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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.

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    $\begingroup$ Are you familiar with Stoyanovsky's twisted D-module formulation of geometric Langlands? It has a symmetry between $G$ and $G^\vee$ similar to the one you propose, but with more or less reciprocal twistings. $\endgroup$ – S. Carnahan Apr 16 '20 at 15:00
  • $\begingroup$ Thanks for the pointer: I'll look into it, but would appreciate a reference if you have one in mind. $\endgroup$ – Andy Sanders Apr 16 '20 at 15:05
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    $\begingroup$ The basic reference is Stoyanovsky's paper arxiv.org/abs/math/0610974 but I think there are more refined versions now. $\endgroup$ – S. Carnahan Apr 16 '20 at 15:51
  • $\begingroup$ This is perhaps the first place I've seen where the misspelling "Langrangians" might actually be appropriate. $\endgroup$ – Thurmond Apr 16 '20 at 17:10
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One answer to your initial question is that the $D$-modules are supposed to actually do something - they're supposed to analogize to automorphic forms under the sheaf-functions dictionary. Therefore various things we know or believe about automorphic forms translate straightforwardly into expectations about these $D$-modules. We could translate that into the Fukaya category but we would have to pass through the Lagrangians-sheaves dictionary at every step.

And the Lagrangian-sheaves dictionary is not in such great shape - a significant problem is that the concrete, but not immediately applicable theorems you mention are not just not immediately applicable - no one knows how to make them apply. There is an approach to constructing geometric Langlands by passing through these Lagrangians - see this paper of Donagi and Pantev, which takes 213 pages to make the construction work in one special case.

Recall here that, the mirror symmetry here is supposed to come from the fact that the moduli spaces of Higgs bundles on $G$ and on $\hat{G}$ are dual abelian fibrations over the Hitchin base, so a skyscraper sheaf on one side is sent to a (Lagrangian) Hitchin fiber on the other side, with the additional data of a line bundle. So the mirror symmetry step is totally straightforward in this case - we know exactly what the object should correspond to. The entire difficulty is handling the transition from the Fukaya category to $D$-modules.

For your claim at the end about the mapping class group, you have to be careful, because the claims of geometric Langlands are really about complex-algebraic sheaves and not complex-analytic sheaves. In this setting Higgs bundles, local systems, and representations of the fundamental group are not all equivalent. Only representations of $\pi_1$ depend only on topology and admit this mapping class group action. The best hope statement of geometric Langlands uses local systems (i.e. vector bundles with flat connections), not representations of $\pi_1$.

More recently, Ben-Zvi and Nadler have defined Betti geometric Langlands, which does use representations of $\pi_1$. However on the other side they have to work with a different category of sheaves - derived constructible sheaves with singular support on the nilpotent cone. This category is not known to be topologically invariant, but is conjectured to be. I don't know what its Fukaya analogue might look like. I have heard that this correspondence is supposed to be a better reflection of Kapustin and Witten's physical construction, because that physical construction is topologically invariant.

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  • $\begingroup$ Thanks for your answer Will. I'm very aware of the difference between local systems, Higgs bundles, and representations with respect to complex and algebraic issues, sorry if I elided that point in my question. You did omit one intermediate object, which is smooth local systems, where the mapping class group does act, but only analytically, not algebraically. A really obscene way to state my question, is to say that any mention of Higgs bundles is a red herring, but I'll also admit I'm just asking a different question at that point... $\endgroup$ – Andy Sanders Apr 16 '20 at 16:05
  • $\begingroup$ @AndySanders Don't you need Higgs bundles to say "this can be made reasonably precise"? Also doesn't the equivalence require doing strange things to the Lagrangians near infinity? $\endgroup$ – Will Sawin Apr 16 '20 at 16:11
  • $\begingroup$ I'm not sure anyone has written it down, but if a co-tangent vector to a flat $C^{\infty}$ unitary connection isn't a flat complex connection, I'm not sure what it could be. But, I totally agree, the current state of the art makes my question seem very naive. In that respect, you've probably answered the main question of: why haven't mathematicians looked into this. $\endgroup$ – Andy Sanders Apr 16 '20 at 16:30
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    $\begingroup$ @AndySanders But the $D$-modules are defined using the complex (algebraic) structure of the moduli space of vector bundles, which $C^{\infty}$ unitary connections don't see. $\endgroup$ – Will Sawin Apr 16 '20 at 17:11
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    $\begingroup$ @AndySanders I'm sure if David Ben-Zvi sees this question, he will have some additional perspective. $\endgroup$ – Will Sawin Apr 16 '20 at 17:12
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To answer [a paraphrase of] your second question first: yes the Kapustin-Witten perspective on geometric Langlands has I think been taken very seriously by a segment of the math community. I find it very misleading though to say (as is often done) that "geometric Langlands is mirror symmetry for the Hitchin space" -- mirror symmetry is a statement about 2d TFTs, while geometric Langlands is one about 4d TFTs which implies a vast amount more structure -- specifically the most important structure for the Langlands story, the action of Hecke operators, is part of the 4d story but not of the mirror symmetry statement.

In any case as Will mentions the Betti Geometric Langlands conjecture formulated in https://arxiv.org/abs/1606.08523 is a direct response to your question - in particular to have a version of geometric Langlands which as you ask should depend only topologically on the Riemann surface (so eg have a mapping class group symmetry). However it is not formulated in Fukaya category language directly. I'm fairly ignorant of Fukaya categories but my impression is that the vast technical difficulties in the subject prevent them currently being rigorously defined on the kind of spaces we are talking about here -- namely both singular and stacky. So the Fukaya-theoretic conjecture you discuss is still more of a guiding principle than a precise question.

Also since the Hitchin space is noncompact you have to decide what KIND of Fukaya category you'd mean (assume say we are dealing with a smooth manifold), i.e. what conditions to put at infinity as Will says - infinitesimal, wrapped, or partially wrapped ("with stops"). The Betti conjecture, which Nadler and I felt captured the spirit of Kapustin-Witten, is morally taking the Fukaya category with stops in the direction of the Hitchin base - i.e. your prototypical Lagrangians allowed are Hitchin fibers, not sections. [By the way for one of probably quite a few papers I can't remember this instant which does treat aspects of GL in a Fukaya perspective there's Nadler's paper on Springer theory https://arxiv.org/abs/0806.4566]

So what to do instead of Fukaya categories? by the microlocal perspective of Nadler, Zaslow, Kontsevich,.... we expect to replace Fukaya categories with categories of microlocal sheaves, eg for cotangent bundles, with constructible sheaves on the base (and you can impose singular support conditions for the "stops" or growth conditions on the Lagrangians). This actually gets you very close to the original characteristic p origin of the geometric Langlands correspondence, which dealt with l-adic sheaves -- via the Grothendieck function-sheaf dictionary those are natural "categorified" substitutes for functions, eg automorphic functions. There are no D-modules in this story.

The beautiful D-module version developed by Beilinson-Drinfeld and Arinkin-Gaitsgory in particular -- the de Rham geometric Langlands correspondence -- has a quite different flavor in many respects, and I would claim is one step further from both the arithmetic (l-adic) origins and from the mirror symmetry story. It is motivated by two (closely related) stories -- the Beilinson-Bernstein realization of representations of Lie algebras as D-modules, which gives it a very close relation to representation theory of affine Kac-Moody algebras; and conformal field theory (eg theory of vertex algebras). This allowed Beilinson-Drinfeld to leverage a crucial result of Feigin-Frenkel to prove a "big chunk" (half-dimensional slice) of the de Rham conjecture, and Gaitsgory and collaborators to develop an amazing program to understand and solve the conjecture in general.

[As a side note I think it's overly dramatic to say the "best hope" is "dashed" -- rather the distance from the original dream to the Arinkin-Gatisgory formulation is technical and not very large and not terribly unexpected - though led to some beautiful math - and is purely about understanding how to match growth conditions on one side with singularity conditions on the other, just as in studying the Fourier transform in different function spaces.]

The de Rham story also has deep relations to physics. The physics I would say is somewhat insensitive to the de Rham vs Betti distinction, which is about different algebraic structures underlying an analytic equivalence, but many of the mathematical questions require you to pick your setting more precisely (except the "core" ones that live in the intersection of the two conjectures). The de Rham story comes up naturally in relation to CFT, to things like gauge theories of Class S and the AGT conjecture, and a whole world that is part of.

OK this is now way too long.

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  • $\begingroup$ It's going to take me some time to digest this answer David, and also the answers of Will. Thank you for your efforts, I truly appreciate it guys. $\endgroup$ – Andy Sanders Apr 16 '20 at 19:36
  • $\begingroup$ Thanks for giving an answer with, unsurprisingly, a lot more detail and background knowledge than mine! If you don’t mind, I want to interrogate a bit your comment about the Betti story being closer to the arithmetic setting. I agree of course that the advantages of $\ell$-adic sheaves because they look closer to the sheaves we use in the sheaf-functions dictionary (and elsewhere in Langlands) are massive, but doesn’t Betti also put heavy restrictions on which sheaves are allowed on the automorphic side? $\endgroup$ – Will Sawin Apr 17 '20 at 0:13
  • $\begingroup$ So in some way Betti Geometric Langlands can see only those parts of the automorphic side that are deformation-invariant in a suitable sense. A concrete example could be sheaves arising from periods as in your work with Venkatesh and Sakellaridis. If we take $i^{G/H}: \operatorname{Bun}_H \to \operatorname{Bun}_G$ or other maps associated to other period varieties, the complex $i^{G/H}_! \mathbb Q_\ell$ does not get to participate in Betti geometric Langlands because it does not have the right singular support. $\endgroup$ – Will Sawin Apr 17 '20 at 0:13
  • $\begingroup$ In some ways this is telling us that if we $\operatorname{Rhom} ( i^X_! \mathbb Q_\ell, i^Y_! \mathbb Q_\ell)$ for spherical varieties $X$ and $Y$, might depend on the curve we're working with and not just the combinatorial data behind $X$ and $Y$, right? I think we can indeed construct examples where this depends, at least if we are allowing our spherical variety to degenerate at some special points. In fact I guess it's probably not possible to give an $\ell$-adic story where these objects participate in a full-featured equivalance of categories... $\endgroup$ – Will Sawin Apr 17 '20 at 0:14
  • $\begingroup$ @WillSawin Thanks for the comments! Maybe the easiest thing to say (though I'm not sure it's the most relevant one) is that you can think of nilpotent support not as a restriction but as an operation -- i.e. you can always project any sheaves to the nilpotent singular support category, so "all sheaves get to participate". $\endgroup$ – David Ben-Zvi Apr 17 '20 at 1:16

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