# Problem 14.4a4

Closed string degeneracies | A First Course in String Theory

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

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Can we create a formula for the number of states?

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$\left[ \frac{(D-2)(D-1)}{2} \right]^2 + \left[ \frac{(D-2)(D-1)}{2} \right](D-2) + (D-2)\left[ \frac{(D-2)(D-1)}{2} \right] + (D-2)^2$
$= ...$
$= (D-2)^2\left\{ \frac{1}{2} \frac{(D-1)^2}{2} + D \right\}$
$= 104976$
$= 324^2$
$= \left[ \frac{(D-2)(D-1)}{2} + (D-2) \right]^2$

The result is the same as the square of the coefficients of $x$ in Equation (14.63) on page 318.

 $\frac{1}{2} \alpha' M^2~|$ $N~|$ $~\bar N~$ $|~\text{Number of states}$ $-2~|$ $0~|$ $~0~$ $|~1$ $0~|$ $1~|$ $~1~$ $|~(D-2)^2$ $2~|$ $2~|$ $~2~$ $|~(D-2)^2\left\{ \frac{1}{2} \frac{(D-1)^2}{2} + D \right\}$ $4~|$ $3~|$ $~3~$ $|~3200^2$ $8~|$ $4~|$ $~4~$ $|~25650^2$

— Me@2018-04-25 05:13:04 PM

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