Ward Identities for Conformal Models

Abstract

The present knowledge concerning the Ward identities which express the symmetry of conformal models is fairly elusive. On the one hand diffeomorphism invariance is invoked, but mostly, locally holomorphic coordinate transformations are used^{[1] [2]}, in clash with the locality principles of all known versions of quantum field theory. Diffeomorphism invariance is on the other hand understood in terms of Riemannian geometry, but not directly in terms of conformal geometry^[3]. Recently, two different sets of Ward identities expressing diffeomorphism invariance in a manifestly conformally invariant way were found for the free bosonic string^{[4] [5]}. It turns out that they are equivalent, modulo the free ghost equations of motion. The purpose of this note is to give a geometrical argument showing that the correct invariance for a large class of conformal models, is that of ref. [4], suitably generalized, the simultaneous validity of the invariance depicted in ref. [5] being restricted to the free field situation.