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 $$\mathcal{W}$$ 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 $$\bar \mathcal{A}$$.

The action is

where

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.