Categorical symmetries in quantum field theories

Thesis

In recent years, the notion of symmetry has greatly expanded, leading to generalised symmetries such as higher-form symmetries, which measure charges of extended operators, and non-invertible symmetries, whose symmetry generators do not have inverses. These are also known as categorical symmetries. This thesis focuses on non-invertible/categorical symmetries, exploring both their abstract mathematical structure and their realisations in physical systems.

The first part develops new methods for constructing such symmetries in space-time dimensions three and higher. We present a general procedure starting from an invertible symmetry and gauging a specific subgroup of it to produce a non-invertible symmetry. This construction is applicable to many gauge theories in any dimension. Specialising to (2 + 1)d theories, we study the effect of gauging all possible subgroups of an ordinary symmetry, generating a symmetry web of related theories.

The second part examines the physical consequences of non-invertible symmetries, asking whether they can organise and classify phases of matter analogously to what ordinary symmetries do in the celebrated Landau symmetry-breaking theory. We introduce a framework to classify all the possible gapped phases for (1 + 1)d systems with a given non-invertible symmetry. The main tool we use is the so-called the symmetry topological field theory (SymTFT), which also tells us what kinds of order parameters exist in a phase. We further extend this to construct phase transitions with non-invertible symmetries by inputting known phase transitions, such as the critical Ising model. This outlines a broadened version of the classic Landau theory, which we call the categorical Landau paradigm. We also demonstrate its possible realisation in quantum lattice models with a concrete example.

Finally, we apply the techniques developed in the rest of this thesis to tackle a long-standing open problem: finding a conformal field theory (CFT) with the unusual Haagerup symmetry.

Authors: Bottini, LE

• DOI: 10.5287/ora-x5b951p5j
• Type of award: DPhil
• Level of award: Doctoral
• Awarding institution: University of Oxford
• Language: English