Title: Free fields and conformal field theory

Lecturer: Ingo Runkel, University of Hamburg

Time:
14:00 -- 17:00 on May 20, May 25 (note the special day), 10:30 -- 12:00 on May 26, June 2, June 9, 15:30 -- 17:00 on May 27, June 3, June 10.

Place:
Room 1218 at BICMR, Resource Plaza, Peking University on May 20, May 27, June 3, June 10.
Room 1213 at BICMR, Resource Plaza, Peking University on May 25
Room 1328 at BICMR, Resource Plaza, Peking University on May 26, June 2, June 9.

Description:
In these lectures we will study some aspects of two-dimesional conformal quantum field theory. The approach will be purely mathematical and will not require a background in physics. The representation theory of certain infinite dimensional Lie algebras (the mode algebras of free bosons and fermions) will be used to construct multivalued functions on the complex plane with interesting monodromy properties. Combining these into single-valued functions leads to examples of so-called correlation functions for free boson and free fermion conformal field theories. Time permitting, we will also investigate conformal field theories with logarithmic singularities.

Plan:

• 1. Background on Lie algebras
  (Lie algebras and representations, super Lie algebras, universal enveloping algebra, induced representations)
• 2. Mode algebra of bosons and fermions
  (Heisenberg Lie algebra and its representations, free fermion modes and representations, Virasoro algebra)
• 3. Vertex operator algebras
  (formal power series, definition of a vertex operator algebra, examples from free bosons and fermions)
• 4. Conformal blocks and correlators from free fields
  (charged vertex operators, conformal blocks on the complex plane, asymptotic behaviour and monodromies, defining properties of correlation functions, solutions via conformal blocks)
• 5. Symplectic fermions
  (definition via commutation relations and vertex operators, representations, conformal blocks and correlators with logarithmic singularities)