A Classification of (2+1)D Topological Phases with Symmetries

Abstract

This thesis aims at concluding the classification results for topological phases with symmetry in 2+1 dimensions. The main result is that topological phases are classified by a triple of unitary braided fusion categories $\mathcal{E}\subset \mathcal{C}\subset \mathcal{M}$ plus the chiral central charge $c$. Here $\mathcal{E}$ is a symmetric fusion category, $\mathcal{E}=\mathrm{Rep}(G)$ for boson systems or $\mathcal{E}=\mathrm{sRep}(G^f)$ for fermion systems, consisting of the representations of the symmetry group and describing the local excitations with symmetry; $\mathcal{C}$ is the category of all the quasiparticle excitations in the bulk, containing $\mathcal{E}$ as its Müger center; $\mathcal{M}$ is a minimal modular extension of $\mathcal{C}$, that also includes the gauged symmetry defects. We also study the stacking of topological phases with symmetry and two types of anyon condensations based on such classification.

PhD thesis