Octonionic Selfduality for SuperMembranes

TL;DR

This paper explores octonionic duality in supermembranes, presenting new forms of self-duality equations with octonionic and quaternionic symmetries, and analyzing their integrability and embeddings.

Contribution

It introduces various symmetric forms of octonionic self-duality equations and demonstrates their invariance under G_2, aiding understanding of supermembrane integrability.

Findings

Multiple symmetric forms of octonionic self-duality equations

Factorization of time dependence via quadratic Poisson algebra

Linear embeddings using G_2 invariance

Abstract

In this work we study the recently introduced octonionic duality for membranes. Restricting the self - duality equations to seven space dimensions, we provide various forms for them which exhibit the symmetries of the octonionic and quaternionic structure. These forms may turn to be useful for the question of the integrability of this system. Introducing a consistent quadratic Poisson algebra of functions on the membrane we are able to factorize the time dependence of the self - duality equations. We further give the general linear embeddings of the three dimensional system into the seven dimensional one using the invariance of the self-duality equations under the exceptional group G_2.