  The low energy effective field theories of $(2+1)$ dimensional topological
phases of matter provide powerful avenues for investigating entanglement in
their ground states. In \cite{Fliss:2017wop} the entanglement between distinct
Abelian topological phases was investigated through Abelian Chern-Simons
theories equipped with a set of topological boundary conditions (TBCs). In the
present paper we extend the notion of a TBC to non-Abelian Chern-Simons
theories, providing an effective description for a class of gapped interfaces
across non-Abelian topological phases. These boundary conditions furnish a
defining relation for the extended Hilbert space of the quantum theory and
allow the calculation of entanglement directly in the gauge theory. Because we
allow for trivial interfaces, this includes a generic construction of the
extended Hilbert space in any (compact) Chern-Simons theory quantized on a
Riemann surface. Additionally, this provides a constructive and principled
definition for the Hilbert space of effective ground states of gapped phases of
matter glued along gapped interfaces. Lastly, we describe a generalized notion
of surgery, adding a powerful tool from topological field theory to the gapped
interface toolbox.
