# 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.

~~~

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

 $\frac{1}{2} \alpha' M^2~|$ $~\text{Number of states}$ $2~|$ $~104976$

.
What does the difference of this part and Section 14.6 come from?

This part is for bosonic closed string, while Section 14.6 is for bosonic open string. There is no $\bar N$ to consider in Section 14.6.

p.290 “A basis vector $| \lambda, \bar \lambda \rangle$ belongs to the state space if and only if it satisfies the level-matching constraint”

$N^\perp = \bar N^\perp$.

.

Can we create a formula for the number of states?

— Me@2018-04-23 11:31:16 AM

.

.