Vertex algebras. 10 June 1997, corrected 19 June, 26 June, 26 September. 

Topological field theory, primitive forms and related topics (Kyoto, 1996), 
35-77, Progr. Math., 160, Birkh\"auser Boston, Boston, MA, 1998. 

Richard E. Borcherds, 
D.P.M.M.S., 16 Mill Lane, Cambridge, CB2 1SB, England. 

Contents. 
1.  Introduction. 
2.  Formal groups and Hopf algebras. 
3.  Vertex groups 
4.  Relaxed multilinear categories 
5.  The representations of a vertex group. 
6.  G vertex algebras 
7.  Vertex operators 
8.  Free field G vertex algebras. 
9.  The main identity 
10. The Yang-Baxter equation 
11. Cohomology of G vertex algebras. 
12. Relation to quantum field theory.