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

The action is $$S_{\mathrm{H}\mathrm{C}\mathrm{S}}[\overline{𝒜}]=\frac{1}{2\pi i}\underset{𝒲}{\int }\mathrm{\Omega }\wedge \mathrm{H}\mathrm{C}\mathrm{S}(\overline{𝒜})$$ where $$\mathrm{H}\mathrm{C}\mathrm{S}(\overline{𝒜})=\mathrm{t}\mathrm{r}(\overline{𝒜}\wedge \overline{\partial }\overline{𝒜}+\frac{2}{3}\overline{𝒜}\wedge \overline{𝒜}\wedge \overline{𝒜})$$ where $\mathrm{\Omega }$ is a holomorphic (3,0)-form and with $\mathrm{t}\mathrm{r}$ denoting a trace functional which as a bilinear form is proportional to the Killing form.

On twistor space P³

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

See also

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

External links

• Holomorphic Chern–Simons theory nLab