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We compute the entanglement entropy in a 2+1 dimensional topological order inthe presence of gapped boundaries. Specifically, we consider entanglement cutsthat cut through the boundaries. We argue that based on general considerationsof the bulk-boundary correspondence, the "twisted characters" feature in theRenyi entropy, and the topological entanglement entropy is controlled by a"half-linking number" in direct analogy to the role played by the S-modularmatrix in the absence of boundaries. We also construct a class of boundarystates based on the half-linking numbers that provides a "closed-string"picture complementing an "open-string" computation of the entanglement entropy.These boundary states do not correspond to diagonal RCFT's in general. Theseare illustrated in specific Abelian Chern-Simons theories with appropriateboundary conditions.

Ishibashi States, Topological Orders with Boundaries and Topological Entanglement Entropy II -- Cutting through the boundary