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，这使得它们自Seiberg和维滕在物理学中的工作以及唐纳森在数学中的工作以来成为数学的一个坚实的组成部分。在这个研究项目中，我将研究一类特别具有挑战性的QFT，除了超对称性之外，它还具有标度不变性-所谓的超共形场论。这些通常在较低维度中得到了相对较好的理解，最近在3d数学和物理学方面都取得了令人兴奋的成果。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来研究：它们本质上是强相互作用的，因此不能用微扰论来研究。在这个项目中，我将使用弦论中的定义，发展5d超共形场论的几何方法。简而言之，弦理论提供了从一类奇异几何（规范卡-丘三重奇点）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。5d SCFT的参数（或模）空间有一个特别漂亮的数学描述-它们是圆锥形的，奇异的，但在较小维度的奇异空间（所谓的辛奇异性）方面有一个叶状结构。从定义5d超共形场论的正则奇点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，从埃米·诺特的作品开始。近年来，人们发现了一种新的对称性概念，其中带电物体不是点状的，而是扩展的算子，这是高维的。这些所谓的高级形式对称性具有深远的影响：它们可能不会形成一个群，而是一个更高级的群，这是一个更高级的类别，可以作用于QFT。在5d超共形场论的背景下，这些对称结构将被研究，并反过来编码在定义的正则奇点。因此，这个项目也将提供一个新的，令人兴奋的奇点几何之间的联系，更高的范畴结构。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，并且与凝聚态系统对称性的最新发展有联系。总之，该计划的目标是提供一个分类的5d超共形场论的规范三重奇点，包括其量子希格斯分支模空间和广义对称性的表征。这个项目涉及量子场论在一个基本的水平，它是在其传统的定义作为一个经典理论和微扰理论的规范量子化方面的挑战。它提供了一类强相互作用的量子场的定义，通过一个纯粹的几何方法。有令人兴奋的连接和影响代数几何和分类的典型奇点，以及代数拓扑的广义对称性有一个自然的描述。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，以开发该项目的几何和拓扑/分类方面。该项目本质上是从各种不同的数学子领域，一个强大的团队，在几何和代数拓扑这两个核心领域的专业知识将是其成功的关键。我将在第四年举办一个大型的研究会议，从而将弦理论和数学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.