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I want to understand the argument being made from equation 6.1 to 6.5 in this paper between pages 27-28

6.2, 6.4 and 6.5 are completely out-of-the-blue to me and I have no clue as to from where they come.

My impression is that this part of the argument in the paper is self-contained and it would be great if someone can help me get these equations.

If people want it then I can try to paraphrase here again as to what is written in the paper but I have actually nothing to add to what is already written in that paper..

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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 Liouville field.

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    $\begingroup$ You had it right the second time: "Liouville" $\endgroup$
    – Ketil Tveiten
    Commented Dec 10, 2013 at 8:33

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