14 | August | 2015 | Physics Town

A First Course in String Theory

14.1 Counting bosonic states

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Let $N(n, k) = {n + k - 1 \choose k - 1}$ , the number of ways to put $n$ indistinguishable balls into $k$ boxes.

p.318 “For open bosonic strings $\alpha' M^2 = N^\perp - 1$ , …”

When $\alpha' M^2 = 3$ , $N^\perp = 4$ , the cases are:

1. four $a_1^\dagger$ ‘s:

$N(4,24) = 17550$

2. one $a_2^\dagger$ and two $a_1^\dagger$ ‘s:

$24 \times N(2,24) = 24 \times 300$

3. two $a_2^\dagger$ ‘s:

$N(2,24) = 300$

4. one $a_3^\dagger$ and one $a_1^\dagger$ :

$24 \times 24 = 576$

5. one $a_4^\dagger$ :

$24$

Total number of possible states for $N^\perp = 4$ is 25650.

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