  We present a mathematical derivation of some of the most important physical
quantities arising in topological bilayer systems with permutation twist
defects as introduced by Barkeshli et al. in cond-mat/1208.4834. A crucial tool
is the theory of permutation equivariant modular functors developed by Barmeier
et al. in math.CT/0812.0986 and math.QA/1004.1825.
