Sigma-models
We will start with a general question.
Consider the sigma-model:
(notice that it does not have ghosts! Just a sigma-model)
Question: which properties should such a sigma-model have to define a string theory
background?
nilpotent symmetry
a nilpotent -ghost such that
It seems that the second property is automatically satisfied locally on the target space. Locally we can always introduce the coordinates such that and put
It seems that it is even nilpotent when the central charge is zero. However, Eq. (1) is a
wrong answer.