Global symmetries: locality, unitarity, and regularity

This paper resolves the apparent tension between locality and unitarity in quantum field theories with non-invertible categorical symmetries by demonstrating that locality enforces specific regularities on symmetry actions, enabling the definition of an observable that quantifies non-locality and encodes fusion algebra data.

The Gist

Imagine you are trying to understand the rules of a massive, complex game played by particles. In physics, these rules are often called "symmetries." Think of a symmetry like a magic trick: you can change the state of the game (rotate it, flip it, or shift it), but the fundamental laws of the game remain exactly the same.

For a long time, physicists believed these magic tricks followed a very strict, simple rulebook: Unitarity. This is the idea that if you perform a trick, you can always perform the exact opposite trick to undo it. It's like a lock and key; if you lock a door, there is always a key to unlock it. In the quantum world, this means every symmetry operator has an inverse.

However, recent discoveries have introduced a new, stranger kind of symmetry called non-invertible symmetry. These are like magic tricks where, once you perform them, you can't simply "undo" them with a single reverse move. It's as if you turn a key, and the door disappears entirely.

This paper tackles a big puzzle: How do these "un-undoable" tricks fit into a universe that is supposed to be "local"?

The Core Conflict: The "Local" Neighborhood vs. The "Global" View

To understand the paper, imagine a city (the universe) made of individual houses (particles).

1. Locality (The Neighborhood Rule): In a local universe, what happens in your house should only depend on what happens in your immediate neighborhood. If you want to check the rules of the city, you should be able to do it by looking at one house at a time and seeing how it connects to its neighbors.
2. Unitarity (The Global Accountant): This is the requirement that the total "energy" or "probability" of the system is conserved. It's like a global accountant who demands that every transaction balances perfectly.

The paper argues that when you have these strange "non-invertible" symmetries, there is a tension between these two views.

• The Local View (Topological): If you look at the symmetry as a "topological" object (like a rubber band stretched around the city), it acts locally. It respects the neighborhood rules. But it is "non-invertible"—you can't just reverse it.
• The Unitary View (The Accountant): If you force the symmetry to be "invertible" (so the accountant is happy and you can undo the trick), you break the "local" rule. The trick now has to reach across the whole city at once, mixing up distant houses in a way that violates the neighborhood rule.

The "Regular" Pattern

The authors discovered a fascinating pattern in how these symmetries behave when the city gets very large (the "thermodynamic limit").

If a symmetry is truly local (it respects the neighborhood rules), the distribution of states in the system follows a very specific, "regular" pattern. Imagine a choir. If the conductor (the symmetry) is local, the choir eventually sings every possible note with a perfectly balanced frequency. The authors call this a Regular Representation. It's like a perfectly mixed salad where every ingredient appears in the exact right proportion.

However, if you try to force a non-invertible symmetry to be "invertible" (to satisfy the Unitary accountant), this perfect balance breaks. The choir starts singing some notes way too often and others too rarely. The pattern becomes "irregular."

The "B-Function": A Lie Detector for Symmetries

To measure this irregularity, the authors invented a new tool called B(g). Think of this as a "Lie Detector Test" for symmetries.

• If B(g) = 0: The symmetry is behaving "locally." It's a topological, non-invertible symmetry. It respects the neighborhood rules, even though it can't be undone.
• If B(g) = 1: The symmetry is the "Identity" (doing nothing).
• If 0 < B(g) < 1: The symmetry is "irregular." It's a unitary symmetry that is trying to act locally but failing. It's a sign that the symmetry is actually a "non-invertible" one that has been forced into an invertible box.

By measuring this "B" value, the authors show that you can actually reverse-engineer the rules of the game. If you look at the shape of the "B" function, you can deduce the hidden "fusion algebra"—the secret rulebook that tells you how these symmetries combine. It's like looking at the ripples in a pond to figure out exactly what kind of stone was thrown in, even if you didn't see the stone.

Real-World Examples

The paper tests this idea on several "games" (theories):

• The Ising Model: A classic model of magnets. They show that the "non-invertible" symmetry here, when forced to be invertible, creates a specific irregular pattern that reveals the underlying rules of the magnet.
• Fibonacci Symmetry: A more exotic rule set. They show that even here, the "B" function reveals the hidden structure, allowing them to calculate the "quantum dimensions" (a measure of the size or weight) of the symmetry objects just by looking at the irregularity.

The Takeaway

In simple terms, this paper says: "If you see a symmetry that doesn't fit the perfect, balanced pattern of a local neighborhood, it's a sign that the symmetry is actually a 'non-invertible' one."

They provide a mathematical tool (the B-function) to detect this. It's a way to tell the difference between a symmetry that is naturally local and one that is a "non-invertible" symmetry pretending to be local. This helps physicists understand the deep structure of quantum field theories by looking at how symmetries behave when they are forced to be "undoable."

Note: The paper focuses entirely on these theoretical mathematical structures and their behavior in quantum field theories. It does not discuss medical applications, engineering uses, or future technologies. It is purely about understanding the fundamental rules of the universe's symmetries.

Technical Summary

Technical Summary: Global Symmetries: Locality, Unitarity, and Regularity

Problem Statement
The paper addresses the apparent tension between two fundamental principles in quantum field theory (QFT): locality and unitarity, specifically in the context of global symmetries. In standard quantum mechanics, Wigner's theorem dictates that symmetries are implemented by (anti)unitary operators forming a group. However, in higher-dimensional QFTs, the requirement of locality (operators acting on local regions) leads to the existence of topological operators that may not be invertible. These "non-invertible" or categorical symmetries form a fusion category rather than a group.

The central problem is understanding how these two perspectives—the "local approach" (topological, potentially non-invertible operators) and the "unitary approach" (invertible operators commuting with the Hamiltonian)—relate to one another. Specifically, the authors investigate how the imposition of locality constrains the decomposition of the Hilbert space into representations of the symmetry group, and how deviations from these constraints in the unitary picture serve as indicators of non-locality and non-invertible categorical structures.

Methodology
The authors employ a comparative analysis between two frameworks for describing symmetries:

1. The Local Approach: Symmetries are defined by topological operators (defects) acting on the Hilbert space. For local QFTs, the authors argue that the Hilbert space must decompose into a regular representation of the symmetry in the high-energy (thermodynamic) limit. They define an observable $C(\alpha)$ as the normalized trace of the symmetry operator in the limit of zero inverse temperature ($\beta \to 0$):
   $$C(\alpha) \equiv \lim_{\beta \to 0} \frac{\text{Tr}_H (U_\alpha e^{-\beta H})}{\text{Tr}_H (e^{-\beta H})}$$
   For local symmetries, $C(\alpha)$ vanishes for all non-identity elements, reflecting the regularity of the representation.

2. The Unitary Approach: The authors consider the set of all unitary operators commuting with the Hamiltonian. Since unitary operators must form a group $G$, non-invertible topological operators are expressed as linear combinations of these unitary generators. The authors introduce a new observable, $B(g)$, defined as the squared magnitude of the trace observable for a unitary operator $g$:
   $$B(g) = |C(g)|^2$$
   This function serves as a measure of "irregularity" or non-locality. If a unitary operator corresponds to a local, topological symmetry, $B(g)$ should vanish (for non-identity elements). If $B(g) \neq 0$, it indicates the operator is a linear combination involving non-local components.

The methodology involves deriving general properties of $B(g)$ based on the structure of the underlying fusion category and then explicitly computing $B(g)$ for various examples (including lattice models and continuum CFTs) to demonstrate how the function encodes the fusion algebra data (quantum dimensions and fusion coefficients).

Key Contributions and Results

• Regularity of Local Symmetries: The paper establishes that for local QFTs with global symmetries (whether group-like or categorical), the Hilbert space asymptotically forms a regular representation. In the limit of large system size or high energy, the trace observable $C(\alpha)$ vanishes for all non-identity symmetry elements. This holds for invertible group symmetries and non-invertible categorical symmetries (e.g., Tambara-Yamagami, Fibonacci categories).
• The Observable $B(g)$: The authors define $B(g)$ as a diagnostic tool. They prove that:
  • $0 \leq B(g) \leq 1$.
  • $B(g) = 1$ corresponds to the identity operator (or a phase).
  • $B(g) = 0$ corresponds to unitary operators that are purely topological (local) and invertible.
  • Critical points where $0 < B(g) < 1$ correspond to unitary operators constructed from "gauging projectors" of subcategories.
• Encoding Fusion Data: A primary result is that the function $B(g)$ encodes the data of the fusion algebra. By analyzing the critical points (minima, maxima, and saddles) of $B(g)$ on the group manifold of unitary operators, one can reconstruct the quantum dimensions of the simple objects in the fusion category and, in many cases, the fusion coefficients themselves.
  • Example: For the Fibonacci category, the minimum of $B(\theta)$ occurs at a specific angle, allowing the derivation of the quantum dimension $d(W) = (1+\sqrt{5})/2$ and the fusion rule $W^2 = e + W$.
  • Example: For the $Z_2 \times Z_2$ group with a 't Hooft anomaly, the absence of certain critical points (saddles) in $B(g)$ distinguishes it from the non-anomalous case, demonstrating that $B(g)$ captures anomaly information.
• Lattice vs. Continuum: The paper analyzes the transverse field Ising model on a lattice. It shows that on the lattice, space-time symmetries (translations) and global symmetries are entangled, preventing a clean separation of the group manifold. However, in the continuum limit, the function $B(g)$ approaches the values predicted by the local categorical symmetry, with the lattice corrections vanishing exponentially.

Significance and Claims
The paper claims that the tension between locality and unitarity is resolved by recognizing that while non-invertible symmetries are not unitary groups, they can be embedded within a larger group of unitary operators. The deviation of these unitary operators from the "regular" behavior (measured by $B(g)$) is a direct signal of the underlying non-local, categorical structure.

The significance of the work lies in providing a concrete, calculable observable ($B(g)$) that bridges the gap between the abstract mathematical structure of fusion categories and the physical properties of unitary operators in QFT. The authors suggest this approach offers a new perspective for:

1. Diagnosing Locality: Determining whether a symmetry in a UV description (like a lattice model) manifests as a local topological operator or a non-invertible symmetry in the IR.
2. S-Matrix Bootstrap: Potentially serving as a diagnostic for the locality of symmetries that commute with the S-matrix, even in the absence of a bulk spacetime description.
3. Reconstructing Symmetry Structures: Inferring the fusion rules and quantum dimensions of a symmetry category solely from the properties of the unitary operators available in the theory.

The authors remain modest regarding future implications, noting that the generalization to continuous symmetries (infinite-dimensional Lie groups) and the precise handling of anomalies and associators in the unitary approach require further investigation. They emphasize that their results rely on the assumption that the unitary operators can be expressed as linear combinations of the simple objects of the local symmetry category.