Volume as an index of a subalgebra

TL;DR

This paper introduces a novel boundary-based approach linking the volume of bulk AdS subregions to the algebraic index of inclusion, offering insights into black hole interior growth, entanglement wedge size, and de Sitter space volume evolution.

Contribution

It establishes a new volume-index relation connecting bulk volume to boundary algebraic index, providing a fresh perspective on holographic complexity and subregion duality.

Findings

Volume of maximal slices equals the exponential of the algebraic index.

Quantifies algebraic size differences between entanglement and causal wedges.

Offers a new boundary interpretation for black hole interior growth.

Abstract

We propose a new way to understand the volume of certain subregions in the bulk of AdS spacetime by relating it to an algebraic quantity known as the index of inclusion. This index heuristically measures the relative size of a subalgebra $\mathcal{N}$ embedded within a larger algebra $\mathcal{M}$. According to subregion-subalgebra duality, bulk subregions are described by von Neumann algebras on the boundary. When a causally complete bulk subregion corresponds to the relative commutant $\mathcal{N}' \cap \mathcal{M}$ -- the set of operators in $\mathcal{M}$ that commute with $\mathcal{N}$ -- of boundary subalgebras, we propose that the exponential of the volume of the maximal volume slice of the subregion equals the index of inclusion. This ``volume-index'' relation provides a new boundary explanation for the growth of interior volume in black holes, reframing it as a change in the relative size of operator algebras. It offers a complementary perspective on complexity growth from the Heisenberg picture, and has a variety of other applications, including quantifying the relative size of algebras dual to the entanglement wedge and the causal wedge of a boundary region, as well as quantifying the violation of additivity of operator algebras in the large $N$ limit. Finally, it may offer insights into the volume growth of de Sitter space through the changes in North and South pole observer algebras in time.