Infinite dimensional Lie algebras appearing in Algebraic Geometry
Student No.:50
Time:Tue/Thu 15:10-17:00, 2015-03-10~ 2015-06-18 (except for public holidays)
Instructor:Eduard Looijenga  [Tsinghua University]
Place:Conference Room 4, Floor 2, Jin Chun Yuan West Building
Starting Date:2015-3-10
Ending Date:2015-6-18

 The lecture on May 26 is cancelled.


This course is set up as an introduction to the theory of conformal blocks (Wess-Zumino-Witten theory), whose ultimate goal is described below. It will lead us to cover some topics that have an interest beyond that as well, such as Heisenberg algebras, Virosoro algebras, Fock representations, affine Lie algebras, the Knizhnik-Zamolodchikov system, Sugawara representation, projectively flat connection and the notion of a modular functor.
The WZW theory attaches to a system consisting of the following data: 1) a compact Riemann surface C together with a finite subset P of C, 2) a simple finite dimensional complex Lie algebra g and 3) an irreducible finite dimensional representation V_p of g for every p in P, a finite dimensional complex projective space. The main result of this theory says that this projective space is independent of the complex structure on C. This is at the origin of the Chern-Simons theory and Jones-Witten theory of knots on 3-manifolds.
Riemann surfaces and the representation theory of finite dimensional Lie algebras. Some basic knowledge of commutative algebra and algebraic geometry will be helpful. We plan to keep a leisurely pace (so that we can make up for some deficiencies as we proceed).
V.G. Kac: Infinite-dimensional Lie algebras, 3rd ed. Cambridge University Press, Cambridge (1990).
V.G. Kac, A.K. Raina: Bombay lectures on highest weight representations of infinite-dimensional Lie algebras, Advanced Series in Mathematical Physics, 2. World Scientific Publishing Co., Inc., Teaneck, NJ (1987).

Eduard Looijenga: From WZW models to Modular Functors, in Handbook of Moduli, G. Farkas, I. Morrison (eds.), 427-166, International Press (2013).