Explicit check of the invariance of the Weil-Petersson form

Question

Using Fenchel-Nielsen coordinates, the Weil-Petersson metric can be written as

$\omega_{WP} = \sum_{i} d\ell_i \wedge d \tau_i,$

where $i$ is an index labelling the curves of a pants decomposition of the surface in question and the pairs $(\ell_i,\tau_i)$ form the associated Fenchel-Nielsen coordinates. Although it is not manifest from the above expression, $\omega_{WP}$ is actual independent on the choice of pants decomposition.

Now, in this question it was asked and answered how the Fenchel-Nielsen coordinates change upon performing the so-called A-move or S-move on the pants decomposition. The answer is provided in this paper (and references there) by explicit transformation laws.

Let us consider the case of a once punctured torus, then a pants decomposition consists of a single curve and therefore we have $\omega_{WP} = d\ell \wedge d\tau$. I tried to check that $\omega_{WP} = d\ell \wedge d\tau = d\ell' \wedge d\tau'$, using the expressions given by Proposition 3.1 of the above paper (and setting $\ell_0 = 1$), but the equality does not seems to hold. Indeed, by putting numerical values one easily finds that the determinant of the transformation is not 1.

Where is the problem?

(Let me also add that it seems to me that there is a typo in the expression for the twist parameter, the last factor should be $\{ \dots \}^{(-1/2)}$ in order to agree with the same expression given in terms of the quantities $A$ and $B$ at pag. 7).

Answer 1

Apparently both the original paper by Okai and Proposition 3.1 in the paper mentioned by the OP contain typos in the expressions. Corrected formulas can be found in equations (5.13) and (5.22) of this recent arXiv submission (see also footnote 9 on page 75):

Jørgen Ellegaard Andersen, Gaëtan Borot, Séverin Charbonnier, Alessandro Giacchetto, Danilo Lewański, Campbell Wheeler, On the Kontsevich geometry of the combinatorial Teichmüller space, arXiv:2010.11806

Starting with their expressions it is easily checked with Mathematica that the determinant of the transformation is equal to 1:

Det@D[{2 ArcCosh[ Cosh[t/2]/Sinh[l/2] Sqrt[(Cosh[l] + Cosh[l0/2])/2]], 
      -2 ArcCosh[ Cosh[l/2] Sqrt[ 
           ((Cosh[t/2]^2 (Cosh[l] + Cosh[l0/2]) - 2 Sinh[l/2]^2) / 
           (Cosh[t/2]^2 (Cosh[l] + Cosh[l0/2]) + Sinh[l/2]^2 (Cosh[l0/2] - 1)))]]},
      {{l, t}}] // FullSimplify[#, t > 0 && l > 0 && l0 > 0] &

1