Categorical Braiding Subfactor

---
name: categorical-braiding-subfactor
description: "Braiding structures on categorical multi-interval Jones-Wassermann subfactor planar algebras from unitary modular fusion categories. Constructs braidings inducing projective unitary representations of balanced superelliptic mapping class groups, encoding higher-genus topological data. New proof of subfactor self-duality. Activation: braiding structures, Jones-Wassermann subfactor, planar algebra, modular fusion category, mapping class group, higher-genus topology, categorical quantum"
metadata:
  arxiv_id: "2607.08296"
  published: "2026-07-09"
  authors: "Multiple authors"
  tags: [quantum, category-theory, subfactor, braiding, topology, modular-fusion]
---

# Categorical Braiding Structures on Subfactors

## Methodology

Constructs braiding structures on multi-interval Jones-Wassermann subfactor planar algebra from any unitary modular fusion category.

### Core Results

1. **Braiding construction**: On multi-interval Jones-Wassermann subfactor planar algebra from unitary modular fusion category
2. **Self-duality proof**: New proof using the braiding construction
3. **Mapping class group representation**: Braidings induce projective unitary representation of balanced superelliptic mapping class group
4. **Higher-genus encoding**: Structures encode non-trivial higher-genus data of underlying category

### Mathematical Framework

- **Unitary modular fusion category**: Input data for construction
- **Planar algebra**: Multi-interval Jones-Wassermann subfactor structure
- **Braiding**: Categorical braiding on the planar algebra
- **Mapping class group**: Balanced superelliptic mapping class group representation

### Workflow

1. Start with unitary modular fusion category
2. Construct multi-interval Jones-Wassermann subfactor planar algebra
3. Define braiding structures on the planar algebra
4. Verify self-duality via braiding
5. Extract projective unitary representation of mapping class group

### Pitfalls

- **Modular fusion requirement**: Input must be a unitary modular fusion category (not just fusion category)
- **Multi-interval complexity**: Construction significantly more complex than single-interval case
- **Higher-genus subtlety**: Encoding higher-genus data requires careful treatment of boundary conditions

### Related Skills

- `topological-quantum-computing` (topological structures)
- `quantum-error-correction-methods` (subfactor theory in QEC)
- `tensor-cookbook-diagrams` (planar algebra diagrams)