The light spectrum

Open String

The mass-shell condition is ${\alpha }^{\prime }{M}^2=N-1$ and thus, the spectrum can be summarised as:

• Ground state $|0;k⟩$ has ${\alpha }^{\prime }{M}^2=-1$. It is a tachyonic scalar.
• Level-one states ${\epsilon }_i{\alpha }_{-1}^i|0;k⟩$ have ${\alpha }^{\prime }{M}^2=0$. They transform in a vector representation $24$ of $SO(24)$ and correspond to the transverse physical polarisations of a massless vector field $A_{\mu }$.
• States at higher levels are massive with ${\alpha }^{\prime }{M}^2>0$. A priori, they transform in tensor representations of $SO(24)$. However, since the states correspond to polarisations of massive tensor fields, they must assemble into representations of $SO(25)$.

Closed Strings

The mass-shell and level-matching conditions are

${\alpha }^{\prime }{M}^2=2(N+\tilde{N}-2)N-\tilde{N}=0$

such that ${\alpha }^{\prime }{M}^2=4(N-1)$. We summarise the spectrum as follows:

• Ground state $|0;k⟩$ has ${\alpha }^{\prime }{M}^2=-4$ and corresponds to a scalar tachyon.
• Level-one states ${\epsilon }_{ij}{\alpha }_{-1}^i{\tilde{\alpha }}_{-1}^j|0;k⟩$ have ${\alpha }^{\prime }{M}^2=0$. They transform in the $24\otimes 24$ representation of $SO(24)$, which decomposes as a symmetric traceless tensor $299$, anti-symmetric tensor $276$ and singlet $1$. They correspond to the transverse polarisations of the graviton $g_{\mu \nu }$, Kalb-Ramond field $B_{\mu \nu }$ and dilaton $\Phi$.
• States at higher levels are massive with ${\alpha }^{\prime }{M}^2>0$. A priori, they transform in tensor representations of $SO(24)$. However, since the states correspond to polarisations of massive tensor fields, they must assemble into representations of $SO(25)$.

We emphasise again, the remarkable fact that the fluctuation of closed strings necessarily include excitations behaving as gravitons and this is the reason for interest in string theory as a theory of quantum gravity.