gh}^0(z)}
  +\VEV{\ln J}=0,
\ee$$ where $J$ is a Jacobian factor associated with the change of variables . However, for , the Jacobian is trivial, i.e., $J=1$ and $\ln J=0$, because the variation of the field is not proportional to the field itself. Therefore $\partialbar\VEV{j_{\rm gh}^0(z)}$ is anomaly free. Similar considerations show other relations in .

A potential anomalous term in a vacuum expectation value is thus a product of the equation of motion and the conjugate field, because such a combination is proportional to the Jacobian factor of a change of variable whose variation is proportional to the field itself. In this sense the final combination $\partialbar\VEV{j_{\rm
 gh}^{+-}(z)}$ is potentially dangerous. Let us consider the following change of variables: $$\eqalign{
  &\delta c=\varepsilon(x)c,\quad\delta b=-\varepsilon(x)b,
\cr
  &\delta\gamma^\pm=\varepsilon(x)\gamma^\pm,\quad
  \delta\beta^\pm=-\varepsilon(x)\beta^\pm,
\cr
  &\delta\xi=\varepsilon(x)\xi,\quad
  \delta\eta=-\varepsilon(x)\eta,
\cr
}
\eqn\twothirteen$$ (more precisely, we should write down the variation of the tilded integration variables, but for , the variation of the tilded variables is proportional to the one of the untilded variables). Then we have $$-{1\over2\pi}\int d^2x\varepsilon(x)
    \partialbar\VEV{j_{\rm gh}^{+-}(z)}+\VEV{\ln J}=0,
\ee$$ where $J$ is the Jacobian factor associated with the variation . From the master formula , we have $$\ln J=-{1\over4\pi}\int d^2x\varepsilon(x)
  (-3+2\times2-1)\partialbar\partial\sigma=0,
\ee$$ where the contributions from the 