nLab Kapustin-Witten TQFT



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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=Σ×CM = \Sigma\times C, where Σ\Sigma is a Riemann surface, possibly with boundary and CC is a Riemann surface of genus g2g\geq 2. Its S-duality is supposed to encode, as a special case, geometric Langlands duality for the Riemann surface CC.

Upon compactification on CC (i.e. CC is taken to be much smaller than Σ\Sigma) down to 2d the resulting effective field theory on Σ\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 H(G)\mathcal{M}_{H}(G), where GG is the gauge group of the theory.

The Hitchin moduli space is a hyperkähler manifold. In complex structure II it can be seen as the moduli space of Higgs bundles on CC, and in complex structures JJ and KK it can be seen as the moduli space of GG-local systems on CC. This is a form of nonabelian Hodge theory.

S-duality manifests as mirror symmetry between H(G)\mathcal{M}_{H}(G) and H( LG)\mathcal{M}_{H}({}^{L}G), where LG{}^{L}G is the Langlands dual group of GG. It exchanges B-branes on H( LG)\mathcal{M}_{H}({}^{L}G) in complex structure JJ with A-branes on H(G)\mathcal{M}_{H}(G) with complex structure KK. 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 H( LG)\mathcal{M}_{H}({}^{L}G) in complex structure JJ, which is an eigenbrane for the Wilson loop operators, corresponding to a local system on CC. S-duality gives us an A-brane 𝒜\mathcal{A} on H(G)\mathcal{M}_{H}(G) in complex structure KK, which is an eigenbrane for the ‘t Hooft operators. Now we perform a hyperkahler rotation and consider 𝒜\mathcal{A} as an A-brane on H(G)\mathcal{M}_{H}(G), in complex structure II. Let 𝒜 cc\mathcal{A}_{cc} be the canonical isotropic brane on H(G)\mathcal{M}_{H}(G). The sheafification of Hom(𝒜 cc,𝒜)\mathrm{Hom}(\mathcal{A}_{cc},\mathcal{A}) gives us a D-module for the sheaf of differential operators given by the sheafification of Hom(𝒜 cc,𝒜 cc)\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-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 Loc GLoc_G, geometric Langlands correspondence


The TQFT was introduced in

Reviews include

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 GLGL-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)

Last revised on July 1, 2022 at 19:40:45. See the history of this page for a list of all contributions to it.