# Problem 14.5b2

Counting states in heterotic SO(32) string theory | A First Course in String Theory

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(b) … Keep only states with $\displaystyle{(-1)^{F_L}=+1}$; this defines the left R’+ sector.

Write explicitly and count the states we keep for the two lowest mass levels, indicating the corresponding values of $\displaystyle{\alpha' M_L^2}$. [This is a shorter list.]

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— This answer is my guess. —

\displaystyle{ \begin{aligned} \alpha' M_L^2 &= 1 + \sum_{n \in \mathbf{Z}^+} \left( \bar \alpha_{-n}^I \bar \alpha_{n}^I + n \lambda_{-n}^A \lambda_{n}^A \right) \\ \end{aligned}}

If we define $N^\perp$ in a way similar to equation (14.37), we have

\displaystyle{ \begin{aligned} \alpha' M_L^2 &= 1 + N^\perp \\ \end{aligned}}

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\displaystyle{\begin{aligned} (-1)^{F_L} |R_\alpha \rangle_L &= + |R_\alpha \rangle_L \\ (-1)^{F_L} |R_\alpha \rangle_R &= - |R_\alpha \rangle_L \\ \end{aligned}}

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\displaystyle{\begin{aligned} \alpha'M^2=1,~~~&N^\perp = 0:~~~~~&|R_\alpha \rangle_L \\ \alpha'M^2=2,~~~&N^\perp = 1:~~~~~&\alpha_{-1} |R_\alpha \rangle_L, \lambda_{-1} |R_\alpha \rangle_R \\ \end{aligned}}

— This answer is my guess. —

— Me@2018-11-06 03:39:15 PM

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