= Lowest levels of a fully transverse ND bosonic string
With 24 physical <half-integer open-string oscillators>, the first three <string level operator> <eigenvalues> are $0,1/2,1$. Their states are the vacuum, $a_{1/2}^{i\dagger}|0\rangle$, and $a_{1/2}^{i\dagger}a_{1/2}^{j\dagger}|0\rangle$. The <creation operators> commute, so the third level is the <symmetric square> of the 24-dimensional transverse vector, of dimension $24\cdot25/2=300$. Its trace is a scalar and its trace-free part has dimension 299. Including the vacuum shift gives $\alpha'M^2=1/2,1,3/2$ when the two longitudinal directions are common directions with <Neumann boundary conditions>. The representation labels refer to the surviving ND rotation group, not full unbroken target Lorentz symmetry.
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