Infinite-dimensional Lie algebras (such as Virasoro algebra or affine Kac-Moody algebras) turn out to be very important
in various areas of modern mathematics and mathematical physics. In particular, they are very useful in the description
of some field theories. In this context one arranges infinite number of the Lie algebra elements into a single object called field.
This idea generalizes to the general theory of vertex operator algebras.
VOAs capture the main properties of the infinite diemensional Lie algebras and have rich additional structure. Vertex operator
algebras proved to be very useful in many situations; the classical example is the KP integrable hierarchy. They are also extensively used
in modern algebraic geometry. Our goal is to give an introduction to the theory of vertex operator algebras from the modern
mathematical point of view. We describe the main definitions, constructions and applications of the theory. The course is aimed at
PhD students and master students.
Prerequisites: basic Lie theory, the theory of affine Kac-Moody algebras.
Preliminary program:
1. Heisenberg algebras and Fock modules.
2. Virasoro algebras, Verma modules.
3. Boson-fermion correspondence, Schur polynomials, KP hierarchy.
4. Vertex operator algebras and Lie algebras.
5. Associativity and operator product expansion.
6. Representation theory of vertex operator algebras.
Literature.
1. Kac, V. Infinite dimensional Lie algebras, Cambridge University Press (1994).
2. Frenkel E., Ben-Zwi D. Vertex algebras and algebraic curves (AMS, 2001)
3. Kac V., Raina A., Rozhkovskaya N., Bombay lectures on Highest weight representations of infinite dimensional Lie algebras.