In gauge theories the presence of constraints can obstruct expressing the global Hilbert space as a tensor product of the Hilbert spaces corresponding to degrees of freedom localized in complementary regions. In algebraic terms, this is due to the presence of a center -a set of operators which commute with all others -in the gauge invariant operator algebra corresponding to finite region. A unique entropy can be assigned to algebras with center, giving place to a local entropy in lattice gauge theories. However, ambiguities arise on the correspondence between algebras and regions. In particular, it is always possible to choose (in many different ways) local algebras with trivial center, and hence a genuine entanglement entropy, for any region. These choices are in correspondence with maximal trees of links on the boundary, which can be interpreted as partial gauge fixings. This interpretation entails a gauge fixing dependence of the entanglement entropy. In the continuum limit however, ambiguities in the entropy are given by terms local on the boundary of the region, in such a way relative entropy and mutual information are finite, universal, and gauge independent quantities.

We discuss the structure of 4D gauged supergravity algebras corresponding to globally nongeometric compactifications of F-theory, admitting a local geometric description in terms of 10D supergravity. By starting with the well known algebra of gauge generators associated to nongeometric type IIB fluxes, we derive a full algebra containing all, closed RR and NSNS, geometric and non-geometric dual fluxes. We achieve this generalization by a systematic application of SL(2, Z) duality transformations and by taking care of the spinorial structure of the fluxes. The resulting algebra encodes much information about the higher dimensional theory. In particular, tadpole equations and Bianchi identities are obtainable as Jacobi identities of the algebra. When a sector of magnetized (p, q) 7-branes is included, certain closed axions are gauged by the U (1) transformations on the branes. We indicate how the diagonal gauge generators of the branes can be incorporated into the full algebra, and show that Freed-Witten constraints and tadpole cancellation conditions for (p, q) 7-branes can be described as Jacobi identities satisfied by the algebra mixing bulk and brane gauge generators.

We consider generalized diffeomorphisms on an extended mega-space associated to the U-duality group of gauged maximal supergravity in four dimensions, E 7(7) . Through the bein for the extended metric we derive dynamical (field-dependent) fluxes taking values in the representations allowed by supersymmetry, and obtain their quadratic constraints from gauge consistency conditions. A covariant generalized Ricci tensor is introduced, defined in terms of a connection for the generalized diffeomorphisms. We show that for any torsionless and metric-compatible generalized connection, the Ricci scalar reproduces the scalar potential of gauged maximal supergravity. We comment on how these results extend to other groups and dimensions.