This post is rather wordy and speculative, but I promise there is a concrete question embedded within. For experts, I'll open with a question:
Question: Given a compact Riemann surface $X$, why does one prefer the category of D-modules on the space/stack $Bun_{X}(G)$ instead of the (Fukaya) category of Langrangians on the space/stack of $G$-local systems $\mathcal{L}_{X}(G).$
Much of what follows is explaining what I mean by this question.
Let $G$ be a connected reductive complex Lie group and $G^{\vee}$ the Langlands dual group. Let $X$ be a compact Riemann surface (smooth projective curve over $\mathbb{C}$). The central players in the geometric Langlands conjecture are the stacks of $G$-bundles on $X,$ denoted $Bun_{X}(G),$ and the stack of $G^{\vee}$-local systems on $X,$ which I'll denote by $\mathcal{L}_{X}(G^{\vee}).$
The best hope (using the words of Drinfeld) is that there is a (derived) equivalence between the category of D-modules on $Bun_{X}(G)$ and the category of quasi-coherent sheaves on $\mathcal{L}_{X}(G^{\vee}).$ This best hope is by now entirely dashed, and, to name one main player, Gaitsgory has expended a tremendous amount of effort to both indicate its failure, and conjecture a solution, along the way writing some very technical and interesting papers to give evidence.
Meanwhile, Witten, Kapustin, and others have made efforts to indicate how to restore some symmetry to this nebulous blob of conjectures, by arguing that they have natural interpretations as dimensional reductions of certain four dimensional super-symmetric gauge theories.
In the course of reading about various aspects of this story, I was struck by a point that Witten has made many times in talks and in print that I want to ask about here. I'm going to speak rather prosaically from here on forward, and welcome an explanation about why my simplifications don't make any sense.
Starting from Hitchin's work in the late 1980's, it became clear that the space of $G$-local systems $\mathcal{L}_{X}(G)$ is basically the cotangent bundle of $Bun_{X}(G).$ If you take stacky language seriously enough, this can be made reasonably precise. Taking this as true, there are many concrete, though not immediately applicable, theorems which indicate that the category of D-modules on $Bun_{X}(G)$ is equivalent to the Fukaya category of Langrangians in its co-tangent bundle: the latter of which is $\mathcal{L}_{X}(G).$
Using this quasi-logic, the best hope geometric Langlands conjecture posits an equivalence of (derived) categories between the Fukaya category of $\mathcal{L}_{X}(G)$ and the category of quasi-coherent sheaves on $\mathcal{L}_{X}(G^{\vee}).$
In other words, geometric Langlands becomes the statement that $\mathcal{L}_{X}(G)$ and $\mathcal{L}_{X}(G^{\vee})$ are mirror partners in the sense of homological mirror symmetry.
Question: Has this interpretation been taken seriously somewhere in the mathematical literature, and if not, is there a good reason it hasn't been?
As one final comment. There is (at least) one major advantage to putting the conjecture into this language. Both the Fukaya category of $\mathcal{L}_{X}(G)$ and the category of quasi-coherent sheaves on $\mathcal{L}_{X}(G^{\vee})$ can be defined without recourse to the complex/algebraic structure on $X.$ This is because both of these spaces/stacks are naturally complex symplectic, and this structure is independent of the complex/algebraic structure on $X.$
With this in mind, it makes more sense to refer to $\mathcal{L}_{\Sigma}(G)$ and $\mathcal{L}_{\Sigma}(G^{\vee})$ where $\Sigma$ is a connected, oriented, smooth surface. An advantage of this is that now topological symmetries (diffeomorphisms) of $\Sigma$ act naturally on these spaces/stacks and the subsequent categories. The study of these symmetries is what people in my field call the study of the mapping class group of $\Sigma,$ and there are many deep open questions about the mapping class group, that might find a natural home in the aforementioned discussion.
