MA 792V – Vertex
Operator Algebras - Spring 2004
Lecture 1. Vertex algebras and the Monster (a short historical
talk). The definition of a vertex algebra.
Lecture 2. An example of vertex algebras: holomorphic vertex
algebras. Formal distributions. Locality of a formal distribution in two
variables. Formal delta function.
Lecture 3. Local formal distributions with values in a Lie
algebra. Examples: Heisenberg, affine Kac-Moody, and Virasoro algebras.
Lecture 4. Properties of local formal distributions. n-th
brackets and lambda-brackets. The definition of a conformal algebra.
Lecture 5. Formal Fourier transform. Examples of conformal
algebras. Lie superalgebras. Free fermions.
Lecture 6. Formal distributions with values in an associative
algebra. Formal expansions. Normally ordered product.
Lecture 7. Locality and bilocality. Normally ordered product
and n-th products of fields. Dong's lemma.
Lecture 8. Linear field algebras. Any local linear field
algebra is a vertex algebra.
Lecture 9. The definition of a vertex algebra. Consequences of
the vacuum and translation axioms. Skew-symmetry.
Lecture 10. Uniqueness Theorem. n-th product identity and
commutator formula. Derivations of vertex algebras.
Lecture 11. Jacobi identity. Equivalence of two definitions of
a vertex algebra.
Lecture 12. Consequences and special cases of the Jacobi
identity. Borcherds identity.
Lecture 13. Generating a vertex algebra from a collection of
local fields: Existence Theorem.
Lecture 14. Representations of the Heisenberg algebra. Free
boson vertex algebra.
Lecture 15. Computations in the free boson vertex algebra. Virasoro
algebra. Graded and conformal vertex algebras.
Lecture 16. Neutral free fermion vertex algebra.
Lecture 17. Charged free fermions. First part of the
boson-fermion correspondence.
Lecture 18. Vertex algebras associated to a 1-dimensional
lattice. Second part of the boson-fermion correspondence.
Lecture 19. Lattice vertex algebras.
Lecture 20. Lattice vertex algebras and affine vertex algebras
at level one.
Lecture 21. Vertex algebras related to a Lie algebra of local
formal distributions. Affine vertex algebras.
Lecture 22. Affine vertex algebras. Sugawara construction.
Lecture 23. Modules over vertex algebras: definition in terms
of n-th product identity.
Lecture 24. Modules over vertex algebras: associativity and
Jacobi identity.
Lecture 25. Consequences of the definition of a module:
translation invariance, locality, generating a module from a subset. Uniqueness
theorem for modules.
Lecture 26. Graded modules over graded vertex algebras. Action
on the lowest weight space and Zhu’s *-product.
Lecture 27. Zhu algebra of a graded vertex algebra.
Lecture 28. Correspondence between vertex algebra modules and
modules over its Zhu algebra.