A Twistor Space Action for Yang-Mills Theory

We consider the twistor space ${\cal P}^6\cong{\mathbb R}^4{\times}{\mathbb C}P^1$of${\mathbb R}^4$withanon-integrablealmostcomplexstructure${\cal J}$suchthatthecanonicalbundleofthealmostcomplexmanifold$({\cal P}^6, {\cal J})$istrivial\mathrm{.}Itisshownthat${\cal J}$-holomorphic Chern-Simons theory on a real $(6\mathrm{\mid }2)$-dimensional graded extension ${\mathcal{P}}^{6\mathrm{\mid }2}$ of the twistor space ${\mathcal{P}}^6$ 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 ${\mathcal{P}}^{6\mathrm{\mid }2}$, one can extend it to a twistor action describing full Yang-Mills theory.