Finite-N algebra for bulk diamonds dual to commutants of time-band algebras

Develop an explicit finite-N (type I) operator-algebraic realization of the boundary algebra corresponding to bulk causal diamond regions D_{ρ_w}—namely the finite-N extension of the commutant of the time-band algebra Y_{I_w}—and establish its relation to generalized entropy for those regions.

Background

For vacuum AdS, the bulk diamond D_{ρw} is dual, in the large-N limit, to the commutant (Y{I_w})′. The generalized entropy considerations suggest there should be a finite-N type I extension of this algebra, but its explicit form is unknown.

Constructing this finite-N algebra would fill a gap in the finite-N formulation of subregion–subalgebra duality for regions not touching the boundary and would clarify how generalized entropies arise there.