Wilson operator algebras and ground states for coupled BF theories

Understand

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.