Non-invertible symmetries and selection rules for RG flows of coset models

This paper introduces a method utilizing local data to classify superselection sectors and topological defect lines in two-dimensional conformal field theories, thereby establishing a unified framework for deriving selection rules governing renormalization group flows in coset and parafermion models.

The Gist

Imagine you are a master architect designing a skyscraper. You have a blueprint for a massive, complex tower (the UV theory, or the "high-energy" state of the universe). Now, imagine you want to know what happens if you start removing specific bricks or changing the materials. Will the building collapse? Will it transform into a smaller, different skyscraper? Or will it settle into a stable, cozy house (the IR theory, or the "low-energy" state)?

This paper is essentially a rulebook for predicting how these cosmic buildings transform when you poke them. The authors, Benedetti, Fendley, and Magan, have developed a new, powerful method to figure out exactly which transformations are allowed and which are forbidden.

Here is a breakdown of their ideas using simple analogies:

1. The Problem: The "Black Box" of Change

In physics, when you change a system (like cooling down a hot gas or stretching a rubber band), it undergoes a Renormalization Group (RG) flow. It moves from a complex state to a simpler one.

• The Old Way: Physicists used to guess these changes by looking at specific symmetries (like mirror symmetry) or by doing very difficult, messy math. It was like trying to predict the weather by only looking at the clouds, without understanding the wind patterns.
• The New Way: This paper says, "Let's look at the blueprint of the building itself." They realized that every theory has a hidden "fingerprint" called Superselection Sectors. Think of these as the different types of rooms that can exist inside the building. Some rooms can only be built if you have specific materials (symmetries).

2. The Key Concept: "Non-Invertible" Symmetries

Usually, in physics, if you do something and then undo it, you get back to where you started (like turning a light switch on and off). This is an "invertible" symmetry.
But the universe has stranger rules. Sometimes, you can do a transformation that cannot be undone.

• The Analogy: Imagine you have a deck of cards.
  • Invertible: You shuffle the deck, then shuffle it back. You get the original order.
  • Non-Invertible: You take the deck, cut it in half, and throw away the top half. You can't get the original deck back. The "symmetry" is that the remaining half still follows a pattern, even though the whole is gone.
    The authors show that these "one-way" transformations are actually the most important clues for predicting how the universe changes.

3. The Method: The "Submodel" Detective

The authors created a systematic way to list every possible "smaller building" (submodel) that can be made from a "big building" (the original theory).

• The Input: They take the local data of the theory (the basic rules of how particles interact).
• The Output: A complete list of all possible "sub-models" or "neighborhoods" that fit inside the original theory.
• The Magic Trick: They found that if you start with a specific "brick" (a particle or field) to trigger a change, the entire neighborhood containing that brick must survive the transformation. The rest of the building might fall away, but that specific neighborhood must remain intact.

4. The "Selection Rules": The Bouncer at the Club

This leads to the paper's main result: Selection Rules.
Think of the universe as a nightclub.

• The UV (High Energy): The VIP section, full of complex, expensive drinks and fancy guests.
• The IR (Low Energy): The regular bar downstairs.
• The Bouncer (The Selection Rule): When the VIPs try to move downstairs, the bouncer checks their ID.
  • Old Rule: "If you have a red hat, you can't go downstairs."
  • New Rule (This Paper): "If you are part of a specific 'club' (a DHR category or fusion ring), the entire club must move downstairs together. You can't leave a member behind."

If the "club" you are trying to move doesn't exist in the downstairs bar, the transformation is impossible. This explains why certain cosmic changes happen and others never do.

5. Testing the Theory: The Coset and Parafermion Models

To prove their method works, they applied it to two famous families of theories:

1. Coset Models: Imagine taking a giant Lego castle (a complex symmetry) and removing a specific smaller castle (a subgroup) from the middle. What's left? They mapped out every possible way this "removal" could happen and found exactly which resulting structures are stable.
2. Parafermion Models: These are like "fractional" versions of standard particles (like having 1/3 of an electron). They used their method to predict how these fractional particles behave when the system changes, confirming known results and finding new possibilities.

Why Does This Matter?

This paper unifies several different ways physicists have been trying to understand the universe for decades.

• It connects algebra (math of groups) with geometry (shapes of space).
• It explains why certain "integrable" systems (systems that are perfectly solvable) work the way they do.
• Most importantly, it gives physicists a complete checklist. Instead of guessing which universe might emerge from a Big Bang or a phase transition, they can now check the "blueprint" and say with certainty: "This transformation is allowed," or "This one is strictly forbidden."

In a nutshell: The authors built a universal translator that converts the complex, abstract math of particle physics into a clear set of "traffic laws." These laws tell us exactly which cosmic journeys are possible and which roads are closed, ensuring that the universe's evolution follows a strict, logical, and predictable path.

Technical Summary

Here is a detailed technical summary of the paper "Non-invertible symmetries and selection rules for RG flows of coset models" by Benedetti, Fendley, and Magan.

1. Problem Statement

The paper addresses the challenge of determining selection rules for Renormalization Group (RG) flows between two-dimensional Rational Conformal Field Theories (RCFT$_2$). While the $c$-theorem provides a global constraint on the central charge, predicting specific IR fixed points triggered by relevant perturbations in the UV requires more refined tools.

• Limitations of existing methods: Traditional approaches rely on conformal perturbation theory, integrability, or specific symmetries (like supersymmetry). However, these often fail to capture the full landscape of possible flows, particularly those involving non-invertible symmetries and superselection sectors.
• The Core Question: How can one systematically classify all possible submodels (extensions) of a given modular-invariant CFT and use this classification to derive rigorous, non-perturbative selection rules for RG flows? Specifically, how do DHR (Doplicher-Haag-Roberts) categories and topological defect lines (TDLs) constrain the flow?

2. Methodology

The authors propose a unified, systematic, and non-perturbative framework based on local algebraic data of the UV CFT.

• Local Data Input: The method starts with a modular-invariant CFT$_2$ ($\mathcal{C}$) characterized by its chiral fusion rules and modular $S$-matrix.
• Submodel Classification: They define a "local submodel" $\mathcal{T} \subset \mathcal{C}$ as a set of conformal families closed under fusion. For diagonal modular invariants, this allows for a finite enumeration of all possible submodels.
• Algebraic Selection Rule: The central hypothesis is that an RG flow triggered by a relevant operator $\phi_{UV}$ is constrained to the smallest closed submodel $\mathcal{T}_{UV}[\phi_{UV}]$ containing that operator. The structure of this submodel (and its extensions) must be preserved along the flow.
• Connection to DHR and TDLs:
  • DHR Categories: The authors link submodels to the category of localized endomorphisms (DHR sectors). A submodel $\mathcal{T}$ lacks certain local operators present in $\mathcal{C}$, thereby creating DHR sectors (charges) in $\mathcal{T}$.
  • Jones Index ($\mu$): They utilize the global Jones index $\mu_{\mathcal{T}}$, calculated via the formula $\mu_{\mathcal{T}} = (\sum d_i^2 / \sum \sum Z_{jk} d_j d_k)^2$, where $d_i$ are quantum dimensions and $Z_{jk}$ are multiplicities. This index serves as a sharp invariant to identify the DHR category.
  • Topological Defect Lines (TDLs): The symmetry category of a submodel is isomorphic to its DHR fusion ring. The authors show that the fusion ring of TDLs commuting with the submodel algebra is preserved along the RG flow.
• Q-Systems: The classification of submodels is mathematically equivalent to the classification of Q-systems (algebraic structures in the DHR category controlling local extensions of chiral algebras).

3. Key Contributions

1. Unified Framework: The paper establishes a direct link between local CFT data, DHR categories, non-invertible symmetries (TDLs), and RG selection rules. It demonstrates that the preservation of the DHR category and the TDL fusion ring are equivalent manifestations of the same underlying algebraic constraint.
2. Systematic Classification: The authors provide a complete classification of all submodels for two major families of RCFTs:
   • Diagonal $su(2)$ Cosets (GKO models): $D(k, l) = \frac{su(2)_k \times su(2)_l}{su(2)_{k+l}}$.
   • $\mathbb{Z}_k$ Parafermion Models: $PF(k) = \frac{su(2)_k}{u(1)_k}$.
3. Non-Invertible Symmetry Analysis: They explicitly identify non-invertible symmetries (fusion rings of TDLs) associated with these submodels, extending previous classifications limited to minimal models.
4. Derivation of Selection Rules: They formulate a rigorous selection rule: The DHR category (and associated TDL fusion ring) of the smallest submodel containing the perturbing operator in the UV must be reproduced in the IR fixed point.

4. Key Results

A. Classification of Submodels

• Coset Models $D(k, l)$: For generic odd $k, l$, the diagonal modular invariant has 16 distinct submodels.
  • Exceptions exist for minimal models ($k=1$ or $l=1$) and supersymmetric minimal models ($k=2$ or $l=2$), where the number reduces to 8, 12, 3, or 5 depending on the specific case.
  • Each submodel is characterized by its DHR category (e.g., $su(2)_{even}$, $\mathbb{Z}_2$, or products thereof) and its global Jones index.
• Parafermion Models $PF(k)$: The number of submodels is $2^S$, where$S$is the number of divisors of$k$.
  • These include invertible symmetries ($\mathbb{Z}_n$ subgroups) and non-invertible symmetries (associated with $su(2)_{even}$ categories).

B. RG Flow Applications

The authors apply their selection rules to re-derive and generalize known integrable flows:

1. Coset Flows:
   • Perturbing $D(k, l)$ by $\phi_{(0,0,2)}$ triggers flows to $D(k-l, l)$ or $D(k, l-k)$.
   • The method confirms that the DHR category $su(2)_{even}^k \times su(2)_{even}^l$ (or similar variants) is preserved, matching the IR submodel structure.
2. Parafermion to Minimal Model Flow:
   • The flow $PF(k) \xrightarrow{\phi_{(0,2)}} D(k-1, 1)$ is recovered. The selection rule confirms that the $su(2)_{even}^k$ DHR category is preserved from the UV parafermion model to the IR minimal model.
3. New Constraints and Possibilities:
   • The analysis suggests that direct flows between theories sharing no non-trivial subsector are forbidden.
   • They discuss a potential flow $PF(k) \to D(k, 1)$ which is not forbidden by symmetry or 't Hooft anomalies but is not observed in known integrable lattice models, suggesting that including other operators with the same symmetry might be necessary to access this fixed point.

C. Specific Flows Analyzed

• $PF(k) \xrightarrow{\phi_{(2,0)}} u(1)_k$: A flow to a compact scalar field at rational radius, preserving a $\mathbb{Z}_k$ fusion ring.
• Consistency Checks: The method successfully reproduces known results for Ising ($D(1,1)$), Tricritical Ising ($D(2,1)$), and 3-state Potts ($PF(3)$) models.

5. Significance

• Beyond Perturbation Theory: The approach provides a non-perturbative method to constrain RG flows, relying solely on the algebraic structure of the UV theory rather than small coupling expansions.
• Unification of Concepts: It bridges the gap between seemingly disparate concepts: Haag duality violations, Jones indices, fusion categories, and RG selection rules. It clarifies that "symmetry" in the modern sense (non-invertible/TDLs) and "superselection sectors" (DHR) are two sides of the same coin in the context of RG flows.
• Completeness: By classifying all submodels (Q-systems) for $c \ge 1$ coset and parafermion models, the paper offers a potentially complete map of allowed RG trajectories for these theories.
• Future Directions: The authors highlight that this method can be extended to non-diagonal modular invariants, mixed IR phases (involving TQFTs), and irrational CFTs (e.g., string theory orbifolds), opening new avenues for understanding quantum field theory landscapes.

In summary, the paper establishes that the classification of local submodels is the fundamental key to understanding selection rules in 2D CFTs, providing a powerful, systematic tool to predict and verify RG flows using non-invertible symmetries and superselection sector preservation.