Canonical Singularities, Generalized Symmetries, and 5d Superconformal Field Theories

超对称量子场论(QFT)具有丰富而复杂的结构，这使它们成为自Seiberg和Witten在物理学以及Donaldson在数学中所做的工作以来数学的坚实组成部分。在这个研究项目中，我将研究一类特别具有挑战性的QFT，它除了具有超对称性外，还具有标度不变性--所谓的超共形场理论。这些通常在较低的维度上相对容易理解，最近在3D的数学和物理方面都有令人兴奋的结果。在更高的维度，如5d或6d，这样的理论基本上不可能从标准方法的角度进行研究：它们本质上是强相互作用的，因此无法用微扰理论来访问。在这个项目中，我将使用弦理论中的定义，发展5D超共形场论的几何方法。简而言之，弦理论提供了一种从一类奇异几何(规范的Calabi-Yau三重奇点)到5D超共形场论的映射。弦理论的一个显著含义是，使用这种方法可以研究这些强耦合理论，以及它们的参数空间和对称性。5DSCFT的参数(或模)空间有一个特别漂亮的数学描述--它们是锥形的，奇异的，但根据更小维度的奇异空间(所谓的辛奇点)，它们具有层状结构。从定义5D超共形场理论的正则奇点发展到这些奇异模空间的映射，是这个项目将需要的令人兴奋的数学联系之一。对称性当然是任何物理系统的核心，从艾美诺特的作品开始。近年来发现了一种新的对称性概念，带电物体不是点状的，而是更高维的扩展算子。这些所谓的更高形式的对称性具有深远的影响：它们可能不会形成一个群体，而是一个更高的群体，这是一个更高的类别，可以作用于QFT。在5D超共形场理论的背景下，我们将研究这些对称结构，并将其编码为定义的正则奇点。因此，这个项目还将在奇点几何和更高范畴结构之间提供一种新的、令人兴奋的联系。这些发展不仅对5D SCFT有影响，而且对低维的相关场论也有影响，并将与最近在凝聚态系统对称性方面的发展有联系。总而言之，本程序的目标是根据正则三重奇点对5D超共形场论进行分类，包括对它们的量子Higgs支模空间和它们的广义对称性的表征。这个项目涉及到基本水平的量子场论，它的传统定义是经典理论和微扰理论的规范量子化，这一点受到了挑战。它通过纯几何方法给出了一类强相互作用的量子场的定义。代数几何和正则奇点的分类以及广义对称具有自然描述的代数拓扑学有着令人兴奋的联系和含义。在这个项目中，我将有两个博士后，他们将带来互补的专业知识，以发展该项目的几何和拓扑/分类方面。该项目本质上来自各种不同的数学子领域，拥有几何和代数拓扑学这两个核心领域的专业知识的强大团队将是其成功的关键。我将在拨款的第四年主持一次大型研究会议，从而将弦理论和数学接口的主要专家带到英国。

Supersymmetric Quantum Field Theories (QFTs) have a rich and intricate structure, which has made them a firm part of mathematics since the works of Seiberg and Witten in physics, and Donaldson in mathematics. In this research program, I will study a particularly challenging class of QFTs, which in addition to supersymmetry have also scale invariance -- so-called superconformal field theories. These are often relatively well-understood in lower dimensions, with exciting recent results in both mathematics and physics in 3d. In higher dimensions such as 5d or 6d such theories become essentially impossible to study from the point of view of standard methods: they are intrinsically strongly interacting, and thus not accessible by perturbation theory. In this project I will develop the geometric approach to 5d superconformal field theories, using their definition in string theory. In a nutshell, string theory provides a map from a class of singular geometries (canonical Calabi-Yau three-fold singularities) to 5d superconformal field theories. It is a remarkable implication of string theory, that using this approach it is possible to study these strongly-coupled theories, as well as their parameter spaces and symmetries. Parameter (or moduli) spaces of 5d SCFTs have a particularly beautiful mathematical description -- they are conical, singular, but have a foliation structure in terms of smaller dimensional singular spaces (so-called symplectic singularities). Developing the map from canonical singularities, which define the 5d superconformal field theory, to these singular moduli spaces, is one of the exciting mathematical connections that this project will entail. Symmetries are of course central in any physical system, starting with the works of Emmy Noether. Recent years have uncovered a new notion of symmetry, where the charged objects are not point-like, but extended operators, which are higher dimensional. These so-called higher-form symmetries have far-reaching implications: they may not form a group, but a higher-group, which is a higher category, that can act on QFTs. In the context of 5d superconformal field theories, these symmetry structures will be studied, and in turn encoded in the defining canonical singularity. Thus, this project will also provide a new, exciting link between geometry of singularities, and higher categorical structures. These developments will have implications not only for 5d SCFTs, but related field theories in lower dimensions, and will have connections to recent developments in symmetries of condensed matter systems alike. In summary, this program has the goal to provide a classification of 5d superconformal field theories in terms of canonical three-fold singularities, including a characterization of their quantum Higgs branch moduli space and their generalized symmetries. This project touches upon quantum field theory at a fundamental level, where it is challenged in terms of its conventional definition as a canonical quantization of a classical theory and perturbation theory. It provides a definition of a class of quantum fields which are strongly interacting, by means of a purely geometric approach. There are exciting connections and implications to algebraic geometry and the classification of canonical singularities, as well as to algebraic topology where the generalized symmetries have a natural description. Within this project I will have two postdocs, who will bring in complementary expertise, to develop the geometric and topological/categorical aspects of the project. The project is intrinsically drawing from a variety of different mathematical sub-fields, and a strong team, with expertise in these two central areas of geometry and algebraic topology will be pivotal for its success. I will host a large research conference in year 4 of the grant, and thereby bring the main experts at the interface of String Theory and Mathematics to the UK.