Why doesn't string theory predict the existence of infinitely many elementary particles? I'm a physicist, but my knowledge of string theory is extremely minimal. My naive conceptual understanding is that the vacuum is modeled as a certain topology (and geometry?) for the spacetime, and fundamental particles are explained as excitations of strings that live on this background. E.g., a certain vibrational state would be the photon, some other state would be the graviton, and so on. The actual spectrum of such vibrations is presumably something we can't calculate, because we don't know the topology of the background (i.e., which possibility it is in the string landscape).
If this is at least qualitatively correct, then why aren't there infinitely many such vibrational states, which would appear at ordinary energy scales as infinitely many elementary particles?
 A: For simplicity take the bosonic string theory in 26 dimensions. When you quantise the open and closed string you find excitations (states) of the string at any level $N$ with masses
\begin{equation}
M^2_\mathrm{open}=\frac{1}{\alpha '}\left(N-1\right),\qquad M^2_\mathrm{closed}=\frac{4}{\alpha '}\left(N-1\right).
\end{equation}
As $N$ takes on any non-negative value, you see that there is indeed an infinite tower of states. The tachyonic state (negative mass) at $N=0$ is a sickness of the purely bosonic string, which goes away for the superstring. At level $N=1$ you find the massless excitations (in 26 dimensions), and from the next level all states are massive. The mass is determined by the pre-factor. You can regard $\alpha '$ as the only free parameter of string theory, and it relates to the tension, $T$, of the string, which is expected to be set by the string scale, which is slightly below the Planck scale
\begin{equation}
\frac{1}{\alpha '}= 2\pi T \lesssim M_\mathrm{Pl}^2 = \left(10^{19} \, \mathrm{GeV} \right)^2 .
\end{equation}
So in conclusion, the infinite tower of massive excited states have masses at the order of the Planck scale, which means they are unobservable, and there is only a finite number of massless excitations. The same thing goes for the superstring.
