W-symmetry in Conformal Field Theory

Abstract

We review various aspects of $\cW$-algebra symmetry in two-dimensional conformal field theory and string theory. We pay particular attention to the construction of $\cW$-algebras through the quantum Drinfeld-Sokolov reduction and through the coset construction.

137 pages (Corrections in Chapter 7 where some incorrect formulas were taken out, see `Note added' for Comments.)

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• Integrable anyon chains: from fusion rules to face models to effective field theories
• Higher spin algebras and large $\mathcal{N}=4$ holography
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• The Operator Product Expansion of the Lowest Higher Spin Current at Finite N
• On the Completeness of the Set of Classical W-Algebras Obtained from DS Reductions
• A semi-local holographic minimal model
• Quantization of a relativistic particle on the SL(2,R) manifold based on Hamiltonian reduction
• A lattice approach to the conformal $\OSp(2S+2|2S)$ supercoset sigma model. Part II: The boundary spectrum
• The structure of the Kac-Wang-Yan algebra
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• 2D CFT blocks for the 4D class ${\mathcal{S}}_k$ theories
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• Free-field realisation of boundary states and boundary correlation functions of minimal models
• Higher Spin Currents in the N=2 Stringy Coset Minimal Model
• Spin-5 Casimir Operator and Its Three-Point Functions with Two Scalars
• The Quantum Spectrum of the Conserved Charges in Affine Toda Theories
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• The Operator Product Expansion between the 16 Lowest Higher Spin Currents in the N=4 Superspace
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• Unitary representations of SW(3/2,2) superconformal algebra
• The Complete structure of the nonlinear $W_4$ and $W_5$ algebras from quantum Miura transformation
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• Complex saddles of three-dimensional de Sitter gravity via holography
• W_k structure of generalized Frenkel-Kac construction for SU(2)-level k Kac-Moody algebra
• Comments on MHV Tree Amplitudes for Conformal Supergravitons from Topological B-Model
• Linearizing W-Algebras
• $d>2$ Stress-Tensor OPE near a Line
• Schwarzian derivative treatment of the quantum second-order supersymmetry anomaly, and coupling-constant metamorphosis
• Chiral Algebras of Two-Dimensional SYK Models
• Conformal invariance and its breaking in a stochastic model of a fluctuating interface
• Double Trace Interfaces
• Holographic Chern-Simons Theories
• Semi-infinite cohomology of W-algebras
• Higher Spin Currents in Orthogonal Wolf Space
• W-algebras from symplectomorphisms
• Quantum line operators from Lax pairs
• Higher Spin Currents in Wolf Space for Generic N
• Quasifinite Highest Weight Modules over Super $W_{1+\infty }$ Algebra
• On the combinatorics of partition functions in AdS3/LCFT2
• Finite W_3 Transformations in a Multi-time Approach
• Supersymmetric W-algebras
• Modularity of Bershadsky-Polyakov minimal models
• N=2 gauge theories, instanton moduli spaces and geometric representation theory
• The vacuum preserving Lie algebra of a classical W-algebra
• Modularity and 4D-2D spectral equivalences for large-N gauge theories with adjoint matter
• Twisted sectors from plane partitions
• Polynomial super-gl(n) algebras
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• Local energy bounds and strong locality in chiral CFT
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• Subregular W-algebras of type A
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• On the mass-coupling relation of multi-scale quantum integrable models
• Little String Origin of Surface Defects
• Quantum criticality in many-body parafermion chains
• The W_N minimal model classification
• KPZ Analysis for $W_3$ Gravity
• Langlands duality in Liouville-H_3^+ WZNW correspondence
• Graph fusion algebras of WLM(p,p')
• Generalized NLS Hierarchies from Rational $W$ Algebras
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• Free Superfield Realization of N=2 Quantum Super W_{3} Algebra
• More on Affine Dynkin Quiver Yangians
• Coset construction of logarithmic minimal models: branching rules and branching functions
• On SW-minimal models and N=1 supersymmetric Quantum Toda-field theories
• How Many $N=4$ Strings Exist ?
• Lectures on conformal field theory and Kac-Moody algebras
• Explicit Construction of $N=2$ $W_3$ Current in the $N=2$ Coset $\frac{SU(3)}{SU(2)\times U(1)}$ Model
• Holographic four-point functions in Toda field theories in $Ad{S}_2$
• Free boson formulation of boundary states in W_3 minimal models and the critical Potts model
• Different facets of the raise and peel model
• A new realization of holography
• Chiral 2D "Strange Metals" from N = 4 SYM
• Consistent anomalies of the induced W gravities
• The BRST quantization of the nonlinear $W{B}_2$ and $W_4$ algebras
• Singularities of nilpotent Slodowy slices and collapsing levels of W-algebras
• Free Field Realization of $WB{C}_n$ and $W{G}_2$ algebras
• Wilson Loop Invariants from $W_N$ Conformal Blocks
• Unitary representations of the ${\mathcal{W}}_3$-algebra with $c\ge 2$
• Jain states on a torus: an unifying description
• Geometry of W-algebras from the affine Lie algebra point of view
• The geometry of W3 algebra: a twofold way for the rebirth of a symmetry
• Metaplectic Covariance of the Weyl-Wigner-Groenewold-Moyal Quantization and Beyond
• Fusion Algebras and Characters of Rational Conformal Field Theories
• Exact Path Integral for 3D Higher Spin Gravity
• Higher-order Galilean contractions
• Symmetry Algebras of Stringy Cosets
• On even spin $W_{\infty }$
• Higher Spin Currents in the Orthogonal Coset Theory
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• Conformal embeddings and higher-spin bulk duals
• The BV-algebra structure of W_3 cohomology
• The Orbifolds of Permutation-Type as Physical String Systems at Multiples of c=26. I.Extended Actions and New Twisted World-Sheet Gravities
• Null Fields Realizations of $W_3$ from $W(sl(4),sl(3))$ and $W(sl(3|1),sl(3))$ Algebras
• A Supersymmetric Enhancement of $\mathcal{N}=1$ Holographic Minimal Model
• On $\widehat{sl}(3)$ reduction, quantum gauge transformations, and $\mathcal{W}-$ algebras singular vectors
• On a class of conformal $\mathcal{E}$-models and their chiral Poisson algebras
• A geometrical approach to super W-induced gravities in two dimensions
• Ward identities in the ${\mathfrak{sl}}_3$ Toda conformal field theory
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• $W_n^{(\ka)}$ algebra associated with the Moyal KdV Hierarchy
• Yangian Symmetries in the $SU(N{)}_1$ WZW Model and the Calogero-Sutherland Model
• Non-perturbative effective field theory for two-leg antiferromagnetic spin ladders
• Supersymmetric Wilson Loops, Instantons, and Deformed $\mathcal{W}$-Algebras
• Central Charge and the Andrews-Bailey Construction
• Singular vectors for the $W_N$ algebras
• Torus structure on graphs and twisted partition functions for minimal and affine models
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• The Higher Spin Rectangle
• SW(3/2,2) subsymmetry in G${}_2$, Spin(7) and N=2 CFTs
• Multi-graded Galilean conformal algebras
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• Adding Complex Fermions to the Grassmannian-like Coset Model
• A new approach to the correlation functions of W-algebras
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• Gauging Spacetime Symmetries on the Worldsheet and the Geometric Langlands Program
• Semi-infinite cohomology in conformal field theory and 2d gravity
• Higher Spin Currents with Manifest $SO(4)$ Symmetry in the Large $\mathcal{N}=4$ Holography
• $\mathcal{S}\mathcal{W}$-algebras and strings with torsion
• Generalized Toda Theory from Six Dimensions and the Conifold
• A Note on Representations of N=2 SW-Algebras
• Integrals of motion in conformal field theory with W-symmetry and the ODE/IM correspondence
• On the Classification of Real Forms of Non-Abelian Toda Theories and W-algebras
• Non-Trivial Non-Canonical W-Algebras from Kac-Moody Reductions
• Miura-Like Free Field Realization Of Fermionic Casimir WB(3) Algebras
• Fermionic realisations of simple Lie algebras
• Generators of Supersymmetric Classical $W$-algebras
• On the large N limit, W_\infty Strings, Star products, AdS/CFT Duality, Nonlinear Sigma Models on AdS spaces and Chern-Simons p-branes
• $A_2^{(2)}$ Parafermions: A New Conformal Field Theory
• Induced quantum gravity on a Riemann Surface
• 4d Chern-Simons Theory as a 3d Toda Theory, and a 3d-2d Correspondence
• The Grassmannian-like Coset Model and the Higher Spin Currents
• Explicit Construction of Spin 4 Casimir Operator in the Coset Model $\widehat{SO}(5{)}_1\times \widehat{SO}(5{)}_m/\widehat{SO}(5{)}_{1+m}$
• The imaginary Toda field theory
• Superconformal theories from Pseudoparticle Mechanics
• Zero-curvature condition in two dimensions. Relativistic particle models and finite \W-transformations
• On the Casimir ${\mathcal{W}\mathcal{A}}_N$ algebras as the truncated ${\mathcal{W}}_{\infty }$ algebra
• Twisted Conformal Field Theories and Morita equivalence
• Consistently Constrained SL(N) WZWN Models and Classical Exchange Algebra
• The W_3 algebra: modules, semi-infinite cohomology and BV-algebras
• Conformally Invariant Sigma Models on Anti de Sitter Spaces, Chern-Simons p-branes and W Geometry
• Some Aspects of c=-2 Theory
• 't Hooft Anomaly Matching Conditions for Generalized Symmetries in 2D
• The Relation between Linear and Non-Linear $N=3,4$ Supergravity Theories
• Correlation Functions of The Tri-critical 3-states Potts Model
• W Algebras, W Gravities and their Moduli Spaces
• $W$-algebras and higher analogs of the Kniznik-Zamolodchikov equations
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• On the Super Field Realization of Super Casimir Wa(n)-Algebras
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• On the Miura and Backlund transformations associated with the supersymmetric Gelfand-Dickey bracket
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• Symmetric deformed 2D/3D Hurwitz-Kontsevich model and affine Yangian of gl(1)
• Extended Non-Abelian Gauge Symmetries in Classical WZNW Model
• The geometry of Casimir W-algebras
• Fermionic Construction in the Supersymmetric Coset Model
• The KP Equation from Plebanski and $SU(\infty )$ Self-Dual Yang-Mills
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• Ward Identities and Anomalies in Pure W_4 Gravity
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• On the free field realization of $WB{C}_n$ algebras
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• Drinfeld--Sokolov Gravity
• SU(m) non-Abelian anyons in the Jain hierarchy of quantum Hall states
• Gelfand-Dickey Algebra and Higher Spin Symmetries On $T^2=S^1\times S^1$
• Extended Nonabelian Symmetries for Free Fermionic Model
• Remarks on A_2 Toda Field Theory
• New approaches in W-gravities
• A new Rational Conformal Field Theory extension of the fully degenerate W_{1+\infty}^{(m)}
• BV-Structure of the Cohomology of Nilpotent Subalgebras and the Geometry of (W-) Strings
• On the Classical $W_4^{(2)}$ Algebra
• A new class of Matrix Models arising from the W-infinity Algebra
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• Boundary critical phenomena in the quantum Ashkin-Teller model
• Brane Symmetries Revisited: Symmetries of Tensile and Tensionless Branes in Possibly Degenerate Metrics and their Manifestations
• W$(2,4)$, Linear and Non-local \W-Algebras in Sp(4) Particle Model
• Two-dimensional conformal field theories with matrix-valued level