A Twistor Space Action for Yang-Mills Theory

We consider the twistor space ${cal P}^6cong{mathbb R}^4{times}{mathbb C}P^1$ 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.