# Problem 14.5c7

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

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c) … Write out the massless states of the theory (bosons and fermions) and describe the fields associated with the bosons.

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— This answer is my guess. —

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spacetime bosons: $NS'+ \otimes NS+$ \displaystyle{\begin{aligned} \left( \{ \bar \alpha_{-1}^I , \lambda_{\frac{-1}{2}}^A \lambda_{\frac{-1}{2}}^B \} |NS' \rangle_L \right) \otimes \left( b_{-1/2}^J~|NS \rangle \otimes |p^+, \vec p_T \rangle \right) \end{aligned}} \displaystyle{\begin{aligned} = \{ \bar \alpha_{-1}^I , \lambda_{\frac{-1}{2}}^A \lambda_{\frac{-1}{2}}^B \} b_{-1/2}^J |NS' \rangle_L \otimes \left( |NS \rangle \otimes |p^+, \vec p_T \rangle \right) \end{aligned}}

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What is the nature of each of the indices $I, J, A, B$?

The vector index $J$ runs over eight values.

— c.f. p.323 A First Course in String Theory (Second Edition) $\displaystyle{I = 2,3,...,9}$ $\displaystyle{A, B = 1, 2, ..., 32}$

— c.f. the blog post Problem 14.5a3

— Me@2018-12-24 10:04:52 PM

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For the states in the form $\displaystyle{ \bar \alpha_{-1}^I b_{-1/2}^J |NS' \rangle_L \otimes \left( |NS \rangle \otimes |p^+, \vec p_T \rangle \right)}$,

they

carry two independent vector indices $I$, $J$ that run over eight values. There are therefore 64 bosonic states. Just like the massless states in bosonic closed string theory[,] they carry two vector indices. We therefore get a graviton, a Kalb-Ramond field, and a dilation:

(NS+, NS+) massless fields: $g_{\mu \nu}, B_{\mu \nu}, \phi$.

— p.323 A First Course in String Theory (Second Edition)

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Then how about the states in the form \displaystyle{\begin{aligned} \lambda_{\frac{-1}{2}}^A \lambda_{\frac{-1}{2}}^B b_{-1/2}^J |NS' \rangle_L \otimes \left( |NS \rangle \otimes |p^+, \vec p_T \rangle \right) \end{aligned}}?

What kinds of fields do they represent?

— Me@2018-12-24 10:42:03 PM

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— This answer is my guess. —

— Me@2018-12-23 11:16:56 PM

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