Particles and Fields [Hardcover]

1 Recent Developments in Affine Toda Quantum Field Theory.- 1 Introduction.- 2 Classical Integrability and Classical Data.- 2.1 Geometry Associated with the Coxeter Element.- 3 Aspects of the Quantum Field Theory.- 4 Dual Pairs.- 5 A Word on Solitons.- 6 Other Matters.- 7 References.- 2 A Class of Fermi Liquids.- 1 Introduction.- 2 Four-Legged Diagrams.- 2.1 The Particle-Particle Bubble.- 2.2 The Particle-Hole Bubble.- 2.3 Higher-Order Diagrams.- 3 A Single-Slice Fermionic Cluster Expansion.- 4 References.- 3 Quantum Groups from Path Integrals.- 1 Classical Field Theory.- 1.1 Classical Actions.- 1.2 The Wess-Zumino-Witten Action.- 2 Categories, Finite Groups, and Covering Spaces.- 2.1 Going Further.- 2.2 Finite-Gauge Theory.- 3 Generalized Path Integrals.- 3.1 Path Integral Quantization.- 3.2 Beyond Quantum Hilbert Spaces.- 3.3 Quantum Finite-Gauge Theory.- 4 The Quantum Group.- 4.1 The 2-Hilbert Space.- 4.2 Locality and Gluing.- 5 References.- 4 Half Transfer Matrices in Solvable Lattice Models.- 1 The Six-Vertex Model.- 2 The Antiferromagnetic Regime.- 3 Corner Transfer Matrix.- 4 Half Transfer Matrix.- 5 Commutation Relations.- 6 Correlation Functions.- 7 Two-Point Functions.- 8 Discussion.- 9 References.- 5 Matrix Models as Integrable Systems.- 1 Introduction.- 2 The Basic Example: Discrete 1-Matrix Model.- 2.1 Ward Identities.- 2.2 CFT Interpretation of 1-Matrix Model.- 2.3 1-Matrix Model in Eigenvalue Representation.- 2.4 Kontsevich-Like Representation of the 1-Matrix Model.- 3 Generalized Kontsevich Model.- 3.1 Kontsevich Integral. The First Step.- 3.2 Itzykson-Zuber Integral and Duistermaat-Heckmann Theorem.- 3.3 Kontsevich Integral. The Second Step.- 3.4 Phases of Kontsevich Integral. GKM as the Quantum Piece of $${\mathcal{F}_V}\{ L\} $$ in the Kontsevich Phase.- 3.5 Relation Between Time-and Potential-Dependencies.- 3.6 Kac-Schwarz Problem.- 3.7 Ward Identities for GKM.- 4 Kp/Toda ?-Function in Terms of Free Fermions.- 4.1 Explicit Definition.- 4.2 Bal]