  Entanglement (Renyi) entropies of spatial regions are a useful tool for
characterizing the ground states of quantum field theories. In this paper we
investigate the extent to which these are universal quantities for a given
theory, and to which they distinguish different theories, by comparing the
entanglement spectra of the massless Dirac fermion and the compact free boson
in two dimensions. We show that the calculation of Renyi entropies via the
replica trick for any orbifold theory includes a sum over orbifold twists on
all cycles. In a modular-invariant theory of fermions, this amounts to a sum
over spin structures. The result is that the Renyi entropies respect the
standard Bose-Fermi duality. Next, we investigate the entanglement spectrum for
the Dirac fermion without a sum over spin structures, and for the compact boson
at the self-dual radius. These are not equivalent theories; nonetheless, we
find that (1) their second Renyi entropies agree for any number of intervals,
(2) their full entanglement spectra agree for two intervals, and (3) the
spectrum generically disagrees otherwise. These results follow from the
equality of the partition functions of the two theories on any Riemann surface
with imaginary period matrix. We also exhibit a map between the operators of
the theories that preserves scaling dimensions (but not spins), as well as OPEs
and correlators of operators placed on the real line. All of these coincidences
can be traced to the fact that the momentum lattice for the bosonized fermion
is related to that of the self-dual boson by a 45 degree rotation that mixes
left- and right-movers.
