18.397 (former 18.276) Mathematical methods in physics (Fall 2017)

Instructor: Victor Kac

Email: kac [at] math . mit .edu

Office: Room 2-176

Lectures: TR 2:30-4, 2-139

COURSE DESCRIPTION

Recent develpments in quantum field theory and the theory of integrable systems produced a number of new algebraic structures. Some of the most remarkable such structures are vertex algebras and their quasiclassical limits, called Poisson vertex algebras. In the course I'll give an exposition of the foundation of the theory of vertex and Poisson vertex algebras, and their applications to representation theory and to the theory of integrable Hamiltonian PDE.

Prerequisites: 18.745 or some familiarity with Lie theory

Text Book: Vertex Algebras for Beginners, Victor Kac, 2nd Ed.

COURSE OUTLINE

  1. Calculus of formal ditributions and the formal Fourier transform. Formal distribution algebras and Lie conformal algebras.
  2. Local quantum fields and the general Wick's formula
  3. Structure theory of vertex algebras. Uniqueness and extension theorems. Borcherds' identity.
  4. Examples: affine, Virasoro, Clifford and lattice vertex algebras. Boson-fermion correspondence and the KP hierarchy.
  5. Representation theory of vertex algebras. Zhu algebra method. Geometric approach: support varieties.
  6. Quasiclassical limit and Poisson vertex algebras.
  7. Lenard-Magri scheme of integrability. Examples: KdV and HD equations
  8. Variational de Rham complex
  9. Non-local Poisson vertex algebras and Dirac reduction. Example: NLS hierarchy
  10. Classical Hamiltonian reduction, classical W-algebras and generalized Drinfeld-Sokolov hierarchies. Lax equations