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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2018.04.25 Wednesday (c) All rights reserved by ACHK