A Two-Index Framework for the Bulk-Boundary Correspondence of Anyon Condensation and Boundary Majorana Statistics

In the paper's words

The bulk-boundary correspondence asserts that topological data of a (2+1)-dimensional phase determine the physics of its boundary, but in the setting of anyon condensation the correspondence is usually stated structurally rather than quantitatively. We propose a set of computable indices that quantify the correspondence between anyon condensation in a parent topological order and the emergence of Majorana statistics at a gapped boundary. The first index, κ = dim(A)², measures the "size" of the condensate, where A is the connected (étale) algebra of condensed anyons inside the parent modular tensor category C.

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

Numbers it reports

1. We therefore report the child/parent dimension ratio $D_{D}/D_{C} = \sqrt{6}/\sqrt{14} = \sqrt{3/7} \approx 0.654654$ (arithmetic: $6/14 = 3/7 \approx 0.428571$; $\sqrt{0.428571} \approx 0.654654$) as a separate diagnostic rather than as an equivalent form of $\kappa$.

   from 4.3 Case III: Doubled Ising category, condensate $A = 1 \oplus (\psi ,\psi )$

2. Doubled Ising, $A = 1 \oplus (\psi ,\psi )$: $D_{C} = \sqrt{14}$; $D_{D} = \sqrt{6}$; $\kappa = \dim(A)^{2} = 4$ (non-normal condensate: $D_{C}^{2}/D_{D}^{2} = 7/3 \neq \kappa$, reported as a separate diagnostic); child/parent dimension ratio $\sqrt{3/7} \approx 0.6547$; $f_{\mathrm{NA}} = 1/3$; boundary carrier ($\sigma ,\sigma$) with four-channel fusion descending from the Ising rule.

   from 5. Results

How it is done

1. We define: The condensation index $\kappa = \dim(A)^{2}$, where A is the condensate algebra in the parent modular tensor category (MTC) C, with $\dim(A) = \Sigma _{a\in A} d_{a}$.

   from 1. Introduction

2. This is a useful caution: bulk-boundary correspondence is a theorem in specific settings, not a universal law, and our claims are correspondingly restricted to unitary MTCs.

   from 2. Background and Related Work

3. A (2+1)d topological order is described by an MTC C with simple objects (anyons) a, quantum dimensions $d_{a}$, topological spins $\theta _{a}$, fusion coefficients $N_{ab}^{c}$, and total quantum dimension $D_{C} = \sqrt{\Sigma _{a} d_{a}^{2}}$.

   from 3. Methods

4. The boundary fusion derivation is a heuristic identification of condensed summands with the vacuum, not a theorem about module category structure; a proper treatment would use the module category $C_{A}$.

   from 6. Discussion

What would falsify it

1. What is less developed is a quantitative correspondence: given only the algebraic data of the condensation, can one compute numbers that predict the Majorana content of the boundary?

   from 1. Introduction

2. Failure modes and falsifiability.

   from 6. Discussion

3. The claims would be falsified by: (i) a gapped, fully non-chiral boundary of a $c = 1/2$ phase obtained by purely bosonic condensation (contradicting the necessity of $\Delta c$ appearing at the boundary); (ii) a Hamiltonian realization along the lines of [9] in which the boundary $\sigma$ defects of the $\psi$-condensed Ising model show single-channel fusion; or (iii) a mixed-state generalization [8] in which the étale-algebra dimension formula $D_{D} = D_{C}/\dim(A)$ fails, breaking $\kappa$.

   from 6. Discussion

Where the claim stops

1. What the indices do and do not capture.

   from 6. Discussion

2. But $\mu$ is defined only mod 16 and only for chiral discrepancies expressible in half-integers; a condensation in a phase with irrational c (e.g., Fibonacci-like) would give non-integer $\mu$, and our interpretation of $\mu$ as a Majorana count fails there.

   from 6. Discussion

3. The fermionic condensation caveat.

   from 6. Discussion

Where to read

1. Abstract342 words · 1 min
2. 1. Introduction412 words · 2 min
3. 2. Background and Related Work732 words · 3 min
4. 3. Methods376 words · 2 min
5. 4. Analysis1,149 words · 5 min
6. 5. Results223 words · 1 min
7. 6. Discussion543 words · 2 min
8. 7. Conclusion124 words · 1 min
9. Appendix A. Divergence report389 words · 2 min