Higher-Spin Self-Dual Yang-Mills and Gravity from the twistor space

TL;DR

This paper extends higher-spin self-dual Yang-Mills and gravity theories to twistor space, establishing correspondences between solutions and geometric structures, thereby deepening the geometric understanding of these theories.

Contribution

It introduces a twistor space formulation for higher-spin SDYM and SDGR, proving a Ward theorem analogue and characterizing solutions via holomorphic bundles and Poisson connections.

Findings

One-to-one correspondence between solutions and holomorphic vector bundles for SDYM.

Solutions of SDGR correspond to Poisson Ehresmann connections with integrable almost complex structures.

The twistor formulation provides a geometric framework for higher-spin self-dual theories.

Abstract

We lift the recently proposed theories of higher-spin self-dual Yang-Mills (SDYM) and gravity (SDGR) to the twistor space. We find that the most natural room for the twistor formulation of these theories is not in the projective, but in the full twistor space, which is the total space of the spinor bundle over the 4-dimensional manifold. In the case of higher-spin extension of the SDYM we prove an analogue of the Ward theorem, and show that there is a one-to-one correspondence between the solutions of the field equations and holomorphic vector bundles over the twistor space. In the case of the higher-spin extension of SDGR we show show that there is a one-to-one correspondence between solutions of the field equations and Ehresmann connections on the twistor space whose horizontal distributions are Poisson, and whose curvature is decomposable. These data then define an almost complex structure on the twistor space that is integrable.