What is a vertex algebra?. Arbeitstagung talk 1997 
(EXPOSITORY) Unpublished 1997 Arbeitstagung talk. 
Richard E. Borcherds,
D.P.M.M.S., 16 Mill Lane, Cambridge, CB2 1SB, England. 
http://www.dpmms.cam.ac.uk/~reb 

The answer to the question in the title is that a vertex algebra is really 
a sort of commutative ring. I will try to explain this in the rest of the 
talk, and show how to use this to generalize the idea of a vertex algebra 
to higher dimensions. The picture to keep in mind is that a commutative ring 
should be thought of as somehow related to quantum field theories in 0 
dimensions, and vertex algebras are related in the same way to 1 dimensional 
quantum field theories, and we want to find out what corresponds to higher 
dimensional field theories. This talk is an exposition of the paper 
q-alg/9706008, which contains (some of) the missing details. There is also 
some overlap with unpublished notes of Soibelman, which have just appeared 
on the q-alg preprint server as q-alg/9709030.