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 ${\displaystyle \mathbb {P} ^{3}}$, viewed as twistor space.

The background manifold ${\displaystyle {\mathcal {W}}}$ 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 ${\displaystyle G.}$ The field content is a partial connection ${\displaystyle {\bar {\mathcal {A}}}}$.

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

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

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

• Holomorphic Chern–Simons theory nLab

^[1]