# Conformal nets V: dualizability

## TL;DR

This paper proves finite-index conformal nets are fully dualizable in a 3-category, enabling the construction of associated topological field theories, and introduces a Peter-Weyl theorem for defects between conformal nets.

## Contribution

It establishes the dualizability of finite-index conformal nets and proves a Peter-Weyl theorem for defects, advancing the mathematical understanding of conformal nets and their applications.

## Key findings

- Finite-index conformal nets are fully dualizable in the 3-category.
- Existence of a local framed topological field theory from finite-index conformal nets.
- A Peter-Weyl theorem for defects between conformal nets.

## Abstract

We prove that finite-index conformal nets are fully dualizable objects in the 3-category of conformal nets. Therefore, assuming the cobordism hypothesis applies, there exists a local framed topological field theory whose value on the point is any finite-index conformal net. Along the way, we prove a Peter-Weyl theorem for defects between conformal nets, namely that the annular sector of a finite defect is the sum of every sector tensor its dual.