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 \( {\mathcal{H}}_{\mathcal{B}} \) of the resulting Hilbert space were then defined for any (d − 2)-dimensional surface \( \mathcal{B} \) , where \( \mathcal{B} \) may be thought of as the boundary ∂Σ of a bulk Cauchy surface in a corresponding Lorentzian description, and where \( \mathcal{B} \) includes the specification of appropriate boundary conditions for bulk fields. Cases where \( \mathcal{B} \) 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 \( {\mathcal{A}}_L^B \) , \( {\mathcal{A}}_R^B \) that act respectively at the left and right copy of B in B ⊔ B.
Below, we consider the case of general \( \mathcal{B} \) , and in particular for \( \mathcal{B} \) = BL ⊔ BR with BL, BR distinct. For any BR, we find that the von Neumann algebra at BL acting on the off-diagonal Hilbert space sector \( {\mathcal{H}}_{B_L\bigsqcup {B}_R} \) is a central projection of the corresponding type-I von Neumann algebra on the ‘diagonal’ Hilbert space \( {\mathcal{H}}_{B_L\bigsqcup {B}_L} \) . As a result, the von Neumann algebras \( {\mathcal{A}}_L^{B_L} \) , \( {\mathcal{A}}_R^{B_L} \) defined in [1] using the diagonal Hilbert space \( {\mathcal{H}}_{B_L\bigsqcup {B}_L} \) turn out to coincide precisely with the analogous algebras defined using the full Hilbert space of the theory (including all sectors \( {\mathcal{H}}_{\mathcal{B}} \) ). A second implication is that, for any \( {\mathcal{H}}_{B_L\bigsqcup {B}_R} \) , 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 BL and BR.
