Local bulk physics from intersecting modular Hamiltonians

We show that bulk quantities localized on a minimal surface homologous to a boundary region correspond in the CFT to operators that commute with the modular Hamiltonian associated with the boundary region. If two such minimal surfaces intersect at a point in the bulk then CFT operators which commute with both extended modular Hamiltonians must be localized at the intersection point. We use this to construct local bulk operators purely from CFT considerations, without knowing the bulk metric, using intersecting modular Hamiltonians. For conformal field theories at zero and finite temperature the appropriate modular Hamiltonians are known explicitly and we recover known expressions for local bulk observables.
Comment: 28 pages, LaTeX. v2: additional discussion of extended modular Hamiltonian. v3: version to appear in JHEP