This isn't an answer. André surely knows what I'm about to say already, but here is some context for guessing what this story should look like. The context is 3d Dijkgraaf-Witten theory: this is a family $Z_{G, k}$ of 3d topological field theories depending on a finite group $G$ and a cocycle $k \in Z^3(BG, \mathbb{C}^{\times})$ (note that $H^3(BG, \mathbb{C}^{\times}) \cong H^4(BG, \mathbb{Z})$).

The value $Z_{G, k}(\Sigma)$ of Dijkgraaf-Witten theory on a closed surface $\Sigma$ is a complex vector space, constructed in the following way. If $X$ and $Y$ are two spaces let $[X, Y]$ denote the mapping space. In particular, $[\Sigma, BG]$ is the space of $G$-bundles on $\Sigma$. The pullback of $k$ along the evaluation map

$$\Sigma \times [\Sigma, BG] \to BG$$

produces a cocycle in $Z^3(\Sigma \times [\Sigma, BG], \mathbb{C}^{\times})$. The transgression of this cocycle along the projection $\Sigma \times [\Sigma, BG] \to [\Sigma, BG]$ produces a cocycle in $Z^1([\Sigma, BG], \mathbb{C}^{\times})$, which one should interpret as a (flat) complex line bundle, and $Z_{G, k}(\Sigma)$ is the space of (flat) global sections of this line bundle.

In the setting where $\Sigma$ is replaced with an elliptic curve $E$ the transgression operation we should interpret cocycles in $Z^3(-, \mathbb{C}^{\times})$ as some kind of 3-line bundles, and "integrated" over $E$ these should produce algebraic line bundles over $\text{Loc}_G(E)$.

In particular, if $R \neq \mathbb{C}$ then $k$ might live in something more like $Z^3(BG, R^{\times})$.

This isn't an answer. André surely knows what I'm about to say already, but here is some context for guessing what this story should look like. The context is 3d Dijkgraaf-Witten theory: this is a family $Z_{G, k}$ of 3d topological field theories depending on a finite group $G$ and a cocycle $k \in Z^3(BG, \mathbb{C}^{\times})$ (note that $H^3(BG, \mathbb{C}^{\times}) \cong H^4(BG, \mathbb{Z})$).

The value $Z_{G, k}(\Sigma)$ of Dijkgraaf-Witten theory on a closed surface $\Sigma$ is a complex vector space, constructed in the following way. If $X$ and $Y$ are two spaces let $[X, Y]$ denote the mapping space. In particular, $[\Sigma, BG]$ is the space of $G$-bundles on $\Sigma$. The pullback of $k$ along the evaluation map

$$\Sigma \times [\Sigma, BG] \to BG$$

produces a cocycle in $Z^3(\Sigma \times [\Sigma, BG], \mathbb{C}^{\times})$. The transgression of this cocycle along the projection $\Sigma \times [\Sigma, BG] \to [\Sigma, BG]$ produces a cocycle in $Z^1([\Sigma, BG], \mathbb{C}^{\times})$, which one should interpret as a (flat) complex line bundle, and $Z_{G, k}(\Sigma)$ is the space of (flat) global sections of this line bundle.

In the setting where $\Sigma$ is replaced with an elliptic curve $E$ (or algebraic curve of higher genus), the transgression operation we're looking for (which I don't know how to define) should interpret cocycles in $Z^3(-, \mathbb{C}^{\times})$ as some kind of 3-line bundles, and "integrate" over $E$ to produce algebraic line bundles over $\text{Loc}_G(E)$.

In particular, if $R \neq \mathbb{C}$ then $k$ might live in something more like $Z^3(BG, R^{\times})$.

This isn't an answer. André surely knows what I'm about to say already, but here is some context for guessing what this story should look like. The context is 3d Dijkgraaf-Witten theory: this is a family $Z_{G, k}$ of 3d topological field theories depending on a finite group $G$ and a cocycle $k \in Z^3(BG, \mathbb{C}^{\times})$ (note that $H^3(BG, \mathbb{C}^{\times}) \cong H^4(BG, \mathbb{Z})$).

The value $Z_{G, k}(\Sigma)$ of Dijkgraaf-Witten theory on a closed surface $\Sigma$ is a complex vector space, constructed in the following way. If $X$ and $Y$ are two spaces let $[X, Y]$ denote the mapping space. In particular, $[\Sigma, BG]$ is the space of $G$-bundles on $\Sigma$. The pullback of $k$ along the evaluation map

$$\Sigma \times [\Sigma, BG] \to BG$$

produces a cocycle in $Z^3(\Sigma \times [\Sigma, BG], \mathbb{C}^{\times})$. The transgression of this cocycle along the projection $\Sigma \times [\Sigma, BG] \to [\Sigma, BG]$ produces a cocycle in $Z^1([\Sigma, BG], \mathbb{C}^{\times})$, which one should interpret as a (flat) complex line bundle, and $Z_{G, k}(\Sigma)$ is the space of (flat) global sections of this line bundle.

In the setting where $\Sigma$ is replaced with an elliptic curve $E$ the transgression operation we need should interpret cocycles in $Z^3(-, \mathbb{C}^{\times})$ as some kind of 3-line bundles, and "integrated" over $E$ these should produce algebraic line bundles over $\text{Loc}_G(E)$.

In particular, if $R \neq \mathbb{C}$ then $k$ might live in something more like $Z^3(BG, R^{\times})$.

This isn't an answer. André surely knows what I'm about to say already, but here is some context for guessing what this story should look like. The context is 3d Dijkgraaf-Witten theory: this is a family $Z_{G, k}$ of 3d topological field theories depending on a finite group $G$ and a cocycle $k \in Z^3(BG, \mathbb{C}^{\times})$ (note that $H^3(BG, \mathbb{C}^{\times}) \cong H^4(BG, \mathbb{Z})$).

The value $Z_{G, k}(\Sigma)$ of Dijkgraaf-Witten theory on a closed surface $\Sigma$ is a complex vector space, constructed in the following way. If $X$ and $Y$ are two spaces let $[X, Y]$ denote the mapping space. In particular, $[\Sigma, BG]$ is the space of $G$-bundles on $\Sigma$. The pullback of $k$ along the evaluation map

$$\Sigma \times [\Sigma, BG] \to BG$$

produces a cocycle in $Z^3(\Sigma \times [\Sigma, BG], \mathbb{C}^{\times})$. The transgression of this cocycle along the projection $\Sigma \times [\Sigma, BG] \to [\Sigma, BG]$ produces a cocycle in $Z^1([\Sigma, BG], \mathbb{C}^{\times})$, which one should interpret as a (flat) complex line bundle, and $Z_{G, k}(\Sigma)$ is the space of (flat) global sections of this line bundle.

In the setting where $\Sigma$ is replaced with an elliptic curve $E$ the transgression operation we should interpret cocycles in $Z^3(-, \mathbb{C}^{\times})$ as some kind of 3-line bundles, and "integrated" over $E$ these should produce algebraic line bundles over $\text{Loc}_G(E)$.

In particular, if $R \neq \mathbb{C}$ then $k$ might live in something more like $Z^3(BG, R^{\times})$.