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Orbifold completion of defect bicategories

Nils Carqueville, Ingo Runkel

TL;DR

The paper develops a universal framework for generalized orbifolds in 2d topological field theories by constructing orbifold, equivariant, and open/closed completions of defect bicategories. It shows that for a pivotal bicategory with adjoints, the orbifold completion B_orb is closed under orbifolding, and every defect X with invertible quantum dimension induces an isomorphism X: (a, X†⊗X) ≅ (b, I_b) in B_orb, linking orbifolds to monadicity-like equivalences. The authors instantiate the theory in Landau-Ginzburg models, recovering equivariant matrix factorisations, proving when orbifolds produce open/closed TFTs, and deriving new orbifold equivalences including Knörrer periodicity and ADE-type minimal-model relations. The results provide a robust, category-theoretic route to generalized orbifolds beyond group actions, with Calabi-Yau/open/closed structures and concrete LG examples. Collectively, this framework unifies existing orbifold constructions, yields new equivalences, and clarifies when defect data yield nondegenerate, physically meaningful theories.

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 "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 B_orb and (B_orb)_orb = 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.

Orbifold completion of defect bicategories

TL;DR

The paper develops a universal framework for generalized orbifolds in 2d topological field theories by constructing orbifold, equivariant, and open/closed completions of defect bicategories. It shows that for a pivotal bicategory with adjoints, the orbifold completion B_orb is closed under orbifolding, and every defect X with invertible quantum dimension induces an isomorphism X: (a, X†⊗X) ≅ (b, I_b) in B_orb, linking orbifolds to monadicity-like equivalences. The authors instantiate the theory in Landau-Ginzburg models, recovering equivariant matrix factorisations, proving when orbifolds produce open/closed TFTs, and deriving new orbifold equivalences including Knörrer periodicity and ADE-type minimal-model relations. The results provide a robust, category-theoretic route to generalized orbifolds beyond group actions, with Calabi-Yau/open/closed structures and concrete LG examples. Collectively, this framework unifies existing orbifold constructions, yields new equivalences, and clarifies when defect data yield nondegenerate, physically meaningful theories.

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 "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 B_orb and (B_orb)_orb = 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.

Paper Structure

This paper contains 26 sections, 22 theorems, 125 equations.

Table of Contents

  1. Introduction and summary
  2. Algebraic Background
  3. Bicategories with adjoints
  4. Algebras and bimodules
  5. Tensor products
  6. Two-dimensional topological field theory with defects
  7. TFTs with defects as symmetric monoidal functors
  8. Bicategory of worldsheet phases
  9. Orbifold TFTs
  10. Domain walls between orbifolded theories
  11. Equivariant bicategory
  12. Definition of the equivariant completion $\boldsymbol{\mathcal{B}_{\mathrm{eq}}}$
  13. Equivalences in $\boldsymbol{\mathcal{B}_{\mathrm{eq}}}$ from adjunctions
  14. $\boldsymbol{\mathcal{B}_{\mathrm{eq}}}$ for pivotal bicategories
  15. Nondegenerate pairings
  16. ...and 11 more sections

Key Result

Lemma 2.2

If $A$ is a separable Frobenius algebra then $\pi_A^2 = \pi_A$ and $\operatorname{im}(\pi_A) = \operatorname{Hom}_A(X,Y)$.

Theorems & Definitions (55)

  • Definition 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Remark 2.4
  • Remark 3.1
  • Theorem 3.2
  • Remark 3.3
  • Proposition 3.4
  • ...and 45 more