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Theories and models

Browse theories and models by their physical question, regime, dependencies, neighboring treatments, and canonical volume owner.

This index is a concise projection, not an independent explanation. Each record keeps the scope needed for use and links to its source page.

Coverage notes
  • A page presented here as a model is a route to its full treatment, not a machine-written definition of the theory or proof of its claims.
  • The index repeats only the scope, preparation, and related context supplied by that treatment.

Coverage note: records included through 2026-08-12

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  • Theory Model

    The Klein–Gordon Field and Its Modes

    How do the Klein–Gordon equation, symplectic normalization, and positive-frequency choices organize free scalar solutions?

    Principal question
    How do the Klein–Gordon equation, symplectic normalization, and positive-frequency choices organize free scalar solutions?
    What the source page covers
    Real scalar action, dispersion relation, positive- and negative-frequency modes, and conserved inner product
    Boundary
    Quantization; Curved-spacetime mode ambiguity
    Scope
    Real scalar action, dispersion relation, positive- and negative-frequency modes, and conserved inner product
    Assumptions
    The Action Principle and Field Equations
  • Theory Model

    What an Interacting Lagrangian Does and Does Not Specify

    Which classical stability, symmetry, scale, regulator, state, and observable data are still needed after an interacting local Lagrangian is written down?

    Principal question
    Which classical stability, symmetry, scale, regulator, state, and observable data are still needed after an interacting local Lagrangian is written down?
    What the source page covers
    The foundational distinction among classical interaction data, formal perturbation, regulator choice, renormalized observables, state specification, and a defined continuum theory.
    Boundary
    Perturbative calculations belong to Scattering; renormalization and continuum EFT control to Renormalization; nonperturbative construction to Lattice and Mathematical QFT.
    Scope
    The foundational distinction among classical interaction data, formal perturbation, regulator choice, renormalized observables, state specification, and a defined continuum theory.
    Assumptions
    The Action Principle and Field Equations
  • Theory Model

    The Dirac Field

    How do Lorentz covariance, the Clifford relation, a first-order action, and positive-energy dynamics define the free Dirac field?

    Principal question
    How do Lorentz covariance, the Clifford relation, a first-order action, and positive-energy dynamics define the free Dirac field?
    What the source page covers
    The physical Dirac action, equation, conserved inner product, and free-field interpretation.
    Boundary
    Clifford algebra and spin representation machinery belong to Mathematical Methods; gauge couplings and chiral anomalies belong to later volumes.
    Canonical treatment
    The Dirac Field
    Scope
    The physical Dirac action, equation, conserved inner product, and free-field interpretation.
    Assumptions
    The Action Principle and Field Equations; One-Particle States: Mass, Spin, and Relativistic Normalization
  • Theory Model

    The Proca Field

    How does the massive vector action propagate exactly three physical spin-one polarizations in four dimensions?

    Principal question
    How does the massive vector action propagate exactly three physical spin-one polarizations in four dimensions?
    What the source page covers
    Proca action, equations, subsidiary condition, mode content, Hamiltonian positivity, and massless-limit caution
    Boundary
    Higgs mechanism; Interacting massive vectors
    Canonical treatment
    The Proca Field
    Scope
    Proca action, equations, subsidiary condition, mode content, Hamiltonian positivity, and massless-limit caution
    Assumptions
    The Action Principle and Field Equations; One-Particle States: Mass, Spin, and Relativistic Normalization
  • Theory Model

    The Free Maxwell Field and Gauge Redundancy

    How do Maxwell dynamics, gauge redundancy, Gauss law, and observable field strengths coexist in the free theory?

    Principal question
    How do Maxwell dynamics, gauge redundancy, Gauss law, and observable field strengths coexist in the free theory?
    What the source page covers
    The free Maxwell model, its equations, redundancy, Gauss law, and field-strength observables.
    Boundary
    The primary definition of gauge structure, global form, gauge fixing, BRST, BV, and generalized symmetries belongs to Symmetry.
    Scope
    The free Maxwell model, its equations, redundancy, Gauss law, and field-strength observables.
    Assumptions
    The Action Principle and Field Equations
  • Theory Model

    Quantizing the Real Scalar Field

    How do mode normalization and canonical commutators construct the free scalar field, Hamiltonian, vacuum, and particle excitations?

    Principal question
    How do mode normalization and canonical commutators construct the free scalar field, Hamiltonian, vacuum, and particle excitations?
    What the source page covers
    Mode expansion, equal-time commutators, creation and annihilation operators, Hamiltonian, and vacuum
    Boundary
    Interacting perturbation theory; Rigorous representation theory
    Scope
    Mode expansion, equal-time commutators, creation and annihilation operators, Hamiltonian, and vacuum
    Assumptions
    The Klein–Gordon Field and Its Modes; Canonical Quantization: Algebra, Representation, and State
  • Theory Model

    Canonical Quantization of the Free Dirac Field

    Why do canonical anticommutation relations give a positive-energy fermionic Fock representation with particle and antiparticle excitations?

    Principal question
    Why do canonical anticommutation relations give a positive-energy fermionic Fock representation with particle and antiparticle excitations?
    What the source page covers
    Mode expansion, canonical anticommutators, particle and antiparticle operators, Hamiltonian, charge, and vacuum
    Boundary
    A proof of spin–statistics; Interacting fermion loops
    Scope
    Mode expansion, canonical anticommutators, particle and antiparticle operators, Hamiltonian, charge, and vacuum
    Assumptions
    Plane Waves, Spin Sums, and Bilinears; Canonical Quantization: Algebra, Representation, and State
  • Theory Model

    Physical-Mode Quantization of the Free Electromagnetic Field

    How does reduced or transverse quantization isolate the two physical photon polarizations and their Fock space?

    Principal question
    How does reduced or transverse quantization isolate the two physical photon polarizations and their Fock space?
    What the source page covers
    Radiation or Coulomb gauge, transverse commutators, photon Fock space, Hamiltonian, and residual conditions
    Boundary
    Charged matter; Non-Abelian fields; BRST quantization
    Scope
    Radiation or Coulomb gauge, transverse commutators, photon Fock space, Hamiltonian, and residual conditions
    Assumptions
    Maxwell Constraints as a Worked Application; Massive and Massless Spin-One Polarizations; Canonical Quantization: Algebra, Representation, and State
  • Theory Model

    Complex Scalars and Conserved Charge

    How does a complex scalar organize independent particle and antiparticle sectors and realize a conserved U(1) charge?

    Principal question
    How does a complex scalar organize independent particle and antiparticle sectors and realize a conserved U(1) charge?
    What the source page covers
    Complex action, global phase symmetry, charge operator, particle and antiparticle modes, and normalization
    Boundary
    Gauging the symmetry; Spontaneous symmetry breaking
    Scope
    Complex action, global phase symmetry, charge operator, particle and antiparticle modes, and normalization
    Assumptions
    Quantizing the Real Scalar Field; Classical Symmetries, Currents, and Stress Tensors
  • Theory Model

    Electric and Magnetic One-Form Symmetries

    How do Maxwell and Yang-Mills examples realize electric and magnetic one-form symmetries, and how do matter, monopoles, and global form break or modify them?

    Principal question
    How do Maxwell and Yang-Mills examples realize electric and magnetic one-form symmetries, and how do matter, monopoles, and global form break or modify them?
    What the source page covers
    Electric and magnetic one-form currents and topological operators, charged Wilson and disorder lines, screening, monopole breaking, global-form dependence, and Abelian versus non-Abelian qualifications.
    Boundary
    Confinement and monopole dynamics belong to Nonperturbative Dynamics; electromagnetic duality belongs to Supersymmetry and Duality.
    Scope
    Electric and magnetic one-form currents and topological operators, charged Wilson and disorder lines, screening, monopole breaking, global-form dependence, and Abelian versus non-Abelian qualifications.
    Assumptions
    Higher-Form Currents, Charges, Backgrounds, and Ward Identities; The Free Maxwell Field and Gauge Redundancy
  • Theory Model

    Abelian Chern–Simons Theory

    How do an integral level or K-matrix determine Wilson-line labels, fusion, braiding, spin, ground-state degeneracy, and global qualifications?

    Principal question
    How do an integral level or K-matrix determine Wilson-line labels, fusion, braiding, spin, ground-state degeneracy, and global qualifications?
    What the source page covers
    Bounded U(1) and integral K-matrix data, line equivalence, fusion, mutual braiding, topological spin, state-space dimension, spin dependence, and nondegeneracy caveats.
    Boundary
    Non-Abelian Chern-Simons and modular-category theorems belong to Mathematical QFT; quantum Hall applications belong to Condensed Matter.
    Scope
    Bounded U(1) and integral K-matrix data, line equivalence, fusion, mutual braiding, topological spin, state-space dimension, spin dependence, and nondegeneracy caveats.
    Assumptions
    Chern–Simons Actions and Level Quantization; Linking, Braiding, and Framing
  • Theory Model

    BF Theory as a Topological Gauge Theory

    What exact operator, linking, state-space, and boundary data follow when a quantized BF coupling is treated as a dynamical topological gauge theory?

    Principal question
    What exact operator, linking, state-space, and boundary data follow when a quantized BF coupling is treated as a dynamical topological gauge theory?
    What the source page covers
    Compact BF gauge fields, gauge redundancies, line and surface operators, linking phases, finite state spaces in bounded examples, and boundary-condition dependence.
    Boundary
    Higher gauge-theory formulation belongs to Mathematical QFT; lattice realizations and phase dynamics belong to Lattice and Condensed Matter.
    Scope
    Compact BF gauge fields, gauge redundancies, line and surface operators, linking phases, finite state spaces in bounded examples, and boundary-condition dependence.
    Assumptions
    BF Couplings and Discrete Topological Data; State Spaces, Cobordisms, and Gluing
  • Theory Model

    Finite Gauge Theory and Dijkgraaf–Witten Twists

    How does summing over finite bundles produce a TQFT, and how do cocycle twists modify amplitudes, line data, and gauging?

    Principal question
    How does summing over finite bundles produce a TQFT, and how do cocycle twists modify amplitudes, line data, and gauging?
    What the source page covers
    Finite-bundle sums, gauge normalization, untwisted and low-dimensional cocycle-twisted examples, state spaces, Wilson lines, and relation to gauging finite symmetry.
    Boundary
    Group-cohomology classification and extended categorical data belong to Mathematical QFT; lattice implementations belong to Lattice.
    Scope
    Finite-bundle sums, gauge normalization, untwisted and low-dimensional cocycle-twisted examples, state spaces, Wilson lines, and relation to gauging finite symmetry.
    Assumptions
    Gauging Continuous and Finite Symmetries; State Spaces, Cobordisms, and Gluing
  • Theory Model

    Symmetry TFT: Encoding Symmetry, Anomaly, and Gauging

    In which controlled settings can a topological theory in one higher dimension encode symmetry defects, anomaly, boundary QFT data, and alternative gauging choices?

    Principal question
    In which controlled settings can a topological theory in one higher dimension encode symmetry defects, anomaly, boundary QFT data, and alternative gauging choices?
    What the source page covers
    A bounded SymTFT interface: bulk topological operators, boundary conditions, relative QFT, anomaly, generalized or non-invertible symmetry data, and gauging as a change of boundary condition.
    Boundary
    Universal constructions, fully extended formulations, and categorical classification belong to Mathematical QFT; holographic applications belong to Holography.
    Scope
    A bounded SymTFT interface: bulk topological operators, boundary conditions, relative QFT, anomaly, generalized or non-invertible symmetry data, and gauging as a change of boundary condition.
    Assumptions
    Operators, Boundaries, and Relative Topological Theories; Anomaly Polynomials and Inflow; Higher-Group Operators, Gauging, and Anomalies
  • Theory Model

    Chiral Effective Theory and Nonlinear Symmetry

    How do Goldstone fields, nonlinear symmetry realization, explicit breaking and derivative counting define chiral EFT before nuclear applications add new scales?

    Principal question
    How do Goldstone fields, nonlinear symmetry realization, explicit breaking and derivative counting define chiral EFT before nuclear applications add new scales?
    What the source page covers
    Goldstone degrees of freedom, nonlinear symmetry, explicit breaking and chiral derivative counting as an EFT architecture.
    Boundary
    Few-body promotion, nuclear forces, regulator questions and many-body applications are handed to the Nuclear and Few-Body architecture page and downstream subject volumes.
    Scope
    Goldstone degrees of freedom, nonlinear symmetry, explicit breaking and chiral derivative counting as an EFT architecture.
    Assumptions
    A Map of Effective-Theory Architectures; Cosets and Nonlinear Realizations
  • Theory Model

    Nuclear and Few-Body EFT Architecture

    How do shallow scales, few-body sectors, force hierarchies and regulator independence organize nuclear and few-body EFT architectures?

    Principal question
    How do shallow scales, few-body sectors, force hierarchies and regulator independence organize nuclear and few-body EFT architectures?
    What the source page covers
    Shallow scales, few-body sectors, promoted interactions and forces, and architecture-level regulator and uncertainty contracts for nuclear EFT.
    Boundary
    QCD and nuclear phenomenology belongs to Volume 6, strong dynamics to Volume 7, lattice and finite-volume evidence to Volume 8 and many-body applications to Volume 12.
    Scope
    Shallow scales, few-body sectors, promoted interactions and forces, and architecture-level regulator and uncertainty contracts for nuclear EFT.
    Assumptions
    Nonperturbative Iteration, Shallow Scales, and Power-Counting Consistency
  • Theory Model

    Heavy-Particle EFT and HQET Architecture

    How does separating a heavy reference momentum from residual motion produce heavy-particle and HQET expansions with reparameterization constraints?

    Principal question
    How does separating a heavy reference momentum from residual motion produce heavy-particle and HQET expansions with reparameterization constraints?
    What the source page covers
    Heavy reference-velocity fields, residual momentum, inverse-mass counting and reparameterization constraints, with HQET as the canonical architecture.
    Boundary
    Small-velocity, potential and quarkonium mode structures are owned by the nonrelativistic/potential architecture page; phenomenology belongs to Volume 6.
    Scope
    Heavy reference-velocity fields, residual momentum, inverse-mass counting and reparameterization constraints, with HQET as the canonical architecture.
    Assumptions
    A Map of Effective-Theory Architectures; The Dirac Field; Integrating Out Heavy Fields
  • Theory Model

    NRQED, NRQCD, and Potential EFT Architecture

    How do mass, momentum and kinetic-energy hierarchies distinguish NRQED, NRQCD and potential EFTs, and where do matching and mode separation enter?

    Principal question
    How do mass, momentum and kinetic-energy hierarchies distinguish NRQED, NRQCD and potential EFTs, and where do matching and mode separation enter?
    What the source page covers
    Mass, momentum and kinetic-energy hierarchies, NRQED and NRQCD field content, potential modes and the matching sequence into potential EFTs.
    Boundary
    Named coefficients and QED, QCD or quarkonium applications belong to Volume 6, nonperturbative bound states to Volume 7 and lattice determinations to Volume 8.
    Scope
    Mass, momentum and kinetic-energy hierarchies, NRQED and NRQCD field content, potential modes and the matching sequence into potential EFTs.
    Assumptions
    Heavy-Particle EFT and HQET Architecture; Modes, Virtualities, and EFT Scale Separation; Integrating Out Heavy Fields
  • Theory Model

    Soft-Collinear Effective Theory: Architecture and Validity

    How do hard, collinear, and soft modes produce factorization and multiscale evolution in SCET?

    Principal question
    How do hard, collinear, and soft modes produce factorization and multiscale evolution in SCET?
    What the source page covers
    SCET as a named architecture: light-cone fields, label and residual momentum, sector gauge symmetries, Wilson-line building blocks, variant selection and validity limits.
    Boundary
    General multipole, factorization, virtuality-evolution and rapidity-evolution methods are owned by the preceding pages; observable factorization and precision calculations belong to Volumes 4 and 6.
    Scope
    SCET as a named architecture: light-cone fields, label and residual momentum, sector gauge symmetries, Wilson-line building blocks, variant selection and validity limits.
    Assumptions
    Modes, Virtualities, and EFT Scale Separation; Matching onto Factorized Operator Structures; Evolution Kernels, Consistency Relations, and Resummation Architecture
  • Theory Model

    SMEFT and HEFT: Architecture and Domain

    How do linear and nonlinear electroweak symmetry realizations lead to distinct effective-theory architectures?

    Principal question
    How do linear and nonlinear electroweak symmetry realizations lead to distinct effective-theory architectures?
    What the source page covers
    Standard Model field content, higher-dimensional gauge-invariant operators, flavor assumptions, linear Higgs-doublet realization, nonlinear electroweak and Higgs realization, basis and input-scheme dependence, matching logic, truncation, and application handoffs
    Boundary
    Current global fits, coefficient bounds, and precision observables owned by Volume 6; A static complete basis catalog
    Scope
    Standard Model field content, higher-dimensional gauge-invariant operators, flavor assumptions, linear Higgs-doublet realization, nonlinear electroweak and Higgs realization, basis and input-scheme dependence, matching logic, truncation, and application handoffs
    Assumptions
    A Map of Effective-Theory Architectures; Representation and Spurion Constraints on Operator Bases
  • Theory Model

    Effective Field Theory of Gravity: Architecture and Power Counting

    How does diffeomorphism invariance organize quantum corrections to gravity below a cutoff?

    Principal question
    How does diffeomorphism invariance organize quantum corrections to gravity below a cutoff?
    What the source page covers
    Low-energy gravitational EFT architecture, diffeomorphism-invariant derivative counting and the bounded four-dimensional Einstein-gravity counterterm benchmark.
    Boundary
    Volume 14 owns quantum fields and renormalization on curved backgrounds and developed gravitational EFT applications; Volume 15 owns ultraviolet quantum-gravity proposals.
    Scope
    Low-energy gravitational EFT architecture, diffeomorphism-invariant derivative counting and the bounded four-dimensional Einstein-gravity counterterm benchmark.
    Assumptions
    A Map of Effective-Theory Architectures; Power Counting and Predictive Order; Levi–Civita Connections, Geodesics, and Riemann Curvature
  • Theory Model

    Unstable-Particle Effective Theory and the Width Expansion

    How does the hierarchy Γ/M define resonant and nonresonant modes and a gauge-consistent expansion without treating an unstable excitation as an external LSZ particle?

    Principal question
    How does the hierarchy Γ/M define resonant and nonresonant modes and a gauge-consistent expansion without treating an unstable excitation as an external LSZ particle?
    What the source page covers
    Complex-pole input, resonant and nonresonant modes, width-to-mass power counting, gauge-consistent matching and the domain of the unstable-particle expansion.
    Boundary
    Pole and line-shape observables belong to Volume 4, W, top and Higgs applications to Volume 6 and nonperturbative resonances to Volume 7.
    Scope
    Complex-pole input, resonant and nonresonant modes, width-to-mass power counting, gauge-consistent matching and the domain of the unstable-particle expansion.
    Assumptions
    Power Counting and Predictive Order; Infrared Cancellation, Regulators, and Matching Consistency; Resonance Poles, Riemann Sheets, and Unstable States; Unstable-Particle Observables and Controlled Resonance Approximations; Modes, Virtualities, and EFT Scale Separation
  • Theory Model

    Hydrodynamic Effective-Theory Architecture

    How do conservation laws, derivative counting, constitutive relations and fluctuation constraints define hydrodynamic EFT before open-system assumptions are added?

    Principal question
    How do conservation laws, derivative counting, constitutive relations and fluctuation constraints define hydrodynamic EFT before open-system assumptions are added?
    What the source page covers
    Hydrodynamic variables, conservation laws, derivative counting, constitutive organization and fluctuation constraints.
    Boundary
    Influence functionals, environment tracing and Lindblad limits are owned by the following open-system page; full dynamics belongs to Volume 11.
    Scope
    Hydrodynamic variables, conservation laws, derivative counting, constitutive organization and fluctuation constraints.
    Assumptions
    A Map of Effective-Theory Architectures; Current Sources and Generating Functionals
  • Theory Model

    Open-System Effective-Theory Architecture and Consistency Conditions

    How do influence functionals, doubled fields, noise and complete-positivity limits organize open-system EFTs, and when is a Lindblad description justified?

    Principal question
    How do influence functionals, doubled fields, noise and complete-positivity limits organize open-system EFTs, and when is a Lindblad description justified?
    What the source page covers
    The architecture-level roles of doubled fields, trace preservation and positivity constraints, tested on a supplied bounded Gaussian influence kernel with its normalization and domain declared.
    Boundary
    Deriving an influence functional by tracing an environment, constructing general Schwinger–Keldysh kernels, and developing noise or Lindblad dynamics and applications belong to Volume 11.
    Scope
    The architecture-level roles of doubled fields, trace preservation and positivity constraints, tested on a supplied bounded Gaussian influence kernel with its normalization and domain declared.
    Assumptions
    Closed-Time-Path Grammar
  • Theory Model

    Electroweak Gauge and Matter Structure

    How are chiral quarks and leptons assigned to the electroweak gauge group consistently?

    Principal question
    How are chiral quarks and leptons assigned to the electroweak gauge group consistently?
    What the source page covers
    SU(2)L times U(1)Y structure, chiral multiplets, hypercharge, covariant derivatives, electric-charge generator, generation replication, and anomaly-cancellation preview
    Boundary
    Full Standard Model parameter ledger; Primary anomaly derivation; Flavor mixing
    Scope
    SU(2)L times U(1)Y structure, chiral multiplets, hypercharge, covariant derivatives, electric-charge generator, generation replication, and anomaly-cancellation preview
    Assumptions
    The Yang–Mills Action and Gauge Self-Interaction; Compact Lie Groups, Roots, Weights, and Weyl Structure
  • Theory Model

    Dynamical Gauge Fields and Matter

    How does a gauge connection become a dynamical quantum field coupled consistently to charged matter?

    Principal question
    How does a gauge connection become a dynamical quantum field coupled consistently to charged matter?
    What the source page covers
    Gauge kinetic term, covariant matter action, field equations, self-interaction orientation, conserved constraints, physical degrees of freedom, and observable content
    Boundary
    Primary definition of gauge redundancy; Gauge fixing and BRST; A particular gauge group or phenomenological model
    Scope
    Gauge kinetic term, covariant matter action, field equations, self-interaction orientation, conserved constraints, physical degrees of freedom, and observable content
    Assumptions
    Gauge Fields, Redundancy, and Observable Content
  • Theory Model

    QCD Fields, Scales, and the Perturbative Domain

    How do color, quark flavors, masses, and asymptotic freedom define the perturbative domain of QCD?

    Principal question
    How do color, quark flavors, masses, and asymptotic freedom define the perturbative domain of QCD?
    What the source page covers
    QCD action, color representations, flavor content as a dated parameter input, quark masses and schemes orientation, conserved currents, perturbative scales, and infrared limitation
    Boundary
    Current numerical parameter table; Confinement and chiral realization; Parton distributions
    Scope
    QCD action, color representations, flavor content as a dated parameter input, quark masses and schemes orientation, conserved currents, perturbative scales, and infrared limitation
    Assumptions
    Non-Abelian Screening and Asymptotic Freedom
  • Theory Model

    The QED Action, Charges, and Observables

    How does a Dirac field coupled to electromagnetism define the quantum theory of electrons, positrons, and photons?

    Principal question
    How does a Dirac field coupled to electromagnetism define the quantum theory of electrons, positrons, and photons?
    What the source page covers
    QED action, local and global symmetries, electric charge, covariant derivative, gauge-invariant observables, gauge fixing as imported structure, and parameters
    Boundary
    Primary gauge-field definition; Detailed quantization derivation; Radiative corrections
    Scope
    QED action, local and global symmetries, electric charge, covariant derivative, gauge-invariant observables, gauge fixing as imported structure, and parameters
    Assumptions
    Dynamical Gauge Fields and Matter; Covariant Free-Photon Quantization and Propagator; Canonical Quantization of the Free Dirac Field
  • Theory Model

    The Yang–Mills Action and Gauge Self-Interaction

    How does non-Abelian curvature generate a self-interacting relativistic gauge theory?

    Principal question
    How does non-Abelian curvature generate a self-interacting relativistic gauge theory?
    What the source page covers
    Compact gauge group, curvature, Yang–Mills action, cubic and quartic self-interaction, matter coupling orientation, classical scale behavior, and parameters
    Boundary
    Primary connection/gauge definition; Quantization; QCD flavor content
    Scope
    Compact gauge group, curvature, Yang–Mills action, cubic and quartic self-interaction, matter coupling orientation, classical scale behavior, and parameters
    Assumptions
    Dynamical Gauge Fields and Matter
  • Theory Model

    The Higgs Doublet and Electroweak Symmetry Breaking

    How does the Higgs field select the electromagnetic subgroup while preserving gauge-invariant physical content?

    Principal question
    How does the Higgs field select the electromagnetic subgroup while preserving gauge-invariant physical content?
    What the source page covers
    Higgs representation and potential, vacuum orbit, unbroken generator, gauge-invariant interpretation, Goldstone directions, radial mode, and parameter relations
    Boundary
    Primary SSB/Higgs distinction; Radiative effective potential; Extended Higgs sectors
    Scope
    Higgs representation and potential, vacuum orbit, unbroken generator, gauge-invariant interpretation, Goldstone directions, radial mode, and parameter relations
    Assumptions
    Electroweak Gauge and Matter Structure; Elitzur's Theorem and the Gauge-Invariant Higgs Mechanism
  • Theory Model

    Deep-Inelastic Scattering and the Parton Model

    How do deep-inelastic structure functions expose pointlike partons and scaling inside a hadron?

    Principal question
    How do deep-inelastic structure functions expose pointlike partons and scaling inside a hadron?
    What the source page covers
    DIS kinematics, hadronic tensor, structure functions, Bjorken scaling, quark-parton model, sum-rule orientation, and target/higher-twist caveats
    Boundary
    Modern global PDF fitting; Full operator-product expansion; Nuclear corrections in depth
    Scope
    DIS kinematics, hadronic tensor, structure functions, Bjorken scaling, quark-parton model, sum-rule orientation, and target/higher-twist caveats
    Assumptions
    Inclusive Annihilation and the Emergence of Jets
  • Theory Model

    Bounce Solutions and False-Vacuum Boundary Conditions

    How does an O(d)-symmetric bounce describe escape from a metastable vacuum?

    Principal question
    How does an O(d)-symmetric bounce describe escape from a metastable vacuum?
    What the source page covers
    Euclidean boundary-value problem, overshoot/undershoot argument, bounce action, thin-wall orientation, and multiple-bounce sum, with assumptions, conventions, observables, and failure modes made explicit.
    Boundary
    Gravitational backreaction; Cosmological phase-transition dynamics; Finite-temperature bounces. Those subjects remain with their canonical volume or Research owner.
    Scope
    Euclidean boundary-value problem, overshoot/undershoot argument, bounce action, thin-wall orientation, and multiple-bounce sum, with assumptions, conventions, observables, and failure modes made explicit.
    Assumptions
    Euclidean Tunneling Saddles and Boundary Conditions; What an Interacting Lagrangian Does and Does Not Specify
  • Theory Model

    Quantum-Mechanical Instantons and Tunnel Splitting

    How does the instanton gas reproduce exponentially small level splitting in a double-well system?

    Principal question
    How does the instanton gas reproduce exponentially small level splitting in a double-well system?
    What the source page covers
    Instanton solution, translation zero mode, determinant ratio, instanton/anti-instanton sum, level splitting, and dilute-gas validity, with assumptions, conventions, observables, and failure modes made explicit.
    Boundary
    Exact WKB; Full resurgent quantization condition; Field-theory instanton moduli. Those subjects remain with their canonical volume or Research owner.
    Scope
    Instanton solution, translation zero mode, determinant ratio, instanton/anti-instanton sum, level splitting, and dilute-gas validity, with assumptions, conventions, observables, and failure modes made explicit.
    Assumptions
    Euclidean Tunneling Saddles and Boundary Conditions; Multi-Saddle Sums and Dilute Ensembles
  • Theory Model

    The O(N) Model as a Strong-Coupling Laboratory

    How does the constrained O(N) field realize asymptotic freedom and nonperturbative mass generation in two dimensions?

    Principal question
    How does the constrained O(N) field realize asymptotic freedom and nonperturbative mass generation in two dimensions?
    What the source page covers
    Constraint action, auxiliary field, beta-function orientation, correlation length, large-N preview, and dimension dependence, with assumptions, conventions, observables, and failure modes made explicit.
    Boundary
    Full perturbative beta-function derivation; Lattice simulation; Critical O(N) CFT. Those subjects remain with their canonical volume or Research owner.
    Scope
    Constraint action, auxiliary field, beta-function orientation, correlation length, large-N preview, and dimension dependence, with assumptions, conventions, observables, and failure modes made explicit.
    Assumptions
    Sigma-Model Dynamics: Target Geometry, Dimension, and Control; Running Couplings and Dimensional Transmutation
  • Theory Model

    Gauge Instantons, Topological Charge, and Moduli

    How does a four-dimensional Euclidean gauge field realize an integer topological charge with finite action?

    Principal question
    How does a four-dimensional Euclidean gauge field realize an integer topological charge with finite action?
    What the source page covers
    Self-duality bound, BPST orientation, topological charge, action, size and orientation moduli, and boundary behavior, with assumptions, conventions, observables, and failure modes made explicit.
    Boundary
    Complete instanton classification; Gauge-model phenomenology; Supersymmetric instanton calculus. Those subjects remain with their canonical volume or Research owner.
    Scope
    Self-duality bound, BPST orientation, topological charge, action, size and orientation moduli, and boundary behavior, with assumptions, conventions, observables, and failure modes made explicit.
    Assumptions
    Euclidean Tunneling Saddles and Boundary Conditions; Topological Sectors, Boundary Data, and Global Form; The Yang–Mills Action and Gauge Self-Interaction
  • Theory Model

    The CP(N-1) Model

    How does a constrained complex field with a redundant phase produce a dynamical gauge-like description and rich theta dependence?

    Principal question
    How does a constrained complex field with a redundant phase produce a dynamical gauge-like description and rich theta dependence?
    What the source page covers
    Projective target, auxiliary gauge field, constraint, topological charge, asymptotic freedom, and large-N orientation, with assumptions, conventions, observables, and failure modes made explicit.
    Boundary
    General gauge-field primary definition; Supersymmetric exact solution; Lattice implementation. Those subjects remain with their canonical volume or Research owner.
    Canonical treatment
    The CP(N-1) Model
    Scope
    Projective target, auxiliary gauge field, constraint, topological charge, asymptotic freedom, and large-N orientation, with assumptions, conventions, observables, and failure modes made explicit.
    Assumptions
    Sigma-Model Dynamics: Target Geometry, Dimension, and Control; Gauge Fields, Redundancy, and Observable Content
  • Theory Model

    Kinks and Domain Walls

    How does a field interpolate between disconnected vacua to form a stable kink or wall?

    Principal question
    How does a field interpolate between disconnected vacua to form a stable kink or wall?
    What the source page covers
    Scalar kink equation, tension, topological charge, translation zero mode, small fluctuations, and higher-dimensional wall interpretation, with assumptions, conventions, observables, and failure modes made explicit.
    Boundary
    Cosmological wall networks; Supersymmetric central charges; One-loop mass renormalization in detail. Those subjects remain with their canonical volume or Research owner.
    Canonical treatment
    Kinks and Domain Walls
    Scope
    Scalar kink equation, tension, topological charge, translation zero mode, small fluctuations, and higher-dimensional wall interpretation, with assumptions, conventions, observables, and failure modes made explicit.
    Assumptions
    Finite-Energy Boundary Data, Charge, and Stability; What an Interacting Lagrangian Does and Does Not Specify
  • Theory Model

    Vortices, Flux Quantization, and Core Scales

    How do winding and gauge fields combine to produce finite-tension vortices and quantized flux?

    Principal question
    How do winding and gauge fields combine to produce finite-tension vortices and quantized flux?
    What the source page covers
    Global versus local vortices, winding, flux quantization, core scales, asymptotic fields, and stability, with assumptions, conventions, observables, and failure modes made explicit.
    Boundary
    Non-Abelian vortex catalog; Superconducting material phenomenology; BPS exact moduli dynamics. Those subjects remain with their canonical volume or Research owner.
    Scope
    Global versus local vortices, winding, flux quantization, core scales, asymptotic fields, and stability, with assumptions, conventions, observables, and failure modes made explicit.
    Assumptions
    Finite-Energy Boundary Data, Charge, and Stability; Gauge Fields, Redundancy, and Observable Content
  • Theory Model

    Monopoles and Dyons

    How does symmetry breaking in a gauge theory support a smooth magnetic monopole?

    Principal question
    How does symmetry breaking in a gauge theory support a smooth magnetic monopole?
    What the source page covers
    Higgs vacuum manifold, magnetic charge, finite-energy ansatz, core structure, asymptotic Abelian field, and charge quantization, with assumptions, conventions, observables, and failure modes made explicit.
    Boundary
    Full gauge-model dynamics; Montonen–Olive duality; Supersymmetric monopole moduli-space results. Those subjects remain with their canonical volume or Research owner.
    Canonical treatment
    Monopoles and Dyons
    Scope
    Higgs vacuum manifold, magnetic charge, finite-energy ansatz, core structure, asymptotic Abelian field, and charge quantization, with assumptions, conventions, observables, and failure modes made explicit.
    Assumptions
    Finite-Energy Boundary Data, Charge, and Stability; Local Potentials and Global Gauge Configurations
  • Theory Model

    Lumps, Textures, and Skyrmions

    How do sigma-model topology and higher-derivative stabilization produce localized textures?

    Principal question
    How do sigma-model topology and higher-derivative stabilization produce localized textures?
    What the source page covers
    Lump charge, Derrick scaling, Skyrme stabilization, collective orientation, and dimensional dependence, with assumptions, conventions, observables, and failure modes made explicit.
    Boundary
    Nuclear phenomenology; Holographic baryons; Complete moduli-space dynamics. Those subjects remain with their canonical volume or Research owner.
    Scope
    Lump charge, Derrick scaling, Skyrme stabilization, collective orientation, and dimensional dependence, with assumptions, conventions, observables, and failure modes made explicit.
    Assumptions
    Finite-Energy Boundary Data, Charge, and Stability; Derrick Scaling, Virial Tests, and Nontopological Stability
  • Theory Model

    The Principal Chiral Model and the Integrability Bridge

    Why do principal chiral models connect non-Abelian sigma-model dynamics to factorized scattering and exact methods?

    Principal question
    Why do principal chiral models connect non-Abelian sigma-model dynamics to factorized scattering and exact methods?
    What the source page covers
    Group-valued field, left/right symmetry, conserved currents, asymptotic freedom orientation, mass-gap evidence, and integrability handoff, with assumptions, conventions, observables, and failure modes made explicit.
    Boundary
    Exact S-matrix derivation; Wess–Zumino–Witten CFT; String worldsheet applications. Those subjects remain with their canonical volume or Research owner.
    Scope
    Group-valued field, left/right symmetry, conserved currents, asymptotic freedom orientation, mass-gap evidence, and integrability handoff, with assumptions, conventions, observables, and failure modes made explicit.
    Assumptions
    Sigma-Model Dynamics: Target Geometry, Dimension, and Control; Quantum Currents, Improvements, and Conservation
  • Theory Model

    Q-Balls, Oscillons, and Sphalerons

    How do Q-balls, oscillons, and sphalerons persist or matter physically without topological stability?

    Principal question
    How do Q-balls, oscillons, and sphalerons persist or matter physically without topological stability?
    What the source page covers
    A comparative taxonomy of fixed-charge minima, long-lived time-dependent configurations, and unstable barrier saddles, with lifetime and control claims separated.
    Boundary
    Electroweak realization belongs to Volume 6, nonequilibrium evolution to Volume 11, and numerical construction must be accompanied by code, inputs, and validation checks.
    Scope
    A comparative taxonomy of fixed-charge minima, long-lived time-dependent configurations, and unstable barrier saddles, with lifetime and control claims separated.
    Assumptions
    Derrick Scaling, Virial Tests, and Nontopological Stability; Continuous Symmetries, Generators, and Charges
  • Theory Model

    Fractional Events, Calorons, and Monopole Constituents

    How do compactification, holonomy, and global data fractionalize topological events into monopole constituents?

    Principal question
    How do compactification, holonomy, and global data fractionalize topological events into monopole constituents?
    What the source page covers
    Constituent topological charge, magnetic charge, action, holonomy dependence, and recombination into an integer-charge caloron under explicit boundary assumptions.
    Boundary
    Finite-temperature dynamics belongs to Volume 11, discretized tests to Volume 8, and bion physics to the controlled-confinement chapter.
    Scope
    Constituent topological charge, magnetic charge, action, holonomy dependence, and recombination into an integer-charge caloron under explicit boundary assumptions.
    Assumptions
    Gauge Instantons, Topological Charge, and Moduli; Topological Sectors, Boundary Data, and Global Form
  • Theory Model

    Free and Generalized Free CFTs

    Which free and generalized-free correlators satisfy conformal crossing, and which additional QFT properties distinguish them?

    Principal question
    Which free and generalized-free correlators satisfy conformal crossing, and which additional QFT properties distinguish them?
    What the source page covers
    Free scalar and fermion CFT data, generalized-free correlators, Wick-like factorization, double-trace spectra and OPE coefficients, equations-of-motion shortening, locality qualifications, and benchmark uses.
    Boundary
    General RG and EFT machinery remains in Volume 5, large-N dynamics in Volume 7, numerical regulators in Volume 8, and mutable information about candidate fixed points in Research.
    Scope
    Free scalar and fermion CFT data, generalized-free correlators, Wick-like factorization, double-trace spectra and OPE coefficients, equations-of-motion shortening, locality qualifications, and benchmark uses.
    Assumptions
    Crossing Equations and Positivity; Unitarity Bounds and Null States
  • Theory Model

    Liouville Theory and the Virasoro Bootstrap

    How does Liouville theory realize a continuous Virasoro spectrum and solve crossing through special structure constants and blocks?

    Principal question
    How does Liouville theory realize a continuous Virasoro spectrum and solve crossing through special structure constants and blocks?
    What the source page covers
    Liouville parametrization, spectrum, reflection relation, DOZZ-type structure constants as data, degenerate-field shift equations, Virasoro blocks, crossing checks, and normalization caveats.
    Boundary
    The rational algebraic core stays in Chapter 5; general distribution theory stays in Volume 1; modular bootstrap synthesis stays in Chapter 13; unsettled classifications and current bounds stay in Research.
    Scope
    Liouville parametrization, spectrum, reflection relation, DOZZ-type structure constants as data, degenerate-field shift equations, Virasoro blocks, crossing checks, and normalization caveats.
    Assumptions
    Noncompact CFTs and Continuous Spectra; Highest-Weight Modules, Null States, and the Kac Determinant
  • Theory Model

    Minimal Models and Fusion Rules

    How do finite Virasoro spectra, null-vector decoupling, and fusion consistency define the minimal models?

    Principal question
    How do finite Virasoro spectra, null-vector decoupling, and fusion consistency define the minimal models?
    What the source page covers
    Minimal-model central charges and Kac tables, field identifications, BPZ equations, fusion rules, modular-invariant pairings at an introductory level, unitarity series, and canonical examples.
    Boundary
    Nonunitary, logarithmic, and noncompact theories move to Chapter 6; general symmetry and anomaly definitions stay in Volume 3; modular and thermal constraints are synthesized in Chapter 13.
    Scope
    Minimal-model central charges and Kac tables, field identifications, BPZ equations, fusion rules, modular-invariant pairings at an introductory level, unitarity series, and canonical examples.
    Assumptions
    Highest-Weight Modules, Null States, and the Kac Determinant; Crossing Equations and Positivity
  • Theory Model

    Generalized-Free and Solvable 1D CFT Data

    Which exactly solvable or generalized-free 1D correlators provide reliable spectra, OPE coefficients, and bootstrap benchmarks?

    Principal question
    Which exactly solvable or generalized-free 1D correlators provide reliable spectra, OPE coefficients, and bootstrap benchmarks?
    What the source page covers
    Generalized-free bosonic and fermionic solutions, disconnected correlators, double-trace towers, OPE-coefficient extraction, contact deformations, and comparison data for analytic and numerical methods.
    Boundary
    This chapter owns the SL(2,R) CFT data problem, not a survey of quantum mechanics; line-defect geometry belongs to the defect chapter, model dynamics to their subject volumes, and mutable solution status to Research.
    Scope
    Generalized-free bosonic and fermionic solutions, disconnected correlators, double-trace towers, OPE-coefficient extraction, contact deformations, and comparison data for analytic and numerical methods.
    Assumptions
    Crossing and Positivity in One Dimension
  • Theory Model

    Free Bosons and Vertex Operators

    How does the compact free boson realize vertex operators, momentum and winding sectors, T-duality, and exactly computable local CFT data?

    Principal question
    How does the compact free boson realize vertex operators, momentum and winding sectors, T-duality, and exactly computable local CFT data?
    What the source page covers
    The compact-boson normalization, local oscillator algebra, vertex operators, normal ordering, momentum and winding lattice, compactification radius, cocycles, neutrality, T-duality at the CFT-data level, and correlators; Chapter 6 owns the noncompact global state space, zero-mode volume, continuous spectrum, and distributional normalization.
    Boundary
    Nonunitary, logarithmic, and noncompact theories move to Chapter 6; general symmetry and anomaly definitions stay in Volume 3; modular and thermal constraints are synthesized in Chapter 13.
    Scope
    The compact-boson normalization, local oscillator algebra, vertex operators, normal ordering, momentum and winding lattice, compactification radius, cocycles, neutrality, T-duality at the CFT-data level, and correlators; Chapter 6 owns the noncompact global state space, zero-mode volume, continuous spectrum, and distributional normalization.
    Assumptions
    The Virasoro Algebra and the Stress Tensor; From the Local OPE to Conformal Data
  • Theory Model

    Affine Current Algebras and WZW Models

    How do affine current algebras and WZW models extend Virasoro symmetry and determine spectra, correlators, and fusion?

    Principal question
    How do affine current algebras and WZW models extend Virasoro symmetry and determine spectra, correlators, and fusion?
    What the source page covers
    Kac–Moody current OPEs, level, Sugawara construction, integrable representations, WZW actions as context, Knizhnik–Zamolodchikov equations, fusion, and normalization conventions.
    Boundary
    Nonunitary, logarithmic, and noncompact theories move to Chapter 6; general symmetry and anomaly definitions stay in Volume 3; modular and thermal constraints are synthesized in Chapter 13.
    Scope
    Kac–Moody current OPEs, level, Sugawara construction, integrable representations, WZW actions as context, Knizhnik–Zamolodchikov equations, fusion, and normalization conventions.
    Assumptions
    The Virasoro Algebra and the Stress Tensor; Lie Groups, Lie Algebras, and Exponential and Adjoint Maps
  • Theory Model

    Seiberg Duality in SQCD

    What are the electric and magnetic theory cards that define the canonical Seiberg-duality claim for four-dimensional N=1 SQCD?

    Principal question
    What are the electric and magnetic theory cards that define the canonical Seiberg-duality claim for four-dimensional N=1 SQCD?
    What the source page covers
    The paired electric and magnetic gauge-theory cards: gauge groups and ranks, matter representations, magnetic singlet, superpotential, holomorphic scale relation, parameter and rank regimes, claim type, and evidence ceiling.
    Boundary
    The following operator-dictionary page owns meson and baryon maps, chiral-ring and moduli-stratum correspondence, anomaly matching, global quotients, generalized-symmetry data, and special-rank audits; this page supplies only the paired theory cards and base duality claim.
    Canonical treatment
    Seiberg Duality in SQCD
    Scope
    The paired electric and magnetic gauge-theory cards: gauge groups and ranks, matter representations, magnetic singlet, superpotential, holomorphic scale relation, parameter and rank regimes, claim type, and evidence ceiling.
    Assumptions
    SQCD Fields, Symmetries, Global Data, and Classical Moduli; Duality Claims, Dictionaries, Regimes, and Evidence
  • Theory Model

    SQCD Fields, Symmetries, Global Data, and Classical Moduli

    What complete theory, symmetry, anomaly, global, and classical-moduli data define four-dimensional N=1 SQCD before strong dynamics is inferred?

    Principal question
    What complete theory, symmetry, anomaly, global, and classical-moduli data define four-dimensional N=1 SQCD before strong dynamics is inferred?
    What the source page covers
    Gauge algebra and global form, quark superfields, flavor and R symmetries, discrete quotients, genuine lines, anomaly tables, mesons and baryons, rank constraints, classical branches, and scale conventions.
    Boundary
    Volume 6 retains generic gauge phases and confinement definitions, Volume 7 retains generic nonperturbative mechanisms, Chapter 10 owns duality, and claims about nonsupersymmetric QCD remain outside this chapter.
    Scope
    Gauge algebra and global form, quark superfields, flavor and R symmetries, discrete quotients, genuine lines, anomaly tables, mesons and baryons, rank constraints, classical branches, and scale conventions.
    Assumptions
    Gauge-Invariant Coordinates and Classical Moduli Varieties; 't Hooft Anomaly Matching
  • Theory Model

    N=4 SYM Field Content, Action, and Superconformal Data

    How are N=4 super-Yang–Mills fields, interactions, supersymmetries, global data, and superconformal observables normalized in one theory card?

    Principal question
    How are N=4 super-Yang–Mills fields, interactions, supersymmetries, global data, and superconformal observables normalized in one theory card?
    What the source page covers
    The N=4 SYM action, field representations, transformations, SU(4) R symmetry, complex coupling, scalar potential, vacuum equations, stress-tensor multiplet, global form, line data interface, and convention dictionary.
    Boundary
    Volume 3 retains generalized symmetries and genuine-line definitions, Volume 9 retains conformal bootstrap, Volume 15 owns brane and holographic interpretations, and Volume 16 owns theorem-level existence questions.
    Scope
    The N=4 SYM action, field representations, transformations, SU(4) R symmetry, complex coupling, scalar potential, vacuum equations, stress-tensor multiplet, global form, line data interface, and convention dictionary.
    Assumptions
    Extended Supersymmetry Gauge Dynamics: Scope and Structure; Superconformal Algebras, Shortening Data, and the CFT Handoff
  • Theory Model

    O’Raifeartaigh and Fayet–Iliopoulos Model Laboratories

    How do O’Raifeartaigh and Fayet–Iliopoulos models provide controlled laboratories for F-term and D-term supersymmetry breaking?

    Principal question
    How do O’Raifeartaigh and Fayet–Iliopoulos models provide controlled laboratories for F-term and D-term supersymmetry breaking?
    What the source page covers
    Rank conditions, tree-level flat directions, spectra, goldstini, FI consistency, gauge symmetry, tachyon regions, classical runaways, and the distinct assumptions of F- and D-term examples.
    Boundary
    Volume 5 retains EFT validity, Volume 7 retains vacuum decay, Volume 6 owns phenomenological models and empirical constraints, and no soft path is promoted into a theorem about nonsupersymmetric QCD.
    Scope
    Rank conditions, tree-level flat directions, spectra, goldstini, FI consistency, gauge symmetry, tachyon regions, classical runaways, and the distinct assumptions of F- and D-term examples.
    Assumptions
    F- and D-Term Breaking, Vacuum Energy, and the Goldstino; Wess–Zumino Models
  • Theory Model

    Yang–Mills and Chern–Simons–Matter Actions

    How are supersymmetric Yang–Mills, Chern–Simons, BF, FI, matter, and superpotential couplings consistently assembled in three dimensions?

    Principal question
    How are supersymmetric Yang–Mills, Chern–Simons, BF, FI, matter, and superpotential couplings consistently assembled in three dimensions?
    What the source page covers
    N=2 Yang–Mills and Chern–Simons multiplets, BF and FI couplings, chiral matter, superpotentials, auxiliary potentials, level quantization, global forms, boundary terms, and classical vacua.
    Boundary
    Volume 3 retains Chern-Simons and anomaly foundations, Volume 5 retains generic fixed points, Volume 12 owns quantum-matter realizations, and this chapter retains the relativistic field-theory dictionary.
    Scope
    N=2 Yang–Mills and Chern–Simons multiplets, BF and FI couplings, chiral matter, superpotentials, auxiliary potentials, level quantization, global forms, boundary terms, and classical vacua.
    Assumptions
    Three-Dimensional Supersymmetry Algebras and Multiplets; Chern–Simons Actions and Level Quantization
  • Theory Model

    Landau–Ginzburg Models and Chiral Rings

    How do supersymmetric Landau–Ginzburg models encode vacua, chiral rings, solitons, orbifolds, and candidate infrared fixed points?

    Principal question
    How do supersymmetric Landau–Ginzburg models encode vacua, chiral rings, solitons, orbifolds, and candidate infrared fixed points?
    What the source page covers
    The (2,2) Landau–Ginzburg action, quasi-homogeneous superpotentials, Jacobi rings, massive vacua, central charges, solitons, orbifold sectors, RG relevance, and infrared-scope qualifications.
    Boundary
    Volume 9 retains general two-dimensional CFT; Volume 1 retains smooth, bundle, symplectic, quotient, and de Rham foundations; this volume owns QFT-specific complex and Kähler target-space constructions; Volume 15 owns string mirror symmetry, and Volume 16 owns theorem-first geometric status.
    Scope
    The (2,2) Landau–Ginzburg action, quasi-homogeneous superpotentials, Jacobi rings, massive vacua, central charges, solitons, orbifold sectors, RG relevance, and infrared-scope qualifications.
    Assumptions
    Two-Dimensional Supersymmetry Algebras, Multiplets, and Superspace; Wess–Zumino Models
  • Theory Model

    Wess–Zumino Models

    What do Wess-Zumino models teach about supersymmetric interactions, vacuum equations, masses, Yukawa couplings, and nonrenormalization?

    Principal question
    What do Wess-Zumino models teach about supersymmetric interactions, vacuum equations, masses, Yukawa couplings, and nonrenormalization?
    What the source page covers
    Canonical and noncanonical chiral models, Kähler potential, superpotential, auxiliary elimination, scalar potentials, mass matrices, Yukawa terms, discrete symmetries, vacua, and perturbative teaching examples.
    Boundary
    Volume 2 retains general action and effective-action definitions, Volume 3 retains ordinary currents and Ward identities, Volume 4 owns amplitudes, and Volume 14 owns dynamical supergravity.
    Canonical treatment
    Wess–Zumino Models
    Scope
    Canonical and noncanonical chiral models, Kähler potential, superpotential, auxiliary elimination, scalar potentials, mass matrices, Yukawa terms, discrete symmetries, vacua, and perturbative teaching examples.
    Assumptions
    Supersymmetric Action Principles and Component Reduction
  • Theory Model

    Supersymmetric Sigma Models and Kähler Geometry

    Why does (2,2) supersymmetry require Kähler target geometry, and which metric, B-field, anomaly, and quantum data may change?

    Principal question
    Why does (2,2) supersymmetry require Kähler target geometry, and which metric, B-field, anomaly, and quantum data may change?
    What the source page covers
    Two-dimensional supersymmetric nonlinear sigma models, complex structures, Kähler metrics, B fields, torsion restrictions, vector and axial R anomalies, beta functions, instanton corrections, and conformal-limit qualifications.
    Boundary
    Volume 9 retains general two-dimensional CFT; Volume 1 retains smooth, bundle, symplectic, quotient, and de Rham foundations; this volume owns QFT-specific complex and Kähler target-space constructions; Volume 15 owns string mirror symmetry, and Volume 16 owns theorem-first geometric status.
    Scope
    Two-dimensional supersymmetric nonlinear sigma models, complex structures, Kähler metrics, B fields, torsion restrictions, vector and axial R anomalies, beta functions, instanton corrections, and conformal-limit qualifications.
    Assumptions
    Two-Dimensional Supersymmetry Algebras, Multiplets, and Superspace; Kähler Sigma Models and Supersymmetric Target Geometry
  • Theory Model

    The Ideal Bose Gas and Bose–Einstein Condensation

    How do Bose statistics, density of states, dimensionality, and the thermodynamic limit produce or obstruct condensation?

    Principal question
    How do Bose statistics, density of states, dimensionality, and the thermodynamic limit produce or obstruct condensation?
    What the source page covers
    Grand-canonical occupations, critical density and temperature, condensate fraction, finite volume, traps as an interface, and distinctions among condensation, coherence, and superfluidity.
    Boundary
    Volume 11 owns equilibrium and hydrodynamic formalism and Volume 5 owns generic criticality. This chapter owns Bose-fluid and lattice-boson phases, observables, and controlled many-body limits.
    Scope
    Grand-canonical occupations, critical density and temperature, condensate fraction, finite volume, traps as an interface, and distinctions among condensation, coherence, and superfluidity.
    Assumptions
    Second-Quantized Bosons and Fermions; Thermal Density Operators and the KMS Condition; Chemical Potentials and Finite-Density Ensembles
  • Theory Model

    The Fermi Gas and Fermi-Surface Kinematics

    How do Pauli statistics and finite density produce a Fermi sea, a codimension-one low-energy surface, and particle–hole excitations?

    Principal question
    How do Pauli statistics and finite density produce a Fermi sea, a codimension-one low-energy surface, and particle–hole excitations?
    What the source page covers
    Fermi momentum and energy, density of states, particle/hole kinematics, linearization, patches, dimensional dependence, and thermodynamic observables.
    Boundary
    Volume 5 owns RG construction and Volume 11 owns generic transport. This chapter owns Fermi-surface kinematics, Landau theory, Luttinger hypotheses, and matter-specific instabilities.
    Scope
    Fermi momentum and energy, density of states, particle/hole kinematics, linearization, patches, dimensional dependence, and thermodynamic observables.
    Assumptions
    Second-Quantized Bosons and Fermions; Chemical Potentials and Finite-Density Ensembles
  • Theory Model

    Hubbard Models: Symmetries and Controlled Limits

    Which exact symmetries and controlled limits of the Hubbard model anchor later claims about Mottness, magnetism, pairing, or pseudogap behavior?

    Principal question
    Which exact symmetries and controlled limits of the Hubbard model anchor later claims about Mottness, magnetism, pairing, or pseudogap behavior?
    What the source page covers
    Hubbard Hamiltonians, particle–hole and spin symmetries, atomic and weak-coupling limits, half filling, charge gap definitions, and controlled strong-coupling entry; spectral transfer and insulator taxonomy have separate owners.
    Boundary
    Volume 8 owns lattice algorithms, sign theory, and numerical certification. This chapter owns correlated-electron Hamiltonians, impurity mappings, phase interpretation, and evidence ceilings.
    Scope
    Hubbard Hamiltonians, particle–hole and spin symmetries, atomic and weak-coupling limits, half filling, charge gap definitions, and controlled strong-coupling entry; spectral transfer and insulator taxonomy have separate owners.
    Assumptions
    Second-Quantized Bosons and Fermions; Dyson Equations and Self-Energies; Emergent Variables and Reorganized Effective Descriptions
  • Theory Model

    The Weakly Interacting Bose Gas

    Which expansion parameter controls a dilute weakly interacting Bose gas, and what belongs to the saddle rather than the fluctuation correction?

    Principal question
    Which expansion parameter controls a dilute weakly interacting Bose gas, and what belongs to the saddle rather than the fluctuation correction?
    What the source page covers
    Dilute-gas action, condensate saddle, Gross–Pitaevskii equation, mean-field equation of state, healing scale, and gas-parameter control; systematic beyond-Bogoliubov terms belong to the successor.
    Boundary
    Volume 11 owns equilibrium and hydrodynamic formalism and Volume 5 owns generic criticality. This chapter owns Bose-fluid and lattice-boson phases, observables, and controlled many-body limits.
    Scope
    Dilute-gas action, condensate saddle, Gross–Pitaevskii equation, mean-field equation of state, healing scale, and gas-parameter control; systematic beyond-Bogoliubov terms belong to the successor.
    Assumptions
    The Ideal Bose Gas and Bose–Einstein Condensation; Short-Range Scattering Data as Many-Body Inputs; Hubbard–Stratonovich Fields and Collective Channels
  • Theory Model

    Two-Channel Resonance Models

    How does a two-channel field model encode resonance position, width, closed-channel fraction, and broad-versus-narrow many-body behavior?

    Principal question
    How does a two-channel field model encode resonance position, width, closed-channel fraction, and broad-versus-narrow many-body behavior?
    What the source page covers
    Open/closed channels, molecular field, detuning, interchannel coupling, resonance width, effective range, dressed molecule, and broad/narrow control.
    Boundary
    Volumes 4 and 5 own scattering and few-body EFT machinery. This chapter owns the use of matched two- and three-body data in quantum gases; current resonance and loss records belong to Research.
    Canonical treatment
    Two-Channel Resonance Models
    Scope
    Open/closed channels, molecular field, detuning, interchannel coupling, resonance width, effective range, dressed molecule, and broad/narrow control.
    Assumptions
    Effective Range, Shallow Poles, and Universality Windows; Hubbard–Stratonovich Fields and Collective Channels
  • Theory Model

    Dimensional Crossover and Confinement-Induced Resonances

    How does transverse confinement renormalize scattering and generate effective lower-dimensional resonances?

    Principal question
    How does transverse confinement renormalize scattering and generate effective lower-dimensional resonances?
    What the source page covers
    Quasi-one- and quasi-two-dimensional scattering, confinement-induced resonances, transverse modes, dimensional anomalies, and crossover validity.
    Boundary
    Volumes 4 and 5 own scattering and few-body EFT machinery. This chapter owns the use of matched two- and three-body data in quantum gases; current resonance and loss records belong to Research.
    Scope
    Quasi-one- and quasi-two-dimensional scattering, confinement-induced resonances, transverse modes, dimensional anomalies, and crossover validity.
    Assumptions
    Effective Range, Shallow Poles, and Universality Windows
  • Theory Model

    Strongly Correlated One-Dimensional Bose Fluids

    How do interaction strength and density interpolate between weakly interacting, Luttinger-liquid, and fermionized one-dimensional Bose fluids?

    Principal question
    How do interaction strength and density interpolate between weakly interacting, Luttinger-liquid, and fermionized one-dimensional Bose fluids?
    What the source page covers
    Lieb–Liniger parameters, Tonks–Girardeau limit, correlation exponents, excitation branches, local correlations, and trap/interface qualifications.
    Boundary
    Volume 11 owns equilibrium and hydrodynamic formalism and Volume 5 owns generic criticality. This chapter owns Bose-fluid and lattice-boson phases, observables, and controlled many-body limits.
    Scope
    Lieb–Liniger parameters, Tonks–Girardeau limit, correlation exponents, excitation branches, local correlations, and trap/interface qualifications.
    Assumptions
    Phase–Density EFT for Bose Superfluids; Dimensional Crossover and Confinement-Induced Resonances
  • Theory Model

    Resonant Bose Matter and Metastable Branches

    Which metastable resonant Bose branches can be defined before three-body loss and equilibration invalidate an equilibrium many-body description?

    Principal question
    Which metastable resonant Bose branches can be defined before three-body loss and equilibration invalidate an equilibrium many-body description?
    What the source page covers
    Upper and lower resonant branches, metastability, three-body loss scales, quench preparation, universal and nonuniversal observables, and evidence-qualified claims.
    Boundary
    Volumes 4 and 5 own scattering and few-body EFT machinery. This chapter owns the use of matched two- and three-body data in quantum gases; current resonance and loss records belong to Research.
    Scope
    Upper and lower resonant branches, metastability, three-body loss scales, quench preparation, universal and nonuniversal observables, and evidence-qualified claims.
    Assumptions
    Efimov Physics and the Three-Body Parameter; Universal Relations and Tan Contact
  • Theory Model

    The Bose–Hubbard Model and Controlled Limits

    Which controlled limits of the Bose–Hubbard model distinguish superfluid, atomic Mott, and compressible regimes before a criticality claim?

    Principal question
    Which controlled limits of the Bose–Hubbard model distinguish superfluid, atomic Mott, and compressible regimes before a criticality claim?
    What the source page covers
    Bose–Hubbard Hamiltonian, filling, atomic limit, hopping expansion, order and compressibility observables, mean-field boundaries, and model-realization corrections; critical scaling belongs to the successor.
    Boundary
    Volume 11 owns equilibrium and hydrodynamic formalism and Volume 5 owns generic criticality. This chapter owns Bose-fluid and lattice-boson phases, observables, and controlled many-body limits.
    Scope
    Bose–Hubbard Hamiltonian, filling, atomic limit, hopping expansion, order and compressibility observables, mean-field boundaries, and model-realization corrections; critical scaling belongs to the successor.
    Assumptions
    Phase–Density EFT for Bose Superfluids; Critical Surfaces, Crossover, and Corrections to Scaling; Universality Classes and Scaling Functions
  • Theory Model

    Electron–Phonon Fields and Retarded Interactions

    How does integrating lattice displacement fields generate retarded electron interactions and competing mass, density-wave, and pairing effects?

    Principal question
    How does integrating lattice displacement fields generate retarded electron interactions and competing mass, density-wave, and pairing effects?
    What the source page covers
    Electron–phonon Hamiltonians, phonon propagators, retarded effective interactions, polaronic renormalization, Peierls/CDW competition, isotope and adiabatic parameters.
    Boundary
    Volume 8 owns lattice algorithms, sign theory, and numerical certification. This chapter owns correlated-electron Hamiltonians, impurity mappings, phase interpretation, and evidence ceilings.
    Scope
    Electron–phonon Hamiltonians, phonon propagators, retarded effective interactions, polaronic renormalization, Peierls/CDW competition, isotope and adiabatic parameters.
    Assumptions
    From Bands and Orbitals to Effective Lattice Hamiltonians; Coherent-State Path Integrals for Many-Body Systems
  • Theory Model

    Multiorbital Correlations and Hund's Metals

    How do orbital degeneracy and Hund exchange reshape coherence, moments, and correlation strength in multiorbital metals?

    Principal question
    How do orbital degeneracy and Hund exchange reshape coherence, moments, and correlation strength in multiorbital metals?
    What the source page covers
    Kanamori interactions, spin/orbital symmetries, Hund coupling, orbital selectivity, coherence scales, local moments, Hund-metal phenomenology, and solver/evidence limits.
    Boundary
    Volume 8 owns lattice algorithms, sign theory, and numerical certification. This chapter owns correlated-electron Hamiltonians, impurity mappings, phase interpretation, and evidence ceilings.
    Scope
    Kanamori interactions, spin/orbital symmetries, Hund coupling, orbital selectivity, coherence scales, local moments, Hund-metal phenomenology, and solver/evidence limits.
    Assumptions
    From Bands and Orbitals to Effective Lattice Hamiltonians; DMFT Impurity Mapping and Self-Consistency
  • Theory Model

    Self-Bound Quantum Droplets

    How can competing mean-field attraction and fluctuation pressure stabilize a self-bound quantum droplet, and when do loss or finite range invalidate the description?

    Principal question
    How can competing mean-field attraction and fluctuation pressure stabilize a self-bound quantum droplet, and when do loss or finite range invalidate the description?
    What the source page covers
    Mixture and dipolar droplet energy functionals, Lee–Huang–Yang stabilization, equilibrium density, surface energy, collective modes, metastability, loss, and beyond-local corrections.
    Boundary
    Volume 11 owns equilibrium and hydrodynamic formalism and Volume 5 owns generic criticality. This chapter owns Bose-fluid and lattice-boson phases, observables, and controlled many-body limits.
    Canonical treatment
    Self-Bound Quantum Droplets
    Scope
    Mixture and dipolar droplet energy functionals, Lee–Huang–Yang stabilization, equilibrium density, surface energy, collective modes, metastability, loss, and beyond-local corrections.
    Assumptions
    Beyond Bogoliubov Theory in the Dilute Expansion; Phase–Density EFT for Bose Superfluids
  • Theory Model

    Localized Probe and Detector Models

    When does a localized probe model define a controlled measurement of a QFT state rather than a formal point interaction?

    Principal question
    When does a localized probe model define a controlled measurement of a QFT state rather than a formal point interaction?
    What the source page covers
    This page owns detector Hilbert space, worldline or worldtube, smearing, switching, coupling, initial state, perturbative order, and recorded observable.
    Boundary
    It does not identify detector excitation with particle number or accept singular point coupling without a regulator and domain analysis.
    Scope
    This page owns detector Hilbert space, worldline or worldtube, smearing, switching, coupling, initial state, perturbative order, and recorded observable.
    Assumptions
    Spurions, Local Counterterms, and Symmetry Response; Microcausality and Relativistic Compatibility; Operational Locality: Couplings, Supports, and Protocols
  • Theory Model

    FLRW Fields and Mode Quantization

    How does a covariant field equation on an FLRW geometry reduce to normalized time-dependent modes without turning a basis choice into a preferred particle interpretation?

    Principal question
    How does a covariant field equation on an FLRW geometry reduce to normalized time-dependent modes without turning a basis choice into a preferred particle interpretation?
    What the source page covers
    This page owns the FLRW metric conventions, scalar mode decomposition, canonical Wronskian normalization, conformal-time evolution, and reconstruction of the field algebra from modes.
    Boundary
    It does not select a vacuum, define an observer-independent particle number, prescribe adiabatic subtraction, or solve the coupled semiclassical Friedmann equations.
    Scope
    This page owns the FLRW metric conventions, scalar mode decomposition, canonical Wronskian normalization, conformal-time evolution, and reconstruction of the field algebra from modes.
    Assumptions
    Covariant Scalar Fields and Curvature Coupling; Covariant Algebraic Quantization and Fock Realizations
  • Theory Model

    The Semiclassical Einstein Equation

    What precisely is asserted by the semiclassical Einstein equation when a renormalized matter stress expectation value sources a classical metric within a controlled approximation?

    Principal question
    What precisely is asserted by the semiclassical Einstein equation when a renormalized matter stress expectation value sources a classical metric within a controlled approximation?
    What the source page covers
    The page owns the mean-field equation, coupling and stress renormalization, state dependence, conservation requirements, approximation ledger, and distinction from operator-valued gravity.
    Boundary
    Stress fluctuations and stochastic sources belong to the next chapter, graviton loops belong to gravity EFT, and ultraviolet-complete quantum spacetime belongs beyond this volume.
    Scope
    The page owns the mean-field equation, coupling and stress renormalization, state dependence, conservation requirements, approximation ledger, and distinction from operator-valued gravity.
    Assumptions
    Conservation, Local Covariance, and the Backreaction Source; Renormalization of Gravitational Couplings by Matter Loops
  • Theory Model

    Conformal and Minimally Coupled Scalars in FLRW

    Which parts of scalar evolution and production in FLRW follow from conformal coupling, minimal coupling, mass, and the chosen scale-factor history?

    Principal question
    Which parts of scalar evolution and production in FLRW follow from conformal coupling, minimal coupling, mass, and the chosen scale-factor history?
    What the source page covers
    This page owns the conformally rescaled scalar equation, effective frequency, conformal-vacuum benchmark, minimally coupled contrast, and curvature-coupling sensitivity ledger.
    Boundary
    It does not promote conformal nonproduction to arbitrary masses or interactions, and it leaves renormalized stress tensors and gravitational backreaction to their dedicated owners.
    Scope
    This page owns the conformally rescaled scalar equation, effective frequency, conformal-vacuum benchmark, minimally coupled contrast, and curvature-coupling sensitivity ledger.
    Assumptions
    FLRW Fields and Mode Quantization; Covariant Scalar Fields and Curvature Coupling
  • Theory Model

    Spinor and Gauge Fields in Expanding Universes

    How are spinor and gauge-field modes defined and constrained on an expanding universe while preserving spin structure, gauge symmetry, and physical polarizations?

    Principal question
    How are spinor and gauge-field modes defined and constrained on an expanding universe while preserving spin structure, gauge symmetry, and physical polarizations?
    What the source page covers
    This page owns FLRW tetrad choices, spinor mode normalization, gauge constraints, ghost or BRST bookkeeping, polarization counting, and conformal benchmarks for non-scalar fields.
    Boundary
    It does not rederive general curved-space spin or gauge geometry, nor does it assign physical particles before a state, asymptotic regime, or detector protocol is specified.
    Scope
    This page owns FLRW tetrad choices, spinor mode normalization, gauge constraints, ghost or BRST bookkeeping, polarization counting, and conformal benchmarks for non-scalar fields.
    Assumptions
    FLRW Fields and Mode Quantization; Spinors, Tetrads, and Spin Connections; Gauge Fields, Gauge Fixing, and Ghosts on Curved Backgrounds
  • Theory Model

    Accelerated Detector Communication Channels

    How do accelerated sender and receiver trajectories, switching clocks, detector response, mode mismatch, and causal delay alter a field-mediated communication protocol?

    Principal question
    How do accelerated sender and receiver trajectories, switching clocks, detector response, mode mismatch, and causal delay alter a field-mediated communication protocol?
    What the source page covers
    Accelerated two-party detector channels with proper-time encodings, trajectory-dependent kernels, finite switching, signal-to-noise separation, and perturbative validation.
    Boundary
    Chapter 4 owns each detector response separately, Volume XIII owns general communication tasks, and this page owns the joint accelerated sender–channel–receiver map.
    Scope
    Accelerated two-party detector channels with proper-time encodings, trajectory-dependent kernels, finite switching, signal-to-noise separation, and perturbative validation.
    Assumptions
    Switching, Smearing, Finite-Time Response, and Transients; Unruh Effect and Uniformly Accelerated Detectors; Curved-Spacetime Channel Deployment Contract
  • Theory Model

    Einstein–Langevin Dynamics

    How does the Einstein–Langevin equation augment a renormalized mean semiclassical equation with a noise source whose covariance is fixed by stress fluctuations?

    Principal question
    How does the Einstein–Langevin equation augment a renormalized mean semiclassical equation with a noise source whose covariance is fixed by stress fluctuations?
    What the source page covers
    The page owns the linear stochastic metric equation, source covariance, retarded solution, induced two-point functions, constraint compatibility, and declared Gaussian-order approximation.
    Boundary
    The stochastic equation does not quantize the metric or replace graviton loops, and its mean must reduce to the causal semiclassical response equation.
    Canonical treatment
    Einstein–Langevin Dynamics
    Scope
    The page owns the linear stochastic metric equation, source covariance, retarded solution, induced two-point functions, constraint compatibility, and declared Gaussian-order approximation.
    Assumptions
    The Stress-Tensor Noise Kernel; Influence Functionals, Dissipation, and Noise; Linear Response and Semiclassical Stability; Langevin Field Equations and Noise
  • Theory Model

    Self-Consistent State–Geometry Solutions

    What makes a state–geometry configuration genuinely self-consistent rather than a quantum stress calculation performed on a background that never satisfies its sourced equation?

    Principal question
    What makes a state–geometry configuration genuinely self-consistent rather than a quantum stress calculation performed on a background that never satisfies its sourced equation?
    What the source page covers
    The page owns fixed-point and iterative solution criteria, state definition on the resulting geometry, boundary and regularity conditions, branch selection, and residual diagnostics.
    Boundary
    Specific cosmological and black-hole benchmarks appear later, existence theorems remain proof-first handoffs, and production solvers must be accompanied by code, inputs, and validation checks.
    Scope
    The page owns fixed-point and iterative solution criteria, state definition on the resulting geometry, boundary and regularity conditions, branch selection, and residual diagnostics.
    Assumptions
    Coupled State–Geometry Initial Data; Constraints, Conservation, and the Bianchi Identity; Constructing Hadamard States by Deformation and Gluing
  • Theory Model

    Entanglement Distribution and State Transfer Through Curved Fields

    Which encodings, wavepackets, trajectories, states, and decoding observables permit entanglement distribution or state transfer through a curved quantum field?

    Principal question
    Which encodings, wavepackets, trajectories, states, and decoding observables permit entanglement distribution or state transfer through a curved quantum field?
    What the source page covers
    Curved implementations of distribution and transfer protocols, including encoding overlap, causal propagation, redshift correction, receiver mode matching, fidelity, and resource accounting.
    Boundary
    Volume XIII owns abstract communication and resource theory, while this page owns geometry-specific implementation and makes no capacity claim from one-shot fidelity alone.
    Scope
    Curved implementations of distribution and transfer protocols, including encoding overlap, causal propagation, redshift correction, receiver mode matching, fidelity, and resource accounting.
    Assumptions
    From Propagators and Response Functions to Channel Maps; Accelerated Detector Communication Channels; Quantum Communication and Entanglement Distribution
  • Theory Model

    Gravitational Production of Massive Relics

    When can expansion alone produce a massive relic abundance, and how do spin, nonadiabaticity, dilution, statistics, and backreaction delimit that prediction?

    Principal question
    When can expansion alone produce a massive relic abundance, and how do spin, nonadiabaticity, dilution, statistics, and backreaction delimit that prediction?
    What the source page covers
    This page owns mode-based relic production, occupation-to-density conversion, spin and coupling dependence, dilution through expansion, and perturbative abundance validity checks.
    Boundary
    It does not own thermal freeze-out, microscopic interactions after production, model-specific dark-matter constraints, or a particle interpretation without a late-time basis.
    Scope
    This page owns mode-based relic production, occupation-to-density conversion, spin and coupling dependence, dilution through expansion, and perturbative abundance validity checks.
    Assumptions
    FLRW Fields and Mode Quantization; Adiabatic Particle Number in Cosmology; Particle Creation in Time-Dependent Backgrounds
  • Theory Model

    Moving Mirrors and the Dynamical Casimir Effect

    How do time-dependent boundaries create quanta and stress flux, and which trajectory, boundary, asymptotic, and renormalization choices determine the result?

    Principal question
    How do time-dependent boundaries create quanta and stress flux, and which trajectory, boundary, asymptotic, and renormalization choices determine the result?
    What the source page covers
    Moving-mirror and dynamical-Casimir kinematics, ray-tracing maps, Bogoliubov coefficients, boundary work, flux observables, and limits of lower-dimensional analogies.
    Boundary
    Chapter 6 owns gravitational collapse and horizons, Chapter 7 owns general stress renormalization, and experimental device modeling belongs to specialist resources.
    Scope
    Moving-mirror and dynamical-Casimir kinematics, ray-tracing maps, Bogoliubov coefficients, boundary work, flux observables, and limits of lower-dimensional analogies.
    Assumptions
    Particle Creation in Time-Dependent Backgrounds; Green Operators, Causal Propagators, and State-Dependent Two-Point Functions
  • Theory Model

    Modified Dispersion, Analogue Horizons, and Universality

    What do modified-dispersion models and analogue horizons reveal about the robustness of horizon mode conversion, and what can they not establish about quantum gravity?

    Principal question
    What do modified-dispersion models and analogue horizons reveal about the robustness of horizon mode conversion, and what can they not establish about quantum gravity?
    What the source page covers
    Controlled dispersive horizon calculations, mode conversion, preferred-frame assumptions, state preparation, universality classes, analogue observables, and evidentiary ceilings.
    Boundary
    Research owns dated analogue experiments, Chapter 19 owns cross-regime validity, and this page does not treat laboratory agreement as confirmation of gravitational ultraviolet physics.
    Scope
    Controlled dispersive horizon calculations, mode conversion, preferred-frame assumptions, state preparation, universality classes, analogue observables, and evidentiary ceilings.
    Assumptions
    Trans-Planckian Sensitivity of the Hawking Derivation; Adiabaticity, Stokes Phenomena, and Production Rates; Effective Field Theory as a Controlled Expansion
  • Theory Model

    Black-Hole Evaporation and Mean Backreaction

    What can a mean semiclassical treatment consistently infer about an evaporating black-hole geometry from renormalized fluxes before adiabaticity or curvature control fails?

    Principal question
    What can a mean semiclassical treatment consistently infer about an evaporating black-hole geometry from renormalized fluxes before adiabaticity or curvature control fails?
    What the source page covers
    The page owns mean mass-loss equations, flux balance, state selection, slowly evolving horizon qualifications, conservation checks, and breakdown diagnostics near uncontrolled regimes.
    Boundary
    Stress fluctuations belong to stochastic gravity, generalized entropy and information claims belong to Chapter 11, and post-semiclassical endpoints remain later-volume handoffs.
    Scope
    The page owns mean mass-loss equations, flux balance, state selection, slowly evolving horizon qualifications, conservation checks, and breakdown diagnostics near uncontrolled regimes.
    Assumptions
    The Semiclassical Einstein Equation; Evaporating Backgrounds and Adiabatic Backreaction; Quantum-State Evolution on Backreacted Backgrounds
  • Theory Model

    BFSS Matrix Quantum Mechanics and M-Theory Conjectures

    What does the BFSS supersymmetric matrix quantum mechanics conjecturally define, in which light-front and large-N limits, and what evidence tests that claim?

    Principal question
    What does the BFSS supersymmetric matrix quantum mechanics conjecturally define, in which light-front and large-N limits, and what evidence tests that claim?
    What the source page covers
    The BFSS degrees of freedom, gauge constraint, Hamiltonian, D0-brane interpretation, longitudinal-momentum scaling, scattering and thermodynamic tests, and explicit conjectural ceiling.
    Boundary
    Volume X owns supersymmetric quantum mechanics, Volume VII owns numerical regulator methods, page 5.11 owns current simulations, and Research owns the evolving M-theory evidence dossier.
    Scope
    The BFSS degrees of freedom, gauge constraint, Hamiltonian, D0-brane interpretation, longitudinal-momentum scaling, scattering and thermodynamic tests, and explicit conjectural ceiling.
    Assumptions
    Nonperturbative Definition Proposals: Objects, Evidence, and Falsifiers; String and M-Theory Duality Webs and Parameter Maps
  • Theory Model

    BMN Plane-Wave Matrix Model and Controlled Sectors

    Which plane-wave M-theory sector is captured by the BMN matrix model, and why does its mass deformation improve control without establishing a universal definition?

    Principal question
    Which plane-wave M-theory sector is captured by the BMN matrix model, and why does its mass deformation improve control without establishing a universal definition?
    What the source page covers
    The BMN Hamiltonian deformation, fuzzy-sphere vacua, supersymmetric spectrum, plane-wave parameter map, perturbative sectors, and relation to rather than replacement of BFSS.
    Boundary
    Volume X owns the supersymmetric representation data, page 5.3 owns BFSS, and Research owns current large-N and numerical evidence for sector recovery.
    Scope
    The BMN Hamiltonian deformation, fuzzy-sphere vacua, supersymmetric spectrum, plane-wave parameter map, perturbative sectors, and relation to rather than replacement of BFSS.
    Assumptions
    BFSS Matrix Quantum Mechanics and M-Theory Conjectures
  • Theory Model

    Rotating and Charged AdS Black Holes

    How do angular momenta, conserved charges, chemical potentials, superradiance, and ensemble choice modify AdS black-hole thermodynamics and stability?

    Principal question
    How do angular momenta, conserved charges, chemical potentials, superradiance, and ensemble choice modify AdS black-hole thermodynamics and stability?
    What the source page covers
    A charged-and-rotating saddle ledger covering horizon potentials, Euclidean identifications, first law, extremality, grand-canonical versus fixed-charge actions, and instability thresholds.
    Boundary
    Volume XI owns finite-density ensembles, Volume XIV owns rotating-horizon QFT and superradiance, Chapter 11 owns charged matter phases, and Chapter 19 owns microscopic BPS counts.
    Scope
    A charged-and-rotating saddle ledger covering horizon potentials, Euclidean identifications, first law, extremality, grand-canonical versus fixed-charge actions, and instability thresholds.
    Assumptions
    AdS Black Branes and Holographic Thermodynamics; Chemical Potentials and Finite-Density Ensembles
  • Theory Model

    IKKT Type-IIB Matrix Model and Emergent-Spacetime Claims

    How is the zero-dimensional IKKT matrix integral proposed to define type-IIB string theory, and what observables would establish emergent Lorentzian spacetime?

    Principal question
    How is the zero-dimensional IKKT matrix integral proposed to define type-IIB string theory, and what observables would establish emergent Lorentzian spacetime?
    What the source page covers
    The IKKT action, gauge and Lorentz symmetries, eigenvalue or matrix-geometry observables, contour and signature choices, large-N scaling, numerical evidence, and unresolved emergence criteria.
    Boundary
    Volume VII owns matrix numerics, page 5.11 owns cross-model evidence, and Research owns live claims about expanding dimensions and Lorentzian contour choices.
    Scope
    The IKKT action, gauge and Lorentz symmetries, eigenvalue or matrix-geometry observables, contour and signature choices, large-N scaling, numerical evidence, and unresolved emergence criteria.
    Assumptions
    Nonperturbative Definition Proposals: Objects, Evidence, and Falsifiers; String and M-Theory Duality Webs and Parameter Maps
  • Theory Model

    Matrix-String Constructions and Second-Quantized Strings

    How do two-dimensional supersymmetric matrix gauge theories recover interacting second-quantized strings in a controlled infrared limit?

    Principal question
    How do two-dimensional supersymmetric matrix gauge theories recover interacting second-quantized strings in a controlled infrared limit?
    What the source page covers
    The matrix-string degrees of freedom, permutation sectors, long-string states, coupling map, string joining and splitting interpretation, and the evidence ceiling outside the infrared construction.
    Boundary
    Volume X owns the supersymmetric gauge theory and duality inputs, page 5.2 owns the duality web, and page 5.3 owns the distinct BFSS conjecture.
    Scope
    The matrix-string degrees of freedom, permutation sectors, long-string states, coupling map, string joining and splitting interpretation, and the evidence ceiling outside the infrared construction.
    Assumptions
    String and M-Theory Duality Webs and Parameter Maps
  • Theory Model

    Open String Field Theory and Tachyon Dynamics

    What off-shell open-string dynamics does string field theory define on a chosen background, and which tachyon-condensation results test that construction?

    Principal question
    What off-shell open-string dynamics does string field theory define on a chosen background, and which tachyon-condensation results test that construction?
    What the source page covers
    The open-string field, star product, BRST kinetic term, gauge structure, level truncation, tachyon potential, D-brane descent tests, and explicit background and convergence limitations.
    Boundary
    Volume III owns BRST/BV foundations, Volume X owns D-brane protected data, page 5.8 owns closed-string formulations, and Research owns convergence and background-independence status.
    Scope
    The open-string field, star product, BRST kinetic term, gauge structure, level truncation, tachyon potential, D-brane descent tests, and explicit background and convergence limitations.
    Assumptions
    Worldsheet Sigma Models and Spacetime Consistency; Nonperturbative Definition Proposals: Objects, Evidence, and Falsifiers
  • Theory Model

    Time-Dependent Geometries and Holographic Thermalization

    How do time-dependent asymptotically AdS geometries model energy injection and approach to a thermal saddle, and which observables diagnose the approach?

    Principal question
    How do time-dependent asymptotically AdS geometries model energy injection and approach to a thermal saddle, and which observables diagnose the approach?
    What the source page covers
    The holographic quench or collapse setup, initial and boundary data, apparent versus event horizons, one-point and nonlocal probes, equilibration scales, and classical-saddle limits.
    Boundary
    Volume XI owns thermalization concepts, Volume XIV owns dynamical horizons, page 10.3 owns initial-state preparation, and page 10.8 owns entanglement growth.
    Scope
    The holographic quench or collapse setup, initial and boundary data, apparent versus event horizons, one-point and nonlocal probes, equilibration scales, and classical-saddle limits.
    Assumptions
    Schwinger–Keldysh Contours and Real-Time Bulk Geometries; Holographic Initial States and Euclidean Caps
  • Theory Model

    Covariant and Closed String Field Theory

    How do covariant and closed string field theories encode gauge-consistent off-shell amplitudes, moduli-space decomposition, and quantum corrections on a specified background?

    Principal question
    How do covariant and closed string field theories encode gauge-consistent off-shell amplitudes, moduli-space decomposition, and quantum corrections on a specified background?
    What the source page covers
    The closed-string field content, BV or L-infinity gauge structure, vertices covering moduli space, loop expansion, background dependence, and current limits on nonperturbative completeness.
    Boundary
    Volume III owns abstract BV machinery, Volume IV owns on-shell amplitudes, page 5.7 owns open-string dynamics, and Volume XVI owns formal homotopy-algebra treatment.
    Scope
    The closed-string field content, BV or L-infinity gauge structure, vertices covering moduli space, loop expansion, background dependence, and current limits on nonperturbative completeness.
    Assumptions
    Worldsheet Sigma Models and Spacetime Consistency; Nonperturbative Definition Proposals: Objects, Evidence, and Falsifiers
  • Theory Model

    D3-Branes and AdS5/CFT4: Parameter-Controlled Regimes and Evidence

    Which sectors and parameter regions of the D3-brane AdS5/CFT4 proposal are calculationally controlled, and what independent evidence supports the broader conjecture?

    Principal question
    Which sectors and parameter regions of the D3-brane AdS5/CFT4 proposal are calculationally controlled, and what independent evidence supports the broader conjecture?
    What the source page covers
    A canonical-example dossier linking N=4 SYM, type-IIB AdS5 times S5, flux, parameter map, protected observables, supergravity regime, string corrections, and unresolved finite-coupling claims.
    Boundary
    Volume X owns N=4 SYM and exact protected data, Chapters 6–10 own holographic calculations, Chapter 5 owns conditional definition claims, and Research owns current evidence updates.
    Scope
    A canonical-example dossier linking N=4 SYM, type-IIB AdS5 times S5, flux, parameter map, protected observables, supergravity regime, string corrections, and unresolved finite-coupling claims.
    Assumptions
    Decoupling Limits and the Original AdS/CFT Proposal; Flux Quantization, Compact Factors, and Kaluza–Klein Towers
  • Theory Model

    Holographic Hydrodynamization, Attractors, and Gradient Asymptotics

    How do holographic far-from-equilibrium solutions approach hydrodynamic attractors before local isotropy or equilibrium, and what controls the divergent gradient expansion?

    Principal question
    How do holographic far-from-equilibrium solutions approach hydrodynamic attractors before local isotropy or equilibrium, and what controls the divergent gradient expansion?
    What the source page covers
    The bulk realization of hydrodynamization, transseries or Borel diagnostics, quasinormal transient sectors, attractor variables, initialization dependence, and model-specific evidence ceiling.
    Boundary
    Volume XI owns hydrodynamic attractors and gradient expansions, Chapter 11 owns phenomenological fluid applications, and this page owns gravitational solution evidence.
    Scope
    The bulk realization of hydrodynamization, transseries or Borel diagnostics, quasinormal transient sectors, attractor variables, initialization dependence, and model-specific evidence ceiling.
    Assumptions
    Bulk Quasinormal Modes and Boundary Hydrodynamic Poles; Time-Dependent Geometries and Holographic Thermalization; Hydrodynamic Attractors and Asymptotic Gradient Expansions
  • Theory Model

    M2, M5, and Higher-Dimensional Brane Examples

    How do M2- and M5-brane near-horizon limits generate AdS4 and AdS7 dictionaries, and which degrees-of-freedom scalings and field-theory definitions are established?

    Principal question
    How do M2- and M5-brane near-horizon limits generate AdS4 and AdS7 dictionaries, and which degrees-of-freedom scalings and field-theory definitions are established?
    What the source page covers
    A comparative M-brane ledger for AdS4 times S7 and AdS7 times S4, flux quantization, N-scaling, supersymmetry, candidate boundary theories, and limits of eleven-dimensional supergravity.
    Boundary
    Volume X owns three- and six-dimensional supersymmetric field-theory data, Volume XIV owns eleven-dimensional supergravity as EFT, and Chapter 5 owns M-theory definition proposals.
    Scope
    A comparative M-brane ledger for AdS4 times S7 and AdS7 times S4, flux quantization, N-scaling, supersymmetry, candidate boundary theories, and limits of eleven-dimensional supergravity.
    Assumptions
    Near-Horizon Brane Geometries and Top-Down Dictionaries; Flux Quantization, Compact Factors, and Kaluza–Klein Towers
  • Theory Model

    D1-D5 Systems and AdS3 Top-Down Data

    Which charge, compactification, moduli, and decoupling data define the D1-D5 AdS3/CFT2 system before its use in three-dimensional gravity or microstate counting?

    Principal question
    Which charge, compactification, moduli, and decoupling data define the D1-D5 AdS3/CFT2 system before its use in three-dimensional gravity or microstate counting?
    What the source page covers
    The D1-D5 top-down construction, AdS3 times S3 compact factors, central-charge scaling, moduli dependence, protected sectors, and separation between symmetric-orbifold and supergravity points.
    Boundary
    Volume IX owns two-dimensional CFT, Volume X owns protected supersymmetry, Chapter 17 owns AdS3 gravity, and Chapter 19 owns microstate entropy and dynamics.
    Scope
    The D1-D5 top-down construction, AdS3 times S3 compact factors, central-charge scaling, moduli dependence, protected sectors, and separation between symmetric-orbifold and supergravity points.
    Assumptions
    Near-Horizon Brane Geometries and Top-Down Dictionaries; Flux Quantization, Compact Factors, and Kaluza–Klein Towers
  • Theory Model

    Algebraic Free Fields on Curved Spacetimes

    How are classical solution spaces, symplectic or inner products, and CCR or CAR relations assembled into a locally covariant free-field algebra on curved spacetime?

    Principal question
    How are classical solution spaces, symplectic or inner products, and CCR or CAR relations assembled into a locally covariant free-field algebra on curved spacetime?
    What the source page covers
    Scalar, Dirac, and selected gauge-free-field algebra construction, quotient by equations of motion, causal propagators, *-relations, time-slice property, functorial embeddings, and choices required by spin or gauge structure.
    Boundary
    State selection, particle interpretation, and curved-spacetime phenomenology remain in Volume XIV; interacting algebras come later and no preferred Fock representation is assumed.
    Scope
    Scalar, Dirac, and selected gauge-free-field algebra construction, quotient by equations of motion, causal propagators, *-relations, time-slice property, functorial embeddings, and choices required by spin or gauge structure.
    Assumptions
    Globally Hyperbolic Spacetimes and the Loc Categories; Green-Hyperbolic Operators and Causal Propagators; Haag–Kastler Nets and Locality
  • Theory Model

    Gaussian Euclidean Fields as Measures

    How does a positive covariance define a Gaussian random distribution, and which support, regularity, reflection, and clustering properties follow from the covariance rather than from formal path-integral notation?

    Principal question
    How does a positive covariance define a Gaussian random distribution, and which support, regularity, reflection, and clustering properties follow from the covariance rather than from formal path-integral notation?
    What the source page covers
    Gaussian measures on distribution spaces, characteristic functionals, covariance operators, Cameron–Martin directions, Wick moments, reflection positivity tests, mass dependence, and the precise topology carrying the measure.
    Boundary
    General measure theory remains in Mathematical Methods, and interacting perturbations begin on the next page; this page does not identify a symbolic determinant with a countably additive field measure.
    Scope
    Gaussian measures on distribution spaces, characteristic functionals, covariance operators, Cameron–Martin directions, Wick moments, reflection positivity tests, mass dependence, and the precise topology carrying the measure.
    Assumptions
    Euclidean Random Fields and Schwinger Hierarchies; Osterwalder–Schrader Axioms and Reflection Positivity
  • Theory Model

    Noninvertible Symmetries, Fusion, and Junction Data

    Which fusion, junction, associator, dual, and dimension data make a system of topological defects a noninvertible symmetry rather than a collection of operators?

    Principal question
    Which fusion, junction, associator, dual, and dimension data make a system of topological defects a noninvertible symmetry rather than a collection of operators?
    What the source page covers
    Non-group-like fusion rules, fusion categories of defects, junction vector spaces, F-symbols, duals, categorical dimensions, pivotal choices, topological action, and consistency conditions from higher associativity.
    Boundary
    Physical construction mechanisms and RG constraints remain in Volume III; this page owns the categorical consistency package and its failure tests.
    Scope
    Non-group-like fusion rules, fusion categories of defects, junction vector spaces, F-symbols, duals, categorical dimensions, pivotal choices, topological action, and consistency conditions from higher associativity.
    Assumptions
    Defects on Stratified Spacetimes and Higher-Categorical Composition; Fusion Categories, Module Categories, and Bimodule Defects; Non-Invertible Topological Defects and Fusion
  • Theory Model

    Edge Modes and Extended Observables at Gauge Boundaries

    When are boundary edge modes required to restore gauge covariance or factorization, and which extended observables do they support without becoming gauge artifacts?

    Principal question
    When are boundary edge modes required to restore gauge covariance or factorization, and which extended observables do they support without becoming gauge artifacts?
    What the source page covers
    The theorem interface among boundary gauge transformations, extended phase spaces, edge-mode fields, dressed observables, boundary charge actions, gluing, and the dependence on chosen boundary conditions.
    Boundary
    Physical subregion factorization, entanglement centers, and edge-mode interpretations remain in Volumes III and XIII; this page supplies the BV–BFV consistency tests.
    Scope
    The theorem interface among boundary gauge transformations, extended phase spaces, edge-mode fields, dressed observables, boundary charge actions, gluing, and the dependence on chosen boundary conditions.
    Assumptions
    Boundary Phase Spaces, Constraints, and the BFV Charge; Gluing, Reduction, and Composition Theorems; Proper and Improper Gauge Transformations
  • Theory Model

    Fewster–Verch Probe Measurements, State Updates, and Causal Composition

    How does a compactly supported system–probe coupling induce a scattering morphism, observable readout, and state update whose causal composition can be proved on a globally hyperbolic spacetime?

    Principal question
    How does a compactly supported system–probe coupling induce a scattering morphism, observable readout, and state update whose causal composition can be proved on a globally hyperbolic spacetime?
    What the source page covers
    The Fewster–Verch measurement scheme, coupling-zone localization, induced observables, pre-instruments, causal factorization for ordered coupling regions, and assumptions needed for probe preparation and readout.
    Boundary
    Phenomenological detector response and experimental modeling remain with Curved Spacetime and Quantum Information; existence of an interacting coupling is assumed or separately constructed rather than hidden in the measurement notation.
    Scope
    The Fewster–Verch measurement scheme, coupling-zone localization, induced observables, pre-instruments, causal factorization for ordered coupling regions, and assumptions needed for probe preparation and readout.
    Assumptions
    Local Operations, Instruments, and AQFT Measurement; Locally Covariant QFT as a Functor
  • Theory Model

    AKSZ Sigma Models as BV–BFV Examples

    How does the AKSZ construction produce BV theories and compatible boundary BFV data from graded symplectic targets and Hamiltonian functions?

    Principal question
    How does the AKSZ construction produce BV theories and compatible boundary BFV data from graded symplectic targets and Hamiltonian functions?
    What the source page covers
    The AKSZ mapping-space construction, source Q-structure, shifted target symplectic form, Hamiltonian master function, transgressed BV action, boundary term, admissible boundary conditions, and controlled sigma-model examples.
    Boundary
    Physical Chern–Simons, BF, and sigma-model dynamics remain in Volumes III and X; this page uses them as mathematical BV–BFV exemplars rather than a universal quantization method.
    Scope
    The AKSZ mapping-space construction, source Q-structure, shifted target symplectic form, Hamiltonian master function, transgressed BV action, boundary term, admissible boundary conditions, and controlled sigma-model examples.
    Assumptions
    The BV Complex and Classical Master Equation; BV–BFV Structures, Boundaries, and Gluing; Bulk–Boundary Master Equations and Anomaly Inflow
  • Theory Model

    Constructive Fermionic and Yukawa Models

    How do Grassmann integration, determinant bounds, multiscale estimates, and bosonic stability combine in constructive fermion and Yukawa models, and in which dimensions and coupling regimes?

    Principal question
    How do Grassmann integration, determinant bounds, multiscale estimates, and bosonic stability combine in constructive fermion and Yukawa models, and in which dimensions and coupling regimes?
    What the source page covers
    Finite Grassmann measures, covariance slicing, Gram bounds, fermionic cluster expansions, Yukawa stability, renormalization conditions, and the exact convergence or analyticity domains of representative low-dimensional models.
    Boundary
    Phenomenological Yukawa sectors and Standard Model parameters remain in Volume VI, while formal fermionic perturbation theory remains in Scattering; no positive bosonic probability measure is presumed for Grassmann fields.
    Scope
    Finite Grassmann measures, covariance slicing, Gram bounds, fermionic cluster expansions, Yukawa stability, renormalization conditions, and the exact convergence or analyticity domains of representative low-dimensional models.
    Assumptions
    The Constructive Program and Cutoff Removal; Cluster Expansions and Correlation Inequalities
  • Theory Model

    Weakly Self-Avoiding Walk and Supersymmetric RG

    How is weakly self-avoiding walk represented by a supersymmetric field integral, and which RG estimates yield its critical two-point and susceptibility asymptotics?

    Principal question
    How is weakly self-avoiding walk represented by a supersymmetric field integral, and which RG estimates yield its critical two-point and susceptibility asymptotics?
    What the source page covers
    The continuous-time weakly self-avoiding walk representation, boson–fermion cancellation, supersymmetric localization identities used in the proof, finite-range RG, critical parameter tuning, and dimension-specific asymptotics.
    Boundary
    Polymer phenomenology remains in Many-Body Quantum Matter and physical spacetime supersymmetry remains in Volume X; the auxiliary supersymmetry here is not a supersymmetric QFT claim.
    Scope
    The continuous-time weakly self-avoiding walk representation, boson–fermion cancellation, supersymmetric localization identities used in the proof, finite-range RG, critical parameter tuning, and dimension-specific asymptotics.
    Assumptions
    Finite-Range Decompositions and Multiscale Integration; Polymer Activities and Normed RG Coordinates; Stable Manifolds and Relevant–Marginal Control
  • Theory Model

    Factorizing S-Matrices, Wedge-Local Fields, and Borchers Constructions

    How can a factorizing two-particle S-matrix seed wedge-local operators, a Borchers triple, and ultimately nontrivial local algebras in two-dimensional integrable QFT?

    Principal question
    How can a factorizing two-particle S-matrix seed wedge-local operators, a Borchers triple, and ultimately nontrivial local algebras in two-dimensional integrable QFT?
    What the source page covers
    The bounded construction chain from analytic unitary crossing-symmetric scattering function through Zamolodchikov–Faddeev operators, wedge locality, Borchers triples, and the locality obligation.
    Boundary
    Volume VII owns exact S-matrix bootstrap and phenomenology; this page treats only operator-algebraic model construction under stated regularity and nuclearity hypotheses.
    Scope
    The bounded construction chain from analytic unitary crossing-symmetric scattering function through Zamolodchikov–Faddeev operators, wedge locality, Borchers triples, and the locality obligation.
    Assumptions
    Scattering Analyticity, Crossing, and Rigorous Bounds; Haag–Kastler Nets and Locality; Isotony, Additivity, Duality, and Primitive Causality
  • Theory Model

    Full Two-Dimensional CFT from Chiral Nets

    How can left- and right-chiral conformal nets be combined into a local full two-dimensional CFT, and which modular-invariant or Q-system data are actually sufficient?

    Principal question
    How can left- and right-chiral conformal nets be combined into a local full two-dimensional CFT, and which modular-invariant or Q-system data are actually sufficient?
    What the source page covers
    Two-dimensional local nets from chiral components, braided product and full-center constructions, locality across light rays, vacuum sector, modular invariants, boundary conditions, and distinctions between full, chiral, and Euclidean formulations.
    Boundary
    Full correlator bootstrap and sewing remain in Volume IX, while VOA full-field constructions remain on their canonical pages; this page owns the net-theoretic route.
    Scope
    Two-dimensional local nets from chiral components, braided product and full-center constructions, locality across light rays, vacuum sector, modular invariants, boundary conditions, and distinctions between full, chiral, and Euclidean formulations.
    Assumptions
    Conformal Nets and Covariance Axioms; Extensions, Orbifolds, Cosets, and Alpha-Induction; Chiral Blocks, Sewing, and Modular Invariance
  • Theory Model

    Global Anomalies, Determinant Lines, and Eta Invariants

    Which global gauge or gravitational transformations produce nontrivial determinant-line holonomy, and how do eta invariants or bordism data detect anomalies invisible to local polynomials?

    Principal question
    Which global gauge or gravitational transformations produce nontrivial determinant-line holonomy, and how do eta invariants or bordism data detect anomalies invisible to local polynomials?
    What the source page covers
    Global anomaly phases, determinant and Pfaffian line holonomy, mapping tori, spectral flow, eta invariants, torsion and bordism refinements, dependence on spin or pin structure, and comparison with local anomalies.
    Boundary
    Phenomenological anomaly constraints remain in Volumes III and VI, while invertible-field-theory packaging is treated in Chapter 23.
    Scope
    Global anomaly phases, determinant and Pfaffian line holonomy, mapping tori, spectral flow, eta invariants, torsion and bordism refinements, dependence on spin or pin structure, and comparison with local anomalies.
    Assumptions
    Determinant Lines, Global Obstructions, and Orientations; Quantum Master Equation and Anomaly Obstructions; Global and Torsion Anomalies
  • Theory Model

    Holomorphic, Topological, and Mixed Factorization Theories

    Which translation invariances and cohomological structures make a factorization theory holomorphic, topological, or mixed, and what analytic data survive in each direction?

    Principal question
    Which translation invariances and cohomological structures make a factorization theory holomorphic, topological, or mixed, and what analytic data survive in each direction?
    What the source page covers
    Holomorphic factorization algebras on complex manifolds, topological local constancy, mixed structures, Dolbeault resolutions, chiral observables, anomaly qualifications, twists, and topology-dependent global observables.
    Boundary
    Physical twisting and supersymmetric dynamics remain in Volume X, while chiral CFT data remain in Volume IX; a cohomological twist is not equivalent to the untwisted unitary theory.
    Scope
    Holomorphic factorization algebras on complex manifolds, topological local constancy, mixed structures, Dolbeault resolutions, chiral observables, anomaly qualifications, twists, and topology-dependent global observables.
    Assumptions
    Prefactorization and Factorization Algebras; BV Quantization and Obstruction–Deformation Complexes