Let $\Sigma$ be a Riemann surface, and let $X:=\operatorname{Hom}(\pi_1(\Sigma),\operatorname{SU}(2))/\operatorname{SU}(2)$ be the $\operatorname{SU}(2)$ character variety of $\pi_1(\Sigma)$.

There is a natural symplectic form on $X$ (by Goldman) coming from the intersection pairing on $H^1(\Sigma; \mathfrak{su}(2)_{\mathrm{Ad}_\rho})$, and its top wedge gives $X$ a natural volume form. Witten (in this paper) calculated the volume of $X$ by decomposing $\Sigma$ into a bunch of copies of $\mathbb S^2-\{3\text{ pts}\}$. The final answer is given by an infinite sum over all irreducible representations of $\operatorname{SU}(2)$.

Now the volume that Witten computes can clearly be written as follows: $$\int_X1\cdot\omega^{\wedge\text{top}}$$ Of course, there are a lot of other interesting functions we could try to integrate! The usual way of representing functions on $X$ is by a spin diagram on $\Sigma$. Alternatively, we could think of the function $\rho\mapsto\prod_i\operatorname{tr}_{V_i}\rho(\alpha_i)$ for some $\alpha_i\in\pi_1(\Sigma)$ and $V_i$ representations of $\operatorname{SU}(2)$.

Given a spin diagram on $\Sigma$, is there any known calculation of the integral: $$\int_Xf\cdot\omega^{\wedge\text{top}}$$ where $f:X\to\mathbb R$ is the function associated to the spin diagram?

I believe this should be calculable using Witten's technique. I'd like to know if anyone has seen the answer in the literature, or at least knows what the answer should be.


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