Wilson operator algebras and ground states for coupled BF theories

TL;DR

The paper develops and quantizes a class of (3+1)D topological field theories built from multiple copies of BF theory with cubic and quartic couplings to realize three-loop and four-loop braiding among loop-like excitations. Through canonical quantization on the three-torus, it constructs the zero-mode and Wilson-operator algebras, builds ground-state multiplets, and extracts modular data via S and T matrices, ensuring consistency with bulk-boundary and previous surface-based analyses. The work also formulates wave-function representations in different operator bases, examines large gauge invariance, and provides a condensation-based interpretation linking these topological orders to UV parent theories. Overall, it offers a comprehensive continuum framework for higher-loop braiding statistics in 3+1 dimensions and connects detailed operator algebras to physically meaningful topological invariants.

Abstract

The multi-flavor $BF$ theories in (3+1) dimensions with cubic or quartic coupling are the simplest topological quantum field theories that can describe fractional braiding statistics between loop-like topological excitations (three-loop or four-loop braiding statistics). In this paper, by canonically quantizing these theories, we study the algebra of Wilson loop and Wilson surface operators, and multiplets of ground states on three torus. In particular, by quantizing these coupled $BF$ theories on the three-torus, we explicitly calculate the $\mathcal{S}$- and $\mathcal{T}$-matrices, which encode fractional braiding statistics and topological spin of loop-like excitations, respectively. In the coupled $BF$ theories with cubic and quartic coupling, the Hopf link and Borromean ring of loop excitations, together with point-like excitations, form composite particles.

Figures (3)

• Figure 1: Hopf links (a) and Borromean rings (b) as an effective quasiparticle. The red dot represents an "ordinary" point-like quasiparticle.
• Figure 2: Four-loop braiding process in $3+1$ dimensions. (a) Loop 1, 2 and 3 form a Borromean ring configuration. Similarly, loop 1, 2 and 4 form a Borromean ring configuration. Alternatively, $L_1$ and $L_2$ form an effective loop $L_{12}$, with loop 3 and 4 are linked to $L_{34}$. In this case, it is the same as the three loop braiding process with $L_{12}$ as the base loop. Braiding $L_3$ around $L_4$ gives rise to a non-trivial phase $2\pi n_1n_2n_3n_4/\mathrm{K}$. (b) Loop $L_1$ is the base loop. $L_2$, $L_3$ and $L_4$ are linked with $L_1$. This braiding process can be understood by dimensional reduction to the $(2+1)$ dimensions in Fig. \ref{['fig:fig1']} (c).
• Figure 3: Three-particle braiding in $2+1$ dimensions. In (a), particle 1, 2 and 3 are labeled by three different colors red, green and blue. We braid particle 2 around 1 and 3 four times. The Wilson loops for particles 1, 2 and 3 are mutually unlinked. For instance, if there is no Wilson loop for particle 3, the braiding between 1 and 2 is trivial. Nevertheless, the three Wilson loops 1, 2 and 3 together form a Borromean ring in $2+1$ dimensions. In (b), we treat particle 2 and 3 as an effective particle and braid it around particle 1. This process is topologically equivalent to (a). (c) is the projection of (a) to the two dimensional spatial plane. The braiding of particle 2 around 1 is trivial if there is no particle 3.