Quantum vertex algebras. 7 March 1999. 
Richard E. Borcherds, 
D.P.M.M.S., 16 Mill Lane, Cambridge, CB2 1SB, England. 
home page: www.dpmms.cam.ac.uk/~reb 

Contents. 
1. Introduction. 
Notation. 
2. Twisted group rings. 
3. Construction of some categories. 
4. Examples of vertex algebras. 
5. Open problems. 

1. Introduction. The purpose of this paper is to make the theory of vertex 
algebras trivial. We do this by setting up some categorical machinery so that 
vertex algebras are just ``singular commutative rings'' in a certain category. 
This makes it easy to construct many examples of vertex algebras, in 
particular by using an analogue of the construction of a twisted group ring 
from a bicharacter of a group. We also define quantum vertex algebras as 
singular braided rings in the same category and construct some examples of 
them. The constructions work just as well for higher dimensional analogues of 
vertex algebras, which have the same relation to higher dimensional quantum 
field theories that vertex algebras have to one dimensional quantum field 
theories.