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  Generalization of Witten's computation of the volume of moduli space

+ 8 like - 0 dislike

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 $H^1(X)$, 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.

This post imported from StackExchange MathOverflow at 2014-09-02 20:36 (UCT), posted by SE-user John Pardon

asked Jan 18, 2012 in Theoretical Physics by John Pardon (75 points) [ revision history ]
edited Sep 2, 2014 by Dilaton

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