Anyon Proliferation and Anyon Superconductivity in Higgsing Transitions via Conformal Embeddings
We discuss (2+1)d Chern-Simons Higgsing transitions based on conformal embeddings and their relation to the proliferation of anyons. These transitions preserve or enlarge the intrinsic topological order despite reducing the gauge group. When the topological order is enlarged, the two phases differ by anyon condensation, and the transition takes an Abelian topological order to a non-Abelian one. In the $SO(N)_{2} \hookrightarrow SU(N)_{1}$ family with $N \geq 3$, the two semiclassical regimes have equivalent intrinsic topological orders but distinct realizations of a global $U(1)$ symmetry. For odd $N$, these are non-Abelian realizations of transitions from fermionic $1/N$ Laughlin states to charge-$2e$ anyon superconductivity coexisting with the same chiral $\mathbb{Z}_{N}$ topological order. The scalar representations are always non-integrable with respect to the UV Chern-Simons level and therefore do not directly label anyons of the un-Higgsed phase, despite defining Wilson lines in the UV. Using heuristic adjoint screening, we propose candidate anyons in the un-Higgsed phase, while branching under the Higgs subgroup reveals channels corresponding to condensable anyons of the Higgs phase. We further construct a $\mathrm{Spin}(16)_{1} \to \mathrm{Spin}(9)_{2}$ transition admitting a condensable algebra that contains a non-Abelian anyon but not all of the allowed gaugable channels in its self-fusion.