Self-Dual Supergravity and Twistor Theory

TL;DR

This work develops a comprehensive twistor-theoretic framework for four-dimensional $\mathcal{N}$-extended self-dual supergravity, unifying complex supermanifold geometry with supersymmetric twistor constructions. By starting from $(4|4\mathcal{N})$ complex superconformal manifolds and passing through complex quaternionic, quaternionic Kähler, and hyper-Kähler RC supermanifolds, it builds supersymmetric Penrose-type correspondences via supertwistor spaces, beta-plane structures, and Gindikin two-forms. The paper provides several equivalent formulations, including a supersymmetric LeBrun Einstein bundle description that accommodates nonzero cosmological constant, and develops the Penrose–Ward transform for both the local supertwistor bundle and the universal line bundle on twistor space. The results also cover the special $\mathcal{N}=4$ case, introduce a real Euclidean version, and connect twistorial data to hyper-Kähler and Calabi–Yau-type properties in the supersymmetric setting. These constructions sharpen the geometric underpinnings of self-dual supergravity and open pathways for twistor-string and related formalisms in supersymmetric gravity.

Abstract

By generalizing and extending some of the earlier results derived by Manin and Merkulov, a twistor description is given of four-dimensional N-extended (gauged) self-dual supergravity with and without cosmological constant. Starting from the category of (4|4N)-dimensional complex superconformal supermanifolds, the categories of (4|2N)-dimensional complex quaternionic, quaternionic Kaehler and hyper-Kaehler right-chiral supermanifolds are introduced and discussed. We then present a detailed twistor description of these types of supermanifolds. In particular, we construct supertwistor spaces associated with complex quaternionic right-chiral supermanifolds, and explain what additional supertwistor data allows for giving those supermanifolds a hyper-Kaehler structure. In this way, we obtain a supersymmetric generalization of Penrose's nonlinear graviton construction. We furthermore give an alternative formulation in terms of a supersymmetric extension of LeBrun's Einstein bundle. This allows us to include the cases with nonvanishing cosmological constant. We also discuss the bundle of local supertwistors and address certain implications thereof. Finally, we comment on a real version of the theory related to Euclidean signature.