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
, viewed as twistor space.
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
The field content is a partial connection
{\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)}
denoting a trace functional which as a bilinear form is proportional to the Killing form.
For application to integrable theory, the three form
must be chosen to be meromorphic.