Six-dimensional holomorphic Chern–Simons theory

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.

The theory has been used to study integrable systems through four-dimensional Chern–Simons theory, which can be viewed as a symmetry reduction of the six-dimensional theory.^[2] For this purpose, the underlying three-dimensional complex manifold is taken to be the three-dimensional complex projective space $\mathbb {P} ^{3}$ , viewed as twistor space.

Formulation

The background manifold ${\mathcal {W}}$ 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 ${\bar {\mathcal {A}}}$ .

The action is $S_{\mathrm {HCS} }[{\bar {\mathcal {A}}}]={\frac {1}{2\pi i}}\int _{\mathcal {W}}\Omega \wedge \mathrm {HCS} ({\bar {\mathcal {A}}})$ where $\mathrm {HCS} ({\bar {\mathcal {A}}})=\mathrm {tr} \left({\bar {\mathcal {A}}}\wedge {\bar {\partial }}{\bar {\mathcal {A}}}+{\frac {2}{3}}{\bar {\mathcal {A}}}\wedge {\bar {\mathcal {A}}}\wedge {\bar {\mathcal {A}}}\right)$ where $\Omega$ is a holomorphic (3,0)-form and with $\mathrm {tr}$ denoting a trace functional which as a bilinear form is proportional to the Killing form.

On twistor space P³

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

External links

Holomorphic Chern–Simons theory nLab

References

^ Chern, Shiing-Shen; Simons, James (September 1996). "Characteristic forms and geometric invariants". World Scientific Series in 20th Century Mathematics. 4: 363–384. doi:10.1142/9789812812834_0026. ISBN 978-981-02-2385-4.
^ ^a ^b Bittleston, Roland; Skinner, David (22 February 2023). "Twistors, the ASD Yang-Mills equations and 4d Chern-Simons theory". Journal of High Energy Physics. 2023 (2): 227. arXiv:2011.04638. Bibcode:2023JHEP...02..227B. doi:10.1007/JHEP02(2023)227. ISSN 1029-8479. S2CID 226281535.

^[1]

^ Cole, Lewis T.; Cullinan, Ryan A.; Hoare, Ben; Liniado, Joaquin; Thompson, Daniel C. (2023-11-29). "Integrable Deformations from Twistor Space". arXiv:2311.17551 [hep-th].

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