}
\eqn\twotwo$$ To get the dependences on the remaining components of the Liouville superpartners, we will use the invariance (or covariance) under . This strategy is the same as the one for a calculation of the super-Liouville action in \[\].

The partition function of the $N=2$ supergravity in the superconformal gauge is defined by $$\eqalign{
  \int\,d\widetilde\mu\,\exp\biggl\{-{1\over2\pi}\int\biggl[\,
  {1\over2}&\left(
  -\partial\Xmu\partialbar\Xmu
  -\partial\Ymu\partialbar\Ymu
  +\psiplusmu\partialbar\psiminusmu+\psiminusmu\partialbar\psiplusmu
  \right)
\cr
  &\qquad+b\partialbar c+\betaplus\partialbar\gammaminus
   +\betaminus\partialbar\gammaplus+\eta\partialbar\xi
   +({\rm c.\ c.})\,\biggr]\biggr\},
\cr
}
\eqn\twofour$$ where we defined the integration measure $d\widetilde\mu$ by $$\eqalign{
  d\widetilde\mu&=
%  \Di\bl e^{\sigma/2}\br
  \Di\bl e^\sigma c\br
  \Di\bl e^\sigma\overline c\br
  \Di\bl e^{-\sigma/2}b\br
  \Di\bl e^{-\sigma/2}\overline b\br
  \Di\bl e^{3\sigma/4}\gammaplus\br
  \Di\bl e^{3\sigma/4}\overline\gammaplus\br
\cr
  &\quad\times
  \Di\bl e^{3\sigma/4}\gammaminus\br
  \Di\bl e^{3\sigma/4}\overline\gammaminus\br
  \Di\bl e^{-\sigma/4}\betaplus\br
  \Di\bl e^{-\sigma/4}\overline\betaplus\br
  \Di\bl e^{-\sigma/4}\betaminus\br
  \Di\bl e^{-\sigma/4}\overline\betaminus\br
\cr
  &\quad\times
  \Di\bl e^{\sigma/2}\xi\br
  \Di\bl e^{\sigma/2}\overline\xi\br
  \Di\eta
  \Di\overline\eta
  \Di\bl e^{\sigma/4}\psiplusmu\br
  \Di\bl e^{\sigma/4}\overline\psiplusmu\br
\cr
  &\quad\times
  \Di\bl e^{\sigma/4}\psiminusmu\br
  \Di\bl e^{\sigma/4}\overline\psiminusmu\br
  \Di\bl e^{\sigma/2}\Xmu\br
  \Di\bl e^{\sigma/2}\Ymu\br
\cr
  &\equiv
%  \Di\bl e^{\sigma/2}\br
  \Di\widetilde c\,
  \Di\widetilde{\overline c}\,
  \Di\widetilde b\,
  \Di\widetilde{\overline b}\,
  \Di\widetilde\gamma^+
  \Di\widetilde{\overline\gamma}^+
  \Di\widetilde\gamma^-
  \Di\widetilde{\overline\gamma}^-
  \Di\widetilde\beta^+
  \Di\widetilde{\overline\beta}^+
  \Di\widetilde\beta^-
  \Di\widetilde{\overline\beta}^-
  \Di\widetilde\xi
  \Di\widetilde{\overline\xi}
  \Di\eta
  \Di\overline\eta
\cr
  &\qquad\times
  \Di\widetilde\psi^{+\mu}
  \Di\widetilde{\overline\psi}^{+\mu}
  \Di\widetilde\psi^{-\mu}
  \Di\widetilde{\overline\psi}^{-\mu}
  \Di\widetilde X^\mu
  \Di\widetilde Y^\mu.
\cr
}
\eqn\twofive$$ The various weight factors ($\exp\sigma$) in the integration measure are determined from the general coordinate (BRST) invariance of the integration measure \[,\]. In the above expression, we 