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While I know and understand the definitions and inputs, I haven’t yet studied the geometric Langlands duality in any detail. But since it is clearly to some extent about higher connections, of course I did wonder a bit about it every now and then in a spare minute.
I wanted to not let me get distracted by this, since I need to be doing other things, but now I couldn’t resist and reminded myself of what I mean here in this comment over on the $n$Café.
Especially with an eye towards earlier discussion here with Domenico and Zoran, I want to note here just for the sake of it the following basic thoughts.
If we ignore the (crucial!) issues of holomorphicity for the moment, there is a nice simple way to think of the basic ingredients that seem to appear in the geometric Langlands duality in terms of the general language of differential cohomology in an (oo,1)-topos, and by just playing around with some abstract structures, one sees something.
So for $G$ a group object in our topos, write $\mathbf{B}G$ for its delooping. This is the “space of $G$bundles” that in geometric Langlands is traditionally writtten $Bun_G$.
Then recall the crucial ingredient structure of our oo-topos $\mathbf{H}$ that allows us to talk about differential cohomology inside it: that’s the adjunctions $(\Pi \dashv LConst \dashv \Gamma) : \mathbf{H} \to \infty Grpd \simeq Top$ and the composite $(\mathbf{\Pi} \dashv \mathbf{\flat}) := (LConst \Pi \dashv LConst \Gamma) : \mathbf{H} \to \mathbf{H}$.
With that notation, the “space of $G$-local systems” $Loc_{G}$ corresponds to $\mathbf{\flat} \mathbf{B}G$. (More properly we ought to use an infinitesimal version $\mathbf{\flat}_{inf}$ here, but let me run with the simple situation for the moment.) In that: morphisms $X \to \mathbf{\flat} \mathbf{B}G$ correspond to $G$-bundles on $X$ with flat connection.
Let me write $Mod$ for some object in $\mathbf{H}$ such that morphisms $X \to Mod$ corresponds to “$\mathcal{O}$-modules” on $X$ (we talked about this object here). Then accordingly $\mathbf{\flat} Mod$ is the coefficient for flat such modules. The infinitesimal version of this would be like $\mathcal{D}$-modules, but again I’ll not get into this here.
Then we have that
$\mathbf{H}(\mathbf{\flat} \mathbf{B}G, Mod)$
is the $\infty$-groupoid of $\mathcal{O}$-modules on something like the space of $G$-local systems,
while
$\mathbf{H}(\mathbf{B}G, \mathbf{\flat} Mod)$is the $\infty$-groupoid of flat modules on the space of $G$-bundles.
(If we refine $\mathbf{H}$ to an $(\infty,2)$-topos, accomodating more truthfully for the fact that $Mod$ wants to be an $(\infty,2)$-sheaf, then this would be two $(\infty,1)$-categories and we’d be yet a bit closer to the standard statement of geometric langlands in terms of derived categories. But for the moment let’s ignore this.)
So it looks like geometric Langlands asserts that these two objects are pretty closely related for Langalnds self-dual groups $G$, at least if some qualifiers (e.g. holomorphic local systems, etc.) are added.
But let’s just see on an abstract nonsense-level, why these things can be related at all.
An observation that accomplishes this is that there is a canonical morphism
$\Gamma \mathbf{B}G \to \Pi \mathbf{B}G \,.$This is given by the composite
$\phi : \Gamma \mathbf{B}G \to \Gamma LConst \Pi \mathbf{B}G \simeq \Pi \mathbf{B}G \,,$where the first morphism is $\Gamma$ applied to the unit of the adjunction $(LConst \dashv \Gamma)$, while the second morphism is a consequence of the “$\infty$-connectedness” of our $\infty$-topos, due to which $\Gamma \circ Lconst \simeq Id$: this says simply that evaluating the constant $\infty$-stack $LConst S$ on the point yields back the $\infty$-groupoid $S$.
So then by precomposition with $\phi$, we obtain a canonical map
$\begin{aligned} \mathbf{H}(\mathbf{B}G , \mathbf{\flat} Mod) &\stackrel{\simeq}{\to} \mathbf{H}(\mathbf{\Pi} \mathbf{B}G, Mod) \\ & := \mathbf{H}(LConst \Pi \mathbf{B}G, Mod) \\ & \stackrel{LConst (\phi^*)}{\to} \mathbf{H}(LConst \Gamma \mathbf{B}G, Mod) \\ & =: \mathbf{H}(\mathbf{\flat} \mathbf{B}G, Mod) \end{aligned}$from flat modules on the space of $G$-bundles to all modules on the space of flat $G$-bundles.
It seems that the content of the geometric Langlands duality conjecture is to say that for Langlands self-dual $G$ and with some fine print added, this becomes an equivalence.
Just a wild guess to complement this beautiful exposition.
There is this automorphic aspect to it; recall that the n-times loop groupoid (physically multisector; alg geom inertia groupoid of inertia groupoid etc. of original groupoid) has a natural action of SL(n,Z). That is for a loop groupoid of a loop groupoid one has the modular group SL(2,Z) in the game. Now when Urs plays with groupoid attack on basic of Langlands, maybe one should see how the behaviour under the modular group, hence automorphic aspect come in that approach.
Good point. I should think about this.
I have to say that I know almost nothing about how the number-theoretic Langlands statement relates to the one of geometric Langlands duality. Is that maybe generally the underlying story, that automorpic-ness on the number-theoretic side corresponds to looking at higher loop spaces on the geometric side?
This must be in Ben-Zvi’s notes. But I don’t really have time for this at the moment.
People say that more or less the geometric Langlands essentially contains number theoretic statements as a part of the picture, but the correspondence is claimed to be quite complicated. Whatever all this means.
Some time ago I created an entry Hitchin fibration…Maybe somebody joins…
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