Ghost number three
Even more alarmingly, there is some nontrivial cohomology at the ghost number three.
The cohomology at the ghost number three is isomorphic (up to some zero mode subtleties) to the cohomology at the ghost number two (i.e. physical states). So, for each physical state there is (besides the corresponding ghost number 2 vertex ) also some ghost number 3 vertex .
It can be obtained by multiplying the ghost number one vertex with the ghost number
two vertex:
Cohomology at the ghost number 3 is alarming because it could potentially obstruct our linearized solutions.