Generalized Wick theorems in conformal field theory and the Borcherds identity

TL;DR

This paper introduces a new contour integral formula for operator product expansions in conformal field theory, extending the generalized Wick theorem and relating it to the Borcherds identity in vertex algebra theory.

Contribution

It presents a novel contour integral expression for the OPE of a normally ordered operator with a single operator, connecting it to Borcherds identity.

Findings

New contour integral formula for operator product expansion

Relation established between Wick theorem and Borcherds identity

Enhanced understanding of operator contractions in conformal field theory

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.