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.
(And the list of requirements is probably incomplete)
(And the list of requirements is probably incomplete)
Because is not globally well-defined.
It seems that being a well-defined string theory is a global property of the target space, there is no such thing as a locally well-defined string theory.
It is essential to have submanifold where has zeroes, and the structure near this submanifold should be somehow a part of the axioms.
In the pure spinor formalism, part of the target space is considered “ghosts” and treated separately from the matter part.