Wilson operator algebras and ground states of coupled BF theories

The multiflavor $\mathit{BF}$ theories in (3+1) dimensions with cubic or quartic coupling are the simplest topological quantum field theories that can describe fractional braiding statistics between looplike 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 the three-torus. In particular, by quantizing these coupled $\mathit{BF}$ theories on the three-torus, we explicitly calculate the $\mathcal{S}$ and $\mathcal{T}$ matrices, which encode fractional braiding statistics and the topological spin of looplike excitations, respectively. In the coupled $\mathit{BF}$ theories with cubic and quartic coupling, the Hopf link and Borromean ring of loop excitations, together with pointlike excitations, form composite particles.