FEINGOLD, RIES, WEINERhas to deal with fusion rules, N (M 1 , M 2 , M 3 ) = dim(I(M 1 , M 2 , M 3 )), which give the dimension of the space of intertwining operators determined by a triple of modules. It is a basic principle of VOA's that there is a one-to-one correspondence between vectors v in a simple VOA V and vertex operators Y M (v, z) acting on an irreducible V -module M . One can think of this as a mapwhich obeys the various axioms defining a V -module. The fusion rule in this case is always N (V, M, M ) = 1 and one axiom normalizes Y M so it is uniquely determined. Given three V -modules, M 1 , M 2 , M 3 , one can think of an intertwining operator as a map Y (ยท, z) : M 1 โ Hom(M 2 , M 3 ){{z, z โ1 }} which obeys the axioms for intertwining operators. (The notation {{z, z โ1 }} indicates rational powers of z.) It is quite possible that the fusion rule is N (M 1 , M 2 , M 3 ) = n > 1, and in that case one does not have a one-to-one correspondence between vectors w in module M 1 and operators Y (w, z) whose components send M 2 to M 3 . It would seem that this is a kind of labeling problem, there not being enough "copies" of the vectors in M 1 to distinguish the n linearly independent intertwiners which could be taken as a basis for the space of all intertwiners. It is also possible to have four modules M 1 , ยท ยท ยท , M 4 with M 3 = M 4 , with fusion rules N (M 1 , M 2 , M 3 ) โฅ 1 and N (M 1 , M 2 , M 4 ) โฅ 1. This also indicates a labeling problem, showing the inadequacy of the notation Y (w 1 , z)w 2 , where knowing that w i โ M i still does not determine which module contains the outcome.Another new feature which appears is the nature of the correlation functions,
