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Langlands program is a web of conjectures about connections between number theory and geometry. Robert Langlands (1967, 1970) 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 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 (mentioned by earlier answers) ---
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.
It may be possible to find 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).
Another direction is the p-adic AdS/CFT may offer help on this problem.
The precise potential of the development is asked as a new question here: Number Theory and Gravity . Experts in the field please feel free to comment/answer this.
