We study the entropy of chiral 2+1-dimensional topological phases, where
there are both gapped bulk excitations and gapless edge modes. We show how the
entanglement entropy of both types of excitations can be encoded in a single
partition function. This partition function is holographic because it can be
expressed entirely in terms of the conformal field theory describing the edge
modes. We give a general expression for the holographic partition function, and
discuss several examples in depth, including abelian and non-abelian fractional
quantum Hall states, and p+ip superconductors. We extend these results to
include a point contact allowing tunneling between two points on the edge,
which causes thermodynamic entropy associated with the point contact to be lost
with decreasing temperature. Such a perturbation effectively breaks the system
in two, and we can identify the thermodynamic entropy loss with the loss of the
edge entanglement entropy. From these results, we obtain a simple
interpretation of the non-integer `ground state degeneracy' which is obtained
in 1+1-dimensional quantum impurity problems: its logarithm is a
2+1-dimensional topological entanglement entropy.Comment: 16 pages, 2 figure