Yesterday morning I thought that I could use some discussions I had a

week before with Markus Reineke to begin to make sense of one

sentence in Kontsevich’ Arbeitstagung talk Non-commutative smooth

spaces :

It seems plausible that Borcherds’ infinite rank

algebras with Monstrous symmetry can be realized inside Hall-Ringel

algebras for some small smooth noncommutative

spaces

However, as I’m running on a 68K RAM-memory, I

didn’t recall the fine details of all connections between the monster,

moonshine, vertex algebras and the like. Fortunately, there is the vast

amount of knowledge buried in the arXiv and a quick search on Borcherds gave me a

list of 17 papers. Among

these there are some delightful short (3 to 8 pages) expository papers

that gave me a quick recap on things I once must have read but forgot.

Moreover, Richard Borcherds has the gift of writing at the same time

readable and informative papers. If you want to get to the essence of

things in 15 minutes I can recommend What

is a vertex algebra? (“The answer to the question in the title is

that a vertex algebra is really a sort of commutative ring.”), What

is moonshine? (“At the time he discovered these relations, several

people thought it so unlikely that there could be a relation between the

monster and the elliptic modular function that they politely told McKay

that he was talking nonsense.”) and What

is the monster? (“3. It is the automorphism group of the monster

vertex algebra. (This is probably the best answer.)”). Borcherds

maintains also his homepage on which I found a few more (longer)

expository papers : Problems in moonshine and Automorphic forms and Lie algebras. After these

preliminaries it was time for the real goodies such as The

fake monster formal group, Quantum vertex algebras and the like.

After a day of enjoyable reading I think I’m again ‘a point’

wrt. vertex algebras. Unfortunately, I completely forgot what all this

could have to do with Kontsevich’ remark…