Killing-Yano equations with torsion, worldline actions and G-structures

TL;DR

This paper explores the geometry of target spaces in supersymmetric particles with torsion, analyzing Killing-Yano equations and G-structures, revealing multiple compatible torsion connections and their geometric implications.

Contribution

It identifies multiple connections with skew-symmetric torsion compatible with G-structures, expanding understanding of worldline action invariance conditions.

Findings

Multiple torsion connections for G=U(n), SU(n), G_2

Killing-Yano equations do not always determine torsion uniquely

Descriptions of geometric nature of torsion couplings

Abstract

We determine the geometry of the target spaces of supersymmetric non-relativistic particles with torsion and magnetic couplings, and with symmetries generated by the fundamental forms of G-structures for $G= U(n), SU(n), Sp(n), Sp(n)\cdot Sp(1), G_2$ and $Spin(7)$. We find that the Killing-Yano equation, which arises as a condition for the invariance of the worldline action, does not always determine the torsion coupling uniquely in terms of the metric and fundamental forms. We show that there are several connections with skew-symmetric torsion for $G=U(n), SU(n)$ and $G_2$ that solve the invariance conditions. We describe all these compatible connections for each of the $G$-structures and explain the geometric nature of the couplings.