Splitting interfaces in 4d $\mathcal{N}=4$ SYM

We discuss entanglement entropies in 4d interface CFTs based on 4d $\mathcal{N}=4$ SYM coupled to 3d $\mathcal{N}=4$ degrees of freedom localized on an interface. Focusing on the entanglement between the two half spaces to either side of the interface, we show that applying the Ryu-Takayanagi prescription in general leads to multiple natural entanglement entropies. We interpret the different entropies as corresponding to different ways of assigning the 3d degrees of freedom localized on the interface to the two half spaces. We contrast these findings with recent discussions of universal relations for entanglement entropies in 2d interface CFTs and formulate generalized relations for 4d interface CFTs which incorporate our results.