Introductory Lectures on Higher-Spin Theories

Preface Contents Contributors 1 From Higher Spins to Strings: A Primer 1.1 Introduction 1.1.1 Motivations 1.1.2 Historical Overview 1.1.3 Outline 1.1.4 Conventions and Some Prerequisites 1.2 Free HS in Flat Space 1.2.1 Poincaré Group and Wigner Classification 1.2.1.1 Massive Case 1.2.1.2 Massless Case 1.2.2 Bargmann-Wigner Program 1.2.2.1 Totally Symmetric Massive Fields 1.2.2.2 Symmetric Massless Fields 1.2.3 Lagrangian Formulation 1.2.3.1 Symmetric Massive Fields 1.2.3.2 Symmetric Massless Fields 1.2.3.3 Massive Lagrangian from Kaluza-Klein Reduction 1.3 Higher Spins in Anti-de Sitter Space 1.3.1 The AdS Geometry 1.3.2 Isometry Group and Unitary Representations 1.3.3 Unitarity Bound, Masslessness and Wave Equations 1.4 Beyond Free Theories 1.4.1 Massless HS Fields and No-Go Theorems 1.4.1.1 Weinberg (1964) 1.4.1.2 Coleman-Mandula (1967) and Extension Thereof 1.4.1.3 Aragone-Deser (1979) 1.4.1.4 Weinberg-Witten (1980) and Its Generalization 1.4.2 Yes-Go Results for HS Interactions 1.4.3 Problems with Interacting Massive HS Fields 1.4.4 Causal Propagation of Massive HS Fields 1.4.4.1 Charged Open Strings in Constant EM Background 1.4.4.2 Physical State Conditions 1.4.4.3 Causal Propagation of String Fields 1.5 Unfolded Formulation and HS Equations 1.5.1 From Riemann to Lopatin-Vasiliev 1.5.1.1 Riemann Normal Coordinates and Riemannian Geometry 1.5.1.2 Lopatin-Vasiliev Formulation of Free Higher Spins 1.5.1.3 Unfolding the Killing Equation: The Gauge Module 1.5.2 Unfolded Equations in Anti-de Sitter Space 1.5.2.1 HS Algebra 1.5.2.2 Non-linear Unfolded Equations 1.5.2.3 Unfolding, Jet Space and BRST-BV Formalism 1.6 From Higher Spins to Strings 1.6.1 Vacuum Solutions, Flat Connections and AdS Space 1.6.2 Linear Order 1.6.3 The Boundary-to-Bulk Propagator Solution 1.6.4 The 1-Form Sector 1.6.5 Asymptotic Symmetries of the AdS Theory 1.6.6 ABJM Triality References 2 Higher-Spin Gauge Theories in Three Spacetime Dimensions 2.1 Introduction 2.2 Higher Spins and Chern–Simons Theory 2.2.1 Three-Dimensional Gravity as a Chern–Simons Theory 2.2.2 Including Higher-Spin Fields 2.2.2.1 Metric-Like Formulation 2.2.2.2 Frame-Like Formulation 2.2.2.3 Chern–Simons Formulation 2.3 Asymptotic Symmetries 2.3.1 Chern–Simons Theory with Boundary 2.3.2 Asymptotically AdS Solutions: The Drinfeld–Sokolov Reduction 2.3.3 The sl(2) Example: Asymptotic Symmetries of Gravity 2.3.4 The sl(3) Case 2.3.4.1 The Highest-Weight Gauge 2.3.4.2 The Single-Row Gauge 2.3.5 Beyond sl(3) 2.4 Quantum W-Algebras and Minimal-Model Holography 2.4.1 Motivation 2.4.2 W-Algebras and Operator Product Expansion 2.4.3 The Quantum W3-Algebra 2.4.4 WN and the Quantum Miura Transform 2.4.5 The W∞[λ] Quantum Algebra and Its Triality 2.4.6 Conformal Field Theories with W-Symmetry 2.4.7 Minimal-Model Holography 2.5 Coupling to Matter 2.5.1 Free Scalar Field in the Unfolded Formulation 2.5.1.1 Matter Fields 2.5.1.2 Oscillator Realisation of hs[λ] 2.5.1.3 Higher-Spin and Matter Fields Using Oscillators 2.5.1.4 Twisted Fields 2.5.2 Non-Linear Theory 2.5.2.1 Master Fields and Vasiliev Equations 2.5.2.2 Higher-Spin Background Solutions 2.5.2.3 Linear Perturbations 2.5.2.4 Subtleties of the Twisted Sector 2.5.2.5 Comments on Higher Orders 2.6 Summary and Further Developments Appendix 1: Frame-Like Formulation in Arbitrary Dimension Appendix 2: More on the Classical Miura Transformation Appendix 3: Constrained Hamiltonian Systems Appendix 4: Solutions to Selected Exercises References 3 Elements of Vasiliev Theory 3.1 Introduction 3.2 Metric-Like Formulation for Free HS Fields 3.2.1 Massless Fields on Minkowski Background 3.2.1.1 Going On-Shell 3.2.1.2 Counting Degrees of Freedom 3.2.1.3 Lagrangian 3.2.1.4 The Long and Winding Road from Representations to Lagrangians 3.2.2 Massless Fields on (Anti)-de Sitter Background 3.2.2.1 Summary 3.3 Gravity as Gauge Theory 3.3.1 Tetrad, Vielbein, Frame, Vierbein,.... 3.3.1.1 Covariant Derivative 3.3.2 Gravity as a Gauge Theory 3.3.2.1 Short Summary on Yang-Mills 3.3.2.2 Back to Gravity 3.3.2.3 Note on First-Order Actions 3.3.2.4 Back to Gravity Again 3.3.2.5 More on Gravity as a Gauge Theory 3.3.2.6 Most Symmetric Background Is Equivalent to flat connection 3.3.2.7 Summary 3.4 Unfolding Gravity 3.5 Unfolding, Spin by Spin 3.5.1 spin-two Retrospectively 3.5.2 higher-spin 3.5.2.1 Unfolded Equations for any spin 3.5.2.2 Fronsdal Equations from Unfolded Ones 3.5.2.3 Rough Structure of HS Connections and Zero-Forms 3.5.2.4 Lower-Spins 3.5.3 scalar 3.5.3.1 Unfolded Equations for spin-zero 3.5.3.2 The Scalar Is Almost Tautological 3.5.4 spin-one 3.5.4.1 Unfolded Equations for spin-one 3.5.5 Zero-Forms 3.5.6 All Spins Together 3.5.6.1 Summary 3.6 Unfolding 3.6.1 Basics 3.6.2 Structure Constants 3.6.2.1 Lie Algebras and Flatness/Zero-Curvature 3.6.2.2 Modules and Covariant Constancy 3.6.2.3 Contractible Cycles and Empty Equations 3.6.2.4 Cocycles and Couplings 3.6.2.5 Zero-Forms and Degrees of Freedom 3.6.2.6 Unfolded Systems of HS Type 3.6.2.7 Summary Higher-Spin Theory in Four Dimensions 3.7 Vector-Spinor Dictionary 3.7.1 so(3,1)-sl(2,C) dictionary 3.7.2 Dictionary 3.8 Free HS Fields in AdS-four 3.8.1 Massless Scalar and HS Zero-Forms on Minkowski 3.8.2 Spinor Version of AdS-four Background 3.8.3 HS Zero-Forms on AdS-four 3.8.4 HS Gauge Potentials on AdS-four 3.8.4.1 Summary 3.9 Higher-Spin Algebras 3.10 Vasiliev Equations 3.10.1 Generalities 3.10.2 Quasi Derivation of Vasiliev Equations 3.10.2.1 Extended Star-Product, Klein Operators 3.10.2.2 Back to HS Equations 3.10.2.3 Deformed Oscillators 3.10.2.4 Vasiliev Equations 3.10.2.5 Reality Conditions 3.10.2.6 Generalizations and Reductions 3.10.3 Perturbation Theory 3.10.4 Manifest Lorentz Symmetry 3.10.4.1 Generalities 3.10.4.2 Implementation 3.10.5 Higher Orders and Gauge Fixing 3.10.5.1 Second-Order Example 3.10.6 Topological Fields and Integrating Flow 3.11 Extras 3.11.1 Fronsdal Operator on Riemannian Manifolds 3.11.2 Other Gravity-Like Actions 3.11.3 MacDowell-Mansouri-Stelle-West 3.11.4 Interplay Between Diffeomorphisms and Gauge Symmetries 3.11.5 Chevalley-Eilenberg Cohomology and Interactions 3.11.5.1 Note on Cohomologies of Lie Algebra 3.11.5.2 Cocycles and Couplings 3.11.6 Universal Enveloping Realization of HS Algebra 3.11.7 Advanced star-Products: Cayley Transform 3.11.8 Poincaré Lemma, Homotopy Integrals 3.11.8.1 Two-Dimensional Case 3.11.8.2 Repeated Homotopy Integrals Appendix 1: Indices Appendix 2: Multi-Indices and Symmetrization Appendix 3: Solving for Spin-Connection Appendix 4: Differential Forms Integration and Einstein-Hilbert Action Appendix 5: Young Diagrams and Tensors Generalities GL(d) SO(d) Why Young Diagrams? Tensor Products Generating Functions Appendix 6: Symplectic Differential Calculus Appendix 7: More on so(3,2) Restriction of so(3,2) to so(3,1) so(3,2) Is sp(4,R) References