  Two types of Carrollian field theories are shown to emerge from finite
current-current deformations of toroidal CFT$_{2}$'s when the deformation
coupling is precisely fixed, up to a sign. In both cases the energy and
momentum densities fulfill the BMS$_{3}$ algebra. Applying these results to the
bosonic string, one finds that the electric-like deformation (positive
coupling) reduces to the standard tensionless string. The magnetic-like
deformation (negative coupling) yields to a new theory, still being
relativistic, devoid of tension and endowed with an "inner Carrollian
structure". Classical solutions describe a sort of "self-interacting null
particle" moving along generic null curves of the original background metric,
not necessarily geodesics. This magnetic-like theory is also shown to be
recovered from inequivalent limits in the tension of the bosonic string.
Electric- and magnetic-like deformations of toroidal CFT$_{2}$'s can be seen to
correspond to limiting cases of continuous exactly marginal (trivial)
deformations spanned by an SO(1,1) automorphism of the current algebra. Thus,
the absolute value of the current-current deformation coupling is shown to be
bounded. When the bound saturates, the deformation ceases to be exactly
marginal, but still retains the full conformal symmetry in two alternative
ultrarelativistic regimes.
