following Jacobian: $$\ln J(t)_{\widetilde X^\mu}={d\over2\pi}dt\int d^2x\sigma\left[
  -{1\over12}(1-t)\partialbar\partial\sigma+{1\over2}M^2e^{(1-t)\sigma}
  \right].
\ee$$ Summing over all the contributions from various fields and by integrating $\ln J(t)$ from $t=0$ to $1$, we have the so-called Liouville action: $$\int_0^1\ln J(t)=-{2-d\over16\pi}
            \int d^2x\partial\sigma\partialbar\sigma.
\eqn\threefour$$ Note that the “Liouville term,” $e^\sigma$, disappears in  because of the supersymmetry of the original model.

As noted in the previous section, the Liouville action should invariant under the global super transformation . Therefore, under a supersymmetric regularization, the Liouville action  should have the form \[\], $$S_{\rm Liouville}\equiv-{2-d\over16\pi}
            \int d^2x\left[\partial\sigma\partialbar\sigma
                           +\partial\phi\partialbar\phi
                           -\phi^+\partialbar\phi^-
                           -\phi^-\partialbar\phi^+
                           +({\rm c.\ c.})\right].
\eqn\threefive$$ In this stage, since we have extracted the $\sigma$-dependences in the integration measure as the Liouville action , our partition function of the matter and the ghost multiplets in a fixed metric background has the following form: $$\eqalign{
  \int\,d\mu\,\exp\biggl\{-{1\over2\pi}\int\biggl[\,
  {1\over2}&\left(
  -\partial\Xmu\partialbar\Xmu
  -\partial\Ymu\partialbar\Ymu
  +\psiplusmu\partialbar\psiminusmu+\psiminusmu\partialbar\psiplusmu
  \right)
\cr
  &\qquad+b\partialbar c+\betaplus\partialbar\gammaminus
   +\betaminus\partialbar\gammaplus+\eta\partialbar\xi
   +({\rm c.\ c.})\,\biggr]\biggr\},
\cr
}
\eqn\threesix$$ where $d\mu$ is a “naive” integration measure: $$\eqalign{
  d\mu&=
  \Di c
  \Di\overline c
  \Di b
  \Di\overline b
  \Di\gammaplus
  \Di\overline\gammaplus
  \Di\gammaminus
  \Di\overline\gammaminus
  \Di\betaplus
  \Di\overline\betaplus
  \Di\betaminus
  \Di\overline\betaminus
\cr
  &\quad\times
  \Di\xi
  \Di\overline\xi
  \Di\eta
  \Di\overline\eta
  \Di\psiplusmu
  \Di\overline\psiplusmu
  \Di\psiminusmu
  \Di\overline\psiminusmu
  \Di\Xmu
  \Di\Ymu.
\cr
}
\eqn\threeseven$$ From  and , we have the following correlation functions of $X^\mu(Z)$, $C(Z)$, 