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arxiv: 2605.21327 · v1 · pith:KNTXP2ILnew · submitted 2026-05-20 · 🧮 math-ph · cond-mat.str-el· hep-th· math.MP· math.OA· math.QA

Universal fusion category symmetries on tensor products of infinite-dimensional Hilbert spaces

Ian Bunner , Corey Jones This is my paper

Pith reviewed 2026-05-21 03:33 UTC · model grok-4.3

classification 🧮 math-ph cond-mat.str-elhep-thmath.MPmath.OAmath.QA
keywords fusion categoriesanyon chainsinfinite-dimensional Hilbert spacestensor productsLevin-Wen modelstopological ordersymmetry realizationslattice models
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The pith

Any unitary fusion category can be realized as symmetries on a tensor product of infinite-dimensional Hilbert spaces after stabilizing anyon chains with infinite-dimensional ancilla spaces.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper shows that anyon chains stabilized by infinite-dimensional ancilla spaces factorize locally as tensor products of infinite-dimensional Hilbert spaces. This factorization lets any unitary fusion category act as symmetries on such tensor product spaces. The authors further prove that any two anyon chains with the same symmetry category become related by a symmetry-compatible locality-preserving unitary after stabilization, establishing a single stable equivalence class for each fixed fusion category. A corollary establishes that physical boundary algebras of Levin-Wen type models are bounded-spread isomorphic after stabilization if and only if they share the same bulk topological order.

Core claim

After stabilizing anyon chains with infinite-dimensional ancilla spaces, these chains factorize locally as tensor products of infinite-dimensional Hilbert spaces. This implies that any unitary fusion category can be realized as symmetries on a tensor product of infinite-dimensional Hilbert spaces. Any two anyon chains with the same symmetry category are related by a symmetry-compatible locality-preserving unitary after stabilization, so that for a fixed fusion category there is a single stable equivalence class of symmetry realizations on the lattice via anyon chains.

What carries the argument

Stabilization of anyon chains with infinite-dimensional ancilla spaces, which induces a local factorization into tensor products of infinite-dimensional Hilbert spaces while preserving the fusion category action.

If this is right

  • Any unitary fusion category admits a realization as symmetries on a tensor product of infinite-dimensional Hilbert spaces.
  • For each fixed fusion category, all stabilized anyon chain realizations belong to one equivalence class under symmetry-compatible locality-preserving unitaries.
  • Physical boundary algebras of Levin-Wen type models are bounded spread isomorphic after stabilization precisely when the models share the same bulk topological order.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The construction supplies a uniform infinite-dimensional lattice representative for each fusion category symmetry, bypassing finite-dimensional restrictions typical in many-body models.
  • Different microscopic anyon chain constructions of the same topological order can be connected through stabilization without altering the symmetry category.
  • The result may enable new model-building techniques that embed given fusion category symmetries into infinite-dimensional degrees of freedom while keeping locality.

Load-bearing premise

Stabilizing anyon chains with infinite-dimensional ancilla spaces produces a well-defined factorization into tensor products of infinite-dimensional Hilbert spaces that preserves the fusion category action.

What would settle it

A concrete unitary fusion category for which no stabilization procedure yields a symmetry-preserving local tensor product factorization into infinite-dimensional Hilbert spaces, or two anyon chains with identical symmetry categories that remain inequivalent under symmetry-compatible locality-preserving unitaries after stabilization.

read the original abstract

We show that anyon chains, after stabilizing with infinite-dimensional ancilla spaces, factorize locally as tensor products of infinite-dimensional Hilbert spaces. This implies that any unitary fusion category can be realized as symmetries on a tensor product of infinite-dimensional Hilbert spaces. We then show that any two anyon chains with the same symmetry category are related by a symmetry-compatible locality-preserving unitary after stabilizing with infinite-dimensional ancilla, showing that for a fixed fusion category, there is a single stable equivalence class of symmetry realizations on the lattice via anyon chains. As a corollary of our proof, we show that the physical boundary algebras of Levin-Wen type models are bounded spread isomorphic after stabilization if and only if they have the same bulk topological order.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 2 minor

Summary. The paper claims that stabilizing anyon chains with infinite-dimensional ancilla spaces yields a local factorization of the total Hilbert space into a tensor product of infinite-dimensional factors, thereby realizing any unitary fusion category as symmetries on such a space. It further asserts that any two anyon chains sharing the same symmetry category become related by a symmetry-compatible locality-preserving unitary after this stabilization, implying a unique stable equivalence class, and derives a corollary that physical boundary algebras of Levin-Wen models are bounded-spread isomorphic after stabilization if and only if they share the same bulk topological order.

Significance. If the stabilization and factorization arguments hold, the result would establish a universal construction embedding arbitrary unitary fusion categories into symmetries on infinite-dimensional tensor-product Hilbert spaces, bridging anyon-chain realizations to standard lattice models. The equivalence-class statement and the Levin-Wen boundary-algebra corollary would clarify when distinct symmetry realizations are stably equivalent, with potential implications for topological order and boundary theories. The technical use of infinite-dimensional ancilla to achieve clean local factorization is a distinctive contribution.

major comments (3)
  1. [Abstract, first paragraph and stabilization construction] Abstract, first paragraph and stabilization construction: the claim that tensoring infinite-dimensional ancilla at each site produces a strict local tensor-product factorization while preserving the full fusion-category action is load-bearing for the central universality result. The argument must explicitly define the dense subspaces, operator domains, and continuity of the fusion-category representation under the stabilization map; without this, residual non-local correlations or domain issues in the infinite-dimensional tensor product could prevent a clean factorization carrying the original action.
  2. [Equivalence of stabilized anyon chains] Equivalence of stabilized anyon chains: the assertion that any two anyon chains with the same symmetry category are related by a symmetry-compatible locality-preserving unitary after stabilization requires an explicit construction of this unitary together with verification that it respects the infinite-dimensional factorization and commutes appropriately with the ancilla degrees of freedom.
  3. [Corollary on Levin-Wen boundary algebras] Corollary on Levin-Wen boundary algebras: the iff statement that physical boundary algebras are bounded-spread isomorphic after stabilization precisely when they share the same bulk topological order depends on the main stabilization result commuting with boundary conditions; any failure of the ancilla to preserve locality at the boundary would undermine the equivalence.
minor comments (2)
  1. The notation for the stabilized local algebras and the precise meaning of 'bounded spread isomorphic' should be introduced with explicit definitions before the corollary is stated.
  2. A brief remark on the choice of dense subspaces in the infinite-dimensional tensor products would improve readability for readers outside functional analysis.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the thorough review and valuable comments, which have helped us strengthen the technical foundations of the manuscript. We have revised the paper to provide the requested explicit definitions and constructions while preserving the original results. Below we address each major comment point by point.

read point-by-point responses

  1. Referee: [Abstract, first paragraph and stabilization construction] Abstract, first paragraph and stabilization construction: the claim that tensoring infinite-dimensional ancilla at each site produces a strict local tensor-product factorization while preserving the full fusion-category action is load-bearing for the central universality result. The argument must explicitly define the dense subspaces, operator domains, and continuity of the fusion-category representation under the stabilization map; without this, residual non-local correlations or domain issues in the infinite-dimensional tensor product could prevent a clean factorization carrying the original action.

    Authors: We agree that the functional-analytic details require explicit treatment to rule out domain or continuity issues. In the revised manuscript we have inserted a new subsection (Section 2.3) that (i) identifies the dense subspace of finite-support vectors in each stabilized factor, (ii) defines the domain of the fusion-category operators as the set of bounded operators that map this dense subspace into itself and are continuous in the strong operator topology, and (iii) proves that the stabilization map intertwines the original anyon-chain representation with the stabilized one without introducing non-local correlations. These additions make the local tensor-product factorization and preservation of the fusion-category action fully rigorous. revision: yes

  2. Referee: [Equivalence of stabilized anyon chains] Equivalence of stabilized anyon chains: the assertion that any two anyon chains with the same symmetry category are related by a symmetry-compatible locality-preserving unitary after stabilization requires an explicit construction of this unitary together with verification that it respects the infinite-dimensional factorization and commutes appropriately with the ancilla degrees of freedom.

    Authors: The original manuscript established existence of the unitary via the uniqueness of the stable equivalence class. To meet the referee’s request for an explicit construction, the revised version now supplies a concrete map: at each site we tensor the original anyon Hilbert space with the infinite-dimensional ancilla and define a local isometry that absorbs the difference between the two chain realizations into the ancilla degrees of freedom. We verify that this isometry commutes with the ancilla projections, preserves the infinite-dimensional tensor-product factorization, and intertwines the two fusion-category actions. The resulting global unitary is therefore locality-preserving and symmetry-compatible by construction. revision: partial

  3. Referee: [Corollary on Levin-Wen boundary algebras] Corollary on Levin-Wen boundary algebras: the iff statement that physical boundary algebras are bounded-spread isomorphic after stabilization precisely when they share the same bulk topological order depends on the main stabilization result commuting with boundary conditions; any failure of the ancilla to preserve locality at the boundary would undermine the equivalence.

    Authors: We have added a dedicated paragraph in the proof of the corollary (Section 4) showing that the stabilization procedure is applied uniformly to boundary sites. The ancilla spaces are chosen to be compatible with the Levin-Wen boundary conditions, and the bounded-spread isomorphism is shown to commute with the stabilization map at the boundary. Consequently, locality is preserved and the iff statement continues to hold after stabilization. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation presented as independent construction from anyon-chain stabilization.

full rationale

The paper states its core results as theorems obtained by stabilizing anyon chains with infinite-dimensional ancilla spaces to obtain local tensor-product factorizations, then realizing fusion-category symmetries on the resulting spaces. No equations, self-definitional loops, fitted inputs renamed as predictions, or load-bearing self-citations appear in the provided abstract or described proof outline. The stabilization step is introduced as an external construction whose properties are then shown to yield the claimed factorization and equivalence classes; it does not reduce to the target statement by definition or prior self-citation. The derivation therefore remains self-contained relative to standard anyon-chain models and fusion-category representations.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The result relies on standard background from fusion category theory and operator algebras on infinite-dimensional Hilbert spaces. No free parameters or new invented entities are mentioned in the abstract. The key unstated premise is that the stabilization procedure is always possible and preserves the relevant algebraic structures.

axioms (2)
  • domain assumption Unitary fusion categories admit consistent anyon-chain realizations on lattices
    Invoked implicitly when the paper starts from anyon chains and stabilizes them.
  • domain assumption Infinite-dimensional ancilla spaces can be added without altering the fusion rules or locality properties
    Central to the stabilization step described in the abstract.

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