Entanglement entropy and duality in AdS 4

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 AdS4 , using gravitational perturbations with general boundary conditions in spherical coordinates. Agreement with the first law of thermodynamics, δS=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

3-cocycles, non-associative star-products and the magnetic paradigm of R -flux string vacua

We consider the geometric and non-geometric faces of closed string vacua arising by T-duality from principal torus bundles with constant H -flux and pay attention to their double phase space description encompassing all toroidal coordinates, momenta and their dual on equal footing. We construct a star-product algebra on functions in phase space that is manifestly duality invariant and substitutes for canonical quantization. The 3-cocycles of the Abelian group of translations in double phase space are seen to account for non-associativity of the star-product. We also provide alternative cohomological descriptions of non-associativity and draw analogies with the quantization of point-particles in the field of a Dirac monopole or other distributions of magnetic charge. The magnetic field analogue of the R -flux string model is provided by a constant uniform distribution of magnetic charge in space and non-associativity manifests as breaking of angular symmetry. The Poincaré vector comes to rescue angular symmetry as well as associativity and also allow for quantization in terms of operators and Hilbert space only in the case of charged particles moving in the field of a single magnetic monopole