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Discrete torsion defects

Ilka Brunner, Nils Carqueville, Daniel Plencner

TL;DR

The paper develops a defect-based framework for discrete torsion in two-dimensional orbifolds, showing that choices in the twisted sectors are captured by a separable Frobenius algebra A_G^c with c ∈ H^2(G,U(1)). It systematically analyzes bulk, boundary, and defect sectors in Landau-Ginzburg models, identifying the bulk with End_{A_G^c,A_G^c}(A_G^c) and boundary conditions as c-projective matrix factorisations, while defects are A_G^c-bimodules with fusion defined over the orbifold algebra. It derives disc correlator invariance rules, RR-charge computations for induced modules, and a generalized Cardy condition, illustrating how discrete torsion modifies projections and charges. Finally, it establishes a broad orbifold-equivalence result in equivariant completion of pivotal bicategories, showing that orbifolding by the quantum-symmetry defect can recover the original theory, thereby unifying abelian and nonabelian cases within a single categorical framework.

Abstract

Orbifolding two-dimensional quantum field theories by a symmetry group can involve a choice of discrete torsion. We apply the general formalism of `orbifolding defects' to study and elucidate discrete torsion for topological field theories. In the case of Landau-Ginzburg models only the bulk sector had been studied previously, and we re-derive all known results. We also introduce the notion of `projective matrix factorisations', show how they naturally describe boundary and defect sectors, and we further illustrate the efficiency of the defect-based approach by explicitly computing RR charges. Roughly half of our results are not restricted to Landau-Ginzburg models but hold more generally, for any topological field theory. In particular we prove that for a pivotal bicategory, any two objects of its orbifold completion that have the same base are orbifold equivalent. Equivalently, from any orbifold theory (including those based on nonabelian groups) the original unorbifolded theory can be obtained by orbifolding via the `quantum symmetry defect'.

Discrete torsion defects

TL;DR

The paper develops a defect-based framework for discrete torsion in two-dimensional orbifolds, showing that choices in the twisted sectors are captured by a separable Frobenius algebra A_G^c with c ∈ H^2(G,U(1)). It systematically analyzes bulk, boundary, and defect sectors in Landau-Ginzburg models, identifying the bulk with End_{A_G^c,A_G^c}(A_G^c) and boundary conditions as c-projective matrix factorisations, while defects are A_G^c-bimodules with fusion defined over the orbifold algebra. It derives disc correlator invariance rules, RR-charge computations for induced modules, and a generalized Cardy condition, illustrating how discrete torsion modifies projections and charges. Finally, it establishes a broad orbifold-equivalence result in equivariant completion of pivotal bicategories, showing that orbifolding by the quantum-symmetry defect can recover the original theory, thereby unifying abelian and nonabelian cases within a single categorical framework.

Abstract

Orbifolding two-dimensional quantum field theories by a symmetry group can involve a choice of discrete torsion. We apply the general formalism of `orbifolding defects' to study and elucidate discrete torsion for topological field theories. In the case of Landau-Ginzburg models only the bulk sector had been studied previously, and we re-derive all known results. We also introduce the notion of `projective matrix factorisations', show how they naturally describe boundary and defect sectors, and we further illustrate the efficiency of the defect-based approach by explicitly computing RR charges. Roughly half of our results are not restricted to Landau-Ginzburg models but hold more generally, for any topological field theory. In particular we prove that for a pivotal bicategory, any two objects of its orbifold completion that have the same base are orbifold equivalent. Equivalently, from any orbifold theory (including those based on nonabelian groups) the original unorbifolded theory can be obtained by orbifolding via the `quantum symmetry defect'.

Paper Structure

This paper contains 12 sections, 5 theorems, 87 equations.

Table of Contents

  1. Introduction
  2. Discrete torsion Landau-Ginzburg orbifolds
  3. Orbifolding defect
  4. Bulk sector
  5. Boundary sector
  6. Defect sector
  7. Disc correlators
  8. Invariance and selection rule
  9. Induced modules
  10. Orbifold equivalence in equivariant completion
  11. Discrete torsion from the perspective of boundary CFT
  12. Quantum symmetry defects for cyclic groups

Key Result

Proposition 2.1

If the graded matrix factorisation $A_G$ satisfies $\text{dim} \operatorname{Hom}^0(I, A_G)=1$, then its separable Frobenius structures are classified by $H^2(G,U(1))$. In particular, the multiplication associated to $c \in H^2(G,U(1))$ is

Theorems & Definitions (13)

  • Proposition 2.1
  • proof
  • Remark 2.2
  • Theorem 2.3
  • proof
  • Example 2.4
  • Example 3.1
  • Proposition 4.1
  • proof
  • Lemma 4.2
  • ...and 3 more