and $B(Z)$: $$\eqalign{
 &\VEV{X^\mu(Z_a)X^\nu(Z_b)}=\eta^{\mu\nu}\ln Z_{ab},
\cr
 &\VEV{C(Z_a)B(Z_b)}=\VEV{B(Z_a)C(Z_b)}=
 {{\theta_{ab}^-\theta_{ab}^+}\over{Z_{ab}}},
\cr
}
\eqn\threeeight$$ where $Z_{ab}$ and $\theta^\pm_{ab}$ are defined by $$\eqalign{
 &Z_{ab}
  =z_a-z_b-\left(\theta_a^+\theta_b^-+\theta_a^-\theta_b^+\right),
\cr
 &\theta^\pm_{ab}=\theta^\pm_a-\theta^\pm_b.
\cr
}
\ee$$ Our next question is the following: What is the correct expression of the ghost number supercurrent and the BRST supercurrent in the partition function ? Note that, in the partition function , we do not have any anomalies and we can always use the naive equations of motion. From the expressions of the BRST anomaly  and the ghost number anomaly , we may take $$\eqalign{
  &j_B(Z)\equiv C\left(T^X+{1\over2}T^{\rm gh}\right)
   +{d-2\over4}C
    \left({1\over2}\Diminus\Phi\Diplus\Phi+i\partial\Phi\right)
\cr
  &\qquad\qquad
  +{1\over4}\Diminus\left[C\left(\Diplus C\right)B\right]
  +{1\over4}\Diplus\left[C\left(\Diminus C\right)B\right],
\cr
  &j_{\rm gh}(Z)\equiv-BC,
\cr
}
\eqn\threeten$$ as the effective covariant supercurrents in the partition function . In , we determined the $\Phi$-dependences as to reproduce  and  under uses of the naive equations of motion of the matter and the ghost fields. This prescription was also applied to $N=0$ and $N=1$ (super-)gravity \[,\]. We emphasize that we obtained the anomalous identities  and  in the BRST invariant path integral framework \[,\], thus those expressions should reflect the (super-)coordinate covariance in the quantum theory.

In order to have a complete description of the $N=2$ quantum supergravity, we should quantize the Liouville 