Problem 14.3b2

A First Course in String Theory

14.3 Massive level in the open superstring

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How come R sector has a factor 16 while NS sector has not?

Equation (14.66):

$f_{NS}(x) = \frac{1}{\sqrt{x}} \prod_{n=1}^\infty \left( \frac{1+x^{n - \frac{1}{2}}}{1-x^n} \right)^8$

Equation (14.68):

$f_R(x) = 16 \prod_{n=1}^\infty \left( \frac{1+x^n}{1-x^n} \right)^8$

p.319 “The overall multiplicative factor appears because each combination of oscillators gives rise to 16 states by acting on each of the available ground states.”

p.319 “We note that the R coefficients are actually double the corresponding NS coefficients. This is not a coincidence, as we will see in the following section.”

p.320 “We have seen that the Ramond sector has world-sheet supersymmetry: there are equal numbers of fermionic and bosonic states at each mass level.”

With the factor 16, how come the R coefficients are only double, but not 16 times as big as the corresponding NS coefficients?

It is caused by the difference of $x^n$ and $x^{n-\frac{1}{2}}$.

— Me@2015.10.06 08:23 AM