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Added
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 $\simeq$ 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 $G$-spaces $X$ (or Hamiltonian $G$-spaces $M$), the various structures of the $X$-relative Langlands program are simultaneously encoded, on the dual side, by a Hamiltonian $\check{G}$-variety $\check{M}$. (p. 10)
These $X$ are “hyperspherical varieties” for a given $G$,
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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