Topics in Quantum Field Theory
and Integrable Systems

(see also a list of introductory literature)

Notes, formulas, etc.  (LaTeX files)

Basic algebra
• Vector spaces
• Algebras
• Hopf algebras
Basic geometry
• Manifolds
• Tensor fields
• Lie derivative
• Lie groups
• Hamiltonian dynamics
QFT: General formulas
• Minkowski vs Euclidean
• Gaussian continual integrals
• Generating functionals
Renormalization
• Generic proofs
QFT: Symmetries
• Noether's theorem
• Energy-momentum tensor
Supersymmetry
• Formulas with SL(2,C) matrices
• Anticommuting spinors
• Superspace
Seiberg-Witten theory
• Nonperturbative solution
BRST
• BRST operator
• BRST symmetry
Curved space-time
• Curved space-time
• Tetrads
Conformal symmetry
• Conformal group: d>2 vs d=2
• Conformally flat metric in complex coordinates
• Stress tensor and conformal symmetry
Operator products in CFT2
• Algebras of chiral conformal fields
• Normal products
• Sugawara construction
• Free scalar field
Vertex algebras
• Formal distributions and locality
• Operator product expansion
• Vertex algebras
• Free fermions
Integrable hierarchies I
• Scalar Ansatz (KP hierarchy)
• Fermionic Fock space
• Fermi-Bose correspondence
• KP hierarchy via free fermions
Integrable hierarchies II
• Adler-Kostant-Symes scheme
• Matrix Ansatz
• Dressing
• Conservation laws
Conformal features of integrable equations
• Virasoro algebra in KdV