nLab Kapustin-Witten TQFT

Idea

The Kapustin-Witten TQFT is the 4d TQFT obtained by topological twisting from N=4 D=4 super Yang-Mills theory on a 4-dimensional manifold of the form $M = \Sigma\times C$, where $\Sigma$ is a Riemann surface, possibly with boundary and $C$ is a Riemann surface of genus $g\geq 2$. Its S-duality is supposed to encode, as a special case, geometric Langlands duality for the Riemann surface $C$.

Upon compactification on $C$ (i.e. $C$ is taken to be much smaller than $\Sigma$) down to 2d the resulting effective field theory on $\Sigma$ reproduces, at certain parameters, the A-model and the B-model topological string theory, whose target as a sigma model is the Hitchin moduli space $\mathcal{M}_{H}(G)$, where $G$ is the gauge group of the theory.

The Hitchin moduli space is a hyperkähler manifold. In complex structure $I$ it can be seen as the moduli space of Higgs bundles on $C$, and in complex structures $J$ and $K$ it can be seen as the moduli space of $G$-local systems on $C$. This is a form of nonabelian Hodge theory.

S-duality manifests as mirror symmetry between $\mathcal{M}_{H}(G)$ and $\mathcal{M}_{H}({}^{L}G)$, where ${}^{L}G$ is the Langlands dual group of $G$. It exchanges B-branes on $\mathcal{M}_{H}({}^{L}G)$ in complex structure $J$ with A-branes on $\mathcal{M}_{H}(G)$ with complex structure $K$. It also exchanges Wilson loop operators and 't Hooft operators (which correspond to Hecke correspondences).

The connection to the geometric Langlands correspondence may now be made more explicit as follows. We start with a zerobrane (a B-brane) $\mathcal{B}$ on $\mathcal{M}_{H}({}^{L}G)$ in complex structure $J$, which is an eigenbrane for the Wilson loop operators, corresponding to a local system on $C$. S-duality gives us an A-brane $\mathcal{A}$ on $\mathcal{M}_{H}(G)$ in complex structure $K$, which is an eigenbrane for the ‘t Hooft operators. Now we perform a hyperkahler rotation and consider $\mathcal{A}$ as an A-brane on $\mathcal{M}_{H}(G)$, in complex structure $I$. Let $\mathcal{A}_{cc}$ be the canonical isotropic brane on $\mathcal{M}_{H}(G)$. The sheafification of $\mathrm{Hom}(\mathcal{A}_{cc},\mathcal{A})$ gives us a D-module for the sheaf of differential operators given by the sheafification of $\mathrm{Hom}(\mathcal{A}_{cc},\mathcal{A}_{cc})$ (see also the related topic of quantization via the A-model). The resulting D-module is expected to be the Hecke eigensheaf predicted by the geometric Langlands correspondence.

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 $Loc_G$, geometric Langlands correspondence

Related concepts

• geometric Langlands duality

• A-model, B-model

• quantization via the A-model

References

The TQFT was introduced in

• Anton Kapustin, Edward Witten, Electric-Magnetic Duality And The Geometric Langlands Program , Communications in number theory and physics, Volume 1, Number 1, 1–236 (2007) (arXiv:hep-th/0604151)

Reviews include

• Anton Kapustin, Langlands Duality and Topological Field Theory (2007) (pdf)

• Edward Frenkel, Gauge Theory and Langlands Duality (2009) (arXiv:0906.2747)

• Edward Witten, Mirror Symmetry, Hitchin’s Equations, and Langlands Duality (2009) (arXiv:0802.0999)

The link between mirror symmetry and geometric Langlands was explored in the earlier paper

• Tamas Hausel?, Michael Thaddeus?, Mirror Symmetry, Langlands Duality, and the Hitchin System (arXiv:0205236)

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

• David Ben-Zvi, David Nadler, The Character Theory of a Complex Group (arXiv:0904.1247)

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)