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Recently, a class of non-Lorentzian supersymmetric Lagrangian field theorieswas considered, in some cases describing M-theory brane configurations, butmore generally found as fixed points of non-Lorentzian RG flows induced uponLorentzian theories. In this paper, we demonstrate how the dynamics of suchtheories can be reduced to motion on the supersymmetric moduli space of BPSsolitons of the parent theory. We focus first on the $\mathcal{N}=(1,1)$$\sigma$-model in $(1+1)$-dimensions with potential, where we produce asupersymmetric extension to the standard geodesic approximation for slow kinkmotion. We then revisit the $(4+1)$-dimensional Yang-Mills-like theory with 24supercharges describing a null compactification of M5-branes. We show that thetheory reduces to a $\sigma$-model on instanton moduli space, extended bycouplings to additional fields from the parent theory, and possessing (8+8)super(conformal) symmetries along with 8 further fermionic shift symmetries. Wederive this model explicitly for the single $SU(2)$ instanton.

Supersymmetric Soliton $\sigma$-models from Non-Lorentzian Field Theories