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).

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).

Consider $X=\operatorname{Hom}(\pi,SL(2,\mathbb C))/\!/SL(2,\mathbb C)$ and $X_{\mathbb R}=\operatorname{Hom}(\pi,SU(2))/\!/SU(2)$, where $\pi$ is a surface group. Note that if we use the right coordinates, then $X_{\mathbb R}$ is exactly the real valued points of $X$.X$ (see the comments below; anyway, it isn't crucial for this question that $X_{\mathbb R}$ is the $\mathbb R$-valued points of $X$).

Since $\pi$ is a surface group, the cup product (intersection pairing) gives rise to a symplectic form $\omega$ on $X_{\mathbb R}$ (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:

$$f\otimes g\mapsto \int_{X_{\mathbb R}}f\cdot g\cdot\wedge^{\operatorname{top}}\omega$$

(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 $X_{\mathbb R}$ in $X$ . . .

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

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

Since $\pi$ is a surface group, the cup product (intersection pairing) gives rise to a symplectic form $\omega$ on $X_{\mathbb R}$ (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:

$$f\otimes g\mapsto \int_{X_{\mathbb R}}f\cdot g\cdot\wedge^{\operatorname{top}}\omega$$

(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 $X_{\mathbb R}$ in $X$ . . .

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

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

Since $\pi$ is a surface group, the cup product (intersection pairing) gives rise to a symplectic form $\omega$ on $X_{\mathbb R}$ (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:

$$f\otimes g\mapsto \int_{X_{\mathbb R}}f\cdot g\cdot\wedge^{\operatorname{top}}\omega$$

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 $X_{\mathbb R}$ in $X$ . . .

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