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Defects between gapped boundaries provide a possible physical realization ofprojective non-abelian braid statistics. A notable example is the projectiveMajorana/parafermion braid statistics of boundary defects in fractional quantumHall/topological insulator and superconductor heterostructures. In this paper,we develop general theories to analyze the topological properties andprojective braiding of boundary defects of topological phases of matter in twospatial dimensions. We present commuting Hamiltonians to realize defectsbetween gapped boundaries in any $(2+1)D$ untwisted Dijkgraaf-Witten theory,and use these to describe their topological properties such as their quantumdimension. By modeling the algebraic structure of boundary defects throughmulti-fusion categories, we establish a bulk-edge correspondence betweencertain boundary defects and symmetry defects in the bulk. Even though it isnot clear how to physically braid the defects, this correspondence elucidatesthe projective braid statistics for many classes of boundary defects, bothamongst themselves and with bulk anyons. Specifically, three such classes ofimportance to condensed matter physics/topological quantum computation arestudied in detail: (1) A boundary defect version of Majorana and parafermionzero modes, (2) a similar version of genons in bilayer theories, and (3)boundary defects in $\mathfrak{D}(S_3)$.

On Defects Between Gapped Boundaries in Two-Dimensional Topological Phases of Matter