This week Yale is hosting a conference on Perspectives in Representation Theory, in honor of Igor Frenkel’s 60th birthday. I’m planning to take the train up there and attend some of the talks tomorrow and Wednesday. Frenkel has been a pioneer in the field of representation theory, especially in the area of infinite-dimensional algebras whose representations are significant for understanding low-dimensional QFT, string theory and topological QFT. He started his career in the early 1980s, with many important results relevant to understanding affine Lie algebras. These became central in the explosion of interest among physicists in 2d conformal QFTs after 1984 due to their importance in string theory. Frenkel’s later work has covered a wide variety of topics, with one theme that of trying to understand higher-dimensional generalizations of affine Lie algebras and their potential application to QFTs in higher space-time dimensions than 2. He has also promoted the themes of “categorification” and geometric incarnation of representations that are now central to much research in this area.

Pavel Etingof and other students and collaborators of Frenkel have put together a wonderful document, On the work of Igor Frenkel, which gives much more detail about the many topics of his mathematical research.

**Update**: Videos of the talks are now available here.

Sigh! I had a look at Pavel Etingof’s paper, and I got seriously depressed, due the following quote:

“In spite of this progress, however, it is still not clear what the representation theory of central extensions of double loop groups should be like. Perhaps we don’t yet have enough imagination to understand what kind of representations (or maybe analogous but more sophisticated objects) we should consider, and this is a problem for future generations of mathematicians.”

On the Lie algebra level, the answer has been around for 15 years or more. Geometrically, the off-shell representations of the multi-dimensional affine algebra (the MF extension is not central) act on trajectories in the space of g-valued p-jets, where g is the finite Lie algebra. The representations of the multi-dimensional Virasoro algebra are completely analogous.

OK, I understand if people ignore an crackpot amateur physicist, but perhaps one should bother to have a look at the work of professional mathematicians, like Billig, Rao or Moody.