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

TL;DR

The paper develops a quantum Hamiltonian formalism for renormalization group flows by linking generalized symmetry to ideals in fusion rings, treating massless RG as a projection with Ker ρ an intrinsic noninvertible ideal. It defines unbroken and emergent symmetries within this algebraic framework and shows how ideal decompositions constrain and generate IR theories and gapped phases in 1+1 and 2+1 dimensions. Through explicit constructions of ring homomorphisms between UV and IR fusion rings, it demonstrates novel RG patterns, including noninteger coefficients and partially solvable flows, across examples like tricritical Ising → Ising and various SU(2) WZW flows. The work argues that ideal structures underpin hierarchical TOs and the sandwich construction, while also outlining open puzzles (e.g., charged-sector mappings in SU(2)^4 → SU(2)_4) and avenues for higher-dimensional generalizations and experimental relevance. Overall, it elevates ideals beyond groups as a fundamental algebraic lens for RG, TO classification, and domain-wall phenomena in quantum many-body systems.

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.

Figures (5)

• Figure 1: Massless renormalization group and its connection to hierarchical structures in TOs. Under the CFT/TQFT, the homomorphism $\rho: \mathbf{A}_{(1)}\rightarrow \mathbf{A}_{(2)}$ induces hierarchical structure of anyons in TOs. Because of the existence of kernel of the homomorphism notified as $\text{Ker}\rho$ (which is intrinsically noninvertible), the fusion rule of a reduced theory $\mathbf{A}_{(2)}$ can become different from the original theory $\mathbf{A}_{(1)}$ and this can be understood as an emergent phenomenonAnderson:1972pca(for more recent example, see the discussions and literature in Kikuchi:2021qxzKikuchi:2022gfiKikuchi:2022biw). In other words, $\mathbf{A}_{(2)}$ can be different from a subring of $\mathbf{A}_{(1)}$. This figure can be seen as an algebraic representation of the hierarchical structure in fractional quantum Hall statesJain:1989txBernevig2008PropertiesONBernevig_2008 and its connection to microscopic descriptions has been studied recently in Yang_2021Yuzhu_2023.
• Figure 2: Construction flow of emergent or enhanced symmetries from coupled symmetries. From the knowledge of existing theories $\mathbf{A}_{(1)}^{[i]}$ and the ideal of the coupled symmetry $\mathbf{A}_{(1)}$, one can obtain system with nontrivial or interacting symmetry $\mathbf{A}_{(2)}$. Only by assuming the existence of the commutativity of the kernel, one can realize the algebraic operation as a projection. Applications of this method to extended symmetry, such as Haagerup symmetry or magma in E-series models Nivesvivat:2025odb are interesting for future research. We note some research on the general construction of ideals in polynomial rings, which might be useful for the general construction of idealsGIANNI1988149KAWAZOE20111158.
• Figure 3: Hierarchical structure of gapped phase induced from massless RG. The homomorphism $\rho$ induces natural hierarchical structures in $1+1$ dimensional gapped phase and corresponding $2+1$ dimensional bulk states of TOs. Assuming Moore-Seiberg data, one can label primary states by the object in $\mathbf{A}_{(1)}$ or $\mathbf{A}_{(2)}$. However, the label of the states in BCFT can be outside of $\mathbf{A}_{(1)}$ and called symmetry breaking boundary states or new boundary states and they will be labeled by extended algebra $\mathbf{A}_{(1)}^{\text{ex}}$ or $\mathbf{A}_{(2)}^{\text{ex}}$. Without such extensions, they will be labelled by objects in an ideal of $\mathbf{A'}_{(1)}$ or $\mathbf{A'}_{(2)}$.
• Figure 4: Picture of the noninvertible object (anyon or symmetry) as a seed to generate a condensable block of a theory. We used blue color for noninvertible objects, black color for fusion between the noninvertible object and other objects, and red color for objects in $\mathbf{A}_{1}$. Because of the noninvertible structure, one can send the ideal generated by $s_{(1)}$ to $0$ only by applying a projection to the theory.
• Figure 5: Domain wall or tri-wire junction problem corresponding to the massless RG. The fields $\varphi$ and $\overline{\varphi}$ are chiral or antichiral fields, respectively. An exact description of the theory appearing in the domain wall is an open problem in general, but there exists respective research verifying this picture (anomaly matching case, in particular)Quella:2006deKimura:2014hvaKimura:2015nka.