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

A bstract Euclidean path integrals for UV-completions of d -dimensional bulk quantum gravity were recently studied in [1] by assuming that they satisfy axioms of finiteness, reality, continuity, reflection-positivity, and factorization. Sectors HB H B of the resulting Hilbert space were then defined for any ( d − 2)-dimensional surface B B , where B B may be thought of as the boundary ∂Σ of a bulk Cauchy surface in a corresponding Lorentzian description, and where B B includes the specification of appropriate boundary conditions for bulk fields. Cases where B B was the disjoint union B ⊔ B of two identical ( d − 2)-dimensional surfaces B 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 A L B , ARB A R B that act respectively at the left and right copy of B in B ⊔ B . Below, we consider the case of general B B , and in particular for B B = B L ⊔ B R with B L , B R distinct. For any B R , we find that the von Neumann algebra at B L acting on the off-diagonal Hilbert space sector H_BL BR H B L ⊔ B R is a central projection of the corresponding type-I von Neumann algebra on the ‘diagonal’ Hilbert space H_BL BL H B L ⊔ B L . As a result, the von Neumann algebras ALBL A L B L , ARBL A R B L defined in [1] using the diagonal Hilbert space H_BL BL H B L ⊔ B L turn out to coincide precisely with the analogous algebras defined using the full Hilbert space of the theory (including all sectors HB H B ). A second implication is that, for any H_BL BR H B L ⊔ B R , 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 B L and B R .