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

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.