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Let $Z_e$ denote the dual edge variable. The PEPO implements the domain-wall mapBecause edge domain walls arise from vertex spins, their product around every plaquette is constrained:Thus the transverse-field Ising operators map to the ${\mathbb{Z}}_2$ lattice-gauge operators, while the image is projected into the positive eigenspace of all $B_p$. At the commuting-projector fixed point, adjoining these automatic projectors givesthe toric code Hamiltonian. Conversely, solving the zero-flux constraint writes $Z_e=Z_u{Z}_v$ locally and recovers the Ising variables, establishing the duality on the supported symmetry sectors.

Solved by gpt-5.6-sol high.

The original Ising symmetry is the global spin flipThe dual system has a one-form symmetry generated by products of edge Pauli operators along closed noncontractible loops; equivalently, closed Wilson loops label its electric and magnetic topological sectors.

The PEPO annihilates the nonsymmetric Ising sector because each virtual index is summed with the global parity constraint. On the dual side it has support only on configurations obeying all contractible zero-flux constraints and, for a fixed untwisted PEPO, one choice of noncontractible loop eigenvalues. Therefore the duality is invertible only after restricting both Hilbert spaces to corresponding symmetry sectors.

On a torus the toric code has four ground states distinguished by two independent noncontractible loop eigenvalues. A single untwisted duality operator reaches only one of them; inserting the two possible Ising twists supplies the other three. This explains why the global-symmetry Ising description and topologically degenerate toric-code description do not contradict each other.

Solved by gpt-5.6-sol high.