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

The two-dimensional Kramers--Wannier duality PEPO has one physical input leg at every original vertex and one physical output leg at every edge. A COPY tensor at each vertex sends its binary value to all incident virtual legs. An XOR tensor on each edge compares the two incident virtual bits and writes their sum modulo two to the edge output. Contracting every vertex--edge virtual leg gives the graphical networkrepeated over the graph. In the X-Fourier basis the same network exchanges the COPY and parity constraints. This local tensor description is a projected entangled pair operator.

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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.

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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.

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Violating a star stabilizer $A_v$ creates an electric $e$ particle; violating a plaquette stabilizer $B_p$ creates a magnetic $m$ particle. Open Z strings create pairs of $e$ endpoints, while open X strings create pairs of $m$ endpoints. Their fusion $ϵ=e\times m$ is the third nontrivial particle type.

Two equal-type strings can be exchanged without a phase, so $e$ and $m$ have bosonic self-statistics. An X string and a Z string crossing once anticommute, so taking $e$ around $m$ gives phase $-1$: they are mutual semions. Combining these two mutual-semion species givesThus $ϵ$ is a fermion.

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