Equation 6.2 is just the Liovelle Action, the action principle for the Liouville Field, which is well-known from the familiar conformal gauge.
$$S_L=\frac{c}{96\pi}\int_\mathcal{M}\left(\dot\varphi^2-\frac{16\varphi}{\left(1-\lvert t\rvert^2\right)^2}\right)\mathrm{d}^2t$$
... along with some trivial facts about partition functions.
You could of course think of it as the $Z_\mathcal{M}$'s (partition functions) of the metrics being related by the $S_L$'s in the same way that the metrics are related by the Liouvelle field.
And yes, I don't know how to spell "Lioivulle" properly.