Wilson operator algebras and ground states of coupled BF theories

The multiflavor 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 BF theories on the three-torus, we explicitly calculate the S and T matrices, which encode fractional braiding statistics and the topological spin of looplike excitations, respectively. In the coupled BF theories with cubic and quartic coupling, the Hopf link and Borromean ring of loop excitations, together with pointlike excitations, form composite particles.