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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: $$(D-2)+(D-2)^2+\binom{D}{3} = \frac{(D-1)(D-2)}{2}+\left(\binom{D+1}{3}-(D-1)\right),$$ where on the LHS are $SO(24)$ representations (a vector, a order 1 tensor and a order 3 symmetrical tensor), and on the RHS are $SO(25)$ representations ( a symmetrical traceless order 3 tensor and an antisymmetrical order 2 tensor). 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.

Nogueira
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some1
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2 Answers2

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To begin with, there is nothing like "the little Lorentz group" just to avoid any terminology confusion. The little group or stabilizer subgroup is the subgroup of the spacetime symmetry group under which a particle's momentum stays invariant.

Particles are classified by their little group and they can be divided into three classes which transform under a different little group: massless particles $(m^2 =0)$, massive particles $(m^2 > 0)$ and tachyonic particles $(m^2< 0)$. If you are not familiar with this, you could either read my answer on a similar question here, or you could just give yourself a quick reminder on the wikipedia page.

All states of the closed bosonic string are massive for $N > 2$ as the mass formula is given by $$ M = \frac{2}{\alpha'}(N + \tilde{N} -2). $$ Therefore they all have the same little group, i.e. belong to the same representation of the Poincaré algebra.

Community
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Moe
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The above floor is not correct. The mass condition only tells us what the little group is, not telling the representation of the little group.

I’m also interested in this problem, below is a lecture note which discusses this topic. https://fse.studenttheses.ub.rug.nl/9451/1/Phys_Bc_2008_H.J._Prins_-_s1565761.pdf

In the lecture note of David Tong, by using Noether’s theorem to write down the charges, we can directly write down the Lorentz generators of charges: $$ M^{\mu\nu} = (p^\mu x^\nu - p^\nu x^\mu ) + i \sum_{n\ge 1} \frac{1}{n} (\alpha^\mu_{-n} \alpha^\nu_{n} - \alpha^\nu_{-n} \alpha^\mu_{n} ) + i \sum_{n\ge 1} \frac{1}{n} (\tilde{\alpha}^\mu_{-n} \tilde{\alpha} ^\nu_{n} - \tilde{\alpha} ^\nu_{-n} \tilde{\alpha} ^\mu_{n} ) $$ This formula is given in ‘David Tong’ Page 47. Above is the Lorentz charge, if one is interested in the representation on Fock space, the generator is $$ i \sum_{n\ge 1} \frac{1}{n} (\alpha^\mu_{-n} \alpha^\nu_{n} - \alpha^\nu_{-n} \alpha^\mu_{n} ) + i \sum_{n\ge 1} \frac{1}{n} (\tilde{\alpha}^\mu_{-n} \tilde{\alpha} ^\nu_{n} - \tilde{\alpha} ^\nu_{-n} \tilde{\alpha} ^\mu_{n} ) $$

A related discussion can be found: What is the spin of stringy excitations?

zixuan feng
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