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: $$ \omega_{h} = \int_{S} \mathrm{trace} (\dot{\nabla} \wedge \dot{\nabla}') $$ where $\dot{\nabla}$ and $\dot{\nabla'}$ 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?

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?

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, as everyone uses different conventions.

Can anyone answer this question by providing a reference and the definition they are using of these two forms?

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: $$ \omega_{h} = \int_{S} \mathrm{trace} (\dot{\nabla} \wedge \dot{\nabla}') $$ where $\dot{\nabla}$ and $\dot{\nabla'}$ 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?