Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 343 / 4 / b

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.