Entanglement entropy and duality in AdS4

Small variations of the entanglement entropy δS and the expectation value of the modular Hamiltonian δE   are computed holographically for circular entangling curves in the boundary of AdS4AdS4, using gravitational perturbations with general boundary conditions in spherical coordinates. Agreement with the first law of thermodynamics, δS=δEδS=δE, requires that the line element of the entangling curve remains constant. In this context, we also find a manifestation of electric–magnetic duality for the entanglement entropy and the corresponding modular Hamiltonian, following from the holographic energy–momentum/Cotton tensor duality.