The Fuschian Monodromy Problem

==Question== 
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. 
 
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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..
 
====Comments====
 
==Answers==
===Answer 1=== 
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-\left| t\right|^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. 
Dimension10 \left(\right. Talk \cdot Edits \left. \right) 10:38, December 13, 2013 (UTC)
====Comments==== 
* You had it right the second time: "Liouville" - Ketil Tweiten 
** Ah, thanks. - Dimension10