We consider the twistor space ${ \cal P}^6\cong{ \mathbb R}^4{ \times}{ \mathbb C}P^1$ of ${ \mathbb R}^4$ with a non-integrable almost complex structure ${ \cal J}$ such that the canonical bundle of the almost complex manifold $({ \cal P}^6, { \cal J})$ is trivial. It is shown that ${ \cal J}$-holomorphic Chern-Simons theory on a real $(6|2)$-dimensional graded extension ${ \cal P}^{ 6|2}$ of the twistor space ${ \cal P}^6$ is equivalent to self-dual Yang-Mills theory on Euclidean space ${ \mathbb R}^4$ with Lorentz invariant action. It is also shown that adding a local term to a Chern-Simons-type action on ${ \cal P}^{ 6|2}$, one can extend it to a twistor action describing full Yang-Mills theory.
Categories: hep-th