Self-dual gravity from higher-spin theory

TL;DR

This paper demonstrates that self-dual gravity with a cosmological constant can be derived from higher-spin symmetries, revealing a closed sector of lower-spin fields governed by Moyal star product interactions.

Contribution

It shows that higher-spin interactions in four dimensions contain a consistent self-dual gravity sector characterized by Moyal star product algebra.

Findings

Self-dual gravity emerges as a unique sector within higher-spin interactions.

Lower helicity fields form a closed sub-sector and act as sources for positive helicities.

The equations of self-dual gravity incorporate the Moyal star product, linking it to higher-spin algebra.

Abstract

Higher-spin symmetry is known to mix lower-spin fields with higher-spin fields, creating a complex interaction picture where no closed finite field sector is expected to exist for dimensions greater than three. By studying the self-dual part of higher-spin interaction vertices in four dimensions, we show that gauge fields of spins greater than two can be consistently set to zero. In this case, the fields with helicities $-2\leq\lambda\leq 0$ form a closed sub-sector and also act as sources for positive helicities. For these lower spin fields, we identify their equations of motion. In particular, we show that self-dual gravity with a cosmological constant emerges as a unique rigid part of higher-spin interactions. Notably, its equations have a form that incorporates the Moyal star product, which is essential for generating the higher-spin algebra. Therefore, we demonstrate that self-dual gravity can be derived from higher-spin symmetries.