Vertex Algebras

Intended audience – Masters and doctoral students, and summer program students.

Pre-requisites: The course is intended to be as self-contained as possible, so there are no formal pre-requisites. Some familiarity with Lie algebras and/or physics.

This minicourse is intended as an introduction to vertex and chiral algebras. These algebras play an important role in modern representation theory, especially the theory of infinite dimensional Lie algebras, and in mathematical physics.

Topics Covered.
In approximate chronological order.

• Motivation for vertex algebras as a toy model of renormalisation in quantum field theory. Normal ordering. Heisenberg vertex algebra.
• Definition of a vertex algebra. Computing in vertex algebras. Operator product expansion and lambda calculus.
• The boson-fermion correspondence. Applications to combinatorics and symmetric functions.
• Lattices and lattice vertex algebras. Affine Lie algebras and affine vertex algebras. Free field realisation.
• The Virasoro algebra and its representations. The Kac determinant formula.
•  Chiral algebras: a geometric reformulation of vertex algebras.