Microscopic Constructions of the $BF+AAB$ Topological Field Theory and Borromean-Rings Braiding in $(3+1)$ Dimensions

Abstract:Topological order was first studied in fractional quantum Hall states, where anyons reflect electron fractionalization. Recent advances in continuum topological field theories have revealed braiding, fusion, shrinking, and diagrammatic consistency relations in higher dimensions [e.g., Huang, Zhang, Ye, JHEP 11 (2023) 210; JHEP 06 (2025) 238]. As the next step of research, these advances call for explicit microscopic lattice constructions that faithfully reproduce the structures and processes described by continuum field theories. Here we construct, step by step, microscopic counterparts of the $(3+1)$-dimensional $BF$ field theory with an $AAB$ twist and gauge group $(\mathbb{Z}_2)^3$. This field theory was introduced in our earlier work on Borromean-Rings braiding [Chan, Ye, Ryu, Phys. Rev. Lett. 121, 061601 (2018)]. We start with the construction of string and thickened membrane operators that create, fuse, shrink, and braid particle and loop excitations, providing microscopic counterparts of Wilson operators in continuum field theories. We derive fusion and shrinking rules and the operators governing particle--loop and Borromean-Rings braiding. We demonstrate selective control of non-Abelian shrinking channels through the internal degrees of freedom of loop operators and establish fusion--shrinking consistency for arbitrary finite groups. An explicit correspondence between all 22 excitations of the $\mathbb{D}_4$ quantum double lattice model and the $BF+AAB$ theory preserves fusion and shrinking rules and particle--loop and Borromean-Rings braiding phases. These constructions place the continuum results on a microscopic lattice footing with tensor-product local Hilbert spaces and short-range interactions.

Strongly Correlated Electrons (cond-mat.str-el); Mesoscale and Nanoscale Physics (cond-mat.mes-hall); High Energy Physics - Theory (hep-th); Quantum Physics (quant-ph)

arXiv:2512.21148 [cond-mat.str-el]

(or arXiv:2512.21148v5 [cond-mat.str-el] for this version)