  The R\'enyi entropy for the $\widehat{\rm SU}(N)_1$ WZW model as described by
$N$ free fermions coupled to a $U(1)$ constraint field is computed on an
$n$-sheeted branched torus. The boundary condition of the harmonic component of
the gauge field on the homology cycles of the genus $g$ Riemann surface is
central to the final result. This calculation is complementary to that of
arXiv:$1510.05993$, which presents the bose side of the bose-fermi equivalence.
