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The spectrum of the Type II string theory (both IIA and IIB) is given by: \begin{array}{*{20}{c}} \hline & {{\text{Sector}}}& & {{\text{Spectrum}}}& & {{\text{Massless Fields}}} & \\ \hline & {{\text{R}} - \operatorname{R} }& & {{{\mathbf{8}}_s} \otimes {{\mathbf{8}}_s}}& & {{C_0},{C_1},{C_2},{C_3}{C_4},...} & \\ \hline & {{\text{NS}} - {\text{NS}}}& & {{{\mathbf{8}}_v} \otimes {{\mathbf{8}}_v}}& & {{g_{\mu \nu }},{F_{\mu \nu }},\Phi ,...} & \\ \hline & {{\text{R}} - {\text{NS}}}& & {{{\mathbf{8}}_s} \otimes {{\mathbf{8}}_v}}& & {{{\Psi '}_\mu },\lambda ',...} & \\ \hline & {{\text{NS}} - {\text{R}}}& & {{{\mathbf{8}}_v} \otimes {{\mathbf{8}}_s}}& & {{\Psi _\mu },\lambda ,...} & \hline \end{array}

I know that for the Ramond-Ramond fields, the even ones belong to the Type IIB string theory and the odd ones belong to the Type IIA string theory.

But what about the rest? Are they there in both Type II string theories? I think it should be the case, because the choice of the GSO projection is only for the R-R sector.

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The NS-NS sector is the same in type IIA and IIB, but the R-NS and NS-R sectors differ. The type IIA theory is non-chiral, so the R-NS and NS-R fields transform in $\mathbf{8}_s \otimes \mathbf{8}_v$ and $\mathbf{8}_v \otimes \mathbf{8}_s'$, where $\mathbf{8}_s$ and $\mathbf{8}_s'$ are the two eight-dimensional spinor representations of $SO(8)$. Type IIB, on the other hand, is a chiral theory where the R-NS and NS-R fields are constructed from the same spinor representation, so $\mathbf{8}_s \otimes \mathbf{8}_v$ and $\mathbf{8}_v \otimes \mathbf{8}_s$.

Similarly, the R-R sector of IIA is given by $\mathbf{8}_s \otimes \mathbf{8}_s'$, while in the IIB case it is given by $\mathbf{8}_s \otimes \mathbf{8}_s$.

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Thanks! So, they simply have different dilatino fields and gravitino fields, right? P.S. I think the 8_v in the first one should be "prime" also. –  Dimensio1n0 May 12 '13 at 14:43
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Yes, the chiralities of the fermions differs in the two theories (and the RR fields). But I'm not sure what you mean by a prime on 8_v. 8_v is the real vector representation of SO(8), and there is only one such irrep. There are two spinors 8_s and 8_s', of different chirality. –  Olof May 12 '13 at 15:59
    
Oops! Forgot that! Sorry –  Dimensio1n0 May 12 '13 at 16:06
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