Carroll fermions in two dimensions

Carroll symmetry is a very powerful characteristic of generic null surfaces, as it replaces the usual Poincar\'e algebra with vanishing speed light version thereof. These symmetries have found universal applications in physics manifolds they arise diverse situations ranging from black hole horizons to condensed matter systems Fermi velocities. In this work, we concentrate on fermions living two-dimensional ($2d$) and explore invariant structure associated field theories systematic manner. The free massless versions these are shown exhibit $2d$ conformal or, equivalently $3d$ Bondi-Metzner-Sachs (BMS) their symmetry. Due degenerate nature manifold, show presence two distinct classes Clifford algebras. We also find that dimensions, there fermion actions. study discrete continuous both quantise them using highest weight representation vacuum. discuss how can be continually deformed by infinite boosts or linear transformations coordinates, leading corresponding BMS theory at singular points.