Twistor Actions for Self-Dual Supergravities
Lionel J. Mason, Martin Wolf
TL;DR
The paper develops holomorphic Chern-Simons–type actions on supertwistor space to encode the self-dual sector of four-dimensional supergravity across $\mathcal{N}=0$ to $\mathcal{N}=8$, including cases with partial $R$-symmetry gauging and with or without a cosmological constant. It shows that finite deformations of the integrable $\bar{\partial}$-operator, constrained by holomorphic Poisson or contact structures, reproduce the SD supergravity field equations and yield explicit actions, notably a holomorphic Chern-Simons action for $\mathcal{N}=8$. A covariant formulation is developed for the $\mathcal{N}=0$ case via a Cap–Eastwood–type six-dimensional geometry, while a fully covariant off-shell action for $\mathcal{N}>0$ remains an open challenge. The work links twistor-geometric structures to space-time dynamics and points toward potential connections with twistor-string theories for Einstein supergravity.
Abstract
We give holomorphic Chern-Simons-like action functionals on supertwistor space for self-dual supergravity theories in four dimensions, dealing with N=0,...,8 supersymmetries, the cases where different parts of the R-symmetry are gauged, and with or without a cosmological constant. The gauge group is formally the group of holomorphic Poisson transformations of supertwistor space where the form of the Poisson structure determines the amount of R-symmetry gauged and the value of the cosmological constant. We give a formulation in terms of a finite deformation of an integrable \dbar-operator on a supertwistor space, i.e., on regions in CP^{3|8}. For N=0, we also give a formulation that does not require the choice of a background.