Critical-vector assumption in the string intercept argument ($M_1^2=(1-a)/{\alpha }^{\prime }$)

In the usual critical Lorentz-invariant bosonic string, level-one transverse vector polarizations fit a massless representation, whereas a massive vector would require an extra polarization. This fixes the intercept to one. Polarization counting alone does not establish that conclusion in every dimension: in three dimensions the one transverse polarization transforms trivially under the transverse rotation group and can be interpreted as a scalar. The quantum consistency assumptions must accompany an argument that the first level must be massless.