Generalization of Witten's computation of the volume of moduli space - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T06:01:19Z http://mathoverflow.net/feeds/question/85948 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/85948/generalization-of-wittens-computation-of-the-volume-of-moduli-space Generalization of Witten's computation of the volume of moduli space unknown (google) 2012-01-18T02:36:39Z 2012-01-18T02:36:39Z <p>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 <a href="http://www.ams.org/mathscinet-getitem?mr=762512" rel="nofollow">Goldman</a>) coming from the intersection pairing on $H^1(X)$, and its top wedge gives $X$ a natural volume form. Witten (in <a href="http://www.ams.org/mathscinet-getitem?mr=1133264" rel="nofollow">this paper</a>) 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 $\operatorname{SU}(2)$.</p> <p>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 <em>spin diagram</em> on $\Sigma$. Alternatively, we could think of the function <code>$\rho\mapsto\prod_i\operatorname{tr}_{V_i}\rho(\alpha_i)$</code> for some $\alpha_i\in\pi_1(\Sigma)$ and $V_i$ representations of $\operatorname{SU}(2)$.</p> <blockquote> <p>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?</p> </blockquote> <p>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 <em>should</em> be.</p>