 For every state "a" in "L" there is a vertex operator "V"("a","z") which creates "a" from the vacuum vector Ω, in the sense that

The vertex operators of energy 1 correspond to the generators of the affine algebra

where "X" ranges over the elements of the underlying finite-dimensional simple complex Lie algebra formula_5.There is an energy 2 eigenvector which give the generators "L""n" of the Virasoro algebra associated to the Kac–Moody algebra by the "Segal–Sugawara construction"

If "a" has energy , then the corresponding vertex operator has the form

The vertex operators satisfy

as well as the locality and associativity relations

These last two relations are understood as analytic continuations