Elias Riedel Gårding: Aspects of generalised symmetries of four-dimensional field theories

Abstract

Symmetry is a foundational concept in quantum field theory, essential for the construction of gauge theories, organising particles into multiplets, and as a link between microscopic and macroscopic physics. Symmetries are related to conserved quantities, or charges, which define certain topological defects. Motivated by this, we view arbitrary topological defects as generalised symmetries. Notions like group structure, multiplets, background fields, 't Hooft anomalies and gauging can be extended to this new kind of symmetry. Generalised symmetries are also preserved by renormalisation group flows, and can exhibit various patterns of spontaneous symmetry breaking. The two main new types of symmetry are higher-form symmetries, which act on extended operators, and non-invertible symmetries, which do not form a group. The algebraic structure of generalised symmetries is captured by higher category theory.

This thesis gives an introduction to generalised symmetries, building up to two key constructions. The first is the symmetry TFT, a topological field theory in one higher dimension than spacetime. This field theory captures the abstract generalised symmetry structure of a quantum field theory, while its topological boundary conditions encode the manifestations of this structure in its different global forms. The second is the construction of a non-invertible symmetry, called a duality symmetry, from a duality between different global forms of the same theory. As a main example, we study the ABJ-anomalous chiral transformation of quantum electrodynamics.

The thesis is based on three papers, which are briefly introduced. They explore various aspects of generalised symmetries of four-dimensional quantum field theories using explicit Lagrangian methods, geometric constructions in string theory and the symmetry TFT viewpoint. We study the 1-form symmetries of class S theories, the spontaneous breaking and anomalies of duality symmetries, and the higher-categorical structure of non-invertible 0-form and 1-form symmetries of the axion–Maxwell theory.