nLab
geometric Langlands correspondence

to be merged with geometric Langlands program

Idea

The conjectured geometric Langlands correspondence asserts that for $G$ a reductive group there is an equivalence of derived categories of D-modules on the moduli stack of $G$-principal bundles over a given curve, and quasi-coherent sheaves on the moduli space of ${}^{L}G$-local systems

$$𝒟\mathrm{Mod}({\mathrm{Bun}}_G)\simeq 𝒪\mathrm{Mod}({\mathrm{Loc}}_{{}^{L}G})$$\mathcal{D} Mod( Bun_G) \simeq \mathcal{O}Mod(Loc_{{}^L G})

for ${}^{L}G$ the Langlands dual group?.

This equivalence is a certain limit of the more general quantum geometric Langlands correspondence that identifies twisted $D$-modules on both sides.

Properties

The Kapustin-Witten TQFT (KapustinWitten 2007) is supposed to exhibit geometric Langlands duality as a special case of S-duality.

Related concepts

• S-duality

  • electro-magnetic duality

    • Montonen-Olive duality

  • Seiberg duality

  • geometric Langlands correspondence

  • quantum geometric Langlands correspondence

References

A classical survey is

• Edward Frenkel, Lectures on the Langlands Program and Conformal Field Theory (arXiv:hep-th/0512172).

Notes on two introductory lecture talks are here:

• Pantev on Langlands I

  Pantev on Langlands II

An interpretation of the geometric Langlands correspondence as describing S-duality of certain twisted reduction of super Yang-Mills theory was given in

• Anton Kapustin, Edward Witten, Electric-Magnetic Duality And The Geometric Langlands Program , Communications in number theory and physics, Volume 1, Number 1, 1–236 (2007) (arXiv:hep-th/0604151)

An exposition of the relation to S-duality and electro-magnetic duality is in

• Edward Frenkel, What Do Fermat’s Last Theorem and Electro-magnetic Duality Have in Common? KITP talk 2011 (web)