# Problem 14.5a3

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

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Paraphrasing the description of heterotic (closed) string theory:

• right-moving part $\sim$ an open superstring theory
• $NS$ sector: $\alpha_{-r}^I, b_{-r}^I,~~~I = 2,3,...,9$
• $R$ sector: $\alpha_{-n}^I, d_{-n}^I,~~~I=2,3,...,9$
• “The standard GSO projection down to NS+ and R- applies.”

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• left-moving part $\sim$ a peculiar bosonic openstring theory
• $I = 2, 3, ..., 23:$ There are totally 24 transverse coordinates
• 8 bosonic coordinates $X^I$ with oscillators $\bar \alpha_{-n}^I$
• 16 peculiar bosonic coordinates

• can be replaced by 32 two-dimensional left-moving fermion fields, $\lambda^A$
• $\lambda^A$ (anti-commuting) fermion fields $\to$ has $NS'$ and $R'$ sectors
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• $NS'$: oscillators $\bar \alpha_{-n}^I, \lambda_{-r}^A$ act on the vacuum $|NS' \rangle$

given $(-1)^{F_L} |NS' \rangle_L = + |NS' \rangle_L \to$ defines the left $NS'+$ sector

$\displaystyle{\alpha' M_L^2 = \frac{1}{2} \sum_{n \ne 0} \bar \alpha_{-n}^I \bar \alpha_{n}^I + \frac{1}{2} \sum_{r \in \mathbb{Z} + \frac{1}{2}} r \lambda_{-r}^A \lambda_r^A}$

• $R'$: oscillators $\bar \alpha_{-n}^I, \lambda_{-n}^A$ act on $R'$ ground states

$\displaystyle{\alpha' M_L^2 = \frac{1}{2} \sum_{n \ne 0} \left( \bar \alpha_{-n}^I \bar \alpha_n^I + n \lambda_{-n}^A \lambda_n^A \right)}$

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— Me@2018-09-20 09:51:17 PM

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