# Massive states of the closed bosonic string fitting into a representation of $SO(D-1)$

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It is usually shown in the literature that massive light-cone gauge states for a closed bosonic string combine at mass level $N=\tilde{N}=2$ into representations of the little group $SO(D-1)$ and then it is claimed (see, for example, David Tong's lecture notes on string theory) that it can be shown that ALL excited states $N=\tilde{N}>2$ fit into representations of $SO(D-1)$. Is there a systematic way of showing this, as well as finding what those representations are? Maybe it was discussed in some papers, but I couldn't find anything remotely useful, everyone just seems to state this fact without giving a proof or a reference.

For OPEN strings at $N=3$ level the counting is:
$$(D-2)+(D-2)^2+\binom{D}{2} = \frac{(D-1)(D-2)}{2}+\left(\binom{D+1}{3}-(D-1)\right),$$
where on the LHS are $SO(24)$ representations, and on the RHS are $SO(25)$ representations. I'd like to find the same type of counting for CLOSED strings at any mass level $N,\tilde{N}>2$, as claimed by Tong above.

asked Jan 21, 2017
edited Jan 21, 2017

String Theory is Lorentz-invariant (in critical dimension), so string states must furnish representations of the full Lorentz group. Isn't it enough?

@AndreyFeldman, I'm guessing OP wants to see an explicit calculation/proof, e,g, these and these string states lump together to form an irrep of the explicitly written down Poincare charges.

@JiaYiyang I think that I cited the proof.

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