# Geometric Langlands correspondence

In mathematics, the **geometric Langlands correspondence** 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 correspondence relates algebraic geometry 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.[1] Establishing the Langlands correspondence in the number theoretic context has proven extremely difficult. As a result, some mathematicians have posed the geometric Langlands correspondence.[1]

## 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.[2]

## Notes

- Frenkel 2007, p. 3
- Kapustin and Witten 2007

## References

- Frenkel, Edward (2007). "Lectures on the Langlands program and conformal field theory".
*Frontiers in number theory, physics, and geometry II*. Springer: 387–533. arXiv:hep-th/0512172. Bibcode:2005hep.th...12172F. - 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.