  We argue that generic field theories defined on null manifolds should have an
emergent BMS or conformal Carrollian structure. We then focus on a simple
interacting conformal Carrollian theory, viz. Carrollian scalar
electrodynamics. We look at weak (on-shell) and strong invariance (off-shell)
of its equations of motion under conformal Carrollian symmetries. Helmholtz
conditions are necessary and sufficient conditions for a set of equations to
arise from a Lagrangian. We investigate whether the equations of motion of
Carrollian scalar electrodynamics satisfy these conditions. Then we proposed an
action for the electric sector of the theory. This action is the first example
for an interacting conformal Carrollian Field Theory. The proposed action
respects the finite and infinite conformal Carrollian symmetries in d = 4. We
calculate conserved charges corresponding to these finite and infinite
symmetries and then rewrite the conserved charges in terms of the canonical
variables. We finally compute the Poisson brackets for these charges and
confirm that infinite Carrollian conformal algebra is satisfied at the level of
charges.
