Content of the course
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 representations, projectively flat connections 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.
If time permits, then we will also make some brief excursions into other chapters that fall under the title of this course, such as the theory of vertex algebras.
Prerequisites
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).
Literature (will be expanded, see also the bibliography in the notes)
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).
Notes
I am writing a set of notes for this course, but these will probably lag behind on the oral lectures. Here is what I did sofar.
Place and time
Tuesdays and Thursdays 15:10-17:00, beginning March 10 2015 (so this is one week later than earlier announced) in Conference Room 4, Floor 2, Jin Chun Yuan West Building.
Material covered
March 10: What we are heading to I: topological interpretation of WZW theory for closed Riemann surfaces.
March 12: What we are heading to II: topological interpretation of WZW theory for punctured Riemann surfaces.
Elementary notions associated to complete local fields of characteristic zero.
March 17: Oscillator algebra and the completion of its universal enveloping algebra.
Construction of the Virasoro Lie algebra as a subalgebra of this completion.
Beginning of the characterization of its topological dual.
March 19: No class.
March 24: The topological dual of the Virasoro Lie algebra and quadratic differentials.
Connection with projective structures on a Riemann surface.
Version for the Lie algebra of vector fields on the circle and its integrated form given by the Bott cocycle.
March 26: Fock representation of the oscillator algebra.
Heisenberg bundle and Fock representation bundle associated with a symplectic local system.
Projective flatness of its curvature.
March 31: Continuation of the curvature discussion.
Level of a representation. Canonical Casimir element of a simple Lie algebra. Dual Coxeter number.
April 2: Central extension of a loop algebra.
Central extension by U(1) of the loop group of a simply connected compact Lie group.
April 7: U(1)-bundle over the moduli space of flat principal bundled over a closed surface. Relation to the Narasimhan-
Sheshadri-Donaldson theorem.
An avatar of Riemann-Roch and Serre duality on a pointed projective nonsingular curve.
April 9: The Fock system associated to a family of pointed projective nonsingular curves.
April 14: Sugawara realization.
April 16: No class.
April 21: Algebraic formulation of the WZW connection. Conversion to a family of pointed curves
The propagation principle.
April 23: The genus zero case: the KZ system.
April 30: Holiday.
May 5: Topological interpretation of the WZW connection.
May 7: Families of curves acquiring ordinary double points.
The factorization property (statement only).
May 12: De Rham complex for families of curves acquiring ordinary double points.
Polydifferentials
May 14: Residue maps for polydifferentials.
Dual of a Kac-Moody module in terms of polydifferentials.