This paper gives a clear, checkable route from a local description of topological order to two important algebraic properties used to study anyons and quantum phases. The authors start from their earlier “local topological order” (LTO) framework, which packages ground-state constraints on finite regions as a net of projections, and they add two new axioms. One axiom forces Haag duality for cone-like regions. The other encodes a form of reflection positivity across a cut. They show these extra axioms hold for the standard toy models of topological order built from commuting projectors.

At a technical level the authors use the canonical pure state produced by the LTO axioms. This state ψ satisfies ψ(pR)=1 for the local ground-state projections pR. On the Hilbert space coming from ψ they form a net of von Neumann algebras A(Λ) associated to cone-shaped regions Λ. The boundary algebra along a cut is written as B(Λ):=pΛA(Λ)pΛ. The Haag duality axiom for LTO (called LTO‑HD in the paper) relates these boundary algebras on the two sides of a cut when restricted to the local ground-space for a large region.

The main mathematical result (stated as Theorem A in the paper) says that when the local algebras are finite dimensional, the LTO‑HD axiom holds, and a couple of technical faithfulness and central-support conditions are met, then Haag duality follows: the von Neumann algebra for a cone equals the commutant of the algebra for its complement. In plain terms, all observables in a cone are exactly the operators that commute with everything outside the cone. The authors check the extra axioms for all the well known commuting-projector models that people use to study topological order, including Kitaev’s toric code, quantum double models, Levin–Wen string-net models, and Walker–Wang models. For Levin–Wen they recover an independent proof of Haag duality previously obtained by other authors.