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 [math]\displaystyle{ \mathbb{P}^3 }[/math], viewed as twistor space.
Formulation
The background manifold [math]\displaystyle{ \mathcal{W} }[/math] 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 [math]\displaystyle{ G. }[/math] The field content is a partial connection [math]\displaystyle{ \bar \mathcal{A} }[/math].
The action is [math]\displaystyle{ S_{\mathrm{HCS}}[\bar\mathcal{A}] = \frac{1}{2\pi i} \int_{\mathcal W} \Omega \wedge \mathrm{HCS}(\bar \mathcal{A}) }[/math] where [math]\displaystyle{ \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) }[/math] where [math]\displaystyle{ \Omega }[/math] is a holomorphic (3,0)-form and with [math]\displaystyle{ \mathrm{tr} }[/math] denoting a trace functional which as a bilinear form is proportional to the Killing form.
On twistor space P3
Here [math]\displaystyle{ \mathcal{W} }[/math] is fixed to be [math]\displaystyle{ \mathbb{P}^3 }[/math]. For application to integrable theory, the three form [math]\displaystyle{ \Omega }[/math] must be chosen to be meromorphic.
See also
- Chern–Simons theory
- Four-dimensional Chern-Simons theory
External links
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.
- ↑ 2.0 2.1 Bittleston, Roland; Skinner, David (22 February 2023). "Twistors, the ASD Yang-Mills equations and 4d Chern-Simons theory" (in en). Journal of High Energy Physics 2023 (2): 227. doi:10.1007/JHEP02(2023)227. ISSN 1029-8479. Bibcode: 2023JHEP...02..227B. https://link.springer.com/article/10.1007/JHEP02(2023)227.
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- ↑ 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].
