  We study the duality between theories of a fundamental scalar or fermion
coupled to $U(N)$ Chern-Simons gauge theory at the level of the three-sphere
partition function, or equivalently entanglement entropy across a circle. The
duality relation between the sphere free energies of the large $N$ bosonic and
fermionic conformal theories is sensitive to certain shifts in the Chern-Simons
level already at order $N$. We first study a similar duality in a ${\cal N}=2$
supersymmetric Chern-Simons matter theory, where it can be checked exactly
using localization. At large $N$, the free energies of supersymmetric and
non-supersymmetric theories are related in a simple way, and we use this fact
to obtain an explicit solution for the scalar and fermion free energies which
obey the duality map and have the expected weak and strong coupling behaviors.
We also suggest that a worldline representation of the free energy allows to
relate its calculation to a sum of Wilson loops in pure Chern-Simons theory. In
the supersymmetric case, we find that this approach precisely reproduces the
localization prediction. In the non-supersymmetric case, we observe that a
result consistent with the boson/fermion duality can be obtained using a
certain framing prescription, which remains to be understood. We also briefly
discuss the case of massive theories, and compare the small mass expansion of
the free energy to previously known results for the correlation functions of
bilinear operators in flat space.
