duality

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 $G$ 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

$\mathcal{D} Mod( Bun_G) \simeq \mathcal{O}Mod(Loc_{{}^L G})$

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.

gauge theory induced via AdS-CFT correspondence

M-theory perspective via AdS7-CFT6F-theory perspective
11d supergravity/M-theory
$\;\;\;\;\downarrow$ Kaluza-Klein compactification on $S^4$compactificationon 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 $Bun_G$, Donaldson theory

$\,$

gauge theory induced via AdS5-CFT4
type II string theory
$\;\;\;\;\downarrow$ Kaluza-Klein compactification on $S^5$
$\;\;\;\; \downarrow$ 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 $Bun_G$ and B-model on $Log_G$, 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)

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:

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)