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Braid Group Representations and Defect Operators in AdS/CFT Correspondence

Tzu-Miao Chou

TL;DR

This work develops a holographic framework connecting bulk braid group representations, Wilson loop observables, and boundary defect operators within AdS/CFT. By leveraging modular tensor categories and their $F$- and $R$-symbols, it shows how bulk anyonic braiding induces corresponding representations on boundary defect algebras, with concrete constructions from Chern-Simons theory, Drinfeld centers, and quantum groups. The central result is a detailed bulk–boundary dictionary that maps bulk Wilson loops to boundary defect operators, preserving braid relations and fusion data across the AdS/CFT correspondence. These insights pave the way for holographic realizations of anyons and topological defects, offering new avenues for studying topological phases and quantum gravity in holographic settings.

Abstract

This paper investigates the connection between braid group representations, defect operators, and holography within the AdS/CFT framework. It focuses on the correspondence between bulk Wilson loops and boundary defect operators, emphasizing how braid group representations map to these operators. The study also explores fusion and braiding operations in modular tensor categories, which are crucial for understanding anyons in topological quantum field theories. By providing a unified framework, this work bridges the gap between bulk and boundary physics and offers insights into the holographic realization of topological defects. The results suggest new avenues for research in holographic anyons and their applications in quantum field theory and condensed matter physics.

Braid Group Representations and Defect Operators in AdS/CFT Correspondence

TL;DR

This work develops a holographic framework connecting bulk braid group representations, Wilson loop observables, and boundary defect operators within AdS/CFT. By leveraging modular tensor categories and their - and -symbols, it shows how bulk anyonic braiding induces corresponding representations on boundary defect algebras, with concrete constructions from Chern-Simons theory, Drinfeld centers, and quantum groups. The central result is a detailed bulk–boundary dictionary that maps bulk Wilson loops to boundary defect operators, preserving braid relations and fusion data across the AdS/CFT correspondence. These insights pave the way for holographic realizations of anyons and topological defects, offering new avenues for studying topological phases and quantum gravity in holographic settings.

Abstract

This paper investigates the connection between braid group representations, defect operators, and holography within the AdS/CFT framework. It focuses on the correspondence between bulk Wilson loops and boundary defect operators, emphasizing how braid group representations map to these operators. The study also explores fusion and braiding operations in modular tensor categories, which are crucial for understanding anyons in topological quantum field theories. By providing a unified framework, this work bridges the gap between bulk and boundary physics and offers insights into the holographic realization of topological defects. The results suggest new avenues for research in holographic anyons and their applications in quantum field theory and condensed matter physics.

Paper Structure

This paper contains 69 sections, 9 theorems, 69 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Motivation
  3. Challenges in Understanding Bulk Braid Structures and Boundary Defect Operators
  4. Review of Braid Groups and Chern-Simons Anyons
  5. Braid Group
  6. Braid Group Definition
  7. Braid Group Representations
  8. Wilson Loops and Defect Operators
  9. Defect Operators Definition
  10. Mapping Braid Group Representations to Defect Operators
  11. Mapping in AdS/CFT
  12. Mapping Defect Operators in AdS/CFT
  13. Challenges and Open Problems
  14. Bulk Anyons and Braid Group Representations in AdS
  15. Introduction
  16. ...and 54 more sections

Key Result

Theorem 2.1

Let $\mathcal{O}_D$ be a defect operator supported on a codimension-$k$ submanifold $\Sigma$ in the boundary CFT, and let $W_\gamma$ be a Wilson line or Wilson surface in the AdS bulk, with $\partial \gamma = \Sigma$. Then, under the AdS/CFT dictionary Haro:2000xnSkenderis:2002wp, the boundary inser

Figures (6)

  • Figure 1: Fusion associativity expressed via the $F$-move.
  • Figure 2: Braiding of $a$ and $b$ defects, depicted via the $R$-move and its compatibility with fusion via $F$-moves.
  • Figure 3: Pentagon identity for fusion, ensuring the consistency of associativity in the fusion process.
  • Figure 4: Hexagon identity relating $F$- and $R$-symbols.
  • Figure 5: Wilson loop $\gamma$ in the AdS bulk mapped holographically to a defect operator $\mathcal{D}_\gamma$ on the boundary CFT.
  • ...and 1 more figures

Theorems & Definitions (22)

  • Theorem 2.1
  • proof
  • Theorem 3.1
  • proof
  • Corollary 3.1.1
  • proof
  • Definition 3.1
  • proof
  • proof
  • Theorem 4.1: Fusion Category of Topological Defects
  • ...and 12 more