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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 groups in algebraic number theory to automorphic forms and representation theory of algebraic groups over local fields and adeles.

Geometric Langlands correspondence is a geometric analog (aim for a reformulation) of the number theoretic Langlands correspondence.

There is a well-known relation between the Geometric Langlands Program and Electric-Magnetic Duality or S-duality in certain quantum field theories ---

N=4 super Yang-Mills theory in four dimensions.

More precisely, the geometric Langlands program can be described in a natural way by compactifying on a Riemann surface a twisted version of N=4 super Yang-Mills theory in four dimensions. See hep-th/0604151, Anton Kapustin, Edward Witten (2006). The key ingredients are

  • the electric-magnetic duality of gauge theory,
  • mirror symmetry of sigma-models,
  • branes,
  • Wilson and 't Hooft operators, and
  • topological field theory.

Hecke eigensheaves and D-modules can be explained from the physics.

Since N=4 super Yang-Mills theory in four dimensions plays a key role in the

gauge-gravity duality

or

the AdS/CFT duality

the duality between

Type IIB string theory on AdS5 × $S^5$ space (a product of 5-dimensional AdS space with a 5-dimensional sphere); or the supergravity

and

N = 4 super Yang–Mills

on the 4-dimensional boundary of AdS5.

My question is that: So far do any researcher finds useful guidance to look at the number theory, or the (Geometric) Langlands correspondence through the gravity theory (like the AdS5 space in Type IIB string theory or the supergravity)?

Does the p-adic AdS/CFT have any help on this problem in math or probably not?

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wonderich
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    That may be an approach of making an already difficult problem (GL) into an almost impossible one of gravity. Also worse if N of gauge group is too small. – AHusain Feb 04 '19 at 06:32
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    There are some calculations in the Langlands program for $GL_n$ that have some kind of stabilization in the large $n$ limit. These have analogues in the geometric Langlands program, thus surely analogues in the Kapustin-Witten field theory picture, thus maybe analogues in some string theory / M-theory. But all the calculations I know of are ones where the expected answer is something simple, so it's not clear whether the gravity analogue will help for the number theory at all. The simplest example is that the cohomology of $\operatorname{Bun}_{GL_n}$ stabilizes as $n\to \infty$. – Will Sawin Feb 06 '19 at 12:26
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    You should say that geometric Langlands is an analogue of the classical Langlands, rather than a reformulation. – Will Sawin Feb 06 '19 at 12:27
  • @Will Sawin, thanks, Will +1, I changed to "geometric analog (aim for a reformulation)" is this better? – wonderich Feb 07 '19 at 16:30
  • @wonderich I don't know if anyone really aims for a reformulation. Why do you think that? – Will Sawin Feb 08 '19 at 03:11
  • because that may be the way physicists like --- the people who write the papers want to understand the math by a "physics reformulated" story? – wonderich Feb 08 '19 at 04:39
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    @WillSawin Is there a reference for that comparing that stabilization on the Langlands side vs simplification by taking planar limit on Kapustin-Witten perspective? – AHusain Feb 17 '19 at 21:23
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    @AHusain I know of nothing about planar limits, nor do I know of any reference. I could give a short description of the stabilization that is observed on the Langlands side and what it might correspond to in the Kapustin-Witten picture, but I don't know anything about what these limits mean in the physical side. – Will Sawin Feb 18 '19 at 15:24
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    @WillSawin I was thinking of what happens with $n \to \infty$ without the GL twist. That might not be as helpful as I thought. – AHusain Feb 18 '19 at 15:42
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    The only significant interaction I'm aware of is the suggestion in https://arxiv.org/abs/1707.01292 of a factorization structure in GL_n geometric Langlands with respect to n (ie we think of the GL_n as coming from a stack of branes and move their locations around in the transverse direction). This is closely related to the Hall-algebraic interpretation of the Langlands correspondence for function fields (associated with Kapranov). – David Ben-Zvi Jun 02 '19 at 14:09

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