Generalized Wick theorems in conformal field theory and the Borcherds identity

Abstract: As a counterpart of the well-known generalized Wick theorem by Bais et. al. in 1988 for interacting fields in two dimensional conformal field theory, we present a new contour integral formula for the operator product expansion of a normally ordered operator and a single operator on its right hand. Quite similar to the original Wick theorem for the opposite order operator product, it expresses the contraction i.e.the singular part of the operator product expansion as a contour integral of only two terms, each of which is a product of a contraction and a single operator. We discuss the relation between these formulas and the Borcherds identity satisfied by the quantum fields associated with the theory of vertex algebras.