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

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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.