Local Mirror Symmetry and Five-dimensional Field Theory

局部镜像对称和五维场论

• 批准号: EP/W021714/1
• 负责人: Cyril Closset
• 金额: $ 46.65万
• 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: 2022
• 资助国家: 英国
• 起止时间: 2022 至 无数据
• 项目状态: 未结题
• 来源: https://gtr.ukri.org/projects?ref=EP%2FW021714%2F1
• 关键词: Local; Mirror; Symmetry; Five; dimensional

The Universe becomes surprisingly simple at very large and at very small scales. Quantum field theory (QFT) is the simplest framework used in modern physics to describe both the very large and the very small. It is a very successful scientific theory, whose predictions have been confirmed experimentally to an astonishing degree, from the realms of particle physics to astrophysics. Nonetheless, one century after its inception, QFT remains a somewhat ad-hoc structure: Firstly, the physicists' understanding of QFT remains somewhat superficial, and breaks down in the so-called strong-coupling regime. Secondly, QFT is very poorly understood mathematically. This uncomfortable position of QFT is less a problem than a motivation to refine our tools, both in physics and in mathematics.My research focusses on supersymmetric QFT and on its mathematical applications. Supersymmetry is an elegant idea from theoretical physics which posits an equivalence between particles of forces (like the photon) and particles of matter (like the electron). It is an incredibly useful tool to study QFT at a more fundamental level, because it allows one to relate many QFT phenomena, such as vacuum degeneracies and particle excitations, to geometric concepts such as algebraic varieties (like the zeros of a polynomial) and enumerative geometry (the counting of various geometric objects), which are of interest to pure mathematicians.This research project consists of two interconnected strands. Firstly, we will study a conjectural map between certain singular geometries, called canonical singularities, and five-dimensional superconformal field theories (5d SCFT), a type of QFT that lives in five space-time dimensions instead of the four we see around us. These theories appear naturally as limits of string theory and of its 11-dimensional completion, M-theory. The main aim is to connect 5d SCFT to local mirror symmetry, which is a string theory relation that has become a very rich area of study in pure mathematics. Mirror symmetry is the statement that two very different geometric objects can be `the same' as far as quantum physics is concerned. This leads to very beautiful and surprising mathematical relations between `shapes' (algebraic geometry) and `volumes' (symplectic geometry), which mathematicians have been working on for many years. This research programme will give a new perspective on some of these mirror symmetry relations, thus furthering the dialogue between string theory and geometry.The second part of the project concerns the computation of quantum observables in 5d SCFT with cutting-edge tools called supersymmetric localisation techniques. These observables give `quantum invariants', which are objects that the QFT assign to a smooth space-time manifold (such as a sphere) which is locally independent of the metric, generalising Donaldson polynomials. We will uncover and explore new relations between these quantum invariants, on the one hand, and the enumerative geometry of the canonical singularities associated to the 5d SCFTs, on the other hand. This will lead to another bridge between QFT and pure mathematics.Thus, both strands of this research project aim to create the groundwork for future dialogues between physics and mathematics. Maintaining this dialogue is crucial for the health of both fields of investigation: physical methods uncover new mathematical relations that would not have been guessed otherwise, opening the way to new mathematical theories, and mathematical approaches in turn lead to a deeper understanding of physics. In due time, this may well lead to a deeper understanding of our own Universe.

宇宙在非常大和非常小的尺度上变得令人惊讶的简单。量子场论（QFT）是现代物理学中用来描述非常大和非常小的最简单的框架。这是一个非常成功的科学理论，其预言已经在实验上得到了惊人的证实，从粒子物理学到天体物理学。尽管如此，在其诞生世纪之后，QFT仍然是一个有点特别的结构：首先，物理学家对QFT的理解仍然有些肤浅，并在所谓的强耦合机制中破裂。其次，QFT在数学上的理解非常少。QFT的这种令人不安的地位与其说是一个问题，不如说是一种改进我们在物理和数学方面的工具的动机。超对称性是理论物理学中的一个优雅的概念，它假定力的粒子（如光子）和物质的粒子（如电子）之间是等价的。它是一个非常有用的工具，可以在更基础的层面上研究QFT，因为它允许人们将许多QFT现象，如真空简并和粒子激发，与几何概念，如代数簇（如多项式的零）和枚举几何（各种几何对象的计数）联系起来，这些概念是纯数学家感兴趣的。首先，我们将研究某些奇异几何（称为正则奇点）和五维超共形场论（5d SCFT）之间的拓扑映射，这是一种存在于五维时空中的QFT，而不是我们周围看到的四维。这些理论很自然地表现为弦理论及其11维完备理论M理论的极限。主要目的是将5d SCFT与局域镜像对称性联系起来，这是一个弦理论关系，已经成为纯数学中非常丰富的研究领域。镜像对称性是指两个非常不同的几何对象可以是“相同的”，就量子物理学而言。这导致了非常美丽和令人惊讶的数学关系之间的“形状”（代数几何）和“体积”（辛几何），数学家们已经工作了多年。该研究计划将为镜像对称关系提供一个新的视角，从而促进弦理论和几何之间的对话。该项目的第二部分涉及使用称为超对称局部化技术的尖端工具计算5d SCFT中的量子可观测量。这些可观测量给出了“量子不变量”，它们是QFT分配给局部独立于度量的光滑时空流形（如球体）的对象，推广了唐纳森多项式。我们将揭示和探索这些量子不变量之间的新关系，一方面，和枚举几何的正则奇点相关联的5D SCFT，另一方面。这将导致QFT和纯数学之间的另一座桥梁。因此，这个研究项目的两个分支都旨在为未来物理和数学之间的对话奠定基础。保持这种对话对于两个研究领域的健康发展至关重要：物理方法揭示了新的数学关系，否则就不会被猜测到，开辟了新的数学理论的道路，而数学方法反过来又导致了对物理学的更深入的理解。在适当的时候，这很可能会导致对我们自己的宇宙更深入的了解。