Lectures on Conformal Field Theory

References:

1. G. Segal, "Definition of Conformal Field Theory".
2. K. Gawedzki, "Lectures on Conformal Field Theory".
3. E. Frenkel, D. Ben-Zvi, "Vertex Algebras and Algebraic Curves".
4. G. Moore, N. Seiberg, "Lectures on RCFT".

Part I: Geometry and Conformal Field Theory

• Lecture I. Introduction. Toy-Model: 2d Topological Field Theory (pdf).
• Lecture II. Segal's axioms and constructive approach. Energy-momentum tensor and its transformations (pdf).
• Lecture III. Ward identities and simplest OPEs (pdf).
• Lecture IV. Space of states and Virasoro Algebra (pdf).
• Lecture V. Back to Segal's axioms. Path integral interpretation (pdf).

Part II: Vertex operator algebras and applications

• Lecture VI. Definition of Vertex Algebra (pdf).
• Lecture VII. Example: Heisenberg Vertex Algebra (pdf).
• Lecture VIII. Dong's Lemma. Reconstruction theorem. More examples of VOAs (pdf).
• Lecture IX. Associativity and Operator Product Expansion (pdf).
• Lecture X. Operator Product Expansion: applications. Correlation Functions(pdf).
• Lecture XI. Further examples. Lattice VOAs, boson-fermion correspondence (pdf).
• Lecture XII. Free fields, path integrals and vertex algebras (pdf).

Rational CFT

• Lecture I. Correlation functions, conformal blocks and intertwining operators (pdf).
• Lecture II. Properties of chiral vertex operators. Degenerate conformal families (pdf).
• Lecture III: Minimal Models and Coulomb gas formalism (pdf).
• Lecture IV. Conformal Field Theory and Quantum Groups (pdf).

Chiral de Rham complex and Mirror symmetry

• Lecture I. Introductory notions. Chiral de Rham complex (pdf).
• Lecture II. Line bundles, zeros of their sections and vertex algebras (pdf).
• Lecture III. Logarithmic coordinates (pdf).