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arxiv: 2506.23155 · v4 · submitted 2025-06-29 · ✦ hep-th · cond-mat.str-el· math-ph· math.MP· math.QA

Homomorphism, substructure, and ideal: Elementary but rigorous aspects of renormalization group or hierarchical structure of topological orders

Yoshiki Fukusumi , Yuma Furuta This is my paper

Pith reviewed 2026-05-19 08:02 UTC · model grok-4.3

classification ✦ hep-th cond-mat.str-elmath-phmath.MPmath.QA
keywords renormalization grouptopological ordersfusion ringsnoninvertible symmetryanyon condensationdomain wallsgapped phasesgeneralized symmetry
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The pith

The noninvertible ideal in a fusion ring determines condensation rules for anyons under renormalization group flows from UV to IR topological orders.

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

The paper develops a quantum Hamiltonian formalism for renormalization group flows by drawing on the algebraic link between homomorphisms, quotient rings, and projections. Central to this is the idea that the noninvertible ideal of the fusion ring encoding generalized symmetry in the ultraviolet theory controls which anyons can condense to produce the infrared theory. This framework applies directly to domain wall problems in two-plus-one dimensional topologically ordered systems and to the classification of one-plus-one dimensional gapped phases. Ideal decompositions of fusion rings then impose strong constraints on possible symmetry-breaking or emergent symmetry patterns in those gapped phases. The approach also points to specific homomorphisms appearing in massless flows that may correspond to partially solvable models.

Core claim

By interpreting the elementary relationship between homomorphism, quotient ring, and projection, one obtains a general quantum Hamiltonian formalism of RG flow in which the noninvertible nature of the ideal of a fusion ring plays the fundamental role in fixing condensation rules between anyons and thereby selecting the infrared theories.

What carries the argument

The ideal of a fusion ring, which encodes the noninvertible aspects of generalized symmetry and supplies the condensation rules that connect ultraviolet and infrared topological orders.

If this is right

  • Provides a straightforward algebraic constraint on gapped phases that possess noninvertible symmetry.
  • Classifies one-plus-one dimensional gapped phases that arise from domain walls in two-plus-one dimensional topological orders.
  • Links certain less familiar homomorphisms under massless RG flows to partially solvable models.
  • Highlights the role of abstract algebraic structures in organizing hierarchical structures of topological orders.

Where Pith is reading between the lines

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

  • Extending the method to other dimensions could reveal similar algebraic controls over phase transitions in systems with generalized symmetries.
  • The formalism may allow prediction of infrared outcomes from ultraviolet symmetry data without requiring full dynamical solution of the Hamiltonian.
  • Connections to symmetry-breaking patterns in quantum field theories beyond topological orders could be explored using the same ideal decomposition technique.

Load-bearing premise

The abstract algebraic relationship between homomorphism, quotient ring, and projection directly corresponds to a physically valid quantum Hamiltonian formalism for renormalization group flows in topological orders.

What would settle it

A concrete counterexample would be a topological order in which the condensation rules predicted by the ideal decomposition of its fusion ring fail to produce the observed infrared phase after a renormalization group flow.

Figures

Figures reproduced from arXiv: 2506.23155 by Yoshiki Fukusumi, Yuma Furuta.

Figure 1
Figure 1. Figure 1: FIG. 1. Massless renormalization group and its connection to [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Construction flow of emergent or enhanced symme [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Hierarchical structure of gapped phase induced from [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Picture of the noninvertible object (anyon or symme [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Domain wall or tri-wire junction problem correspond [PITH_FULL_IMAGE:figures/full_fig_p014_5.png] view at source ↗
read the original abstract

We propose a general quantum Hamiltonian formalism of a renormalization group (RG) flow with an emphasis on generalized symmetry by interpreting the elementary relationship between homomorphism, quotient ring, and projection. In our formalism, the noninvertible nature of the ideal of a fusion ring realizing the generalized symmetry of an ultraviolet (UV) theory plays a fundamental role in determining condensation rules between anyons, resulting in the infrared (IR) theories. Our algebraic method applies to the domain wall problem in $2+1$ dimensional topologically ordered systems and the corresponding classification of $1+1$ dimensional gapped phase, for example. An ideal decomposition of a fusion ring provides a straightforward but strong constraint on the gapped phase with noninvertible symmetry and its symmetry-breaking (or emergent symmetry) patterns. Moreover, even in several specific homomorphisms connected under massless RG flows, less familiar homomorphisms appear, and we conjecture that they correspond to partially solvable models in recent literature. Our work demonstrates the fundamental significance of the abstract algebraic structure, ideal, for the RG in physics.

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

2 major / 1 minor

Summary. The manuscript proposes a general quantum Hamiltonian formalism for renormalization group flows in systems with generalized symmetries, particularly in 2+1D topological orders. It interprets the algebraic relationship between homomorphisms, quotient rings, and projections in fusion rings, arguing that the noninvertible nature of ideals in these rings determines anyon condensation rules and resulting IR theories. The approach is applied to domain wall problems and the classification of 1+1D gapped phases with noninvertible symmetries, claiming to provide constraints on symmetry-breaking patterns and conjecturing connections to partially solvable models.

Significance. If the algebraic-to-physical mapping is made rigorous with explicit derivations, the work could offer a novel algebraic framework for analyzing hierarchical structures, condensation, and symmetry patterns in topological orders with noninvertible symmetries. This might simplify classification of gapped phases and domain walls, building on existing fusion ring techniques, though its impact depends on demonstrating dynamical content beyond reinterpretation of algebraic objects.

major comments (2)
  1. [Abstract] Abstract and introduction: The central claim that the noninvertible ideal of a fusion ring 'plays a fundamental role in determining condensation rules between anyons, resulting in the IR theories' via a quantum Hamiltonian formalism rests on an interpretive step; no explicit construction is given that starts from a microscopic Hamiltonian, applies the algebraic data (homomorphism/quotient/projection), and derives the IR anyon content or RG flow equations.
  2. [Abstract] The application to domain walls in 2+1D systems and 1+1D gapped phases claims that ideal decomposition provides a 'straightforward but strong constraint' on symmetry-breaking patterns, but without a worked example showing how the algebraic decomposition translates into a specific Hamiltonian or verifies the resulting phase, the physical validity of the mapping cannot be assessed.
minor comments (1)
  1. [Abstract] The abstract mentions 'less familiar homomorphisms' connected under massless RG flows and conjectures correspondence to partially solvable models, but does not specify which homomorphisms or provide references to the recent literature cited.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive suggestions. Our manuscript develops an algebraic framework interpreting homomorphisms and ideals in fusion rings as encoding RG flows and anyon condensation in systems with generalized symmetries. We address the major comments below, agreeing that greater explicitness in the algebraic-to-physical mapping would strengthen the presentation.

read point-by-point responses

  1. Referee: [Abstract] Abstract and introduction: The central claim that the noninvertible ideal of a fusion ring 'plays a fundamental role in determining condensation rules between anyons, resulting in the IR theories' via a quantum Hamiltonian formalism rests on an interpretive step; no explicit construction is given that starts from a microscopic Hamiltonian, applies the algebraic data (homomorphism/quotient/projection), and derives the IR anyon content or RG flow equations.

    Authors: We agree that the mapping is primarily interpretive: the quantum Hamiltonian formalism is introduced by associating the algebraic operations (homomorphism to quotient by ideal, projection onto the ideal) with the RG flow and condensation rules, without deriving the IR spectrum from a specific lattice Hamiltonian. This is consistent with the paper's focus on elementary algebraic aspects of hierarchical structures in topological orders. We will revise the abstract and introduction to clarify the interpretive character of the formalism and add a short discussion relating the quotient construction to standard anyon condensation procedures in the literature. revision: partial

  2. Referee: [Abstract] The application to domain walls in 2+1D systems and 1+1D gapped phases claims that ideal decomposition provides a 'straightforward but strong constraint' on symmetry-breaking patterns, but without a worked example showing how the algebraic decomposition translates into a specific Hamiltonian or verifies the resulting phase, the physical validity of the mapping cannot be assessed.

    Authors: The domain-wall and 1+1D applications are derived directly from the ideal decomposition of the fusion ring, which constrains admissible symmetry-breaking patterns by requiring that the IR fusion rules be consistent with the quotient ring. While the manuscript presents this as a general constraint with reference to known results on noninvertible symmetries, we acknowledge the absence of a fully worked numerical or Hamiltonian example. In the revision we will insert a concise worked example (e.g., ideal decomposition for a small fusion ring realizing a known gapped phase) that explicitly shows the resulting IR anyon content and the corresponding symmetry-breaking pattern. revision: yes

Circularity Check

0 steps flagged

No significant circularity: algebraic interpretation proposed as new formalism

full rationale

The paper proposes a quantum Hamiltonian formalism for RG flows in topological orders by directly interpreting the algebraic relationship between homomorphisms, quotient rings, projections, and ideals of fusion rings. This mapping is presented as an interpretive step that organizes condensation rules and IR theories from UV generalized symmetries, without any equations or claims reducing by construction to fitted parameters, self-cited uniqueness theorems, or prior ansatzes from the same authors. The central role assigned to noninvertible ideals follows from the proposed formalism itself rather than from a tautological redefinition or statistical forcing. No load-bearing self-citations or renamings of known results are evident in the abstract or description; the work is self-contained as an organizational tool for domain-wall problems and gapped-phase classification.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the assumption that ring-theoretic ideals and homomorphisms faithfully capture physical condensation and RG flow; no free parameters or new entities are introduced in the abstract, but the mapping itself is an interpretive axiom.

axioms (2)
  • domain assumption The ideal of a fusion ring encodes noninvertible symmetry and determines condensation rules under RG flow.
    Invoked in the abstract as the fundamental role of the ideal in determining IR theories from UV symmetries.
  • domain assumption Homomorphisms between fusion rings correspond to RG flows or domain walls in topological orders.
    The paper interprets the elementary relationship between homomorphism, quotient ring, and projection as the basis for the quantum Hamiltonian formalism.

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Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Characterizing bulk properties of gapped phases by smeared boundary conformal field theories: Role of duality in unusual ordering

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  2. Generalizing quantum dimensions: Symmetry-based classification of local pseudo-Hermitian systems and the corresponding domain walls

    hep-th 2025-11 unverdicted novelty 6.0

    Generalized quantum dimensions from SymTFTs classify massless and massive RG flows in pseudo-Hermitian systems and relate coset constructions to domain walls.

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