In mathematics, thegeometric Langlands correspondence relatesalgebraic geometry andrepresentation theory. It is a reformulation of theLanglands correspondence obtained by replacing thenumber fields appearing in the originalnumber theoretic version byfunction fields and applying techniques fromalgebraic geometry.^[1] The correspondence is named for the Canadian mathematicianRobert Langlands, who formulated the original form of it in the late 1960s.

Thegeometric Langlands conjecture asserts the existence of the geometric Langlands correspondence.

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 theTaniyama–Shimura conjecture, which includesFermat's Last Theorem as a special case.^[1]

Langlands correspondences can be formulated forglobal fields (as well aslocal fields), which are classified intonumber fields orglobal 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.^[2]

The geometric Langlands conjecture forgeneral linear groups${\displaystyle GL(n,K)}$ over a function field${\displaystyle K}$ was formulated byVladimir Drinfeld andGérard Laumon in 1987.^[3][4]

The geometric Langlands conjecture was proved for${\displaystyle GL(1)}$ byPierre Deligne and for${\displaystyle GL(2)}$ by Drinfeld in 1983.^[5][6]

A claimed proof of the categorical unramified geometric Langlands conjecture was announced on May 6, 2024 by a team of mathematicians includingDennis Gaitsgory.^[7][8] 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.^[9]

In a paper from 2007,Anton Kapustin andEdward Witten described a connection between the geometric Langlands correspondence andS-duality, a property of certainquantum field theories.^[10]

In 2018, when accepting the Abel Prize, Langlands delivered a paper reformulating the geometric program using tools similar to his original Langlands correspondence.^[11][12] Langlands' ideas were further developed by Etingof, Frenkel, and Kazhdan.^[13]

• 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.

• Quotations related toGeometric Langlands correspondence at Wikiquote
• Quantum geometric Langlands correspondence atnLab

• Algebraic geometry
• Langlands program
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