Algebras and Hilbert spaces from gravitational path integrals: Understanding Ryu-Takayanagi/HRT as entropy without AdS/CFT

TL;DR

The paper shows that in any UV-complete bulk gravity with a Euclidean path integral satisfying simple axioms, boundary observables organize into type I von Neumann algebras that are mutual commutants, yielding a Hilbert-space interpretation of the Ryu-Takayanagi entropy. By constructing left/right surface algebras and their representations on a diagonal boundary Hilbert space, the authors derive a discrete sector decomposition indexed by μ and demonstrate a tensor product factorization of the corresponding Hilbert spaces, enabling RT/HRT to be computed as standard traces with quantum corrections in semiclassical limits. A key innovation is the introduction of hidden sectors, quantization of projection traces, and a gravitational replica trick that recovers RT entropy while remaining fully within a bulk, holography-free framework. The framework is illustrated with explicit 2d models (topological gravity and JT gravity, with and without matter) and clarifies how islands and entropy mixing arise from algebraic structures, while also showing how semiclassical limits can violate assumptions if the axioms are not maintained. Overall, this approach provides a robust, UV-complete, bulk-based interpretation of gravitational entropy and suggests broad applicability to various UV completions of quantum gravity beyond AdS/CFT.

Abstract

Recent works by Chandrasekaran, Penington, and Witten have shown in various special contexts that the quantum-corrected Ryu-Takayanagi (RT) entropy (or its covariant Hubeny-Rangamani-Takayanagi (HRT) generalization) can be understood as computing an entropy on an algebra of bulk observables. These arguments do not rely on the existence of a local holographic dual field theory. We show that analogous-but-stronger results hold in any UV-completion of asymptotically AdS quantum gravity with a Euclidean path integral satisfying a simple and (largely) familiar set of axioms. We consider a quantum context in which a standard Lorentz-signature classical bulk limit would have Cauchy slices with asymptotic boundaries $B_L \sqcup B_R$ where both $B_L$ and $B_R$ are compact manifolds without boundary. Our main result is that (the UV-completion of) the quantum gravity path integral defines type I von Neumann algebras ${\cal A}^{B_L}_L$, ${\cal A}^{B_R}_{R}$ of observables acting respectively at $B_L$, $B_R$ such that ${\cal A}^{B_L}_L$, ${\cal A}^{B_R}_{R}$ are commutants. The path integral also defines entropies on ${\cal A}^{B_L}_L, {\cal A}^{B_R}_R$. Positivity of the Hilbert space inner product then turns out to require the entropy of any projection operator to be quantized in the form $\ln N$ for some $N \in {\mathbb Z}^+$ (unless it is infinite). As a result, our entropies can be written in terms of standard density matrices and standard Hilbert space traces. Furthermore, in appropriate semiclassical limits our entropies are computed by the RT-formula with quantum corrections. Our work thus provides a Hilbert space interpretation of the RT entropy. Since our axioms do not severely constrain UV bulk structures, it is plausible that they hold equally well for successful formulations of string field theory, spin-foam models, or any other approach to constructing a UV-complete theory of gravity.

Figures (13)

• Figure 1: We consider boundaries $B_L,B_R$ that are complete in the sense that $\partial B_L = \emptyset = \partial \bar{B}_R$. We also require $B_L, B_R$ to be compact.
• Figure 2: A slice $\Sigma_{bulk}$ (red) of the path integral intersects the (here, spherical) AlAdS boundary $M$ at a codimension-2 surface $\partial \Sigma_{bulk}$ (red circle) which splits $M$ into two hemispheres $N_1$ and $N_2^*$. Each half of the path integral defines a quantum state $|\psi_i\rangle$ by computing the wavefunction of $\psi_i$ on $\Sigma_{bulk}$. These wavefunctions can be thought of as the result of Euclidean evolution from the boundary conditions $N_i$, and the full path integral defined by $M$ can then be regarded as computing the inner product $\langle\psi_2|\psi_1 \rangle$.
• Figure 3: A reflection-symmetric linear combination $M\in {\underline{X}}^d$ of smooth source manifolds. The representation on the left side of the equality makes reflection-symmetry manifest. On the right side of the equality, the same $M$ is shown as an explicit sum of terms, each proportional to a source manifold $M_{I,J}$ that can be cut into $N_I^*$ and $N_J$, and with coefficients of form $\gamma_I^* \gamma_J$. Here and in all figures below, we make no attempt to distinguish a given source from its complex conjugate. Thus $N_I^*$ appears as simply a reflected version of $N_I$. The 'diagonal' manifolds $M_{I,I}$ are individually reflection-symmetric. Axiom \ref{['ax:RP']} requires that such reflection-symmetric $M$ have $\zeta(M) \ge 0$.
• Figure 4: The source manifold $M_{\epsilon_0}$ contains a cylinder $C_{\epsilon_0}$ of length $\epsilon_0$. Changing the length of this cylinder to $\epsilon$ defines a new source manifold $M_\epsilon$. Here and in all figures below, the symbols $\times$ denote potential features of smooth sources whose details are not shown but which serve to distinguish certain boundary points. For example, such features might be local peaks in the Kretchman scalar of the boundary metric, extrema of a smooth scalar source, or points where a fermionic source becomes large. The main role of these features in our figures is to provide a simple and clean visualization of effects that arise when boundary-sources break symmetries of the simple cases that we choose to depict.
• Figure 5: Gluing $N^*_1$ to $N_2$ and gluing $N_2^*$ to $N_1$ defines source-manifolds-without-boundary $M_{N_1^* N_2}$ and $M_{N_2^* N_1}$ that are related by a diffeomorphism that complex-conjugates sources. As depicted here, the relevant diffeomorphism acts as a reflection across the shaded plane. Thus $(M_{N_1^* N_2})^* = M_{N_1 N_2^*}$, where "=" means that the two are related by a source-preserving diffeomorphism.

Theorems & Definitions (24)

• Definition 1
• Definition 2
• Lemma 1
• proof
• Corollary 1
• proof
• Corollary 2
• proof
• Corollary 3
• proof