Higher Structures from Simple Lattice Models

open_access
Open Access
Author:
Ringrose, Noah
Area of Honors:
Mathematics
Degree:
Bachelor of Science
Document Type:
Thesis
Thesis Supervisors:
Adrian Ocneanu, Thesis Supervisor
Luen-Chau Li, Thesis Honors Advisor
Keywords:
Fusion categories
Quantum field theory
Topological Defects
Non-invertible symmetries
TQFT
Condensed matter physics
Conformal Field Theory
Gauge symmetry
Lattice models
Statistical mechanics
Abstract:
Recent developments have revealed that global symmetries of physical systems can be described and reconstructed from topological defects of various codimensions, which constrain the dynamics of the theory \cite{Gaiotto_2015}. These defects can be locally deformed and fused according to the discrete combinatorial data of a structure known as a fusion category—without affecting the underlying physics. This realization has sparked a surge of new results and unprecedented interdisciplinary collaboration across mathematics \cite{décoppet2024classificationfusion2categories,freed2024topological}, high-energy theory \cite{gagliano2025higherrepresentationsquarkconfinement,cordova2024particlesolitondegeneracy2dquantum}, and condensed matter physics \cite{kawagoe2024levinwengaugetheoryentanglement,seifnashri2025gaugingnoninvertiblesymmetrieslattice,Tan:2022vaz}. Structures and symmetries that were once seen as niche or exotic are now being understood as part of a broader unifying mathematical framework. While category theory and higher structures often lie outside the standard training of physicists \textemdash and many of the first examples of categorical symmetries are very unfamiliar to mathematicians \textemdash these modern methods emerge naturally in some of the simplest physical models, and the underlying mathematics is surprisingly intuitive. In this thesis, we explore how the language of topological defects and fusion categories arises directly from studying the simplest and most fundamental classical lattice model: the 2D Ising model. Beginning with the physics of the early nineteenth century, we develop from the ground up the theories of classical and quantum statistical mechanics, duality, gauge symmetry, conformal field theory, and non-invertible symmetries. My goal is to demonstrate how topological and categorical structures are not merely abstract and esoteric mathematical formalisms, but are in fact necessary to study even the most familiar and ubiquitous physical theories.