Quantum Theory and Symmetries; Proceedings of the 11th International Symposium, Montréal, Canada

This volume of the CRM Conference Series is based on a carefully refereed selection of contributions presented at the "11th International Symposium on Quantum Theory and Symmetries", held in Montréal, Canada from July 1-5, 2019. The main objective of the meeting was to share and make accessible new research and recent results in several branches of Theoretical and Mathematical Physics, including Algebraic Methods, Condensed Matter Physics, Cosmology and Gravitation, Integrability, Non-perturbative Quantum Field Theory, Particle Physics, Quantum Computing and Quantum Information Theory, and String/ADS-CFT. There was also a special session in honour of Decio Levi. The volume is divided into sections corresponding to the sessions held during the symposium, allowing the reader to appreciate both the homogeneity and the diversity of mathematical tools that have been applied in these subject areas. Several of the plenary speakers, who are internationally recognized experts in their fields, have contributed reviews of the main topics to complement the original contributions. Preface Organization Organizing Committee International Advisory Board QTS Conference Board Plenary Speakers Session Organizers Sponsors Statement of Principles Contents Part I Special Session in Honour of Decio Levi: Integrability: Continuous and Discrete, Classical & Quantum Spin Chains of Haldane–Shastry Type: A Bird's Eye View 1 Introduction 2 Calogero–Sutherland Models 2.1 The Calogero Model 2.2 The Sutherland Model 3 Spin Dynamical Models 3.1 The AN-1 Spin Calogero Model 4 Spin Chains of Haldane–Shastry Type 4.1 The Chain Sites 4.2 The Partition Function 5 Supersymmetric Spin Chains 5.1 The su(m|n) Supersymmetric Exchange Operator 5.2 Chemical Potential 5.3 Partition Function 6 Associated Vertex Models 7 Thermodynamics 7.1 Example: The su(1|1) Case 8 Conclusions References Features of Discrete Integrability 1 Introduction 2 What Is a Quad Equation? 3 A Few Standard Features of Integrable Quad Equations 3.1 Discrete Lax Pairs 3.2 Consistency Around the Cube (CAC) 3.3 Lax Pair from the Consistency Around the Cube 3.4 Symmetries 3.5 Singularity Analysis 3.6 Complexity Analysis: Algebraic Entropy 3.7 What About Quad Equations? 4 A Case Study 4.1 Symmetries 4.2 A Lax Pair 4.3 What About Singularities? 4.4 What About Entropy? 5 Comments and Perspectives References Darboux-BäcklundTransformationsforSpin-ValuedLinearProblems 1 Introduction 2 Clifford Algebras, Spin Groups, and Isometries of Rp,q 3 Spin-Valued Linear Problems 3.1 Geometric Interpretation: Soliton Surfaces Approach 4 Darboux-Bäcklund Transformations 4.1 The Darboux-Bäcklund Transformation in the Case of Spin Groups 5 Conclusions and Open Problems References Painlevé IV Transcendents Generated from the Complex Oscillator 1 Introduction 2 Supersymmetric Quantum Mechanics 3 Polynomial Heisenberg Algebras 3.1 Second-Degree PHA 4 Complex Oscillator 5 SUSY Partners of the Complex Oscillator 6 PIV Transcendents 7 Conclusions References The Veronese Sequence of Analytic Solutions of the CP2s Sigma Model Equations Described via Krawtchouk Polynomials 1 The CP2s Sigma Model 2 Projective Formalism 3 Solutions of the CP2s Sigma Model 4 The su(2) Spin-s Representation 5 Geometrical Aspects of Surfaces References A Novel Integrable Fourth-Order Difference Equation Admitting Three Invariants 1 Introduction 2 Integrability of Difference Equations 3 A Novel Example 4 Final Remarks References Weighted Hurwitz Numbers, τ-Functions, and Matrix Integrals 1 Hurwitz Numbers: Classical and Weighted 1.1 Geometric Meaning of Classical Hurwitz Numbers 1.2 Weighted Hurwitz Numbers GH1, HO1, H1, GH2 2 Hypergeometric τ-Functions as Generating Functions for Weighted Hurwitz Numbers GH1, HO1, H1, GH2 3 Wronskian and Matrix Integral Representation of τ(G,β)([X]) 3.1 Adapted Basis, Recursion Operators, Quantum Spectral Curve 3.2 Quantum and Classical Spectral Curve Rational Weighting Case Mellin-Barnes Integral Representation: Meijer G-Functions BH, HR Quantum Case Expressed as Mellin-Barnes Integrals HR 3.3 Determinantal Representation of τ(G,β)(t) Eulerian Wronskian Representation 3.4 Matrix Integral Representation of τ(G,β)([X]) BH, HR Wronskian Representation: Rational Case Matrix Integral Representation of τ(G,β)([X]): Rational Case Wronskian Representation: Quantum Case Matrix Integral Representation of τ(G,β)([X]): Quantum Case References Constant Curvature Holomorphic Solutions of the Supersymmetric G(2,4) Sigma Model 1 Introduction 2 CCH Solutions of the Susy G(M,N) σ-Model 2.1 Susy Invariant Solutions 3 CCH Solutions of the Susy G(2,4) σ-Model 3.1 The Case of Z1 3.2 The Case of Z2 3.3 The Case of Z3 3.4 The Case of Z4 4 Conclusions and Final Comments References How to Deal with Nonlocality and Pseudodifferential Operators. An Example: The Salpeter Equation 1 The Spinless Salpeter Equation: An Introduction 2 Solutions of the Salpeter Equation 3 A Recursive Solution: Nonlocality via Laplace Transform References A New Approach to Analysis of 2D Higher Order Quantum Superintegrable Systems 1 Introduction 2 The Canonical Form for a Symmetry Operator 3 Third Order Superintegrability 4 Some Examples (Mostly New) References Ladder Operators and Rational Extensions 1 Introduction 2 Maya Diagrams 3 Rational Extensions 4 Categorical Structure 5 Ladder Operators 6 Examples References Tachyons and Representations of Sp(2,R) 1 Introduction 2 SO0(2,3), Sp(2,R), the Poincaré Group and Their Lie Algebras 3 Algebraic Results 4 Representations References A Confined Quasi-Maximally Superintegrable N-dimensional System, Classical and Quantum, in a Space with Variable Curvature 1 The Classical Model: General Features 1.1 Metrics and Scalar Curvature 1.2 Solution of the Radial Equation of Motion 2 The Quantum Model 3 Concluding Remarks and Future Perspectives References Conditional Discretization of a Generalized Reaction–DiffusionEquation 1 Introduction 2 Case 1 2.1 Symmetry Reduced Equations and Solutions 2.2 Construction of the Discretized Equation 3 Case 4 3.1 Symmetry Reduced Equations and Solutions 3.2 Construction of the Discretized Equation 4 Conclusions References Discrete Curve Flows in Two-Dimensional Cayley–Klein Geometries 1 Introduction 2 Cayley–Klein Planes 3 Moving Frames 4 Recurrence Relations 4.1 Exterior Derivative 4.2 Shift Map 5 Invariant Linearization Operators 6 Geometric Flows References Zernike System Stems from Free Motion on the 3-Sphere 1 Introduction: The so(4) Algebra 2 Realization of SO(4) on the 3-Sphere 3 Two Coordinate Systems for S3 4 Projection on the 2-Sphere S2 5 Interbasis Expansion Coefficients 6 Concluding Remarks References Part II Algebraic and Non-perturbative Methods W-Algebras via Lax Type Operators 1 Introduction 2 What Is a W-Algebra? 2.1 Fundamental Physical Theories and Corresponding Fundamental Algebraic Structures (Poisson) Vertex Algebras Vertex Algebras 2.2 A Toy Model 2.3 Hamiltonian Reduction 2.4 From the Toy Model to W-Algebras Classical Finite W-Algebras Finite W-Algebras Classical Affine W-Algebras Quantum Affine W-Algebras 3 Linear Algebra Intermezzo 3.1 Set Up 3.2 The ``Identity'' Notation 3.3 Generalized Quasideterminants 4 Quantum Finite W-Algebras and (Twisted) Yangians 4.1 Lax Type Operators for Quantum Finite W-Algebras 4.2 The Generalized Yangian Identity 4.3 Quantum Finite W-Algebras and (Extended) Twisted Yangians 5 Classical Affine W-Algebras and Integrable Hierarchies of Lax Type Equations 5.1 Lax Type Operators for Classical Affine W-Algebras 5.2 Integrable Hierarchies of Lax Type Equation References Color Algebraic Extension of Supersymmetric Quantum Mechanics 1 Introduction 2 Z2n-Graded Color Superalgebra and Z22-Graded SQM 3 Extensions of Bruce–Duplij Model 3.1 From Superalgebra to Z22-Graded Color Superalgebra 3.2 N Extension of Z22-Graded SQM 3.3 Z22-Graded SCM 4 Z23-Graded SQM References The Racah Algebra and sln 1 Introduction 2 Definition of the Higher Rank Racah Algebra 3 Realizing sln in n-1 Variables 3.1 Some Operators in U(Dn) 4 Realization of Rn in n-2 Variables 4.1 The Differential Embedding 5 Conclusions References On Reducible Verma Modules over Jacobi Algebra 1 Introduction 2 Preliminaries 3 Case G2 4 Verma Modules and Singular Vectors 4.1 Definitions 4.2 Case G2 Weight 2δ1 Weight 2δ2 Weight δ1+δ2 Weight δ1-δ2 Weight δ1 Weight δ2 Weight 3δ2 References Howe Duality and Algebras of the Askey–Wilson Type: An Overview 1 Introduction 2 General Framework 3 The Dual Presentations of the Racah Algebra 4 More Dual Pictures: An Overview 4.1 The Racah Family 4.2 The Bannai–Ito Ensemble 4.3 The Askey–Wilson Class 5 Conclusion References Second-Order Supersymmetric Partners of the Trigonometric Rosen–Morse Potential 1 Introduction 2 Supersymmetric Quantum Mechanics 3 Trigonometric Rosen–Morse Potential 4 SUSY Partners of the Trigonometric Rosen–Morse Potential 5 Conclusions References A Noncommutative Geometric Approach to the Batalin–Vilkovisky Construction 1 Introduction: Why Noncommative Geometry 2 Finite Spectral Triples and Induced Gauge Theories 3 The BV Construction in the Algebraic Context 4 BV-Spectral Triple: The Notion and the Relevance 5 Conclusions and Outlooks References A New Method for Constructing Squeezed States for the Isotropic 2D Harmonic Oscillator 1 Introduction 2 Squeezed States of the 1D Harmonic Oscillator 3 The 2D Oscillator 4 SU(2) Coherent States 5 2D Squeezed States 6 Conclusion References Electron in Bilayer Graphene with Magnetic Fields Associated with Solvable Potentials 1 Introduction 2 Bilayer Graphene: A Brief Overview 2.1 Effective Hamiltonian 2.2 Effective Hamiltonian with an Applied Magnetic Field 2.3 Stationary States of H 3 Second-Order SUSY QM 4 Solvable Cases 4.1 Case with j=0 4.2 Case with j≠0 5 Conclusions References Twist Knot Invariants and Volume Conjecture 1 Introduction 2 Chern–Simons Theory and Knot Invariants 3 Twist Knots 4 Volume Conjectures and Super-A-Polynomial References Demazure Formulas for Weight Polytopes 1 Introduction 2 Polytope Expansion of Lie Characters 2.1 Weyl Character Formula 2.2 Brion Lattice-Polytope Sum Formula 2.3 Polytope Expansion 3 Demazure Character Formulas 4 Lattice-Polytope Formulas of Demazure Type 5 Conclusion References Point Transformations: Exact Solutions of the Quantum Time-Dependent Mass Nonstationary Oscillator 1 Introduction 2 Nonstationary Oscillator with Time-Dependent Mass 3 Point Transformation 3.1 Orthogonal Set of Solutions and the Related Spectral Problem 4 Concluding Remarks References Influence of the Electron–Phonon Interaction on the Topological Phase Transition in BiTeI 1 Introduction 1.1 Electron–Phonon Interaction in Semiconductors 1.2 Phonon-Induced Topological Insulation 2 Electron–Phonon Interaction from First-Principles 2.1 A Competition Between Intraband and Interband Couplings 3 Topological Phase Transition in BiTeI 3.1 Static Lattice 3.2 Phonon-Induced Gap Renormalization 4 Conclusion and Outlook References Part III Condensed Matter Physics Nonlinear Coherent States for Anisotropic 2D Dirac Materials 1 Introduction 2 Anisotropic 2D Dirac Hamiltonian 3 Annhilation Operator 3.1 Nonlinear Coherent States 3.2 Some Examples 4 Conclusions References Monopole Operators and Their Symmetries in QED3-Gross–Neveu Models 1 Confinement of a Dirac Spin Liquid 2 Scaling Dimension of a Monopole Operator 3 Hierarchy Among Monopole Operators References Critical Exponents for the Valence-Bond-Solid Transition in Lattice Quantum Electrodynamics 1 The U(1)-VBS Transition 2 The Gauged NJL Model and Critical Exponents 3 Fermion Bilinear Scaling Dimensions 4 Discussion References Emergent Geometry from Entanglement Structure 1 Introduction 2 Emergent Geometry 2.1 Graphical Representation 2.2 Exact Examples 2.3 Numerical Computation 3 Entanglement Contour 4 Entanglement Current 5 Conclusion References Interplay of Coulomb Repulsion and Spin–Orbit Coupling in Superconducting 3D Quadratic Band Touching Luttinger Semimetals 1 Introduction 2 Screening and Electronic Properties of Luttinger Semimetals 3 Superconductivity in Luttinger Semimetals 4 Pairing Beyond s-Wave from Spin–Orbit Coupling 5 Conclusion Bibliography Soft Degrees of Freedom, Gibbons–Hawking Contribution and Entropy from Casimir Effect 1 Introduction 2 Capacitor Thermodynamics: Textbook Approach 3 Capacitor Thermodynamics: Euclidean Approach 4 Boundary Conditions 5 Physical Degrees of Freedom 6 Dynamics and Charge 7 Extra Contributions to Partition Function 8 Charged Black Body Partition Function Appendix: Details on Quantum Coulomb Solution Bibliography Part IV Cosmology and Gravitation & String Theory Probes in AdS3 Quantum Gravity 1 Introduction 2 AdS3 Higher Spin Gravity 3 Euclidean Black Holes 3.1 Example 4 Extremal Black Holes 4.1 A Practical Definition of Extremality 4.2 Example: Extremal sl(3) Black Holes 5 Wilson Lines 5.1 Wilson Lines in AdS3/CFT2 Bibliography Fundamental Physics, the Swampland of Effective Field Theory and Early Universe Cosmology 1 Introduction 2 Early Universe Scenarios 3 Constraints from Fundamental Physics 4 Conclusions and Discussion References Scale-Invariant Scalar Field Dark Matter Through the Higgs-Portal 1 Introduction 2 Dynamics Before Electroweak Symmetry Breaking 2.1 Inflation 2.2 Radiation Era 3 Dynamics After the Electroweak Symmetry Breaking 4 Phenomenology 4.1 Dark Matter Decay 4.2 Invisible Higgs Decays into Dark Scalars 5 Conclusions References The Moduli Portal to Dark Matter Particles 1 Introduction 2 The Model 3 Results 3.1 Production Rate Densities and Evolution of FIMP Relic Density 3.2 Agreement with Planck Results 4 Concluding Remarks References Unified Superfluid Dark Sector 1 Introduction 2 Review of Dark Matter Superfluid 3 Unified Dark Superfluid 4 Growth of Density Inhomogeneities 5 Observational Signatures References de Sitter Vacua in the String Landscape: La Petite Version 1 Introduction 2 How Hard Is to Get a de Sitter Vacuum in String Theory? 3 Discussion and Conclusion References Intensity Mapping: A New Window into the Cosmos 1 Introduction 2 Forecasts for Cosmology and Astrophysics 3 Conclusions References Aberration in Gravito-Electromagnetism 1 Gravito-electromagnetism 2 Lagrange Inversion Theorem and Analysis 3 Final Expression for the Force 4 LIGO-Type Experiment References Stable, Thin Wall, Negative Mass Bubbles in de Sitter Space-Time 1 Introduction 2 Thin Wall Bubbles 3 Stable Negative Mass de Sitter Bubbles 4 Conclusion References Ferromagnetic Instability in PAAI in the Sky 1 Introduction 2 Cosmic Relics and the Origin of Negative Pressure 3 Ferromagnetic Instability of PAAI 3.1 Evolution and Stability of Domain Walls 4 A Minimal Model for Dark Energy 5 Origin of Cosmic Magnetic Fields 6 Conclusions 7 Questions from the Audience References Three Partial Differential Equations in Curved Space and their Respective Solutions 1 Introduction 2 The Solutions to Eqs. (46Equ44) and (46Equ77) 2.1 A Third Example References Part V Integrability What Does the Central Limit Theorem Have to Say About General Relativity? 1 Introducing a Probabilistic Context in General Relativity 2 Emergence of a Weighted Newton's Law of Gravitation 3 Conclusion References Dressing for a Vector Modified KdV Hierarchy 1 Introduction 2 Higher Vector mKdV Flows 3 Soliton Solutions for the Vector mKdV Hierarchy 3.1 Soliton Solution: Dressing and Bäcklund Transformations 3.2 Breather Solution 4 Conclusions References Time Evolution in Quantum Systems and Stochastics 1 Time Evolution and the Feynman–Kac Formula 1.1 The Quantum Canonical Transformation 1.2 The Path Integral: Feynman–Kac Formula 2 The Quantum (D)NLS and a Hierarchy of S(P)DEs 2.1 The Continuum Models and SPDEs References Part VI Particle Physics Solvable Models of Magnetic Skyrmions 1 Introduction 2 Magnetic Skyrmions and Gauged Sigma Models 2.1 Formulating Magnetic Skyrme Models as Gauged Sigma Models 2.2 A Bogomol'nyi Equation for Gauged Sigma Models 2.3 Boundary Contributions to the Energy 3 Exact Magnetic Skyrmions 3.1 The General Solution in Four Easy Steps 3.2 Examples 4 Conclusion References Applications of Symmetry to the Large Scale Structure of the Universe (Scale Invariance) and to the Hadronic Spectrum (Heavy Diquark Symmetry) 1 Scale Invariance and Inflationary Cosmology 1.1 Symmetries of Space Time Metrics 1.2 Mass Density Perturbations in the Inflationary Cosmology 1.3 Non-Gaussianity and the Galaxy Power Spectrum at Small Wave Vectors 2 Heavy Diquark and Quark Spin Flavor Symmetry 2.1 Evidence for the Applicability of Heavy Quark Symmetry 2.2 T Q Q Tetraquarks References Leptophobic Z in Supersymmetry and Where to Find Them 1 Introduction 2 Theoretical Framework and Z Mass Bounds 3 Leptophobic Z in HL- & HE-LHC 4 Conclusion References Axion-Like Particles, Magnetars, and X-ray Astronomy 1 Introduction 2 Oscillation Equations 3 Stokes Parameters 4 ALP-to-Photon Conversion Probability 5 Discussion and Conclusion References Anomalies in B Decays: A Sign of New Physics? 1 Introduction 2 B-Decay Anomalies 2.1 b →s μ+ μ- 2.2 b →c τ- 3 Models of New Physics 3.1 Distinguishing NP Explanations 3.2 Simultaneous Explanations of b →s μ+ μ- and b →c τ- 4 Summary References Loopholes in WR Searches at the LHC References t-t-h, Top & Bottom Partners, and the Brane Higgs Limit 1 Introduction 2 Top and Bottom Partners: A Doublet and Two Singlets 2.1 Higgs Production in the Brane Higgs Limit 2.2 Higgs Phenomenology 2.3 Higgs Phenomenology 3 Outlook References Part VII Quantum Information Mirror Dirac Leptogenesis 1 The Model 1.1 Heavy N Decay and CP Violation 2 Flavor Enhancement 3 Notes and Comments References Fast Tests for Probing the Causal Structure of Quantum Processes 1 Introduction 2 Preliminaries 2.1 Notation 2.2 Cause-Effect Relations Induced by Quantum Processes 2.3 Discrimination of Causal Hypotheses 3 Optimal Classical Strategy for Finding a Causal Intermediary 4 Quantum Advantages 4.1 A Benefit of Quantum Coherence 4.2 General Quantum Scenario 4.3 Parallel Strategies with an External Reference System 4.4 The Ultimate Quantum Limit 5 Other Examples of Speedups in the Identification of Causal Hypotheses 5.1 Multiple Candidates for the Causal Intermediary 5.2 Detection of Causal Link Between Two Variables 5.3 Identification of the Cause of a Variable 6 Discussion and Conclusions References Qubits as Edge State Detectors: Illustration Using the SSH Model 1 Introduction 2 Two-Level System: A Rapid Review 3 Su–Schrieffer–Heeger Model and Edge States 4 Tripartite System: TLS-SSH-Chain 5 Conclusions References RepLAB: A Computational/Numerical Approach to Representation Theory 1 A Tour of RepLAB Features 1.1 Unique Features 2 Algorithm 2.1 Sampling from the Commutant Algebra 2.2 Computing the Eigendecomposition of a Hermitian Matrix 2.3 Grouping Equivalent Representations 2.4 Post-processing Real Representations 3 Other Approaches to Solve Problem 1 4 Conclusion References Index This volume of the CRM Conference Series is based on a carefully refereed selection of contributions presented at the "11th International Symposium on Quantum Theory and Symmetries", held in Montreal, Canada from July 1-5, 2019. The main objective of the meeting was to share and make accessible new research and recent results in several branches of Theoretical and Mathematical Physics, including Algebraic Methods, Condensed Matter Physics, Cosmology and Gravitation, Integrability, Non-perturbative Quantum Field Theory, Particle Physics, Quantum Computing and Quantum Information Theory, and String/ADS-CFT. There was also a special session in honour of Decio Levi. The volume is divided into sections corresponding to the sessions held during the symposium, allowing the reader to appreciate both the homogeneity and the diversity of mathematical tools that have been applied in these subject areas. Several of the plenary speakers, who are internationally recognized experts in their fields, have contributed reviews of the main topics to complement the original contributions