Vertex algebras for beginners

Abstract

Preface. 1: Wightman axioms and vertex algebras. 1.1: Wightman axioms of a QFT. 1.2: d = 2 QFT and chiral algebras. 1.3: Definition of a vertex algebra. 1.4: Holomorphic vertex algebras. 2: Calculus of formal distributions. 2.1: Formal delta-function. 2.2: An expansion of a formal distribution a(z,w). 2.3: Locality. 2.4: Taylor's formula. 2.5: Current algebras. 2.6: Conformal weight and the Virasoro algebra. 2.7: Lie superalgebras of formal distributions and conformal superalgebras. 3: Local fields. 3.1: Normally ordered product. 3.2: Dong's lemma. 3.3: Wick's theorem and a non-commutative generalization. 3.4: Restricted and field representations of Lie superalgebras of formal distributions. 3.5: Free (super)bosoms. 3.5: Free (super)fermions. 4: Structure theory of vertex algebras. 4.1: Consequences of translation covariance. 4.2: Quasisymmetry. 4.3: Superalgebras, ideals, and tensor products. 4.4: Uniqueness theorem. 4.5: Existence theorem. 4.6: Borcherds OPE formula. 4.7: Vertex algebras associated to Lie superalgebras of formal distributions. 4.8: Borcherds identity. 4.9: Graded and Mobius conformal vertex algebras. 4.10: Conformal vertex algebras. 4.11: Field algebras. 5: Examples of vertex algebras and their applications. 5.1: Charged free fermions. 5.2: Boson-fermion correspondence and KP hierarchy. 5.3: gl and W. 5.4: Lattice vertex algebras. 5.5: Simple lattice vertex algebras. 5.6: Root lattice vertex algebras and affine vertex algebras. 5.7: Conformal structure for affine vertex algebras. 5.8: Superconformal vertex algebras. 5.9: On classification of conformal superalgebras. Bibliography. Index