Problem 14.4a3

Closed string degeneracies | A First Course in String Theory

(a) State the values of $\alpha' M^2$ and give the degeneracies for the first five mass levels of the closed bosonic string theory.

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But for non-massless states, this probably is not true anymore:

$\|$	$~\text{Number of states}$
${a_1^I}^\dagger {a_1^J}^\dagger \bar a_1^{K\dagger} \bar a_1^{L\dagger} \| p^+, \vec p_T \rangle~\|$	$~\left[ \frac{(D-2)(D-1)}{2} \right]^2$
${a_1^I}^\dagger {a_1^J}^\dagger \bar a_2^{K\dagger} \| p^+, \vec p_T \rangle~\|$	$~\left[ \frac{(D-2)(D-1)}{2} \right](D-2)$
${a_2^I}^\dagger \bar a_1^{K\dagger} \bar a_1^{L\dagger} \| p^+, \vec p_T \rangle~\|$	$~(D-2)\left[ \frac{(D-2)(D-1)}{2} \right]$
${a_2^I}^\dagger \bar a_2^{K\dagger} \| p^+, \vec p_T \rangle~\|$	$(D-2)^2$

So the total number of states for $\frac{1}{2} \alpha' M^2 = 2$ ( $N = \bar N = 2$ ) is

$\left[ \frac{(D-2)(D-1)}{2} \right]^2 + \left[ \frac{(D-2)(D-1)}{2} \right](D-2)$
$+ (D-2)\left[ \frac{(D-2)(D-1)}{2} \right] + (D-2)^2$

Should $D$ be 10 or 26?

p.324 “Out of 26 left-moving bosonic coordinates of the bosonic factor only ten of them are matched by the right-moving bosonic coordinates of the superstring factor.”

$D$ should be 26 for bosonic strings. So the total number of states is