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A note on permutation twist defects in topological bilayer phases

Jürgen Fuchs, Christoph Schweigert

TL;DR

This work provides a model-independent, category-theoretic derivation of key quantities for topological bilayer phases with permutation twist defects, using permutation-equivariant modular functors and a ${\mathbb Z}_2$-equivariant TFT framework. It identifies the permutation twist defect with the module category ${\mathcal{P}}_{\mathcal{D}}$ over ${\mathcal{D}}={\mathcal{C}}{\boxtimes}{\mathcal{C}}$, realized as ${\rm mod}-{A_{\mathcal{P}}}$ for an Azumaya algebra ${A_{\mathcal{P}}}$, and analyzes defect-Wilson line categories, fusion, and generalizations to networks of defects. The generalized conformal blocks on surfaces with genons are shown to coincide with ordinary conformal blocks of the underlying theory on two-fold covers, enabling Verlinde-type dimension computations and highlighting the genon-type dependence of ground-state degeneracies. Braiding in the ${\mathbb Z}_2$-equivariant setting is formulated, pointing to mapping-class-group representations that are relevant for universal quantum computation, and a speculative orbifold perspective is offered for gauging the permutation symmetry. Overall, the paper provides a rigorous, structure-theoretic path to quantify defect-induced degrees of freedom and their algebraic/ topological properties in bilayer topological phases.

Abstract

We present a mathematical derivation of some of the most important physical quantities arising in topological bilayer systems with permutation twist defects as introduced by Barkeshli et al. in cond-mat/1208.4834. A crucial tool is the theory of permutation equivariant modular functors developed by Barmeier et al. in math.CT/0812.0986 and math.QA/1004.1825.

A note on permutation twist defects in topological bilayer phases

TL;DR

This work provides a model-independent, category-theoretic derivation of key quantities for topological bilayer phases with permutation twist defects, using permutation-equivariant modular functors and a -equivariant TFT framework. It identifies the permutation twist defect with the module category over , realized as for an Azumaya algebra , and analyzes defect-Wilson line categories, fusion, and generalizations to networks of defects. The generalized conformal blocks on surfaces with genons are shown to coincide with ordinary conformal blocks of the underlying theory on two-fold covers, enabling Verlinde-type dimension computations and highlighting the genon-type dependence of ground-state degeneracies. Braiding in the -equivariant setting is formulated, pointing to mapping-class-group representations that are relevant for universal quantum computation, and a speculative orbifold perspective is offered for gauging the permutation symmetry. Overall, the paper provides a rigorous, structure-theoretic path to quantify defect-induced degrees of freedom and their algebraic/ topological properties in bilayer topological phases.

Abstract

We present a mathematical derivation of some of the most important physical quantities arising in topological bilayer systems with permutation twist defects as introduced by Barkeshli et al. in cond-mat/1208.4834. A crucial tool is the theory of permutation equivariant modular functors developed by Barmeier et al. in math.CT/0812.0986 and math.QA/1004.1825.

Paper Structure

This paper contains 18 sections, 3 theorems, 41 equations.

Table of Contents

  1. Introduction
  2. The module category $\mathcal{P}_{\mathcal{D}}$ and categories of defect Wilson lines
  3. Module categories.
  4. A module category for bilayer systems.
  5. $\mathcal{P}_{\mathcal{D}}$ as part of an equivariant topological field theory.
  6. The Azumaya algebra $A_{\mathcal{P}}$.
  7. Braided induction for tensor products of algebras.
  8. The Azumaya algebra ${A_{\mathcal{P}}}\,{\otimes}\,{A_{\mathcal{P}}}$.
  9. Categories of defect Wilson lines.
  10. Fusion of surface defects.
  11. More general Wilson lines.
  12. Generalized conformal blocks and their dimensions
  13. Mixed tensor products.
  14. Spaces of conformal blocks.
  15. A generalization of the Verlinde formula.
  16. ...and 3 more sections

Key Result

Proposition 2

The following diagram of monoidal functors and monoidal natural isomorphisms commutes: \xymatrix@R+6pt{ && ~~~(A_1{\otimes}A_2)\mbox{\rm -bimod}_{\mathcal{C}}~~~ && \\ {}\save[]*\txt{ \begin{picture}(0,0) \put(54,-7) {\begin{turn}{118}\xyRightarroww\end{turn}} \put(89,32) {$\scripts

Theorems & Definitions (7)

  • Remark 1
  • Proposition 2
  • Remark 3
  • Lemma 4
  • proof
  • Proposition 5
  • proof