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a beginning at geometric Langlands correspondence
have finally merged the very stubby geometric Langlands program with the still stubby geometric Langlands correspondence
At the moment Kapustin-Witten TQFT and topologically twisted D=4 super Yang-Mills theory don’t talk to each other. The former is just the $N = 4$ case of the latter?
Thanks for noticing! I guess I had forgotten about the entry Kapustin-Witten TQFT. Yes, this is just the case for $N=4$ of the general mechanism discussed at topologically twisted D=4 super Yang-Mills theory (and in that entry really the discussion is more general than D=4 sYM, really). I have briefly interlinked the two entries now, thanks again for noticing. Of course, eventually much more could be said here.
Added pointers to the references here pointers to
Dennis Gaitsgory, Nick Rozenblyum, Notes on Geometric Langlans – A study in derived algebraic geometry (web)
Robert Langlands, The Search for a Mathematically Satisfying Geometric Theory of Automorphic Forms, Notes for a lecture at Mostow-Fest, Yale 2013 (IAS page, video, pdf)
I have polished and expanded a good bit the Idea section at geometric Langlands correspondence.
So here is an observation:
Consider that canonical covering of a complex curve by
the formal disks around finitely many points
together with the curve itself with these points removed
which leads to that double-coset description of the moduli stack of bundles on it.
If we are working in the synthetic differential complex analytic $\infty$-topos $\mathbf{H}$ ($\infty$-stacks over the site of formal complex manifolds), then we may take not just a finite number of points but all the points.
If we consider $\mathbf{H}$ as cohesive over $\infty$-stacks over formal points (essentially: “formal moduli problems”) then the object of “all formal disks around all points of $\Sigma$” is just $\flat \Sigma$.
On the other hand, the curve $\Sigma$ with all its points removed is what I used to write $\Omega \Pi_{dR}\Sigma$ and which more recently (such as at differential cohomology hexagon) I changed to writing just $\Pi_{dR}\Sigma$).
Considered in stable objects in the complex analytic $\infty$-topos, then that differential cohomology hexagon says that we have a Cartesian fracture square
$\array{ && \Pi_{dR} \Sigma \\ & \nearrow & & \searrow \\ \Pi_{dR} \flat \Sigma && && \Sigma \\ & \searrow & & \nearrow \\ && \flat \Sigma } \,.$So in particular this means that the union of all the formal disks around all points $\flat \Sigma$ and the “complement of all points” $\Pi_{dR} \Sigma$ is a cover of $\Sigma$.
Moreover, under the function field analogy then $\flat\Sigma$ plays the role of the product of all the formal completions of the function algebra on $\Sigma$. And $\Pi_{dR}\Sigma$ is at least intuitively the “rationalization” of $\Sigma$. So the above looks like the correct kind of fracture theorem one would want to see.
Moreover, it’s just the kind of covering as it governs the Langlands story. Given any coefficient object $E$, then $E$-bundles modulated by
$\Sigma \longrightarrow E$will be equivalently given by $E$-bundles on all formal disks around all points of $\Sigma$ together with one $E$-bundle on $\Sigma$ “without its points”, subject to a transition funciton on their intersection $\flat \Pi_{dR} \Sigma$, which is like all punctured formal disks around all points of $\Sigma$.
This is the analogy of the diagram in your other comment?
Yes, thanks, I should have pointed to that.
Well, the question there in itself still stands, whether we have a dual such fracture hexagon for spectra and Bousfield localization, but what I am suggesting here now is that the right geometic persepctive is as above.
Of course with any sufficiently well-behaved “function theory” functor that takes cohesive homotopy types to spectra of functions, it would take the above hexagon to a dual hexagon of spectra wich ought to involve some kind of algebraic completion. I should try to find a context in which this is exactly true…
I have expanded the Idea-section at geometric Langlands correspondence with a bit of information all from page 4 of Arinkin-Gaitsgory 12:
gave more citations to the proof of the conjectured equivalence in the abelian case:
gave citation for the proof that the conjectured equivalence is in fact FALSE in general;
slightly expanded the pointer to the proposal by Arinkin-Gaitsgory 12 for how to go about fixing this.
(this is prompted by discussion here)
added brief pointer to the recent back-and-forth:
Langlands’s doubts about or dissatifaction with the “geometric Langlands program” expressed in these talks (where he suggests that his name not be associated with the “geometric” part of the program) eventually led to
This in turn led to the reaction
added pointer to
added pointer to the recent announcement:
Dennis Gaitsgory, Proof of the geometric Langlands conjecture [web]
Dennis Gaitsgory, Sam Raskin, Proof of the geometric Langlands conjecture I: construction of the functor [arXiv:2405.03599]
Dima Arinkin, D. Beraldo, J. Campbell, L. Chen, Dennis Gaitsgory, J. Faergeman, K. Lin, Sam Raskin, Nick Rozenblyum, Proof of the geometric Langlands conjecture II: Kac-Moody localization and the FLE [arXiv:2405.03648]
The first paper says:
The category $D-mod_{\frac{1}{2}} (Bun G)$ is the de Rham incarnation of the automorphic category. It is the primary object of study in the geometric Langlands theory.
A pointer on this page as to the nature of these half-twisted $D$-modules would be helpful.
More on this on p. 9.
I suppose, with D-modules understood as (sections of) flat vector bundles, their half-twisted version should correspond to tensoring with half-densities, as familar from geometric quantization (here).
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