Subregion-subalgebra duality: emergence of space and time in holography

TL;DR

The paper advances a boundary-algebraic account of bulk emergence by formulating subregion-subalgebra duality, linking bulk subregions to boundary type $\mathrm{III}_1$ subalgebras in the large $N$ limit. It defines $\mathcal{X}_R$ (the entanglement-wedge dual) and $\mathcal{Y}_{\hat{R}}$ (the causal-domain dual) via a state-dependent large $N$ limit, showing that bulk locality and causal structure arise as algebraic geometrization of boundary operator structure. It then demonstrates, through explicit examples, how entanglement wedges can be reconstructed without entropy, and how superadditivity and commutants encode bulk-geometric relations such as islands, interior diamonds, and mirror operators in black hole spacetimes. The framework generalizes subregion-subregion duality to general bulk regions, including those not touching the boundary, and provides a principled route to understand holographic quantum error correction, time bands, and finite-temperature geometries. Overall, it offers a rigorous, entropy-free map between boundary algebras and bulk spacetime regions, with broad implications for bulk reconstruction, causal structure, and the emergence of spacetime.

Abstract

In holographic duality, a higher dimensional quantum gravity system emerges from a lower dimensional conformal field theory (CFT) with a large number of degrees of freedom. We propose a formulation of duality for a general causally complete bulk spacetime region, called subregion-subalgebra duality, which provides a framework to describe how geometric notions in the gravity system, such as spacetime subregions, different notions of times, and causal structure, emerge from the dual CFT. Subregion-subalgebra duality generalizes and brings new insights into subregion-subregion duality (or equivalently entanglement wedge reconstruction). It provides a mathematically precise definition of subregion-subregion duality and gives an independent definition of entanglement wedges without using entropy. Geometric properties of entanglement wedges, including those that play a crucial role in interpreting the bulk as a quantum error correcting code, can be understood from the duality as the geometrization of the superadditivity of certain algebras. Using general boundary subalgebras rather than those associated with geometric subregions makes it possible to find duals for general bulk spacetime regions, including those not touching the boundary. Applying subregion-subalgebra duality to a boundary state describing a single-sided black hole also provides a precise way to define mirror operators.

Figures (28)

• Figure 1: Some examples of subregion-subalgebra duality. (a) The right region of an eternal black hole (shaded) is dual to the single-trace operator algebra ${{\mathcal{M}}}_R$ of CFT$_R$ (i.e. the one defined on the right boundary) in a thermal field double state. (b) Duality between the entanglement wedge of a boundary spatial region $R$ and an algebra defined in ${\mathfrak{b}}_R$ (only a time slice of the bulk is shown). (c) A wedge region (shaded) in an eternal black hole geometry dual to the boundary subalgebra associated with a time band indicated by $T$ in the CFT$_R$ in a thermal field double state. (d) A radial wedge region (between the red surfaces and the conformal boundary) in global AdS dual to the boundary subalgebra ${{\mathcal{M}}}_T$ associated with a time band $T$ for the CFT in the vacuum state. (e) A causal diamond in global AdS that does not touch the boundary is dual to the commutant of ${{\mathcal{M}}}_T$ where $T$ is a time band.
• Figure 2: Starting with the algebra ${\mathcal{X}}_R$ for the boundary region $R= R_1 \cup R_2$ (where $R_{1},~R_2$ are half-lines separated from each other by an interval) in the vacuum state, it is possible (see Sec. \ref{['sec:expD']}) to reconstruct the bulk spacetime region dual to ${\mathcal{X}}_R$, i.e. the entanglement wedge of $R,$ using equivalence of algebras alone. The shaded regions are time-slices of the reconstructed entanglement wedges, and the union of the regions bounded by dashed lines gives the causal domain. (a): a single time slice in the Poincare patch. The region outside the dashed lines is the causal wedge. (b) a single time slice in global AdS. The union of regions between the dashed lines and the boundary is the causal domain.
• Figure 3: The superadditivity \ref{['hen03']} is realized on the gravity side through geometric properties of entanglement wedges, i.e. $E_{R_1} \cup E_{R_2} \subseteq E_{R_1 \cup R_2}$ and $E_{R_1 \cap R_2} \subseteq E_{R_1} \cap E_{R_2}$. These properties played important roles in interpreting the bulk gravity system as a quantum error correcting code.
• Figure 4: For the boundary CFT in a state dual to a bulk single-sided black hole. The operator algebra ${{\mathcal{M}}}_T$ associated with a half-infinite time band $T$ should be dual to the shaded bulk wedge region ${\mathfrak{c}}_T$. The commutant of ${{\mathcal{M}}}_T$ gives a precise definition of mirror operators advocated for in Papadodimas:2012aqPapadodimas:2013jku, and is dual to the shaded bulk region labeled by $({\mathfrak{c}}_T)'$.
• Figure 5: (a) Algebras ${{\mathcal{M}}}_1$ and ${{\mathcal{M}}}_2$ of single-trace operators on two different Cauchy slices are inequivalent. (b) At finite $N$, the algebra associated with the time band indicated in the plot is equivalent to that on a single Cauchy slice. But in the large $N$ limit, it is inequivalent and is a nontrivial subalgebra of the full algebra.