Global symmetries: locality, unitarity, and regularity

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Global symmetries: locality, unitarity, and regularity — explained simply

What is this paper about?

This paper looks at a puzzle about symmetries in quantum field theory (QFT):

• In ordinary quantum mechanics, symmetries are described by unitary operators (they preserve probabilities) and form a group.
• In QFT, we also care about locality (roughly, that things far apart don’t affect each other instantly). Some QFT symmetries act “locally” and behave like special, bendable “topological” operators in spacetime. These can be non-invertible (you can’t always undo them), which seems to clash with the “everything is unitary and invertible” picture.

The authors show that locality forces very specific patterns in how symmetries appear in the space of states, and they introduce a simple test to measure how “non-local” a unitary symmetry is. They then test this idea in several examples and show it even captures deep algebraic data about the symmetry.

The big questions

The paper asks, in simple terms:

• How does locality constrain the way symmetries act in QFT?
• Can we detect, just from unitary operators, whether a symmetry is genuinely local (topological) or secretly non-local?
• Can a simple “thermometer-like” observable tell us about the hidden structure (fusion rules) of non-invertible symmetries?

How do they study it? The main ideas and tools

First, some quick translations:

• Hilbert space: the set of all possible quantum states.
• Locality: physics in one place shouldn’t instantly depend on faraway places.
• Topological operator: an operator you can slide around in spacetime without changing answers (so it “acts locally” in the QFT sense).
• Non-invertible symmetry: a symmetry-like operation you can’t always undo.
• Fusion category: think of a “catalog of symmetry building blocks” and rules for how they combine, like LEGO bricks that can fuse into others.

The authors define a simple, physical “test” for a symmetry operator U:

• Heat the system up to extremely high temperature (so all states contribute).
• Insert the symmetry operator and compare to not inserting it.

Formally, they look at

• C(α) = a normalized trace of a symmetry operator at very high temperature, and
• B(g) = |C(g)|², which is a number between 0 and 1.

You can think of C and B as a “symmetry detector”:

• If a symmetry is truly local/topological, C(g) becomes 0 for every non-identity element g, and 1 only for the identity.
• If a symmetry is unitary but not locally acting, C(g) can be nonzero for non-identity g. That’s a sign of non-locality.

They study this detector in several ways:

• By cutting space into many, weakly interacting “cells” and using group characters (like a fingerprint of a representation).
• By using exact calculations in free fields (bosons and fermions).
• By using supersymmetric indices (which count special states in a controlled way).
• By using 2D conformal field theory (CFT) and modular transformations (a change of viewpoint on the torus) to show the same behavior holds for non-invertible, categorical symmetries.

What did they find?

1) Local symmetries force “regularity”

For any locally acting symmetry (including non-invertible, topological ones), the high-temperature detector behaves like this:

• C(g) = 0 for every non-identity g, and C(identity) = 1.
• This means the space of states “looks regular”: every charge type shows up in the expected, balanced proportions at high energies. This is called the regular representation.

They show this in:

• Lattice spin chains: as you add more sites, the pattern of spins approaches the regular pattern.
• Free fields: explicit formulas show C(g) → 0 unless g is the identity.
• Supersymmetric theories: a similar effect appears in the Schur index limit.
• 2D CFTs with non-invertible (categorical) symmetries: the same “only identity survives” result follows from modular methods.

Big message: locality implies regularity.

2) A unitary viewpoint reveals “irregularity” and measures non-locality

If you forget about locality and only ask for “unitary and commutes with the Hamiltonian,” you can build many symmetry-like operators. Some of these are non-local in QFT. The number

• B(g) = |C(g)|² measures how non-local a unitary symmetry is:
• B(g) = 1 only for the identity (or a phase times identity).
• B(g) = 0 for genuine, invertible topological symmetries (besides the identity).
• 0 < B(g) < 1 can happen for non-invertible structures and encodes useful information.

They prove general properties of B(g), and show that special “critical points” of B(g) are tied to “gauging projectors” (think: averaging over part of the symmetry), which produce simple Z2-like unitary operators with predictable B-values that depend on “quantum dimensions” (numbers measuring the size of objects in the fusion category).

3) Concrete examples

• Ising model (2D CFT): it has a non-invertible line D and an invertible line η. If you build certain unitary combinations from D, you get B ≠ 0 and a symmetry that acts non-locally on operators (for example, it turns the energy operator into itself with an extra η-line attached). This irregularity signals non-locality.
• Fibonacci category: generated by W with W² = e + W. The authors find
  • B(θ) = (3 + 2 cos(√5 θ)) / 5
  • The minimum is B = 1/5 at a special angle, and the associated unitary squares to the identity. From just this B-function, you can reconstruct the Fibonacci fusion rules and quantum dimension of W. That’s powerful: the unitary “detector” recovers the categorical data.

Why is this important?

• It bridges two viewpoints on symmetry:
  • The local/topological viewpoint (categorical symmetries), and
  • The unitary viewpoint (ordinary quantum-mechanical symmetries as groups).
• It gives a simple, physical test (the high-temperature trace) to see whether a symmetry is locally acting or not.
• It shows that the “shape” of B(g) can encode the fusion algebra and quantum dimensions, i.e., deep structural information about non-invertible symmetries.

What could this change or help with?

• Classifying phases of matter: Non-invertible symmetries appear in 2D systems and topological phases (anyons). This test could help identify them in models or simulations.
• Building and checking QFTs: When designing theories (in high-energy or condensed matter), this gives a diagnostic for whether a supposed symmetry is truly local/topological.
• Understanding dualities and gauging: The gauging-projector viewpoint connects B(g) to operations like “gauging a subgroup,” helping map out families of related theories.
• Quantum information: Non-invertible symmetries and fusion rules matter for topological quantum computing. A simple way to read off fusion data from dynamics is useful.

In one sentence

The paper shows that locality forces a “regular” pattern of symmetry representations in QFT, and introduces a simple, high-temperature observable whose behavior not only detects when a unitary symmetry is non-local, but also encodes the hidden categorical (fusion) structure of non-invertible symmetries.

Knowledge gaps, limitations, and open questions

Below is a single, consolidated list of what the paper leaves missing, uncertain, or unexplored, phrased to be actionable for future research:

• Rigorous existence and regulator-independence of the observable C(α) in general QFTs: specify conditions (e.g., spectrum growth, UV behavior, boundary conditions) under which the β→0 limit exists and is independent of the regulator used in defining the thermal trace.
• Precise locality and factorization assumptions: replace heuristic cell decompositions with a formal framework (e.g., algebraic QFT split property) and prove regularity under minimal and necessary locality assumptions (including in theories with gauge constraints and fermionic spin structures).
• Gapless and non-finite-correlation-length theories: extend the regularity results beyond gapped systems with finite correlation length to interacting, gapless QFTs in d>2, and clarify whether C(α)=0 for g≠e holds in generic CFTs (outside 2d RCFTs).
• Gauge theories beyond supersymmetric indices: develop non-supersymmetric methods to compute C(α) (or alternatives) in gauge theories, systematically accounting for boundary charges, Gauss constraints, and contributions of non-local operators (e.g., Wilson/’t Hooft lines).
• Order-of-limits subtleties in index-based arguments: resolve the q→1 versus infinite sum over representations λ in class S computations; present a rigorous treatment of the limit and its dependence on flavor fugacities, including “second sheet” issues.
• General proof of regularity for categorical symmetries: provide a theorem establishing eq. (regularity condition) C(gα)=δgα,e for arbitrary fusion categories (beyond diagonal 2d RCFTs), including non-modular, non-unitary, and higher-dimensional settings.
• Modular transformation argument constraints: justify and generalize the claim Δg>Δe used in the modular transformation proof (in 2d) and characterize exceptions (e.g., defects that do not raise the ground-state energy, degeneracies, or protected sectors).
• Extension to higher-form and higher-categorical symmetries: define analogues of C(α) and B(g) for p-form symmetries and higher-category defects in d≥3, and test whether regularity and irregularity measures persist.
• Non-diagonal modular invariants and extended chiral algebras: analyze whether regularity holds for 2d CFTs with non-diagonal modular invariants or extended algebras where Verlinde lines need not act diagonally.
• Treatment of anomalies and projective phases: systematically classify how ’t Hooft anomalies (including those of non-invertible symmetries) modify the properties of C(α) and B(g), and derive corrected bounds and selection rules.
• Reconstruction of fusion data from B(g): formalize an algorithm to extract quantum dimensions and fusion coefficients from the critical points of B(g), assess uniqueness/ambiguity, and test on categories beyond the Fibonacci and Ising examples.
• Proofs of B(g) properties: supply the missing proofs for the stated properties of B(g) (boundedness, dimensional relation to number of simples, correspondence of minima to locally acting operators, value at gauging projectors) and delineate all required assumptions.
• Dimensionality claim (group manifold vs. number of simples): rigorously justify or refine the statement that an underlying categorical symmetry with n simple objects induces an (n−1)-dimensional Lie group of unitary operators; identify counterexamples or necessary qualifiers.
• Systematic construction/classification of unitary completions: develop a general method to construct all unitary operators from a given categorical symmetry (including non-invertible lines) and classify their group structure, redundancies, and physical equivalence classes.
• Identifiability and robustness of B(g): study whether distinct non-local unitary operators can share the same B(g), determine the sensitivity of B(g) to UV regulator choices and deformations preserving the categorical symmetry, and propose complementary observables.
• Quantitative rates to regularity: derive finite-size/finite-N bounds controlling how fast C(α)→0 for g≠e in lattice and continuum models, and connect these rates to correlation length, spectral gaps, and operator algebra features.
• Spontaneous symmetry breaking: analyze how C(α) and B(g) behave when the symmetry is spontaneously broken (non-singlet vacua, degenerate ground states), including finite temperature and finite volume effects.
• Operator-level non-locality diagnostics: augment B(g) with direct, quantitative criteria for non-local action (e.g., how unitary symmetries attach lines to local operators), and relate these to selection rules and correlator constraints in dynamical setups.
• Beyond finite simple-object categories: relax the assumption of a finite number of simples (e.g., non-compact CFTs, logarithmic CFTs) and investigate whether C(α) and B(g) remain meaningful or require redefinition.
• Real-world computability and measurement: propose computational schemes (e.g., Monte Carlo, tensor networks) or experimental protocols (in condensed matter or quantum simulators) to estimate C(α) and B(g), including error analyses and finite-size scaling.
• Tube algebra/Drinfeld center connection: make explicit the connection between B(g) and categorical structures (tube algebra, Drinfeld double), and use these tools to derive regularity and irregularity systematically.
• RG flow and deformation dependence: study how C(α) and B(g) evolve under renormalization, whether irregularity is monotone along flows, and how emergent IR categorical symmetries influence these observables.
• Boundary, defect, and topology dependence: examine how spatial topology, boundaries, and additional defects modify C(α) and B(g), and whether regularity statements depend sensitively on these geometric choices.
• Completeness of examples: broaden tests beyond Ising, Fibonacci, free fields, and class S (e.g., non-supersymmetric interacting theories in 3+1d) and provide explicit computations that confirm or falsify the proposed framework.
• Completion of arguments and appendices: include the missing sections and appendices referenced (e.g., proofs in Section “simple proofs,” details in Appendix on η+ defect), and ensure the derivations supporting key claims are accessible and reproducible.

Practical Applications of “Global symmetries: locality, unitarity, and regularity”

This paper introduces an operational bridge between local (topological) and unitary perspectives on symmetries in quantum field theory, centered on two computable observables:

• C(α): a high-temperature, symmetry-twisted thermal trace that detects “regularity” of symmetry action on the Hilbert space.
• B(g) = |C(g)|²: a bounded [0,1] measure of non-locality/irregularity for unitary symmetry actions, whose critical points encode fusion data (quantum dimensions, gauging projectors, and—in favorable cases—fusion rules).

These tools yield actionable diagnostics for non-invertible/categorical symmetries and their compatibility with locality, providing concrete workflows for numerics, experiments, and theory.

Below are applications grouped into immediate and long-term categories. Each bullet lists the use case, sector(s), potential tools/workflows, and key assumptions/dependencies.

Immediate Applications

The following applications can be prototyped now using existing numerical, analytical, and experimental toolchains.

• Detect non-invertible/categorical symmetries in lattice and continuum models
  • Sectors: condensed matter, quantum materials, statistical mechanics, high-energy theory
  • Workflow: compute C(α) ≡ lim_{β→0} Tr[U(α)e^{{-βH}]/Tr[e{-βH}]} via high-T extrapolation of symmetry-twisted partition functions; evaluate B(g)=|C(g)|² for unitary combinations U(g) constructed from symmetry generators (including non-invertible/topological defects).
  • Tools: exact diagonalization (ED), Monte Carlo with symmetry twists, transfer-matrix methods, tensor networks (MPO/PEPS with defect insertions), CFT partition-function codes using modular S-matrices.
  • Assumptions/dependencies: finite correlation length at the scales probed; access to symmetry-twisted boundary conditions/defect insertions; control of anomalies (projective phases modify expectations); sufficiently large system sizes for high-T asymptotics.
• Phase and anomaly diagnostics via regularity tests
  • Sectors: condensed matter (SPT/SET/topological order), CFT/RCFT, lattice gauge theory
  • Use case: verify locality-compatible global symmetries by checking that C(α)→0 for non-identity and that spectra approach regular representations. Flag symmetry actions that violate regularity as non-local or anomalous.
  • Tools: character expansions, Schur/superconformal index computations for SUSY theories, modular transformations of torus partition functions, automated detection of regularity in numerics.
  • Assumptions: diagonal RCFT or known modular data for closed-form tests; for SUSY indices, the q→1 limit and correct regulator choice; no spontaneous symmetry breaking in the sector tested.
• Reconstruct categorical data from B(g) critical points
  • Sectors: mathematical physics, condensed matter theory, CFT/RCFT
  • Use case: infer quantum dimensions and constrain fusion rules by locating critical points of B(g) and matching to gauging projectors P_{C′} and associated Z₂ generators η_{C′}; for 1D group manifolds (e.g., Ising/Fibonacci), fit B(θ) to recover d’s and candidate fusions.
  • Tools: symbolic regression/curve fitting of B(g); libraries for fusion categories and modular data; consistency checks (e.g., Verlinde).
  • Assumptions: finite number of simples; availability of B(g) along continuous deformations; anomaly-free or anomaly-accounted actions.
• Symmetry-aware upgrades to tensor-network and Monte Carlo pipelines
  • Sectors: computational physics/software
  • Use case: incorporate regularity/irregularity constraints as diagnostics or priors during optimization; MPO algebra plugins to insert non-invertible defects and measure B(g); automated alerts when extracted spectra violate locality-compatibility.
  • Tools/products: “B-Scanner” module for TeNPy/ITensor; MPO defect insertion templates; twisted-trace high-T extrapolation utilities.
  • Assumptions: efficient representation of defects/symmetry operators; stable extrapolations in finite-size/finite-β settings.
• Benchmarks and validation for dualities and gauging
  • Sectors: high-energy theory, CFT/RCFT, lattice field theory
  • Use case: distinguish duality defects (non-invertible) from unitary invertible symmetries; validate gauging procedures by verifying predicted B-values at critical points (B = (1 − 2/∑ d_a²)² for subcategory gauging).
  • Tools: RCFT toolkits (Verlinde, S-matrix), lattice duality implementations, SUSY index computations.
  • Assumptions: accurate modular/character data; correct identification of subcategories; control of global structures (e.g., centers, spin structures).
• Curriculum and research training on non-invertible symmetries
  • Sectors: academia/education
  • Use case: hands-on modules using Ising and Fibonacci examples to teach locality vs unitarity, projectors, gauging, and how B(g) decodes category data.
  • Tools: Jupyter notebooks with ED/tensor-network demos; symbolic algebra for small categories.
  • Assumptions: small-scale computational resources; curated model libraries.

Long-Term Applications

These opportunities require further research, scaling, or new experimental/theoretical developments.

• Experimental probes of categorical symmetries via twisted thermodynamics and dynamics
  • Sectors: quantum simulation (cold atoms, Rydberg arrays, superconducting qubits), materials
  • Use case: design protocols to implement symmetry/defect insertions and measure high-T twisted traces or dynamical correlators whose selection rules reflect non-invertible symmetries; use B(g) as an experimental diagnostic for topological defects and gauging-induced criticalities.
  • Tools: programmable defects via Floquet engineering, ancilla-mediated symmetry twists, quench protocols mapping twisted traces to interferometric signals.
  • Dependencies: reliable implementation of non-invertible defects; precise control over boundary conditions; readout fidelity; mitigation of finite-temperature and finite-size errors.
• Automated discovery of fusion categories from many-body data
  • Sectors: mathematical/computational physics, ML for science
  • Use case: given numerical spectra and twisted partition-function data, fit B(g) manifolds, extract quantum dimensions, and infer consistent fusion algebras/categories; rank candidate categories for a model.
  • Tools/products: “CatSym-Discover” ML pipeline combining symbolic regression and category-theory constraints (positivity/integrality/Verlinde).
  • Dependencies: curated training sets; robust uncertainty quantification; handling anomalies and projective actions.
• Symmetry-enriched quantum error-correcting codes and gates from defects
  • Sectors: quantum information/quantum computing
  • Use case: map non-invertible/categorical symmetries to logical operations and syndromes; use B(g) to certify locality/non-locality of logical gates; design protected, defect-based gate sets (e.g., Fibonacci-like constructions) and gauge-induced Z₂ generators for fault-tolerant operations.
  • Tools: code-capacity simulators with defect insertion; compilation frameworks that enforce regularity constraints; protocols for gauging-induced logical operations.
  • Dependencies: physical platforms supporting topological/anyon-like excitations; coherence times sufficient for defect manipulation; integration with syndrome extraction.
• Bootstrap and classification constraints using regularity
  • Sectors: conformal bootstrap, S-matrix bootstrap, HEP theory
  • Use case: encode regularity (C(α)→δ_{α,e}) and B(g)-based constraints into bootstrap functionals to shrink allowed theory spaces; classify phases by categorical symmetry fingerprints inferred from B(g).
  • Tools: bootstrap solvers with symmetry modules; constraint libraries for categorical data.
  • Dependencies: rigorous bounds connecting high-T twisted traces to bootstrap observables; handling of anomalies and large-volume limits.
• Standardized benchmarks and certification for topological phases
  • Sectors: policy/standards, quantum technology
  • Use case: develop community standards for reporting B(g)-based diagnostics in numerical and experimental studies of topological phases; certify presence/absence of non-invertible symmetries and gauging signatures.
  • Tools: benchmark suites, reference datasets (Ising, Fibonacci, Tambara–Yamagami, etc.).
  • Dependencies: consensus on protocols; open-source reference implementations; inter-platform reproducibility.
• Materials-by-design workflows for symmetry-enriched phases
  • Sectors: quantum materials/chemistry
  • Use case: use irregularity targets (B(g) profiles) as design objectives when searching for lattice Hamiltonians realizing desired categorical symmetries; inverse-design of couplings that yield specified gauging critical points.
  • Tools: optimization over Hamiltonian parameter spaces with B(g) as objective; ab initio-informed effective models.
  • Dependencies: computational scale; mapping from microscopic chemistry to effective low-energy models with computable twisted traces.
• Educational outreach and workforce development in categorical symmetries
  • Sectors: academia/education/policy
  • Use case: build interdisciplinary programs (physics, math, CS) centered around practical diagnostics (C, B) to train a workforce for quantum technologies where non-invertible symmetries/topological defects matter.
  • Tools: modular courseware, collaborative problem banks, open-source codebases.
  • Dependencies: sustained funding; cross-disciplinary coordination.

Cross-cutting assumptions and dependencies

• Locality and factorization: key regularity results rely on locality (topological/defect operators act locally) and factorization up to known caveats (gauge constraints, spin structures).
• High-temperature/large-volume limits: C(α) is defined via β→0; finite-size scaling and extrapolation must be controlled. For 2D CFTs, modular transformations relate β→0 to β→∞ with defects.
• Finite simple-object sets: many reconstructions assume a finite number of simples (finite fusion categories).
• Anomalies/projective actions: modify expectations for C(α), B(g); must be identified and incorporated.
• No spontaneous symmetry breaking (for the symmetry under test): the vacuum should be a singlet to apply the paper’s unitarity-based group construction cleanly.
• Model accessibility: ability to implement symmetry/defect insertions numerically or experimentally.

In sum, the paper’s observables C(α) and B(g) provide a practical, quantifiable interface between abstract categorical symmetry structures and concrete computations/measurements. They enable immediate diagnostics in numerical/theoretical studies and chart clear paths toward experimental certification, automated category inference, and symmetry-aware design of quantum phases and devices.

• 0-form symmetry: A symmetry that acts on local (pointlike) operators rather than extended objects, typically represented by codimension-one operators in spacetime. "Let us consider operators ${\cal O}_{\cal M}$ which are labeled by co-dimension one surfaces ${\cal M}$ in space time: the zero-form symmetries"
• ADE algebra: A classification of simply-laced Lie algebras (types A, D, E) often used to label families of theories. "Theories in class ${\cal S}$ with regular punctures are labeled by an $ADE$ algebra $H$"
• anomaly: An obstruction to realizing a symmetry in a strictly local or non-projective way, often seen as a phase ambiguity or inconsistency. "If projective phases are present then some of these statements will be modified manifesting the {\it anomaly}"
• Cartan generator: A generator of the maximal abelian (Cartan) subalgebra of a Lie algebra, used to define charges and fugacities. "Here $Q$ is the Cartan generator of an $SU(2)$ global symmetry."
• categorical symmetries: Symmetries described by higher-categorical structures (e.g., fusion categories), allowing non-invertible symmetry operators. "This emerges in the context of categorical symmetries where symmetry operators are generically non-invertible."
• co-dimension: The difference between the dimension of ambient space and the dimension of a submanifold on which an operator is supported. "operators ${\cal O}_{\cal M}$ in this sub-class can be associated to co-dimension $n$ surfaces, ${\cal M}$, in space-time."
• class ${\cal S}$: A class of 4d N=2 theories constructed from 6d (2,0) theories compactified on Riemann surfaces with punctures. "Concretely, we consider theories of class ${\cal S}$ with regular punctures"
• diagonal RCFT: A rational conformal field theory whose torus partition function pairs identical holomorphic and antiholomorphic representations. "We can see this for the simple case of a diagonal RCFT in two dimensions."
• Drinfeld double: A construction in category theory and quantum algebra producing a braided (modular) category from a fusion category, relevant for excitations and defects. "the tube algebra and the Drinfeld double of a category."
• duality defect: A topological defect implementing a duality transformation, often non-invertible as an operator on the Hilbert space. "Example of the former is the duality defect of the $2d$ Ising model"
• Eisenstein series: Special modular forms appearing in expansions and exact expressions for indices and partition functions. "expressions in terms of Eisenstein series for the flavored Schur index"
• Fibonacci category: A fusion category with two simple objects e and W obeying W² = e + W; appears in non-invertible symmetry contexts. "The Fibonacci category is generated by a hermitian operator $W$ such that, $W^2=e+W$."
• fusion algebra: The algebraic structure encoding how symmetry lines/defects fuse, given by nonnegative integer coefficients. "demonstrate that this observable can encode data associated to the fusion algebra of symmetries."
• fusion category: A semisimple rigid tensor category with finitely many simple objects and fusion rules; models non-invertible symmetries. "The structure of a fusion category has many important physical implication."
• Gauss' law: A constraint in gauge theories that ties charges to flux through boundaries, affecting Hilbert space factorization. "one needs to impose Gauss' law on the boundary"
• gauging: The procedure of promoting a global symmetry to a gauge symmetry, often producing categorical symmetries in the resulting theory. "gauging symmetry groups leads in general to theories with categorical symmetries"
• Haar measure: The unique invariant measure on a compact group used for averaging/integration over group elements. "for Lie groups we replace the sum with the Haar measure."
• Heisenberg chain: A spin-chain model with SU(2) symmetry, used as a lattice example of representation growth and regularity. "An example of such a Hamiltonian is the Heisenberg chain."
• Lie group: A group with a differentiable manifold structure, allowing continuous parameters for unitary symmetries. "Thus, the set $G$ will be in general a Lie group."
• maximal torus: The largest abelian subgroup of a compact Lie group, generated by Cartan elements; used in character expansions. "We conclude again that the maximal torus of $G$ has regular representations."
• modular fusion categories: Braided fusion categories with non-degenerate braiding; arise in rational CFT and topological phases. "This is the direct analogue of expression \eqref{eq:Cwithcharacters} for modular fusion categories."
• modular invariance: Invariance of 2d CFT partition functions under modular transformations of the torus. "It is useful to use the modular invariance of the partition function"
• modular transformation: A transformation of the torus parameter τ by SL(2,ℤ), exchanging temporal and spatial cycles. "exchanging the roles of the temporal and the spatial circles using a modular transformation."
• moment map operators: Operators in supersymmetric theories that sit with conserved currents in multiplets and source flavor symmetries. "This pre-factor comes from the moment map operators of the theory"
• N-ality: A congruence class of representations under the center, used to group charges; often controls selection rules. "same N-ality"
• non-invertible symmetry: A symmetry whose topological operators do not have inverses under fusion, unlike group-like symmetries. "Such symmetries are often called {\it non-invertible}."
• projective phases: Phases indicating projective (rather than linear) representations, related to anomalies. "If projective phases are present then some of these statements will be modified manifesting the {\it anomaly}"
• quantum dimension: A positive number assigned to a simple object in a fusion category, generalizing representation dimension. "The quantity $d_g$ is called the {\it quantum dimension} of element $g$."
• radial quantization: A quantization scheme in CFT mapping states to local operators via cylinders/planes, used in indices. "the trace in the Schur index is computed in the radial quantization"
• regular punctures: Punctures on the Riemann surface defining class S theories that lead to flavor symmetries. "theories of class ${\cal S}$ with regular punctures"
• regular representation: The representation containing each irrep with multiplicity equal to its dimension; character vanishes away from identity. "it is the character of the regular representation of $G$."
• Riemann surface: A one-complex-dimensional manifold used as the compactification space in class S constructions. "and a choice of a Riemann surface"
• Schur index: A protected supersymmetric index counting BPS operators in 4d N=2 SCFTs with specific fugacity assignments. "This is the ratio between the Schur index"
• splitability property: A locality-related property ensuring actions of symmetries factorize across regions. "This perspective on locality is related to the splitability property discussed in \cite{BuchholzDoplicherLongo1986}."
• Tambara–Yamagami fusion rules: Fusion rules of categories with a non-invertible object D and an abelian group (here ℤ2), e.g., D² = e + η. "satisfy the Tambara-Yamagami ${\mathbb Z}_2$ fusion rules"
• theta-function: A special function with modular properties; indices can be expressed as theta-functions. "the index is a theta-function"
• topological defects: Extended operators modifying the theory along submanifolds, topological under deformations. "we can also consider topological defects by Wick rotating the lines"
• topological operators: Operators whose correlation functions are invariant under smooth deformations of their support. "characterized by the properties of topological operators in quantum systems"
• Topological Field Theory (TQFT): A field theory where observables depend only on topology, governing fusion and deformation rules of topological operators. "The action of the operators is purely dictated by Topological Field Theory (TQFT) rules"
• torus partition function: The partition function of a 2d CFT on a torus, used to probe modular properties and defects. "the torus partition function"
• tube algebra: An algebra associated with a fusion category capturing defect/line operator composition in 2d CFTs. "constructions involving notions of the tube algebra and the Drinfeld double of a category."
• Verlinde lines: Topological lines in 2d CFT associated to primary fields, whose fusion follows the Verlinde algebra. "We use the fact that the Verlinde lines act diagonally"
• Wick rotating: Analytic continuation between Euclidean and Lorentzian signatures, here exchanging roles of space and time for lines/defects. "by Wick rotating the lines to be localized in space"
• Wilson lines: Line operators in gauge theory obtained by exponentiating the gauge field along a path, often ending on charged operators. "gauge theories contain natural non-local operators: {\it e.g.} Wilson lines ending on charged local operators."
• Wigner's theorem: The statement that symmetries in quantum mechanics are represented by unitary or antiunitary operators preserving transition probabilities. "By Wigner's theorem this is a set of natural operators in QM"
• Witten index: A supersymmetric index Tr(−1)^F counting net ground states, insensitive to continuous deformations. "the usual Witten index \cite{Witten:1982df} $Tr(-1)^F$"
• WZW model: A two-dimensional conformal field theory with affine Lie algebra symmetry, e.g., at level k. "the $({\frak g}_2)_1$ WZW model."

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