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Presumably the “V1” of the link address will change, but there doesn’t seem to be an anchor for the paper.
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
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 twisted relative cohesive cycles
Is the ’relative’ of ’relative cohomology’ relevant?
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.
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 -spaces (or Hamiltonian -spaces ), the various structures of the -relative Langlands program are simultaneously encoded, on the dual side, by a Hamiltonian -variety . (p. 10)
These are “hyperspherical varieties” for a given ,
a convenient class of graded Hamiltonian G-spaces that is closely related to the class of cotangent bundles of spherical varieties.
added some hyperlinks, such as for Hamiltonian action.
Where you ask for a link “Weil representation”, should we redirect to the entry Weil-Deligne representation?
all right, so Segal-Shale-Weil representation
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