Non-Lorentzian Field Theories with Maximal Supersymmetry and Moduli Space Dynamics

We present gauge theories in 2+1 and 4+1 dimensions with 16 supersymmetries which are invariant under rotations and translations but not boosts. The on-shell conditions reduce the dynamics to motion on a moduli space of BPS states graded by a topologically conserved quantity. On each component of the moduli space only half the supersymmetry is realised. We argue that these theories describe M2-branes and M5-branes which have been infinitely boosted so that their worldvolume `time' has become null.