nLab
geometric Langlands correspondence

Contents

Idea

The geometric Langlands program is the analogue of the Langlands program with number fields replaced by function fields.

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

𝒟Mod(Bun G)𝒪Mod(Loc LG) \mathcal{D} Mod( Bun_G) \simeq \mathcal{O}Mod(Loc_{{}^L G})

for LG{}^L G the Langlands dual group.

This equivalence is a certain limit of the more general quantum geometric Langlands correspondence that identifies twisted DD-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.

gauge theory induced via AdS-CFT correspondence

M-theory perspective via AdS7-CFT6F-theory perspective
11d supergravity/M-theory
\;\;\;\;\downarrow Kaluza-Klein compactification on S 4S^4compactificationon elliptic fibration followed by T-duality
7-dimensional supergravity
\;\;\;\;\downarrow topological sector
7-dimensional Chern-Simons theory
\;\;\;\;\downarrow AdS7-CFT6 holographic duality
6d (2,0)-superconformal QFT on the M5-brane with conformal invarianceM5-brane worldvolume theory
\;\;\;\; \downarrow KK-compactification on Riemann surfacedouble dimensional reduction on M-theory/F-theory elliptic fibration
N=2 D=4 super Yang-Mills theory with Montonen-Olive S-duality invariance; AGT correspondenceD3-brane worldvolume theory with type IIB S-duality
\;\;\;\;\; \downarrow topological twist
topologically twisted N=2 D=4 super Yang-Mills theory
\;\;\;\; \downarrow KK-compactification on Riemann surface
A-model on Bun GBun_G, Donaldson theory

\,

gauge theory induced via AdS5-CFT4
type II string theory
\;\;\;\;\downarrow Kaluza-Klein compactification on S 5S^5
\;\;\;\; \downarrow topological sector
5-dimensional Chern-Simons theory
\;\;\;\;\downarrow AdS5-CFT4 holographic duality
N=4 D=4 super Yang-Mills theory
\;\;\;\;\; \downarrow topological twist
topologically twisted N=4 D=4 super Yang-Mills theory
\;\;\;\; \downarrow KK-compactification on Riemann surface
A-model on Bun GBun_G and B-model on Log GLog_G, geometric Langlands correspondence

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)

duality in physics, duality in string theory

References

General

A classical survey is

Another set of lecture notes on geometric Langlands and nonabelian Hodge theory is

Collections of resources are here;

Notes on two introductory lecture talks are here:

See also

Interpretation in string theory

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)

  • Edward Frenkel, Gauge theory and Langlands duality, Séminaire Bourbaki, June 2009 (pdf)

More recent developments are surveyed in

Further discussion is also in

  • Tamas Hausel, Global topology of the Hitchin system (arXiv:1102.1717, pdf slides)

  • Kevin Setter, Topological quantum field theory and the geometric Langlands correspondence. Dissertation (Ph.D.), California Institute of Technology 2013 (web)

Discussion from the point of view of M-theory is in

  • Meng-Chwan Tan, M-Theoretic Derivations of 4d-2d Dualities: From a Geometric Langlands Duality for Surfaces, to the AGT Correspondence, to Integrable Systems (arXiv:1301.1977)

A relation to T-duality (of the group manifolds!) is discussed in

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

Revised on March 23, 2014 04:23:50 by Urs Schreiber (89.204.138.46)