How many massless particles are there for closed strings in bosonic string theory? Page 53 of String Theory and $M$-Theory by Becker and Schwarz states that there are 576 massless states for the $N = 1$ level of closed string, but only identifies the graviton, dilaton and antisymmetric part. So are there 3 massless particles for the closed bosonic string, or 576 or some other number?
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These $576$ degrees of freedom are distributed among the three fields. As explained on page 53, graviton is the traceless symmetric part of $\Omega^{ij}$. This means \begin{equation} d(d+1)/2-1 \end{equation} degrees of freedom in $d$ dimensions. The trace (dilaton) is one degree of freedom. The antisymmetric part is another $$ d(d-1)/2 $$ degrees of freedom. Set $d=24$ and you get $576$ total degrees of freedom at $N=1$ level.
To avoid confusion, $d$ here is not spacetime dimension, but spacetime dimension minus two.
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$\begingroup$ Ah thank you, so there are 3 then. I suspected this was the case but the wording confused me. $\endgroup$ Commented May 11, 2022 at 9:29