Fixing a Math Error in Anyon Condensation Rules

In the paper's words

A companion preprint (https://papers.qnfo.org/papers/a-two-index-framework-for-the-bulk-boundary-correspondence-of-anyon-condensation/) introduced computable indices for anyon condensation: the dimension ratio κ = D_C²/D_D², the chirality-sensitive index μ = 2 Δ c, and the non-Abelian fraction f_NA. We correct an arithmetic error in that work's Case III, the condensation of Ising × overlineIsing: the parent total quantum dimension satisfies D_C² = 16, not $14$, because the non-Abelian anyon (σ,barσ) has quantum dimension d_(σ,barσ) = √2·√2 = 2, contributing 2² = 4 to the sum. The condensing boson (ψ,barψ) fixes (σ,barσ), which splits into two Abelian child anyons, so the child is the toric code with D_D² = 4, giving the normal condensate ratio κ = 16/4 = 4 and a child-side non-Abelian fraction f_NA = 0, not 1/3. [...]

The abstract, verbatim; every line below is quoted from the text.

What it claims

within the sixteen-fold way series, μ ≠ 0 (equivalently ν odd) implies multi-channel bulk fusion structure. Proof from stated inputs: if ν is odd, the theory contains σ with σ × σ = 1 + ψ (Input 3, reducing to Input 1 at ν = 1); this fusion has two channels ($1$ and ψ), hence is multi-channel. If ν is even, the theory is Abelian (Input 3), all fusions are single-channel, and f_NA = 0 trivially since there are no non-Abelian anyons. blacksquare

Numbers it reports

1. The condensing boson $(\psi,\bar\psi)$ fixes $(\sigma,\bar\sigma)$, which splits into two Abelian child anyons, so the child is the toric code with $D_D^2 = 4$, giving the normal condensate ratio $\kappa = 16/4 = 4$ and a child-side non-Abelian fraction $f_{\mathrm{NA}} = 0$, not $1/3$.

   from Abstract

2. The companion work [10] proposed to make the correspondence quantitative by attaching computable numbers to a condensation: the dimension ratio $\kappa$, the chirality index $\mu = 2\,\Delta c$ (where $\Delta c$ is the change in chiral central charge across the condensation), and the non-Abelian fraction $f_{\mathrm{NA}}$.

   from 1. Introduction

3. This is verified for Case III: replacing Ising by $\overline{\mathrm{Ising}}$ and vice versa leaves $d_{(\sigma,\bar\sigma)} = 2$ and $D_C^2 = 16$ unchanged, while flipping the sign of each factor's central charge.

   from 4.7 Mirror-blindness of $\kappa$ and $f_{\mathrm{NA}}$

How it is done

1. The splitting of the fixed-point anyon $(\sigma,\bar\sigma)$ into Abelian child anyons is the standard fixed-point resolution required by the bootstrap, and we use this formalism to replace the heuristic "$\psi \sim 1$" boundary rule of [10].

   from 2. Background and Related Work

2. For each case we: (i) list the parent simple objects and compute $D_C^2$ by direct summation; (ii) identify the condensate algebra $A$ and compute the orbits of the fusion action of $A$'s invertible components; (iii) resolve fixed-point orbits by splitting into simple child objects, checking consistency with $D_D^2$ and with the bootstrap requirements of [2]; (iv) compute $\kappa$, $\mu$, $f_{\mathrm{NA}}$ from the definitions.

   from 3.3 Computational procedure

3. Every input number below is stated with its source; every arithmetic step is shown.

   from 4. Analysis

4. Proof from stated inputs: if $\nu$ is odd, the theory contains $\sigma$ with $\sigma \times \sigma = 1 + \psi$ (Input 3, reducing to Input 1 at $\nu = 1$); this fusion has two channels ($1$ and $\psi$), hence is multi-channel.

   from 4.8 Restricted completeness test

What would falsify it

1. Outside this series the analogous statement is a conjecture supported only by the small sample of R1–R2; no counterexample was found among the cases examined.

   from 5. Results

2. The identification $\mu = \nu \bmod 16$ (R3) would be falsified by any sixteen-fold way member whose chiral central charge deviates from $\nu/2 \bmod 8$ in a way that makes $2\,\Delta c \ne \nu \bmod 16$; more seriously, it would be falsified if the sixteen-fold way label were not determined by $\Delta c$ alone (e.g., if two members with the same $\Delta c$ carried different $\nu$).

   from 6. Discussion

3. The completeness conjecture (R4, outside the series) would be falsified by a single condensate with $\mu \gt 0$ or $f_{\mathrm{NA}} \gt 0$ whose boundary defect fusion is single-channel; our evidence covers only the sixteen-fold way series and the three computed families, so the general claim should be treated as a conjecture.

   from 6. Discussion

Where the claim stops

1. First, $\kappa$ and $f_{\mathrm{NA}}$ are mirror-blind (Section 4.7): they cannot distinguish a theory from its conjugate, so any chirality-dependent boundary physics is invisible to them; only $\mu$ is signed, and it can vanish even when parent and child have distinct braiding structures.

   from 6. Discussion

2. Second, computing $f_{\mathrm{NA}}$ requires knowledge of the full simple-object set of the child; where the child is not fully classified, the index cannot be evaluated.

   from 6. Discussion

3. Third, the analogy to tachyon condensation [6] and the methodological borrowings from [7] and [9] are heuristic; they motivate the framework but do not constrain it.

   from 6. Discussion

Where to read

1. Abstract219 words · 1 min
2. 1. Introduction372 words · 2 min
3. 2. Background and Related Work568 words · 2 min
4. 3. Methods280 words · 1 min
5. 4. Analysis1,029 words · 4 min
6. 5. Results225 words · 1 min
7. 6. Discussion728 words · 3 min
8. 7. Conclusion178 words · 1 min