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...

64 pages. Length limit in Abstract metadata. The complete version of Abstract is shown in PDF. In this version, braiding is added