Spectra

Summary of lowest excitations of open and closed strings when $D=26$ and $a=1$.

Open Strings

Requirement: $L_0-a$ annihilates physical states. This provides a quantisation of the mass-shell condition and determines the mass of excitations in terms of the number and type of raising operators.

${\alpha }^{\prime }{M}^2=N-1$ where $M^2$ is the eigenvalues of $-p^2=p^{\mu }{p}_{\mu }$ and N is the eigenvalue of the level operator. The remaining constraints that $L_m$ annihilates physical states for all $m>0$ must be solved for.

• Ground state $|0;k⟩$ has ${\alpha }^{\prime }{M}^2=-1$ and therefore corresponds to a Tachyon that transforms as a scalar under D-dimensional Lorentz transformations.
• Level-one states ${\epsilon }_{\mu }{\alpha }_{-1}^{\mu }|0;k⟩$ have ${\alpha }^{\prime }{M}^2=0$ and transform as vectors under D-dimensional Lorentz transformations. They correspond to massless gauge bosons with polarisation vector ${\epsilon }^{\mu }$.
• States at higher levels have ${\alpha }^{\prime }{M}^2>0$ and transform as tensor powers of the vector representation under D-dimensional Lorentz transformations.

The tachyonic ground state indicates that we are perturbing around an unstable vacuum of bosonic string theory. In fact, the reason to move onto superstring theory, apart from getting the critical dimension down to $D=10$, is that there are no tachyons in the spectrum.

Closed Strings

Requirement: $L_0-a$ and ${\tilde{L}}_0-a$ annihilate physical states.

We consider $H=L_0+{\tilde{L}}_0-2a$ and $L_0-{\tilde{L}}_0$, which provide a quantisation of the mass-shell and level-matching conditions respectively. We find,

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

and combining these produces ${\alpha }^{\prime }{M}^2=4(N-1)$.

• Ground state $|0;k⟩$ has ${\alpha }^{\prime }{M}^2=-4$ and therefore corresponds to a tachyon that transforms as a scalar under D-dimensional Lorentz transformations.
• Level-one states now take the form ${\epsilon }_{\mu \nu }{\alpha }_{-1}^{\mu }{\tilde{\alpha }}_{-1}^{\nu }|0;k⟩$ for some polarisation tensor ${\epsilon }_{\mu \nu }$ and have mass-squared ${\alpha }^{\prime }{M}^2=0$. The polarisation tensor decomposes into three irreducible representations of D-dimensional Lorentz transformations. They are the trace, symmetric traceless and anti-symmetric tensor parts. The corresponding excitations are the dilaton $\Phi$, the graviton $h_{\mu \nu }$ and B-field $B_{\mu \nu }$ respectively.
• States at higher levels have ${\alpha }^{\prime }{M}^2>0$ and transform in various tensor representations of D-dimensional Lorentz transformations.

String theory is considered as a theory of quantum gravity since the symmetric traceless tensor excitation in the closed string is interpreted as fluctuations in the D-dimensional space-time metric $g_{\mu \nu }={\eta }_{\mu \nu }+h_{\mu \nu }$ or graviton.