Braiding statistics of loop excitations in three dimensions

While it is well known that three dimensional quantum many-body systems can support nontrivial braiding statistics between particlelike and looplike excitations, or between two looplike excitations, we argue that a more fundamental quantity is the statistical phase associated with braiding one loop α around another loop β, while both are linked to a third loop γ. We study this three-loop braiding in the context of (Z_N) ^K gauge theories which are obtained by gauging a gapped, short-range entangled lattice boson model with (Z_N) ^K symmetry. We find that different short-range entangled bosonic states with the same (Z_N) ^K  symmetry (i.e., different symmetry-protected topological phases) can be distinguished by their three-loop braiding statistics.