String theory maps certain singular geometric shapes—canonical Calabi-Yau three-fold singularities—directly to five-dimensional superconformal field theories, which are otherwise impossible to study using standard physics methods. These theories are fundamental to understanding quantum field theory at its limits, where conventional tools like perturbation theory break down entirely. The project addresses a gap in knowledge: how to define and classify strongly interacting quantum systems that exist in higher dimensions, where even basic properties like symmetries become complex categorical structures rather than simple groups. If successful, this research will provide a geometric classification of 5d superconformal field theories, linking singular geometry to higher-form symmetries and moduli spaces. The work is purely fundamental science—it does not target any immediate practical application. However, similar foundational research into quantum field theory and string theory has historically underpinned advances in condensed matter physics, including topological phases of matter that now inform quantum computing research. Deeper understanding of these geometric structures could eventually open new pathways in materials science or quantum information, though that is not the project’s aim.
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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.
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