Topological aspects of the gapless and the gapped

Symmetry is a fundamental organising principle in theoretical physics, underpinning the conservation of physical quantities and the characterisation of phases of matter. It is closely related to topology, leading to generalisations such as higher-form and non-invertible symmetries, with applications in electromagnetism, the standard model of particle physics, condensed matter, and quantum computation. This thesis harnesses generalised symmetries to derive universal results in gapped and gapless phases of matter, focusing particularly on topologically ordered systems and conformal field theories (CFTs).
Topological order is characterised by long-range entangled gapped phases, namely ground states that cannot be disentangled by finite-depth quantum circuits. At low energies it is described via topological quantum field theories (TQFTs). One part of this thesis focusses on extracting patterns of long-range entanglement from the low-energy TQFT for a class of topological orders described by abelian BF theory. By considering algebras of topological operators restricted to subregions, a new measure of entanglement, termed essential topological entanglement, is introduced to refine topological entanglement entropy. This is complemented by a detailed analysis of the edge-mode spectrum of abelian BF theory and its contribution to entanglement entropy across different topologies. A key mathematical result, enabling the above analysis, is the generalisation of Kac–Moody current algebras and their representation theory to higher dimensions.
On the gapless front, the thesis considers CFTs with continuous higher-form symmetries, invertible or non-invertible. It is shown that they also possess infinite-dimensional current algebras akin to those described above. Their representation theory allows for a complete characterisation of the quantum states of these theories across arbitrary spatial topologies. In four dimensions, CFTs with continuous one-form symmetries can be described in terms of free photons. Combining this with the previous analysis leads to a one-to-one correspondence between states on particular topologies and line operators, i.e. wires carrying distributions of electric and magnetic charge.