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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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    $\begingroup$ 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. $\endgroup$ – AHusain Feb 4 at 6:32
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    $\begingroup$ 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 $n\to \infty$. $\endgroup$ – Will Sawin Feb 6 at 12:26
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    $\begingroup$ You should say that geometric Langlands is an analogue of the classical Langlands, rather than a reformulation. $\endgroup$ – Will Sawin Feb 6 at 12:27
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    $\begingroup$ @WillSawin Is there a reference for that comparing that stabilization on the Langlands side vs simplification by taking planar limit on Kapustin-Witten perspective? $\endgroup$ – AHusain Feb 17 at 21:23
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    $\begingroup$ The only significant interaction I'm aware of is the suggestion in 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). $\endgroup$ – David Ben-Zvi Jun 2 at 14:09

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