When left and right disagree: Entropy and von Neumann algebras in quantum gravity with general AlAdS boundary conditions

Euclidean path integrals for UV-completions of d-dimensional bulk quantum gravity were studied in [1] by assuming that they satisfy axioms of finiteness, reality, continuity, reflection-positivity, and factorization. Sectors HB of the resulting Hilbert space were defined for any $(d-2)$-dimensional surface B, where B may be thought of as the boundary ∂Σ of a bulk Cauchy surface in a corresponding Lorentzian description, and where B includes the specification of boundary conditions for bulk fields. Cases where B was the disjoint union B B of two identical $(d-2)$-dimensional surfaces were studied in detail and, after the inclusion of finite-dimensional `hidden sectors,' were shown to provide a Hilbert space interpretation of the associated Ryu-Takayanagi entropy. The analysis was performed by constructing type-I von Neumann algebras ALB, ARB that act respectively at the left and right copy of B in B B. Below, we consider the case of general B = BL BR with BL,BR distinct. For any BR, we find that the von Neumann algebra at BL acting on HBL BR is a central projection of the corresponding type-I von Neumann algebra on the `diagonal' Hilbert space HBL BL. As a result, the von Neumann algebras ALBL, ARBL defined in [1] using the diagonal Hilbert space coincide precisely with those defined using the full Hilbert space of the theory. A second implication is that, for any HBL BR, including the same hidden sectors as in the diagonal case again provides a Hilbert space interpretation of the Ryu-Takayanagi entropy. We also show the above central projections to satisfy consistency conditions that lead to a universal central algebra relevant to all choices of BL,BR.