Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 342 / 1 / c

The block matrix isIf both $q\in \mathcal{L}$ and $q^{\prime }$ condense, choose their boundary excursions so that the two string operators cross once. Their commutator is their mutual full-braiding phase:Both operators act as the identity on every ground state, so consistency requiresWriting $q^{\prime }=(q_{\mathrm{top}}^{\prime },q_{\mathrm{bot}}^{\prime })$, this saysSet $l_{\mathrm{top}}=M^{-1}{q}_{\mathrm{bot}}^{\prime }$ and choose $l_{\mathrm{bot}}=0$. Then $Kl=(0,q_{\mathrm{bot}}^{\prime }{)}^T$, andTherefore every additionally condensable anyon has the formThe $Kl$ term is a local particle in the anyon lattice of an Abelian Chern--Simons theory, so the condensate is maximal modulo local excitations, as required for a Lagrangian subgroup of Abelian anyons.