# A Defect Verlinde Formula

## TL;DR

This paper derives a Verlinde-like formula for boundary excitations in 2+1D topological orders, relating fusion rules and half-linking numbers, with explicit computations for Abelian and non-Abelian cases, aiding quantum computing platform design.

## Contribution

It introduces a novel Verlinde-like formula connecting boundary fusion rules and half-linking numbers in topological orders, with practical computation methods for various anyon models.

## Key findings

- Derived a boundary Verlinde formula for topological orders.
- Explicit calculations of half-linking numbers in Abelian and non-Abelian cases.
- Potential applications in designing quantum computing platforms.

## Abstract

We revisit the problem of boundary excitations at a topological boundary or junction defects between topological boundaries in non-chiral bosonic topological orders in 2+1 dimensions. Based on physical considerations, we derive a formula that relates the fusion rules of the boundary excitations, and the "half-linking" number between condensed anyons and confined boundary excitations. This formula is a direct analogue of the Verlinde formula. We also demonstrate how these half-linking numbers can be computed in explicit Abelian and non-Abelian examples. As a fundamental property of topological orders and their allowed boundaries, this should also find applications in finding suitable platforms realizing quantum computing devices.

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Source: https://tomesphere.com/paper/1901.08285