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

Abstract

In this paper, we bridge this gap systematically by establishing an explicit correspondence between continuum topological field theory and microscopic lattice constructions of three-dimensional non-Abelian topological orders. While Wilson operators defined by gauge-field holonomies represent topological excitations at long distances, we explicitly construct microscopic lattice operators that create, fuse, and shrink particle and loop excitations, derive their fusion and shrinking rules microscopically, and show how non-Abelian shrinking channels are selectively controlled through the internal degrees of freedom of loop creation operators. Crucially, we demonstrate that the lattice shrinking rules satisfy the \textit{fusion-shrinking consistency} relations previously identified at long distances, thereby establishing these relations as a general, microscopically verifiable principle. Moreover, by computing the complete excitation spectrum alongside all fusion and shrinking data microscopically, we establish a correspondence between the $\mathbb{D}_4$ quantum double lattice model and the $BF$ field theory with an $AAB$ twist and gauge group $(\mathbb{Z}_2)^3$. Through the precise microscopic construction of the $BF+AAB$ field theory, this work definitively addresses the long-standing skepticism regarding its microscopic realizability, which has persisted over the past years. Finally, our work bridges the gap between theories at long and short distances, taking a crucial step toward a more complete microscopic construction of topological data previously identified at long distances, including braiding, pentagon relations, and fusion-shrinking hexagon relations. This progress holds the potential to deepen our understanding of quantum matter across all length scales and foster collaboration between researchers from diverse fields and approaches.