# nLab geometric Langlands correspondence

to be merged with geometric Langlands program

duality

# Contents

## 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

$𝒟\mathrm{Mod}\left({\mathrm{Bun}}_{G}\right)\simeq 𝒪\mathrm{Mod}\left({\mathrm{Loc}}_{{}^{L}G}\right)$\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

### Relation to S-duality

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

### Relation to T-duality

In some cases the passage between a Lie group and its Langlands dual group can be understood as a special case of T-duality. (Daenzer-vanErp)

## References

A classical survey is

Notes on two introductory lecture talks are here:

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

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)

The relation to T-duality is discussed in

• Calder Daenzer, Erik Van Erp, T-Duality for Langlands Dual Groups (arXiv:1211.0763)

Revised on March 31, 2013 18:40:31 by Urs Schreiber (89.204.138.121)