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
    • .

    • 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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2018.09.21 Friday (c) All rights reserved by ACHK