R\'enyi Entropy for a $\mathbf{2}\mathbf{d}$ CFT with a gauge field: $\widehat{\mathrm{S}\mathrm{U}}(\mathbf{N}{)}_{\mathbf{1}}$ WZW theory on a branched torus

The R'enyi entropy for the $\widehat{\mathrm{S}\mathrm{U}}(N{)}_1$ 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.