= Critical-vector assumption in the string intercept argument
{title2=$M_1^2=(1-a)/\alpha'$}
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.
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