Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 342 / 1 / d / Solution

Forthe vectors $q_e=(1,0{)}^T$ and $q_m=(0,1{)}^T$ haveThe K-matrix formula therefore gives bosonic self-exchange for $e,m$ and mutual full-braiding phaseFusion adds vectors. Becauseboth double fusions lie in $K{\mathbb{Z}}^2$ and braid trivially with every vector; hence they represent the vacuum in the anyon lattice of an Abelian Chern--Simons theory. Finally $q_f=q_e+q_m$ has $q_f^T{K}^{-1}{q}_f=1$, so its exchange phase is $e^{i\pi }=-1$. These are exactly the fusion and braiding data of the surface-code anyon model.