Geometric Langlands correspondence

In mathematics, the geometric Langlands correspondence is a geometric reformulation of the number theoretic Langlands correspondence.

Overview

In mathematics, the classical Langlands correspondence is a collection of results and conjectures relating number theory to the branch of mathematics known as 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]

In spite of its importance in number theory, establishing the Langlands correspondence in the number theoretic context has proved extremely difficult. As a result, some mathematicians have worked on a related conjecture, the geometric Langlands correspondence. This is a geometric reformulation of the classical Langlands correspondence which is obtained by replacing the number fields appearing in the original version by function fields and applying techniques from algebraic geometry.^[1]

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.^[2]

Notes

References

• Frenkel, Edward (2007). "Lectures on the Langlands program and conformal field theory". Frontiers in number theory, physics, and geometry II. Springer: 387–533. Bibcode:2005hep.th...12172F. arXiv:hep-th/0512172 . 
• Kapustin, Anton; Witten, Edward (2007). "Electric-magnetic duality and the geometric Langlands program". Communications in Number Theory and Physics. 1 (1): 1–236. Bibcode:2007CNTP....1....1K. arXiv:hep-th/0604151 . doi:10.4310/cntp.2007.v1.n1.a1.