Problem 14.4a4

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

Can we create a formula for the number of states?

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

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

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

Physics Town

continues

Problem 14.4a4

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