Quick Calculation 14.4

A First Course in String Theory

How many states are there at $N^\perp = \frac{3}{2}$ ?

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This answer is my guess.

There are $(D-2)$ transverse (i.e. non-lightcone) coordinates. When $D = 10$ , the number of transverse coordinates is 8. So there are 8 states with $N^\perp = \frac{1}{2}$ .

For $N^\perp = \frac{3}{2}$ , there are 3 indices $I$ , $J$ , and $K$ . So there are totally 24 states.

— Me@2015.05.26 03:56 PM

The above answer is incorrect.

Equation (14.38) $\left(\alpha'M^2 = 1,~~N^\perp = \frac{3}{2}\right)$ :

$\{ \alpha_{-1}^I b_{-1/2}^J, b_{-3/2}^I, b_{-1/2}^I b_{-1/2}^J b_{-1/2}^K \} |NS \rangle \otimes |p^+, \vec p_T \rangle$

There are $8 \times 8$ states for