Bridging Microscopic Constructions and Continuum Topological Field Theory of Three-Dimensional Non-Abelian Topological Order

Abstract

Continuum field-theoretical descriptions of topological order are often constructed at long distances without direct reference to microscopic short-distance realizations, guided instead by general principles such as gauge invariance, locality, symmetry, response, and topological invariance. A classic example is provided by Chern--Simons-type topological field theories for two-dimensional anyon systems. Recently, this framework has been extended to three-dimensional topological orders, where particle and loop excitations exhibit highly nontrivial phenomena, including braiding, fusion, and shrinking. Field-theoretical approaches have further led to diagrammatic representations, pentagon and hexagon relations, and \textit{fusion--shrinking consistency} conditions governing these processes. Despite these advances, a long-standing question remains: do such long-distance field-theoretical structures admit faithful microscopic counterparts with tensor-product local Hilbert spaces and short-range interactions? In this work, we answer this question by establishing an explicit correspondence between continuum topological field theory and microscopic lattice constructions of three-dimensional non-Abelian topological order. While Wilson operators encode long-distance topological excitations, we construct microscopic lattice operators that create, fuse, shrink, and braid particles and loops. Using these operators, we compute fusion and shrinking rules, particle--loop and Borromean-Rings braiding phases, and show how non-Abelian shrinking channels can be selectively controlled by the internal degrees of freedom of loop operators. We further show that the lattice shrinking rules satisfy the \textit{fusion--shrinking consistency} relations previously obtained from field theory, establishing these relations as a microscopically verifiable organizing principle for 3D topological order. Remarkably, by...