supermultiplet. We define the partition function of the Liouville supermultiplet as $$\eqalign{
  &\int\Di\left(e^{\sigma/2}\right)
      \Di\left(e^{\sigma/4}\phi^+\right)
      \Di\left(e^{\sigma/4}\overline\phi^+\right)
      \Di\left(e^{\sigma/4}\phi^-\right)
      \Di\left(e^{\sigma/4}\overline\phi^-\right)
      \Di\left(e^{\sigma/2}\phi\right)
\cr
  &\times\exp\left\{
   -{2-d\over16\pi}\int d^2x\left[\partial\sigma\partialbar\sigma
                           +\partial\phi\partialbar\phi
                           -\phi^+\partialbar\phi^-
                           -\phi^-\partialbar\phi^+
                           +({c.\ c.})\right]
   \right\},
\cr
}
\eqn\threeeleven$$ where we have chosen the weight factors ($\exp\sigma$) following the prescription in \[,\]. The full partition function is given by a product of  and .

If we apply the same procedure of the derivation of  also to the gravitinos $\phi^\pm$ and the gauge field $\phi$, the partition function  changes to $$\eqalign{
  &\int\Di\left(e^{\sigma/2}\right)
      \Di\phi^+\Di\overline\phi^+\Di\phi^-\Di\overline\phi^-\Di\phi
\cr
  &\times\exp\left[
   -\left({2-d\over16\pi}-{1\over24\pi}\right)
    \int d^2x\partial\sigma\partialbar\sigma
   -{1\over4\pi}M^2\int d^2xe^\sigma
   +\cdots
   \right],
\cr
}
\eqn\threetwelve$$ where we only indicated the $\sigma$-dependence of the action. The integration of the Liouville field in  is, on the other hand, highly non-linear because the integration variable is $e^{\sigma/2}$, not simply $\sigma$. To avoid this problem, here we apply the background field method \[,\] and include the one-loop renormalization effect arising from the non-trivial measure $e^{\sigma/2}$.

To do this, we set $e^{\sigma/2}\equiv e^{\sigma_0/2}+\varphi$ and expand the Liouville action in  with respect to $\varphi$ up to the second order. If we assume the $M^2e^\sigma$ term in  is canceled by a suitable counter term, the resulting action for the quantum fluctuation $\varphi$ has the same form 