REPRESENTATION
THEORY AND RELATED TOPICS SEMINAR
Jethro van Ekeren
ABSTRACT: Modular
forms crop up unexpectedly in several places in representation theory.
One very well known example, called monstrous moonshine, is the
appearance of coefficients of the J function as dimensions of
irreducible representations of the monster group. Another is the
modular invariance of graded dimensions of certain modules over an
affine Kac-Moody algebra.
These examples find a common generalisation in the theory of vertex
algebras, specifically in a theorem of Zhu which implies that the
graded dimensions of irreducible modules over a fixed vertex algebra
(subject to certain technical requirements) span a finite dimensional
space of functions invariant under the modular group. Zhu's proof is
intriguing because it is very different than the original proofs of
either of the special cases mentioned above, and it uses ideas from
geometry and theoretical physics.
In this (purely expository) talk I will introduce the notion of a
vertex algebra, explain the statement of Zhu's theorem, outline its
proof, and try to motivate the ideas underlying the proof.
For further information visit http://www.northeastern.edu/martsinkovsky/p/rtrt.html or contact Alex Martsinkovsky alexmart >at< neu >dot< edu