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In holographic duality, a higher dimensional quantum gravity system emergesfrom a lower dimensional conformal field theory (CFT) with a large number ofdegrees of freedom. We propose a formulation of duality for a general causallycomplete bulk spacetime region, called subalgebra-subregion duality, whichprovides 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. Subalgebra-subregion duality generalizes and bringsnew insights into subregion-subregion duality (or equivalently entanglementwedge reconstruction). It provides a mathematically precise definition ofsubregion-subregion duality and gives an independent definition of entanglementwedges without using entropy. Geometric properties of entanglement wedges,including those that play a crucial role in interpreting the bulk as a quantumerror correcting code, can be understood from the duality as the geometrizationof the additivity anomaly of certain algebras. Using general boundarysubalgebras rather than those associated with geometric subregions makes itpossible to find duals for general bulk spacetime regions, including those nottouching the boundary. Applying subalgebra-subregion duality to a boundarystate describing a single-sided black hole also provides a precise way todefine mirror operators.

Subalgebra-subregion duality: emergence of space and time in holography