BMS$_{3}$ (Carrollian) field theories from a bound in the coupling of current-current deformations of CFT$_{2}$

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.