Supersymmetric Rényi entropy in two dimensions

We compute the exact partition function on the branched two-sphere by the localization technique. it is found that it does not depend on a branching parameter q, which means that supersymmetric rényi entropy defined by utilizing it is equivalent to the usual entanglement entropy. we also provide the interpretation of the conical singularities on the branched sphere as defects sit on the poles of the nonsingular two-sphere.