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nLab > Latest Changes: geometric Langlands correspondence

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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeSep 2nd 2010
    • (edited Sep 29th 2013)
    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeSep 29th 2013

    have finally merged the very stubby geometric Langlands program with the still stubby geometric Langlands correspondence

    • CommentRowNumber3.
    • CommentAuthorDavid_Corfield
    • CommentTimeSep 30th 2013

    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=4N = 4 case of the latter?

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeSep 30th 2013

    Thanks for noticing! I guess I had forgotten about the entry Kapustin-Witten TQFT. Yes, this is just the case for N=4N=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.

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeJul 6th 2014

    Added pointers to the references here pointers to

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeJul 17th 2014

    I have polished and expanded a good bit the Idea section at geometric Langlands correspondence.

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeJul 19th 2014
    • (edited Aug 23rd 2014)

    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 H\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 H\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 Σ\Sigma” is just ♭Σ\flat \Sigma.

    On the other hand, the curve Σ\Sigma with all its points removed is what I used to write ΩΠ dRΣ\Omega \Pi_{dR}\Sigma and which more recently (such as at differential cohomology hexagon) I changed to writing just Π dRΣ\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

    Π dRΣ ↗ ↘ Π dR♭Σ Σ ↘ ↗ ♭Σ. \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 ♭Σ\flat \Sigma and the “complement of all points” Π dRΣ\Pi_{dR} \Sigma is a cover of Σ\Sigma.

    Moreover, under the function field analogy then ♭Σ\flat\Sigma plays the role of the product of all the formal completions of the function algebra on Σ\Sigma. And Π dRΣ\Pi_{dR}\Sigma is at least intuitively the “rationalization” of Σ\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 EE, then EE-bundles modulated by

    Σ⟶E \Sigma \longrightarrow E

    will be equivalently given by EE-bundles on all formal disks around all points of Σ\Sigma together with one EE-bundle on Σ\Sigma “without its points”, subject to a transition funciton on their intersection ♭Π dRΣ\flat \Pi_{dR} \Sigma, which is like all punctured formal disks around all points of Σ\Sigma.

    • CommentRowNumber8.
    • CommentAuthorDavid_Corfield
    • CommentTimeJul 19th 2014

    This is the analogy of the diagram in your other comment?

    • CommentRowNumber9.
    • CommentAuthorUrs
    • CommentTimeJul 19th 2014
    • (edited Jul 19th 2014)

    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…

    • CommentRowNumber10.
    • CommentAuthorUrs
    • CommentTimeAug 23rd 2014
    • (edited Aug 23rd 2014)

    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)

    • CommentRowNumber11.
    • CommentAuthorUrs
    • CommentTimeJan 20th 2019

    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

    • Robert Langlands, Об аналитическом виде геометрической теории автоморфных форм, IAS 2018 (ias:2678, pdf)

    This in turn led to the reaction

    diff, v56, current

    • CommentRowNumber12.
    • CommentAuthorUrs
    • CommentTimeJan 30th 2020

    added pointer to

    diff, v57, current

    • CommentRowNumber13.
    • CommentAuthorUrs
    • CommentTimeMay 10th 2024

    added pointer to the recent announcement:

    diff, v60, current

    • CommentRowNumber14.
    • CommentAuthorDavid_Corfield
    • CommentTimeMay 10th 2024

    The first paper says:

    The category D−mod 12(BunG)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 DD-modules would be helpful.

    • CommentRowNumber15.
    • CommentAuthorUrs
    • CommentTimeMay 10th 2024
    • (edited May 10th 2024)

    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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