Boundary topological entanglement entropy in two and three dimensions

The topological entanglement entropy is used to measure long-range quantum correlations in the ground state of topological phases. Here we obtain closed form expressions for topological entropy of (2+1)- and (3+1)-dimensional loop gas models, both in the bulk and at their boundaries, in terms of the data of their input fusion categories and algebra objects. Central to the formulation of our results are generalized $\mathcal{S}$-matrices. We conjecture a general property of these $\mathcal{S}$-matrices, with proofs provided in many special cases. This includes constructive proofs for categories up to rank 5.