Verlinde Dimension Formula for the Space of Conformal Blocks and the moduli of G-bundles V

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  derived from the Propogation of Vacua and the Factorization Theorem among others. 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.

The course will be based upon some parts of my book ‘Conformal Blocks, Generalized Theta Functions and the Verlinde Formula’ published by the Cambridge Ubiversity Press this year. This course should be suitable for any one interested in interaction between algebraic geometry, representation theory, topology and mathematical physics.



University of North Carolina; Member, School of Mathematics