Six-dimensional holomorphic Chern–Simons theory explained

In mathematical physics, six-dimensional holomorphic Chern–Simons theory or sometimes holomorphic Chern–Simons theory is a gauge theory on a three-dimensional complex manifold. It is a complex analogue of Chern–Simons theory, named after Shiing-Shen Chern and James Simons who first studied Chern–Simons forms which appear in the action of Chern–Simons theory.^[1] The theory is referred to as six-dimensional as the underlying manifold of the theory is three-dimensional as a complex manifold, hence six-dimensional as a real manifold.

P ^{3} , viewed as twistor space.

Formulation

l{W} on which the theory is defined is a complex manifold which has three complex dimensions and therefore six real dimensions.^[2] The theory is a gauge theory with gauge group a complex, simple Lie group G. The field content is a partial connection \bar l{A} .

The action is$$S\_[\backslash bar\backslash mathcal\{A\}]\; =\; \backslash frac\; \backslash int\_\; \backslash Omega\; \backslash wedge\; \backslash mathrm(\backslash bar\; \backslash mathcal)$$where $$\backslash mathrm(\backslash bar\; \backslash mathcal)\; =\; \backslash mathrm\backslash left(\backslash bar\; \backslash mathcal\; \backslash wedge\; \backslash bar\; \backslash partial\; \backslash bar\; \backslash mathcal\; +\; \backslash frac\; \backslash bar\; \backslash mathcal\; \backslash wedge\; \backslash bar\; \backslash mathcal\; \backslash wedge\; \backslash bar\; \backslash mathcal\backslash right)$$where

\Omega is a holomorphic (3,0)-form and with tr denoting a trace functional which as a bilinear form is proportional to the Killing form.

On twistor space P³

Here

l{W} is fixed to be P ^{3} . For application to integrable theory, the three form \Omega must be chosen to be meromorphic.

See also

• Chern–Simons theory
• Four-dimensional Chern-Simons theory

External links

• Holomorphic Chern–Simons theory nLab

References

^[3]

Notes and References

1. Chern . Shiing-Shen . Simons . James . Characteristic forms and geometric invariants . World Scientific Series in 20th Century Mathematics . September 1996 . 4 . 363–384 . 10.1142/9789812812834_0026. 978-981-02-2385-4 .
2. Bittleston . Roland . Skinner . David . Twistors, the ASD Yang-Mills equations and 4d Chern-Simons theory . Journal of High Energy Physics . 22 February 2023 . 2023 . 2 . 227 . 10.1007/JHEP02(2023)227 . 2023JHEP...02..227B . 226281535 . en . 1029-8479. 2011.04638 .
3. Cole . Lewis T. . Integrable Deformations from Twistor Space . 2023-11-29 . 2311.17551 . Cullinan . Ryan A. . Hoare . Ben . Liniado . Joaquin . Thompson . Daniel C.. hep-th .