abstract duality: opposite category,
concrete duality: dual object, dualizable object, fully dualizable object, dualizing object
between higher geometry/higher algebra
Langlands duality, geometric Langlands duality, quantum geometric Langlands duality
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
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.
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-CFT6 | F-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 invariance | M5-brane worldvolume theory |
$\;\;\;\; \downarrow$ KK-compactification on Riemann surface | double dimensional reduction on M-theory/F-theory elliptic fibration |
N=2 D=4 super Yang-Mills theory with Montonen-Olive S-duality invariance; AGT correspondence | D3-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 |
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
geometric Langlands correspondence
A classical survey is
Another set of lecture notes on geometric Langlands and nonabelian Hodge theory is
Collections of resources are here;
David Ben-Zvi, Geometric Langlands – Lectures and Resources (web)
geometric Laglands page
Notes on two introductory lecture talks are here:
See also
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
A relation to T-duality (of the group manifolds!) is discussed in