Braid Group Representations, Modular Data, and the Classification of Majorana Zero Mode Fusion Rules in 2D Topological Superconductors

In the paper's words

We investigate whether the braid group representation on N anyons uniquely determines the modular data (S and T matrices) of a non-Abelian topological order, and whether this framework classifies all fusion rules for Majorana zero modes (MZMs) in 2D topological superconductors. We show that the local N-anyon braid group representation constrains the modular data up to finite gauge ambiguity but does not uniquely fix the global S-matrix without specifying the underlying spin structure. For MZMs, the braid generators satisfy the Ising anyon R-matrix with the asymptotic Hecke relation σ_i² = 1 (up to phase), which rigidly fixes the local F and R symbols of the Ising category.

The abstract, verbatim, and its opening sentence is the claim. Every line below is quoted from the text.

Numbers it reports

1. The Verlinde formula [4] expresses the fusion coefficients as: $$N_{ab}^c = \sum_x \frac{S_{ax} S_{bx} S_{cx}^*}{S_{1x}}$$ where $1$ denotes the vacuum.

   from 2.2 Modular Data and Fusion Categories

2. MZMs are zero-energy bound states localized at vortex cores in 2D $p+ip$ TSCs [2,3].

   from 2.4 Majorana Zero Modes and Ising Anyons

How it is done

1. MZMs are zero-energy bound states localized at vortex cores in 2D $p+ip$ TSCs [2,3].

   from 2.4 Majorana Zero Modes and Ising Anyons

What would falsify it

The paper's text states no explicit falsification criterion; the limits below are what it says about where the claim stops.

Where the claim stops

1. We have shown that the braid group representation on $N$ anyons constrains the modular data of a non-Abelian topological order up to finite gauge ambiguity, but does not uniquely determine the global $S$-matrix without specifying the underlying spin structure.

   from 6. Conclusion

2. The global $S$-matrix, however, depends on the spin structure of the underlying manifold, reflecting the fermionic nature of the topological order.

   from 6. Conclusion

3. We show that the local $N$-anyon braid group representation constrains the modular data up to finite gauge ambiguity but does not uniquely fix the global $S$-matrix without specifying the underlying spin structure.

   from Abstract

Where to read

1. Abstract155 words · 1 min
2. 1. Introduction305 words · 1 min
3. 2. Background413 words · 2 min
4. 3. Analysis886 words · 4 min
5. 4. Results340 words · 1 min
6. 5. Discussion477 words · 2 min
7. 6. Conclusion157 words · 1 min
8. Changelog9 words · 1 min