# A Defect Verlinde Formula

**Authors:** Ce Shen, Ling-Yan Hung  
**arXiv:** [1901.08285](https://arxiv.org/abs/1901.08285)  
**Date:** 2019-01-24  
**Summary page:** https://www.summarizepaper.com/en/paper/1901.08285v1/

## Key points

- Authors explore boundary excitations at topological boundaries or junction defects in non-chiral bosonic topological orders in 2+1 dimensions
- They derive a formula relating fusion rules of boundary excitations to "half-linking" number between condensed anyons and confined boundary excitations
- Formula is analogous to the Verlinde formula in conformal field theory
- Findings deepen understanding of behavior of boundary excitations in non-chiral bosonic topological orders
- Half-linking numbers can be computed in explicit Abelian and non-Abelian examples
- Results have implications for quantum computing devices, aiding in designing suitable platforms for realizing quantum computing systems
- Derived formula provides insights into fusion rules and interactions at topological boundaries, enhancing efficiency and robustness of quantum computing architectures
- Paper contributes to understanding of topological orders in 2+1 dimensions and offers a powerful tool for studying these systems with potential applications beyond theoretical physics.

## Summary

In the paper "A Defect Verlinde Formula," authors Ce Shen and Ling-Yan Hung explore the problem of boundary excitations at topological boundaries or junction defects between topological boundaries in non-chiral bosonic topological orders in 2+1 dimensions. They derive a formula that relates the fusion rules of these boundary excitations to the "half-linking" number between condensed anyons and confined boundary excitations. The derived formula is a direct analogue of the Verlinde formula, which has been extensively studied in conformal field theory. By establishing this connection, Shen and Hung provide a deeper understanding of the behavior of boundary excitations in non-chiral bosonic topological orders. To illustrate their findings, the authors demonstrate how these half-linking numbers can be computed in explicit Abelian and non-Abelian examples. This computational approach allows for practical applications and further exploration of the properties of topological orders and their allowed boundaries. The results presented in this paper have significant implications for quantum computing devices as understanding the behavior of boundary excitations is crucial for designing suitable platforms for realizing quantum computing systems. The derived formula provides valuable insights into the fusion rules and interactions between different types of excitations at topological boundaries, paving the way for more efficient and robust quantum computing architectures. Overall, this paper contributes to our understanding of topological orders in 2+1 dimensions by providing a new perspective on boundary excitations. The defect Verlinde formula offers a powerful tool for studying these systems and has potential applications beyond theoretical physics, particularly in the development of quantum computing technologies.