Explicit example de Rham moduli space of connections Let $\Sigma$ be a Riemann surface and let $n,d$ be two relatively prime integers.  We can consider different moduli spaces related to those. On one hand we have:
-$M_{Dol}$ the moduli space of stable Higgs field of rank $n$ and degree $d$
-$M_B$ moduli space of (twisted) representations of fundamental group of $\Sigma$ with central monodromy $e^{\frac{2 \pi d}{n}}$
-$M_{dR}$ moduli space of stable meromorphic connections with one pole with residue $-\frac{2 \pi i d}{n}$.
In the case $n=1$ there is an explicit description of the first twos:
$$M_{Dol}=Jac(\Sigma) \times \mathbb{C}^g $$ and $$M_B=(\mathbb{C}^*)^g .$$
I struggled to find an explicit description for de Rham moduli space however.
I know this is biholomorphic to $M_B$ but it's not the same as an algebraic variety.
Is there a well known algebrogeometric description of it? Is there a concrete description of the Riemann Hilbert map too in this case?
EDIT: The general question was motivated by the following way more specific one. What happens at the moduli space if we consider a map (not necessarily holomorphic) $f:\Sigma \to \Sigma$ and we look at the map on the moduli space $$(\mathcal{E},\nabla) \to (f^*\mathcal{E},f^*\nabla) .$$
I was trying to get a glimpse of the geometrical properties of this kind of morphism(for example I think it should be holomorphic but I can't see why) via the identifications in the answer(which are very well written anyway).
 A: Yes, there is a description of the rank 1 de Rham moduli space, and you can find it for example in Goldman's notes: Higgs Bundles and Geometric Structures on Surfaces.
The deRham moduli space is given by an affine holomorphic bundle over the Jacobian with underlying holomorphic bundle the cotangent bundle, where the Atiyah class is determined by the natural symplectic form (see Tyurins book on Theta functions,..).
You can proof all this by applying classical Hodge theory of harmonic 1-forms on Riemann surfaces. As opposed to the higher rank case, the gauge group has different connected components classified by $H^1(\Sigma,\mathbb Z).$ Flat line bundle are given by closed complex 1-forms. By factoring out the identity component of gauge transformations you obtain the first deRham cohomology group $H^1(\Sigma,\mathbb C)$, but taking the quotient by the full gauge group you obtain $H^1(\Sigma,\mathbb C)/H^1(\Sigma,\mathbb Z).$ The Riemann Hilbert map is given by integration and exponentiation.
A: The difference of two connections on a line bundle $L$ is an $\mathcal{O}_X$-linear homomorphism $L\to L\otimes\Omega^1_X$, i.e. an element of $H^0(X, \Omega^1_X)$. Therefore the space of line bundles of with connection on an elliptic curve $E$ is a $H^0(E, \Omega^1_E)$-torsor over $\operatorname{Jac}(E)=E$, where we treat $H^0(E, \Omega^1_E)$ as an algebraic group abstractly isomorphic to $\mathbf{G}_a$. Such torsors are classified by the group
$$ H^1(E, H^0(E, \Omega^1_E)) \simeq H^1(E, \mathcal{O}_E)\otimes H^0(E, \Omega^1_E) \simeq \operatorname{Hom}(H^1(E, \mathcal{O}_E), H^1(E, \mathcal{O}_E)). $$
The last isomorphism is due to the fact that $H^1(E, \mathcal{O}_E)$ $H^0(E, \Omega^1_E)$ are dual to each other. It should be easy to check that the class of the torsor corresponding to the de Rham moduli space corresponds to $\pm {\rm id}$ under the above isomorphism.
