# Questions tagged [geometric-langlands]

The geometric-langlands tag has no usage guidance.

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### What d.o. $\sum_i f_i(z)\partial_z^i$ correspond to subalgebras $M$ in polynoms $C[x_i]$ being Langlands dual to motive of $Spec(M) \to X$?

Briefly: The question is about presenting explicit examples of the construction discussed in the recent MO question "Relation between motives and geometric Langlands" and Will Sawin's asnwer ...

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### Relation between motives and geometric Langlands

When working over a number field (or a function field over a finite field), one predicts that the Langlands program is related to the theory of motives over this field. There are several ways I have ...

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### Kapustin-Witten branes and the derived moduli stack of Higgs bundles

A lot has been discussed on overflow regarding geometric Langlands and the physics of Kapustin and Witten's groundbreaking paper https://arxiv.org/abs/hep-th/0604151. I would like to add my two cents ...

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### Definition of nearby cycle over an affine line

In some famous papers like Gaitsgory's "Construction of central elements in the affine Hecke algebra via nearby cycles" and Beilinson-Bernstein's "A proof of Jantzen conjectures", ...

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### In which sense affine Grassmannian is "affine"

A pretty naïve question: Which meaning has the term "affine" in the notion of affine Grassmanian. Especially, I do not see any immediate connection to the concept of an "affine scheme&...

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### Functoriality of Feigin–Frenkel duality

For a simple Lie algebra $\mathfrak{g}$, we have the W-algebra of level $k$, denoted by $\mathcal{W}^k(\mathfrak{g})$. Using Wakimoto free field realization and screening operators, Feigin and Frenkel ...

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### Is this construction related to the geometric Langlands program perhaps?

Given a complex Lie algebra $\mathfrak{g}$, a choice of Cartan subalgebra $\mathfrak{h}$ of $\mathfrak{g}$ and a dominant integral weight $\lambda$ of $\mathfrak{g}$, there is a natural construction ...

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### Confusion about definition of crystals

In the notes by Lurie there seems to be two possible definition for crystals which both makes sense for arbitrary functors $X : \mathrm{CRing}_k \to \mathrm{Set}$. ($k$ here is a field. We probably ...

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### What is the sum operation on torsors induced by Weil uniformization?

Let $k$ be an algebraically closed field, $G$ a reductive group, and $C$ a curve. The algebraic version of the Weil uniformization theorem (see e.g. arXiv:1511.06271v2) says that groupoid of $G$-...

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### Roadmap to geometric Langlands for a mathematical physics student

I am a student of both mathematics and physics, who has recently been studying string theory. My mathematics background is mostly differential geometry (principal bundles, Lie groups, etc.), although ...

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### Why are they called reductive groups? [duplicate]

The reductive groups play a central role in the Langlands correspondence. Why are these groups called reductive? Does this name suggest something conceptual about these groups?

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### Dual Coxeter numbers, Langlands dual groups, black holes and twisted compactification of 6d (2,0) A D E theories on a circle

A 6-dimensional (2,0) superconformal quantum field theory comes in Lie algebra A, D, E types. These theories do not have classical Lagrangian and are purely quantum.These theories on a torus ...

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### From Galois representations to automorphic forms for $\mathfrak{sl}_2$ (via Drinfeld's shtukas)

Drinfeld-Lafforgue have proven function fields Langlands conjectures in type A: see https://arxiv.org/pdf/math/0212417.pdf (Laumon's survey in English), https://arxiv.org/pdf/math/0212399.pdf (...

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### Duality of Hitchin fibrations in type A

For $G = GL_n$, it is known that the generic fibers of the Hitchin fibration are the Picard stacks of line bundles on the corresponding spectral curves and the duality of Hitchin fibrations in this ...

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### Generation of trace fields of Frobenii on local systems

Let $\overline{X}$ be a smooth proper curve over $\mathbb{F}_q$, for some $q$, $S$ a collection of $\mathbb{F}_q$ points of $\overline{X}$, and set $X=\overline{X}-S$.
For a rank $n$ $\overline{\...

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### Relationship between the TQFTs in Kapustin-Witten and Ben-Zvi-Sakellaridis-Venkatesh

In upcoming work of Ben-Zvi-Sakellaridis-Venkatesh, (see for instance these notes or this lecture) some important aspects of the Langlands correspondence are stated in the language of topological ...

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### Beilinson-Drinfeld local geometric class field theory

There is the following version of categorical local geometric class field theory:
Let $\mathbb{D}=\operatorname{Spec} \mathbb{C}((t))$, $L\mathbb{G}_m$: the loop group of $\mathbb{G}_m$ over $\mathbb{...

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### Statement of local geometric Langlands

A precise statement of the global geometric Langlands conjecture is well-known.
However, I am unable to find a statement of the local Langlands conjecture. Does anyone have a modern statement or a ...

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### Categorical-geometric Langlands for tori

Fix a "nice" curve $X$ (smooth, projective, proper, geometrically connected, what-have-you) and an algebraic torus $G$, both over a field of characteristic $0$ (possibly algebraically closed?...

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### Elementary questions on the geometric Langlands program for the orthogonal and symplectic families

If $G^\vee = SL(n,\mathbb{C})$ and $V = \mathbb{C}^n$, then the Langlands dual of $G^\vee$ is $G = PGL(n,\mathbb{C})$. Denote by $T$ and $T^\vee$ maximal tori in $G$ and $G^\vee$ respectively. The ...

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### Representation theory of Chevalley groups as a categorical trace

Dennis Gaitsgory's 2016 preprint, From Geometric to Function-Theoretic Langlands (or How to Invent Shtukas) includes in the third section a very compressed but suggestive discussion of the ...

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### Equivalence between $\mathcal{D}_\lambda$ modules and $\mathcal{D}_{0}$ modules

Fix $G$ a finite dimensional reductive group and $\lambda$ a weight. Apparently the category of $\mathcal{D}_\lambda$ modules on $G/B$ is equivalent to the category of $\mathcal{D}_0$ modules on $G'/B'...

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### On a remark of Langlands

I'm been wondering about this for a while and hope someone can enlighten me.
In this interview of Robert Langlands's from 2010, on pg 21 (Question 8) he states "At one point, when fairly young, I ...

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### Langlands dual group in math vs. Goddard-Nyuts-Olive dual group in physics

Given a group $G$, there is a so-called Langlands dual group $G^{∨}$.
Given a group $G$, there is also a so-called Goddard-Nyuts-Olive dual group $G^{'}$ that relates to the magnetic charge.
Are ...

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### Understanding moduli of shtukas of non-minuscule cocharacter

I have kind of a soft question. I've studied the basics of L. Lafforgue's proof of function field Langlands for GLn, and its use of the moduli of shtukas with two legs, and the cocharacters $[1,0,\...

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### Geometric Langlands: From D-mod to Fukaya

This post is rather wordy and speculative, but I promise there is a concrete question embedded within. For experts, I'll open with a question:
Question: Given a compact Riemann surface $X$, why ...

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### Integral kernels for geometric langlands

My apologies for the imprecise question(s), it should be clear enough that I´m a complete beginner in this subject.
The (de Rham) Geometric Langlands Conjecture over $\mathbb{C}$ takes as input a ...

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### Does $\text{Bun}_G$ have the homotopy type of a classifying space in positive characteristic?

In these lecture notes by Jacob Lurie, he identifies the homotopy type of $\text{Bun}_G$ with that of a certain classifying space $B\mathcal{P}_{sm}$ when the group scheme $G$ is over $\mathbb{C}$. ...

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### Ramified Geometric Langlands

Is the following a reasonable formulation of (a part of) the geometric Langlands conjecture for $\mathrm{GL}_n$ over a curve $X$?
(*) Let $\mathcal{L}$ be an irreducible rank $n$ $\ell$-adic local ...

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### What is the relationship between the sheaf-function dictionary and cohomology of moduli spaces of shtukas?

I'm a newcomer to the geometric Langlands setting, and have mostly consulted surveys like Laumon's overview of L. Lafforgue's proof or Frenkel's recent advances survey, so apologies if this is ...

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### What is the analogy between the moduli of shtukas and Shimura varieties?

I have heard that moduli spaces of shtukas are supposed to be the analogue of Shimura varieties in the setting of function fields. Could someone more knowledgeable about these objects explain how this ...

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### Number Theory and Gravity

Langlands program is a web of far-reaching and influential conjectures about connections between number theory and geometry. Proposed by Robert Langlands at IAS (1967, 1970), it seeks to relate Galois ...

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### Analog of Ramanujan-Petersson conjecture in Geometric Langlands

The Ramanujan conjecture asserts that
\begin{align}
|\tau(p)|\leq 2p^{11/2}
\end{align}
where $\tau(p)$ is the $p^{th}$ Fourier coeffecient in the q-expansion of the weight 12 cusp form $\Delta(z)$. ...

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### Remark 12.8.8 in Arinkin--Gaitsgory

I can not understand Remark 12.8.8 in the preprint "SINGULAR SUPPORT OF COHERENT SHEAVES AND THE GEOMETRIC LANGLANDS CONJECTURE". I am somewhat embarrased by the degree of my confusion, hopefully ...

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### Beilinson-Drinfeld quantization and stable bundles

To motivate this question, I'm going to try and explain some background notions. This won't be absolutely necessary for experts, but I want to be vaguely honest about where this question comes from. ...

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### Implications of gauge symmetry breaking on the spectral side of geometric Langlands?

Let $G$ be a complex reductive algebraic group and $X$ be a smooth compact complex curve. It's easy to see that the space of vacua in B-twisted $N=4$ SUSY Yang--Mills theory is $\mathfrak{h}^*[2]/W$ (...

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### Compact generation of the category of D-modules on moduli stack of principal bundles for algebraic groups?

Let $k$ be an algebraically closed field of characteristic 0. Let $X$ be a connected smooth complete curve over $k$. Consider the moduli stack $\mathrm{Bun}_G$ of principal $G$-bundles on $X$ for ...

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### Langlands dual and integrable representations

Assume I successfully classified the integrable representations of a certain semi-simple Lie group $G$. Given this information, what do I know about the integrable representations of $G^\vee$, the ...

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### Examples of function fields Langlands for small genus (<= 2)

See Edward Frenkel's article "Lectures on the Langlands program and conformal field theory" for an exposition of the function fields Langlands correspondence (now a theorem of Drinfel'd, L.Lafforgue &...

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### Bi-Whittaker functions and local Langlands compatibility

I'm trying to figure out the arithmetic analogue of a key conjecture in the geometric local Langlands correspondence. Briefly, one expects for $K=\mathbb{C}((t))$ an equivalence of dg categories $$\...

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### References for Langlands classification

I kindly ask about some references concerning the representation theory of the Langlands dual of a compact Lie group, and how it relates to things related to the original compact Lie group.
My ...

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### LMS Lectures on Geometric Langlands

Everybody knows how insightful are David Ben-Zvi talks (and comments/answers here on mathoverflow). I was trying to watch the LMS 2007 Lecture Series on Geometric Langlands by David, supposedly made ...

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### Global Langlands function fields

Has V. Lafforgue proved the automorphic-to-Galois direction in the Global Langlands conjectures for general reductive groups over function fields?
What is the current status, more generally?
Related ...

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### Any progress on Strominger-Yau and Zaslow conjecture?

In 2002 Hausel - Thaddeus interpreted SYZ conjecture in the context of Hitchin system and Langlands duality. Let briefly explain it
Let $\pi : E \to Σ$ a complex vector bundle of rank $r$ and ...

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### Feigin-Frenkel centre and opers for reductive Lie algebras

Edward Frenkel (together with Boris Feigin and others) has proven many interesting results connecting the representation theory of an affine Kac-Moody algebra at the critical level with the geometry ...

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### A question related to the semisimplification of a Weil-Deligne representation

I have been trying to find the answer to this question, I think it must not be hard but I don't get it.
I have a Weil-Deligne representation ($\rho,N$) of the Weil group $W$ of $Q_p$, that is $\rho$ ...

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### What do Hecke eigensheaves actually look like?

Let $\mathbb F_q$ be a finite field, $C$ a curve over $\mathbb F_q$ of genus $g\geq 2$, $\rho: \pi_1(C) \to GL_2(\overline{\mathbb Q}_\ell)$ an irreducible local system. The geometric Langlands ...

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### Geometric Satake and Restriction

The Geometric Satake correspondence (due to Lusztig, Ginzburg, Mirkovic-Vilonen) relates perverse sheaves on the Loop Group $\hat{G}$ (with their convolution product) to the Representations of the ...

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### Homological contractibility of a prestack

This question is in reference to Gaitsgory's preprint Contractibility of the space of rational maps. On p. 5 of the preprint, Gaitsgory defines a prestack $\mathscr{Y}$ (say over affine $\mathbb{C}$-...

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### Meaning of topological tensor products in Frenkel-Gaitsgory

The appendix to http://arxiv.org/abs/math/0508382 by Frenkel & Gaitsgory (following an earlier work of Beilinson) describes three different monoidal structures, denoted by $\otimes^!,\otimes^*,$ ...