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Consider $X=\operatorname{Hom}(\pi,SL(2,\mathbb C))/\!/SL(2,\mathbb C)$ and $Y=\operatorname{Hom}(\pi,SU(2))/\!/SU(2)$, where $\pi$ is a surface group. Note that if we use the right coordinates, then $Y$ is exactly the real valued points of $X$ (for the purposes of this question, though, this is irrelevant).

For those reading the comments below, I used to use $X_{\mathbb R}$ instead of $Y$.

Since $\pi$ is a surface group, the cup product (intersection pairing) gives rise to a symplectic form $\omega$ on $Y$ (which can also be naturally viewed as a holomorphic symplectic form on $X$). Now for $f,g\in\Gamma(X,\mathcal f\in\Gamma(X,\mathcal O_X)$, we have a natural bilinear pairinglinear map:

$$f\otimes g\mapsto f\mapsto \int_{Y}f\cdot g\cdot\wedge^{\operatorname{top}}\omega$$int_{Y}f\cdot\wedge^{\operatorname{top}}\omega$$(I believe it is more or less correct to call this the Yang-Mills measure, e.g. see this paper by Bullock, Frohman, and Kania-Bartoszynska) Question: Is there a completely algebraic description of this pairingintegral? i.e. one that just works with input \pi and the algebraic group SL(2), and avoids the notion of real and complex points? Note: there is certainly a simple algebraic description of \omega, so perhaps we just need to algebraically define the homology class of Y in X . . . A paper of Witten's is perhaps related (and great to read even if it isn't). 4 deleted 39 characters in body Consider X=\operatorname{Hom}(\pi,SL(2,\mathbb C))/\!/SL(2,\mathbb C) and X_{\mathbb R}=\operatorname{Hom}(\pi,SU(2))/\!/SU(2), Y=\operatorname{Hom}(\pi,SU(2))/\!/SU(2), where \pi is a surface group. Note that if we use the right coordinates, then X_{\mathbb R} Y is exactly the real valued points of X (see for the purposes of this question, though, this is irrelevant). For those reading the comments below; anyway, it isn't crucial for this question that I used to use X_{\mathbb R} is the \mathbb R-valued points instead of X).Y. Since \pi is a surface group, the cup product (intersection pairing) gives rise to a symplectic form \omega on X_{\mathbb R} Y (which can also be naturally viewed as a holomorphic symplectic form on X). Now for f,g\in\Gamma(X,\mathcal O_X), we have a natural bilinear pairing:$$f\otimes g\mapsto \int_{X_{\mathbb R}}f\cdot int_{Y}f\cdot g\cdot\wedge^{\operatorname{top}}\omega

(I believe it is more or less correct to call this the Yang-Mills measure, e.g. see this paper by Bullock, Frohman, and Kania-Bartoszynska)

Question: Is there a completely algebraic description of this pairing? i.e. one that just works with input $\pi$ and the algebraic group $SL(2)$, and avoids the notion of real and complex points?

Note: there is certainly a simple algebraic description of $\omega$, so perhaps we just need to algebraically define the homology class of $X_{\mathbb R}$ Y$in$X\$ . . .

A paper of Witten's is perhaps related (and great to read even if it isn't).

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