Geometric Langlands correspondence

In mathematics, the geometric Langlands correspondence relates algebraic geometry and representation theory. It is a reformulation of the Langlands correspondence obtained by replacing the number fields appearing in the original number theoretic version by function fields and applying techniques from algebraic geometry.^[1] The geometric Langlands conjecture asserts the existence of the geometric Langlands correspondence.

The existence of the geometric Langlands correspondence in the specific case of general linear groups over function fields was proven by Laurent Lafforgue in 2002, where it follows as a consequence of Lafforgue's theorem.^[2]

Background

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.^[1]

Langlands correspondences can be formulated for global fields (as well as local fields), which are classified into number fields or global function fields. Establishing the classical Langlands correspondence, for number fields, has proven extremely difficult. As a result, some mathematicians posed the geometric Langlands correspondence for global function fields, which in some sense have proven easier to deal with.^[3]

The geometric Langlands conjecture for general linear groups $GL(n,K)$ over a function field $K$ was formulated by Vladimir Drinfeld and Gérard Laumon in 1987.^[4]^[5]

Status

The geometric Langlands conjecture was proved for $GL(1)$ by Pierre Deligne and for $GL(2)$ by Drinfeld in 1983.^[6]^[7]

Laurent Lafforgue proved the geometric Langlands conjecture for $GL(n,K)$ over a function field $K$ in 2002.^[2]

A claimed proof of the categorical unramified geometric Langlands conjecture was announced on May 6, 2024 by a team of mathematicians including Dennis Gaitsgory.^[8]^[9] The claimed proof is contained in more than 1,000 pages across five papers and has been called "so complex that almost no one can explain it". Even conveying the significance of the result to other mathematicians was described as "very hard, almost impossible" by Drinfeld.^[10]

Connection to physics

In a paper from 2007, Anton Kapustin and Edward Witten described a connection between the geometric Langlands correspondence and S-duality, a property of certain quantum field theories.^[11]

In 2018, when accepting the Abel Prize, Langlands delivered a paper reformulating the geometric program using tools similar to his original Langlands correspondence.^[12]^[13] Langlands' ideas ideas were further developed by Etingof, Frenkel, and Kazhdan.^[14]

Notes

^ ^a ^b Frenkel 2007, p. 3.
^ ^a ^b Lafforgue, Laurent (2002). "Chtoucas de Drinfeld, formule des traces d'Arthur–Selberg et correspondance de Langlands". arXiv:math/0212399.
^ Frenkel 2007, p. 3,24.
^ Frenkel 2007, p. 46.
^ Laumon, Gérard (1987). "Correspondance de Langlands géométrique pour les corps de fonctions". Duke Mathematical Journal. 54: 309–359.
^ Frenkel 2007, p. 31,46.
^ Drinfeld, Vladimir G. (1983). "Two-dimensional ℓ–adic representations of the fundamental group of a curve over a finite field and automorphic forms on GL(2)". American Journal of Mathematics. 105: 85–114.
^ "Proof of the geometric Langlands conjecture". people.mpim-bonn.mpg.de. Retrieved 2024-07-09.
^ Klarreich, Erica (2024-07-19). "Monumental Proof Settles Geometric Langlands Conjecture". Quanta Magazine. Retrieved 2024-07-20.
^ Wilkins, Alex (May 20, 2024). "Incredible maths proof is so complex that almost no one can explain it". New Scientist. Retrieved 2024-07-09.
^ Kapustin and Witten 2007
^ "The Greatest Mathematician You've Never Heard Of". The Walrus. 2018-11-15. Retrieved 2020-02-17.
^ Langlands, Robert (2018). "Об аналитическом виде геометрической теории автоморфных форм1" (PDF). Institute of Advanced Studies.
^ Etingof, Pavel and Frenkel, Edward and Kazhdan, David (2019). "An analytic version of the Langlands correspondence for complex curves". arXiv:1908.09677.{{cite arXiv}}: CS1 maint: multiple names: authors list (link)

References

Frenkel, Edward (2007). "Lectures on the Langlands Program and Conformal Field Theory". Frontiers in Number Theory, Physics, and Geometry II. Springer. pp. 387–533. arXiv:hep-th/0512172. Bibcode:2005hep.th...12172F. doi:10.1007/978-3-540-30308-4_11. ISBN 978-3-540-30307-7. S2CID 119611071.
Kapustin, Anton; Witten, Edward (2007). "Electric-magnetic duality and the geometric Langlands program". Communications in Number Theory and Physics. 1 (1): 1–236. arXiv:hep-th/0604151. Bibcode:2007CNTP....1....1K. doi:10.4310/cntp.2007.v1.n1.a1. S2CID 30505126.

External links

Quotations related to Geometric Langlands correspondence at Wikiquote
Quantum geometric Langlands correspondence at nLab

[FOOTNOTEFrenkel20073-1] Frenkel 2007, p. 3.

[:0-2] Lafforgue, Laurent (2002). "Chtoucas de Drinfeld, formule des traces d'Arthur–Selberg et correspondance de Langlands". arXiv:math/0212399.

[FOOTNOTEFrenkel20073,24-3] Frenkel 2007, p. 3,24.

[FOOTNOTEFrenkel200746-4] Frenkel 2007, p. 46.

[5] Laumon, Gérard (1987). "Correspondance de Langlands géométrique pour les corps de fonctions". Duke Mathematical Journal. 54: 309–359.

[FOOTNOTEFrenkel200731,46-6] Frenkel 2007, p. 31,46.

[7] Drinfeld, Vladimir G. (1983). "Two-dimensional ℓ–adic representations of the fundamental group of a curve over a finite field and automorphic forms on GL(2)". American Journal of Mathematics. 105: 85–114.

[:1-8] "Proof of the geometric Langlands conjecture". people.mpim-bonn.mpg.de. Retrieved 2024-07-09.

[9] Klarreich, Erica (2024-07-19). "Monumental Proof Settles Geometric Langlands Conjecture". Quanta Magazine. Retrieved 2024-07-20.

[10] Wilkins, Alex (May 20, 2024). "Incredible maths proof is so complex that almost no one can explain it". New Scientist. Retrieved 2024-07-09.

[11] Kapustin and Witten 2007

[12] "The Greatest Mathematician You've Never Heard Of". The Walrus. 2018-11-15. Retrieved 2020-02-17.

[13] Langlands, Robert (2018). "Об аналитическом виде геометрической теории автоморфных форм1" (PDF). Institute of Advanced Studies.

[14] Etingof, Pavel and Frenkel, Edward and Kazhdan, David (2019). "An analytic version of the Langlands correspondence for complex curves". arXiv:1908.09677.{{cite arXiv}}: CS1 maint: multiple names: authors list (link)

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