Holographic R\'enyi Entropies at Finite Coupling

We compute R\'enyi entropies for a spherical entangling surface in four-dimensional N=4 super-Yang-Mills at strong coupling using the AdS/CFT correspondence. Incorporating the effects of the leading \alpha' corrections to the low energy effective action of type IIB string theory, we calculate the leading corrections in inverse powers of the 't Hooft coupling (and the number of colours). The results are compared with known weak coupling calculations. Setting the order of the R\'enyi entropy q to one, it reduces to the entanglement entropy and the strong and weak coupling results match without any corrections, as expected. In the limit of q going to 0, the relation between the strong and weak coupling entropies is connected to the known corrections for the thermal free energy in flat space. We also compute the correction to the scaling dimension of twist operators.