertfordshire
• 依托单位国家: 英国
• 项目类别: Research Grant
• 财政年份: 2023
• 资助国家: 英国
• 起止时间: 2023 至 无数据
• 项目状态: 未结题
• 来源: https://gtr.ukri.org/projects?ref=EP%2FX014959%2F1
• 关键词: Geometric; structures; twisted; supersymmetry

Since the era of Newton, there has been a close relationship between developments in mathematics and theoretical physics. One of the greatest challenges in the modern age is to formulate a unified theory combining the two pillars of our current understanding of the universe: the Standard Model of particle physics and Einstein's General Relativity. Despite their extraordinary successes, these two theories are fundamentally incompatible and it seems that, once again, a resolution of this physical problem will require great advances in mathematics.Superstring theory is a promising candidate for a unified theory, but it exists only in ten-dimensional spacetime. One way to explain the four-dimensional world that we observe via string theory is to view the extra six dimensions as a tiny compact geometric space (like the surface of a sphere, but with more dimensions) whose presence can only be directly detected by particles of such high energy that they are not created in current experiments. In this construction, the shape of this compact space determines the four-dimensional laws of physics we can see around us in experiments. Further, understanding these spaces is not only important for physical models in string theory - they are also the subject of deep questions in pure mathematics research.This project aims to develop new understanding of their geometric properties, via a new physical approach to the mathematics. In the simplest scenarios, the spaces which one requires turn out to be objects, called special holonomy manifolds, that have been studied by mathematicians for decades. Many researchers have focused on spaces in this class called Calabi-Yau manifolds, as they are physically promising and relatively easy to construct. However, more general solutions exist, with additional physical fields called fluxes, and these have physically desirable features. Recently, a new mathematical notion of geometry, "generalised geometry", has been developed which includes these fluxes naturally, and this provides an elegant description of the more general solutions. This is just one of many examples of physics leading to new mathematics and the mixing of these two disciplines has often led to astounding progress on both sides.An important observation, which linked string theory and geometry even more closely, was that on Calabi-Yau manifolds there exist simplified string theories, called topological string theories, whose physics are more tractable and directly encode interesting mathematical features of the geometries. In particular, they encode quantities which are invariant under smooth deformations of the geometry. These invariants are regarded as key mathematical properties of the spaces and are used, for example, to determine if constructed examples of such spaces are really different or not, which can be very hard to ascertain otherwise. Conversely, the invariants encode information about the physics of topological strings, which in turn can provide exact answers to calculations in the full string theory.However, all of this has only been understood in the cases of Calabi-Yau manifolds. There are signs that such theories exist in more general situations, and that generalised geometry is the natural way to approach their constructions. One scenario where this is particularly relevant is actually another type of special holonomy manifold: seven-dimensional spaces called G2 manifolds. These have become a particular focus of the geometry community, and even though they still have zero flux, generalised geometry has been seen to give an elegant "physical" approach to these spaces.This project aims to construct analogues of topological string theories in this wider context and to use these to discover new invariants, which will become key objects in both the mathematical and physical understanding of these spaces.

自牛顿时代以来，数学发展与理论物理学之间一直存在密切的关系。现代最大的挑战之一是制定一个统一理论，结合了我们当前对宇宙的两个支柱：粒子物理学的标准模型和爱因斯坦的一般相对论。尽管取得了非凡的成功，但这两种理论从根本上是不兼容的，似乎再次解决了这个物理问题的解决方案将需要在数学方面取得重大进步。统一理论是统一理论的有前途的候选人，但它仅存在于十维时段。解释我们通过弦理论观察到的四维世界的一种方法是将额外的六个维度视为一个微小的紧凑几何空间（例如球体的表面，但具有更大的尺寸），其存在只能被如此高能量的粒子直接检测到如此高能量，以至于它们在当前实验中没有产生。在这种结构中，这个紧凑空间的形状决定了我们在实验中可以看到的物理学的四维定律。此外，了解这些空间不仅对于弦理论中的物理模型很重要 - 它们也是纯数学研究中深层问题的主题。本项目旨在通过对数学的新物理方法来发展对其几何特性的新理解。在最简单的场景中，人们所要求的空间被证明是数十年来数学家已经研究的对象，称为特殊的自动歧管。许多研究人员都专注于这个班级的空间，称为卡拉比（Calabi-Yau）歧管，因为它们在身体上很有前途且相对易于构建。但是，存在更一般的解决方案，具有称为通量的其他物理领域，并且具有物理上理想的特征。最近，已经开发了一种新的数学数学概念“广义几何”，它自然包含这些通量，这提供了对更通用的解决方案的优雅描述。 This is just one of many examples of physics leading to new mathematics and the mixing of these two disciplines has often led to astounding progress on both sides.An important observation, which linked string theory and geometry even more closely, was that on Calabi-Yau manifolds there exist simplified string theories, called topological string theories, whose physics are more tractable and directly encode interesting mathematical features of the geometries.特别是，它们编码在几何形状平滑变形下不变的数量。这些不变性被认为是空间的关键数学属性，例如用于确定此类空间的构造示例是否真的不同，这可能很难确定。相反，不变性编码有关拓扑字符串物理学的信息，进而可以在完整的字符串理论中为计算提供确切的答案。有迹象表明这种理论存在于更普遍的情况下，并且广义几何形状是处理其结构的自然方法。一种特别相关的方案实际上是另一种特殊的全体歧管：七维空间，称为G2歧管。这些已成为几何社区的特别重点，尽管它们仍然具有零通量，但已被认为是对这些空间的优雅“物理”方法。该项目旨在在此更广泛的上下文中构建拓扑弦理论的类似物，并使用这些方法来发现这些新不变的对象，这将成为这些空间的数学和物理理解中的关键对象。

项目成果

A heterotic Kodaira-Spencer theory at one-loop

• DOI: 10.1007/jhep10(2023)130
• 发表时间: 2023-06
• 期刊: Journal of High Energy Physics
• 影响因子: 5.4
• 作者: A. Ashmore;Javier Jos'e Murgas Ibarra;D. McNutt;C. Strickland‐Constable;Eirik Eik Svanes;David Tennyson;Sander Winje