Contents

Idea

The Kapustin-Witten TQFT is the 4d TQFT obtained by topological twisting from N=4 D=4 super Yang-Mills theory. Its S-duality is supposed to encode, as a special case, geometric Langlands duality.

Upon compactification down to 2d it reproduces, at certain parameters, the A-model and the B-model.

gauge theory induced via AdS-CFT correspondence

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

$\,$

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 $Loc_G$, geometric Langlands correspondence

References

The TQFT was introduced in

Reviews include

The 0-1-2 extended QFT version of $GL$-twisted N=4 D=4 super Yang-Mills theory is considered in

A discussion formalized in BV quantization of factorization algebras is in

• Kevin Costello, Notes on supersymmetric and holomorphic field theories in dimension 2 and 4 (pdf)
Revised on September 30, 2013 12:40:04 by Urs Schreiber (82.113.99.90)