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I'm confused about the exact multiplicative factor that relates Goldman symplectic form on the $\operatorname{SL}(2,\mathbb R)$-character variety and the Weil–Petersson symplectic form on Teichmüller space.

One way to define Goldman symplectic form is the following: $$ \omega_{h} = \int_{S} \operatorname{trace} (\dot{\nabla} \wedge \smash{\dot{\nabla}}') $$ where $\dot{\nabla}$ and $\smash{\dot{\nabla}}'$ are variations of the Levi-Civita connection of the hyperbolic metric $h$. If those variations are induced by divergence-free, $h$-self-adjoint, traceless endomorphisms $\dot{J}$ and $\dot{J}'$, then it is easy to see that $$ \dot{\nabla} = -\frac{1}{2} J\nabla\dot{J} \ . $$ Now, Goldman claims that this form is $-8$ times the Weil-Petersson symplectic form, but, integrating by parts, I get instead a multiplicative factor of $+4$. What am I doing wrong?

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  • $\begingroup$ This is spelled out in the paper by Goldman, The symplectic nature of fundamental groups of surfaces., Adv. in Math. 54 (1984), no. 2, 200-225. In each case there is a natural choice, and there is no multiplicative factor. $\endgroup$
    – abx
    Commented Feb 25 at 8:04
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    $\begingroup$ I'm not so sure: there he says the multiplicative factor is -8. I guess it's not so clear even for you... $\endgroup$
    – AMath91
    Commented Feb 25 at 10:28
  • $\begingroup$ Oops, you are right, I missed that. But what is the problem? Do you have doubts on Goldman's computation? $\endgroup$
    – abx
    Commented Feb 25 at 13:27
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    $\begingroup$ TeX note: $\dot{\nabla}'$ \dot{\nabla}' puts the dot too high. You can force TeX to forget the extra height of the dot by using \smash: $\smash{\dot{\nabla}}'$ \smash{\dot{\nabla}}'. I have edited accordingly. $\endgroup$
    – LSpice
    Commented Feb 27 at 14:35

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The factor of four and why Goldman has a different factor is explained on page 6 of Generalization of a formula of Wolpert for balanced geodesic graphs on closed hyperbolic surfaces.
A derivation with prefactor four is given in section 2 of The complex symplectic geometry of the deformation space of complex projective structures.

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  • $\begingroup$ I know the reference you are referring to. The problem is that I don't really agree with what they say. They say that Goldman uses $\mathrm{trace}(XY)$ as coupling in the Lie algebra but at page 211 of "The symplectic nature of fundamental groups of surfaces", Adv. in Math. 54 (1984), no. 2, 200-225, Goldman says that he is considering the Killing form on the Lie algebra. Moreover, the second paragraph of what the authors say is very unsatisfactory. $\endgroup$
    – AMath91
    Commented Feb 27 at 15:28
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    $\begingroup$ Goldman defines the Killing form on page 214; that definition is a factor of 4 larger than the conventional definition (see for example math.stackexchange.com/a/2833737/87355 ), as pointed out in the first reference I cited. $\endgroup$
    – Carlo Beenakker
    Commented Feb 27 at 15:54
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    $\begingroup$ Ok, you're right. But still, if with Goldman's conventions, Goldman symplectic form is (-8) times the Weil-Petersson symplectic form (as he claims in the paper), then when you take the standard killing form on the Lie algebra, you divide Goldman symplectic form by $4$, which means that the multiplicative factor should become $-2$. $\endgroup$
    – AMath91
    Commented Feb 27 at 16:15

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