22
votes
Accepted
LMS Lectures on Geometric Langlands
The videos from the LMS lectures and all of the GRASP videos are now available again from the links you gave (for download, not streaming). Many apologies for their long hiatus offline and many thanks ...
- 24k
19
votes
Accepted
Implications and consequences of the recent proof of the geometric Langlands conjecture
To avoid writing a long essay I'll be very telegraphic.
Here are some of the many open problems that are reasonably "next" in the area. I'll use [GLC] to refer to the recent papers proving ...
- 24k
14
votes
Accepted
Relationship between the TQFTs in Kapustin-Witten and Ben-Zvi-Sakellaridis-Venkatesh
A curve $C$ over $\mathbb F_q$ has dimension $3$ in this perspective (which is why you get a vector space) and a local field has dimension $2$ (which is why you get a category. So one only has to go ...
- 148k
14
votes
Accepted
Relation between motives and geometric Langlands
First note that the formulation of geometric Langlands (the eigensheaf part, not the full conjectured equivalence of categories) is that associated to a local system on $X$ there exists an eigensheaf ...
- 148k
13
votes
Relationship between the TQFTs in Kapustin-Witten and Ben-Zvi-Sakellaridis-Venkatesh
The (fairly poetic and ill-formed) idea in this story is that the Kapustin-Witten story and the Langlands program are about the SAME four-dimensional TQFTs, but evaluated on different "manifolds&...
- 24k
13
votes
Accepted
References for Langlands classification
The first source in which I really discovered quite explicitly the archimedean local Langlands classification is in this beautiful article of Knapp, reviewing it in some pages. Moreover, it has the ...
- 5,424
13
votes
Geometric Langlands: From D-mod to Fukaya
To answer [a paraphrase of] your second question first: yes the Kapustin-Witten perspective on geometric Langlands has I think been taken very seriously by a segment of the math community. I find it ...
- 24k
11
votes
Beilinson-Drinfeld local geometric class field theory
As pointed out in the comments, this is Theorem 6.3.1.2 of Hilburn-Raskin. (It certainly was known much earlier, but I'm not sure what to give as a reference.)
Their proof is stated quite elegantly, ...
- 5,958
11
votes
Examples of function fields Langlands for small genus (<= 2)
If you really want an example of a representation, there's something funny you will find. Any irreducible cuspidal automorphic representation of $GL_n(\mathbb A_F)$ factors as a restricted tensor ...
- 148k
9
votes
Accepted
Global Langlands function fields
The abstract of V. Lafforgue's paper https://arxiv.org/abs/1404.6416 says
For any reductive group G over a global function field, we use the cohomology of G-shtukas with multiple modifications and ...
- 148k
9
votes
Statement of local geometric Langlands
There is no precise formulation of local geometric Langlands in the literature, but the rough outline is known and goes back to the papers of Frenkel-Gaitsgory starting with https://arxiv.org/abs/math/...
- 24k
8
votes
Langlands dual group in math vs. Goddard-Nyuts-Olive dual group in physics
The groups $G^{\vee}$ and $G'$ are the same. It is clear from the description in the paper you refer to.
Their definitions are the same as well, just swapping roots and coroots. You can do it in ...
- 12.3k
8
votes
Accepted
Understanding moduli of shtukas of non-minuscule cocharacter
Yes, in general you need to consider all cocharacters.
$\mathrm{GL}_n$ has the special property that the dominant coweights are all sums of minuscule cocharacters, i.e. ones of the form $[1,\cdots,1,...
- 3,058
8
votes
Geometric Langlands: From D-mod to Fukaya
One answer to your initial question is that the $D$-modules are supposed to actually do something - they're supposed to analogize to automorphic forms under the sheaf-functions dictionary. Therefore ...
- 148k
7
votes
Accepted
Beilinson-Drinfeld quantization and stable bundles
Answer is Yes for all. These commuting differential operators can be defined,
on the moduli space of stable bundles and written in theta functions terms (but it might not be illuminating or ...
- 24.8k
7
votes
Accepted
Implications of gauge symmetry breaking on the spectral side of geometric Langlands?
We discussed some conjectural implications in Section 4.2 of the paper. I wouldn't say that the category of sheaves with nilpotent singular support was necessarily the "right" category to consider ...
6
votes
Accepted
Duality of Hitchin fibrations in type A
The relevant open subset is simply the open subset where the spectral curve is smooth. Thus it is the nonvanishing locus of some discriminant polynomial.
The Hitchin fiber is the moduli space of ...
- 148k
5
votes
The affine Grassmannian and the Bogomolny equations
For those who could be interested, I worked out a formal construction of the E3-structure on the derived Satake category here, following the arguments hinted at by Lurie.
- 455
5
votes
From Galois representations to automorphic forms for $\mathfrak{sl}_2$ (via Drinfeld's shtukas)
For (1), shtukas are not needed for the construction of automorphic forms from Galois representations. Rather, this is done using the converse theorem. Shtukas are used to check the hypothesis of the ...
- 148k
5
votes
Geometric Langlands: From D-mod to Fukaya
More a comment than an answer:
I think the lagrangian-to-sheaf dictionary is more “immediately applicable” than is commonly supposed. In particular, it’s possible to immediately apply the dictionary ...
- 8,723
4
votes
Accepted
Kapustin-Witten branes and the derived moduli stack of Higgs bundles
The short answer is yes.
The sigma model to $M_G(X)$ is a low energy approximation to a 2d gauge theory with gauge group $Maps(X, G)$. Studying the sigma model to the stack $Higg_G(X)$ is basically ...
4
votes
What is the relationship between the sheaf-function dictionary and cohomology of moduli spaces of shtukas?
I don't understand the question well but the notes of Gaitsgory "FROM GEOMETRIC TO FUNCTION-THEORETIC LANGLANDS (OR HOW TO INVENT SHTUKAS)" might be useful (link), specifically Section 2.3. The ...
- 41
3
votes
Accepted
Categorical-geometric Langlands for tori
In case $G=T$ is a torus and $G^\vee=T^\vee$ is the dual torus, the geometric Langlands conjecture — or “categorical geometric class field theory” (for a smooth projective curve $C$ over $\mathbb C$) ...
- 24k
3
votes
Accepted
Remark 12.8.8 in Arinkin--Gaitsgory
The answers to your questions can be found in this article: https://arxiv.org/abs/1108.5351. I highly recommend reading it before trying to understand Arinkin-Gaitsgory.
Let me try to resolve your ...
- 5,958
3
votes
What is an Oper?
For reviews, one can try the articles of Frenkel and Teschner.
In the physics context, a fairly recent article which uses opers is this one, where Gaiotto and Witten define an oper for $G = \text{SU}(...
- 5,146
2
votes
Number theory and physics
My comment follows a possible relation between
Number Theory and Gravity (physics).
Langlands program is a web of conjectures about connections between number theory and geometry. Robert Langlands (...
- 10.5k
2
votes
Accepted
Generation of trace fields of Frobenii on local systems
Your proof in the case of four points assumes that the local monodromies at those four points are unipotent.
In general, one needs a bound not just on the set of ramification points but on the breaks/...
- 148k
1
vote
Accepted
What is the sum operation on torsors induced by Weil uniformization?
As pointed out by Jesse Silliman in the comments, this question is ill-posed. In fact, adelic automorphic forms are global sections of the structure sheaf of the moduli of $G$-torsors, not elements of ...
- 1,969
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