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I am beginning to think that all these things are secretly the same:
Langlands’ conjecture 3;
the modular functor, hence quantization of 3dCS/2dWZW.
Read number-theoretic Langlands from the CS/WZW-theoretic perspective (using this and this) then one has: on the collection of flat connections (Galois representations) we assign the theta functions which are the partition functions (essentially: L-functions) with “source fields” these flat connections. That’s the definition of the conformal blocks.
But that’s also just the interpretation of equivariant elliptic cohomology in QFT language, which is a refined picture of the modular functor.
From this perspective Langlands’s “functoriality” is Lurie’s “global equivariance”, the fact that the construction is natural in the gauge group.
In this picture the automorphic forms don’t necessarily appear prominently, they rather seem a technical means to express the theta functions (their L-functions), analogous to how for writing down the standard modular functor it may be useful, but not necessary, to express the conformal blocks in certain preferred coordinates.
That Langlands’ conjecture 3 only spoke of a homomorphism ${}^L G \to {}^L G$, where there is normally a different group ${}^L G \to {}^L G^{'}$. There are typos in Gelbart’s conjecture 3’, where primes are missing, which got copied in. Hopefully it’s OK now.
But actually, it says on our page that this is Gelbart’s full conjecture 3, when we only have his conjecture 3’, where groups are split. Is it worth giving the full story?
Right, thanks! (In fact Joost pointed that out the missing primes to me, too, yesterday, when I told him about how I think “functoriality” is analogous to “global equivariance”. But I forgot to go and fix it.)
Yes, and the missing details about splitness should be mentioned. Sorry for the glitches. Thanks for your help.
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