In mathematics, the geometric Langlands correspondence is a reformulation of the Langlands correspondence obtained by replacing the

number fields In mathematics, an algebraic number field (or simply number field) is an extension field K of the field of rational numbers such that the field extension K / \mathbb has finite degree (and hence is an algebraic field extension). Thus K is a ...

appearing in the original

number theoretic Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Math ...

version by function fields and applying techniques from

algebraic geometry Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...

. The geometric Langlands correspondence relates

and representation theory.

History

In mathematics, the classical Langlands correspondence is a collection of results and conjectures relating number theory and representation theory. Formulated by Robert Langlands in the late 1960s, the Langlands correspondence is related to important conjectures in number theory such as the Taniyama–Shimura conjecture, which includes Fermat's Last Theorem as a special case.Frenkel 2007, p. 3 Establishing the Langlands correspondence in the number theoretic context has proven extremely difficult. As a result, some mathematicians have posed the geometric Langlands correspondence.

Connection to physics

In a paper from 2007,

Anton Kapustin Anton Nikolayevich Kapustin (born November 10, 1971, Moscow) is a Russian-American theoretical physicist and the Earle C. Anthony Professor of Theoretical Physics at the California Institute of Technology. His interests lie in quantum field th ...

and Edward Witten described a connection between the geometric Langlands correspondence and S-duality, a property of certain

quantum field theories In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines classical field theory, special relativity, and quantum mechanics. QFT is used in particle physics to construct physical models of subatomic particles ...

. In 2018, when accepting the Abel Prize, Langlands delivered a paper reformulating the geometric program using tools similar to his original Langlands correspondence.

Notes

References

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External links

*{{wikiquote-inline
Quantum geometric Langlands correspondence
at nLab Algebraic geometry Langlands program Representation theory