In Tongs' "String Theory" lectures chapters 5.4.1-5.4.2 Tong referred to alternative way to find the mass of the different string states using suitable operators, and then integrating the relevant operators and requiring the new operator which we will call the vertex operator to be conformally dimensionless this would give eventually the mass squared of the state.
He shows that for example the ground state that is given by a tachyon in the free closed bosonic string theory have a suitable operator that is given by: $$:e^{ikX}:$$ The conformal dimension of this operator (or weight) is given by the OPE with the corresponding component of the stress tensor, we will choose for simplicity the left side $T_{zz}=T$ and get: $$h = \frac{\alpha'k^2}{4}$$
Now we define the vertex operator to be:
$$V_{tachyon}=\int d^2z:e^{ikX}:$$ because the conformal dimension of $dz$ is equal to $-1$ this will yields the mass squared of the tachyon:
$$h-1=0\Rightarrow M^2=-k^2=-\frac{4}{\alpha'}$$
my question is what is the way to find the operator, conformal dimension and the mass squared of the ground state eventually, through the same procedures in different superstring theories (Type I, Type II, Heterotic)?
