Orbifold completion of defect bicategories

Abstract

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 "world sheet phases" and defects between them. We develop a general framework which takes such a bicategory $\mathcal{B}$ as input and returns its "orbifold completion" ${\mathcal{B}}_{\mathrm{orb}}$. The completion satisfies the natural properties $\mathcal{B}\subset {\mathcal{B}}_{\mathrm{orb}}$ and $({\mathcal{B}}_{\mathrm{orb}}{)}_{\mathrm{orb}}\cong {\mathcal{B}}_{\mathrm{orb}}$, and it gives rise to various new equivalences and nondegeneracy results. When applied to TFTs, the objects in ${\mathcal{B}}_{\mathrm{orb}}$ correspond to generalised orbifolds of the theories in $\mathcal{B}$. 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.