A Two-Index Framework for the Bulk-Boundary Correspondence of Anyon Condensation and Boundary Majorana Statistics
Abstract
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. For normal condensates — those in which every local A-module sector contributes once to the child, as in Cases I and II — κ coincides with D_C²/D_D², the ratio of squared total quantum dimensions of parent and child; for non-normal condensates (Case III) the two expressions differ, and we report both. The second index, μ = Δc/(1/2) = 2Δc, counts chiral Majorana edge modes forced by the mismatch Δc between the chiral central charges of parent and child phases. A third diagnostic, the non-Abelian excess fraction f_NA, measures the share of boundary quantum dimension carried by sectors with d > 1. We evaluate all indices explicitly for three canonical cases: condensing A = 1 ⊕ ψ in the Ising category (κ = 4, μ = 1, boundary σ defects with two fusion channels — the fingerprint of Majorana zero modes); condensing A = 1 ⊕ e in the toric code (κ = 4, μ = 0, single-channel fusion, no Majorana structure); and condensing A = 1 ⊕ (ψ,ψ) in the doubled Ising category (κ = dim(A)² = 4, child/parent dimension ratio D_D/D_C = √(3/7), f_NA = 1/3, non-Abelian boundary carrier (σ,σ)). The contrast between the first two cases shows that κ alone does not detect Majorana statistics; μ and f_NA do. All numerical results are derived by explicit arithmetic from standard category data, divergences between independent drafts are documented in Appendix A, and limitations of the framework Full text and updates: papers.qnfo.org/papers/a-two-index-framework-for-the-bulk-boundary-correspondence-of-anyon-condensation/
