Orbifolds of two-dimensional quantum field theories have a natural formulation in terms of defects or domain walls. This perspective allows for a rich generalisation of the orbifolding procedure, which we study in detail for the case of topological field theories. Namely, a TFT with defects gives rise to a pivotal bicategory of ``worldsheet phases{''} and defects between them. We develop a general framework which takes such a bicategory B as input and returns its ``orbifold completion{''} B-orb. The completion satisfies the natural properties B subset of B-orb and (Borb)(orb) S congruent to B-orb, and it gives rise to various new equivalences and nondegeneracy results. When applied to TFTs, the objects in B-orb correspond to generalised orbifolds of the theories in B. In the example of Landau-Ginzburg models we recover and unify conventional equivariant matrix factorisations, prove when and how (generalised) orbifolds again produce open/closed TFTs, and give nontrivial examples of new orbifold equivalences.
Orbifolds of two-dimensional quantum field theories have a natural formulation in terms of defects or domain walls. This perspective allows for a rich generalisation of the orbifolding procedure, which we study in detail for the case of topological field theories. Namely, a TFT with defects gives rise to a pivotal bicategory of “worldsheet phases” and defects between them. We develop a general framework which takes such a bicategory B as input and returns its “orbifold completion” Borb. The completion satisfies the natural properties B⊂Borb and.(Borb)orb @ Borb, and it gives rise to various new equivalences and nondegeneracy results. When applied to TFTs, the objects in Borb correspond to generalised orbifolds of the theories in B. In the example of Landau-Ginzburg models we recover and unify conventional equivariant matrix factorisations, prove when and how (generalised) orbifolds again produce open/closed TFTs, and give nontrivial examples of new orbifold equivalences.