Neumann-Dirichlet string zero-point energy ($E_0=d/48$)

Each of $d$ physical transverse bosons with a Neumann-Dirichlet open-string boundary condition contributes $\frac{1}{2}{\sum }_{r>0}r=1/48$ by the half-integer zeta-regularized mode sum. Thus $L_0^{\perp }=N+d/48$ and the convention $L_0={\alpha }^{\prime }{p}^2+N-a$ has $a=-d/48$. For $d=24$, the vacuum shift is $+1/2$. The ordinary integer-moded open-string shift is instead $-d/24$; these endpoint sectors have different normal-ordering constants.

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, $a_{1/2}^{i†}|0⟩$, and $a_{1/2}^{i†}{a}_{1/2}^{j†}|0⟩$. The creation operators commute, so the third level is the symmetric square of the 24-dimensional transverse vector, of dimension $24\cdot 25/2=300$. Its trace is a scalar and its trace-free part has dimension 299. Including the vacuum shift gives ${\alpha }^{\prime }{M}^2=1/2,1,3/2$ 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.