Verlinde Formula for the moduli of G-bundles

Let G be a simply-connected complex semisimple algebraic group and let C be a smooth projective curve of any genus. Then, the moduli space of semistable G-bundles on C admits so called determinant line bundles. E. Verlinde conjectured a remarkable formula to calculate the dimension of the space of generalized theta functions, which is by definition the space of global sections of a determinant line bundle. This space is also identified with the space of conformal blocks arising in Conformal Field Theory, which is by definition the space of coinvariants in integrable highest weight modules of affine Kac-Moody Lie algebras. Various works notably by Tsuchiya-Ueno-Yamada, Kumar-Narasimhan-Ramanathan, Faltings, Beauville-Laszlo, Sorger and Teleman culminated into a proof of the Verlinde formula.

The main aim of this course will be to give a complete and self contained proof of this formula. The proof requires techniques from algebraic geometry, geometric invariant theory, representation theory of affine Kac-Moody Lie algebras, topology and Lie algebra cohomology. Some basic knowledge of algebraic geometry and representation theory of semisimple Lie algebras will be helpful; but not required. I will develop the course from scratch recalling results from different areas as we need them.

There is no text book available which is suitable for our course. We will mainly rely on research papers. This course should be suitable for graduate students interested in interaction between algebraic geometry, representation theory, topology and mathematical physics.