is defined by $$H\equiv-\Dslash^\dagger\Dslash
  \equiv-e^{a\sigma}\partial e^{b\sigma}\partialbar e^{a\sigma}.
\eqn\twonines$$ where $\Dslash$ is the kinetic operator of the field $\tilde\phi$. By rewriting the action in  in terms of the integration variables in , we can read off the various values of $a$ and $b$ in  for the each fields (see Table 1).

To see how our present formulation works, let us first consider the ghost number anomaly \[\]. In $N=2$ case, the ghost number current is known to be anomaly free, due to a cancellation of the background charge \[\]. The ghost number supercurrent \[\] is defined by $$\eqalign{
  j_{\rm gh}(Z)&\equiv-BC(Z)
\cr
  &=i\eta c+\thetaminus\left(-\eta\gammaplus+i\betaplus c\right)
   +\thetaplus\left(\eta\gammaminus+i\betaminus c\right)
\cr
  &\quad+\thetaminus\thetaplus\left(-\eta\xi-\betaminus\gammaplus
   -\betaplus\gammaminus-bc\right)
\cr
  &\equiv
  j_{\rm gh}^0(z)+\thetaminus j_{\rm gh}^+(z)
  +\thetaplus j_{\rm gh}^-(z)+\thetaminus\thetaplus
  j_{\rm gh}^{+-}(z).
\cr
}
\eqn\twonineprime$$ We can immediately see $$\partialbar\VEV{j_{\rm gh}^0(z)}
  =\partialbar\VEV{j_{\rm gh}^+(z)}
  =\partialbar\VEV{j_{\rm gh}^-(z)}=0,
\eqn\twooneone$$ i.e., these currents are anomaly free. To show this, let us consider the following infinitesimal change of variables in : $$b\rightarrow b+i\varepsilon(x)\eta,\quad
  \xi\rightarrow\xi-i\varepsilon(x)c.
\eqn\twoeleven$$ Note that the partition function itself does not change under a change of the integration variables. Therefore the variation of the action and the variation of the integration measure should be canceled each other, and we have the following identity: $$-{1\over2\pi}\int d^2x\varepsilon(x)
    \partialbar\VEV{j_{\rm 