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

Abstract

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