Conformal weights of space-time and ghosts I'm studying the paper of Kaplunovsky  https://arxiv.org/abs/hep-th/9205070
In particular in page 12 he says

Hamiltonians $H$ and ̄$\bar{H}$ are totals of space-time, ghost and internal
components; evaluating the space-time and ghost components, we find
that massless vectors have $(H,\bar{H})_{int} = ( 1/2 , −3/8 )$, massless
fermions have $( 1/12 , 0)$ and massless scalars have $( 1/12 , 1/8 )$

As far as i know the conformal weights were $(3/2, -1/2)$ for $\beta$$\gamma$ ghosts, $(2,-1)$ for $bc$ ghosts, $(0, 1/2)$ for fermions (in heterotic string), $(0,0)$ scalars and $(1,1)$ vectors, and $H = L_0 -c/24$  with $c/24 = (1/12, 1/8)$.
Can someone help me find where i am wrong?
 A: You have given the right conformal weights for the worldsheet fields. Their mode expansions consist of oscillators and zero modes.
I don't have a good understanding of the paper. But when people talk about the conformal weights of string states they are talking about vertex operators. These are built from the two types of modes above in order to ensure that $(H, \tilde{H}) = (0, 0)$ (no matter what worldsheet fields are involved) which reflects the fact that Weyl symmetry is gauged. Kaplunovsky seems to be dividing this conformal weight of zero between "internal" and "external" contributions which are equal and opposite. Something that's internal in this context would mean it's associated with the compact dimensions instead of the extended ones.
Another confusing thing is that he says $(H, \tilde{H}) = (0, 0)$ applies only to the massless states. This must mean it's a statement about the zero modes (like $e^{ipX}$) where oscillators are ignored. But even then it's common to see different fractional contributions that cancel when you have spin structures. This is because the R vacuum is created from the NS vacuum by a twist operator which gives it a higher conformal weight.
Definitely not a full answer but maybe it can lead you to a jumping off point.
