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    • CommentRowNumber1.
    • CommentAuthorAnton Hilado
    • CommentTimeNov 29th 2022

    Created page. Will fill out more later.

    v1, current

    • CommentRowNumber2.
    • CommentAuthorAnton Hilado
    • CommentTimeNov 29th 2022

    Added Sakellaridis’ survey to the references.

    v1, current

    • CommentRowNumber3.
    • CommentAuthorDavid_Corfield
    • CommentTimeJul 13th 2023

    Added

    Presumably the “V1” of the link address will change, but there doesn’t seem to be an anchor for the paper.

    diff, v3, current

    • CommentRowNumber4.
    • CommentAuthorDavid_Corfield
    • CommentTimeJul 14th 2023

    I was trying to get a sense of what the ’relative’ in ’relative Langlands program’ means.

    I can’t see it directly addressed in the 451-page article in #3, but in a brief note

    • David Ben-Zvi, Relative Langlands (pdf),

    he writes

    Slogan: the relative Langlands program can be explained via relative TFT

    Does he mean as in

    If so, I see the note at twisted differential c-structure on this article

    it is proposed to call such twisted structures “relative fields”.

    So we might have “twisted Langlands program”?

    But then what’s this at field(physics)?

    Fields \simeq twisted relative cohesive cycles

    Is the ’relative’ of ’relative cohomology’ relevant?

    • CommentRowNumber5.
    • CommentAuthorDavid_Corfield
    • CommentTimeJul 14th 2023

    Added

    • David Ben-Zvi, Relative Langlands (pdf)

    diff, v4, current

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeJul 14th 2023
    • (edited Jul 14th 2023)

    What Freed & Teleman call “relative” field theory is what Stolz & Teichner call “twisted” field theory, namely (e.g. p. 51 in this pdf, using an observation that I gave them in 2005 when we met in Hamburg) not a cobordism representation as such, but a morphism from a fixed one, hence what one might call a “generalized pointed field theory” or an “object in a co-slice of field theories” or the like.

    This describes field theories with a kind of anomalies: Because, by the “holographic principle of higher category theory”, such a morphism between two cobordism representations, is itself a kind of cobordism representation, but failing to strictly preserve composition in a way measured by the non-triviality of the cobordism representations that it is a morphism between.

    (I used to draw the corresponding “tin-can diagrams” a lot in the old days, I think there is a series of posts called “D-branes from tin-cans” on the nCafe, following my Fields Institute talk from 2007 here)

    Now, browsing through Ben-Zvi’s brief note that you link to, I get the impression that by “relative” he just means “with boundary” and/or “extended”, hence in any case: with higher codim data.

    • CommentRowNumber7.
    • CommentAuthorDavid_Corfield
    • CommentTimeJul 14th 2023

    I get the impression that by “relative” he just means “with boundary” and/or “extended”, hence in any case: with higher codim data.

    That looks right. From the long article in #3, it’s about extended “4-dimensional arithmetic quantum field theory” (p. 15).

    Odd that quoted remark in #4 then. It seems one source of “relative” in “relative Langlands” is Jacquet’s “relative trace formula”.

    Ultimately here it seems it’s relative in the sense that:

    For favorable GG-spaces XX (or Hamiltonian GG-spaces MM), the various structures of the XX-relative Langlands program are simultaneously encoded, on the dual side, by a Hamiltonian Gˇ\check{G}-variety Mˇ\check{M}. (p. 10)

    These XX are “hyperspherical varieties” for a given GG,

    a convenient class of graded Hamiltonian G-spaces that is closely related to the class of cotangent bundles of spherical varieties.

    • CommentRowNumber8.
    • CommentAuthorAnton Hilado
    • CommentTimeJul 22nd 2023
    I don't really know at the moment where the term "relative Langlands" comes from, but it seems to go at least as far back as this 2012 book of Sakellaridis-Venkatesh (https://arxiv.org/abs/1203.0039), where I think connections to TQFT were not yet apparent. It could be that the way one of the big conjectures of the program is formulated is in terms of a relative trace formula (see also page 11 of these slides https://math.jhu.edu/~sakellar/KAST.pdf). Now that the draft of Ben-Zvi-Sakellaridis-Venkatesh is out, I'm hoping to be able to flesh out a bit more of this nLab page soon.
    • CommentRowNumber9.
    • CommentAuthorAnton Hilado
    • CommentTimeJul 22nd 2023

    Added Rapahael Beuzart-Plessis’ lectures at the 2022 IHES summer school.

    diff, v5, current

    • CommentRowNumber10.
    • CommentAuthorAnton Hilado
    • CommentTimeFeb 6th 2024

    Added mention of the Hamiltonian GG-space. Added a brief mention of the role of quantization in the relative Langlands program. Will add examples later.

    diff, v6, current

    • CommentRowNumber11.
    • CommentAuthorAnton Hilado
    • CommentTimeFeb 6th 2024

    Brief mention of Weil representation and theta series.

    diff, v6, current

    • CommentRowNumber12.
    • CommentAuthorUrs
    • CommentTimeFeb 6th 2024

    added some hyperlinks, such as for Hamiltonian action.

    diff, v7, current

    • CommentRowNumber13.
    • CommentAuthorUrs
    • CommentTimeFeb 6th 2024

    Where you ask for a link “Weil representation”, should we redirect to the entry Weil-Deligne representation?

    • CommentRowNumber14.
    • CommentAuthorAnton Hilado
    • CommentTimeFeb 6th 2024
    I think it should be "Segal-Shale-Weil representation" or "metaplectic representation".
    • CommentRowNumber15.
    • CommentAuthorUrs
    • CommentTimeFeb 6th 2024
    • CommentRowNumber16.
    • CommentAuthorAnton Hilado
    • CommentTimeFeb 12th 2024

    Added reference to Sakellaridis’ KAST slides.

    diff, v9, current

    • CommentRowNumber17.
    • CommentAuthorAnton Hilado
    • CommentTimeFeb 12th 2024

    Added more material from Sakellaridis’ KAST lectures.

    diff, v9, current